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RATS Programming Manual, By Walter Enders

1. reglist end dofor 1 The first time through the DOFOR loop i the integer assigned to the series dirgdp Hence dirgdp 1 to 12 is added to reglist Notice that the supplementary card on the ENTER instruction also contains reglist This is because we want the new list of regressors to include ENTER can also be used when passing information to a procedure Here we consider only the use of ENTER to create a variable list Vector and Matrix Manipulations 183 everything in the previous list 7 e a constant and the variable we are adding to the list Next i the integer assigned to dirm2 and a new list is created The new vector reglist contains everything in the previous list 7 e a constant and dirgdp 1 to 12 and dlrm2 1 to 12 In this way the program estimates the regression equations If you do not want a constant in any of the regressions you can replace the first line of the routine with dec vector integer reglist 0 Now reglist has been dimensioned such that it contains no entries 3 1 Automating Model Selection in a VAR You can use ENTER to modify the list of deterministic variables in a VAR Suppose that we want to determine whether to include seasonal dummy variables in the 3 variable 12 lag VAR We want a routine that will make three loops In the first loop the system is estimated without the seasonal dummies and the seasonal dummies are used in the second loop The VAR output is not displayed at this sta
2. system model chap2 var dlrgdp dirm2 drs lags 1 to 8 det constant end system estimate noprint residuals resids8 1962 2 compute aic Ynobs logdet 2 25 3 compute sbc Ynobs logdet 25 3 log nobs dis aic aic sbc sbe aic 3197 47118 sbc 2968 73198 The A C selects the 12 lag model whereas the SBC selects the 8 lag model We can also determine lag length using a likelihood ratio test Under the null hypothesis we can restrict lags 9 12 of all coefficients in all three equations to be zero If this restriction is binding we reject the null hypothesis Consider the following set of instructions ratio degrees 4 3 3 mcorr 37 1962 2 resids12 resids8 Covariance Correlation Matrices RESIDS12 1 RESIDS12 2 RES DONAS 00004258409 ODIT AAT aS 0 2104634745 0 0 BSD Saas OF TODS oF TODS oF 0000843189 98 000 G9 16 RS OAS 9 Sic odor OOP OCHA Sa Ol OO OG D 6319 9 oil 030509775570 RESIDS8 1 RESIDS8 2 RESIDS8 3 RESIDS8 1 0 00004674920 ORPEOSZ SS Saal 0 2567983642 ESIDS8 2 0 00000726876 0 00003026766 0 2174072492 RESP S S1 S O 0 ONSE2S 4777 5 0 OOOR Ga 2 927 5 0 40938101386 Log Determinants are 21 923113 21 458149 Chi Squared 36 55 330714 with Significance Level 0 02068921 The RATIO instruction uses DEGREES 4 3 3 since the 8 lag model eliminates four lags of three variables in each of the three equations MCORR 37 since there are 3
3. Hence the first COMPUTE instruction creates a matrix C corresponding to the matrix C 1 in the discussion above The function 7MQFORM X Y creates a matrix equal to Y XY for Y xn and X xm INV C creates C and TR INV C creates the transpose of C Hence the second instruction creates CY cy The function 7DECOMP S1 creates the matrix 2 equal to Choleski decomposition of C YXC C G Multiplication by C yields the factorization containing the desired form of G At this point it is possible to obtain the impulse responses and variance decompositions using the DECOMP G option on an ERRORS or IMPULSES instruction VARs and Error Correction Models 85 Example The 3 variable VAR with dirgdp dlrm2 and drs is in the appropriate form since all of the variables appear to be difference stationary Although there is strong evidence that a 12 lag model is appropriate it is instructive to estimate the system using only 1 lag The goal here is to illustrate the creation of the desired matrices and coefficient sums these sums are trivial to calculate in a l lag model As such re estimate the 3 variable VAR using the following instructions system model chap2 var dlrgdp dirm2 drs lags 1 det constant end system estimate outsigma v noftests VAR System Estimation by Least Squares Dependent Variable DLRGDP Variable Coe fer Std Error PS Sret Signif CK CK Ck Ck CK CK CI CI CK CK C KKK KKK KKK KKK KKK KKK KKK KKK KKK KKK KKK KK
4. MAXIMIZE is able to find the value s of p that solve T ee Yox B t 1 The syntax and principal options of MAXIMIZE are maximize options frml start end funcval where frml A previously defined formula start end The range of the series to use in the estimation funcval Optional The series name for the computed values of fys x D The key options for our purposes are METHOD RATS is able to use any one of four different algorithms to find the maximum BFGS BHHH GENETIC or SIMPLEX ITERATIONS The upper limit of the number of iterations used in the maximization ROBUSTERRORS Computes a consistent estimate of the covariance matrix that NOROBUST corrects for heteroscadesticity CVCRIT Convergence limit 0 00001 Also note that NLPAR affects the MAXIMIZE instruction TRACE NOTRACE Prints the intermediate results including the values of the estimated coefficients and function values This is useful for tracking convergence problems MAXIMIZE produces a number of internal variables including BETA TSTATS NOBS Zo NREG and 7 CONVERGED In addition the internal variable FUNCVAL is equal to final value of the function being maximized Note You can use TEST and RESTRICT with the BFGS and BHHH options Coefficients are numbered by their position in the NONLIN statement 34 Walter Enders Examples 1 To estimate the model y ax use NONLIN a b var FRML L log var y a x beta 2 var C
5. PROCEDURE procedure name list of parameters TYPE instructions OPTION instructions LOCAL instructions INQUIRE instructions set of instructions to be performed END procedure name The procedure is executed by procedure name list of parameters Note that the order of the parameter list must match that used within the procedure Writing Your Own Procedures 213 4 Writing a Procedure to Test for Unit Roots We will illustrate the process by writing a procedure called UNIT SRC that will perform an augmented Dickey Fuller test The final procedure will be similar to that in DFUNIT SRC that comes with RATS However by writing such a procedure from scratch it will be possible to illustrate the structure and key instructions of any procedure We begin simply At first the procedure will only estimate a model of the form Ay Bo pyci BiAyzi The entire regression output will not be displayed Instead the procedure will only display the estimate of p and the t statistic for the null hypothesis p 0 After compiling the procedure can be invoked using unit series For example you can use to procedure to test for a unit root in the variable rgdp using Q unit rgdp We begin by writing UNIT SRC as procedure unit y type series y set dy 2 y y 1 lin noprint dy y 1 dy 1 constant dis The estimate of rho beta 1 dis the t statistic for the null hypothesis rho 0 is tstats 1 end unit The interpretati
6. TRANSFORM SRC is called only if the user clicks the OK button if menuchoice eq 1 f select series status cancel depvar if cancel eq 1 transform depvar 44 Of course you could modify the procedure to allow all entries to be negative Writing Your Own Procedures 247 The file SEASONS SRC includes a similar check in case the user cancels the choice Select a Variable to Estimate Program 6 2 in the file CHAPTER6 PRG EXECUTES the procedure using source noecho c winrats seasons src seasons 248 Walter Enders Index Akaike information criterion 19 ARIMA models esseeeeene iv automating model selection 182 Blanchard Quah decomposition 71 82 86 BOXJENK iv 35 103 104 178 192 207 Box Jenkins methodology 104 192 BRANCGH ier iv 133 147 148 CDE 5 eue ede ees iv 40 41 CHOICE options esee 207 Choleski decomposition 53 59 63 69 70 71 72 75 83 84 176 180 COMPUTE iv 21 22 30 32 34 37 38 44 71 84 98 110 115 116 134 147 151 152 168 171 173 174 176 179 190 191 230 241 CORRELATE eee iv 7 30 135 CVMODEL eee iv 75 76 78 DECLARE iv 66 75 168 169 170 171 174 182 190 191 195 203 215 230 Dickey Fuller tests 136 141 142 143 144 146
7. 0 020 di LUI ll Il MI 0 01 HIT JI I BI T 0 015 4 i 0 00 2 4 A i WAL Y Jl IP 0 010 Kf NI IM 0 01 I l M I l 0 005 0 02 4 0 03 UUUTTTUUTDPTPPUTTTUTDPDPUDUDPTTUTTTPPTTPTDUTDDPDPTTTTUTTTT 0 000 TUDUTTUUDPTDIUTUDUDPDDDUDPDUDUIPDUUDTDTUTTTTITDUDUDTTTTTITTUUDTT T T T T T 1959 1963 1967 1971 1975 1979 1983 1987 1991 1995 1999 1959 1963 1967 1971 1975 1979 1983 1987 1991 1995 1999 Linear and Nonlinear Estimation 5 Recall that the typical syntax of the GRAPH instruction is graph options number series start end where number The number of series to graph Here the number 2 tells RATS that two series are to be graphed The names of the series are indicated on the supplementary cards there is one such card for each series series The name of the series to graph Remember there is one supplementary card for each series start end Range to plot If omitted RATS uses the current sample range The graph created here illustrates only a few of the available options The commonly used options are HEADER A string of characters placed in quotes KEY The location of the KEY You can use KEY NONE UPLEFT UPRIGHT LOLEFT LORIGHT ABOVE BELOW LEFT RIGHT The default is NONE DATES RATS will label the horizontal axis unless the NODATES option is NODATES specified STYLE The default style is a line graph The full set of styles is STYLE L
8. 055961728 2 38588 01703837 BO 001382754 000804149 Le 1359192 08551898 Bl Taoa LA Ses 040165933 Do OSs 00000037 Cal 806864043 028403811 28 40689 00000000 Notice that the estimated values of b and c are such that their sum exceeds unity It is possible to estimate an GARCH 1 1 model by restricting b c1 1 The only modification needed is to replace the NONLIN instruction above with nonlin a0 al a2 a3 bO b1 c1 b0 ge 0 bl ge 0 cl ge 0 bl cl eq 1 If you re estimate the model you will find max iters 200 L 7 MAXIMIZE Estimation by BFGS Convergence in 25 Iterations Final criterion was 0 0000027 0 0000100 Quarterly Data From 1960 03 To 2001 01 Usable Observations Gs Function Value SS2 09657368 Variable Coeff Std Error Muss Sume hake CK UKCK UK CK AK kk CK CK CAUCA CIC ck ke ok OUO COCOA CK KC kak ok ke kak Ay ke oka ak kak a ka Ke AO 310 1912 9150 7216 OW 013536159 3 49324 0 00047720 Al 561910 HS OS 056049098 Los also 00000000 A2 o 24 5 11 2 929 707525339624 g 2G OOH EAN OMEN A3 GIL SCS FSO CML T0 S SuI9 oV T5 A SSH 01124748 BO UOMO SD OOS 000591367 2 69034 2 0 1 3 5 1 9A Bl g LQ SOO SIN 029293592 01921215 7 00000000 al 806006898 029293592 2 ALA TS 00000000 4 An ARMA 1 1 IGARCH 1 1 Model of the Spread As a final example for this section suppose you want to estimate the spread as an ARMA 1 1 model with JGARCH 1 1 errors spread Oo Ospread
9. 213 215 222 231 differencing 6 36 38 103 105 129 144 180 DIMENSION eene 169 170 171 DISPLAY 70 91 152 171 172 173 198 DOE ett 19 99 100 103 105 106 107 108 109 110 111 113 115 116 120 121 124 125 126 127 131 133 134 138 140 141 143 144 146 147 148 152 155 156 157 159 161 162 166 172 176 177 181 182 183 185 186 187 222 227 230 DOFOR eee 110 111 113 116 125 140 141 143 172 182 185 186 187 Downward Bias c ccccccccessccceesseeeesseeeeens 139 ENABLE eere 235 243 244 Engle Granger procedure 18 ENTER iv 114 182 183 185 186 ENTRIES s iv 12 100 113 114 EQUATION iv 22 150 151 152 197 EQV Instruction iv 96 97 109 Error correction models 64 65 73 149 ERRORS iv 53 58 62 66 711 75 83 84 196 203 ESTIMATE nnana arien iv 48 52 53 58 65 67 71 84 85 105 168 239 240 243 244 EWISE naiara iv 176 177 179 195 EXCLUDE 8 9 10 116 186 201 202 EXECUTE iens 100 234 FOREGCKXAST eeu iv 60 62 63 FERMI eher 21 22 24 30 32 33 34 35 36 37 38 41 42 44 46 94 155 Generalized ARCH GARCH 40 41 44 46 Granger causality sseseeee 48 GRAPH ice
10. BETA 1000 0 94693484752 0 04471912585 0 70122076370 1 04215477194 Note your output will be a bit different from mine in that your computer probably used different random numbers than mine to generate the 1000 random walk sequences An issue that may arise concerns the use of random numbers in the program We might want to ensure that the random numbers selected by RATS actually appear to be random Towards this end we can modify the routine as follows IF Statements and Monte Carlo Experiments 135 do i 1 1000 redraw set epsilon ran 1 cor number 4 qstats noprint epsilon if signif lt 0 005 branch redraw set y 2 y 1 epsilon lin noprint y constant y 1 com beta i beta 2 end do Now the first instruction inside the loop is a label called REDRAW The program will jump back to this point and redraw a new set of random numbers if the current series does not appear to be random The second instruction in the loop fills the series epsilon with 100 pseudo random numbers The next line obtains the first four autocorrelations of the epsilon series The CORRELATE instruction creates the internal variable signif that holds the significance level of the for the null hypothesis that these first four autocorrelations are all equal to zero In the next line we compare this significance level to 0 005 If the significance level for the Ljung Box Q 4 statistic is less than 0 001 the program jumps back to RED
11. If the field is not specified defined 0 and the SET instruction is not performed No error message is created since the series corrs is used only when the field is specified by the user 3 gra max 1 0 min 1 0 style bar number 0 nodates header Residual ACF 1 xx If the GRAPH option is selected a graph of xx is produced Examples 1 Suppose you want to perform three unit root tests on tb yr The first does not have any deterministic regressors the second has an intercept and the third has an intercept and a trend Use do det 1 3 unit det det lags 7 tblyr end do det The procedure produces the desired result since the first time through the loop det 1 the second time det 2 and the third time det 3 2 To perform the test with an intercept in the equations for tb yr and tb3mo use dofor j tblyr tb3mo unit lags 7 j end dofor Writing Your Own Procedures 225 3 unit lags 7 graph tblyr 1980 1 corrs A unit root test containing an intercept and seven lagged values of tblyr is performed A graph of the residuals is displayed and the autocorrelations of the residuals are returned in the series corrs Note that the test is performed beginning with 1980 1 The results for the t test are identical to those obtained from dif tblyr drl lin drl 1980 1 constant tblyr 1 drl 1 to 7 226 Walter Enders 6 A Procedure for Computing Lag Lengths Anyone working with time series data routinely selec
12. Note that the series rss holds the residual sum of squares associated with each regression The pattern of these values can be seen for every potential threshold value if we create a scatter diagram using scatter Header Residual Sums of Squares style lines hlabel Threshold 1 thresh test rss low high Residual Sums of Squares 0 01170 0 01165 0 01160 M 0 001155 0 01150 0 01145 T T T T 0 005 0 010 0 015 0 020 Threshold It is the case that the threshold of 0 01724 does result in the lowest residual sum of squares However it is clear from the scatter diagram that there is a sharp local minimum around 0 0025 This suggests that there may be two thresholds implying the existence of a low medium and high state IF Statements and Monte Carlo Experiments 133 4 Branching IF THEN ELSE blocks allow you to have some control over the flow of your program It is possible to have a set of instructions executed or completely skipped depending on a set of criteria that you specify Nevertheless program execution necessarily resumes with the first instruction following END IF Note that in the example estimating the threshold model RATS always displays We have found the attractor Another flexible way to control the execution of your program is the BRANCH instruction BRANCH enables you to jump to the particular point in your program that you specify You can jump out of
13. RATS encounters a USERMENU instruction enabling choice 2 240 Walter Enders procedure seasons local integer depvar usermenu action define title Trends 1 gt gt Select a Variable to Estimate 2 gt gt Estimate the Trend 3 gt gt Done usermenu action modify enable no 2 If the user chooses Select a Variable to Estimate menuchoice 1 so that the instructions in the first IF block are executed The SELECT series depvar instruction presents the user a list of all series in memory The integer value corresponding to the selected series is assigned to depvar and the label of depvar is stored in the string variable a The remaining instructions in the first IF block create a graph of depvar and enable the choice Estimate the Trend loop usermenu if menuchoice eq 1 f select series depvar compute a l depvar gra header Time Path of a 1 depvar usermenu action modify enable yes 2 If the user selects Estimate the Trend SEASONS invokes the procedure ESTIMATE This second procedure interacts with the user to request information concerning the type of time trend to estimate Once ESTIMATE has completed its functions program control returns to LOOP ENDLOOP block The instructions within this block are continually executed until the BREAK instruction is encountered As long as the user does not select DONE i e as long as Ymenuchoice does not equal 3 the user can continually select variables and estima
14. Thus if df lt 2 58 the value of hits 0O is increased by one Similarly if df lt 2 89 hits5 is increased by one and if df lt 3 51 hits is increased by one After the 10 000 DO j loops we can use these three numbers to indicate the number of times that the test correctly rejected the null hypothesis of a unit root if df 2 58 compute hits10 hits10 1 if df 2 89 compute hits5 hits5 1 if df lt 3 51 compute hits hits 1 end do j The last section of the program displays the key output for each value of p hits10 hits5 and hits are shown display rho rho display 10 5 1 display HHHHEF hits 10 HHHH hitsS THHHHHE hits display end dofor 5 The values used are those reported in Dickey and Fuller 1979 not those obtained in Program 4 7 above IF Statements and Monte Carlo Experiments 145 When p 0 2 the test does reasonably well at the 5 significance level the false null hypothesis of a unit root is rejected in 87 30 of the Monte Carlo replications However when p 0 05 the probability of correctly rejecting the null hypothesis of a unit root is estimated to be only 12 72 Jazzing Up the Program A Monte Carlo analysis can take a very long time to execute We could for example run the program for a number of sample sizes and for additional values of alphal Your computer screen can remain blank for a very long time before you see any output You have proba
15. amp Bg 46 Walter Enders h b b teh Now it is necessary to use two placeholders one for e and another for h In the program below templ is the placeholder in the equation for spread Oo ouspread i Bg As such the first FRML instruction creates e as spread a0 al spread 1 beta templ 1 Similarly the second FRML instruction uses the placeholder temp2 to create the conditional variance as h bO bl e 1 2 cl temp2 1 The third FRML instruction uses two SUBFORMULAS the first equates templ with e and the second equates temp2 with h set temp 0 set temp2 0 nonlin a0 al beta bO bl cl b0 ge 0 bl ge 0 cl ge 0 bl cl 1 frml e spread a0 al spread 1 beta temp1 1 frml h b0 bl e 1 2 cl temp2 1 frml L templ e temp2 h log temp2 temp1 2 h The next two instructions are used to initialize the parameters For now it is sufficient to note that the BOX constant ar 1 ma 1 noprint spread instruction estimates an ARMA 1 1 model of the spread without GARCH errors The initial guesses for a0 al beta and bO are taken from this ARMA 1 1 model Finally the MAXIMIZE instructions are used to obtain the maximum likelihood estimates box constant ar 1 ma 1 noprint spread com a0 beta 1 al beta 2 beta obeta 3 bO 96seesq b1 0 2 cl 0 5 max method simplex iters 5 L 7 max iters 200 L 7 MAXIMIZE Estimation by BFGS Con
16. beta 2 end do i sta fractiles alpha star Statistics on Series ALPHA STAR Observations 1000 Sample Mean 0 4179067424 Variance omo Zey Sisamelarcdur rore i5 209212997 SE of Sample Mean 0 004810 Minimum 0878398657 Maximum 0489024933 01 ile 0706360305 99 Sile 8022149869 05 ile adt THES CO SNES S 95 Sile 169 0 6 89 9 5 99 10 ile SALAD PSA 90 ile 2 6102 0 55181 05707 25 ile SOES 5O22 75 Sile 81505225715 Median PANTS SAC Aras sta fractiles beta_star Statistics on Series BETA_STAR Observations 1000 Sample Mean EAO O ZASS Variance 0 198084 Standard Error 0 44506578336 SE of Sample Mean 0 014074 Minimum 26104747424 Maximum 5152 5 NIBINSIMIS QE Ssse 48336068574 9S ak ILS MOIS OS OSS 9H OS al IS 64414171297 SaL lle 08050680611 3 0 anle 75625461184 OS als 94 69 VIO AT Hales BOSAL MAZ NS VISS Tarile 48805437120 Median 17862968423 IF Statements and Monte Carlo Experiments 163 We can use the distribution for alpha star to form a symmetric 90 confidence interval for Since 5 of the values lie below 0 1715603199 and 5 lie above 0 6706139537 the 9096 confidence is the range 0 1715603199 to 0 6706139537 Similarly 9096 confidence interval for B using the bootstrap distribution is 0 64414171297 to 2 08050680611 Notice that this confidence interval is much wider than that obtained using the normal approximation 164 Walter Enders 7 2 The AR Coefficients of Real GDP Growth Step
17. com beta2 hat 1 96 se 2 u beta2_hat 1 96 se_2 dis 95 lh com beta2_hat 2 57 se_2 u beta2_hat 2 57 se_2 dis 99 lh Normal Approximation 90 0 01094 0 26151 95 0 01315 0 28560 99 0 05963 0 33208 In order to see if the residuals appear normal we can use the STATISTICS instruction sta resids Statistics on Series RESIDS Quarterly Data From 1959 04 To 2001 01 Observations 166 Sample Mean 0 00000000000 Variance 709552 98 05 Standard Error 0 00842349604 SE of Sample Mean 0 000654 1 00000000 0 78455071 0 50000229 5 TZ 0 00103121 E Step LS LE 0 00000 Signif Level Mean 0 Skewness 0 05245 Signif Level Sk 0 Kurtosis 1 40625 Signif Level Ku 0 Jarque Bera 1 9 s 7H9 41015 Signif Level JB 0 Although the residuals do not appear to be skewed there is excess kurtosis and the Jarque Bera test clearly rejects normality In preparation for the construction of bootstrap confidence intervals a bit more bookkeeping is necessary The integer boot num containing the number of replications is set equal to 1000 If you want to alter the number of replications simply change this single program statement The next instruction creates the series beta2 star the 1000 bootstrap estimates of D i e the values of B will be stored in this series The third instruction creates the values of the y sequence 166 Walter Enders com boot_num 1000 set beta2_star 1 boot_num 0 se
18. e g 1991 3 corresponding to the entry value To find the date label of the observation 100 periods before 2001 1 use dis 7 DATELABEL 2001 1 100 Your output will be 1976 01 Note In performing date arithmetic RATS does not allow you use spaces inside the 7DATELABEL function hence in the instruction above you cannot place a space on either side of the minus sign You can also use 7DATELABEL with any integer value For example if you insist on putting spaces next to the minus sign you can use com i 2001 1 100 dis 7DATELABEL i Similarly dis cal 1976 1 yields 69 94 Walter Enders 2 Series as Integers Just as each calendar date has an associated entry value each series has its own sequence number If continue to use Program 3 2 and type TABLE you will see Series Obs Mean Std Error Minimum Maximum DATE 169 1979 876331 12 232185 1959 100000 2001 100000 GDP 169 3572 739053 2873 158128 496 100000 10243 600000 RGDP 169 5142 364497 1950 840494 2273 000000 9439 900000 M2 169 1904 835266 1399 706717 287 800000 5043 710000 M3 169 2414 462229 1916 764710 290 053333 7260 136667 TB3MO 169 5 915148 2 590483 2 303333 15 053333 TB1YR 167 6 153872 2 393622 2113333 14 380000 DLRGDP 168 0 008475 0 008960 0 020598 0 037804 RESIDS 101 0 000000 0 007737 0 027190 0 031366 Notice that the series in MONEY_DEM XLS are in order followed by the dlrgdp series you created The series numbers are such that date is
19. i It is possible to write a simple program to measure the size of the bias and to see how it is affected by the magnitude of o Towards this end we could perform the following tasks Step 1 Generate a series in the form of the AR 1 model using a value of 0 2 Step 2 Estimate the series using OLS and calculate the discrepancy between the estimated value of o and 0 2 140 Walter Enders Step 3 Repeat Steps 1 and 2 a total of 1000 times Display the average value of the discrepancy Step 4 Repeat Steps 1 to 3 using alternative values of 1 Consider the first three lines of Program 4 6 in the file CHAPTER4_1 all 100 set discrep 1 1000 0 set y 0 Since the program uses only simulated data we do not include a CALENDAR instruction The ALLOCATE instruction sets the default length of any series we create equal to 100 For each value of o we will store the 1000 discrepancies between the actual and estimated values in the series called discrep The third instruction initializes all 100 observations in the simulated y sequence to be zero Now we have to decide which values of to include It might make sense to include 0 5 since it is midway between 0 and 1 0 and two values near 1 0 say 0 9 and 0 99 Finally we might want to include 0 0 for comparison purposes Consider the next two instructions com alphal 0 dofor alphal 2 5 9 99 0 The DOFOR instruction acts in a way that is similar to the DO instruct
20. i 1 and an AR 1 model is estimated The value of the SBC is compared to that of the AR 0 if the AR 1 has the smaller SBC bestlag is equated to i and sbcmin is replaced by sbc Thus if the AR 1 provides a better fit than the AR 0 bestlag 1 and sbcmin contains the SBC of the AR 1 The next time through the loop i 2 The SBC from the AR 2 is compared to that of the previously selected best fitting model If the AR 2 fits better than the model indicated by bestlag the value of bestlag is set equal to 2 On exiting the loop bestlag contains the lag length of the model providing the best fit do i 1 maxlag lin noprint y v1 maxlag v2 constant y 1 to i compute sbc nobs log rss Vonreg log 96nobs if sbc lt sbcmin compute bestlag i sbcmin sbc end do i If bestlag 0 the regression is not estimated and the procedure displays DO NOT USE LAGS If bestlag is greater than zero i e if any of the AR p models fit better than the AR 0 the AR bestlag model is estimated and the output is displayed Notice that the sample period runs from v bestlag to v2 if bestlag EQ 0 dis DO NOT USE LAGS if bestlag GT O lin y vl bestlag v2 constant y 1 to bestlag 228 Walter Enders If the user specifies laglength the procedure returns the value of bestlag Consider if defined laglength com laglength bestlag end laglength Writing Your Own Procedures 229 7 Interacting With Procedures RATS contai
21. run equilibrium relationship Thus in the interest rate example if you want to estimate a model with 5 lagged changes in each series use 17 If you do your own hypothesis testing you will find that a 6 lag specification seems quite reasonable VARs and Error Correction Models 65 system model term var tblyr tb3mo lags 1 to 6 det constant ect spread lt lt NOTE We used DEFINE spread on the LINREG instruction end system NOTICE THAT WE SET UP THE MODEL IN LEVELS NOT IN FIRST DIFFERENCES RATS will report the results in first differences along with the error correction term Since we want 5 lags in the first differences we use 6 lags of the level Step 3 Enter the appropriate ESTIMATE instruction For the interest rate example we can use estimate outsigma s residuals resid Depe Vari Depe Vari KRR R able D_TB1YR 1 De DEI Di IHE AR CS D TBIYR 4 D APIS NIRE S D TB3MO 1 D TB3MO 2 D TB3MO 3 D TB3MO 4 D TB3MO 5 Constant BCA able DESEE TDYRACID DESEE D TB1YR 3 D TBIYR 4 DISCI SIBYRESS ED D TB3MO 1 D TB3MO 2 D TB3MO 3 D TB3MO 4 D TB3MO 5 Constant TIGE ndent Variable TB1YR Coeff Ck ck Ck Ck C Ck C Ck Ck 00k Ck 00k Ck Ck Ck Ck ck Ck Ck Ck 0k Ck Ck Ck Ck Ck Ck c Ck ck Ck ck ck Ck Ck Ck Ck ck Ck ck ck ck Ck Ck Sk Ck kk Sk ko ko kv Sk ko ko ko ko kx koX 528 1G 7 OILS SADIE TOS 244713790 223004947 22 2
22. set dirgdp log rgdp log rgdp 1 Instead of estimating the series as an AR 1 suppose that we want to ascertain the number of lags to use in an AR p representation Suppose that you believe that the maximum possible lag length is 12 do p 1 12 lin dlrgdp constant dlrgdp 1 to p end do p RATS will estimate the regression exactly 12 times The first time through the loop p 1 so that RATS estimates an AR 1 model i e lags 1 to p is simply lag 1 The second time through the loop p 2 so that RATS includes lags 1 and 2 in the autoregression Each time through the loop p is increased by 1 hence the number of lags used in the autoregression is increased by 1 As it stands the routine has a number of flaws First if we want to perform lag length tests we need to estimate each autoregression over the same sample period Since one usable observation is lost for each lag included in the model we can estimate all of the autoregressions over the same sample period by modifying the LINREG instruction such that lin dlrgdp 14 Now all autoregressions begin with observation 14 since one usable observation is lost by taking first differences and 12 are lost when estimating the AR 12 regression The second flaw is that we get too much output Oftentimes you will want to calculate the aic or sbc from each equation and then to examine the output of the model with the smallest aic and or sbc For each regression you can compu
23. signif lt 05 enter varying reglist reglist i 1 to 4 188 Walter Enders enter varying templist reglist end dofor 1 lin j NOTE the change in this line reglist end dofor j NOTE the addition of this line If you execute the program you will obtain four regression equations Instead of reproducing a rather large amount of output simply note that Equation Causal Variables dirgdp dirgdp drs dim3 dim3 drs dirgdp drs dlp dlp dirgdp dim3 drs dlp Vector and Matrix Manipulations 189 4 Example Moving Average Representations Suppose that we have an ARMA p q model and want to calculate the coefficients of its infinite order moving average representation Specifically suppose that we are working with the model yr Qo Oayri Ooyra OpYr p i BiEr1 Borg As long as the equation is invertible it is possible to express the y sequence in terms of the sequence as y o 66 i 0 One way to obtain the values of the sequence is to use the Method of Undetermined Coefficients It should be clear that o Oo 1 0 0o p and that 1 However finding the remaining values of the sequence can be complicated In particular the formulas for the coefficients are given by do 1 1 Bi o0 2 Bo 104 bode 3 B3 Q201 100 90 Thus with knowledge of the a and B the values of the various can be found by iteration us
24. x 1 x 1 2 It is possible to SET entry i of one series based on the value of a corresponding value of a second series Consider SET plus if resids 1 lt 0 1 0 The series plus is equal to 1 if the lagged value of resids is negative and is 0 when the lagged resids is greater than or equal to zero To better understand the output of the IF function suppose that resids contains the residuals from an AR 2 model As such the first two entries of resids are missing since two usable observations are lost in estimating the AR 2 ESIDS NA NA 41281771461 11837667346 3210 09 52 OTAOIGS 67954385844 GROSSISSO 741 78068210129 LASMINI o WIDZ AT TENZO 1 2 3 4 5 6 7 8 9 10 GS Se CGS 15 O a SS In RATS all series begin with entry 1 The first entry of plus i e plus 1 is zero since the previous entry of resids is undefined i e it is not true that resids for period O is negative Both plus 2 and plus 3 are equal to zero since the first two entries of resids are NA hence they are NOT positive Since resids 3 and resids 4 are both negative the fourth and fifth entries of plus are set equal to 1 The same logic prevails throughout plus 10 equals zero since resids 9 is positive In essence the single statement above replaces the more complex and much slower set plus 0 do t 2 N if resids t 0 com plus t 1 1 end dot 128 Walter Enders 3 Estimating a Threshold
25. 0915 92195185 231514842 227888340 22 27 95 3B 2 0 IASON 198144931 OLS MOA a 22 9 074105591 ndent Variable TB3MO Coeff de oW deco eoe e ok dece coe e e dece Coe eec deck RRE o eec RON Pe ERRER RRE R ook KNOCK Koo e Bi Re i e 290217816 578900068 409504186 084036114 467464611 637950862 073016809 28714 2 3 0 8 9 919 085007689 2059 07S 7i 006499189 566413670 003371434 544498028 243383900 076416728 446724622 Sil Clam Jue Tene e 294864692 144076253 492870264 p ALS QOS ALLA 5 DLA ASL 002972876 1098399589 Oy OE GS ey SG LOMO E P E E A E E col I I ORE e cox ex ie 1S ey e de ey en 280102775 oZ TASOS 72010959492 TZA PATA 7241092909 12525500 72140590998 TZA SONARO 220078579 216152434 10556816S SOL SAAT SS US Sicehe O cS 0 0 EU NISI 63222 21686 78858 3598162 05824 TAPISO OSS 16335 99457 34267 02888 e Sa gres 98954030 03200757 32145878 s 7320059 04413868 20465261 52814921 eS Siecle eal 03611 2 a ils s i 752599 owes 06064 38626 02138 15156772 oO OS 10842 5359S 34544 94620 amp 3 ey Goo Gee RE D e cox Cox fe oO GQ Oy QB Cm 102805167 ASUS SZ ILE 01055885 95362828 67010686 xil naa SUONI 03656203 SIE TOL SA 5 LSOZLIZAG 105999912 SYORIS2 S SIES ST 76936005 100259935 69982004 3201071999210 90722
26. 1 0 1 F 4 153 9 16377 with Significance Level 0 00000116 RESTRICT is the most powerful of the hypothesis testing instructions RESTRICT can test multiple linear restrictions on the coefficients and estimate the restricted model Although RESTRICT is a bit difficult to use it can perform the tasks of SUMMARIZE EXCLUDE and TEST Each restriction is entered in the form Bio t Bjo cens Bi ou r where Qu are the coefficients of the estimated model 1 e each coefficient is referred to by its assigned number D are weights you assign to each coefficient and r represents the restricted value of the sum which may be zero To implement the test you type RESTRICT followed by the number of restrictions you want to impose Each restriction entails the use of two supplementary cards The first lists the coefficients to be restricted by their number and the second lists the values of the D and r Examples 1 To test the restriction that the constant equals zero use restrict 1 1 10 The first line instructs RATS to prepare for one restriction The second line is the supplementary card indicating that coefficient 1 i e the constant is to be restricted The third line imposes the restriction 1 0 0 0 2 To test the restriction that Q Qy i e amp O 0 use restrict 1 23 1 10 Again the first line instructs RATS to prepare for one restriction The second line is the supplementary card indicating that co
27. 141 142 143 145 146 147 150 151 152 154 155 158 NEES assunti iv 1 21 22 23 25 30 31 34 35 120 154 155 157 161 162 NEPAR 5 senapsituemieies iv 23 25 33 NONLIN iv 21 24 30 32 33 34 36 37 38 41 42 44 45 75 76 78 155 161 162 OPTION estet rer 47 48 205 207 208 209 211 212 215 216 221 229 244 ORDER daten eo 130 lom E 141 143 Procedures eere eerie 204 229 Q statistiCs rere 7 17 A1 124 QUERY 2 eunte EVER iv 229 230 RATIO ee III iv 51 56 ReQreSSION ceseeeeeeseeereeeeeees 6 160 198 200 RESTRICT ARR iv 8 10 11 33 201 ROBUSTERRORS iv 12 13 14 23 33 100 Schwartz Bayesian Criterion 19 SCRATCH een iv 97 SEED eneenII NIB 139 142 151 SIMULATE ne iv 150 151 152 SPGRAPBE cer re eene 5 110 Structural VARS ccccccccccccessssececececeesenseeees 71 SUBFORMULA 35 36 38 44 SUMMARIZE pooni ai 8 10 SUR ieu RET iv 47 61 62 SYSTEM 47 48 49 53 60 64 105 TES Tee eee er tue 8 9 10 33 Unit root tests Indx 249 Dickey Fuller 136 141 142 143 144 146 213 215 222 231 UNTIG een iv 115 117 118 USERMENU nern iv 235 236 237 239 240 242 243 244 Vector autoregression V AR 1 35 37 38 39 47 48 49 50 51 52 53 55 57 58 60 61 62 63 64 69 70 75 77 82 83
28. 4 Writing a Procedure to Test for Unit Roots eeeesseeeeneeenenn eene emen mener nenne 213 hu MM PM 214 4 1 Creating Local Variables 222i tet tbe t t bete Bie Pe ttes b eda a Pede eu Pea buie 214 ZEN IIPAB IOTER 215 5 Retrieving START and END entry values eene ene e enne 219 lociim EE 220 5 1 Passing Information by Address ieii aii iat nene nennen ene en en nennen 222 5 2 Optional Fieldsi Per iaae ate LE AE O LE A VENEA E ATE 223 locu 224 6 A Procedure for Computing Lag Lengths ssssseseeeeeeee eene ene eene nennen 226 7 Interactitig Watli PFoCed tesicu uii cete ding ee eap aee o auae aede eei agro ade uu oc eaa ae ges 229 Ic ce S 230 S Creating a Menu e ERI gru t ede de d de me dee Fe doe uod E 233 Example MEETUPS 233 8 1 Creating USERMENU 5 itai repeti nen ap eoe Se EEER RR nna e ado ERE RA DXRE caasududa hdd dag Yu a SAHEN RAN deg ace 235 Jazzing up the Procedure iiec iie erre sen eR Fe ecu nx a Ch scduageadaaaenea Fa nera Een uA ER RA nA EE EYE na 237 9 An Interactive Procedure with Menu and USERMENU seen eene emen 239 Jazzing up the Procedure iiie d open ek Fen een na i eame exa nu TERR rai esiin Era aa 242 Jazzing up the Proced re 2 iato Lagen aen secun Ei a C4 seh ne cea XR S URRRFS ae ua ERR a AKA ea nd een ERR aoa 244 Clam Up IEEE 246 DAN CES aie osc TRE T e Relerenc sis
29. 63 3 87 4 36 1960 2 527 10 2379 20 301 11 303 23 2 99 3 65 1960 3 529 90 2383 60 306 48 308 89 2 36 2 90 1960 4 524 60 2352 90 310 93 313 66 2 31 2 81 To help you understand the output from the sample programs several conventions are used concerning typefaces Boldface Within a sample program a RATS instruction set of instructions or a procedure in boldface produces the subsequent sample output Instructions and procedures not in boldface either produce no output or the output is not shown Courier 10 point RATS output is shown in Courier 10 point font The output is either indented or contained in a highlighted box In addition the Courier 10 point font is sometimes used to separate references to a particular program statement from the remainder of the text Italics Many RATS instructions are used with parameters and options that you need to specify The fields that you should specify are italicized For example the ALLOCATE instruction can be used to indicate the terminal date in a data set Since you need to input See the RATS User s Guide for details about working with data sets that are not in an EXCEL format and with variables that are not quarterly Linear and Nonlinear Estimation 3 the terminal value the description of the instruction is written as ALLOCATE date UPPERCASE All text references to file names contained on the data disk are in UPPERCASE In addition textual references to the proper names of R
30. 71 for the initial observation and 168 for the last observation Thus each regression has 97 usable observations Question two of the quiz concerns the use of date notation in a DO loop Since the use of date notation is equivalent to the use of integers it is possible to use date notation for the indices of the DO loop In MONEY_DEM XLS observation 100 is equivalent to date 1988 1 and observation 168 is 2004 1 How could you rewrite lines 3 10 of the program above using date labels Answer do i 2 1988 1 2004 1 lin noprint define ar1 dlrgdp 1959 3 i constant dlrgdp 1 forecast 1 1 arl f arl lin noprint define ar2 dlrgdp 1959 3 i constant dlrgdp 1 to 2 forecast 1 1 ar2f ar2 end do i set pe 1 1988 2 2001 1 dlrgdp f ar1 2 set pe 2 1988 2 2001 1 dlrgdp f ar2 2 Loops over Dates and Series 109 5 Loops for Series One of the most powerful features of RATS is that it allows you to perform a DO loop such that the index refers to a series Here is a routine to create the growth rates of the variables in MONEY DEM XLS Recall that gdp is series 2 rgdp is series 3 m2 is series 4 m3 is series 5 TB3mo is series 6 and TB lyr is series 7 The first part of Program 3 5 reads in the data set MONEY DEM XLS but does not create the variable dlrgdp Instead we can create the growth rate of each series using a DO loop Consider scratch 6 scr no doi 1 6 set scr noci log i 1 0 log i 1 1 end
31. Autoregression The threshold autoregressive TAR model has become popular in that it allows for different degrees of autoregressive decay Consider a two regime version of the threshold autoregressive TAR model developed by Tong 1983 p p Y zl bans a 1 ME i l i where y is the series of interest the ot and D are coefficients to be estimated t is the value of the threshold p is the order of the TAR model and 7 is the Heaviside indicator function lif y T i6 if y lt T The nature of the system is that there are two states of the world In the one state of the world y exceeds the value of the threshold T so that J 1 and 1 J 0 As such y follows the autoregressive process Oo LOtiy i Similarly in the low state y falls short of the threshold c so that J 0 1 J 1 and y follows the autoregressive process Bo Xpiy In a sense there are two attractors or potential equilibrium values In the high state the system is drawn toward Oo 1 L0 and in the low state the system is drawn toward Do 1 XB Moreover the degree of autoregressive decay will differ across the two states if for any value of i a B The key feature of the TAR model is that a sufficiently large shock can cause the system to switch between states PROGRAM 4 3 in the file CHAPTERA 1 PRG illustrates the estimation of a TAR model for the dirgdp series As usual the first five lines of the pr
32. DO loop Since the data are quarterly it seems plausible to perform leg length tests for lags 16 versus 12 12 versus 8 and 8 versus 4 The following instruction can be found in Program 3 4 on the file labeled CHAPTER3 2 PRG The program reads in the data set and creates the three variables dlrgdp dirm2 and drs The next instruction creates a DO loop such that the variable lags runs from 16 to 8 in steps of minus 4 i e lags will equal 16 12 and 8 do lags 16 8 4 Next the SYSTEM END SYSTEM block sets up the three variable VAR using a lag length of 1 to lags system model chap3 vars dlrgdp dlrm2 drs lags to lags det constant end system The ESTIMATE instruction below is the key to the program The system is estimated beginning with observation 1959 2 lags since one observation is lost as a result of differencing and lags observations are lost by incorporating the lagged variables The residuals are stored in unrestrict unrestrict 1 contains the residuals from the dlrgdp equation unrestrict 2 contains the residuals from the dlrm2 equation and unrestrict 3 contains the residuals from the drs equation The following two statements calculate the AIC and SBC for the unrestricted model Note that there are 3 lags 1 coefficients in each of the three equations estimate resids unrestrict noprint 1959 2 lags com aic_u nobs logdet 2 3 lags 1 3 com sbc_u nobs logdet log Ynobs 3 lags 1 3 The second SYSTEM END
33. If you write the procedure using WORD or some other word processing program be sure to save the program in ASCII i e txt format Good programming style dictates that similar files all have the same extension As such most RATS programmers use the extension SRC to indicate a file containing source code Say that the five lines are saved as c winrats bic src To compile the procedure use source c winrats bic src or if you do not want the source code and location numbers displayed to the screen use source noecho c winrats bic src Once you have estimated a regression you can obtain the aic and sbc by simply typing Q bic Writing Your Own Procedures 207 2 Using SWITCH Options The procedure BIC SRC is quite simple since we did not need to pass any information to the procedure Unlike BJIDENT SRC the procedure BIC SRC contains all of the necessary information to compute the aic and the sbc The RATS instruction LINREG or BOXJENK creates the regression variables 96nobs rss and nreg These are the only three pieces of information needed to create the aic and sbc Since all are stored internally in RATS we do not need to send or pass them to the procedure However the most useful procedures perform far more complicated tasks on variables matrices and or entire series or set of series Since a procedure is external to RATS we need a mechanism to pass information from RATS to the procedure itself As discussed in C
34. L can be compared to a X distribution with degrees of freedom equal to the number of restrictions in the system If L exceeds this critical value reject the null hypothesis that the restriction is not binding i e conclude that the restriction is binding The usual form of the RATIO instruction is ratio degrees df mcorr c other options start end series containing the residuals from the unrestricted system series containing the residuals from the restricted system The first supplemental card lists the series containing the residuals from the unrestricted system and the second supplemental card lists the series containing the residuals from the restricted system RATS uses these lists to construct and I l where start end The range over which the test is to be performed degrees dfc The number of degrees of freedom equal to the number of restrictions in the system mcorr c Sims small sample correction for likelihood ratio tests i e the value of C Set mcorr equal to the largest number of parameters estimated in any one of the equations usually equal to the number of parameters estimated in each of the unrestricted equations The other principal option NOPRINT suppresses the printing of the covariance matrices and the marginal significance level of the test It is possible to obtain the marginal significance level with the instruction display signif The likelihood ratio test is based on asymptotic theory
35. LOGDET we can display the multivariate AJC and SBC using compute aic Ynobs logdet 2 2 254 37 compute sbc nobs logdet 2 25 37 log nobs dis aic aic sbc sbc aic 3217 51381 sbc 2952 17634 VARs and Error Correction Models 63 Note that there are 25 regressors in the first two equations and 37 regressors in the third equation If you compare these values of multivariate AJC and SBC to those from the 3 variable VAR you will find that the near VAR has the better fit Now GROUP these three equations into a model called chap2_sur You can obtain 24 impulse responses and 1 step to 24 step ahead forecast error variances from a Choleski decomposition of v using For brevity only a partial list of the impulses is shown group chap2 sur eq eq2 eq3 errors impulses model chap2 sur 3 24 v Responses to Shock in DLRGDP Entry DLRGDP 1 0 006850793364 2 0 000558731002 3 0 000366973202 Responses to Shock in DLRM2 Entry DLRGDP OOS OSHS a8 01465609436 L iL 000000000000 0 0 2 5 3 Responses to Shock in DRS Entry DLRGDP DLRM2 001122502049 000268039828 220100599 0827 50 LRM2 05388227762 04584256661 02529968390 DLRM2 DRS 126303901431 222089855433 089690963848 DRS EC OMS 09367 92 gL SIGS 7TLSA SS 045146339746 DRS 1 0 000000000000 2 0 001322010498 3 051901152595 7 1L 3 1 0 000000000000 0 OR 0022 102 1 5 53210 9 O O OULOVGAZ
36. N replications Efron 1979 shows that such bootstrap estimates converge to the population standard deviation as N goes to infinity Program 4 12 will estimate the standard deviation of the mean using the bootstrap The first two instructions set the default size of a series to 10 and seed the random number generator The third line generates x Note that x is not quite normally distributed since fix 10 96ran 2 generates the integer value of 10 times an i i d normally distributed random number with a standard deviation of 2 Dividing this result by 10 and adding 2 yields the x series listed above 0 8 3 5 0 5 1 7 7 0 0 6 1 3 2 0 1 8 and 0 5 The series mean is created the 100 entries of mean will hold the bootstrap mean values u through Loo all 10 seed 2002 set x 24 f1x 10 ran 2 10 set mean 1 100 0 The instructions in the DO loop will be performed 100 times A series y of length 10 is created Note that fix uniform 1 11 draws the integer value of a uniformly distributed random variable on the interval 1 11 Thus fix uniform 1 11 will be one of the integers from 1 to 10 each selected with a probability of 0 1 Suppose these integers happen to be 2 1 5 6 10 10 4 5 7 and 2 The values of y will be such that y 1 x 2 y 2 x 1 y 3 x 5 y 10 x 2 Hence y is a bootstrap sample it consists of randomly selected values of x drawn with replacement the probability of selecting any part
37. Residuals from dlm3 and Residuals from drs 4 Note that if you use brackets to explicitly create a vector or symmetric matrix you should also specify the data type 1 e integer string series For example vector string 172 Walter Enders 2 The syntax of the supplementary card in LINREG is list of explanatory variables The list of explanatory variables can include labels and or series numbers Suppose that you want to regress the log of M2 Im2 on a constant the log of RGDP Irgdp the log of the GDP deflator lp and a short term interest rate Program 5 1 in the file CHAPTERS PRG reads in the seven series from the data set MONEY DEM XLS and creates these variables If you enter the TABLE instruction you will see that tb3mo is series 6 tb yr is series 7 Irgdp is series 8 m2 is series 9 and lp is series 10 All of the following produce the identical regression output lin Im2 constant Irgdp lp tb3mo lin Im2 08 106 NOTE 0 is equivalent to constant If you type pri 0 you get constant com a ll 0 8 10 6 ll lin Im2 a You could embed this routine in a DOFOR loop to obtain a regression of m2 on each of the short term interest rates dofor i 2 tb3mo tblyr com a ll 0 8 10 1 Il lin Im2 a end do for The first time though the loop i 6 and the regressor list uses tb3mo The second time through the loop i 6 and the regressor list uses tb yr 3 Program 4 1 also illustrates the di
38. SYSTEM block shown below estimates the VAR using four fewer lags 1 e lags 4 You should be careful not to leave any spaces on either side of the plus sign This restricted system is estimated over the same sample period as the unrestricted system 1959 2 lags and the residual series are saved in restrict The AIC and the SBC for the restricted model are calculated and denoted by aic r and sbc r respectively Notice that there are 3 lags 4 1 3 estimated coefficients in the system each of the three equations contains lags 4 lags of each variable plus an intercept 106 Walter Enders system model chap3 vars dlrgdp dlrm2 drs lags 1 to lags 4 det constant end system estimate resids restrict noprint 1959 2 lags com aic_r nobs logdet 2 3 lags 4 1 3 com sbc_r nobs logdet log nobs 3 lags 4 1 3 We display the AIC and SBC for the unrestricted and restricted systems using dis Lags lags aic_u aic_u sbc_u sbc_u dis Lags lags 4 aic_r aic_r sbc_r sbc_r Notice that there are 36 3 4 3 restrictions four lags of three variables in three equations and each unrestricted equation contains 3 lags 1 parameters Hence the next four instructions calculate and display the significance level for the test The final line ends the DO loop the counter lags is decreased by four and the process continues for the next value of lags ratio degrees 3 4 3 mcorr 3 lags 1 noprint 1959 2 lags unr
39. The experience taught me the importance of anticipating every possible choice a user can make My recommendation for debugging the procedure is to experiment with various combinations of choices that might seem implausible to you For example SEASONS SRC will catch a mistaken attempt to take the log of a series containing negative numbers However this might not deter someone from trying to take the growth rate of the same series If you fool around with the procedure you will discover the glitch One way to remedy the problem is to modify TRANSFORM SRC such that portion that calculates the growth rate requires all entries to be positive To make the change you do not have to do much more than copy the instructions from the logarithm choice and paste them into the Growth Rate section choice Growth Rate sta noprint fractiles y if minimum gt 0 com a G label y set os a y y 1 1 if minimum le 0 mes style alert I CANNOT PERFORM THE DESIRED TRANSFORMATION A user might also select CANCEL when selecting a variable to transform or estimate In no other series has been selected previously SEASONS SRC will attempt transform or estimate a constant To clean up the program you can use the STATUS option on the two SELECT instructions The option STATUS INTEGER returns the integer value 0 if the user clicks the Cancel button and a 1 if the user clicks the OK button Thus in the segment below
40. Yuniform 5 15 set y x 0 5 ran 1 The trick to understanding the program is to pretend that we do not know the actual data generating process The next four lines of the program use the NONLIN NLLS block to estimate a and B Note that the NLLS instruction saves the residuals in series e The estimated values of o and p i e and B are saved as alpha_hat and beta_hat nonlin alpha beta frml monte y beta x alpha com alpha 0 48 beta 0 98 nils frml monte y e Variable Coeff Std Error T Stat Signif CK CK CK Ck CK CK CI CI CC CC CK CI CI CIC CK CC CI CI CIC CCS CC CK CI CI CI CCS CK CK CK CI CI CI CC C C CK A M A KG KA x Kk A Ax M 1 ALPHA 0 4194759184 0 1538968309 2 72570 0 00892917 2 BETA 1 1695671363 0 4194023378 2 78865 0 00756525 The parameter estimates are reasonable Note that stopping at this point would require us to use a normal approximation to obtain a confidence interval for the coefficients For a 9096 confidence interval 1 64 standard deviations on either side of alpha gives us an interval of 0 16709 to 0 67187 and two standard deviations on either side of beta gives us an interval of 0 48175 to 1 85739 Thus Step 1 has been completed In preparation for Step 2 the series alpha star and beta star are initialized to contain 1000 entries com alpha hat beta 1 beta hat beta 2 set alpha star 1 1000 0 set beta star 1 1000 0 The first instruction inside the DO loop constructs the bo
41. allowing for an interaction among the AR and MA terms Consider the following bilinear 1 1 specification for dIm3 38 Walter Enders dlm3 o4dlm3 4 Big cidlm3 48 4 The bilinear specification is a way to allow for nonlinear adjustment In a period with 0 the autoregressive coefficient is amp and the moving average coefficient is Bi However the presence of the interaction term cidIm3 means that the degree of autoregressive decay and the coefficients of the moving average will change over time To estimate the bilinear model you can use set temp 0 nonlin al b1 cl var frml e dim3 al dlm3 1 b1 temp 1 cl dlm3 1 temp 1 frml L temp e log var e 2 var Since the bilinear specification contains an MA term it is necessary to use the placeholder temp The NONLIN instruction prepares RATS to estimate the four parameters a1 b1 c and var The first FRML instruction creates the formula e as dlm3 o dlm3 Bye 1 cidlm3 4e 1 The second FRML instruction uses a SUBFORMULA to equate temp and e and to create the log likelihood function The following COMPUTE instruction provides the initial guesses For this example the BFGS method does not work well unless the initial guesses are quite good As such the initial guesses are refined using the SIMPLEX method and the final estimates are reported using the default BFGS method Notice that the maximization begins with observation three on
42. also need an initial guess for the estimated variance var The LINREG instruction creates the internal variable SEESQ equal to the standard error of estimate squared This value is used as the initial guess for var I had difficulty finding a solution using the BFGS method As such two different MAXIMIZE instructions are used The first maximize instruction uses the SIMPLEX method to find the maximum likelihood estimates of a0 al bO bl gamma var The second MAXIMIZE instruction uses these estimates as its initial guesses lin noprint dlm3 constant dlm3 1 com a0 beta 1 al beta 2 b0 1 bl 1 gamma 500 var seesq maximize iters 20 method simplex Istar maximize Istar All of the examples in the remainder of this chapter are in Program 1 4 of the file CHAPTERI PRG Linear and Nonlinear Estimation 35 MAXIMIZE Estimation by Simplex Quarterly Data From 1959 01 To 2001 01 Usable Observations 167 Total Observations 169 Skipped Missing Function Value GOS ARES 5 7 5 Variable Coeff dkok We eoe oe RRR REE SEO eee Hee eoe ERK REE REREE AO 00277148 Al sl P62 TWS BO 00035734 B1 64739304 GAMMA 87711434 VAR 00002487 MAXIMIZE Estimation by BFGS Convergence in 36 Iterations Final criterion was 0 0000000 lt 0 0000100 Quarterly Data From 1959 01 To 2001 01 Usable Observations 167 Total Observations 169 Skipped Missing Function Value MOT Vis HO SIZ Variable Coeff Std Er
43. an IF THEN ELSE block BRANCH back to some key instruction or to jump to almost any other point in the program The key rules are that you cannot jump back to the ALLOCATE instruction into the middle of a nested block such as a DO loop or an IF THEN ELSE block Otherwise there are only two rules about branching The first is that BRANCH works only within a compiled section of a program Second to use BRANCH you must first label the point in the program where you want to branch to You label this point by beginning that line with a colon and then immediately supplying the label Consider the following three labels Examples 100 jump here exit Any one of these three labels is a legitimate jump point labels can be line numbers multiple words or a word that conveys information about the execution of the program The unconditional jump to abel is performed by the instruction BRANCH label 4 1 Sample Program Using BRANCH Suppose that you want to generate an AR 1 process for a series with 100 observations In particular suppose you want to generate a random walk model in the form yr yia i where is an i d d random number with mean zero and variance equal to unity For each series you generate you want to estimate an equation in the form 134 Walter Enders v Bo Biy T i Your goal is to save the estimate of B and repeat the entire process 1000 times Your aim is to obtain information about the dist
44. counter will be decreased by 1 each time through the WHILE loop The variable sign is used to store the significance level of the t statistic for the coefficient for drl lags COMPUTE initializes the variable sign to be 0 5 Note sign can be initialized to be any real number greater than 0 05 As long as sign exceeds 0 05 RATS will loop through the WHILE END WHILE block below com lags 13 sign 5 while sign 0 05 com lags lags lin noprint drl constant drl 1 to lags exclude noprint drl lags com sign signif dis Significance of lag lags sign end while 116 Walter Enders Significance of lag 12 0 92989 Significance of lag 11 0 60897 Significance of lag 10 0 62405 Significance of lag 9 0 32442 Significance of lag 8 0 43783 Significance of lag 7 0 01255 The first time through the loop the variable sign is compared to 0 05 Since sign exceeds 0 05 i e since the condition sign gt 0 05 is true all of the instructions within the block are executed Hence lags is decreased from 13 to 12 and drl is estimated as an AR 12 EXCLUDE calculates but does not display the value of the F statistic for the exclusion restriction that the coefficient on drl 2 0 Of course with only one restriction an F test is equivalent to a t test Note that EXCLUDE creates the internal variable SIGNIF containing the significance level of the restriction The key instruction is COMPUTE sign signif Th
45. diagonal elements are unity yields 1 0 1 023 1 Finally you can obtain the impulse responses with the instruction impulse model term results impulses decomp g 24 Responses to Shock in TB1YR Entry SIWSSIBYAES TB3MO L 06427986990792 Q 6577221963170 2 0 8384894480582 0 9133375666674 3 0 6580803688245 0 6962100440585 Responses to Shock in TB3MO Entry SESE YAIR TB3MO it 0 000000000000 0 243115941753 2 0 0497702790072 0 271975395949 3 00753591399240 OSSIFGIO G6 SIBI COS Hence the impulse responses are identical to those obtained using the instruction impulse model term result impulses 24 s 2 In the 3 variable VAR with dirgdp dlrm2 and drs we obtained impulse responses using system model chap2 var dlrgdp dirm2 drs lags 1 to 12 det constant end system estimate outsigma v errors impulses model chap2 3 24 v 78 Walter Enders Now suppose we want the contemporaneous relationships among the variables to be Cyt Eyr emt 821 yt 237E rt Emt er Ert where ey m and en are the regression residual from the dlrgdp dlrm2 and drs equations and Em and are the pure shocks i e the structural innovations to dlrgdp dlrm2 and drs respectively The economic interpretation is that the unforecastable change in the log of real M2 i e Emr is due to the pure shocks in dirgdp drs and dirm2 Hence we have imposed a standard money demand function on the contemporan
46. doi The first line of the routine creates six new series The series numbers begin at 8 since a total of 7 series reside in memory and run through 13 The variable scr_no contains the integer value 7 Notice that the indices of the DO loop range from 1 through 6 The first time through the loop 1 1 so that series 8 8 scr_no 1 is set equal to the log of series 2 minus the log of series 2 lagged one period The next time through the loop i 2 so that series 9 is set equal to the log of series 3 minus the log of series 3 lagged one period In this fashion series 8 13 contain the logarithmic changes of gdp rgdp m2 m3 tb3mo and tblyr respectively Now we can jazz up the program a bit We know that we can read the label assigned to a series using L variable We can also assign a label to a variable In fact we can make our routine more user friendly by assigning labels to series 8 through 13 Each label will begin with dl and end with the name of the original series being changed e g dlrgdp for the change in the log of rdgp As discussed in Section 2 1 we cannot use EQV in a loop so we will want to use LABELS here Now consider the following modification of our program doi 1 6 setscr noci log i 1 0 log i 1 1 labels scr no 4 DL 1 i 1 end do i tab scr no 1 to scr_no 6 Series Obs Mean Std Error Minimum Maximum DLGDP 168 0 018022 0 009419 0 010052 0 057029 DLRGDP 168 0 008475 0 008960
47. drs Variable Coeff Std Error T Stat Signif KKEKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKK 1 CONSTANT 0 012981373 0 053771900 0 24142 0 80955008 2 AR 2 0 268737166 0 076271837 3 52341 0 00055917 3 AR 7 0 335461768 0 073223473 4 58134 0 00000940 4 MALLI 0 385063544 0 078041223 4 93410 0 00000205 5 MA 3 0 183914603 0 074495456 2 46880 0 01463518 Formulating the model in this form makes our programming problem a bit more complicated than estimating an ARMA 7 3 model The first autoregressive coefficient estimated is 02 and the second is 7 Similarly the first MA coefficient estimated in D and the second is Ds In fact the five estimated coefficients are contained in the vector 7 BETA Note that BETA has only 5 elements BETA 1 0 012981373 BETA 2 0 268737166 BETA 3 0 335461768 BETA 4 0 385063544 and BETA 5 0 183914603 We need to transfer these five values to the appropriate elements of the A and B matrices in the formula used to calculate phi The next two instructions use the variable p to store the number of AR coefficients and q to store the number of MA coefficients The third instruction sets up an indicator called flag equal to the number of coefficients estimated minus p q Note that BETA is a column vector so that ROWS BETA indicates the total number of coefficients estimated Thus flag equals 1 if there is an intercept since the number of coefficients estimated will excee
48. drs 3 Since DATES is the GRAPH default all of the following have the same effect gra nodates 1 gra dates 0 1 com dates 0 graph dates dates Linear and Nonlinear Estimation 15 3 2 Using Choice Options Notice that GRAPH has a number of options requiring you to make choices For example the full set of choices for the location of the key is KEY none upleft upright loleft loright above below left right Similarly STYLE line polygon bar stackbar overlap vertical step symbol For all RATS choice options you can select the choice by its integer value in the choice list Examples 1 Since NONE is the default value and the first choice for KEY and BAR is the third choice for STYLE the following will all produce a bar graph without a key gra style bar 1 gra sty 3 1 com j 23 gra key 1 sty j 1 2 Suppose you saved the residuals from a regression in a series called resids You can obtain the first 12 autocorrelations in a series called corrs and the partial correlations in a series called partial by using cor number 12 partial partial resids cors You can plot an overlapping bar graph of the ACF and PACF without dates containing a header called ACF and PACF and a key at the bottom of the graph either of the following gra nodates sty overlap key below header ACF and PACF 2 f cors partial gra nodates sty 5 key 7 header ACF and PACF 2 cors partial 16 Walter Enders 3 3 Us
49. econometric studies This is when Monte Carlo analysis is most valuable There are two RATS functions that are especially useful for generating random numbers RAN X A random draw from a Normal 0 x distribution UNIFORM L H A random draw from a uniform distribution ranging from lower bound L to upper bound H Examples 1 set x ran 1 This instruction equates each value of x with an i i d random number drawn from a normal distribution with a mean of zero and a standard deviation equal to unity 2 set x uniform 1 1 This instruction equates each value of x with an i i d random number drawn from a uniform distribution with a lower bound of 1 and an upper bound of 1 3 Set y 6 ran 1 set y 2 3 0 5 y 1 sqrt 0 75 ran 1 You might use these two statements to generate an AR 1 sequence with a mean of 6 i e 6 3 1 0 5 an autoregressive parameter of 0 5 and a long run variance of 1 0 The first statement initializes the series to equal the long run mean plus a normally distributed random number with a variance equal to unity The second equates the entries beginning with entry 2 equal to the desired AR 1 process such that the unconditional variance of the y is unity 5 1 A Simple Monte Carlo Experiment A Modified Coin Tossing Problem Suppose you toss a coin and a tetrahedron For the coin you get 1 point for a tail and 2 points for a head The faces of the tetrahedron are labele
50. first EQUATION instruction creates an equation in the form y 1 Over O1Brxe1 ei Notice that NOCONSTANT is used Hence the value of 1 is assigned to the AR 1 coefficient and the value of o is assigned to the coefficient for x 1 Note MORE indicates that an explanatory variable appears on a supplementary card The VARIANCE option is not used Similarly the second EQUATION instruction creates an equation of the form x 1 o0pi xii ayer eo GROUP creates a model called unit using the equations eq and eq2 The first COMPUTE instruction creates the symmetric matrix v the elements of v represent the elements of X Here O1 Ox 1 and O2 05 0 The second COMPUTE instruction initializes the variable success to be zero and the SET instruction initializes the 2000 entries of the series betahat to be zero betahat is used to store the estimated values of B group unit eql eq2 com symmetric v Il 1 0 0 01 0 0 1 0 II com success 0 set betahat 1 2000 0 The instructions within the DO i loop are performed 2000 times SIMULATE creates a Monte Carlo replication of the model unit using the error structure specified by v The simulated series contain 149 observations and begin with entry 2 Note that sims 1 contains the simulated valued of y and sims 2 contains the simulated values of x The LINREG instruction estimates a regression of sims 1 on a constant and sims 2 Only the last 100 observations are u
51. forecasters often estimate a nonlinear series using a quadratic or cubic trend Alternatively the series can be linearized with a logarithmic or a square root transformation The appropriately transformed series can then be estimated with a linear time trend The issue is to compare the fit of the various models The first four lines of Program 1 2 on the file CHAPTER1 PRG read in the data set MONEY_DEM XLS If you plot the series rgdp you will notice that the trend appears to be nonlinear The next four instructions of the program plot the time paths of log rgdp and rgdp log rgdp Irgdp Log transformation sqrt rgdp rgdp rt Square root transformation gra overlay line header Linearized RGDP key below patterns 2 Irgdp rgdp rt 26 Walter Enders Linearized RGDP 92 1 90 P 88 acd e 86 fo Pd 8 2 80 7 6 p ropHper n Ey L 1959 1963 1967 1971 1975 1979 1983 1987 1991 1995 1999 LRGDP moe RGDP RT Each series seems to be linear as such the most appropriate transformation may not be clear Hence you might try estimating rgdp using the following three forms rgdp Oo Ojtime atime Quadratic trend Irgdp Qo Qtime Log transformation rgdp rt Oo Q4time Square root transformation A problem arises in comparing the three estimates since each uses a different dependent variable Instead it is possible to estimate the
52. form the autocorrelations Suppose you want the user of your procedure to supply a series along with the start and end dates The typical format will be procname series start end Your procedure should have the following structure Writing Your Own Procedures 221 procedure procname seriesname start end type series seriesname type integer start end option instructions local instructions inquire series series vl gt gt start v2 gt gt end As in the previous examples the PROCEDURE instruction contains the name of the procedure procname along with a list of all of the parameters what will be passed to the procedure Here we pass a Series seriesname and the start and end entries The second instruction declares seriesname to be a SERIES and the third declares start and end to be integers The fourth instruction equates v and v2 with the start and end values selected by the user If the user does not provide the values start and end v and v2 will automatically default to the first and last dates for which the series is defined To alter UNIT SRC such that we can pass starting and ending dates we need to make the following five modifications to the procedure 1 We need to list all parameters passed to the procedure Since we pass start and end values the first line of the UNIT SRC should be procedure unit y start end 2 We need to instruct RATS that start and end are integers This is done using type integer star
53. in num 5 Once we exit the loop the values of num are printed to the screen If you run the program yourself you will obtain something like ENTRY SUM 129 262 241 252 116 NOP WN rp A score of 1 was never observed a 2 was observed 12 9 of the time a 3 was observed 26 2 of the time and so on In fact the sample proportions are reasonably close to the true probabilities If you increase the number of simulated rolls you should obtain output that is closer to the true probabilities Of course your answers will be somewhat different from mine Your computer will not draw precisely the same random numbers as mine However this can be an undesirable feature of a program Suppose we are debugging a program or want to perform a sensitivity analysis such that we want to use precisely the same random numbers from one computer run to the next This can be done by seeding RATS random number generator in the identical fashion from one run to the next The instruction that does this is SEED Integer If you include SEED 2001 immediately after the allocate statement you should get the same output as that shown above 5 2 Downward Bias in an AR Model It is well known that the OLS estimates of a first order autoregressive process are biased towards zero The size of the bias increases as the magnitude of the autoregressive coefficient increases from zero to unity Consider the following autoregressive model yr Ao Quy
54. instructions ifx dis The value of x equals 2 end if This set of instructions will display The value of x equals 2 only when the value of x equals 2 You need to be careful about the use of the double equal sign since you are not equating x with the value of 2 If by mistake you type IF x 2 RATS will not interpret this as any of the 120 Walter Enders relational statements above Instead the equals sign will cause RATS set the value of x equal to 2 since the condition is TRUE the message always be displayed Note that you do not use the END IF statement if you are within a compiled section of a RATS program For example an IF statement within a DO loop or within a RATS procedure does not need the END IF statement Otherwise you should include the END IF instruction Examples 1 dot 2 2000 4 if valid x t 220 com x t 0 5 x t 1 x t 1 end do t I do not recommend it but suppose you want to fix the missing values in your data set by averaging The valid function returns zero if the argument is a missing value If the data runs from periods 1 through 2001 1 this routine will replace a missing value of x t with the average of the two adjacent values Example 3 in the Section IF x y z below discusses a faster way to perform the averaging 2 sta noprint fractiles drs if minimum gt 0 log drs ldrs if minimum le 0 dis I CAN ONLY TAKE THE LOG OF POSITIVE NUMBERS The STAT
55. last two equations in the form rgdp exp amp Q time e Exponential trend and rgdp time e Squared trend You can estimate the quadratic specification using set time t set t2 t t The squared trend model is equivalent to rgdp Qt 20 0 time a time In fact this is a quadratic trend with only two free coefficients 00 80 60 40 Linear and Nonlinear Estimation 27 lin noprint rgdp constant time t2 compute aic Y nobs log rss 2 nreg compute sbc nobs log rss Yonreg log Ynobs dis The aic aic and sbc sbc The aic 2596 41409 and sbc 2605 80379 To estimate the exponential specification use the following four instructions nonlin a0 a1 frml model 2 rgdp exp a0 al time com 302 1 a1 2 0 1 nlis frml2model 2 rgdp Variable Coeff Std Error T Stat Signif Ck CK Ck Ck Ck CK CI Ck CK C CK CIC CK CC CK CI CK CK CK CIC KC CI C CK C CI C I S KK Kx M KG x ko ko ko ko 1 AO 7 8099164840 0 0064018809 1219 94092 0 00000000 2 Al 0 0078135227 0 0000509728 153 28810 0 00000000 compute aic nobs log rss 2 nreg compute sbc nobs log rss Vonreg log 96nobs dis The aic aic and sbc sbc The aic 2566 27519 and sbc 2572 53499 As such the both the A C and SBC select the exponential specification over the quadratic specification The square root specification can be estimated using nonlin a0 a1 frml model 3
56. level the bracketed instructions are executed The bracketed ENTER instruction adds lags 1 to 4 of dlpgdp to reglist If the null hypothesis cannot be rejected the bracketed ENTER instruction is skipped so that the lagged values of dlpgdp are not added to reglist exclude noprint i 1 to 4 if signif lt 05 enter varying reglist reglist i 1 to 4 Next templist is updated such that the regressors in templist are identical to that in reglist As it turns out the null hypothesis cannot be rejected so that we conclude that the dirgdp sequence does not Granger cause dlm3 At the end of the DOFOR loop reglist contains only the constant enter varying templist reglist end dofor 1 On completing the first pass through the DOFOR loop i dlm3 To be more precise the entire process is repeated for i the integer corresponding to series number of d m3 As such four lags of dlm3 are added to templist and a regression of dlm3 on its own four lags and a constant is estimated The entire process discussed above is repeated for dm3 and subsequently for dr and dlp It turns out that only the lags of dlm3 help to forecast the current value of dIm3 On completing the four loops the final two instructions below estimate a regression containing only those variables passing the Granger causality test lin dlm3 reglist If you execute the program your output will be Vector and Matrix Manipulations 187 Varia
57. models to additional diagnostic checks Modifying the Program The idea of looping over lags is easily extended to selecting the order p and q of an ARMA model The remaining portion of Program 3 3 illustrates the process of fitting an ARMA p q model to the change in the 1 year T bill rate The first line creates drl as the first difference of tblyr Then two DO loops are created The program loops over all values of p and q from 0 to 4 so that a total of 25 ARMA models are estimated Inside the DO loops the BOXJENK instruction estimates an ARMA model without an intercept using the current values of p and q The start date for all models is fixed at 1960 4 in order to ensure that all equations are estimated over the same sample period Note The first two observations of tblyr are NA one observation is lost be differencing and four more are lost due to the four autoregressive lags For each model the value of p and q the AIC and SBC are displayed dif tblyr drl do p 2 0 4 do q 0 4 box ar p ma q noprint drl 1960 4 compute aic Ynobs log rss 2 9onreg compute sbc nobs log rss Ynreg log nobs dis Order p q The aic aic and sbc sbc end do q end do p 104 Walter Enders 0 0 0 0 0 i 1 1 1 il 2 2 2 2 2 3 S 3 3 3 4 4 4 4 4 I CO MEIN fS cedes CON sore ce Ies 99 INS 5 Ce Se COMED fee eo qe 99 di E59 6 ells diss dee ts teh te teh ess e Dra tel tel dea es rs tel fe te
58. obtained the first set of estimations Hence it seems reasonable to conclude that initial guesses near zero lead to a local not a global maximum 82 Walter Enders 6 The Blanchard Quah Decomposition Blanchard and Quah 1989 provide an alternative way to obtain a structural identification Let y be a difference stationary series and let z be stationary Ignoring any deterministic regressors we can estimate a 2 variable VAR of the form p P Ay yan Ay _ nO es i l ih p Dp Lee N a DAY Y 5 02 5 5 i l i l In order to use the Blanchard Quah technique both variables must be in a stationary form Since Ly is 111 we use the first difference of the series If in your own work you find z is also 1 use its first difference in the VAR In contrast to the Sims Bernanke procedure Blanchard and Quah do not directly associate the structural variables 21 and ex with pure shocks to y and z Instead the y and zi sequences are the endogenous variables and the 1 and e2 sequences represent what an economic theorist would call the exogenous variables The structural variables are assumed to be uncorrelated with each other and to have unit variances Although the structural variables are unobserved they are related to the regression residuals by ME ad ezt 2 Ez 2 Changes in x will have no long run effect on the y sequence if En i a 82 an 0 This long run restricti
59. of the data The following instruction returns the seasonal frequency in the variable value INQUIRE SEASONAL value Since we used CALENDAR to instruct RATS that MONEY DEM XLS contains quarterly data we can store the seasonal frequency in the variable v using 220 Walter Enders inq seasonal v dis v 4 It is unlikely that you will ever need to use INQUIRE outside of a procedure In fact the more useful form of INQUIRE is INQUIRE SERIES series name vl gt gt start v2 gt gt end where start and end are the start and end integer values supplied by the user when the procedure is executed The expressions gt gt start and gt gt end instruct RATS to equate v with start and to equate v2 with end if explicit values are given by the user Examples BJIDENT SRC contains the instruction inquire series series nbeg gt gt start nend gt gt end 1 Gbjident rgdp 1959 1 2001 1 Here start 1 and end 2001 1 Similarly nbeg 1 and nend 161 If the user does not provide the values start and end nbeg and nend will automatically default to the first and last dates for which the series 1s defined 2 bjident rgdp 2001 1 Here start is undefined so nbeg becomes the first defined value and nend is 169 Since the first entry for rgdp is 1 nbeg 1 3 bjident tblyr 1959 1 1999 Now start 1 and end 161 so that nbeg 1 and nend 161 Since the first entry of tb yr is 3 RATS uses entries 3 through 161 to
60. procedure shown below uses OPTION SWITCH with the name altform notice that the default value of altform is 0 The IF ELSE block uses altform to determine which form to calculate and display If altform 0 the procedure will calculate and display the aic and the sbc as in our original procedure Otherwise the procedure will calculate and display aic and sbc procedure bic option switch altform 0 if altform 0 compute aic nobs log rss 2 9onreg compute sbc nobs log rss Vonreg log 96nobs display aic aic bic sbc else com aic nobs log rss 2 9onreg ZYnobs log nobs com sbc nobs log rss Ynreg log nobs Ynobs log nobs display aic 2 aic bic sbc end bic 99 Suppose you have just estimated drs using LIN drs constant drs 1 to 7 Since the default value of altform is 0 all of the following will cause the procedure to display the aic and sbc in logarithmic form bic bic noaltform bic altform 0 com ii 0 bic altform ii Similarly all of the following will display the alternate form of the aic and sbc bic altform 1 Since altform 1 the ELSE Block is executed bic altform Turns ON the altform OPTION bic altform 2 Since altform 0 the ELSE portion of the procedure is executed 38 Note that aic and aic will necessarily select the same model since aic is a monotonic transformation of aic Similarly sbc and sbc will sel
61. procedure should contain instructions that are similar to 210 Walter Enders if trans eq 1 gral y if trans eq 2 dif y dy gra 1 dy if trans eq 3 log y ly dif ly dly gra 1 dly If the option trans is left unspecified or set equal to 1 the routine will produce a graph of y If the user sets trans 2 the time path of the first difference of y will be shown and if the user sets trans 3 the time path of the growth rate of y will be shown Writing Your Own Procedures 211 3 Passing Series to a Procedure Usually you will want a procedure to perform one or more operations on a series As an experienced RATS user you will have passed a series along with its start and end dates to a procedure For example we can use BJIDENT SRC to construct the ACF and the PACF for the series digdp over the sample period 1959 1 through 1985 4 using bjident dlrgdp 1985 4 In essence you are passing the sample values of the series gdp along with the integer values of the start and end dates to the procedure There must be a way for the procedure to recognize the type of information it is being sent This is done by listing all of the parameters series integer values and matrices that can be passed on the first line of the procedure If you actually open BJIDENT SRC you will see that the first thirteen lines are proc bjident series start end type series series type integer start end option integer diff 0 opti
62. rgdp a0 al time 2 com a0 1 al 0 1 nlls frml model_3 rgdp compute aic gonobs log 4orss 2 nreg compute sbc nobs log rss Vonreg log 96nobs dis The aic aic and sbc sbc The aic 2649 14346 and sbc 2655 40326 Note that the A C and SBC rate this model as the one with the poorest fit 5 The Logistic Smooth Transition Autoregressive LSTAR Model The LSTAR model generalizes the standard autoregressive model to allow for a varying degree of autoregressive decay In its simplest form the LSTAR model can be represented by To allow the threshold T to be something other than zero use 0 1 exp Yos oJ 28 Walter Enders p p y Ay ay 6 B By 5 i l i l where 0 1 exp yy _ and y gt 0 is a scale parameter In the limit as y gt 0 or the LSTAR model becomes an AR p model since each value of 0 is constant As y oo 0 0 so that the behavior of y is given by Oto Gyr OpYep Similarly as y 0 1 so that the behavior of y is given by Qt Bo o B1 Yi Ap Bp yc For intermediate values of y the degree of autoregressive decay depends on the value of y You can see the effects of y using Program 1 3 on the file CHAPTERI PRG Since we are not interested in using calendar dates we omit the CAL instruction The first instruction sets the default length of a series to 101 entries The second instruction crea
63. sample mean is 1 87 and the standard deviation is 2 098 Next we will draw 100 bootstrap samples in order to estimate the standard deviation of the mean Each bootstrap sample consists of 10 randomly selected values of x drawn with replacement each of the 10 values listed above is drawn with a probability of 0 1 It might seem that this resampling repeatedly selects the same sample However by sampling with replacement some elements of x will appear more than once in the bootstrap sample The first three bootstrap samples might be t p 3 4 5 6 7 8 9 10 W xi 35 17 0 5 0 5 18 20 17 0 6 0 6 7 0 1 89 x 0 5 06 06 08 17 70 18 3 18 0 8 1 81 X3 0 5 0 6 70 13 13 70 13 L8 35 0 6 2 49 where x denotes bootstrap sample i IF Statements and Monte Carlo Experiments 159 Notice that 0 6 and 1 7 appear twice in the first bootstrap sample 0 6 0 8 and 1 8 appear twice in the second and that 1 3 appears three times in the third bootstrap sample As such the sample means are not identical For 100 such bootstrap samples the standard deviation of the 100 values of u is the estimate of the standard deviation of the mean Ou The algorithm is as follows 1 Select N 100 bootstrap samples each consisting of 10 data points N should be between 25 and 200 for estimating standard errors 2 Evaluate the parameter of interest Uu for each bootstrap sample 3 Estimate the standard deviation of the mean by the sample standard deviation of the
64. series 1 gdp is series 2 rgdp is series 3 The series dlrgdp created with the SET instruction is series 8 As you create additional series RATS stores each consecutively Thus the series resids created by the LINREG instruction is series 9 As discussed above you can just use the series number instead of its label Anywhere RATS expects a series name you can simply use the sequence number You do need to make sure that you reference sequence numbers as integers and not floating point numbers Thus you can print out the first values of rgdp using pril43 Recall that the syntax for the PRINT instruction is PRINT start end series list Thus pri 1 4 3 instructs RATS to print from entry 1 through 4 the values of series 3 If you follow the logic you know that it is possible to print the first four values of rgdp and dlrgdp using pril438 ENTRY RGDP DLRGDP 1959 01 2273 0 NA 1959 02 2332 4 0 025797235495 1959703 2331 4 0 000428834862 1959 04 2339 1 0 003297294498 Care must be taken if a series is on the right hand side of a FRML SET or COM instruction since RATS will interpret the integer as a scalar In fact whenever it is ambiguous you can force RATS to use the series instead of an integer if you use series number Examples 1 To print the last two years of the rgdp series you can use pri 2001 7 3 Loops over Dates and Series 95 2 Estimate an AR 1 autoregression of the logarithmic change in rgdp using the first
65. set alpha hat 1 1000 0 set beta hat 1 1000 0 The next three instructions prepare RATS to perform a nonlinear estimation of the model and provide initial guesses of and f The statements within the DO loop will be performed 1000 times The 50 values of y are created as xp plus an i i d disturbance drawn from a standardized Normal distribution The NLLS instruction actually perform the estimation NONLIN alpha beta FRML monte y beta x alpha COM alpha 0 48 beta 0 98 do i 1 1000 set y x 0 5 ran 1 NLLS frml monte noprint y The next instruction stores the estimated value of in the i th entry of alpha hat and the estimated value of D as the i th entry of beta hat On exiting the DO loop the TABLE instruction is used to provide the summary statistics for alpha hat and beta hat com alpha hat 1 alpha beta hat 1 beta end do i table alpha hat beta hat 156 Walter Enders Series Obs Mean Std Error Minimum Maximum ALPHA_HAT 1000 0 50061759618 0 14924811080 0 00760025999 1 00199740576 BETA_HAT 1000 1 06013032231 0 37423173774 0 30586651854 2 97669650556 The first half of the sample is quite similar to the second half of the sample Consider table 500 alpha_hat beta_hat Series Obs Mean Std Error Minimum Maximum ALPHA_HAT 500 0 50526653814 0 14860337610 0 00760025999 0 93395005105 BETA_HAT 500 1 04927811557 0 38210832158 0 33895348276 2 97669650556 table 501 alpha hat beta hat S
66. tet dis score To this point we have done nothing but replicate the possible outcome of the game However it is simple to modify our program to replicate the game 1000 times We can then calculate the proportion of instances in which we get scores of 2 3 4 5 and 6 If the Monte Carlo method works these sample proportions should come close to the true probabilities We will use the series num to hold the number of times we roll a 1 2 3 4 5 and 6 Of course the number of times a 1 is obtained will be zero For example num 4 will equal the number of times a score of 4 is obtained Consider the instructions from Program 4 5 on the file labeled CHAPTERA 1 PRG all 6 set num 0 do j 1 1000 compute coin fix if uniform 1 1 gt 0 2 1 com tet fix uniform 1 0 5 0 compute num coin tet num coin tet 1 add 1 to sum total faces end do j print num The ALLOCATE instruction sets the default series length equal to 6 we need only six entries to hold the series num Line 2 initializes all entries of num to equal zero The DO loop indicates that the lines 4 through 6 will be performed 1000 times Line 4 is the simulated coin toss and line 5 is the simulated tetrahedron toss To understand line 6 recall that coin tet is equal to the score Thus line 6 increments the appropriate entry of score by 1 For example if coin tet 5 IF Statements and Monte Carlo Experiments 139 line 6 adds 1 to the value stored
67. that may not be very useful in the small samples available to time series econometricians Moreover the likelihood ratio test is only 52 Walter Enders applicable when one model is a restricted version of the other Alternative test criteria are the multivariate generalizations of the AJC and SBC AIC T loglXl 2 N SBC T logl amp N log T where X determinant of the variance covariance matrix of the residuals and N total number of parameters estimated in all equations Thus if each equation in an n variable VAR has p lags and an intercept N n p n each of the n equations has np lagged regressors and an intercept Note that fora VAR ESTIMATE creates the following variables NOBS Number of usable observations LOGDET Log determinant of the estimate of 2 SIGMA Covariance matrix of residuals When you use the OUTSIGMA option on the ESTIMATE statement RATS computes the covariance matrix of the residuals You can fetch the logarithmic determinant of this covariance matrix using LOGDET The following three statements will compute and display the multivariate versions of the A C and SBC respectively compute aic Ynobs logdet 2 N compute sbc Ynobs logdet N log Zonobs dis aic aic sbc sbc where you must set N to equal the number of parameters estimated in the system VARs and Error Correction Models 53 1 1 Innovation Accounting The variance decomposition and impulse response functi
68. the loop i 1 and RATS performs the operation COM a 1 1 The second time through the loop i 2 and RATS performs the operation COM a 2 2 This procedure continues through i 10 A more efficient way to perform the same task is to use declare vector integer a 10 ewise a 1 2 i Here EWISE sets element a 1 1 element a 2 2 The index i runs from 1 though the dimension of the array For RECTANGULAR and SYMMETRIC arrays EWISE is equivalent to the use of multiple DO loops Consider EWISE a i j formula in i and j Here EWISE sets element i j according to the specified formula For example declare rect integer ix 3 3 ewise ix 1 j z1 j dis ix 0ST 2 i 0 c 2 Xx nm Note that you do not need to be too careful about the fact that i and j are integers while the elements of the matrix ix might be real Consider 178 Walter Enders declare rect real ix 3 3 ewise ix i j zi j dis ix 0 00000 1 00000 2 00000 1 00000 0 00000 1 0000 2 00000 1 00000 0 00000 2 2 Selecting ARMA Coefficients The BOXJENK instruction also allows you to use vectors to input information The syntax and principal options of the BOXJENK instruction are boxjenk options depvar start end residuals where depvar The dependent variable start end The range to use in the estimation residuals The name of the series to call the residuals The important options for our purposes are CONSTANT You must specify constant if y
69. the program is recall that 6l 1 is a string equal to the label of series i Hence dl 9 61 1 refers to the label dl plus the label of series i The first time through the loop i 3 Since the label dirgdp does not exist s dl l i creates a series with the name dlrgdp as the logarithmic change in series i The second time through the loop i is equal to the integer value of m2 and l i is the string m2 Since there is no series named dlm2 s dl 1 i creates this series as the logarithmic change in m2 Similarly the third time through the loop the logarithmic change in tb3mo is created Unlike the LABELS instruction the names creates by S label can be used for input and for output Loops over Dates and Series 113 6 2 DOFOR and ENTRIES In Section 3 1 it was indicated that any RATS instruction containing supplementary cards in a regression format allows you to use the ENTRIES option However any object you include on the supplementary card except or a space is counted as an entry Thus dirgdp 1 is counted as four entries dlrgdp 1 and Similarly dlrgdp 1 to p is counted as 6 entries dirgdp 1 to and p Suppose you want to determine whether the contemporaneous and lagged growth rate of M3 dim3 affected the growth rate of real GDP You want to include contemporaneous money growth and the possibility that lags 1 to 4 of money growth are important Since it is unclear whether the AR 1 or A
70. the spread The second FRML instruction creates the conditional variance as h bO b1 e 1 2 cl temp 1 The third FRML statement uses a SUBFORMULA to equate temp with h and to define the log likelihood Ly set temp 0 nonlin a0 al a2 a3 bO bl cl b0 ge 0 bl ge 0 cl ge 0 frml e spread a0 al spread 1 a2 spread 2 a3 spread 3 frml h bO b1 e 1 2 cl temp 1 frml L temp h log temp e 2 temp After initializing the parameters with the LINREG and COMPUTE instructions the first MAXIMIZE instruction refines the initial guesses with the SIMPLEX method The second obtains the final estimates using to BFGS method lin noprint spread constant spread 1 to 3 com a0 beta 1 al beta 2 a2 9obeta 3 a3 9obeta 4 bO 96seesq bl 0 2 cl 0 5 max method simplex iters 5 L 7 Linear and Nonlinear Estimation 45 max iters 200 L 7 MAXIMIZE Estimation by BFGS Convergence in 26 Iterations Final criterion was 0 0000017 0 0000100 Quarterly Data From 1960 03 To 2001 01 Usable Observations 13 Function Value 332 16269201 Variable Coelo Swe ol iatis L Sic etn Signif Ck CK CK CK CI Ck CK CI CK CI CK C CK CI CK CC CC CK C C CI CK CC C CI CK Ck CI Ck I C I x KK Kk x Kx Mk Kk ko ko ko ko ko AO 3107497108419 16 2 MOTOTS Saks S167 35193616 2298 0810 0 0 00 5219 Al sel SES 5105957 7 0 5 3 15 81048 00000000 A2 242706983 075485128 RAPEN 00130310 A3 SES SNNT 00
71. the system using estimate noprint residuals resids 12 Notice that we used the NOPRINT option a three variable VAR with twelve lags produces a substantial amount of output Since we are not sure if we actually want the 12 lag model we suppress the output The residuals are saved in the vector of series resids12 resids12 1 contains the residuals from the first regression resids12 2 contains the residuals from the second regression and resids12 3 contains the residuals from the third regression The next three lines are used to compute and display the multivariate AJC and SBC Notice that N 37 3 since there are thirty seven estimated coefficients in each of the three equations of the 14 system compute aic Ynobs logdet 2 37 3 compute sbc Ynobs logdet 37 3 log nobs dis aic aic sbc sbe aic 3198 00556 sbc 2859 47155 In order to perform a lag length test we re estimate the system using only 8 lags Notice that we restrict the estimation to begin in 1962 2 so that both systems are estimated over the same sample period The residuals are saved in the vector resids Hence 14 The function EQNSIZE number returns the number of regressors in equation number and EQNSIZE O returns the number of regressors the most recently estimated equation Since equations 1 2 and 3 each contain the same number of regressions an alternative is to use compute aic Ynobs logdet 2 3 eqnsize 1 56 Walter Enders
72. time to complete the 50000 replications TASK 4 Calculate the Sample F statistic exclude noprint zplus zminus compute f j cdstat end do j TASK 5 statistics fractiles f IF Statements and Monte Carlo Experiments 149 Statistics on Series F Observations 50000 Sample Mean 1 6524588263 Variance LASHES 97 Standard Error 1 1741364996 SE of Sample Mean OR OOS Zor ise Sis aL Se ake 314 70023 Signif Level Mean 0 0 00000000 Skewness is FOS Signif Level Sk 0 0 00000000 Kurtosis 5 01475 Signif Level Ku 0 0 00000000 Jarque Bera WES 2E 2 27 Signif Level JB 0 0 00000000 Minimum 0 0013651902 Maximum 14 4743025649 01 ile 0 1977081049 99 ile OINGOISIGNIOPI9IODIO 05 iie 0 3681509883 95 ile 3 9233591336 l 10 ile 0 5025774653 90 Sile Sor 908038 25 Sile Og SiS PATA AL 75 Sile Zy A SILAS OIS Median Ib BSP IO MSG ILS Suppose that you estimated a series as a threshold process Ay LPY 1 D payri lif y 20 where J 0 if Ju 0 If the sample F statistic of the null hypothesis p p2 0 was 3 5 you would be able to reject the null hypothesis at the 10 significance level but not the 5 level You can modify the program by i including an INFOBOX ii obtaining the critical values for additional sample sizes and iii obtaining the critical values when Chan s 1993 method see Section 3 1 in this chapter is used to estimate the threshold 5 5 Inference in a Cointegrat
73. to example 1 the SET command applies the 96IF function operates to the entire series In fact this single statement works just like the following DO loop set dummy 0 dot 1 2001 1 if t ge 1992 1 com dummy t 1 end do 1 Although both yield the identical values for dummy there is one major difference The actual DO loop works many times slower than the SET with IF The following two programs both create a dummy variable equal to zero for observations 1 to 249 and equal to one for observations 250 through 500 In order to ascertain the differential speed of the DO loop versus the implied DO loop if the SET with IF each program performs this task 10000 times Program 1 took 1 minute and 18 seconds to run while Program 2 took only 12 seconds Program 1 Program 2 all 500 all 500 set dummy 0 set dummy 0 do i 1 10000 do i 1 10000 do t 1 500 set dummy if t ge 250 1 0 if t ge 250 com dummy t 1 end do i end dot end do i minute 18 seconds 12 seconds IF Statements and Monte Carlo Experiments 127 The point is to avoid DO loops if possible since they are slow The implied DO loop of the IF instruction used in conjunction with SET is an efficient programming technique Reconsider the example where we fixed a missing value by averaging do t 2 2000 4 if valid x t 2 2 0 com x t 0 5 x t 1 x t 1 end do t The following is much faster set x 2 2000 4 if Yvalid x x
74. to use matrix manipulations on your data set it is important to realize that RATS does not treat a series in the same way as a vector For example a vector is a one dimensional array such that the elements have subscripts that run from 1 to N In order to manipulate the vector each element needs to be defined Unlike a vector you can manipulate a series even if it has ranges that are undefined or NA The key point is that RATS treats the two differently to perform any matrix manipulations on your data set you need to create vectors from your series You can also create a RECTANGULAR matrix of series such that you can refer to each element by its row and column In a RECTANGULAR matrix columns represent variables and rows represent observations Thus element i j is the element in the i th row of the j th column Similarly element i j is the i th observation of variable j You can create matrices from series using the MAKE instruction The syntax is MAKE array start end numobs numvars list of variables numobs INTEGER used by RATS to return the number of observations i e the number of rows numvars INTEGER used by RATS to return the number of variables i e the number of columns Options EQUATION equation supplying variables LASTREG use regressors from last regression NOTE Omit the supplementary card with either of the above TRANS set up the transpose of the observation array Examples For all of the examples below r
75. used the same SEED integer Notice that 90 of the f statistics exceeded 2 57 95 exceeded 2 88 and 99 exceeded 3 49 IF Statements and Monte Carlo Experiments 143 If you look at the Dickey Fuller table you will see that the at the 10 5 and 1 significance levels are 2 58 2 89 and 3 51 respectively Thus suppose you had a sample with 100 observations and estimated the series in question as an AR 1 process If it turned out that the estimated value of p 05 so that the estimated value of Qt 0 95 with a standard error of 0 02 could you reject the null hypothesis p 0 The answer is no Although the estimate 6 is 2 5 standard deviations away from unity it is not permissible to use a traditional t statistic Instead the Monte Carlo results show that when the true value of p 0 i e 0 1 so that the true data generating process is a random walk we would obtain a t statistic that exceeds 2 5748048837 i e is less than 2 5748048837 in absolute value more than 90 of the time As such at the 1 5 and 10 significance levels we cannot reject the null hypothesis p 0 Power of the Test Now that we know critical values for the Dickey Fuller test it is instructive to show how to ascertain its power Since the Dickey Fuller confidence intervals exceed those for the usual t test it is to be expected that the power of the Dickey Fuller test is low Program 4 8 on the file labeled CHAPTERA 1 illustrates a way to det
76. v tn emen periit e e Eder Ferret tee ke aa e iia painiar ia 114 7 oops with While and Until estea eie tenerte he eate hen da ge edge inen dde re aga ken eas da su Roanne 115 Frequently Asked QuestiOnsz 1 nro egent i Pete aee testen id EE tkd seus ipee e 117 Chapter 4 IF Statements and Monte Carlo Experiments 119 Examples ee rine 120 I TIfzThen Else I Oe A eiie Itn tede eden biet EEE E 121 lacu sees 121 1 1 Sample Program Lag Lengths Again sessseeeeeeeenen eene eene e enne 123 2 The 96IF X y z Punctlon c ceeds iioc ca ce desee cundo cut epu ev eee reca daaa cede vv pines 126 locnm 126 3 Estimating a Threshold Autoregression sssssseseeeeeeeeeee eene nennen nn en nennen 128 3 1 Estimating the Threshold iss uie edet rer er aao Penn eig aea douane a a 130 A BEACH NE RN 133 locii p E 133 4 1 Sample Program Using BRANCH eene eene nee rne nennen ene enne 133 5 Morite Carlo Experiments 5 2 2er eee dd dee needs eid eg Eu erre a rea e epu egens 136 ERAN OS me RM 137 5 1 A Simple Monte Carlo Experiment seeeeeeeeseeeee enne enne en eterne ennt nennen 137 5 2 Downward Bias in an AR Mod l rnision aieeaa eene ener ene e enne 139 5 3 Powerof the Dickey Fuller Test uie iit dena ad nn n ti a un Reo E eau genre e ERR MARRY eda 141 Power of the Pests iiie aito eei p
77. you run the program your output should be The aic 725 43529 and sbc 119 23556 The aic 727 49692 and sbc 718 19732 The aic 728 35931 and sbc 719 05971 The aic 729 80200 and sbc 717 40253 The aic 724 17933 and sbc 702 48027 The aic 725 39124 and sbc 700 59230 The SBC selects the model with only one lag of dirgdp However if you estimate the model selected by the AIC your output will look like lin dirgdp 6 constant dlrgdp 1 to 2 dlm3 Variable Coeff Std Error T Stat Signif OC ck C KKK KKK KKK KKK KKK KKK KKK KKK KKK KKK KKK KKK KKK KKK KKK KKK KK KK KKK KK KK KK KKK ko ko 1 Constant 0 0023202979 0 0016286184 1 42470 0 15619062 2 DLRGDP 1 0 2347026242 0 0766291214 3 06284 0 00257298 3 DLRGDP 2 0 1412306952 0 0766582885 1 84234 0 06727619 4 DLM3 0 1475857849 0 0715417775 2 06293 0 04073517 Jazzing Up the Program The problem with ENTRIES is that we have to count the number of entries to use from the supplementary card As illustrated above this can be tricky when you use braces However RATS allows you to replace the individual entries on a supplemental card with a vector You use the ENTER instruction to manipulate the items in the vector By changing the contents of the vector you modify the information on the supplementary card The first few examples in the chapter on matrices Chapter 5 cover this technique in great detail Loops over Dates and Series 115
78. yr are stored in impulses 1 1 and impulses 2 1 respectively The response of tblyr and tb3mo to innovations in tb3mo are stored in impulses 1 2 and impulses 2 2 respectively The following two graphs were created using spgraph hfields 2 vfi 1 header Impulse Response Functions graph header Shocks to the One Year Rate key upright number 0 patterns klabels implabels 2 impulses 1 1 impulses 2 1 graph header Shocks to the 3 month Rate key upright number 0 patterns klabels implabels 2 impulses 1 2 impulses 2 2 spgraph done 18 Note that the impulses are not scaled since all are in the same units 68 Walter Enders Impulse Response Functions Shocks to the One Year Rate Shocks to the 3 Month Rate 0 30 Notice that both graphs use the KLABELS option This option instructs RATS to use the vector implabels to label each of the series The first element in implabels is 1 year and the second is 3 month As such the series on the first supplemental card is labeled year and the series on the second supplemental card is labeled tb3mo The program MONTEVAR PRG allows you to place confidence bands around your impulse response functions The program is distributed with RATS so that it should in the same directory as RATS itself Now that you know how to work with VARs it should be trivial for you to modify MONTEV AR PRG so as obtain confidence intervals for the impulse
79. 0 020598 0 037804 DLRM2 168 0 017045 0 008553 0 002633 0 053028 DLM3 168 0 019167 0 009285 0 005379 0 038578 DLTB3MO 168 0 003286 0 109786 0 332247 0 396932 DLTB1YR 166 0 000104 0 102851 0 329450 0 288489 Note that LABELS does not allow you to refer to a series by its label For example you will obtain an error message if you type TABLE dlrgdp 110 Walter Enders 6 The DOFOR Instruction The DO instruction forces a particular relationship between subsequent values of the index The index of a DO instruction must be an integer and the index is increased by the same amount from one loop to the next However there are many instances in which we do not want one value of the index to bear any precise relationship to the adjacent values In such circumstances DOFOR is particularly helpful Program 3 6 illustrates a simple way to create multiple graphs on a page After reading in MONEY DEM XLS the program creates the logarithmic changes in real gdp the gdp deflator and m3 and the difference in the 3 month T bill rate drs using set dlrgdp log rgdp log rgdp 1 set dim3 log m3 log m3 1 dif tb3mo drs set price gdp rgdp set dlp log price log price 1 The SPGRAPH instruction below indicates that we want to create a Special Graph with the header Graphs of Four Principal Series The graphs are to be arranged into two horizontal fields and two vertical fields spgraph hea Graph of Four Principal Series hfi 2
80. 00000000 compute aic Ynobs log rss 2 9onreg compute sbc nobs log rss gVonreg log 96nobs dis The aic aic and sbc sbc The aic 908 23350 and sbc 901 99752 Diagnostic checking of the residuals reveals no evidence of any significant autocorrelations However as correlation coefficients are measures of linear association it is possible that money supply growth displays nonlinear adjustment As such we might want to estimate dlm3 as the following LSTAR model B B dlm3 _ E dlm3 Q a dlm3 _ 1 exp ydlm3 _ t 30 Walter Enders Consider the following instructions nonlin a0 al bO bl gamma gamma ge 0 frml Istar dlm3 a0 al dlm3 1 bO b1 dlm3 1 1 exp gamma dlm3 1 lin noprint dlm3 constant dlm3 1 com a0 beta 1 al beta 2 b0 1 bl 1 gamma 500 The NONLIN instruction indicates that we want to estimate the five parameters a0 al bO b1 and gamma Moreover the value of gamma is constrained to be greater or equal to zero The FRML instruction creates the desired formula and assigns it the name star Next we need the initial guesses The LINREG instruction estimates dlm3 as an AR 1 model The COMPUTE instruction uses these estimates to obtain the initial guesses for a0 and al The initial guesses for b0 and b1 were obtained by trial and error The NLLS instruction below instructs RATS to estimate dIm3 using the formula named star nlis frml I
81. 004559080262 002463374844 DLRM2 002312008411 000947945412 000666465186 DLRM2 000606481672 001450397016 QIONINISOIORINIKQNIROS DRS SZ GSNPATSIOSSIBEIOIS 063402066434 128481679716 DRS SEISSLOISIG OU ONIIO 20 69 52 9280 29 10 9755910 9162916 DRS 463405065412 296041401790 038876550562 Since we normalized the shocks to have unit variances the interpretation of the absolute magnitudes of the impulse responses is unclear We can scale the responses of each variable in terms of standard deviations The scaled responses to a fiscal policy shock are obtained using set r7 1 12 impulses 1 3 sigma 1 1 5 set r8 1 12 impulses 2 3 sigma 2 2 5 set r9 1 12 impulses 3 3 sigma 3 3 5 88 Walter Enders A graph of the scaled responses is obtained from com implabels II dlrgdp dlrm2 drs ll graph header Responses to a Fiscal Shock key upright number 1 klabels implabels patterns 3 r7 r8 r9 Initially the fiscal shock acts to increase dirgdp however by the third quarter dirgdp is negative By construction the cumulated change in d rgdp zero Similarly the fiscal shock is estimated to create a sharp increase in the short term interest rate Note that drs is positive for the first two quarters Thereafter drs seems to fluctuate around zero so that the cumulated change in the 3 month bill rate is positive This is possible since we did not impose any restriction concer
82. 036283 3078523550 Sa EOR Se CASS Do cs E 64532 64532 61654 61654 10485 10485 SSMO Ey CP fex e y ey SV Oe 00000008 00000008 00040262 00040262 03690688 03690688 01229020 Notice that the F statistic with three degrees of freedom in the numerator and 153 in the denominator is 2 25276 with a significance level of 0 08449735 Hence at the 596 significance it is possible to reject the null hypothesis and conclude that the restriction is binding At the 1046 significance we accept the null hypothesis 12 Walter Enders 3 The LINREG Options LINREG has many options that will be illustrated in the following chapters The usual syntax of LINREG is linreg options depvar start end residuals list where depvar The dependent variable start end The range to use in the regression The default is the largest common range of all variables in the regression residuals Series name for the residuals Omit if you do not want to save the regression residuals list The list of explanatory variables The most useful options for our purposes are DEFINE You can name the equation by setting DEFINE equal to the name you choose Later you can refer to the equation by its name PRINT NOPRINT Print the regression output VCV NOVCV Print the covariance correlation matrix of the coefficients ENTRIES Number of entries to use from the supplementary card all Note that LINGEG also contains options for
83. 05 boot num beta2 star fix 995 boot num Confidence intervals for beta2 10 0 01137 0 24317 5 0 03765 0 25895 1 0 09205 0 3096 An alternative way to construct confidence intervals it to use the bootstrapped f statistics for the null hypothesis B Program 4 14a not discussed here shows how to construct confidence intervals using the Bootstrapped T Statistics Chapter 5 Vector and Matrix Manipulations Although RATS is not intended to be a matrix programming language it has evolved to the point where you can use it to perform very complicated econometric tasks entirely in matrix notation In fact you can create various vectors and matrices from your data set For example you can create a Y vector an X matrix and obtain the ordinary least squares OLS coefficient estimates from X X x v I will show you how to do that and more in the section entitled Making Matrices from Your Data However most RATS users will not want their programs to consist entirely of matrix manipulations One of the strengths of RATS is that you can use vectors and matrices to complement the existing instruction set Programming in a pure matrix language can be cumbersome it is a lot easier to let RATS perform some of the matrix manipulations for you For example the LINREG instruction creates the coefficient vector beta and the XO matrix You can call and manipulate both of these matrices without having to construct them yourself
84. 10 48573 0 00000000 2 TB3MO Q9 osh2slto 2 0 O OS Q2 6 5449 89 30172 0 00000000 We can obtain the first 12 residual autocorrelations and partial autocorrelations using cor number 12 partial partial resids cors Correlations of Series RESIDS Quarterly Data From 1959 03 To 2001 01 Autocorrelations is 6820700 075663894 0 243 2555 Q1 3 34 99 0 9 5190 49 0 1102 41514 J3 0 00965375 0 0551392 00 0106175 O 0239909 O 01S2915 O 0564s74 Partial Autocorrelations Le 06820700 ORES ACOA CR OPSIESS OI SX NE S ORBI ICI ME Oral OS gt 980 MEL R07119 2 2 5 Ta ONSISO S NEL OO 4AM eA i6m 00550733 0 0300 TOI 3 367M OM OA OS 685 DJUNG Om SCEL SELOS QVZ 118 3055 Significance Level 0 00000000 As expected the residual autocorrelations seem to decay at a geometric rate Notice that we used the QSTATS option this option produces the Ljung Box Q statistic for the null hypothesis that all 12 autocorrelations are zero Clearly this null is rejected at any conventional significance level If we wanted additional Q statistics we could have also used the SPAN option For example if we wanted to produce the Q statistics for lags 4 8 and 12 we could use cor number 12 partial partial qstats span 4 resids cors Since we are quite sure that the autocorrelations differ from zero we dispense with this option As indicated in the example above we can graph the ACF and PACF using gra nodates number 0 sty 5 key 7 heade
85. 100 observations set 8 log rgdp log rgdp 1 lin 8 100 constant 8 1 As before the first line creates the logarithmic change of rgdp However there is no label attached to the series it is just referred to as 8 The second line prepares RATS to estimate a regression with series 8 as the dependent variable such that the last sample point is observation 100 The third line instructs RATS to include a constant and the lagged value of series 8 in the regression 3 set y log 2 versus set y log series 2 The first instruction sets each entry of y equal to the natural log of 2 hence all values of y are 0 69315 The second statement sets each entry of y equal to the natural log of the corresponding entry of series 2 Suppose that the first four entries of series 2 are 1 4 2 and 6 The second statement sets the first four values of y to be 0 1 38629 0 69315 and 1 79176 4 Suppose that the series y is the second series in RATS memory All of the following create the growth rate of y set gy log y log y 1 set gy log series 2 log series 2 1 set gy log 2 0 log 2 1 The first instruction creates gy as the log of the current value of y less the lag of the previous period s log of y The second instruction uses square brackets to distinguish between the number 2 and series number 2 Notice that it is necessary to use the construction log series 2 1 log series 2 1 creates an error mes
86. 122 Walter Enders do p 1 12 lin noprint dlrgdp 13 constant dlrgdp 1 to pj compute aic Ynobs log rss 2 Yonreg if p eq l or aic lt aic min com p_opt p com aic min aic end do p The ELSE Block If the condition for the IF Block fails you may want the RATS to execute an alternative set of statements The syntax is If condition program statements Else Alternate program statements j End if The first set of statements will be executed if the condition is true otherwise the alternate set of program statements will be executed Again you do not need the braces for a single statement You should not use the END IF instruction if you are within a compiled section of a program ELSE IF Blocks Oftentimes in your programming you will encounter a situation where there is not an either or choice You might have a number of possible states and want to execute different sets of instructions for each possible state In such circumstances you can use ELSE IF blocks each consists of a set of instructions that is executed if the appropriate condition is met The typical structure of a complex IF instruction is IF condition I first set of statements Else if condition 2 second set of statements Else last set of statements End if if not within a compiled program segment IF Statements and Monte Carlo Experiments 123 Note RATS performs the first IF or ELSE IF condition that is true If a
87. 1369878 SE of Sample Mean 0 50 0 59 55 11 ESIC ee sk Sica LOS 57 9 6L Signif Level Mean 0 0 00000000 Skewness 0 46258 Signif Level Sk 0 0 00000000 0 47716964 0 00000000 Kurtosis 0 07801 Signif Level Ku 0 JB 0 Jarque Bera WLS 19S dL 7 Signif Level Minimum 2399 SASS Maximum 3107230084 01 ile 0732848428 99 Sile 1432856243 05 ile 1554922902 95 Sile 0196162470 10 ile 2617774667 90 Sile 9400523351 Byres 4552469550 Voile 8244537550 Median 6507239420 The percentage of successes is 16 85000 Notice that 16 896 of the confidence intervals contained the true value of B Moreover the average value was 0 6271085179 whereas the actual value was 1 0 Clearly it is inappropriate to use the usual confidence intervals for B Now if you rerun the program using p 0 5 you will obtain The percentage of successes is 33 30000 Hence the level of D affects the accuracy of the 95 confidence interval Moreover the value of the elements of 2 are also important If you replace the elements of v with com symmetric v zl 1 51 5 1 Il you will obtain The percentage of successes is 56 95000 154 Walter Enders 6 Antithetic Random Variables A complete Monte Carlo experiment involves many replications and a substantial amount of computer time It is typical to obtain a sampling distribution over a number of different parameter values and sample sizes In the Enders Granger 1998 ex
88. 2 needs to be modified for a time series model due to the presence of lagged dependent variables As such the bootstrap y is constructed in a slightly different manner Consider the simple AR 1 model yr Bo Biy E As in Step 2 we can construct a bootstrap sample of the error terms e containing the elements e e irs e Now the bootstrap y sequence using this sample of error terms In particular given the estimates of Bo and B and an initial condition for yi the remaining values of the y sequence can be constructed using yi By Pi Vex e For higher order AR p models initial conditions for y through y are needed Typically these values are selected by random draws from the y sequence Program 4 14 uses this method to bootstrap the coefficients of an AR 2 model of the dirgdp in the form dirgdp Bo Bidlrgdpia Bodirgdp z The issue is that the coefficient of dlrgdp 2 has a significance level of 0 0756 However since the residuals are not normally distributed it might be wise to use the bootstrap to obtain confidence intervals for B The first six lines of Program 4 14 read in the data set MONEY_DEM XLS seed the random number generator with 2001 and estimate the model lin dirgdp resids dlirgdp 1 to 2 constant 31 A second although less common bootstrapping technique used in time series models is called Moving Blocks For an AR p process select a length L that is longer than p L
89. 2212237 Thus 0 004968068719 is the first defined entry of the series called resids 2 and 0 000793836267 is the second defined entry of the same series You can reference the individual elements as follows dis resids 2 14 0 00497 Next we want to decompose the variance covariance matrix using a Choleski decomposition You can display the variance covariance matrix v and the Choleski decomposition of v using dis v 4 25841e 05 8 43189e 06 2 83336e 05 7 56122e 04 6 56400e 04 0 30310 dis decomp v 0 00653 0 00000 0 00000 0 00129 0 00516 0 00000 0 11587 0 15611 0 51507 Vector and Matrix Manipulations 181 Hence v B B where B decomp v To obtain impulse responses it is desirable to use the transpose of this matrix You can compute the matrix G as the transpose of decomp v with com g tr decomp s dis Decomposed Matrix g Decomposed Matrix 0 00653 0 00129 0 11587 0 00000 0 00516 0 15611 0 00000 0 00000 0 51507 You can obtain the autocorrelations of the residuals using the DO loop below The first time through the loop i 1 so that the autocorrelations of resids 1 are displayed doi 1 3 cor qstats span 4 resids 1 end do Sections 4 through 6 of Chapter 2 contain a number of examples involving structural VARs In a structural VAR you model the individual elements of the matrix g 182 Walter Enders 3 Example ENTER and Supplementary Cards RATS allows you to replace the individu
90. 3mo comj j l set scr no j log i 0 log 1 labels scr_no j DL l i end dofor table scr_no 1 scr_no 2 scr_no 3 Series Obs Mean Std Error Minimum Maximum DLRGDP 168 0 0084752669 0 0089595106 0 0205982814 0 0378040869 DLM2 168 0 0170454250 0 0085532148 0 0026327924 0 0530276961 DLTB3MO 168 0 0032859057 0 1097860047 0 3322470533 0 3969315930 Now only three series are created from scratch beginning with scr no 1l The first time through the loop i is the integer value of rdgp i e 1 3 and j 1 The SET instruction creates series 8 note the scr no 1 8 as the logarithmic change in rgdp series 3 The second time through the loop j is incremented by 1 and series 9 is created as the logarithmic change in M2 Similarly the last time through the loop series 10 is created as the logarithmic change in tb3mo One final way to improve the program is to use the S L function As mentioned earlier S L returns the series number corresponding to the label L You can verify that rgdp is the third series in RATS memory by entering the instructions com a s rgdp dis a 3 A very useful feature of S L is that it will create a series with the name L if it does not already exist Suppose you have read in MONEY DEM XLS An alternative way to create the logarithmic changes in rgdp m2 and tb3mo is dofor i 2 rgdp m2 tb3mo set s dl 1 i log 1 0 loga 1 end dofor The key to understanding
91. 55 211296040285 3 0 000473985466 0 000059458005 088298485503 Responses to Shock in DLRM2 LEONE es DLRGDP DLRM2 DRS al 0 000000000000 0 005163725798 SIESIGSIBIBIPZ 9I65 59 2 0 000686946710 0 004308977026 ERE OWAIEO REI S OQ OOLGILSAI9ISA Qs 00 20 2 0102191126 2102 9 9 9 69 9 081 95 Responses to Shock in DRS Entry DLRGDP DLRM2 DRS dL 0 000000000000 0 000000000000 oO LO 7 2 1L 117 Tis 2 0 001437493942 0 002210813500 200554826822 3 0 001143540867 0 001410062827 132047370809 Decomposition of Variance for Series DLRGDP Step Std Error DLRGDP DLRM2 DRS 1 0 006525649 100 000 0 000 0 000 8 0 008207901 69 424 15 408 ILS LGS 12 0 008465552 66 607 16 607 LG FSS 24 0 008657572 65 424 16 476 18 100 Decomposition of Variance for Series DLRM2 sep S Sc M ETTSTSONS DLRGDP DLRM2 DRS 1 0 005322934 Hewes 94 107 0 000 8 0 008874702 Do Sas SZIOISIO 11415 SE 12 0 009251525 Sa ILS TE B9S 1 IJS 24 0 009627022 6 524 TE TALS 18 761 Decomposition of Variance for Series DRS Step Std Error DLRGDP DLRM2 1 0 550543146 4 429 8 041 8 0 692450601 15 969 9 a2 Jj 12 079995922 2c 50 1L db FO 24 0 824410351 SAS OS 12 064 60 Walter Enders The variance decompositions suggest a rich interaction among the variables particularly at the longer forecast horizons For example dirgdp explains all of its 1 step ahead forecast error variance but dlrm2 and drs explain 15 408 and 15 168 percent of the 8 step ahead forecast
92. 6 so that the loop is not performed a fourth time doi 6 3 2 Here i begins at 6 and is decreased by 2 every time the loop is completed At the end of the first loop i 4 and the loop is completed a second time At the end of the 5th loop i 4 so the loop is not completed a sixth time 100 Walter Enders 3 1 DO Loops Switches and Choices Since all Switch and Choice options have an integer representation they can be selected by the index of a DO loop Consider the following Examples 1 doi 0 1 lin robusterrors i drs constant drs 1 to 7 end do i In Chapter 1 the change in the short term interest rate drs was estimated as an AR 7 with and without the ROBUSTERRORS option The program segment estimates an AR 7 model with and without ROBUSTERRORS Recall that ROBUSTERRORS is OFF when its value equals 0 and is ON when its value equals 1 2 Any RATS instruction containing supplementary cards in a regression format allows you to use the ENTRIES option The syntax for the option is entries 2 number of entries to process For example the instructions below will produce three regression equations The first will regress y on a constant the second will regress y on a constant and x and the third will regress y on a constant x and m doi 1 3 lin entries i y constant x z end doi 3 The RATS procedure BJIDENT SRC will display the autocorrelation and_ partial autocorrelation functions for the series you specify
93. 7 Loops with While and Until The DO loop is appropriate if you know exactly how many loops you want to make However there are circumstances in which the number of repetitions is unclear For example a common way to select the a lag length in an AR p model is to estimate the autoregression equation using the largest value of p deemed reasonable If the t statistic on the coefficient for lag p is insignificant at some pre specified level estimate an AR p 1 and repeat the process until the last lag is statistically significant If you used a DO loop it would be necessary to estimate every autoregression from p to 1 A more efficient procedure is to stop the process once a significant lag is found The syntax for a WHILE block is while condition block of statements executed as long as condition is true end while omit if the WHILE block is nested inside another compiled section The syntax for an UNTIL block is until condition block of statements executed as long as condition is true end until omit if the WHILE block is nested inside another compiled section The remaining portion of Program 3 8 uses the WHILE instruction to perform the type of lag length test discussed above Suppose that we want to fit an AR p model for drl with no more than 12 lags and that we want the t statistic for the last coefficient to be significant at the 5 level The COMPUTE instruction initializes the variable lags to be 13 This
94. 7 coefficients in the unrestricted equations of the system The elements along the principal diagonal of the 15 Note that resids 2 contains the series resids1 2 1 resids12 2 and resids12 3 and that resids8 contains the series resids 1 resids 2 and resids8 3 Thus it is sufficient to use resids 2 on the first supplementary card and resids8 on the second supplementary card The identical output is obtained using ratio degrees 4 3 3 mcorr 37 1962 2 resids12 1 resids12 2 resids12 3 resids8 1 resids8 2 resids8 3 VARs and Error Correction Models 57 Covariance Correlation Matrices are the autocovariances of the residuals For example in the 12 lag model the variance of the residuals from the drs equation is 0 30309775570 The residual covariances are in the lower portion of the matrices i e below the diagonal and the residual correlations are in the upper portion of the matrices The log determinants of the unrestricted and restricted models are 21 923113 and 21 458149 respectively Given the calculated value of x 55 330714 the restriction is binding at the 5 but not the 1 significance level Thus the AJC and likelihood ratio test both select the 12 lag model Block Exogeneity We can perform a block exogeneity test to determine whether lags of drs enter the equations for dirgdp and dirm2 The name is a bit misleading I prefer to use the term block exclusion test If lags of drs can be excluded
95. 84 85 86 87 105 171 179 181 182 183 184 185 187 195 MeCIOTS S Ee eU T RECETAS E 168 249 250 Walter Enders References Bernanke B 1986 Alternative Explanations of Money Income Correlation Carnegie Rochester Conference Series on Public Policy 25 pp 49 100 Blanchard O and D Quah 1989 The Dynamic Effects of Aggregate Demand and Supply Disturbances American Economic Review 79 pp 655 673 Chan K S 1993 Consistent and Limiting Distribution of the Least Squares Estimator of a Threshold Autoregressive Model Annals of Statistics pp 520 533 Davidson R and J G MacKinnon 1993 Estimation and Inference in Econometrics Oxford University Press Oxford Dickey D A and W A Fuller 1979 Distribution of the Estimates for Autoregressive time Series With a Unit Root Journal of the American Statistical Association 74 pp 427 431 Efron B 1979 Bootstrap Methods Another Look at the Jackknife Annals of Statistics 7 pp 1 26 Enders W and C W J Granger 1998 Unit Root Tests and Asymmetric Adjustment With an Example Using the Term Structure of Interest Rates Journal of Business and Economic Statistics 16 pp 304 11 Engle R 1982 Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation Econometrica 50 pp 987 1007 Engle R D Lilien and R Robbins 1987 Estimating Time Varying Risk Premium in
96. 864 SLOPE SZ 66 Walter Enders Step 4 To obtain the impulse responses and variance decompositions use the instruction ERRORS IMPULSES MODEL model For the interest rate example we can use errors impulses model term 2 24 s Responses to Shock in TBlYR Entry TBLYR TB3MO L Os G427 986990792 O65 71227963055 2 O OSOd ood 49055 oM O GiLIas 7 One aAS 3 0 6580803688211 0 6962100440500 Responses to Shock in TB3MO Entry SMBs IL NOIR TB3MO 1 0 000000000000 0 243115941739 2 Quo en 0 59 MEO SIONIS XS EU S 0 073515992403 0 180662886845 Decomposition of Variance for Series TBlYR ge Me TORERO TBLYR TB3MO 1 0 642798699 100 000 0 000 CUM PEIECIS ZIP NNIRS 98 828 iL 1072 IA 2 LOOEIOIONIKOE 99125 POS 24 4 042565038 CIO Oe 0 900 Decomposition of Variance for Series TB3MO Step Std Error Amey IL YOR TB3MO POR ONSZAIRG 5t S76 978 1172 5 OZ AL 8 2 463471405 D6 Vall 21 599 IA EOS EIESO SOIT 97 004 2129916 Zal AUGS 98 009 iL 9 9l As in the previous example RATS produces the impulse responses for periods 1 through 24 only the first three impulses are shown here RATS displays the variance decompositions for all forecast horizons through 24 we display only the 1 step 8 step 12 step and 24 step ahead forecast error variances To create graphs of the impulse responses it is helpful to know a bit about matrices Chapter 5 considers the construction and manipulation of matrices in great detail For now it is sufficien
97. A n x n diagonal matrix from an n x 1 or 1 x n SOLVE A B Solves the problem Ax B where A is an NXN array of known values B is an Nx1 array of known values and x is an Nxl array of unknowns The function returns x as an Nx1 RECTANGULAR array In addition to the usual conformability restrictions you need to be sure that the matrices contain the same type of variables For example if A is a 2 x 2 matrix of real numbers and is a 2 x 2 identity matrix of integers A is not permissible 2 1 Operations on Subcomponents of a Matrix One way to manipulate an element is through the COMPUTE instruction Consider the following examples 1 com a 1 5 3 0 Here the COMPUTE instruction equates the element in the first row of the fifth column of the matrix A with the real number 3 0 2 com x a 1 5 Here the COMPUTE instruction equates the variable x with the element in the first row of the fifth of the matrix A Sometimes there is a particular relationship among the elements of a matrix You can exploit this relationship using the EWISE instruction To use EWISE you must dimension the matrix For a vector EWISE contains an implied DO loop Consider Vector and Matrix Manipulations 177 EWISE a i 2 formula in i Suppose you wanted to construct a vector of integers running from 1 to 10 One way to do this would be to declare vector integer a 10 doi 1 10 com a i i end doi 12345678910 The first time through
98. A TAY TAY TAS TAS e a N GAY ToS AX DAS TAS To Ta a A KARA a To a E l a E r NE hah e TAS a A SK MG hahaha AeA MAG Ae ha Am hn AET 1324 0 247417925 0 062562 114 s DIATE 0 00007661 Za 923 0 002782858 0 000749196 3 71446 0 00020364 Notice that RATS displays the log likelihood of the restricted and the unrestricted models Hence we might want to relax one of the four restrictions since the difference between the log likelihoods is significant at the 0 00784853 level Nevertheless we can obtain the impulse responses using impulse model chap2 decomp g 24 Responses to Shock in DLRGDP Entry DLRGDP DLRM2 DRS Hi 0 006525648724 001614562466 000000000000 2 0 000674747862 sOOL SOUSA 7 5 1E 1632828312602 S 0 000810331530 000383503057 o MISOLINI SLG Responses to Shock in DLRM2 Entry DLRGDP DLRM2 DRS AL 0 000000000000 004941736423 000000000000 2 0 001074367700 003482474955 044400414535 S OR OOM ZEA 258 4 95 001927339970 007632342684 Responses to Shock in DRS Entry DLRGDP DLRM2 DRS 1 0 000000000000 0 001532083485 0 550543146080 2 Ts 001203397095 ME OPI Id L2 24 10 O ZZS LST GEGA Sa OOO Sao Wm 0 002 100925 4455 0 IA S0 o 772 3E To graph the impulse responses we need to create a matrix to hold the nine series there are three responses to each of the three shocks The next two instructions creates a 3 x 3 matrix called impulses Be aware that each element of impulses is a series The RESULT option info
99. ATS instructions and procedures are all in upper case RATS itself does not distinguish between UPPERCASE and lowercase characters Within the sample programs all names are in lower case If you have the file in your a drive you can read in the data set using the following four lines contained in Program 1 1 in the file CHAPTER1 PRG cal 195914 all 2001 1 open data a money_dem xls Alter this line if the data set is not on drive a data org obs format xls Note that only the first three letters of the CALENDAR and ALLOCATE instructions have been used in fact any RATS instruction can be called using only the first three letters of its name If you use the TABLE instruction your output should be table Series Obs Mean Std Error Minimum Maximum DATE 169 1979 876331 12 232185 1959 100000 2001 100000 GDP 169 3572 739053 2873 158128 496 100000 10243 600000 RGDP 169 5142 364497 1950 840494 2273 000000 9439 900000 M2 169 1904 835266 1399 706717 287 800000 5043 710000 M3 169 2414 462229 1916 764710 290 053333 7260 136667 TB3MO 169 5 915148 2 590483 2 303333 152053333 TB1YR 167 6 153872 2 393622 2 713333 14 380000 Many of the examples presented will use the growth rates of M2 M3 and real GDP the first differences of the 3 month and 1 year T bill rates and the rate of inflation as measured by the growth rate of the GDP deflator You can create these six variables using set dlrgdp log rgdp log rgdp 1 set dim2 l
100. After compiling the procedure the syntax to EXECUTE the procedure is execute bjident options series start end or if you use as a shortcut for EXE 26 You need to be a bit careful since any object you include on the supplementary card except or a space is counted as an entry Thus dirgdp 1 is counted as four entries dlrgdp 1 and Similarly dlrgdp 1 to pj is counted as 6 entries dlrgdp 1 to and p Loops over Dates and Series 101 bjident options series start end where start end The range of the series to use for constructing the autocorrelations and partial autocorrelations The most useful options are DIFF Maximum regular differencings 0 SDIFFS Maximum seasonal differencings 0 TRANS NONE LOG ROOT Transformation to apply to data GRAPH NOGRAPH Do High resolution graphs SPAN Seasonal span Since TRANS has three choices you can obtain the ACF and PACF of a series y the log of y and the square root of y using doi 1 3 bjident trans 1 y end do i Similarly you can obtain the ACF and PACF of y and its first and second differences using do i 2 0 2 bjident diff i end do i You can also manipulate the start or end entry For example to obtain the ACF and PACF of y using observations 1 though 100 150 and 200 use do i 100 200 50 bjident y 1 end doi 102 Walter Enders 3 2 Lag Length Tests Now scroll down Program 3 3 and form the growth rate of real GDP using
101. Bau 5E There are two cases to consider In the first case you completely specify all of the numerical values of the gj In such circumstances you simply create the G matrix and enter the appropriate values for all of the gj In the second case you fix at least n ny2 elements of G However since the remaining values of the gj are free parameters they need to be estimated from the data This second case is the most typical and forms the basis of the Sims Bernanke and Blanchard Quah decompositions Nevertheless we begin with simple case where G is known 4 1 Structural VARs with a Known G Matrix It is straightforward to perform a structural decomposition when G is known Chapter 5 describes how to work with matrices in RATS For now it is sufficient to know that you can use the COMPUTE to construct the matrix G and enter the desired numerical values for the g Then use the DECOMP G option on the ERRORS or IMPULSE instruction Since you are not performing a decomposition using the covariance matrix from the ESTIMATE instruction do not specify the covariance matrix in the name field of the ERRORS of IMPULSE instruction Examples 1 In the error correcting model of the term structure relationship variance decomposition and impulse responses were obtained using a Choleski decomposition Moreover 24 impulse responses were obtained and stored in the matrix impulses using 72 Walter Enders impulse model term result impulses 24 s R
102. CE 1 The two USERMENU instructions enable the Estimate the Trend and the Estimate the Trend and Seasonals choices If the user selects Estimate the Trend MENUCHOICE 22 s_ 0 and if the user selects Estimate the Trend and Seasonals MENUCHOICE 23 s 1 Both of these choices pass depvar and the value of s to the procedure ESTIMATE If Done is selected 7MENUCHOICE 4 and the procedure exits the LOOP loop usermenu if menuchoice eq 1 f Original Instructions usermenu action modify enable yes 2 usermenu action modify enable yes 3 if Zbmenuchoice eq 2 coms 0 estimate depvar s_ if menuchoice eq 3 f coms 1 estimate depvar s_ j 244 Walter Enders if menuchoice eq 4 break end loop It is also necessary to modify ESTIMATE Note that s_ is included on the parameter list and is defined as an integer Moreover span is a local integer that will be set equal to the seasonal span of the data i e span 4 for quarterly data 12 for monthly data The seasonal span is determined on the CALENDAR instruction proc estimate depvar s_ s_ is included on the parameter list type integer s_ s is an integer type series depvar local integer span Span is an integer representing the seasonal span of the data local series time t2 t3 fitted dec vector int reglist compute reglist llconstantll The following IF block is added to the procedure If the user selected Estimate the Trend s 0 so th
103. CI CIC CC C CI CI CI CCS C C CK CI CI CI CSS C Ck Ck SK A M A KG KG x Kk ko ko ko ko koX 1 Constant 0 327483556 0 148546781 2 20458 0 02892885 2 TB1YR 1 1 259723680 0 077483059 16 25805 0 00000000 3 TBIYR 2 0 608960910 0 119220538 5 10785 0 00000093 4 TB1YR 3 0 552438699 0 119096075 4 63860 0 00000731 5 TBIYR 4 0 255732154 0 077249623 3 31046 0 00115404 With 3 lags the estimate of rho 0 05253 The t statistic for the null hypothesis rho 0 is 2 33526 LAGLENGTH SRC selects a lag length of 4 if you selected the YES button you see the output from UNIT SRC such that the augmented Dickey Fuller test contains bestlags 1 lags If you want to include these two procedures in your research you could make several small modifications a Notice that LAGLENGTH SRC displays regression output with 4 lags and UNIT SRC displays with 3 lags the estimate of rho You can modify LAGLENGTH SRC such that it displays the message In the augmented Dickey Fuller test The output will look like In the augmented Dickey Fuller test With 3 lags the estimate of rho 0 05253 The t statistic for the null hypothesis rho 0 is 2 33526 b It is important that the user compile UNIT SRC prior to compiling LAGLENGTH SRC One way to ensure this is done correctly is to include both procedures on a single file in the desired order Save the file using a descriptive name and the extension SRC When the user compiles the file all of the procedu
104. Creates a matrix or vector from data series Vector and Matrix Manipulations 169 Whenever you create a matrix using the DECLARE instruction you need to inform RATS about three features of the matrix First you can create different types of matrices The types will usually be rectangular an r x c matrix where r rows and c columns vector a one dimensional array symmetric a symmetric r x r matrix Second you need to DIMENSION the matrix by indicating the number of rows and columns it contains You can DIMENSION the matrix directly on the DECLARE instruction or on a DIM instruction It is useful to know that matrices can be redimensioned within any program However you cannot DECLARE a matrix as a different data type For example if A is a vector of integers you cannot DECLARE A to be a vector of real numbers or RECTANGULAR matrix of integers Third you need to instruct RATS concerning the type of information contained in the matrix A matrix might contain one or more series or a set of integers real numbers or string variables 1 1 Declare DECLARE is the most general instruction for creating a matrix DECLARE allows you to completely specify the type of the matrix i e rectangular vector or symmetric along with its dimension and the contents of the elements The syntax for DECLARE is DECLARE matrix type list of names where matrix type Will usually be rectangular vector or symmetric By default the elemen
105. Examples of Nonlinear Least Squares eseeeesssseeeeeeenen eene en eene nnns 24 5 Maximum Likelihood Estimation sseeseeeeeeeeeeene enne nen enne en enne rine enters 32 lacus me EMITE 34 GSGARCH Models 5 nr enituit a Sut EC ict ECL ILE is DI ESL EN ERE 40 6 1 Examples of GARCH Processes aiana aa ener irri rrr riri rri 41 Chapter 2 VARs and Error Correction Models 47 Examples o as a0 uctus Oatodbsae sin aS ud South Ute in dl en cac 48 1 Hypothesis Testing and Model Selection sssseeeenenen eene nennen 51 1 1 Innovati n od DIPL 53 EXAMPLES LLLI 53 2 Example Estimation of a 3 Equation VAR sssssssee eene ener rne e enne enne 55 Bl ck EXo5enetty niteat dti e RR HRS S ORE RR ERRNER SEE REC iaia EAR Ea SEE RENI a RECETAS TRE TR 57 Innovation ACCOUDLT 1 cea da e ea enge k ua nihi aan RB cue EXE Ee xe kB as SER Ra esae za Me RR Pe He 4T ea ena Dna rara 58 EXTENSIONS REQUESTED LIPS 60 2 LE N6ais ARS ines ctp tS E erem x sh fac A inte cee deeat Mabe dae eevee 61 lcu RR 62 3 Brror Correc onModels3 55 either eher oe Ferte neret Medea eec et adaa aeia 64 4 Structural De compoSItI Ds s ern ro dene a a Pes xor eerie cec ete pend denda vers 69 4 1 Structural VARs with a Known G Matrix cseeeeeeeeseeee eene emere ennemi nnns 71 Example Sona o sime asit tU rete secedere ferire fe ofc Uer SLM LA Loft b aati trent cabo eoa 71 5 The Si
106. INE POLY GON BAR STACKEDBAR OVERLAPBAR VERTICAL STEP S YMBOL PATTERNS By default graphs use colors to distinguish series if possible NOPATTERNS and dashed lines and hatched patterns otherwise PATTERNS forces the latter to be used The program illustrates the use of the SPGRAPH instruction to place multiple graphs on a single page The first time SPGRAPH is encountered RATS is told to expect a total of four graphs The layout is such that two will go in the horizontal field HFIELD and two in the vertical field VFIELD The option HEADER produces Graphs of Five Principal Series as the header for the full page The individual headers on the four GRAPH instructions produce the headers on the individual panels The instruction SPGRAPH DONE instructs RATS to produce the output shown on the next page 6 Walter Enders 2 Linear Regression and Hypothesis Testing The LINREG instruction is the backbone of RATS and it is necessary to review its use As such suppose you want to estimate the first difference of the 3 month T bill rate i e drs as the autoregressive process 7 drs Q Yajdrs t i l The next two lines of the program estimate the model over the entire sample period less the seven usable observations lost as a result of the lags and the additional usable observation lost as a result of differencing and save the residuals in a series called resids linreg drs resids constant drs 1 to 7 Linear Re
107. ISTICS instruction with the FRACTILES option stores the smallest value of drs in the internal variable minimum If the smallest value of drs exceeds zero RATS forms the new series drs as the log of drs If the smallest value of drs is negative or zero RATS will display a warning message 3 if converged lt gt 1 dis Model did not converge A number of RATS instructions including NLLS and MAXIMIZE produce the internal variable converged Note that converged 1 if the nonlinear estimation converged within the allowable number of iterations and is equal to zero otherwise Here the warning message Model did not converge is displayed if the previous nonlinear estimation did not converge 4 if converged 0 dis Model did not converge The program will display Model did not converge if convergence is not obtained However the following contains a serious mistake if converged 0 dis Model did not converge IF Statements and Monte Carlo Experiments 121 The program will never display Model did not converge since the equal sign instead of a double equal sign was used In my own programs I try to avoid this possible source of error and use EQ instead of 1 If Then Else Blocks If you want to execute a block of statements when a condition is met simply include them in braces as illustrated below IFx 0 program statements END IF omit the END when the IF occurs inside a compiled sections Al
108. K KKK KK KKK KK KKK KK KKK KKK KKKEKK 1 DLRGDP 1 0 1006658743 OOS S977 3 56 128403 OO 20092557 Pre DEEST UNIT 0s 3695359027 7 0707193955893 Se Ceia S OCTO OO ORTRQ RIS Se DRS 0 0015984145 0 0008299146 Ls 9219000 00558237V 4 Constant 0 0047686768 0 0009110037 E92 2 5 NEG EQUO OOO Dependent Variable DLRM2 Variable Coeff Std Error 1L SE ETE Signif Ck CK CK Ck Ck CK CI Ck CK CC CK CI C CK CC CC C CK CC CK CI CK CC CC CK CC CI CK IC CI Ck CI CI Ck x x KK Kk Kk x Mk ko ko ko E LRGDP 1 0 060434793 QOGAN 059752 0 33084820 LRM2 1 00291102216 0 056862167 11 06504 0 00000000 RS 1 OmUO SCS O27 0 000655978 5 5 5 7030 0 00000010 Constant 0 002338028 0 000720072 SZ 0 00141302 ndent Variable DRS Coc isis Ste CINE ETSI ORIS JS EE Signif ACkCkck KKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKK LRGDP 1 12 09953395 To 729 15245 256128 OFS EA OB TSO LRM2 1 VoBVIIOLLOS PELUAUSIOG 7 LZS 0 26300908 ESSE ORGIES S 607727318 0 08203892 STSL 0 06281438 Constant m ORBIS UO OP AD 0 09005476 s THESIS 0 08858167 The ESTIMATE instruction creates the 3 x 3 matrix e VARLAGSUMS We can display this matrix using dis varlagsums 0 89933 0 36554 0 00160 0 06043 0 37082 0 00366 12 09953 7 98709 0 84633 86 Walter Enders Notice that the coefficient on di rgdp in the first equation i e ai1 1 equals 0 1006658743 and the first element of varlagsums is 0 89933 Hence varlagsu
109. LTZSS O SHES EAA MOS SS 198307071178 148405838091 At this point you could produce 12 out of sample forecasts beginning with 2001 2 with FORECAST model chap2_sur results fores 12 2001 2 64 Walter Enders 3 Error Correction Models RATS works a bit differently if you want to estimate an error correction model In Chapter 1 we established a long run relationship between the 1 year and 3 month T bill rates Recall that the estimated long run relationship is tbl yr 0 6980794657 0 9167216207 tb3mo As such we might want to estimate an error correction model of the form drl Qioltb 1yr 0 6980794657 0 9167216207 tb3mo Aui D drlii Ai2 L drsy 1 ei drs Obo tb LTyr 0 6080794657 0 9167216207 tb3mo Ao L drl Ao L drs eo where A L are polynomials in the lag operator L The steps in setting up the VAR including the error correction term are a bit different Step 1 Estimate the long run equilibrium relationship using the DEFINE option on the LINREG instruction This step allows you to pass the estimated coefficients from LINREG to the VAR system Thus in the interest rate example we can use lin define spread tblyr resids constant tb3mo Step 2 Set up the VAR system using the MODEL option on the SYSTEM instruction Moreover in the SYSTEM END SYSTEM block include the instruction ECT name where name comes from the LINREG DEFINE name instruction used to estimate the long
110. MENU is activated it controls the flow of the program until a selection is made Typically you will use the USERMENU instruction at least three times within a procedure The first use is to DEFINE the structure of the menu the second causes the menu to appear on the menubar and the third is to REMOVE the menu The typical syntax to DEFINE the structure a USERMENU is usermenu action define title Title 1 gt gt string 1 2 gt gt string 2 n gt gt string n The ACTION DEFINE option is used to create a pull down menu to the right of the RATS Help menu with the title determined by the string Title The choices on the USERMENU appear as string 1 string 2 through string n The menu does not actually appear on the menubar until the simple instruction USERMENU is encountered The user can now select one of the choices Once a choice is selected the variable 7 MENUCHOICE is set equal to integer value of the selection At this point a series of IF instructions can be used to control program execution Finally the menu is removed using The next section discusses how to turn on and off choices using the ENABLE option 236 Walter Enders USERMENU action remove It is possible to rewrite the procedure TRANSFORM so that it uses the USERMENU instruction instead of the MENU instruction Consider the modified procedure called TRANSFORM_USER The procedure is executed using transform_user series As in the origin
111. NT instruction allows you to compare the difference between the errors 1 and the resids series dec vector series errors 1 set errors 1 e t 1 pri 1 5 errors 1 resids ENTRY ERRORS 1 RESIDS 1959 01 0 005265949134 NA 1959 02 0 016019975946 NA 1959 03 0 016059236473 NA 1959 04 0 005057184696 0 005265949134 1960 01 0 017909559980 0 016019975946 Notice that the first three entries of resids are NA one observation was lost because we used the first difference of dlrgdp and two more were lost because we used two AR coefficients in the LINREG instruction Notice that the first entry of errors 1 corresponds to fourth entry of resids To explain note that x was constructed with no missing observations Hence first element of e i e e 1 1 contains the difference between y3 and Except for the two period shift the resids and errors 1 series are identical Chapter 6 Writing Your Own Procedures In RATS a procedure is a set of instructions that resides outside of the program itself In writing a procedure you are effectively writing your own RATS instruction along with options supplementary cards and choices concerning the series and variables to use It is clear why you might want to write a procedure Suppose that there is a particular task involving a set of instructions that you frequently invoke It would be desirable if you could simply type a few keystrokes that instructed RATS to perform this more c
112. OMPUTE a guess b guess var guess MAXIMIZE L 2 Maximum Likelihood Estimates of the LSTAR Model In constructing your model it is often helpful to define the log likelihood function using several FRML statements instead of one complicated expression To illustrate the procedure recall that in the previous section we estimated dlm3 as LSTAR model D B dlm3 dim3 at at dim3 2 1 exp ydlm3 t We can prepare to obtain the maximum likelihood estimates using the three instructions below Notice that the NONLIN instruction is identical to the one we used for the NLLS estimation As before y is restricted to be positive However there is no need to restrict var the form of the log likelihood function is such that RATS will not find a negative value for var The first FRML instruction creates the formula expression note that expression is used in the second FRML instruction In either case you can obtain the desired formula for the log likelihood However breaking down a complicated formula into several smaller expressions is a useful way to prevent errors in your programs nonlin a0 al bO bl gamma var gamma ge 0 frml expression a0 al dIm3 1 b0 b1 dIm3 1 1xexp gamma dlm3 1 frml Istar log var dlm3 expression 2 var Next a linear regression is estimated The estimated intercept and slope coefficients are used as the initial guesses for a0 and a Unlike NLLS we
113. R 2 model for dirgdp is most appropriate you might want to estimate all your equations using both lag lengths You could read in MONEY_DEM XLS and construct dirgdp and dlm3 as follows set dlrgdp log rgdp log rgdp 1 set dim3 log m3 log m3 1 Then you could estimate six regressions using the following supplementary cards constant dlrgdp 1 j constant dlrgdp 1 to 2 constant dlrgdp 1j dlm3 constant dlrgdp 1 to 2 dlm3 constant dlrgdp 1j dlm3 dlm3 1 to 4 constant dlrgdp 1 to 2 dlm3 dlm3 1 to 4 Alternatively you could use the following set of instructions contained in Program 3 8 dofori 78 14 do p 1 2 lin noprint entries 1 dlrgdp 6 constant dlrgdp 1 to p dlm3 dlm3 1 to 4 compute aic Ynobs log rss 2 9onreg compute sbc nobs log rss Yonreg log Znobs dis The aic aic and sbc sbc end do p end dofor i The first time through the DOFOR loop i 7 so that the ENTRIES option reads the constant and dirgdp 1 to p entries on the supplementary card Hence inside the DO loop as p changes from 1 to 2 the pure AR 1 and AR 2 are estimated Next i 8 so that ENTRIES option reads the constant dlrgdp 1 to p and dim3 entries on the supplementary card Again two regressions are estimated using one and two lags of dlrgdp Finally i 14 so that the 114 Walter Enders ENTRIES option reads the constant dlrgdp 1 to p dlm3 and dlm3 1 to 4 entries on the supplementary card If
114. RATS Programming Language Walter Enders Department of Economics Finance amp Legal Studies University of Alabama Tuscaloosa AL 35487 wenders cba ua edu 2003 Walter Enders All Rights Reserved Distributed by Estima www estima com enders This book is distributed free of charge and is intended for personal non commercial use only You may view print or copy this document for your own personal use You may not modify redistribute republish sell or translate this material without the express permission of the copyright holder Table of Contents Quick Index of Selected Functions and Instructions iv Chapter 1 Linear and Nonlinear Estimation 1 Te The Data NIAE 2 2 Linear Regression and Hypothesis Testing ssssseseeeseenene eene ere e emen 6 I acini EE 10 8 The LINREG Options ERE 12 3 1 Usiie Switch On ERR E 13 IDOL PEE 14 3 2 Use Choice Optlons eres ctt ntes ce tn inedia np eee d apros cu eng od od daas on cana edad da de ca Eau edad a reg 15 Examples ice EE 15 3 3 Using Switches Choices and Internal Variables sese eene menn 16 4 Nonlinear Least Squares inert rend e eR RE rae node Sexe ka cus iaa cyangeviaannecdacuaadeaiecnuets 21 4 1 Changing the Convergence Criteria ssseseeseeeee eene eme nne enemies 23 Examplesso susto tte ti p Ear ren rr rer pacers D LI I n UI LM S E Cx eret 24 4 2
115. RAW Whenever such a jump occurs RATS draws a new set of pseudo random numbers 136 Walter Enders 5 Monte Carlo Experiments A Monte Carlo experiment attempts to replicate an actual data generating process DGP using experimental means You can use RATS to generate one or more series that characterize those produced by the actual data generating process in all key respects A Monte Carlo experiment will generate a random sample of size T and the parameters and or sample statistics of interest are calculated This process in repeated N times where N is a large number so that the distribution of the desired parameters and or sample statistics can be tabulated These empirical distributions are used as estimates of the actual distributions In fact the previous Sample Program was a Monte Carlo experiment that can answer the following question If the y sequence is actually a random walk what is the sampling distribution of the OLS estimate of pi Notice that the sample mean of the estimated values of B 0 94693484752 is below the true value of unity and that the minimum value 0 70122076370 is further away from unity than the maximum value 1 04215477194 As it turns out Dickey and Fuller 1979 show that it is quite correct to infer that the estimate of B is biased to be below unity We will examine the Dickey Fuller distribution in detail in two of the sample programs below For now suffice it to say that the use of the Monte Carlo met
116. RMA coefficients The next six lines of code are identical to that of the program above dec vect phi number com phi 1 1 do j 1 number 1 com phi j 1 b j dok 1 j com phi j 1 phi j 1 phi j 1 k a k end do k end do j The remainder of the program is unchanged set response phi t gra nodates style bar Header Impulse Responses 1 response Impulse Responses 08 0 6 0 4 7 0 2 7 0 0 7 0 2 0 4 py a a a e a a a a a Vector and Matrix Manipulations 195 4 1 Impulse Responses in a First Order VAR Suppose you estimated a dirgdp dlrm2 and drs as a first order VAR using 7 estimate sigma outsigma v residuals resids coeffs ca You could obtain the impulse responses using errors impulses model modelname 3 24 v or impulse model modelname 3 24 V An alternative way to obtain the identical answers is to use RATS matrix instructions The matrix ca contains the slope coefficients and the intercept terms Since the impulse responses represent deviations from equilibrium you will want to create a new matrix called A containing only the slope coefficients Consider the next three instructions dec rect a 3 3 ewise a i j ca i j dis A a The DECLARE instruction creates A as a 3 x 3 rectangular matrix The EWISE instruction equates each element of A with the corresponding element of ca The third instruction allows you to view the newly created matrix A Next c
117. S and NOBS Moreover NLLS defines the internal variable CONVERGED Note that CONVERGED 1 if the estimation converged and otherwise is equal to 0 4 1 Changing the Convergence Criteria Numerical optimization algorithms use iteration routines that cannot guarantee precise solutions for the estimated coefficients Various types of hill climbing methods are used to find the parameter values that maximize a function or minimize the sum of squared residuals If the partial derivatives of the function are near zero for a wide range of parameter values RATS may not be able to converge to the optimum point Moreover you should also be cautious of results indicating that convergence takes place in one iteration the resulting parameter values and associated f statistics are often unreliable NLPAR allows you to select the various criteria RATS uses to determine when and if the solution converges As such the NLPAR instruction allows you to control the precision of your answers There are three principal options for NLPAR the syntax 1S nlpar options CRITERION In the default mode CRITERION COEFFICIENTS Here convergence is determined using the change in the numerical value of the coefficients between iterations Setting CRITERION VALUE means that convergence is determined using the change in the value of the function being maximized CVCRIT Converge is assumed to occur if the change in the COEFFICIENTS or VALUE is less than the nu
118. Similarly many RATS instructions accept a vector or a matrix as inputs In essence you pass information to the instructions using matrices In fact there are so many RATS instructions that utilize or create matrices that advanced RATS programmers often incorporate some matrix manipulations into their overall program 1 Creating Matrices and Vectors In traditional matrix algebra the elements of a matrix are numbers RATS allows you to be much more flexible Not only can the elements of a matrix be real numbers complex numbers or integers they can also be strings labels or series Be aware that many RATS instructions have options allow you to create matrices For example the ESTIMATE instruction discussed in Chapter 2 can have the form ESTIMATE OUTSIGMA V residuals resids coefficients coef where OUTSIGMA Computes and saves the covariance matrix of the residuals COEFFICIENTS coef Creates a matrix of the coefficients Column i contains the coefficients of the i th equation RESIDUALS resids Creates a vector of series The residuals from the first equation are stored in the series called resids 1 the residuals from the second equation are stored in the series called resids 2 and so forth Also you can create vectors and matrices in RATS using the DECLARE COMPUTE and MAKE instructions The main use of each is DECLARE The most general instruction for creating matrices COMPUTE Useful for creating small matrices or vectors MAKE
119. Use the bootstrap sample to calculate a bootstrapped y series called y For each value of i running from 1 to T calculate y as yi B B xit e Note that the estimated values of the coefficients are treated as fixed Moreover the values of x are treated as fixed quantities so that they remain the same in the bootstrap sample Step 3 Use the bootstrap sample to estimate new values of Bo and i calling the resulting values B and Bj Step 4 Repeat Steps 2 and 3 many times and calculate the sample statistics for the estimated values B and P using the sample properties of f and f Of course it is possible to apply the identical procedure to a nonlinear regression model as well Program 4 13 of the file CHAPTERA 2 PRG finds bootstrap confidence intervals for the nonlinear regression model discussed in conjunction with antithetic variates v Bx 0 The alternative method is to bootstrap the paired y x combinations IF Statements and Monte Carlo Experiments 161 Instead of using two actual series from MONEY_DEM XLS the program uses artificially generated data for x and y As such we will know how the two series data are actually generated and the true values of and B As in Program 6 the first two lines set the default length of a series to 50 seed the random number generator and generate the x and y sequences Hence the true parameter values are B 1 and a 0 5 all 50 seed 2001 set x
120. a variable has been selected the user is allowed to select any of the three choices from the Trends menu Hence it is possible to select an alternative variable by returning to Select a Variable to Estimate estimate the selected variable with Estimate the Variable or to exit the procedure with Done 3 If the user selects Estimate the Variable another menu appears However this second menu is actually a dialog box with buttons that allow the user to select one of four choices The user can select whether to estimate the series without a trend with a linear trend with a quadratic trend or with a cubic trend Whichever choice is made the program displays the regression output and shows a graph of the actual and the fitted values of the series The program consists of two separate procedures with the names ESTIMATE and SEASONS SEASONS creates the USERMENU called Trends No parameters are passed to SEASONS and the procedure contains no OPTIONS CHOICES or SWITCHES As such the first instruction defines a LOCAL INTEGER variable called depvar Depvar is the integer value corresponding to the series selected by the user i e the dependent variable in the regression equation The first USERMENU instruction defines the menu Trends and creates the three possible selections The second USERMENU instruction illustrates the use of the ACTION MODIFY option Notice that choice 2 is not enabled the user will not be able to select Estimate the Trend until
121. ags 1 through i and displays the results Regardless of whether we use the 6 lag model selected by the AZC or the 1 lag model selected by the SBC the t statistic is sufficiently negative that we reject the null hypothesis Oto equals zero As such we conclude that the two interest rates are cointegrated Linear and Nonlinear Estimation 21 4 Nonlinear Least Squares Given that many economic variables display asymmetric adjustment nonlinear estimation methods have become quite popular RATS allows you to perform nonlinear estimations in a number of ways We will focus on nonlinear least squares and maximum likelihood estimation Suppose that you want to estimate the following model using nonlinear least squares w ox E Since the disturbance term is additive you cannot simply take the log of each side and estimate the equation using OLS However nonlinear least squares allows you to estimate a and p without transforming the variables In RATS you use the following structure to estimate a model using nonlinear least squares NONLIN list of parameters to be estimated FRML formula name the equation to be estimated COMPUTE initial guesses for the parameters NLLS FRML formula name dependent variable For the example at hand you could use nonlin alpha beta frml equation 1 y alpha x beta com alpha 1 0 beta 2 0 5 nlis frml equation 1 y The first instruction informs RATS that two parameters named alpha and beta are to
122. aic with p opt lags is aic end if If you run the program your output will be Variable Coeff Std Error T Stat Signif KKKKKK KKK KKK KKK KKK KKK KKK KKK KKK KKK KKK KKK KKK KKK KKK KKK KKK KKK KKK KK KG KG KA Kx ko ko ko 1 Constant 0 0051566068 0 0010217954 5 04661 0 00000119 2 DLRGDP 1 0 2508977521 0 0769801061 3 25925 0 00135976 3 DLRGDP 2 0 1362250820 0 0762100846 1 78749 0 07571568 The aic with 2 lags is 732 28752 Jazzing Up the Program 1 You might want to use a routine such as this in a number of your programs However you might not always want to use a maximum possible lag length of 12 To generalize the program you can create a variable called max_lag and a variable called diffs right before the DO p 1 12 instruction Max_lag contains the maximum lag length you are willing to consider and the value of diffs equals 1 if the variable has been differenced You will also need to modify the DO instruction such that the index p runs through max_lag Hence if you want to consider a maximum of 16 lags using a variable that has been differenced once use com max_lag 16 diffs 1 do p 1 max_lag lin noprint dlrgdp max_lag diffs 1 IF Statements and Monte Carlo Experiments 125 These instructions cause RATS to use a maximum of 16 lags Since the variable has been differenced once diffs 1 the estimation for all 16 regressions will begin with observation 18 You can easily modify the program so tha
123. ake three changes to the basic program The user should be able to select the deterministic regressors to include in the model select the lag length to use in the augmented form of the Dickey Fuller test and to choose whether to display a graph of the residuals The Choice of Lag Length Since the number of lags is an integer we need to make only two modifications to the procedure 1 Include the instruction OPTION INTEGER LAGS 1 2 Replace the supplementary card in the LINREG instructions with y 1 dy 1 to lags constant 216 Walter Enders Since the default value of LAGS is 1 if you use unit rgdp or unit lags 1 lrgdp you estimate a regression only one lag of A rgdp If you invoke the procedure using unit lags 4 Irgdp you estimate a regression with four lags of Alrgdp The Choice of Deterministic Regressors The selection of the deterministic regressors is a clearly a CHOICE option the deterministic regressors can be NONE CONSTANT or TREND i e constant plus trend As such we need to make four modifications to the program 1 Include the instruction OPTION CHOICE DET 2 NONE INTERCEPT TREND Notice that default value of DET is 2 If you use unit det none rgdp the value of DET will be 1 unit rgdp or unit det intercept rgdp the value of DET will be 2 and unit det trend rgdp the value of DET will equal 3 2 In order to create the time trend we need to include the instruction set time t 3 Since the pr
124. al entries on a supplemental card with a vector You use the ENTER instruction to manipulate the items in the vector By changing the contents of the vector it is easy to add subtract or in almost any reasonable way modify the information on the supplementary card You first have to create the vector that will hold the information If we are going to modify the various series listed on the card it is necessary to create a vector of integers Recall that series can be referenced by their labels or integers The syntax of ENTER needed to perform this task 1s ENTER varying vector variables for vector Use the VARYING option since the length of the vector will vary as you add or delete variables We will return to the issue of automating model selection in a VAR in Sections 3 1 and 3 2 below Before considering the full VAR system suppose you want to estimate the three regression equations dirgdp Qo A11 L dlrgdp 1 dlrgdp Qo A1 L dlrgdp 1 A12 L dlrm2 1 dlrgdp Qo A11 L dlredp A12 L dlrm2 1 A13 L drs 1 The next instruction in Program 5 2 is used to DECLARE the integer vector reglist This vector will hold the list of regressors we want to place on the supplementary card The second line above places constant in the vector reglist dec vector integer reglist compute reglist II constant Il dofor i dlrgdp dlrm2 drs enter varying reglist reglist i 1 to 12 lin dlrgdp
125. al procedure the user passes a series to the procedure the TYPE instruction defines this parameter as a series The first USERMENU instruction defines a menu with the title Transform Clicking on this pull down menu will reveal the three choices Logarithmic First Difference and Growth Rate Notice that the instruction is completely contained on a single program statement the use of the sign as a continuation is simply to make the program easier to read procedure transform_user y type series y usermenu action define title Transform 1 gt gt Logarithmic 2 gt gt First Difference 3 gt gt Growth Rate The menu does not appear until USERMENU with the default option ACTION RUN is encountered If the user selects Logarithmic Ymenuchoice 1 and the instructions in the first IF block are executed Again there is a check that prevents the user from trying to take the log of a number that is not positive usermenu if menuchoice eq 1 f sta noprint fractiles y if minimum gt 0 com a L label y set Yos a log y if minimum le 0 mes style alert I CAN ONLY TAKE THE LOG OF POSITIVE NUMBERS The integer variable menuchoice equals 2 if the user selects First Difference and equals 3 if the user selects Growth Rate After the appropriate IF block is completed a graph of the transformed series is created The instruction USERMENU action REMOVE removes the menu from the menubar In order to make ano
126. ample discussed above we examined various types of TAR models using sample sizes ranging from 50 to 1000 with 100 000 Monte Carlo replications for each I would turn on my computer go to lunch and hope it would be done when I returned The problem is the critical values will change from one replication to the next we wanted to use a number of replications such that the critical values stabilized in the second decimal place One technique for reducing the sampling variance inherent in a Monte Carlo study and reduce the number of replications is to use antithetic random numbers The basic idea is to pool two different unbiased estimates of the parameters of interest so as to obtain an estimate with a small variance Suppose that a 1 and 2 are the two different estimates of the parameter amp from replications 1 and 2 Consider the pooled estimator that is the simple average of the two 0 5 a 1 a 2 The variance of is var 0 25 var a 1 var a 2 2 cov a 1 a 2 If the two estimates have a negative covariance the pooled estimator will have a far smaller variance than those from the two independent draws The problem is to find a way to ensure that the estimates have a negative covariance i e are antithetic In certain circumstances the solution is remarkably simple use the same set of numbers with the sign reversed for the second replication Intuitively if the first replication gives an estima
127. ample1 Vt dy t Gu Gun te Zr dag F yy ui T A2221 The first instruction below prepares RATS to create a system of equations with the name examplel The VARIABLES instruction names the two variables in the system and the LAGS instruction indicates that one lag of each is to be included in the model The DETERMINISTIC instruction informs RATS to include a constant term in each regression equation END closes the system and ESTIMATE produces the coefficient estimates and the F statistics for the Granger causality tests The option RESIDUALS resids instructs RATS to save the residuals from the y VARs and Error Correction Models 49 equation in a series called resids 1 and the residuals from the z equation in a series called resids 2 system model example1 var y Z lags 1 det constant end system estimate residuals resids Modification of the SYSTEM END SYSTEM block is straightforward If you want to 1 Include 4 lags of y and z in each equation replace the LAGS instruction with lags 1 to 4 2 Include lags 1 2 3 4 and 8 of y and z in each equation replace the LAGS instruction with lags 1 to 4 8 3 Include an exogenous variable w such that the VAR is y aio uy a XE el d13Wt A14Wr 1 i Zt 59 GAY 1 A22Z 023W d2AW 1 replace line 4 with det constant w 0 to 1 4 Estimate a 3 variable VAR using y z and w replace the VARIABLES instructions with varyzw 5 Include quarte
128. ancel The selection will affect the string variable b and the vector reglist For example if NO Time Trend is selected b No Time Trend and the vector reglist is unaltered hence reglist contains only the constant Instead if Cubic Trend is selected b Cubic Time Trend and reglist contains a constant time t2 and 13 242 Walter Enders menu Select the degree of the Polynomial choice NO trend com b No Time Trend choice Linear trend enter varying reglist reglist time com b Linear Time Trend choice Quadratic trend enter varying reglist reglist time t2 com b Quadratic Time Trend choice Cubic trend enter varying reglist reglist time t2 t3 com b Cubic Time Trend end menu Next the linear regression is estimated using the depvar as the dependent variable and the variables in reglist as the independent variables The PRJ instruction is used to create a series of the fitted values The final three instructions obtain the label assigned to depvar and create a graph of the series and the fitted values from the regression On completion of the graph control passes back to the LOOP ENDLOOP block in the procedure SEASONS lin depvar reglist prj fitted compute a l depvar gra header Trend Estimate of a subheader b key below patterns klabelzll Fitted Actual ll 2 fitted depvar end estimate Jazzing up the Proce
129. ariance of the residuals i e the squared standard error of the estimate as eel T 3 com v scalar tr e e 163 dis v An alternative is to use com v dot e e 163 dis 96sqrt v Vector and Matrix Manipulations 201 Note that we need to covert the 1 x 1 matrix tr e e into a scalar before dividing by the scalar value 163 Alternatively we could use dot e e to produce the inner product If you compare the answer to that from the regression model you should find the same answer Next we can find the standard errors of the coefficients using that fact that var B var e X X com v beta 46scalar v xx inv dis v beta dis sqrt v_beta 1 1 sqrt v_beta 2 2 sqrt v_beta 3 3 0 00102 0 07698 0 07621 Notice that these are the same as the standard errors of the coefficients that we obtained using the LINREG instruction Typically we estimate regressions in order to perform a Wald test on the regression coefficients The simplest way to perform such a test is to use an EXCLUDE or RESTRICT instruction However to further illustrate matrix manipulations consider a set of linear restrictions of the form RB c where R qxk B estimated coefficients c constants and q number of restrictions and k number of estimated parameters The F statistic is F q T k c R BE ROCX R c R By q6 Now consider the null hypothesis that the intercept term is zero We can write this restriction a
130. at the four parameters a0 al bO and b1 are to be estimated The three FRML instructions create the appropriate log likelihood The first FRML instruction defines e as y bO b1x The second defines the conditional variance h as an ARCH 1 process The third uses the definitions of e and h to define the log likelihood function L The MAXIMIZE command instructs RATS to find the maximum likelihood estimates of a0 al bO and b1 nonlin a0 al bO bl frml e y b0 b1 x frml h a0 al e t 1 2 frml L log h log e 2 h com initial guesses maxL2 6 1 Examples of GARCH Processes 1 An ARCH Model of the Spread If you continue to enter the instructions on Program 1 4 you can form the difference between the 1 year rate and the 3 month rate as set spread tblyr tb3mo It appears that an AR 3 model of the spread is quite reasonable Consider lin spread resids constant spread 1 to 3 Variable Coeff Std Error T Stat Signif Ck CK Ck Ck CK CI Ck CK CC CC CK CC CK CIC CC CK CC CC CK CC CC CK C CI Ck KC CI C SK x Kk x Kk Mk Kk kx ko ko kokok 1 Constant 0 047619130 0 024652597 1 93161 0 05517527 2 SPREAD 1 0 890042258 0 078014946 11 40861 0 00000000 3 SPREAD 2 0 318602488 0 101791701 3 12995 0 00207856 4 SPREAD 3 0 161545395 0 077536269 2 08348 0 03879731 The individual autocorrelations and Ljung Box Q statistics of the residuals indicate that there is no serial correlation in the residual series The first
131. at this new section of the procedure is bypassed If the user selected Estimate the Trend and Seasonals s_ 1 As such the procedure uses the INQUIRE instruction to obtain the seasonal span Next a seasonal dummy variable called seasons is created The current value of seasons plus span 2 leads are added to the regressor list For example if span 4 there will be three seasonal dummy variables in addition to the intercept The remaining portions of the ESTIMATE procedure are unaltered if s eq 1 f inquire seasonal span seasons seasons enter varying reglist reglist seasons 0 to span 2 j Jazzing up the Procedure 2 It is straightforward to modify the procedure so that it incorporates the features of TRANSFORM The source code below indicates the necessary modifications of SEASONS Notice that a new choice called Transform a Variable has been added to the USERMENU instruction As such Estimate the Trend moves to choice 3 and Estimate the Trend and Seasonals moves to choice 4 As such it is necessary to change the instructions using the ENABLE option to reflect the new position usermenu action define title Trends 1 gt gt Transform a Variable New Choice 2 Select a Variable to Estimate 4 Another possible way to write the procedure is to use s as an OPTION in ESTIMATE Writing Your Own Procedures 245 3 gt gt Estimate the Trend 4 gt gt Estimate the Trend and Seasonals 5 gt gt Done us
132. ate y Q yf you can use nonlin alpha beta frml equation 1 y alpha y 1 beta com alpha 1 0 beta 0 5 nlls frml equation_1 y 2 The NONLIN instruction allows you to impose restrictions on the parameters To estimate y 2 x X5 x E your first two instructions should be nonlin alpha alphal alpha2 alpha3 frml y alpha x1 alphal x2 alpha2 x3 alpha3 To ensure 0 065 modify the NONLIN instruction such that nonlin alpha alphal alpha2 alpha3 alphal alpha2 To ensure that alpha2 in not negative use Linear and Nonlinear Estimation 25 nonlin alpha alphal alpha2 alpha3 alpha2 gt 0 or nonlin alpha alphal alpha2 alpha3 alpha2 ge 0 Suppose you are having difficulty estimating y a y If NLLS does not converge you can increase the number of iterations from the default value of 40 provide better initial guesses use NLPAR or use an alternative estimation method A useful way to obtain satisfactory initial guesses is to use the simplex or genetic estimation method for a few iterations and then switch to the GAUSS method To apply this technique to the equation from example 1 use nonlin alpha beta frml equation 1 y alpha y 1 beta com alpha 1 0 beta 0 5 nlls frml equation_1 method simplex iters 5 y nlls frml equation_1 y The first NLLS instruction uses the simplex method to perform the estimation and the second uses these results as initial guesses Time series
133. atrix inum consisting of four integers com inum ll 1 4 3 2 Il In contrast com vect inum Il 1 4 3 2 Il creates a vector of real numbers since the default data type for vector is real numbers com vect integer inum Il 1 4 3 2 II creates a vector of integers since both types are specified Note the appropriate use of double brackets vect integer must be enclosed in brackets 7 You can use COMPUTE in association with previously defined variables Consider comill 1 i12 2 i113 3 i14 4 com rect integer inum lli11 112 1113 i14ll dis inum T 2 3 4 Vector and Matrix Manipulations 175 Note that once the data type is made explicit you must take care not to use a different type If you let j 3 equal 3 9 and enter com j11 2 1 j1222 j132 39 j1424 com rect integer inum Ilj11 j12 1 j13 j14ll SX22 Expected Type INT gt gt gt gt j12 313 j14 EG ER Got RI FAL Instead 176 Walter Enders 2 Matrix Operations Suppose that A and B are conformable matrices The following are just some of the matrix operations you are allowed to perform COM C A B or A B Addition and subtraction COM C A B Multiplication A and B TR A Transpose of A INV A Inverse of A DECOMP A Choleski decomposition of SYMMETRIC only KRONEKER A B Kroneker product of A and B CORR A B Correlation of A and B DET A Determinant DOT A B Dot product DIAG
134. ausality tests Suppose that we have determined that a four lag model is most appropriate and that a tentative forecasting equation for y is yr Oo aiiyra ai2yr2 aiayes A14Yt 4 A21Xt 1 022X62 023Xi3 G24Xp4 E where x is one of the series that we might want to include in our final forecasting equation Since there is likely to be a fair amount of correlation among the regressors it is standard to rely on F tests to determine whether or not to include a series in the forecasting equation It is said that x Granger causes y if it is possible to reject the null hypothesis a2 an az 024 0 Hence if we cannot reject this null hypothesis we exclude all values of x from the forecasting equation Of course we can also determine if y Granger causes itself by testing the null hypothesis a1 a1 a13 a14 0 One way to proceed is to estimate a very general forecasting equation using all variables in the data set and then to eliminate variables based on Granger causality tests The other way is to begin with only an intercept and sequentially add variables keeping only those that pass the causality test The purpose of this section is to illustrate the use of the ENTER instruction not to determine which of the two methods is best As such we will use the second method to obtain a forecasting equation for the logarithmic change in M3 As in the previous section it is necessary to keep track of two regress
135. be estimated The FRML instruction sets up a formula named equation 1 the form of the equation is y alpha x beta The COMPUTE instruction provides the initial guesses for alpha and beta The last line instructs RATS to use nonlinear least squares NLLS to estimate the series y using the formula previously defined as equation 1l In fact all nonlinear least squares estimations use this four step procedure Specifically Step 1 Specify the parameter set to be estimated using the NONLIN instruction The syntax for NONLIN is NONLIN parameter list In most instances the parameter list will be a simple list of the coefficients to be estimated You can also include equality constraints such as a b or a b 1 0 note the double 7 Tf the model had the form y axe where i is log normal it would be appropriate to estimate the regression in logs using LINREG 22 Walter Enders equal sign or weak inequality constraints of the form b ge 0 or b le 0 In some instances you might find it helpful to use a previously defined VECTOR for the parameter list Step 2 The formula needs to be specified This is accomplished using the FRML instruction The syntax for FRML is FRML options formulaname depvar function t where formula name the name you choose to give to the formula depvar dependent variable function t the function to be estimated In the example above equation is the name of the formula to be estimated y
136. be manipulated using the various string handling instruction provided with RATS The function s label also returns the series number whose name is label Be sure to include the label in single or double quotation marks To display the series number associated with rdgp use com ii s rgdp dis ii Loops over Dates and Series 99 3 Do Loops The DO loop is a very simple way to automate many of your repetitive programming tasks The usual structure of a DO loop is doi 1 n program statements end do i The structure is such that RATS will perform all program statements within the DO loop exactly n times The first time the program statements are executed the counter i contains the integer value 1 On reaching the end of the loop the counter i is incremented by 1 i e i 2 If i is less than or equal to n the block of program statements is executed again On reaching the end of the loop the value of i is incremented by 1 compared to n and if i n the loop is executed once more After the n th loop i n 1 and the instruction following the loop is executed Consider the example doi 1 5 dis i end doi 1 Un ULP More generally you can use do integer n1 n2 increment program statements end do integer where nl n2 and increment are integers To understand how the loop performs consider doi 1 5 2 Here i begins at and is incremented by 2 every time the loop is completed After three loops i
137. ble Coeff Std Error T Stat Signif c CK CC CK C Ck CK KKK KKK KKK KKK KKK KKK KKK KKK KKK KKK KKK KKK KKK KEK KKK KKK KKK Ck Ck Ck Ck Ck Ck Sk ko M 1 Constant 0 002906841 0 000993077 2 92711 0 00392329 2 DLM3 1 0 824726885 0 080688781 10 22108 0 00000000 3 DLM3 2 0 004028644 0 104551220 0 03853 0 96931127 4 DLM3 3 0 079967942 0 104224480 0 76727 0 44406097 5 DLM3 4 0 109141754 0 079475991 1 37327 0 17160277 Jazzing Up the Program It is straightforward to modify the program to use each of the four variables as the dependent variable in the regression equation Thus the final output will be a four variable near VAR in the sense that only the variables that are causal remain in the system Towards this end we will let the index j loop over dirgdp dlm3 drs and dlp Consider dec vector integer templist reglist dofor j dlrgdp dlm3 drs and dlp compute templist llconstantll compute reglist llconstantll Now each time through the DOFOR j loop templist and reglist will be initialized to contain only a constant There are only three other required modifications to the program As indicated below the two LINREG instructions are changed to indicate that the dependent variable in the regression is series j The final line in the program closed the DOFOR j loop dofor i dlrgdp dlm3 drs dlp enter varying templist reglist i 1 to 4 lin noprint j NOTE the change in this line templist exclude noprint i 1 to 4 if
138. ble parameter set The way to avoid this problem is to use the method shown below to create your own integer vector Although it is not especially efficient to use three lines of code instead of one you could estimate the model above using declare vector integer mas com mas Il 2 All box constant ar 2 ma mas y 3 100 resids In other circumstances the method can be particularly useful A friend of mine working with energy price tick data wanted to estimate an MA model with coefficients at lags 1 60 120 180 and 240 A simple way to create the vector is declare vector integer mas 63 ewise mas i i com mas 61 120 mas 62 180 mas 63 240 box constant ar 2 ma mas y 3 100 resids Here you declare the vector mas and fill the vector with the coefficient values you want in the estimation The EWISE instruction fills all 63 entries with integer values running from 1 to 63 The COMPUTE instruction corrects entries 61 62 and 63 such that they equal 120 180 and 240 respectively The final instruction estimates a model with 2 autoregressive coefficients and the 63 moving average coefficients the MA terms at lags 1 60 120 180 and 240 2 3 Manipulating the Output of a VAR In Chapter 2 we were able to create a 3 variable VAR using system model chap2 var dlrgdp dirm2 drs lags 1 to 12 det constant end system The necessary instructions to set up the VAR are repeated in Program 5 2 The next statement instructs RATS t
139. bly noticed an INFOBOX that quickly appears and disappears when you read in a data set Similarly it is possible to keep track of the number of iterations completed by using the INFOBOX instruction It is necessary to use INFOBOX three times in the program once to DEFINE the box a second time to update or MODIFY the box and a third time to REMOVE the box The syntax for INFOBOX is INFOBOX ACTION2 other options messagestring where ACTION DEFINE MODIFY REMOVE With ACTION DEFINE the key options are PROGRESS NOPROG This option determines whether the box will include a progress bar You must specify values for LOWER and UPPER when you use PROGRESS LOWER Integer value for the lower bound UPPER Integer value for the upper bound With ACTION MODIFY use the option CURRENT The current integer value for the progress bar 146 Walter Enders Thus to keep track of the number of iterations over j that have been completed immediately before the instruction DO j 1 10000 include infobox action define progress lower 1 upper 10000 Replications Completed Immediately after the instruction DO j 1 10000 include infobox current j Immediately after the instruction END DO j include infobox action remove If you now run the program you should see something that looks like Progress Box Replications Completed 5 4 The Enders Granger Statistic In addition to shamelessly promoting my own work the f
140. changed If you compare these results to those obtained earlier you will see that the option had very little effect on standard errors and the t statistics Note that some versions of the Reference Manual incorrectly refer to this as 7STDERRORS 14 Walter Enders Notice that the ROBUSTERRORS option can be ON or OFF you can enter ROBUSTERRORS or NOROBUSTERRORS Similarly you can use either PRINT or NOPRINT for the regression output or VCV or NOVCYV for displaying the covariance matrix It is helpful to think of these options as a switch The option will be executed if the switch is ON and will be bypassed if the switch is OFF If neither choice is indicated in the options field of an instruction RATS will resort to the default value of that particular option For all RATS switching options you can also turn on the switch by equating its value to 1 and turn off the switch by equating its value to zero This can be very useful in writing procedures see Chapter 6 since it is simple for the user to pass the appropriate switch value to the procedure Examples 1 Since the PRINT option is the LINREG default all of the following have equivalent effects lin drs lin print drs lin print 1 drs comj 1 where j is an integer lin print j drs 2 Since NOROBUSTERRORS is the LINREG default all of the following have equivalent effects lin robusterrors drs lin robusterrors 1 drs comj 1 where j is an integer lin robusterrors j
141. cols A returns the number of columns in matrix A Hence we can obtain p and q using compute p cols alpha compute q cols beta Unless otherwise specified creation of a literal vector using COMPUTE causes RATS to create a row vector Hence alpha has 1 row and 3 columns while beta has 1 row and 2 columns Next we need to set up a vector that we call phi to hold the twenty four values of the 6j However there is a small problem in that a vector cannot have an element zero Thus we cannot use the notation o because we cannot have an element of a vector designated as phi 0 Instead we need to store the first value of in phi 1 the second value in phi 2 Thus phi 1 will equal 1 phi 2 will equal Bi ouphi 1 In essence we need to create the phi vector such that phi 1 o phi 2 1 Hence we will replace the usual formula for calculating impulse responses with oi 1 j gt BAY mos TELS k 1 We can DECLARE the phi vector to contain 24 elements and initialize the first value phi 1 1 using Vector and Matrix Manipulations 191 dec vect phi number com phi 1 1 We cannot directly use the formula above since O is undefined for k gt p and is undefined for j gt q The final bookkeeping task concerns treatment of these values One straightforward method is to define two new vectors of dimension 24 We can let the first p elements of vector A hold the elements of alpha and
142. considered line 2 sets the default series length to 2001 1 However the value 4 in the series field instructs RATS to create a block of series numbered 1 to 4 Notice that the integer values assigned to the seven series contained on MONEY DEM XLS now begin with 5 and end with 11 At this point in the program series 1 through 4 have no label You can assign a name to each using the EQV instruction The syntax of EQV for EquiValance is EQV integer values of series list of names for series For example we can assign series 1 2 3 and 4 the names resids1 resids2 resids3 and resids4 X DA using eqv 1 to 4 residsl resids2 resids3 resids4 After entering these two lines a table instruction produces the names of the four series tab 1 to 4 Notice there are no commas between the labels and there is no symbol on the list of labels Loops over Dates and Series 97 Series Obs Mean Std Error Minimum Maximum RESIDS1 0 RESIDS2 0 RESIDS3 0 RESIDS4 0 You can use the names created with EQV for input and for output An alternative is to assign a label to a series using the LABELS instruction The syntax for LABELS is labels list of series numbers labels for the series each label in quotation marks Hence to assign the labels resids1 resids2 resids3 and resids4 to series 1 through 4 use labels 1 to 4 residsl resids2 resids3 resids4 Notice that each label is enclosed in single or doub
143. correcting standard errors and t statistics for hypothesis testing The ROBUSTERRORS option computes a consistent estimate of the covariance matrix that corrects for heteroscedasticity as in White 1980 ROBUSTERRORS and LAGS can produce various types of Newey West estimates of the coefficient matrix Moreover SPREAD is used for weighted least squares and INSTRUMENTS is used for instrumental variables The appropriate use of these options is described in Chapter 5 of the RATS User s Guide LINREG creates a number of variables that you can use in subsequent computations A partial list of these variables is BETA The coefficient vector The first coefficient estimated is BETA 1 the second BETA 2 and so on For example in the output for dr above the constant is BETA 1 the coefficient for dr 1 is BETA 2 and so forth XX The X X matrix Note that XX ij contains the estimated covariance of coefficient 1 with coefficient j Linear and Nonlinear Estimation 13 TSTATS The vector containing the t stats for the coefficients The first t statistic is 7TSTATS 1 the second is TSTATS 2 and so on STDERRS Vector of coefficient standard errors NDF Degrees of freedom NOBS Number of observations NREG Number of regressors RSS Residual sum of squares RSQUARED Centered R2 JSEESQ Standard error of estimate squared DURBIN Durbin Watson statistic QSTAT Ljung Box Q statistic QSIGNIF Significance level o
144. cs for p or p2 is zero or undefined the simulated series needs to be eliminated from the study Thus after estimating the TAR model we check to see is the absolute value of the product of the two f statistics is less than 0 0000001 If so we branch to step 1 to obtain a replacement series Program 4 9 of the file CHAPTERA 2 PRG contains the program that we used to calculate the Monte Carlo values The first four lines perform some bookkeeping tasks The default length of a series is 200 Also the random number generator is seeded and the series f is initialized with 50000 values of zero The sample values of the F statistic for the null hypothesis p p2 0 will be stored in f Line 4 initializes the value of the y sequence to zero all 200 seed 2001 set f 1 50000 0 set y 0 The next line of the program begins the DO loop and the following line is labeled reset The program jumps back to this line if the simulated series does not cross the threshold i e the program will BRANCH to reset if the two t statistics are too small do j 1 50000 reset Task 1 is performed by the next three lines of the program The next two lines in the program simulate a random walk sequence of 200 observations The third instruction takes the first difference of the simulated series TASK 1 COMPUTE the y series set y 2 200 y 1 ran 1 diff y 2 200 dy Next the IF instruction is used to set the indicator called plus to equa
145. ction estimates the nonlinear equation using the formula previously defined as monte Next the values of alpha bar i and beta bar i are updated 50 of the estimated value of alpha is added to alpha bar i and 5096 of the estimated value of beta is added to beta bar i At the end of the DO loop entry i of each of these series contains the desired pooled estimate After the process is performed 500 times the DO i loop is completed The TABLE instruction is used to obtain the distribution of alpha bar and beta bar com alpha bar i alpha bar 1 0 5 alpha com beta bar i beta bar 1 0 5 beta end do j end do i tab alpha bar beta bar Series Obs Mean Std Error Minimum Maximum ALPHA BAR 500 0 50192972485 0 00880740910 0 45061006127 0 56465375430 BETA BAR 500 1 05368592894 0 07719055534 0 99439278963 1 65755600806 The mean values are quite similar to those of alpha hat and beta hat However the standard errors of alpha bar and beta bar are about 17 times and 5 times smaller than those of alpha hat and beta hat respectively 158 Walter Enders 7 Bootstrapping Programming for bootstrapping is similar to that for a Monte Carlo experiment However there is an essential difference In a Monte Carlo study you generate the random variables from a given distribution such as the Normal The bootstrap takes a different approach the random variables are drawn from their observed distribution This is quite useful if you are work
146. d 1 through 4 For the tetrahedron you get the number of points shown on the downward face Your total score equals the number of points received for the coin and the tetrahedron Of course it is impossible to have a score of zero or 1 It is straightforward to calculate that the probabilities of scores 3 4 and 5 equal 0 25 while the probabilities of scores 2 and 6 equal 0 125 138 Walter Enders It is possible to simulate a roll of the coin and tetrahedron on the computer Since the probability of a head is 0 5 we can use the following program statement compute coin if uniform 0 1 gt 5 2 1 The instruction equates the variable coin with 2 if a uniformly drawn random number over the interval 0 1 exceeds 0 5 If the value of uniform 0 1 is less than or equal to 0 5 the value of coin is 1 In this way a head is equal to 2 while a tail is equal to 1 Similarly we can simulate the toss of the tetrahedron com tet fix uniform 1 0 5 0 The first line equates x with a integer value of uniformly distributed random number over the interval 1 5 Note that FIX transforms a floating point number to an integer As such if the random number is between 1 and 2 the value of tet is the integer 1 Similarly if the number is between 4 and 5 the value of tet is the integer 4 Thus the probabilities that tet will equal 1 2 3 or 4 are all equal to 0 25 We can obtain the total score using com score coin
147. d p q by 1 and is Zero if no intercept is present If this was the only time you were going to use the program this step would be unnecessary However you might want to use the same routine for other estimations that might not include an intercept As such it becomes convenient to allow the program to determine whether of not an intercept was included in the estimation compute p cols ar compute q 96cols ma com flag rows beta p q As in the original program above the next two instructions fill the vectors A and B with zeroes dec vect a number b number com a const 0 b const 0 Now we need to transfer the values of 7BETA 2 to A 2 BETA 3 to A 7 BETA 4 to B 1 and BETA S to B 3 One way to do this would be to use com a 2 beta 2 a 7 beta 3 b 1 beta 4 and b 3 9obeta 5 However a more general way to do this is to recall that the first element of AR is the integer 2 the second element of AR is the integer 7 the first element of MA is the integer 1 and the second element of MA is the integer 3 Hence we can write 194 Walter Enders do i 1 p com a ar 1 i beta i flag end do i do i 1 q com b ma 1 1 beta itptflag end doi Since flag 1 the first loop equates A 2 with BETA 2 and A 7 with BETA 3 and the second loop equates B 1 with BETA 4 and B 3 with BETA S The key point to note is that this set of instructions will work for any pattern of A
148. d prediction errors from the AR 1 and the AR 2 each prediction error is the difference between the actual and forecasted values of dlrgdp Lines 3 and 4 calculate and display the mean square prediction error for the AR 1 and lines 5 and 6 calculate and display the mean square prediction error for the AR 2 set pe 1 101 169 dlrgdp f ar1 2 set pe 2 101 169 dlrgdp f ar2 2 sta noprint pe 1 dis The MSPE from the AR 1 is mean 108 Walter Enders sta noprint pe_2 dis The MSPE from the AR 2 is mean The MSPI The MSPI from the AR 1 is 2 44821e 05 from the AR 2 is 2 31782e 05 p P F P A Small Quiz Now take a little quiz Some might recommend eliminating an initial observation every time through the loop In this way you would estimate a model with 98 observations each time through the loop How would you rewrite the DO loop to perform this task Answer Rewrite the LINREG instructions such that the initial observation increases by 1 each time through the loop The two LINREG instructions should be lin noprint define ar1 dlrgdp 1 97 i constant dlrgdp 1 lin noprint define ar2 dlrgdp i 97 i constant dlrgdp 1 to 2 Now the first time through the loop the initial observation is 3 i e 3 100 97 and the last observation is 100 The second time through the loop the initial observation is 4 and the last observation is 101 Continuing through i 168 yields an entry value of
149. d reward for holding the asset If r is the one period excess return from holding the asset and h the conditional volatility they estimate a model in the form ri Bo Bil 44 Walter Enders h Oo 01 0 4 0 3 7 0 2 7 4 0 1 7 The appropriate FRML instructions to estimate this model are frml e r b0 b1 h frml h a0 a1 0 4 8 1 2 0 3 g 21 2 0 2 e 3 2 0 1 e 4 2 The first statement defines e as r Bo Di i If Bi is positive increases in risk as measured by h increase the expected return The second statement defines the conditional variance 3 An IGARCH Model of the Spread The specification for the spread used above required four coefficients to estimate the conditional variance h A more parsimonious specification is the GARCH 1 1 model spread Oo Ospread oospread ospread s x 2 h PUE b bg ch Notice that it is not possible to define h using a FRML statement As in the case of a MA model the following is an illegal statement because h is defined in terms of its own lagged value frml h bO b1 e 1 2 cl h 1 The appropriate solution is to use a placeholder for h 1 As such the SET statement initializes the series temp to be zero NONLIN prepares RATS to estimate the parameters a0 al a2 a3 bO b1 and cl The NONLIN instruction restricts b0 b1 and c1 to be positive The first FRML instruction creates e as the AR 3 model of
150. ds 2 header RESIDUAL ANALYSIS gra header Time Path of the Residuals 1 resids gra max 1 0 min 1 0 style bar number 0 nodates header Residual ACF 1 corrs spgraph done 5 Notice that the procedure creates the series corrs and resids We want to include these newly created series on the list of local series using local series dy resids corrs time The complete procedure is listed below procedure unit y type series y option switch graph 0 option integer lags 1 option choice det 2 none intercept trend local series dy resids corrs time set dy y y 1 set time t if det eq 1 lin noprint dy resids y 1 dy 1 to lags if det eq 2 lin noprint dy resids y 1 dy 1 to lags constant if det eq 3 lin noprint dy resids y 1 dy 1 to lags constant time dis The estimate of Rho beta 1 dis The t statistic for the null hypothesis Rho 0 is tstats 1 cor noprint number 24 resids corrs if graph 1 spgraph hfields 1 vfields 2 header Residual Analysis At this point in the construction of the procedure you might want to place the COR instruction inside of the IF block As such the correlations would be computed only if the GRAPH option is selected However when we extend the program in later sections we will want to place COR outside of the IF block 218 Walter Enders gra header Time Path 1 resids gra max 1 0 min 1 0 style bar number 0 nodates header Resid
151. dure 1 The procedure is called SEASONS because we are going to modify it to allow for seasonal dummy variables If the seasonal span of the data is s the procedure will estimate a model of the form yr do atime atime aztime b D b2D2 b D 14 The first step is to add the choice Estimate the Trend and Seasonals to the USERMENU The procedure will set the variable s_ 0 if the user selects Estimate the Trend and will set s_ 1 if the user selects Estimate the Trend and Seasonals As such the modified procedure adds s_ to Writing Your Own Procedures 243 the list of local integer variables Moreover note that the third choice on the USERMENU is Estimate the Trend and Seasonals and that Done has been moved to the fourth position procedure seasons local integer depvar s_ usermenu action define title Trends 1 gt gt Select a Variable to Estimate 2 gt gt Estimate the Trend 3 gt gt Estimate the Trend and Seasonals 4 gt gt Done We do not want to ENABLE either of the Estimate choices unless the user has chosen Select a Variable to Estimate As such the following two lines are used to prevent the user from selecting choice 2 or choice 3 usermenu action modify enable no 2 usermenu action modify enable no 3 Next it is necessary to modify the statements that are executed with each MENUCHOICE Consider the program segment below If Select a Variable to Estimate 1s selected MENUCHOI
152. dure using unit tblyr you can print the ACF of the regression residuals by entering the instruction print corrs The two potential problems with this method are 1 the user needs to know the name of the series containing the correlations and 2 you might not want the name corrs to refer to a global variable A way to rectify this problem is to modify the PROCEDURE TYPE SERIES and LOCAL SERIES instructions such that procedure unit y start end corrs Add corrs to the parameter list type series y corrs Define corrs to be a series that can be passed back to the main program local series dy resids time Remove corrs as from the list of LOCAL series No other instructions need to be modified or added Now you can invoke the procedure and print the ACF of the residuals using unit series start end name for the correlations Note that the user can select any valid series name for the correlations corrs is simply a placeholder for the name specified by the user For example the following three instructions all yield the identical output unit tblyr 3 169 corrs pri corrs unit tblyr 1 2000 1 zz pri zz unit tblyr x pri x 5 2 Optional Fields Notice that the following three instructions produce quite different results unit tblyr The estimate of rho with 1 lags 20 05255 The t statistic for the null hypothesis rho 0 is 2 31170 unit graph tblyr corrs The estimate of rho with 1 lags 0 05355 The t stat
153. e TABLE instruction shows the order of the series in RATS memory Since b 11 the summary statistics for tblyr and the two newly created series are displayed At this point the EQV or LABELS instructions can be used to assign labels to the two new series Note that any additional series created will begin with integer number 14 98 Walter Enders scratch 2 b tab b to b 2 Series Obs Mean Std Error Minimum Maximum TB1YR 167 6 1538722555 2 3936221961 2 7133333333 14 3800000000 No Label 12 0 No Label 13 0 Retrieving Labels and Integer Numbers The simplest way to retrieve the integer value assigned to a series is to use the COMPUTE instruction You COMPUTE an integer value to be equal to the name of a series com integer series name For example to display the integer value assigned to rgdp use com num rgdp dis num At first this instruction makes no sense since num is a number and rgdp is a series recall that COMPUTE is not used to set entire series Since RATS expects a number on the right hand side it will equate num with the series number assigned to rgdp Thus after entering dis num in Program 3 3 you will see the integer 7 displayed on the screen The function L number returns the LABEL attached to the specified variable Hence dis 1 7 RGDP dis cl rdgp RGDP Also note that you can use com a 1 7 to assign the string RGDP to the string variable a Once a has been computed it can
154. e cointegrating vector and with all values of A L 0 Hence the simulated model is yr2 yu OL Yer Bix eu Xp Xia Oo yer Bixe1 ex Step 2 Estimate the cointegrating vector y By Bx e and use the f distribution to form a 95 confidence interval around B The point of the exercise is to determine how well or poorly the distribution works for cointegrated variables If this confidence interval includes the actual value of B used in STEP 1 add 1 to the variable success If the f distribution is appropriate success should be increased in 95 of the Monte Carlo replications Also store the value of B so that it can be compared to the actual value of B4 Step 3 Repeat the experiment 2000 times and examine the variable success Step 4 Repeat Steps 1 3 for different values of 0 05 B and the elements of X The program relies on the EQUATION and the SIMULATE instructions If you are familiar with these instructions you can skip the remainder of this section As used in the program the syntax for the EQUATION instruction is Edi you have an equation with no AR or MA terms omit these fields and the MORE option IF Statements and Monte Carlo Experiments 151 EQUATION COEFFS coeffs other options equation depvar ARlags MAlags list of explanatory variables where equation Equation name depvar Dependent variable ARlags Number of AR lags MAlags Number of AR lags coeffs Vector of coeffic
155. e examples will use a single data set MONEY DEM XLS and all examples are compatible with RATS 5 0 This chapter begins with a quick overview of some of the basic RATS instructions and options we will be using in the later chapters It is intended to refresh your memory and to introduce the use of switches options choices and internal variables It then shows how to estimate nonlinear models using nonlinear least squares NLLS and maximum likelihood techniques Thomas Doan Thomas Maycock and Mark Wohar made many helpful comments on the earlier versions of the manuscript All errors that remain are my own 2 Walter Enders 1 The Data Set The file labeled MONEY DEM XLS contains quarterly values of seasonally adjusted U S nominal GDP real GDP in 1996 dollars RGDP the money supply as measured by M2 and M3 and the 3 month and 1 year treasury bill rates for the period 1959 1 2001 1 Both interest rates are expressed as annual rates and the other variables are in billions of dollars The data was obtained from the website of the Federal Reserve Bank of St Louis www stls frb org index html and saved in Excel format If you open the file you will see that the first eight observations are DATE GDP RGDP M2 M3 TB3mo TBlyr 1959 1 496 10 2273 00 287 80 290 05 2 77 NA 1959 2 509 20 2332 40 292 12 294 35 3 00 NA 1959 3 510 20 2331 40 296 12 298 24 3 54 4 49 1959 4 514 20 2339 10 297 14 299 10 4 23 4 74 1960 1 527 90 2391 00 298 66 300
156. e maximum value of p to use in the estimated models autoregressive models option integer maxlag 1 The third section of procedure declares the LOCAL variables Here v v2 bestlag and i are all declared as local integers and sbe sbcmin are local real numbers local integer v1 v2 bestlag i local real sbc sbcmin Note that LAGLENGTH SRC does not create any series as such we do not define any local series The series y is not created by the procedure y is just a placeholder for the series passed to LAGLENGTH SRC by the user Writing Your Own Procedures 227 In any procedure the executable instructions follow the declaration of the local variables The first executable instruction is INQUIRE The integers v and v2 will contain the start and end dates selected by the user The next three lines initialize several key variables The LINREG regresses y on a constant calculates the SBC called sbcmin for this regression and computes bestlag 0 In essence the procedure estimates and calculates the SBC for an AR O model inq series y v1 gt gt start v2 gt gt end lin noprint y v1 maxlag v2 constant compute sbcmin 9onobs log 96rss Ynreg log Znobs bestlag 0 The next section of the procedure estimates an AR p model for lag length running from 1 to maxlag Notice that all estimations are performed over the same sample period the start date for each is v maglag and the end date is v2 The first time through the DO loop
157. e ordering dlrgdp dirm2 drs with the following set of instructions 6 16 Although RATS produces the impulse responses for periods 1 through 24 only the first three impulses are shown here RATS displays the variance decompositions for all forecast horizons through 24 we display only the l step 8 step 12 step and 24 step ahead forecast error variances VARs and Error Correction Models 59 errors impulses model chap2 3 24 v As shown on the next page a one standard deviation shock to dirgdp approximately equal to 0 00653 units induces a contemporaneous increase in dlrm2 by 0 00129 units and a contemporaneous increase in drs by 0 11587 units After one period dirgdp is still 0 00093 units above its initial value while dirm2 and drs are 0 00058 and 0 21130 units away from their initial values On the other hand a one standard deviation shock to d rm2 equal to 0 00516 units has no contemporaneous effect on d rgdp but induces a contemporaneous decrease of 0 15611 units on drs After one period dirgdp 0 00069 dlrm2 0 00430 and drs 0 10718 Given the ordering of the Choleski decomposition a one standard deviation drs shock equal to 0 51507 has no contemporaneous effect on the other variables in the system After one period dirgdp 0 00144 dlrm2 0 00221 and drs 0 20055 Responses to Shock in DLRGDP Entry DLRGDP DLRM2 DRS 1 0 006525648724 0 001292115488 EIBIES 96922 o rau 2 0100019219 ONES ZI E 0 5 7 COS S
158. e usable observation is lost as a result of differencing and another is lost as a result to the term dlm3 1 com al 8 bl 0 cl 0 1 var 0 01 max method simplex iters 4 L 3 max L 3 MAXIMIZE Estimation by BFGS Convergence in 32 Iterations Final criterion was 0 0000095 lt 0 0000100 Quarterly Data From 1959 03 To 2001 01 Usable Observations dL Function Value 1590 80067422 Variable Coeff Stc ETT GI Ttar Signif Ck Ck ck Ck ck kk KKK KKK KKK KKK KKK KKK KKK KK KKK KKK KKK KKK KKK KKK KKK KKK KKK KKK ko kx ko ko ko dks Al omoro OW TENE OP OSES SOS 73 81048 0 00000000 oe Bil 092759 009 9 11 549 01 0 603610 O MALTAS Se 9 12L520605 9860399296 L59647 0711959666 4 VAR 0 000026837 0 000002077 E70 21915 ae OP 000001000 You can see that the bilinear coefficient c is not significant at conventional levels As such it does not appear that the bilateral model is a satisfactory representation of the dlm3 sequence One word of caution is in order since it appears that D is significant at the 5 level However it does not follow that dlm3 follows an ARMA 1 1 process Since dlm3 1 1 is correlated with the individual t statistics can be misleading In fact if you eliminate the Linear and Nonlinear Estimation 39 bilinear coefficient c and estimate dlm3 as a pure ARMA 1 1 model the MA 1 coefficient is insignificant To illustrate the point consider set temp 0 nonlin al b1
159. e variable sign is equated to the significance level of the restriction If at the end of any loop sign lt 0 05 the WHILE END loop is terminated Notice that sign exceeds 0 05 for lags 12 through 8 so that looping over the instructions within the WHILE END block continues However in the AR 7 model the significance level of the coefficient on dlr is less than 0 05 and RATS exits the loop i e the condition sign gt 0 05 is not true The next two lines produce the AR 7 model lin drl constant drl 1 to lags Variable Coeff Std Error T Stat Signif KK ck C KKK KKK KKK KKK KKK KKK KKK KKK KKK KKK KKK KKK KKK KK KK KKK KK KK KK KKK ko ko ko kx kx kx Xo 1 Constant 0 007234002 0 053259079 0 13583 0 89213923 2 DRL 1 0 281822834 0 080465922 3 50239 0 00060674 3 DRL 2 0 343129641 0 083606666 4 10409 0 00006620 4 DRL 3 0 274614957 0 088185213 3 11407 0 00220824 5 DRL 4 0 035831951 0 090593547 0 39552 0 69301422 6 DRL 5 0 016548074 0 087971257 0 18811 0 85104475 7 DRL 6 0 030787141 0 083329087 0 36946 0 71229913 8 DRL 7 0 202414362 0 080118649 2 52643 0 01255294 Jazzing Up the Program The program can be made much more useful by incorporating it into a DOFOR loop You can allow DOFOR to loop over each series Within each of these loops the WHILE END block determines the lag length The remaining portion of Program 3 8 constructs the variable dlp The counter i in the DOFOR instruction loops over each for these fi
160. ead in the data set MONEY_DEM XLS and create dirgdp using PROGRAM 5 of CHAPTERS PRG Next estimate dirdgp as an AR 2 process using lin dirgdp resids constant dlrgdp 1 to 2 Variable Coeff Std Error T Stat Signif CK CK Ck Ck CK CI CI CIC CC CK CI CIC CC C CI CI CI CCS CC CK CI CI CIC CC C CI CI CI CI CC C Ck CK A E M KA KG Kx Kk Kk ko kx ko koX 1 Constant 0 0051566068 0 0010217954 5 04661 0 00000119 2 DLRGDP 1 0 2508977521 0 0769801061 3 25925 000135976 3 DLRGDP 2 0 1362250820 0 0762100846 1 78749 0 07571568 Now we can perform the identical estimation using matrices First we can make the matrix x containing the regressors from the AR 2 using make lastreg x 4 2001 1 198 Walter Enders Unlike a series you can show the contents of a matrix using the DISPLAY instruction If you DISPLAY x you will see that the first 5 rows of matrix x are dis x 1 00000 4 28835e 04 0 02580 1 00000 0 00330 4 28835e 04 1 00000 0 02195 0 00330 1 00000 0 00495 0 02195 1 00000 0 00185 0 00495 You can see that the first column consists of all 1 s the second column contains dirgdp 1 and the second column contains dirgdp 2 You can display any particular element of x by referring to its row and column For example dis x 2 3 4 28835e 04 Next create the matrix y containing the dependent variable i e the contemporaneous values of dirgdp Care needs to be taken about the conformability of the x and y matrices It is necessary t
161. ecall that the first few impulses from the model are Responses to Shock in TB1YR Entry TBlYR TB3MO Lo 06427986990792 051551 11222 1 965055 2 O GAS PEOX NOS SSWIOIGIGIG ZI 3 0 6580803688211 0 6962100440500 Responses to Shock in TB3MO Entry TRIR TB3MO 1 0 000000000000 0 243115941739 2 009770291006 9m0 RZ ASS SD SS 3 0 073515992403 0 180662886845 Instead suppose you want to force the regression residual from the tblyr equation to be identical to the pure innovation in tb yr and the regression residual from the tb3mo equation to be identical to the pure innovation in tb3mo This is equivalent to setting gi g21 0 Consider e 1 016 e 0 1 amp To equate e with 1 and e with z use com g l 1 0 10 1 II To obtain 24 impulses using g as opposed to the Choleski decomposition use impulse model term results impulses decomp g 24 Arne dE SC TB3MO 0000000000000 0 0000000000000 0949641548808 0 2761958544234 7143622610318 0 3227246002156 Responses to Shock in TB3MO Entry TBLYR TB3MO 1 0 000000000000 1 000000000000 2 ORV ASTE S SOUS NEMESIS OT 087710 1 2 2 5 3 ORS 02890 C695 08s One AS STESSO TAI For comparison purposes only the first three impulse responses are shown Now an innovation to tb yr has no contemporaneous effect on tb3mo Also note that an innovation in tb3mo has a 1 unit effect on tb3mo This follows since G is normalized such thato and 07 both e
162. ecification spread Op OSpread oospread O3spread 3 E v Jb b b bE and v WN O 1 As such E amp 0 and E h b bE b b 2 The NONLIN instruction prepares RATS to estimate the six parameters a0 al a2 a3 bO bl b2 and b3 Since the coefficients in h cannot be negative b0 bl b2 and b3 are constrained to be non negative The first FRML creates e as spread a0 al spread a2 spread 2 a3 spread The 12 Tf any of these coefficients is zero it becomes possible to estimate a negative value for the conditional variance Linear and Nonlinear Estimation 43 second FRML creates the ARCH 3 model for the conditional variance and the third creates the log likelihood nonlin a0 al a2 a3 bO b1 b2 b3 b0 ge 0 b1 ge 0 b2 ge 0 b3 ge 0 frml e spread a0 al spread 1 a2 spread 2 a3 spread 3 frml h bO b1 e 1 2 b2 e 2 2 b3 e 3 2 frml L log h e 2 h A linear regression without ARCH errors is used to obtain the initial guesses for a0 al a2 a3 and bO The initial guesses are refined using the SIMPLEX method and the final estimates are reported using the default BFGS method Notice that we do not need to specify the start end dates here The maximization begins with observation nine three usable observations are lost as a result of the term spread 3 another three are lost as a result of the term e 3 and two are missing since tbly
163. ect the same model Writing Your Own Procedures 209 2 1 Integer and Choice Options Oftentimes you will want something more complicated than an ON OFF switch At times it will be convenient to pass a particular number to the procedure For example BJIDENT SRC allows you to obtain the ACF and the PACF using first differences of the data using DIFF Similarly BJIDENT SRC allows you to select a logarithmic transformation of the data transformation using TRANS log The syntax for an integer choice such as the number of lags to use in the ACF and PACF is OPTION INTEGER option name default value To allow for a non numerical set of choices such as TRANS log use OPTION CHOICE option name default number list of choices Examples 1 Suppose you want your procedure to estimate the series y as an AR p model where p is selected by the user Since the number of lags is an integer use the INTEGER option To set the default number of lags equal to use OPTION INTEGER lags 1 You should protect the user from inadvertently entering lags 0 Somewhere in your procedure you could use the following set of instructions if lags ge 0 lin y constant y 1 to lags 2 Suppose you want the user to determine whether to graph series y in levels first differences or in logarithmic first differences If you want the default to be such that the series is displayed in levels you can use OPTION CHOICE trans 1 levels diff growth Your
164. ed System This is one of my favorite programs It illustrates a number of RATS advanced features and the folly of using traditional distribution theory to perform hypothesis tests on a cointegrating vector Suppose that x and y are two non stationary time series variables that are cointegrated of order The error correction representation of the system is Yr yi1 Of Bo Yer Bux An D Ayr Ab D Axia ei Xp Xr 1 Ol Bo Yer Dix A21 L Ay 1 Az D Ax a ez It is assumed that e and e are serially uncorrelated but the covariance Ee e need not be zero As in Chapter 2 if the variances and covariance are time invariant we can write the variance covariance matrix as 150 Walter Enders Oi On Y 11 O21 05 where Var ei Ox and Cov e15e 01 62 The nature of the system is such that y and x are unit root processes that are linked by the cointegrating vector Bo y i Dix i Suppose you estimate the regression equation Y Bo Bx te Stock 1987 proves that the OLS estimates of the coefficient of a cointegrating vector converge faster than similar estimates for stationary variables As such it is inappropriate to use standard t statistics and confidence intervals to perform inference on B The following Monte Carlo experiment illustrates the problem Step 1 Generate x and y as a cointegrated system For simplicity we will generate the series without an intercept in th
165. efficients 2 and 3 are to be restricted The third line imposes the restriction 1 0 1 002 0 Linear and Nonlinear Estimation 11 3 If you reexamine the regression output it seems as if Q O5 0 O 04 0 and Qs Q6 0 To test these three restrictions use restrict 3 23 110 45 110 56 110 There are 3 restrictions and one set of supplemental cards for each restriction Note that RESTRICT can be used with the CREATE option to test and estimate the restricted form of the regression Moreover whenever CREATE is used you can save the regression residuals simply by providing RATS with the name of the series in which to store the residuals In the example below RATS displays the output of the restrictions from Example 3 above and stores the regression residuals in the series resids NOTE Only a portion of the output is shown restrict create 3 resids 23 110 45 110 56 110 i 15 LoS Level 0 08449735 2 25276 with Significance Variable Coeff STRETTON P Site lite SALG met KEKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKK EAER 1 Constant 015301284 056541363 002570162 78704020 D ESSI RS 2 RS 3 RS 4 RS 5 RS 6 RSH 73 5 SS AIEO OI gaol iii 332 STOS a2 aG S 0925 o db oos SHLiE SEIES S I9IStS SIMI SEINO GIO EIE Gy ey fex Se ey cox e 068502626 068502626 0797736678 OTITIS GOLE 074036283 074
166. eous relationship among the three variables Moreover the unforecastable portions of dirgdp and drs i e ey and e are due only to their own pure shocks We can model these contemporaneous relationships as n 1 0 Offe x Ea 821 1 8a Ever e 0 0 1 fle rt rt Notice that we have an over identified system in that we have restricted four elements of G to be zero To perform this alternative decomposition use the following instructions nonlin g21 g23 dec frml rect g form frml g form II 1 0 0 1 g21 1 g2310 0 1 Il com g21 2 g23 20 3 cvmodel factor g csigma g form The NONLIN instruction informs RATS that we want to estimate the parameters g2 and g23 The next instruction DECLARES the FORMULA g form The third instruction is used to specify the form of g form The fourth instruction provides initial guesses for the two parameters Next we use CVMODEL to perform the estimation cvmodel factor g sigma g form Since n 3 exact identification entails n n 2 3 restrictions VARs and Error Correction Models 79 Covariance Model Estimation by BFGS Convergence in 12 Iterations Final criterion was 0 0000000 0 0000100 Observations ESIC Log Likelihood 1706 46892511 Log Likelihood Unrestricted 3L 3 2 OOZ TEZIS Chi Squared 1 7 06771404 Significance Level 0 00784853 Variable Coeff Site Clas Mutants Ore Site cite SLEALE VASE RS aT a AT PATE TAS TAS PAY eT Tay E E STATUT
167. er causality tests The variance covariance of the residuals is saved as the matrix v The ERRORS instruction produces forecast error variances from 1 step ahead through 24 step ahead horizons and impulse responses for each of the three variables in the system The ordering of the Choleski decomposition is that used on the VARIABLES instruction Hence the errors statement below uses the ordering dlrgdp dlrm2 gt drs 54 Walter Enders estimate outsigma v errors impulses model chap2 3 24 v 2 Suppose that the VARIABLES instruction in the example above was replaced with var drs dlrm2 dirgdp The forecast error variances from 1 step ahead through 12 step ahead horizons will be displayed for each of the three variables in the system Now the errors model chap2 3 12 V statement uses the ordering drs dirm2 dlrgdp Note that IMPULSE allows you to experiment with different orderings without having to re estimate the model VARs and Error Correction Models 55 2 Example Estimation of a 3 Equation VAR Program 2 1 in the file CHAPTER2 PRG contains all of the instructions in the sample program discussed below Suppose you want to estimate a VAR using the three variables dipgdp dirm2 and drs After reading in the data set MONEY DEM XLS and constructing the variables set up the VAR system with 12 lags of each variable using system model chap2 var dlrgdp dirm2 drs lags 1 to 12 det constant end system Next estimate
168. eries Obs Mean Std Error Minimum Maximum ALPHA HAT 500 0 49596865421 0 14989449989 0 07604389005 1 00199740576 BETA HAT 500 1 07098252906 0 36624672687 0 30586651854 2 74200765785 Notice the mean values of the estimates are quite close to their true values However the standard errors of alpha hat and beta hat are very large Results with Antithetic Variables The remaining portion of the program takes advantage of antithetic random numbers The next two instructions create the series alpha bar and beta bar these two series will hold the averaged estimates of o and p set alpha bar 1 500 0 set beta bar 1 500 0 Again the next three instructions prepare RATS to perform a nonlinear estimation of the model and provide initial guesses of o and f nonlin alpha beta frml monte y beta x alpha com alpha 0 48 beta 0 98 Next two DO loops are created The instructions within the DO i loop are executed 500 times The series eps will contain 50 i i d draws from a standardized normal distribution The next portion of the program is the critical part do i 1 500 set eps Y ran 1 do j 2 0 1 set y x 0 5 1 j eps j eps nlls frml monte noprint y Each time through the DO j loop y is set equal to x 1 j eps j eps The first time through the DO j loop j 0 so that IF Statements and Monte Carlo Experiments 157 yt i The second time through the DO j loop j 1 so that y xg The NLLS instru
169. ermenu action modify enable no 3 Estimate the Trend is choice 3 usermenu action modify enable no 4 Estimate the Trend and Seasonals is choice 4 If Transform a Variable is selected 7 MENUCHOICE 1 and the user is presented with a list of variables Selection of the variable invokes the procedure TRANSFORM Once the graph of the transformed variable is displayed program control returns to the LOOP block of instructions The user can make another transformation Select a Variable to Estimate or exit the procedure by selecting Done Given the new positions for the menu choices only the remaining IF MENUCHOICE instructions need be modified Consider loop usermenu if menuchoice eq 1 f select series depvar C transform depvar if menuchoice eq 2 Select a Variable to Estimate is now choice 2 select series depvar compute a l depvar gra header Time Path of a 1 depvar usermenu action modify enable yes 3 Enable Estimate the Trend usermenu action modify enable yes 4 Enable Estimate the Trend and Seasonals if menuchoice eq 3 Estimate the Trend is choice 3 coms 0 estimate depvar s_ if menuchoice eq 4 Estimate the Trend and Seasonals coms 1 estimate depvar s_ j 246 Walter Enders if menuchoice eq 5 Done is now choice 5 break end loop usermenubreak action remove end seasons Cleaning up I have used SEASONS SRC in one portion of my undergraduate forecasting class
170. ermine how often the Dickey Fuller test rejects a false null hypothesis To be more specific given that the true data generating process is such that p lt 0 i e lt 1 the program calculates the probability of rejecting the false null hypothesis p 0 and accepting the correct alternative hypothesis p lt 0 The first two lines of the program set the default sample size of any series to 200 and seed the random number generator all 200 seed 237 Suppose we want to determine the power of the Dickey Fuller test for various values of 0 close to unity The program uses the four values 0 8 0 9 0 95 and 0 99 DOFOR will loop over real values if the index has been defined as a real number This is accomplished using com aplhal 0 0 The first time through the DOFOR loop alpha 0 8 and rho 2 The next line below initializes the variables hits10 hits5 and hits to equal zero we will use these variables to hold the number of times the Dickey Fuller test correctly rejects the null hypothesis of a unit root at the 10 5 and 1 significance levels respectively The fifth line below initializes the y sequence to equal its unconditional mean value of zero with a variance of 1 0 com alphal 0 0 dofor alphal 0 80 0 90 0 95 0 99 com rho alphal 1 0 compute hits10 hits5 hits 0 set y cran 1l The DO j loop prepares RATS to do 10 000 Monte Carlo replications Within each loop the y sequence is constructed as a
171. error variance in d rgdp respectively Extensions 1 If you want to reverse the ordering of the variables such that drs dlrm2 dirgdp use system model chap2 var drs dlrm2 dlrgdp lags 1 to 12 det constant end system estimate outsigma v errors impulses model chap2 3 24 v 2 You can produce multivariate forecasts using the FORECAST instruction The most useful form of the instruction is FORECAST model modelname results forecasts steps start where modelname The model name used on the SYSTEM instruction results forecasts Creates the series forecasts 1 forecasts n which contain the forecasts of the n variables in the system steps Number of periods to forecast start First period to forecast The following FORECAST instruction uses the VAR model chap2 to produce 12 out of sample forecasts beginning with 2001 2 forecast model chap2 results fores 12 2001 2 pri fores ENTRY FORES 1 FORES 2 FORES 3 2001 02 0 007005058389 0 018494401728 0 194920330700 2001 03 0 012225055977 0 015287643122 0 242193105882 2001 04 0 011144421777 0 015551352314 0 624105358190 2002 01 0 010444864257 0 015556006874 0 017371770409 2002 02 0 016462341748 0 014234534497 0 382157540684 2002 03 0 013156260294 0 012041121844 0 184462465867 2002 04 0 011033652617 0 013449326547 0 446338917263 eto The 12 forecasts for dirgdp dirm2 and drs are contained in the series fores 1 fores 2 and fores 3 res
172. es 1 response If you run the program as shown your output will be 6 You cannot use com a 0 this instruction would cause A to equal the number 0 CONSTANT sets each element of A to zero 192 Walter Enders Impulse Responses Jazzing Up the Program It is likely that you might want to find the impulse responses to an ARMA p q model that you estimated using the BOXJENK instruction BOXJENK estimates a model and stores the resulting coefficients in a vector called BETA The first element of BETA is always the constant if any then the AR p coefficients and finally the MA q coefficients We need to make the following modifications to the program 1 Program 5 4 illustrates the process by estimating drs from the data set MONEY_DEM XLS Consider cal 1959 1 4 all 2001 1 open data a money_dem xls data org obs format xls dif tb3mo drs com number 24 The first five lines instruct RATS to read the data set and create the variable drs The sixth line indicates that we want 24 impulse responses If you use the Box Jenkins methodology you can convince yourself that a plausible model for the drs sequence is drs Oodrs a3 O4drs 3 i Big pse5 Nevertheless in order to illustrate some vector manipulations we will estimate this model including an intercept We can estimate the model with Vector and Matrix Manipulations 193 com ar I2 7I com ma II1 3ll box constant ar ar ma ma
173. esired transformation from the menu Note that you cannot execute any other instructions until you select Done Writing Your Own Procedures 239 9 An Interactive Procedure with Menu and USERMENU This section will illustrate a simple program that uses both the MENU and the USERMENU instructions The procedure allows the user to estimate a regression equation with seasonal dummy variables and a nonlinear time trend of the form yr ao atime atime astime b D b2D2 bs 1D 1 amp where D through D are seasonal dummy variables and s is the seasonal span of the series The USERMENU instruction allows the user to select the dependent variable The MENU instruction is used to create a dialog box that allows the user to enter the degree of the polynomial After the procedure is executed RATS creates a menu called Trends on the menubar If the user clicks the mouse on this Trends menu a list of three potential selections appears Select a Variable to Estimate Estimate the Trend and Done The flow of the program is as follows 1 If the user chooses Select a Variable to Estimate a list of all of the series in RATS memory is displayed Once the selection is made a graph of the time path of the series is displayed Initially the user is not allowed to select Estimate the Variable The program requires the user to choose Select a Variable to Estimate before the selection Estimate the Variable is enabled 2 Once
174. estimation procedure so as to eliminate a0 nonlin al bO bl gamma gamma ge 0O frml Istar dim3 al diIm3 1 bO b1 dIm3 1 1 exp gamma dlm3 1 Linear and Nonlinear Estimation 31 Now the NLLS estimation instruction yields nils frml Istar dim3 Variable Coeff Std Error T Stat Signif KKEKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKK i A 1 3256892 0 5077631 2 61084 0 00987429 2 BO 0 0053999 0 0014724 3 66727 0 00033150 3 BL 2 0864844 0 5009183 4 16532 0 00005029 4 GAMMA 256 7862707 61 7734749 4 15690 0 00005199 compute aic Ynobs log rss 2 9onreg compute sbc nobs log rss gVonreg log 96nobs dis The aic aic and sbc sbc The aic 911 59844 and sbc 899 12647 All of the coefficients are significant at conventional levels As measured by the AJC and SBC there is little to choose between the alternative LSTAR models Moreover the SBC continues to select the AR 1 model As such there is only mild evidence that dlm3 displays LSTAR adjustment 32 Walter Enders 5 Maximum Likelihood Estimation Suppose you wanted to estimate 2 Yr Bxr amp 5 amp N 0 o Of course the most straightforward technique is to use OLS However you could obtain the maximum likelihood maximum likelihood estimate of B and o using the following instructions NONLIN b var FRML log var y b x 2 var COMPUTE b initial guess var initial guess MAXIMIZE L Notice t
175. estrict restrict dis Significance level signif dis end do Lags 16 aic r 45335 sbc Lags 12 aic 63138 sbc Significance level 0 76164 Lags 12 aic_r 00556 sbc Lags 8 aic ATIS SoC Significance level 002069 Lags 8 aic_r SOOO At COIs SE 3058 63806 Lags 4 aic 926m DSK GIS SiLSG I 7t Significance level 3 87731e 04 Thus at conventional significance levels the restriction from 16 to 12 lags is not binding However restricting the system from 12 to 8 lags has a prob value of 0 02069 and further restriction the system from 8 to 4 lags has a prob value of 3 87731e 04 The multivariate AIC and SBC both select the 12 lag model over the 16 lag model However in the other two cases the AIC selects the model with the long lag and the SBC selects the model with the short lag Loops over Dates and Series 107 4 Loops for Dates As suggested by the last program one important use of a DO loop is to perform a set of RATS instructions such that the start and or end date is altered each time through the loop Since dates have an equivalent integer representation it is simple to use date manipulations in the loop Suppose that you wanted to compare the l step ahead mean square prediction errors for an AR 1 and an AR 2 model of dlrgdp One way to do this is to estimate each model over the first 100 observations and then obtain the 1 step ahead forecast for each Since the value of dlrgdp fo
176. f Q statistic RHO First lag correlation coefficient of the residuals The internal variables can be called from anywhere in a RATS program including a procedure You need to be a bit careful since the internal variables are recalculated every time you estimate a regression 3 1 Using Switch Options You can see how to work with RATS options by re estimating the unconstrained AR 7 model of the change in the 3 month interest rate lin robusterrors drs constant drs 1 to 7 Variable Coeff Sicel JUseseione Site cite Seeger is Ck CK Ck Ck CK CI Ck CK C CK CI CK CC CI CK CC CC CK CI CC C CK CC CI Ck KC CI Ck KC CI Ck x A x KK Kk KK x Kk ko ko ko ko Constant 013585280 0 052823866 odd 4 9 0 79 0 5192 7 JL s 2s Sia 4 Sa So ds Se VI Ie ies AS duy ASF Is Silat RS 2 RS seo RS 4 RS 5 RS 6 SES TH 361765228 SALIS STALO SIUS TSZ OAAS 201542364 096578440 214686434 096834548 7169560699169 123604815 SEIESSINISIRG SINT 51111229 32 95 SUIESIO SI9I9ISIOIS 092486533 SS SION o TASOS IZZ S 22603 OATS 70806 32127 coy e e e e a ey 00018704 o 052110 5 00146116 552201 918 1L 2 07268790 ASO 2072 0287 2 8IES Notice that the LINREG instruction now uses the ROBUSTERRORS option to correct for possible heteroscedasticity Since ROBUSTERRORS corrects only the covariance matrix the point estimates of the coefficients are necessarily un
177. f econometrics will recall that P is idempotent By definition the square matrix A is idempotent if A A A You can verify that P is idempotent using com test p p p dis test Since P has the dimensions 166 x 166 you will see a tremendous amount of output displayed to the screen Nevertheless all of the values of test are approximately equal to zero Although each value should be exactly equal to zero rounding errors are present As another exercise you might recall that the rank of an idempotent matrix equals the trace of the matrix You can display the trace of P using dis trace p 3 00000 You can use the projection matrix to obtain the predicted values of y and the error terms using 200 Walter Enders com y hat p y dis y hat com e y y_hat dis e If you display e and print resids you should obtain exactly the same results Additionally you can obtain the orthogonal complement M of P by forming MzI P Notice that M is another useful way to calculate the residuals Since e yi it follows that e y Py I Py My We can calculate M using com m identity 166 P Exercises 1 Verify that identity 166 is an identity matrix with 166 rows and columns dis identity 166 2 Verify that M and P are orthogonal by entering dis m p 3 Verify that M is idempotent by entering com test m m m dis test 5 2 Hypothesis Testing in the Regression Model We can calculate the v
178. fference between different data types Consider com names Il rgdp tb3mo Il dis names 1 2 tb3mo The 1 x 2 rectangular matrix names consists of the string variables rgdp and tb3mo DISPLAY element a 1 2 produces the string tb3mo PRINT operates on one or more series As such if you try to PRINT names or an element of names you will get an error message pri names SX22 Expected Type SERIES Got RECTANGULAR STRING Instead gt gt gt gt pri names lt lt lt lt Vector and Matrix Manipulations 173 Now create a x 2 rectangular matrix b that contains the series numbers of series rgdp and tb3mo com b Il rgdp tb3mo II pri b ENTRY RGDP TB3MO 1959 01 2273 0 2 7733 33333333 1959 02 2332 4 3 000000000000 2000 04 9393 7 6 016666666667 2001 01 9439 9 4 816666666667 pri b 1 2 ENTRY TB3MO 1959 01 24 413333333333 1959 02 3 000000000000 1959 03 3 540000000000 2000 03 6 016666666667 2000 04 6 016666666667 2001 01 4 816666666667 Because b consists of a integer for which there are corresponding series you can PRINT b or an element of b If you wanted to DISPLAY the second ENTRY of tb3mo i e entry tb3mo 1959 02 you could enter dis b 1 2 2 3 00000 Finally create the VECTOR c consisting of the strings rgdp and tb3mo Note that we needed to use vector string c The default for COMPUTE is RECTANGULAR and the default for VECTOR is REAL The double bracket vector str
179. from both the dirgdp and dirm2 equations we can model these two variables using a simple 2 variable VAR The way to perform the test is to estimate a VAR with the lags of drs and a second without the lags Consider system model unrestricted var dlrgdp dirm2 lags 1 to 12 det constant drs 1 to 12 end system estimate noprint residuals unrest system model restricted var dlrgdp dirm2 lags 1 to 12 det constant end system estimate noprint residuals rest The first block of instructions estimates a VAR for dirgdp and dirm2 that includes the 12 lags of drs Even though drs is not deterministic the DETERMINISTIC instruction allows you include deterministic regressors and variables that are not estimated within the system The residuals of this unrestricted VAR are saved in unrest The second block of instructions estimates a 2 variable VAR without the lags of drs and saves the residuals in rest The likelihood ratio test has 24 degrees of freedom 12 lags of drs are excluded from each equation and mcorr 37 each regression in the unrestricted model has 37 regressors ratio degrees 24 mcorr 37 unrest rest Log Determinants are 20 596224 20 059977 Chi Squared 24 63 813364 with Significance Level 0 00001814 58 Walter Enders The restriction is clearly binding Since the lags of drs should be included in the dirgdp and dirm2 equations so that drs is not block exogenous we need to return to the 3 variable VAR Yo
180. ge In the third loop the program estimates the best fitting of the two models and displays the output The first line of the program creates a seasonal dummy variable that is 1 in the 4 th quarter of each year and zero in all other quarters The next instruction of PROGRAM on the file CHAPTERS PRG initializes a switch called print that is OFF i e 0 in the first two loops and is ON i e 1 in the third loop It also initializes the variable aic_min this variable will be used to hold the aic for the best fitting model sea seasons com print 0 aic min 100000000 Next we create two vectors one will hold the deterministic regressors we want to keep in the VAR perm det and the other holds the current regressor list temp det These two integer vectors are initialized to hold only a constant dec vector integer temp det perm det com temp det llconstantll com perm det Ilconstantll Next we begin the three DO loops On the first loop i 1 and the bracketed conditional statement is ignored Hence the only deterministic regressor is the constant On the second loop i 2 and three seasonal dummy variables are added to the list of temporary variables On the third loop i 3 and the first set of conditional statements is ignored but the PRINT switch is turned ON doi 1 3 ifi 2 enter varying temp_det 184 Walter Enders perm_det seasons 1 to 3 if i 3 com print 1 Next the VAR system
181. gh lag 8 One way to do this to enter the following program instructions The Engle Granger test for cointegration requires that 2 lt o lt 0 Note that an intercept is not necessary since regression residuals necessarily have a mean of zero Linear and Nonlinear Estimation 19 dif resids dresids lin noprint dresids 1961 4 resids 1 dresids 1 to 8 compute aic Y nobs log rss 2 nreg compute sbc nobs log rss Ynreg log nobs dis T stat tstats 1 The aic aic and sbc sbc The first line creates the first difference of resids as the series dresids The next two lines instruct RATS to regress dresids on resids and on dresids 1 to 8 In order to ensure that all eight regressions are estimated over the sample period the start date on the LINREG instruction is fixed at 1961 4 Since we are interested in only the t statistic on resids we suppress the output using the NOPRINT option Next we calculate the Akaike Information Criterion A C and the Schwartz Bayesian Criterion SBC as AIC T In residual sum of squares 2n SBC T In residual sum of squares n In T where n number of parameters estimated including the intercept term if any and T number of usable observations We can use the internal variables constructed by LINREG to create AJC and SBC since Yonobs The number of usable observations in the previously estimated model rss The residual sum of squares in the previous
182. gma X frml is the FORMULA you created in Step 2 and other options include iters maximum number of iterations to use in the nonlinear estimation method BFGS SIMPLEX GENETIC The BFGS algorithm can be quite sensitive to the initial guess However only the BFGS estimation method can display standard errors and t statistics Step 4 Now that G has been created you can use the ERRORS or IMPULSES instruction to obtain the impulse responses Examples 1 In the estimation of the term structure relationship we used the IMPULSES instruction to obtain the impulse responses from a Choleski decomposition of the variance covariance matrix s It is instructive to use the methodology described in Steps 1 to 4 above to perform the same decomposition Hence we want G to have the form G 2 76 Walter Enders where g2 is a free parameter to be estimated from the data Consider the following instructions nonlin g21 dec frml rect g form frml g form I 1 0 1 g21 1 Il com g21 0 01 The NONLIN instruction informs RATS that we want to estimate a single parameter called g21 The next instruction DECLARES the FORMULA g form The third instruction is used to specify the form of g form Unlike our previous examples there is a free parameter Since this parameter is estimated using non linear estimation methods we need to provide RATS with an initial value In the example the value 0 01 is used Next we use CVMODEL to pe
183. gression Estimation by Least Squares Dependent Variable DRS Quarterly Data From 1961 01 To 2001 01 Usable Observations 161 Degrees of Freedom 153 Centered R 2 0 284460 REBATI OS Zo 28 Uncentered R 2 0 284720 JD 5X JR 45 840 Mean of Dependent Variable PSOHES 5962s Std Error of Dependent Variable 0 8194146456 Standard Error of Estimate 0 7088184693 Sum of Squared Residuals 76 870814220 Regression F 7 153 6892 Significance Level of F 00000001 Durbin Watson Statistic s 95510 7 Variable Coeff Cote c MUBUTOTS HS Jr Sis Synt gmasts Ck CK CK Ck CK CI C CK C CK CI C CK CC CK CI Ck CK CC CC CK CI CC Ck CK CI CI C CK C CK CI Ck C Ck I Ck x KK Kx x Kk Kk ko ko ko ko ko ko ko Le 2s 3 4 Sa 6 Js Sie Constant 2013585280 RS 1 361765228 RS 2 EASIEST RS 3 2393346293 RS 4 185273449 RS 5 72 OMS 42364 RS 6 096578440 ESTA 214686434 SOS oO SIN OIG 5202 91 20795710494 23561207 084224961 S STETS 089250834 40720 2019 9526 ni 90055 SIOIOIOSIRO IOS Oo On 084095745 14843 s 07S J907 gt 2E 795 80835102 00001102 00000171 00001961 04943488 02525340 2219715 SHE OS e 00732 QLI GL OGY 9 Coy Gro G amp G ooo Notice that the estimation begins in 1961 1 RATS automatically adjusts for the eight observations lost due to differencing and the use of seven lags As such there are 161 usable observations 169 8 161 given the five parameters e
184. h te tS tS te res pmo momo momomomomommomomomonmomouomotmnontnuomonmommonmonmonmovngm The AIC selects an ARMA 1 4 model and the SBC selects an ARMA 0 2 i e the SBC selects an MA 2 specification The final instruction in the program estimates an MA 2 model over the full sample period box ma 2 drl Box Jenkins Estimation by Gauss Newton Convergence in 16 Iterations Final criterion was 0 0000062 0 0000100 Dependent Variable DRL Quarterly Data From 1959 04 To 2001 01 Variable SO eS SEOMRA TSEN Signif KKEKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKK ds MATI 0 293888856 0 076814701 95192059 52g QI00 08582 90 Ba MATZ OI 2208999206 BORO G8 54345 Loos 50904 72551 7 RATS uses the Gauss Newton algorithm to estimate coefficients of an ARMA model The default number of iterations for the BOXJENK instruction is 40 Thus if you use this type of automated procedure to select the order of an ARMA model you must check to ensure that the estimation process converged Line 2 of the printed output for the MA 2 indicates that the process converged in 16 iterations Loops over Dates and Series 105 3 3 Lag Length Tests ina VAR We can use a similar procedure to perform lag length tests in a VAR In Chapter 2 we performed some likelihood ratio tests to determine the lag length in a VAR using the variables dlrgdp dlrm2 and drs It is possible to automate the procedure using a
185. hapter 1 a number of RATS instructions such as LINREG have SWITCH and CHOICE options It is straightforward to include such options within your own procedure Recall that a SWITCH option allows only two choices ON or OFF For all RATS switching options you can turn on the switch by equating its value to 1 and turn off the switch by equating its value to 0 The appropriate syntax to include a SWITCH option in a procedure is OPTION SWITCH option name default value Note The default value must be 0 or 1 If you turn back to Section 3 1 of Chapter 1 you will see that LINREG has a SWITCH option named PRINT Since PRINT 1 is the default all of the following will cause the regression output to be displayed lin drs constant drs 1 to 7 lin print drs constant drs 1 to 7 lin print 1 drs constant drs 1 to 7 com ii 1 lin print 11 drs constant drs 1 to 7 Similarly all of the following will suppress printing the regression output lin noprint drs constant drs 1 to 7 lin print 0 drs constant drs 1 to 7 com ii 0 lin print 11 drs constant drs 1 to 7 208 Walter Enders We can illustrate the use of a SWITCH option in a procedure by returning to BIC SRC Note that a number of authors calculate the values of the aic and the sbc using the following two formulas T T aic TW Ye e2 rior and sbc srm Se etinm rir t 1 t 1 It is simple to modify BIC SRC to allow us to select the desired form The modified
186. hat the right hand sides of the VAR equations contain only pre determined variables Since the error terms are serially uncorrelated with constant variances each equation in the system can be estimated using OLS Moreover OLS estimates are consistent and asymptotically efficient Even though the errors are correlated across equations estimation using seemingly unrelated regressions SUR does not add to the efficiency of the estimation procedure since both regressions have identical right hand side variables If one or more of the equations is constrained so as to have different right hand side variables than the others including the possibility of differing lag lengths the system is called a near VAR A near VAR can be estimated using RATS SUR instruction In this case SUR improves the efficiency of the estimates Preparing RATS to perform a VAR analysis consists of the following two steps Step 1 After making the necessary data transformations you must define the equations to use in the VAR Typically you will use the following five instructions to set up a VAR P Don t let the saying You can t teach an old dog new tricks apply to you Users of RATS 4 3 and earlier will find that all of their VAR programs are compatible with version 5 0 However to take advantage of some of the new features in RATS you must use the MODEL modelname OPTION with the SYSTEM instruction 48 Walter Enders SYSTEM MODEL modelname VARIABLES lis
187. hat the steps to perform maximum likelihood estimation are very similar to those for nonlinear least squares To obtain maximum likelihood estimates Step 1 Specify the parameters to be estimated on a NONLIN instruction In the example above the NONLIN instruction prepares RATS to estimate the parameters b and var Since RATS cannot process Greek characters b and var are used to denote B and o respectively Step 2 Define the likelihood or support for observation t using a FRML statement In the example above if we are willing to assume that the values of are assumed to be normally distributed random variables that are independent of each other the log likelihood of observation f is 1 c A 2 0 2 In 27 01 2 In o 2 y Bxy T The values of B and var that maximize Y are identical to those maximizing t l T T Tlno l0 By t l Hence it is appropriate to use FRML L log var y b x 2 var Step 3 Set the initial values of the parameters using the COMPUTE command Step 4 Use the MAXIMIZE instruction to maximize the formula defined in Step 2 Linear and Nonlinear Estimation 33 The MAXIMIZE instruction is the key to performing any maximum likelihood estimation Suppose your data set contains 7 observations of the variables y and x and you have used the FRML instruction to define the function L f y xs D where x and B can be vectors and x can represent a lagged value of y
188. he problem is to identify the unobserved values of and z from the regression residuals ei and ez If we knew the four values 211 21 213 and g14 we could obtain all of the structural shocks for the regression residuals Of course we do have some information about the values of the gi Consider the variance covariance matrix of the regression residuals 70 Walter Enders Eee 2 X We know the four elements of this matrix in fact you can display the elements of the matrix using DISPLAY SIGMA As in the earlier sections of this chapter denote the elements of X as OF Oo O0 Y 2 11 12 Oz 05 Although the values of the gj are unknown and and x are unobserved we know that e Ge Hence it must be the case that Eee EG e G Since Ee e X and E Xe it follows that Oi 0 GX G Oz 05 where X is the diagonal matrix defined above consisting of var 1 0 and var 0 If it is assumed that o 0 we can write 2 2 On On j u 81 811801 58 2 2 On On 811821 58 Ea Since the four values of 6 are known it would appear that there are four equations to determine the four unknown values gii 812 221 and goo However the symmetry of the system is such that O51 O12 So that there are only three independent equations to determine the four elements of G The Choleski decomposition adds an additional restriction If the pure shock to z is to have no c
189. his chapter will develop the syntax allowing you to pass information concerning series names entries and options to a procedure All of the procedures developed here are available on the file labeled CHAPTER6 PRG 206 Walter Enders 1 A Procedure to Display the AIC and SBC It is likely that your RATS sessions require you to calculate and display the aic and sbc a number of times After estimating a regression equation you need to enter the following three lines compute aic nobs log rss 2 nreg compute sbc nobs log rss Yonreg log Znobs display aic aic bic sbc Once you have typed the three lines you never need to type them again Instead of retyping you probably scroll upward in your program and re execute the three lines However this can be a bit of a hassle and if you are like me you might even lose your place in a complicated program A simple way to avoid this is to write a procedure containing the lines that you want to execute Every procedure should begin and end with its own name For example you can write a procedure called BIC as the following five lines procedure bic compute aic nobs log rss 2 nreg compute bic nobs log rss nreg log nobs display aic aic bic sbc end bic If you write the procedure in RATS you can just save the procedure in a separate file somewhere on your hard drive I save all of my procedures in the same directory containing RATS32S EXE
190. hock in TBlYR ENGEY TB1YR 1 1 0000000000000 2 1 0949641548808 5 OPMMEDS G2 oukOe ses Responses to Shock in TB3MO Entry TB1YR L 939357211920 TSS 2 1 2084956466838 3 0 9572619992645 TB3MO 0000000000000 ei Oe9 gt Odd 3227246002156 TB3MO 0000000000000 so LYONS 7307 310 9 8 9197 TAZ Wilk VARs and Error Correction Models 75 5 The Sims Bernanke Decomposition Suppose that you want to fix at least n n 2 but not all of the elements of G RATS allows you to estimate an exactly identified or an over identified structural VAR The procedure to obtain a Sims Bernanke decomposition consists of the following four steps Step 1 Use NONLIN to enumerate the elements of G that you want to estimate This list of free parameters informs RATS of the names of the parameters that will be estimated Step 2 DECLARE a FORMULA containing a rectangular matrix and use the formula to construct the G matrix Note that you need to provide RATS with an initial guess for each free parameter to be estimated Step 3 Use the CVMODEL instruction to estimate the G matrix The standard syntax you will use is CVMODEL factor output matrix other options Ysigma frml where output matrix is the name of the matrix used to store the estimate of G This will be the matrix you use to obtain the impulse responses on the IMPULSES instruction sigma is the variance covariance matrix obtained from estimating the VAR i e Yosi
191. hod is justified by the Law of Large Numbers and various forms of the Central Limit Theorem Consider the simplest case where v is an identically and independently distributed i i d random number with mean u and variance o i e Ve u o Consider the sample mean using T observations T V 1 T v t l As the sample size T grows sufficiently large a EV gu b EV uy 5o T c the distribution of V approaches a normal distribution with mean ji and variance 6 T Hence the Monte Carlo mean V is an unbiased estimate of the population mean with a variance of o T Note that V is normally distributed around the true mean u when T is large The sample variance is an unbiased estimate of the population variance Dividing the sample variance by T yields an unbiased estimate of the variance of the sample mean around the true mean The point is that population means can be estimated using means of simple random samples the accuracy is decreasing in o T so that the greater T the greater the accuracy Note IF Statements and Monte Carlo Experiments 137 that it is difficult to obtain a high degree of accuracy for the standard deviation since it is a function of T not Ty When the distribution for v is more complicated it is often not possible to obtain results in the form of a b and c Moreover it is often difficult to derive the properties of the distribution of the sample mean for the sample sizes typically used in
192. icular value is 0 1 The STATISTICS instruction generates the mean of y and this value is stored in entry i of mean At the end of the DO loop mean contains the 100 bootstrap means The standard deviation of these means is obtained using the TABLE instruction do i 1 100 set y x fix uniform 1 11 sta noprint y com mean i mean end do i table mean Series Obs Mean Std Error Minimum Maximum MEAN 100 1 90330000000 0 61347998082 0 78000000000 4 31000000000 160 Walter Enders Thus the bootstrap estimate of the standard deviation of the mean is about 0 61348 Notice this is quite similar to the standard deviation of x 2 098 divided by the square root of the number of observations 2 098 10 5 0 663 7 1 Bootstrapping Regression Coefficients Suppose you have a data set with T observations and want to estimate the effects of variable x on variable y Towards this end you might estimate the linear regression v Bo Bix E Although the properties of the estimators are well known you might not be confident using standard t tests if the estimated residuals do not appear to be normally distributed One way to estimate the sample properties of P and is to use the method of Bootstrapped Residuals After estimating the model you perform the following steps Step 1 Calculate the residuals as e y Be pas Step 2 Generate a bootstrap sample of the error terms e containing the elements E ot e
193. ient values assigned to the constant AR terms MA terms and explanatory variables respectively Other Principal Options CONSTANT Include a constant in the equation CONSTANT is the default for NOCONSTANT ARMA models MORE NOMORE For ARMA equations MORE indicates that other explanatory variables are on a supplementary card VARIANCE Value for the variance of the residual series SIMULATE creates a Monte Carlo replication of a model using normally distributed error terms The syntax used in the program is simulate model modelname results results number start V where number The number of observations in the simulated model start The start date for the first entry V The variance covariance matrix of the residuals results The simulated values of the first equation are stored in results 1 the simulated values of the second equation are stored in results 2 and so on The Program The first four instructions of Program 4 10 on the file CHAPTERA 2 PRG set the default size of a series to 150 entries SEED the random number generator and initialize the y and x series to equal zero The COMPUTE instruction sets D 1 0 and o 0 1 allocate 150 seed 2002 set y 0 0 set x 0 com betal 1 0 alphal 0 1 alpha2 0 1 equation noconstant more coeffs ll 1 alphal alphal betal Il eq1 y y 1 x 1 equation noconstant more coeffs ll 1 alpha2 betal alpha2 ll eq2 x x 1 y 1 152 Walter Enders The
194. ing overrides both of these defaults We can display either the entire vector or a single element of the vector Note that a VECTOR uses only a single subscript com vector string c Il rgdp tb3mo II dis c rgdp tb3mo dis c 2 tb3mo com d 1 0 213 4ll ez I 1 2I disde 1 00000 2 00000 3 00000 4 00000 l2 174 Walter Enders A single COMPUTE instruction can be used to create several matrices and or vectors Here a is a 2 x 2 matrix of real numbers and b is a 1 x 2 rectangular matrix of integers Note that COM vect b Il 1 2 ll creates a vector of real numbers com a 1 0 213 All vect integer b II 1 2 II If you are going to explicitly declare the type of matrix and the type of data on the COMPUTE instruction it might be simpler to use the DECLARE instruction Nevertheless the next several examples might prove instructive 5 com vect x II 1 0 4 0 3 9 2 0 II Since vect is specified the COMPUTE instruction creates the vector x consisting of four real numbers Since the default for a matrix argument is a RECTANGULAR matrix com y Il 1 0 4 0 3 9 2 0 Il creates a 1 x 4 rectangular matrix of real numbers The difference is that a matrix has double subscripts that reference the row and column while a vector has a single subscript For example you reference the value 4 0 in each using dis x 2 y 1 2 4 00000 4 00000 6 The following instruction creates the 1 x 4 RECTANGULAR m
195. ing Switches Choices and Internal Variables The best way to illustrate the use of switches choices and internal variables is to work with another example Economic theory suggests that long term and short term interest rates bear a long run equilibrium relationship Suppose that we try to estimate this relationship using the variables tb3mo and tblyr from the file MONEY DEM XLS As a first step we might want to plot the two variables Consider gra header 3 month and 1 yr T bill rates vlabel annual percentage rate patterns key upleft 2 tblyr tb3mo 3 month and 1 yr T bill rates m TB1YR PESCA TB3MO annual percentage rate MEZ PA row Lek tee ee EE ole Te 3 T ee oe Eee aT Ge ale T PT toe ho Ta Le Te TT 3T nete E T 1959 196 1967 1971 197 1979 198 1987 1991 199 1999 It appears that both variables are non stationary and there are periods of time during which they move apart Nevertheless they do seem to bear a strong long run relationship with each other We can estimate this relationship using lin tblyr resids Note that a indicates that the instruction continues on the next line The option VLABEL label allows you to supply a header for the vertical axis Linear and Nonlinear Estimation 17 constant tb3mo Variable Coeff Site Clam Tea TS Pe Sites ntes KKEKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKK 1 Constant 0 6980794657 0 0665742554
196. ing the formula j o B a0 forj 1 2 3 24 k 1 We want the program to contain the following steps 1 Allow the user to enter any number of autoregressive coefficients and or moving average coefficients 2 Calculate the 6 iteratively using the formula above 3 Create a bar graph of the first 24 values of the sequence 190 Walter Enders There are many ways to perform these three tasks the program developed below is designed to illustrate matrix manipulations Consider the first five lines of Program 5 3 on the file CHAPTERS PRG compute number 24 all number com alpha Il1 1 4 2 Il com beta Il 7 3ll The first two lines set the default length of any series to 24 You can change the value of number to obtain a smaller or larger number of impulse responses The third and fourth lines create the vectors alpha and beta The two vectors hold the coefficients of the ARMA model Here you can enter as many or as few values for a and D as desired In the example o 1 1 o 0 4 3 0 2 D 7 and B 0 3 Hence these first four lines have accomplished task 1 However we did not DECLARE either alpha or beta to be vectors trefer to the individual elements of each as alpha 1 1 alpha 1 2 alpha 1 3 beta 1 1 and beta 1 2 Now we need to do some bookkeeping Our first bookkeeping task is to compute values of p and q i e the dimensions of the alpha and beta vectors The function
197. ing with data that is not Normally distributed In essence the bootstrap uses the plug in principle the observed distribution of the random variables is the best estimate of their actual distribution The idea of the bootstrap was developed in Efron 1979 Suppose you have a data set of size T and want to estimate the mean u and the standard deviation of the mean o The key point made by Efron is that the observed data set is a random sample of size T drawn from the actual probability distribution generating the data As such the empirical distribution function is defined to be the discrete distribution that places a probability of 1 T on each of the observed values It is the empirical distribution function and not some pre specified distribution such as the Normal that is used to generate the random variables The bootstrap sample is a random sample of size T drawn with replacement from the observed data putting a probability of 1 T on each of the observed values Once the bootstrap sample has been drawn it is possible to estimate the mean of the bootstrap sample Call this estimate using the first bootstrap sample u If N bootstrap samples are drawn there are N bootstrap estimates of u that we denote by u through LL The bootstrap estimate of 6 is the standard deviation of the u7 sequence To be more specific suppose that we have the following 10 values of x t 1 2 3 4 5 6 7 8 9 10 x 0 8 35 05 17 7 0 06 13 2 0 1 8 05 The
198. ion Notice however that the index is not an integer and does not increase in any precise way from one loop to the next In fact for the last iteration the index decreases from 0 99 to 0 The instruction com alphal 0 is necessary since we need to initialize the index to be a real number Otherwise RATS would have expected a list of integers for alphal Note that we could have used any real number e g com alphal 5 for the initialization The next instruction reseeds the random number generator each time 0 takes on a new value This way the random numbers do not change as the values of the various values for o change The first instruction inside the DO i loop generates y as the current value of amp multiplied by y plus an i d normally distributed random number with mean zero and variance equal to 1 Since we are interested only in the discrepancy between the actual and estimated values of o the output is suppressed The last line in the DO i loop equates entry i of discrep with the difference between the actual and estimated values of o On exiting the DO i loop discrep will contain 1000 values of the discrepancy seed 2001 do i 1 1000 set y 2 alphal y 1 ran 1 lin noprint y constant y 1 com discrep i alphal beta 2 end 1 IF Statements and Monte Carlo Experiments 141 The sample statistics of discrep including the mean are displayed using the TABLE instruction If you want you can display ma
199. ird restriction it might be argued that fiscal shocks have no long run effect on real money balances VARs and Error Correction Models 87 The relationship among the regression residuals and the structural shocks is 8i 8p 83 Em 5 821 En 5 831 8 823 u 833 En Now restricting the elements of G such that C 1 G has all elements above the principal diagonal equal to zero is identical to assuming that monetary and fiscal policy shocks have no long run effect on the dirgdp sequence and that fiscal shocks have no long run effect on the dirm2 sequence Estimate the 3 variable VAR using all 12 lags You can create the appropriately restricted G matrix using compute g varlagsums decomp maform sigma tr inv Zovarlagsums As in earlier programs the impulse responses will be saved in a 3 x 3 matrix called impulses The next two lines instruct RATS to create this matrix and to obtain the impulse responses using the decomposition of G declare rectangular series impulses 3 3 impulses model chap2 result impulses decomp g 24 Responses to Shock in DLRGDP Entry iL 2 3 DLRGDP 0 003010662081 0 000447294073 0 001943388494 Responses to Shock in DLRM2 Entry 1 2 S DLRGDP 0 003711849678 0 000701092521 0 000027167701 Responses to Shock in DRS Entry 1 2 3 DLRGDP 0 004443216973 OR O so 902 OF 05 000597985372 Q DLRM2 004756093316
200. is estimated On the first loop only the constant is included in the deterministic regressors On the second loop the list will include the seasonal dummy variables system 1 to 3 vars dlrgdp dlrm2 drs lags 1 to 12 det temp det end system estimate print print After the system is estimated the multivariate AJC is computed The function eqnsize equation returns the number of regressors in the specified equation To obtain the number of coefficients estimated in the system find the number of regressors in equation 1 i e the first equation in the system and multiply by 3 If the resulting value of the A C is less than aic min the list of permanent regressors is equated to the current regressor list and aic min is replaced by the current AJC Otherwise these two instructions are skipped com aic nobs logdet 2 9ceqnsize 1 3 dis aic if aic aic min enter varying perm det temp det com aic min aic j Next the temporary list is replaced by the permanent list and the loop is completed enter varying temp det perm det end do If you run the program you will find that the model without the seasonal dummy variables is selected Vector and Matrix Manipulations 185 3 2 Creating a Near VAR Using ENTER Suppose that we want to forecast a series y using lagged values of a number of macroeconomic variables One way to determine which variables to include in the forecasting equation is to use Granger c
201. is line if your data is not on drive a data org obs format xls Next print out the four values of real GDP rgdp from 1959 1 through 1959 4 using pri 1959 1 1959 4 rgdp ENTRY RGDP 1959 01 2273 0 1959 02 2332 4 1959 03 2331 4 1959 04 2339 1 Now try using pri nodates rgdp ENTRY RGDP 1 2273 0 2 2399 3 2331 4 4 2339 1 165 9191 8 166 9318 9 167 9369 5 168 9393 7 169 9439 9 Thus rdgp is one dimensional array containing 169 observations or entries As you can see entry 1 is equivalent to 1959 1 entry 2 is equivalent to 1959 2 and 2001 1 is entry 169 In fact you can substitute the integers for the date labels whenever you find it convenient For example you can obtain first four values using pri 1 4 rgdp ENTRY RGDP 1959 01 2273 1959 02 2332 1959 03 2331 1959 04 2339 Pee auc Loops over Dates and Series 91 If you do not want the date labels use pri nodates 1 4 rgdp ENTRY RGDP 1 2273 0 2 2332 4 3 2331 4 4 2339 1 What about the series tb yr Recall from Chapter 1 that the first two observations for this series were NA or missing In the language of econometricians tb yr contains only 167 observations Nevertheless in RATS Programming Language tb yr contains the same number of entries i e 169 as all of the other series in the data set Entries 1 and 2 are simply recorded as NA If you PRINT entries 1 to 4 and entries 165 to 169 you will obtain print nodates 1 4 tbly
202. is the dependent variable and the equation to be estimated is alpha x beta It would also be possible to omit the depvar field and use FRML equation 1 alpha x beta The most useful options for nonlinear least squares estimate are LASTREG NOLASTREG Converts the last regression estimated into a formula EQUATION equation to convert Converts the indicated linear equation to a formula Use only after estimating the equation Step 3 RATS requires initial guesses for the parameters to be estimated This is accomplished using COMPUTE Step 4 Instruct RATS to estimate the FRML using the NLLS instruction The syntax is NLLS frml formulaname other options depvar start end residuals coeffs where depvar Dependent variable used on the FRML instruction start end Range to estimate residuals Series to store the residuals Optional coeffs Series to store the estimated coefficients Optional The principal options are METHOD GAUSS SIMPLEX GENETIC GAUSS requires a twice differential function USE SIMPLEX if you have convergence problems It is possible to use SIMPLEX or GENETIC to refine the initial guesses before using GAUSS ITERATIONS Maximum number of iterations to use Linear and Nonlinear Estimation 23 ROBUSTERRORS As in LINREG this option calculates a NOROBUST consistent estimate of the covariance matrix Note that NLLS defines most of the same internal variables as LINREG including RSS BETA TSTAT
203. is the length of the block To construct the bootstrap y series randomly select a group of L adjacent data points to represent the first L observations of the bootstrap sample In total you need to select T L of these samples to form a bootstrap sample with T observations The idea of selecting a block is to preserve the time dependence of the data However observations more than L apart will be nearly independent Use this bootstrap sample to estimate the bootstrap coefficients 9 and f IF Statements and Monte Carlo Experiments 165 Variable Coeff Std Error T Stat Signif KKEKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKK 1 DLRGDP 1 0 2508977521 0 0769801061 3 25925 0 00135976 2 DLRGDP 2 0 1362250820 0 0762100846 1 78749 0 07571568 3 Constant 0 0051566068 0 0010217954 5 04661 0 00000119 The next two instructions do some bookkeeping The first line below stores the estimates B D and B in the variables betal_hat beta2_hat and betaO hat respectively The second line stores the standard error and t statistic of B in the variables se_2 and t2 respectively com betal_hat beta 1 beta2 hat beta 2 beta0_hat beta 3 com se_2 7STDERRS 2 t2 tstats 2 If we use the normal approximation we can obtain the lower and upper u limits of the 90 95 and 99 confidence intervals for Busing dis Normal Approximation com beta2 hat 1 644 se 2 u beta2_hat 1 644 se_2 dis 90 Ih
204. istic for the null hypothesis rho 0 is 2 31170 40 The version of UNIT SRC distributed with this book contains the fix described below Hence it will not produce the error message 224 Walter Enders unit graph tblyr The estimate of rho with 1 lags 0 05355 The t statistic for the null hypothesis rho 0 is 2 31170 SX22 Expected Type SERIES Got Function Instead The Error Occurred At Location 1060 of UNIT The reason is that the field for the correlations i e name for the correlations must be specified if a graph of the ACF is to be displayed Recall that the GRAPH instruction uses gra max 1 0 min 1 0 style bar number 0 nodates header Residual ACF 1 corrs Hence the graph cannot be created unless a name for the field is specified by the user The way to fix the problem is to use one series to pass the correlations and another series on the COR instruction Consider the following modifications to the procedure 1 LOCAL SERIES dy resids xx time A variable xx is defined to be a local series This series will hold the correlations 2 cor noprint number 24 resids v1 v2 xx if defined corrs set corrs 1 25 xx The autocorrelations of resids are now stored in the series xx Note that the RATS function DEFINED name returns the status of a procedure parameter or option called name Here if the field for corrs is specified by the user defined 1 and the SET instruction is performed
205. ita nennen 179 3 Example ENTER and Supplementary Cards eene eene eren 182 3 1 Automating Model Selection in a VAR ssseseeeeenee eene emen emere nennen 183 3 2 Creating a Neat VAR Using ENTER enean rhet teri Rhen ln ea ah Aene s REY ER ERR A EE 185 Jazzing Up the IMPR O 187 4 Example Moving Average Representations esessseeeeeen ene eene emen enne 189 iii 4 1 Impulse Responses in a First Order VAR seesessesee eene eee emen renes 195 5 Creating Matrices from Your Data si ratonera aietara ad an ene enne enters 197 IS cdm CETT 197 5 1 Estimating the Regression Coefficients sssssseseesee eee eene enne ene e enne 198 EX6ICISES Tune entitatem E The Gendt cree na he teva ip eret td p tte ec E Seuss ye ee EU eens 200 5 2 Hypothesis Testing in the Regression Model eeessseeeeenen eene ener 200 5 3 Creating Series from a Matrix iiic ete nsns aane e n Eae TENE PAEK EEEE ER eser se 203 Chapter 6 Writing Your Own Procedures 204 1 A Procedure to Display the AIC and SBC esses nene enne 206 2 Using SWITCH Options MERE 207 2 1 Inte ser and Choice OPON S rrisin tette etna itte eer eite ER Rn aer iu Ee dh ate eda e RES IRR RA dn 209 3 Passing Series to a PIOCed re s ete ter tei e eh hd t E eer etd n oed Enn ka Ee eda EY ER ck RE da 211
206. itive plus 1959 1 is set equal to zero However dirgdp 1959 2 is positive As such plus 1959 3 is set equal to 1 Continuing through the observations you can verify that plus is 1 when dirgdp is positive Next we create 1 as the series minus using set minus plus Now we can create the variables dlrgdp 1 L dlrgdp o 1 I dlrgdp and lI dlrgdp Consider set yl plus plus dlrgdp 1 set y2 plus plus dlrgdp 2 set yl minus minus dlrgdp 1 set y2 minus minus dlrgdp 2 Now we can estimate the regression using lin dirgdp plus y1 plus y2 plus minus yl minus y2 minus Variable Coeff Std Error 1 SEES Signif KKEKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKK x IPOS 005600681 0 001464144 Su O29 OW WOOUSIGIs Y1 PLUS 192048274 108292518 5 1 1 93402 07806129 Y2 PLUS 1744277730 086316981 02078 04497008 Y1_MINUS sIAGGZOE IS SSC IS ATZ Y2 MINUS 010018863 odio 5557 99 US DIO SOS TINS 06048 95184946 1 2 MINUS 003903606 OOS SEA S aaa 1 24014 ALOTTI JSA 0 0 130 Walter Enders At this point you might want to pare down the number of coefficients since a number have t statistics that are quite low Nevertheless our goal here is to illustrate programming techniques not to obtain the best fitting TAR model for real GDP 3 1 Estimating the Threshold One problem with the above model is that the threshold may not be zero When t is unknown Chan 1993 sho
207. l of the program statement in braces will be executed if the value of x is equal to zero Otherwise the entire set of statements will be ignored If you have a single instruction with or without supplementary cards you can place them on the same line as the IF statement Consider if 1 ge 4 lin y constant y 1 to i The IF block will estimate an AR 7 model so long as the value of i is greater than or equal to 4 If i 1s less than 4 the regression will not be estimated Examples 1 if lags gt 0 dis The lag length is lags lin noprint y constant y 1 to lags com aic nobs log Yrss 2 Yonreg dis The aic is aic If the value of lags is greater than zero the routine will display The lag length is lags Moreover an AR p model with p lags will be estimated and the value of the AJC will be calculated and displayed 2 Consider the following routine to determine the optimal lag length for an AR process for dirgdp The first time through the DO loop p 1 and an AR 1 model is estimated The resulting AJC is compared to aic min On this first loop p 1 so that the lag length p 1 is stored in the variable p opt and the value of the AJC replaces aic min The next time through the loop an AR 2 model is estimated and the AJC for this model is compared to aic min After the DO loop is complete p opt contains the lag length of the best fitting model and aic min contains the AIC for that model
208. l zero if y 1 is negative and to equal otherwise The second and third instructions shown below create zplus the indicator multiplied by the lagged value y 1 and zminus one minus the indicator multiplied by the lagged value y 1 148 Walter Enders TASK 2 Estimate the TAR Model set plus if y 1 lt 0 0 1 set zplus plus y 1 set zminus 1 plus y 1 In the two instructions below RATS estimates the TAR model using only observations 102 through 200 Thus as in the previous program this mimics a data set with 100 total observations linreg noprint dy 102 200 zplus zminus The next two instructions are used to check the two f statistics to see if they are both different from zero If the product of the two is sufficiently close to zero the program branches back to the line labeled reset Hence if the two f statistics are sufficiently low the program branches back to the point where a new sequence is generated TASK 3 compute tl tstats 1 t2 tstats 2 if abs t1 t2 lt 0 0000001 branch reset If the program does not BRANCH to reset the next three lines of the program obtain the sample F value for the null hypothesis p p2 0 The last line within the loop stores this sample F value as entry j of the series f At the end of 50 000 replications the program exits the END DO j loop The STATISTICS instruction with the FRACTILES option produces the output shown below Be aware that the program takes a long
209. le quotation marks and that you use the symbol to begin the supplementary card for LABELS There is an important distinction between EQV and LABELS EQV produces a name that can be used for manipulations within a program However EQV cannot be used within a compiled section of a program LABELS attaches an output label to a series that RATS displays when printing a series or writing a series to a data file However you cannot manipulate the series using its output label The main reason to use an output label is to display strings that cannot be created with the SET or EQV instructions e g Spaces or a mix of upper case and lower case letters The SCRATCH instruction provides an alternative way to create consecutively numbered series Unlike ALLOCATE the SCRATCH instruction assigns series numbers beginning with the highest unused integer value Since tb yr is assigned the integer value of 11 any new series created by SCRATCH will begin with the integer value of 12 The simplest way to create series from SCRATCH is to use scratch number start end scr no where number The number of series to create start end The range of entries to allocate to the series scr no An integer variable equal to the number of existing series prior to the execution of SCRATCH Hence scr no 1 contains the integer value of the first series created by SCRATCH The next line of Program 3 3 creates two additional series and uses b to hold the value of scr no Th
210. likelihood function may find a local not a global maximum It is always wise to repeat the estimations using various initial guesses of the parameters to be estimated In working through this example I tried a number of initial guesses It turns out that initial guesses very near zero lead to an unsatisfactory result Consider the output from using initial guesses g2 0 02 and go 0 01 com g21 02 g23 0 1 cvmodel factor g sigma g form VARs and Error Correction Models 81 Covariance Model Estimation by BFGS Convergence in 3 Iterations Final criterion was 0 0000000 lt 0 0000100 Observations 156 Log Likelihood 1696 18382404 og Likelihood Unrestricted 710 002 TisxZi03 Chi Squared 1 ZT SSI SAMS ALY Significance Level 1 46283041e 07 Variable Coeff STARETE T Site clits Signif PAS ToS TAS aT TAR TUTOR TA TAS IAT eT Tay TE a To TAS YAT TAS THES TS UIST TS TAS TIS TUS TS TAS Ta Te Taf TAS ASTA THT TTS TE TS A TAS TAS TAS TRE TD TS TOPS TAS TITS US TAS TIS To Ta Y TOR TAS US TAA TTS TS AT TAS TAS TAY OS TOTS La 2 0 0431096354 0 0000000000 0 00000 0 00000000 Za 923 0 0022092447 0 0007865452 2 80880 0 00497272 Notice that the routine did converge However the standard error and t statistic of g5 are both shown to be zero Moreover the log likelihood of the restricted model is smaller than that for initial guesses g2 2 g23 0 3 I experimented with a wide range of initial guesses and usually
211. linear least squares 22 NLPAR Select convergence criteria 23 NONLIN Lists parameters for a nonlinear estimation 21 QUERY Prompts user to input variables 229 RATIO Performs a likelihood ratio test 51 RESTRICT Test linear restrictions 10 ROBUSTERRORS Corrects for heteroscedasticity 12 SCRATCH Creates consecutively numbered series 97 SIMULATE Creates a simulated series 151 SUBFORMULAS Used in creating formulas 46 SUR Seemingly unrelated regressions 61 UNTIL Conditional control of program execution 115 USERMENU Presents user with a list of choices 235 WHILE Conditional control of program execution 115 Some instructions can perform multiple tasks The descriptions refer to the task performed in the text The pages are the locations containing the primary explanation or illustration Chapter 1 Linear and Nonlinear Estimation This book is not for you if you are just getting familiar with RATS Instead it is designed to be helpful if you want to simplify the repetitive tasks you perform in most of your RATS sessions Performing lag length tests finding the best fitting ARMA model finding the most appropriate set of regressors and setting up and estimating a VAR can all be automated using RATS programming language As such you will not find a complete discussion of the RATS instruction set It is assumed that you know how to enter your data into RATS and how to make the standard data transformations If you are interested in learning about any par
212. ll are false RATS will perform ELSE if present Nested IF Statements It is possible to nest IF statements Consider the following example If condition I and condition 2 hold then statement 1 is executed Statement 2 is executed if condition I does not hold if condition I if condition 2 statement 1 else statement 2 With the appropriate use of braces you can write programs with many conditional statements embedded within others If you want to use nested IF statements you should refer to the RATS Reference Manual My experience indicates that it is best to avoid nested IF statements since they are very difficult to debug It is always possible to program precisely the same conditional statements without nested IF statements using the relational OR and AND conditions 1 1 Sample Program Lag Lengths Again Reconsider the problem of using the AIC to select the lag length for an autoregressive model of dlrgdp After reading in the data set MONEY DEM XLS Program 4 1 on the file labeled PROGRAMA PRG constructs the variables dlrgdp drs and dlrm2 Since these should be familiar to you we can jump to the next part of the program lin noprint dlrgdp 14 constant com aic min nobs log Yrss 2 96nreg p opt 0 do p 1 12 lin noprint dlrgdp 14 constant dlrgdp 1 to pj compute aic nobs log rss 2 9onreg if aic lt aic_min com p_opt p com aic min aic j end do p As discussed in Example 2 abo
213. long with the message Do you want to enter the lag length If the YES button is selected the user will see the prompt Enter the lag length The variable integer variable bestlag is set equal to the value input by the user 2 You can add the following two lines to LAGLENGTH SRC messagebox style yesno status status Do you want to run a unit root test if status eq 1 unit lags bestlag 1 y v1 v2 If you position the two lines after the IF DEFINED laglength instruction the variable bestlag contains the lag length selected by the SBC The MESSAGEBOX instruction prompts the user with the question Do you want to run a unit root test If the YES button in the MESSAGEBOX is selected the variable status 1 Since the condition on the IF instruction is TRUE LAGLENGTH SRC calls UNIT SRC Hence UNIT SRC performs a unit root test on the variable y over the sample period v to v2 using the lag length contained in bestlag If the NO button is selected status 0 and the unit root test is not performed Writing Your Own Procedures 231 Read in the data set MONEY_DEM XLS and compile the procedures UNIT SRC and LAGLENGTH SRC It is important to compile the procedures in this order If procedure A makes reference to procedure B B must be compiled before A Now enter the following instruction and select the YES button laglength maxlag 8 tblyr Variable Coeff Std Error T Stat Signif CK CK Ck Ck Ck CK CI CIC CC C CK CI CI CIC CC C CK CI
214. ly estimated model Yonreg The number of regressors in the previously estimated model Since TSTATS 1 contains the t statistic for the coefficient on resids the last line displays this t statistic the AJC and the SBC Now you could go back and rerun the routine after editing the supplementary card for the LINREG instruction such that resids 1 dresids 1 to 7 However to preview some of the material in the next chapter it is more efficient to use RATS programming language We can embed the routine in a DO loop The formulas reported here and in the RATS User s Manual are easily computable monotonic transformations of the AJC and SBC They will select the same model as the actual AJC and or SBC 20 Walter Enders doi 1 8 lin noprint dresids 1961 4 resids 1 dresids 1 to i lt lt Note the modification 1 to i compute aic nobs log rss 2 nreg compute sbc nobs log rss nreg log nobs Note the modification to the next line dis Lags i T stat tstats 1 The aic aic and sbc sbc end doI 90523 94474 T2529 TAPIS 11804 41748 Torpa 59437 S S st S S S S st Co SP epy nh SS GO NOY eal Gal Ural dp teh db Fa 0 pu er nr nr qr nr n ab Gal Gah esp oup mE Cc 9pomoomomnuootostv Now each time RATS cycles through the LOOP i increases from 1 to 8 Each time the supplementary card in encountered RATS estimates the regression using a l
215. mate of p and line 8 displays the t statistic for the null hypothesis p 0 Thus if we use unit rgdp we will obtain the estimate of p the t statistic for the null p 0 for the equation Alrgdp Bo pirgdpi i BiAlrgdpia The procedure seems to work as intended However there is a potential problem that needs to be addressed Just because a procedure works within your particular program does not mean it will work in all possible programs Suppose that you posted a copy of UNIT SRC on your class webpage and one of your students had a data set that included a variable called dy Every time she invoked UNIT SRC the procedure would write over her variable In order to separate variables used in the main body of a program from those used in a procedure RATS allows you to designate variables used in a procedure as local variables 4 1 Creating Local Variables Unless otherwise stated all variables in RATS can be used in the main program or in a procedure Thus by default all variables are global If you modify a variable within a procedure it will be modified when control returns to the main body of the program In many circumstances this can be desirable In fact you might want a procedure to perform a particular transformation on a variable and then return control to the main program Other times you might create a variable within a procedure that is already named within the main body of the program Thus the procedure would over
216. mber specified The default is 0 00001 SUBITERATIONS Subiteration limit 30 Limits the number of new coefficient vectors examined once a direction has been chosen You should increase the limit only if requested by RATS Note CVCRIT and SUBITERATIONS are options for the NLLS and MAXIMIZE instructions Unlike NLPAR the convergence criterion is altered for this estimation only 24 Walter Enders Examples 1 nlpar cvcrit 0 0001 Setting CVCRIT 0 0001 means that RATS will continue to search for the values of the coefficients that minimize the sum of squared residuals until the change in the coefficients between iterations is not more than 0 0001 2 nils frml equation_1 cvcrit 0 0001 y RATS will continue to search for the values of the coefficients that minimize the sum of squared residuals until the change in the coefficients between iterations is not more than 0 0001 Unlike example 1 the convergence criterion is 0 0001 for this estimation only The instruction nlpar cvcrit 0 0001 changes the default criterion for all subsequent estimations 3 nlpar criterion value cvcrit 0 0000001 Setting CVCRIT 0 0000001 and CRITERION VALUE means that RATS will continue to search for the values of the coefficients that minimize the sum of squared residuals until the change between iterations is less than 0 0000001 4 2 Examples of Nonlinear Least Squares 1 The FRML instruction can include lagged dependent variables For example to estim
217. ms 1 1 1 a 1 Similarly varlagsums 2 2 and 96varlagsums 3 3 correspond to 1 az 1 and 1 a33 1 respectively The off diagonal elements Yovarlagsums i j equal a 1 Next create the matrices C s and s2 using the following three instructions com c SJ VARLAGSUMS com sl 7 MQFORM 96SIGMA TR INV c com s2 DECOMP S1 Notice that s2 corresponds to C 1 G we want each element above the principle diagonal to be zero We can display this matrix using dis s2 0 01222 0 00000 0 00000 0 01210 0 01518 0 00000 0 18724 0 51647 0 63922 Next we can compute and display G using com g C S2 dis g 0 00626 0 00472 0 00102 0 00443 0 00374 0 00234 0 08599 0 55838 0 54099 The impulses responses can be obtained using impulses decomp g model chap2 24 6 2 Decomposing GDP Real M2 and the Interest Rate The neoclassical macroeconomic model suggests that aggregate demand shocks can have short run but not long run effects on economic real variables As such the Blanchard Quah decomposition is ideally suited for analyzing the effects of various shocks on key macroeconomic variables Let j Em and represent a fiscal policy shock a monetary policy shock and a productivity shock respectively In terms of our 3 variable VAR we might suppose that fiscal shocks and monetary shocks have no long run effects on real GDP Thus we have two of the requisite three restrictions To obtain a th
218. ms Bernanke Decomposition eeeeeeeeeeeeeeeenee eene enne rne e enne r Ea e en entren E 75 EXAMPLE MEE TM 75 6 The Blanchard Quah Decomposition eeeeeseeeeeeeeneeeeen enne em ene aieiaa ea nene entere nns 82 6T The Technical Details 525 reet ete re RE eese nat cevexten edebant riae rri erento 83 Ic 85 6 2 Decomposing GDP Real M2 and the Interest Rate essen eem 86 Chapter 3 Loops Over Dates and Series 09 TeDates as Tite Seis RD 90 1 leOmitting CALENDAR a a iiie i i t pene er URSI detuned ata teeta ro ee aieo a 91 I oci e 92 VAN TDIISAE MITUNTER 94 EXAMPlES iire cus 94 Retrieving Labels and Integer Numbers eesseeee eene eene nennen ener nnns 98 3 DO LOODS 99 ii 3 1 DO Loops Switches and CHOICES ie Ine rra ehe np erbe te ha Led ere ra Een 100 locii cp E E 100 ESAE PSI MEE 102 Modifying the Presta iy iii eret re topped Pere ne aU dela pred n in DEA Hr rep Erde PR n i 103 3 3 Las M Lensth Tests Ina VAR QR 105 4 TlOOpS FOr Dates ER 107 Ss LOOPS fOr SCH eS TOPIC E ID IC IDOL 105 E A 109 6 Th DOFOR Instruction x2 epe anada a aa ge e PE cvs Crece eve Rb ceraie ety 110 62 DOFORand ENTRIES iniret iei eei peri cente iia tah nahin 113 Jazzing Up the Program
219. n AR 1 process with i i d normally distributed errors Note that the value of alphal is determined by the number of times the program has completed the DOFOR 144 Walter Enders loop As constructed the variance of the y sequence is 1 0 If you take the variance of each side of the SET instruction you obtain var y a vary 1 0 Hence var y var y7 1 1 do j 1 10000 set y 2 alphal y 1 sqrt 1 alphal 2 ran 1 The next instruction creates the first difference of y and calls the resulting series dy Next dy is regressed on a constant and y Notice that we use only the last 99 observations of the simulated dy sequence for the regression This technique is used to eliminate the problem of choosing the initial condition for the y sequence Since we do not want to impose any particular initial condition on the simulated sequence we generate 200 values for y and use only the last 100 Since we lose one additional observation by differencing the regression uses 99 observations so as to replicate a data set containing 100 observations The t statistic for the estimated value of 0 is stored in the variable df dif y dy linreg noprint dy 102 constant y 1 compute df tstats 2 The three IF statements below compare the calculated t statistic to the Dickey Fuller critical values If the t statistic is less than 2 58 the null hypothesis of p 0 is rejected at the 10 significance level
220. n in the system The information on each supplementary card contains the equation name used for the DEFINE option on LINREG instruction SUR creates the variables XX covariance matrix of coefficients NOBS number of observations 7 NREG number of regressors 7 LOGDET log determinant of the estimate of X and SIGMA final estimate of Y Step 3 To create a model GROUP the equations and provide a modelname GROUP modelname equation equation2 Step 4 As in a VAR obtain the impulse responses and variance decompositions with ERRORS IMPULSES MODEL modelname equations steps name Similarly the forecasts can be obtained with FORECAST MODEL modelname RESULTSzforecasts steps start Example The 12 lag VAR for dirgdp dirm2 and drs indicated that it was possible to eliminate dirgdp 1 to 12 from the dirgdp and dirm2 equations To impose these restrictions set up the following three equations Note that eq regresses dlrgdp on 12 lags of dirm2 and drs eq2 regresses dlrm2 on 12 lags of dlrm2 and drs and eq3 regresses drs on 12 lags of all three variables lin define eq1 dlrgdp constant dlrm2 1 to 12 drs 1 to 12 lin define eq2 dlrm2 constant dlrm2 1 to 12 drs 1 to 12 lin define eq3 drs constant dlrgdp 1 to 12 dlrm2 1 to 12 drs 1 to 12 Next estimate the system of equations saving the covariance matrix of the residuals using sur outsigma v 3 eql eq2 eq3 Since SUR creates NOBS and
221. nd Matrix Manipulations 171 1 2 COMPUTE In many ways you create and manipulate matrices just as scalars You PRINT a series but you DISPLAY or WRITE a scalar and a matrix Similarly you manipulate a series using SET but you can manipulate scalars and matrices using COMPUTE In fact the simplest way to create and manipulate matrices is with the COMPUTE instruction COMPUTE allows you to implicitly DECLARE and DIMENSION a matrix so that there is no need to have an explicit DECLARE instruction Moreover COMPUTE also allows you to completely specify the data type There are two rules you need to know when using compute 1 COMPUTE creates aRECTANGULAR matrix if you do not specify vector or symmetric You can specify i e explicitly DECLARE the type of the matrix using braces 2 COMPUTE determines the type of the elements e g integer real string from the expression you enter Examples 1 com a ll dlrgdp dim3 drs Il dis a dlrgdp dlm3 drs The COMPUTE instruction creates a 1 x 3 RECTANGULAR matrix containing the string variables dirgdp dlm3 and drs You can refer to each element by its position in the matrix For example dis a 1 2 dlm3 Suppose that the residuals from a VAR using dlrgdp dlm3 and drs are stored in series 1 2 and 3 You could graph each of the residuals series using doi 1 3 graph header Residuals from a 1 i 1 i end do The headers of the graphs would be Residuals from dlrgdp
222. ne Graphs of Four Principal Series a Panel 1 Time Path of DLM3 amp Panel 3 Time Path of DRS 0 03 4 0 02 4 i 0 01 4 0 00 0 01 TTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTITTTTEFTTTTTTTTETTTTTIEITTTTTTTT 4 ITTTTCDTDTDTTTTTTETTTTTTTTTTTTT ECT e r ae RO 1959 1963 1967 1971 1975 1979 1983 1987 1991 1995 1999 1959 1963 1967 1971 1975 1979 1983 1987 1991 1995 1999 Panel 2 Time Path of DLRGDP Panel 4 Time Path of DLP 0 04 0 030 0 03 4 0 025 4 0 02 4 0 020 4 0 01 0 015 4 0 00 0 010 4 MN 0 01 4 0 005 4 0 02 4 0 03 Let CIE L DZEUBOeDPETUDEDUPUDOETERTATUEUDEDETUEUREDEUSERDOTE TUNE D E t TIT PU tte 0 000 T T T T T TTTTTTETETTTTTTETTTTTTT ETT TIT ETETTTTTTEETTTTT ETTTTTTT ET ETTITTTTTITTTT ET ET TTTT TT TT 1959 1963 1967 1971 1975 1979 1983 1987 1991 1995 1999 1959 1963 1967 1971 1975 1979 1983 1987 1991 1995 1999 6 1 DOFOR and Loops for Series In Program 3 5 we created the logarithmic change of series 2 through 7 and stored the results in series 8 through 13 It is quite possible that you do not want to create a variable in the form Aln X for every variable in our data set this is especially true if a series has one or more negative values Suppose that you wanted to create only the logarithmic changes in real gdp series 3 m2 series 4 and tb3mo series 6 Program 3 7 illustrates how you can perform this task using the DOFOR instruction 112 Walter Enders scratch 3 scr_no dofor i rgdp m2 tb
223. ning the effect of the fiscal shock on drs Responses to a Fiscal Shock 0 90 gt dirgdp a as dirm2 iG drs 072 4 0 54 0 36 0 18 iO e Na bee ue d Ys Pd A m V Wu 3 Sx 0 00 G S A A v N IS gt a 0 18 ee x lm M x ur 35 F SS f p acra 0 36 i l l l l l l Chapter 3 Loops Over Dates and Series Try this little program all 10 scratch 1 set 1 5 pri 1 Notice that the program does not use the RATS CALENDAR Instruction Instead line 1 instructs RATS to set the default length of a series to 10 by default any series will have 10 observations The second instruction instructs to create one series The third line instructs RATS to set 1 equal to 5 This may seem nonsensical at first and I will explain the meaning in more detail below However since RATS does not display an error message it must somehow set 1 equal to 5 Perhaps you can figure out the dilemma by entering line 4 line 4 instructs RATS to PRINT over the default range Your output should look like ENTRY No Label 1 1 5 2 5 3 5 4 5 5 5 6 5 etc Now you can make sense of the program There is a series called 1 that has a length of 10 Each value of the series i e entries 1 through 10 is equal to the number 5 The point is that the series called 1 does not have a label like y gdp or inflation It is series number 1 becau
224. ns a number of instructions that allow you to interact with a procedure The most straightforward of these instructions is MESSAGEBOX You can use MESSAGEBOX to halt the execution of the procedure and display an Alert that contains a message Execution of the procedure will resume when the user enters the appropriate button The most commonly used syntax for MESSAGEBOX is messagebox status value style style Your Message where STATUS value The integer variable value will set equal to a particular number corresponding to the button selected by the user STYLE ALERT Displays a MESSAGEBOX that will look like the following Alert Your Message STYLE YESNO Halts execution of the program and sets value 1 if YES is selected and value 0 if NO is selected STYLE OKCANCEL Halts execution of the program and sets value 1 if OK is selected and value 0 if CANCEL is selected STYLE YNCANCEL Halts execution of the program and sets value 1 if YES is selected value 0 if NO is selected and value 1 if CANCEL is selected In most instances you will use a SWITCH or CHOICE OPTION to allow the user to make a selection The advantage of MESSAGEBOX is that it allows the user to make a choice in the midst of the execution of the procedure The QUERY instruction works in a similar fashion QUERY halts the execution of the procedure and prompts the user to to input the values for a list of variables The most commonly used synta
225. ny interesting sample statistics using the command statistics fractiles discrep The final instruction closes the DOFOR loop As such changes from 0 2 to 0 5 to 0 9 to 0 99 and finally to 0 0 table discrep end dofor Series Obs Mean Std Error Minimum Maximum DISCREP 1000 0 0166418436 0 0972591579 0 2935443023 0 3218321838 DISCREP 1000 0 0267019830 0 0905765140 0 2170420309 0 3415792684 DISCREP 1000 0 0443060778 0 0615724835 0 0715945104 0 3328153050 DISCREP 1000 0 0556321214 0 0456496779 0 0321881468 0 2691217328 DISCREP 1000 0 0103641243 0 0974089395 0 3157692119 0 2998128902 You can see that the mean value of discrep increases as Qu increases The smallest mean value of the discrepancy i e 0 0103641243 occurs for amp 0 Why didn t this number turn out to be exactly zero doesn t this show a bit of an upward bias in the discrepancy The answer involves the fact that we used only 1000 replications of the AR 1 process In many Monte Carlo experiments 100 000 replications are used in order to get a more precise estimate of the true sampling distribution 5 3 Power of the Dickey Fuller Test The last example used the first order AR 1 process Yt Oto Oy i where is generated from a white noise process In spite of the downward bias in the estimate of most applied econometricians would still use a standard t test to determine whether 0 is significantly different from zero The bias i
226. o The EXCLUDE SUMMARIZE TEST and RESTRICT instructions allow you to perform hypothesis tests on several coefficients at once EXCLUDE is followed by a supplementary card listing the variables to exclude from the most recently estimated regression RATS produces the F statistic and the significance level for the null hypothesis that the coefficients of all excluded variables equal zero Consider the following two exclusion restrictions exclude drs 5 to 7 Null Hypothesis The Following Coefficients Are Zero DRS Lag s 5 to 7 F 3 153 7 69621 with Significance Level 0 00008002 exclude constant drs 5 to 7 Null Hypothesis The Following Coefficients Are Zero Constant DRS Lag s 5 to 7 F 4 153 5 78538 with Significance Level 0 00023113 The first exclusion restriction tests the joint hypothesis s ots 0c 0 and the second tests the joint hypothesis Q Qs Q6 7 0 The results support those of the t tests both of these null hypotheses can be rejected at conventional significance levels SUMMARIZE has the same syntax as EXCLUDE but is used to test the null hypothesis that the sum of the coefficients indicated on the supplementary card is equal to zero In the following example the value of t for the null hypothesis 5 Qt O7 0 is 0 88256 Linear and Nonlinear Estimation 9 summarize drs 5 to 7 Summary of Linear Combination of Coefficients DRS Lag s 5 to 7 Value 0 1097225 t Statistic 0 88256 Standard Er
227. o begin with observation 4 since one usable observation is lost by using first differences and two more are lost as a result of estimating a model with two lags Hence make y 4 2001 1 dlrgdp If you enter DISPLAY y you will see that the first five rows are dis y 0 00330 0 02195 0 00495 0 00185 0 01296 You can display the individual elements of y by referring to their row and column For example dis y 3 1 0 00495 5 1 Estimating the Regression Coefficients We want to compute and display the matrix of coefficients as x x x y Consider the following program statements com Xx tr x x com xx inv inv xx Vector and Matrix Manipulations 199 com xy tr x y com beta xx_inv xy dis beta 0 00516 0 25090 0 13623 We could have written all of the above in one step However it is instructive to consider each of the program statements The first line creates xx as x x The second line takes the inverse xx and the third creates x y The fourth line creates B as x x x y and the fifth displays B Here is how you could have written all of the above in the single step dis inv tr x x tr x y Next call the predicted value of y The matrix x x x x is often called the projection matrix P since X p and p x x x y Moreover since the error term e y it follows that yr I P y My We can calculate the projection matrix P as com p X xx_inv tr x Students o
228. o estimate the VAR and create the variance covariance matrix v The regression residuals are stored in the series resids 1 resids 2 and resids 3 and the coefficients are stored in the matrix ca estimate sigma outsigma v residuals resids coeffs ca Note that ca is a37 x 3 RECTANGULAR matrix You can view the coefficients using dis ca 180 Walter Enders 0 04961 0 02504 27 24470 0 02046 0 06033 6 04229 0 16255 0 05159 12 42336 0 12126 0 07211 9 71846 0 15415 0 00161 2 79501 Element ca 1 1 is the regression coefficient of dlpgdp on its own first lag and element ca 37 2 is the intercept in the d rm2 equation Similarly PRINT the residuals using pri 16 resids ENTRY RESIDS 1 RESIDS 2 RESIDS 3 1962 02 0 004738155830 0 004968068719 0 651980315507 1962 03 0 001610217811 0 000793836267 0 117657508925 1962 04 0 010007743363 0 005422212237 0 101551774645 Notice that the residual series begin with 1962 02 since one usable observation is lost by differencing and twelve are lost by using lags in the regression equations The PRINT instruction caused RATS to print three series resids 1 resids 2 and resids 3 In essence resids is a vector containing three series You can print any one of the three series using resids i For example you can print the residuals from the second regression equation using pri 16 resids 2 ENTRY RESIDS 2 1962 02 0 004968068719 1962 03 0 000793836267 1962 04 0 00542
229. o k At this point beta2 star contains the 1000 values of B We can use the STATISTICS instruction to obtain the FRACTILES of beta2 star sta fractiles beta2 star Statistics on Series BETA2 STAR Minimum 0 155 IRESE 53S LOA SS 01 Sile 0 0558588474 2886134655 05 Sile 0 0113366577 CASTS ISS 10 ile 2 027 5 15 8 55 2141146300 25 ile 2 016 63189 53189237 1684400633 Median 31204612933 From the output it is clear that a symmetric 90 confidence interval includes zero the lower and upper boundaries are 0 0113366577 and 0 2431975939 respectively This suggests that we can exclude the second lag of dirgdp The FRACTILES option may not provide all of the information you would like concerning the distribution of beta2_star One way to obtain IF Statements and Monte Carlo Experiments 167 whatever set of FRACTILES you want is to order beta2_star from the smallest to the largest value order beta2_ star Now the 50 th value 5 of 1000 replications and the 950 th 95 of 1000 can be used to obtain the lower and upper bounds of a 90 confidence interval Similarly the same logic can be used to construct 95 and 99 confidence intervals The following four instructions produce these three confidence intervals dis Confidence intervals for beta2 dis 10 beta2 star fix 05 boot num beta2 star fix 95 boot num dis 5 beta2 star fix 025 boot num beta2 star fix 975 boot num dis 1 beta2 star fix 0
230. ocedure creates the series time we should include this in the list of local series using LOCAL SERIES dy time 4 We need to include a set of IF statements to select the regressors If det eq 1 lin noprint dy y 1 dy 1 to lags If det eq 2 lin noprint dy y 1 dy 1 to lags constant If det eq 3 lin noprint dy y 1 dy 1 to lags constant time The Choice to Display a Graph of the Residuals We can use a SWITCH OPTION to allow the user to turn ON or OFF a display of the graph We need to make the following modifications to the procedure 1 Include the instruction OPTION SWITCH graph 0 As formulated the default the value of GRAPH is zero In order to turn the option on i e in order to set GRAPH 1 we can use unit graph Irgdp unit graph 1 lrgdp 2 We need to save the residuals from the selected regression equation Hence we modify our IF instructions such that the residuals are saved in the series called resids If det eq 1 lin noprint dy resids y 1 dy 1 to lags If det eq 2 lin noprint dy resids y 1 dy 1 to lags constant If det eq 3 lin noprint dy resids y 1 dy 1 to lags constant time Writing Your Own Procedures 217 3 We obtain the ACF of resids using cor noprint number 24 resids corrs 4 If GRAPH 1 we create a time series plot of the residuals and the ACF of the residuals 39 using cor noprint number 24 resids corrs if graph 1 spgraph hfields 1 vfiel
231. og m2 log m2 1 set dim3 log m3 log m3 1 dif tb3mo drs Note that instructs RATS to use the default range dif tblyr drl of the data set price gdp rgdp set dlp log price log price 1 Notice that the logarithmic growth rate of a variable is preceded by dl the suffixes s and refer to the short term and long term interest rates and that price i e the GDP deflator is computed as the ratio of nominal to real U S GDP The logarithmic change in price called dlp is the quarterly inflation rate We can create graphs of five of these series using 4 Walter Enders spgraph hea Graphs of the Five Principal Series hfi 2 vfi 2 key upleft gra hea Panel 1 Time path of dlm3 2 dlm2 t dlm3 gra hea Panel 2 Time path of dlrgdp 1 dlrgdp gra hea Panel 3 Time path of drs 1 drs gra hea Panel 4 Time path of dlp 1 dlp spgraph done Panel 1 Time path of dlm2 and dim3 Panel 3 Time path of drs 0 056 5 DLM3 0 048 du 0 040 4 0 032 0 024 4 0 016 4 0 008 4 0 000 0 008 TPT ee ee ae on eee TET eT eT eo TT TETNT ME ore TREE Lo eh ee TO TO TAE VATE 1959 1963 1967 1971 1975 1979 1983 1987 1991 1995 1999 1959 1963 1967 1971 1975 1979 1983 1987 1991 1995 1999 Panel 2 Time path of dirgdp Panel 4 Time path of dlp 0 04 0 030 0 03 0 025 8e aed d M l li Il
232. ogram read in the data set and construct the variable dipgdp using set dlrgdp log rgdp log rgdp 1 First suppose that we want the value of the threshold t to equal zero This might be the case if we were certain the positive real GDP growth behaved differently from negative growth We have already determined that it is reasonable to use a lag length of 2 Hence we need to construct a number of variables First we can construct the indicator function using set plus if dlrgdp 1 gt 0 1 0 Note first that we denote the indicator 7 by a variable called plus we cannot use the symbol i since i is a reserved integer variable to represent a series For each possible entry in the data set the SET instruction compares d rgdp to the value 0 IF dprgdp is greater than zero the value IF Statements and Monte Carlo Experiments 129 of plus is equal to 1 otherwise plus is zero You can see how the program works by printing the first 10 values of plus and dirgdp pri 1 10 dlrgdp plus DLRGDP NA 10257912 39495 000428834862 003297294498 021945448475 SOWA SOUS 001847653167 SOSEZIDIOSSSIQOISIO 005763460460 018546572263 lC om om m onmnpooco Note that p us 1959 1 is zero since dirgdp 1958 1 is not in the data set Since this value was not positive plus 1959 1 is set equal to zero Similarly the initial value of dlrgdp is NA we lost an observation by differencing since d rgdp 1959 1 is not pos
233. older series Finally use a SUBFORMULA to equate the placeholder and the desired series For example a simple way to create the formula for the MA 1 process is set temp 0 0 nonlin bl var frml e y b1 temp 1 frml L temp e log var e 2 var The SET instruction generates the placeholder series temp containing all zeros The first FRML instruction defines the desired relationship such that e is equal to y bitemp 17 The second FRML statement uses the SUBFORMULA to equate temp with e so that e y bie 1 and creates the log likelihood L The way to conceptualize the process is to suppose you knew p and var and wanted to construct the log likelihood function T A log var y Be Y var t 2 One way to construct the sum would be to loop over the following three instructions beginning with f 2 amp y Bitempia temp amp L log var e var The first time through the loop t 2 so that 2 y Bitempi temp and L log var 2 var The next time through the loop t 3 and ys Bitemp y3 Bi 2 Hence temp 3 and L log var e3 hvar Continuing through t T yields the values L2 L5 Lr The sum of these values yields the desired log likelihood function A To illustrate the procedure recall that the change in the 3 month interest rate drs was estimated as an AR 7 process It turns out that a more parsimoni
234. ollowing program illustrates the use of several other BRANCHING and IF statements within a Monte Carlo study In Enders and Granger 1998 we generalized the Dickey Fuller methodology to consider the null hypothesis of a unit root against the alternative hypothesis of a threshold autoregressive TAR model The simple version TAR model is Ay iyi 1 D payri li 20 where J f Ped Oif y 0 The basic idea is that autoregressive decay might not be symmetric If y lt 0 the indicator function 0 so that Ay p2y 1 and if y 20 Z 1 so that Ay piy Notice that if p P2 0 the process is a random walk However as in the Dickey Fuller test it is not possible IF Statements and Monte Carlo Experiments 147 to use a classical F statistic to test the null hypothesis p p 0 Instead the following program can be used to accomplish the following tasks 1 Generate a random walk sequence 2 Estimate the sequence as a threshold autoregressive model 3 BRANCH if the simulated sequence does not cross the threshold 4 Calculate the sample F statistic for the null hypothesis p p 0 5 Obtain the distribution of the sample F statistics There is one technical task that must be done It is necessary to ensure that there are a sufficient number of observations on each side of the threshold to estimate the TAR model One way to check is immediately after task 3 If either of the r statisti
235. omplicated set of instructions In this way you could automate your task within any particular program More importantly by writing a procedure you can call up this set of instructions in a number of programs Since the procedure resides in a file you can send the file to your co author students or post it on your website You might think of a procedure as a macro in WORD Older programmers like myself might want to think of a procedure as an external function in FORTRAN Now the advantage is that you can customize RATS by writing or downloading a procedure RATS comes with a number of useful procedures and you can download many more from the Estima website www estima com Even if you do not want to write your own procedures from scratch the material in this chapter should be useful to the advanced RATS user It is often quite simple to edit existing procedures to tailor them to your needs In order to use a procedure it must be complied The syntax for compiling a procedure stored on an external file is source name The sole option is noecho If you use the noecho option you will not see anything displayed on the screen after compiling the procedure If you omit noecho you will see each line of the source code preceded by the location in memory where the beginning of the code resides This is useful for two reasons If you need to debug a procedure knowing the location in computer memory can be useful Also the fact
236. on integer sdiff 0 option choice trans 1 none log root option switch graph 1 option integer span local integer nbeg nend spanl i j local series corrs partials local series transfrm diffed upper lower local integer number inquire series series nbeg gt gt start nend gt gt end As in BIC SRC the first line names the procedure However the first line also contains the three parameters SERIES START and END The second line declares that SERIES is a series The third line declares START and END to be integers This illustrates the general organization of a procedure The first set of instructions classifies the parameters being passed by the procedure Next the various options used in the procedure are enumerated Note that the number of differences DIFF seasonal differences SDIFF and the seasonal span SPAN are all INTEGER OPTIONS The form of the data transformations TRANS is a CHOICE OPTION and whether or not to display a graph is a SWITCH OPTION The third set of instructions indicates whether or not the variables will be local defined only within the procedure or global accessible throughout the main program and procedures Fourth you might want to inquire about the nature of the series being passed to the procedure it is especially useful to obtain the 212 Walter Enders starting and ending entries Finally come the set of instructions you want the procedure to perform The general structure of any procedure is
237. on of lines 1 and 8 is straightforward these are simply the starting and ending lines of the procedure In line 1 we inform RATS that we will pass a parameter to UNIT SRC Line 2 needs a bit of explanation In general you will want to pass series integers real variables matrices etc to the procedure RATS needs a mechanism by which to determine the type of parameter that is being passed Line 2 indicates that the parameter we pass 1 e y is a series This is done using the TYPE instruction The syntax is TYPE data type parameter list where data type can be any RATS date type such as SERIES INTEGER REAL SYMMETRIC parameter list is the list of parameters you which to declare as this particular date type 214 Walter Enders Notes 1 Use one TYPE instruction for each data type 2 The TYPE instructions should begin in line 2 of a procedure i e immediately after the PROC name parameter list instruction 3 We cannot include dy on a TYPE instruction since we do not pass this series to the procedure Instead dy is created within the procedure If you are not familiar with programming it might seem troublesome that we want to pass gdp to the procedure but the procedure seems to use only the series denoted by y Actually y is only a placeholder it will accept any series that is passed to it Line 3 creates the first difference of y and the regression is estimated in lines 4 and 5 Line 7 displays the point esti
238. on provides the extra piece of information that allows us to identify the four elements G matrix Given the relationship between the regression residuals and the structural variables it follows that var e1 gui T g1 var e 21 g22 cov e1e2 211821 821822 where the time subscripts have been omitted since Ay and z are assumed to be covariance stationary VARs and Error Correction Models 83 Estimation of the VAR provides you with var ei var e2 cov e1e2 and the coefficient sums 1 Yaj k and XajXK Hence there are four equations that allow you to solve for the four unknowns 211 212 21 and g22 Once the G matrix is identified it is possible to obtain the impulse responses and variance decompositions using the ERRORS and IMPULSES instructions 6 1 The Technical Details In the n variable case n n 2 restrictions are needed for the exact identification of G RATS contains a very simple mechanism that allows you to impose n n 2 long run restrictions Let X e and gj be the n x 1 vectors of variables regression residuals and structural shocks respectively The estimated VAR has the form x A L Xi 1 Or 1 AQ D x ei where A L n x n matrix with elements A L and A j L is p th order polynomial in the lag operator L Given that the variables in x are stationary we know there exists a moving average representation of the form x 2 C L e C L Ge where C L i
239. ons are easily obtained using the ERRORS instruction To obtain the impulse responses and variance decompositions using a Choleski decomposition use errors IMPULSES MODEL modelname equations steps name where modelname The name of the model as defined on the SYSTEM instruction equations Number of equations in the VAR steps The forecast horizon and the number of impulse responses name The name of the covariance matrix used on the ESTIMATE instruction If you exclude IMPULSES RATS calculates and prints only the variance decompositions Note that the IMPULSE instruction discussed below gives you more control over calculation and display of the impulse response functions Examples 1 The sample program used in Section 2 below estimates a 3 equation 12 lag VAR using the variables dlrgdp dlrm2 and drs The first two instructions create the growth rate of the real value of the money supply as measured by M2 The SYSTEM instruction creates a MODEL called chap2 The VARIABLES command instructs RATS to create a 3 variable VAR using dirgdp dirm2 and drs Note that 12 lags of each variable and a constant are to be included in each equation set Irm2 log m2 price Creates the log of the real value of m2 dif Irm2 dlrm2 Creates the first difference of Irm2 system model chap2 var dlrgdp dirm2 drs lags 1 to 12 det constant end system ESTIMATE produces the estimates of the three equations and the F statistics for the Grang
240. ontemporaneous effect on y it must be the case that gi 0 Similarly if the pure shock to y is to have no contemporaneous effect on z it must be the case that go 0 In either case there is a fourth equation that can be used to solve for the other three values of the G matrix To generalize the argument to an n th order VAR systems we have L GG where amp and G are n x n matrices Using the same logic it is possible to show that it is necessary to impose n n 2 additional restrictions on G to identify completely identify the P This normalization assumption is innocuous because it simply scales the magnitudes of gi 812 13 and gia VARs and Error Correction Models 71 system Regardless of the size of the system the Choleski decomposition is recursive in that it sets amp 0 nm 0 G Ej 82 O0 Ent 52 SY Enn Since each element above the principle diagonal is zero there is exactly the number of restrictions needed to identify all of the remaining elements of G However many other possibilities exist RATS allows you to select the form of the G matrix so that you can impose a far richer set of restrictions on the G matrix Moreover it is possible to impose overidentifying restrictions so that you can test hypotheses concerning the restrictions Suppose we normalize the elements on the principle diagonal to be unity To keep the notation simple suppose we let the G matrix be Ioa ae i Gs m ae ES
241. or lists The first called reglist holds only the variables that we want to keep for the final forecasting equation These are the variables that have already passed the Granger causality test The second called templist holds the variables in reglist plus the variable that is currently under consideration for inclusion Only if this variable passes the causality test will it be added to reglist The final section of Program 5 2 creates the variables dlm3 and dlp and the two integer vectors templist and reglist dec vector integer templist reglist compute templist llconstantll compute reglist llconstantll Next the program loops over the series d rgdp dlm3 drs and dlp The DOFOR instruction below uses the fact that RATS allows you to refer to a series by its name or by its number The first time through the loop templist and reglist contain only the constant term The ENTER instruction creates templist as the constant i e the contents of reglist plus the first four lags of dirgdp Next the LINREG instruction estimates a regression of dlm3 on the contents of templist 186 Walter Enders dofor i dlrgdp dlm3 drs dlp enter varying templist reglist i 1 to 4 lin noprint dim3 templist The EXCLUDE instruction is used to perform the F test that the coefficients on the four lags of dlrgdp are statistically significant from zero If the null hypothesis that the coefficients are jointly equal to zero is significant at the 596
242. otstrap sample of the residuals e star note that the BOOT instruction can also be useful for these situations The series e star will consist of 50 values drawn from the regression residuals e Each value of e star is a random draw with replacement from e Note that each element of e has a 1 50 chance of being selected since fix Yuniform 1 51 generates an integer on the interval 1 50 The next instruction uses the bootstrap residuals to construct the bootstrap y series y Notice that the coefficient values 162 Walter Enders used in the construction are those originally estimated from the data Moreover the values of x have not been modified or resampled Thus Step 2 is completed do i 1 1000 set e_star e fix uniform 1 51 set y_star beta_hat x alpha_hat e_star Step 3 requires us to use the bootstrapped y_star sequence to obtain new coefficient estimates This is accomplished in the four instructions in the NONLIN NLLS block below The coefficient estimates for and p are stored in entry i of alpha star and beta star respectively On exiting the DO loop these two series will each contain 1000 values of bootstrapped coefficients The sample properties of these coefficients are obtained using the STATISTICS instruction with the FRACTILES option nonlin alpha beta frml monte y star beta x alpha com alpha alpha hat beta beta hat nlls frml monte noprint y star com alpha star 1 beta 1 beta star 1
243. ou want to include an NOCONSTANT intercept AR List of autoregressive coefficients Default 0 MA List of moving average coefficients Default 0 Thus the program line below will estimate the model y ao a1y 1 aoyro amp Big Bog over the sample period 3 through 100 since two observations are lost due to the 2 AR coefficients and saves the residuals in a series called resids box constant ar 2 ma 2 y 3 100 resids It is important to know that the options ar p and ma q include AR coefficients for lags 1 through p and MA coefficients 1 through q In contrast you can use ar Il list ll and ma I list Il to include only those coefficients enumerated in ist For example ar 4 calls for the inclusion of autoregressive coefficients a1 a a3 and a4 Instead ar Ill 4ll calls for the inclusion of autoregressive coefficients a and a4 but not a or a3 Thus to estimate the model y ao ayr a2ye2 amp B282 Bases use box constant ar 2 ma ll2 4l y 3 100 resids What you are doing is entering a vector of integers i e the vector 2 4 into the instruction This method works well if the vector list is short Since you are creating a vector you must type each integer value that you want to include Hence you cannot enter ma Il 1 to 4 SII Vector and Matrix Manipulations 179 Moreover if you embed the instruction in a procedure you want to give the user the flexibility to input any possi
244. ous representation of the series is drs Qidrs 1 O4drs 3 Bier This ARMA model can be estimated using the six instructions listed below The SET instruction is used to create the series temp containing all zeros The NONLIN instruction prepares RATS to estimate al a7 b and var The first FRML instruction creates the desired formula such that e is equal to drs ajdrs a drs bytemp The second FRML statement uses a SUBFORMULA to equate temp with e so that the next time through the loop temp e and creates the formula L log var elvar When you use a placeholder you need to properly specify the start end dates on the MAXIMIZE instruction In the case at hand one usable observation is lost as a result of differencing and another seven usable Linear and Nonlinear Estimation 37 observations are lost as a result of the term drs Since eight usable observations are lost the maximization must begin in period 9 If you omit the start end dates or begin with a starting date less than 9 you will obtain the error message SR10 Missing Values And Or SMPL Options Leave No Usable Data Points set temp 0 nonlin a1 a7 b1 var frml e drs al drs 1 a7 drs 7 bl temp 1 frml L temp e log var e 2 var com al 0 4 a7 3 bl 5 var 1 max L 9 MAXIMIZE Estimation by BFGS Convergence in 14 Iterations Final criterion was 0 0000025 0 0000100 Quarterly Da
245. pectively VARs and Error Correction Models 61 2 1 Near VARs In a near VAR the right hand sides of the equations in the system are not identical Examples include i Different lag lengths Yr Ani ye1 a11 2 y1 2 aizia ei Zt A21Yt 1 A22Z1 1 enr ii The z series does not Granger cause y Yt 411 1 i Zt a21yr 1 422Z1 1 ezt iii A third variable w affects only zi Ve auye dizi ei Zt doiyri A22Z 1 023W er Since the equations have different right hand side variables the efficiency of the estimates can be improved using Seemingly Unrelated Regressions Use the following method to estimate a near VAR using RATS SUR instruction Step 1 You must define the equations to use in the near VAR The simplest way to set up your equations is using the DEFINE option of the LINREG instruction Examples 1 To set up the first near VAR system above use linreg define equation1 y y 1to2 z 1 linreg define equation2 z y 1 z 1 2 To set up the third near VAR system above use linreg define equation y y 1 z 1 linreg define equation2 z y 1 z 1 w Step 2 Use the SUR instruction to estimate the system The typical syntax of SUR is SUR OUTSIGMA V equations start end equation where equations The number of equations in the system you want to estimate start end The range of entries to use 62 Walter Enders There is one supplementary card for each equatio
246. qual unity VARs and Error Correction Models 73 2 Recall that the long run equilibrium relationship between the two interest rates is such that tblyr 0 6980794657 0 9167216207 tb3mo so that A tblyr 0 9167216207 Atb3mo or drl 0 9167216207 drs For illustration purposes suppose we wanted to impose a similar restriction on the innovations In particular suppose we wanted to let innovations in tb3mo be unaffected by innovations in tb yr but we wanted innovations in tb3mo to change the contemporaneous value of tblyr by 0 9167216207 units Since the residuals from the tblyr and tb3mo equations are the e and ez sequence respectively e 1 091 e eal qq X18 Now a pure shock to tb yr i e an j shock affects the contemporaneous value of tb yr but not tb3mo A pure shock to tb3mo i e an shock has a l unit effect on tb3mo and a 0 9167216207 unit effect on tblyr To obtain the impulse responses using this G matrix recall that we estimated the long run relationship using lin define spread tblyr resids constant tb3mo Immediately after this LINREG instruction insert the line com x beta 2 Now the variable x contains the desired slope coefficient After estimating the error correction model construct G using com g 1 0 x 10 1 0 II To obtain 24 impulses using this G matrix use impulse model term results impulses decomp g 24 74 Walter Enders Responses to S
247. quals zero T Stat and the marginal significance level of the t test Signif For example the coefficient of the first lag of drs is estimated to be 0 361765228 with a standard error equal to 0 079570494 The associated t test for the null hypothesis a 0 is 4 54647 If you use a f table you can verify that the significance level for this value of t is 0 00001102 It is always important to determine whether there is any serial correlation in the regression residuals The CORRELATE instruction calculates the autocorrelations and the partial autocorrelations of a specified series The syntax and principal options are correlate options series start end where series The series used to compute the correlations start end The range of entries to use The default is the entire series corrs Series used to save the autocorrelations Optional The principal options are NUMBER The number of autocorrelations to compute The default is the integer value of one fourth the total number of observations PARTIAL Series for the partial autocorrelations If you omit this option the PACF will not be calculated QSTATS Use this option if you want the Ljung Box Q statistics SPAN Use with qstats to set the width of the intervals tested For example with quarterly data you can set span 4 to obtain Q 4 Q 8 Q 12 and so forth In the example at hand we can obtain the first twelve autocorrelations partial autocorrelations and the a
248. r ENTRY TBlYR 1 NA 2 NA 3 4 493333333333 4 4 740000000000 print nodates 165 169 tblyr ENTRY TBlYR 165 5 816666666667 166 5 856666666667 167 5803333333333 168 5 630000000000 169 4 416666666667 Thus you can refer to value of the 1 year T bill rate in 2001 1 using either tblyr 2000 1 or tblyr 169 You can DISPLAY the equivalent specifications if you enter dis tb1yr 2001 1 tblyr 169 4 41667 4 41667 1 1 Omitting CALENDAR Given that all of the series in MONEY DEM XLS all have 169 entries you could read in the data set using all 169 open data a money_dem xls data org obs format xls 92 Walter Enders In fact if you want to refer to entries only by number rather than by date label you never use the CALENDAR instruction However you would not be able to use date labels Thus using CALENDAR gives you the choice of using date labels or entry values You cannot use date labels if you omit CALENDAR Examples 1 Date arithmetic Since 2001 1 169 it follows that 2001 1 8 161 is equivalent to 1999 1 Hence to print the last two years of the rgdp series you can use pri 2001 1 8 rgdp Note that RATS will perform the date arithmetic 2001 1 8 if you do not use spaces adjacent to the minus sign 2 Estimate an AR 1 autoregression of the logarithmic change in rgdp using the first 100 observations set dlrgdp log rgdp log rgdp 1 lin dirgdp 100 constant dlrgdp 1 The first line creates
249. r ACF and PACF patterns 0 2 cors partial 18 Walter Enders ACF and PACF l MM S Li 0 2 T T T T T T T T T T T T T EN CORS PARTIAL Notice that we used the PATTERNS option The default value is NOPATTERNS i e the default is PATTERNS 0 With this option OFF your monitor will show the ACF in black and the PACF in blue Since it is clear that the residuals decay over time we can estimate the dynamic process Take the first difference of the resids and call the result dresids dif resids dresids Now estimate the dynamic adjustment process as P dresids Q resids _ X a dresids E i If we can conclude that o is less than zero we can conclude that the resids sequence is a convergent process However it is not straightforward to estimate the regression equation and then test the null hypothesis o 0 One problem is that under the null hypothesis of no equilibrium long run relationship i e under the null of no cointegration between the two rates we cannot use the usual f statistics Secondly we do not know the appropriate lag length p to use when estimating the regression equation The ACF suggests that we can look at a relatively short lag lengths although the partial autocorrelation coefficient at lag 7 appears to be significant As such it seems prudent to estimate the regression equation using all lags throu
250. r period 101 is known it is possible to obtain the squared prediction error for the AR 1 and the AR 2 specifications Next estimate each model over the first 101 observations and obtain the squared 1 step ahead prediction error for period 102 Repeat the entire process up through observations 168 so that you have the 1 step ahead prediction error for period 169 and compare the means of the sum of the squared prediction errors Continue to scroll down Program 3 4 and enter the following instructions set f arl 0 set f ar2 0 do i 100 168 lin noprint define ar1 dlrgdp 3 i constant dirgdp 1 forecast 1 1 arl f arl lin noprint define ar2 dlrgdp 3 i constant dlrgdp 1 to 2 forecast 1 1 ar2f ar2 end do i Lines 1 and 2 initialize the two series f arl and f ar2 these series are used to store the forecasts from the two models The third line instructs RATS to execute the DO loop 69 times beginning with i 100 and terminating after i 168 Within each loop RATS estimates an AR 1 model of dlrgdp over the sample period 3 through i Lines 5 and 6 instruct RATS to make a 1 step ahead forecast for period i 1 and store the forecast in f arl Lines 7 9 repeat the procedure for the AR 2 specification and stores the forecast in f ar2 After the 69 loops are completed f arl and f ar2 each contain the 69 1 step ahead forecasts The remainder of the program is straightforward The first two lines shown below calculate the square
251. r begins in period 3 lin noprint spread constant spread 1 to 3 com a0 beta 1 al beta 2 a2 beta 3 a3 9obeta 4 b0 seesq bl 0 2 b2 0 2 b3 0 2 max method simplex iters 4 L max iters 200 L MAXIMIZE Estimation by BFGS Convergence in 26 Iterations Final criterion was Quarterly Data From 1959 01 To 2001 01 Usable Observations 161 Total Observations 169 Function Value 0 0000058 0 0000100 Skipped Missing 317 68938352 Variable Coeff STORE TEO Me STEE Signif KKK KK KKK KKK KKK KKK KKK Ck Ck KKK KKK KKK KKK KKK KKK KKK KKK KKK KKK KKK KKK KKK ko Mk Mk ko ko AO 00570536155 0 014792 70 JGS OQ QOOLLSS0 Al A2 A3 BO B1 B2 B3 iba 2o S 4 Be On Te So 0 144647372 5 3100 312 51 5 0 0 OR 0 0 0 793283630 0228779390 203326143 o LOZSL4A ZA 446546881 066602420 085452808 5102 5 3412 511 003589145 o L1L95 2 612 5L sORlAge Tay 112246500 allel 6 db iL S o SLOTS ils ike 69272 90967 37461 sO DZ 29099 SONO 00000000 09050946 sODGILY SLO 00000000 08839543 20817858 00006942 2 The ARCH M model Engle Lilien and Robbins 1987 developed the ARCH in Mean ARCH M model to allow the conditional variance of an asset s return to affect the expected return The idea is that an increase in risk as measured by an increase in the conditional volatility of the asset s returns should increase the expecte
252. reate a matrix to contain the impulse responses We will create an 8 x 3 matrix called imp to hold the eight impulse responses of a shock to the first variable on the time path of the three variables in the system dec rect imp 8 3 Thus imp 4 2 will contain the impulse response of a shock to dirgdp on the value of dlm2 44 Next we create the impulse responses themselves Since we consider only a first order VAR after the first period the system evolves as imp imp 1 a where the 1 x 3 vector imp are the impulse responses of dirgdp dlm dr to a real gdp shock and A is the coefficient matrix created above 37 Although the 12 order VAR is more appropriate the example here is intended to be illustrative 196 Walter Enders ewise imp 1 g 1 j doi 2 8 com c tr xrow imp i 1 a do j 1 3 com imp i j c 1 j end doj end do i dis Ht JHEHHHHHE HETHHHBHBR TEE AHHBHBHBE imp The logic of the program is to initialize the impulses with a 1 unit shock to d rgdp Within the first loop c is computed as the transposed value of the i 1 row of imp i e the impulses for period i 1 multiplied by the coefficient matrix a Then the current value of imp for real gdp money and interest rates are computed as the associated value of the 1 x 3 vector c Displaying imp yields identical answers to the ERRORS impulses instruction Vector and Matrix Manipulations 197 5 Creating Matrices from Your Data If you are going
253. res will be compiled in the appropriate order For example you might you have a single file structured as follows 232 Walter Enders procedure unit y start end corrs_save type series y corrs_save type integer start end other program statements end unit KKK KK KK K K K procedure laglength y start end laglength type series y type integer start end laglength other program statements end laglength Writing Your Own Procedures 233 8 Creating a Menu RATS allows you to present the user with menu of choices in one of two ways The first presents the user with a box listing the choices After the user selects a choice from the list a specified block of instructions can be performed The typical syntax for the MENU block is MENU Message to Display CHOICE Name of First Choice Instruction block for first choice CHOICE Name of Second Choice Instruction block for second choice END MENU Example The following procedure called TRANSFORM uses the MENU instruction to prompt the user for a particular type of data transformation After compiling TRANSFORM SRC the procedure is executed using transform series The user will see a dialog box that looks something like Select the Transformation Logarithmic C First Difference C Growth Rate Cancel Once the desired transformation is selected the procedure will create a graph of the transformed series The label of the transformed series
254. responses VARs and Error Correction Models 69 4 Structural Decompositions A Choleski decomposition is not the only way to obtain the impulse responses In fact it is straightforward to show that the impulse response function is not identified unless additional restrictions are imposed on the VAR system The Choleski decomposition is only one way to impose the necessary number of identifying restrictions Consider a 2 variable model n n y Ya Dya 2 220 2 Tp i l i n n Y a0 as te i l i The impulse response function is obtained using the moving average representation y Vb De Vb De t i l i l Cr Y ba Der b Dezi i l i l The issue is that the regression residuals e and ez are linear combinations of the pure innovations in y and z If we call these pure innovations and 5 we have eit g11 1 12 2 C21 22114 22 2 or ey GE The nature of the system is such that the pure innovations are serially uncorrelated and orthogonal to each other Nevertheless a pure innovation in y will have a contemporaneous effect on z if 221 O and a pure innovation in z will have a contemporaneous effect on y if g12 0 Even though and x are serially uncorrelated their effects have some persistence since the values of aj i are not all equal to zero If we let var 1 0 and var it follows that o 0 Eg amp j y2Y 0 o0 T
255. rform the estimation cvmodel factor g sigma g form Covariance Model Estimation by BFGS Convergence in DeLeercttons een male eret rl Onmwas O 10 00 O08 09 E9000 Observations 167 Log Likelihood 309 97546215 og Likelihood Unrestricted 309 97546215 Variable Coeff Scl Jnsese owe pss Statua Ck CK CK Ck OK CI C CK CI CK CI C CK CI CK CI C CK C CK CI C CK CK CI CCCII C CK CI CK I C CK CI Ck kk KK Kk KK x Mk ko ko ko ko ko la x21 023217913 07029267200 34 96123 0 00000000 Notice that the estimation converged in only five iterations If an estimation does not converge you can increase the number of iterations from the default value of 40 provide better initial guesses or use an alternative estimation method A useful way to obtain satisfactory initial guesses is to use the simplex or genetic estimation method for a few iterations and then switch to the BFGS method Consider cvmodel factor g iters 4 method simplex sigma g form cvmodel factor g osigma g form Also note that the reported value of g2 is estimated to be more than 34 standard deviations from zero To interpret this estimate recall that RATS estimates the matrix G such that G G sigma You can display G using VARs and Error Correction Models 77 dis g 0 64280 0 00000 0 65772 0 24312 For comparison purposes you can display the variance covariance matrix using dis sigma 0 41319 0 42278 0 49170 Standardizing G such that the
256. ri iv 5 14 15 80 101 110 203 205 216 217 221 224 GROUP tg dint aint iv 62 63 152 Heteroskedasticity eese 249 IF E EE E iv 119 120 121 122 123 124 125 126 127 128 133 144 146 147 152 208 216 217 230 235 236 237 240 244 245 IGARCH sunina epo 44 45 impulse response function iv 53 68 69 IMPULSES 53 62 66 75 83 84 Inf reneez see er ie vea 149 249 INFOBOX asinna rere iv 145 149 INQUIRE iv 212 219 220 221 227 244 INTEGER option eere 209 InteractiVe a aen acie te ea ea lee ends 235 TEA BEES 5 ense iv 97 109 112 169 lag length tests esssssss 1 102 105 likelihood ratio tests ssss 51 105 LINREQG iv 6 9 12 13 14 19 21 23 30 34 35 40 44 61 62 64 65 73 94 102 108 116 152 168 172 185 187 201 202 203 207 215 221 227 LOCAL ettet iv 212 215 216 221 223 224 226 235 239 local variables ee iv 214 227 EOOPD eius 20 237 240 242 243 245 LSTAR model 27 28 29 30 31 34 MAKE ts ien EE iv 168 197 MAXIMIZE iv 23 32 33 34 35 36 37 38 41 43 44 45 46 120 MENU eee iv 233 236 239 241 MESSAGEBOX iv 229 230 Monte Carlo methods 119 136 137 138
257. ribution of the estimated values of Bi A program that will perform this task is all 100 set y 0 set beta 1 1000 0 do i 1 1000 set y 2 y 1 ran 1 lin noprint y constant y 1 com beta i beta 2 end do table beta The first line of Program 4 4 sets the default length of all series to 100 The second line initializes all 100 values of y to equal zero The estimated values of B will be stored in the series called beta The third line initializes all 1000 values of beta to be zero The initialization is necessary since RATS cannot create a series using COMPUTE Instead you create the series with SET so that later instructions can manipulate the individual entries of the series There is a second interesting feature of the SET instruction Although the default length for beta is 100 SET allows us to override this default by specifying the START and END values The first instruction in the DO loop below generates a simulated random walk the current value of y is equal to the previous value plus a random error term drawn from a normal distribution with mean zero and variance equal to 1 The second line in the loop estimates the model and the third equates the i th value of the series beta with the estimated value of Bi On exiting the loop the TABLE instruction displays the sample statistics on the series beta If you run the program your output will look something like Series Obs Mean Std Error Minimum Maximum
258. rly seasonal dummy variables so that the system becomes Y a o buDi bD biDs a Y az Zr A b51D b53D bo3Ds Gy a221 First create the seasonal dummy variables using seasonal dummy Next modify the DET instruction such that det constant dummy 1 to 3 50 Walter Enders In some instances you might want to create centered seasonal dummy variables Centered seasonal dummies are normalized to have a mean of zero For example instead of taking on the values 0 0 0 and 1 a centered seasonal dummy variable for quarterly data has the values 0 25 0 25 0 25 and 0 75 Centered seasonal dummy variables are useful in situations such as unit root tests where you do not want to shift the magnitude of the intercept term To estimate the VAR with centered seasonal dummy variables use seasonal centered dummy det constant dummy 1 to 3 VARs and Error Correction Models 51 1 Hypothesis Testing and Model Selection Most hypothesis tests in a VAR involve cross equation restrictions The RATIO instruction can easily perform such tests Let 2 and X be the variance covariance matrices of the unrestricted and restricted systems respectively Form the test statistic L L T c log 122 log I where X and X are the determinants of 2 and X c is the maximum number of regressors contained in the longest equation of either VAR system and T is the number of usable Observations
259. rms RATS to store the nine impulse responses in impulses Note that impulses 1 1 contains the responses of dirgdp to an shock and impulses 3 1 contains the responses of dirgdp to an shock declare rectangular series impulses 3 3 impulse model chap2 result impulses decomp g 24 Since the variables have different units it is useful to plot the standardized responses Note that the first entry of impulses 1 1 contains the standard deviation of dirgdp first entry of 80 Walter Enders impulses 2 2 contains the standard deviation of dirm2 and the first entry of impulses 3 3 contains the standard deviation of drs Hence we can standardize the responses of each variable to an shock using set rl 1 12 impulses 1 3 impulses 1 1 1 set r2 1 12 impulses 2 3 impulses 2 2 1 set r3 1 12 impulses 3 3 impulses 3 3 1 We can graph the three series using com implabels II dlrgdp dlrm2 drs ll GRAPH HEADER Responses to an Interest Rate Shock KEY upright patterns number 1 klabels implabels vlabel standard deviations 3 r1 r2 r3 Responses to an Interest Rate Shock standard deviations 0 75 T T T T i T i T T T T T Hence a one standard deviation innovation in the 3 month 7 bill rate is predicted to reduce real money balances and after the second period real GDP Even though these results seem plausible one caution is in order Nonlinear estimations of a
260. ror 0 1243225 Signif Level 0 3788566 EXCLUDE can only test whether a group of coefficients is jointly equal to zero The TEST instruction has a great deal more flexibility it is able to test joint restrictions on particular values of the coefficients Suppose you have estimated a model and want to perform a significance test of the joint hypothesis restricting the values of coefficients 0j Oj and Og to equal rj rj and rg respectively Formally Q rij Qj r j and Q rp To perform the test you first type TEST followed by two supplementary cards The first supplementary card lists the coefficients by their number in the LINREG output list that you want to restrict and the second lists the restricted value of each Suppose you want to restrict the coefficients of the last three lags of drs to all be 0 1 i e amp s 0 1 O 0 1 and oc 0 1 To test this restriction use test 678 0 1 0 1 0 1 F 3 153 12 20650 with Significance Level 0 00000033 RATS displays the F value and the significance level of the joint test o5 0 1 og 0 1 and O7 0 1 If the restriction is binding the value of F should be high and the significance level should be low Hence we can be quite confident in rejecting the restriction that each of the three coefficients equals 0 1 To test the restriction that the constant equals zero and that a 0 1 amp 0 1 and a amp 0 1 use 10 Walter Enders test 1678 0 0 1 0
261. ror p SEGUE Sa Guise KKEKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKK T AO 0 0026446 0 0032940 Ex 0280715 Sime O AZZ OG 2 Al 31L519 7 610 2 015 0 6332808 2 510419 0 00605571 Su Be 0 0094796 OROO S559 OAS CHO RM OS 4 3 837 4 Bl 2erS SOO SHES Of BALE SNS 4 08475 0 00004412 The estimated coefficients and their associated f statistics are very similar to those obtained from NLLS Notice that we also have an estimate of the variance equal to 0 0000243 3 Maximum Likelihood Estimation of Moving Average Processes It is straightforward to use maximum likelihood estimation to estimate a model with unobserved components Consider the simple MA 1 model y Ee Big Since the sequence is unobserved it is not possible to use LINREG or NLLS to estimate the process To estimate B using maximum likelihood techniques it is necessary to construct a formula of the form y By However the following is an illegal statement because e is defined in terms of its own lagged value a nonresolvable recursive expression frml e y bl e 1 1 The RATS instruction BOXJENK is the most useful way to estimate standard ARMA p q models The aim of this example is to illustrate the use of a SUBFORMULA on the FRML instruction 36 Walter Enders The way to circumvent this problem is to create a placeholder series using the SET instruction Then define the desired formula in terms of the placeh
262. s By 1 0 0 20 B comq 1 com R z Il 1 O O ll com c Il 0 ll com f tr c R beta inv r xx inv tr r c r beta com fl scalar f q v dis f1 We can obtain the t statistic for the null hypothesis using 202 Walter Enders dis sqrt f1 5 04661 You can easily verify this value as the same as that obtained using the LINREG instruction We can test the more complicated hypothesis D B2 0 using b o 112 0 0 I B 0 com n 2 com R 21 0 1 0 10 0 1 II com c Il O I O II com f tr c R beta inv r xx_inv tr r c r beta com f1 scalar f 2 v dis f1 9 17622 We obtain precisely the same value using LINREG ad EXCLUDE Consider exclude dlrgdp 1 2 Null Hypothesis The Following Coefficients Are Zero DLRGDP Lag s 1 to 2 F 2 163 9 17622 with Significance Level 0 00016735 Vector and Matrix Manipulations 203 5 3 Creating Series from a Matrix There are some instructions that operate only on series As such you may want to create a series from a vector or several series from matrix For example suppose that you want to crate a graph of the regression residuals contained in the 166 x 1 vector e Since GRAPH requires a series we need to DECLARE a series that can be used to hold the residuals The first instruction creates the vector errors containing a single series The second instruction sets each element of errors 1 equal to the corresponding element of e The PRI
263. s 1 A L L and G is the n x n matrix relating the regression residuals and the structural shocks Now let the variables in x be arranged such that only i has a long run effect on x n only 1 and x have a long run effect on x only x and 3 have a long run effect on x3 and so on Notice that there are exactly n n 2 such restrictions Since the coefficient sums are obtained from C 1 G these restrictions translate into the assumption that each element above the principle diagonal in C 1 G be zero The key point to note is that we can impose these restrictions on C 1 G from a Choleski decomposition of C 1 GG C 1 y If we evaluate Ai L a11 0 ai D L ai 2 L a11 3 L at L 1 we obtain the coefficient sum Xa K 84 Walter Enders Given the relationship between the regression residuals and the structural variables it follows that Eee GG Yet Eee is precisely the variance covariance matrix of the regression residuals that we have called X Thus to obtain cay G we need only to obtain the Choleski decomposition of cay cay Once the VAR has been estimated you can obtain the desired matrix using COMPUTE C VARLAGSUMS COMPUTE S1 MQFORM SIGMA TR INV C COMPUTE S2 DECOMP S1 COMPUTE G C S2 To explain when you use the ESTIMATE instruction RATS creates the matrix 26V ARLAGSUMS containing the n x n matrix of the appropriate sums of the lag coefficients
264. s only large when Q is large The situation is quite different if we want to test the hypothesis 1 Now under the null hypothesis the y sequence is a non stationary process As such it is inappropriate to use classical statistical methods to estimate and perform significance tests on the coefficient Qt Dickey and Fuller 1979 used Monte Carlo methods to obtain the appropriate critical values to test for the presence of a unit root The following program produces results that mimic their tj distribution The logic of the method is identical to that for the AR 1 process used in the previous example The difference is that we will generate random walk sequences instead of mean reverting autoregressive processes The ALLOCATE instruction in Program 4 7 of the file labeled CHAPTERA 1 PRG uses a sample size of 100 Line 2 initializes a series called tstat this series will hold all of the statistics generated in the Monte Carlo experiment Notice that we will perform the experiment 10000 times and will have 10000 statistics As such we want tstat to 142 Walter Enders have a length of 10000 Line 3 initializes the y sequence to zero and sets the first entry equal to a normal distributed random number with a mean of zero and standard deviation of unity Lines 5 9 are performed 10000 times Line 5 generates a sequence that mimics the random walk yr Yet Line 6 takes the first difference of the sequence and calls the result d
265. sage The third instruction is correct because the use of lag notation ensures that there is no ambiguity in the meaning of 2 0 and 2 1 However log 2 log 2 1 would not produce the desired effect 5 The very first program in this chapter contained the line set 1 5 This set all entries of series 1 equal to the number 5 2 Creating Numbered Series and Labels Since RATS allows you work with a series using its label or its integer value you will want to become familiar with creating numbered series assigning a label to a series fetching the integer value of a series from its label and fetching the label of a series from its integer value 96 Walter Enders In principal you do not need to assign a label to a series However labels make it easier to remember recall the various steps in your program and to interpret your output One way to create series is to use ALLOCATE instruction with the optional series field Consider the first five lines of Program 3 3 on the file labeled CHAPTER3_1 PRG cal 1959 1 4 all 4 2001 1 lt lt lt Note that line 2 is modified open data a money_dem xls data org obs format xls tab Series Minimum Maximum No I No I No I No I DATE 8763314 2321846 1000000 1000000 GDP F3IO 538 JL 27 6 1000000 6000000 RGDE 3644970 8404937 0000000 9000000 TB3MO SPL SLATS OO ASSIS 30353 NODS Sess TBlYR SES S OS pS ISGAZZ SIUSISSISISJOIS 3800000 As in all the other programs
266. se it is the first series that has been created Similarly the entries of the series are not dates like 99 2 or 2001 3 Instead of using dates RATS numbers each value 1 through 10 You might want to think of series 1 as a 10 x 1 vector Each element in the vector has the value of 5 The point is not that RATS can represent a series name by a number and a calendar date by a number Instead RATS always represents a series by a number This is true regardless of whether or not you attach a name or label to the series You are allowed to attach a label to a series for convenience Similarly RATS always represents each calendar date by a number You are allowed to attach a date label like 99 2 or 2001 3 for your own convenience This is true regardless of whether or not you use the CALENDAR instruction For your convenience the programs illustrated in this section can all be found on the file labeled CHAPTER3 1 PRG 5 As explained when we discuss matrices within RATS Programming Language there is a difference between a series and a declared vector 90 Walter Enders 1 Dates as Integers In case you skipped over the last two chapters note that the file labeled MONEY_DEM XLS contains a number of variables in Excel format If you open the file labeled CHAPTER3_1 PRG you will find Program 3 2 open the file and read in the data set by entering the following four lines cal 1959 1 4 all 2001 1 open data a money_dem xls Modify th
267. sed to avoid any problems concerning the initial conditions used for y and xj COMPUTE stores the estimated value of D in betahat i do i 1 2000 sim model unit results sims 149 2 v lin noprint sims 1 51 constant sims 2 com betahat i beta 2 if betal gt beta 2 1 96 stderrs 2 and betal le beta 2 1 96 stderrs 2 com success success 1 dis betal 1 96 stderrs 2 betal 1 96 stderrs 2 success end doi The IF instruction needs a bit of explanation Recall that 7STDERRS 2 holds the standard error of BETA 2 Hence beta 2 1 96 STDERRS 2 is the lower bound of the 95 confidence interval constructed using the f distribution Similarly beta 2 1 96 STDERRS 2 is the upper bound of the confidence interval If the actual value pi exceeds the lower bound and is less than the upper bound p is within the confidence interval Thus if the condition is TRUE success is increased by 1 The DISPLAY instruction displays the lower and the upper bound for each replication along with the current value of success IF Statements and Monte Carlo Experiments 153 After the loop is completed the summary STATISTICS for betahat are displayed along with the percentage of instances in which D fell within the confidence interval sta fractiles betahat dis The percentage of successes is success 20 Statistics on Series BETAHAT Observations 2000 Sample Mean OsGA LOSS iL 7S Variance POT QS S Standard Error 0 266
268. sedvecesexvvevaeuckweceeuvescutsducvevM vU ces ds Vv vVvE Es ERR E paP Ku WE RvE PEE UR T LOO iv Quick Index of Selected Functions and Instructions NAME Description as used in test Approx Page PIF Evaluate a conditional statement 126 PLG Returns the label of series i 98 S L Returns series number corresponding to L 182 BOXJENK Estimates an ARIMA model 178 BRANCH Program execution jumps to a new location 133 CDF Cumulative density function 41 COMPUTE Evaluates an expression 171 CORRELATE Creates a correlogram 7 CVMODEL Covariance matrix modeling 75 DECLARE Creates a matrix 169 ENTER Manipulates items on supplemental cards 182 ENTRIES Number of entries to process on supplementary card 113 EQUATION Creates an equation 151 EQV Assigns labels to series 96 ERRORS Creates forecast error variances and impulse responses 53 ESTIMATE Estimates a system of equations 48 EWISE Operates on elements of a matrix 177 FORECAST Creates dynamic forecasts 60 GRAPH Creates a high resolution graph 5 GROUP Combines equations into a system 62 IMPULSE Creates impulse response functions 67 INFOBOX Displays dialog box 145 INQUIRE Selects the starting and ending values of a series 211 LABELS Assigns labels to series 97 LINREG Estimates a linear regression 6 LOCAL Creates local variables in a procedure 214 MAKE Creates a matrix from data 197 MAXIMIZE Finds the maximum of a function 33 MENU Displays a set of choices 233 MESSAGEBOX Displays a message 229 NLLS Non
269. set the remaining values to zero Similarly we can let the first g elements of vector B hold the elements of beta and set the remaining values to zero dec vect a number b number com a const 0 b const 0 ewise a i if i lt p alpha 1 i 0 0 ewise b i if i lt q beta 1 i 0 0 Hence the DECLARE instruction in the program segment above creates the vectors A and B and sets the dimension of each to 24 COMPUTE uses the CONST x instruction to set all values of A and B to equal the constant zero Next looping over p equates the first p values of A with the corresponding elements of alpha Similarly looping over q sets the first q elements of B equal to the corresponding elements of beta Finally we can write a routine that calculates the remaining values of phi do j 1 number 1 com phi j 1 b j do k 1 j com phi j 1 phig 1 phig 1 k a k end do k end do j The first loop initializes phi j 1 equal to Bj Each time through the inner loop i e for each value of k Oxj 1 is added to phi j 1 Exiting this inner loop yields the desired sum i PN B Mr m k l This process is repeated for each value of j up to and including j 23 The last task is to create a bar graph of the phi sequence Since phi is a vector it cannot be graphed directly However we can convert phi into a series called response and graph response using set response phi t gra style bar header Impulse Respons
270. ssociated Q statistics of the residuals with Let the dependent variable be denoted by y Uncentered R is 1 sum of squared regression residuals sum of squared values of y Centered R is 1 sum of squared regression residuals sum of squared deviations of y from the mean of y 8 Walter Enders cor number 12 partial partial qstats span 4 resids Correlations of Series RESIDS Quests erly aD atom Omer 59103 MTM 0515 QA Autocorrelations EOP ORIS AS OM OTRO 16S Ole 01578332 0 0125602 0346243 0 0536726 7 0 0680555 0 1041120 0 1047616 0 0499244 o dior Ted 022 99161 Partial Autocorrelations ls OMS A SiO PR ORO0O2 69316 M 015874 0 S182 180 2187 0341736 0 0546344 ys 050702992 031065979 0 1 330995 070575503 Sdbals oS 0 OSSIA gg Boc St atsites 0 1125 Significance Level 0 99847633 So 67S Sem e OENES OVEL ORS 989 0219 9 5558 Significance Level 0 65486726 All of the autocorrelation and partial autocorrelations are small and the Ljung Box Q 4 Q 8 and Q 12 statistics do not indicate the values are statistically significant Moreover the individual f statistics suggest that only one of the autocorrelation coefficients is insignificant at conventional significance levels However the lags of drs are correlated with each other so that the individual r statistics can be misleading We might be concerned that the model is over parameterized since sum of coefficients s O6 O7 is approximately zer
271. st eade epe hase a ee EHE N aie p ee sa ea d SEPA KENT eux RHET QR ESAE iaa adi aaia 143 SA The Enders Granger StatstlC ien iati aree enu dna Ra en s po d Rea iaai eraai orendi in 146 5 5 Inference in a Cointegrated System sseeesssssseeseeese ener e e enne nnns 149 The Program idiarena ai aaea e a aiaa aeaa araa ad iini aioa aiai aia 151 6 An thetic Random Variables ertt dienen ena sa th aai eun e aeu Ee MARK RR an yu d REN IRR Ra ia 154 6 1 Biasin NILS Estimates SEPTIES 154 T Booftstrapping n eiecti acean eei eiis RE EEKEREN EERTE ENTNER RENERE EE SANIA 158 7 1 Bootstrapping Regression Coefficients ccc eee eee eee sened KAE enenatis 160 7 2 The AR Coefficients of Real GDP Growth seeeessssssssseeeeee eee nennen nnne nnns nna 164 Chapter 5 Vector and Matrix Manipulations 168 1 Creating Matrices and VeetOps 125321 rer ceva oid d eb rr ore deb nag ane erga FER ca casts nU rk kPa d NR tada ERR dada 168 ISBPTOEu pU LET ET 169 locii vu DHT 170 1 22 GOMPBUTE i iar onere tet santa acces cuits ceret XE ed sedo roe das as FI abre tnu coe E eee are AA aE 171 IC T ERE 171 PASE D SO SCUDI CES 176 2 1 Operations on Subcomponents of a Matrix ssssseeeeeeene eene ene n enne 176 2 2 selecting ARMA Coefficients TIL 178 2 3 Manipulating the Output of a VAR sess ene m enn deaa aan
272. star dlm3 resids Dependent Variable DLM3 Variable Coeff Std Error T Stat Signif Ck CK Ck Ck KICK C CK CI CK C CC CK CC CC Ck CK CC CC CK C CI Ck S kk x KK Kk Kk kx Kk Mk ko kock ko 1 AO 0 0026210 0 0031396 0 83479 0 40506392 2 A1 1 6729950 0 6362132 2 62961 0 00937179 3 BO 0 0094409 0 0054145 1 74363 0 08312034 4 Bl 2 3815524 0 6026591 3 95174 0 00011560 5 GAMMA 240 2373406 65 4646481 3 66973 0 00032915 compute aic nobs log rss 2 9onreg compute sbc nobs log rss Ynreg log Ynobs dis The aic aic and sbc sbc The aic 909 99076 and sbc 894 40079 The model appears satisfactory The estimated values of al b and gamma are significant at conventional levels and the estimated value of 50 has a prob value of 0 083 If you examine the autocorrelations of the residuals using CORRELATE you will find that they are all insignificant at conventional levels Moreover in a linear AR 1 model we require that the absolute value of the autoregressive coefficient be less than unity in absolute value Here the maximum and minimum values of dlm3 are 0 00538 and 0 03858 respectively As such the autoregressive coefficient o Bi 1 exp yy 1 has a range of 0 90542 to 0 76053 The AIC selects the LSTAR model while the SBC selects the linear AR 1 model Since a0 is not significant at conventional levels we can re estimate the model without this intercept term Hence modify the first two lines of the
273. stimated there are 161 8 153 degrees of freedom Next RATS reports four Goodness of Fit measures centered R R bar square centered R adjusted for degrees of freedom uncentered R and TR number of Linear and Nonlinear Estimation 7 observations multiplied by the uncentered R The mean and standard error of drs over the 161 observations used in the regression are reported to be 0 0155900621 and 0 8194146456 respectively Note This will differ from the mean and standard error over all 168 observations These are followed by the standard error of the estimate i e the square root of the sum of the squared regression residuals divided by the degrees of freedom and the sum of squared residuals The F statistic and its significance level can be used to test the hypothesis that all coefficients in the regression other than the constant are zero Here the sample value of F for the joint test a a3 a7 01s 8 6892 Given that there are 7 restrictions and 153 degrees of freedom this F value is significant at the 0 00000001 level The Durbin Watson test for first order serial correlation in the residuals is 1 956107 2 0 is the theoretical value of this statistic in the absence of serial correlation For each right hand side variable the next portion of the output reports the coefficient estimate Coeff the standard error of the coefficient Std Error the t statistic for the null hypothesis that the coefficient e
274. t end 3 We need to use the INQUIRE instruction to store the start and end values in the variables v and v2 Note that INQUIRE is an executable instruction it should be placed after all TYPE OPTION and LOCAL instructions inquire series y v start v2 end 4 UNIT SRC creates the two integer variables v and v2 We can declare then as LOCAL variables using local integer v1 v2 5 Modify the LINREG GRAPH and CORRS instructions such that v and v2 are used as the starting and ending entries if det eq 1 lin noprint dy v1 v2 resids y 1 dy 1 to lags if det eq 2 lin noprint dy v1 v2 resids y 1 dy 1 to lags constant if det eq 3 lin noprint dy v1 v2 resids y 1 dy 1 to lags constant time 222 Walter Enders The graph and autocorrelations can be obtained from gra header Time Path 1 resids v2 cor noprint number 24 resids v2 corrs The source code for the first nine lines of the procedure is procedure unit y start end type series y type integer start end option switch graph 0 option integer lags 1 option choice det 2 none intercept trend local series dy resids corrs time local integer v1 v2 inquire serieszy v1 gt gt start v2 gt gt end Now the procedure can be quite useful For example open the file CHAPTER6 PRG and read in the data set using Program 6 1 As the program illustrates you can embed the UNIT SRC in a DO loop so that you that you perform augmented Dickey Fuller
275. t it can be used with any variable Simply replace every occurrence of dirgdp with the symbol x For example the first portion of the program will become com max_lag 16 diffs 1 lin noprint x max_lag diffs 1 constant com aic min nobs log rss 2 nreg p opt 0 do p 1 max_lag lin noprint x max_lag diffs 1 constant x 1 to p compute aic nobs log rss 2 9onreg if aic lt aic_min com p_opt p com aic min aic end do p If you eliminate the END IF instruction you can use the routine to find the optimal lag length of dirgdp dirm2 and drs using dofor x dlrgdp dirm2 drs The modified program End DOFOR The complete program is on the final portion of Program 4 1 126 Walter Enders 2 The IF x y z Function There is a second type of conditional statement that is especially useful when working with a series Consider IF condition y z This function returns y if the condition is true and returns z if the condition is false Note that y and z can be REAL or INTEGER Examples 1 com x if i ge 5 1 0 1 0 Here the IF function is used with a real variable x The value of x will be 1 0 if i is greater than or equal to 5 and will be 1 0 if i is less than 5 2 set dummy if t ge 1992 1 1 0 In conjunction with SET the IF function works on each entry of a series Here each value of DUMMY is equal to 1 for t 2 1992 1 and is equal to zero otherwise Thus in contrast
276. t of dependent variables LAGS 1 to lag length DETERMINISTIC list of deterministic constant seasonals and exogenous variables END SYSTEM Step 2 You instruct RATS to estimate the system using ESTIMATE The typical form of the ESTIMATE instruction is ESTIMATE OUTSIGMA V residuals resids other options start end For an n equation VAR the OPTION RESIDUALS resids creates n series of residuals The residuals from the first equation are stored in the series called resids 1 the residuals from the second equation are stored in the series called resids 2 and so forth Other Options OUTSIGMA matrix Computes and saves the covariance matrix of the residuals NOPRINT Suppresses printing of the OLS estimation of each equation NOFTESTS Suppresses printing the results of all Granger causality tests SIGMA Displays but does not save the covariance matrix of the residuals Use both OUTSIGMA and SIGMA if you want to compute save and print the variance covariance matrix COEFFICIENTS coef Creates a matrix of the coefficients Column i contains the coefficients of the i th equation Note that RATS 5 0 recognizes the older form of the ESTIMATE instruction As such you can still use ESTIMATE OUTSIGMASYV other options start end residuals where residuals is the first series in a block of series used to store the residuals Examples It is straightforward to set up and estimate the following 2 variable 1 lagVAR as a MODEL called ex
277. t to know that it is necessary to create a 2 x 2 matrix of series to hold the response functions there are two sets of responses for each of the two variables This is accomplished by using the DECLARE instruction to create the 2 x 2 rectangular matrix impulses declare rectangular series impulses 2 2 In most circumstances it is also necessary to create a matrix of labels A graph that labels the variables IMPULSES 2 1 or IMPULSES 2 2 is not very helpful com implabels I 1 year 3 month ll VARs and Error Correction Models 67 The impulse responses are created by the IMPULSE instruction If we use the MODEL option the form of the IMPULSE instruction the we need is impulse MODEL rmodelname RESULTS 2matrix equations steps shock to name where modelname The model name used on the STSTEM instruction equations Number of equations in the system Use with the MODEL option matrix Name of the matrix used to store the impulses steps The forecast horizon and the number of impulse responses shock_to The component to be shocked Use with the MODEL option name The name of the covariance matrix used on the ESTIMATE instruction To create 24 impulses from the model term that are stored in the matrix impulses use impulse model term result impulses noprint 24 s The first is a placeholder for the number of equations and the second tells RATS to shock all equations The responses of tb yr and tb3mo to innovations in tb
278. t y star 0 The instructions in the DO loop will be executed 1000 times The first instruction in the loop randomly selects an integer in the range 1959 2 to 2001 1 The value of ii serves as a randomly selected entry value The COM instruction uses this value to equate the first two values of y star Le y and y with two consecutive values of dirgdp Next the bootstrapped e series is created from residuals of the AR 2 model Each value e e Veram e is a random draw with replacement from resids Note that each element of resids has a 1 166 chance of being selected since fix uniform 1959 4 2001 1 1 generates integers on the interval 4 169 do k 1 boot_num com ii fix uniform 1959 2 2001 1 com y star 1 dlrgdp ii y_star 2 dlrgdp ii 1 set e star resids fix uniform 1959 4 2001 1 1 The first instruction below creates the series y star for entries 1959 4 through 2001 1 using the bootstrapped residuals e star As such the first two entries of y star remain the values SET by the first instruction inside the DO loop Notice that the estimated values of B B and B are used in the construction of y star The second instruction below uses the bootstrap sample to estimate B B and p The value of B is stored as the i th entry of beta2 star set y star 3 betal hat y star 1 beta2_hat y_star 2 betaO hat e star lin noprint y star y star 1to 2 constant compute beta2 star k beta 2 end d
279. t yields the lowest residual sum of squares do i low high set plus if dirgdp 1 lt thresh_test i 0 1 set minus plus set yl plus plus dlrgdp 1 set y2 plus plus dlrgdp 2 set yl minus minus dlrgdp 1 set y2 minus minus dlrgdp 2 lin noprint dlrgdp plus yl plus y2 plus minus yl minus y2 minus com rss 1 rss if rss lt rss test compute rss test rss compute thresh thresh test 1 end doi Once the program loop exits the loop we can display the consistent estimate of the threshold if we use dis We have found the attractor dis Threshold thresh We have found the attractor Threshold 0 01724 Finally we can estimate the TAR model with the consistent estimate of the threshold using set plus if dlrgdp 1 lt thresh 0 1 set minus plus set yl plus plus dlrgdp 1 set y2 plus plus dlrgdp 2 132 Walter Enders set yl minus minus dlrgdp 1 set y2 minus minus dlrgdp 2 lin dlrgdp plus y1l plus y2 plus minus yl minus y2 minus Variable oes Std Error iSt Signif KKEKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKK PLUS 022981605 0 009875828 SOA HOM R 2821 E5 5 6 Y1_PLUS 443888007 406708488 09142 27673082 NETUS 036693896 208404259 o dE 2459 7 86046101 MINUS 005019041 SEO SIR ZI EOIO RT a TASI 00000379 Yl MINUS 237965600 104676378 SZ MESSIS 02433550 Y2 MINUS od5412 302 55 915 OO SOUSIO a HOODS AL 08922107
280. ta From 1961 01 To 2001 01 Usable Observations IOL Function Value Ae a VLO COOL Variable Coeff SEORE RLO dP Seen Signif vy gU net TD TAL spe es TOL TAF Ud E du e A TOS TAS TE de s tue s e US TO TS TAS AY DAT AES OF TARTS Ta Y AY THE Da dba AIL O SS TIZ OSLO 10 5 laa LSU A MASS 9510 00027 9 ZI INT 0392906296 OTIO SS SIEOSIEZ 810 iore 9 715 MEM I OO OOOO So Bil 0 817502905 0 044594888 18 33176 0 00000000 4 VAR OA SATHOS0ST On 088606437 2 ED SH RIO 000100010 The same technique is used for estimating a higher order MA q process Suppose you wanted to estimate the spread as drs Qidrs 1 O4drs 3 i Bier Bog Now the NONLIN instruction contains the coefficient b2 The first FRML instruction uses temp 1 and temp 2 as placeholders for e and e 2 The second FRML instruction creates the desired log likelihood and the COMPUTE instruction provides the initial guesses Notice that the start date can remain at 9 since no usable observations are lost from the MA terms set temp 0 nonlin al a7 b1 b2 var frml e drs al drs 1 a7 drs 7 b1 temp 1 b2 temp 2 frml L temp e log var e 2 var com al 0 4 a7 3 bl 5 b2 0 3 var 1 max L 9 4 A Bilinear 1 1 Model of the Money Supply Given that dlm3 seems to exhibit nonlinear behavior it might be desirable to estimate an alternative nonlinear specification The bilinear model generalizes the standard ARMA p q model by
281. tant r2 1 to 4 If there are no ARCH or GARCH effects this regression will have little explanatory power so that the coefficient of determination i e the usual R statistic will be quite low With a sample of T residuals under the null hypothesis of no ARCH errors the test statistic TR converges to a X distribution with n degrees of freedom If TR is sufficiently large rejection of the null hypothesis is equivalent to rejecting the null hypothesis of no GARCH errors On the other hand if TR is sufficiently low it is possible to conclude that there are no ARCH effects Since LINREG creates the internal variables NOBS i e T and S7RSQUARED R you can easily compute TR and the significance of trsq as y with 4 degrees of freedom with compute trsq Yonobs rsquared cdf chisqr trsq 4 Here the CDF instruction calculates the marginal significance of trsq using a y distribution with 4 degrees of freedom The syntax for CDF is Linear and Nonlinear Estimation 41 CDF distribution statistic degreel degree2 where distribution The desired F t Y or normal distribution is selected using FTEST TTEST CHISQ or NORMAL statistic The value of the test statistic degreel Degrees of freedom for TTEST and CHISQ or numerator degrees of freedom for FTEST degree2 Denominator degrees of freedom for FTEST The six instructions below can be used to estimate a regression with ARCH 1 errors The NONLIN instruction indicates th
282. te Carlo Experiments There are many instances in which we want to perform a set of instructions only if a particular condition is met For example in the last chapter we used the WHILE instruction to reduce the order of an AR p model if the last coefficient was not significant at the 5 level The process continued until a significant lag was found At that point the program produced the output of the final autoregressive model In fact RATS enables you to control the flow of a program using many types of conditional statements Suppose that A and B are two numbers of the same type real integer RATS is able to check the following conditions involving the standard relational operators equality A or A EQ B not equal A lt gt B or A NE B greater than AB or A GE B less than A lt B or A LT B greater or equal to A gt B or A GE B less than or equal to A lt B or A LE B Moreover RATS can create compound conditional statements that use AND NOT and OR A AND B A NOT B A OR B The simplest way to use relational operators is with the IF instruction The basic syntax of IF is IF condition instruction to be executed if the condition is true where condition is any of the relational conditions listed above and instruction is any valid RATS instruction If the specified condition is met RATS will execute the next instruction If the condition is not met the next instruction will be skipped For example consider the following
283. te and display the aic and sbc using compute aic Y nobs log rss 2 nreg compute sbc nobs log rss Yonreg log Znobs It is also common to determine a lag length based on the outcome of t tests This methodology picks the lag length such that the t statistic for the last lag is significant at some pre specified level Given the presence of an intercept each regression will have p 1 coefficients and the t statistic for the last lag can be obtained from TSTATS p 1 dis Lags p AIC aic SBC sbe t tstats p 1 Hence we can put these instructions in the loop and use the NOPRINT option in LINREG to obtain Loops over Dates and Series 103 do p 1 12 lin noprint dlrgdp 14 constant dlrgdp 1 to p compute aic Ynobs log rss 2 9onreg compute sbc nobs log rss gVonreg log 96nobs dis Lags p AlC aic and SBC sbc t H otstats p 1 end do p Lags 1 AIC 703 21209 SBC 697 11238t 3 79455 Lags 2 AIC 704 98313 SBC 695 83356 t 1 93483 Lags 3 AIC 703 03417 SBC 690 83475t 0 22304 Lags 11 AIC 2 693 50814SBC 656 90987t 0 38318 Lags 12 AIC 2 695 48673 SBC 655 83860t 1 92196 Here the AIC selects the model with two lags while the SBC selects the model with one lag In this example the choice is unclear since t statistic on lag 2 has a prob value that is slightly greater than 0 05 At this point a careful researcher would subject the
284. te of that is too high the second should give an estimate that is too low 6 1 Bias in NLLS Estimates Antithetic random variables can be quite effective when working with nonlinear models The sample program illustrated below replicates the results of a Monte Carlo experiment reported in Davidson and MacKinnon 1993 The issue is to find the bias if any of the exponent a in the nonlinear regression model y 2px e IF Statements and Monte Carlo Experiments 155 Davidson and MacKinnon 1993 use a sample size of 50 with a single set of the x sequence drawn from a uniform distribution on the interval 5 15 using parameter values p 1 a 0 5 and drawn from an i i d standardized normal distribution The program will use the following four instructions to set up the nonlinear estimation NONLIN alpha beta FRML monte y beta x alpha COM alpha 0 48 beta 0 98 NLLS frml monte noprint y Program 4 11 of the file labeled CHAPTERA 2 performs the Monte Carlo experiment without using antithetic variates The first three lines define the default size of a series to equal 50 seed the random number generator and create the 50 entries of the sequence x as random draws from a uniform distribution on the interval 5 15 The next two lines initialize 1000 values of alpha hat and beta hat to zero These two series will be used to store the estimates of and p respectively all 50 seed 2001 set x uniform 5 15
285. te regression equations Once BREAK is encountered the program exits the loop and the USERMENU is removed from the menubar if menuchoice eq 2 estimate depvar if menuchoice eq 3 break end loop usermenu action remove end seasons The procedure ESTIMATE is invoked from SEASONS Hence it is necessary to compile ESTIMATE before SEASONS Since SEASONS passes the parameter depvar to ESTIMATE it is necessary to declare depvar as a series The program needs to create variables for the linear Writing Your Own Procedures 241 quadratic and cubic time trends and for the fitted values from the regression The next instruction declares time t2 t3 and fitted to be local series proc estimate depvar type series depvar local series time t2 t3 fitted The next instruction creates an integer vector called reglist that will hold the variables to be used in the regression equation The second instruction below uses COMPUTE to include a constant in the list of regressors The linear quadratic and cubic time trends are created by the subsequent SET instructions dec vector int reglist compute reglist llconstantll set time t set t2 t t sett3 t t t The MENU instruction below creates a dialog box with the title Select the Degree of the Polynomial The user will be presented with four choices in a dialog box that looks like Select the degree of the Polynomial NO trend C Linear trend C Quadratic trend C Cubic trend C
286. ted with the label D series label The final instruction creates the graph of the transformed series Notice that the choice Growth Rate does not check for the possibility of values that equal zero If any value of series does equal Zero the associated entry value for the transformed series will be NA The next two lines of Program 6 1 compile the procedure and EXECUTE the procedure using rgdp source noecho c winrats transform src transform rgdp Select Growth Rate and use the instruction Writing Your Own Procedures 235 table Series Obs Mean Sieel daieicovie Minimum Maximum DATE 169 LOTI 8763311 112 6232135 1959 100000 2001 100000 GDP 169 15 7 2 5 FSIOS3 PONO LSENG 496 100000 10243 600000 RGDP 169 5142 364497 1950 840494 2273 000000 9439 900000 M2 169 1904 835266 11999 OG 73L7 287 800000 5043 710000 M3 169 2414 462229 12916 ROAM ALO 2 9 PISIS 7260 136667 TB3MO 169 5 9d 2 590483 2 XOS S93 15 0539339 TB1YR 167 GRRIESISIONI 25393622 2 TISSI 14 380000 GRGDP 168 07009 5SM 999027 0 020388 0 038528 Notice that the series GRGDP has been created Since this variable has not been declared as a LOCAL SERIES it is accessible from the main body of the program 8 1 Creating a USERMENU A second way to create an interactive procedure is to use the USERMENU instruction USERMENU allows you to add a user defined menu to the RATS menu bar The user clicks on the pull down menu and the list of choices appears Once a USER
287. tes the series y since t runs from 1 to 101 y runs from 5 0 to 45 0 Each of the next four SET instructions creates a series representing as a function of yy for y 0 5 1 2 0 and 5 0 all 101 set y 5 1 O 1 t set thetal 1 exp 0 5 y 1 set theta2 1 exp y 1 set theta3 1 exp 2 y 1 set theta4 1 exp 5 y 1 The remainder of the program creates a scatter diagram of the various functions of 0 Note that y acts as a smoothness parameter for y near zero movements in y have small effects on 0 when y is small com labels Il 0 5 1 0 2 0 5 0 II scatter header Effects of Gamma on Theta style lines patterns klabels labels key below vlabel THETA hlabel GAMMA 4 y thetal y theta2 y theta3 y theta4 Linear and Nonlinear Estimation 29 Effects of Gamma on Theta THETA E 8 Now use Program 1 4 to read in MONEY DEM XLS Form the logarithmic change in M3 and call the resulting series dIm3 set dim3 log m3 log m3 1 Next estimate dIm3 as an AR 1 process and obtain the AJC and SBC using lin dlm3 resids constant dIm3 1 Variable Coeff Std Error T Stat Signif Ck Ck ck ck ck kk Ck Ck Ck Ck Ck Ck Ck Ck Ck Ck Ck kk Ck Ck Ck Ck Ck Sk Ck Ck Sk Ck Ck kk Ck CK kk Ck Sk kk Sk Sk Sk kk Sk Sk Sk Sk Mk ko kx kx ko ko 1 Constant 0 0030875133 0 0009012005 3 42600 0 00077312 2 DLM3 1 0 8436040010 0 0424922667 19 85312 0
288. tests on tb yr using lags 1 through 12 doi 1 12 unit lags i tblyr end doi estimate of rho with 1 lags 0106 21510 t statistic for the null hypothesis rho 0 is estimate of rho with 2 lags 0 2PerSpE t statistic for the null hypothesis rho 0 is estimate of with 11 lags 0 0S ESIC ec asic ALS the null hypothesis rho 0 is estimate of with 12 lags O 05406 C Sie She SIEG the null hypothesis rho 0 is 5 1 Passing Information by Address To this point we have thought in terms of passing information to a procedure However in many circumstances you will want to pass a variable from a procedure to the main program This is not particularly difficult if you are working with a global variable a global variable is accessible from any point in RATS Of course if you sent the procedure to someone else that person would need to know how you named the variable in question It would be most convenient if you Writing Your Own Procedures 223 allowed the user to fetch the variable in question using any name she selected The way to do this is to pass the information by address not by name This is done by placing an asterisk immediately preceding the variable name on the TYPE instruction Suppose for example that you wanted to pass the residual autocorrelations created by CORR back to the main program To illustrate suppose we modify UNIT SRC by removing corrs from the list of LOCAL series Once you invoke the proce
289. that you get to see each line of the procedure allows you to see the instructions and comments contained in the procedure Good programmers will include a set of comments in the procedure usually at the beginning that describe the proper use of the procedure If you are an experienced RATS user you certainly have used procedures many times You know that a procedure needs to be compiled only once within any program Once it has been compiled the procedure can be used any number of times In Chapter 2 we discussed the procedure BJIDENT SRC Recall that you can use the procedure to construct the autocorrelations and partial autocorrelations of a series using Writing Your Own Procedures 205 bjident options series start end where start end The range of the series to use for constructing the autocorrelations and partial autocorrelations Some of the options for the procedure are DIFF Maximum regular differencings 0 TRANS NONE LOG ROOT Transformation to apply to data GRAPH NOGRAPH Do High resolution graphs The key point to note is that the procedure allows you to make a number of important choices You choose the series to use along with the start and end dates Moreover the procedure contains three different types of options The DIFF option uses an integer value the TRANS option is an OPTION CHOICE allowing you to select a particular type of data transformation and GRAPH NOGRAPH is a SWITCH option A good portion of t
290. the Term Structure The ARCH M Model Econometrica 55 pp 391 407 Engle R and C W J Granger 1987 Cointegration and Error Correction Representation Estimation and Testing Econometrica 55 pp 251 76 Sims C 1986 Are Forecasting Models Usable for Policy Analysis Federal Reserve Bank of Minneapolis Quarterly Review pp 3 16 Stock J 1987 Asymptotic Properties of Least Squares Estimators of Cointegrating Vectors Econometrica 55 pp 1035 56 Ter svirta T and H M Anderson 1992 Characterizing Nonlinearities in Business Cycles using Smooth Transition Autoregressive Models Journal of Applied Econometrics 7 pp S119 139 Tong Howell 1983 Threshold Models in Non Linear Time Series Analysis Springer Verlag New York White H 1980 A Heteroskedasticity Consistent Covariance Matrix Estimator and Direct Test for Heteroskedasticity Econometrica 48 pp 817 838
291. the logarithmic change of rgdp The second line prepares RATS to estimate a regression with dirgdp as the dependent variable such that the last sample point is observation 100 Note A second observation will be lost as a result of the lagged change The asterisk instructs RATS to use the default value for the start entry 3 Estimate an AR 1 autoregression of the logarithmic change in rgdp using the last 100 observations and save the residuals in the series resids lin dlrgdp 2001 1 99 resids constant dlrgdp 1 Now line 2 instructs RATS to estimate the regression using the start date beginning at 100 observations from the end of the data set Note that if you want RATS to perform the date arithmetic 2001 1 100 you cannot put a space on either side of the minus sign Also note that you will get precisely the same output using lin dlrgdp 69 resids constant dlrgdp 1 The reason that the two are equivalent is that there are 169 observations for rgdp Observation 69 is identical to 2001 1 100 If you take the time to count you can verify that both of these entries are equivalent to 1976 1 Fortunately you never have to actually count the number that is equivalent to a particular date label In fact RATS provides a number of instructions that are helpful for using date manipulation The two most useful ones are Loops over Dates and Series 93 CAL YEAR PERIOD The entry number PERIOD of YEAR DATELABEL T The date string
292. the series on itself so that the resulting F statistic is necessarily significant at the 5 level R will be 1 Here dumb luck yields the correct lag length of zero Since you can t count on being this lucky all of the time it is important to be careful that you do not get caught in an infinite loop or a loop that produces an error 2 What is the difference between WHILE and UNTIL Answer The WHILE instruction checks to determine if condition is true at the beginning of a loop i e when WHILE condition is encountered The UNTIL instruction checks at the end of the loop i e when END UNTIL or the closing brace is encountered Thus the UNTIL block of instructions is always executed at least once If PROGRAM 5 had used UNTIL instead of WHILE the first three lines could have been dofor i dlrgdp dlm3 drs drl dlp com lags 13 until sign lt 0 05 Note the reversed inequality The first time UNTIL is encountered sign is not defined so the condition sign 0 05 is not true The program continues until a significance level less than 0 05 is encountered 118 Walter Enders 3 What other conditions are allowed in the WHILE and UNTIL instructions Answer As detailed in the very beginning of the next chapter any of the standard conditional relationships are allowed including lt gt i e not equal and i e equality as well as compound conditions created with AND and OR Chapter 4 IF Statements and Mon
293. ther transform or to transform another variable it is necessary to execute the procedure a second time Writing Your Own Procedures 237 if menuchoice eq 2 f com a D label y set s a y y 1 if menuchoice eq 3 f com a G label y set s a y y 1 1 graph header Time Path of a 1 s a usermenu action remove end transform_user Jazzing up the Procedure To allow several transformations on the same variable it is possible to use a LOOP Place the LOOP instruction immediately preceding USERMENU and an END LOOP instruction before USERMENU action remove Of course it is necessary to have a way to break out of the loop This is easily accomplished by adding a fourth menu choice and a fourth IF block The key changes to the procedure are usermenu action define title Transform 1 gt gt Logarithmic 2 gt gt First Difference 3 gt gt Growth Rate New continuation sign 4 gt gt Done New choice loop Begin loop here usermenu THE ORIGINAL INSTRUCTION SET BELONGS HERE if menuchoice eq 4 break If Done is selected terminate the loop graph header Time Path of a 1 s a end loop usermenu action remove end transform_user Program 6 1 compiles and EXECUTES the procedure using source noecho c winrats transform_user src C transform user rgdp 238 Walter Enders Notice the new selection on the Menu bar entitled Transform You can select the d
294. ticular RATS instruction you can use RATS Help Menu or refer to the Reference Manual and User s Guide The emphasis here is on what I call RATS programming language These are the instructions and options that enable you to write your own advanced programs and procedures and to work with vectors and matrices The book is intended for applied econometricians conducting the type of research that is suitable for the professional journals As I tell my students to do state of the art research it is often necessary to go off the menu I m being a bit facetious but by the time a procedure is on the menu of an econometric software package it s not new This book is especially for those of you who want to start the process of going off the menu Of course it will be impossible to illustrate even a small portion of the vast number of potential programs you can write My intent is to give you the tools to write your own programs Towards that end I will discuss a number of the key instructions and options in RATS programming language and illustrate their use in some straightforward programs I hope that the examples provided here will enable you to improve your programming technique This book is definitely not an econometrics text If you are like me it is too difficult to learn econometrics and the programming tools at the same time As such I will try not to introduce any sophisticated econometric methods or techniques Moreover all of th
295. ts are real numbers You indicate other element types by the use of brackets The most typical element types are INTEGER SERIES LABELS or STRING list of names The names of the matrices you wish to create Note that you can include the DIMENSION in parentheses It is important to note that within a compiled procedure you cannot dimension a matrix on the DECLARE instruction Instead you must use a separate DIMENSION statement 33 Within a procedure you cannot DECLARE and DIMENSION a matrix on the same instruction See Chapter 6 for details on writing your own procedures 170 Walter Enders Examples 1 To create a vector x that can hold 100 real numbers you can use declare vector x 100 or declare vector x dim x 100 NOTE Within a procedure you must use the second set of instructions 2 To create a vector x that can hold 100 integers and a vector y that can hold 50 integers you can use declare vector integer x 100 y 50 or declare vector integer x y dim x 100 y 50 3 To create a vector x that can hold 10 series you can use declare vector series x 10 4 To create a vector x that can hold 100 real numbers and a 10 x 20 matrix b containing real numbers you can use dec vector x 100 dec rectangular b 10 20 Or dec vector x dec rectangular B dim x 100 B 10 20 Note You can include different types of matrices on a DIMENSION instruction but not on a DECLARE instruction Vector a
296. ts the optimal lag length to use in an autoregressive model It is straightforward to write a procedure to automate this process The procedure below called LAGLENGTH SRC estimates a number of AR p models and reports the one selected as best by the SBC The procedure is executed using laglength option y start end laglength where y Series to estimate as an AR p model start and end The range to use in the estimation Note that one observation will be lost for each lag laglength Optional If laglength is specified the procedure returns the integer value of the lag selected by the SBC The sole option for the procedure is MAXLAG Maximum number of lags to use in the lag length test In the first line below the PROCEDURE instruction contains the name of the procedure and a parameter list containing y start end laglength The next two lines define the TYPE of these parameters Line 2 indicates that y is a series and line 3 indicates that start end and laglength are all integers Notice the asterisk preceding aglength If this field is specified by the user laglength will return the integer value of the optimal lag to the main body of the program procedure laglength y start end laglength type series y type integer start end laglength The next section of a procedure should contain the OPTIONS Here there is a single option called MAXLAG with a default value equal to 1 The user can use the option maxlag p to select th
297. twelve autocorrelations and the associated Q statistics can be obtained from 42 Walter Enders cor number 12 span 4 qstats resids Correlations of Series RESIDS Quarterly Data From 1959 03 To 2001 01 Autocorrelations 1 0 0177090 0 0072871 0 0877679 0 0372146 0 0197141 0 1939013 7 0 0416785 0 0473398 0 0010301 0 0979544 0 1078251 0 0908990 Ljung Box Q Statistics Q 4 1 6279 Significance Level 0 80377726 Q 8 8 9906 Significance Level 0 34309102 Q 12 14 3235 Significance Level 0 28052925 Nevertheless ARCH errors manifest themselves in the autocorrelations of the squared residuals You can form the squared residuals and perform the Lagrange multiplier test using set r2 resids 2 lin r2 constant r2 1 to 3 Variable Coeff Std Error T Stat Signif CK CK Ck Ck CK CK CI CIC CK CC CK CI KKK CK C KKK KKK KKK CK CI CI KKK C C KKK KKK KKK KKK I SK C KKK KKK KG KG X ko ko ko 1 Constant 0 046242482 0 018049071 2 56204 0 01132890 2 R2113 0 217038776 0 078030863 2 78145 0 00606210 3 R242 0 025838287 0 079861347 0 32354 0 74670932 4 R2 3 0 159631091 0 077999507 2 04657 0 04233716 To test the restriction that the coefficients for the 3 lagged values of r2 all equal zero use compute trsq nobs rsquared cdf chisqr trsq 3 Chi Squared 3 11 981860 with Significance Level 0 00744556 Since we reject the null hypothesis of no ARCH errors we can try to estimate the spread using the following sp
298. u can confirm that the multivariate AJC and SBC also indicate that drs is not block exogenous Innovation Accounting To obtain the variance decompositions and impulse responses it is necessary to re estimate the system in order to save the variance covariance matrix Use the OUTSIGMA option on the ESTIMATE instruction to save the covariance matrix as V system model chap2 var dlrgdp dirm2 drs lags 1 to 12 det constant end system estimate outsigma v F Tests Dependent Variable DLRGDP Variable IAS She ake ale SHG meets DLRGDP bo S277 2 WII IXS DLRM2 2 1065 00193254 DRS So SLIS 0 0002667 F Tests Dependent Variable DLRM2 Variable In Sheeneat ene ale SALONIE DLRGDP 0 8990 0 5494694 DLRM2 8 5958 0 0000000 DRS 36 IZING 0 0000285 F Tests Dependent Variable DRS Variable T ren tatie siters LRGDP 5 4620 0 0000001 LRM2 2 5 9 015 1 0 0097938 RS 8 0504 0 0000000 The ESTIMATE instruction will produce the equivalent output of three OLS regression estimates for each of the three equations in the system To save a considerable amount of space the output box above reports only the Granger causality tests At conventional significance levels dirm2 and drs Granger cause dirgdp dlrgdp does not Granger cause dirm2 and all variables Granger cause drs It is straightforward to obtain the impulse responses using the ERRORS instruction As indicated above we can obtain the impulse responses and variance decompositions using th
299. ual ACF 1 corrs spgraph done end unit Writing Your Own Procedures 219 5 Retrieving START and END entry values If you pass a series to a procedure you might not want the procedure to operate on the entire range of entries The INQUIRE instruction is specifically designed to allow the user to select the starting and ending values of any series passed to a procedure It is important to note that procedures are often designed to work with a variety of data sets As such it is most useful to write procedures that use integers as opposed to dates to represent the entries of a series You can use the INQUIRE instruction to obtain the defined range of a series as follows INQUIRE SERIES series name value value2 After execution value will contain the starting entry value of series name and value2 will contain the ending entry value To illustrate suppose you read in the data set MONEY DEM XLS and enter the following TABLE instruction Both series are quarterly running from 1959 1 to 2001 1 as such 1959 1 is entry 1 and 2001 1 is entry 169 Recall that the first two values of tb yr are NA Next enter inquire series rgdp v1 v2 inq series tblyr v3 v4 dis v1 v2 v3 v4 1 169 3 169 Hence v and v2 contain the first and last entries of rgdp respectively However tblyr 1 and tbl1yr 2 are NA as such v3 3 and v4 169 You can also use INQUIRE with the SEASONAL option to obtain the seasonal frequency
300. values of dirgdp The first instruction below creates the series thresh_test that will hold the potential thresholds Initially this series is SET equal to dlrgdp The ORDER instruction orders the series from lowest to highest Thresh test now contains the ordered values of dirgdp set thresh test dlrgdp order thresh test Now we can use each value of thresh test as a potential threshold We begin by equating thresh with the very first value of thresh_test compute thresh thresh test low IF Statements and Monte Carlo Experiments 131 Next we begin the loop For each value of i running from low to high we take the associated value of thresh test and use it as a potential threshold Inside the DO loop the program creates the series plus minus y1 plus y2 plus yl minus y2 minus and estimates a TAR model The residual sum of squares is compared to rss test If the resulting residual sum of squares exceeds this value two instructions in brackets are not executed the value of i is incremented by 1 and the loop is repeated However if rss is lower than rss test i e if the residual sum of squares from the current regression is lower than any from the prior regressions the bracketed instructions are executed The value of rss test is replaced by the value of 96rss and the value of thresh is equated to the thresh_test i e current value of the test threshold Once the loop is completed thresh will hold the value of the threshold tha
301. var frml e dim3 al dlm3 1 b1 temp 1 frml L temp e log var e 2 var com al 8 bl 0 var 0 01 max method simplex iters 4 L 3 max L 3 Variable Coeff Std Error T Stat Signif KKKKKKKK KK KKK KKK KKK KKK KKK KKK KKK KKK KKK KKK KKK KKK KKK KKK KKK KKK KKK kA Kk kA A XX o 1 Al Q9799 0 0125 78 12362 0 00000000 De aBT 0 0965 0 0653 1 47713 4 0 13964074 3 VAR 2 7033e 05 2 0527e 06 13 16948 0 00000000 40 Walter Enders 6 GARCH Models Suppose you want to estimate a simple regression model with an ARCH 1 error process Y Bo E Bix E where 2 v 0 0 7 and v WN O 1 Since v is white noise E 0 and E 1 7 h Qo 0 Hence the desired formula for the log likelihood of y can be written in the form log h log h The autocorrelation function of the residuals is not satisfactory for detecting ARCH errors Correlations measure linear association and ARCH errors manifest themselves in the autocorrelations of the squared residuals The Lagrange Multiplier LM test for ARCH disturbances has been proposed by Engle 1982 After you have estimated the most appropriate model for y save the residuals Suppose you have estimated the model lin y resids constant x Then obtain the square of the residuals and regress these squared residuals on a constant and on n lagged values of the squared residuals For example if n 4 set r2 resids 2 lin r2 cons
302. ve the variable aic min will be used to contain the smallest value of the AJC and the integer p opt will be used to contain the value of p yielding the lowest AIC However to allow for the case where the optimal lag length is zero the initial values of aic min and p opt are obtained from regressing d rgdp on a constant In the second line of the program the variables aic min and p hold the A C and lag length for this regression The first time 124 Walter Enders through the loop p 1 and the AR 1 model will be estimated The value of the resulting AJC will be compared to aic min 3 f the calculated AIC is less than aic min the two instructions in brackets will be executed If the optimal lag is zero the condition on the IF instruction below is true As such RATS displays the message An AR model is inappropriate The autocorrelations are The COR instruction then produces the first twelve autocorrelations of dirgdp and the associated Q statistics If p opt is greater than zero the ELSE block is executed Hence the autoregression using p opt lags is displayed along with the value of the AJC and p opt END IF instruction is used since the IF ELSE block is not within a compiled section of the program if p_opt 0 dis An AR model is inappropriate The autocorrelations are cor number 12 span 4 qstats dlrgdp else lin dlrgdp resids constant dlrgdp 1 to p opt compute aic nobs log rss 2 96nreg dis The
303. ve variables Within the DOFOR loop the WHILE END block determines the lag length for series i At the end of each WHILE loop the label of series i and the lag length is displayed If you want the AR p model for each series remove the comment label 7 e remove from the final LINREG instruction set price gdp rgdp set dlp log price log price 1 dofor i dlrgdp dlm3 drs drl dlp com lags 13 sign 5 while sign gt 0 05 com lags lags 1 Loops over Dates and Series 117 lin noprint i constant i 1 to lags exclude noprint i lags com sign signif 3 Note that END WHILE is removed since we are in a compiled section dis label series i Lag length lags lin 1 constant i1 1 to lags end dofor DLRGDP Lag length 1 DLM3 Lag length 1 DRS Lag length 11 DRL Lag length 7 DLP Lag length 3 Frequently Asked Questions 1 What would happen if condition was never true Answer Big problems One possibility is that looping continues indefinitely until you click the HALT icon on RATS Menu Bar The other possibility is that you produce an error that causes 1 RATS stop execution and display an error message or 2 an inadvertent condition that is true For example suppose you used the program above on a series that behaves as a white noise process so that none of the lags is significant at the 5 level Hence looping will continue until lags equals zero At this point the program regresses
304. vergence in 31 Iterations Final criterion was 0 0000050 lt 0 0000100 Quarterly Data From 1960 03 To 2001 01 Usable Observations 13 Function Value 330 04737868 Variable Coeff Sit Gi PISIS TIS DSi xem DX E ToS ToS TAL TAS Ta TAS EE THe TE TE TOE TAS CELERE EE EC EUER CHS TS TAL TN EOS RER kkk TE ESTOS TA TAD TAL AE TE TE LASS THE TAPAS OOF TS kkk kkk kk AO ONO CT SAST SSI MORO Z0S0S0899 SIS G7 o MEN OO OOZES 9 A1 6930927974 0 0663996430 10 43820 00000000 BETA 2087641734 0 0987429689 TEZ 03449664 BO 0015160176 0 0006060467 o DOIL ALS 01236732 Bl SUIEOZN HIE GIONE 0 03183551475 04985 00000000 sl eoo ONONIOSSESSSSAN E 942715 00000000 Chapter 2 VARs and Error Correction Models A vector autoregression VAR is a multivariate generalization of the single equation autoregressive model In the two variable case we can let the time path of the y be affected by current and past realizations of the z sequence and let the time path of the z sequence be affected by current and past realizations of the y sequence Consider the following 2 variable 1 lag VAR in standard form Vt dy t Gy aun te Zr dy FAY 022214 It is assumed that e1 and e are serially uncorrelated but the covariance Ee e need not be zero If the variances and covariance are time invariant we can write the variance covariance matrix as Y Ax Oi O 12 O21 On where Var e Oj and Cov e 1 2 O12 021 Note t
305. vfi 2 We want to loop over the series dlm3 dlrgdp dr and dlp Since the series numbers bear no particular relationship to each other it is simplest to use a DOFOR loop However we need to use a counter j to indicate how many loops have been completed Before entering the DOFOR loop the value of j is initialized to zero and is increased by one every pass through the DOFOR loop The first time through the loop i equals the integer representing the series dm3 and j 1 Consider com j 0 dofor i dlm3 dlrgdp drs dlp comj j 1 The next instruction below creates the header Panel 1 Time path of dlm3 To understand note that the variable called header is necessarily a string variable The string is equal to Panel plus the value of j 1 plus the string Time Path of plus the label of dlm3 i e the label of series i The GRAPH instruction creates a graph of series i using the header created in the previous COMPUTE instruction After the first pass i becomes the integer value of dlrgdp and j 2 Next the header Panel 2 Time Path of dlrgdp is created and the GRAPH instruction creates a graph of series dlrgdp using this header The process continues until i has taken on the values corresponding to dr and dlp The final instruction SPGRAPH done displays the output Loops over Dates and Series 111 com header Panel j Time Path of l i gra hea header 1 i end doi spgraph do
306. will be the series label appended with an L for the logarithmic transformation D for the first difference or G for the growth rate Notice that the TYPE instruction listed below declares the parameter y to be a series The instructions in the MENU END MENU block create the dialog box shown above 234 Walter Enders procedure transform y type series y menu Select the Transformation choice Logarithmic sta noprint fractiles y if minimum gt 0 com a L label y set s a log y if minimum le 0 mes style alert I CAN TAKE THE LOG OF POSITIVE NUMBERS ONLY choice First Difference com a D label y set s a 2 y y 1 choice Growth Rate com a G label y set os a y y 1 1 end menu graph header Time Path of a 1 s a end transform If Logarithmic is selected a simple check is performed before the routine attempts to create the log of series The STATISTICS instruction is used to obtain the minimum value of the series If this value is negative an ALERT is displayed informing the user that it is not possible to obtain the log of a negative number Only if the smallest value of series is positive will the procedure create the new series The label attached to the newly created series is L series label Similarly if First Difference 1s selected the instructions in the second CHOICE block are executed The first difference of series is crea
307. write the values of the variable A local variable is one that is used only in the procedure that defines it Thus changing the value of a local variable changes its value only within the procedure There is another reason to use local variables Suppose a procedure includes a series that is not defined as a local series If you attempt to compile the procedure prior to the ALLOCATE instruction RATS will display the error message Writing Your Own Procedures 215 SRL ALLOCATE Instruction Needed Before Series or Equations Can Be Used The point is that RATS will have no way of knowing the length of the series appearing in the procedure By declaring the variable as a local variable the procedure can be compiled before the ALLOCATE instruction is encountered To define a variable as local use LOCAL SERIES series name or LOCAL INTEGER name Thus we want to include the following line in UNIT SRC LOCAL SERIES dy The precedence of statements is that DECLARE instructions should precede TYPE LOCAL and OPTION instructions All of these must precede the executable instructions Hence we want the first three lines of UNIT SRC to be procedure unit y type series y local series dy Notice that we do not need to define y as a local variable since y is a parameter 1 e the symbol y acts only as a placeholder 4 2 Adding Options To turn UNIT SRC into a procedure that you might want to use in your own research we need to m
308. ws how to obtain a super consistent estimate of the threshold parameter For a TAR model the procedure is to order the observations from smallest to largest such that y ey ey ley For each value of y let t y set the Heaviside indicator according to this potential threshold and estimate a TAR model The regression equation with the smallest residual sum of squares contains the consistent estimate of the threshold In practice the highest and lowest 15 of the y values are excluded from the grid search so as to ensure an adequate number of observations on each side of the threshold We can conduct this procedure using the following program segment Consider compute low 1959 2 fix 15 nobs high 2001 1 fix 15 nobs This first instruction creates two variables low is equal to the integer corresponding to the 15 of the observations following 1959 2 and high is equal to the integer corresponding to the last observation less 1596 of the total number of observations The first instruction below creates a variable rss test that will be used to hold the residual sum of squares for the best fitting model This value is initially set to be higher than any possible value of the estimated residual sum of squares The second instruction creates the series that will hold the calculated residual sum of squares from each regression estimated compute rss test 1000000 0 set rss 0 Next we need to create a series for the ordered
309. x is query prompt Your message variable list 230 Walter Enders where variable list The list of variables whose values are input by the user The individual variables can be any combination of integer real string or label variables Note that QUERY uses a dialog box similar to that shown above The user s response is entered such that each value is separated by commas or spaces Notice that QUERY does not display a separate line for each value to be entered As such I recommend using a separate QUERY instruction for each value to be input Also the TYPE of variable to be input e g real string or integer must be specified prior to the QUERY instruction COMPUTE or DECLARE for example can be used to determine the variable TYPE Examples 1 Suppose you reach the end of LAGLENGTH SRC and see the message DO NOT USE LAGS You might be concerned that the SBC selected a model that was too parsimonious You could insert a MESSAGEBOX to prompt the user for further instructions Consider the following modifications to the procedure a if bestlag EQ 0 mes style alert DO NOT USE LAGS This modification will inform the user of the problem by halting execution of the procedure and display an ALERT b if bestlag EQ 0 mes style YESNO status status Do you want to enter the lag length if status eq 1 query prompt Enter the lag length bestlag The procedure will display a MESSAGEBOX with YES and NO buttons a
310. y The usual form of the Dickey Fuller test is to estimate the AR 1 equation in the form Ay Qo Pyr 1 i As such lines 7 and 8 estimate dy on a constant and y 1 Since p o 1 we want to know whether it is possible to reject the null hypothesis p 0 The sampling distribution of the t statistic for the null hypothesis p 0 is stored as entry i of tstat Once the 10000 replications of this Monte Carlo experiment are completed statistics fractiles tstat produces the distribution of the r statistics all 100 set tstat 1 10000 0 set y 0 0 compute y 1 ran 1 do i 2 1 10000 set y 2 1002 y 1 ran 1 diff y dy linreg noprint dy 100 constant y 1 compute tstat i tstats 2 end do i statistics fractiles tstat Statistics on Series TSTAT Observations 10000 Sample Mean i 41795979 Variance 0 5 32 512 7 Standard Error 0 8515412362 SE of Sample Mean 0 008515 ESIC ie LS aL E910 5 SESS 7 Signif Level Mean 0 00000000 Skewness SIRO ORE Signif Level Sk 0 00000000 Kurtosis 07 S3538 Signif Level Ku 0 00000000 Jarque Bera TEAL o ALL S Signif Level JB 0 00000000 OF OR 0 0 Minimum 8214558640 Maximum DIES NROVNQSIQSIS 01 ile 4919869782 99 Sile 0 6158486194 05 ile 8861478731 95 Sile 200669050993 10 ile 5748048837 90 ile MAZ ATO ISS 25 Sile 20199 8 970 295 75 Sile 01037 7954 Median 5835459078 Your output will look a bit different from mine since we have not

RATS Programming Manual, By Walter Enders

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