Time Series Modelling using TSMod 3.24
Contents
1. can have fixed variances hy On a Red Hat system a symbolic link in the usr local bin directory created using ln s usr java j2re1 4 2 bin java does the trick or follow the asymmetric power autoregressive conditional heteroskedasticity APARCH or exponential generalised ARCH EGARCH specifications of the table The standard GARCH specification is a special case of the APARCH specification with 7 2 When using maximum likelihood estimation the underlying densities can be specified as either Gaussian or possibly skewed Student t Regressors xj j 1 6 contain explanatory variables influencing the model at different levels It is possible to include h or hy 2 in Tit j 1 2 3 Table 1 Model specifications General model 1 L O L yo nt T u You TZL O L 75232 u Bilinear alternative L w yo2 7x2 A L uy_ 1 VU L wy_1 O L uy we 1 L Y yi0 nt Ti T1 Vt 14 034 Ut APARCH B L hn w W404 TETs B L 1 a 1 L 1 8 L 1 ust uel Teze EGARCH B L log ht w mvat These BE 1 a 1 1 6 L hy lul pre 1h The manual Davidson 2003a contains full details on the possible models but the struc ture is rich enough to comprise ARIMA and ARFIMA models with APARCH FIAPARCH Hyperbolic APARCH threshold ARCH GARCH M resp EGARCH based versions Further more linea2. are the parameter estimates and standard deviations TSMod can report either standard or robust standard deviations The results indicate that even for a model which is notoriously hard to estimate the three slightly different models result in very similar outcomes For most practical purposes these estimates can be considered equal Note that some difference was expected as the likelihood functions are not equal Gauss estimates in the frequency domain whereas Ox or the ARFIMA package used through Ox and TSMod differ in the manner in which the first observations are used for conditioning Note that TSMod defaults to reporting standard errors based on the combination of the Hessian and the outer product of the gradient which is more robust in the case of misspeci fication of the likelihood function The last column of Table 2 reports these robust standard deviations Alternatively TSMod can provide heteroskedasticity consistent estimates of the covariance matrix and standard deviation according to the formula of Newey and West 1987 Source Bureau of Labor Statistics series SAO 1957 1 1995 12 Inflation series were constructed using 100A In s and the main effect of seasonality was taken out of the series by regressing on a set of seasonal dummies Equation for dLPs 0 3 0 25 0 2 0 15 0 1 0 05 Conditional Variances 1 1 1 1 1 1964 1970 1976 1982 1988 1994 Figure 2 Conditio
3. 0 2728 fod tare pies la a Kc z 8207 0 53633 0 102507 e ARFIMA d 0 3895 0 0558371 a 4 E 9 z ARI 0 20999 0 0667187 F ARFIMA 0 E AR2 0 Fixed dLPsorg ARS 0 Fixed lType 1 Intercept 0 E AR4 0 Fixed Trend ARS 0 Fixed lType 2 Intercept o ARG 0 Fixed AR 0 Ete Select Regressors o E ARB 0 Fixed ARS D Fixed Type l vlype2 vi 0532316 E ARI 0 EEren 4292151 d x i 3 Bilinear Order 0 R12 Gaie 0 058805 0 045211 i ETTER 4 X O output and Diagnostics Options x 3 954520 a Correlogram Points o E od farfima6 excl sa0 in7 3295129 gj Q Test Order 12 ol Lacaupiet _ Parameter Constraints A n ML 4 gt Box Pierce Lju Equation for dP Refresh Clear st ML ILM Tests of Parameter Restrictig ua Actual Student ML Covariance Matrix Formu Fed Poman Standard Robust 1 Print in Results Window Observation 958 Month 1 F Correlograms Covari h Ell Forecasts amp MA Coeffs 0 8 lI Dbservation 1995 Month 12 Retrieve Generated Series Hal Aoo Ja Residuals i VW Once _ Wariance Adjusted f j Differencing Always Conditional Varian o yr j l j i N 4No Auto Yes Regime Probs ST N j N SS Simulations i Refresh m Export Listings to Files of T 0 5 FI i Figure 1 A typical desktop with TSMod option dialogs and graphical output Menus are mostly self explanatory The user manual Davidson 2003c provides further details and its information will not be copied here No
4. TI 2003 091 4 Tinbergen Institute Discussion Paper Time Series Modelling using TSMod 3 24 Charles S Bos Department of Econometrics and Operations Research Vrije Universiteit Amsterdam and Tinbergen Institute Tinbergen Institute The Tinbergen Institute is the institute for economic research of the Erasmus Universiteit Rotterdam Universiteit van Amsterdam and Vrije Universiteit Amsterdam Tinbergen Institute Amsterdam Roetersstraat 31 1018 WB Amsterdam The Netherlands Tel 31 0 20 551 3500 Fax 31 0 20551 3555 Tinbergen Institute Rotterdam Burg Oudlaan 50 3062 PA Rotterdam The Netherlands Tel 31 0 10 408 8900 Fax 31 0 10 408 9031 Please send questions and or remarks of non scientific nature to driessen tinbergen nl Most TI discussion papers can be downloaded at http www tinbergen nl Time Series Modelling using TSMod 3 24 Charles S Bos Tinbergen Institute and Department of Econometrics amp O R Vrije Universiteit Amsterdam De Boelelaan 1105 NL 1081 HV Amsterdam The Netherlands E mail cbos feweb vu nl 3 November 2003 Abstract TSMod is an interactive program which allows the user to estimate a broad range of univariate models This review describes the possibilities of the package from a user s perspective and with a secondary focus on the numerical accuracy of the program Keywords Time series software econometrics 1 Introduction Performing applied econometr
5. ction definition as in the listing in Figure 3 Starting TSMod from this file with ox1 s5000 5000 TSMod_Ecker ox was enough to implement the above model in TSMod Figure 3 Specifying a user defined function in TSMod_Ecker ox TIME SERIES MODELLING v3 24 RUN SPECIFICATION 1111111111111 TATTLE TT define USER_FUNCTION import lt packages tsmod32 tsmgui32 gt 1 11111 111 11111 1 1111 11111 111 11111 1 1 1 1 1 1111 1 111 1 11111 1 11 1 1111 1 1111 111111 UserFunction const mcX const cStart const cEnd const vP const sName decl vE iX iY iX VarNum X iY VarNum Y Compute residuals vE mcX iY vP 0 vP 1 exp 0 5 sqr mcX iX vP 2 vP 1 return vE cStart cEnd With the data set two alternative starting vectors of Ba 1 10 500 or 6 1 5 5 450 are provided From the first set of points or from TSMod s defaults 3 0 no convergence is found The second set closer to the optimum leads to the results in Table 3 Up to all digits specified by TSMod resulting parameter values are equal to the reference results as is the case for the sum of squared residuals The non robust standard errors differ slightly from the reference results It should be noticed that this is quite an accomplishment as the model combined with this data set is graded to have a higher level of difficulty Also the ease with which the residual function of the model can be add
6. ed to TSMod is remarkable Table 3 Estimation results on Ecker nonlinear regression model TSMod Reference By 1 5544 0 0149915 1 55444 0 0154080 Bo 4 0888 0 0459662 4 08883 0 0468030 b3 451 54 0 0453601 451 541 0 0468005 SSR 0 00146359 0 001463589 Results from TSMod for the Ecker data set with reference results The ref erence parameter estimates are reported with one extra digit compared to the TSMod results TSMod standard deviations are computed using the non robust formula 4 Drawbacks and minor problems Even though TSMod is a nice program as is it is not without some minor problems It is not a commercial product with the backing of an entire company and a long history in bug hunting and therefore some issues can be found with the program Less problematic are bugs like an incorrect i e illogical model specification not being handled gracefully or if the program stops with an error message if the sample size was moved around so much that TSMod got confused about where to start In such cases a restart of the program helps a lot which is not much of a hassle as the model specification and estimated parameter values are saved between sessions In an earlier version of TSMod there was a problem that only non linear least squares was available as an estimation method If the model was purely linear the basic regression model which is found so often in applied work estimation was not fully efficient as t
7. en nl cbos gnudraw html Bos C S Franses P H and Ooms M 1999 Long memory and level shifts Re analyzing inflation rates Empirical Economics 24 427 449 Choirat C and Seri R 2002 OxJapt an Ox version of Merten Joost s Java Application Programming Interface http site voila fr choirat software oxjapi oxjapi html Davidson J 2003a Time Series Modelling Version 3 24 http www cf ac uk carbs econ davidsonje Davidson J 2003b Time Series Modelling Version 3 24 Programming Reference http www cf ac uk carbs econ davidsonje Davidson J 2003c Time Series Modelling Version 3 24 User s Manual http www cf ac uk carbs econ davidsonje Doornik J A 1999 Object Oriented Matrix Programming using Ox 3rd edn Timberlake Consultants Ltd London See http www nuff ox ac uk Users Doornik Doornik J A and Ooms M 2001 A Package for Estimating Forecasting and Simulating Arfima Models Arfima Package 1 01 for Ox Package manual Doornik J A and Ooms M 2003 Computational aspects of maximum likelihood esti mation of autoregressive fractionally integrated moving average models Computational Statistics amp Data Analysis 42 3 333 348 Hamilton J D 1989 A new approach to the economic analysis of nonstationary time series and the business cycle Econometrica 57 357 384 Newey W K and West K D 1987 A simple positive semi definite heteros
8. he full gradient method was applied instead of solving the regression equation directly by ordinary least squares This problem may serve as a good example of the speed with which TSMod is improving While writing the review a new version of the program appeared which filled this gap Likewise a more serious bug in an earlier version of the program which surfaced while working on this report was taken out by the author of TSMod in the course of hours not days New versions of the program come out regularly either to improve on a bug or to elaborate further on the features of the program When TSMod solves a model by optimising a criterion function be it the sum of squared residuals or a likelihood estimate a general purpose gradient method is used This optimising routine behind it all is the MaxBFGS routine of Ox which is a high quality gradient type optimising routine However with the inflation data set in Section 3 1 it was quite possible to specify a model which was of a richer structure than the data could support e g by specifying a varying variance structure when in effect the variance is constant In such cases the likelihood function may well be very flat or multimodal and it was possible to get to situations where TSMod had to be helped with correct starting values for the optimisation 5 Conclusions In this review part of the possibilities of TSMod was investigated The wealth of models incorporated in the package is impres
9. ics is partly science partly an art One tends to start with a data series and wonder what the internal relations between observations could be And so start the artistic part of the analysis trying out different types of models judging what might work and what is not useful For this purpose you rummage through the economet ric toolbox which over the last decades got filled with modelling tools like the Box Jenkins ARIMA structure extensions allowing for fractional integration ideas on where regressors might come into the system different disturbance structures switching models non linearity heteroskedasticity of various types etc etc For the more theoretically minded there is often no other solution but to implement the exact method which is needed to have full control while for many applications it suffices to have available a general program which is capable of estimating a series of models easily and in a comparable fashion to quickly track down the model which would fit the data at hand TSMod short for Time Series Modelling is a program written by James Davidson which can be used for such a purpose In this review the capabilities of TSMod background and relation to other programs are described in Section 2 Section 3 describes numerical results for a real world example and for two reference datasets and is followed by Section 4 describing a few points that could be improved upon in future versions of the program Section 5 c
10. kedasticity and autocorrelation consistent covariance matrix Econometrica 55 703 708 Sowell F 1992 Maximum likelihood estimation of stationary univariate fractionally inte grated time series models Journal of Econometrics 53 165 188 10
11. nal variance estimate for an ARFIMA 12 d 0 GARCH model on inflation With these results the user can continue specifying e g GARCH type heteroskedasticity Easily a graph like Figure 2 is extracted displaying the jump in volatility in the beginning of the first oil crisis TSMod works very well in quickly trying out different specifications combinations of breaks with GARCH or long memory and can be a very useful tool 3 2 Two reference data sets The Statistical Engineering and the Mathematical and Computational Sciences divisions of the National Institute of Standards and Technology provides a collection of statistical refer ence datasets at http www itl nist gov div898 strd For these data sets the model and the optimum values are reported up to a high degree of precision As a test of the TSMod package two of the datasets are tried here Linear regression with unbalanced regressors The first one is a linear regression model where the data is chosen such that estimation is cumbersome The model itself is simply 10 y gt piz i 0 see http www itl nist gov div898 strd 1lls data Filip shtml for details The dataset is most easily prepared in another program like Excel or Ox as the editing capabilities of TSMod only allow for the usual transformations as squaring or cubing the data not for computing zt i gt 3 3 With the data loaded in TSMod estimating the model is not a problem After specifying the regressi
12. on model estimation is immediate The results from TSMod using the non robust formula for calculating standard deviations are exactly equal to the results as Ox would report them Also the fact that the matrix of regressors is unbalanced and that scaling is advised is mentioned in the TSMod output The parameter values and standard deviations correspond with the reference results for all digits reported by TSMod only in the residual standard deviation a small difference is found 3When the author heard of this limitation it was addressed quickly in the first upgrade to TSMod which was released after completing the review Note that equal results are only found when using the ordinary least squares estimation method if the model is estimated using the iterative optimisation routine and the nonlinear least squares criterion function the correct solution is not attained as the criterion function is too flat Nonlinear regression In the previous reference data set a standard linear model was used A second reference data set at http www itl nist gov div898 strd nls data eckerle4 shtml concerns a study involving circular interference transmittance as a response variable depending on the wavelength The data set provided contains 35 observations The model is _ 2 2 v Gow Soe 6 j which can be estimated in TSMod using a user specified function For this purpose an adapted version of the file TSMod_Run ox was created with the fun
13. oncludes 2 Overview 2 1 General TSMod is program program developed by James Davidson of Cardiff University It evolved from an earlier Ox Doornik 1999 package for long memory modelling intended to provide a framework from which to teach students the basics of Econometrics and which could serve at the same time to estimate forecast and analyse in many ways a range of econometric univariate time series models It is available free of charge for use in academic research and teaching provided that the usage of TSMod is acknowledged through a reference to the accompanying manual Davidson 2003a The latest version of the program can be downloaded from Davidson s homepagehttp www cf ac uk carbs econ davidsonje At time of writing version 3 24 is available though the program is clearly under ongoing development with minor updates and new revisions appearing regularly 2 2 Technicalities Time Series Modelling can be considered a program in the sense that it comes with its own Graphical User Interface GUI but could also be considered as an Ox package TSMod consists of a series of routines written in Ox with an OxJapi Choirat and Seri 2002 or Java shell around them and graphical output using GnuPlot http www gnuplot info through GnuDraw Bos 2003 Since TSMod itself is written in Ox it runs on an any platform where Ox can be used and where OxJapi is available as well The program has been tested on Windows and Linux pla
14. price level see the article for details The inflation series derived from the price levels is modelled using a long memory model accounting for level shifts around the oil crises in the article break points at 1973 07 1976 07 1979 01 and 1982 07 are used Originally the ARFIMA X 12 d 0 model with breaks and restrictions on AR parameters h2 _ ou 0 is estimated using Gauss with an approximative Whittle likelihood function in the frequency domain Alternatively it is possible to use the ARFIMA package Doornik and Ooms 2001 Doornik and Ooms 2003 for Ox which implements the exact maximum likelihood EML in the time domain using methods of Sowell 1992 And the third option is to use TSMod with the conditional maximum likelihood CML procedure Results of these three estimation methods are provided in Table 2 Table 2 Estimation results on inflation data Gauss Whittle Ox EML TSM CML 0 Oo 0 To 0 To og robust d 0 3808 0 057 0 3805 0 050 0 3895 0 056 0 066 0 057 0 050 0 056 Q 0 2020 0 2104 0 2100 0 089 P12 0 0593 0 0597 0 0588 0 046 73 07 0 4212 0 102 0 2990 0 102 0 2758 0 101 0 255 76 07 0 0631 0 0361 0 0433 0 131 79 01 0 2480 0 2770 0 2728 0 104 ys2 07 0 5342 0 100 0 5322 0 102 0 5363 0 103 0 103 Results of estimating an ARFIMA X 12 d 0 model on U S inflation data using three different estimation methods Reported
15. r and nonlinear equations can be specified and all kind of exogenous regressors taken up into the analysis The parameters in the models can be made dependent upon unob served regimes allowing for Markov Switching models Hamilton 1989 or smooth transition regime switching Apart from prespecified models with a little bit of Ox programming ex perience the user can also specify a residual function to optimise whatever other nonlinear function using the TSMod package see the example on nonlinear regression in Section 3 2 Estimation options include Nonlinear Least Squares Generalised Method of Moments Conditional Maximum Likelihood and Whittle estimation in the frequency domain The maximum likelihood estimators condition on the first observations for applying the ARMA filter whereas for fractional integration the finite approximation to the infinite lag fractional difference operator is applied Filtering the data delivers residuals whose likelihood is com puted according densities described above When no dynamic specification is present the TSMod can be triggered to use ordinary least squares for a regression model instead of the nonlinear least squares used otherwise The program allows the residuals to be stored for further analysis 2 4 Usage After starting TSMod the program comes up with the main window Initially the user is presented with the options to load old settings data change some general estimation and output op
16. sive the user interface sufficiently friendly and full proof that even less experienced users or students should have little problem in estimating their first models It works well as a tool for comparing different model components in an explanatory analysis of a data set and even non standard models could easily be implemented using a user specified function In Section 3 it was found that the numerical accuracy of TSMod is good as it is able to replicate quickly and to a high precision results found in the literature or in reference data sets This overview of the program is limited in the sense that only the interactive mode of TSMod was discussed with a small side tour to include a user specified residual function in section 3 2 A separate document Davidson 20036 describes the programming interface of TSMod as all the routines implemented in TSMod can also be used from a user s own Ox application The programming interface would however be a topic on its own and is left for the reader to explore TSMod is a useful program it is nice to see how it keeps evolving over time and I definitely plan to keep a version around in order to quickly estimate a model on a data set The fact that it is freely available for academic use makes it a good candidate for the more advanced Econometrics classes as well as students tend to appreciate to be allowed to use a program at home References Bos C S 2003 GnuDraw http www tinberg
17. te that many menus are sticky meaning that separate option dialogs can be left open on the desktop for easily changing the settings See Figure 1 for an example desktop with many open dialogs Given a data set it is easy to specify a model by selecting the number of ARMA lags choosing whether the parameter d of fractional integration should be estimated and specifying the error structure The model is estimated at the press of a button and tends to be quick after estimation a large selection of test statistics is printed and several diagnostic plots can be chosen from the menu The usage is simple enough for students to use in class and offers enough possibilities for practitioners to use TSMod as a valuable tool However it should be noted that the package is not commercial which is noted from the occasional bug See also Section 4 2 5 Competitors As TSMod is a program to analyse time series it has several competitors In the commercial range the most well known are EViews TSP and PcGive Among the free or open source programs Gretl comes to mind which has more extensive data editing capabilities but is mostly intended for undergraduate use with regression models less specifically targeted at time series models 3 Numerical details 3 1 Inflation and long memory To test the functionality and numerical accuracy the dataset analysed in Bos Franses and Ooms 1999 is loaded into TSMod The series concerns the U S
18. tforms but other operating systems should not pose any problems The user is responsible for getting Ox and Java working and GnuPlot if the operating system is of the Unix family The GnuPlot executable for Windows is included with the TSMod installation package As such installation is easy On Windows it s a matter of unzipping the installation file into the directory for Ox packages and possibly creating a shortcut to the batch file starting TSMod On Unix or more specifically Linux the user has to assure that the Java executable can be found by the operating system and that GnuPlot is installed Again a desktop shortcut starting the program is easily created Without the shortcut TSMod can be started with the magic command oxl s5000 5000 lt path to TSMod gt TSMod_Run ox but the use of the shortcut is advisable The installation comes with extensive documentation in the form of PDF manuals David son 2003a 20036 2003c a version in HTML and internally in the program itself help is available on the menus and on the meaning of the myriad of options 2 3 Model features and estimation The main model is built up around the Box Jenkins structure for Autoregressive Integrated Moving Average ARIMA models see the General model in Table 1 or alternatively a bilinear model can be used Instead of an integer order of integration also fractional integration FI 1 is implemented The disturbance terms u h ex e 7 1 d 0 1
19. tions or work through the help menu The real work starts after loading a dataset The data should be prepared to correspond to the Ox standard for data files which in general means that basic ASCII Excel Lotus or GiveWin files can be read without problem TSMod provides basic data editing capabilities including the possibility to add basic log difference power etc transformations of existing variables Individual variables can be plotted and summary statistics are available as well to check that the data is read correctly Time Series Modelling 3 24 z OML and GARCH Options Equation i Value Fixed File Setup Model Values Actions Options Help FE GARCH Coefficients in standard form else ARMA in S Cleat ARFIMA d zi Dependent Variable is dLPs Root of ML Variance GARCH Intercept to be Estimated int fo s894985927 Observations 1 456 1958 Month 1 to 1995 Month 12 A ARI 0 20998580108 J used for estimation Number of Gauss Seidel Iterations to Compute GARCH_M ARQ r Conditiona ML Time Domain 0 aussian Likelihoo l M i n Se ay GARCH_M Trimming Factor zs E Oane ealain TE E trong convergence Compute EGARCH Likelihood e o m ba i elect Dependent Variable E b7307 0 27584 0 101498 Number of Gauss Seidel Its t 0 b7607 0 04328 0 11177 r bzani
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