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Working Paper Series Forecasts of Inflation From Var Models WP 94-08 This paper can be downloaded without charge from: http://www.richmondfed.org/publications/ Roy H. Webb Federal Reserve Bank of Richmond Working Paper 94-8 FORECASTS OF INFLATION FROM VAR MODELS Roy H. Webb* Federal Research Department Reserve Bank of Richmond July 1994 This is a preprint of an article published in the Journal of Forecasting , vol. 14, May 1995, pp. 267-85. *The author is grateful to James Lothian, Stephen McNees, Yash Mehra, Radha referees for helpful comments. The views and opinions expressed in this paper are solely those of the author and should be attributed to any other person or the Federal Reserve Bank of Richmond. Murthy, and two anonymous not FORECASTS OF INFLATION FROM VAR MODELS ABSTRACT Why are forecasts of inflation from VAR models forecasts of performance, real variables? This paper Accounting actions, two changes of that relatively a VAR model poor fitted to U.S. Statistical work by other authors has found that coefficients in such price equations monetary documents and finds that the price equation postwar data is poorly specified. so much worse than their in may not be constant. monetary policy for those two shift5 yields significantly more Based on specific regimes are proposed. accurate forecasts and lessens the evidence of misspecification. Key words: regimes. Inflation forecasts, vector autoregressive models, monetary policy FORECASTS OF INFLATION Forecasts of real macroeconomic FROM VAR MODELS statistics for the United State5 have been successfully produced by vector autoregressive (VAR) models. McNees (1986), for example, compared a series of published forecasts produced in the early 1980s by Robert Litterman's VAR model with forecasts made by conventional macroeconomic forecasters. McNees’s comparisons showed Litterman's forecasts to be more accurate than six competitors for real GNP growth and the unemployment ahead. rate for forecasts varying in length from one to eight quarters In addition, Webb (1991) compared simulated forecasts of business cycle turning points generated by a VAR model adjusted by individually set lag lengths (AVAR) with actual forecasts from three large forecasting services. That study found little difference for one-quarter-ahead superior performance by the AVAR for four-quarter-ahead forecasts but forecasts. VAR forecasts of inflation have been less successful relative to other forecasts. McNees's data shows that the average error of the series of Litterman's inflation forecasts was typically the worst of several that were compared, with his errors often double those of the best forecaster. Zarno- wit2 and Braun (1991) also simulated VAR forecasts from a model with restrictions similar to Litterman's, which is often referred to as a BVAR model. In comparison with actual forecasts included in the National Bureau of Economic Research-American Statistical Association survey, they found "The BVAR forecasts of [real GNP] perform relatively well . . . . but the BVAR forecasts of ... [the GNP implicit price deflator) are apparently much weaker." fOreCaSt Simulated of inflation from models that are presented below are often less accurate than simply predicting no change. This paper attempts to diagnose the source of inaccurate inflation forecasts and to suggest a direction for improvement. performance of several models is first documented. evidence of misspecification from two models. The relatively poor In the next section is presented for representative Possible sources of misspecification price equations are next examined; 2 evidence is presented that the price equations' coefficient5 to two monetary policy regime changes. are unstable due A possible remedy is evaluated by means of simulated forecasts. RELATIVELY POOR PERFORMANCE OF VAR INFLATION FORECASTS VAR MODELS A VAR model uses historical data to predict future values. unrestricted X= VAR (WAR) p + p(L)X+ where X is a terms, f3(L) a vector kxl An model can be written e (1) vector of variables, kxl u is a vector of constant kxl is a polynomial of degree m in the lag operator of error terms. (L), and e is In practice, a k-variable model is often simply k separate equations for which the coefficients can be estimated by ordinary least squares. pt = PLp+ gg P&L-, The price equation in the WAR considered below is + et (2) where p is the inflation rate, t indexes time, v indexes the particular variable, j indexes the lag number, and pI, and the fl,,@ are coefficients. Note that even for a relatively small VAR model such as the one used in this paper, with the number of variables k=5 and the lag length m=6, the price equation contains 31 coefficients to be estimated. It is often believed desirable to limit the number of estimated coefficients. As Doan put it, "Forecasts made using unrestricted vector autoregressions often suffer from the overparameterization forecast errors. n’ of the models ... [which] causes large out-of-sample In addition, Hafet and Sheehan (1989) presented evidence that led them to conclude that "relatively short-lagged be more [VAR] models accurate, on average, than longer-lagged specifications." typically, performance [tend] to More Engle and Yoo (1987) have simply asserted that "The forecasting of unrestricted VARs has not been particularly good . ..." 3 In order to limit the number of estimated parameters, this paper's strategy is to reduce many lag lengths in an adjusted VAP (AVAP) model. For each equation, set two lag lengths -- one for the dependent variable and another for all independent variables -- in order to optimize criterion. Criterion some statistical The particular criterion used in this paper is the Schwarz (SC), which typically gives a parsimonious specification.' It is defined as SC = T In a2 + Nln T (3) where T is the number of observations, N is the number of estimated coefficients, and a2 is the estimated residual variance. By minimizing the Schwarz Criterion one is trading off the lower residual variance from adding an additional coefficient against a penalty term that rises with the number of estimated coefficients. The strategy for selecting lag lengths in this paper is to (1) specify a maximum lag length; (2) assume that all independent variables in each equation have the same lag length; and (3) compute the Schwarz value for all possible lag length combinations. lengths for several variants of the Appendix. a basic five variable The resulting lag model are presented in Of special interest is the price equation for the AVAP model, which is PC=Pp+g bP% +c, g b,J”,t-j + Et (4) where the number of lags m, of the dependent variable is three and the number of lags q of each independent variable is one. Forecast error statistics from representative VAR models are presented in Tables 1 and 2; a detailed account of the models' construction used is given in the Appendix. five U.S. time series: base, the manufacturing bill rate. real and the data Unless otherwise noted, each model contains GNP, the GNP implicit price deflator, the monetary capacity utilization rate, and the go-day Treasury- Two models illustrate the basic result of relatively poor infla- 4 tion forecasts. The model labeled AVAB is identical to one described in Webb (19851, except that the lag lengths are set by the simpler procedure described above. The model labeled BVAB uses Bayesian "priors" that have been advocated by Litterman (1984) to effectively limit the model's number of parameters that need to be estimated. FORECASTING PROCEDURE A rolling regression procedure was used to simulate forecasts to compare with data actually observed through 1990 Q4. Coefficients in each model were first estimated based on actual data from 1952 Q2 to 1977 Ql; one-quarterahead forecasts were made for each variable, which were then used to prepare forecasts for the next quarter, and 50 forth up to eight quarters ahead.3 Each model's coefficients were then reestimated based on data from 1952 Q2 to 1977 Q2, and forecasts were again produced for each variable up to eight quarters ahead. accordingly The procedure was repeated through 1990 Q3. 100 observations quarter-ahead There were on which the first estimate was based, 55 one- forecasts, and 48 eight-quarter-ahead forecasts. Each forecast was compared with actual values and error statistics were calculated. error is simply the actual value minus the predicted value. An The square root of the mean squared error (RMSE) was then calculated for all available forecasts. Exhibit 1 {place here} contains BMSE values for the levels of the interest rate and percentage changes of real GNP and the implicit deflator. For the latter two variables the columns labeled "Single Quarter were derived from percentage Forecasts” changes from the preceding quarter stated as a compound annual rate; the column labeled "Cumulative Forecasts" was derived from the percentage change over four quarters. The forecast horizon is the number of quarters beyond the latest data that were used to estimate the coefficients for the model. Thus for a one-quarter-ahead forecast for 1977 Q2, only data referring to 1977 Ql and previous quarters would be used; that 5 same data would also produce a two-quarter-ahead forecast for 1977 Q3, and so forth. RMSEs are useful for giving a quick view of model accuracy. Another way to view the same data is presented with Theil U statistics in Exhibit 2. {Place here} Those values simply represent the ratio of the RMSE described above to the RMSE of a forecast of no change from the previous period. with RMSEs, smaller U values represent more As accurate forecasts; in addition, values greater than 1.0 indicate a forecast of little value under most loss functions. BASIC RESULTS The U Statistics in Exhibit 2 confirm that the inflation forecasts from the AVAR model are relatively inaccurate. While forecast5 of the T-bill rate are similar or slightly worse than a no-change forecast, real GNP forecasts are substantially better. worse. Inflation forecasts, however, are substantially The BVAR model embodies a substantially different strategy for reducing a VAR model's parameterization. While its inflation forecasts are more accurate than the AVAP model, its single-quarter inflation forecasts are either worse than or not significantly better than a no-change forecast at each horizon.4 change forecast. Its cumulative inflation forecast is much better than a noThe BVAR is less accurate than the AVAR, however, for GNP and interest rate forecasts.5 A more formal comparison of model and no-change forecasts can be made by testing the hypothesis that a series of forecast errors from a model is not significantly different from the series of errors that would result from a nochange forecast. Diebold and Mariano (1991) have proposed such a test that has several desirable properties; perhaps valid even if the two series most important for this paper, it is of forecast errors are contemporaneously lated or if either series has significant autocorrelation. corre- The DM test statistic is calculated from autocovariances of the difference in squared errors6 from two forecasts, and is asymptotically normally distributed. 6 For each model listed in Exhibits 1 and 2, the inflation forecasts at each horizon were compared with no-change forecasts. DM statistics were calculated and are presented in Exhibit 3 {place here}; negative values indicate that the average squared error from the model was larger than the squared error of a no-change ,forecast. For the AVAE model, inflation fore- casts at each horizon were significantly worse than the no-change forecasts. For the BVAE, forecasts one quarter ahead were also significantly worse than the no-change forecast, but were not significantly different at two, three, and four quarter horizons. The four-quarter cumulative forecast was significantly more accurate than the no-change comparison. There are many possible explanations few can be examined in a single article. for the poor performance; Any small macroeconomic only a model omits variables that could be important, and there is certainly a long list of omitted variables that might affect prices. While examination of potential additions will be left for future research, poor choices may have been made in choosing the five variables included in the model. To examine that possibili- ty, two additional models were used to produce simulated fOreCastS. labeled AVAB-M2, The rationale substitutes the M2 monetary aggregate for the monetary base. is that many researchers have documented a relationship M2 and prices.' One, Another, between labeled AVAR-CPI, substitutes the Consumer Price Index for the GNP implicit price deflator, on the grounds that movements in the deflator reflect not only price changes but also changes in the relative quantities of goods produced. lengthy intervals and is a ClOBer In contrast, the CPI has fixed weights for approximation to the type of price index most analysts would prefer. As indicated in Exhibit 1, substituting M2 for the monetary base failed to improve inflation forecasts and also produced less accurate forecasts for other variables. significantly At each forecasting horizon the inflation forecasts were worse than the no-change comparison. The results from substituting the CPI for the implicit deflator are slightly better. The U statistic for the inflation term fell below unity at 7 the shorter horizons in the table, as well as for the four-quarter forecast. cumulative The GNP and interest rate forecasts were slightly worse than in the original model. A final variation is to simulate forecasts from a six-lag WAR as a check on the conventional wisdom that substantially reducing the number of estimated coefficients will normally improve the accuracy of VAR forecasts. The results for prices were surprisingly accurate (to the author, at least); these results should not have been too surprising, however, in light of Lupoletti and Webb (1986), who found simulated WAR inflation forecasts at all but a one-quarter horizon to be competitive with those from major forecasting services. Attempting to isolate the reason for the WAR a hybrid model was simulated. rates are highly persistent, model's greater accuracy, The intuition here is that since inflation long lags might be especially useful. The AVAR specification was adopted for every equation except the price equation, and the Unrestricted six-lag version was used for prices. roughly between the AVAR and WAR in forecast accuracy. was then simulated, using unrestricted the monetary base. The results were Another hybrid model lags in the equations for prices and The results are shown in the table, labeled A/WAR. This model produced the most accurate longer-term inflation forecasts, with little penalty for other variables. The main result of this section is that the VAR models generated price forecasts that, for most horizons, were significantly worse than simply predicting no change. In order to isolate potential problems, the price equation from the AVAR and A/WAR models is examined more closely below. SPECIFICATION TESTS FOR PRICE EQUATIONS The price equations from two VAR models will be examined more closely in this section. Equation (5) is from the AVAR model with shorter lag lengths, and the other is equation reprinted below. (6) with longer lag lengths. Both equations are 8 Pt = b + k$ v-1 01 Pj,GG,,-j + et Residuals from both equations were first tested for serial correlation. The specific test' is a Lagrange multiplier test that takes into account the presence of lagged values of the dependent variable and little prior knowledge of the exact form of serial correlation. The procedure is to first estimate the price equation and its residuals and then add residuals to the equation before reestimating it. m lagged values of the An F test on the signifi- cance of the lagged values is asymptotically valid for a null hypothesis of no serial correlation and an alternative hypothesis of either AR(m) or MA(m) errors. As the first line in Exhibit 4 {place here) indicates, the null hypothesis of no serial correlation For equation is rejected for equation (6) with unrestricted when the alternative (5) with short lags. lags, the null hypothesis is not rejected is first order correlation, an unsurprising six lagged values included in the equation. fifth order autocorrelation, With an alternative hypothesis of however, the null hypothesis can be rejected. The next test, for autoregressive conditional hetereoskedasticity (ARCH) I is also a two-step procedure; the first step price equation and its residuals. is also to estimate the The second step is to regress the squared residual on a constant term and m lags of the squared residual. observations times the R2 statistic is chi-squared with under the null hypothesis of no ARCH. equation (5) with m=l, log-level price was accepted. series, m The number of degrees of freedom The null hypothesis is rejected for and is also rejected for equation A final test is whether it is appropriate in first differences. result with to estimate (6) with m=5. the price equation A standard Dickey-Fuller test was used to examine the and the null hypothesis of the presence of a unit root Similar test results for the series of first differences were 9 ambiguous, however. shown in Exhibit 4, either acceptance or rejection of As the null hypothesis could occur, depending on the lag length of the differenced inflation rates.g Since there is no conclusive reason a priori to prefer one lag length to another, the choice of working with first or second differences of the price data is a close call. Since overdifferencing risks losing information that could be useful in forecasting, the basic models in this paper use the first difference of the implicit deflator. FORECAST PERFORMANCE WITH MODIFIED MODELS STATIONARITY AND POSSIBLE COINTEGRATION The VAR models discussed above contained three variables in differences: the price level, the monetary base, and real GNP. Formal tests for stationar- ity have already been discussed for the price series. Dickey-Fuller also failed to reject the null hypothesis of nonstationarity tests for the monetary base, real GNP, and the interest rate, but did reject the null when data were differenced one time. The capacity utilization rate, which is stationary by construction, was not tested. ARE THE VARIABLES COINTEGRATED? It is also possible that the equations in differences may be misspecified of those variables that is stationary. the variables if there is a linear combination It is especially important to check in this model since (1) other authors, for example Mehra (1991), have found that closely related series are cointegrated, and (2) theory suggests that the variables could be related by the quantity equation with interest-sensitive velocity and a money-multiplier monetary base and the money supply. relation between the If cointegration were to be found, then the use of an error correcting model might well improve forecast accuracy. A cointegrating equation was therefore used to estimate the parameters of such a linear combination. found to be nonstationary, A cointegrating If the residuals from that equation are not the hypothesis of cointegration can be accepted." equation for the four nonstationary variables was therefore estimated, and the results are displayed in Exhibit 4. An augmented Dickey- 10 Fuller test was used to test the null hypothesis that the residuals were nonstationary, which would imply that the series are not cointegrated. While the test result could possibly be affected by the lag length m, the value shown in the table is the one that maximizes the ? statistic, and is also the shortest lag length for which the Ljung-Box Q statistic does not indicate significant serial correlation. The failure to reject the null hypothesis is consistent with no cointegration. SHOULD_THE? At this point it is useful to question the economic significance of the finding that the interest rate is nonstationary. It is well known that the Dickey-Fuller test is not powerful against a highly persistent alternative, such as a root of 0.98. And there is a reason that one would expect the interest rate series to be highly persistent. Goodfriend (1991) discusses theoretical and empirical work on central bank smoothing of nominal interest rates -- that is, daily changes in reserve supply that keep the federal funds rate within a narrow band that is only changed infrequently. Such smoothing could impart a high degree of persis- tence to even a quarterly average of a daily rate, as is used in this paper. It may therefore be appropriate to treat the interest rate as stationary, as was done in the models presented earlier. A regression with the interest rate in levels would not be spurious, and differencing valuable information. could lose But if that assumption were wrong and the interest rate were actually a random walk, the regression in levels would be spurious and differencing would be appropriate. In order to empirically examine which specification lag lengths were reset in the A/WAR is more appropriate, model with the interest rates in differ- ences; series of forecasts were then generated. The somewhat ambiguous results are shown in Exhibit 5 for the model labeled A/WAF+l?D. The interest rate forecasts from the latter model were slightly more accurate, but forecasts of GNP growth and inflation were less accurate. Those results are consistent with the notion that the information 1055 from differencing may be large enough to justify entering the interest rate in level form. 11 ANOTHER POSSIBILITY OF COINTEGRATION: Admitting the possibility the interest rate and inflation series are nonstationary, that could there be a stable "real" rate, that is, the nominal rate minus the quarterly inflation rate?ll To test this possibility, it is not necessary to estimate a cointe- grating equation since the coefficients are known to be one and minus one. real rate series was therefore constructed; a Dickey-Fuller A test, presented in Exhibit 4, rejects the null hypothesis of the presence of a unit root. Given the stationary combination of two arguably nonstationary series, the next step is to construct a vector error correcting version of the A/WAR model (VECM). The strategy used here was to difference the nominal rate and inflation series, and add the first lag of the constructed real rate series to each equation. Aapt = IQ, + kg v=l The resulting price equation is Pj,A,t-j + y==t-1 + ‘lt (7) =l where p is the log of the implicit deflator, rr is the error correcting term, that is the nominal rate minus the inflation rate, and the variables in X are altered to include the first differences of the interest rate and inflation rate. Lag lengths were reset to minimize the Schwarz Criterion. were then generated, with summary statistics given in Exhibit 5. tion values indicate that the VECM A/WAR model. Forecasts The infla- forecasts are much worse than the standard It appears that the information lost from differencing the inflation series was substantial, while the value of including the level of the real rate was not large. One final check of possible cointegration was made using the multivariate test proposed by Watson (1992). In this test the null hypothesis is the existence of one cointegrating vector, the real interest rate; the alternative is the existence of an additional cointegrating vector. \ The test is a likelihood ratio, and in this case involves comparing the largest eigenvalue of a matrix computed from the appropriate VECM with critical values that were 12 computed from the statistics' asymptotic chi-squared distributions. In this case, the largest eigenvalue was 9.7, well below the critical value of 20.4 that would indicate rejection of the null hypothesis at the 10% level. This failure to reject the null is consistent with the results of the univariate tests above, and does not indicate that further analysis of possible cointegration would prove useful. In short, reconsidering cointegration the stationarity of the variables and possible did not indicate a direction for improving the accuracy of inflation forecasts from the VAB models examined here. POSSIBLE REGIME CHANGES It is common to think of nonstationary behavior of an economic variable by analogy to a random walk with frequent small, permanent disturbances. Indeed, widely used statistical tests for nonstationarity are derived under the assumption of such a random walk as the null hypothesis. nonstationarity are also possible, however. Other types of An alternative that is pursued in this section involves large, infrequent shocks. Several authors have found that U.S. postwar inflation data appear to have been generated by a process with one or more discrete shifts. Evans and Wachtel (1993) proposed a two-state Markov switching process for the CPI, and found that it explained some of the puzzles that are raised by the ex post bias often found in series of inflationary anticipations. Boschen and Talbot (1991) found evidence of unstable coefficients in regres- sions of inflation on several variables, notably including the growth of the monetary base, growth of real GNP, and the differenced T-bill rate. Balke and Fomby (1991) studied the GNP deflator from 1870 to 1988 and found four "level shifts" to the inflation rate; the two in postwar data were in 1968 and in 1983.12 The last finding provides statistical evidence that something important that affects inflation changed around the mid 1960s and early 1980s. An obvious possibility I find is that something about monetary policy changed. it plausible that (1) the President and Congress chose to tolerate higher 13 inflation during the Viet Nam buildup for a variety of reasons, including raising revenues via 'bracket creep' and the personal income tax;13 (2) as long as this consensus held, monetary policy had an inflationary bias;14 (3) this consensus was destroyed by the early 1980s;15 and (4) monetary policy then included low inflation as an important goal. It is therefore possible to view monetary policy as making a discrete move toward more inflation at some point in the middle of the 19605, with the monetary effects on prices becoming apparent by 1968; a discrete move toward disinflation occurred somewhat later and the inflation effects became apparent by 1983. DATING REGIME CHANGES: moves? What were the exact dates of these discrete The strategy here will be to look for specific actions that do not fit the pattern of the Fed's usual behavior. It is possible to loosely character- ize monetary policy since the Federal Reserve - Treasury Accord in 1951 as "leaning against the wind.' That is, the Fed's day-to-day actions involved open market operations that varied the quantity of bank reserves in'order to keep the federal funds rate within a narrow band. The funds rate band, in turn, was set in order to respond to pressing economic conditions. This often involved raising the funds rate when inflation was rising and unemployment relatively low, and lowering the funds rate when inflation was low and real growth weak. slow In addition, there were times during which unusually rapid or money growth would bolster the case for changing the funds rate target. The Fed departed from that strategy in the 1960s; the background account in this paragraph substantially is taken from Kettl (1986). In 1965 President Johnson increased military action in Viet Nam, proposed domestic social legislation requiring increased spending, but resisted increasing nominal interest rates. In June 1965 the Fed's chairman William McChesney Martin learned that the Viet Nam buildup was larger than the administration publicly admitting. was In October Martin wrote to the President arguing for an immediate increase in interest rates. On December 5 the Fed raised the discount rate by 50 basis points, and 'Johnson was furious. advisers believed that the decision was precipitous . ... He and his The President saw 14 The President the discount rate increase as a personally vindictive act."16 and the Chairman met at the President's ranch in Texas later that month. In 1966 real growth was a rapid 6.0 percent, the unemployment rate was a very low 3.7 percent, and the inflation rate (the change in the annual average CPI) rose from 1.6 in 1965 to 2.9 percent. A consistent application of leaning against the wind would have required increasing the funds rate until inflation was clearly checked. Although interest rates rose during the first part of the year, in the fourth quarter "Monetary policy promptly moved to relax the degree of reserve restraint."" The inflation rate was 3.1 percent in 1967, 4.2 percent in 1968, and 5.5 percent in 1969. departure Here is a clear from the previous leaning against the wind strategy, namely lowering interest rates, beginning in the fourth quarter of 1966, while inflation was rising and unemployment low. The fourth quarter of 1966 was therefore the end of a low inflation period. enemy By the end of the 1970s inflation was being described as "public number one." That sentiment did not translate immediately into presi- dential support for a substantial change in monetary policy. In the lead paragraph of his 1980 Economic Report to Congress, President Carter asserted that higher oil prices were the major reason for rising inflation in 1979. In that twelve page statement, there are only two sentences that mention monetary policy. The most relevant is, "Monetary policy will have to continue in support of the same anti-inflationary goals.111E [Emphasis mine] firmly In January 1981 the political party controlling the Presidency and the Senate changed. The discussion of monetary policy also changed substantially. example, in the first annual report of the Reagan Administration's Economic Advisers essentially (1982), one encounters the following phrases: Council of "Inflation is a monetary phenomenon." p.75; "The appropriate policy for reducing the inflation rate is a decrease in the rate of money growth." p.76; "The Administration reduction For expects that the Federal Reserve will achieve an orderly in the trend of money growth to a noninflationary [Emphasis mine]. rate." p.64 15 Monetary growth was high and example, Ml variable in the 1970s and also in 1980; for grew at a 16 percent annual rate between May and November of 1980. In 1981, however, "growth in Ml-B adjusted for shifts into NOW accounts was about 2 l/4 percent -- 1 l/4 percentage points below the lower end of its targeted range."" That behavior of the main intermediate target of Fed policy, shift-adjusted Ml-B, was substantially different from what had been seen in the past decade. Turning to specific actions, in the first quarter of 1981 the unemployment rate was 7.4 percent, compared to an average 7.0 percent in 1980 and 5.8 percent in 1979. The trend in real activity was difficult to interpret in light of the monetary volatility in the last year as well as the imposition and removal of credit controls in 1980. shift-adjusted During the second quarter of 1981 Ml-B was at or below the lower bound of the target range that the Fed had announced in February. Yet the federal funds rate rose by five full percentage points from April 1 to July 8.20 Here is an extraordinary departure from previous behavior; the second quarter of 1981 is therefore the last of the inflationary period.21 ECONOMETRIC RESULTS: Based on the evidence discussed above, three periods are studied: an early low inflation period from 1952 Q2 to 1966 44; a middle inflationary period from 1967 Ql to 1981 42; and a disinflationary period from 1981 Q3 to 1990 44. Separate regressions were run for each subperiod and the entire sample; the results are displayed in Exhibit 6. Perhaps most striking is the complete failure of the restricted equation for the early period, with essentially no explanatory power from its regressors. Regressions for the middle and late periods appear to work somewhat better, at least as indicated by the usual summary statistics. For the whole period the sum of the coefficients on nominal variables, that is lagged prices, the interest rate, and the monetary base, is approximately unity. the middle period, however, it is In 1.26 while in the late period it is 0.80. The sum of coefficients on lagged prices rises substantially in the two later periods; the strong persistence of inflation is not evident in the early 16 period. The coefficient on the monetary base is significantly different from zero only in the middle period. These statistical results are consistent with the story of monetary regime changes.22 regimes. There are several options when forecasting under changing Perhaps the simplest is to use a few dummy variables to allow some coefficients to abruptly shift. O-l dummies were constructed Dl, respectively. Based on the results summarized in Exhibit 6, for the middle and late periods, labeled Dm and In addition, the product of Dm and the monetary base is labeled Db, and the product of the implicit deflator and sum of Dm and Dl is labeled Dp. equation Dm, Dl, and one lagged value each of Db and Dp were added to (5), with the results also displayed in Exhibit 6. {place here) As might be expected, the apparent fit of the equation improved and the four dummy variables were each significant at conventional Diagnostic levels. tests were again repeated and are displayed in Exhibit 6. The F test for serial correlation was repeated on the price equation for the entire sample. In contrast to the results presented in Exhibit 4, the null hypothesis of no serial correlation is not rejected at conventional levels. The test statistic for ARCH is also presented, and the hypothesis of no ARCH is again rejected at the one percent level. statistics were constructed Finally, Dickey-Fuller test for the inflation rate over the three intervals. In the early and middle periods the null hypothesis of a unit root is rejected at the one percent level. For the late period, the null is rejected at either the five or ten percent level, again depending on a lag length used in the test. Adding the dummy variables to the price equation of the AVAB model substantially improved its forecasting performance, The modified AVAB model has by far the most as shown in Exhibit 5. accurate inflation forecasts at each horizon of any of the models considered in the paper, with the GNP forecasts remaining as accurate as any of the models considered. same dummy variables to the A/WAR Adding the model slightly improved its inflation 17 forecasts, although the AVAR-D model remained significantly more accurate. Neither model forecasts the interest rate well. To put these figures in perspective, consider the root-mean-square errors of the ASA-NBER inflation forecasts from 1968 to 1990, taken from Zarnowitz and Braun, Table 5. forecasts for the same period. The AVAR-D model was used to produce simulated One-quarter-ahead forecasts from the VAR model were less accurate, with an RMSE of 1.48 percent versus 1.00 percent. Four- quarter cumulative forecasts from the VAR model were much more accurate, with an RMSE of 0.88 percent versus 1.92 percent. VAR's performance If taken at face value, the is competitive with actual forecasters of inflation over that interval. The major reason not to take the results at face value is the amount of experimentation that was used to construct the AVAR-D model. In addition, the model had the benefit of using revisions of GNP and implicit deflator data that were unavailable to the real-time forecasters. CONCLUSION Several small VAR models have been examined in this paper, with particular emphasis on inflation forecasts and the inflation equation of the models. Two models are of particular interest. stricted lags in two equations. One has unfashionably long, unre- The other uses dummy variables to represent the political decision made in the 1960s to allow higher rates of inflation and the subsequent decision made in the early 1980s to lower the inflation rate. The latter model produced the most accurate inflation forecasts of the models examined while its GNP forecasts retained their accuracy. That result suggests a resolution to the puzzle of relatively inaccurate inflation forecasts from VAR models; two monetary policy regime changes led to inaccurate inflation forecasts from models with constant coefficients. Moreover, either adding dummy variables or dividing the entire sample into three subperiods also removed serial correlation from the residuals of the price equation. 18 These results may be of interest to several groups. To anyone inter- ested in price behavior, the improved performance of the price equation after an adjustment for temporal instability points out a danger of using constant coefficients over most of the postwar period. the finding that long, unrestricted To users of small macro models, lags led to an equation that worked well relative to a highly restricted equation suggests that long, unrestricted lengths should not be rejected out of hand. Students of monetary policy may find the dating of regime changes of interest. nonstationarity lag The finding that possible of the inflation rate over the postwar period is not apparent in the three subperiods of consistent monetary policy may be of interest to empirical macroeconomists contexts. concerned about potential nonstationarity in other Finally, if one wishes to forecast inflation, using the admittedly ad hoc dummy variables may be preferable to ignoring the monetary regime changes altogether. Several directions for future research are also evident. First, do the results that are presented above overstate the probable accuracy of current inflation forecasts, due to too much experimentation? Of course, new data over time will reveal whether the post-sample performance of these models is in line with those results. Also, despite the experimentation a basic stability in the model structure. there has been Since 1984 the author has published results from VAR models with the same variables, with the same starting date for regressions, and with the same choices for differencing.23 Changes have been limited to different lag lengths and the addition of the dummy variables for the two shifts in monetary policy. This paper views monetary policy changes as being responsible persistent fied. shifts in the inflation process that several authors have identi- Are there other candidates such as energy plausible? for large, prices that are equally And if the monetary policy explanation suggested dates of regime change robust? is accepted, are the One approach would be to estimate a structural relation between instruments and goals of monetary policy and 19 examine its stability in the neighborhood of the dates of the suggested changes. Residuals from the best performing price equation still displayed ARCH, and that specification failed to explain inflation during the early subperiod. It is therefore apparent that this model, like many other small macro models, might benefit from additional variables that help predict inflation rates. The challenge will be to narrow the field, given the large number of variables that analysts have used with apparent success in price equations in the past. 20 APPENDIX: TEE DATA AND MODELS The following data series, with used in this paper: Citibase mnemonic in parentheses, were P, the implicit price deflator for gross national product (GD); Y, real gross national product (GNP82); M, the monetary base as estimated by the Federal Reserve Bank of St. Louis (FMBASE); CU, the capacity utilization rate in manufacturing three month treasury bills (FYGM3); urban consumers (IPXMCA); R, the secondary market yield on CPI, the consumer price index for all (PUNEW), and the M2 monetary aggregate after 1959 (FM2). All except the interest rate were seasonally adjusted by the agency compiling the data. Natural logarithms were taken of each data series except the interest rate and the capacity utilization rate. maintained by Citibase on July 19, 1991. Robert Hetzel; see Hetzel The table collection below Data were the latest revisions Pre-1959 M2 data were obtained from (1989) for further information on its construction. describes each model in the paper. of equations that are individually described. DEP VAR contains the dependent variable for each eguation. Each model is a The column labeled The column labeled DIFF? contains a Y for differenced series, N for levels, and 2 for a series differenced twice. The number of lagged terms for the dependent variable and the common lag length for all the independent variables are given in the next two columns. as described The final column notes if an equation contained dummy variables in the final section, or RR, the difference between the nominal interest rate and the inflation rate; a constant term was also included in each equation. 21 Model Descriptions MODEL DEP VAR DIFF? # OWN LAGS # LAGS OF IND VARS A/WAR P Y t4 C R A/WAR-D P Y 6 6 1 1 M 6 2 6 6 C R A/WAR-RD P Y M C R AVAR P Y M C R AVAR-CPI OTHER TERMS Y Y Y N N 4. 4 d, % CPI Y M C R AVAR-D P Y d, d, d, 4 M C R AVAR-M2 P Y M2 C R BVAR" P Y M C R VECM P Y M C R RR RR RR RR RR 22 EXHIBIT 1 Forecast Error Statistics from Several VAR Models Root Mean Square Errors Model Variable Single Quarter Forecasts Horizon: IQ 4Q x2 Cumulative Forecasts As2 AVAR IPD GNP RTB 1.57 3.40 1.20 1.92 3.07 1.95 2.62 3.50 2.44 1.84 1.58 IPD GNP RTB 1.58 3.81 1.24 1.59 3.75 2.03 1.79 4.21 2.69 1.11 1.92 IPD GNP RTB 1.86 3.75 1.26 2.32 3.41 2.05 3.17 3.84 2.60 2.30 2.15 CPI GNP RTB 1.91 3.40 1.24 2.56 3.24 2.04 3.22 3.58 2.60 2.16 1.80 IPD GNP RTB 1.69 3.40 1.20 1.64 3.09 1.97 1.73 3.60 2.50 1.11 1.60 BVAR AVAR-M2 AVAR-CPI A/WAR Note : Model construction and data sources are described in detail in the Appendix. Variable abbreviations'include the quarterly average level of RTB, the go-day Treasury-bill rate, and annualized rates of change from the previous quarter of GNP, real gross national product; IPD, the GNP implicit price deflator; and CPI, the consumer price index. The numerical entries are root-mean-squared error statistics described in the text, covering synthetic forecasts from 1977 Q2 to 1990 43. 23 EXHIBIT 2 Forecast Error Statistics from Several VAR Models Theil U Statistics Model Variable Single Quarter Forecasts Horizon: u 22 Cumulative Forecasts AC2 AVAR IPD GNP RTB 1.05 0.75 1.00 1.22 0.63 1.08 1.44 0.66 1.04 1.40 0.47 IPD GNP RTB 1.06 0.84 1.03 1.01 0.77 1.12 0.98 0.80 1.15 0.85 0.58 IPD GNP RTB 1.25 0.83 1.05 1.46 0.70 1.13 1.74 0.73 1.11 1.74 0.65 CPI GNP RTB 0.85 0.75 1.03 0.92 0.66 1.13 1.02 0.68 1.11 0.96 0.54 IPD GNP RTB 1.13 0.75 1.00 1.04 0.63 1.09 0.95 0.68 1.06 0.84 0.48 BVAR AVAR-M2 AVAR-CPI A/WAR Note: Model construction and data sources are described in detail in the Appendix. Variable abbreviations include the quarterly average level of RTB, the go-day Treasury-bill rate, and annualized rates of change from the previous quarter of GNP, real gross national product; IPD, the GNP implicit price deflator; and CPI, the consumer price index. The numerical entries are Theil U statistics described in the text, covering synthetic forecasts from 1977 Q2 to 1990 43. 24 EXHIBIT 3 VAR Inflation Forecasts Compared to No-Change Forecasts Model Variable Single Quarter Forecasts Horizon: u 2sL !?Q Cumulative Forecasts !a AVAR U Statistic DM Statistic Significance, % 1.05 -3.7 eo.1 1.22 -11.7 CO.1 1.44 -11.6 co.1 1.40 -5.2 co.1 % 1.06 -3.2 0.1 1.01 -0.2 >lO 0.98 0.63 >lO 0.85 2.4 1.7 % 1.25 -12.8 co.1 1.46 -18.0 <O.l 1.74 -10.0 co.1 1.74 -4.9 CO.1 % 0.85 14.5 co.1 0.92 7.3 <O.l 1.02 -1.0 >lO 0.96 0.9 >lO 1.13 -6.1 co.1 1.04 -1.2 >lO 0.95 1.6 >lO 0.84 2.2 2.9 BVAR U Statistic DM Statistic Significance, AVAR-M2 U Statistic DM Statistic Significance, AVAR-CPI U Statistic DM Statistic Significance, A/WAR U Statistic DM Statistic Significance, % : Model construction and data sources are described in detail in the Appendix. Synthetic forecasts inflation from 1977 42 to 1990 44 are compared with no-change forecasts. The U Statistic ie the ratio the root-mean-squared The DM statistic error of the model forecast to that of a no-change forecast. tests the null hypothesis that the model and no-change forecast have equal accuracy; negative values indicate that the sum of squared forecast errors is larger for the model forecast. Note 25 EXHIBIT 4 Specification Tests for Price Equations Equation (5) pc = pP + Serial correlation test: ARCH test: Equation x2 (1) F(1,145) = 3.93; Serial correlation test: ARCH test: Dickey-Fuller x2(5) level = .049. = 5.13; Significance level = .024. pt = cr,+ ej? GL-j v-1 -1 (6) Significance + et F(5,114) = 2.48; Significance level = .036. = 14.64; Significance level = .012. test for a unit root Z, = p + m P,A= t-j; P&-l + et F-1 e,tp=11 = 0.80; 10% sig. level = -2.58; Price Level ( Z, = P, ): m=2; Differences ( Z, = AP, ): m=l; 9,(p=l) = -3.42; 5% sig. level = -2.89; Differences ( Z, = AP, ): m=5; $,(p=l) = -2.06. "Real" Rate ( Z, = Rt- (pt-ptel) ): m=2 ; Cointegrating equation Q,(p=l) = 14.1. P, = 1.29 + .83&l,- .15Y, + .016R, + I?, Augmented Dickey-Fuller test on residuals from cointegrating AQ, = -XI?,-, + * PjAQt-j F-1 t (x=0) = 3.12; Note: This significance 10% level = 3.89 level is from Engle and Yoo (1987). equation: 26 EXHIBIT 5 Forecast Error Statistics from Several Models Theil U Statistics Model Variable Single Quarter Forecasts Horizon: lJJ !T!Q 22 Cumulative Forecasts 22 A/WAR IPD GNP RTB 1.13 0.75 1.00 0.95 0.68 1.06 1.06 0.65 1.03 0.84 0.48 IPD GNP RTB 1.19 0.81 0.98 1.02 0.75 0.98 0.98 0.61 1.03 0.94 0.69 IPD GNP RTB 1.30 0.80 0.95 1.55 0.73 1.08 2.09 0.61 1.20 1.49 0.71 IPD GNP RTB 1.16 0.75 1.00 0.69 0.68 1.05 0.65 0.65 1.02 0.65 0.48 IPD GNP RTB 1.05 0.75 1.00 1.44 0.66 1.04 1.54 0.64 1.00 1.40 0.47 IPD GNP RTB 0.83 0.75 1.00 0.67 0.66 1.04 0.56 0.64 1.00 0.58 0.47 A/WAR-RD VBCM A/WAR-D AVAR AVAR-D Note: Model construction and data sources are described in detail in the Appendix. Variables include the quarterly average level of RTB, the go-day Treasury-bill rate, and annualized rates of change from the previous quarter of GNP, real gross national product; and IPD, the GNP implicit price deflator. The numerical entries are Theil U statistics described in the text, covering synthetic forecasts from 1977 Q2 to 1990 Q3. 27 EXHIBIT 6 Regression Results for Several Time Periods 1952 Q2 to 1966 Q4 fit= 0.28 - 0.08~~~, + O.OBp,-, + O.llp,-, + 0.07r,-, + O.O2c,-, - o.Olm,-, + O.OSy,-, (0.76) (-0.01) (0.51) (0.72) (0.28) (0.24) (0.06) (-0.54) 2 = -0.08 8 =1.85 F(1,149) = 0.17 Q, = -7.90' (m=O) x2 (1) = 3.48"' 1967 Ql to 1981 Q2 $, = -2.78 + 0.30&s, - o.o4p,-, + o.o4p,-, + 0.33r,,, + 0.02~~~, + 0.6Om,-, - 0.08~~~, (-1.52) (4.87) (-0.54) (2.30) (-0.28) (0.27) (0.26) (2.56) Rz = 0.57 a = 1.59 F(1,148) = 2.42 x2(1)=2.31 Q, = -3.64' (m=O) 1981 43 to 1990 44 8t = -8.87 + 0.21P,-, + O.O9p,-, + 0.20~~~, + 0.20r,-, + O.lOc,-, + O.lOm,-, - 0.15~~~~ (-0.21) (0.53) (1.02) (-1.54) (1.16) (1.15) (1.07) (1.68) Ra = 0.51 6 = 1.06 9, = -2.63 9, = -2.63"' (m=l) (m=l) F(1,28) = 0.45 9, = -2.95" x2(1) = 0.46 (m-2) 1952 Q2 to 1990 44 rr,= -3.84 + 0.3Op,-, + 0.23~~~~ + 0.22p,+ + o.o05r,-, + o.o5c,-, + 0.17m,-, - 0.22~~~ (-0.59) (-1.42) (6.38) (2.89) (2.71) (0.07) (2.95) (1.54) R1 = 0.59 8 = 1.75 F(1,145) = 3.93" x2(1) = 5.13.' 28 EXHIBIT 6 (Continued) 1952 Q2 to 1990 44 with Dummies 3, = -1.29 - 0.05pcw1 + 0.02~~-~ + O.O6p,-, + 0,2lr,-, + 0.04~~~, + O.O2m,-, - O.O2y,-, (-0.51) (-0.45) (0.29) (0.30) (-0.44) (0.82) (2.34) .(1.16) - 2.11d, - 1.32d, + o.38dD + 0.42d, (-2.66) (-2.00) (2.53) (3.61) R'"= 0.68 Significance levels: 6 = 1.56 l%, *; F(1,141) 5%, *; = 0.25 X2(l) = 23.4' lO%, - Note: For each equation t-statistics are in parentheses. Symbol definitions are as follows: p is the inflation rate, t is a time subscript, r is the level of the 3-month T-bill rate, c is the manufacturing capacity utilization rate, m is the percentage change of the quarterly average monetary base, y is the percentage change of real GNP, dm is a dummy variable that is unity from 1967 Ql to 1981 42 and zero otherwise, dl is a dummy variable that is unity from 1981 Q3 to 1990 44 and zero otherwise, dp is the product of p and dm+dl, and db is the product of dm and m. The 9, statistic allows a Dickey-Fuller test The F statistic of the hypothesis of a unit root in the dependent variable. allows a test of the null hypothesis that serial correlation is zero. The chi-squared statistic allows a test of the null hypothesis of no ARCH. 29 REFERENCES Balke, Nathan S. and Thomas B. Fomby. 1991. Shifting Trends, Segmented Trends, and Infrequent Permanent Shocks. Journal of Monetary Economics 28: 6185. 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Working Paper 3965, National Bureau of Economic Research, Cambridge, Massachusetts. - 31 ENDNOTES 1. Doan (1990). p. 8-16. One 2. Why was the Schwarz Criterion used instead of some other statistic? reason, which is only applicable for this paper, is simply to have a sharp difference between the restricted and unrestricted models. More generally, Yi and Judge (1988) studied the asymptotic performance of three widely used statistics for model selection. Only the Schwarz Criterion excluded irrelevant variables with a probability approaching one as the sample size increased. 3. These would be referred to as early quarter forecasts, since they assume knowledge of the previous quarter's implicit deflator and GNP values, which are now first released near the end of the first month of a quarter. In real time, a forecaster would also have other valuable information by the time those values are released, such as some values in the current quarter for interest rates and the monetary base. 4. Forecasts from one to eight quarters ahead were examined, although only a few horizons are shown in the table. The discussion in the text refers to all periods, whether or not included in the table. 5. These BVAR model results are based on a model in which real GNP, the implicit price deflator, and the monetary base are differenced. Many users of BVAEs, however, prefer to use levels and trend terms. In other words, they prefer to assume trend stationarity rather than difference stationarity. To see if these results are sensitive to that assumption, the BVAR model was also simulated with all variables in levels and with a time trend in the equations. The results of simulated forecasts were generally worse, especially for the inflation rate. 6. While this paper uses squared errors, other functions of the error series would also be valid. 7. A particularly 8. For a discussion of this particular test see Godfrey well-known example is Hallman, Porter, and Small (1989). (19881, section 4.4. The shorter lag length was chosen to minimize the Schwarz Criterion. A longer lag was also examined to check the robustness of the test to the lag length. This was particularly important for prices since earlier work, Webb (1988), had found that the results of Dickey-Fuller tests for unit roots in the CPI series were also dependent on the lag length. 9. 10. Engle and Granger (1987) is the classic reference on the subject. 11. For a proper ex post real rate, the 3-month nominal rate R would be the value at the beginning of the quarter, the log price level P would be the value at the end of the quarter, and the real rate would be R, - (P,-P,.,).The measures for the nominal rate and the price level that are actually used in this paper are 32 quarterly averages. would be preferable. Of course, one could also argue that an ex ante real rate 12. It is interesting to note that they also applied their procedure to real GNP growth over the same period and did not find any level shift. 13. See, for example, and a nonindexed tax inflation rate raised falling real personal Hetzel (19901, who emphasizes the interaction of inflation code. In 1974 alone he estimates that.aneleven percent the growth in tax revenues to seventeen percent despite income. 14. While the Federal Reserve has considerable independence in its day to day operations, if it were to pursue goals outside a political consensus for very long, legislation putting permanent restrictions on the Fed could be enacted. Since officials would most likely not wish to see such legislation enacted, as a practical matter one would not expect to see the Fed continue to pursue goals that the President and Congress opposed. 15. The Economic Recovery Tax Act of 1981 included substantial indexing of the personal income tax for inflation, thereby reducing the government's revenue gains from inflation. According to Steurle (19911, "The major individual reform instituted in 1981 was not the direct reduction in tax rates, but the establishment of indexing of tax brackets .... Eventually, indexing will dominate all other provisions of the 1981 Act." 16. Kettl, p. 104. 17. Martin (19671, p. 215. The phrase "lowering the degree restraint" can be translated as lowering the fed funds rate. of reserve 18. The other is "The October actions of the Federal Reserve Board to change the techniques of monetary policy helpedmoderate inflationary expectations whichhad been partly responsible for the pressure on the [foreign exchange value of the1 dollar." 19. Volker (1982) P. 129. 20. For an account of specific Fed actions during this period see Cook (1989). 21. Some analysts prefer to date the monetary regime change as October 6, 1979, the time of a Fed announcement of a major change in operating procedures. Arguing against that date are (1) at the time, the President did not publicly support a strong, disinflationary monetary policy, preferring instead to focus on wage-price guidelines, energy conservation, and the risks of monetary policy being too tight; (2) an important political incentive for inflation, the lack of indexing of the personal income tax, was not removed until 1981; and (31 monetary growth in 1980 was not consistent with a shift to a disinflationary monetary policy. 22. The statistical hypothesis of structural stability could of course be formally tested with a standard Chow test. That test, however, is invalid when the full-sample regression has residuals that display ARCH, as in this case. It 33 is however possible to construct a likelihood ratio test of the joint hypothesis of constant coefficients and constant residual variance against the alternative of changing coefficients and/or changing residual variance. The LR statistic in this case has a chi-squared (18) distribution; the calculated LR value, 43.45, indicates the null hypothesis is rejected at the 1% level. 23. See Webb (1984, 1985, 1991) and Lupoletti and Webb (1986). An important part of the BVAR specification procedure is to set "hyperzirameters" that could in principle be used to impose prior beliefs on the data. Here these parameters were chosen to minimize the log-determinant of the variance-covariancematrixconstructedfromone-quarter-aheadforecasterrors for the model. The resulting choices are summarized in the RATS statement SPECIFY(type=symmetric,decay=.5,tight=,6) .9 (see Doan, ch. 8.8 for further details).