Predicting Excess Stock Returns Out of Sample: Can

Predicting Excess Stock Returns Out of Sample: Can Anything
Beat the Historical Average?
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Citation
Campbell, John Y. and Samuel B. Thompson. 2008. Predicting
excess stock returns out of sample: Can anything beat the
historical average? The Review of Financial Studies 21(4): 15091531.
Published Version
doi:10.1093/rfs/hhm055
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February 6, 2015 9:37:43 AM EST
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http://nrs.harvard.edu/urn-3:HUL.InstRepos:2622619
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Predicting Excess Stock Returns Out of Sample:
Can Anything Beat the Historical Average?
John Y. Campbell and Samuel B. Thompson!
* Campbell: Corresponding author. Department of Economics, Littauer Center,
Harvard University, Cambridge MA 02138, USA, and NBER. Tel 617-496-6448. Email
john_campbe[email protected] Thompson: Arrowstreet Capital, LP, 44 Brattle Street,
5th oor, Cambridge MA 02138, USA. Email [email protected]
Abstract
Goyal and Welch (2006) argue that the historical average excess stock return forecasts future excess stock returns better than regressions of excess returns on predictor
variables. In this paper, we show that many predictive regressions beat the historical average return, once weak restrictions are imposed on the signs of coe!cients
and return forecasts. The out-of-sample explanatory power is small, but nonetheless
is economically meaningful for mean-variance investors. Even better results can be
obtained by imposing the restrictions of steady-state valuation models, thereby removing the need to estimate the average from a short sample of volatile stock returns.
Towards the end of the last century, academic nance economists came to take
seriously the view that aggregate stock returns are predictable. During the 1980s, a
number of papers studied valuation ratios, such as the dividend-price ratio, earningsprice ratio, or smoothed earnings-price ratio. Value-oriented investors in the tradition
of Graham and Dodd (1934) had always asserted that high valuation ratios are an
indication of an undervalued stock market and should predict high subsequent returns,
but these ideas did not carry much weight in the academic literature until authors
such as Roze" (1984), Fama and French (1988), and Campbell and Shiller (1988a,b)
found that valuation ratios are positively correlated with subsequent returns and that
the implied predictability of returns is substantial at longer horizons. Around the
same time, several papers pointed out that yields on short- and long-term Treasury
and corporate bonds are correlated with subsequent stock returns [Fama and Schwert
(1977), Keim and Stambaugh (1986), Campbell (1987), and Fama and French (1989)].
During the 1990s and early 2000s, research continued on the prediction of stock
returns from valuation ratios [Kothari and Shanken (1997) and Ponti" and Schall
(1998)] and interest rates [Hodrick (1992)]. Several papers suggested new predictor
variables exploiting information in corporate payout and nancing activity [Lamont
(1998) and Baker and Wurgler (2000)], the level of consumption in relation to wealth
[Lettau and Ludvigson (2001)], and the relative valuations of high- and low-beta
stocks [Polk, Thompson, and Vuolteenaho (2006)]. At the same time, several authors
expressed concern that the apparent predictability of stock returns might be spurious.
Many of the predictor variables in the literature are highly persistent: Nelson and Kim
(1993) and Stambaugh (1999) pointed out that persistence leads to biased coe!cients
in predictive regressions if innovations in the predictor variable are correlated with
returns (as is strongly the case for valuation ratios, although not for interest rates).
Under the same conditions, the standard t-test for predictability has incorrect size
[Cavanagh, Elliott, and Stock (1995)]. These problems are exacerbated if researchers
are data mining, considering large numbers of variables, and reporting only those
results that are apparently statistically signicant [Foster, Smith, and Whaley (1997)
and Ferson, Sarkissian, and Simin (2003)]. An active recent literature discusses alternative econometric methods for correcting the bias and conducting valid inference
[Cavanagh, Elliott, and Stock (1995), Mark (1995), Kilian (1999), Lewellen (2004),
Torous, Valkanov, and Yan (2004), Campbell and Yogo (2006), Jansson and Moreira
(2006), Polk, Thompson, and Vuolteenaho (2006), and Ang and Bekaert (2007)].
A somewhat di"erent critique emphasizes that predictive regressions have often
performed poorly out-of-sample [Goyal and Welch (2003, 2006) and Butler, Grullon,
1
and Weston (2005)]. This critique had particular force during the bull market of the
late 1990s, when low valuation ratios predicted extraordinarily low stock returns that
did not materialize until the early 2000s [Campbell and Shiller (1998)]. Goyal and
Welch (2006) argue that the poor out-of-sample performance of predictive regressions
is a systemic problem, not conned to any one decade. They compare predictive
regressions with historical average returns and nd that historical average returns
almost always generate superior return forecasts. They conclude that “the profession
has yet to nd some variable that has meaningful and robust empirical equity premium
forecasting power.”
While it is not clear how much weight should be placed on out-of-sample statistics
in judging the predictability of stock returns [Inoue and Kilian (2004)], in this paper
we take up Goyal and Welch’s (2006) challenge and ask whether standard variables
could have been used in real time to forecast twentieth and early twenty-rst century
stock returns. We show that simple restrictions on predictive regressions, suggested by
investment theory, improve the out-of-sample performance of key forecasting variables
and imply that investors could have proted by using market timing strategies.
We begin in Section 2 by comparing the in-sample and out-of-sample forecasting
power of standard predictor variables through the end of 2005. We use at least 20
years of data to obtain initial coe!cient estimates and restrict the forecast evaluation
period to the period since 1927 when high-quality total returns data are available
from CRSP. We calculate an out-of-sample !2 statistic that can be compared with
the usual in-sample !2 statistic. Like Goyal and Welch (2006), we nd unimpressive
in-sample results and poor out-of-sample performance for many of the usual linear
regressions.
Our contribution is to show that restricted regressions perform considerably better
than these unrestricted regressions. We begin by considering two alternative restrictions that can be imposed on any theoretically motivated forecasting regression: rst,
that the regression coe!cient has the theoretically expected sign; and second, that the
tted value of the equity premium is positive. We impose these restrictions sequentially and then together, and nd that they substantially improve the out-of-sample
performance of predictive regressions.
Next we look more closely at valuation ratios, for which theory restricts not only
the sign but also the magnitude of the regression coe!cients. We nd that out-ofsample predictability can be improved even more by restricting the magnitude of the
coe!cients to the values implied by a simple steady-state model, such as the Gordon
2
(1962) growth model in the case of the dividend-price ratio.
Section 1 shows that several commonly used forecasting variables do have some
ability to predict stock returns out-of-sample. The out-of-sample !2 statistics are
positive, but small. This raises the question of whether the predictive power is economically meaningful. In Section 2, we show that even very small !2 statistics are
relevant for investors because they can generate large improvements in portfolio performance. In a related exercise, we calculate the fees that investors would be willing
to pay to exploit the information in each of our forecasting variables. Section 3 briey
concludes.
1
Theoretical Restrictions on Predictive Regressions
In this section, we conduct an out-of-sample forecasting exercise inspired by Goyal
and Welch (2006), with modications that reveal the e"ectiveness of theoretically
motivated restrictions. We use monthly data and predict simple monthly or annual
stock returns on the S&P 500 Index. This immediately creates a tradeo" between
the length of the data sample and the quality of the available data. High-quality
total return data are available monthly from CRSP since 1927, while total monthly
returns before that time are constructed by interpolation of lower-frequency dividend
payments and therefore may be suspect. Accordingly, we use the CRSP data period
as our out-of-sample forecast evaluation period, but use earlier data to estimate an
initial regression.
Table 1, whose format is based on the tables in Goyal and Welch (2006), reports
the results. We begin by discussing monthly return forecasts reported in Panel A, and
then discuss annual forecasts reported in Panel B. Each row of the table considers
a di"erent forecasting variable. The rst four rows consider valuation ratios: the
dividend price ratio, earnings price ratio, smoothed earnings price ratio, and book-tomarket ratio. Each of these ratios is measured in levels, rather than logs, with some
accounting measure of corporate value in the numerator, and market value in the
denominator. The smoothed earnings price ratio, proposed by Campbell and Shiller
(1988b, 1998) is the ratio of a 10-year moving average of real earnings to current
real prices. Campbell and Shiller argue that this ratio should have better forecasting
3
power than the current earnings price ratio because aggregate corporate earnings
display short-run cyclical noise; in particular, earnings drop close to zero in recession
years, such as 1934 and 1992, creating spikes in the current earnings price ratio that
have nothing to do with stock market valuation levels.1
The next row of Table 1 uses a smoothed measure of accounting real return on
equity (ROE) as the predictor variable. Real ROE is measured as the level of real
earnings divided by lagged real book equity and the gross one-year ination rate. This
gives a measure of the total resources available to be divided between real payouts
and real growth in book equity.2 We smooth real ROE over ten years to remove
short-run cyclical noise in corporate protability. We include this variable in Table
1 not because it has been claimed to predict aggregate stock returns by itself, but
because we will use it in combination with valuation ratios in Table 2.
The next ve rows of Table 1 consider nominal interest rates and ination: the
short-term interest rate, the long-term bond yield, the term spread between long- and
short-term Treasury yields, the default spread between corporate and Treasury bond
yields, and the lagged rate of ination.
The last two rows of Table 1 evaluate forecasting variables that have been proposed more recently: the equity share of new issues proposed by Baker and Wurgler
(2000) and the consumption-wealth ratio of Lettau and Ludvigson (2001). Lettau
and Ludvigson’s variable is based on a cointegrating relationship between consumption, aggregate labor income, and aggregate nancial wealth. Rather than estimate
a separate cointegrating regression, we simply include the three variables directly in
the forecasting equation for stock returns.3
The rst column of Table 1 reports the rst date for which we have data on
each forecasting variable. For dividends, earnings, stock returns, long-term bond
yields, and ination we have data, originally assembled by Robert Shiller, back to
the early 1870s. Other data series typically begin shortly after the end of World War
I, although book equity does not become available until 1926, which means that the
book-to-market ratio begins then and the ten-year smoothed ROE begins in 1936.
All data series continue to the end of 2005. The second column reports the date at
which we begin the out-of-sample forecast evaluation. This is the beginning of 1927,
when accurate data on total monthly stock returns become available from CRSP, or
20 years after the date in column 1, whichever comes later.
The third and fourth columns of Table 1 report the full-sample "-statistic for
4
the signicance of each variable in forecasting stock returns, and the adjusted !2
statistic of the full-sample regression.4 It is immediately obvious from the column of "statistics that many of the valuation ratios and interest-rate variables are statistically
insignicant in predicting stock returns over the long sample periods considered here.
The most successful variables are the earnings yield, the Treasury bill rate, and the
term spread. Of the two recently proposed variables, net equity issuance is modestly
successful and the consumption-wealth ratio is strikingly successful in-sample.
The remaining columns of Table 1 evaluate the out-of-sample performance of these
forecasts, using an out-of-sample !2 statistic that can be compared with the in-sample
!2 statistic. This is computed as:
P#
(#$ ! #b$ )2
2
$
(1)
!!" = 1 ! P#$=1
2
$=1 (#$ ! # $ )
where #b$ is the tted value from a predictive regression estimated through period "!1,
and #$ is the historical average return estimated through period " ! 1. If the out-ofsample !2 is positive, then the predictive regression has lower average mean squared
prediction error than the historical average return.5 We use the entire available
history of stock returns, back to 1871, to estimate the historical average return. This
gives the historical mean an advantage over predictive regressions with variables that
have become available more recently, because more data are available to estimate
the historical mean than to estimate such predictive regressions. However, this is a
real-world advantage of the historical mean that should be taken into account in our
tests.
The out-of-sample performance of the predictor variables is mixed. The fth column of Table 1 shows that only two out of four valuation ratios (the earnings yield
and smoothed earnings yield) and two out of ve interest-rate variables (the Treasury
bill rate and term spread) deliver positive out-of-sample !2 statistics. The interest
rate results are consistent with the conclusion of Ang and Bekaert (2007) that the
Treasury bill rate and term spread are robust return predictors. The performance of
these variables would be stronger if we started the sample period later, because the
interest rate process changed dramatically at the time of the Federal Reserve-Treasury
Accord in 1951. Of the two recently proposed variables, net issuance performs reasonably well but the consumption-wealth ratio does not. The di!culty of estimating
coe!cients in a short sample is particularly severe for this series because it includes
three separately estimated components.6
5
All the regressions we have reported predict simple stock returns rather than log
stock returns. The use of simple returns makes little di"erence to the comparison
of predictive regressions with historical mean forecasts, but all forecasts tend to underpredict returns when log returns are used. The reason is that high stock market
volatility in the 1920s and 1930s depressed log returns relative to simple returns in
this period. Thus the gap between average stock returns in the late twentieth century
and the early twentieth century is greater in logs than in levels.
1.1
Sign restrictions on slope coe!cients and forecasts
Despite the mixed performance of unrestricted predictive regressions out of sample,
it is premature to conclude with Goyal and Welch (2006) that predictive regressions
cannot protably be used by investors in real time. A regression estimated over a
short sample period can easily generate perverse results, such as a negative coe!cient
when theory suggests that the coe!cient should be positive. Since out-of-sample
forecast evaluation begins as little as 20 years after a predictor variable rst becomes
available, while the historical mean return is estimated from the beginning of our
data set, this can be an important problem in practice. For example, in the early
1930s the earnings-price ratio was very high, but the coe!cient on the predictor was
estimated to be negative. This led to a negative forecast of the equity premium in
the early 1930s and subsequent poor forecast performance. In practice, an investor
would not use a perverse coe!cient but would likely conclude that the coe!cient is
zero, in e"ect imposing prior knowledge on the output of the regression.
In the remaining columns of Table 1, we explore the impact of imposing sensible
restrictions on the out-of-sample forecasting exercise. In the sixth column, we set
the regression coe!cient to zero whenever it has the “wrong” sign (di"erent from the
theoretically expected sign estimated over the full sample). In the seventh column,
we assume that investors rule out a negative equity premium, and set the forecast to
zero whenever it is negative. In the eighth and nal column, we impose rst the sign
restriction on the coe!cient, and then the sign restriction on the forecast.
These restrictions never worsen and almost always improve the out-of-sample performance of our predictive regressions. With no restrictions, as we noted above,
only earnings-based valuation ratios have a positive out-of-sample !2 , but the slope
restriction delivers a positive out-of-sample !2 for the dividend yield and the sign
restriction brings the out-of-sample !2 close to zero for the book-to-market ratio.
6
The sign restriction also delivers a positive out-of-sample !2 for the long-term bond
yield and the consumption-wealth ratio.
Figure 1 illustrates the e"ect of the restrictions for the dividend-price ratio. The
top panel shows annualized excess return forecasts based on the full-sample OLS
regression coe!cient, the rolling (“out-of-sample”) OLS regression coe!cient without
restrictions, and the out-of-sample coe!cient with both coe!cient and forecast sign
restrictions. The bottom panel shows the cumulative out-of-sample !2 for these three
forecasts. The coe!cient sign restriction signicantly improves the forecasts in the
1930s, when the coe!cient was estimated to be negative. The forecast restrictions
bind periodically during the 1960s and 1990s. Valuation ratios were unusually low
during these periods, leading to unprecedentedly low forecasts. Campbell and Shiller
(2001) also noted the unusually low valuation ratios of the 1990s, and wrote “We
do not nd this extreme forecast credible; when the independent variable has moved
so far from the historically observed range, we cannot trust a linear regression line.”
Our forecast restrictions are a simple way to avoid such incredible forecasts.
Panel B of Table 1 reports comparable results for annual regressions, estimated
using overlapping monthly data. In-sample "-statistics are corrected for serial correlation and are even lower than those reported in Panel A for monthly data. Despite this
weak in-sample predictive power, these regressions perform quite well out-of-sample.
When both our theoretical restrictions are imposed, all four valuation ratios and the
three variables based on the Treasury yield curve have out-of-sample !2 statistics of
at least 2%. Net equity issuance and the consumption-wealth ratio also benet from
our restrictions but do not beat the historical mean return at the annual frequency.
1.2
Quantitative coe!cient restrictions for valuation ratios
So far we have used theory only to impose weak sign restrictions on coe!cients and
excess return forecasts. We now go further and impose the restrictions of a simple
steady-state model on the coe!cients for valuation ratios. This approach can be
interpreted as a conditional version of the procedure advocated by Fama and French
(2002) to estimate the unconditional average mean return. It can also be interpreted
as a way to combine the information from multiple data sources, for example, on
market valuations, corporate protability, and payout policy, without increasing the
number of free coe!cients that must be estimated.
7
We start from the Gordon (1962) growth formula,
%
= ! ! '$
&
(2)
which describes the dividend-price ratio in a steady state with a constant discount rate
and dividend growth rate. We combine this formula with the steady-state relation
between growth and accounting return on equity,
' = (1 !
%
)!)($
(
(3)
where %*( is the payout ratio, to obtain a growth-adjusted return forecast:
b%& = % + (1 ! % )!)(+
!
&
(
(4)
The same approach can be used to obtain growth forecasts from the earnings yield.
Using the fact that %*& = (%*()((*& ) we have:
µ ¶
b'& = % ( + (1 ! % )!)($
!
(5)
( &
(
a payout-ratio-weighted average of the earnings yield and the accounting return on
equity. When return on equity equals the expected return, as might be the case in
b'& = (*& .
long-run equilibrium, then this implies that !
Finally, since (*& = (,*-)!)($ we have:
µ
¶¸
!
,
%
b() = !)( 1 +
!1 +
!
( -
(6)
To use these formulas in practice, one must decide how to combine historical and
contemporaneous data on the right-hand side variables. We follow Fama and French
(2002) by using historical average data on payouts and protability, but di"er from
them by using current rather than historical average data on valuation ratios to obtain
a return forecast conditional on the market’s current valuation level. This procedure
assumes that movements in valuation ratios, relative to historical cash ows, are
explained by permanent changes in expected returns. It is a compromise between the
view that valuation ratios are driven by changing forecasts of protability, in which
8
case the implied movements in returns would be smaller, and the view that valuation
ratios are driven by temporary changes in discount rates, in which case the implied
return movements would be larger [Campbell and Shiller (1988a)].7
Table 2 reports regression forecasts using both the unadjusted valuation ratios
from Table 1, and two variants of growth-adjusted ratios. Our rst set of growthadjusted ratios uses Equations (4), (5), and (6) with current data on the valuation
ratios and historical data on the payout ratio (an average from the beginning of
the sample) and the return on equity (a ten-year moving average as in Table 1).8
Our second set of growth-adjusted ratios in addition subtracts the historical average
real interest rate from the beginning of the sample period in order to convert a
theoretical real return forecast into a theoretical excess return forecast. The historical
average real interest rate is extremely stable, so this nal step is close to an intercept
adjustment.9
We use these ratios in four di"erent ways. First, we report unrestricted regressions
of returns on the ratios. Second, we report regressions restricted as in Table 1 to have
positive slope coe!cients and positive return forecasts. Third, given that valuation
ratios are positive, we can obtain reasonable return forecasts merely by bounding
the intercept above zero and the slope coe!cient to lie between zero and one. This
is similar to the second approach but can be implemented by coe!cient restrictions
rather than by restricting forecasts directly. Finally, we impose the restrictions of the
steady-state theory by restricting the intercept of the regression to be zero and the
slope coe!cient to be one. That is, we estimate the return forecast directly from the
data without using any historical information on the covariance between returns and
valuation ratios.10
Table 2 shows that the last and most restrictive approach delivers the best outof-sample performance in monthly data. The out-of-sample !2 statistics range from
-0.66% to 0.32% when no restrictions are imposed, from -0.45 to 0.43% when the
restrictions of Table 1 are imposed, and from 0.24% to 0.97% when the zero-intercept
and unit-slope restrictions are imposed. These restrictions improve the out-of-sample
monthly forecasting power of every predictive regression we consider. The out-ofsample !2 statistics are also reliably positive in annual regressions that impose zerointercept and unit-slope restrictions, ranging from 1.85% to 7.99%, but here the theoretical restrictions worsen out-of-sample predictive power in a few cases. The most
striking example is the dividend-price ratio with no growth adjustment, where the
zero-intercept restriction is the least theoretically appealing as it e"ectively assumes
9
zero real growth in dividends.
Goyal and Welch (2006) emphasize that out-of-sample performance has been particularly poor for predictive regressions in the last few decades. To investigate this
issue, Table 3 repeats Table 2 for three subsamples: 1927-1956, 1956-1980, and 19802005. We choose the rst subsample to end twenty years after we are rst able to
calculate ten-year smoothed ROE, so that growth adjustments in the second and
third subsamples are entirely based on ROE and not on historical earnings growth.
We omit the book-to-market ratio in the rst subsample, as the series does not become available until 1926 and thus we cannot estimate coe!cients before the rst
subsample begins.
As Goyal and Welch (2006) have noted, the ability of valuation ratios to forecast
stock returns is strongest in the rst subsample. which includes the Great Depression
and World War II, slightly weaker in the second subsample, which includes the postwar boom and the oil shocks of the 1970s, and weakest in the third subsample, which
includes the great equity bull market at the end of the twentieth century. Even in the
third subsample, however, earnings-based valuation ratios outperform the historical
mean return in forecasting future monthly stock returns. The outperformance is particularly convincing when the ratios are adjusted for growth, in which case it can be
achieved either by imposing a zero intercept and a unit slope or by merely imposing
a positive intercept and a slope between zero and one. Results are more mixed in
annual data during the third subsample, but here too earnings-based valuation ratios
perform reasonably well. The relatively weak performance of the dividend-price ratio
in this period is consistent with the observation of Boudoukh, Michaely, Richardson,
and Roberts (2007), who nd that in recent years share repurchases have played a
more important role in total payouts to shareholders.
Figure 2 compares the historical mean stock return with four forecasts based on
the smoothed earnings yield: the unrestricted rolling regression forecast, the zerointercept unit-slope forecast based on the smoothed earnings yield with no adjustments, and the zero-intercept unit-slope forecasts based on the two growth-adjusted
versions of the smoothed earnings yield. The theoretically restricted forecasts are
more stable over time and have superior out-of-sample !2 statistics as illustrated in
the bottom panel of the gure. Although there is a very gradual decline over time in
the out-of-sample !2 statistics of these forecasts, this does not mean that the forecasts are not working in the later years of the sample; it simply means that they are
not beating the historical mean forecast as decisively as they did in earlier years.
10
2
How Large an !2 Should We Expect?
In the previous section, we showed that many of the forecasting variables that have
been discussed in the literature do have positive out-of-sample predictive power for
aggregate U.S. stock returns, when reasonable restrictions are imposed on the predictive regression. However, the !2 statistics are small in magnitude. This raises the
important question of whether they are economically meaningful.
To explore this issue, consider the following example:
#$+1 = . + /$ + 0$+1 $
(7)
where #$+1 is the excess simple return on a risky asset over the riskless interest rate, .
is the unconditional average excess return, /$ is a predictor variable with mean zero
and constant variance 12* , and 0$+1 is a random shock with mean zero and constant
variance 1 2+ . For tractability, consider an investor with a single-period horizon and
mean-variance preferences. The investor’s objective function is expected portfolio
return less (2*2) times portfolio variance, where 2 can be interpreted as the coe!cient
of relative risk aversion.11 If the investor does not observe /$ , the investor chooses a
portfolio weight in the risky asset:
¶
µ ¶µ
.
1
(8)
3$ = 3 =
2
1 2* + 1 2+
and earns an average excess return of:
¶
µ ¶µ
.2
42
1
$
=
2
1 2* + 1 2+
2
(9)
where 4 is the unconditional Sharpe ratio of the risky asset.
If the investor observes /$ , the investor sets:
¶
µ ¶µ
. + /$
1
$
3$ =
2
1 2+
(10)
where the denominator is now 12+ rather than 1 2* + 12+ because the variation in the
predictor variable /$ is now expected and does not contribute to risk. The investor
earns an average excess return of:
¶ µ ¶µ 2
¶
µ ¶µ 2
. + 1 2*
4 + !2
1
1
=
$
(11)
2
1 2+
2
1 ! !2
11
where
12*
! = 2
1 * + 1 2+
2
(12)
is the !2 statistic for the regression of excess return on the predictor variable /$ .
The di"erence between the two expected returns is:
¶
µ ¶µ
!2
1
(1 + 4 2 )$
2
1 ! !2
(13)
which is always larger than !2 *2, and is close to !2 *2 when the time interval is short
and !2 and 4 2 are both small. The proportional increase in the expected return from
observing /$ is:
µ
¶µ
¶
!2
1 + 42
$
(14)
1 ! !2
42
which is always larger than !2 *4 2 and is close to !2 *4 2 when the time interval is
short and !2 and 4 2 are both small.
This analysis shows that the right way to judge the magnitude of !2 is to compare
it with the squared Sharpe ratio 4 2 . If !2 is large relative to 4 2 , then an investor
can use the information in the predictive regression to obtain a large proportional
increase in portfolio return. In our monthly data since 1871, the monthly Sharpe
ratio for stocks is 0.108, corresponding to an annual Sharpe ratio of 0.374. The
squared monthly Sharpe ratio is 4 2 = 0+012 = 1+2%. This can be compared with
the monthly out-of-sample !2 statistic for, say, the smoothed earnings-price ratio of
0.43% in the last column of Panel A of Table 1. A mean-variance investor can use
the smoothed earnings-price ratio to increase average monthly portfolio return by
a proportional factor of 0.43/1.2 = 36%. The absolute increase in portfolio return
depends on risk aversion, but is about 43 basis points per month or 5.2% per year for
an investor with unit risk aversion, and about 1.7% per year for an investor with a
risk aversion coe!cient of three. The calculation can also be done in the annual data
shown in Panel B of Table 1, comparing the squared annual Sharpe ratio of 11.8%
to the annual out-of-sample !2 statistic for, say, the smoothed earnings-price ratio of
7.9%. A mean-variance investor can use the smoothed earnings-price ratio in annual
data to increase average annual portfolio return by a proportional factor of 7+9*11+8 =
67%. The absolute increase in portfolio return is 9.6% per year for an investor with
unit risk aversion, and 3.2% per year for an investor with a risk aversion coe!cient
of three.
12
The investor who observes /$ gets a higher portfolio return in part by taking on
greater risk. Thus the increase in the average return is not pure welfare gain for
a risk-averse investor. To take account of this, in Table 4 we calculate the welfare
benets generated by optimally trading on each predictor variable for an investor with
a relative risk aversion coe!cient of three. We impose realistic portfolio constraints,
preventing the investor from shorting stocks or taking more than 50% leverage: that
is, conning the portfolio weight on stocks to lie between 0% and 150%. The investor’s
optimal portfolio depends on an estimate of stock return variance at each point in
time, and we assume that the investor estimates variance using a rolling ve-year
window of monthly data.12 We report the change in utility, relative to investing with
the historical mean forecast of the equity premium, for each of the models considered
in Table 3. These utility di"erences have the units of expected annualized return,
so they can be interpreted as the transactions costs or portfolio management fees
that investors would be prepared to pay each year to exploit the information in the
predictor variable.
The utility gains in Table 4 are broadly consistent with the out-of-sample !2 statistics reported in Table 3. Utility gains are reliably positive using annual returns and
coe!cients xed at theoretical levels: The only utility loss occurs for the unadjusted
dividend-price ratio in the 1980-2005 sample, when share repurchases become important. The results are somewhat more mixed for monthly returns, but here too are
generally positive. The utility gains tend to be greatest in the 1927-1956 subsample,
but by contrast with Table 3, are not always smaller in 1980-2005 than in 1956-1980.
The utility gains reported in Table 4 are limited by the leverage constraint, together with the high average equity premium. Predictable variations in stock returns
do not generate portfolio gains when there is a binding upper limit on equity investment. Utility gains would be larger if we relaxed the portfolio constraint or included
additional assets, with higher average returns than Treasury bills, in the portfolio
choice problem. On the other hand, Table 4 does not take any account of transactions costs. Modest gains from market timing strategies could be o"set by the
additional costs implied by those strategies. Optimal trading strategies in the presence of transactions costs are complex, and so we do not explore this issue further
here. We note that even the baseline strategy based on a historical-mean forecast
incurs rebalancing costs and that utility gains of 50 basis points per year, which are
commonly achieved in Table 4, are su!cient to cover substantial additional costs.
Since small !2 statistics can generate large benets for investors, we should expect
13
predictive regressions to have only modest explanatory power. Regressions with large
!2 statistics would be too protable to believe. The saying “If you’re so smart, why
aren’t you rich?” applies with great force here, and should lead investors to suspect
that highly successful predictive regressions are spurious. Since the squared Sharpe
ratio and average real interest rate increase in proportion with the investment horizon,
however, larger !2 statistics are believable at longer horizons. Authors such as Fama
and French (1988) have found that !2 statistics increase strongly with the horizon
when the predictor variable is persistent, a nding that is analyzed in Campbell, Lo,
and MacKinlay (1997, Chapter 7) and Campbell (2001). This behavior is completely
consistent with our analysis and empirical results.
3
Conclusion
A number of variables are correlated with subsequent returns on the aggregate U.S.
stock market in the twentieth century. Some of these variables are stock market valuation ratios, others reect the levels of short- and long-term interest rates, patterns
in corporate nance or the cross-sectional pricing of individual stocks, or the level
of consumption in relation to wealth. Goyal and Welch (2006) argue that in-sample
correlations conceal a systematic failure of these variables out of sample: None have
been able to beat a simple forecast based on the historical average stock return.
In this paper, we have shown that most of these predictor variables perform better out-of-sample than the historical average return forecast, once weak restrictions
are imposed on the signs of coe!cients and return forecasts. The out-of-sample explanatory power is small, but nonetheless is economically meaningful for investors.
We have also explored theoretical restrictions on the coe!cients relating valuation
ratios to future returns. Although these restrictions are based on steady-state models, which do not fully describe asset prices in an economy with changing growth
rates and discount rates, they improve the out-of-sample performance of valuation
forecasts by removing the need to estimate coe!cients from short samples of volatile
stock returns. We nd that even in the period 1980-2005, where return prediction is
most challenging, theoretically restricted valuation models often outperform return
forecasts based on the long-run historical mean of stock returns. These models also
generate meaningful utility gains for mean-variance investors. Importantly, the performance of the models does not depend sensitively on the particular valuation ratio
that is used or the manner in which it is adjusted for long-run growth. We inter14
pret this result as illustrating the well-known econometric principle that even false
theoretical restrictions can be helpful in forecasting if they reduce the variance of a
predictor more than they increase its bias.
We have deliberately kept our methods simple to avoid the charge that we have
tested a wide variety of methods and have only reported those that succeed. The results suggest that further improvements in performance can be achieved by exploring
alternative methods to combine theory with historical data. One obvious possibility is
a Bayesian approach with priors centered either on a model with a constant expected
stock return, as in Wachter and Warusawitharana (2006), or on a valuation model
with constant or slowly changing protability and payout ratios of the sort we have
explored in this paper. Of course, given the quantity of historical data that are now
available, optimal forecasts of stock returns going forward may place greater weight
on the data, and less weight on theoretical restrictions, than those methods that most
successfully predicted stock returns during the twentieth century.
15
Endnotes
Acknowledgements: We are grateful to Jan Szilagyi for able research assistance,
to Amit Goyal and Ivo Welch for sharing their data, and to Malcolm Baker, Lutz
Kilian, Martin Lettau, Sydney Ludvigson, Rossen Valkanov, the editor, and two
anonymous referees for helpful comments on an earlier draft entitled “Predicting the
Equity Premium Out of Sample: Can Anything Beat the Historical Average?” This
material is based upon work supported by the National Science Foundation under
Grant No. 0214061 to Campbell.
1. Goyal and Welch (2006) consider these variables, along with the ratio of lagged
dividends to lagged prices (the “dividend yield” in Goyal and Welch’s terminology).
We drop this variable as there is no reason to believe that it should be a better
predictor than the ratio of lagged dividends to current prices (the “dividend price
ratio”).
2. If earnings obey the clean-surplus relation, the growth rate of real book equity
equals real ROE minus the payout ratio. ROE can alternatively be measured from
the growth rate of book equity rather than from reported accounting earnings, as in
Cohen, Polk, and Vuolteenaho (2006).
3. In 2003, the BEA revised the denitions of several variables used by Lettau
and Ludvigson (2001) to construct their series. We use the updated data available on
Martin Lettau’s website, not the original series used in Lettau and Ludvigson (2001).
Data revisions raise the deeper problem that the series may not have been available
to investors in real time, but we do not try to deal with this issue here.
4. The adjustment of the !2 statistic for degrees of freedom makes only a very
small di"erence in samples of the size used here. The adjustment is about 5 basis
points for a regression starting in 1871, and about 10 basis points for a regression
starting in 1927.
5. Clark and West (2005) point out that if the return series is truly unpredictable,
then in a nite sample the predictive regression will on average have a higher mean
squared prediction error because it must estimate an additional coe!cient. Thus the
expected out-of-sample !2 under the null of unpredictability is negative, and a zero
out-of-sample !2 can be interpreted as weak evidence for predictability. We do not
pursue this point here because, like Goyal and Welch (2006), we ask whether predictive
regressions or historical average return forecasts have delivered better out-of-sample
16
forecasts, not whether stock returns are truly predictable.
6. Earlier drafts of this paper reported better out-of-sample performance for this
series. BEA data revisions in 2003 and the addition of recent years to the forecast
evaluation period are responsible for the change in results.
7. Cochrane (2006) also emphasizes the importance of the fact that valuation
ratios do not forecast growth rates of cash ows. The ability of the theoretical models
in this section to predict stock returns is consistent with Cochrane’s results.
8. Before 1926, we do not have data on ROE and thus we cannot calculate tenyear smoothed ROE until 1936. Before that date, we use real earnings growth to
estimate '.
9. One could consider forecasting the short-term real interest rate using recent
data on ination and nominal interest rates. There are two reasons not to pursue this
approach. First, the volatility of ination makes it di!cult in practice. Second, the
steady-state growth model delivers a long-run forecast of stock returns that should be
matched to a long-run forecast of real bond returns, such as the TIPS yield or (since
this is not available before 1997) the long-run historical average real interest rate. If
the short-term real interest rate exceeds such a long-run real rate forecast, then it is
quite possible that the short-term real stock return shifts upwards in parallel, leaving
the excess stock return forecast unchanged.
10. A subtle but important issue is that we use the steady-state model to forecast
the arithmetic average stock return, and take arithmetic averages of ROE and other
historical data. Some authors, such as Siegel (1994), take geometric averages of
historical data and forecast the geometric average stock return. These two approaches
are equivalent if the volatility of dividend growth and stock returns are the same, as
implied by the steady-state model, but are di"erent in the data because stock returns
are much more volatile than dividend growth and ROE. A full exploration of the two
approaches is beyond the scope of this paper, but we note that the Siegel approach
would generate higher forecasts of arithmetic average stock returns, and thus would
perform even better in the late twentieth century than the approach we use here.
11. Merton (1969) presents the analogous portfolio solution for the case where the
investor has power utility with relative risk aversion 2, asset returns are lognormally
distributed, and the portfolio can be continuously rebalanced. Campbell and Viceira (2002, Chapter 2) use a discrete-time approximate version of Merton’s solution.
17
Sentana (2005) also explores the relation between regression forecasts and optimal
portfolio construction.
12. In an earlier draft we reported similar results for the case where the investor
estimates variance using all available historical data. If these variance estimates are
incorrect, for example because the predictor variable forecasts variance as well as
return, this will reduce the utility generated by trading on the predictive variable.
18
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22
Table 1: Excess return prediction with regression constraints
Sample
Begin
Forecast
Begin
In-Sample
t-statistic
In-Sample
R-squared
Out-of-Sample R-squared with Different Constraints
Unconstrained
Positive Slope
Pos. Forecast
Both
Panel A: Monthly Returns
Dividend-price ratio
Earnings-price ratio
Smooth earnings-price ratio
Book-to-Market
ROE
T-Bill rate
Long-term yield
Term spread
Default spread
Inflation
Net equity issuance
Consumption-wealth ratio
1872m2
1872m2
1881m2
1926m6
1936m6
1920m1
1870m1
1920m1
1919m1
1871m5
1927m12
1951m12
1927m1
1927m1
1927m1
1946m6
1956m6
1940m1
1927m1
1940m1
1939m1
1927m1
1947m12
1971m12
1.25
2.29
1.85
1.96
0.36
2.44
1.46
2.16
0.74
0.39
1.74
4.57
1.13%
0.71
1.36
0.61
0.02
0.86
0.19
0.65
0.10
0.06
0.48
2.60
-0.65%
0.12
0.33
-0.43
-0.93
0.52
-0.19
0.46
-0.19
-0.22
0.34
-1.36
0.05%
0.18
0.42
-0.43
-0.06
0.51
-0.19
0.47
-0.19
-0.21
0.34
-1.36
0.07%
0.14
0.38
0.00
-0.93
0.57
0.20
0.45
-0.19
-0.18
0.50
0.27
0.08%
0.18
0.43
0.00
-0.06
0.55
0.20
0.46
-0.19
-0.17
0.50
0.27
Panel B: Annual Returns
Dividend-price ratio
Earnings-price ratio
Smooth earnings-price ratio
Book-to-Market
ROE
T-Bill rate
Long-term yield
Term spread
Default spread
Inflation
Net equity issuance
Consumption-wealth ratio
1872m2
1872m2
1881m2
1926m6
1936m6
1920m1
1870m1
1920m1
1919m1
1871m5
1927m12
1951m12
1927m1
1927m1
1927m1
1946m6
1956m6
1940m1
1927m1
1940m1
1939m1
1927m1
1947m12
1971m12
2.69
2.84
3.01
1.98
0.35
1.77
0.91
1.72
0.07
0.17
0.54
3.76
10.89%
6.78
13.57
8.26
0.32
4.26
0.77
3.10
0.01
0.07
0.35
19.87
5.53%
4.93
7.89
-3.38
-8.60
5.54
-0.15
4.79
-3.81
-0.71
-4.27
-7.75
5.53%
4.93
7.89
-3.38
-0.03
5.54
-0.15
4.79
-3.81
-0.71
-4.27
-7.75
5.63%
4.94
7.85
1.39
-8.35
7.47
2.26
4.74
-3.81
-0.71
-2.38
-1.48
5.63%
4.94
7.85
1.39
-0.03
7.47
2.26
4.74
-3.81
-0.71
-2.38
-1.48
This table presents statistics on forecast errors for stock returns. We use S&P500 total returns (dividend included) where data prior to January 1927 was obtained from
Robert Shiller's website. "Sample Begin" denotes when the predictor was first available. All statistics are for the period that starts at "Forecast Begin" and ends on
December 31, 2005 (for monthly forecasts) or December 31, 2004 (for annual forecasts). The "Unconstrained" out-of-sample R-squared compares the forecast error of the
historical mean versus the forecast from unconstrained ordinary least squares. "Positive Slope" introduces the restriction that the coefficient on the predictor must be of the right
sign, otherwise the historical mean is used as a predictor instead. "Pos. Forecast" requires that the prediction be positive, otherwise we use zero as the forecast. "Both"
indicates that we impose both restrictions. The in-sample t-statistic and R-squared both come from the last regression (i.e., over the whole sample from the "Sample Begin"
date to December 2005 for monthly data and December 2004 for annual data). The "In-Sample t statistic" is heteroskedasticity robust. The Annual forecasts in Panel B are
based on 12 overlapping annual returns per year. The t-statistics in Panel B are corrected to take into account correlation induced by the overlapping nature of the
dependent variable. For the consumption-wealth ratio, we report the robust F-statistic that all the slope coefficients are zero.
Table 2: Excess return prediction with valuation constraints
Out-of-Sample R-squared with Different Constraints
Positive Slope,
Pos. Intercept,
Unconstrained
Pos. Forecast
Bounded Slope
Fixed Coefs
Sample
Begin
Forecast
Begin
In-Sample
t-statistic
In-Sample
R-squared
Panel A: Monthly Returns
Dividend/price
Earnings/price
Smooth earnings/price
Dividend/price + growth
Earnings/price + growth
Smooth earnings/price + growth
Book-To-Market + growth
Dividend/price + growth - real rate
Earnings/price + growth - real rate
Smooth earnings/price + growth - real rate
Book-To-Market + growth - real rate
1872m2
1872m2
1881m2
1891m2
1892m2
1892m2
1936m6
1891m5
1892m2
1892m2
1936m6
1927m1
1927m1
1927m1
1927m1
1927m1
1927m1
1956m6
1927m1
1927m1
1927m1
1956m6
1.25
2.28
1.85
1.40
1.82
2.00
1.61
1.47
1.53
1.97
1.68
1.12%
0.71
1.35
1.03
0.49
1.10
0.33
0.86
0.36
0.84
0.36
-0.66%
0.12
0.32
-0.05
-0.05
0.11
-0.35
-0.02
0.00
0.15
-0.45
0.08%
0.18
0.43
0.20
0.08
0.25
-0.34
0.21
0.12
0.26
-0.45
0.19%
0.25
0.43
0.17
0.07
0.21
-0.34
0.18
0.09
0.23
-0.42
0.42%
0.76
0.97
0.63
0.57
0.72
0.33
0.41
0.39
0.52
0.24
Panel B: Annual Returns
Dividend/price
Earnings/price
Smooth earnings/price
Dividend/price + growth
Earnings/price + growth
Smooth earnings/price + growth
Book-To-Market + growth
Dividend/price + growth - real rate
Earnings/price + growth - real rate
Smooth earnings/price + growth - real rate
Book-To-Market + growth - real rate
1872m2
1872m2
1881m2
1891m2
1892m2
1892m2
1936m6
1891m5
1892m2
1892m2
1936m6
1927m1
1927m1
1927m1
1927m1
1927m1
1927m1
1956m6
1927m1
1927m1
1927m1
1956m6
2.69
2.84
3.01
1.77
1.42
1.75
1.97
1.46
1.13
1.53
2.03
10.89%
6.78
13.57
9.30
4.44
10.45
5.45
7.69
3.27
7.90
5.77
5.53%
4.93
7.89
2.49
1.69
3.16
-3.53
2.87
2.01
3.35
-1.73
5.63%
4.94
7.85
2.99
2.11
3.33
-0.64
3.24
2.05
3.35
-1.12
3.76%
4.34
6.44
2.67
1.80
3.23
-2.39
2.95
2.04
3.38
-1.82
2.20%
5.87
7.99
4.35
3.89
5.39
3.63
1.89
1.85
3.22
2.33
The table presents forecast statistics for value predictors under various constraints. The predictor label "+ growth" indicates that we add an earnings growth forecast to the
predictor. The predictor label "- real rate" indicates that we subtract a forecast of the real risk-free rate from the predictor. See the text for details. The "In-Sample" statistics are
defined as in Table 1. The "Unconstrained" and "Positive Slope, Pos. Forecast" columns are described in Table 1. "Pos. Intercept, Bounded Slope" indicates that we constrain
the intercept to be positive and the slope to be between zero and one. "Fixed Coefs" indicates that we fix the intercept at zero and the slope at one.
Table 3: Subsample stability
Sample: 1927-1956
Pos.
Intercept,
Bounded
Unconstrained
Slope
Panel A: Monthly Returns
Dividend/price
Earnings/price
Smooth earnings/price
Dividend/price + growth
Earnings/price + growth
Smooth earnings/price + growth
Book-To-Market + growth
Dividend/price + growth - real rate
Earnings/price + growth - real rate
Smooth earnings/price + growth - real
rate
Book-To-Market + growth - real rate
Panel B: Annual Returns
Dividend/price
Earnings/price
Smooth earnings/price
Dividend/price + growth
Earnings/price + growth
Smooth earnings/price + growth
Book-To-Market + growth
Dividend/price + growth - real rate
Earnings/price + growth - real rate
Smooth earnings/price + growth - real
rate
Book-To-Market + growth - real rate
Fixed
Coefs
-0.86%
0.16
0.56
-0.15
-0.06
0.09
0.21%
0.28
0.53
0.18
0.12
0.25
0.63%
1.04
1.33
0.78
0.73
0.93
-0.01
0.06
0.30
0.20
0.45
0.41
0.27
0.39
9.95
7.45
12.51
2.77
2.21
3.73
Sample: 1956-1980
Pos.
Intercept,
Bounded
Unconstrained
Slope
Fixed
Coefs
Sample: 1980-2005
Pos.
Intercept,
Bounded
Unconstrained
Slope
Fixed
Coefs
0.88%
0.56
0.80
0.18
-0.12
0.19
-0.62
-0.24
-0.34
0.57%
0.45
0.48
0.18
-0.12
0.19
-0.73
-0.24
-0.34
0.67%
0.30
0.51
0.59
0.33
0.47
0.73
0.76
0.66
-1.30%
-0.53
-1.06
0.11
0.05
0.06
-0.12
0.11
0.06
-0.21%
-0.09
-0.06
0.11
0.05
0.06
-0.02
0.11
0.06
-0.54%
0.07
0.01
0.14
0.16
0.16
0.00
-0.08
0.03
0.60
-0.28
-0.82
-0.28
-0.91
0.74
0.89
0.04
-0.14
0.04
-0.02
0.02
-0.27
4.53
5.34
8.22
3.05
2.38
3.87
3.67
7.58
10.49
4.83
4.37
6.38
4.40
3.44
4.51
3.49
1.67
1.25
9.46
5.08
4.93
1.76
-0.85
1.27
-7.09
-3.57
-4.68
5.99
3.25
3.71
1.74
-0.85
1.19
-5.16
-3.56
-4.68
6.88
2.56
3.71
6.61
3.97
4.65
10.34
8.28
7.16
-16.19
-6.06
-8.86
1.87
1.63
2.30
-0.24
2.19
1.88
-1.38
0.88
1.33
1.82
1.63
2.23
0.14
2.19
1.88
-7.98
1.47
1.33
0.28
1.60
1.81
-2.43
-2.95
-0.64
5.34
5.37
3.19
-4.91
-3.36
-4.84
-4.22
7.32
11.85
2.36
-0.25
2.36
0.35
-0.47
-6.20
The table provides out-of-sample R-squared statistics from predicting the equity premium with a forecasting variable versus the historical mean. The subsamples roughly but not
exactly divide the data into thirds. The column labels "Unconstrained," "Pos. Intercept, Bounded Slope," and "Fixed Coefs" are all defined in Tables 1 and 2. We do not provide
results for the "Book-To-Market + Growth" predictor in the 1927-1956 subsample because we do not have 20 years of data until 1956.
Table 4: Portfolio choice
Sample: 1927-1956
Pos.
Intercept,
Bounded
Unconstrained
Slope
Panel A: Monthly Returns
Dividend/price
Earnings/price
Smooth earnings/price
Dividend/price + growth
Earnings/price + growth
Smooth earnings/price + growth
Book-To-Market + growth
Dividend/price + growth - real rate
Earnings/price + growth - real rate
Smooth earnings/price + growth - real
rate
Book-To-Market + growth - real rate
Panel B: Annual Returns
Dividend/price
Earnings/price
Smooth earnings/price
Dividend/price + growth
Earnings/price + growth
Smooth earnings/price + growth
Book-To-Market + growth
Dividend/price + growth - real rate
Earnings/price + growth - real rate
Smooth earnings/price + growth - real
rate
Book-To-Market + growth - real rate
Fixed
Coefs
-0.03%
0.86
0.53
-0.28
-1.05
-0.47
0.43%
1.42
1.14
0.64
0.36
0.77
0.11%
1.80
1.89
0.43
1.60
0.89
0.42
-0.31
0.96
0.61
-0.74
0.79
0.36
1.14
0.55
1.29
1.81
-0.18
-0.14
0.05
Sample: 1956-1980
Pos.
Intercept,
Bounded
Unconstrained
Slope
Fixed
Coefs
Sample: 1980-2005
Pos.
Intercept,
Bounded
Unconstrained
Slope
Fixed
Coefs
1.93%
0.28
0.75
0.46
0.08
0.04
0.43
0.01
-0.17
0.92%
0.12
0.05
0.46
0.08
0.04
0.15
0.01
-0.17
1.46%
-0.51
-0.04
0.64
-0.21
0.13
1.38
1.49
0.73
-3.69%
-0.80
-2.73
0.22
0.00
0.04
0.08
0.04
-0.09
-0.22%
0.07
0.13
0.22
0.00
0.06
-0.02
0.04
-0.09
-1.32%
0.74
0.46
0.34
0.19
0.24
0.23
0.18
0.17
-0.20
-0.38
0.38
-0.38
0.05
1.07
1.94
0.04
-0.04
0.04
-0.10
0.21
-0.11
0.76
1.30
1.91
0.34
0.28
0.53
0.34
0.78
1.42
1.04
1.52
1.34
0.44
0.34
0.66
0.49
0.02
0.60
1.56
1.41
1.53
0.35
0.15
0.28
-0.01
0.01
-0.09
0.59
0.41
0.44
0.35
0.15
0.25
0.14
0.01
-0.09
0.95
0.40
0.44
0.70
0.49
0.51
1.27
1.06
0.85
-4.74
-1.44
-3.06
0.19
0.04
0.41
-0.10
0.06
-0.02
0.34
0.53
0.81
0.17
0.04
0.36
-0.13
0.06
-0.02
-0.95
0.68
0.81
0.34
0.24
0.50
0.26
0.23
0.15
0.68
0.87
0.39
-0.20
0.25
-0.18
0.13
0.94
1.93
0.24
-0.17
0.24
-0.14
0.36
0.26
This table presents out-of-sample portfolio choice results. The numbers are the change in average utility from forecasting the market with the predictor instead of the historical mean.
All numbers are annualized, so we multiply the monthly numbers by 12. "Unconstrained" indicates that we use the unconstrained OLS predictor of the equity premium. "Pos.
Intercept, Bounded Slope" indicates that we use the forecast with the intercept bounded above zero and the slope bounded between zero and one. "Fixed Coefs" indicates that we
use the forecast that sets the intercept to zero and the slope to one. The utility function is E(Rp) - (!/2)Var(Rp), where Rp is the portfolio return and !=3. All utility changes are
annualized, so we multiply monthly utility changes by 12.
Figure legends
Figure 1: Forecasting excess returns with the dividend yield. The top panel shows the
historical annualized excess return forecasts for the historical mean and three different
regression models, as labeled in the figure, where “oos” refers to “out-of-sample” and
“ols” refers to ordinary least squares. The bottom panel shows the cumulative out-ofsample R-squared up to each point in the historical sample, for the three regression
models relative to the historical mean.
Figure 2: Forecasting excess returns with the smoothed earnings yield. The top panel
shows the historical annualized excess return forecasts for the historical mean, an
unrestricted regression on the smoothed earnings yield, and three theoretically restricted
forecasts from the smoothed earnings yield, as labeled in the figure. The bottom panel
shows the cumulative out-of-sample R-squared up to each point in the historical sample,
for the four models that use the earnings yield relative to the historical mean.