Permanent and temporary components of stock prices: evidence from assessing macroeconomic shocks.

Taylor, Mark P.

1. Introduction

This paper examines the interaction between macroeconomic shocks and stock price movements, from both a theoretical and an empirical perspective.

The mean reversion and predictability of stock returns is probably the most well researched topic in the empirical literature on financial economics, dating back at least to Cowles and Jones (1937). Numerous empirical studies have been unable to reject the hypothesis that returns are unpredictable and that stock prices follow a random-walk, or martingale, process (e.g., Granger and Morgenstern 1963; Fama 1965, 1970; Le Roy 1982). In the past decade, however, various studies have challenged this conventional view and reexamined the predictability of stock returns: Contrary to the random-walk hypothesis, recent empirical evidence has lent strong support to the hypothesis of mean reversion in stock prices.

The influential work of Fama and French (1988) reports impressive findings that U.S. stock prices are mean reverting (i.e., contain a slowly decaying temporary component) and induce returns characterized by a large negative autocorrelation process for long return horizons of several years. Moreover, Fama and French show that between 25% and 45% of the variation of three- to five-year U.S. stock returns appears to be predictable from past returns. The Fama and French study has been corroborated by a number of other studies that report similar findings that stock returns contain large predictable components (Lo and Mackinaly 1988, 1989; Poterba and Summers 1988; Frennberg and Hansson 1993; Cochrane 1994; Lee 1995).

In this paper, the information contained in macroeconomic variables is used to investigate whether stock prices contain a temporary component and are, therefore, mean reverting. In order to illustrate and identify the relationship between macroeconomic and financial time series, we outline a simple log-linear macro model with overlapping nominal wage contracts and a real stock price determination equation. Apart from the inertia introduced by overlapping contracts, the model is essentially neoclassical and Fisherian in structure and allows reasonably complex dynamics. (1) We use this model to demonstrate a number of issues.

First, we show that changes in log real stock prices may be serially correlated even under the assumption of fully efficient markets in the sense that there are no profitable arbitrage opportunities between current and expected stock price movements. Second, we show how the temporary and permanent components of stock price movements may be related to aggregate macroeconomic supply and demand disturbances. In particular, in the context of the same macro model, we show that aggregate demand shocks have only temporary effects on real stock prices, while supply shocks may affect the level of real stock prices permanently. These results are not specific to the particular model analyzed, however, and apply to any standard macro model with a long-run vertical supply curve and short-run nominal inertia.

We then go on to investigate the size and significance of this mean-reverting component in U.S. stock prices, for the January 1949 through December 1997 (1949:1-1997:12) period, by placing appropriate structural restrictions on a vector autoregressive system in real stock prices and consumer prices, corresponding to a long-run vertical aggregate supply curve framework in which, in line with our illustrative macro model, only aggregate supply shocks have a long-run effect on real stock prices. In contrast, in Lee (1995), the interpretation of the shocks is not based on an underlying macroeconomic model. A further contribution to the literature of this paper is to examine the sensitivity of the mean-reverting (temporary) component to the choice of variable to extract the temporary component from the vector autoregressive analysis. This complements Lee (1995), who uses dividends to extract the temporary component. We elicit the temporary component of U.S. stock prices using consumer prices, interest rate, output , the wage rate, and consumption. Further, we investigate the robustness of the results to the periodicity of the data.

The remainder of the paper is set out as follows. The next section provides a brief overview of the literature of mean reversion in stock prices. Section 3 outlines a simple macro model with overlapping nominal wage contracts and shows how aggregate macroeconomic supply and demand disturbances may be related to the temporary and permanent components of stock price movements. Section 4 outlines the econometric method used to decompose real stock prices into temporary and permanent components. The data, preliminary tests, and the empirical findings for U.S. stock prices are reported in section 5. Section 6 concludes the study.

2. Mean Reversion in Stock Prices

The essence of the mean-reversion hypothesis is that stock prices contain a temporary component. Thus, the market value of stocks deviates from the fundamental value but will revert to its mean. There exists a number of competing theories that explain the deviation of the actual market prices of stocks from their fundamental values, including noise trading (De Long et al. 1990), fads (Shiller 1984), speculative bubbles (Blanchard and Watson 1982), and limits of arbitrage (Shleifer and Vishny 1997). Furthermore, since stock prices that deviate from fundamentals in a highly persistent way may look as if they are following a random walk, arbitrageurs would find it difficult to detect such a deviation (Summers 1986; Cuthbertson 1996). Studies of mean reversion and the associated predictable component of stock prices tend to rely on one of two related testing methodologies: the test of autoregression on multiperiod returns (the regression-based test; Fama and French 1988) and the variance-ratio test (Cochrane 1988 ; Cochrarie and Sbordone 1988; Poterba and Summers 1988; Lo and MacKinlay 1988). More recently, vector autoregressive analysis has also been used to identify the permanent and temporary components of stock prices (e.g., Cochrane 1994; Lee 1995; Cuthbertson, Hayes, and Nitzsche 1997).

The regression-based and variance-ratio tests of mean reversion have been subject to recent criticism. Kim, Nelson, and Startz (1991) suggest that mean reversion is a feature of the pre-World War II environment but not the postwar environment. Moreover, there is evidence of poor small-sample performance of the test statistics. The small-sample problem arises because, even though the sample period may be very large, the number of nonoverlapping return observations is necessarily small, and therefore there is not much independent information in the return series. Thus, the reliability of inferences drawn from individual point estimates of long-horizon autocorrelations has recently been questioned (Richardson and Stock 1989; Jegadeesh 1990; Kim, Nelson, and Startz 1991; Mankiw, Romer, and Shapiro 1991; Richardson 1993). The difficulty in drawing inferences from t-statistics based on overlapping data arises because the approximating asymptotic distributions perform poorly and long-horizon t-statistics tend to ove rstate the degree of mean reversion. Using an alternative asymptotic distribution theory for statistics involving multiyear returns, Richardson and Stock (1989) and Richardson (1993) show that empirical inference does not easily reject the hypothesis of no mean reversion--the number of significant negative autocorrelations at long return horizons is reduced substantially. Mankiw, Romer, and Shapiro (1991) find only moderate evidence against the random-walk hypothesis. In fact, Cecchetti, Lam, and Mark (1990) and Richardson (1993) show that the U-shaped pattern is consistent with stock prices following a random-walk process.

An alternative perspective on the mean-reversion literature is given by Cochrane (1994) and Lee (1995). They argue that univariate estimation of stock prices will not reject the random-walk hypothesis for short autoregressions (e.g., AR(1)) and that mean reversion is evident in univariate analysis only from long return horizons. However, evidence from mean reversion in stock prices comes when one isolates a transitory multivariate shock.

Cochrane (1994) estimates a vector autoregression (VAR) of annual changes in the natural logarithm of stock prices and changes in the natural logarithm of dividends for the 1927-1988 period. Furthermore, since stock prices and dividends are cointegrated, the (one-period lag of the) natural logarithm of the dividend/price ratio is included in the VAR. Two shocks on stock prices (and dividends) are isolated--a dividend ("permanent") shock causes stock prices to move immediately to their long-run values, and a price ("temporary") shock has only a transitory effect on stock prices. Furthermore, the temporary shock is persistent with a half life of about five years. The size of the transitory component is large and consistent with the long return horizon analysis--some 57% of the variance of returns is explained by temporary shocks.

Employing a restricted two-variable autoregression involving stock price-dividend spreads and real stock prices, Lee (1995) reports similar results for quarterly data. The distinguishing feature of Lee (1995) is that permanent and temporary shocks to stock prices are identified using the present value hypothesis (i.e., a stationary dividend/price ratio) and assuming dividends to be a nonstationary, I(1) process. Lee (1995) estimates a restricted bivariate VAR of the price-dividend spread and stock returns and identifies the temporary and permanent shocks to stock prices by restricting the long-run response of the temporary shock to stock prices to equal zero. The permanent and temporary shocks are attributed to the dividend series--the random-walk component generates the permanent innovations (shocks), and the stationary component generates the temporary innovations. The two dividend innovations are related to stock prices through the present value model. (2) These multivariate studies strongly support the me an-reversion hypothesis and suggest a large mean-reverting component, around 50% to 60%, in U.S. (and international) stock prices, at least for studies that include the prewar period.

The mean-reversion hypothesis implies that lagged information helps predict stock returns. Many recent studies find that stock returns can be predicted by lagged information, with the predictable component in stock returns related to the business cycle (Fama and French 1989; Balvers, Cosimano, and McDonald 1990; Breen, Glosten, and Jagannathan 1990; Cochrane 1991; McQueen and Roley 1993). Moreover, Pesaran and Timmermann (1995) show that stock returns are predictable to a magnitude that is economically exploitable and that the degree of predictability is related not only to the business cycle but also to the magnitude of the macroeconomic shocks. Thus, Pesaran and Timmermann (1995) reinforce other multivariate studies that stock returns are predictable using a relatively small number of independent variables.

In the present paper, we seek to contribute to this literature in two ways. First, in the next section, we set out a simple macroeconomic model to illustrate the contributions made to the permanent and temporary components of stock price movements due to macroeconomic factors, in particular, aggregate supply and demand shocks. Intuitively, it is clear that the broad qualitative implications of our analysis--namely, that demand shocks will have a temporary effect on real stock prices while supply shocks will affect them permanently--holds not only for our illustrative model but also for any standard macroeconomic model with a long-run vertical supply curve and short-run nominal inertia. In section 4, we apply this broad macroeconomic framework empirically to decompose real stock price movements into their permanent and temporary components arising from supply-side and demand-side shocks, respectively, and measure and test the statistical significance of each component in explaining U.S. stock price movements o ver the postwar period.

3. A Simple Macro Model

In order to illustrate the relationship between macroeconomic and financial time series, we outline a simple macro model that includes a stock price determination equation. We demonstrate that changes in real stock prices may be serially correlated even under the assumption of fully efficient markets in the sense that there are no profitable arbitrage opportunities between current and expected stock price movements. Also, in the context of the same macro model, we show that aggregate demand shocks have only temporary effects on real stock prices, while supply shocks may affect the level of real stock prices permanently.

In the traditional aggregate demand--aggregate supply (ADAS) model with a long-run vertical supply curve, aggregate demand innovations result in only a temporary rise in output, while aggregate supply innovations permanently affect the level of aggregate output. (3) That is, in the long run, aggregate demand innovations raise the price level but not output. The model developed in this section is in this tradition.

Consider a simple log-linear macro model with overlapping nominal wage contracts that is essentially neoclassical and Fisherian in structure (in the sense that agents are always able to distinguish between real and nominal shocks) and that allows reasonably complex dynamics. The model incorporates the salient features of the models of Fischer (1977), Blanchard (1981), and Blanchard and Quah (1989):

[y.sub.t] = [m.sub.t] - [p.sub.t] + a[[theta].sub.t] + [alpha][[pi].sub.t]. (1)

[y.sub.t] = [n.sub.t] + [[theta].sub.t]. (2)

[p.sub.t] = [w.sub.t] + [[pi].sub.t] - [[theta].sub.t]. (3)

[w.sub.t] = w \ {[E.sub.t-2][n.sub.t] = n}. (4)

[[pi].sub.t] = [phi][y.sub.t]. (5)

[q.sub.t] = [[pi].sub.t] + [summation over ([infinity]/j=0)] [[rho].sup.j][E.sub.t][DELTA][[pi].sub.t+1+j] + [k.sup.*]. (6)

where the permissible range of the parameter space is governed by

a > 0, 0 < [alpha] < 1, 0 < [phi] < 1, 0 < [rho] [less than or equal to] 1. (7)

The variables, y, m, p, w, n, and [theta] denote, respectively, the log of output, the money supply, the price level, the nominal wage, employment, and technology. The log of real dividends on equities is represented by [pi]; n represents full employment, q is the log of the real price of equities, and [delta] denotes the first-difference operator.

Equation 1 represents the aggregate demand side of the economy, with aggregate demand a function of real money balances, productivity or technology, [theta], and distributed profits. For generality, we follow Blanchard and Quah (1989) in allowing productivity to affect aggregate demand on the grounds that it is likely to affect investment, so that we expect a > 0, although unless a is very large, the results are qualitatively unaffected. The production function, Equation 2, relates output to the level of employment and technology. (4) Equation 3 states that prices are set using a markup pricing rule. The nominal wage (Eqn. 4), chosen two periods ahead, is set at the expected full employment level in a two-period overlapping contracts framework (Fischer 1977). (5) Equation 5 expresses log of real dividends (distributed profit) as a function of real output. (6)

Equation 6 specifies the log of real stock prices as a linear function of current expected future log real dividends. Following Campbell and Shiller (1988a, b), this follows from a log-linear approximation of the standard present value model of stock prices. (7) The equation says that the log real stock price at time t is determined by the log real dividend at time t, expected real dividend growth into the infinite future, and a constant. Future real dividend growth rates are discounted at the rate [[rho].sup.j], for j = 0, ..., 4, where [rho] is close to but smaller than (positive) unity.

To close the model, we follow Blanchard and Quah (1989) in assuming that m and [theta] are determined as follows:

[[theta].sub.t] = [[theta].sub.t-1] + [e.sub.s,t] (8)

[m.sub.t] = [m.sub.t-1] + [e.sub.d,t] (9)

where [e.sub.d] and [e.sub.s] are serially uncorrelated and pairwise orthogonal demand and supply disturbances.

We solve the previous model for the variables of interest ([DELTA][p.sub.t], [DELTA][q.sub.t]) in terms of the two disturbances ([e.sub.d,t] and [e.sub.s,t]). Real stock returns expressed in terms of supply and demand disturbances are given by

[DELTA][q.sub.t] = [phi](1 -[rho]) )([1 + [phi] - [alpha][phi]).sup.-1] X[([e.sub.d,t] - [e.sub.d,t-1]) + (1 + a) ([e.sub.s,t] - [e.sub.s,t-1])] + [phi][rho]([e.sub.s,t] - [e.sub.s,t-1]) + [phi][e.sub.s,t-1]. (10)

From Equation 10, we see that demand disturbances have short-run effects on real stock prices and that these effects disappear over time: In this overlapping wage contracts model, a demand disturbance has no long-run effects after one period. Supply disturbances have both short- and long-run positive effects on real stock prices. The long-run impact of a supply shock on real stock prices is to raise them by a factor [phi] of the original shock. Intuitively, this is because the supply shock has a permanent, one-for-one impact on real output, and, from the dividend-setting Equation 5, dividends rise permanently by a factor of this permanent effect. (8)

Similarly, solving for inflation,

[DELTA][p.sub.t] = [e.sub.d,t] - (1 - [alpha][phi])([l - [alpha][phi]+ [phi]).sup.-1] [([e.sub.d,t] - [e.sub.d,t-2]) + (1 + a)([e.sub.s,t] - [e.sub.s,t-2])] + a[e.sub.s,t]-(1 - [alpha][phi]) [e.sub.s,t-2]. (11)

From Equation 11, demand and supply disturbances have both short- and long-run effects on prices. We interpret Equation 11 using the parameter space (Eqn. 7). A supply disturbance decreases prices in the short run. However, in the long run, the net effect of a supply disturbance depends on the value of (a + [alpha][phi]). (a + [alpha][phi]) > 1, a supply disturbance will increase prices in the long run, whereas with (a + [alpha][phi]) < 1, prices will decrease in the long run. This is because supply shocks, through their effect on investment, may raise aggregate demand (Eqn. 1). if this effect is weak (a is small), the traditional supply-side effects will dominate (since [alpha][phi] < 1), and a supply shock will depress prices in the long run.

In summary, demand and supply shocks have both short- and long-run effects on inflation. However, demand shocks have short-run effects on real stock prices, and these effects disappear over time. In contrast, supply shocks have both short- and long-run effects on real stock prices. Also, Equation 10 demonstrates that changes in real stock prices may be serially correlated even under the assumption that there are no profitable arbitrage opportunities between current and expected stock price movements, that is, fully efficient markets.

This theoretical analysis suggests, moreover, a rationale for investigating empirically the mean-reverting component of real stock prices from a macroeconomic perspective. In particular, if we were able to identify the empirical analogues of Equations 10 and 11, then we could calculate the component of real stock price movements that is mean reverting and, in the context of a simple long-run vertical supply curve macroeconomic model of the kind we have outlined, is due to demand-side shocks (essentially by setting the supply-side shocks to zero in Eqn. 10) and the component that is permanent and due to supply-side shocks (by setting the demand-side shocks to zero in Eqn. 10). (9) In the next section, we discuss an empirical method for doing precisely this.

4. VAR Decomposition

Blanchard and Quah (1989) suggest an econometric technique to decompose a series into its temporary and permanent components. One advantage of the Blanchard-Quah decomposition is that it identifies permanent and temporary shocks in a multivariate time-series context. A number of recent studies have applied the Blanchard-Quah decomposition to macroeconomic and financial variables (Gali 1992; Gamber and Joutz 1993; Bayoumi and Eichengreen 1994; Bayoumi and Taylor 1995; Lee 1995; Gamber 1996; Falk and Lee 1998; Peel and Taylor 1998). For this reason, only a brief outline of the theoretical underpinnings of the decomposition is presented here. The fundamental feature of the Blanchard-Quah technique is that it imposes a long-run restriction (making use of economic theory) on the VAR to identify the decomposition. Consider a 2 X 1 vector of time series [x.sub.t] = [[DELTA][q.sub.t] [DELTA][p.sub.t]]', where L is the lag operator. Both [DELTA][q.sub.t] and [DELTA][p.sub.t], are assumed to be realizations at time t from a stationary stochastic process with its deterministic components removed. The variables [q.sub.t] and [p.sub.t] are thus assumed to be realizations of first-difference stationary or I(1) processes. By the multivariate form of Wold's decomposition (Hannan 1970), [x.sub.t] will have a moving average representation. (10) Consider a transformation of the Wold representation given by (11)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (12)

where [a.sub.mn](j) (m, n = 1, 2) represents the impulse response of the mth element of xt to the nth element of [[e.sub.T,t] [e.sub.P,t]]' after j periods and [e.sub.T,t] and [e.sub.P,t] represent white-noise innovations. By imposing restrictions on the coefficients of Equation 12 and on the covariance matrix of the innovations, we can identify [e.sub.T,t] and [e.sub.P,t] as temporary and permanent innovations to real stock prices, respectively. By assumption, p is affected by the same two innovations, although a permanent or temporary innovation to q need not necessarily affect p in the same way. In effect, we will recover the empirical counterparts to Equations 10 and 11 and, therefore, interpret the temporary component as the response of stock prices due to aggregate demand innovations and the permanent component as the response of stock prices due to aggregate supply innovations. In this context, the interrelationship between macroeconomic and financial variables allows us to estimate a temporary compone nt in real stock prices that is mean reverting.

We identify [e.sub.T,t] and [e.sub.P,t] and recover the VAR residuals in the following way. Write [e.sub.t.] = [e.sub.T,t] [e.sub.P,t]' and denote the bivariate vector of innovations recovered from the vector autoregressive representation for [x.sub.t] as [v.sub.t]. Since the VAR representation is simply an inversion of the Wold representation (Eqn. 12), [v.sub.t] will in general be a linear function of [e.sub.t], [v.sub.t] = A(0) [e.sub.t], say, where A(0) is a 2 X 2 matrix of constants. To recover the underlying demand and supply innovations from the VAR residuals then requires that the four elements in A be identified, which requires four identifying restrictions. Three restrictions can be obtained by normalizing the variances of [e.sub.T,t] and [e.sub.P,t] to unity and setting their covariances to zero. The fourth, crucial identifying restriction, which effectively identifies [e.sub.T,t] as the temporary stock price innovation, is the requirement that [e.sub.T,t] has no long-run effect on the level of real stock prices, although it may affect the long-run aggregate price level. This latter restriction on the Wold representation (Eqn. 12) may be written as

[summation over ([infinity]/j=0)][a.sub.11](j) = 0. (13)

These four restrictions are then sufficient to recover the underlying temporary and permanent innovations to real stock prices (for further details, see Blanchard and Quah 1989). That part of the Wold decomposition corresponding to, respectively, permanent and temporary past innovations in [DELTA][q.sub.t] can then be cumulated and taken as the permanent and temporary components of [q.sub.t]. Note that once these restrictions are imposed on Equation 12, they become the analogues of Equations 10 and 11. We can use these equations to identify the temporary and permanent components of real stock prices arising from macroeconomic factors, as discussed in the previous section. (12)

5. Estimating Macroeconomic Shocks to Stock Prices

Monthly data for the United States were obtained from the data bank of the Center for Research in Securities Prices (CRSP) of the University of Chicago. The sample period is 1949:1-1997:12. (13) The data series of interest are the real stock price index and the consumer price index. The real stock price index is constructed by deflating the stock price index by the consumer price index. The stock price index is the Standard and Poor's (S&P) 500 index obtained from the CRSP stock files indices and the consumer price index obtained from the stocks, bonds, bills, and inflation (SBBI) files (Ibbotson and Associates). The logarithms of the real stock price index and the consumer price index are denoted by [q.sub.t] and [p.sub.t], respectively.

Table 1 reports some summary statistics on the series of interest. The sample autocorrelations reveal some degree of persistence in both series, as they tend to die off slowly. The first-order autocorrelations values close to one suggest that the series are nonstationary. This impression is borne out by Table 2, which reports the unit root and cointegration tests for each series. The sequential procedure employed in testing for unit roots follows Dickey and Pantula (1987) in order to ensure that only one unit root is present in the series. The unit root tests are the Augmented Dickey-Fuller (ADF) and the Phillips-Perron (PP) [Z.sub.t]-tests for the null hypothesis that the series in question is I(1) (see Dickey and Fuller 1979, 1981; Perron 1988). A lag length of six was chosen. Using either test, we cannot reject the hypothesis that the series are first-difference stationary, that is, I(l). (14)

As a test for cointegration, the results of the ADF test for a unit root in the least-squares residual from a regression of [p.sub.t] onto [q.sub.t] and a constant are reported in Table 2 (final row). As in the case of unit root tests, a lag length of six was chosen. The null hypothesis of no cointegration cannot be rejected at the 5% level of significance.

Estimating the Vector Autoregressive Process

A VAR of [[DELTA][q.sub.t] [DELTA][p.sub.t]]' was estimated prior to effecting the decomposition. (15) The lag length for the VAR was chosen as follows. First, using the Bayes Information Criterion (BIC), the initial lag length was determined. (16) Second, using the Ljung-Box Q-statistic, we tested for the whiteness of the residuals, and the lag depth increased (if necessary) until they were approximately white noise. The chosen lag depth was nine.

Given the estimates of the VAR parameters and the covariance matrix of VAR residuals, we then carried out the decomposition as described in section 4. The generated cumulative impulse response functions illustrate the effect of a one-unit (standard deviation) shock to the level of real stock prices and the level of consumer prices and are presented in Figure 1.

There are a number of interesting features that are worth noting. Real stock prices increase as a result of both positive aggregate demand (temporary) and aggregate supply (permanent) shocks. By construction, an aggregate demand shock results in only a temporary rise in real stock prices.

However, as is evident in Figure 1, the temporary shock to real stock prices is itself quite persistent and relatively large with a half-life of about 18 months. An aggregate supply shock has a negative short- and long-run effect on consumer prices, implying that the feedback of productivity shocks to the demand side is small (in the context of our simple macro model outlined in section 3, the parameter a is small). Consistent with our simple macro model, an aggregate demand shock has a positive short- and long-run effect on consumer prices.

The decomposition generates three components of real stock prices: temporary (aggregate demand), permanent (aggregate supply), and deterministic (due to drift and seasonals). (17) The temporary (permanent) component is estimated by cumulating the temporary (permanent) innovations in real stock prices over time. The deterministic component is the difference between the real stock price series and the sum of the temporary and permanent components. The resulting series, depicted in Figure 2, illustrate the log real stock prices and permanent and temporary components of real stock prices. The size of the temporary component appears small in magnitude relative to the permanent component. As shown in the following, however, temporary innovations in real stock prices explain a statistically significant proportion of the total variation in real stock price movements.

Our decomposition suggests that temporary shocks to real stock prices during the 1960s and early 1970s caused stock prices to be lower than they otherwise would have been in the absence of such shocks (see Figure 2). Although these findings contrast to Lee (1995), they can be explained in terms of the interpretation of the shocks. Lee (1995) attributes the temporary innovations to dividends, whereas we attribute them to aggregate demand. For this reason, a more useful comparison is to consider the components of stock prices in the context of the business cycle (see Blanchard and Quah 1989).

The results support a story that is similar to Blanchard and Quah (1989). Table 3 presents the estimated values of the temporary (aggregate demand) and permanent (aggregate supply) innovations around the periods of the 1974-1975 recession and the October 1987 stock market crash. The period leading up to the 1974-1975 recession exhibits a string of negative permanent shocks to real stock prices. (18)

The permanent component depicted in Figure 2 is highly consistent with Lam (1990). He identifies three postwar periods as the main episodes of negative permanent shocks: the recession in the fourth quarter of 1957 through the first quarter of 1958, the first major oil shock in the third quarter of 1974 through the first quarter of 1975, and the severe recession in the fourth quarter of 1981 through the first quarter of 1982. (19) Figure 2 also identifies these three periods when the permanent component went down quite dramatically.

Interestingly, the late 1970s and 1 980s is a period in which stock prices are associated with a positive temporary component, and it is this component that suddenly turns negative that is associated with the 1987 stock market crash--the temporary component of real stock prices caused stock prices to fall by 15% in October 1987, whereas the permanent component caused stock prices to fall by 10% for the same month. More recently, the upward movements in stock prices is attributed to a positive temporary component.

Forecast Error Variance Decomposition

Table 4 reports the forecast error variance decompositions of real stock prices and consumer prices to the contributions of permanent innovations and temporary innovations (Hamilton 1994, chap. 11). At the 12-month horizon, over 42% of the forecast error variance in real stock prices is due to temporary (aggregate demand) innovations. The size of the temporary component in real stock prices is of similar magnitude to that of other studies, for example, Fama and French (1988), Cochrane (1994), and Lee (1995). Furthermore, from Monte Carlo simulations, the temporary component is statistically significant at the standard level of significance. At the 95 percentile, some 38% of the forecast error variance in real stock prices is due to temporary innovations.

Evidence of a temporary component in U.S. real stock prices is robust to the frequency of the data employed. Following a similar decomposition, as explained previously, quarterly consumer price and real stock price data reveal that, at the one-year horizon, more than 46% of the forecast error variance in real stock prices is due to temporary shocks. The forecast error variance decomposition for the quarterly analysis is presented in Table 5.

Sensitivity Analysis

The choice of inflation in the VAR to estimate the temporary component in real stock prices is motivated by the macroeconomic model, outlined in section 3. The decomposition methodology does not restrict the choice of variable used with the first difference of log real stock prices in the VAR. A temporary component in real stock prices can be estimated from a VAR of [[DELTA][q.sub.t]' [z.sub.t]]' with [z.sub.t] assumed to be realizations at time t from a stationary stochastic process with its deterministic component removed and that [[DELTA][q.sub.t] [z.sub.t]]' is not cointegrated. We consider four variables in defining [z.sub.t]--the interest rate (20) (Treasury-bill rate), log output (GDP at current prices), log wage rate (index of hourly manufacturing earnings), and log consumption. All data were obtained from the International Monetary Fund's International Financial Statistics (IFS) database for the period 1957-1997.

We estimated six bivariate VARs, and given their parameter estimates and covariance matrices of VAR residuals, we then carried out the VAR decomposition as outlined in section 4. (21) Table 6 reports the resulting temporary component in real stock prices from these six VARs. Remarkably, the results are very similar to the earlier findings that used first difference of consumer prices in the VAR. In summary, at the one-year horizon, all the temporary components fall within the range of 38% to 66%. Quarterly findings generate a slightly higher temporary component than those generated using monthly data. This is likely to be the result of less noisy quarterly data.

6. Conclusion

In this paper, we have sought to measure and test the significance of a mean-reverting component in U.S. real stock prices using a restricted VAR motivated by a simple macro model that includes a stock price determination equation. The macro model demonstrates that changes in real stock prices may be serially correlated even under the assumption of fully efficient markets in the sense that there are no profitable arbitrage opportunities between current and expected stock price movements. Furthermore, aggregate demand shocks have only temporary effects on real stock prices, while aggregate supply shocks may affect the level of real stock prices permanently. Although we demonstrated this in the context of a particular macro model, it is clear that the general results must hold true in any standard model with a long-run vertical supply curve and nominal inertia. Intuitively, this is because real stock prices are related to real economic fundamentals, and these can be affected in the long run only by supply-side shocks.

Monthly U.S. real stock prices reveal a significant temporary component due to aggregate demand shocks that is mean reverting and that accounts for some 42% of the total stock price variation. Since the temporary component is found to be statistically significant, we reject the pure random-walk hypothesis in favor of the mean-reversion hypothesis. The finding of this significant mean-reverting component from assessing macroeconomic shocks to real stock prices supports the hypothesis that, even if stock markets are efficient, real stock returns are serially correlated. This evidence supports Fama and French's (1988) argument that mean reversion in stock prices may result from the workings of efficient markets and may not be only the result of ill-informed speculation or other noise trader activity (see, e.g., De Long et al. 1990).

The size of the temporary component is robust to the periodicity chosen and the variables employed in the underlying VAR used to carry out the decomposition of real stock prices. Employing quarterly data generates a slightly higher temporary component in real stock prices than does monthly data.

The results are consistent with those of more recent studies that have used vector autoregressive techniques to decompose stock prices into their temporary and permanent components (Cochrane 1994; Lee 1995). Moreover, the results are not subject to the recent controversy associated with long return horizon analysis (Richardson and Stock 1989; Cecchetti, Lam, and Mark 1990; Kim, Nelson, and Startz 1991; Mankiw, Romer, and Shapiro 1991; Richardson 1993).

The association between a significant mean-reverting component and predictability of stock returns has several potential implications for the practical investor. The evidence of mean reversion implies that real stock returns are to some degree predictable. Also, in the presence of mean reversion, an investor with a relative risk aversion coefficient of less (greater) than unity will invest less (more) in equities as his investment horizon increases (Samuelson 1991). Further, the presence of a mean-reverting component suggests using a portfolio strategy of going long in equities that have recently declined in value.

[FIGURE 1 OMITTED]

[FIGURE 2 OMITTED]

Table 1

Summary Statistics

 [q.sub.t] [[DELTA]q.sub.t] [p.sub.t]

Mean -- 0.33 --
Standard deviation -- 4.08 --
Autocorrelation
 coefficient at lag k
k= 1 0.99 * 0.05 1.00 *
 2 0.97 * -0.03 0.99 *
 3 0.95 * 0.03 0.99 *
 4 0.94 * 0.04 0.98 *
 5 0.92 * 0.11 0.98 *
 6 0.91 * -0.05 0.98 *
 7 0.89 * -0.02 0.97 *
 8 0.87 * -0.03 0.97 *
 9 0.86 * 0.00 0.96 *
 10 0.85 * -0.00 0.96 *
 11 0.83 * 0.01 0.95 *
 12 0.82 * 0.03 0.95 *

 [[DELTA]p.sub.t]

Mean 0.33
Standard deviation 0.35
Autocorrelation
 coefficient at lag k
k= 1 0.62 *
 2 0.54 *
 3 0.48 *
 4 0.44 *
 5 0.45 *
 6 0.41 *
 7 0.42 *
 8 0.45 *
 9 0.45 *
 10 0.44 *
 11 0.41 *
 12 0.39 *

The sample period is 1949:1-1997:12. The term [p.sub.t] is the natural
logarithm of the consurmer price index, [q.sub.t] is the natural
logarithm of real stock prices, and [DELTA] denotes the first
difference. An asterisk denotes that the sample autocorrelation is at
least two standard deviations to the left or right of its expected value
under the hypothesis that the true autocorrelation is zero.
Table 2

Unit Root and Cointegration Tests: Consumer Prices and Real Stock Prices

 ADF PP

[q.sub.t] -2.15 -2.02
[DELTA][q.sub.t] -8.40 -22.60
[[DELTA].sup.2][q.sub.t] -14.05 -61.83
[p.sub.t] 0.65 2.50
[DELTA][p.sub.t] -4.12 -12.37
[[DELTA].sup.2][p.sub.t] -14.88 -48.89
[[mu].sub.t] -0.46 --

See Table 1. The term [[mu].sub.t] is the residual from the OLS
regression of [p.sub.t] onto [q.sub.t] and a constant. The unit root
tests are the Augmented Dickey--Fuller (ADF) test statistic and the
Phillips- Perron (PP) [Z.sub.t]-test statistic, without time trend and
with constant, for the null hypothesis that the series contains a unit
root (see Dickey and Fuller 1979, 1981; Perron 1988); the lag truncation
was set at six. For a 5% a significance level, the critical value for PP
and ADF is -2.88. The cointegration test is the ADF; for a 5%
significance level, the critical value is -2.87 (see MacKinnon 1991).
Table 3

Permanent and Temporary Innovations: Selected Dates

Year:Month Temporary (%) Permanent (%)

1973:8 0.09 -0.15
1973:9 -0.03 0.08
1973:11 -0.04 -0.10
1974:1 0.02 -0.04
1974:2 0.04 -0.05
1974:3 0.00 -0.04
1974:4 -0.06 0.01
1974:5 0.01 -0.05
1974:7 -0.05 -0.04
1974:8 0.00 -0.10
1974:9 -0.05 -0.08
1974:10 0.07 0.08
1974:11 -0.03 -0.04
1975:1 0.02 0.08
1975:2 0.04 0.02

1986:2 -0.01 0.09
1986:3 -0.02 0.07
1986:7 -0.07 0.00
1986:8 0.03 0.04
1986:9 -0.01 -0.07
1987:1 0.08 0.03
1987:10 -0.15 -0.10
1987:11 -0.05 -0.05
1987:12 0.01 0.04
1988:2 0.00 0.04

The selected dates refer to the 1974-1975 recession and the October 1987
stock market crash. The innovations are identified using the procedure
set out in section 4 and are innovations to real stock prices. By
construction, the standard deviations of these innovations are equal to
1%. To conserve space, only the months that exhibit an innovation
greater than (absolute) 0.35 are reported.
Table 4

Forecast Error Variance Decomposition

 Percentage of Variance Due to:
Horizon Temporary Shocks Permanent Shocks
(Months) Consumer Prices Stock Prices Consumer Prices Stock Prices

 1 44.63 42.15 55.37 57.85
 (12.2; 77.7) (37.7; 47.0) (22.3; 87.8) (53.0; 62.3)
 2 43.49 41.95 56.51 58.05
 (11.7; 77.2) (37.6; 47.2) (22.8; 88.3) (52.8; 62.4)
 3 43.07 42.00 56.93 58.00
 (11.4; 76.8) (37.6; 47.4) (23.2; 88.6) (52.6; 62.4)
 4 41.93 41.79 58.07 58.21
 (10.9; 76.1) (37.4; 47.7) (23.9; 89.1) (52.3; 62.6)
 5 41.53 42.07 58.47 57.93
 (10.5; 75.9) (37.7; 48.5) (24.1; 89.5) (51.5; 62.3)
 6 40.58 42.12 59.42 57.88
 (10.0; 75.5) (37.7; 48.7) (24.5; 90.0) (51.3; 62.3)
12 42.60 42.50 57.40 57.50
 (12.5; 76.4) (38.1; 49.7) (23.6; 87.5) (50.3; 61.9)
24 42.82 42.46 57.18 57.54
 (13.5; 76.6) (38.0; 49.6) (23.4; 86.5) (50.4; 62.0)
36 42.93 42.47 57.07 57.53
 (13.4; 76.7) (38.1; 49.6) (23.3; 86.6) (50.4; 61.9)

See Table 1. Figures are the percentage of total variation in real stock
prices and consumer prices (all in logs) explained by temporary and
permanent shocks. The sample period is 1949:1-1997:12. The figures in
parentheses are the 5th and 95th percentiles (95% confidence intervals)
computed using Monte Carlo integration and drawing from a normal
distribution for the estimated coefficients and a Wishart distribution
for the inverse of the estimated error covariance matrix, with 1000
replications.
Table 5

Quarterly Analysis: Forecast Error Variance Decomposition


Horizon Temporary Shocks Permanent Shocks
(Quarters) Consumer Prices Stock Prices Consumer Prices Stock Prices

 1 28.93 44.17 71.07 55.83
 (1.7; 74.6) (36.5; 52.8) (25.4; 98.3) (47.2; 63.5)
 2 25.20 43.35 74.80 56.65
 (2.6; 73.5) (35.9; 52.9) (26.5; 97.4) (47.1; 64.1)
 3 28.07 44.26 71.93 55.74
 (4.5; 74.4) (36.5; 55.1) (25.6; 95.5) (44.9; 63.5)
 4 29.70 46.20 70.30 53.80
 (5.4; 74.9) (38.2; 60.8) (25.1; 94.6) (39.2; 61.8)
 5 30.64 45.86 69.36 54.14
 (6.2; 74.7) (37.9; 60.6) (25.3; 93.8) (39.4; 62.1)
 6 31.39 45.85 68.61 54.15
 (7.1; 75.0) (37.8; 60.8) (25.0; 92.9) (39.2; 62.2)
12 31.72 46.17 68.28 53.83
 (7.4; 74.9) (38.0; 63.6) (25.1; 92.6) (36.4; 62.0)
24 31.95 46.14 68.05 1 53.86
 (7.2; 74.8) (37.7; 64.5) (25.2; 92.8) (35.5; 62.3)
36 31.97 46.14 68.03 53.86
 (7.2; 74.9) (37.6; 64.5) (25.1; 92.8) (35.5; 62.4)

See Table 4.In generating the forecast error variance decomposition, a
Var of length of 7 was chosen. The lag length for the VAR was chosen as
follows. First, using the Bayes Information Criterion, the initial lag
length was determined. Second, using the Ljung-Box Q-statistic, we
tested for the whiteness of the residuals and the lag depth increased
(if necessary) until they were approximately white noise. The period of
analysis was the first quarter of 1949 through the fourth quarter of
1997. Figures are the percentage of total variation in real stock prices
and consumer prices (all in logs) explained by temporary and permanent
shocks. The sample period is 1949:1-1997:12. The figures in parentheses
are the 5th and 95th percentiles (95% confidence intervals) computed
using Monte Carlo integration and drawing from a normal distribution for
the estimated coefficients and a Wishart distribution for the inverse of
the estimated error covariance matrix, with 1000 replications.
Table 6

Estimation of the Temporary Component in Real Stock Prices Using
Alternative Variables in the VAR

Percentage of Variance in Real Stock Prices Due to Temporary Shocks

 Variables with Real Stock Prices in the VAR
 Interest Rates Interest Rates Output Wage Rate
Horizon (Monthly) (Quarterly) (Quarterly) (Monthly)

 1 37.73 59.4 42.45 59.5
 (33.4; 42.7) (49.0; 69.7) (33.4; 52.2) (53.8; 65.4)
 2 37.72 57.8 42.47 59.1
 (33.4; 43.3) (47.8; 67.6) (33.7; 52.3) (53.5; 65.4)
 3 37.75 59.8 44.44 59.2
 (33.4; 43.2) (48.5; 68.7) (35.3; 55.2) (53.5; 65.5)
 4 37.76 60.0 45.95 59.1
 (33.6; 43.5) (49.3; 69.4) (36.6; 57.0) (53.5; 65.6)
 5 37.82 60.7 46.36 59.0
 (33.5; 43.8) (49.6; 69.9) (37.2; 57.8) (53.4; 65.6)
 6 37.16 60.7 46.59 59.3
 (32.9; 43.1) (49.9; 69.4) (37.2; 58.4) (53.7; 66.3)
12 37.72 60.6 46.53 59.4
 (33.5; 44.7) (49.8; 69.1) (37.1; 58.8) (53.3; 66.3)
24 37.73 60.6 46.52 59.3
 (33.6; 45.5) (49.7; 69.2) (37.0; 59.0) (53.1; 66.4)
36 37.75 60.6 46.53 59.2
 (33.6; 45.6) (49.7; 69.2) (36.9; 59.0) (53.0; 66.5)

 Variables with Real Stock
 Prices in the VAR
 Wage Rate Consumption
Horizon (Quarterly) (Quarterly)

 1 65.3 59.4
 (54.5; 75.2) (48.5; 69.7)
 2 64.9 58.72
 (53.9; 74.7) (48.0; 69.9)
 3 65.2 60.6
 (54.1; 75.3) (49.5; 71.9)
 4 65.6 59.5
 (54.2; 75.7) (48.5; 71.5)
 5 65.7 60.0
 (54.3; 75.9) (49.6; 71.9)
 6 65.7 60.0
 (54.1; 75.6) (49.6; 72.0)
12 64.5 60.2
 (52.2; 74.5) (48.8; 72.0)
24 64.3 60.1
 (51.6; 74.3) (48.3; 72.1)
36 64.2 60.0
 (51.2; 74.1) (47.5; 72.1)

In generating the forecast error variance decomposition, the lag length
for the VAR was chosen as follows. First, using the Bayes Information
Criterion, the initial lag length was determined. Second, using the
Ljung--Box Q-statistic, we tested for the whiteness of the residuals and
the lag depth increased (if necessary) until they were approximately
white noise. The following lag lengths were chosen: For interest rates,
a lag of 14 was chosen for the monthly analysis and a lag of 5 for the
quarterly analysis; for output (nominal GDP), a lag of 8 was chosen; for
wage rates, a lag of 12 was chosen for the monthly analysis and a lag of
8 for the quarterly analysis; and for consumption, a lag of 8 was
chosen. The periods of analysis were 1957:1-1997:12 for monthly and the
first quarter of 1957 through the fourth quarter of 1997 for quarterly
analysis. Figures are the percentage of total variation in real stock
prices (all in logs) explained by temporary and permanent shocks. The
sample period is 1949:1- 1997:12. The percentage of the forecast error
variance explained by the permanent shock is the difference between 100%
and the corresponding entry above. The figures in parentheses are the
5th and 95th percentiles (95% confidence intervals) computed using Monte
Carlo integration and drawing from a normal distribution for the
estimated coefficients and a Wishart distribution for the inverse of the
estimated error covariance matrix, with 100 replications.

Received October 1999; accepted November 2001.

(1.) Fisherian in the sense that agents are able always to distinguish between real and nominal shocks.

(2.) As identified by Cochrane (1994), the dividend-price ratio helps predict stock returns (see also Fama and French 1988; Campbell and Shiller 1988a, b; Hodrick 1992).

(3.) See, e.g., Gordon (1978, chap. 7) and Branson (1979, chap. 7).

(4.) Note that, for simplicity, there is no physical capital stock in the model--or else it is normalized to unity and held constant--and we assume constant returns to labor. These assumptions should not affect the broad implications of the model relevant to our purposes.

(5.) The choice of two-period overlapping contracts is arbitrary--the analytical results are qualitatively unaffected as long as there is some overlap in the nominal wage contracts, although the time taken to reach long-mn equilibrium obviously increases with the length of the contract.

(6.) This dividend-setting equation is in the spirit of Lintner's classic (1956) analysis of dividend-setting behavior. Lintner introduces additional dynamics through a partial adjustment of actual to target dividends, however. While this could be done in the present analysis, it would add nothing to the model of interest in the present context, and we therefore assume that the target level of dividends--a function of the level of real output--is achieved in each period.

(7.) We assume that the dividend enters the log dividend-price ratio, [[delta].sub.t] [equivalent to] [[pi].sub.t] - [q.sub.t], in the current period t; that is, the dividend in period t is also known in period t. Given a constant level of the population, equal to the labor force, we can assume that each member of the population holds one share and receives a single dividend per period.

(8.) Solving for changes in real output, we have

[DELTA][y.sub.t] = ([1 + [phi] - [alpha][phi]).sup.-1] [([e.sub.d,t] - [e.sub.d,t-2]) + (1+a)([e.sub.s,t] - [e.sub.s,t-2])] + [e.sub.s,t-2].

Since the entire first term on the right-hand side of this equation is zero after two periods following either a demand or a supply shock, the only remaining term is the lagged supply shock. Thus, demand shocks have no long-run impact on real output, while a supply shock has a one-for-one long-run impact.

(9.) Note that the model we have outlined is for illustrative purposes only. However, any standard macroeconomic model with a long-run supply curve, some degree of nominal inertia, and incorporating a standard forward-looking stock price equation where dividends are a function of real activity will generate qualitatively similar results.

(10.) If [[q.sub.t] [p.sub.t]]' is cointegrated of order 1, 1, then the vector [x.sub.t] is not well behaved in that the moving average representation of that vector is noninvertible (Engle and Granger 1987). Therefore, a necessary condition for the Blanchard-Quab (1989) decomposition is that [q, [p.sub.t]]' is not cointegrated. Later we test this proposition on our data and are unable to reject the hypothesis of noncointegration at standard significance levels. In the presence of cointegrating variables, an alternative decomposition technique is the Stock and Watson (1988) common trends representation. Cochrane (1994) examines the relationship between the Sims (1980), Beveridge-Nelson (1981), and Blanchard-Quah (1989) decompositions and cointegration, and Crowder (1995) examines the relationship between the Blanchard-Qush decomposition, the Stock and Watson (1988) common trends representation, and cointegration.

(11.) For expositional purposes, we assume that the deterministic component of the vector time series has been removed. In our empirical analysis, we always include drift and seasonal dummy terms.

(12.) Note, however, that Equations 10 and 11 are derived from a highly stylized model, so that we would not wish to test soy exact parameter restrictions implied by those equations on actual data. See footnote 9.

(13.) We choose to limit the period of analysis to postwar for the reasons identified by Schwert (1989). The war and prewar periods (including the Great Depression) were times of extreme volatility, and thus data tend to he noisy. Since the VAR decomposition that we employ requires stable parameter estimates, the results that employ the war and prewar periods are likely to be fragile.

(14.) Throughout this paper, unless otherwise specified, we use a nominal significance level of 5% in hypothesis testing.

(15.) Seasonal dummies were included in the VAR.

(16.) The BIC chose a lag length of two, and the Akaike Information Criterion (AIC) chose nine.

(17.) For ease of exposition in sections 3 and 4, we ignored any deterministic components of the series. They are easily handled in the empirical analysis by including seasonal dummies and intercepts in the estimated VARs.

(18.) The permanent component in stock prices could be thought of as the fundamental price since by definition the temporary component is taken out of the actual price. In this case, the fundamental stock price is acting as a predictor of the 1974-1975 recession.

(19.) Lam (1990) decomposed GNP growth into permanent and transitory components using a univariate breaking trend model. The permanent component in his model is the cumulated effect of trend breaks.

(20.) The monthly rate of return on Treasury bills is calculated geometrically by [r.sub.t] = [([r.sup.*.sub.t] + 1).sup.(1/12)]-1, where [r.sup.*.sub.t] is the annualized rate of return on Treasury bills in period t.

(21.) We tested each variable for Unit roots and cointegration. All variables were found to be first-difference stationary and were not cointegrated with real stock prices. These are not reported but are available by request from the authors.

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Liam A. Gallagher * and Mark P. Taylor +

* Department of Economics, University College Cork, Cork, Ireland and Business School, Dublin City University, Dublin, Ireland; E-mail l.gallagher@ucc.ie; corresponding author.

+ Department of Economics, University of Warwick, Coventry CV4 7AL, United Kingdom and CEPR; E-mail mark. taylor@warwick.ac.uk.

We wish to thank two anonymous referees and Kent Kimbrough (coeditor) for their helpful comments and suggestions.