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## THE COMOVEMENTS BETWEEN REAL ACTIVITY AND PRICES

## AT DIFFERENT BUSINESS CYCLE FREQUENCIES

WOUTER J. DEN HAAN

University of California at San Diego and NBER

April 1996

In this paper, I present two different methods that can be used to obtain a concise set of descriptive results about the comovement of variables. The statistics are easy to interpret and capture important information about the dynamics in the system that would be lost if one focused only on the unconditional correlation coefficient of detrended data. The methods do not require assumptions about the order of integration. That is, the methods can be used for stationary as well as integrated processes. They do not require the types of assumptions needed for VAR decompositions either. Both methods give similar results. In the postwar period, the comovement between output and prices is positive in the “short run” and negative in the “long run”. During the same period, the comovement between hours and real wages is negative in the “short run” and positive in the “long run”. I show that a model in which demand shocks dominate in the short run and supply shocks dominate in the long run can explain the empirical results, while standard sticky-price models with only demand shocks cannot.

I am especially grateful to Valerie Ramey for her insightful ideas and the many fruitful discussions on this topic. I also would like to thank Jeff Campbell, Timothy Cogley, Graham Elliott, Marjorie Flavin, Clive Granger, Bob Hall, Jim Hamilton, Tom Sargent, John Taylor, and Bharat Trehan for useful comments. This project is supported by NSF grant SBR-9514813.

### 1. INTRODUCTION.

The observed correlation between macroeconomic variables plays an important role in the development and testing of business cycle theories. Macroeconomic models are judged on their ability to reproduce key correlations in the data, such as the comovement of output with wages, prices, and hours. Using these kind of empirical results to judge theories presupposes that there is a set of correlations upon which everyone can agree. In fact, the strength and even the sign of these empirical correlations are often very sensitive to the methods used to calculate them.

Nowhere is this problem more evident than in the case of the comovement between output and prices. For decades, it was generally believed that prices and output exhibited a positive correlation.[1] Often, some type of Phillips-curve effect was mentioned to rationalize a positive comovement between real activity and either prices or inflation. Recently, however, the sign of the relationship between output and prices has been called into question. In a noteworthy study, Cooley and Ohanian (1991) find that, although the correlation between output and prices is positive from 1870 to 1975, it appears to be negative during the postwar period. The correlation they study is the unconditional correlation coefficient of detrended GNP and prices, using various methods of detrending. In contrast, support for a positive correlation is given in Chadha and Prasad (1993), who argue that prices and GNP have to be detrended using different methods. The correlation between hours and real wages has also received considerable attention in the literature. Most empirical studies in the literature have shown that real wages are, if anything, procyclical rather than countercyclical.[2]

In this paper, I argue that an important source of the disagreement in the literature is the focus on only one correlation coefficient. By focusing on only the unconditional correlation, one is losing valuable information about the dynamic aspects of the comovement of variables. Moreover, since the unconditional correlation coefficient is only defined for stationary variables, the researcher has to transform the data to render it stationary, and there are many ways of doing this. I present two different types of methods that can be used to obtain a concise set of descriptive results about the comovement of variables. The statistics are easy to interpret and capture important information about the dynamics in the system that would be lost if one focused on only the unconditional correlation coefficient of detrended data. The first method calculates the correlations of VAR forecast errors at different horizons.[3] Identifying assumptions, usually needed for VAR decompositions, or assumptions about the order of integration of the variables are not required. That is, the methods can be used for stationary as well as integrated processes. The second method uses band-pass filters from the frequency domain proposed by Baxter and King (1994). In this paper, I prove, using spectral analysis for integrated processes, that the properties of these type of frequency filters are equivalent for stationary and first and second-order integrated processes. This result contradicts several claims in the literature that the properties of frequency domain filters depend on the order of integration of the input series.

I apply these methods to two examples: the comovement of output with the general price level, and the comovement of hours with real wages. Both methods give very similar results. For the postwar period, the comovement between GNP and prices is positive in the “short run” and negative in the “long run”. In the complete sample from 1875 to 1993, the correlation between prices and GNP is positive at most frequencies. For the comovements between real wages and hours, I find a similar difference between the short run and the long run during the postwar period. In the “short run”, the correlation between hours and real wages is negative, and in the “long run”, it is positive. The results are robust to changes in the specification of the VAR, the data frequency, the particular price index used, and the particular postwar period considered.

Hansen and Heckman (1996) criticize the calibration approach for not matching a full set of dynamics of the model to the dynamics in the data. The statistics that are proposed in this paper do incorporate information about the dynamics but are still very intuitive. I argue that the type of statistics proposed in this paper is very effective in providing information to aid a researcher who wants to build a structural model. I support this claim by showing that the reported empirical findings have important implications for economic theories that have not been mentioned in the literature. For example, Lucas (1977) concluded from the empirical finding that real wages are acyclical that “*any attempt to assign systematic real wage movements a central role in an explanation of business cycles is doomed to failure”.* Boldrin and Horvath (1995) use the same empirical finding to motivate a model of real wage contracts. However, I find significant cyclical patterns when I distinguish between short-run and long-run movements, which suggest that cyclical movements in real wages should play an important role. The dynamic pattern found in the comovement of prices and output also has important implications for economic theories. Kydland and Prescott (1990) conclude that “*any theory in which procyclical prices figure crucially in accounting for postwar business cycle fluctuations is doomed to failure*”. My results indicate that a theory in which prices do not have some procyclical feature is, at best, missing an important part of the explanation of US business cycle fluctuations. A related question is whether the sign of the correlation coefficients can reveal the importance of demand versus supply shocks. One might think that a negative correlation between prices and output implies the presence of supply shocks. However, Chadha and Prasad (1993) and Judd and Trehan (1995) document that a standard “sticky-price” model with only demand shocks is capable of generating a negative unconditional correlation coefficient between prices and output for some detrending methods. The methods proposed in this paper reveal important information that is relevant to this question. In particular, I show that, in contrast to the results in Chadha and Prasad (1993) and Judd and Trehan (1995), a standard sticky-price model with only demand shocks cannot explain the complete set of empirical results reported in this paper. I argue that a model in which demand shocks dominate in the short run and supply shocks dominate in the long run is a plausible explanation for the empirical results.

The remainder of this paper is organized as follows. In section 2, I show how to use a VAR to measure the correlation between output and prices at different forecast horizons. In section 3, I show how to use band-pass filters to measure the correlation between output and prices at different frequencies. In section 4, I present the empirical results for the US economy regarding the comovements between prices and output. In section 5, I present the empirical results regarding the comovements between real wages and hours. In section 6, I discuss the implications of the empirical results for economic theories. The last section concludes.

### 2. MEASURING CORRELATIONS AT DIFFERENT FORECAST HORIZONS.

In section 2.1, I show how to use forecast errors to calculate correlation coefficients at different forecast horizons. In section 2.2, I analyze the relationship between this procedure and impulse response functions.

#### 2.1 Using forecast errors to calculate correlation coefficients.

Consider an N-vector of random variables, Xt. The vector Xt is allowed to contain any combination of stationary processes and processes that are integrated of arbitrary order. If one wants to describe the comovement between prices, *Pt, and output , Yt, then Xt has to include at least Pt and Yt*. Estimating the VAR involves estimating:

(2.1)

where Al is an NN matrix of regression coefficients, and B are N-vectors of constants, is an N-vector of innovations, and the total number of lags included is equal to L. The estimated VAR can be used to construct a time series of k-period ahead forecast errors for each of the elements of Xt. I denote the k-period ahead forecast and the k-period ahead forecast error of the variable Yt by , and , respectively. I do the same for Pt. The forecast errors are used to calculate the covariances or correlation coefficients between the two series. I denote the covariance between the two random variables and by COV(k) and the correlation coefficient between these two variables by COR(k). If the series are stationary, then the correlation coefficient of the forecast errors will converge to the unconditional correlation coefficient of the two series as k goes to infinity. In appendix E I show that a consistent estimate of COV(k) and COR(k) does not require any assumptions on the order of integration of Xt. For example, it is possible that Xt contains stationary as well as integrated processes. An important assumption for the derivation of this consistency results is that equation (2.1) is correctly specified. In particular, the lag order must be large enough to guarantee that t is not integrated. That is, if Xt contains I(1) stochastic processes, then the lag order has to be at least equal to 1, and when Xt contains I(2) stochastic processes, then the lag order has to be at least equal to 2.

Using an unrestricted VAR in levels leads to consistent estimates of the covariances of the k-period ahead forecast errors both when Xt does and when Xt does not include integrated processes. Alternatively, one can estimate a VAR in first differences or an error-correction system. When the restrictions that lead to these systems are correct, then imposing them may or may not lead to more efficient forecasts in a finite sample.[4] Asymptotically, there is no efficiency gain.[5] However, if they are not correct, then they lead to inconsistent estimates of the correlation coefficients. This suggests that, in practice, one would want to estimate the VAR in levels.

#### 2.2 The relationship with impulse response functions.

There is an alternative way to use the VAR to construct measures of comovements at different forecast horizons. The alternative method clarifies the relationship between this procedure and impulse response functions. The k-period ahead forecast error, , can be written as follows:

(2.2)

In this equation, the k-period ahead forecast error is written as the sum of the updates in the forecast of Yt+k , starting at period t+1. The first term on the right hand side is just the one-period ahead forecast error realized at period t+k. The second term is the update of the two-period ahead forecast. I denote the update in the k-period ahead forecast at period t by . Also, I denote the covariance between and by COV(k). Since the terms on the right hand side of equation 2.1 are serially uncorrelated, there is a simple relation between COV(k) and COV(k). That is,

(2.3).

When k is equal to one, the two covariances are identical. The “COV(k)” covariances, therefore, contain the same information as the “COV(k)” covariances. The “COV(k)” covariances clarify the relation between these statistics and impulse response functions. Suppose that Xt = A Zt, where A is an NM matrix of coefficients and Zt is an M-vector of (independent) fundamental shocks. Let be the effect on output of a one standard deviation shock in the mth element of Zt after k periods. Thus, is the impulse response of the mth element of Zt on Yt. . I define in the same way. Then, COV(k) is equal to the sum of the products of the k-step impulse responses across all fundamental shocks. That is,

(2.4)

Thus, COV(k) measures the comovement of output and prices after *k periods in response to a typical shock. COV(k*) accumulates the effects over k periods. The impulse response functions give complete information about the comovements of output and prices after any type of shock. Estimating impulse response functions, however, requires making identifying assumptions. The results often depend on the particular type of identifying assumptions, and the assumptions are often ad hoc. The advantage of the procedure proposed in this paper is that it does not require making these type of assumptions. The disadvantage of this procedure is that it only gives the dynamic effect of a typical shock.

### 3. MEASURING CORRELATIONS AT DIFFERENT FREQUENCIES.

In this section, I show how to use spectral analysis to decompose series by frequency and to measure the correlations of two series at different frequencies. In section 3.1, I assume that the variables are stationary. In section 3.2, I show that the procedures remain valid if the series are first or second-order integrated processes, or when the series contain a deterministic linear or quadratic time trend.

#### 3.1 Frequency-domain filters for stationary processes.

From the Wold-theorem, I know that any covariance stationary series has a time-domain representation. Equivalently, any covariance stationary series has a frequency-domain representation. Informally, the variable xt can be represented as a weighted sum of periodic functions of the form cos(t) and sin(t), where denotes a particular frequency. The frequency domain representation is given by

(3.1)

Here, () and () are random processes. The spectrum of a series xt is given by

(3.2)

where j is the jth autocovariance and i2 = -1. The spectrum is useful in determining which frequencies are important for the behavior of a stochastic variable. If the spectrum has a peak at frequency = /3, then the cycle with periodicity 6 (= 2 / (/3) ) periods is quantitatively important for the behavior of this stochastic variable. Consider the following examples. If xt is white noise, then the spectrum is flat. A flat spectrum means that all cycles are equally important for the behavior of the variable xt. This is intuitive since the existence of cycles implies forecastibility, and white noise is, by definition, unforcastable. Next, suppose that xt is an AR(1) with coefficient , where 0 < < 1. The spectrum for this random variable has a peak at = 0 and is monotonically decreasing with ||. Since the periodicity of a cycle with zero frequency is infinite, this stochastic process does not have an observable cycle. If the stochastic variable xt has a unit root, then the spectrum would be infinite at frequency zero.

Baxter and King (1994) show how to construct filters that isolate specific frequency bands, while removing stochastic and deterministic trends. Suppose one wants to isolate that part of a stochastic variable xt that is associated with frequencies between 1 and 2, with 0 1 2 . If 2 = , then the filter is called a high-pass filter since all frequencies higher than 1 are included. If 2 , then the filter is called a band-pass filter. The filters are two-sided symmetric linear filters and can be expressed as follows.

(3.3)

where is the filtered series, L is the lag operator, and

(3.4)

Let the Wold-representation for xt be given by

(3.5)

Then,

(3.6)

A useful result in spectral analysis is that the spectrum of is given by

(3.7)

where is the gain of the filter B(L). The spectrum of the filtered series has to be equal to Sx if || [1, 2] and equal to zero if is outside this set. Therefore, the gain of the filter has to be equal to one if || [1,2] and equal to zero otherwise. Using the converse of the Riesz-Fischer theorem, one can find the time-series representation, i.e. B(L), that corresponds to these conditions on the gain of the filter. The formulas are as follows

(3.8)

The ideal filter is an infinite moving average and cannot be applied in practice. In practice one has to truncate B(L). This gives an approximate filter A(L), where

(3.9)