# Policy inference using VAR models.

POLICY INFERENCE USING VAR MODELS

I. INTRODUCTION

Macroeconomics has a long tradition of examining the response of macroeconomic variables to shocks in alternative policy measures. Much recent work in this area has been based on the empirical regularities between variables found in vector autoregressive (VAR) models. The VAR results in turn have been used to investigate the impact of changes in different policy variables. VAR modeling procedures have been used to test the relative effects of changes in money, interest rates, broad credit measures and fiscal actions on economic activity (1). Moreover, because VAR analysis does not require the specification of an underlying theoretical model, it also has been used to study the linkages among macroeconomic variables across national borders. (2) Although debate over the alleged atheoretical nature of VARs continues--for example, see Cooley and LeRoy [1985!, Leamer [1985! and Braun and Mittnik [1985! -- VAR models are frequently used for policy analysis.

How the dynamic specification of a VAR model is generated and how the dynamics may influence the resulting policy recommendation is the subject of this study. While modeling the dynamic relationship among macro variables is a longstanding subject, relatively few studies have tested for the effects of alternative lag structures on a given model. To our knowledge, there has been little systematic analysis of the sensitivity of policy effects to changes in the lag structure of a given VAR model. (3)

In this paper we investigate this specification problem using a simple four variable VAR model similar in spirit to Sims's [1980b!. (4) This model consists of quarterly observations for money, real output, prices and nominal short-term interest rates for the period 1960 through 1985. In selecting this model, we opt to focus on the impacts of lag specification on policy inferences derived from a given VAR model. In doing so, we ignore the effects of altering the specification in terms of variables included or excluded. This approach seems reasonable, however, since our purpose is to investigate the effects of lag structure, not specification problems in general.

To compare the policy conclusions derived from the alternative dynamic structures, we use six different lag selection criteria. These criteria, briefly described in section II, vary the bias-efficiency tradeoff inherent in such statistical procedures. Section III discusses the estimation of the VAR model using the alternative dynamic structures. Comparison of policy recommendations derived from the models is presented in section IV using variance decompositions. Anticipating that discussion, the evidence from our model suggests that policy recommendations are very sensitive to changes in the dynamic structure. Conclusions close the paper in section V.

II. LAG LENGTH SELECTION PROCEDURES

Many previous researchers have chosen to impose equal lag lengths on each of the variables in their models. (5) This approach produces parameter estimates that may suffer from bias or inefficiency, depending on whether the model is under or over parameterized. We use six different approaches to determine the lag specification: five are based on statistical criteria, with the exact criteria listed in Table I. For comparability, we also employ the nonstatistical, ad hoc procedure of arbitrarily fixing the maximum lag on all variables to be four quarters.

One of the statistical approaches used is the standard F-test. (6) The F-test imposes zero restrictions on lags beyond some point and tests this restricted model against a more general lag structure. The F-statistic is calculated sequentially until adding another lag does not statistically improve the fit of the model. Two of the selection procedures use a mean squared error criterion: these are Mallows's [1973! [C.sub.p! statistic and Akaike's [1970! Final Prediction Error (FPE) criterion. both [C.sub.p! and FPE trade off bias from selecting lag lengths that are too short against a loss in efficiency caused by selecting lag lengths that are too long. The FPE procedure was suggested by Hsian [1979 1981! for use in Granger-type causality tests and more recently has been used by McMillin and Fackler [1984! and Chowdhury, Fackler and McMillin [1986! in estimating VAR models. We also use two approaches grounded in Bayesian rules of lag choice. One is Schwarz's [1978! Bayesian Information Criterion (BIC) and the other, due to Geweke and Meese [1981!, is called the Bayesian Estimation Criterion (BEC). These criteria offer a useful contrast to [C.sub.p! and FPE, because the Bayesian procedures are characterized as trading off bias for more efficiency. Indeed, as noted by Geweke and Meese, both the BIC and BEC criteria tend to underfit in small samples, a function of placing relatively more weight on efficiency. Their Monte Carlo experiments indicate that the probability of underfitting the model is relatively large (50 percent) in small samples. (7)

III. LAG LENGTH SELECTION RESULTS

The six lag length selection procedures are used to determine the structure of a representative four-variable VAR system. Because our expressed purpose is to examine the sensitivity of policy inferences from our VAR model to alternative lag lengths, we do not experiment with changing the variables included in the model or the estimation period. Moreover, because the algorithm used here searches the lags for the entire model simultaneously, restricting the number of variables to four was not only one of choice, but also of necessity. (8) Thus, the estimated VAR models differ only in terms of the lag lengths selected.

The specific variables included in our model are the money stock (M1), real GNP (in 1982 dollars), the implicit GNP price deflator (1982 = 100) and the three-month Treasury bill rate (hereafter T-bill). (9) All variables (except the T-bill) are seasonally adjusted, quarterly values and are transformed into stationary series by taking first-differences in the logarithms. The T-bill measure is calculated as the first difference in its quarterly average level. The estimation period is 1960:I to 1985:IV. (10)

Application of the various statistical criteria requires the (arbitrary) selection of a maximum allowed lag length. Geweke [1978! has argued that specifying a maximum lag that is too short may impose unwarranted zero restrictions. To simultaneously keep the lost degrees of freedom manageable and allow a reasonable lag set for the procedures to choose from, the maximum lag length was set at twelve.

The results from using the lag length selection procedures on the data are reported in Table II. The results dramatically illustrate the fact that different statistical procedures often produce a wide range of lag structures. For example, the lags found on money (DM1) with inflation (DP) as the dependent variable range from five using the F-test criterion to zero based on the [C.sub.p!, BIC and BEC procedures. The influence of real GNP (DRGNP) on the T-bill rate (DTB) also varies substantially, from long lags (ten or more) in the F, [C.sub.p! and FPE determined models to none in the Bayesian models. (11) This variety of dynamic structures reflects the different bias-efficiency tradeoffs inherent in each procedure. The results generally are consistent with previous findings in that the Bayesian criteria yield parsimonious lag structures, the F the longest and [C.sub.p! and FPE somewhere in between.

The variety of lag lengths for each variable is reduced somewhat by comparing lag selections within procedures of similar bias-efficiency tradeoff. Lag lengths within such groups generally are similar that is, in only three instances do the [C.sub.p! and FPE results differ, while the BIC and BEC criteria produce different lags in only three cases. Given the different bias-efficiency tradeoff between these two groups, however, differences in lag selection between, say, the FPE and BEC outnumber similar choices by about two to one. While this outcome should not be surprising, it reinforces our concern that the choice of lag lengths in previous VAr analyses may account for the reported divergence in results and subsequent policy conclusions.

IV. VARIANCE DECOMPOSITION RESULTS

We compare policy outcomes from our VAR model by estimating each of the five lag structures in Table II and a model that arbitrarily imposes four lags on each variable. To compare the outcomes, we use the variance decompositions for a twenty-quarter horizon. (12)

To cast the discussion in a style similar to several previous analyses, we report the variance decompositions based on the presumption that money is the policy variable. In doing so, let us be clear that our purpose is not to discover whether, say, the money or the interest rate variable better explains the behavior of other variables and therefore should be the policymakers' choice variable. Instead, presuming that M1 is the policy variable allows us to focus exclusively on the sensitivity of relevant policy implications to alternative dynamic structures. It should be noted that our concern about the usefulness of VARs for policy analysis is not sensitive to the a priori choice of a policy variable. Following suggestions from previous research, we report the outcome for two orderings. Results obtained from ordering M1 first are reported in Table III, while Table IV presents the outcomes when M1 is ordered last. (13)

The results in Table III are striking. Looking down the DM1 column, the impact of money on prices, output and interest rates vary wildly across the different lag structures. For example, shocks in M1 explain 12 percent of the variation in inflation using the FPE lag structure but none of the variation when the BIC and BEC lag structures are used. Such differences between lag choices are important in light of previous work on lag length selection. For instance, Geweke and Meese [1981! and Lutkepohl [1985! have demonstrated that the Bayesian rules tend to select the correct lag specification more often than model based procedures, such as the Cp. Thus, the results for the F, FPE and Cp criterion, all of which tend asymptoitically to choose longer lag models, may not reflect the economic relationship inherent in the data.

An ongoing debate has focused on the relative roles of money and interest rates in explaining the behavior of real output. The relative effect of M1 and the T-bill rate on real output is, therefore, of interest. (14) From Table III we see that shocks to the T-bill rate dominate M1 shocks in explaining the variation of output for three lag structures: the fixed-four (20 percent vs. 6 percent), the F (41 percent vs. 13 percent) and the FPE (14 percent vs. 9 percent). The outcome using the [C.sub.p! lag structure is close, with the edge going to the T-bill rate (13 percent vs. 10 percent). Recall, however, that these lag selection processes are characterized by selecting a model that is never too small, but also one whose estimates may be inefficient.

If one uses the BIC criterion, however, shocks to M1 explain more of the variation in output than do interest rates (12 percent vs. 0 percent). In contrast, the BEC results suggest that money (3 percent) and interest rates (0 percent) both play no apparent role in explaining the behavior of real output. The difference between the BIC and BEC results is especially troublesome because these selection procedures use similar criteria.

The evidence in Table III indicates, given our model, that if one attempted to explain output growth based on the evidence from only one lag length selection procedure, empirical support is found for the mutually exclusive hypotheses that (a) interest rates dominate money, (b) money dominates interest rates or (c) neither money or interest rates play a significant role in explaining variations in real output growth. Different implications follow even if one focuses only on the results derived from the models constructed using the Bayesian lag selection procedures. The qualitative and quantitative differences in M1's effect on output (as well as inflation and interest rates) are large enough to rule out a reliable conclusion about M1's role without prior specification of the lags.

The variance decompositions from a M1-last ordering are presented in Table IV. Although there are some instances where changing M1's ordering greatly influences its role in explaining the variation of the dependent variable, most changes from Table III are relatively small. More importantly, reordering the variables does not reduce the uncertainty about M1's role in explaining variation in the other variables. Again contrasting M1 with the T-bill, the variance decompositions in Table IV show M1 shocks dominating T-bill shocks in explaining inflation for the fixed-four and FPE lag structures and near equality for the other lag structures. Turning to the results for output, again we discover that

across the six alternative lag structures, interest rate shocks dominate money shocks in explaining output three times, money dominates once and twice the outcomes are close. As in Table III, the relative importance of money or interest rate shocks in explaining output behavior hinges on the choice of the lag structure used.

Finally, we calculated 90 percent confidence intervals for the different variance decompositions following the procedure in Runkle [1987!. (15) These results do not change our conclusions and, therefore, are not reported. Alternative lag structures yield non-overlapping confidence intervals suggesting different policy conclusions. An interesting aspect emerges from such calculations. Runkle [1987! calculates the confidence intervals for the variance decompositions of a VAR model that imposes equal lag lengths on each variable. If lag lengths in fact are not equal on all variables, the estimated confidence intervals imposing equal lag lengths may be relatively wide. Our estimated confidence intervals using the relatively short Bayesian lag structures were much tighter than those found for the longer lag structures, in particular for the fixed-four results. This finding suggests that Runkle's results may stem in part from imposing the same lag structures across all variables. We plan to pursue this issue in future research.

V. CONCLUSIONS

The conclusions derived from the evidence presented here can be stated succinctly. First, because no hard rule exists for the selection of lags in a multivariate setting, VAR modelers must explicitly concern themselves with the choice of dynamic structure. Second, within the confines of our model, policy inferences are shown to be quite sensitive to different statistical procedures for choosing lag structures. In addition, our evidence suggests that selecting one policy recommendation over another can result purely from relatively minor differences in the dynamic structure. While previous researchers have argued that one should check the sensitivity of VAR results to alternative orderings of the variables, our evidence indicates that VAR model builders should also realize that a different lag structure could easily alter policy conclusions.

(*1) Professor of Economics, Southern Illinois University at Edwardsville. Associate Professor of Finance, University of Notre Dame. We would like to thank Stuart Allen, Sandy Batten, James Fackler, Benjamin Friedman, Scott Hein, James Kennedy, Stefan Mittnik, Dan Thornton, Roy Webb, Arnold Zellner, two anonymous referees and Richard Sweeney for useful comments and criticisms on earlier drafts. The usual caveat applies. We also thank Tom Pollmann for his research assistance. An earlier version of this paper was completed while Hafer was a Research Officer at the Federal Reserve Bank of St. Louis.

(1.) See, for example, Sims [1980a 1980b 1982 1986!, King [1983!, Friedman [1983!, McMillin and Fackler [1984!, Chowdhury, Fackler and McMillin [1986! and Evans [1986!.

(2.) Kuszczak and Murray [1986! and Genberg, Salemi and Swoboda [1987! provide two recent examples of such analysis.

(3.) Batten and Thornton [1984! examine the robustness of policy conclusions from the reduced-form "St. Louis" equation to changes in lag structure. Nickelsburg [1985! investigates the question of how well alternative lag specification procedures select the structure of a VAR model when that structure is known. Based on his Monte Carlo simulations, he concludes that no criterion is consistently able to select the true lag structure.

(4.) This model has also been used by Ohanian [1988! to examine the impact of a random walk on a VAR model.

(5.) For example, Fischer [1981! fixes lags at three quarters for each variable Sims [1980b! fixes lags at four for each variable based on a test between four or eight King [1983! fixes the lags at six quarters for each variable and Friedman [1983! fixes the lags at eight quarters.

(6.) The following discussion draws from Batten and Thornton [1984!.

(7.) This feature of the Bayesian procedures also was noted in tests of the St. Louis equation and of the money-income relationship (Batten and Thornton [1984! and Thornton and Batten [1985!). Nicklesburg [1985! also finds that the Bayesian criteria tend to underfit the true lag structure.

(8.) The algorithm uses the Gram-Schmidt decomposition to reduce the total number of regressions from [(L+1).sup.v! to [(L+1).sup.v-1! where L is the maximum allowed lag length, and v is the number of variables. As the reader easily can see, the addition of one more variable would increase the number of regressions from 2,197 to 28,561 and would make the estimation, even at a Federal Reserve Bank, prohibitively costly.

(9.) The data are taken from the Federal Reserve Bank of St. Louis data bank and are available upon request.

(10.) Use of the 1960-85 sample period may bother some readers concerned about the effects of estimating the models across the pre- and post-1979 periods. Such a concern is valid if we were attempting to establish some policy recommendation. Under current circumstances, however, we are not as concerned about any specific policy outcome, but the sensitivity of representative policy recommendations derived from a given VAR model.

(11.) Some may argue that the shorter lag length models are nested within the larger and, therefore, are subject to testing against the larger model. Thornton and Batten [1985! have noted, however, that application of a standard F or [X.sup.2! test for lag restrictions imposes a different criterion that confounds the bias-efficiency tradeoff characterizing each lag selection procedure. Consequently, we treat the different criteria's lag selections as if they are the "true" lag structures.

(12.) VAR models that impose equal lag lengths across the variables are estimated using OLS. Because other models use various lag lengths in different equations, the estimation procedure is GLS. The estimated equations are used to generate both impulse response functions and variance decompositions. To conserve space, only the variance decompositions are reported below. Our basic conclusion that the choice of a lag length selection procedure may substantially influence policy inference follows both from the variance decompositions and the impulse response functions. In addition, since our quantitative results are not sensitive to horizon length, we report only the outcomes for the twenty-quarter horizon.

(13.) To conserve space, the correlation matrices among the innovations are not reported. In all cases, these correlations are relatively low.

(14.) See, among others, Sims [1980a 1980b!, King [1983! and Litterman and Weiss [1985!.

(15.) We would like to thank David Runkle for providing us with his algorithm.

REFERENCES

Akaike, Hirotugu. "Statistical Predictor Identification." Annals of the Institute of Statistical Mathematics 22(1), 1970, 203-17.

Batten, Dallas S. and Daniel L. Thornton. "How Robust are the Policy Conclusions of the St. Louis Equation?: Some Further Evidence." Federal Reserve Bank of St. Louis Review, June/July 1984, 26-32.

Braun, Phillip A. and Stefan Mittnik. "Structural Analysis with Vector Autoregressive Models: Some Experimental Evidence." Working Paper No. 84, Department of Economics, Washington University, December 1985.

Chowdhury, Abdur R., James S. Fackler and W. Douglas McMillin. "Monetary Policy, Fiscal Policy, and Investment Spending: An Empirical Analysis." Southern Economic Journal, January 1986, 794-806.

Cooley, Thomas F. and Steven F. LeRoy." Atheoretical Macroeconometrics: A Critique." Journal of Monetary Economics, November 1985, 283-308.

Evans, Paul. "It the Dollar Higher Because of Large Budget Deficits?" Journal of Monetary Economics, November 1986, 227-49.

Fischer, Stanley. "Relative Shocks, Relative Price Variability, and Inflation." Brookings Papers on Economic Activity 2, 1981, 381-431.

Friedman, Benjamin M. "The Roles of Money and Credit in Macroeconomic Analysis," in Macroeconomics, Prices and Quantities: Essays in Memory of Arthur M. Okun, edited by J. Tobin. Washington, D.C.: The Brookings Institution, 1983, 161-99.

Genberg, Hans, Michael K. Salemi, and Alexander Swoboda. "The Relative Importance of Foreign and Domestic Disturbances for Aggregate Fluctuations in the Open Economy: Switzerland, 1964-81." Journal of Monetary Economics, January 1987, 45-67.

Geweke, John. "Testing the Exogeneity Specification in the Complete Dynamic Simultaneous Equation Model." Journal of Econometrics, 7(1), 1978, 163-85.

Geweke, John and Richard Meese. "Estimating Regression Models of Finite But Unknown Order." International Economic Review, February 1981, 55-70.

Gordon, Robert J. and Stephen R. King. "The Output Cost of Disinflation in Traditional and Vector Autoregressive Models." Brookings Papers on Economic Activity 1, 1982, 205-42.

Hsiao, Cheng. "Autoregressive Modeling and Money Income Causality Detection." Journal of Monetary Economics, January 1981, 85-106.

_____. "Autoregressive Modeling of Canadian Money and Income Data." Journal of the American Statistical Association, September 1979, 533-60.

King, Stephen R. "Interest Rates and the Influence of Money on Output and Prices." Unpublished manuscript, Stanford University, August 1983.

Kuszczak, John and John D. Murray. "A VAR Analysis of Economic Interdependence: Canada, the United States and the Rest of the World," in How Open is the U.S. Economy? edited by R.W. Hafer. Lexington: Lexington Books, 1986, 77-131.

Leamer, Edward E. "Vector Autoregressions for Causal Inference?" Carnegie-Rochester Conference Series on Public Policy: Understanding Monetary Regimes, Spring 1985, 255-304.

Litterman, Robert B. and Laurence Weiss. "Money, Real Interest Rates, and Output: A Reinterpretation of Post-War U.S. Data." Econometrica, January 1985, 129-56.

Lutkepohl, Helmut. "Comparison of Criteria for Estimating the Order of a Vector Autoregressive Process." Journal of Time Series Analysis 6(1), 1985, 35-52.

Mallows, C. L. "Some Comments on [C.sub.p!. Technometrics, November 1973, 661-75.

McMillin, W. Douglas and James S. Fackler. "Monetary vs. Credit Aggregates: An Evaluation of Monetary Policy Targets." Southern Economic Journal, January 1984, 711-23.

Nickelsburg, Gerald. "Small-Sample Properties of Dimensionality Statistics for Fitting VAR Models to Aggregate Economic Data." Journal of Econometrics 28(2), 1985, 183-92.

Ohanian, Lee E. "The Spurious Effects of Unit Roots on Vector Autoregressions: A Monte Carlo Study." Journal of Econometrics, November 1988, 251-66.

Runkle, David E. "Vector Autoregressions and Reality." Journal of Business and Economic Statistic, October 1987, 437-42.

Schwarz, Gideon. "Estimating the Dimension of a Model." The Annals of Statistics, March 1978, 461-64.

Sims, Christopher A. "Are Forecasting Models Useful for Policy Analysis?" Federal Reserve Bank of Minneapolis Quarterly Review, Winter 1986, 2-16.

_____. "Policy Analysis with Econometric Models," Brookings Papers on Economic Activity 1, 1982, 107-52.

_____. "Comparison of Interwar and Postwar Business Cycles: Monetarism Reconsidered." American Economic Review, Papers and Proceedings, May 1980b, 250-57.

_____. "Macroeconomics and Reality." Econometrica, January 1980a, 1-48.

Thornton, Daniel L. and Dallas S. Batten. "Lag-Length Selection and Tests of Granger Causality Between Money and Income." Journal of Money, Credit and Banking, May 1985, 164-78.

I. INTRODUCTION

Macroeconomics has a long tradition of examining the response of macroeconomic variables to shocks in alternative policy measures. Much recent work in this area has been based on the empirical regularities between variables found in vector autoregressive (VAR) models. The VAR results in turn have been used to investigate the impact of changes in different policy variables. VAR modeling procedures have been used to test the relative effects of changes in money, interest rates, broad credit measures and fiscal actions on economic activity (1). Moreover, because VAR analysis does not require the specification of an underlying theoretical model, it also has been used to study the linkages among macroeconomic variables across national borders. (2) Although debate over the alleged atheoretical nature of VARs continues--for example, see Cooley and LeRoy [1985!, Leamer [1985! and Braun and Mittnik [1985! -- VAR models are frequently used for policy analysis.

How the dynamic specification of a VAR model is generated and how the dynamics may influence the resulting policy recommendation is the subject of this study. While modeling the dynamic relationship among macro variables is a longstanding subject, relatively few studies have tested for the effects of alternative lag structures on a given model. To our knowledge, there has been little systematic analysis of the sensitivity of policy effects to changes in the lag structure of a given VAR model. (3)

In this paper we investigate this specification problem using a simple four variable VAR model similar in spirit to Sims's [1980b!. (4) This model consists of quarterly observations for money, real output, prices and nominal short-term interest rates for the period 1960 through 1985. In selecting this model, we opt to focus on the impacts of lag specification on policy inferences derived from a given VAR model. In doing so, we ignore the effects of altering the specification in terms of variables included or excluded. This approach seems reasonable, however, since our purpose is to investigate the effects of lag structure, not specification problems in general.

To compare the policy conclusions derived from the alternative dynamic structures, we use six different lag selection criteria. These criteria, briefly described in section II, vary the bias-efficiency tradeoff inherent in such statistical procedures. Section III discusses the estimation of the VAR model using the alternative dynamic structures. Comparison of policy recommendations derived from the models is presented in section IV using variance decompositions. Anticipating that discussion, the evidence from our model suggests that policy recommendations are very sensitive to changes in the dynamic structure. Conclusions close the paper in section V.

II. LAG LENGTH SELECTION PROCEDURES

Many previous researchers have chosen to impose equal lag lengths on each of the variables in their models. (5) This approach produces parameter estimates that may suffer from bias or inefficiency, depending on whether the model is under or over parameterized. We use six different approaches to determine the lag specification: five are based on statistical criteria, with the exact criteria listed in Table I. For comparability, we also employ the nonstatistical, ad hoc procedure of arbitrarily fixing the maximum lag on all variables to be four quarters.

One of the statistical approaches used is the standard F-test. (6) The F-test imposes zero restrictions on lags beyond some point and tests this restricted model against a more general lag structure. The F-statistic is calculated sequentially until adding another lag does not statistically improve the fit of the model. Two of the selection procedures use a mean squared error criterion: these are Mallows's [1973! [C.sub.p! statistic and Akaike's [1970! Final Prediction Error (FPE) criterion. both [C.sub.p! and FPE trade off bias from selecting lag lengths that are too short against a loss in efficiency caused by selecting lag lengths that are too long. The FPE procedure was suggested by Hsian [1979 1981! for use in Granger-type causality tests and more recently has been used by McMillin and Fackler [1984! and Chowdhury, Fackler and McMillin [1986! in estimating VAR models. We also use two approaches grounded in Bayesian rules of lag choice. One is Schwarz's [1978! Bayesian Information Criterion (BIC) and the other, due to Geweke and Meese [1981!, is called the Bayesian Estimation Criterion (BEC). These criteria offer a useful contrast to [C.sub.p! and FPE, because the Bayesian procedures are characterized as trading off bias for more efficiency. Indeed, as noted by Geweke and Meese, both the BIC and BEC criteria tend to underfit in small samples, a function of placing relatively more weight on efficiency. Their Monte Carlo experiments indicate that the probability of underfitting the model is relatively large (50 percent) in small samples. (7)

III. LAG LENGTH SELECTION RESULTS

The six lag length selection procedures are used to determine the structure of a representative four-variable VAR system. Because our expressed purpose is to examine the sensitivity of policy inferences from our VAR model to alternative lag lengths, we do not experiment with changing the variables included in the model or the estimation period. Moreover, because the algorithm used here searches the lags for the entire model simultaneously, restricting the number of variables to four was not only one of choice, but also of necessity. (8) Thus, the estimated VAR models differ only in terms of the lag lengths selected.

The specific variables included in our model are the money stock (M1), real GNP (in 1982 dollars), the implicit GNP price deflator (1982 = 100) and the three-month Treasury bill rate (hereafter T-bill). (9) All variables (except the T-bill) are seasonally adjusted, quarterly values and are transformed into stationary series by taking first-differences in the logarithms. The T-bill measure is calculated as the first difference in its quarterly average level. The estimation period is 1960:I to 1985:IV. (10)

Application of the various statistical criteria requires the (arbitrary) selection of a maximum allowed lag length. Geweke [1978! has argued that specifying a maximum lag that is too short may impose unwarranted zero restrictions. To simultaneously keep the lost degrees of freedom manageable and allow a reasonable lag set for the procedures to choose from, the maximum lag length was set at twelve.

The results from using the lag length selection procedures on the data are reported in Table II. The results dramatically illustrate the fact that different statistical procedures often produce a wide range of lag structures. For example, the lags found on money (DM1) with inflation (DP) as the dependent variable range from five using the F-test criterion to zero based on the [C.sub.p!, BIC and BEC procedures. The influence of real GNP (DRGNP) on the T-bill rate (DTB) also varies substantially, from long lags (ten or more) in the F, [C.sub.p! and FPE determined models to none in the Bayesian models. (11) This variety of dynamic structures reflects the different bias-efficiency tradeoffs inherent in each procedure. The results generally are consistent with previous findings in that the Bayesian criteria yield parsimonious lag structures, the F the longest and [C.sub.p! and FPE somewhere in between.

The variety of lag lengths for each variable is reduced somewhat by comparing lag selections within procedures of similar bias-efficiency tradeoff. Lag lengths within such groups generally are similar that is, in only three instances do the [C.sub.p! and FPE results differ, while the BIC and BEC criteria produce different lags in only three cases. Given the different bias-efficiency tradeoff between these two groups, however, differences in lag selection between, say, the FPE and BEC outnumber similar choices by about two to one. While this outcome should not be surprising, it reinforces our concern that the choice of lag lengths in previous VAr analyses may account for the reported divergence in results and subsequent policy conclusions.

IV. VARIANCE DECOMPOSITION RESULTS

We compare policy outcomes from our VAR model by estimating each of the five lag structures in Table II and a model that arbitrarily imposes four lags on each variable. To compare the outcomes, we use the variance decompositions for a twenty-quarter horizon. (12)

To cast the discussion in a style similar to several previous analyses, we report the variance decompositions based on the presumption that money is the policy variable. In doing so, let us be clear that our purpose is not to discover whether, say, the money or the interest rate variable better explains the behavior of other variables and therefore should be the policymakers' choice variable. Instead, presuming that M1 is the policy variable allows us to focus exclusively on the sensitivity of relevant policy implications to alternative dynamic structures. It should be noted that our concern about the usefulness of VARs for policy analysis is not sensitive to the a priori choice of a policy variable. Following suggestions from previous research, we report the outcome for two orderings. Results obtained from ordering M1 first are reported in Table III, while Table IV presents the outcomes when M1 is ordered last. (13)

The results in Table III are striking. Looking down the DM1 column, the impact of money on prices, output and interest rates vary wildly across the different lag structures. For example, shocks in M1 explain 12 percent of the variation in inflation using the FPE lag structure but none of the variation when the BIC and BEC lag structures are used. Such differences between lag choices are important in light of previous work on lag length selection. For instance, Geweke and Meese [1981! and Lutkepohl [1985! have demonstrated that the Bayesian rules tend to select the correct lag specification more often than model based procedures, such as the Cp. Thus, the results for the F, FPE and Cp criterion, all of which tend asymptoitically to choose longer lag models, may not reflect the economic relationship inherent in the data.

An ongoing debate has focused on the relative roles of money and interest rates in explaining the behavior of real output. The relative effect of M1 and the T-bill rate on real output is, therefore, of interest. (14) From Table III we see that shocks to the T-bill rate dominate M1 shocks in explaining the variation of output for three lag structures: the fixed-four (20 percent vs. 6 percent), the F (41 percent vs. 13 percent) and the FPE (14 percent vs. 9 percent). The outcome using the [C.sub.p! lag structure is close, with the edge going to the T-bill rate (13 percent vs. 10 percent). Recall, however, that these lag selection processes are characterized by selecting a model that is never too small, but also one whose estimates may be inefficient.

If one uses the BIC criterion, however, shocks to M1 explain more of the variation in output than do interest rates (12 percent vs. 0 percent). In contrast, the BEC results suggest that money (3 percent) and interest rates (0 percent) both play no apparent role in explaining the behavior of real output. The difference between the BIC and BEC results is especially troublesome because these selection procedures use similar criteria.

The evidence in Table III indicates, given our model, that if one attempted to explain output growth based on the evidence from only one lag length selection procedure, empirical support is found for the mutually exclusive hypotheses that (a) interest rates dominate money, (b) money dominates interest rates or (c) neither money or interest rates play a significant role in explaining variations in real output growth. Different implications follow even if one focuses only on the results derived from the models constructed using the Bayesian lag selection procedures. The qualitative and quantitative differences in M1's effect on output (as well as inflation and interest rates) are large enough to rule out a reliable conclusion about M1's role without prior specification of the lags.

The variance decompositions from a M1-last ordering are presented in Table IV. Although there are some instances where changing M1's ordering greatly influences its role in explaining the variation of the dependent variable, most changes from Table III are relatively small. More importantly, reordering the variables does not reduce the uncertainty about M1's role in explaining variation in the other variables. Again contrasting M1 with the T-bill, the variance decompositions in Table IV show M1 shocks dominating T-bill shocks in explaining inflation for the fixed-four and FPE lag structures and near equality for the other lag structures. Turning to the results for output, again we discover that

across the six alternative lag structures, interest rate shocks dominate money shocks in explaining output three times, money dominates once and twice the outcomes are close. As in Table III, the relative importance of money or interest rate shocks in explaining output behavior hinges on the choice of the lag structure used.

Finally, we calculated 90 percent confidence intervals for the different variance decompositions following the procedure in Runkle [1987!. (15) These results do not change our conclusions and, therefore, are not reported. Alternative lag structures yield non-overlapping confidence intervals suggesting different policy conclusions. An interesting aspect emerges from such calculations. Runkle [1987! calculates the confidence intervals for the variance decompositions of a VAR model that imposes equal lag lengths on each variable. If lag lengths in fact are not equal on all variables, the estimated confidence intervals imposing equal lag lengths may be relatively wide. Our estimated confidence intervals using the relatively short Bayesian lag structures were much tighter than those found for the longer lag structures, in particular for the fixed-four results. This finding suggests that Runkle's results may stem in part from imposing the same lag structures across all variables. We plan to pursue this issue in future research.

V. CONCLUSIONS

The conclusions derived from the evidence presented here can be stated succinctly. First, because no hard rule exists for the selection of lags in a multivariate setting, VAR modelers must explicitly concern themselves with the choice of dynamic structure. Second, within the confines of our model, policy inferences are shown to be quite sensitive to different statistical procedures for choosing lag structures. In addition, our evidence suggests that selecting one policy recommendation over another can result purely from relatively minor differences in the dynamic structure. While previous researchers have argued that one should check the sensitivity of VAR results to alternative orderings of the variables, our evidence indicates that VAR model builders should also realize that a different lag structure could easily alter policy conclusions.

(*1) Professor of Economics, Southern Illinois University at Edwardsville. Associate Professor of Finance, University of Notre Dame. We would like to thank Stuart Allen, Sandy Batten, James Fackler, Benjamin Friedman, Scott Hein, James Kennedy, Stefan Mittnik, Dan Thornton, Roy Webb, Arnold Zellner, two anonymous referees and Richard Sweeney for useful comments and criticisms on earlier drafts. The usual caveat applies. We also thank Tom Pollmann for his research assistance. An earlier version of this paper was completed while Hafer was a Research Officer at the Federal Reserve Bank of St. Louis.

(1.) See, for example, Sims [1980a 1980b 1982 1986!, King [1983!, Friedman [1983!, McMillin and Fackler [1984!, Chowdhury, Fackler and McMillin [1986! and Evans [1986!.

(2.) Kuszczak and Murray [1986! and Genberg, Salemi and Swoboda [1987! provide two recent examples of such analysis.

(3.) Batten and Thornton [1984! examine the robustness of policy conclusions from the reduced-form "St. Louis" equation to changes in lag structure. Nickelsburg [1985! investigates the question of how well alternative lag specification procedures select the structure of a VAR model when that structure is known. Based on his Monte Carlo simulations, he concludes that no criterion is consistently able to select the true lag structure.

(4.) This model has also been used by Ohanian [1988! to examine the impact of a random walk on a VAR model.

(5.) For example, Fischer [1981! fixes lags at three quarters for each variable Sims [1980b! fixes lags at four for each variable based on a test between four or eight King [1983! fixes the lags at six quarters for each variable and Friedman [1983! fixes the lags at eight quarters.

(6.) The following discussion draws from Batten and Thornton [1984!.

(7.) This feature of the Bayesian procedures also was noted in tests of the St. Louis equation and of the money-income relationship (Batten and Thornton [1984! and Thornton and Batten [1985!). Nicklesburg [1985! also finds that the Bayesian criteria tend to underfit the true lag structure.

(8.) The algorithm uses the Gram-Schmidt decomposition to reduce the total number of regressions from [(L+1).sup.v! to [(L+1).sup.v-1! where L is the maximum allowed lag length, and v is the number of variables. As the reader easily can see, the addition of one more variable would increase the number of regressions from 2,197 to 28,561 and would make the estimation, even at a Federal Reserve Bank, prohibitively costly.

(9.) The data are taken from the Federal Reserve Bank of St. Louis data bank and are available upon request.

(10.) Use of the 1960-85 sample period may bother some readers concerned about the effects of estimating the models across the pre- and post-1979 periods. Such a concern is valid if we were attempting to establish some policy recommendation. Under current circumstances, however, we are not as concerned about any specific policy outcome, but the sensitivity of representative policy recommendations derived from a given VAR model.

(11.) Some may argue that the shorter lag length models are nested within the larger and, therefore, are subject to testing against the larger model. Thornton and Batten [1985! have noted, however, that application of a standard F or [X.sup.2! test for lag restrictions imposes a different criterion that confounds the bias-efficiency tradeoff characterizing each lag selection procedure. Consequently, we treat the different criteria's lag selections as if they are the "true" lag structures.

(12.) VAR models that impose equal lag lengths across the variables are estimated using OLS. Because other models use various lag lengths in different equations, the estimation procedure is GLS. The estimated equations are used to generate both impulse response functions and variance decompositions. To conserve space, only the variance decompositions are reported below. Our basic conclusion that the choice of a lag length selection procedure may substantially influence policy inference follows both from the variance decompositions and the impulse response functions. In addition, since our quantitative results are not sensitive to horizon length, we report only the outcomes for the twenty-quarter horizon.

(13.) To conserve space, the correlation matrices among the innovations are not reported. In all cases, these correlations are relatively low.

(14.) See, among others, Sims [1980a 1980b!, King [1983! and Litterman and Weiss [1985!.

(15.) We would like to thank David Runkle for providing us with his algorithm.

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Title Annotation: | variables in policy changes; vector autoregressive models |
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Author: | Hafer, R.W.; Sheehan, Richard G. |

Publication: | Economic Inquiry |

Date: | Jan 1, 1991 |

Words: | 3755 |

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