Building and testing the reduced form VAR

The lag length of the reduced form VAR model was examined firstly by the LR tests to detect the possibility to sequentially drop out lags from the model, secondly by the AIC, SBC and HQ information criteria²⁹ and thirdly, by examining the statistical properties of residuals of the equations. (To save space, the details of the information criteria and the LR tests are not reported.) The outcomes of the two different methods for testing the lag length conflicted sharply with each other, however, since the LR tests proposed using long lag structures, while the information criteria suggest more parsimonious specifications with two (AIC) or three (SBC and HQ) lags.³⁰

The examination of the residuals also suggests estimating a model with long lag structure, because the residuals of the equations for the short interest rates tend to show signs of autocorrelation when short lag structures are used³¹. The desire to save degrees of freedom, however, finally led us to adopt a short lag structure of four lags³². With this lag structure, the stability of the equations and the statistical properties of the residuals are improved, compared to the alternative of three lags recommended by the information criteria³³.

28 For a proof of the consistency of VAR representation in levels when there are cointegrated variables in the system, see Hamilton (1994), p. 579.

29 Paulsen (1984) has shown that both the SBC and HQ information criteria are weakly consistent in the case of variables with unit roots. AIC, however, has been shown to overestimate the true lag order with positive propability irrespective of the existence of any unit roots. (see also Jansson 1994, pp. 35-36.)

30 Clarida and Gertler (1996) solved the problems in choosing the lag length, due to a large number of parameters relative to the length of sample period, by including only the lags 1,2,3,6,9 and 12 in their model. For comparison, this kind of staggered lag structure was adopted also here. Only the AIC criteria preferred the models with staggered lags to models with full lag structures of three or four lags, however.

31 The normality of the residuals was examined with the Jarque-Bera statistics and the results showed non-normality in most cases irrespective of the lag structure

32 One reason for rejecting the long lag structures was that the impulse responses estimated with the long lag structure of the lags 1-8 and 12 overall yielded impulse responses that were more difficult to interpret than those computed from models with more parsimonious lag structures.

The diagnostic tests for the VAR(4) residuals are summarized in Table III.1 in the Appendix III. (The significance levels are in brackets.) Autocorrelation was detected by the LM(1) and Ljung-Box Q(20) tests. The LM(1) test rejects the null hypothesis of no autocorrelation only in the cases of industrial productions of the countries (at 5% level).

According to the Q(20) statistics, six of the residual series are, however, autocorrelated, when the horizon for detecting the autocorrelation is extended beyond one lag. Luckily, the autocorrelated residuals leave the OLS estimator unbiased, although its efficiency is reduced.

The ARCH(1) statistic measures the heteroscedasticity of the residuals which was found at 1% significance level in the series of short term interest rates and monetary growths of both countries, the German inflation rate and the US government bond rate. Visual inspection suggests that the reported heteroscedasticity seems to be due to some outliers among the observations.

The Jarque-Bera statistics maintains the null hypothesis of normality in the cases of the US inflation, M3 growth, industrial production and DEM/USD exchange rate. The residuals of call money, federal funds’ rate, German inflation rate, German money growth and German production however all appear to be non-normally distributed at the 1% significance level. In addition, the residuals of the government bond rates of the countries seem to be non-normal at the 5% level. Luckily, again, the non-normality appeared to be mainly due to kurtosis, which is a less serious problem than skewness, because the kurtosis leaves the OLS estimator unbiased.

A possible source for the observed non-normality of the residuals is provided by the few outlier observations in the data. Thus, the model was re-estimated by introducing some observation specific dummy variables into the model³⁴. The dummies were set on three periods. The first one was set just at the beginning of the sample period to take into account the fluctuations in the German short-term interest rate. The regime change in

33 The normality of residuals is not a critical assumption in VAR modeling, because the properties of OLS estimator are not affected by nonnormality. The autocorrelation in error terms leaves the OLS estimator unbiased but reduces its efficiency.

US monetary policy at the turn of the 1970’s and 1980’s was taken into account with a whole set of dummies. Finally, the unification of Germany was considered in the modeling with a single observation specific dummy for period 1990:6. The non-normality of the residuals was reduced in some cases when the dummies were introduced (results not reported in detail). Surprisingly, however, the residuals showed fewer signs of autocorrelation without than with the dummies. Thus, only the dummy representing the currency exchange of Germany in the period 1990:6 was included into the final model.

The stability of the model was preliminary examined by the Cusum-tests, while more formal testing of the VAR residuals was performed by the RESET, Chow and the Goldfeld–Quandt tests. The line graphs of the Cusum tests are shown in Appendix III. It is seen that the V-mask representing 5% confidence levels is slightly crossed in the cases of only call money rate and German industrial production, when overall, most of the equations seems to be quite stable.

The results of the formal stability tests are presented in Table III.2 in the Appendix III.

The F-statistics of RESET(3) tests show a possibility of misspecification at the 5 % significance level only for the US government bond rate. F-values of Chow tests, in turn, find signs of a structural break in the cases of German inflation and growth rate of M3. Chow tests, could not, however, be computed for the DEM/USD exchange rate, because the computer program reported near singular matrix.

The Goldfeld-Quandt tests for the VAR equations were performed as LR tests which are used to find the most possible date for the regime switch. It is seen that the regime switch dates suggested by the Goldfeld-Quandt tests differ from those proposed by visual inspection of the Cusum-test figures, since now the most likely date of a regime switch is at the turn of the 1970’s and the 1980’s for most of the variables. For the US variables these results get a natural explanation for the change that occurred in the operating procedure of the Federal Reserve at that time. Unfortunately, the large number

34 There are some drawbacks in introducing dummy variables into the model, however, since the outlier observations may include valuable information about the dynamics of the model. Also some degrees of freedom are lost when using the dummies.

of parameters in the model prevents us from taking the possible regime switch into account by estimating separate models for both regimes.

Overall, we see that some signs of misspecifications or instability were found in some of the equations. It has to be noted however that when the coefficients from different sub-periods are estimated with very high precision, statistically significant differences are easily found, although the differences are necessarily not large enough to also imply economically significant differences between the periods. Further, the changes in the coefficients of a dynamic model with large number of parameters may offset each other, thus leaving many important dynamic relationships unaffected.³⁵