Grocer 1.0, an Econometric Toolbox for Scilab: an
Contents
1. O 0 2674627 0 0213193 0 0623181 0 0493728 22 0 0470486 0 0430857 0 0397769 0 0366611 0 0338010 A one off increase of ly is shown to increase lm1 by 0 the first period you can easily check that it is in fact the case since ly enters with a lag in the equation 0 2674627 the second period this is minus the sum of the coefficients of the variable delts lagts 1 lm1 lp ly and of the variable lagts 1 lm1 lp ly and so on 2674627 0213193 0623181 0493728 0470486 0430857 0397769 0366611 0338010 O O OO OO OOG If you want to calculate the contribution of the variable ly to the growth rate of Im1 then it can be done as follows gt ly_mainf mainf 1 coef 2 coef 4 coef 2 0 coef 2 coef 4 coef 2 200 gt ly_contrib contrib delts ly ly_mainf And the result is 1964q3 0 1964q4 0 0002158 1965q1 0 0061336 1965q2 0 0029902 1965q3 0 0029388 TOnce again we have performed the estimation of the equation but it can be made once and the results saved in a results typed list as done here in rhe note also that by the option noprint in ols we have chosen not to print the estimation results we have recovered the coefficients in the vector coef and asked to calculate only the 10 first coefficients of the impulse function 23 1965q4 0 0030687 1966q1 0 0040551 1966q2 0 0021385 1966q3 0 0023299 1966q4 0 0034330 19672. scilab 3 0 Copyright c 1989 2004 Consortium Scilab INRIA ENPC Startup execution loading initial environment loading Grocer 1 0 Copyright Eric Dubois et al 2004 gt Scilab commands are executed at the prompt gt In what follows every time an example will be presented all the necessary commands will be exposed as they are run in Scilab The results will be shown as they appear in Scilab window with the font verbatim For example showing how to add the numbers 4 and 5 in Scilab should be done as follows gt 4 5 And the result should be presented as follows ans 2 2 The time series type Grocer can deal with matrices or vectors of data but also with time series Time series can be created through the importation of an Excel database by Grocer function impecx2bd that can also be used to import vectors or matrices by any operation on other time series or with the Grocer command reshape which transforms a vector into a time series Any frequency is allowed but annual quarterly and monthly time series have a specific more user friendly representation than other frequencies Standard operations addition subtraction multiplication and division and functions log arithm exponential sin cosin can be performed on time series as if they were vectors For instance tsl ts2 denotes the addition of the 2 time series ts1 and ts2 NA Non Available values are authorized and the 2 time series ts1 and ts
3. gt delts ly delts lagts 1 ly delts lagts 2 ly delts lagts 3 ly delts lagts 4 ly gt delts delts lp delts delts lagts 1 1lp delts delts lagts 2 1p gt delts delts lagts 3 1p delts delts lagts 4 lp gt delts rnet delts lagts 1 rnet delts lagts 2 rnet gt delts lagts 3 rnet delts lagts 4 rnet gt cte Grocer results are then final model ending reason only one model selected by stage 1 Figure 1 estimation and reduction of the initial model General model with N potential exogenous variables size n woe At least one is rejected adjust the size of the corresponding tests Sp cification tests accepted OK a Elimination of the least significant variables Fisher test with size Model with No variables with p non significant ones 10 Drop the first non significant variable size a y Sp cification tests size n v rejected gt stop and don t keep the model All remaining variables significant gt stop and store the model Drop the 2 non significant variable size a Idem Drop the p th non significant variable size a Idem 2 t At least one variable is not significant size a Drop the least significant variable that rema
4. Available specification tests include the Lagrange multiplier autocorrelation test Grocer function arl1m the ARCH test Grocer function archz White heteroskedascity test Grocer function white as well as a function allowing any heteroskedascity test of Breusch and Pagan type Grocer func tion bpagan in sample and out of sample Chow stability tests Grocer functions chowtest and predfail Brown Durbin and Evans cusum backward and forward stability tests Grocer function cusumb and cusumf Ramsey RESET test of linearity Grocer function reset Jarque and Bera and Doornik and Hansen normality tests Grocer functions jbnorm and doornhans Grocer provides also a function performing Wald restriction tests Grocer function waldf As an example performing the Lagrange multiplier autocorrelation test on the equation estimated above is done that way gt arlm rhe 4 Lagrange multiplier 1 4 autocorrelation test chi2 4 7 1563181 p value 0 1278545 Lagrange multiplier 1 4 autocorrelation test F 4 91 1 941783 p value 0 1102067 The reader can check that it is exactly the result presented by Hendry and Ericsson 1991 themselves 3 3 The function automatic Grocer contains the most recent developments of the LSE methodology that is a function automatic that mimics Hendry and Krolzig package pc gets 3 This package aims at replicating automatically the approach followed by Hendry and his fellows when estimating a mod
5. Grocer provides four among the most popular unit root tests the augmented Dickey Fuller test function adf the Phillips Perron test function phil_perr the Schmidt Phillips test function schmiphi and the Kwiatkowski Phillips Schmidt Shin stationarity test func tion kpss Two cointegration tests are currently available the corrected Augmented Dickey Fuller test with its Phillips Perron variant and the Johansen cointegration test They are performed by the functions cadf and johansen 6 5 Filters Grocer provides three usual filters the Hodrick Prescott filter npfilter the Baxter King one bkfilter and the more recent Christiano Fitzgerald one cffilter 7 Conclusion This paper has proposed an insight into Grocer functions Once again the interested reader can have a more precise view of Grocer with a look at the user manual the on line help embodied in Grocer package and even better by using it Grocer is due to evolve regularly and some new features are already underway A few procedures existing in some commercial packages and some more rare but nevertheless useful methods notably in the bayesian field remain to be implemented The matrix oriented nature of Scilab will make it easy to stay at the forefront of the econometric science And the great similarity between Scilab and Matlab or Gauss will help as well Acknowledgements I wish to thank the Scilab team for their precious advices James Le Sage for havin
6. it contains most standard econometric procedures And third it has a few more or less common features that can make its use fruitful it implements the so called LSE methodology as regards univariate econometric estimation in particular Grocer implements the more recent advances of this methodology namely a pc gets like program called automatic in grocer it contains a host of multivariate methods from standard old fashioned simultaneous equations methods to many up to date VAR methods such as VARMA Vector Error Correc tion Models Bayesian Vector Autoregressions it proposes a calculation of the contributions of exogenous variables to an endogenous one for any dynamic equation The aim of this paper is to provide the reader with an insight into Grocer most interesting features from an econometrician point of view The interested reader can find a detailed de scription of all econometric functions available in Grocer in Dubois 2004a and explanations about their implementation in Dubois 2004b Part 2 of this paper presents Grocer basic features Part 3 discusses how the LSE method ology has been implemented Part 4 presents the multivariate methods Part 5 explains the contributions method Part 6 presents quickly the other functions available in Grocer Part 7 concludes 2 Grocer basic features Grocer is basically a toolbox for Scilab So a few words about Scilab are necessary the user can find a sligthly more detailed descrip
7. 1 0 6887091 0 1363550 5 050854 AR1 2 2 1 2679083 0 1098543 11 541726 AR2 1 1 0 5941061 0 1402932 4 2347476 AR2 2 2 0 5592538 0 0921302 6 0702529 AR3 1 1 0 0681945 0 1171420 0 5821526 AR4 1 1 0 2815736 0 0855952 3 2895965 MA1 1 1 0 2965995 0 1606131 1 8466709 MA1 2 1 0 602702 0 0803646 7 4995981 MA1 1 2 0 8641752 0 1387191 6 2296768 MA1 2 2 0 8351471 0 1529343 5 4608236 V 1 1 0 2492625 0 0225682 11 044879 V 2 1 0 0424334 0 0147947 2 8681376 V 2 2 0 2013021 0 0182819 11 011036 5 Contributions Grocer proposes functions to calculate the contributions of the exogenous variables to an endogenous one that is the part of the current evolution of the endogenous variable that can be explained by the evolutions of each variable current and past Part 5 1 exposes the theoretical background and 5 2 presents an example 5 1 Theoretical background Assume that an endogenous variable y is described by the following equation A L yz By L ai By L ree ut With L the lag operator Lyt ye 1 Then one has 20 ut Ba Pee Ba Ayu and A yt FRA 21 BRA us aay A we This formulation shows how the variation of the endogenous variable can be explained by the contribution of the exogenous variables 1 amp t and of the residuals with contribution of variable j defined as Bi L A L Vit The calculation of these contributions involves the infinite Moving Average deco
8. VAR and its ecm form VAR forecasting Lastly Grocer contains a program performing ARMA and VARMA estimations 4 1 Old fashionned simultaneous equation methods Grocer Function sur is rather general it can be applied to any system of equations provided it is linear in its coefficients So you can impose constraints to the coefficients in the system provided they remain linear Take as an example the sur estimation on Grunfeld s investment data presented in Greene s 2003 textbook in chapter 14 eql igm al fem a2 cgm a3 eq2 ich a4 fch a5 cch a6 eq3 ige a7 fge a8 cge a9 eq4 iwest al0 fwest all cwest al2 eqd iuss al3 fuss al4 cuss al5 Then the sur estimation is performed by Grocer instruction gt r sur eq1 eq2 eq3 eq4 eq5 If you want to impose for instance the constraint that all constants are equal for sure a meaningless constraint on economic grounds then you should write eqi igm al fgmta2 cgmta3 eq2 ich a4 fchtad5 cchta3 eq3 ige a6 fgeta7 cgeta3 5These programs as well as most univariate econometric methods all multicollienarity tests and outlier diag nostics the Kalman filter some unit root and cointegration tests rest mainly on the translation and adaptation of parts of LeSage Matlab toolbox see LeSage 1999 14 eq4 iwest a8 fwestta9 cwestta3 eqo iuss al0 fusstall cussta3 r sur eqi1 eq2 eq3 eq4 eq5 Functions per
9. able to impose her own prior e Error Correction estimation ecm and a bayesian version becm in each case the cointe gration relationships are estimated by a Johansen estimation see section 6 the user can either impose her own number of cointegration relationships or let the program determine the number of cointegration relationships at a prespecified level 1 5 or 10 e lastly the user can forecast any of these VAR with the function varf As an example here is the result of the estimation of the bayesian error correction model presented in LeSage 1999 chapter 5 pp 128 129 First load the database gt load SCI macros grocer db datajpl dat 15 Then perform the estimation gt results becm 2 0 1 0 5 1 endo illinos indiana kentucky michigan gt ohio pennsyvlania tennesse westvirginia Johansen estimation results for variables illinos indiana kentucky michigan ohio pennsyvlania tennesse westvirginia time order 0 of lags 2 NULL Trace Statistic Crit 90 Crit 95 Crit 99 r lt 0 214 38998 153 6341 159 529 171 0905 r lt 1 141 48161 120 3673 125 6185 135 9825 r lt 2 90 363179 91 109 95 7542 104 9637 r lt 3 61 554975 65 8202 69 8189 77 8202 r lt 4 37 103415 44 4929 47 8545 54 6815 r lt 5 21 069871 27 0669 29 7961 35 4628 r lt 6 10 605086 13 4294 15 4943 19 9349 r lt 7 3 1924592 2 7055 3 8415 6 6349 conclusions from the trace statistics at a 1
10. extrap of another one 3 The application of the LSE methodology ordinary least squa res specification tests and the pc gets like function automatic Grocer is oriented towards the implementation of the LSE methodology Grocer contains many specification tests part 3 2 and implements the pc gets methodology that represents the most thorough expression of this methodology part 3 3 Before that a short presentation of ols the function performing ordinary least squares in Grocer is required part 3 1 3 1 Ordinary least squares There are several functions performing ordinary least squares in Grocer that are more or less user friendly and complete The most user friendly and complete one is ols As any high level Grocer econometric function ols can be applied to various data types matrices vectors time series as well as strings Two kinds of strings are allowed first the strings const or cte that avoid the user to create the constant vector or time series relevant to her problem second the name of an existing variable between quotes The variable 1m1 which is the log of the monetary aggregate m1 can for instance be entered as such e g 1m1 or between quotes 1m1 In the last case Grocer function is able to display the names of its inputs For instance estimating Hendry and Erisson 1991 equation 6 can be performed as follows 1The string cte that is an abbreviation of the French world const
11. for the interface with Excel not presented here and many standard functions that are now listed 6 1 Univariate econometric procedures Besides ols there are numerous functions applying econometric methods to a single equation model instrumental variables function iv least absolute deviations regression function lad non linear least squares function nls logit function logit probit function probit tobit function tobit Cochrane Orcutt ols function olsc and maximum likelihood ols regression for AR1 errors function olsar1 ols with t distributed errors function olst Hoerl Kennard Ridge Regression function ridge Huber Ramsay Andrew or Tukey robust regression us ing iteratively reweighted least squares function robust Theil Goldberger mixed estimation function theil A function performing estimation of a linear model with Garch p q residuals is also available garch 6 2 The Kalman filter A function performing a Kalman filter estimation function kalman is available An appli cation of the Kalman filter is also explicitly programmed the time varying parameters function tvp 6 3 Multicollinearity tests and outlier diagnostic Grocer provides Belsey Kuh and Welsch 1980 standard multicollineraity tests through the function bkw Outlier diagnostic tests dfbeta dffits studentized residuals hat matrix diagonals are implemented by the function dfbeta 26 6 4 Unit roots and cointegration
12. 0 level there are 8 cointegration relation s at a 54 level there are 2 cointegration relation s at a 1 level there are 2 cointegration relation s NULL Max Eigenvalues Statistic Crit 90 Crit 95 Crit 99 1 lt 0 72 908372 49 2855 52 3622 58 6634 1 lt 1 51 118434 43 2947 46 2299 52 3069 1 lt 2 28 808204 37 2786 40 0763 45 8662 1 lt 3 24 45156 31 2379 33 8777 39 3693 l1 lt 4 16 033544 25 1236 27 5858 32 7172 1 lt 5 10 464785 18 8928 21 1314 25 865 1 lt 6 7 4126267 12 2971 14 2639 18 52 1 lt 7 3 1924592 2 7055 3 8415 6 6349 conclusions from the maximal eigenvalues statistics at a 10 level there are 8 cointegration relation s at a 54 level there are 2 cointegration relation s at a 1 level there are 1 cointegration relation s 16 Cointegrating vectors from johansen estimation variable vector 1 vector 2 illinos 0 0131635 0 0099366 indiana 0 3050443 0 0399855 kentucky 0 0525625 0 0273951 michigan 0 0417678 0 1871467 ohio 0 0176776 0 0199591 pennsyvlania 0 0526335 0 0250484 tennesse 0 0823386 0 2737890 westvirginia 0 0201478 0 0216022 x becm estimation results for variables del illinos del indiana del kentucky del michigan del ohio del pennsyvlania del tennesse and del westvirginia PRIOR hyperparameters tightness 0 1 decay 1 symetric weights based on 0 5 estimation results for dependent variable del illinos number of observations 170 nu
13. 2 need not cover the same period but need to have the same frequency in that case the new time series is defined over the time period common to both time series It should be noted that the time period is an attribute of the time series itself not of the database where it possibly comes from provided that they have the same frequency time series from different databases can be added subtracted without any difficulty Time series allow the user to apply all econometric procedures to data with different lengths without having to transform them manually The user can choose the time period over which to perform the estimation this is done by setting the time bounds with the Grocer command bounds For instance setting the bounds from the 3rd month of 1990 to the 10th month of 2003 involves the following Grocer command bounds 1990m3 2003m10 Bounds can also be discontinuous such as bounds 1990m3 1995m12 2003m10 Lastly the user can choose not to provide her own time bounds in that case the econometric procedure will choose the greatest possible time bounds starting from the first non NA values A few specific functions have been programmed to lag function lagts differentiate func tion delts time series compute the growth rate growthr of a time series aggregate monthly time series to quarterly m2q or quarterly to annual m2a extrapolate a time series by the value overlay or the growth rate
14. 9 R 0 1970140 adjusted R 0 1012939 standard error of the regression 17 478053 sum of squared residuals 341 89747 DW O 2 0311438 variable coeff t de student p value del illinos 1 0 1095202 0 2204470 0 8257891 del indiana 1 1 5753669 1 591901 0 1132757 del kentucky 1 0 3796700 1 4551827 0 1474736 del michigan 1 0 0409345 0 1170027 0 9069969 del ohio 1 0 7621534 1 4819112 0 1402261 del pennsyvlania 1 0 5749257 2 2692817 0 0245155 del tennesse 1 0 0356746 0 0352836 0 9718952 del westvirginia 1 0 0166022 0 1428371 0 8865890 del illinos 2 0 1730335 0 3750635 0 7080835 del indiana 2 1 3777092 1 420519 0 1572990 del kentucky 2 0 1541704 0 6121892 0 5412353 del michigan 2 0 2511154 0 7332133 0 4644442 del ohio 2 0 3895893 0 8455066 0 3990235 del pennsyvlania 2 0 0116675 0 0461857 0 9632167 del tennesse 2 0 6676482 0 6941234 0 4885578 del westvirginia 2 0 1099663 1 0124952 0 3127487 lag 1 of coint vec 1 2 2584975 1 7072336 0 0896151 18 lag 1 of coint vec 2 2 3590674 1 7774235 0 0772971 cte 8 0747728 0 4674689 0 6407666 4 3 VARMA Grocer also provides ARMA and VARMA they are implemented by the same function varma As an example here is how one can estimate the model proposed by Jenkins and Alavi 1981 to describe the relationship between the minks and the muskrats skins traded by the Hudson s Bay Company between 1948 and 1909 F
15. Grocer 1 0 an Econometric Toolbox for Scilab an Econometrician Point of View Eric Dubois Direction G n rale du Tr sor et de la Politique Economique Tel doc 679 139 rue de Bercy 75012 PARIS e mail Grocer toolbox free fr Abstract Grocer is an econometric toolbox for Scilab a free opensource matrix oriented toolbox similar to Matlab and Gauss It contains more than 50 econometric different methods with many variants and extensions Most standard econometric tools are available Grocer contains also two original econometric tools a function allowing the automatic estimation of the true model starting from a more general and bigger one a method that provides the most thorough expression of the so called LSE econometric methodology a function calculating the contributions of exogenous variables to an endogenous one JEL 870 c130 220 c320 keywords econometric software estimation general to specific contributions 1 Introduction Grocer available at http dubois ensae net grocer html is an econometric toolbox for Sci lab a free opensource matrix oriented software very similar to Matlab and Gauss and available at http scilabsoft inria fr It is built both upon the translation of existing programs in Matlab or Gauss and upon native programs Grocer has some properties that can make it appealing to an applied econometrician First it is free and opensource and therefore portable and very flexible Second
16. ante was first implemented to maintain compatibility with earlier versions of Grocer used by some forerunners both abbreviations are available in the 1 0 version Some other usual variables such as trends could also be treated this way this is left for further releases First load the corresponding database that is given with Grocer package that contains the time series lm1 ly lp and rnet gt load SCI macros grocer db bdhenderic dat Then set the time bounds gt bounds 1964q3 1989q2 And lastly perform the estimation gt rhe ols delts 1m1 lp delts 1p delts lagts 1 1m1 lp ly rnet lagts 1 1m1 lp ly const The result is then ols estimation results for dependent variable delts 1m1 lp estimation period 1964q3 1989q2 number of observations 100 number of variables 5 R 0 7616185 adjusted R 0 7515814 Overall F test F 4 95 75 880204 p value 0 standard error of the regression 0 0131293 sum of squared residuals 0 0163761 DW 0 2 1774376 Belsley Kuh Welsch Condition index 114 variable coeff t statistic p value delts 1p 0 6870384 5 4783422 3 509D 07 delts lagts 1 1lm1 lp ly 0 1746071 3 0101342 0 0033444 rnet 0 6296264 10 46405 0 lagts 1 1m1i lp ly 0 0928556 10 873398 0 const 0 0234367 5 818553 7 987D 08 x 3 2 Specification tests In the spirit of the LSE methodology Grocer can compute many specification tests
17. coeff t statistic p value delts delts lagts 2 1p 0 3821030 2 4121215 0 0178669 delts lagts 3 1lm1 lp ly 0 1884416 2 8830509 0 0049142 delts lagts 1m1 lp ly 0 2149036 3 6961388 0 0003737 delts rnet 0 3198695 2 680324 0 0087312 delts lagts 3 ly 0 3393305 3 0482243 0 0030136 delts 1p 0 7653748 6 2359835 1 386D 08 cte 0 0300893 6 9787739 4 690D 10 lagts 1m1 lp ly 0 1104120 10 974 0 rnet 0 7746108 10 886597 0 tests results FAK test test value p value Chow pred fail 50 0 4727966 0 9939980 Chow pred fail 90 0 8364555 0 5951258 Doornik amp Hansen 2 248449 0 3249043 AR 1 4 0 6915387 0 5997762 hetero x_squared 1 1282701 0 3459758 One can note that the program finds new significant variables besides the ones found by Hendry and Ericsson variables delts delts lagts 2 lp delts lagts 3 lm1 lp ly delts delts lagts 2 lp 13 and delts lagts 3 ly and that the standard error of the regression is lower which proves the usefulness of the program whether these variables are economically meaningful is another ques tion beyond the scope of this paper 4 Multivariate methods Grocer contains the old simultaneous equations estimation methods Seemingly Unrelated Regressions two and three stages least squares as well as the more recent VAR ones As regards the VAR methodology the following procedures are currently available VAR estimation impulse response calculation ecm estimation bayesian
18. ctober available at http www spatial econometrics com Mrkaic M 2001 Scilab as an Econometric Programming System Journal of Applied Econometrics vol 16 n 4 July August pp 553 559 Scilab 1993 Introduction to Scilab INRIA available at http scilabsoft inria fr product index_product php page Terceiro J J M Casals M Jerez G R Serrano and S Sotoca 2000 Time Series Analysis using MATLAB Including a complete MATLAB Toolbox available at http www ucm es info icae e4 e4download htm 28
19. el As far as I know Grocer is with pc gets the only econometric package today to provide such a program 3 3 1 Theoretical background Suppose that a researcher wants to recover a data generating process DGP starting from a data set with some variables that are relevant and other not Let the DGP be E y 2t T X b where 2 is a 1 x p vector of variables x xj and ba p x 1 vector of parameters The approach proposed by Hendry and Krozlig 2000 based upon Hoover and Perez 1999 is the following one see also the presentation in figure 1 Let Z 21 2n be a set of variables which are potentially relevant with z 1 k belonging to Z and let E yt zt zt x B be the corresponding postulated general model 3Informations about pc gets are available at http www doornik com pcgive pcgets index html Let a n p y be 4 real numbers between 0 and 1 these are significance levels usually they should be equal to 0 05 or 0 01 except for 7 Let UY be a real number greater than 1 The algorithm consists in the following potential 5 steps the algorithm stopping when it has found the final model First step Estimation of the general model i The general model is estimated which leads to a set of estimated coefficients If all variables are significant at the a level then the general model is the final model ii If at least one coefficient is non significant at the pre specified a level then perform k si
20. forming two stage function twosls and three stage function threes1s least squares have the same syntax Note that the function twosls is here reserved to two stage least squares in a system of equations two stage least squares applied on a single equation is performed through function iv As with sur the only compulsory arguments are the texts of the equations of the system In that case the program determines what are the coefficients and what are in each equation the endogenous variables This imposes however constraints on the way equations are written as with sur the coefficients must be named al a2 an without any discontinuity and the names of the endogenous variables must be exactly equal to the left hand sides of the equations The command below illustrates the simplest way of performing a Two Stage Least Squares es timation see Dubois 2004a for the results of estimation It is contained in function twosls_d which belongs to the library macros grocer multi in the library where the user has chosen to install Scilab gt rt twosls yl atb x1 y2 d e y1 x2 f x2 coef a b d e f y y y 4 2 VAR methods Grocer provides several tools to perform VAR estimations e VAR estimation itself function var and its low level counterpart var1 e Impulse response functions function irf calculated either with an asymptotic formula or with a Monte Carlo simulation Bayesian VARs function bvar the user being
21. g Monte Carlo simulations show that this method leads indeed to very satisfactory results the average inclusion rate of a non relevant variable can be set at a low level while retaining significant power 3 3 2 An example Hendry and Ericsson 1991 revisited To show the power of this method we plug Hendry and Ericsson 1991 equation in a much bigger set of potential explanatory variables we add 2 to 4 lags of the variable A m1 ly Ip1 variables called here delts lagts 2 lm1 lp ly delts lagts 3 lm1 lp ly delts lagts 4 1m1 lp ly 0 to 4 lags of variable A ly variables called here delts ly to delts lagts 4 ly 0 to 4 lags of the variable A A Ip and 1 to 4 lags of the variable A rnet variables called here delts rnet to delts lagts 4 rnet this seems a sensible parametrisation of a set containing 0 to 4 lags of every exogenous variable the log of GDP ly the inflation rate A lp the interest rate rnet and lags of the log of m1 Im1 Load the database again gt load SCI macros grocer db bdhenderic dat Set the same time bounds gt bounds 1964q3 1989q2 And use the function automatic with until 4 lags of each regressor r1i automatic delts 1m1 lp lagts 1mi lp ly delts 1p rnet delts lagts 1lm1 lp ly delts lagts 2 1m1 lp ly delts lagts 3 1lm1 lp ly delts lagts 4 1lm1 lp ly
22. g provided the basis of so many grocer functions and Emmanuel Michaux for his faithful use and testing of Grocer The usual disclaimer applies 27 References Dubois E 2004a Grocer 1 0 An Econometric Toolbox For Scilab user manual available at http dubois ensae net grocer html Dubois E 2004b Grocer 1 0 an Econometric Toolbox For Scilab a Scilab point of view presented at the first Scilab International conference 2 amp 3 of november Greene W 2003 Econmetric Analysis fifth edition Prentice Hall New Jersey Hendry D F and N R Ericsson 1991 Modelling the demand for narrow money in the United Kingdom and the United States European Economic Review p833 886 Hendry D F and H M Krozlig 1999 Improving on data mining reconsidered by K D Hoover and S J Perez Econometrics Journal n 2 pp 41 58 Hendry D F and H M Krozlig 2001 Computer Automation of General to Specific Model Se lection Procedures Journal of Economic Dynamics and Control 25 6 7 pp 831 866 Hoover K D and S J Perez 1999 Data mining reconsidered a general to specific approach to specification search Econometrics Journal n 2 pp 167 191 Jenkins G M and A S Alavi 1981 Some aspects of Modelling and Forecasting Multivariate Time Series Journal of Time Series Analysis vol 2 n 1 pp 1 47 LeSage J 1999 Applied Econometrics using MATLAB University of Toledo O
23. gnificance tests If these specification tests do not all pass the pre specified level 4 then adjust the significance level of the corresponding tests by a factor Y the fact that at least one of the specification test is rejected should lead the user to add other variables to the information set but this is clearly a task that the computer is not able to do so the process goes on but the significance level is adjusted to avoid rejecting too many subsequent models if not all Let then s s 87 be the corresponding vector of significance levels s n if the corresponding test has been successfully passed s 7 if the corresponding test has not been successfully passed Second step Elimination of globally insignificant variables programmed in func tion auto_stage0 i Range all from the lowest absolute value of Student s t to the greatest Set for initialisation sake j 1 F Oands s ii while F lt Y and s lt s then estimate the model with the j least significant variables withdrawn calculate the Fisher test F of this model against the general model calculate the vector s of p values of the k specification tests iii call the second step model the last model that verifies F lt Y and s lt s Third step Multiple reduction paths programmed in function auto_stage1 For all insignificant coefficients at level a of the second step model i Drop the corresponding variable from the list of exogenous variable
24. ins size a 4 Specification tests size n i t dropped variable l Re introduce the 11 Number of selected models of h The initial model is the final one All models are rejected gt the union model is the final model This is the final model Construction of the union model Ad Only one model is accepted against the union model this is the final The selected model is then union model All models are rejected gt the new union model is the final model model umber of selected models v This is the final model Only one model is accepted against the union model gt this is the final model 12 Several models are accepted against the union model gt restart the process with the union model as a new general model Construction of the new union model Several models are accepted against the union model gt select the final model by the mean of an information criterion AIC BIC ols estimation results for dependent variable delts 1m1 lp estimation period 1964q3 1989q2 number of observations 100 number of variables 9 R 0 7941498 adjusted R 0 7760531 Overall F test F 8 91 43 883636 p value 0 standard error of the regression 0 0124659 sum of squared residuals DW O 2 1391056 0 0141413 Belsley Kuh Welsch Condition index 166 variable
25. irst load the Matlab data base gt mtlb_load SCI macros grocer db mink dat Then transform the data gt z2 mink 2 mean mink 2 gt z z1 z2 2 62 gt muskrat zi1 gt mink z2 2 62 One of the model estimated by Jenkins and Alavi takes the following form 1 11 1 L 21 1 L7 631 1 L3 641 1 L4 0 muskrat 7 0 1 dlo2 L 211 L7 mink E 1 614 1 L 611 2 L Elt 0121 L 1 O12 2 L 24 Its estimation in Grocer involves the definition of the matrices phil to phi4 theta and sigma This is done by the following instructions where the sign in the matrices indicates that the coefficients are constrained to the given value d gt phil 70 5 07 47 07 707 gt phi2 0 0 0 0 gt phi3 0 0 0 2 gt phi4 0 0 0 gt theta 0 0 0 0 gt sigma 0 0 0 0 2 3 07 07 This function rests on a translation and adaptation of the Matlab package E4 developed by Terceiro et al 2000 19 gt result varma muskrat mink phil phi2 phi3 phi4 theta sigma 1 And here is the result given by Grocer xxkkkkkkkkkkkkk VARMA estimation Results for model k x Y e with muskrat Y mink V e V Log likelihood 9 0059411 Information criteria AIC 0 1309527 BIC 0 5808111 Parameter Estimate Std Dev t test AR1 1
26. le while the underlying equation is estimated on the quarterly logarithm of the variable The corresponding first order approximation is the following one 4 4 contrib 4 wA 1 kcontribi A 1 k Wa kcontribi Ak k 2 k 1 21 4 4 a II exp yA 1 j gt II exp yA j j 2 Opel k 1 j 1 WA 1 i OYA 1 4 4 k 1 X exp ya 13 k 1 j 1 4 a 4 k ezp va X J expan X TT erplua j 2 katja k i j 2 4 i 2 SJ clon k 1 j 1 YA O Oyama l _ Wie eep ya 13 Xk Fan erplya WAi Oyay 1 5 Nes exp yA 1 with YA j contribution of the logarithm of variable Y in quarter j of year A Y A level of variable A for year A contrib a contribution of variable i in quarter k of year A contrib 4 contribution of variable i in year A 5 2 An example As an example take again Hendry and Ericsson equation presented in section 3 1 This is a dynamic equation which makes the contributions procedure especially fruitful One can for instance estimate the impulse function from a one off increase of 1 of ly This can be done by the following instructions gt load SCI macros grocer db bdhenderic dat bounds 1964q3 1989q2 gt rhe ols delts 1m1 lp delts 1p delts lagts 1 1mi lp ly rnet gt lagts 1 1lm1 lp ly cte noprint gt coef rhe beta gt mainf 1 coef 2 coef 4 coef 2 0 coef 2 coef 4 coef 2 10
27. mber of variables 19 R 0 3004672 adjusted R 0 2170792 standard error of the regression 3 6390646 sum of squared residuals 14 821402 DW 0 2 0273897 variable coeff t de student p value del illinos 1 0 1439182 1 3753802 0 1708340 del indiana 1 0 4953484 2 3979857 0 0175751 del kentucky 1 0 0380825 0 7010216 0 4842536 del michigan 1 0 0176187 0 2418778 0 8091681 del ohio 1 0 1991313 1 8615471 0 0644036 del pennsyvlania 1 0 1826746 3 48336 0 0006305 del tennesse 1 0 1300340 0 6176380 0 5376453 del westvirginia 1 0 0349045 1 4624539 0 1454740 17 del illinos 2 0 0556921 0 5565481 0 5785724 del indiana 2 0 0201131 0 0987109 0 9214848 del kentucky 2 0 0399308 0 7617436 0 4472748 del michigan 2 0 0290325 0 4069044 0 6845931 del ohio 2 0 0995948 1 0386543 0 3004493 del pennsyvlania 2 0 0557372 1 0787899 0 2822187 del tennesse 2 0 0717670 0 3583999 0 7204913 del westvirginia 2 0 0045788 0 2121682 0 8322315 lag 1 of coint vec 1 0 1081266 0 3911411 0 6961856 lag 1 of coint vec 2 0 2929133 1 0600059 0 2906545 cte 3 4135785 0 9487682 0 3440932 x results for the variables indiana kentucky michigan ohio pennsyvlania and tennesse are here omitted for the sake of brevity they are of course given by the function becm estimation results for dependent variable del westvirginia number of observations 170 number of variables 1
28. mposition of an ARMA model It can be performed by standard methods For instance if p and q are the respective degrees of the polynoms A L and B L and C L the corresponding infinite moving average representation then one can calculate the coefficients of C with the corresponding algorithm C 0 By 0 for i 1 to q kat C i By i NRO Al Ci j for i q 1 to N in p i 1 VS C i ye A G x C i j This algorithm has been written in Scilab this is Grocer function mainf Once the infinite moving average representation has been calculated Grocer function contrib allows the calcu lation of the contribution of the corresponding variable However the decomposition above is not exact for 2 reasons e the algorithm can only provide a finite number of terms e the data do not go back to minus infinite at best they can only go back to the big bang In practice you can ignore the discrepancy or distribute it one way or another on all the contributions Grocer proposes a method programmed in function balance_identity that distributes the discrepancy proportionally to the absolute value of each contribution The method can be rigorously applied only to linear models It can however be extended to any non linear problem by a first order approximation In Grocer one such problem is dealt through function contrib_logq2gra This function calculates the annual contributions to the growth rate of an endogenous variab
29. odel and restart the process at step 3 ii a At the end of step 3 the list of third step models contains zero one or a few models Fourth step Encompassing i if the list of third step models is empty then the final model is the general model ii if the list of third step models contains only one model then this model is the final model iii if the list of third step models contains more than one model then build the union of all these models and tests at significance level y all third step models against this union model a if all third step models are rejected against the union model then the union model is the final model b if only one third step models is not rejected against the union model then this third step model is the final model Fifth step redoing the process with the union model Fifth step occurs only when there are more than one third step model not rejected against the union model In that case call the union model the new general model and restart the process with this new general model at the beginning of third step If at the end of the process there are still more than one third step model not rejected against the new union model then select the final model on the basis of an information criterion The exploration of several paths and the use of specifications tests cover against the risk of eliminating a relevant variable but only the more relevant paths are indeed explored Hendry and Krozli
30. q1 0 0016583 1967q2 0 0088530 1967q3 0 0024139 1967q4 0 0056231 1968q1 0 0029065 1968q2 0 0136596 1968q3 0 0005435 1968q4 0 0113783 1969q1 0 0061560 1969q2 0 0021782 1969q3 0 0065130 1969q4 0 0061694 results for the dates 1970q1 to 1983q4 are here omitted for the sake of brevity they are of course given by the function contrib 1984q1 0 0050476 1984q2 0 0066269 1984q3 0 0045817 1984q4 0 0063316 1985q1 0 0083387 1985q2 0 0080658 1985q3 0 0060436 1985q4 0 0044149 1986q1 0 0066662 1986q2 0 0086946 1986q3 0 0078957 1986q4 0 0097245 1987q1 0 0080935 1987q2 0 0066234 1987q3 0 0117735 1987q4 0 0129108 1988q1 0 0096213 1988q2 0 0097046 1988q3 0 0117405 1988q4 0 0123862 1989q1 0 0100042 1989q2 0 0122195 24 One should note that the contribution of the variable is calculated to be 0 in the first period this is because the variable ly has an impact on lm1 only after one period and because data before 1963q1 are not available in the data base This means that the first calculated contribu tions will be sometimes very far from what should be their unknown true value Indeed the discrepancy between Im1 and the sum of the contributions amounts to 0 0041 around 0 4 in 1964q1 but decreases quickly to less than 0 0001 0 01 from 1975q3 on Lastly from this equation one can calculate the contributions of each variable to the an nual growth rate This entails the transformation from contributions of quarterly variation
31. s and estimate the corresponding model M1 ii Calculate the vector s of p values of the k specification tests iii if it exists s lt s then stop the process for this variable do not store the corresponding model and go to the following insignificant variable of the second step model if any else In practice parameter 7 can be differentiated with respect to the specification tests in that case it is better to interpret 7 as a vector this is indeed the way 7 is treated in Grocer implementation a in the estimation of M1 search an insignificant variable at level a starting from the vari able with the lowest Student s t to the one with the greatest Student s t until the corresponding model verifies s gt s for all that is a model which passes all specification tests b if there is no such variable then stop the process for this variable store model M1 in the list of third step models and do again all step 3 process with the following insignificant variable of the third step model if any c if there is such a variable and if the corresponding model has already been encountered on a preceding exploration in step 3 then stop the process for this variable and restart all step 3 process with the following insignificant variable of the second step model if any d if there is such a variable and if the corresponding model has not already been encountered on a preceding exploration in step 3 then call M1 this new m
32. s of logarithms into contributions in annual growth rates as explained in the previous section and is done through the function contrib_logq2gra This is done as follows gt ml exp 1m1 gt he_lp_list list lp 1 coef 2 coef 4 coef 2 1 coef 1 gt 1 coef 1 coef 2 coef 4 coef 2 J gt he_ly_list list ly 1 coef 2 coef 4 coef 2 gt 0 coef 2 coef 4 coef 2 J gt he_rnet_list list rnet 1 coef 2 coef 4 coef 2 coef 3 gt he_resid_list list resid 1 tcoef 2 coef 4 coef 2 1 gt listcont_unbal listcont_bal contrib_logq2gra m1 he_lp_list he_ly_list gt he_rnet_list he_resid_list The contribution of for instance ly can be recovered that way gt listcont_bal 2 With the following result ans 1965a 0 0225424 1966a 0 0213406 1967a 0 0231886 1968a 0 0296916 1969a 0 0280491 1970a 0 0269894 1971a 0 0306605 1972a 0 0308545 1973a 0 0492482 1974a 0 0335330 1975a 0 0192224 1976a 0 0213143 1977a 0 0237160 1978a 0 0269834 25 1979a 0 0286258 1980a 0 0214260 1981a 0 0015827 1982a 0 0112075 1983a 0 0181204 1984a 0 0251375 1985a 0 0294632 1986a 0 0312292 1987a 0 0396425 1988a 0 0468648 6 The list of other Grocer econometric tools Besides the functions presented previously Grocer contains some utilities functions mim icking Gauss or Matlab ones for the manipulation of strings and matrices
33. tion in Dubois 2004a and a more detailed presentation in Scilab 1993 for an old but still useful tutorial or on Scilab web site the on line help at http scilabsoft inria fr product man html eng index htm or 2 1 lt A few words about Scilab Scilab developed by INRIA in France is a matrix oriented software and it is therefore very convenient for the programming of econometric functions It looks much like Matlab in fact Scilab contains a translator from Matlab that works well even if it is not perfect Commands can be run directly on the command window they can also be gathered in scripts or in func tions Scilab functions can be compiled notably to be added to the Scilab distribution Grocer is in fact built upon more than 300 Scilab functions that are compiled the first time the user runs Scilab after Grocer has been installed which is made very simply by unzipping the Grocer distribution in the Scilab directory see Dubois 2004a for details As Gauss and Matlab Scilab contains a robust numerical optimisation program and there fore allows to implement any likelihood maximisation Scilab contains numerous data types real and complex matrices booleans strings lists functions It contains also a specific type the typed list which is at the basis of Grocer time series type Once you have installed Scilab on your computer and thereafter Grocer then running the software will lead the user to the following window
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