The correlation between shocks to output and the price level: evidence from a multivariate GARCH model.

Hueng, C. James

1. Introduction

In what way has The price level in the United States been related to the level of output? At one time it was generally believed that the correlation between the price level and output is positive and to be useful a macroeconomic model should be able to predict a positive price-output correlation. (1) Hence, Mankiw (1989, p. 88) criticizes real business cycle theory because it cannot explain the fact that "inflation tends to rise during booms and fall during recessions." This general belief in procyclical prices began to change in the early 1990s after several authors reported finding that detrended measures of the price level and output have a negative correlation during the postwar period. One important early study is that of Kydland and Prescott (1990, p. 17), who conclude that "any theory in which procyclical prices figure crucially in accounting for postwar business cycle fluctuations is doomed to failure. The facts we report indicate that the price level since the Korean War moves countercyclically." (2) This apparent change in the sign of the price-output correlation has resulted in a literature on the cyclical behavior of the price level. Although this literature is concerned largely with measuring the correlation between fluctuations in the price level (or inflation) and fluctuations in output (the price-output correlation), it is also concerned with what determines whether an economy has a positive or a negative price-output correlation.

This paper contributes to the literature on the cyclical behavior of prices by presenting time-varying estimates of the price-output correlation. The motivation for this research is the dependence of the estimated price-output correlation on sample period. The difference between pre- and postwar estimates is only one example. There also appears to be some disagreement about when the change in the sign of the correlation occurred. For example, Cooley and Ohanian (1991) report a negative correlation for their entire post-1945 sample, Kydland and Prescott (1990) argue that it has been negative since the end of the Korean War, but Wolf (1991) presents evidence that it did not become negative until after 1973. (3) One advantage of our methodology is that it can estimate when the change in the sign occurred. Another is that it is capable of finding other changes in the sign of the correlation that until now have not been reported in the literature.

Although the previous literature clearly recognizes that the price-output correlation is sample dependent, the typical study implicitly assumes that the correlation is fixed over any given sample period. By focusing only on the constant correlation within arbitrarily chosen periods, one may be losing important information about the dynamics of the comovement of output and the price level within and across regimes. Furthermore, there is a risk that the regimes might have been misspecified. This paper therefore presents time-varying estimates of the price-output correlation for the United States. This is done by estimating a two-variable vector autoregression (VAR) model in which it is assumed that the disturbances follow a bivariate generalized autoregressive conditional heteroskedasticity (GARCH) process. In the GARCH process, the conditional variance-covariance matrix of the residuals changes over time, allowing quarterly estimates of the contemporaneous price-output correlation coefficient. These estimates allow the identification of periods during which the price-output correlation was generally positive, those during which it was essentially zero, and those during which it was generally negative. It also allows a comparison of whether the price-output correlation is systemically different during periods of recession than it is during periods of recovery. (4)

This paper defines the price-output correlation to be the contemporaneous correlation coefficient between unexpected changes in output and the price level, a definition supported by the work of den Haan (2000). Assuming a fixed residual variance-covariance matrix, he shows that the forecast errors from a VAR can be used to obtain consistent estimates of the price-output correlation as long as the number of lags in the VAR is sufficient to cause the disturbance to be stationary. It does not matter whether the variables in the VAR are stationary and what filter is used to detrend the data.

This paper finds that the price-output correlation is usually positive before 1945 and nearly always close to zero from 1945 through 1963. Beginning in 1963, the frequency of negative correlations increases dramatically. In addition, we find that a zero (or at least statistically insignificant) price-output correlation has been much more common than realized by previous writers. For the entire sample period (1876:IV-1999:IV), this paper reports a statistically insignificant price-output correlation 57% of the time. For the post-1944 part of the sample, the correlation is zero 70% of the time, significantly positive only 11% of the time, and significantly negative only 19% of the time. The results are supportive of macroeconomic models in which both aggregate demand and aggregate supply shocks can be important in both the short and the long run--that is, models that allow the price-output correlation to be positive, negative, or zero for extended periods of time at forecast horizons up to four years in the fut ure.

Section 2 explains why one should expect the price-output correlation to be time-varying and in doing so offers a brief survey of previous literature on the price-output correlation. Section 3 describes the data and presents the methodology employed by this paper. Section 4 presents and discusses the results, while section 5 offers some conclusions.

2. Reasons for a Time-Varying Price-Output Correlation

There is some value in asking why one should expect the price-output correlation to be time-varying. The simplest possible explanation for a time-varying price-output correlation is that the relative frequency and sizes of shocks to aggregate supply and aggregate demand change. In a model with flexible prices, during periods when shocks to aggregate supply dominate, the correlation is negative, while during those when shocks to aggregate demand dominate, the correlation is positive.

Several writers, however, confront such a naive connection between the sign of the price-output correlation and the relative importance of supply and demand shocks. Chada and Prasad (1993), Ball and Mankiw (1994), and Judd and Trehan (1995) show that sticky-price models with only demand shocks can yield a negative price-output correlation. (5) These models therefore imply that the price-output correlation could be time varying because the degree of price stickiness changes over time, den Haan (2000), however, shows that models with only a demand shock are not capable of generating the pattern of price-output correlations across different forecast horizons that he obtains. Hence, his results imply that a changing degree of price stickiness by itself cannot be the only cause of a time-varying price-output correlation.

A more interesting explanation for a time-varying price-output correlation comes from the observation that monetary policy affects the price-output correlation. Cover and Pecorino (2003) and Pakko (2000) present models in which monetary policy can change the sign of the price-output correlation. (6) Since monetary policy is likely to be systematically different during recessions and expansions, their models imply that the price-output correlation is time varying.

Pakko (2000) obtains his results by simulating a shopping time monetary model with endogenous monetary policy. He finds that the price-output cospectrum is negative at all frequencies in response to a productivity shock if there is a constant growth money rule. But if it is assumed that monetary policy is implemented in a way that makes the money stock procyclical, then Pakko finds that the price-output cospectrum can be made positive at all frequencies. Since the degree to which monetary policy is procyclical changes with economic conditions, Pakko's model implies a time-varying price-output correlation.

Cover and Pecorino (2003) examine the effect of optimal monetary policy on the price-output correlation in an IS-LM model augmented with an upward-sloping aggregate supply curve. They find that the more successful monetary policy is in offsetting aggregate demand shocks, the less likely it is that the price-output correlation is positive. Whether or not successfully offsetting aggregate demand shocks causes the price-output correlation to be close to zero or to become negative depends on how the monetary authority responds to supply shocks. The greater the degree to which the monetary authority allows supply shocks to affect output, the more likely it is that the correlation is close to zero. But if the monetary authority tries to prevent temporary supply shocks from affecting output, the more likely it is that the price-output correlation is negative, if the degree to which the monetary authority has been successful at offsetting aggregate demand shocks, as well as the degree to which the monetary authority has allowed temporary supply shocks to affect output has changed over time, then Cover and Pecorino's model implies that the price-output correlation is time varying.

3. Data and Methodology

The U.S. quarterly time-series data on real gross domestic product (GDP) and nominal GDP are from the U.S. Department of Commerce for the period 1959:I-1999:IV and Balke and Gordon (1986) for the period 1875:I-1959:I. The data sets were spliced together so that the quarterly growth rates of real and nominal GDP through the first quarter of 1959 are equal to those in the Balke-Gordon data, while those beginning with the second quarter of 1959 are equal to those in the Department of Commerce data. (7) The price level is defined to be the GDP deflator.

In order to show that the correlation coefficient is time varying, we start our analysis with constant conditional correlations between output and price over a variety of sample periods, as has been assumed in all previous studies. The constant conditional correlations are calculated from the residuals of an unrestricted VAR(p) process governing quarterly measures of output ([Y.sub.t]) and the price level ([P.sub.t]):

[X.sub.t] = [B.sub.0] + [summation over (p/j=1)] [B.sub.j][X.sub.t-j] + [e.sub.t], (1)

where [X.sub.t] = [[[y.sub.t] [P.sub.t]].sup.'] and the order of the VAR, p, is set to be the minimum lag length that renders the residual vector [e.sub.t] to be serially uncorrelated and stationary. Equation 1 is estimated by ordinary least squares (OLS). The Ljung-Box Q-test is used to test for up to 12 lags of serial correlation in [e.sub.t], and the Dickey-Fuller test is used to test the stationarity of [e.sub.t]. Because the lag length employed (p) is sufficiently long to eliminate serial correlation, the particular correlation coefficient we are examining is that between changes in output and the price level that cannot be explained by the past behavior of either variable. Hence, the correlation coefficient examined here is similar to the one examined by den Haan (2000) and is a consistent estimate of the output-price correlation.

Previous studies (e.g., Kydland and Prescott 1990; Cooley and Ohanian 1991; Wolf 1991; Backus and Kehoe 1992; Smith 1992; Chada and Prasad 1993) have tried different measures of variables to calculate the price-output correlation and different filtering methods to obtain stationarity for the variables. The advantage of using the VAR prediction errors to examine price-output correlations is that, as proved by den Haan (2000), it does not require assumptions about the order of integration or the types of assumptions needed for VAR decompositions, as long as Equation 1 has sufficient lags so that [e.sub.t] is not integrated. Nevertheless, as den Haan points out, if the variables in the VAR are correctly filtered, then the resulting estimates of the price-output correlation are more efficient. Hence, for cautiousness and to check the robustness of our results, we estimated VARs using several filters commonly seen in the literature. (8) Table 1 reports the results from two specifications. The "least restricted mod el" uses the logarithms of output and the price level and includes five lags. The "best model" is selected by the augmented Dickey-Fuller and KPSS (see Kwiatkowski et al. 1992) tests. The results of these tests, presented in Appendix A, show that output growth is better modeled as level stationary and that inflation is better modeled as trend stationary. Therefore, the best model uses demeaned and detrended output growth and inflation and includes four lags. The results for the other filtered data sets are very similar to those presented in Table i and do not qualitatively change any of our conclusions. (9)

The first row of Table I shows that the estimated correlation coefficient for the entire sample period is positive and is approximately equal to 0.21. Following Friedman and Schwartz (1982), we further divide the sample into four subperiods. Prior to 1901, the estimated correlation coefficient is positive but small enough to be essentially zero; during the period 1901-1928, it increases to approximately 0.31; during the period 1929-1946, it increases further to approximately 0.41; and finally, during the postwar period, it falls below 0.05, once again essentially zero. If (following Cooley and Ohanian 1991) we look at two subperiods after World War 11, we find that the correlation coefficient is negative (approximately -0.08) for the period 1954:I-1973:I but essentially zero for the period 1 966:I-1999:IV

The results reported in Table 1, as do the results of the previous literature (summarized in Appendix B), clearly show that the price-output correlation is time variant and therefore lend support to the idea that the price-output correlation should be estimated in a manner that allows it to be time varying. The most widely used time-varying covariance model is the multivariate GARCH model. Several parsimonious specifications of the multivariate GARCH model are available (see Kroner and Ng 1998). This paper employs a specification originally proposed by Baba, Engle, Kraft, and Kroner, called the BEKK model (see Engle and Kroner 1995). The important feature of this specification is that it builds in sufficient generality, and at the same time, it does not require estimation of many parameters. More important, the BEKK model guarantees by construction that the variancecovariance matrix in the system is positive definite.

Specifically, assume that the disturbance, e,, in Equation 1 is a conditionally zero-mean Gaussian process with time-varying conditional variance-covariance matrix [H.sub.t]: [e.sub.t] \ [[OMEGA].sub.t-1] ~ N(0, [H.sub.t]), where [[OMEGA].sub.t-1] is the information set at time t - 1. Consider the following GARCH (1, 1) process,

[H.sub.t] = [GAMA]'[GAMA] + G'[H.sub.t-1]G + A'[e.sub.t-1][e'.sub.t-1]A, (2)

where the constant matrix [TAU] is restricted to be upper triangular so that all the terms on the right-hand side of the equation are in quadratic form, which guarantees a positive definite [H.sub.t].

In order to restrict the specification for [H.sub.t] further to obtain a numerically tractable formulation, we set the constant matrices G and A to be diagonal. These restrictions assume that the conditional covariance (the off-diagonal terms in [H.sub.t]) depends only on the past covariances and not on the variances. Finally, in order to obtain estimates that made economic sense, it was necessary to restrict the diagonal elements of G so that their product would always be positive. (10)

After Equations 1 and 2 have been estimated jointly by maximum likelihood estimation (MLE), it is straightforward to obtain a time series of estimated correlation coefficients from the series of [H.sub.t], the estimated [H.sub.t] obtained from the MLE. For inference, we obtain confidence intervals by a Monte Carlo simulation. This was done in the following way. Consider the following alternative expression of the GARCH process:

[e.sub.t] = [h'.sub.t][v.sub.t], (3)

where [v.sub.t] ~ N(0, 1) and [h.sub.t] is an upper triangular matrix obtained from the Cholesky decomposition of [H.sub.t] such that [h'.sub.t][h.sub.t] = [H.sub.t]. Therefore, [e.sub.t] [[OMEGA].sub.t-1] ~ N(0, [H.sub.t]), where [H.sub.t] follows the GARCH process (Eqn. 2). Let [h.sub.t] be the Cholesky decomposition of [H.sub.t]. One thousand sets of random matrices [v.sub.it] are drawn from the bivariate normal distribution N(0, 1), i = 1, ..., 1000. The generated residuals [e.sub.it] = [h.sub.t][v.sub.it] and the estimated coefficients in Equation 1 are then used to generate the artificial data [X^.sub.it]. Finally, Equations 1 and 2 are estimated by the MLE using the artificial data. The resulting [H^.sub.it], i = 1, ..., 1000, from these 1000 simulations are used to construct the one-standard-deviation band for the original [H.sub.t], which in turn includes the one-standard-deviation hand for the estimated correlation coefficients.

4. Results

Results for Price-Output Correlation from One-Step-Ahead Forecast Errors

The series of correlation coefficients obtained directly from estimating Equations 1 and 2 are estimates of the time series of correlation coefficients for the stochastic error term in Equation 1. Because the residuals obtained from estimating Equation 1 are one-step-ahead forecast errors, the results discussed in this section could be called results for one-step-ahead forecast errors, but here they are referred to simply as the correlation coefficient. The last section discusses results for other forecast horizons.

Figure 1 presents the estimates of the time series of correlation coefficients along with the one-standard-deviation band obtained from the simulations obtained using the best model." As can be seen from Figure 1, there are many periods during which the correlation coefficient is not significantly different from zero. In order to make it easy to identify periods during which the correlation coefficient is either positive or negative, Figure 2 presents a graph in which statistically insignificant values of the correlation coefficient are represented by gray squares connected by a gray line, while statistically significant values continue to be represented by black squares. A black line connects consecutive observations of the estimated correlation only if both observations are significantly different from zero.

Figure 2 shows that the correlation coefficient was largely positive during four distinct periods: (i) the early 1880s, (ii) the long period from 1907 through 1944, (iii) 1958-1959, and (iv) 1981-1985. One also can see that there are five periods during which the correlation coefficient was largely negative: (i) 1877:IV-1878:III, (ii) 1892-1893, (iii) 1945:IV-1946:III, (iv) 1964:I-1978:II, and (v) the last seven years of the sample, 1993:I-1999:IV. The estimated price-output correlation coefficient is not significantly different from zero during four relatively long periods. These are the periods 1883- 1891, 1894-1906, 1947-1957, and 1986-1992. Additionally, the correlation is not significantly different from zero during the following shorter periods: 1928-1929, 1959-1963, 1967:IV-1969:I, and 1971"II-1974:III.

Table 2 provides a summary of the information in Figure 2 by presenting for various sample periods the frequency of significantly positive, not significantly different from zero, and significantly negative correlations. For the entire sample (1876:IV-1999:IV), the estimated correlation coefficient is not significantly different from zero over one-half of the time, significantly positive about one-third of the time, and significantly negative only 12% of the time. But the majority of positive observations occur prior to 1945. Hence, for the 1945:I-1999:IV period, the correlation is significantly positive only 11% of the time using the best model and 14% of the time using the least restrictive model. It is notable that Table 2 implies that during the postwar period, the price-output correlation has not been largely negative; rather, it has been largely not significantly different from zero since 1945. Only about 20% of quarters yield a significantly negative correlation, while 70% of quarters have a correlation coefficient not significandy different from zero.

Table 2 clearly shows that negative correlations also are largely a post-1963 phenomenon. During the period 1945:I-1963:IV, the correlation was significantly negative only 5% of the time but significantly negative over one-fourth of the time after 1963. Table 2 suggests that rather than asking why the correlation changed from being positive before the war to being negative after the war, macroeconomists should be asking why it was nearly always zero during 1945-1963 and why negative correlations rarely appeared before 1964.

One possible criticism of the conclusion that the price-output correlation is usually zero is that it is based on simulated standard deviations that are relatively large. Nearly all the standard deviations are greater than 0.15, about 90% are greater than 0.17, and over one-half of them are greater than 0.2. This means that the absolute value of an estimated correlation coefficient must be greater than 0.17 to be considered significant. What sort of results do we obtain if we use more modest criteria to determine whether an estimated correlation coefficient is economically significant?

The more modest criteria used here are as follows. (i) A period with a clearly positive correlation is one in which the estimated price-output correlation is continuously positive and includes at least four consecutive quarters during which the estimated correlation coefficient is greater than 0.15. (ii) A period with a clearly negative correlation is one in which the estimated correlation is always negative but includes at least four consecutive quarters during which the estimated correlation coefficient is less than -0.15. (iii) Any period of at least four quarters with an estimated correlation that is continuously less than 0.15 in absolute value, as well as any period that does not meet the previous two criteria, is considered a period with an essentially zero correlation. Tables 3 and 4 present average values of the estimated correlation coefficient for periods with positive and negative price-output correlation coefficients based on these modest criteria.

As shown in Table 3, the previously mentioned criteria yield eight periods with a clearly positive price-output correlation. The three longest periods are within the years 1907-1944, and each is longer than seven years. (12) The fourth longest period with a positive correlation is 1981:I-1985:III, only one quarter shy of being five years long, while the fifth longest is the 1879:IV-1883:III period. During each of these five periods, the average value of the price-output correlation is greater than 0.3.

As can be seen from Table 4, the previously mentioned criteria continue to imply that the price-output correlation was rarely negative before 1964. The period with the longest continuously negative price-output correlation (by these criteria) is the 27-quarter period that closes the sample (1993:II-1999:IV). For the least restrictive model, the second and third longest periods (1964:I-1967:III and 1974:IV-1978:II) are both one-quarter shy of being four years long. During each of the post-1963 periods with a clearly negative price-output correlation, the average value is between -0.22 and -0.27.

Table 5 presents the shares of quarters with clearly positive, clearly negative, and essentially zero correlations. For the period 1876:IV-1944:IV, 57% of quarters have a clearly positive correlation, but only 5% have a clearly negative correlation according to the results obtained from the best model. For the sample period 1945:I-1963:IV, the share of quarters with a clearly positive correlation declines to 8%, while 87% of quarters have an essentially zero correlation. Finally, for the post-1963 sample, the share of quarters with a negative correlation jumps to 35%, the share with positive correlations increases to 17%, while the share of quarters with an essentially zero correlation falls dramatically to 49% according to the results obtained from the best model.

Even though the results in Tables 2 and 5 are somewhat in line with the findings in previous studies with constant correlations, namely, positive output-price correlations for the prewar sample and negative for the postwar (or post-1973) period, the methodology used in this paper provides more information about the dynamics of the comovements of output and the price level. For example, we find that negative correlations are largely a post-1963, instead of postwar or post-1973, phenomenon and that the correlation has been insignificantly different from zero much more often than realized by previous studies.

Table 5 raises questions that are similar to those suggested by Table 2. In particular, why did the frequency of positive correlations decline after 1944, why is the correlation almost always zero during 1945-1963, and why did negative correlations suddenly become more important after 1963? Like Table 2, Table 5 suggests that macroeconomists should look for models that allow the price-output correlation to be positive, zero, or negative for long periods.

Comparison of Results for Recessions and Expansions

Since one purpose of macroeconomics is to explain the existence of business cycles, it is natural to ask whether the price-output correlation tends to have different values during recessions and expansions. Since the bars in Figure 2 represent periods of recession, a cursory glance at that figure shows that the correlation is rarely negative and significant during recessions. Table 6, which presents average values of the estimated price-output correlation for periods of recession and recovery, helps make sense of this observation. Using the results for the best model, for the entire sample the average correlation is 0.134 during recessions and only 0.087 during expansions. But these averages mean nothing because of the tendency for the correlation to be positive during pre-1945 expansions and negative during post-1945 expansions. For the pre-1945 sample, the average correlation is 0.150 during recessions and 0.228 during expansions.

The results are somewhat different for the period after 1944. Notice from Figure 1 that during postwar recessions, the estimated correlation is typically positive (though not always significant). Figure 2 shows that during the postwar period, significantly negative price-output correlations usually appear only during expansions. Indeed, they appear only twice during postwar recessions: during the last two quarters of 1970 and during the fourth quarter of 1974 and the first quarter of 1975. The last two rows of Table 6 show that during the postwar period, the correlation coefficient is more likely to be positive during recessions and more likely to be negative during expansions. The average value of the correlation during 1945:I-1963:IV is 0.084 during recessions and --0.036 during expansions. Because the correlation coefficient was essentially zero for most of this period, the small size of these numbers is not surprising. After 1963, however, the average value of the price-output correlation increases to 0.1 09 during recessions and falls to --0.059 during expansions.

Combining the results in Table 6 with those in Tables 2 and 5, it is reasonable to conjecture that the main reason for changes in the price-output correlation is changes in its value during periods of expansion. Although after 1944 the average value of the correlation during recessions declined, it continues to be mostly positive during periods of recession. On the other hand, the average value of the correlation during expansions not only declined after 1944 but also changed from being mostly positive to mostly negative after 1963.

Table 7 drives home this point by presenting for various periods the shares of quarters in which the point estimates of the price-output correlation are positive during recessions and expansions. Notice that for the best model, the share of quarters with a positive price-output correlation during recessions varies only from 71% to 74%. But for expansions, the share of quarters with a positive price-output correlation declines from 83% before 1945 to 58% for 1945-1963 and then to 43% for 1964-1999. Both before and after the war, more than two-thirds of recessionary quarters have a positive price-output correlation. The results in Table 7 clearly suggest that the tendency of the price-output correlation to be positive during recessions has changed very little, while its tendency to be positive during expansions has declined dramatically since 1945.

Results for Other Forecast Horizons

As mentioned previously, den Haan (2000) shows that examining the price-output correlation at different forecast horizons can provide information about what types of macroeconomic models are consistent with the data. When using quarterly data for the period 1960:11-1997:11, he finds that the price-output correlation is approximately zero at relatively short forecast horizons and negative at relatively distant forecast horizons. (13) To see if den Haan's results hold up under our methodology, in this section we present estimates of the price-output correlation at two additional forecast horizons. Appendix C derives and presents the formula for calculating the price-output correlation, which depends on both the VAR coefficients and the parameters in the GARCH process, at various forecast horizons.

Figure 3 presents the eight-step-ahead and 16-step-ahead price-output correlations along with the one-step-ahead price-output correlations previously presented in Figure 1. Figure 3 shows that the 16-step ahead correlation is nearly always less than the eight-step-ahead correlation, which in turn is nearly always less than the one-step-ahead correlation. Hence, a negative correlation is much more prevalent for both the eight- and the 16-step-ahead forecast periods than for the one-step-ahead period. This result is broadly consistent with the findings of den Haan (2000) and Pakko (2000), both of whom find that a positive correlation tends to be a short-run or high-frequency phenomenon, while negative correlations tend to have a longer-run or lower-frequency manifestation. But it is also clear from Figure 3 that the eight- and 16-step-ahead correlations are not always less than the one-step-ahead correlation. This is particularly true during the period after 1963, when all three correlations are often approxima tely equal. Indeed, during much of the post-1963 period, there is very little difference among the three correlations. This implies that since 1963, the relative importance of demand and supply shocks often has been similar in the long and short runs.

5. Summary and Conclusions

This paper finds that the price-output correlation has changed often since 1876. The estimates suggest that the correlation is usually positive before 1945 and nearly always close to zero from 1945 through 1963. After 1963, however, the frequency of negative correlations increases dramatically. Although periods with a consistently positive or negative price-output correlation do not coincide in any obvious way with the business cycle, the paper does find that after 1944, the price-output correlation is rarely negative during recessions and rarely positive and significant during expansions. This suggests that it is reasonable to conjecture that the price-output correlation behaves differently after 1944 mainly because it is behaving differently during periods of expansion.

These apparent changes in the price-output correlation coefficient suggest that either the relative importance of demand-side and supply-side shocks has changed over time, the reaction of policymakers to such shocks has changed, or both. The results also suggest that economists have overemphasized the idea that there has been a negative correlation between prices and output during the postwar period. Macroeconomists should not look for models that limit the sign or size of the price-output correlation.

Appendix A

We identify the order of integration for each variable by using the augmented Dickey-Fuller (ADF) tests, which test the null hypothesis of a Unit root, and the generalized KPSS test (Hobijn, Franses, and Ooms 1998), which tests the null hypothesis of stationarisy against the alternatives of nonstationarity. Following a suggestion by Dickey and Pantula (1987), the unit root tests are first conducted with two roots, and if two roots are rejected, then single unit root is tested. Column 1 of Table Al shows that the ADF test rejects the null of a Unit root for both second-differenced series and that the KPSS test fails to reject the null of zeromean stationary for both series. That is, the second-differenced series are better modeled as zero-mean stationary processes. Column 2 shows that the ADF test rejects the null of a unit root for both first-differenced series. The KPSS test fails to reject the null of level stationary for output growth and the null of trend stationary for inflation. That is, output growth i s better modeled as a stationary process with a drift, while inflation is better modeled as a stationary with a drift and a deterministic time trend. In column 3, the results show that for the log-price level, the ADF test fails to reject the null of unit root and that the KPSS tests reject the null of stationarity; that is, the log price is better modeled as I(1). For the log-output level, the ADF test rejects the null of unit root at the 5% level but not at the 1% level. However, based on the facts that the KPSS tests reject the null of stationarity and that the estimated coefficient on the first lag is 0.968, we model the log-output as an I(1) series.

Table A1

Unit Root Tests

 (1) (2)
 Differenced Differenced Output
 Output Growth Inflation Growth Inflation

ADF (a) -13.747 -12.518 -10.423 -8.400
KPSS (null: 0.047 0.090 11.825 2.618
 zero-mean stationary)
KPSS (null: - - 0.042 0.692
 level stationary)
KPSS (null: - - - 0.075
 trend stationary)

 (3)
 Log Log 5% Critical
 Output Price Value

ADF (a) -3.629 -2.394 -3.432
KPSS (null: 360.742 60.156 1.656
 zero-mean stationary)
KPSS (null: 3.188 2.966 0.460
 level stationary)
KPSS (null: 0.174 0.588 0.148
 trend stationary)

(a)This is the ADF t-test. The ADF [rho]-test yields the same
conclusion. The lags in the ADF test are selected by the AIC criterion.
We also use Schwartz's criterion and a sequence of 5% t-tests for the
significance of coefficients on additional lags, as suggested by Ng and
Perron (1995). The results with these two criteria do not change the
conclusions here.

Appendix B

Results of Preivous Writers

Author Filter Sample [[rho].sup.a]

Kydland and Hodrick-Prescott 1954-1989, -0.55
 Prescott (1990) quarterly
Wolf (1991) Hodrick-Prescott 1957-1989, 0.09
 quarterly
 1973-1989, -0.40
 quarterly
Cooley and Hodrick-Prescott Annual
 Ohanian (1991) 1870-1900 0.24
 1900-1928 0.24
 1928-1946 0.77
 1949-1975 -0.58
 Quarterly
 1948:II-1987:II -0.57
 1954:I-1973:I -0.36
 1966:I-1987:II -0.68
 Log differencing Annual
 1870-1900 0.05
 1900-1928 0.12
 1928-1946 0.67
 1949-1975 -0.07
 Quarterly
 1948:II-1987:II -0.06
 1954:I-1973:I -0.05
 1966:I-1987:II -0.23
 Linear detrending Annual
 1870-1900 -0.17
 1900-1928 -0.12
 1928-1946 0.73
 1949-1975 -0.53
 Quarterly
 1948:II-1987:II -0.67
 1954:I-1973:I -0.69
 1966:I-1987:II -0.87
 First-differenced Annual
 output and 1871-1975 -0.03
 second- 1871-1910 -0.13
 differenced 1928-1946 0.28
 price
 Quarterly
 1948:II-1987:I 0.03
 1954:I-1973:I -0.03
 1966:I-1987:I 0.05
Smith Hodrick-Prescott Annual
 (1992) 1869-1909 0.23
 1910-1929 0.03
 1930-1945 0.49
 1946-1983 -0.68
 First differencing Annual
 1869-1909 0.04
 1910-1929 -0.02
 1930-1945 0.84
 1946-1983 -0.54
Backus and Hodrick-Prescott Annual
 Kehoe Prewar 0.22
 (1992) Interwar 0.72
 Postwar -0.30
 Log differencing Prewar 0.13
 Interwar 0.37
 Postwar -0.25
Chada and Log differencing 1947-1989, -0.07
 Prasad Linear detrending quarterly -0.69
 (1993) Linear detrending -0.10
 with 1973
 break
 Hodrick-Prescolt -0.19
 Inflation and 0.21
 detrended
 output
 Inflation and 0.13
 detrended
 output with
 1973 break
 Inflation and 0.16
 Hodrick-
 Prescott
 filtered output
 Inflation and From
 Beveridge- 0.01 to
 Nelson 0.20 for
 stationary different
 component ARMA
 of output models
 Inflation and 0.67
 Blanchard-
 Quah
 stationary
 component of
 output
den Haan Forecast errors 1948-1997, Up to 0.4
 (2000) from VAR quarterly in short
 and monthly forecasting
 horizons
 and down
 to -0.6
 in long
 forecasting
 horizons
Pakko Cospectrum of Quarterly Positive at
 (2000) real GNP and 1875:I-1914:IV low, near
 GNP deflator zero at high
 after applying frequencies
 H-P filter 1920:I-1940:IV Positive at all
 and first- frequencies
 differencing 1950:I-1994:IV Negative
 to Gordon- at low
 Balke data set frequencies

Author Main Conclusion

Kydland and Negative correlation
 Prescott (1990) since Korean War
Wolf (1991) Positive correlation
 before 1973,
 negative after 1973

Cooley and The only consistent
 Ohanian (1991) positive correlation
 is between two
 world wars
































Smith Positive correlation
 (1992) from the late 19th
 century until World
 War 11, except period
 around World
 War I; negative
 correlation for the
 post-Depression
 period, except a
 period the 1950s or 1960s
Backus and Positive correlation
 Kehoe before World
 (1992) War 11, negative
 after World War II



Chada and Similarly filtered
 Prasad price and output
 (1993) have negative
 correlations;
 inflation rate has
 positive correlation
 with output using
 other filters





















den Haan Correlation is
 (2000) approximately zero
 at relatively short
 forecast horizons
 and negative
 forecast horizons



Pakko Correlation changes
 (2000) sign after the war
 because of changes
 in cospectrum
 at low frequencies





(a)Estimate of contemporaneous correlation coefficient.

Appendix C

To find the time-varying conditional forecast errors, consider the Wald representation of Equation 1:

[X.sub.t] = [[alpha].sub.0] + [summation over ([alpha]/j=1)] [[alpha].sub.j][e.sub.t+1-j], [[alpha].sub.1] = I.

Let the s-step-ahead forecast error of [X.sub.t] be [F.sup.s.sub.t]. Then

[F.sup.s.sub.t] = [X.sub.t] - E([X.sub.t]\t-s) = [e.sub.t] + [[alpha].sub.2][e.sub.t-1] + [[alpha].sub.3][e.sub.t-2] + ... + [[alpha].sub.s][e.sub.t-s+1], and E([F.sup.s.sub.t]/t - s) = 0. The conditional variance-convariance matrix of [F.sup.s.sub.t] is

V([F.sup.s.sub.t]/t - s) = E(([e.sub.t] + [[alpha].sub.2][e.sub.t-1] + [[alpha].sub.3][e.sub.t-2] + ... + [[alpha].sub.s][e.sub.t-s+1])([e.sub.t] + [[alpha].sub.2][e.sub.t-1] + [[alpha].sub.3][e.sub.t-2] + ... + [[alpha].sub.s][e.sub.t-s+1]) / t - s]

= E[([e.sub.t][e.sub.t] + [[alpha].sub.2][e.sub.t-1][e.sub.t-1][[alpha].sub.2] + ... + [[alpha].sub.s] [e.sub.t-s+1] [e.sub.t-s+1][[alpha].sub.2]) / t - s],

where the second equality uses the fact that [e.sub.t] is serially uncorrelated. Lte [e.sub.t][e.sub.t] = [H.sub.t] + [w.sub.t], where [w.sub.t] is a white-noise process. Then

V([F.sup.s.sub.t]/t - s) = E[([H.sub.t] + [w.sub.t]) + [[alpha].sub.2]([H.sub.t-1] + [w.sub.t-1])[[alpha].sub.2] + ... + [[alpha].sub.s]([H.sub.t-s+1] + [w.sub.t-s+1])[[alpha].sub.2] / t - s]

= E[([H.sub.t] + [[alpha].sub.2][H.tub.t-1][[alpha].sub.2] + ... + [[alpha].sub.s][H.sub.t-s+1][[alpha].sub.s] / t - s]

= E([H.sub.t] / t - s + [[alpha].sub.2]L * E([H.sub.t] / t - s + 1)[[alpha].sub.2] + [[alpha].sub.3][L.sup.2] * E([H.sub.t] / t - s + 2) [[alpha].sub.3] ... + [[alpha].sub.s][L.sup.s-1]E([H.sub.t] / t - 1)[[alpha].sub.s],

where L is the lag operator. Taking the conditional expectation of Equation 2, we have

E([H.sub.t] / t - k) = ([GAMMA][GAMMA]) + GL * E([H.sub.t] / t - k + 1)G + AL * E([H.sub.t] / t - k + 1)A.

Using the fact that E([H.sub.t] / t - 1) = [H.sub.t] and solving the previous equation recursively for k = 2,...,s, V([S.sup.s.sub.t] / t - s) can be derived from [H.sub.t].

[FIGURE 1 OMITTED]

[FIGURE 2 OMITTED]

[FIGURE 3 OMITTED]

Table 1

Constant Conditional Correlation Coefficients between the Logarithms
of Output and the Price Level

Sample Period Best Model Least Restrictive Model

1876:LV-1999:IV 0.213 0.218
1876:IV-1900:IV 0.056 0.065
1901:I-1928:IV 0.306 0.307
1929:I-1946:IV 0.407 0.414
1948:ll-1999:IV 0.046 0.032
1954:I-1973:I -0.077 -0.076
1966:I-1999:JV 0.004 -0.003

Table 2

Share of Quarters When Correlation Coefficient is Positive, Zero, and
Negative

 Share of
 Quarters Not
 Share of Quarters Significantly
 Different
 Significantly Positive (%) from Zero (%)
 Least
 Best Restricitive Best
Sample Period Model Model Model

1876:IV-1999:IV 31 34 57
1876:IV-1944:IV 47 51 47
1945:I-1999:IV 11 14 70
1945:I-1963:IV 8 9 87
1964:I-1999:IV 14 16 61

 Share of
 Quarters Not
 Significantly Share of Quarters
 Different
 from Zero (%) Significantly Negative (%)
 Least
 Restrictive Best Restrictive
Sample Period Model Model Model

1876:IV-1999:IV 54 12 12
1876:IV-1944:IV 44 6 5
1945:I-1999:IV 65 19 21
1945:I-1963:IV 86 5 5
1964:I-1999:IV 55 26 29

Table 3

Average Value of Estimated Price-Output Correlation Coefficient for
Various Sample Periods during Which the Correlation is Clearly Positive
(a)

 Average Estimated Correlation
 Least Length of
Sample Period Best Model Restrictive Model Period

(1) 1879:IV-1883:III 0.352 0.354 16
(2) 1893:IV-1895:II 0.236 0.217 7
(3) 1907:I-1914:III 0.317 0.333 31
(4) 1915:I-1927:IV 0.328 0.319 52
(5) 1930:I-1944:III 0.384 0.443 59
(6) 1958:I-1959:II 0.295 0.269 6
(7) 1978:III-1980:II 0.203 0.203 12
(8) 1981:I-1985:III 0.323 0.258 19

(a)A clearly positive is one in which the correlation coefficient is
continuously positive and includes at least four consecutive
observations greater than 0.15.

Table 4

Average Value of Estimated Price-Output Correlation Coefficient for
Various Sample Periods in Which the Correlation is Clearly Negative (a)

 Average Estimated Correlation
 Least Length of
Sample Period Best Model Restrictive Model Period

 (1) 1877:IV-1878:III -0.295 -0.257 4
 (2) 1891:IV-1893:II -0.388 -0.377 7
 (3) 1903:I-1904:I -0.190 -0.203 4
 (4) 1945:IV-1946:III -0.437 -0.423 4
(5a) 1964:I-1965:I -0.235 -- 5
(5b) 1964:I-1967:III -- -0.223 15
 (6) 1969:II-1971:1I -0.209 -0.246 8
 (7) 1974:IV-1978:II -0.267 -0.267 15
 (8) 1993:II-1999:IV -0.244 -0.221 27

(a)A clearly negative period is one in which the correlation coefficient
is continuously negative and includes at least four consecutive
observations less than -0.15.

Table 5

Share of Observationa When Correlation is Clearly Positive, Essentially
Zero, and Clerly Negative (a)

 Share of Quarters Share of Quarters
 Clearly Positive (%) Essentially Zero (%)
 Least Least
 Best Restrictive Best Restrictive
Sample Period Model Model Model Model

1876:IV-1999:IV 38 40 48 43
1876:IV-1944:IV 57 60 38 34
1945:I-1999:IV 14 15 62 54
1945:I-1963:IV 8 8 87 87
1964:I-1999:IV 17 19 49 37

 Share of Quarters
 Clearly negative (%)
 Least
 Best Restrictive
Sample Period Model Model

1876:IV-1999:IV 14 17
1876:IV-1944:IV 5 5
1945:I-1999:IV 24 31
1945:I-1963:IV 5 5
1964:I-1999:IV 35 44

(a)A clearly positive period is one in which the correlation coefficient
is continuously positive and includes at least four consecutive
observations greater than 0.15. A clearly negative period is defined
analogously. All other periods have an essentially zero correlation

Table 6

Average Values of Price-Output Correlation during Recessions and
Expansions

 Recession Expansions
 Best Least Best
Sample Model Restrictive Model Model

1876:IV-1999:IV 0.134 0.149 0.087
1876:IV-1944:IV 0.150 0.167 0.228
1945:I-1963:IV 0.084 0.072 -0.036
1964:I-1999:IV 0.109 0.104 -0.059

 Expansions
 Least
Sample Restrictive Model

1876:IV-1999:IV 0.097
1876:IV-1944:IV 0.250
1945:I-1963:IV -0.016
1964:I-1999:IV -0.045

Table 7

Share of Quarters with a Positive Price-Output Correlation during
Recessions and Expansions (a)

 Recession (%)
Sample Best Model Least Restrictive Model

1876:IV-1999:IV 74 77
1876:IV-1944:IV 74 78
1945:I-1963:IV 73 80
1964:I-1999:IV 71 70

 Expansion (%)
Sample Best Model Least Restrictive Model

1876:IV-1999:IV 64 63
1876:IV-1944:IV 83 84
1945:I-1963:IV 58 52
1964:I-1999:IV 43 43

(a)In this table, a quarter is defined as having a positive price-output
correlation if its point estimate is positive.

Received September 2001; accepted October 2002.

(1.) See, for example, Bums and Mitchell (1946, p. 101), Hansen (1951, pp. 4-5), and Blanchard and Fischer (1989, p. 20). It should be noted that before 1990, not all economists thought that prices are always procyclical. The most important exception is Milton Friedman (1977), who in his Nobel Prize lecture explores the possibility of countercyclical prices. See also Friedman and Schwartz (1982, p. 402), who report a negative correlation between output and the price level for both the United States and the United Kingdom.

(2.) Other early contributors to this literature are Cooley and Ohanian (1991). Wolf (1991), Backus and Kehoe (1992), and Smith (1992).

(3.) There is also ample evidence that the signs and sizes of other correlations depend on sample period. See, for example, Hartley (1999).

(4.) It is worth noting that by allowing the residuals to follow a GARCH process, the empirical model employed here implicitly takes into account changes in the variance of output, such as that discussed by McConnell and Perez-Quiros (2000).

(5.) Spencer (1996) uses similar arguments to show that a negative price-output correlation is consistent within an aggregate demand-aggregate supply model in which demand shocks affect the price level permanently and output temporarily, while supply shocks have permanent effects on both output and the price level.

(6.) Gavin and Kydland (1999) and Floden (2000) also present models in which monetary policy affects the price-output correlation.

(7.) This is equivalent to multiplying the logarithms of the Balke-Gordon data by a constant so that the Balke-Gordon observation for 1959:I equals that for the Department of Commerce data. Michael Pakko provided the authors with the Balke-Gordon data.

(8.) The complete set of filters considered were (i) demeaning and detrending. (ii) firs-differenced, (iii) demeaning and detrending the first-differenced data, (iv) using first-differenced output and the second-differenced price level, and (v) demeaning and detrending first-differenced output and second-differenced price level. As is stated in the text, only the third specification was not rejected by the data. The results for the filtered data are too similar to those reported in the text to be worth reporting here. we chose to report only the results for the third specification (the "best model") and the unfiltered data ("least restrictive model") because each of the other filters is implicitly imposing a constraint on the data rejected by our specification tests.

(9.) In addition to the fitters mentioned in footnote 8, we also filtered the data with the Hodrick-Prescott filter. Again, the results obtained using this filter are very similar to those reported in the text.

(10.) If the product of the diagonal elements in G is allowed to be negative, it tends to dominate the other terms in Equation 2, causing the covariance term in Equation 2 to be negatively related to its lagged value. Hence, any quarter with a positive covariance is almost always followed by one with a negative covariance, which in turn is followed by one with a positive covariance. The authors do not believe that such a pattern makes economic sense. Furthermore, estimates in which the diagonal elements of G were not restricted to have a positive product often did not converge.

(11.) The plot of the correlation coefficients obtained from the least restrictive model (i.e., using only the logarithms of the data) is very similar to Figure 1. As is made clear from Tables 2 to 6, the best model is slightly more likely to yield a price-output correlation that is insignificantly different from zero.

(12.) The period 1907:I-1914:III is separated from the period 1915:I-1927:IV in Table 3 because the estimated correlation for 1914:IV is -0.154 using the least restrictive model and -0.213 using the best model.

(13.) With monthly data, den Haan (2000) finds that correlations are positive at short forecast horizons and gradually become negative as the horizon lengthens.

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James Peery Cover *

C. James Hueng +

* Department of Economics, Finance, sad Legal Studies, University of Alabama, P.O. Box 870224, Tuscaloosa, AL 35487-0224, USA; E-mail jcover@cba.ua.edu; corresponding author.

+ Department of Economics, Finance, and Legal Studies, University of Alabama, P.O. Box 870224, Tuscaloosa, AL 35487-0224, USA; E-mail chueng@cba.ua.edu.

The authors wish to thank two anonymous referees for many helpful comments and discussion.