Do forecasts of stock returns also forecast covariances? ABSTRACT

Transcription

1 Do forecasts of stock returns also forecast covariances? Gregory R. Duffee Haas School of Business University of California Berkeley This Draft: February 12, 2002 ABSTRACT I construct forecasts of aggregate stock returns with standard variables, then use the forecasts to predict covariances between stock returns and real variables such as consumption and GDP growth. The results are strong and surprising. Return forecasts contain substantial information about covariances, but the sign of the relation is opposite that predicted in standard asset-pricing models. Higher expected returns correspond to substantially smaller covariances. Part of this pattern is driven by the fact that forecasts of stock returns also forecast business cycle dynamics. When expected returns are low, inflation and output growth are negatively correlated. When expected returns are high, this negative relation disappears. Voice , fax , duffee@haas.berkeley.edu. Address correspondence to 545 Student Services Building #1900, Berkeley, CA This is a much-revised version of Why does the slope of the term structure forecast excess returns? I thank seminar participants at many universities and the NBER 2001 Summer Institute, and in particular John Campbell, Doug Elmendorf, Blake LeBaron, and Sydney Ludvigson for helpful discussions and/or comments. The most recent version of this paper is at duffee.

2 1 Introduction Decades of research has tentatively concluded that there are predictable variations in aggregate stock returns. The uncertainty in this conclusion is partially driven by our inability to interpret comfortably these variations in an agreed-upon model of asset pricing. No-arbitrage implies that expected excess returns are determined by the covariance between returns and the state-price deflator. Thus we would like evidence that this covariance changes over time. However, there is little direct evidence of variation in the covariance of aggregate stock returns with any plausible proxy for the state-price deflator, except for stock returns themselves (i.e., stock returns are heteroskedastic). In particular, there is no evidence of time-varying covariances between stock returns and real variables such as aggregate consumption. One interpretation of this fact is that we are unable to observe the true time-variation in covariances because we are working with poorly and infrequently measured real variables. Researchers commonly use ARCH-type models to look for time-varying second moments, where the variables used to forecast second moments are lagged products of shocks. Even with perfectly measured data, it is hard to notice year-to-year swings in the volatility of aggregate consumption growth using four observations of quarterly shocks. Measurement error compounds the difficulty. Some higher-frequency data are available, such as monthly aggregate consumption or industrial production, but these are subject to greater measurement error than quarterly data. In this paper I rely on finance theory and use time-variation in expected stock returns to look for time-variation in covariances. In other words, can covariances between stock returns and real variables be predicted with expected stock returns? The basic approach is straightforward. I use variables identified in earlier research to form a time-t forecast of future excess aggregate stock returns. I then examine whether this return forecast also predicts covariances between these stock returns and, say, consumption growth. I document an economically strong and statistically robust link between expected excess stock returns and covariances involving aggregate consumption and GDP. This result is based on quarterly U.S. data from 1953 through Unfortunately, at least for those desiring a simple explanation for time-varying expected stock returns, the pattern goes the wrong way. When expected excess returns are lower than average, subsequent stock returns and macroeconomic growth are closely tied. For example, the correlation between innovations in annual stock returns and GDP growth is about 0.6. By contrast, when expected excess returns are higher than average, the relation between stock returns and macroeconomic growth is much weaker the correlation between innovations in annual stock returns and GDP growth is less than 0.3. In addition, both stock returns and macroeconomic growth 1

3 are less volatile when expected excess returns are high. Similar patterns are documented for different return horizons, across subsamples, and for alternative methods of forming forecasts of stock returns and covariances. This puzzling evidence raises a number of questions. First, what kinds of news moves stock prices in low-expected-return periods versus high-expected-return periods? Second, since expected excess returns are probably just proxying for fundamental forces driving timevarying covariances, what are these fundamental forces? Third, why do investors require higher expected excess returns in periods when stock returns are only weakly related to aggregate consumption growth? I shed some light on these questions, but I cannot offer a clearcut answer to any of them. I find that the kinds of shocks that hit the economy when expected returns are low are qualitatively different from those that hit when expected returns are high. When expected returns are low, inflation and output growth are inversely related. When expected returns are high, inflation and output growth have a greater tendency to move together. A superficial interpretation of this pattern is that aggregate demand shocks are relatively more important when expected returns are high, but there are doubtless other interpretations that are consistent with the evidence presented here. This inflation-output relation helps explain the link between expected returns and covariances between stock returns and macroeconomic growth. In low-expected-return periods, contemporaneous shocks to real dividends and inflation are negatively correlated. Because investors react positively to both news of higher dividends and news of lower inflation, this negative correlation magnifies the univariate relation between stock returns and real activity. In high-expected-return periods, shocks to dividends and inflation are positively correlated, dampening the univariate relation between stock returns and real activity. I have less success in identifying fundamental variables that drive conditional covariances. Since expected returns are countercylical, a natural hypothesis is that the state of the business cycle determines both expected returns and conditional covariances. I find mixed evidence for this conclusion. Covariances appear to be driven by turning points in business cycles. In other words, recent output growth and forecasts of near-term output growth help predict covariances between stock returns and the macroeconomy; higher growth (or higher forecasts) corresponds to lower covariances. But expected excess returns contain substantial incremental predictive power for covariances. I am also unsuccessful at explaining why investors appear to require high returns when the conditional covariance between stock returns and aggregate consumption growth is low. One potential explanation is that stockholders consumption may be decoupled from aggregate consumption in recessions. In recessions, a larger share of aggregate consumption may 2

4 be driven by liquidity-constrained agents. The consumption of stockholders, who are not constrained, may still be closely linked to stock returns even though aggregate consumption is not. In the absence of consumption data for stockholders, I can test this story only indirectly by extending the instrumental variables approach in Campbell and Mankiw (1990). The results are not supportive of the potential explanation. If the results are taken literally, they suggest a larger proportion of liquidity-constrained agents in low-expected-return periods. The next section presents the theoretical motivation behind the empirical analysis in the paper. Section 3 presents the main empirical evidence. It describes the construction of expected stock returns, measures of the covariance between stock returns and real variables, and then reports regression evidence. Section 4 is devoted to convincing the reader that these results are robust. Section 5 considers the questions that are raised by the first set of results. Concluding comments are offered in Section 6. The Appendix contains a description of the data. 2 The theoretical motivation 2.1 Expected returns and covariances The stochastic discount factor is at the center of standard asset-pricing theory. Denote this strictly positive random variable by exp(m(t)). In a discrete-time framework, the excess return to any asset i during period t + 1 satisfies [ ] E t exp(m(t +1))=E t exp(r i (t + 1))) exp(m(t +1)). (1) R i (t + 1) denotes the log excess return (log real return less log riskfree rate) to the asset. Equation (1) is a requirement of no arbitrage. In a utility-based framework, we can think of exp(m(t + 1)) as the ratio of the marginal utility of real wealth at time t + 1 to the marginal utility of real wealth at time t. If we assume conditional joint normality of R t (t) andm(t), (1) implies E t R i (t +1)+ 1 2 Var t(r i (t +1))= Cov t (R i (t +1),M(t + 1)) (2) Equation (2) says that excess returns are linear in covariances with the stochastic discount factor. The variance term adjusts for Jensen s inequality created by using log returns instead of returns. Asset-pricing models restrict M(t+1). The intuition used in this paper is that of recursive 3

5 utility, developed by Epstein and Zin (1989) and Weil (1989). In recursive utility M(t +1) can be expressed as M(t +1)=θ[log δ (1/Λ) C(t +1)] (1 θ)r w (t +1). (3) In (3), C(t + 1) is the change in log real consumption from t to t +1,R w (t +1)isthelog return to total real wealth, including human capital, and the parameters are determined by investors rate of time preference, elasticity of intertemporal substitution, and relative risk aversion. Substituting (3) into (2) yields E t R i (t +1)+ 1 2 Var t(r i (t +1))= (4) ( ) ( ) θ Λ Cov t R i (t +1), C(t +1) +(1 θ)cov t R i (t +1),R w (t +1) In principle we can test (4) by looking for variables that capture time-variation in the conditional covariances on the right-hand-side, and then checking whether this time-variation is accompanied by time-variation in expected excess returns. The history of such tests is not encouraging. The standard approach is to use information in the residuals of asset returns and consumption growth to detect changes in variances and covariances. These methods detect strong evidence of time-variation in the volatility of returns to financial assets (which is a component of R w (t + 1)), but this variation appears unrelated to variation in expected excess returns. 1 There is no evidence of substantial timevariation in either the volatility of aggregate consumption growth or in covariances of asset returns with aggregate consumption growth. 2 It is not clear what to make of this evidence. An important limitation on our ability to test (4) is poor data. Aggregate consumption is measured infrequently and with noise. Thus ARCH or GARCH methods have low power for detecting changes in volatility over time. Similarly, a two-dimensional ARCH has low power for detecting changes in the covariance between an asset s return and consumption. In addition, total wealth is poorly measured. Financial wealth is measured reasonably well, but human wealth is unobserved. In this paper I reverse the logic of previous tests of (4). If there is some variation in the investors conditional covariances over time, (4) tells us that expected excess returns will also vary. Thus we can use expected excess returns to predict covariances. In this recursive utility framework the only reason for variations in expected excess returns is variations in 1 See Campbell (1999) and Whitelaw (2000) for discussions and references. 2 See, e.g., Boudoukh (1993). 4

6 conditional covariances with consumption or wealth. In alternative models for the stochastic discount factor, such as the habit formation framework used by Campbell and Cochrane (1999), expected excess returns can vary independently of these covariances. Nonetheless, it is difficult to write down a consumption-based asset-pricing model where conditional covariances with consumption or wealth do not affect expected excess returns. The power of this approach depends on the magnitude of the variation in expected excess returns. Thus to implement it, we want to focus on assets for which there is evidence of substantial time-variation in expected excess returns. The most obvious choice is the aggregate stock market, owing to the large literature documenting time-variation in expected excess returns. Denote the excess log return to the stock market during time t as R(t). (This return has no subscript, for consistency with the empirical work that follows.) Then (4) suggests the following hypothetical regressions. ( R(t +1) Et R(t +1) )( C(t +1) E t C(t +1) ) = b 0,c + b 1,c E t R(t +1)+e c (t +1) (5) ( R(t +1) Et R(t +1) )( R w (t +1) E t R w (t +1) ) = b 0,w + b 1,w E t R(t +1)+e w (t +1) (6) The time-t expectations of the left-hand-sides of (5) and (6) are Cov t (R, C)andCov t (R, R w ) respectively. The coefficients b 1,c and b 1,w will be nonzero if expected excess returns are correlated with these conditional covariances. Risk aversion in a recursive utility model implies that either b 1,c or b 1,w will be positive if expected excess returns vary over time. 3 These regressions might appear similar to tests for constant correlations conditional on the sizes of the shocks. Boyer, Gibson, and Loretan (1999) note that when one-period-ahead correlations are constant, correlations conditioned on the size of either shock will be nonconstant. But here, the right-hand-side variable is not a shock; it is known at time t. Thus (5) and (6) are not subject to the effect described in Boyer et al. 2.2 The testing framework Two barriers prevent us as econometricians from estimating (5) and (6) directly. First, we do not perfectly observe C or R w. Second, we do not know investors time-t expectations. I overcome these barriers in the usual fashion. I use the log of aggregate per capita real consumption on nondurables and services to proxy for C. We do not observe the return to total wealth but we do observe dividends to wealth. Campbell (1996) and others 3 They need not both be positive, or even nonnegative. For example, if σ t (R, C) andσ t (R, W )are negatively correlated, an increase in σ t (R, C) could correspond to both a large increase in expected excess returns and a decrease in σ t (R, W ). In this (admittedly unlikely) situation, b 1 < 0 is possible. 5

7 interpret labor income as the dividend to the human wealth component of total wealth. More broadly, we could think of GDP as the dividend to total wealth. Shocks to dividends should correspond to shocks to returns, both because current dividends are a component of the current return and because a shock to current dividends conveys information about future dividends. Thus I use innovations in log changes in (real, per capita) aggregate labor income and GDP as proxies for the innovation in the return to total wealth. Estimation follows a two-step procedure. First, investor expectations are proxied by forecasts from OLS regressions on a set of variables known to investors at time t. Denote these predetermined variables by Z t. The regressions are R t+1 = a 0,r + a 1,rZ t + R t+1, (7) y t+1 = a 0,y + a 1,yZ t +ỹ t+1. (8) Regression (7) is used to produce the forecasted return R t+1. R t+1 a 0,r + a 1,rZ t (9) Second, regress the product of residuals from this first step onto the OLS forecast of excess returns. R t+1 ỹ t+1 = b 0 + b 1 Rt+1 + e t+1 (10) The econometric properties of this two-step process are nonstandard. issues, assume the true model is To clarify the R t+1 = E t (R t+1 )+q t+1, (11) y t+1 = E t (y t+1 )+v t+1, (12) E t (R t+1 ) = α 0,r + α 1,rZ t, (13) E t (y t+1 ) = α 0,y + α 1,yZ t, (14) Cov t (q t+1,v t+1 ) = β 0 + β 1 E t (R t+1 ), (15) q t+1 v t+1 = Cov t (q t+1,v t+1 )+ɛ t+1. (16) We can plug the true model into (10) to reveal (after some substitutions) R t+1 ỹ t+1 = β 0 + β 1 Rt+1 + [ β 1 ( R t+1 q t+1 )+q t+1 (ỹ t+1 v t+1 )+ v t+1 ( R t+1 q t+1 )+(ỹ t+1 v t+1 )( R t+1 q t+1 )+ɛ t+1 ]. The term in square brackets is the true residual for the OLS regression of R t+1 ỹ t+1 on R t+1. 6

8 The first term in this residual arises because R t+1 is a generated regressor, as discussed in Pagan (1984). The hypothesis we are interested in testing is β 1 = 0. Under this null, the presence of a generated regressor has no effect on either the parameter estimates or their standard errors. The next three terms in square brackets cause inference problems even under the null. They all depend, in part, on fitted values from the first step. OLS standard errors from the second step will not capture the uncertainty in these fitted values, thus under the null we expect the second-step standard errors to be biased downward. Under alternative hypotheses these terms are also likely to be correlated with the explanatory variable R t+1. Note that these terms are all related to the joint behavior of innovations in R and y. Under an alternative hypothesis that the covariance between R and y varies with E t (R t+1 ), these terms are likely to be correlated with R t+1. To get a sense of the importance of these features of the econometric approach, I use Monte Carlo simulations. 2.3 Monte Carlo evidence The variables that exhibit the greatest forecasting power for stock returns are typically correlated with lagged stock returns. Forecasts produced with these variables (such as D/P ratios) are subject to the predictive regressions bias of Stambaugh (1999). Because the implications of this bias on the second-stage regressions used here are unclear, the simulated model should include lagged dependent variables among the forecasting instruments. I therefore assume the vector Z t is given by Z t =(R t y t h t ) where h t is an independent variable distributed as N(0, 1). The conditional means of R t+1 and y t+1 are γ r h t and γ y h t, respectively. Thus α 0,r = α 0,y =0,α 1,r =(00γ r ),andα 1,y =(00γ y ). The zero coefficients on R t and y t imply that the explanatory power of these lagged values is an artifact of finite sample sizes. Only h t truly forecasts R t+1 and y t+1. In order to model the relation between Cov t (q t+1,v t+1 )ande t (R t+1 ), we need to specify how the variance of R, the variance of y, and their correlation depend on E t (R t+1 ). In the empirical results we will see later in the paper, much of the action is in time-varying correlations. This fact motivates the following specification choice. The vector of shocks (q t v t ) is distributed as a bivariate normal with unit variances. The correlation coefficient is 1 if β 0 + β 1 E t (R t+1 ) < 1; ρ = 1 if β 0 + β 1 E t (R t+1 ) > 1; β 0 + β 1 E t (R t+1 ) otherwise. Thus the correlation is linear in E t (R t+1 ) unless this linear relation produces a value outside 7

9 of [ 1, 1]. Under the null hypothesis, 1 β 0 1andβ 1 = 0. The alternative values of β 0 and β 1 that we will consider imply that the boundary is exceeded only in extremely rare events. Thus under the alternative hypotheses, the covariance is almost exactly linear in E t (R t+1 ). An econometrician observes a sample of length N and follows the two-step procedure described above to estimate the model. Two methods of implementing step one are considered. The first method uses a single regression. This produces in-sample forecasts of R t+1 and corresponding in-sample residuals. Because lagged values of R and y are used in forecasting, these regressions are subject to the predictive regressions bias. It seems plausible that the second-step estimates may be improved by using forecast errors that are not subject to this bias. This motivates the second method. It uses step-ahead forecasts from expanding regressions. The forecasts of R τ+1 and y τ+1 are constructed from OLS regressions estimated with observations one through τ. The out-of-sample forecast errors are the differences between the realized values at τ + 1 and the forecasts. The expanding regressions use a minimum of 30 observations, so the first forecast is made for observation 31. Table 1 reports the mean results across 1000 simulations. The baseline uses N = 160 (the number of quarters in 40 years), δ r =0.2 andδ y =0.4 (which produce R 2 s roughly in line with the observed predictability of stock returns and consumption growth), β 0 =0.3 (roughly the unconditional correlation between residuals of quarterly stock returns and consumption growth), and β 1 = 0 (the null hypothesis). The simulations indicate that this estimation procedure is well-behaved when in-sample residuals are used. The estimate of β 1 is unbiased and five percent of the absolute t-statistics exceed When out-of-sample residuals are used, the estimate of β 1 has a minimal bias, but the standard errors are too small. The five percent critical value for the absolute t-statistics is The first alternative hypothesis we consider is β 1 = 1. The magnitude of this coefficient is motivated by the empirical evidence presented in the next section, although the sign is reversed. Under this alternative, the relation between Cov(q t+1,v t+1 )ande t (R t+1 ) is linear for h t [ 6.5, 3.5]. Since h t is distributed as a standard normal, the likelihood of values outside of this range is vanishingly small. The simulation results indicate substantial bias towards zero in the estimate of β 1. The mean estimate using in-sample residuals is 0.76 and the mean estimate using out-of-sample residuals is only The power of the test is much larger when in-sample residuals are used than when out-of-sample residuals are used. When the five percent critical value is used to reject the null hypothesis (1.96 for the in-sample residuals and 2.15 for the out-of-sample residuals), the hypothesis is rejected in half of the simulations using in-sample residuals but in only a quarter of the simulations using out-ofsample residuals. When the alternative hypothesis is changed to β 1 = 1, the results are 8

10 similar (aside from the sign change in the estimate of β 1 ). Table 1 also reports simulation results under the null hypothesis for various changes to the baseline assumptions. First, the (constant) correlation between q t+1 and v t+1 is increased from 0.3 to 0.5. The main effect of this change is to increase the critical value for the t-test. Second, mean of R t is made time-invariant. This is designed to capture the idea that any forecastability of stock returns is spurious. This change increases the critical value for the t-test with out-of-sample residuals, but has little effect on the in-sample results. Third, the number of observations is cut in half. The effect of this change is similar to the effect of the second change. There are three main points to take from this Monte Carlo evidence. First, residuals should be constructed in-sample rather than with expanding regressions. Second, given reasonable parameter values under the null, this two-stage regression procedure using insample residuals produces unbiased point estimates and reasonably accurate rejection rates at a five percent critical value. Third, under plausible alternative hypotheses, inference is somewhat problematic. Point estimates are biased toward zero and the likelihood of failing to reject the null hypothesis is fairly high. 3 The evidence for predictable covariances 3.1 Some timing issues Asset prices are measured at points in time while real variables are time-averaged. Thus we need a timing convention to align asset returns with growth in real variables. I follow Campbell (1999) by defining an asset return from quarter t to quarter t + k as the return from the beginning of quarter t (the end of t 1) to the beginning of quarter t + k (the end of quarter t + k 1). The growth in a real variable over the same period is defined as the log change from quarter t to quarter t + k. Covariances between observed asset returns and real variables are determined largely by economic fundamentals. But they are also affected by time-averaging, measurement error, and the frequency at which individuals and firms adjust their behavior. Lynch (1996) notes that when decision costs lead to infrequent adjustments by agents, covariances between stock returns and real variables such as consumption growth will be lower than if agents reacted instantaneously. There is no reason to believe that the influences of time-averaging or measurement error on covariances will vary systematically over time. But a case can be made for a systematic relation between decision costs and the business cycle. For example, it wouldn t be hard to construct a model in which the utility cost of slow adjustment is larger 9

11 in recessions. Although the relation between decision costs and the macroeconomy is interesting, it is not the focus of this paper. Therefore I want to minimize the potential influence of decision costs on the results here. Lynch s model implies that as the measurement horizon for returns and growth in real variables is increased, the influence of infrequent decisionmaking declines. The econometric cost of longer horizons is fewer independent observations. I consider both quarterly and annual horizons because there is no clear way to evaluate this tradeoff. 3.2 The data All of the data are quarterly beginning in 1953:4, the first quarter after the end of the Korean War. The real variables I consider are those discussed in Section 2: log real per capita GDP, denoted Y (t), log consumption on nondurables and services, denoted C(t), and log labor income L(t). Changes from quarter t to quarter t + k are denoted C(t, t + k), Y (t, t + k), and L(t, t + k). My measure of aggregate excess stock returns in quarter t is the log return to the CRSP value-weighted index during quarter t less the continuously-compounded threemonth Treasury bill yield at the end of quarter t 1. The excess return from the beginning of quarter t to the beginning of quarter t + k is denoted R(t, t + k). This notation uses twotimeindexesinsteadofthesingletimeindexusedinsection2becausewewillconsider different return horizons. In the remainder of this paper, we will use only excess returns, not raw returns. Thus when the paper refers to stock returns, it should be read as excess returns. The number of variables used in the literature to forecast stock returns is impressively large. To limit the potential for data-mining, I focus on variables that are plausibly linked to the macroeconomy. My motivation is that the business cycle is the most likely source of variability in covariances between returns and real variables. 4 Fama and French (1989) identify three variables that forecast returns and are linked to the business cycle: The slope of the term structure TERM, the aggregate dividend/price ratio D/P, and the default premium (i.e., the credit spread) DEF. More recently, Lettau and Ludvigson (2001) find that the trend deviation in the log consumption-wealth ratio is a strong forecaster of expected stock returns. This deviation, denoted CAY, is a natural candidate for my purposes. The variables TERM, D/P, DEF, andcay are chosen because of their ability to forecast stock returns. I also need to forecast C(t, t+k), Y (t, t+k), and L(t, t+k) for the purpose of constructing innovations to real variables. Because there is some evidence of serial correlation in log changes in these variables, I use lagged changes of GDP and consumption 4 There is, however, somewhat sparse evidence connecting the business cycle to volatilities. See Schwert (1989). 10

12 as additional forecasting variables. I also include lagged aggregate stock returns, which tend to lead consumption and output. Thus I use a total of seven forecasting variables. 5 This set of variables is used to construct expectations of both stock returns and macroeconomic growth. 3.3 Constructing forecasts, shocks, and covariance estimates The variables used at the end of quarter t to form forecasts of future stock returns and macroeconomic growth are TERM(t) D/P(t) DEF(t) Z(t) = CAY (t 1). C(t 2,t 1) Y (t 2,t 1) R(t, t +1) The second lags of CAY and the growth rates of consumption and output are used because the macro variables are announced with a lag. The stock return R(t, t + 1) is known at the end of t. Stock returns from the beginning of quarter t + 1 to the beginning of quarter t + k are forecast with the regression R(t +1,t+ k) =a r,k,0 + a r,k,1z(t)+ R(t +1,t+ k). This regression is estimated for both quarterly (k = 2) and annual (k = 5) stock returns. Both regressions are estimated with quarterly data, which produces overlapping residuals for annual stock returns. The regression-produced forecast of R(t + 1,t + k) is denoted R(t + 1,t + k). Forecasts and residuals for changes in log consumption, GDP, and labor income are produced in the same way. The residuals for these series are denoted C(t +1,t+ k), Ỹ (t +1,t+ k), and L(t +1,t+ k). The sample period for the quarterly regressions is 1954Q2 through 2000Q1. Because of the additional three quarters needed for annual horizons, the sample period for the annual regressions ends in 1999Q2. The top panel of Figure 1 displays the forecasts of annual returns. The remaining panels in the figure display the corresponding annual stock return 5 I examined no other forecasting variables in this study. 11

13 residuals and annual GDP growth residuals. All variables are expressed in percent. The stock return forecast made at time t, the error in this forecast, and the contemporaneous error in forecasted GDP growth are all plotted at time t on the horizontal axis. It is hard to believe that these expected returns really correspond to the expectations of typical investors. For example, the return forecast produced for mid-1999 is about negative ten percent. It strains the imagination to come up with reasons why investors would buy stocks if they truly believed this forecast. A more plausible view is that these regression-based forecasts are noisy measures of investors true expectations. If our forecasting regression is correctly specified, then asymptotically the product of the OLS residual to stock returns and the OLS residual to, say, log consumption growth is drawn from a distribution with mean zero and covariance σ t (R, C). (For notational convenience, the dependence of this covariance on the horizon k is suppressed.) This product is therefore used as an ex-post estimate of the covariance. The notation I use is σ t ex (R, C) R(t +1,t+ k) C(t +1,t+ k). The superscript ex refers to an ex-post estimate. Ex-post covariances between returns and either GDP or labor income are similarly defined. Table 2 reports summary statistics for these covariance estimates, as well as for stock return forecasts. All of the variables are measured in percent. Note that the annual covariances are approximately eight times the corresponding quarterly covariances. This is the standard result (see, e.g., Campbell (1999)) that stock returns and macroeconomic growth are more closely tied at the annual frequency than at the quarterly frequency. 3.4 Explaining conditional covariances We now examine whether stock return forecasts also forecast covariances. The basic regression is σ ex t (R, X) =b 0 + b 1 R(t +1,t+ k)+e(t +1,t+ k), X {C, Y, L}. (17) The results for quarterly and annual horizons are displayed in Table 3. The table reports results for the full sample and for two subsamples created by splitting the full sample period in half. The significance level of a test that the coefficients are constant across the two subsamples is also reported. The test statistics use a Newey-West adjustment with one MA lag for quarterly horizons and four MA lags for annual horizons. There are four broad conclusions to take from this table. First, covariances between returns and both consumption and GDP growth are sensitive to forecasted stock returns. 12

14 But in stark contrast to the predictions of the model in Section 2, in the data covariances are higher when expected returns are lower. For both quarterly and annual horizons, in the full sample we can statistically reject the null hypothesis of no relation. The covariance between labor income and stock returns does not appear to be related to expected stock returns. The second conclusion is that the economic significance of this predictability is large. To put these results into perspective, consider what the point estimates from the consumption regressions imply at two different levels of expected returns: the 25th and 75th percentiles of the distribution of R(t, t + k). The values reported in Table 2 in combination with the parameter estimates in Table 3 imply the following. At the quarterly horizon, the covariance between stock returns and consumption growth conditional on the 75th percentile of expected returns is The quarterly covariance conditioned on the 75th percentile is 1.39, almost three times as large. Put differently, the ratio expected excess return RRA = 100 conditional covariance is 8.5 at the 25th percentile and 162 at the 75th percentile. This ratio is approximately the coefficient of relative risk aversion in a power utility model. 6 At the annual horizon, the corresponding conditional covariances are 3.74 and 10.70, with implied RRA coefficients of 8.4 and 299 respectively. These results are very surprising on two counts. First, research prior to this paper found no evidence not even weak evidence for time-varying covariances between stock returns and real variables. Second, the sign of this time-variation is at odds with the intuition of consumption-based asset-pricing models. It is hard to understand why investors demand high risk premia during periods when stock returns are largely decoupled from the real economic growth. One interpretation of these results is that they are spurious. Although there is no way to completely rule out this explanation, the third conclusion drawn from Table 3 is that there is no evidence that these results are an artifact of a particular subsample. The subperiod results indicate that the negative relation is somewhat stronger in the period than in the period, but the only statistical evidence of instability is for labor income for which there is no clear evidence in any period between conditional covariances and expected stock returns. In Section 4 I consider other tests of robustness. The fourth conclusion is that the negative relation is stronger at the annual horizon than at the quarterly horizon. This suggests that expected returns contain information about covariances between quarterly stock returns and subsequent macroeconomic growth, as 6 The ratio is missing the Jensen s inequality term in (4). 13

15 well as information about covariances between returns and contemporaneous macroeconomic growth. I take a closer look at this issue later. For now, I focus on the annual horizon. Although these linear regressions were motivated by the asset-pricing logic in Section 2, the signs of the estimated coefficients cannot be explained by this logic. Therefore we have no reason to continue to assume that the true relation between expected returns and covariances is linear. A more robust way to examine the data is to simply split the sample in half based on the level of the stock return forecast at time t. Let the dummy variable DUM A (t) be defined as { 1 if DUM A (t) = R(t +1,t+5)> median( R(t +1,t+ 5)); 0 otherwise. The subscript A refers to forecasts of annual returns. (In Section 4 a similar dummy is introduced based on quarterly return forecasts.) The split-sample results are determined by estimating σ t ex (R, X) =b 0 (1 DUM A (t)) + b 1 DUM A (t)+e(t), X {C, Y, L}. (18) In (18), b 0 represents the covariance conditional on low expected returns and b 1 represents the covariance conditional on high expected returns. The results of estimating (18) are displayed in Table 4. For completeness, I also report the split-sample covariances between the real variables. Where three numbers appear as a group in the table, the first is the estimated coefficient, the second (in parentheses) is the p-value of the asymptotic χ 2 (1) test that the coefficient equals zero, and the third (in brackets) is the split-sample correlation. The final set of numbers are the p-values of the asymptotic χ 2 (1) test that the covariances are equal across the two samples. The test statistics use a Newey-West adjustment (four MA lags). Table 4 confirms that covariances between returns and both consumption and output are higher when expected returns are lower. We can reject at the 5% level the hypothesis that the covariance with consumption growth is constant across the two samples, and at the 1% level the corresponding hypothesis for output growth. The covariances in the low-expectedreturn sample are about three times the covariances in the high-expected return sample. The table also confirms that the covariance with labor income is insensitive to expected returns. Changes in covariances can be decomposed into changes in standard deviations and changes in correlations. Table 4 indicates that the standard deviations of both stock returns and the real variables in the low expected return sample are about 1.2 to 1.3 times the corresponding standard deviations in the high expected return sample. This pattern in stock 14

16 return variances is consistent with earlier evidence that stock return volatilities and expected returns exhibit a modest negative relation. 7 This pattern for real variables is documented here for the first time, but it is not particularly dramatic, and the statistical significance is marginal. Correlations between stock returns and real variables are more sensitive to expected returns. They are about twice as large in the low expected return sample than in the high expected return sample. (Again, labor income is an exception.) The correlations are so low in the high-expected-return period that we cannot reject at the 1% level the hypothesis that stock returns have a zero covariance with either consumption growth or GDP growth. Visual evidence is helpful here. Figure 2 displays four scatter plots. The left column contains scatter plots of stock return shocks and either consumption (top) or GDP (bottom) shocks in high-expected-return periods. The right column corresponds to low-expectedreturn periods. It is clear from this figure that the correlation patterns documented with regression evidence are not artifacts of a few data points. The results in this section document a strong, stable negative relation between the covariances and the regression-produced forecasts of expected returns. In the next section we take a closer look at various aspects of this relation. 4 Additional evidence 4.1 Out-of-sample results The use of in-sample regressions means that we are using information that investors do not have in forming forecasts of conditional variances. This is innocuous as long as investors forecasts are time-invariant linear functions of the state vector Z(t). Then the econometrician is simply inferring what investors already know. Since this assumption of investors forecasts is likely overly strong, here I adopt a procedure that could be implemented in real time. For each quarter-end τ, I replicate the first-step regressions described in Section 3.3 using data available through τ. (For example, I assume the most recent available observation of aggregate consumption is that for τ 1.) 8 Predicted returns and residuals of returns, consumption growth, GDP growth, and labor income growth from these first-step regressions are then used to estimate (17) at the annual horizon. The first-step regressions are also used to form out-of-sample, time-τ forecasts of R(τ +1,τ +5), C(τ +1,τ +5), Y (τ +1,τ +5), and L(τ +1,τ +5). The forecast of R(τ +1,τ +5) is combined with the parameters from the quarter-τ estimate 7 See, e.g., Whitelaw (2000). 8 Data revisions are ignored. 15

17 of (17) to forecast out-of-sample covariances σ τ(r, X),X {C, Y, L}. The asterisk refers to forecasts made using this out-of-sample approach. The process is repeated for each τ {1965Q1, 1999Q2}. (I choose the first date so that each regression equation has at least 30 degrees of freedom.) To evaluate the accuracy of these forecasted covariances, I compare them to the product of realized forecast errors. Realizations of R(τ + 1,τ + 5) and the contemporaneous growth rates in real variables are used to construct forecast errors. Products of the forecast errors correspond to ex-post estimates of the covariances, denoted στ ex, (R, X),X {C, Y, L}. I then estimate the regression σ t ex, (R, X) =c 0 + c 1 σ t (R, X)+η(t), X {C, Y, L}. If the forecasts are unbiased, we should find c 0 =0andc 1 = 1. Results from this regression are displayed in Table 5. Of course, this methodology is not truly out of sample; the choice of variables included in the forecasting vector Z(t) is determined by our knowledge of the entire sample path. In addition, the forecasting variable CAY defined by Lettau and Ludvigson uses an in-sample cointegrating regression. I do not attempt to address the first of these two problems, but to address the second I reproduce the entire exercise dropping CAY as a forecasting variable. These results are also displayed in Table 5. The main message of Table 5 is that the out-of-sample forecasts for covariances between stock returns and either consumption or GDP contain substantial information about actual covariances. I focus on the results using CAY ; those excluding CAY are similar. The first column of numbers reports the mean estimates of the regression coefficient b 1 in (17). The means for consumption and GDP are about half their in-sample counterparts reported in Table 3. This is not surprising; the errors-in-variables problem in (17) is more pronounced with smaller sample sizes. Excluding the results for labor income, the estimates of c 0 are statistically indistinguishable from zero and the estimates of c 1 are statistically indistinguishable from one. (The standard errors use a Newey-West adjustment (four MA lags)). We can reject at the 5% level the hypothesis that c 1 = When do stock returns forecast the macroeconomy? As we saw in Section 3.4, contemporaneous covariances at the annual horizon are more sensitive to expected returns than are covariances at the quarterly horizon. This suggests that non-contemporaneous covariances at the quarterly horizon are also sensitive to expected returns. I take a look at this issue here. The approach I take is a little different from that used earlier. Instead of constructing shocks to returns and economic growth, I use the 16

18 stock return R(t + 1,t + 2) to predict contemporaneous and three future quarters of changes in consumption and output. The coefficient on the stock return is allowed to depend on DUM Q (t), an indicator variable for the magnitude of the quarter-t forecast R(t +1,t+2). If this forecast is greater than the median forecast, the dummy equals one. The regression for consumption is C(t +1+k, t +2+k) = b 0 + b 1 DUM Q (t)+ (b 2 + b 3 DUM Q (t))r(t +1,t+2), k =0,...,3. I also estimate this regression using GDP. The sample period is 1954Q2 through 2000Q1. Because the residuals are correlated, I use a Newey-West adjustment for the test statistics (three MA lags). Table 6 contains the results, which can be summarized with two observations. First, the relation between stock returns and both contemporaneous and future macroeconomic growth is strong when expected returns are low. All of the estimates of b 2 exceed zero and six of the eight are statistically significant at the 5% level. This evidence is consistent with earlier work, which found that (unconditionally) increases in stock prices correspond to contemporaneous and future increases in aggregate consumption and output. 9 However, the second observation is that when expected returns are high, the relation between stock returns and macroeconomic growth disappears. The column labeled p-val of sum reports the significance level of the test that b 2 +b 3 =0. Thishypothesisisnotrejected at the 10% level in any of the regressions. Thus when expected returns are high, quarterly stock returns are largely decoupled from both contemporaneous and future macroeconomic growth. The well-documented link between stock returns and macroeconomic growth holds only in periods of low expected returns. As an aside, this result implies that the forecasting equation used to produce residuals of consumption and GDP growth in Section 3.3 is misspecified. In that regression, a linear relation is assumed to hold between stock returns and future macroeconomic growth, which is inconsistent with the evidence here. 4.3 Other assets The theory motivating the empirical tests in this paper does not imply anything special about stocks. Any risky asset will work, as long as it has predictable variations in returns. An earlier version of this paper looked at both stocks and long-term bonds. The detailed results are not included in this paper because they can be summarized easily. There is 9 See, e.g., Fischer and Merton (1984) or Barro (1989). 17

19 no significant relation between forecasts of excess bond returns and conditional covariances between bond returns and real variables. Given the other results presented here, this insignificance does not tell us much. If I had documented a positive relation between expected stock returns and conditional covariances with stock returns, the insignificance of the bond results would be more worrisome. We would be tempted to interpret a positive relation in the stock market as evidence that risk premia varied because investors were reacting to conditional covariances. Since risk premia vary in the bond market without corresponding variations in conditional covariances, this would be a more difficult story to tell. However, since expected stock returns and covariances move inversely, we cannot tell this story anyway. The next section considers some alternative explanations for these empirical patterns. 5 Interpretations This paper documents a strong relation between forecasted stock returns and conditional covariances between stock returns and economic growth. This evidence begs a number of questions that I examine in this section. 5.1 What moves stock prices? We saw in Section 4.2 that the relation between stock returns and macroeconomic growth is predictably strong or weak, depending on expected returns to stocks. This suggests that the kinds of shocks that hit the economy in periods of high expected stock returns are qualitatively different from the kinds of shocks that hit in periods of low expected stock returns. Can we say anything more about the nature of these shocks? Put differently, can we say what stock prices are reacting to across these regimes? Stock prices should respond to news about cash flows and discount rates. Realizations of cash flows, discount rates, and future interest rates are noisy measures of these expectations. Thus a standard test of models of asset-price determination is to regress asset returns on current and future realizations of relevant variables. 10 I follow the same approach here. A natural starting point is dividends. I also use inflation, based on earlier research that documents a strong inverse relation between inflation and stock prices. 11 In addition, long-term bond yields should contain information about future discount rates. Denote by DIV (t + 4) the log of one year of trailing dividends (sum of dividends during quarters t See Fama (1990) for an implementation of this approach and references to earlier research. 11 See, e.g., Fama (1981). 18

20 through t + 4). This variable is known at the beginning of quarter t + 5. The appendix describes the construction of the series. The one-year trailing inflation rate in quarter t + 4 (log change in the CPI from t to t+4) is denoted π(t+4). I assume this variable is known at the beginning of quarter t + 5. The ten-year Treasury bond yield observed at the beginning of quarter t + 5 is denoted 10YR(t +5). I want to use these three variables to explain the innovation R(t+1,t+5). Therefore I construct innovations in the explanatory variables by regressing them on information available at the end of quarter t. The explanatory variables DIV (t),div(t 4), π(t),π(t 4), 10YR(t), and 10Y R(t 4). Denote the in-sample residuals as DIV (t+4), π(t+4), and 10YR(t+5). I performed some extensive data-mining to end up with this short list of explanatory variables. Below I briefly mention results using other variables of interest. The main regression is R(t +1,t+5)=b 0 + b 1 DIV (t +4)+b 2 π(t +4)+b 3 10YR(t +5)+e(t +5). Following the methodology adopted earlier, I estimate this regression on two samples, based on whether the expectation of R(t +1,t+ 5) formed at the end of quarter t isgreaterorless than its median value. 12 The results are displayed in Table 7. When dividend innovations are used to explain innovations in stock returns, we see a dramatic difference across the two samples. When expected stock returns are low, dividend shocks and stock returns are closely linked. The adjusted R 2 using this single variable is By contrast, when expected stock returns are high, there is, at best, a weak relation between shocks to dividends and stock returns. The estimated coefficient is statistically indistinguishable from zero at the 5% level, and the adjusted R 2 is only (The standard error of the estimated residuals (SEE) is higher in the latter sample. The reason is that, as documented in Table 4, the variance of stock returns is larger when expected returns are high than when they are low.) This dichotomy is consistent with the evidence presented elsewhere in this paper. There is an apparent disconnect between real activity and the stock market in high-expected-return periods. What explains this? Part of the answer lies with the dynamics of dividends and inflation. When the inflation innovation is included in the high-expected-return regression, the adjusted R 2 in the highexpected-return sample jumps from 0.07 to In addition, the coefficient on the dividend innovation is larger than when inflation is not included, and is strongly significant. Adding inflation to the regression in the low-expected-return sample also raises the adjusted R 2, but 12 The regressions used to construct innovations in dividends, inflation, and ten-year bond yields are also estimated separately on these samples, to allow for the possibility that their dynamics depend systematically on expected returns. 19