the value-weighted CRSIindex of slocks traded on die NYSK, thcAMF.X, and (lie NASDAQ. The dividend-price ratio is measured as the sum of dividends paid on the index over ihe previous year, divided by the current level of the index; .summing dividends over a full year removes any seasonal patterns in dividend payments, but the ntrreni slock index is used lo incorporate the most recent information in slock prices.

The inlet cst-i ale variable is a transformation of the one-month nominal US Treasury bill rale motivated by the fact that unit-root tests often fail lo reject the hypothesis that the bill rate has a unit root. We subtract a backward one-year moving average of pasl bill rates from the current bill rate to get a stochastically detrended interest rale that is equivalent to a triangularly weighted moving average of past changes in bill rates, where the weights (let line as one moves back in lime. Accordingly the detreuded interest rale is stationary if changes in bill rates are stationary. This stochastic detrending method has been used by Campbell (КИП) and Hodrick (1992).

fable 7.1 shows a typical set of results when the dividend-price ratio is used lo forecast returns. The* table reports monthly regressions of log real slock returns onto the log of the dividend-price ratio at the start of the holding period. Returns arc measured over a holding period of A. months, which ranges from one month lo 48 months (four years); whenever A > 1, the regressions use overlapping monthly data. Results are reported for the period 1927 to 1994 and also for subsamples 1927 to 1951 and 1952 to 1994. For each regression fable 7.1 reports the R- statistic and the /-statistic for Ihe hypothesis that the coefficient on ihe log dividend-price ratio is zero. The /-statistic is corrected for heteroskedasticity and serial correlation in the equation error using the asymptotic theory discussed in the Appendix. Table 7.1 follows Fama and French (1988a) except that the regressor is tne log dividend-price ratio rather than the level of the dividend-price ratio (a change which makes very little difference to the results), overlapping monthly data are used lot all horizons, and the sample periods are updated. Although the results in the table are for real stock returns, almost identical results are obtained for excess returns over the one-month Treasury bill rale.

At a horizon ol One month, the regression results in Table 7.1 are rather unimpressive: lhe /{- statistics never exceed 2%, and the /-statistics exceed 2 only in the post-World War II subsample. The striking fact about the table is how much stronger the results become when one increases the horizon A. At a two-year horizon the R- statistic is 14% for the full sample, 22% for the prewar subsample, and 32% for the postwar subsample; al a four-year horizon the if statistic is 2f>% for the lull sample and 42% for each of ihe subsamples. In die full sample and the prewar subsample the regression /-

1,1 this wav til uii-.isuiiiig ilie iliviilrtul-[ii ii с 1.1 iio is Manila id in the academic literature, and il is also t onunotilv used in ilie (iuaiictal iudusiiv-

Forecast Horizon (A)

r is the log real return on a value-weighted index of NYSE, AMEX, and NASDAQ slocks, td-p) is the log ratio ol dividends over the last year to the current price. Regressions are estimated by OI.S, with Hansen and Hodrick (1980) standard errors, calculated from equation (A.3.3) in the Appendix setting autocovat iauces beyond lag К - I to zero. Newey and West (1987) standard errors with q - К - 1 or q - 1{K - 1) are very similar and typically are slightly smaller iban those reported in the table.

statistics also increase dramatically with the forecast horizon, although they are fairly stable within the range 3.0 to 3.5 in the postwar subsample.

Il is interesting to compare the results in Table 7.1 with those obtained when stock returns are regressed onto the stochastically detrended short-term interest rate in Table 7.2. The regressions reported in Table-7.2 are run in just the same way as those in Table 7.1. Once again almost identical results are obtained if real returns are replaced by excess returns over the one-month Treasury bill rate.

fable 7.2 shows that, like the dividend-price ratio, the stochastically detrended short rate has some ability to forecast stock returns. However this forecasting power is very different in two respects. First, it is concentrated in the postwar subsample; this is not surprising since short-term interest rates were pegged by the Federal Reserve during much of the 1930s and 1940s, and so the detrended short rate hardly varies in these years. Second, the forecasting power of the short rale is al much shorter horizons than the

Table 7.1. hnig-horizan regressions of log .stock relurns on the log dividend-price ratio. )+!+ + )+л = Р1Ю01. - Il) + Пи-к.к

Table 7.2. I .onghoriwii recessions of tog sloth returns on Ihe stochastically tletrentletl short-term interest rate.

Forecast Пшют (A.)

1927 lo 1994 P(K)

t<p{K)) Л 1927 to 1951

\ KPiK)) ! 1952 to 1994 , P(l<)

np{K))

-5.408 0.005 -2.292

3.144 0.000 0.222

-(3.547 0.019 -3.263

-17.181 0.0 Hi -2.582

-6.1 КЗ 0.000 -0.165

-18.621 0.047 -3.200

-41.003 0.023 -1.504

73.712 0.012 0.520

-50.400 0.103 -2.741

-4.492 ().()()() -0.104

158.989 0.031 1.062

-20.115 0.013 -1.354

-20.148 -20.129 0.004 (1.002 -1.341* -0.838*

-07.505 -50.900 0.005 0.002 - .037* -0.580*

-20.573 0.010 -1.555*

-25.894 0.008 -1.092*

is the log real return on a value-weighted index of NYSE. ЛМЕХ. and NASDAQ storks, yi., i: the 1-month nominal Treasury bill rate. Regressions are estimated by OlS. with Hansen and Hodrick (1980) standard errors, calculated from equation (Л.З.З) in the Appendix setting aiilocovariances beyond lag K-l to zero. Newey and West (1987) standard errors, with ц = \K - 1). are used when the Hansen and Hodrick (1980) covariance matrix estimator is not positive definite. The cases where this occurs are marked .

f recasting power of the dividend-price ratio. The postwar R2 statistics arc comparable to those in Table 7.1 at horizons of one or three months, but t icy peak at 0.10 at a horizon of one year and then rapidly decline. The regression -statistics are likewise insignificant beyond a one-year horizon.

How can wc understand the hump-shaped pattern of R2 statistics and /-statistics in Table 7.2 and the strongly increasing pattern in Table 7.1? At one level, the results in Table 7.1 can be understood by recalling the formula relating the log dividend-price ratio to expectations of future returns and dividend growth rales, given above as (7.1.24):

pi = K(

l+t + ; + r,+ i+;]

This expression shows that the log dividend-price ratio will be a good proxy for market expectations of future stock returns, provided that expectations of future dividend growth rates are not too variable. Moreover, in general

the log dividend-price ralio will be a belter proxy for expectations of long-horizon returns than for expectations of short-horizon returns, because the expectations on the right-hand side of (7.1.24) are of a discounted value of all returns into the infinite future. This may help to explain the improvement in forecast power as the horizon increases in Table 7.1.

К veil in the absence of this effect, however, it is possible lo obtain results like those in Tables 7.1 and 7.2. To see this we now relurn lo our AR(I) example in which the variable a perfect proxy for the expected stock return at any horizon, is observable and can be used as a regressor by the economeuician. Problem 7.4 develops a structural model of slock prices and dividends in which a multiple of the log dividend-price ralio has the properties of the variable x, in the AR(1) example.

We use the AR(1) example to show that when .v, is persistent, the R2 ol a return regression on x, is very small al a short horizon; as the horizon increases, the R2 first increases and then eventually decreases. Wc also discuss finite-sample difficulties with statistical inference in long-horizon regressions.

R1 Statistics

First consider regressing the one-period return r,+ , on the variable x,. For simplicity, we will ignore constant terms since these are not the objects of interest; constants could be included in the regression, or we could simply work with demeaned data. In population, 0(1) = 1, so the lilted value is just x, itself, with variance a2, while the variance of the return is given by equation (7.1.31) above. It follows that die one-period regression R2 statistic, which we write as R2(\), is

Var[r,+ i] \a; 1-0/

where for simplicity wc arc using die approximate version of (7.1.31) dial holds when ф . p and p is close lo one. /< -( 1) teaches an upper bound of (1 -ф)/2 when the variability of dividend news, a2, is zero. Thus even when a slock is effectively a real consol bond with known real dividends, so that all variation in its price is due lo changing expected returns, the one-period R2 statistic will be small when ф is large. The reason is thai innovations to expected returns cause large unforecastable changes in stock prices when expected returns arc persistent.

The behavior of the fi2 statistic in a long-horizon regression is somewhat more complicated. A regression with horizon К lakes the form

r,+ , + + r,+K = p{K)x, + iiм л.л. (7.2.2)

In the AR( I) example, the best forecast of the one-period relurn j periods ahead is always F.,.v/+/ ) = ,\,. The best forecast of the < itmiilalive

return over л iiioullis is lound by summing the forecasts of one-period

returns up to horizon K,snfi(K) = (1+04-----Ь</>А~) = (I-фк)/( 1 -ф).

The R- statistic lor the A-period regression is given by

,?{K) * Val 11 и I H + Ky[r,4.A-l) ,

Var >,. + + rl+K\ Dividing by the one-period R- statistic and rearranging, we obtain

R(K) /V; r >nl + --- + K/fr,+A-l\

(7.2.-1)

И) V Vari K,[rl+,\

Varr(+,1

VarrM, + + rl+K\

The first ratio on the right-hand side of (7.2.4) is just the square of the A-pcriod regression coefficient divided by the square of the one-period regression cocllii ieut. In die AR( 1) example this is (1 -фл)-/(1 -ф)~, whit b is approximately equal lo A- for large ф and small A. The second ratio on the right-hand side of (7.2.1) is closely related lo the variance ratio discussed in Chapter 2. In fact, it can he rewritten as 1/(AV(A)), where V(A) is the A-period variance ratio for stock returns. In thcAR(l) example, Problem 7.3 shows that the autocorrelations of stock returns are all negative. It follows that V(A) < I,so l/(AV(A)) > 1/A.

Pulling the two terms on the tight-hand side of (7.2.4) together, we find thai if expected slock returns arc very persistent, the multiperiod R- statistic grows at first approximately in proportion to the horizon A. This behavior is well illustrated by the results in Table 7.1. Intuitively, it occurs because forecasts of expected relurns several periods ahead are only slightly less variable than the forecast of the next periods expected return, and they are perfectly correlated with il. Successive realized returns, on the other hand, are slightly negatively correlated with one another. Thus at fust the variance of the multiperiod lilted value grows more rapidly than the variance of the multiperiod realized return, increasing lite multiperiod R2 .statistic. Eventually, of course, forecasts of returns in the distant future die out so die fust ratio on the right-hand side of (7.2.1) converges to a fixed limit; but Ihe variability of realized niulliperiod returns continues to increase, so the second ratio on the right-hand side of (7.2.4) becomes proportional to I /А. Thus eventually multiperiod R2 statistics go to zero as the horizon increases.

It may be helpful to give an even more explicit formula for the AR( 1) example in die case where die long horizon is just Iwo periods, lhat is where A = 2. In this case tedious but straightforward calculations and the

simplifying approximation that holds when ф < p and p is close to one yield

RHl)

2 + (\-ф)<о1/о1) 2(1+0) + 2(1 -ф )(cr2 /а2)

(7.2.5)

The ratio in (7.2.5) approaches (1 + ф) as a2 fa2 approaches zero, so a two period regression may have an R2 statistic almost twice that of a one-periocj regression if expected returns are persistent and highly variable. On the other hand the ratio approaches (1 + <#)2/2 as a%/o2 approaches infinity, soj a two-period regression may have an R2 statistic only half that of a one-period regression if expected returns have only small and transitory variation.

Calculations for horizons beyond two periods become very messy, but Campbell (1993b) reports some numerical results. When ф = 0.98, p = 0.995, and a2/a2 = 0, for example, a one-period regression has an Я2 statistic of only 1.5%, but the maximum R2 is 63% for a 152-period regression. When the forecasting variable is highly persistent, the R2 statistic can j continue to rise out to extremely long horizons. j

Difficulties with Inference in Finite Samples \

The /-statistics reported in Tables 7.1 and 7.2 are based on the asymptotic i theory summarized in the Appendix. There are however a number of pitfalls i in applying this theory to regressions of returns onto the information variable x,.

A first problem arises from the fact that in the regression of the one-period return r,+i on x r,+i = B(l)x, + ц,, the regressor x, is correlated wilh past error terms for i > 0, even though it is not correlated with contemporaneous or future error terms *),+]+,. These correlations exist because shocks to the state variable x, are correlated with shocks to returns, and the variable x, is persistent. In the language of econometrics, the regressor x, is predetermined, but it is not exogenous. This leads to finite-sample bias in the coefficient of a regression of returns on xt. In the AR(1) example, there is a simple formula for the bias when the regression horizon is one period:

The term -(1 + ?>ф)/Т is the Kendall (1954) expression for the bias in the OI S estimate of ihe persistence parameter ф obtained by regressing on x,. As Stambaugh (1986) has shown, this bias leads to a bias in the OLS estimate of the coefficient /9(1) when the return innovation nl+\ covaries with the innovation in the forecasting variable %,+\. In our simple example with uncorrelated news about dividends and future returns driving current returns, the ratio an(/a2 = - p/(l - рф), which produces the second