International North-Holland Journal A FORECASTING Robert KUNST Instituie for Aduanced of Forecasting 2 (1986) COMPARISON 447 447-456 OF SOME VAR TECHNIQUES and Klaus NEUSSER * Studies, A-1060 Vienna, Austria Abstract: Higher dimensional multivariate time series models suffer from the problem of overparametrisation which impairs their forecasting performance. Starting from such unrestricted vector autoregressive models the paper discusses two ways to cope’ with this difficulty. The first approach reduces the number of free parameters by applying a subset modelling strategy. The second approach takes a Bayesian point of view by formulating ‘priors’ which are then combined with sample information, but leaving the original specification unaltered. Using Austrian quarterly macroeconomic time series a comparative study is undertaken by running alternative forecasting exercises. Both methods improve out-of-sample forecasting performance substantially at the cost of some bias in ex-post simulations. Comparing the ex-ante predictions of the two approaches, the former does better at short horizons whereas the latter gains as the forecast horizon lengthens. Keywords: Multivariate time-series methods, Forecasting evaluation, Empirical study, Subset modeling, Bayesian analysis. 1. Introduction Building up big structural models incrementally equation by equation has been criticized on theoretical as well as on empirical grounds [see Sims (1980) Lucas and Sargent (1981)]. Of special concern to these authors are the often arbitrary exclusion restrictions used to identify the models. As far as the forecasting performance of these models is concerned, simple univariate ARIMA models seem to be superior or at least equivalent [see McNees (1981)]. An alternative proposed by Sims (1980) is to build vector autoregressive models with no a priori exclusion restrictions. This methodology however quickly exhausts the available degrees of freedom, since each variable has to appear in each equation with the same lag specification. The scarcity of observations relative to the number of parameters to be estimated brings up the problem of overparametrisation, and the test statistics as well as the forecasts of these models tend to deteriorate rapidly [see Fair (1979)]. This paper examines two ways to overcome this difficulty. The first one is to start out with the unrestricted vector autoregressive model and to use the concept of Wiener-Granger causality [see Granger (1969)] to reduce the dimensionality of the model. This procedure is similar in spirit to the ‘general-to-specific’ approach of Hendry et al. (1984) in that it tests which variables contribute significantly to the precision of forecasting and excludes all other variables. Since the pre-test estimator has been shown to be statistically inferior to Stein rules [see Judge and Brock (1983)], it is tempting to contrast this approach in its forecasting performance with Bayesian point of view. This is * The authors 0169-2070/86/$3.50 would like to thank 0 1986, Elsevier Manfred Science Deistler Publishers and two referees for their B.V. (North-Holland) comments and criticism 448 R. Kunst, K. Neuwer / Forecasting comparison of VA R techniques done by taking the unrestricted vector autoregressive system and combining the sample information with prior information on the parameters and their standard errors. This method has been successfully applied by Litterman (1980), and Doan, Litterman and Sims (1984) to macroeconomic data of the United States. These alternative modeling techniques are evaluated by actually building models of the Austrian economy and by using them to generate different forecasting scenarii. First, an ex-ante forecast from 1981 to 1984 is computed, so that a comparison with actual outcomes is possible. Second, an ex-post forecast is performed to see how well the models are able to track reality. The evaluation of alternative forecasting techniques has been the subject of extensive research which are well summarized by Fildes (1985). Especially, it may be interesting to compare the results obtained in this paper with the ones of Kling and Bessler (1985), who also tried to evaluate alternative multivariate forecasting procedures - including among a dozen different techniques Litterman’s procedure. The plan of the paper is as follows. Section 2 presents the data and their transformations. The next two sections present the construction of the restricted and the Bayesian vector autoregressive model which are then compared by their forecasting performance in section 5. The last section finally summarizes the findings and outlines future research. 2. The data The paper proposes to analyze a small quarterly forecast model of the Austrian economy consisting of 8 variables over the period 1964.1-1984.4: real gross domestic product (GDP), employment (LIZ), monetary base (MB), the deflator of gross domestic product (PGDP), the average bond yield in the secondary market (R), the terms-of-trade (TT) calculated with respect to goods exports and imports, average earnings of employees (WAGE) and real goods exports (XG). Although the selection of variables is somewhat arbitrary, they represent nonetheless key economic indicators for any economy. ’ Before starting the actual model building process, the logarithm of each variable, with the exception of the interest rate, is taken. This transformation is supposed to be variance stabilizing, but it does not take out the trend. For the restricted vector autoregressive model (RVAR), which is built in the philosophy of Box/Jenkins, all variables were therefore differenced once *. For the other techniques no additional data transformations are carried out in order to avoid the problems associated with overdifferencing, which may be relevant for the interest rate and the terms of trade, and to allow for more flexibility. Since testing is not an issue in these approaches, and since OLS will give consistent estimates, the relative forecasting performance should not depend upon the handling of the trend. The seasonal adjustment was done by adding seasonal dummies to each of the eight equations in each model. Although this method is simple in postulating repeating fixed patterns, it allows to treat the seasonality alike in each approach. 3. A restricted vector autoregressive model Building an ‘unrestricted’ VAR model, as proposed by Sims (1980), is likely to lead very many separately insignificant parameters, so that the search for restrictions which reduce the dimensional’ The data are available upon request refered to Amdt (1982). ’ Nelson and Plosser (1982) presented than a deterministic one. from the authors. evidence that The reader a stochastic especially trend model interested is more in the Austrian appropriate economic for economic situation is time series R. Kunst, K. Neurser / Forecasting compurison of VA R techniques 449 ity of the model seems to be a rewarding task. This is especially motivated by a possible disturbing influence of insignificant parameters on forecasting. 3 There is no unique way for setting up such restrictions. In many cases, they are based on information extracted from theory concerning the structure and interaction of the time series. This approach leads to some kind of approximation of a usual structural econometric model. An alternative is the specification of restrictions on empirical grounds, which is based on the following procedure: - An unrestricted time series model is estimated. F- and t-statistics are noted. Those influence of one variable and a specific lag on an other variable, respectively. - Insignificant variables and insignificant lags of variables are eliminated. The significance set to loosely to 15% to reduce the possibility of neglecting significant regressors which increased due to multicollinearity. - The resulting model, that is, an autoregressive model with zero restrictions, is estimated for forecasting. test the level was might be and used To cope with possible effects of multicollinearity and to retain a reasonable amount of degrees of freedom, the total number of regressors is restricted to 20. Thus, stepwise selection and elimination of variables was necessary. This procedure, which is almost identical with the one used in Kunst (1985) where it was applied successfully to a 1Zdimensional system, started with univariate autoregressive modelling of each of the eight time series. The maximal autoregressive lag was chosen by the AIC criterion, and only in cases where AIC suggested an excessive number of autoregressive lags BIC was preferred [see Akaike (1974)]. Now, by the method of OLS regression the first four lags of one or two of the other variables were added to the p own lags identified by AIC or alternatively by BIC. Additionally, a constant and three seasonal dummies were inserted as regressors to correct for nonstationary seasonal behavior of the data. The resulting equation is x, = &,Xtpj + 2PjyIej+ Xp, + c + e,., Applying the procedure outlined above reduces the number of parameters from 320 to only 84. The set of regressors in the restricted model (RVAR) depends on the search strategy. A comparison of all possible subset models, as suggested by Haggan and Oyetunji (1984) would be rather tedious and time consuming. Anyway, the RVAR should not be ‘far away’ from the optimal model. The significance of explaining variables offers a rough causality test in the spirit of Granger (1969) and Sims (1972). With Granger and Sims, a variable is said to cause another one if and only if its lags reduce the forecasting variance additionally to the proper lags of the variable to be forecasted. Accurate significance levels, however, are not valid due to the iterative elimination process. Influences which are quicker than one quarter of a year are not captured by the lags model. However, according to Pierce and Haugh (1977), they are reflected in the residual correlations. The direction cannot be determined from the data. This phenomenon is known as ‘instantaneous causality’ and the, partially substantial, correlations are given in table 1. Within the framework of a forecasting model, instantaneous causality could be interpreted as an indicator for flaws in the model, that is, information which could be used for improving forecasts but is not. On the other hand, if the model is viewed as correct, it is impossible to improve point forecasts by using these correlations which however do affect stochastic forecasting. 3 This fact is well documented in the paper by Fair (1979) 450 R. Kunst, Table 1 Cross-correlations of residuals from K. Neusser RVAR / Forecasting comparison of VA R techniques model. GDP LE MB PGDP R TT WA GE XG 1.00 0.23 - 0.04 - 0.00 - 0.07 0.06 0.04 0.42 0.23 1.00 -0.18 - 0.01 - 0.07 - 0.02 -0.15 0.07 - 0.04 -0.18 1.00 0.04 -0.25 0.16 0.04 0.09 -0.00 - 0.01 0.04 1.00 -0.12 0.15 0.33 - 0.04 - 0.07 - 0.07 -0.25 - 0.12 1.00 - 0.39 - 0.07 - 0.03 0.06 - 0.02 0.16 0.15 PO.39 1.00 - 0.11 ~ 0.28 0.04 - 0.15 0.04 0.33 ~ 0.07 -0.11 1.00 0.26 0.42 0.07 0.09 - 0.04 - 0.03 - 0.28 0.26 1.00 GDP LE MB PGDP R TT WAGE XG 4. Bayesian vector autoregression The Bayesian approach starts with the presumption that the given data set does not contain information in every dimension. This means that by fitting an overparameterized system some coefficients turn out to be non-zero just by pure chance. Since the influence of the corresponding variables is just accidental and does not correspond to a stable relationship inherent in the data, the out-of-sample forecasting performance of such models deteriorates quickly. The role of the Bayesian prior can therefore be described as prohibiting coefficients to be nonzero ‘too easily’. Only if the data really provide information will the barrier raised by the prior be broken through. The actual model building procedure’ starts by postulating a VAR with 6 lags for each of the 8 variables, which in contrast to the RVAR model are not differenced. The next step consists in the specification of a prior distribution for the coefficients. In this paper the so called ‘Minnesota prior’ is used [see Doan, Litterman and Sims (1984)]. It specifies a random walk process with drift for each of the variables involved and does not allow for influences of own lags beyond the first one and of other variables. This specification does not represent a genuine Bayesian prior, since it does not characterize the beliefs of an investigator, who usually postulates some relationships among those variables. The Minnesota prior could however be regarded as the intersection of the a priori beliefs of many economists. In this sense it represents an improvement over the so called ‘diffuse prior’, which is often used to represent the notion of ‘knowing little’. Furthermore, this specification may be justified on the grounds that most macroeconomic time series seem to follow a stochastic trend model [see Nelson and Plosser (1982)]. Since the prior distribution is specified as a multivariate normal, it is necessary to set, besides the mean, the standard deviations of the coefficient of variable j with lag 1 in equation i. 4 Instead of specifying each standard deviation separately, they are set as a function of 3 ‘meta-parameters’ r, w and d: s(i, j, 1) =Tf(i, j)g(l)‘j/‘j, f(i,j)=l.O =w g(l)=d’-‘, if i=j, if i#j, where OIw11, Osdsl, and where the scaling factors si and sj adjust for the relative size of the variables. 4 The covariances are set equal to zero. R. Kunst, K. Nezmer / Forecasting comparison of VAR techniques 451 The parameter r stands for the overall tightness of the prior; a smaller value indicating that more weight is given to the prior. The function g(l) sets the form of the lag pattern, which in this case has a geometrically declining shape with decay parameter d; the higher the lag the smaller the corresponding standard deviation. The symmetric function f(i, j) controls the interaction among the different variables; a higher w allows for more interaction by setting a higher a priori standard deviation for cross effects. In this investigation two specifications of the parameters 7, d, and w are considered: ‘loose prior’ (BVAR-loose) : 7 = 0.2 ‘tight prior’ (BVAR-tight) : 7 = 0.1 d = 1 .O w = 1 .O d = 0.5 w = 0.5 In the first case the lag pattern does not decay and own lags and lags of other variables are treated alike. in the second case the overall tightness is high, the lag pattern decays rapidly, and cross influences are given less weight. The specification also includes four seasonal dummies for each equation. Since no constant term then enters the equations, the coefficients of these variables will determine the drift in the time series. These coefficients are not restricted a priori and are fully determined by the data. 5. Forecasts One of the main purposes of an econometric model is forecasting. With structural models it is possible to evaluate economic theories or to gain insight into economic processes. With time series models, this goal is more or less abandoned. Thus, the acid test for an econometric time series model is its forecasting performance. The best model will be the one which forecasts best. The comparative investigation in this paper relies on two different forecasting scenarii and several criteria for the goodness of fit over the forecasting intervals. Furthermore, the two approaches are contrasted to an ‘unrestricted’ VAR model with 6 lags for each variable 5 in each equation (UVAR-6) to demonstrate how overparametrisation affects forecasts performance. The first scenario consists of an ‘ex-ante’ forecast over the period 1981.1-1984.4. For this exercise each model is estimated over the period 1964.1-1980.4 and used to generate forecasts of each variable from 1981.1 up to 1984.4. Then the information of the next quarter, 1981.1, is incorporated into the model by updating the parameters through Kalman filtering and a new set of forecasts is generated, now going from 1981.2 to 1984.4. This procedure is repeated until 1984.4 is reached when all the available information is used. ’ In this way 16 one period, 15 two period, 14 three period, . . . , and 1 sixteen period forecast are generated, which can be checked against the actual realizations. The statistics used to evaluate the different forecasts are the root mean square error (RMSE), the U-statistic of Theil (1966), 7 and the mean absolute percentage error (MAPE). The first two criteria are based on quadratic loss functions. Regrettably, the true forecaster’s loss function is unknown. It even may be skew, with different costs for optimistic and pessimistic mistakes [see Jager (1985)]. However, as the modelling procedures rely on quadratic criteria, quadratic loss is a useful technical assumption. ’ The variables were not differenced. 6 During this updating the parameter vector has been kept fixed. This is especially relevant for the RVAR model, since in this case the search procedure should have been repeated in each step. ’ Theil’s (i-statistic compares the improvement of a particular forecast against a ‘no-change’ forecast. Values over one indicate that a ‘no-change’ forecast would have been better. 452 R. Kunst, K. Neusser / Forecasting of VA R techniques comparison Whereas ex-ante forecasting is methodologically closer to the actual situation of the forecaster who iteratively adapts his model to the data, ex-post forecasting is compatible with asymptotic theory which postulates a true model that is completely identifiable in the long run. The second exercise therefore computes ex-post forecasts over 1981.1-1984.4, where the whole sample period is used for estimation. The statistics in this case are the mean forecast error (the bias), the standard deviation, and the value of the T-statistic against the null hypothesis that the mean forecast error is zero. The general conclusion to be drawn from the models’ ex-ante performance is documented in tables 2a-c which show a marked superiority of RVAR and BVAR-tight over UVAR-6 and BVAR-loose which is not shown here to save space. For short forecast horizons up to 3 quarters ahead, RVAR almost dominates. For longer horizons, RVAR loses ground relative to BVAR-tight which predicts four of the series better at step 8, but only one at step 1. RVAR and BVAR-loose are the only specifications with all U-statistics for one period ahead smaller than one, however, BVAR-loose quickly deteriorates as the forecast horizon becomes longer. Steps 4 and 8, which reflect the treatment of seasonality, put up severe problems for all models, but BVAR-tight seems to be most robust in this direction. The seasonal patterns show the worst influences in the GDP, LE and Table 2a &-ante forecasts from the UVAR-6 model for the period 1981.1-1984.4. a Variable Forecast step GDP 0.0270 0.31 0.39 0.0398 0.38 0.58 0.0525 0.56 0.79 0.0704 3.63 1.08 0.0979 1.08 1.60 0.1142 1.05 1.72 0.1241 1.19 1.99 0.1148 3.18 1.74 LE 0.0106 0.56 0.12 0.0204 0.78 0.22 0.0305 1.51 0.33 0.0381 3.86 0.41 0.0551 2.48 0.57 0.0685 2.40 0.74 0.0802 3.34 0.87 0.0900 5.00 1 .Ol MB 0.0287 1.13 0.46 0.0448 1.13 0.84 0.0540 0.99 0.94 0.0616 0.89 1.06 0.0640 0.74 1.17 0.0807 0.78 1.35 0.0817 0.67 1.31 0.0943 0.68 1.69 PGDP 0.0246 0.93 6.48 0.0226 0.66 5.18 0.0273 0.63 7.01 0.0220 0.42 4.66 0.0368 0.54 8.84 0.0255 0.33 5.36 0.0483 0.56 10.58 0.0581 0.60 11.69 R 0.633 1.55 5.90 1.246 1.66 10.95 1.792 1.74 16.40 2.061 1.63 31.26 2.540 1.68 27.84 3.009 1.72 34.06 3.420 1.74 38.70 3.445 1.60 39.01 TT 0.483 1.08 61.23 0.0611 1.40 82.67 0.0756 1.46 110.9 0.0985 3.33 159.7 0.1039 2.24 155.1 0.1213 2.89 223.6 0.1447 2.69 255.5 0.1575 4.30 299.4 WAGE 0.0287 0.33 0.61 0.0357 0.98 0.76 0.0430 0.46 0.92 0.0445 0.85 1.06 0.0595 0.55 1.31 0.0655 0.82 1.49 0.0837 0.67 1.88 0.1037 1.00 2.41 XG 0.0992 1.60 2.01 0.1160 2.22 2.11 0.2094 2.97 4.01 0.2466 4.41 4.62 0.2763 3.41 5.48 0.2868 3.32 5.85 0.3093 3.11 6.52 0.2676 2.84 5.61 1 a Rows one to three 2 show 3 the RMSE, 4 Theil’s U-statistic, 5 6 and the MAPE, respectively. I 8 R. Kunst, Table 2b &-ante forecasts from the RVAR model K. Neusser / Forecasting for the period comparison 1981.1-1984.4. of VAR techniques 453 a Variable Forecast 1 2 GDP 0.0142 0.16 0.22 0.0176 0.17 0.26 0.0192 0.20 0.30 0.225 1.16 0.37 0.0311 0.34 0.52 0.0389 0.36 0.65 0.0464 0.45 0.78 0.0478 1.32 0.80 LE 0.0033 0.18 0.03 0.0059 0.23 0.06 0.0086 0.43 0.10 0.0110 1.11 0.13 0.0144 0.65 0.17 0.0178 0.62 0.20 0.0216 0.90 0.26 0.0242 1.34 0.29 MB 0.0198 0.78 0.34 0.0295 0.74 0.51 0.0335 0.61 0.56 0.0362 0.52 0.56 0.0371 0.43 0.63 0.0400 0.38 0.74 0.0390 0.32 0.74 0.0324 0.23 0.63 PGDP 0.0120 0.45 2.86 0.0166 0.49 4.37 0.0171 0.39 4.58 0.0210 0.40 5.20 0.0225 0.33 5.71 0.0236 0.31 6.12 0.0204 0.24 5.06 0.0218 0.22 5.27 R 0.3066 0.75 2.75 0.5467 0.73 4.74 0.7991 0.77 76.33 0.9691 0.77 9.49 1.231 0.81 11.82 1.501 0.86 14.39 1.741 0.89 17.71 1.916 0.89 20.88 TT 0.0311 0.69 38.01 0.0398 0.91 51.74 0.0480 0.93 61.33 0.0519 1.76 71.51 0.0590 1.27 86.97 0.0639 1.52 95.52 0.0709 1.32 108.9 0.0708 1.94 118.1 WA GE 0.0121 0.14 0.26 0.0111 0.30 0.22 0.0137 0.15 0.31 0.0152 0.29 0.35 0.0240 0.22 0.52 0.0233 0.29 0.53 0.0314 0.25 0.72 0.0343 0.33 0.77 XG 0.0389 0.63 0.76 0.0450 0.86 0.82 0.0600 0.85 1.12 0.0691 1.23 1.37 0.0775 0.96 1.72 0.0889 1.03 2.02 0.0949 0.96 2.20 0.1136 1.21 2.52 a Rows one to three step show the RMSE, 4 3 Theil’s U-statistic, 5 6 and the MAPE, I 8 respectively XG series, an indicator for possible seasonal trends rather than seasonal constants to be present in the data. The performance of UVAR-6 can be improved by setting all lags of order 5 and 6 to zero. This agrees with the supposed deteriorating influence of insignificant parameters. It is worth to contrast this results with the one obtained by Kling and Bessler (1985) who arrive at exactly the opposite conclusion, that is, an improvement in forecasting with an increasing number of free parameters. The forecasts of the terms of trade TT are unsatisfactory with all models. LE and XG show tendencies to get out of control in the long run, especially with UVAR-6 and BVAR-loose. However, it should be mentioned that 1981-84 represents a difficult time range for forecasting with widely unexpected changes in employment levels and foreign trade. It is possible to improve upon these forecasts by scanning over different values of the ‘metaparameters’ 7, d and w. Using the log-determinant of the matrix composed by cross products of 8 quarters-ahead ex-ante forecast errors during the period 1981.1-1984.4 as a criterion function, optimal values of these parameters can be found [Doan, Litterman and Sims (1984, p. 12)]. Applying this method to the data at hand, values of d close to one and w close to zero - leaving T unchanged at 0.1 - have been obtained. This means the ‘optimal data based prior’ would be a univariate 454 R. Kunst, Table 2c E.x-ante forecasts from K. Neusser the BVAR-tight model / Forecasting for the period comparison 1981.1-1984.4. of VAR techniques a Variable Forecast 1 2 3 GDP 0.0147 0.17 0.23 0.0229 0.22 0.36 0.0240 0.26 0.37 0.0221 1.14 0.36 0.0304 0.33 0.47 0.0390 0.36 0.65 0.0372 0.36 0.57 0.0312 0.86 0.55 IE 0.0061 0.33 0.06 0.0075 0.29 0.09 0.0122 0.60 0.14 0.0154 1.56 0.17 0.0202 0.91 0.24 0.0243 0.85 0.30 0.0294 1.22 0.36 0.0325 1.81 0.40 MB 0.0261 1.03 0.45 0.0378 0.95 0.62 0.0385 0.71 0.67 0.0300 0.43 0.47 0.0405 0.47 0.69 0.0502 0.48 0.84 0.0509 0.42 0.93 0.0380 0.27 0.60 PGDP 0.0191 0.72 5.06 0.0241 0.71 6.26 0.0253 0.58 6.00 0.0277 0.52 7.54 0.0354 0.52 8.57 0.0397 0.52 9.34 0.0372 0.43 8.73 0.0368 0.38 9.44 R 0.3569 0.87 3.09 0.6686 0.89 6.09 0.9274 0.90 8.28 1.123 0.89 10.40 1.376 0.91 13.18 1.597 0.91 15.80 1.775 0.91 18.59 1.931 0.90 21.63 TT 0.0409 0.91 58.22 0.0437 1.00 54.70 0.0471 0.91 67.70 0.0319 1.08 51.12 0.0450 0.97 73.31 0.0461 1.10 73.09 0.0554 1.03 95.03 0.0452 1.23 79.97 WA GE 0.0264 0.30 0.57 0.0359 0.98 0.85 0.0293 0.31 0.62 0.0195 0.37 0.45 0.0337 0.31 0.76 0.0466 0.59 1.02 0.0413 0.33 0.89 0.0266 0.26 0.53 XG 0.0419 0.68 0.83 0.0372 0.71 0.74 0.0460 0.865 0.82 0.0443 0.79 0.88 0.0560 0.69 1.09 0.0610 0.71 1.34 0.0533 0.54 1.16 0.0688 0.733 1.50 a Rows one to three step show the RMSE, Theil’s 4 U-statistic, 5 6 and the MAPE, 7 8 respectively. autoregressive model for each of the variables with no cross effects between variables. * This unsatisfactory result is the consequence of a symmetric f(i, j) matrix which treats each of the eight variables in the system alike and could be remedied by putting a weak economic structure on the prior standard deviations. This is done by dividing the set of variables into core variables (in this case GDP, LE, and WAGE) which are thought to be important in explaining all the variables of the system and into the rest (PGDP, MB, R, TT, XG) which are thought to be of lesser importance. This method results in a considerable improvement in the forecasting performance, especially in the variables which have proven to be hard to forecast (R, TT, and XG). Table 3 gives the results from ex-post forecasting and shows severe biases, especially with the ‘good’ models BVAR-tight and RVAR. It might be concluded that UVAR forecasts provide no information relative to no-change but do not show any systematic tendency towards over- or underestimation. As mentioned before, ex-post forecasting methodologically favors non-Bayesian VAR, so the RVAR biases for all but one of the series are even more surprising. This can, however, be explained by regarding the RVAR estimates as pre-test estimates, whose bias is a well-known fact.’ * This result corresponds to the conclusion of Kling and Bessler (1985, b. 15) that univariate AR models are difficult to beat. R. Kunst, K. Neuwr / Forecasting comparison 455 of VA R technrques 6. Summary and conclusions The results in the last sections have demonstrated that the RVAR and BVAR-tight are superior in most respects to the other model specifications. This evidence suggests that the problems associated with an overparametrised model can be avoided by either reducing the number of parameters through exclusion restrictions or by placing prior restrictions on the parameters in a Bayesian way. Using either of these techniques the forecasting performance can be considerably improved. It therefore pays to trade a small bias against more precision in forecasting. RVAR dominates BVAR-tight for shorter forecasting horizons but not over longer ones. It is worth emphasizing that both approaches can be improved. First of all, there is the problem of seasonality. The results showed that fixed seasonal factors are not the best way to handle it. The relative forecasting performance is, however, not affected by this deficiency, since each technique has used the same method. In the case of RVAR, it seems worthwhile to apply several test procedures to Table 3 &-post forecasts Variables of the different models for the period 1981.1 to 1984.4. a Models Bayesian vector autoregression tight Unrestricted vector autoregression loose (6 lags) Restricted vector autoreg. GDP 0.01233 0.01029 4.790 0.00220 0.00601 1.463 - 0.01089 0.00669 0.651 0.03224 0.01715 7.520 LE 0.1258 0.01057 4.760 0.00555 0.00615 3.607 0.00210 0.00458 1.836 0.01522 0.01206 5.048 MB - 0.01202 0.02451 1.962 - 0.00619 0.01910 1.296 - 0.00203 0.01586 0.511 0.00244 0.02278 0.427 PGDP - 0.01584 0.01098 5.769 - 0.00276 0.00833 1.324 0.00058 0.00803 0.290 - 0.00901 0.01050 3.435 R 0.03204 0.89012 0.144 -0.13098 0.40098 1.307 -0.11350 0.29172 1.556 0.78934 1.191 2.650 TT 0.00096 0.03138 0.126 - 0.00126 0.02199 0.230 0.00062 0.01816 0.136 - 0.02980 0.03308 3.603 WA GE 0.00389 0.01885 0.826 o.oOQ49 0.00704 0.279 0.00122 0.00696 0.705 0.02356 0.02330 4.044 XG 0.00543 0.02954 0.736 - 0.00489 Q.02215 0.883 - 0.00172 0.01978 0.347 0.07242 0.05062 5.723 a The first entry for each variable denotes the average forecast reports the value of the T-statistics against the null hypothesis error, the second entry its standard that the mean of the forecast error deviation, is zero. and the third 456 R. Kunst, K. Neusser / Forecasting comparison of VA R techniques refine the specification. This may well lead to a further reduction in the number of parameters, thereby approaching more and more a structural model. Furthermore, alternative subset modeling strategies should be tried, since the final model may depend on the search procedure. A possible disadvantage with this methodology is that as new information becomes available one has, in principle, to start over again with the search procedure since in the meantime the causality structure may have changed. As has been mentioned earlier, the Bayesian approach can be improved by putting more structure based on economic theory into the prior thereby abandoning the symmetric treatment of all variables. This would make the approach more Bayesian in spirit since the prior can now reflect better the a priori beliefs of the investigator. On the other hand the greater flexibility makes it more difficult to find the optimal forecasting model. References Akaike, H., 1974, A new look at statistical model identification, IEEE Transactions on Automatic Control, AC-19, 716-723. Amdt, SW., 1982, The political economy of Austria (American Enterprise Institute, Washington, DC). Doan, T., R. Litterman and C. Sims, 1984, Forecasting and conditional projection using realistic prior distributions, Econometric Reviews 3, l-100. Fair, R., 1979, An analysis of the accuracy of four macroeconometric models, Journal of Political Economy 87, 701-718. Fildes, R., 1985, Quantitative forecasting - the state of the art: Econometric models, Journal of the Operational Research Society 36, no. 7, 549-580. Granger, C.W.J., 1969, Investing causal relations by econometric models and cross-spectral methods, Econometrica 37, 424-438. Haggan, V. and O.B. Oyetunji, 1984, On the selection of subset autoregressive time series models, Journal of Time Series Analysis 5, no. 2, 103-113. Hendry, D.F., A.R. Pagan and J.D. Sargan, 1984, Dynamic specification, in: Z. Griliches and M.D. Intriligator, eds., Handbook of econometrics, Vol. II (North-Holland, Amsterdam) 1023-1100. Jlger, A., 1985, A note on the informational efficiency of Austrian economic forecasts, Empirica 12, no. 2, 247-260. Judge, G.G. and M.E. Brock, 1983, Biased estimation, in: Z. Griliches and M.D. Intriligator, eds., Handbook of econometrics, Vol. I (North-Holland, Amsterdam) 599-649. Kling, J.L. and D.A. Bessler, 1985, A comparison of multivariate procedures for economic time series, International Journal of Forecasting 1, 5-24. Kunst, R.M., 1985, Ein Zeitreihenmodell fur die iistereichische Wirtschaft (A time series model for the Austrian economy), Research memorandum no. 211 (IHS, Vienna). Litterman, R.B., 1980, A Bayesian procedure for forecasting with vector autoregressions, Working paper (Massachusetts Institute of Technology, Cambridge, MA). Lucas, R. and T. Sargent, 1981, After Keynesian macroeconomics, in: R.E. Lucas, Jr. and T.J. Sargent, eds., Rational expectations and econometric practice, Vol. 1 (University of Minnesota Press, Minneapolis, MN) 295-319. McNees, SK., 1981, The methodology of macroeconometric model comparisons, in: J. Kmenta and J.B. Ramsey, eds., Large-scale macroeconometric models (North-Holland, Amsterdam) 397-422. Nelson, C.R. and C.I. Plosser, 1982, Trends and random walks in macroeconomic time series, Journal of Monetary Economics 10,139-162. Pierce and Haugh, 1977, Causality in temporal systems, Journal of Econometrics 5, 265-293. Sims, C.A., 1980, Macroeconomics and reality, Econometrica 48, l-48. Theil, H., 1966, Applied economic forecasting (North-Holland, Amsterdam).