自向量回归模型var

Evaluation of Vector Error Correction Models in comparison with Simkins: Forecasting with Vector Autoregressive (VAR) Models Subject to Business Cycle Restrictions Stefan Zeugner 9851051 Seminar paper for: A. Geyer: Seminar aus Operations Research 3622 Stefan Zeugner Student at Vienna University of Economics and B.A. April 2002 e-mail: h9851051@wu.edu Stefan Zeugner, 9851051 2 Index Introduction...............................................................................................3 Simkins’ unrestricted VAR ........................................................................4 A VAR in logs ...........................................................................................5 VEC models..............................................................................................7 Seasonality – A VEC in the Differences .................................................11 Conclusion..............................................................................................15 References: ............................................................................................16 Appendix 1: Forecasting Results Of the Considered Models .................17 Appendix 2: Model Forecasts for Three Periods in Diagrams ................18 Abstract: This paper is based on the cited article of Scott Simkins (1995): In order of producing macroeconomic forecasts, he constructed a 5-varibale VAR restricted by common characteristics of business cycles in a Monte Carlo procedure. Simkins then evaluated its performance against an unrestricted VAR and a Bayesian VAR and concluded that his procedure was only marginally superior to an unrestricted VAR and that a BVAR analysis performed much better in predicting GNP, unemployment and inflation. In this paper I will show that a slight improvement in the specification of the unrestricted VAR and, even more, Vector Error Correction models produce forecasts able to compete with the BVAR mentioned above. Stefan Zeugner, 9851051 3 Introduction Originally, I was supposed to reproduce the basis of this paper, the article “Forecasting with Vector Autoregressive (VAR) Models Subject to Business Cycle Restrictions”. The author, Prof. Simkins, was primarily interested in an application of business cycle theory to forecasting of five US quarterly macroeconomic time series: Real GNP, GNP deflator, unemployment rate, real fixed investment and money supply (M1) from 1948 to 1990. By applying a turning point procedure (Bry and Boschan), he then distinguished seven completed business cycles in the data. These cycles were normalized by their mean and divided into nine stages – the first trough (start), the peak and the second trough (end) and three successive thirds for the expansion and the contraction phase – and finally the mean of these stages (plus/minus standard deviation bounds) was computed for each of the five variables. Then Simkins estimated an ordinary, 6-lag level VAR and applied multinormal drawing procedures to its parameters and errors in order to conduct a Monte Carlo simulation for the whole sample period. Then the same turning point and stage procedure as above was applied to the simulation outcomes and only those corresponding to the “historical” business cycle patterns (for each variable – this resulted in a selection of about 10% of the simulated paths) were selected as good enough for conducting dynamic forecasts (1 to 8 steps ahead) for three arbitrarily chosen periods (1987:3-1989:2, 1988:2-1990:1, 1989:1-1990:4). The author then evaluated the “fit” of these predictions by Theil’s U-Statistic of GNP, Deflator and unemployment rate forecasts. Besides his own restricted VAR, Simkins did an evaluation of well-known, more established VAR techniques: an unrestricted, normal VAR and a Bayesian VAR (BVAR) with Minnesota Priors (this is a technique of imposing prior distributions near to random walk to the VAR parameters and obtaining the “best” distribution by successive re-estimation of the VAR’s final distribution by Monte Carlo Methods). As I mentioned above, I was supposed to reproduce the paper and I invested a lot of energy in understanding the theory of Bayesian and Monte Carlo techniques in order Stefan Zeugner, 9851051 4 to calculate the restricted VAR model and the BVAR. But the means I had were inadequate: no courses and experts regarding the matter at university, and the software Eviews, Mathematica, MS Excel and VBA. After three days I concluded that accomplishing my task would be a matter of weeks, rather than days. Therefore I chose a different path: Simkins wrote his paper in 1994, when Vector Error Correction (VEC) models did already exist: Nevertheless he did not consider them for evaluating the performance of his own model or for applying his procedure to them. In the following pages, I will show how I estimated VEC models that even beat the Theil U Statistic for BVARs. Moreover I will demonstrate that a slight change in the specification of the VAR (not to levels, but to their logarithms) improves its performance considerably. These types of models are much more simple to estimate than a BVAR or Simkins’ variant (and they could provide a better basis for applying Simkins’ or Bayesian procedures). I will first introduce Simkins’ unrestricted VARs and consider some improvements. Then I will evaluate certain variants of VECs and their different performance regarding different questions. Simkins’ unrestricted VAR Simkins estimated a simple 6-lag VAR with constants, corresponding to a macroeconomic model by Litterman. Figure 1 shows its estimation output. The model was evaluated versus the two others by Theil’s U Statistic: This measure divides the root mean squared error (RMSE) of the models forecast by the RMSE of the naïve forecast: the naïve forecast is simply taking the last value in the sample Deflator Investment M1 GNP unemployment R-squared 0.999950 0.999396 0.999834 0.996594 0.975708 Adj. R-squared 0.999937 0.999247 0.999793 0.995750 0.969685 Sum sq. resids 6.444210 53068.20 610.8618 9375.864 10.92836 S.E. equation 0.230777 20.94230 2.246874 8.802640 0.300528 F-statistic 80186.42 6677.206 24289.37 1180.301 161.9992 Log likelihood 24.53443 -660.6930 -321.3939 -528.9517 -15.60716 Akaike AIC 0.085073 9.101224 4.636762 7.367785 0.613252 Schwarz SC 0.701786 9.717937 5.253474 7.984498 1.229965 Mean dependent 51.99013 2337.464 256.1728 375.7796 5.724013 S.D. dependent 29.12900 763.0011 156.0972 135.0275 1.726051 Determinant Residual Covariance 102.4292 Log Likelihood -1430.210 Akaike Information Criteria 20.85803 Schwarz Criteria 23.94159 Figure 1: Estimation output of Simins’ VAR Stefan Zeugner, 9851051 5 (the forecast’s starting point) as a prediction for the time series. The more the value of Theil’s U Statistic is close to zero, the better is the fit of the underlying forecast. A value below 1 indicates that the model performs better than the naive forecast. This measure is applied to Real GNP, GNP Deflator and unemployment rate forecasts – the performance of Simkins’ three models is shown in Figure 2. The Theil U Statistics for the unrestricted VAR were computed by seven 1 to 8-step ahead forecasts in the period 1987:3 to 1990:4. Thus the starting periods for the seven forecasts are from 1987:2 to 1988:4. It can easily be seen that the dynamic forecasts become the less accurate, the more they are ahead of their starting period. Simkin’s methods are only marginally superior to the predictions by an unrestricted VAR, whereas the Bayesian VAR performs much better than the simple and the “theoretical” VAR. A VAR in logs However, some problems were not considered in this approach: First, a VAR is a linear model, i.e. it does not capture non-linear elements, elements existing certainly in level series of GNP, deflator, money supply and investment (especially concerning Variable Steps ahead (k) Unrestricted VAR model Restricted VAR model Bayesian VAR model Real GNP 1 1.062 1.043 0.303 2 1.227 1.186 0.298 3 1.198 1.132 0.311 4 1.158 1.060 0.366 5 1.124 0.993 0.445 6 1.116 0.977 0.549 7 1.135 1.007 0.648 8 1.157 1.039 0.794 GNP Deflator 1 0.551 0.510 0.290 2 0.651 0.582 0.289 3 0.721 0.605 0.284 4 0.786 0.621 0.274 5 0.838 0.634 0.262 6 0.869 0.631 0.253 7 0.900 0.634 0.252 8 0.941 0.646 0.266 Unemployment Rate 1 3.152 3.117 0.656 2 4.332 4.212 0.635 3 5.601 5.342 0.779 4 6.569 6.124 0.939 5 7.861 7.160 1.302 6 8.227 7.406 1.523 7 8.606 7.750 1.945 8 8.529 7.775 2.458 Figure 2: Theil U statistics of Simkins’ three models’ 1-8-step-ahead Stefan Zeugner, 9851051 6 their exponential growth). The easiest way to respond to this problem is to linearize the data by taking the logs of the levels. Second, the lag length of 6 is not the optimal choice, if one considers selection criteria based on log likelihood. A log-estimation of different lag lengths and choice by the Schwarz criterion (SC) and the Akaike info criterion (AIC) lead to the conclusion that a VAR with lag three would be optimal: SC and AIC are the highest for a 2-lag VAR1, plus one lag for being sure to capture additional information (for the case that the minimum lies between lag two and lag three). One might consider a lag selection by an LR statistic, too: This is a measure of testing the null-hypothesis of adding parameters to the model does not change it significantly towards the “good” direction. Given the high number of added restrictions (parameters of the equations) per lag (25, the number of degrees of freedom in a Chi2-distribution per adding one lag) the resulting p-values prefer the 3-lag model to every higher-lag model2. But Simkins wanted to conduct dynamic forecasts up to eight periods ahead. Considering this aim, more lags would certainly add a bit more “real” information into far-ahead forecasts, even if their short-term performance would suffer. Concerning that, a 6-lag (and maybe an 8-lag) VAR seem to be the best choice because there log-likelihoods increase considerably over lag 5 and 7. Nevertheless, for the sake of a short paper I will only analyze the three lag VAR: The estimation output of such a VAR is shown in Figure 3. 1 The AIC is even higher with higher lag number (6 and 8), but very slightly. 2 The p-values are computed by the following procedure: one minus the Chi-squared distribution of two times the log-likelihood of the lower-lag VAR minus log-likelihood of the higher-lag VAR, with the as many degrees of freedom as the difference of parameters between the two. The resulting p-values are 0.99, 0.567 and 0.687 for the four-, six, and eight-lag VARs versus the three-lag VAR, respectively. Deflator Investment M1 GNP Unemployment R-squared 0.999911 0.997028 0.999874 0.999419 0.967683 Adj. R-squared 0.999902 0.996721 0.999861 0.999359 0.964340 Sum sq. resides 0.004090 0.072261 0.006796 0.012190 0.476889 S.E. equation 0.005311 0.022324 0.006846 0.009169 0.057349 F-statistic 109195.5 3243.435 76727.68 16637.24 289.4563 Log likelihood 623.2854 392.1148 582.4173 535.3801 240.2119 Akaike AIC -7.543918 -4.672234 -7.036240 -6.451926 -2.785241 Schwarz SC -7.237691 -4.366007 -6.730013 -6.145700 -2.479014 Mean dependent 3.837678 5.875014 5.425485 7.710055 1.696497 S.D. dependent 0.537401 0.389856 0.580659 0.362215 0.303693 Determinant Residual Covariance 4.22E-20 Log Likelihood 2448.957 Akaike Information Criteria -29.42804 Schwarz Criteria -27.89691 Figure 3: Estimation output for a VAR in logs (3 lags) Stefan Zeugner, 9851051 7 As seen in the Theil U table in the appendix3, this log-VAR provides a much better fit then the original one, a feature that also marks the covariance matrix of the residuals: A comparison of the two determinants of each’s residual covariance matrix shows a value >100 for the original VAR and a value near to zero for the log-VAR. Since a linearly dependent covariance matrix seems unlikely, the zero-value must be due to very small covariances – but these are caused by the transformation into log-units, and must not be due to a real improvement of the model. The same goes for the “criterions”: The lower AIC and SC values of the log-VAR can not be considered as an improvement since the dependent variable has changed. VEC models As seen by a closer look at Figure 1 and 3, the high R2s of the VAR models in (log- linearized) levels hint at a spurious regression problem. This does not mean that there is no relationship between our five variables, but part of the R2 might only be due to the correlation of integrated data. A unit-root test on the five variables confirms this suspicion: not even the coefficient4 of the unemployment rate is negative enough to reject the null hypothesis of an integrated time series! The residuals of the level- VAR are also integrated, whereas the residuals of the log-VAR are not. 3 The Theil U values for the unrestricted VAR in the appendix differ slightly from its Theil U values in Figure 2. This is due to the fact that Theil U statistics in the appendix are from the VAR I reproduced relying on Simkins‘ paper (and computed in MS Excel), and Figure 1 is copied from Simkins. The difference may be attributed to MS Excel’s minor accuracy concerning matrix operations. 4 The coefficient of a LS regression of the first difference of the unemployment rate versus its value lagged by one period (plus 4 lagged first differences). The unit-root test carried out was an Augmented-Dickey-Fuller Test. Covariance matrix of the VAR in levels Deflator GNP Investment M1 Unempl. Deflator 0.046707 0.321017 -0.155511 -0.045697 -0.007788 GNP 0.321017 62.17747 87.38736 3.884951 -0.859825 Investment -0.155511 87.38736 362.9078 4.918358 -2.739014 M1 -0.045697 3.884951 4.918358 5.019088 -0.107527 Unempl. -0.007788 -0.859825 -2.739014 -0.107527 0.073740 Covariance matrix of the VAR in logs Deflator GNP Investment M1 Unempl. Deflator 2.54E-05 1.25E-05 6.94E-07 -6.41E-07 -4.84E-05 GNP 1.25E-05 0.000449 4.32E-05 9.43E-05 -0.000508 Investment 6.94E-07 4.32E-05 4.22E-05 1.37E-05 -7.15E-05 M1 -6.41E-07 9.43E-05 1.37E-05 7.57E-05 -0.000276 Unempl. -4.84E-05 -0.000508 -7.15E-05 -0.000276 0.002962 Figure 4: Covariance matrices for Simkins’ VAR and the log-VAR Stefan Zeugner, 9851051 8 The first response to this problem would be to estimate a VAR in the first differences (resp. The returns) of our five variables. Nevertheless, some important information may also be contained in the levels of the data: e.g. the so-called “productivity slowdown” in US after-war growth rates (the higher the GNP, the less its growth). Regarding that, a Vector Error Correction (VEC) Model would be the right response, principally a VAR in first differences but with correction restrictions based on the cointegration concept. In order to know if a VEC is appropriate, a cointegration test has to be conducted. Figure 5 summarizes such a test for the number of cointegration relations, and the columns correspond to the five different assumptions concerning the structure of the VEC equations (the number of lags does not change the outcome significantly). According to its output, assumption 45 is selected because it sounds reasonable that imbalances in our four integrated (without the unemployment rate) variables may grow or fall with respect to time. In addition, a number of two cointegration equations may be more realistic regarding the character of our time series and the SC and AIC of assumption four seem more convincing. 5 The cointegration equation contains constants and a linear trend. Series: LOG(DEFL) LOG(GNP) LOG(INV) LOG(MONE) LOG(UNEMP) Lags interval: 1 to 6 Data Trend: None None Linear Linear Quadratic Rank or No Intercept Intercept Intercept Intercept Intercept No. of CEs No Trend No Trend No Trend Trend Trend Akaike Information Criteria by Model and Rank 0 -29.24852 -29.24852 -29.34955 -29.34955 -29.34029 1 -29.34006 -29.46143 -29.55635 -29.55415 -29.55694 2 -29.38920 -29.50464 -29.59161 -29.62532 -29.61359 3 -29.42351 -29.52707 -29.58078 -29.61583 -29.59574 4 -29.40625 -29.49769 -29.48565 -29.55768 -29.54971 5 -29.28650 -29.38928 -29.38928 -29.44938 -29.44938 Schwarz Criteria by Model and Rank 0 -26.42494 -26.42494 -26.43184 -26.43184 -26.32847 1 -26.32823 -26.43078 -26.45041 -26.42938 -26.35687 2 -26.18913 -26.26693 -26.29743 -26.29349 -26.22529 3 -26.03521 -26.08229 -26.09836 -26.07694 -26.01920 4 -25.82971 -25.84585 -25.81499 -25.81172 -25.78493 5 -25.52172 -25.53038 -25.53038 -25.49635 -25.49635 L.R. Test: Rank = 4 Rank = 4 Rank = 2 Rank = 2 Rank = 2 Figure 5: Johansen Cointegration test summarizing five assumptions Stefan Zeugner, 9851051 9 Therefore a VEC with two cointegration equations under assumption four is estimated, one for 3 lags and one for 6 lags. Figure 7 shows their estimation outputs. The two cointegration equations yield the same output regardless of which variables are included in each of them, since they can be transformed linearly. An short look at the two lower tables shows that almost all of the variables depend significantly on at least one cointegration equation. It seems that the trend variable is significant although it has only a slight impact on the outcome (eliminating it worsens the results only slightly) – and at least one cointegration equation is justified. In addition the cointegration relationships provide an opportunity of economic interpretation: If one looks, e.g., at the 6-lag VEC, what effect does the level of GNP have on GNP 3-lag-VEC 6-lag-VEC Standard errors & t-statistics in parentheses Standard errors & t-statistics in parentheses Cointegrating Eq: CointEq1 CointEq2 CointEq1 CointEq2 LOG(DEFLATOR(-1)) 1.000000 0.000000 1.000000 0.000000 LOG(INVESTMENT(-1)) 0.000000 1.000000 0.000000 1.000000 LOG(M_ONE(-1)) -0.210937 -0.575160 -0.466773 -0.062476 (0.29787) (0.23871) (0.14714) (0.08027) (-0.70815) (-2.40943) (-3.17220) (-0.77833) LOG(REALGNP(-1)) 6.397353 -6.304075 3.968452 -0.932555 (2.88127) (2.30905) (1.20802) (0.65899) (2.22033) (-2.73016) (3.28508) (-1.41513) LOG(UNEMPL(-1)) 0.648705 -0.789048 0.323661 -0.028291 (0.40834) (0.32724) (0.16503) (0.09003) (1.58864) (-2.41119) (1.96119) (-0.31425) @TREND(48:1) -0.057139 0.048589 -0.035204 -1.13E-05 (0.02594) (0.02079) (0.01102) (0.00601) (-2.20279) (2.33740) (-3.19345) (-0.00187) C -48.31831 43.10782 -29.50128 1.703781 The cointegration equations in the 3-lag-VEC model Error Correction: D(LOG(DEFLAT OR)) D(LOG(INVEST MENT)) D(LOG(M_ONE)) D(LOG(REALGN P)) D(LOG(UNEMPL )) CointEq1 0.012441 -0.074175 0.022298 -0.039480 0.180855 (0.00656) (0.02735) (0.00871) (0.01198) (0.07367) (1.89511) (-2.71165) (2.56138) (-3.29632) (2.45480) CointEq2 0.002520 -0.172149 0.002547 -0.037476 0.320545 (0.00766) (0.03193) (0.01016) (0.01398) (0.08601) (0.32881) (-5.39081) (0.25066) (-2.68022) (3.72691) The cointegration equations in the 6-lag-VEC model Error Correction: D(LOG(DEFLAT OR)) D(LOG