EtsEVtER Journal of Financial Economics 41(1997) 16% 203

Book-to-market, dividend yield, and expected market returns: A time-series analysis

S.P. Kothari, Jay Shanken*

WC find reliable evidence I~JI both book-to-market (9 Nt and dividend yield track time-series variation in expect& real stock returns over the priod IX!6 91 (in which B M is stronger) and the subperti 1941-91 (In which dividend yieid is stronger). A Bay&an bootsmap procedure implies hat an investor with prior belief 0.5 that expected re:ums on the equal-weigh’ed index are never negative comes away from the full-period B. hi evidencxz with posterior pro&hi@ O.oII for the hypothesistO.I4 with the impact of the I933 outher temperedI. Altnough this raises doubts about market c%cicnq. the post-t940 evidence is consr$tent with expected returns always heins positive.

There is considerable evidentle that stock returns are predictable. The aca- demic, literature on return predictability goes back at Ieast to the late 1970s.

*Corresponding author.

We thank Ray Bail. Peter Bossaerts. Stwe Caulej-. Gene Fama. K~I French (the r&met. Burton HoMieN John Long &iii Schwert (the editorl Rob Stambaugh. Jerr: Warner. Robert Whitelaw. and espectill~ Johnathan Lewellen. Aiso. thanks to participants in semtnars al NBER. the Southern Finance Association. Virginia T&, London Business School. Dartmouth. Washington University. Yale. the University of California at Irvine and Los Angelex Camhn~. and Clniversily of Rochester. We gratefully acknmvledgfinancial wpport from the John M. Ol*n Foundation and the Bradley Policy Research Center at the Simon School. University of Rochester. Assistant: from K Em Goetz and Jane Muellner on manuscript preparation is appreciated.

0304-80SXA7,Sl7.CIO 4‘ 1997 Elsevier Science S.A. All rights reserted PIf s0304-405x(97)00002-0

170 S. P. Kothari, J. Shanken /Journal of Financial Economics 44 (I 997) 169-203

Several studies (e.g., Fama and Schwert, 1977; Keim and Stambaugh, 19886; and Fama and French, 1988) regress returns on predetermined variabks, includ- ing past returns, to infer the existence of statistically significant time-variation in expected returns. Among the predetermined variables, many studies have found dividend yield to have predictive power in both cross-section (L&en- berger and Ramaswamy, 1979) and time series (Rozeg, 19@ Shifter, 19w, Fama and French, 1988a, 1989; and Campbell and Shilfer, 1988). Since the publication of Fama and French (1992, 1993), the book-to-market ratio (B/M) has emerged as a strong contender as a determinant of expected returns. Fama and French, Davis (MM), and Chan et al. (1995). among others+ provide evidence that B/M significantly explains cross-sectional variation in average returns, although Kothari et al. (1995) show that the effect is weaker in large firms (also see fought-an, 1996) and argue that the magnitude and signiGcance of the effect may be overstated due to data mining and a variety of selection biases in the COMPUSTAT data base (also see Breen and Korz,, czyk. 1994

The literature emphasizes two views of the predictive power of financial ratios that are relevant in both time series and cross-section. The first (e@cient markets) view has been referred to by Fama and French (1988b) as the ‘discount-rate effect (also see Keim and Stambaugh, 1986). Holding expected cash flows roughly constant, anything that incmases the rate at which cash Sows are discounted (e.g., changes in risk or liquidity) lowers market value and irtcmss both eqxcted return and financial ratios. This assumes+ ofcoum that the most rarmt value in the numerator (dividend or book value of equity) does an adequate job of controlling for expected future cash f?ow. An alternative (ineffi&nt markets) view is that financial ratios refkct the extent to which the market is overvalued (low ratios) or undervalued (high ratios) at a given point in time. fn the &se of underpricing. for example, future returns (and thus ?rue’ expected mums) will be high insofar as the undervaluation is likely to be corrected over the given horizon.*

In this research we evaluate the ability of an aggregate B/M ratio to track time-series variation in expected market index retunrs, and compare its forecast- ing ability to that of divrdend yield. We employ a vector-autoregressive (VAR) framework to evaluate the statistical signiRcance of the regression evidem~ of predictability of expected returns. The bootstrap simulation procedure is used to test the null hypothesis of no predictability as well as to assess the magnitude and economic significance of the relation between expected returns and its detetminants, such as B/M and dividend yield. Finally. we extend the traditional bootstrap procedure by developing a Rayesian-bootstrap simulation. With the

‘A third motivation for the cro6s-JeFtjo nal investigation of dbidcnd yield is taxes. This study. howvcr. does not focus on the tax motivation.

S.P. Ko~hwi, J. Sbanken/Joumai ojFinanciai Economics 44 (1997) 169--X83’ 171

exceptions of MacBcth and Emanuel (1993) and independent work by Pontiff and SchalI(1995), B/M has not been considered in time series and the relative abilities of B/M and dividend yield have not been evaluated.”

Our time-s&es investigation at the market level employs the Center for Research in Security Prices (CRSP) equal-weighted and value-weighted port- folio annual returns data. Real returns ate computed using the observed CPI inflation rate. Corresponding yields 4qual to dividends paid in the preceding year divided by the price level at the end of that year are also computed from CRSP data as in Mama and French (198&I). Time-series data on B/M are for the stocks included in calculating the Dow Jones IndustriaJ Average. Return and B/M data are availabic since 1926, permitting us to study B/M over a longer period than has been possible in the cross+eot ional literature. Use of the Dow B/M variable along with broader index returns is in the spirit of Fama and French (1989), who regress bond returns as well as stock index returns on both stock yields and bond-yW spreads, viewing the explanatory variables as proxies for the overall state of the aggregate economy.

Although the use of ordinary least squares (OLS) regressions to assess the ability d dividend yield rnd B/M to forecast one-year returns might appear to be straightforward time-series regressions of returns on beginning-of-period yield measures are subject to sma&sampk biases (Starnbaugh, 1986; Mankiw and Shapiro. 19%). The bias arises from the fact that the mgre&on disturbances are correlated with future vahrcs of the i&epe&ent variable. A large imzasc in price tends to product both iitrge positive returns and a ooatemporaneous d4mease in yield. Since the yield only gradually reverts to a normal tevsel, large returns tend to follow what. in same appear to be relatively high yields. This imparts an upward bias to the regression slope co&Gent. The bias is opposite to that which arises when the laggcd dependent variable is inch&d as a re- gressor, since the beginning-of-period yickl is highly negatively cornfated with the lagged return, In large samples, the sample mean of the yield measure converges to the true mean (under ftiriy general a-ssumptions), eliminating the source of the spurious correlation between yield and return.

-____ -_--~ zMac&eth and Emanuel(1993)prcscntsevidcmz that price-t&rook hassomsfomcastingabilityfor anodii @-year mums on the standard and poor’s IRdwrial index from I!246 to 1982 tt is not dear (to us) from the discussion in tFie paper. however, wh&cr TiK p&2 and book valuer simulated under the null hypothesii were t&ted in a manner similar tv tha4 obsmvd in the haiotil data. If ov(, there is some question as to the reliability of the rest&a We hare similar conarns about the hvvtstmp pra.tculu~r vf Gvetzmann and Jorion (1993), in which them ix no relation betwcm raodvti Etums and tbe (actual) dividends used in computing divihttd yield. It is natural, uttdcr the null hyphcsb, to have returns iodepcnknt d past divider&, but returns sbotdd anticpate futorr: dividends in an e&ient market. even if expected returns are constant (e.& see the evidence in Kothari and !Shankeu, 1992L The VAR framework discussed below allurn this titure ofthe data to be relleucd in the simulated modd.

172 S.P. Korhari. J. Shanken/Joumal of Financial Economics 44 (1997) 169-203

Since large-sample inference is suspect in this context, we resort to simulation methods to evaluate the statistical significance of the regression evidence on return predictability, as in recent studies by Hodrick (1992), Goetzmann and Jorion (1993), and Nelson and Kim (1993). In particular, we employ a VAR framework to simulate an empirical distribution for the OLS slope coefficient. As previous authors have noted, the VAR simulation approach is appealing in that it explicitly models the process relating regression disturbances to future values of the independent variable. Moreover, the complications intro- duced by using overlapping multiyear returns (the usual standard errors are too low) can be addressed, along with conditional heteroskedastitiiy of the disturbances. The existing simulation evidence indicates that conventional asymptotic methods are not reliable, even in the nonoverlapping return appiiea- tions considered here.

Thus far, the simulation literature has focused on testing the point null hypothesis of no pfCdiCtdbi!ity in expected returns, i.e., a slope coe&ieut of zero in the rrtum t-egres+,n While this is ,L natural place to start, ;& wouid be useful to be abie to say something more about the magnitude and economic sign& cance of the true cmeBicht if the null hypothesis is rejezted. We explore the issue of economic signibcance by evaluating the power function of the test in simula- tions. This permits a formal test of the composite null hypothesis that the slope co&e&t is less than some prespeeified value. Fama and French (1988a) provide some indication of economic sig&cance and the impaet of bias by looking at the out-of-sample forecasting performance of their estimated re- turn/yield relation Degrees of freedom are a scarce resource when using mubi- period returns. however, limiting the usefulness of this approach. Hodriek (1992) also simulates a particular a&native to explore the power of his tests

In addition+ we deveLop a Bayesian-bootstrap simulation procedure that estimates the likeli~ of d&rent slope values, conditional on the historical OLS estimate. The information contained in the likelihood function nieeiy comptements that conveyed by conventional pvalue analysis and provides a richer framework in which to assess economic significance. Moreover, tire Etayesian interpretation of the likelihood function allows us to easily incorporate prior belie& about phursible expected return variation and to explore the impact of different views of market et%iency on the conclusions drawn from the given sample evidence.

Overall, we find evidence of economically and statistically significant vari- ation in one-year returns forecast by the B/M ratio over the 192691 period. Tbe slope codficients, especially for the equal-weighted index, are so large as to suggest that expeeted real returns are sometimes negative, raising doubts about market elBciency. The evidence for dividend yield is comparable ‘for the value- weighted index returns and weaker for the equal-weighted index. In the 1941-91 subperiod, both B/M and dividend yield track expected return variation and the evidence is eon&tent with positive expected returns. The B/M and dividend

S. P. Kothari. J. Shanken 2 Jownal of Financial Economics 44 /1997) 169 203 173

yield results are comparable for the equal-weighted index, but the dividend yield evidence is stronger for the value-weighted index in the subperiod.

We report descriptive statistics in Section 2 and initial regression results in 8ection 3. Section 4 describes our bootstrap simulation methodology, while Section 5 describes the computation of bootstrap p-values and likelihoods based on the simulations. Section 6 d&cusses the robe of prior information about expected returns in the context of testing the sensitivity of expected returns to determinants like B/M and dividend yield. Bootstrap simulation results are presented in Sections 7 and 8. Section 7 focuses on the magnitude of the slope coe@cients on the finaucial ratios in the expected return refation, while Section 8 provides tests of I?K hypothesis that expected returns are always nonnegative. section 9 summarizes our main findings

This section begins with a brief description of the data ~sI, fohowed by ‘descriptive statistics for the time series of B/M. dividend yield, and market index returns To ensure that the B/M ratio is known to investors at the beginning of the return year, we measure annual returns from April of year t to March of year r~1aad~thesefetumsontheratioofbookvaluefor~eart-ltopricc at the end of Match of year t. The corresponding dividend yiefd equals dividends paid from April of year t - 1 tohlarchofyearrdividedbypriceattheendof March of year r. These tlttios and returns an t&erred to as the ‘year r’ observations

The hook value ‘per share’ that corresponds to the Dow Jones industrial Average is published in The Valve Line fnmamenf +xy. Time series data are available since 1920. Book value data for 1920-35 are based on estimates by The V&e Line. The book value for year t - 1 is deflated by the Dow Jones industrial Average at the end of March of year t + 1 to obt&n the B/M ratio, which is used to forecast returns for year t.

We begin by presenting some descriptive statistics in Table I for returns, B,IM, and dividend yield over the sample Period 1926-9! and subperiod 3941-91. Retcm and dividend yield information is for both equal- and value-weighted CJSP indexes. All three series are positively skew& with the equal-weighted returns being the most highly skewed. For the whole period, the average real return on the equal-weighted (value-weightad) index is 15.3% (9.4%) with standard deviation 38.8% (23.9%). The minimum is, - 61% ( - 48.4%) in 1937 and the maximum is 214% (85.7%) in 1933. The returns on both indexes are lower and less vohtile in the frost-1940 subperiod.

The mean of the B/M series for 1926,-91 is 70% with standard deviation 23%. The minimum is 27% for 1929 and the maximum is 148% for 1933, the same year in which returns achieve their maximum for both indexes. The mean of the

Wpfi

ve

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tics

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ual-w

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Mean

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ness

dv

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38

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9.4

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1.42

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- 48

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85.7

27.2

52.8

66.5

80.5

147.

7 i

I.41

2.71

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4.31

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2 2.6

6 3.3

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1 5.0

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Pane

l B:

Pat

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n (%

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Value

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(%)

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9 42

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.0 81

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ld (%

I 3.6

7 1.3

5 0.9

4 I.8

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cightc

d div

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ld W

) 4.2

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-----

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April

of

year

t to

March

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year

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I to

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year

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Book

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the

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of

the

book

va

lue

per

share

at

the

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of ye

ar I

- I

of the

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in the

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age

to the

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nes

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age

at the

en

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year

I. $

Equa

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paid

from

April

of

year

t -

I IO

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r inv

ested

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2

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cq

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for

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year

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Ma

rch

3 I

of ye

ar 1.

2 P Va

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eight

ed

divide

nd

yield

is the

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tio

ofdivi

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ta Ma

rch

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o the

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S.P. Kotltari, J. ShankenlJonmal of Financial Economics 44 (1999) I@-201 175

equal-weighted dividend yield series is 3.65% with standard deviation 1.44%. The minimum is 1.41% for 1934 and the maximum is 8.21% for 1938. Dividend yields for the value-weighted index are higher with a mean of 4.40%. AU ratios are Iess volatile over the post-1940 subperiod.

Both financial ratios are highly autocorrelated, with 1926-91 first-order autocorrel?tions of 0.68 for B/M and 0.56 (0.55) for the equal-weighted (value- weighted) index dividend yield. We cannot reject the null hypothesis that each series is generated by a first-order autoregressive process. The correlation between B/M and equal-weighted (value-weighted) dividend yield is 0.42 (0.66). Tbe higher comlatioll with the value-weighted yield is reasonable, since the B/M ratio is for the relatively large firms in the Dow index. The financial ratio autocorrelations are all higher in the post-1940 perid close to 0.80. Ako, B/M is lO-15% more highly correlated with each dividend yield series.

3OLSTt?pSfhrrafts

This section summarizes OLS regression results, dckrring an assessment of small-sample biases to later slctions. Table 2 presents the results of regressing annual returns on B/M (panel A) or dividend yiekl (panel 8). Tbe table reports rest&s using both equal- and value-weighted index returns tar the entire sample period 1926-91. We also explore t+?e sensitivity of resuJt.s to tbe 1933 outiier by running tbe regres&ons with rhL: outlier deleted or ‘trunca&&. In the latter case, the regressions are run as if the extreme 1933 values for rez~ra (214% equal- weighted, 86% value-weighted) and B/M (148%) had been the next highest SampIe values (129% equal-weighted retura, 74% VahJe-weighted return and 119% B/M). Sinoe systematic deletion of the single observation most supportive of a mtive regression slope is obviously biased against such a finding, trunc- ation is adopted as a man neutral means ‘of evaluating sensitivity. This approach is in the spirit of the statistical technique known as Winsorization (see Dixon and Massey, 1969).

Tbe standard errors and r-statistics (except White t) aie the usual OU quantities. The B/M and dividend yield variabies are scaied by ttir respective standard deviitions so that the dopes can be interpreted as the change in expected return for a one-standard deviation change in, the values of the financial ratios This he@ in assessing the economic significance of a given beta and facilitates the formulation of o&s prior beliefs. Thr full-sample (‘All’) slope estimates in Table 2 are all large in economic terms and most ate more than two or three standard errors above zero in these simpie regressions.

The 192691 slope coe&ient on B/M using the equaCweighted index is 20.2%. with OLS standard error 4.1% and t-statistic 4.87 (pvalue O.OQO). Tbe adjusted R2 of nearly 26% is quite large relative to past findings for l-year returns, and the estimated slope is enormous. If B/M increases from its historical

176 S.P. Korhari, J. ShankeniJournal qfl;inoncial Economccs 44 (1497) 169.-203

S.P. Korhari. J. ShankenlJoumal of Financial Ectmomics 44 (1997) 169- 203 f77

average of 70% by one standard deviation (23%) the expected real return, which averages 15.3% is forecast to rise by 20.2 percentage points As we shall see later, adjustments for small-sample bias do not materially alter these con- clusions. Deleting the 1933 outlier cuts the B/M coefhcient in half, but the estimate is still very large and highly statistically significant. Truncation also has a substantud, but smaller. efkct. Turning to the regression results for the value-weighted index reveafs that the Dow B/M ratio explains a much smaller fraction of the time-series variation in these returns than for the CBSP equal- weighted index.

Dividenf? yieM, in contrast, is somewhat better at explaining variation in returns on the value-weighted index than on the equal-weighted index. For the full sample, dividend yield explains less of the variation in value-weighted in&x returns than does B/M, but explanatory power is similar with deletion or truncation. Although equal-weighted dividend yield steadily incrzases from 1929 to 1932, it drops nearly by half, to 3.44%. at the beginning of 1933. fience, deleting the 1933 observation actually strengthens the telation between equal- weighted returns and dividend yiefd. Although value-weighted dividend yield also begins to drop, at 694% it is still well above its mean 4.40% at the beginning of 1933 and hence oeietion weakens the value-weighted rdation.

The slightly better abiJity of dividend yield to explain value-weighted index rctumslnayberrlatcdtothttactt:latalarl4efractiaaoitbcfirmsintbeindexdo not pay dividends Thedone, these firms’ dividend yields cannot refkct in expected market return. Since firms that do not pay dividends are, on average, smaller in market capilaiization, they have a relatively small efkt on the dividend yield of the value-weighted index.

Multiple regressions of returns on B/M and dividend yiekl over the 192691 period, shown in Table 3, teU essentiahy the same story as the univariate regre&ons B/M dominates dividend yield in explaining equal-weighted index returns B/M also outperforms dividend yield for value-weighted ir&x returns when the 1933 outlier is inch&d, but otherwise the explanatory power of the variables is similar.

Given the concerns about small-sample biases mentioned earher, we can hardly have much confidena in the White (19f8tJt asymptotic adjustment for heteroskedasticity shown in Table 2. We use it as a heuristic for (we hope) identifying situations in which results may be particnfarly sensitive to he&m- skedasticity problems or other misspecifkations. This appears to be the case for the full-sample equal-weighted index regressions, in which the White r-statistic is about 45% smaller (larger) than the usual OLS t-statistic for the B/M (dividend yield) slope. Exploratory regzssions (not reported) of squared or absolute return residuals on B/M suggest a signifkantly positive conditional variance relation, whereas the refation for dividend yield is negative and statistically insignificant. We examine weighted least squares and nonlinear regression specikations for B/M, w*ith little impact on our basic conclusions. There is no

Table

3

OLS

muitip

ic re

gres

sion

msuit

s: 19

26

91

R I*,

=

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25.0

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I,99

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115

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0.65

10.3

q 19

33

trunc

ated

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2.41

14.9

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2

1933

tru

ncate

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45

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urn

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of div

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s fro

m Ap

ril

of ye

ar t

to Ma

rch

of ye

ar t

+ 1.

i:

R/M,

is

the

ratio

of

the b

ook

value

pe

r sh

are

at the

end

of

year

’ -

1 ot

the

stock

s in

the

Dow

Jone

s ind

ustri

al Av

erag

e to

the

Dow

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ustrL

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erag

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3

at the

en

d of

March

of

year

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M,

ir de

flate

d by

its

sta

ndar

d de

viatio

n ov

er the

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180 S.P. Koihari. J. Shanken Journal of Financial Economicr 44 (1991) 169 203

significant association between B/M and squared or absolute residuals (t- statistics less than 0.5) with the 1933 outlier deleted, and both methods of treating the outlier greatly reduce the huge gap bktween conventional and White r-statistics.

Table4contains regression results for the 1941--91 subperiod. The motivation for examining this subperiod is that, as noted by Officer (1973) and !Schwert (1989). stock return variability was unusually high during the 1929-39 Great Depression. Dividend yield modestly outperforms B/M over the subperiod with equal-weighted index returns and substantially outperforms R/M with value weighted returns. Analysis of post- 1962 data reveals that dividend yield tracks variation in expected returns through time. while the evidence for B/M is inconclusive. Our full-period and subperiod results are similar to those reported by Pontiff and Schall(l995). who also include default and term premiums and three-month T-bill yield in their regression model and whose subperiod begins in 1959.

O..eral;. WC find +r;!i bc+b B/hi and Gvidend yield appear tc track variation in market index returns, but full- and subperiod analyses suggest that neither variable consistently dominates the other in its explanatory power. Apart from the dramatic full-period, equal-weighted index (BjM case), the OLS slope estimates range from 4.4% to 9.3%. still large in economic terms. These values of beta imply that a one-standard-deviation change in tbe value of B/N or dividend yield is associated with a shift of about 49% in the expected real return on the index.

We now consider simulation methods to examine whether the condusions above are materially afTected by small-sample biases. Like Nelson lurd Kim ( 1993). we posit a simple VAR model of returns and determinants of expected returns. In this model. the expected value at time t of the return realized at t + I, r, + t, is a Iinear function of the variable X, (dividend yield or B/M), which follows a simple autoregmssive process:

rt + I = x i- fix, + u, + l; u * i.i.d. (0. & (1)

-‘It+1 =I’+~xr+rr+~;r W i.i.d. (0. tr:). (21

where u and c are white noise error terms with contemporaneous covariance a,.. The parameters in Eqs. (1) and (2) are estimated for annual real returns data by OLS.

Stambaugh (1986) shows that the bias in the OLS estimate b of fl in Eq. (I) is

w - 8) = &+/u,!)E(p - 4, (3)

SP. Korhari. J. Shankm/Jownal of Financial Economics 44 (IQ971 MY--203

where p is the OLS estimate of 4 in Eq. (2). As expected, the residuals in the return regression equation are strongly negatively contemporaneously corre- lated with the residuals in the financial ratio autoregression (2). For the 1926-9 I period, the correlation is - 0.80 in the case of B/M and - O-65 ( - 0.79) for equal-weighted (value-weighted) dividend yield. Over tbe post-1940 period, residuals in the two dividend yield VAR’s an more strongly contemporaneously correlate4about -O.%Sineach~The~ti~correlationisiMtucedbythe fact that price increases tend to reduce the financial ratios wbik resulting in positive returns. Thus, by (3), tbe OLS slope estimates are biased upwards.

Combining (3) with the fact that the bias in p is approximately - (1 + 34) divided by tbe sampk size T (Kendall 1954) suggests the following bias- adjusted estimator of 8:

b A = b + &r/&l + 3pA)/7-. (4

where s, and s,” are sample moments of the OLS residuals from Eqs. ( I ) and (2) and PA = (rp + l)/(r - 3) is a bias-adjusted estimator for 9. After computing the bias-adjusted estimates hA and PA, the intercepts are specitkd by setting tbe avm fitted vahres in (1) and (2) equal to the sampk means. Corresponding adjust& (~rf residual pairs are tben obtained from (It and (2), with each rtsiciual taxies averaging to zero over time. Tbeae residuals are used in the bootstrap shdatiom which we describe below.

The system is started up with the historical value for x at tbe heginning of the sampk lreriod. Given a hypothetical value of /3 and substituting tbe estimator pAforAtimetieriesofreturnsaadeewvalucsoCxanthengatcratedfnmr(l) and (2) by randomly selecting adjusted (a. r) residual pairs with mt. Finally, tbe simulated time series of returns, which has the same length as the original data used to estimate Eq. (1 L is regressed on tbe simulated time series of X*S. The estimated slope from this regmssion is saved. This procedure is repeati 2StMl times with the hypothetical @value fixed and each time, starting tbe system with the historical vale: for .Y at the beginning of tbe sampk period- This yields an empirical distribution of slope estimates ihat is employ& in a variety of’ inkrence procedures3

The entire process (2500 iterations) is repeated for each of a discrete set of beta values starting from zero. Thus, we have a differcat empirical distribution of slope estimates for each hypothetical alternative. Wc focus on ‘nonnegative slope

‘Since the distribution of the r&da& is fiti i~foss simulatioas. as the Iruc slope c&kkttt vat&% cbestrnittatalvaiianceofmunts inarilscr wil tht prvdkfahk var%atim k~ mturns Ifanything, this wordd appear to make it more diikuh to lad eAknce of strottg ptcdrctability aad thus might impart a eonsl?lvatiye hias to our lesub Likewise, with rqwd to the mx! of bias-adjusted rcsidualr It would be interesting to exphzmz the sensitivity of inferences to altemak pmaxlurcs which attempt to maintain a fixed total variability of mums.

I82 S.P. &&an’, J. Shankn/Joumal of Financial Economics 44 (1997) 169-203

values since the two prominent competing hypotheses concerning predictability, which are market inetliciency (temporary mispricing) and time-variation in equilibrium expected returns (and the associated discount-rate effect on market value), both suggest a positive relation between returns and the financial ratios. The set of beta values considered allows for the possibility that expected returns are negative when the market is overvalued; the maximum beta is speciRed so as to generate a minimum expected return equal to the negative of the sample mean return. This specification of the domain of the prior distribution is not ‘pure’ in that it makes some use of the realized data. Obviously, other ranges could be considered, but some heuristic is needed to limit the analysis and this one strikes us as reasonable.

To test +he nuil hypo?hesis that d eqL ‘1s zero. we look at the proportion of the 256~ simuhued siope esumates, generated under the /? = 0 hypothesis, that exceed the historical slope. The latter is the estimate obtained from the actual data. This gives us an empirical pvalue for the historical slope. The procedure is similar to that employed by Nelson and Kim (1993), except for our use of bias-adjusted estimates and residuals. and two other diR&ncesz they randomly select an initial start up value from the unconditional distribution of x, rather than using the initial historical value (our approach is motivated by the oondi- tionality prineipk of i&renee (Rerger, 19sS. Section 1.6)). and they sample without replaccmmt while we sample with replaaement.

If the null hypothesis j3 = 0 is reju%d, it is of further interest to see whether valuesofBuptosome~ven~tive~canbe~edasw~l.Totestthis composite null hypothesis, we look at the empirical distribution of estimates generated under the assumption that /I eqttab the preapccifiecl Iewd. Again, we compute the proportion of estimates that exceed the historical B estimate and report tbis proportion as an empirical gvalut for the composite bypotbesis. Since larger slope estimates should be generated when the true siope is larger, we expect to see the empirical pvabres increase (weakerrejectiorl)asthe~- ftedlevelof~imzases. Note that tbe pvalue for testing the composite null hypothesis that 6 is less than or equal to a given value is just the Rower function evaluated at that value of betrm, for a test with size (type I error probability) equal to tire pvalue for fJ = 0. Shenken (1987a)develops a similar approach to testing a composite hypothesis of approximate porHolio efRciency.

It is wet1 known (but sometimes forgotten) that the p-value is not the probability that the given null hypothesis is true (see Lindley, 1957). Rather, it is just one ‘metric’ for assessing the extent to which the sample evidence hAcs unusual under the null hypothesis. Spt&ically, it is the probability under the null of obtaining a historical slope as big as (or bigger than) that observed. The

S.P. Kothari, J. ShkenjJounaoI of Financid Economics 44 (1997) 169-203 183

distribution of the sample statistic under the alternative plays no role in this context.

In order to obtain probabiMes for the diRerent hypotheses, conditional on tbe historical evidence, the Bayesian approach ,focuses on the %kelihood func- tion’, in conjunction with prior (subjective) probabilities for the hypotheses. With a continuous density function for the sample evidemz (here, the historical slope), the relative likelihood of one hypothesis as compared to aDOther can Ioosey be described as follows: it is the ratio of the respective probabilities (under the different h.qotheses or models) of generating sample evidence within an infinitesimal aeigh5orhood of the historical slope.

If the VAR disturbances in Eqs (I) and (2) are assumed to be jointly normally distributed, a dosed-form expression for the iikelihood function can be obtained (see Kanded and Stambaugh, 19!&). Here, we take a d&rent approach, intro- ducing what we call B bootstrap likelihood, is, the proportion of simulated samples (under a given hypothesis) for which the observed slope is in some fixed neighborhood of the historical slope Although lacking the mathematical el- egance of more formal Bavesian approack our strategy, like the ‘da&al bodstrap, is simple and pragmatic, and does not requi= much in the way of assumptions.* For example, the b&trap procedure aI- for the positive skewness that we sometimes observe in the VAR residuala A similar approach c&d be adapted to a variety of situations that an commonly viewed as analyticaNy intractable an& the&~ not BmenaMe to Bayesian analysis. While there is a literature on nonparametric density estimation, we are not aware of any pqxm that have used bootstrapping techniq~ in the manner dthis paper.

Bootstrap likelihoods arr: computed for the same set offi values cons&r4 in the class&d p-value analysis. These proportions are scakd so that the sum of the likelihoods over all values of lp conside& is one. As in any Bayesian application, scaling does not a&ct the relative tikdihoods or the associated posterior probabilities derived from the likelihoods and a given prior. Moreover, the scaled proportions can be interpreted directly as posterior probabilities for the diffireat slope values when prior belie& arc uniformly distributed over the various slopes. Although we do not claim that this is the ‘right’ prior, it provides a convenient &&ace point for inteqxeting the data. Addirionally, if the likelihood is (approximately) zero for the largest slope consider4 posterior probabiirties would be unaffected by expanding the uniform prior to larger values of beta. This is the case in mauy of our simuiatious.

*One sboukJ OO( push this point loo iu. ils any boo~rap tcdmique is limitd by the inf’ormativc- mfss oftbe obseme4l sample fmu which the simulations IVC dcvhpcd Also. in princ+ple we wouid waattocoaditionona~~esetof~tslabisriFslurthe~~~asouroimpMied procedure may entail wme loss of information, although it is easily exterded to acwotmodate amditioning on several statistics

184 S.P. Kothari. J. Shanken/Journal of Finuncial Economirl 44 (1997) 169 203

We have explored the impact of varying the width of the neighborhood of the historical slope used in computing the likelihoods. The sensitivity‘ to this alteration is minimal (usually identical to two decimals). This robustness is reassuring for our approach and presumably reflects the fact that the true density function is approximately linear within the small neighborhoods considered.

In this section we consider the role of prior information about expected returns and the implications of this information for hypotheses about /I. In our linear regression framework, the implied expected return or conditional mean for a given level of the financial ratio, X. depends on both regression parameters, z and B. Intuitively, WC tend to think about priors on these parameters by s:nr?Ing t. i? h prior +I:&- “I!VX* :he gb.leral level of expected returns and x, as well as the extent of variation in expected returns for given movements in .x. The latter is our prior on fl. The prior for x is then implied by these more funda- mental beliefs about the level and variability of expected returns.

In some Rayesian applications. researchers impose so-&led ‘diffuse* or ‘non- informative’ priors intended to represent initial ignorana about the parameters of a model; See Section 3.3 of Berger (1985) for a general introduction and DeGroot (1982) for an insightful critique of the use of diffuse priors. This attempt to achieve *objectivity’ avoids the often difficult, but rewarding activity of assessing the economic (or other) meaning of alternative parameter values and structuring the prior beliefs accordingly. Here, we take the view that the prior distribution is a valuable tool for exploring the economic signibcance of the likelihood function and that it is important to consider a range of inforrna- tive prior b&is rather than some abstract notion of the ‘right’ prior. While this will entail some ad hoc assumptions. we believe that the insights obtained justify the structure adopted. Shanken (1987b) entertains a similar Bayesian approach to testing portfolio efbciency. The limitations of diffuse priors in this context are clarified in Kandel et al. (1995L5

Recall that z is speciIied as P - /R in our simulations. Given ( 1 k this entails an assumption that the sun& mean, $ of the return surprises is zero. This is a natural simptifying assumption when the focus is on the slope estimator and its sampling distribution. The condition has important consequences for analysis of the lert# of expected returns. however. As we now argue, the assumption about Ii amounts to an indirect method of characterizing beliefs about 2.

‘See Ma*Owitiz and Usmen (1996) fur additional discussion of thee issues in another financiaf

context.

S.P. Kothari. J. Shank IJownal of Financial Economics 44 (1997) 169~ 203 185

Given a value of /? and the sample observations w and F, the condition ii = 0 amounts to an assertion that one’s initial belief about x is consistent with fi and the r&tive values of the mean outcomes. in contrast, since P = x + /?K + g, a belief that ‘ii > 0 (Ti < 0) implies that, for the given 8. the observed mean return, 7, is higher (lower) than would have been expected, given one’s prior on z and the sample mean K. In this way, we can think of assertions about pi as implicit statements about one’s conditional prior for ~1, given p.

This turns out to be a particularly convenient perspective, given our simula- tion methodology and the hypotheses of interest. For example. individuab with strong priors that expected return variation is an equilibrium response to changes in risk would likely have strong priors that expected real stock returns are always positivc6 On the other hand, one often sees statements in the financial press, and sometimes in the academic literature, that the market is substantiafly ‘overvalued’. Such individuals would presumably have strong priors that expected returns are sometimes negative. Given the apparent diver- sity of prior beliefs about this issue, it is interesting to investigate the conclusions that wouM be drawn from the given empirical evi&nce for a range of beliefs

Specifying x as we have simplifies the analys& in that it allows us to formulate the hypothesis of nonnegative expected returns in terms of an upper bound on 6. To see this, substitute T - J? K for z in the expected return expression, x + /Xx. to get iF + p(x - iii). Assuming /Y ; 0. the minimum expected return in a given sample period is F + j?(xni, -a) getting this expression equal to zero and solving lor p yields #I0 = F,!(X - x,,, ). Thedote, the hypothesis that expected returns are nonnegative during the given period is equivalent to j? s /&,. Fur- thermore, the expected return coming to a slope of 2&, is - 7. IMore generally. ifii#O, then (F-@-bg-r hq tbe analysis is modified by substituting F - t for F in the expresstons for &,. Except where noted otherwise. we assume that 0 6 0 in the analysis beiow.

A limitation of this approach is that it treats & as if it were an exogenous constant. A fuller treatment of the endogeneity ofexpected returns in a dynamic context is ieft to future research.

7. Tea!aforexpeeWlremlretnmnriatfen

This section presents bootstrap evidence for the regressions of real equai- weighted and value-weighted index returns on B;M and dividend yield for the entire period, 1926-91. and the 1941-91 and 1963-91 subperiods.

bi: is possible to devise models in which the expected real market pwWwn is negative IBoudoukh et al., 1996). Since real T-bill returns excccdcd 4.5% in the deflationary period of the late ISZOS and early 1930s. however. the negative premium required to make total exptxted real return acgatiw durirtg that period wouid have to bc quite large.

186 S.P. Kothari. J. Shanken/Jounzal of Financiul Economics 44 (1997) 169-203

Table 5 Regmsion of one-year CRSP equal-weighted index real returns on book-to-market or dividend yield: 1926-1991

__ ____ _c_I__-- -~_..-.--- -.-- .-- .-“.-_..----_---.----_._--______

f3 % Mean h(%) p-value LKLHD Mean h(%) pvalue LKLHD ---- ~-

0 1.88 0.001 I 2.77 0.002

2 3.83 0.004 3 4.91 0.002 4 5.91 0.007 5 6.83 0.004 6 7.85 O.OlO 7 8.79 0.014 8 9.77 0.022 9 10.81 O.OF

10 11.80 0.040 11 \- -‘lr \l.< - 12 13.84 0.075 13 14.64 0.103 14 15.81 0.149 15 16.80 16 17.77 0257 17 _ 18 -.

19 -

Book-to-market tesults Dividend yield results ~---

0.001 1.23 0.127 0.041 o.m3 2.20 a167 0.047 0.004 3.32 0.213 0.067 0.003 4.37 0.279 0.075 O.&-i8 5.39 0.352 0.084 0.010 6.27 0.426 0.095 0.012 7.26 0.518 0.100 0.017 8.28 0.617 0.101 0.03 1 9.31 a692 0.092 p-036 10.24 0.77 I OB80 (?NG 11.10 0.8-36 0.069 O.W+ : 2.43 0.902 0.051 0.089 13.29 0.937 0.036 0.W 14.24 0.955 0.025 0.148 15.23 0.974 0.016 0.198 16.49 0.9R8 a009 0.227 17.26 0.990 a005

18.25 0.996 o.aI3 19.30 0.998 0.002

- 20.36 0.999 a001

S.P. Korhari, J. Shanken/Jmmal of Financial Ecpunics 44 (1997) 169-203 187

the minimum expected return is zero. This ensures that the lowest implied expected return over 1926-91 is the negative of the sample mean return or - 1X3%, as discussed above.

‘The bootstrap standard error is 4.3% under the null hypothesis that fl = 0, and it varies very little as the true value of /? is altered in the simulations. It is higher than, but close to, tbe OLS standard error, 4.1%. Tbe OLS estimate is biased upward, however, as discus& earlier. The averap simulated vabre of the OLS estimator given in the second cohmn tends to exoeed the true vahre by M-1.9%, a bit higher than the 1.75% predi&d using Stambaugh’s formula.

The bootstrap revalue in the third column of Table 5, which is actually a simulation estimate of the p-vaJue, is 0.001. This is the proportion of tbe 2500 simulated (under the null) estimates that exceed the historical slope of 20.2%. The standard error of this estimate, based 00 a binomial analys& is the square root of (0.001~0999~/2500, which is less than 0.001. The standard error for a p-value do.01 (/3 = 6) is 0.002. Having strongly mjected the nuU hypothesis of a zero slope, we would like additional information tqarding the true value offi. Examining the p-values in column 3, we see tbat values of /3 up to 6% can be rejectd at the 0.01 levei, whik vahres up to 10% can be r&cted at the 0.05 levd. At this slope value, a move done-standard deviation in B/M implies an &tease in expected return of a whopping 1000 basii points.

WenowshiCtthefocusfmm~~ngwhicbvaluePoCBcanbe~~at conventional significanot leveks to which values have highest likelihood, amdi- tional on the one-year histori& slope eatima&. The likdi reported in cohunn4ofTablc5starta.0.001 foraslopeofzeroand - (almost) monotonically to 0.23 at fl = 16% the maximum co- These likelihoods (or posterior probabilities under the uniform prior) an computed using a neigh- borhood consisting of points within 1% of the historical skope, and are plotted in Fii 1 (see triangles). The values of beta with high likelihood cormspond to huge levels of expected return variation. Under the (naive) uniform prior. most of the posterior probability exceeds /I = 8%. which is the largest value of beta that ensures that conditional expected returns (fitted values) are always non- negative over the 1926-91 period.

In genera& the ratio of posterior probability for a nul! hypothesis to posterior probability for an alternative (the ‘poste&r sjds ratio’) equals the prior odds ratio, p/(1 - p)t times another ratio often r&erred to as the l ayes factor’ (see Je&eys, 1957). This f&ctor is tbe weighted average likelihood under the null bypotbesis divided by the weighted average tikelilrood under the alternative, with weights determined by the prior. In a test of the null hypothesis tbat there is no variation in expected returns r&ted to B/‘&4, the Bayes faclor is just tbe likelihood at /3 = 0 divided by a weighted average of the likelihoods for positive betas. From Fig. 1 it is clear that the Bayes factor wi’tl always be less than one (over the range of slopes considered) and it will be close to zero for any prior that puts substantial weight on the larger values of beta. Thus we have a sharp shift

188 S.P. Kolhari. J. Shonken iJournu1 of Finonciol Economics 44 (1997) 11% 203

in beliefs away from the hypothesis /3 = 0, particularly for those individuals who believe that expected returns are sometimes negative.

The bootstrap evidence from the regression of equal-weighted index returns on dividend yield is reported in columns 5-7 of Table 5. The maximum /I values considered correspond to a minimum expected return of - 15.3% in each case. The range of slopes for B/M is smaller than that for dividend yield since the minimum value of B/M is more extreme whea measured in units of standard deviations from the mean. The bootstrap standard error is 4.9% under the nufl hypothesis fi = 0. It is higher than, but again close to, the OLS standard error of 4.8% in Table 2. The OLS bias (difference between the first two columns) ranges

S.P. Krdwi. J. Shot&en, Journal oj Financial Economics 44 (I 997) 169 203 189

from 1.2% to ! 4%. comparable to the 1.3% predicted using Stambaugh’s bias formula. The bias is roughly 20% of the value of the historical slope, 6.7.

The bootstrap pvaluil for @ = 0 is 0.13, close to the pvalue of 0.12 based on the t-statistic of 1.39 in Table 2. Although we cannot reject the null hypothesis of a zero slope at conventional levels. the likelihood function in Fig. I (see squares) rises ini:ially, peaks at /3 = 7%. and declines steadily after that. Thus, the Bayes factor will be less than one, i.e, will favor the conclusion that expected returns do vary, except for priors that place most of the weight on very large values of j!I(greater than 11 o/6) Therefore, anyone who believes that expected returns were positive over the pziod 1926-91 (B 5 9%) will become more confident that expected returns vary with dividend yield. The maximum possible shift in odds is by a factor of 2.5 = O.lOl~WMl. the ratio of the maximum hkelihood to the likelihood at zero. On the other hand, an investor with prior beUs heavily weighted on #I values greater than 11% will become less confident that expected returns vary with yield. This is a relative statement, though; such an investor will still be confident (high posterior probability) that /3 # 0, but this will be due to the prior dominating the sample evidence.

7.2. Vaiue-weighted rtwm rqpssed on B,“Af rw di&kvtd yield

Bootstrap evidence for the l:alue-weighted index is given in Table 6. The bootstrap pvalws for /II = 0 ale close to 0.02 for both B,‘M and dividend yield, rejecting the null of no expeclcd return variation. The likelihood functions plotted in Fig 2 tell a subtler story, however. As earlier, the B/M likelihood function never drops below the fl= 0 likelihood over the range of slopes considere& although it does turn downward. Thus, the evidence implies a shift in beliefs away from the hypothesis of no expected return variation for the entire class of priors consider+. In contrast. while the likelihood function for dividend yield generally favors vitive values of& an investor with a prior concentrated on very large values of /I would come away from the evidence less confident that expected return varies with dividend yield. This follows from the fact that the likelihood function for dividend yield eventually drops below the likelihood for fl = 0, as earlier in Fig. 1.

7.3. Suhperiod ana!sses

This section discusses bootstrap simulation results for the 1941-91 and 1963-91 subperiods. The Ml-91 subperiod excludes the Great Depression period of high return volatility; the more recent ml-l%2 subperiod is of interest because it coincides with the period over which much crossaectional examination of book-to-market ‘ratios has been undertaken.

‘Not surprisingly, given the smaller time-series sample and greater persistence in the independent variables over the subperiods. the estimated slope biases

190 S.P. Kolhari, J. ShankeniJournal of Financial Economics 44 (1997) 169-203

Table 6 Regression of one-year CRSP value-weighted +dex real returns on book-to-market or dividend yield 19261991 --I__-.-__--_- .-.- ._--.- -- ..- ~_- . --.- - _-- -.-.-- -.-_, ..-- __._ .-.--._-._-l__

p % Mean b( %) p-value LKLHD Mean h(%l pvalue LKLHD --------- -----II__

Book-to-market results Dividend yield results

0 1131 0.016 0.013 0.94 0.022 0.020 1 2.43 0.032 0.023 1.84 0.043 0.029

* ~ 3136 0.050 0.035 ‘86 0.076 0.047 3 4.41 0.085 0.050 3.90 0.140 0.067 4 5.35 0.134 0.073 4.89 0.212 0.090 5 6.36 0.203 0.103 5.90 0.313 0.115 6 7.40 0.306 0.129 6.72 0.417 0.131 7 8.35 0.429 0.156 7.81 0.571 0.139 8 9.39 0.5n 0.155 8.91 0.720 0.122 9 10.35 O.7? I 0.!46 9.90 0.836 0.0%

IO 1 16 nr.$ri Q 112 10.82 0.912 0.068 !I i i.92 0.965 0.041 12 12.79 0.987 0.023 13 13.85 0.9% 0.09 14 I4.W 0.998 0.003 15 15.94 l.atO OBOI 16 16.86 l.alO O.OlX)

.~ ._.

Che-year returns are regrwd 011 Lwginningd-yrw B.M or dividend yield (scaled by standard deviation) to obtain the historical slope estimate. and a first-order autoqn+on is estimaiai tar B/M. Bias-adjustadastimatcsdthercgr&on parameten are used to cated adjusted regndon residuals. Given a by~othatical value d p, the return regnshn siopc, and the bias-adjusted autcrcwreLation cstinwe. adiciai time series of annual returns and B, M or dividend yieki ratios (7 - 661 arr generated by randomly sampling from the joint disttibution Wendar year Pairs) of adjusteddwlrr ftxe (anificid) returns are regrewd on the tartiticialr B+M 01 dividand yield ratios to obtain an estimate of @. The proad~ is rePeated 2500 times and b. tbc average of Ibe regression co&&nt escimales. iseimpkue& The bootstrappvalue for a given /I is the proportion of the 25OOcstimatea that exa)od the historical slope. LKLHD (the bootstraP iikdibuod) is the scaled proportion of the Zsoo estimates that lia within a smat! interval antered at the historical slo~c The proporrions arc~40thatthesumof likctihooQoverall~sisnne.

from Eq. (3) are larger for the subperiod estimators. For example, the simulated bias for the B/M slope (not rqorted in tables) exceeds 4.0 for both indexes over 196391. &s-adjusted t-statistics are computed by subtracting the simulated mean value (under the null fi = 0) from the estimator b and dividing by the standard deviation of the bootstrap slopes.

Table 7 presents bootstrap p-values for /? = 0, along with pvalues based on the OiS r-statistics and bias-adjusted t-statistics. ksults for 1941-91 are given

S.P. Kothori. J. Shanken/Joumai of Financial Eronomics 44 (1997) 169 -203

in panel A, while results for 1963-91 are in panei B. As expected, the bias- adjusted pvalues are often much larger than the conventional CXS pvalues. However, when the values are relatively low, the bias-adjusted pvaks are smaller than the bootstrap p-values. This apparently reflects nonnormality of the bootstrap distribution for the slope estimator, with the bootstrap having a fatter upper tail. For example. the adjusted p-value is OJlNi in the equal- weighted dividend yield case (Ml-91), while the bootstrap’pvalue is 0.048. The binomial standard errors are 0.002 for the 0.00s pvafue aad 0.004 for the 0,048 p-VillUe.

We focus on the more conservative bootstrap pvalua. In the post-1940 period, the hypothesis of no related expected return variation is rejected at the

132 S.P. Kothari. J. Shanken /Journal 01 Financial Economics 44 (19%‘) 169. 203

Table 7 P-values for the null hypothesis fi = 0: subperiods 1941 91 and 1963-31 -- _____---. _-___--__----

index Financial ratio OIS Bias-adjusted Bootstrap

Panel A: 1941 91

Equal-weighted E&al-weighted

Value-wn’ghted Value-weighted

fSook-to-market Dividend yield

Book-m-market Dkdend yield

0.002 0.014 0.040 0.001 O.OOS 0.048

0.03 I 0.127 0.156 0.001 O.lUl 0.024

Panel B: 1963-91

E&p&weighted Book-to-market 0.036 0.295 0.250 E4pd-WCigkd DiVidcnd,yidd 0.001 0.012 0.023

Value-weighted Book-to-market 0.156 0.629 0.571 Vahu-weighted Dividend yidd 0.008 0.102 0.115

..- _- ..-. I-

F~xII-ud~’ ~EC? index IS :h- CWP equal *+ XI arnual real return inclusi. z of dividends from April tti~y~ f to hla~b Oc yur f + I. V&e-wcightnf rndex is tbc CRSP value-*eighted annlrztl red return inc!Gvc of dividends fron, AprilofyearrtoMarchofyurr+l. Book4o-markd is the ralio of the book value per share at the end of year r - I of the stocks in the Dow Jones hdusirid Average to the I%w ha Induplrial Awragc zt the end of March of year t. Equxr-~~(eddividadyiddkfhenriooCdividcn&rpzidfronAprilofymrt-I~oMard,of yew I IO the aetmwdatcd valwe ofoacdobr invcstaj in tbc CRSPcqual-weighted index exclusive of divi&~&foroneycareedingonMardt31ofy~~r. V~uc-~rddividadyidd~tbcntioddivikadrpiridfromApritoCyearr-ltoMarebd year I to the cwxtumbtal vplucofone ddlar iavcs&d in tbc CRSP vabc-eighted index exdusiw of divideads l&r l-year ending 08t Mat& 31 Or year 1.

0.05 I+, except far the combination of value-weighted index and ?M. The smallest pvalue is 0.024 for the regression of value-weighted returns on dividend yield. In the past-1962 period, rejection at the 0.05 level is only observed for the eqtutCwe&hted index and dividend yield.

The bootstrap likelihood functions provide additional insights into these results as well as the potential contrasts between classical and Bayesian infmnce in all but one case, an investor who 6rmly believes that expected returns are never negative will come away from the regression evidem~ with greater conti that expected returns vary with B/M or dividend yield. This is true .~ven for the post-l%2 scenario with B/M and the equal-weighted index (with bootstrap pvalue 0.2Sk although the maximum shift in odds (Bayes factor) is only 1.7 in this case. On the other hand, in most q the direction of the shift in beliefs for an investor who firmly believes that expected returns are sometimes negative depends on the specifics of their prior distribution.

S.P. Korhari. J. Shanken ‘Journal of l;inonciai Economics 44 (1997) 169-203 193

The post-1962 likelihood function for the equal-weighted index and dividend yield rises steeply and stays well above the likelihood at j3 = 0 for the entire range of j? value& This will be viewed by ail investors (for the class of priors considered) as evidence in support of time-varying expected returns. At the other end of the spectrum, the value-weighted, B/M, post-1862 iikelihood function for the value- weighted index and B/M steadily deciines as 6 increases, shifting all priors toward the con&&n that B/M does not track variation in expected mturns for the v&re+veighted index. One final case wortb highlighting is the post-1962 scenario for the value-weighted index and dividend yield. Here, the likelihood function drops below the likelihood at fl= 0 only at the largest value co&&red (j? = 9%). Thus vittuaUy all investors considered wiii view the evidence as supporting a relation between dividend yield and expected return, even though the c&s&al test fails to mject :he null hypothesis of no relation at the 0.10 level.

8 Tests for momaegatire expmed wmms

The regre&on evident: examined above suggests considerable variation in expected returns as a function of both B; M and dividend ytetd. ktg. 3 plots the

Fig. 3. Expected equabeightcd index returns: 1926-1991 Expeaed returns arc fltcd vows from the rcgessbn of real returns oa prior book-to-market ratio or dividend yield over the 1926-1991 perio& The fitted values are evaluated al the maximum EkeMwud beta v&es. 16% for book-to-nwket and 6% for dividend yield. with the interczpts spdied so as to equate the avemp expected returns to the sampk mean return.

194 S.P. Kothari. J. Shunken/Jourwal of Financial Economics 44 (1997) 169-203

Fig. 4. Expedcd vaiuc-wciphtd index rctums 1926-1991 Expracd returns ate fitted rdues from the mgrcsion of real mwns on prior book-to-market ratio or lividend yi&l over tb 1926- 1991 period. l’be fitted values are cvaloated at tlw maximum lidibood beta values. 16% for both book-to-markci and divided yiekf, with the interc+x xpcciliod so as to cqwtc tbc tlKwgc rxpxtcd rcwm.5 to tbc xanlpk mcatl rctum.

1926-91 equal-weighted expected re~ums (fitted values) using historical BjM or dividend yield and the maximum likelihood slope values /? = 16% and fi = 7%, respectively. Consistent with the market being predictably overvalued, the expected returns based on BfM are negative for sever8) years at Both the beginning and end of the s8mpk and two ye8rs in between. The most neg8tive expected return is - 14.5% in 1929 when the market value for the Dow 30 is about 3.7 (l/0.27) times book value. Expected returns bed on dividend yield are 8lwap positive, however. Vahte-weighted index expected icturns are plotted in Fig 4. Here, negative expected returns based on B/M are smalkr in magni- tude 8ad &Ilit4?d t0 the 192&30 PlZi’iOd.

Perhaps as impktusibie 8s nCgatiVe expected returns, at hzast to 8 believer of

market efB&ncy, are the Iarge positive expected returns impiied by the high B/M vaiuns, particu$rly from the mid-1970s to the early 1980s. B/M values in this period imply expected real returns ranging from 20% to 50% per year. While these rates are extremely high, an interpretation that the market was on the brink of’ a near-mrm upward cormctiou is too simplistic Anyone attempting to c8pit8.b on the bm.krvalu8tion’ in the 1970s wotdd have had to endure 8 prolonged wait before the market began to rise in the early198os. A similar statement c8n be made regarding the market’s petformalwx since the late 19&os.

S P. Kohari, J. Shanken i Journal c/Financial Economics 44 (I 997) I69 -203 195

The Dow Jones B/M has been about 0.4 during this period, fairly low, by historical standards. However, expectations of a low market return have been foiled by the stellar market performance in the 1990s to date. The Dow Jones B/M stood at its historical lowest, 0.27, on April 1, 1995, but the market performance since then has been one of the best in the post-Work! War II period. pe Dow Jones B/M is currently (August 1996) only 0.23.

In the remainder of this section, we consider formal tests of the hypothesis that expected real returns are always non-negative over the given period of interest. By the law of iterated expectations, if expected returns are non-negative conditional on all in’ormation availabIe at each point in time, then expected returns must also be non-negative conditional on any subset, in particular when conditioning on B,‘M or dividend yield.

From Table 5, we see that the hypothesis that expected returns are non- negative over the 1926-91 period is rejected at the 0.02 Ieve! (p-value for j? 5 8) for B/M and equal-weighted index returns. The corrqumdiag pvaiue for B/M and value-weighted index miums in Table 6 is about 020 (for jf I 5). Thus, we cannot rejeck at conventional signiticance &eve4 the hypothesis that the nega- the fitted vahres for the value -weighted index, in Fig. 4, am due to error in estimating /I. There is no evidence of negative expected returns in the dividend yidd results.

As explained in Seaion 6 these non-negativity tests implicitly assume that the sample mean of the return disturbances+ i& is zero for the given period. We now compare this test to the conventional regmssion procedure for making infer- ences about a conditional mean. The eslimated expected return @ted v&e) at the minimum sample value of B/M (27.2% for 1929) is - 22.3% for the fzqual-weighted index. The corresponding standard error, based on the usual regression formula, is 17.3%. This yields a t-statistii of - 1.3 and a pvaiue of 0.10 for the hypothesis that the minimum expected return is positive over the 1926-91 period.

The hirly large increase in p-value (0.02 to 0.10) re&cts the fact that the conventional procedure incorporates uucertaint~ about the sample mean distur- bance term, whereas our ~-based test con&ions on a t&&about tbe magnitude of i&l Due to the substantial variability in return surprises, the conventional test for non-negative expected returns is not powerful. Conditioning on the mean disturbance permits us to concentrate on the slope cxxfkieut and the impha- tions of time variation for nonnegative expected returns, holding the average level of expected return constant at the sample mean. since we focus on a narrower issue, the power of our test is, not surprisingiy. enhanced.

‘A more formal statistical appendix is available from the authors on request.

196 S.P. Kothari, J. ShankenlJoumal oJ’Finrmcial Economics 44 (IWi) IS-- 203

Now, suppose that (relative to one’s priors) the data indicate that the mean return surprise, ri, is positive.8 For example, the literature on the equity premium puzzle(Mehra and Prescott, 1985) suggests a lower prior expected real return on the equal-weighted index than the sample mean of 15.3%. Alternatively, one might believe that the observed sample of US. returns is subject to a selection bias related to survival and grdwth of equity markets at the country level (Brown et al., 1995).

Suppose, far exarnk~ that half of the mean return is viewed as an ex post surp* i.e, H = 7.65%. Given the analysis in -ion 6, this has the eflkct of cutting in half the maximum value offi copsistent with non-negative expested tiurns The associated p-value (for /I 5 4) is 0.002 as compared to 0.02 (for /I I 8) earlier. Naturally. incorporating a belief that the return surprises are positive, on avm leads to a stronger rejection of non-negativity. since our belief about the general level of expected returns is lowered relative to that of the previous scenario.

X2. Bqwzsian tests

We now turn to Bayesian tests of the nonnegativity restriction. The equal- weighred B,‘M regression is examined first. In order to analyze the expected return hypothesis in a manageable way that is readily communicaterl to a var- iety of ti some additional structure is needed. We consider priors that assign probability p to betas from zero to 8% and 1 - p to betas from 9% to 16%, the prior probability distributed uniformly within each subset. With this prior, p is the probability for the composite null hypothesis that expected returns are always non-negatives conditional on this hypothesis, no variation in expected returns (#I = 0%) is considered as plausible as anything up to an 800- basis-point (fi = 8%) variation in expected returns for every one-standard- deviation change ia B/M. Prior weight on beta values between 9% and 16% entails a belief that expected returns can be negative (i.e., when B/M is relatively low) and that expected returns vary greatly through time.

The assumption that probability within each subset is uniformly distributed is adopted for simplicity. While it violates desirable conditions like continuity of the prior density, we believe the general conclusions that emerge are su@ciently rdbust that the benefits of simplification outweigh the costs. Given the uniform- ity assumption, the Bayes factor $ just the ratio of average likelihood under the null to average likelihood under the altemativc, which is imlependcnt of p. The Bayes factor for our equal-weighted index B/M regreGon is 0.086. Thus, in light of the B/l&l eGdencz there is roughly a 12-&A shift in odds against the hypothesis that expected returns are atways non-negative over the period 192691.

In Fig. 5, we plot the posterior probability that expected returns are always nonnegative as a function of the prior probability p. The *no-change’ line with a slope of one is a natural benchmark for which the prior and posterior probabilities are equal. Thus, the straight line in Fig. 5 corresponds to a scenario

S. P. Korhari. J. Shanken ‘Journal qf Financial Economics 44 (!997) 169-203

1.m

0.90

0.80

0.70

OM

oxl

OAO

O.Xl

0.20

QIO

om

Fig 5. Posterior probability that cxpeded real rcturxs arc ~-negative 1926 1991 Poskrior prob&ilitics arc derived from tk like&xxi timctionr plotted IB Pigs. I and L with prior distributiom spedid over the ranges of beta values in those Crgurcs. Beta r&es in the first half of each range aIri cxuwistcnl with nonlltgrtivc expected ITtumi. v-h& betas in tbc z3cxmod half yield negative expeucd mtums at some point in the 1926-1991 period The prior distributions arc uniformly distributed within each MF. prior probability is the wet&t assigned to the first ItaIfofthe

b-raagp.

irt which the Bayes factor is one. The low va!ue of the actual ibyes factor in the equal-weighted B/M case ensures that the posterior probability curve iies beiow the no-change tine for p between zero and one. This means that. regardless of the prior (p # O,l), one ends up less confident that expected returns are always non-negative (i.e., 0 I /I I 8%). In particular, an individual who is initially

198 S.P. Kofhari, J. Shanken/Joumal of Finaaciai Economics 44 (1997) 169-203

uncommitted @ = 0.5) with regard to the hypothesis that expected returns are non-negative has posterior probability 0.08, while an individual with prior confidence in the non-negativity condition, say p = 0.8, has posterior probabil- ity 0.26.

Now, as we did for the classical test of non-negativity in the previous subsection, we consider the alternative assumption that iJ, the mean surprise component of return, is 7.65%. As before, the maximum /I consistent with non-negative expected returns is reduced to 4%. In addition, the range of frs entertained as plausible a priori is likewise cut in half, with a maximum fl of 8%. At this value, the most negative expected return over the 192691 period is now - (F -‘ii)=- 7.65%. Given the likelihoods reported in Tabie 5, the resulting Bayes factor (ratio of average likelihoods from 0 to 4 and 5 to 8) is 0.226, almost three times the Bayes factor obtained under the si = 0 assumption. The corres- poading posterior probability for non-negative expected returns of the l corn- mitted’ (I, = 0.5) investor is now 0.18. as compared to 0.08 above. Truncation :td\. a mLl;n?al Pffr?! ia *&i; rec;LIx.

The increase in posterior probabihty may be surprising, since the classical test rt$cctd non-negativity more strongly when U = 7.65*/s. The reason is that the prior associated with I? = 7.65% eliminates the weight previously placed on the very large values of fl(> 8) that have extremely huge likehhood, given the historical slope of 20.2% (see Fig. I). Thus, the increase is a consequence of the change in prior implied by the given belief about Ti (maximum 8 reduced from 16% to 8%), in conjunction with the particuhtr behavior of’ the hkelihood bmction in this example.

in the value-weighted B/M case, the posterior probability curve again lies below the no-change tine, despite the relatively large pvalue of020 (for /? s 5% in TaMe 6) fof the null hypothesis of non-negative expected returns. The posterior probabilities are kss extreme than those for the equal-weighted index, however. For exam& an observer with p = 0.5 comes away from the B/M evidemz with posterior probability 0.25 that expected returns are non-negative. Posterior probability cutves for dividend yield in Fig. 5 he above the no-change line, mflectittg &teased confidence in the non-negativity hypothesis. Curves for the 1941-W subperiod, given in Fig. 6, all he above the no-change line as we!1

8.3. Tests with truncath

One view of the 1933 outher is that i: is an observation that just happened to occur in one year out of 66 in our sample, but which has an en ante @opuiation) probabihty far Iess than this From this perspective, conventional weighted least squares ptocedures do not adequately address the proMem since they still give too much weight to the 1933 data point. To gain additional insight into the problem, we rerun the 1926-91 B/M bootstrap analysis with 1933 truncated, as discussed earlier in Section 3.

S.P. Knthari, J. Shonken~Jo~rnal of Financial Economics 44 (1997) 169-203 199

l.CQ

0.90

om

am

0.60

o.50

04

OS

o.20

a10

a00

F*tX Post&or pmbability that crpadcd teal tcWntb ate nonacyUrcc: IW-1991 PostetCprobabiiities on derived ftuttt the IikeIibood functions aad prior &tributiot~ spedal overthecomspondiagraagsofbctaval~Becav~afintfwhr;t~oreachoreach~arrcorYsirteot witb non-negative expdcd returns. while betas in the second ltd+d ibegMive expatcd teturtt5 at sotne point in the WI-1991 period. The prior distribrttions ate uttironnly distribulcd with crh Ita& prior probabiiity is the weight assigned to the lirst half of the beta range.

With the equal-weighted stock in&z, the bootstrap pvalue for the null hypothesis that expected returns are unrelated to B/M @ = 0) rises to 0.005 with tmncatioo, from G.001 earlier, the pvalue for the hypothesis of non-negative expected rptums (6 I 8) increases to 0.084, from 0.022 when 1933 is included

(Table 5). Corresponding p-values for the value-weighted index are 0.045 (0.016 earlier) for no relation. and 0.27 (0.20 earlier) for non-negative expected returns.

Perhaps more informative, given the limitations of p-values. are the posterior probabilities. Now, an uncommitted, i.e., p = 0.5, observer comes away from the equal-weighted B/M evidence with posterior probability 0.14 (0.08 earlier) that expected returns were non-negative over the 192691 period. For the vafue- weighted index. the posterior probability is 0.34 (0.25 earlier). Thus there is still a shift in beliefs, albeit a weaker one. away from the non-negative expected returns hypothesis.

9. Summary ad coaehasioas

We find reliable evidence that both dividend yield and B,,M track time-series variation in eawted real one-year qtock returns over the period 1926-91 and Lb< sub~“-~iJd ??‘! 141 The P X r&lion is stronger over the full period. while the dividend yield relation is stronger in the subperiod.

Jhe estimated 1926-91 equal-weighted return slope on B/M implies chat expected return moves by an astounding 20 percentage points ( 18 bias-adjusted) for a one-standard-deviation drarrge in B/M. The coefRcient is so large that implied expected returns are sometimes negative. particularly in the late 1920s when the B/M ratio is relatively low. The results are quite sensitive to the 1933 data point, however, when both B/M and market index returns attain their maximum values. Using a +truncation’ method that reduces the impact of this outher. the pvahte for non-negative expected returns is 0.08 (0.02 without truncation). ASthough the pvalue of O.Og might be interpreted by some as ‘insignificant’, the posterior probability for an investor who assigns prior pro& ability 0.5 to the hypothesis of non-negative expected returns is only 0.14 (O.Orr without truncationt Results for the vahte-weighted index and B,/M are qualitat- ively simihtr, though the &i&s in beliefs are far less dramatic.

Thus, the B/M results suggest that expected return variation over the 192691 period was not driven entireiy by equilibrium changes in compensation for risk. Rather, It appears that the market may have been iocfficieat, particularly in the late 1920s and early 193oa a period of great economic volatility. Ihe stronger results for the equal-weighted index, which is infiueneed more by small-firm returns, is consistent with this conclusion, although greater variation in risk- rehcd predictabiity is also piausible for smaller firms.

The market ineffciency conclusion should be tempered by several observa- tions. First, them is the familiar element of data mining which implies that the sign&ance ofextreme rest&s is overstated, although by how much it is difhcult to say. Whiie the 6nancial ratios considered here can be motivated theuretically, dividend yield is examined in part because of its prominence as compared to, say, earnings yield in pas: time-series studies (elg. Fama and French, 1988). B/M

.s.i? Kothari. J. ShankenlJomnal of Fincmcial Economics 44 (1997) 169-203 201

is considered primarily hecause of its recently acquired celebrity status in explaining cross-sectional stock return variation, although the extent to which this biases time-series results at the market level is less clear. These concerns and the fati that the forecasting power of B/N and dividend yield varies consider- aMy over dif%rent subperiods give further credence to t!re idea that data mining &&atiUOS should teIllpcr One’s Bt of the value of t!tese ratios in making futUrC iOVCSttOCOt decisions

The &&kOd Yk!d S!Op esthetes for the 1926-91 period and al! of the estimates for the 1941-91 subperiod are IIXOR ‘reasonable’ than the B/M results, hut still srsbsrantia!, implying expected rCaum movements from three to eight percentage points for one-standard-devisation changes in the financial ratios. w estjsx-tates all result in varying degrees of inrfeus4 contidence in the proposition that expected returns were Rever negative. While the considerable estimated variation in expbcted returns might he di!!!cult to recxmci!e with market dticjeucy~ the hypothesis that the slope is greater than t!trae is never med at the 0.05 kvd. Definitive statesnents concerning the extent to which d-tad variation in t qectu! returns is related to risk or systematic mispric- kg (or both! RX!tlirC further analysis.

one step toward reconciliation with srtarkct e&iency mig!tt bc hased on & argument of Brown et a!. (1995k who em t!te rok of survivor hiascs, even at the country !eW!. IruSe arguments gain credibility in that t!xe evi- w of inefficieacy is tied tc a historic period of considerahk uncertainty. However, urbi!e survival of the U.S. market could impart an upward hias to the -on s;Eope coefWent, it might a!so &as tbe interctpp uparanl. Adjusting for the e@Ct Of SlUViVOr bias 00 the Slope WOUld t&uOe the statiStiCi! signi- ficanee (larger p-value) of our evidence of oegative expected returnq but the inten;ept &as would imply a stronger rejection of noo-neg+ivity, as shown in sactioo 8.1.

violatjon of the homoskedastic linear rvqmssioo speci!ieatiou used in most of our anal+ might give a misleading impression as to the likeli!rood of negative expected t-eturns, but our initial analysis suggests otherwise. Tbe first-order autorclpr#isive pr- postulated for the f$atiial ratios could easi!y be genera!- although we suspect this would have !ittle impact on tbe main ~!usio~s- It worth! also he interesting to compare the performance of the bootstrap likelihood methodology with tlae bkellbood function generated under t!te assumption of hivariate normality of the vector automgressive disturbances tsee ~&t and Stambaugh, 1996). Although extension of the latter approach to overlapping multiyear returns would appear di!Iicult, use of the bootstrap methodology in tbis context is straightforward and should provide further insights jnto the behavior of expected return variation.

10 &j&30 to our empirical conclus!cxns, we note several examples of the limited jnformativeness of standard p-value analysis and the benefits of explor- ing the likelihood function. One interesting scenario illustrates the important

202 S.P. KotAori, J. Shanken fJoutna1 of Financial Economics 44 (1997) 169-203

role that prior beliefs can sometimes play in interprc ting data. An investor who is confident that expected returns are always positive views the regression results as supporting the hypothesis of expected return variation, while an investor strongly inclined to believe that expected returns are sometimes negative views the regression as favoring the constant expected return conclusion. In another scenario, investors spanning 3 wide variety of prior beliefs view the regression evidence as supporting a relation between dividend yield and expected return, even though WC ,fail to reject the null hypothesis of no relation at the 0.10 ievei (Fvaiue 0.12).

Finalfy, we note that the bootstrap approach to likelihood/Bayesian analysis is quite flexible and could prove useful in a variety of empirical contexts. This is likely to be the case in accounting and corporate finance applications, where the variables of interest are often far less well-behaved statistically than stock retums.6

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