NBER WORKING PAPER SERIES WEAK AND SEMI-STRONG FORM STOCK RETURN PREDICTABILITY, REVISITED. Wayne E. Ferson Andrea Heuson Tie Su

Transcription

1 NBER WORKING PAPER SERIES WEAK AND SEMI-STRONG FORM STOCK RETURN PREDICTABILITY, REVISITED Wayne E. Ferson Andrea Heuson Tie Su Working Paper NATIONAL BUREAU OF ECONOMIC RESEARCH 1050 Massachusetts Avenue Cabridge, MA August 2004 The views expressed herein are those of the author(s) and not necessarily those of the National Bureau of Econoic Research by Wayne E. Ferson, Andrea Heuson, and Tie Su. All rights reserved. Short sections of text, not to exceed two paragraphs, ay be quoted without explicit perission provided that full credit, including notice, is given to the source.

2 Weak and Sei-Strong For Stock Return Predictability, Revisited Wayne E. Ferson, Andrea Heuson, and Tie Su NBER Working Paper No August 2004 JEL No. G0, G1 ABSTRACT This paper akes indirect inference about the tie-variation in expected stock returns by coparing unconditional saple variances to estiates of expected conditional variances. The evidence reveals ore predictability as ore inforation is used, and no evidence that predictability has diinished in recent years. Sei-strong for evidence suggests that tie-variation in expected returns reains econoically iportant. Wayne E. Ferson Boston College 140 Coonwealth Avenue Chestnut Hill, MA and NBER Andrea Heuson University of Miai 5250 University Drive Coral Gables, FL Tie Su University of Miai 5250 University Drive Coral Gables, FL

3 1. Introduction The epirical evidence for predictability in coon stock returns reains abiguous, even after any years of research. This paper akes indirect inference about the tie-variation in expected stock returns by coparing unconditional saple variances of returns to estiates of expected conditional variances. The key to our approach is a su-of-squares decoposition: Var{R} = E{Var(R Ω)} + Var{E(R Ω)}, (1) where R is the rate of return of a stock and Ω is the public inforation set. E(. Ω ) and Var(. Ω ) are the conditional ean and variance and Var{.} and E{.}, without the conditioning notation, are the unconditional oents. We are interested in the ter Var{ E(R Ω ) }; that is, the aount of variation through tie in conditionally expected stock returns. We infer this quantity by subtracting estiates of the expected conditional variance fro estiates of the unconditional variance. We focus on the predictability in onthly stock returns. This is otivated by the epirical literature on asset pricing, which ost coonly studies onthly returns. We use two approaches to estiate the average conditional variances. These correspond to the classical description of increasing arket inforation sets described by Faa (1970). Weak-for inforation considers only the inforation contained in past stock prices. This analysis, suarized in Table 1, builds on a coparison of daily and onthly saple variances, and is related to the variance ratios studied by Lo and MacKinlay (1988) and others. Sei-strong for inforation relates to other lagged variables that are clearly publicly available. Our analysis uses 2

4 regressions for individual stock returns, on lagged fir-specific characteristics. Our approach is unique in that it relies on the covariation of the predictable coponents of individual stocks to draw inferences about index predictability. These results are reported in Table 2. Studies of predictability in stock index returns typically report regressions with sall R-squares, as the fraction of the variance in returns that can be predicted with lagged variables is sall. The R-squares are larger for longer-horizon returns, because expected returns are considered to be ore persistent than returns theselves. 1 However, because stock returns are very volatile, sall R-squares can ask econoically iportant variation in the expected returns. Stocks are long "duration" assets, so a sall change in the expected return can lead to a large fluctuation in the asset value. To illustrate, consider the siple Gordon (1962) constant-growth odel for a stock price: P = ke/(r-g), where P is the stock price, E is the earnings per share, k is the dividend payout ratio, g is the future growth rate of earnings and r is the discount rate. The discount rate is the required or expected return of the stock. Consider an exaple where the price/earnings ratio, P/E = 15, the payout ratio, k = 0.6, and the expected growth rate, g = 3%. The expected return is 7%. Suppose there is a shock to the expected return, ceteris paribus. In this exaple a change of one percent in r leads to approxiately a 20% change in the asset value. Of course, it is unrealistic to hold everything else fixed, but the exaple 1 Thus, the variance of the expected returns accuulates with longer horizons faster than the variance of returns, and the R-squares increase (see, e.g. Faa and French, 1988, 1989). 3

5 suggests that sall changes in expected returns can produce large and econoically significant changes in asset values. Consistent with this arguent, studies such as Kandel and Stabaugh (1996), Capbell and Viceira (2001) and Fleing, Kirby and Ostdiek (2001) show that optial portfolio decisions can be affected to an econoically significant degree by return predictability, even when the aount of predictability, as easured by R-squared, is sall. Generalizing the Gordon odel to allow for changes in growth rates, Capbell (1991) estiates that changes in expected returns through tie ay account for half of the variance of equity index values. Our weak-for tests find no reliable evidence of predictability in odern data. Even so, a siulation study shows that the tests have the power to detect odest aounts of predictability. On the other hand, our sei-strong for tests find sall but statistically and econoically significant predictability. In contrast to recent studies that rely on aggregate predictor variables, we find no evidence that the predictability has diinished over tie. The rest of the paper is organized as follows. Section 2 discusses our approaches to easuring the variance of conditional expected stock returns. Section 3 presents the data and ain results. Section 4 studies the power of our approach using siulations and the robustness of our epirical findings. Conclusions are offered in Section 5. Two Appendices discuss data, estiation issues and technical details. 4

6 2. Measuring Average Conditional Variances 2.1. Weak For Inforation Equation (1) says that we can estiate the variance of conditional expected stock returns by first estiating the unconditional variance and then subtracting an estiate of the average conditional variance. Thus, in order to use Equation (1) we need to estiate the average variance of the returns around the conditional ean, E{Var(R Ω )} = E{[R-E(R Ω )] 2 }. The proble is that we don't know the conditional ean, E(R Ω ). Our approach in this section follows Merton (1980), who showed that while the ean of a stock return is hard to estiate, it is nearly irrelevant for estiating the conditional variance, when the tie between observations is short. We use high frequency returns to estiate the conditional variance, subtract its average fro the onthly unconditional variance, and the difference -- according to Equation (1) -- is the variance of the onthly conditional ean. Nelson (1990, 1992) further develops Merton's idea. Suppose that the stock value can be approxiated by a continuous process fored as a step function, with tie intervals of length h between the steps. Take the interval [T-h,T), chop it into D pieces, and consider the average of the D squared log price changes as an estiator for the conditional variance of the returns over the interval. Nelson proves the estiator is consistent, in the sense that it approaches the conditional variance in the "continuous record" liit, as h approaches zero and D becoes infinite. The intuition is that for sall h, the conditional ean is effectively constant, so the saple variance approaches the conditional variance as D grows. By siilar logic, Nelson (1992) shows that isspecification of the conditional ean function washes out as h gets sall. 5

7 Evidence fro Nelson (1991) supports the idea that for onthly stock returns, chopping the onth into days should work well. He finds that daily returns easured with versus without dividends, or with versus without a siple adjustent for risk-free interest rates, produce virtually the sae estiates of conditional variances. Siilarly, Schwert (1990) finds that different dividend series have alost no effect on the easured daily variances of a long historical stock return series that we use in our analysis. We estiate E{Var(R Ω )} by the tie series average of the daily return variances for each onth. Using onthly returns data, we estiate the unconditional variance, Var(R). Then, we infer the variance of the conditional expected returns by Equation (1). Let the return for onth be R = ln(v /V -1 ) Σ, where V is the value of the stock at tie and j is the daily log value = j j change for day j. Assue that the conditional ean for onth is µ = E( Σ j ), with E( j j ) = µ /D, D being the nuber of days in the j j onth. The unconditional ean onthly return is E( µ ) = µ, and we are interested in Var( µ ), the variance of the onthly expected returns. Define the average daily variance, ADV = E{E[( -µ /D) 2 j ]}, and the unconditional onthly variance, j MV = E{(R -µ ) 2 }. Siple calculations show that Var( µ ) = MV - D(ADV). The odel sketched above uses the approxiation that the eans shift onthly, while daily returns fluctuate independently around the conditional eans. However, there is weak serial dependence in daily stock returns. The question is whether or not to attribute this serial dependence to changes in the conditional expected return. On the one hand, uch of the literature on predictability allows that serial 6

8 dependence ay reflect changing conditional eans. Faa and French (1988) use rate-of-return autoregressions to study predictability. Lo and MacKinlay (1988) and Conrad and Kaul (1988) odel expected returns within the onth as autoregressive processes. On the other hand, serial dependence in daily returns can arise fro endof-day price quotes that fluctuate between bid and ask (Roll, 1984) or fro nonsynchronous trading of the stocks in an index. These effects should not be attributed to tie-variation in the expected discount rate for stocks. We estiate Var( µ ) with and without adjustents for serial dependence. To illustrate the adjustent, let θ = E{ E[( - µ /D)( j - µ /D) j, j, j-j' =1] }. Assuing that j the first order daily serial dependence reflects arket icrostructure effects unrelated to discount rates, we estiate Var( µ ) = MV - D(ADV + 2 θ ). In the Appendix B we describe how the calculations are adjusted to obtain unbiased estiators in finite saples. Biases in the finite saple variances and autocovariances arise due to estiation error in the saple eans. In addition, there is a "finite record" bias, which arises because h>0 and D<. To address these biases we use Monte Carlo siulations. The calculations described above do not ipose the requireent that variance estiates can't be negative. The Appendix B also describes how this restriction is iposed. Section 4 further explores the power and robustness of the ethods Sei-strong For Inforation: Using Individual Stock Regressions Much of the epirical literature on asset-return predictability uses regressions of stock-index returns on lagged, arket-wide inforation variables. This approach raises two types of concerns. First, there are statistical probles associated with the 7

9 regressions, especially when the data are heteroskedastic, the right-hand side variables are highly persistent or the left-hand side returns are overlapping in tie. 2 The second issue is data ining. If the lagged variables arise fro any researchers sifting through the sae data sets, there is a risk of finding spurious predictability (Lo and MacKinlay 1990; Foster, Sith and Whaley 1997). We use individual stocks to estiate the su-of-squares decoposition in Equation (1), focusing on the aggregate predictability. Basic portfolio theory iplies that individual-stock expected returns teach us about index predictability, only to the extent that they are correlated across the stocks. Consider the N x N covariance atrix of the conditional ean returns for N stocks, Cov{E(R Z)}, where Z stands for the lagged, public inforation regressors. Letting 1 be an N-vector of ones, the variance of the conditional expected returns on an equally-weighted portfolio, R p is: Var{E(R p Z)} = (1/N 2 ) 1'Cov{E(R Z)}1. (2) Since there are N(N-1) covariance ters, but only N variances in this expression, the expected return variance for the portfolio approaches the average of the firs' expected return covariances, while the individual stock predictability vanishes as N gets large. To estiate the predictability of the index we odel Cov{E(R Z)} fro individual-stock regressions on lagged, fir-specific variables. We use Monte Carlo 2 Boudoukh and Richardson (1994) provide an overview of the statistical issues. Stabaugh (1999) and Ferson, Sarkissian and Siin (2003) provide ore recent analyses and references. 8

10 ethods to handle the statistical issues, as described in Appendix B. There is soe correlation between our fir-specific variables and the instruents selected in previous studies of aggregate predictability, so we are not copletely iune to data ining bias. However, our easure does not rely on the direct index predictability that so any previous studies have explored, and the nuber of studies that exaine individual-stock return predictability with tie-series regressions is still relatively sall. Using only fir-specific instruents we probably understate the correlations aong the expected returns. If we use arket-wide instruents for individual stocks, we are likely to overstate the correlations. Coparing the two cases we estiate a range of plausible values. 3. Epirical Results 3.1. Results using Weak-for Inforation Table 1 presents estiates of predictability based on the coparison of onthly and daily return variances. Appendix A describes the data. Panel A presents results for the Standard and Poor's index over different subsaples and Panel B suarizes the results for the individual coon stocks of twenty six large firs. Three estiators of predictability are shown. The estiator denoted as σ µ ) includes no ( adjustent for daily serial dependence, while the estiator σ µ ) ( adjusts for any first order autocorrelation, taking the view that daily serial correlation reflects icrostructure issues unrelated to changes in discount rates. The estiator # σ(µ ) takes the view that -- for individual stocks -- the ain icrostructure effect is the bid-ask bounce, which produces negative autocorrelation. Thus, negative autocorrelations are reoved each onth fro the predictability calculation while 9

11 no adjustent is ade for positive ones. (The average autocorrelation paraeter, θ, taken over the onths is positive for the Standard and Poor's 500 index, and negative for 16 of the 26 firs.) For coparison, the second colun contains the unconditional standard deviations of the onthly returns, expressed as annual percentages. 3 We first discuss the estiates in coluns 3-5, labeled GMM estiates. These are the three estiates with analytical adjustents for finite saple biases, as derived in Appendix B. 4 The estiates of predictability for the stock index range fro 1.9% to 5.4% using Schwert's (1990) data for the period. However, over the period where the CRSP daily data are available, the estiated volatility of the expected returns is 2% using either σ µ ) or ( # σ (µ ). After adjusting for positive serial dependence, which ay arise fro nonsynchronous trading of the stocks in the index, the estiator σ µ ) ( delivers a value of zero. Over the ost recent 120 onths of the saple all of the estiates of predictability are zero. We estiate the finite saple biases in the predictability estiates using siulations. The siulations also provide epirical p-values for assessing the statistical significance of the results. 5 The adjusted estiates are shown in coluns 3 The onthly variance is ultiplied by 12, then the square root of this result is ultiplied by 100. All of the nubers in the tables are annualized this way. 4 We also exaine results using the estiators without the analytical adjustents for finite saple biases. When we rely on the siulations described below to control the finite saple biases, the results are siilar. 5 We resaple fro the actual data for a given stock or index, randoly with replaceent. For each siulation trial we generate an artificial tie series with the sae nuber of daily observations as the original data series. The artificial data satisfy the null hypothesis that the expected return is constant. We copute the estiators on 10

12 6-8 of Table 1. They tell a siilar story. The estiates for the index range fro zero to 5.1% for the period, and the larger figures appear statistically significant. However, using CRSP data for , the estiates are 1.44% or less, and none are statistically significant. Over the last 120 onths the adjusted estiates are all equal to zero. Panel B suarizes results for a saple of 26 large firs' individual coon stocks. These cover the post-1962 period where CRSP daily data are available. Averaged across the stocks, the predictability estiates range fro zero to 2.2%, depending on the choice of estiator and saple period. The epirical p- values range fro 0.23 to 0.35, thus providing no evidence of predictability. Even the extree cases present no reliable evidence of predictability. Taking the stock with the axiu value of σ µ ), its epirical p-value is 0.07 or 0.08, depending on the ( subperiod. Accounting for the fact that this is the axiu of 26 cases, the results are insignificant. 6 the artificial saple in exactly the sae way as on the original saples. We repeat this for 1,000 trials. The average across the trials is the expected finite saple bias. We use the distribution of the siulated estiates to generate epirical p-values. These are the fraction of the siulations where the variance estiates are larger than the saple values. A sall p-value eans that the saple estiate is unlikely to occur if expected returns are constant. 6 Let u i be the epirical p-value for the i-th experient, i=1,...,n. Under the null hypothesis of no predictability the epirical p-value is uniforly distributed on [0,1]. Assuing independent experients, Prob{Min i u i q} = 1 - Prob{u i >q; i=1,...,n} = 1 - Π i Prob{u i >q} = 1 - (1-q) n. For q=0.07 and n=26, the probability of finding the iniu p-value to be 0.07 or 11

13 In suary, while the older historical data suggests econoically significant predictability in the arket index, there is little evidence of weak-for predictability in odern data. In particular, the evidence for the ost recent ten years suggests that any weak-for predictability in the index has vanished Sei-Strong For Tests Table 2 presents our estiates of predictability based on the covariances of individual stock regressions on lagged variables. The regressions use onthly data fro 1969 through 2001 and twenty large firs' coon stocks. We estiate predictability for an equally weighted portfolio. The data are described in Appendix A. The three rows for each saple period contain saple values of the iplied predictability using either all the eleents of the covariance atrix of the fitted expected returns, the off-diagonal eleents only, or the diagonal eleents only. Excluding the diagonals provides soe inforation on what would be expected to happen as the nuber of siilar stocks used in the calculation becoes large. Excluding the diagonals and using fir-specific regressors, the first colun of figures shows that the iplied predictability estiates fall in a narrow range, fro 2.37% to 2.67%, depending on the subperiod. saller in 26 trials is 85%. Of course, the trials are not independent, so the correct probability is soewhere between 7% and 85%. 7 Lo and MacKinlay (1999) present weak for tests with less evidence for predictability in ore recent data, and suggest that the deise of such predictability ay be related to "statistical arbitrage" trading by Wall Street firs. Nelson and Ki (1993) also find that weak-for evidence for stock index predictability is thin in post World War II data. 12

14 The regressions behind Table 2 are subject to statistical biases, which we control via siulation as discussed in Appendix B. The bias-adjusted estiates are suarized in the second colun of figures. Using only off-diagonal ters, the estiates are 1.63% to 1.84% annualized. Using the full covariance the figures are 2.41% to 2.59%. These estiates are statistically significant according to the epirical p-values. It is interesting that the easures of predictability based on the diagonals only, as suarized in the third row, behave differently fro estiates ephasizing the off diagonal ters. The diagonal-only estiates are nuerically larger but not significantly different fro zero; their right-tail p-values are larger than 20% in each experient. This reflects the relatively large sapling variability of the regression estiates of expected returns, copared with their relatively sall sapling covariability. This is one of the advantages of our approach, copared with previous studies that rely on direct regressions of stock indexes on lagged variables. Because the sapling covariability of the expected returns with fir-specific lagged variables is relatively sall, we are able to estiate the predictability with relatively high precision. Thus, based on the covariability, we can say that 1.8% to 2.6% is strongly statistically significant. The estiates of predictability in Table 2 ephasizing the off diagonals, are siilar whether we use the full saple or concentrate on subsaples of the last 120 onths or the ost recent 60 onths. 8 This is interesting in view of recent epirical 8 In additional experients not reported in the table we include an additional lagged predictor, a easure of the fir's dividend yield. The results with this additional regressor (which is not available for all of the firs) are very siilar to those in the table. 13

15 studies that find index predictability, easured directly using lagged variables, has weakened in recent saples. It ay be that the predictability was "real" when first publicized, but diinished as traders attepted to exploit it. 9 Alternatively, the predictability ay have been spurious in the first place, as a result of statistical biases and/or naive data ining. If the predictability is spurious we would expect lagged instruents to appear in the epirical literature, then fail to work with fresh data (e.g., Ferson, Sarkissian and Siin, 2003). But Table 2 presents no evidence that predictability is weaker in the recent subperiods. As our fir-instruent-only easures do not rely on aggregate predictor variables, Table 2 provides new and interesting evidence that the underlying predictability has not diinished. This is consistent with the "efficient arkets' view of predictability, as reflected in uch of the conditional asset pricing literature (see reviews by Ferson (1985) and Cochrane, 2001). According to this view returns ay be predictable if required expected returns vary over tie in association with changing risk or risk aversion. If required expected returns vary over tie there ay be no abnoral trading profits and thus, no incentive to exploit the predictability. Predictability ay therefore persist in an efficient arket. The two right-hand coluns of Table 2 show predictability estiates when each regression uses both fir-specific and arket index ("Macro") characteristics as the lagged regressors, or alternatively, when only the Macro variables are used. These calculations show the effects of using the econoy-wide lagged regressors. The estiated standard deviations of the onthly expected returns for the full 9 See Schwert (2003) for a recent review of this evidence. 14

16 saple period are between 4.2% and 8.6%, depending on whether or not we exclude the diagonals fro the calculation, and even higher when we focus on the diagonals alone. The econoy-wide regressors increase the covariability of the expected return estiates. The "Macro only" exaples in the far right colun suggest that predictability is diinished soewhat in the ore recent subperiods, which is consistent with the evidence cited earlier in studies that rely on econoy-wide regressors. Our sei-strong for estiates of predictability provide ore reliable evidence of tie-variation in onthly stock returns than our weak-for tests. This is expected if returns are ore easily predicted using ore inforation. But is 2-3% on an annual basis an econoically significant effect? The siple Gordon odel exaple fro the introduction provides an illustration. Consider a onth in which the required expected return jups by roughly two standard deviations, say fro 7% to 11%. Other things held fixed, the stock price would fall to half of its forer value in response. Of course, this overstates the effect, to the extent that a shock that changes the required return also changes the expected cash flows and future growth rates, but the exaple suggests the econoic significance of predictability. 4. Power and Robustness This section presents soe results on the statistical power of our ethods and on the robustness of our epirical findings. Concerns about power focus on the weak-for tests where we do not find significant predictability. Our approach contains several steps where approxiations are introduced or estiation is required. Each step is prone to soe error, and the cuulative effect of the errors ay result in low power. 15

17 To evaluate the power we use siulations that include each of the steps. We also copare various estiators to isolate the ipact of the different steps on power. In the siulations we generate data following the odel described in Section 2.1. The conditional eans fluctuate each onth as draws fro a noral distribution, whose standard deviation controls the aount of variation in the expected returns. Daily returns fluctuate randoly around the onthly eans, with variances chosen so that the first and second oents of the siulated returns atch the saple returns. By setting the expected return variation to equal zero, we get critical values for a 5% test, defined such that 5% of the siulations produce statistics larger than the critical value. Setting the expected return variation to larger values, we trace out the power curves illustrated in Figures 1 and 2. The power of a particular test is the fraction of the siulation trials that produce a statistic larger than the critical value, given that an alternative hypothesis with tie-variation in the expected returns generates the data. Figure 1 takes the variances of expected returns as the statistics. Figure 2 takes the epirical p-values of the expected return variances as the test statistics. The approach is the sae, except that, because the epirical p-value is the result of a siulation, we now have to conduct siulations within the siulations. Various estiators are displayed in figures 1 and 2. Auto is the saple autocorrelation statistic, and the others are the iplied expected return estiators with various adjustents. We use the saple autocorrelation statistic as a "straw an" for coparison, because ost of the statistics used in the literature on weakfor predictability are transforations of the saple autocorrelation (see, e.g. Cochrane, 1991). Of course, we expect the saple autocorrelation to perfor poorly 16

18 under our odel of the return dynaics, and the siulations bear this out. The power of the autocorrelation is nearly level at about 5%, independent of the aount of expected return variation. The estiator ABB refers to analytical and bootstrap bias adjustents, ABMB refers to analytical bias, icrostructure bias and bootstrap adjustents. ABMNB refers to analytical bias, icrostructure bias for negative autocorrelations only and bootstrap adjustents. Boot has only bootstrap bias adjustents and Unadj. refers to the estiators with no bias adjustents. The differences between the power of these estiators parse out the separate effects on power of the various steps in our estiation strategy. The figures show that the ost coplicated procedure, ABMB suffers lower power in soe parts of the curve, but otherwise the various estiators give siilar results. Figures 1 and 2 suggest that our weak for tests have the power to detect odest aounts of predictability. For exaple, if the annual standard deviation of the expected returns is 3%, the power of the various statistics is 14% to 19%. Using 15% as the standard deviation of the onthly index returns, an expected return standard deviation of 3% eans than a regression of the return on its expected return would produce an R-squared of only (0.03/0.15) 2 =4%. That is, the expected returns account for only 4% of the variance of the returns. With an annual standard deviation of 5%, the R-squared would be about 11%, which is siilar to the values reported in soe stock return regression studies. At this level the power of our tests is near 60%. At a standard deviation of 6% or ore for the expected returns, the power of our tests is 80% or higher. In tables 3 and 4 we explore the robustness of our epirical results. One of the questions to be further addressed relates to the treatent of the autocorrelations 17

19 of daily returns. Panel A of Table 3 presents results for the Standard and Poor's index, replacing the stock index with index futures prices. Following Boudoukh et al. (1993) and Blue, MacKinlay and Terker (1989), there should be no issues with icrostructure-related autocorrelation in the index futures returns. The daily index futures are available fro June of 1982 through Septeber of The table shows that our ain results are robust to the use of futures over this period, and over the ore recent ten-year subsaple. All of the point estiates of weak-for predictability are zero. We use large firs' coon stocks in the ain experients of tables 1 and 2, because such stocks should be representative of the arket. However, sall firs' stocks ay have ore serial correlation related to arket icrostructure effects, so the contrast between the two should provide soe indication as to the iportance of these effects. In addition, sall-fir stocks ay display ore predictability related to arket inefficiencies, as it is ore costly to trade the stocks of saller copanies to exploit inefficiencies. Panel B of Table 3 repeats the tests of Table 1, using 26 sall-fir stocks in place of the large-fir stocks. The returns data are available fro January of 1980 through Septeber of The panel shows that the average weak-for predictability estiates and the point estiates for soe of the sall firs are soewhat higher than we found for the large fir stocks. The average bias-adjusted estiates are 0.25% to 1.35% over the full saple, and 1.25% to 1.32% for the ost recent decade, depending on the estiator. The standard deviations of the onthly returns are also arkedly higher for saller firs. Thus, the average predictability estiate of 1.35% corresponds to only (.0135/.305) 2 = 0.2% of the variance explained 18

20 by the expected return. Given this low signal-to-noise ratio, the average right-tail p- values are all above 25%. The axiu values across the 26 firs see ore ipressive; for exaple, bias-adjusted predictability values of 16-17% are found for one fir in the ost recent decade. However, accounting for the ultiple coparisons, the chances of finding nubers this large are as high as 7.5% to 18.8%. 10 Table 4 presents the results of repeating the analysis of Table 2 using our saple of sall fir stocks. For coparability with Table 2, the first twenty stocks are used. The available saple covers the period. The results are very siilar to those of Table 2. Using fir-specific lagged predictor variables the biasadjusted estiates of sei-strong for predictability are between 1.78% and 3.62%, depending on how the diagonal eleents of the expected return covariance atrix are treated. The diagonal eleents iply ore predictability than the off-diagonal eleents, as we saw before, but the covariability is soewhat lower aong the sall stocks. With the lower sapling covariability, the diagonal-only estiates of predictability becoe statistically significant. Again, there is no evidence that seistrong for predictability has diinished in the ost recent decade. 5. Conclusions Sall changes in expected returns can produce large and econoically significant changes in asset values. This paper presents new estiates of tievariation in the expected returns of stocks, using indirect ethods. Weak-for tests 10 Following footnote 6, 1-( ) 26 =18.8% and 1-( ) 26 =7.5%. 19

21 find no reliable evidence of predictability in odern data. Sei-strong for tests find sall but econoically significant predictability. In contrast to recent studies that rely on aggregate predictor variables, we find no evidence that the predictability has diinished in recent saples. 1 The authors are grateful to Gurdip Bakshi, Hendrick Bessebinder, Charles Cao, John Cochrane, Pat Fishe, Bruce Grundy, Ravi Jagannathan, Herb Johnson, Avi Kaara, Terence Li, Stewart Mayhew, Sion Pak, Mark Rubinstein, Robert Stabaugh, and Willia Zieba for discussions and coents. The advice of the Editor, David Hsieh, and an anonyous referee were especially helpful. Su acknowledges financial support fro the Research Council at the University of Miai. Appendix A: Sei-strong Variables Our sei-strong for tests use data on lagged, fir-specific instruents and econoy-wide, acro instruents for the index. The fir-specific instruents are obtained fro CRSP and COMPUSTAT. They include for each stock onth, (1) the average return over the previous twelve onths; (2), the book-to-arket ratio, defined as the ost recently-available book value per share (quarterly Copustat data ite #60 divided by ite #61) divided by the one-onth-lagged price per share; and (3) the earnings-to-price ratio, defined as the trailing four quarters' earnings (Copustat data ite #11) divided by the one-onth lagged stock price. The stock prices and returns are fro CRSP. The acro instruents include: (1) the lagged three-onth Treasury Bill 20

22 secondary arket yield; (2) the lagged, one-onth holding period return on a three-onth Treasury Bill; (3) the spread between the Treasury Bill yield and the ten-year constant aturity Treasury Bond yield to aturity; and (4) the spread between Moody's Seasoned AAA and BAA corporate bond yields, to proxy for corporate default risk. All of the yield series are easured as the lagged onthly average of daily values, fro the Federal Reserve. In addition, we use the one-onth lagged holding period return on the Standard and Poor's Index, excluding dividends, and the lagged 12-onth holding period return on the S&P 500 Index, fro CRSP. We also use the dividend yield on the SPX index. The dividend yield is coputed as the trailing 12-onth dividends divided by the SPX index level. Monthly dividends are obtained fro Blooberg. Appendix B: Estiation Issues B.1 Weak-for Estiators In our weak-for tests the objective is to estiate: Var( µ ) = MV - D(ADV + 2 θ ), (B.1) where MV = E{(R -µ ) 2 }, ADV = E{E[( - µ /D) 2 j ]} and θ = E{ E[( j -µ /D)( j - j µ /D) j,j', j-j' =1] }. The finite saple variances, Mˆ V and  DV, and the onthly average of the daily saple autocorrelations, have expectations that differ fro the true values. Siple calculations show that 21

23 E( Mˆ V ) = MV - (D/M)(ADV + 2θ), (B.2) E( Â DV ) = [(D-1)/D] ADV - 2 θ /D, E(θˆ ) = θ - (ADV + 2 θ )/D, where M is the nuber of onths in the saple. The syste (A.2) provides three equations in three unknowns, and ay be solved for unbiased estiators of MV, ADV and θ. These are given as: θ = {D Â DV + D(D-1) θˆ }/{(D-1)(D-2) - 2}, (B.3) ADV = {2D θˆ + D(D-2) Â DV }/{(D-1)(D-2) - 2}, MV = Mˆ V + (D/M) ADV + (2D/M) θ. Using these estiators, we for the unbiased estiator of the variance of the onthly expected returns as: Var( µ ) = MV - D( ADV + θ 2 ). (B.4) B.2 Iposing Positivity Boudoukh, Richardson and Sith (1993) describe estiators for risk preius, iposing the restriction that the expected risk preiu is positive. Their tests involve the iniization of a quadratic for in the restricted and unrestricted estiates, which we adapt to the current setting as follows. Let φˆ be an N-vector of unrestricted estiates of Var( µ ) for N assets, whose saple values ay be 22

24 negative. Consider the estiator φ = Arg Min φ ( φ φ ˆ )'Cov( φˆ ) -1 ( φ φ ˆ ) subject to φ 0, (B.5) where the inequality in the constraint applies eleent-by-eleent. The Kuhn-Tucker conditions for this proble specify satisfy φ and an N-vector of ultipliers U = {u i } i, which ( φ φˆ )' Cov( φˆ )-1 - U = 0, (B.6) φ i u i = 0, i=1,...,n, φ i 0, i=1,...,n, and u i 0, i=1,...,n. We solve the syste (B.6) nuerically to obtain the restricted estiators, that when N=1 this aounts to setting negative. φ. Note φ i = 0 whenever the unrestricted value is B.3 Sei-strong For Estiates In our sei-strong for analysis, we use siulations based on a paraetric for of the bootstrap to control statistical biases in the predictive regressions. The following regression syste is estiated for each asset: R t+1 = µ + t+ 1 (B.7) Z t+1 = µ z + A (Z t - µ z ) + v t+1, 23

25 where A is an L x L atrix and there are L instruents in Z. The saple eans estiate µ and µ z, and the OLS coefficient estiates A. These estiates are taken as paraeters of the siulation. Then, we resaple fro the vector of residuals }, randoly with replaceent, and use these as the shocks in the { t+ 1, v t+ 1 siulation. We build up the tie series of Z t recursively in each siulated saple, along with the conteporaneous returns, which satisfy the null hypothesis of constant expected returns,µ. This approach preserves the first order autocorrelation of the instruents and accoodates the finite saple bias discussed by Stabaugh (1999), which arises when t+ 1and v t + 1are correlated. We regress R t+1 on Z t, using the siulated data saples, for each of the 1,000 siulation trials, and estiate the predictability exactly as in the original data. Since the covariance of the expected returns is zero, the true predictability is zero in the siulations. The average easured predictability, taken across the siulation trials, is our estiate of the bias, and the fraction of the siulations in which the actual data estiate exceeds the siulated value, is the epirical p-value. References Boudoukh, Jacob, and Matthew Richardson, "The Statistics of Long-horizon Regressions," Matheatical Finance 4 (1994), Boudoukh, Jacob, Matthew Richardson and To Sith, "Is the Ex ante Risk Preiu Always Positive? A New Approach to Testing Conditional Asset Pricing," Journal of Financial Econoics 34 (1993), Blue, Marshall, A.C. MacKinlay and Bruce Terker, "Order Ibalances and Stock Price Moveents on October 19 and 20, 1987," Journal of Finance 44 (1989),

26 Capbell, John Y., "A Variance Decoposition for Stock Returns," Econoic Journal, 101 (1991), Capbell, John Y. and Luis M. Viceira. "Who Should Buy Long-Ter Bonds?," Aerican Econoic Review, 91 (2001), Cochrane, John H., "Asset Pricing," Princeton, N.J: Princeton University Press (2001). Pp ISBN Cochrane, John H. "Volatility Tests And Efficient Markets: A Review Essay," Journal of Monetary Econoics, 27 (1991), Conrad, Jennifer and Gauta Kaul. "Tie-Variation In Expected Returns," Journal of Business, 61 (1988), Faa, Eugene F., "Efficient Capital Markets: A Review of Theory and Epirical Work," Journal of Finance 25 (1970), Faa, E. and K. French, "Peranent and Teporary Coponents of Stock Prices," Journal of Political Econoy 96 (1988), Faa, Eugene F., and Kenneth R. French, "Business Conditions and Expected Returns on Stocks and Bonds," Journal of Financial Econoics 25 (1989), Ferson, Wayne E., "Theory and Epirical Testing of Asset Pricing Models," Chapter 5 in Finance, Handbooks in Operations Research and Manageent Science, by Jarrow, Maksiovic and Zieba (editors), Elsevier, (1995), Ferson, Wayne E., Sergei Sarkissian and Tiothy Siin, "Spurious regressions in Financial Econoics?" Journal of Finance 58 (2003), Fleing, Jeff, Chris Kirby and Barbara Ostdiek, "The Econoic Value of Volatility Tiing," Journal of Finance 61 (2001), Foster, F. Douglas, To Sith and Robert E. Whaley, "Assessing Goodness-of-fit of Asset Pricing Models: The Distribution of the Maxial R-squared," Journal of Finance 52 (1997), Gordon, Myron 1962, "The Investent, Financing and Valuation of the Fir," Irwin, Hoewood, IL. 25

27 26 Hansen, Lars P., "Large Saple Properties of the Generalized Method of Moents Estiators," Econoetrica 50 (1982), Kandel, Shuel A., and Robert F. Stabaugh, " On the Predictability of Stock Returns: An Asset Allocation Perspective," Journal of Finance 51 (1996), Lo, Andrew and A. C. MacKinlay, "Stock Prices Do Not Follow Rando Walks: Evidence fro a Siple Specification Test," Review of Financial Studies 1 (1988). Lo, Andrew and A. C. MacKinlay, "Data Snooping Biases in Tests of Financial Models," Review of Financial Studies 3 (1990), Lo, Andrew and A. C. MacKinlay, "A Nonrando Walk Down Wall Street," Princeton University Press, Princeton, NJ (1999). Lo, Andrew and A. C. MacKinlay, "The Econoetrics of Financial Markets," Princeton University Press, Princeton, NJ (1997). Merton, Robert, "On Estiating the Expected Return of the Market," Journal of Financial Econoics 8 (1980), Nelson, Daniel B., "ARCH Models as Diffusion Approxiations," Journal of Econoetrics 45 (1990), Nelson, Daniel B., "Conditional Heteroskedasticity in Asset Returns: A New Approach," Econoetrica 59 (1991), Nelson, Daniel B., "Filtering and Forecasting with Misspecified ARCH Models I: Getting the Right Variance with the Wrong Model," Journal of Econoetrics 52 (1992), Nelson, C. and Myung J. Ki, "Predictable Stock Returns: The Role of Sall Saple Bias," Journal of Finance 48 (1993), Roll, Richard., "A Siple Iplicit Measure Of The Effective Bid-Ask Spread In An Efficient Market," Journal of Finance, 39 (1984), Schwert, G. W., "Indexes of US Coon Stock Prices," Journal of Business 63 (1990), Schwert, G. W., 2003, in George M. Constantinides, Milton Harris and Rene M. Stulz,

28 27 Editors: Handbook of the Econoics of Finance, Elsevier Science Publishers, North Holland, pp ISBN: Stabaugh, Robert F., "Predictive Regressions," Journal of Financial Econoics 54 (1999),

29 28 Table 1 Return Predictability Based on Weak-For Inforation The predictability easures are the standard deviation of the expected returns, in percent per onth, on an annualized basis. * The easures copare onthly unconditional return variances with average conditional variances, estiated fro daily data. Panel A presents results for the Standard and Poor's stock index return over different subperiods. Panel B contains siilar statistics for a saple of twenty six large firs' individual coon stocks. For individual stocks the average statistics (Avg) are shown, along with the cases that produce the largest (Max) and sallest (Min) estiate of expected return variation. Within each panel, the coluns present different estiators, and the onthly return standard deviations are included for coparison. The estiator σ (µ ) is the annualized standard deviation of the onthly expected returns, in percent. The estiator σ (µ ) * is adjusted to reove the effects of all first order serial dependence in daily returns. The estiator σ (µ ) # is adjusted to reove only the effects of negative first order serial dependence. The GMM estiates are found using the Generalized Method of Moents. The Finite Saple Adjusted estiates subtract a bootstrapped estiate of finite saple bias, under the null hypothesis of independent and identically distributed returns. Right-tail epirical p-values are on the second line of each case with bootstrapped finite saple adjustents Tie Monthly Return GMM Estiates: Finite Saple Adjusted: Period Standard Deviation σ ( µ ) σ ( µ ) * σ ( µ ) # σ ( µ ) σ ( µ ) * σ ( µ ) # Panel A: Standard and Poor's Stock Index Panel B: Twenty-six Large-capitalization Coon Stocks Avg Max Min

30 29 table 1, continued Tie Monthly Return GMM Estiates: Finite Saple Adjusted: Period Standard Deviation σ ( µ ) σ ( µ ) * σ ( µ ) # σ ( µ ) σ ( µ ) * σ ( µ ) # Panel B: Twenty-six Large-capitalization Coon Stocks, continued Avg Max Min * Annualized figures are derived by ultiplying the onthly decial variance estiate of the expected returns by 12, taking the square root and ultiplying the result by 100.

31 30 Table 2 Sei-strong For Predictability The iplied predictability is the annualized standard deviation of the tie-varying onthly expected returns, in percent, for an equally-weighted portfolio of 20 large coon stocks. The figures are estiated fro the covariance atrix of the individual stock regressions. The rows report alternative calculations where either all of the eleents of the covariance atrix are used, or when the diagonal and off-diagonal eleents are ephasized. The coluns report experients where different predictor variables are in the regressions: Either fir-specific variables only, firspecific and Macro variables, or Macro variables only. The Bias-adjusted estiates subtract a bootstrapped estiate of finite saple bias, under the null hypothesis of constant expected returns. Epirical right-tail p-values for this estiate are denoted as p-value Covariances Fir Variables Only Fir and Macro Macro Variables Used Iplied Bias-Adjusted P-value Variables Only Predictability Panel A: October, Deceber 2001 All eleents Off-diagonal Diagonal Panel B: January, Deceber, 2001 All eleents Off-diagonal Diagonal Panel C: January, Deceber, 2001 All eleents Off-diagonal Diagonal

32 31 Table 3 Robustness of Weak-for Predictability Results The predictability easures are the sae as in Table 1. Panel A presents results for Standard and Poor's stock index futures returns over different subperiods. Panel B contains siilar statistics for a saple of 26 sall firs' individual coon stocks Tie Monthly Return GMM Estiates: Finite Saple Adjusted: Period Standard Deviation σ ( µ ) σ ( µ ) * σ ( µ ) # σ ( µ ) σ ( µ ) * σ ( µ ) # Panel A: Stock Index Futures Panel B: Twenty-six Sall-capitalization Coon Stocks Avg Max Avg Max

33 32 Table 4 Sei-strong For Predictability of Sall-capitalization Stocks The iplied predictability is the annualized standard deviation of the tie-varying onthly expected returns, in percent, for an equally-weighted portfolio of 20 sall-fir coon stocks. The ethodology and sybology is otherwise the sae as in Table Covariances Fir Variables Only Fir and Macro Macro Used Iplied Bias-AdjustedP-value Variables Only Predictability All eleents Off-diagonal Diagonal All eleents Off-diagonal Diagonal