Measuring tbe Predictable Variation in Stock and Bond Returns matrix of this estimator warrants further discussion.The monthly, quarterly,and annual regressions employ nonoverlapping returns,so it quite common to use (21) for these models.7 For the two-,three-,and four-year regressions,the disturbance vectoris serially correlated.Thus,it is not imme- diately obvious how to go about constructing a consistent estimate of V.One possibility,which is discussed in more detail below,is to use a weighting scheme that yields a consistent,positive semidefinite estimator.An alternative approach,which is adopted by Fama and French (1989),is to rely on a truncated estimator.The subsequent analysis makes use of both approaches. To implement the Fama and French (1989)approach we assume that the nonoverlapping errors for the regression are serially uncorre- lated.Under these circumstances,the V matrix of Theorem 1.1 takes the form: V-2 [空小 (22) As a consequence,we can follow Hansen (1982)and compute a con- sistent estimate of V from the sample autocovariances of and the sample covariance matrix of the instruments.The main drawback of using this truncated estimator is that it is not guaranteed to be positive semidefinite. A heteroscedasticity and autocorrelation consistent estimator of V that is guaranteed to be positive semidefinite can be obtained by employing an appropriate weighting scheme.The estimator used in empirical analysis is constructed using the quadratic spectral kernel and automatic bandwidth selection procedure of Andrews (1991).8 First,the optimal bandwidth is estimated by fitting a first-order au- toregressive model to each element of the sample disturbance vector This bandwidth controls the rate at which the weights decline 7 This estimator for V is identical to the heteroscedasticity-consistent covariance matrix estimator of White (1980). 8Andrews (1991)shows that quadratic spectral weights are optimal;that is,they minimize the asymptotic truncated mean-squared error of the estimator. 593
The Review of Financial Studies/v 10 n 3 1997 as the lag length increases.Then V is computed as +[2成+成 (23) where w is the weight at lag i given by the quadratic spectral kernel. 2.5 Standard errors for the sample R2 Theorem 1.4 provides a way to compute consistent standard errors for the sample R2.Recall that,under the alternative hypothesis where returns are to some extent predictable over time,the variance of the sample R2 is a function of the autocovariance structure of.In particular,it is given by [+k5+- (24) Since the variablewill in general be serially correlated,the stan- dard errors for the sample R2are computed by applying the automatic bandwidth selection procedure and quadratic spectral kernel of An- drews (1991)to the sample analog of 3.Empirical Results The empirical section of this article attempts to answer two basic questions.First,are the findings of Fama and French (1989)robust to minor changes in the data used to construct the instrumental variables? Second,and more importantly,can their interpretation of the long- horizon results be justified given the distributional properties of the test statistics and sample R-?The analysis begins with an overview of the evidence from the long-horizon regressions. 3.1 An initial look at the regression evidence Tables 2 and 3 report the results of GMM estimation of the multiple regression models.The estimates in Table 2 are for the dividend yield and term spread.Those in Table 3 are for the default spread and term spread.Panel A of each table contains the estimated slopes and their associated t-ratios.As mentioned earlier,these estimates are identical to the ones that would have been obtained by using OLS to fit the models to the data.Panel B of each table presents Wald tests of the 594
Measuring the Predictable Variation in Stock and Bond Returns null hypothesis that the slopes for the model are equal to zero and gives the sample R for each regression.The t-ratios and Wald tests are corrected for autocorrelation and conditional heteroscedasticity using the truncated estimator of Section 2.4. Upon initial inspection,the results shown in Tables 2 and 3 would seem to bolster claims that long-horizon stock and bond returns are highly predictable.Almost all of the point estimates of slope coef- ficients are positive,and many of them are more than two standard errors away from zero.The dividend yield-a stock market variable- seems to have the ability to forecast bond returns.The bond market variables-a term spread and default spread-seem to have the ability to forecast stock returns.Overall,the point estimates of slope coeffi- cients appear to document a clear pattern of time-series variation in expected returns that is common across the stock and bond markets. The Wald tests and sample R2 in panel B of Tables 2 and 3 also suggest strong predictability at long horizons.Most of the test statistics for the one-,two-,three-,and four-year returns appear to be highly significant in light of their limiting distribution under the null hypothe- sis that returns are unpredictable.Moreover,the pattern in the sample R2 documented by Fama and French (1989)is clearly evident.The sample R2 increases from less than 3%for monthly returns to well over 25%for many of the four-year returns.In short,the regression results appear to support the conclusion that predictability increases with the length of the return horizon.But a more thorough examina- tion of the regression evidence raises serious questions about whether such a conclusion is actually justified. 3.2 The regression evidence revisited The fact that the sample R2increases with the length of the return horizon does not,in and of itself,signal that long-horizon returns are more predictable than those at short horizons.Indeed,the distribu- tional theory developed earlier suggests that we would expect to see an increase in the sample R2 at long horizons,even if the long-horizon returns are unpredictable.This can easily be illustrated by consider- ing the case where the assumptions of classical regression analysis are satisfied.Under such circumstances,the limiting distribution of the sample R2 is TR院号X, (25) where denotes a chi-square distribution with m degrees of free- dom.If,as in the case of monthly returns,there are two instruments In the regressions where the returns are nonoverlapping,the lag truncation parameter is set equal to zero to obtain White's (1980)estimator of the variance-covariance matrix. 595