Expected Returns and Expected Dividend Growth?
Martin Lettau
New York University and CEPR
Sydney C. Ludvigson
New York University
Comments Welcome
First draft: July 24, 2001
This draft: October 31, 2002
?Lettau: Department of Finance, Stern School of Business, New York University, 44 West Fourth Street,
New York, NY 10012-1126; Email: mlettau@stern.nyu.edu, Tel: (212) 998-0378; Fax: (212) 995-4233;
http://www.stern.nyu.edu/? mlettau. Ludvigson: Department of Economics, New York University, 269
Mercer Street, 7th Floor, New York, NY 10003; Email: sydney.ludvigson@nyu.edu, Tel: (212) 998-8927;
Fax: (212) 995-4186; http://www.econ.nyu.edu/user/ludvigsons/. Lettau acknowledges ?nancial sup-
port from the National Science Foundation. Ludvigson acknowledges ?nancial support from the Alfred P.
Sloan Foundation and from the National Science Foundation. We thank John Y. Campbell, Mark Gertler,
Anthony Lynch, Lucrezia Reichlin, Peter Schotman, Charles Steindel and seminar participants at Duke
University, Ohio State University, the New School for Social Research, New York University, the University
of Iowa, the University of Maryland, the University of Montreal, Yale, the SITE 2001 summer conference,
and the CEPR Summer 2002 Finance Symposium for helpful comments, and Nathan Barczi for excellent
research assistance. Any errors or omissions are the responsibility of the authors.
Expected Returns and Expected Dividend Growth
Abstract
We develop a consumption-based present value relation that is a function of future div-
idend growth. Using data on aggregate consumption and measures of the dividend pay-
ments from aggregate wealth, we show that changing forecasts of dividend growth make
an important contribution to ?uctuations in the U.S. stock market, despite the failure of
the dividend-price ratio to uncover such variation. In addition, these dividend forecasts
are found to covary with changing forecasts of excess stock returns. The variation in ex-
pected dividend growth we uncover is positively correlated with ?business cycle?variation
in expected returns, and the results suggest that a substantial fraction of the variation in
expected dividend growth is common to variation in expected excess returns. Movements in
expected dividend growth that are entirely common to movements in expected returns have
no e?ect on the log dividend-price ratio.
An implication of these ?ndings is that the log dividend-price ratio will have di?culty
predicting both dividend growth and excess returns at business cycle frequencies. Such a
failure of predictive power is not an indication that risk-premia are constant, however. On
the contrary, the results presented here imply that the log dividend-price ratio will have
di?culty revealing business cycle variation in both the equity risk-premium and expected
dividend growth precisely because expected returns ?uctuate at those frequencies, and covary
with changing forecasts of dividend growth. The ?ndings imply that both the market risk-
premium and expected dividend growth vary considerably more than what can be revealed
using the log dividend-price ratio alone as a predictive variable.
JEL: G12, G10.
1 Introduction
One does not have to delve far into recent surveys of the asset pricing literature to uncover
a few key empirical results that have come to represent stylized facts, part of the ?standard
view? of U.S. aggregate stock market behavior.
1. Large predictable movements in dividends are not apparent in U.S. stock market data.
In particular, the log dividend-price ratio does not have important long horizon fore-
casting power for the growth in dividend payments.1
2. Returns on aggregate stock market indexes in excess of a short term interest rate
are highly forecastable over long horizons. The log dividend-price ratio is extremely
persistent and forecasts excess returns over horizons of many years.2
3. Variance decompositions of dividend-price ratios show that changing forecasts of future
excess returns comprise almost all of the variation in dividend-price ratios. These ?nd-
ings form the basis for the conclusion that expected dividend growth is approximately
constant.3
Empirical evidence on the behavior of the dividend-price ratio has transformed the way
?nancial economists perceive asset markets. It has replaced the age-old view that expected
returns are approximately constant, with the modern-day view that time-variation in ex-
pected returns constitutes an important part of aggregate stock market variability. Even the
extraordinary behavior of stock prices during the late 1990s has not unraveled this trans-
formation. Academic researchers have responded to this episode by emphasizing that?in
contrast to stock market dividends?movements in aggregate stock prices have always played
an important role historically in restoring the dividend-price ratio to its mean, even though
the persistence of the dividend-price ratio implies that such restorations can sometimes
take many years to materialize (Heaton and Lucas (1999); Campbell and Shiller (2001);
Cochrane (2001), Ch. 20; Fama and French (2002); Campbell (2001); Lewellen (2001)).
1A large literature documents the poor predictability of dividend growth by the dividend-price ratio over
long horizons, for example, Campbell (1991); Cochrane (1991); Cochrane (1994); Cochrane (1997); Campbell
(2001); Cochrane (2001). Ang and Bekaert (2001) ?nd somewhat stronger predictability; we discuss their
results further below.
2See Fama and French (1988), Campbell and Shiller (1988); Hodrick (1992); Campbell, Lo, and MacKinlay
(1997); Cochrane (1997); Cochrane (2001), Ch. 20; Campbell (2001).
3See Campbell (1991); Cochrane (1991); Hodrick (1992); Campbell, Lo, and MacKinlay (1997), Ch. 7;
Campbell (2001); Cochrane (2001), Ch. 20.
3
These researchers maintain that, despite the market's unusual behavior recently, changing
forecasts of excess returns make important contributions to ?uctuations in the aggregate
stock market.
Yet there are noticeable cracks in the standard academic paradigm of predictability based
on the dividend-price ratio. On the one hand, several researchers, focusing primarily on fore-
casting horizons less than a few years, have argued that careful statistical analysis provides
little evidence that the log dividend-price ratio forecasts returns (for example, Nelson and
Kim (1993); Stambaugh (1999); Ang and Bekaert (2001); Valkanov (2001)). These ?nd-
ings have led some to conclude that changing forecasts of excess returns make a negligible
contribution to ?uctuations in the aggregate stock market.
On the other hand, other researchers have found that excess returns on the aggregate
stock market are strongly forecastable at horizons far shorter than those over which the
persistent dividend-price ratio predominantly varies. Lettau and Ludvigson (2001a) ?nd
that excess stock returns are forecastable at horizons over which the dividend-price ratio has
comparably weak forecasting power, namely at ?business cycle? frequencies, those ranging
from a few quarters to several years. Such predictable variation in returns is revealed not
by the dividend-price ratio, but instead by an empirical proxy for the log consumption-
wealth ratio, denoted cayt, a variable that captures deviations from the common trend in
consumption, asset (nonhuman) wealth and labor income. The consumption-wealth variable
cayt is less persistent than the dividend-price ratio, consistent with the ?nding that the
former forecasts returns over shorter horizons than latter.
Taken together, these empirical ?ndings raise a series of puzzling questions. Do the
intermediate horizon statistical analyses using the dividend-price ratio imply that expected
excess returns are approximately constant? If so, then why does cayt have predictive power
for excess returns at horizons ranging from a few quarters to several years? Moreover, if
business cycle variation in expected returns is present, why does the dividend-price ratio
have di?culty capturing this variation?
This paper argues that it is possible to make sense of these seemingly contradictory
?ndings and in the process provide empirical answers to the questions posed above. We study
a consumption-based present value relation that is a function of future dividend growth. The
evidence we present has two key elements. First, using data on aggregate consumption and
dividend payments from aggregate (human and nonhuman) wealth, we show that changing
forecasts of stock market dividend growth do make an important contribution to ?uctuations
in the U.S. stock market, despite the failure of the dividend-price ratio to uncover such
4
variation. Although U.S. dividend growth is known to have some short-run forecastability
arising from the seasonality of dividend payments, to our knowledge this study is one of
the few to ?nd important predictability at longer horizons, and in particular at horizons
over which excess stock returns have been found to be forecastable. Second, these dividend
forecasts are found to covary with changing forecasts of excess stock returns. The variation
in expected dividend growth we uncover is positively correlated with business cycle variation
in expected returns, the latter captured by movements in cayt, and our results suggest that a
substantial fraction of its variation is common to variation in expected excess stock returns.
These ?ndings help resolve the puzzles discussed above, for several reasons. First, they
can explain why business cycle variation in expected excess returns is captured by cayt, but
not well captured by the dividend-price ratio. Movements in expected dividend growth that
are common to movements in expected returns have o?setting e?ects on the log dividend-
price ratio. Second, the results help explain why the dividend-price ratio has consistently
been found to be a relatively weak predictor of US dividend growth, despite the evidence pre-
sented here that dividend growth is highly forecastable. Again the reason is that movements
in expected dividend growth that are entirely common to movements in expected returns
have no in?uence on the log dividend-price ratio. Third, although common movements in
expected returns and expected dividend growth have o?setting e?ects on the dividend-price
ratio, such movements will not have o?setting e?ects on the log consumption-wealth ratio, or
on cayt. It follows that cayt should be marked by both business cycle and very long horizon
variation in expected returns, a phenomenon that would make cayt less persistent than the
dividend-price ratio, consistent with the data.
To understand intuitively why dividend growth might be forecastable, it is useful to recall
the interpretation o?ered in Lettau and Ludvigson (2001a) for why cayt might forecast re-
turns. According to that interpretation, forward-looking investors who want to maintain ?at
consumption paths over time will exhibit consumption that ?smooths out? transitory vari-
ation in wealth arising from time-variation in expected returns. Thus, consumption will be
low relative to its long-run trend relation with at and yt in advance of low future returns, and
high relative this common trend in advance of high future returns. The same logic suggests
that aggregate consumption may also contain information about future dividend payments
from aggregate wealth. We refer to the total dividend payments from aggregate wealth as
aggregate dividends. Consumption and components of aggregate dividends (including stock
market dividends) are likely to share a common trend, and deviations from this common
trend should be a function of expected future dividend growth, just as deviations from the
5
common trend in consumption and aggregate wealth should be a function of expected fu-
ture returns to aggregate wealth. Consistent with this hypothesis, we ?nd that ?nd log
consumption is cointegrated with empirical measures of dividend payments from aggregate
wealth, and that deviations from their common stochastic trend reveal changing forecasts of
dividend growth to the stock market component of aggregate wealth. This result is directly
analogous to the ?nding that cayt reveals changing forecasts of future returns to the stock
market component of aggregate wealth (Lettau and Ludvigson (2001a)).
Our approach represents a departure from the standard one of studying models of ex-
pected dividend variation, without considering how those expectations are determined in
relation to aggregate consumption. In a classic paper, Miller and Modigliani (1961) pro-
vided assumptions under which the value of the ?rm is independent of dividend payout
policy, with the striking conclusion that ?nance theory provides no prediction as to the
behavior of dividends paid on equity. Here we argue that, although ?nance theory pro-
vides no prediction about dividends, consumer theory provides a prediction about optimal
consumption, which is inextricably tied to aggregate dividends in the long-run. Thus, the
long-run behavior of aggregate dividends is pinned down by theory, and may be inferred
from observable consumption behavior.
We emphasize four implications of our ?ndings. First, the log dividend-price ratio is
likely to fail statistical tests of return predictability at anything but extremely long hori-
zons, consistent with the evidence reported in Nelson and Kim (1993), Stambaugh (1999),
Ang and Bekaert (2001) and Valkanov (2001)). Such a failure is not an indication that
expected returns are constant, however. On the contrary, the log dividend-price ratio will
have di?culty revealing business cycle variation in the equity risk-premium precisely be-
cause expected returns ?uctuate at those frequencies, and covary with changing forecasts
of dividend growth. Once we entertain the possibility that there is substantial comovement
in expected returns and expected dividend growth at some frequencies, the logic that the
dividend-price ratio should reveal time-variation in expected returns over those horizons is
stood on its head. Movements in expected returns that are entirely common to movements
in expected dividend growth have no e?ect on the log dividend-price ratio. These ?ndings
therefore suggest not only that expected returns vary, but that they vary by far more (over
shorter horizons) than what can be revealed using the log dividend-price ratio alone as a
predictive variable.
Second, time-varying investment opportunities will be poorly captured by variation in
the log dividend-price ratio, because it fails to reveal signi?cant movements in the invest-
6
ment opportunity set that occur over business cycle horizons. This implication is especially
relevant for the growing body of literature on strategic asset allocation, in which the log
dividend-price ratio is used as a proxy for time-variation in the investment opportunity set,
and as an input into the optimal asset allocation decision of a long-horizon investor.4
Third, the results presented here imply that modeling stocks as if dividends pay a multi-
ple ? > 1 of log aggregate consumption is unlikely, by itself, to provide a complete resolution
of the equity premium puzzle. Several researchers have noted that introducing such au-
tonomous variation in dividend growth in equilibrium asset pricing models may o?er one
way of rationalizing the observed equity premium, since those models can in principle make
the dividend claim far more risky than the consumption claim. Campbell (1986), Abel
(1999), Bansal and Yaron (2000) and Campbell (2001) develop models of this form. If these
models are modi?ed so that consumption and aggregate dividends are cointegrated, however,
the dividend claim cannot be more risky than the consumption claim in the long-run, no
matter how volatile dividends are relative to consumption in the short-run. We illustrate this
point using a simple example, which shows that if the adjustment parameter governing the
error correction representation for consumption and dividends takes on empirically plausible
values, the dividend claim will be only slightly more risky than the consumption claim.
Finally, our ?ndings imply that the log dividend-price ratio and the log consumption-
wealth ratio may, at times, give very di?erent signals about the future path of stock prices.
The most recent episode in history provides an example. The level of aggregate stock market
valuation at the end of 2000 would still require a 75 percent decline in stock prices to restore
the dividend-price ratio to its historical mean; by contrast, the empirical measure of the
consumption-wealth ratio was largely restored to its sample mean after the broad market
declines in late 2000 and in 2001 (Lettau and Ludvigson (2002)).
The rest of this paper is organized as follows. In the next section, we lay out the the-
oretical framework linking aggregate consumption and dividend payments from aggregate
wealth, to the expected future growth rates of dividends, and show how we express a present
value relation for future dividend growth in terms of observable variables. We then move
on in Section 3 to discuss the data, and present results from estimating the common trend
in log consumption and measures of the dividend payments from aggregate wealth. For the
main part of our analysis, we focus on ?ndings using the growth in dividends paid from the
CRSP value-weighted stock market index, in order to make our results directly comparable
with those from the existing literature which has typically found little forecastability in this
4For a lucid summary of this literature, see Campbell and Viceira (2001).
7
series. Nevertheless, in Section 5.3 we show that our main conclusions are not altered by
including aggregate share repurchases in the measure of dividends. In Section 3 we also
emphasize our use of annual data to insure that any forecastability of dividend growth we
uncover is not attributable to the seasonality of dividend payments. Section 4 presents an
illustrative example which shows that entirely common variation in expected returns and ex-
pected dividend growth will not be revealed by the log dividend-price ratio. We then move
on in Section 5 to the empirical results and present the outcome of forecasting regressions for
dividend growth on the US stock market. Section 6 aims to quantify the common variation
in expected dividend growth and expected returns by modeling expected returns and ex-
pected dividend growth in a simple principal components framework, and by employing the
frequency-domain measures of comovement developed in Croux, Forni, and Reichlin (2002).
These ?ndings reinforce the conclusion that persistent variation in the log dividend-price ra-
tio is better described as low frequency variation in forecasts of excess stock market returns
than in forecasts of dividend growth, consistent with the arguments in Heaton and Lucas
(1999), Campbell and Shiller (2001), Cochrane (2001), Fama and French (2002), Campbell
(2001), and Lewellen (2001). In Section 7 we discuss some implications of our ?ndings for
the equity premium puzzle, mentioned above. Finally, Section 8 discusses one possible ex-
planation for why expected returns might be positively correlated with expected dividend
growth on US stock markets, even though ?rms may have an incentive to smooth dividend
payments if shareholders themselves desire smooth consumption paths. Section 9 concludes.
2 A Consumption-Based Present Value Relation for
Dividend Growth
This section develops a consumption-based present value relation for future dividend growth.
We consider a representative agent economy in which all wealth, including human capital,
is tradable. Let Wt be beginning of period aggregate wealth (de?ned as the sum of human
capital, Ht, and nonhuman, or asset wealth, At) in period t; Rw;t+1 is the net return on
aggregate wealth. For expositional convenience, we consider a simple accumulation equation
for aggregate wealth, written
Wt+1 = (1 + Rw;t+1)(Wt   Ct): (1)
8
Labor income Yt does not appear explicitly in this equation because of the assumption that
the market value of tradable human capital is included in aggregate wealth.5 Throughout
this paper we use lower case letters to denote log variables, e.g., ct ? log(Ct).
De?ning r ? log(1 + R); Campbell and Mankiw (1989) derive an expression for the log
consumption-aggregate wealth ratio by taking a ?rst-order Taylor expansion of (1), solv-
ing the resulting di?erence equation for log wealth forward, and imposing a transversality
condition.6 The resulting expression is:7
ct   wt = Et
1X
i=1
?iw(rw;t+i   ?ct+i); (2)
where ?w ? 1   exp(c   w). This expression says that the log consumption-wealth ratio
embodies rational forecasts of returns and consumption growth.
Equation(2) is of little use in empirical work because aggregate wealth includes human
capital, which is not observable. Lettau and Ludvigson (2001a) address this problem by
reformulating the bivariate cointegrating relation between ct and wt as a trivariate cointe-
grating relation involving three observable variables, namely ct, at, and yt;where at is the log
of nonhuman, or asset, wealth, and yt is log labor income. The resulting empirical ?proxy?
for the log consumption-aggregate wealth ratio is a consumption-based present value relation
involving future returns to asset wealth8
cayt ? ct   ?aat   ?yyt = Et
1X
i=1
?iw (!ra;t+i   ?ct+i + (1   !) ?yt+1+i) ; (3)
where ! is the average share of asset wealth, At; in aggregate wealth, Wt, ra;t is the log return
to asset wealth, At, and ?a and ?y are parameters to be estimated, discussed further below.
Under the maintained hypothesis that asset returns, consumption growth and labor income
5None of the derivations below are dependent on this assumption. In particular, equation (3), below,
can be derived from the analogous budget constraint in which human capital is nontradeable: At+1 =
(1 + Ra;t+1)(At + Yt   Ct); where, Ht = Et P1j=0 Qji=0(1 + Ra;t+i) iYt+j.
6This transversality condition rules out rational bubbles.
7We omit unimportant linearization constants in the equations throughout the paper.
8We will often refer loosely to cayt as a proxy for the log consumption-aggregate wealth ratio, ct wt. More
precisely, Lettau and Ludvigson (2001a) ?nd that cayt is a proxy for the important predictive components
of ct   wt for future returns to asset wealth. Nevertheless, the left-hand-side of (3) will be proportional to
ct   wt under the following two conditions: ?rst, expected labor income is constant, and second, the return
to human capital is either constant or proportional to the return to nonhuman capital. Although we do not
observe the return to human capital, Lettau and Ludvigson (2002) ?nd that expected future labor income
growth does not appear to vary much in aggregate data.
9
growth are covariance stationary, (3) says that consumption, asset wealth and labor income
are cointegrated, and that deviations from the common trend in ct; at; and yt summarize
expectations of returns to asset wealth, consumption growth, labor income growth, or some
combination of all three. The cointegrating residual on the left-hand-side of (3) is denoted
cayt for short. The cointegration framework says that, if risk premia vary over time (for any
reason), cayt is a likely candidate for predicting future excess returns. Both (2) and (3) are
consumption-based present-value relations involving future returns to wealth.
In this paper we use the same accounting framework to construct a consumption-based
present value relation involving future dividend growth. The objective of this paper is to
study the behavior of a particular component of aggregate dividends, namely dividends
to stock market wealth, which we denote dt. We can move from the consumption-based
present value relation involving future returns, (3), to one involving future dividend growth
by expressing the market value of assets in terms of expected future returns and expected
future income ?ows. A complete derivation is given in Appendix A. This delivers a present-
value relation involving the log of consumption and the logs of dividends from stock market
wealth and nonstock dividends including primarily labor income, the dividend from human
capital. Rather than creating additional notation, we denote nonstock dividends as yt, since
estimates of national income shares suggest that labor income is by far the most important
component of nonstock income produced by private factors of production. The resulting
present value relation takes the form
cdyt ? ct   ?ddt   ?yyt = Et
1X
i=1
?iw(??dt+i + (1   ?)?yt+i   ?ct+i); (4)
where ? is the average share of stock market wealth in aggregate wealth, and ?a and ?y are
parameters to be estimated, discussed further below.
Equation (4) is a consumption-based present value relation involving future dividend
growth. Under the maintained hypothesis that ?dt; ?yt, and ?ct are covariance stationary,
equation (4) says that consumption, stock market dividends, and dividends from other forms
of aggregate wealth (primarily human capital) should be cointegrated, and that deviations
from their common trend (given by the left-hand-side of (4)) provide a rational forecast of
either dividend growth, labor income growth, consumption growth, or some combination of
all three. We denote the cointegrating residual on the left-hand-side of (4) as cdyt, for short.
It is instructive to compare equation (4) with the proxy for the consumption-aggregate
wealth ratio, (3), studied in Lettau and Ludvigson (2001a). Equation (3) says that if in-
vestors want to maintain ?at consumption paths (i.e., expected consumption growth is ap-
10
proximately constant), ?uctuations in cayt reveal expectations of future returns to asset
wealth, provided that expected labor income growth is not too volatile. This implication
was studied in Lettau and Ludvigson (2001a). Those results indicate that cayt has little pre-
dictive power for consumption and labor income growth, but instead forecasts excess returns
on the aggregate stock market. Notice that if cayt forecasts only asset returns, (3) says that
it is a state variable that summarizes changing forecasts of future returns to asset wealth.
Analogously, equation (4) says that if investors want to maintain ?at consumption paths,
?uctuations in cdyt summarize expectations of the growth in future dividends to aggregate
wealth. This implication of the framework is studied here. Investors with ?at consumption
paths will appear to smooth out transitory ?uctuations in dividend income stemming from
time-variation in expected dividend growth. Thus, consumption should be high relative to its
long-run trend relation with dt and yt in anticipation of high dividend growth in the future,
and low in anticipation of low future dividend growth. Moreover, if expected consumption
growth and expected labor income growth do not vary much, as previous research suggests,
cdyt should display relatively little predictive power for future consumption and labor income
growth, but may forecast stock market dividend growth, if in fact expected dividend growth
varies over time. In this case, (4) says that cdyt is a state variable that summarizes changing
forecasts of stock market dividend growth.
The framework developed above, with its approximate consumption identities, serves
merely to motivate and interpret an investigation of whether consumption-based present
value relations might be informative about the future path of dividend growth, asset returns,
labor income growth or consumption growth. The empirical investigation itself, discussed
in the next section, is not dependent on these approximations. Nevertheless, we may assess
the implications of framework presented above by investigating whether such present-value
relations are informative about the future path of consumption growth, labor income growth
or dividend growth from the aggregate stock market. We do so next.
3 The Common Trend in Consumption, Dividends and
Labor Income
3.1 Data and Preliminary Analysis
Before we can estimate a common trend between consumption and measures of aggregate
dividends, we need to address two data issues that arise from the use of aggregate consump-
11
tion and dividend/income data. First, we use nondurables and services expenditure as a
measure of aggregate consumption. This measure is a subset of unobservable total consump-
tion which also includes the service ?ow from the stock of durable goods. Note that it would
be inappropriate to use total personal consumption expenditures (PCE), available from the
Bureau of Economic Analysis, as a measure of consumption in this framework. This total
PCE series includes expenditures on durable goods, which represent replacements and ad-
ditions to the capital stock (investment). Durables expenditures are properly accounted for
as part of nonhuman wealth, At, a component of aggregate wealth, Wt, in (1).9
Second, we have experimented with constructing various empirical measures of nonstock
dividends by adding forms of non-equity income from private capital, the largest component
of which is interest income, to labor income. In our sample, however, we ?nd the strongest
evidence of a common trend between log consumption, log stock market dividends, and log
labor income. A likely explanation is that the in?ationary component of nominal interest
income, along with the explicit in?ation tax on interest income inherent in the U.S. tax
code, creates peculiar trends in interest income that have nothing in particular to do with
the evolution of permanent real interest income. These problems are especially evident in
our sample during the 1970s and 1980s when nominal interest income grew rapidly because
9Treating durables purchases purely as an expenditure (by, e.g., removing them from At and including
them in Ct) is also improper, for the same reason that it would be improper to treat the stock of housing or
nonequity ?nancial wealth purely as an expenditure, namely because doing so ignores the evolution of the
asset over time, which must be accounted for by multiplying the stock by a gross return. (In the case of
many durable goods this gross return would be less than one and consist primarily of depreciation.) What
should be used in this budget constraint for Ct is not total expenditures but total consumption, of which
the service ?ow from the durables stock is one part. But the service ?ow is unobservable, and is not the
same as the investment expenditures on consumption goods. An assumption of some sort is necessary, and
we follow Lettau and Ludvigson (2001a) by assuming that the log of unobservable real total consumption,
cTt , is a multiple, ? > 1 of the log of real nondurables and services expenditure, ct, plus a stationary random
component, ?t. Under this assumption, the observable log of real nondurables and services expenditures, ct,
appears in the cointegrating relation (3). The evidence is supportive of this assumption, since test results
strongly reject the null of no cointegration for ct, at and yt. As a robustness check, we performed our
empirical analysis using total expenditures for cTt and found that our conclusions are not altered by the use
of this measure in place of nondurables and services expenditures.
12
of in?ation.10;11 In addition, we do not directly observe dividend payments from some forms
of nonhuman, non?nancial household net worth, primarily physical capital.12 Nevertheless,
it is not necessary to include every dividend component from aggregate wealth to obtain
a consumption-based present value relation that is a function of future stock market divi-
dend growth, the object of interest in this study. As long as the excluded forms of dividend
payments are cointegrated with the included forms (as models with balanced growth would
suggest), the framework above implies that the included dividend measures may be com-
bined with consumption to obtain a valid cointegrating relation. In the application here, if
nonstock/nonlabor forms of dividend income are cointegrated with the dividend payments
from stock market wealth, dt, and from human capital, yt, the framework above implies a
cointegrating relation among ct; dt and labor income yt, and the resulting cointegrating resid-
ual should in principle reveal expectations over long-horizons of either future ?dt, ?yt or
?ct, or some combination of all three. For these reasons, we focus in this paper on empirical
results based on using consumption, ct; stock market dividends, dt; and labor income, yt; to
form an estimate of a cointegrating residual cdyt.
Taken together, these data issues imply that the cointegrating coe?cients in both (3) and
(4) should not sum to one. As discussed in Lettau and Ludvigson (2001a), the cointegrating
10The real component of nominal interest income is not directly observable. Nominal interest income can
be put in real terms by de?ating by a price level to get the component which should be associated with
real consumption, but one would still need to subtract the product of some in?ation rate and the stock of
?nancial assets from this amount. Measurement is complicated because the stock data are in the ?ow of
funds while the nominal interest data are in the National Income and Product Accounts, and the components
do not match perfectly.
11Some researchers have documented a signi?cant decline in the percentage of ?rms paying tax-ine?cient
dividends in data since 1978 (e.g., Fama and French (2001)). It might seem that such a phenomenon
would create problem with trends in stock market dividend income similar to those for interest income.
An inspection of the dividend data from the CRSP value-weighted index, however, reveals that?with the
exception of the unusually large one-year decline in dividends in 2000, discussed below?the total dollar value
of CRSP value-weighted dividends (in real, per capita terms) has not declined precipitously over the period
since 1978 or over the full sample. In fact, the average annual growth rate of real, per capita dividends is
5.6% from 1978 through 1999, greater than the growth rate for the period 1948 to 1978. The annual growth
rate for the whole sample (1948-2001) is 4.2%.
12One response to this point is to use the product side of the national income accounts to estimate income
?ows of such components of wealth as the residual from GDP less reported dividend and labor income.
This approach requires that the income and product sides of the national accounts be combined, however,
a procedure that creates its own measurement di?culties since there is no way to know how much of the
statistical discrepancy between the two is attributable to underestimates of income versus overestimates of
output.
13
parameters ?a and ?y in (3) are, in principle, equal to the shares ! and (1   !); in practice,
the estimated values of these parameters are likely to sum to a number less than one be-
cause only a fraction of total consumption based on nondurables and services expenditure is
observable (see Lettau and Ludvigson (2001a)). The same issues apply to the cointegrating
parameters ?d and ?y, which are in principle equal to the shares ? and 1 ?. In addition, the
sums of estimated coe?cients (where ?hats? denote estimated values), ba + by and bd + by,
are unlikely to be identical, since a component of aggregate dividends is omitted in (4).
The parameters ba; by, bd, and by may be estimated using either single equation or system
methods. The estimated values of the cointegrating residuals cayt and cdyt are denoted dt
and c t, respectively.
The data used in this study are annual, per capita variables, measured in 1996 dollars,
and span the period 1948 to 2001. We use annual data in order to insure that any fore-
castability of dividend growth we uncover is not attributable to the seasonality of dividend
payments. Annual data is also used because the outcome of both tests for, and estimation of,
cointegrating relations can be highly sensitive to seasonal adjustments. Stock market divi-
dends are measured as dividends on the CRSP value-weighted index and are scaled to match
the units of consumption and labor income. Appendix B provides a detailed description of
the sources and de?nitions of all the data used in this study.
We begin by testing for both the presence and number of cointegrating relations in the
system of variables x0t ? [ct;dt;yt]0. Such tests have already been performed for the system
v0t = [ct;at;yt]0 in Lettau and Ludvigson (2001a) and Lettau and Ludvigson (2002), and
those tests suggest the presence of a single cointegrating relation among those variables. We
refer the reader to those papers for details and simply note here that there is strong evidence
of cointegration among ct; at; and yt.13 The results of cointegration tests for ct; dt; and yt
are contained in Appendix C of this paper. We assume all of the variables contained in xt
are ?rst order integrated, or I(1), an assumption veri?ed by unit root tests. In addition, the
?ndings presented in the Appendix C suggest the presence of a single cointegrating vector for
the three variables in xt. The cointegrating coe?cient on consumption is normalized to one,
and we denote the single cointegrating relation for x0t = [ct;at;yt]0 as ?0 = (1; ?d; ?y)0,
and for x0t = [ct;dt;yt]0 as ?0 = (1; ?d; ?y)0.
The cointegrating parameters ?d, ?y and ?d, ?y must be estimated. We use a dy-
namic least squares procedure which generates ?superconsistent?estimates (Stock and Wat-
13Cointegration tests in Lettau and Ludvigson (2001a) and Lettau and Ludvigson (2002) were based on
quarterly data. The outcome of these tests is not altered by using the annual data in this study.
14
son (1993)).14 This procedure estimates b0 = (1; 0:13; 0:68)0: The Newey-West corrected
t-statistics (Newey and West (1987)) for these estimates are -10.49 and -34.82, respectively.
We denote the estimated cointegrating residual b0xt as c t. The estimated cointegrating
vector for v0t = [ct;at;yt]0 is b0 = (1; 0:29; 0:60)0; very similar to that obtained in Lettau
and Ludvigson (2001a) using quarterly data. The Newey-West corrected t-statistics for these
estimates are -14.32 and -30.48, respectively.
4 An Illustrative Example
In the next section we discuss empirical results which suggest that expected dividend growth
and expected excess returns contain a common component. Before considering those ?ndings,
it is useful to consider a simple example that illustrates how common variation in expected
dividend growth and expected returns a?ects the log dividend-price ratio. We will argue
below that this example is empirically relevant.
Let pt be the log price of stock market wealth, which pays the dividend, dt. Following
Campbell and Shiller (1988) the log dividend-price ratio may be written (up to a ?rst-order
approximation) as
dt   pt = k + Et
1X
i=1
?i(rt+i   ?dt+i); (5)
where ? = 1= 1 + exp(d   p)?, and rt is the log return to stock market wealth: This equation
says that if the log dividend-price ratio is high, agents must be expecting high future returns
on stock market wealth, or low dividend growth rates. Many studies, cited in the intro-
duction, have documented that dt   pt explains virtually none of the variability of dividend
growth over long-horizons, and as a consequence, expected dividend growth is often modelled
as constant. Instead, what has been found to be forecastable over long-horizons is excess re-
turns, suggesting that risk premia vary slowly over time. Notice that the consumption-based
present value relation for future dividend growth, (4), is an alternative to (5) for capturing
possible time-variation in expected dividend growth.
Equation (5) can be simpli?ed if we assume that expected stock returns follow a ?rst-order
autoregressive process, Etrt+1 ? xt = ?xt 1 + ?t: If expected dividend growth is constant,
14Two leads and lags of the ?rst di?erences of ?yt and ?dt are used in the dynamic least squares regression.
15
the log dividend-price ratio takes the form
dt   pt = Et
1X
i=1
?i(rt+i   ?dt+i) = xt1   ??: (6)
In this case the log dividend-price ratio does not forecast dividend growth at any horizon but
instead forecasts long-horizon stock returns. Forecasting regressions of long-horizon returns
on the log dividend-price ratio will display R2 statistics that are hump-shaped in the horizon,
depending on the value of ?:15 Thus, under the standard view that expected dividend growth
is approximately constant, any and all variation in expected returns must be revealed by
variation in the dividend-price ratio. It follows that long-horizon forecasting regressions of
excess returns provide one way of assessing whether risk-premia vary.
Now suppose, in contrast to the standard view, that expected dividend growth is not
constant but instead varies according to the ?rst-order autoregressive process,
Et?dt+1 ? gt = ?gt 1 + ?t: (7)
The e?ect of such variation on the dividend-price ratio depends on the correlation between
expected dividend growth and expected returns. For example, if variation in expected divi-
dend growth is common to variation in expected returns, expected returns may contain two
components, Etrt+1 = gt + xt; so that the log dividend-price ratio becomes
dt   pt = Et
1X
i=1
?i(rt+i   ?dt+i) (8)
=
? g
t
1   ?? +
xt
1   ??
?
  gt1   ?? = xt1   ??: (9)
This expression is precisely the same as (6) for the case in which expected dividend growth is
constant. The log dividend-price ratio still forecasts returns because it captures a component
of excess returns, xt, that is independent of expected dividend growth, but it does not forecast
dividend growth even though, by construction, expected dividend growth varies over time.
Moreover, variation in dt   pt does not capture one component of time-varying expected
returns, gt, because that component is common to time-varying expected dividend growth.
In this case, the variation in expected dividend growth is entirely common to variation
expected returns and therefore has completely o?setting e?ects on the log dividend-price
ratio. It follows that ?uctuations in the log dividend-price ratio will be driven only by one of
the two components of expected returns, xt. Thus, if we relax the assumption that expected
15Campbell, Lo, and MacKinlay (1997), Chapter 7, provides an illustration of this point.
16
dividend growth is approximately constant, it no longer follows that any and all variation in
expected returns must be revealed by variation in the dividend-price ratio.
This simple example conveys two important messages. First, to uncover changing fore-
casts of future dividend growth, we may be required to employ a present value relation that
does not involve expected returns. Second, to uncover all variation in expected returns (in-
cluding that which is common to variation in expected dividend growth), we may be required
to use a present value relation that does not involve expected dividend growth. Thus, we
need distinct present value relations for future returns and future stock market dividend
growth, rather than a single present value relation involving the future values of both, as
in the dividend-price ratio. In this paper, those distinct present value relations are the two
consumption-based ratios discussed above.
To see why, we show that the ability of cayt to reveal variation in expected returns and the
ability of cdyt to reveal variation in expected dividend growth in this example is not a?ected
by whether expected returns and expected dividend growth are correlated with one another.
Suppose for expositional clarity that consumption growth is i.i.d., i.e., ?ct+1 = ?t+1, where
?t+1 is an unforecastable error, and consider a simple example in which aggregate wealth is
equal to stock market wealth and aggregate dividends are equal to stock market dividends.
In this case, the share of stock market wealth in aggregate wealth is unity, implying that cayt
in (3) collapses to the log consumption-wealth ratio, ct wt = ct pt, and cdyt in (4) collapses
to the log consumption-dividend ratio ct   dt. Combining (4) and (7), it is straightforward
to show that the log consumption-dividend ratio forecasts dividend growth and the common
component of expected returns:16
ct   dt = Et
1X
i=1
?i(?dt+i   ?ct+i) = gt1   ??; (10)
while, using (3), the log consumption-wealth ratio captures both components of expected
returns:
ct   wt = Et
1X
i=1
?i(rt+i   ?ct+i) = gt1   ?? + xt1   ??: (11)
Equation (11) says that the log consumption-wealth ratio is a better predictor of returns
than dt   pt because it captures both components of time-varying expected returns, gt and
16In this illustrative example, ? = ?w since all wealth is invested in the stock market. More generally, we
assume that the steady state stock market dividend-price ratio and consumption-aggregate wealth ratio are
su?ciently similar so that ? ? ?w and this simple example captures the essence of the problem.
17
xt. If the former is less persistent than the latter, ct   wt will also be less persistent than
dt   pt and will forecast returns over shorter horizons than dt   pt.
5 Long-Horizon Forecasting Regressions
This section presents forecasting results using dt and c t as predictive variables. Table
1 ?rst presents summary statistics for log of real, per capita consumption growth, labor in-
come growth, dividend growth, the change in the log of the CRSP price index, ?pt, and the
change in the log of household net worth, ?at, all in annual data. Real dividend growth is
considerably more volatile than consumption and labor income, having a standard deviation
of 12 percent compared to 1.1 and 1.8 for consumption and labor income growth, respec-
tively. It is somewhat less volatile than the log di?erence in the CRSP value weighted price
index, which has a standard deviation of 16 percent, but still more volatile than the log
di?erence in networth, which has a standard deviation of 4 percent. Consumption growth
and labor income growth are strongly positively correlated, as are ?pt and ?at. Annual real
consumption growth and real dividend growth have a weak correlation of -0.16.
How does the persistence of c t compare to dt   pt and dt? Table 2 displays au-
tocorrelation coe?cients. It is well-known that the dividend-price ratio is very persistent.
In annual data from 1948 to 2000 it has a ?rst order autocorrelation 0.88, a second order
autocorrelation of 0.72 and a third order autocorrelation of 0.60. The autocorrelations of
ccdyt and dcayt are much lower and die out more quickly. Their ?rst order autocorrelation
coe?cients are 0.48 and 0.55, respectively; their second order autocorrelation coe?cients are
0.13 and 0.22 respectively.
To better understand the time-series properties of dt   pt, dt, and c t in our annual
data set, it is useful to examine estimates of error-correction representations for (dt;pt)0,
(ct;at;yt)0 and (ct;dt;yt)0. Table 3 presents the results of estimating ?rst-order cointegrated
vector autoregressions (VARs) for dt and pt; for ct, at and yt, and for ct, dt; and yt.17 For
dividends and prices, the theoretical cointegrating vector (1; 1)0 is imposed; for the other
two systems, the cointegrating vectors are estimated as discussed above. The table reveals
several noteworthy properties of the data on consumption, household wealth, stock market
dividends, and labor income.
First, Panel A shows that the log dividend-price ratio has little ability to forecast future
17The VAR lag lengths were chosen in accordance with ?ndings from Akaike and Schwartz tests. The
second system is also studied in Ludvigson and Steindel (1999).
18
dividend growth or price growth in the cointegrated VAR. Variation in the log dividend-price
ratio is too persistent to display statistical evidence of cointegration in this sample, a result
made apparent by the absence of a statistically signi?cant error-correction representation
in Panel A (although see the discussion below of the ?ndings in Lewellen (2001)). Second,
Panel B shows that estimation of the cointegrating residual dt 1 is a strong predictor
of wealth growth. Both consumption and labor income growth are somewhat predictable
by lags of either consumption growth and/or wealth growth, as noted elsewhere (Flavin
(1981); Campbell and Mankiw (1989)), but the adjusted R2 statistics (especially for the
labor income equation) are lower than those for the asset regression. More importantly, the
cointegrating residual dt 1 is an economically and statistically signi?cant determinant of
next period's asset growth, but not next period's consumption or labor income growth. This
?nding implies that asset wealth is mean-reverting, and adjusts over long-horizons to match
the smoothness of consumption and labor income. These results are consistent with those
in Lettau and Ludvigson (2001a).
Panel C displays estimates from a cointegrated VAR for ct; dt; and yt: The results are
analogous to those for the cointegrated VAR involving ct, at, and yt. Consumption and labor
income are predictable by lagged consumption and wealth growth, but not by the cointegrat-
ing residual c t 1. What is strongly predictable by the cointegrating residual is dividend
growth: c t 1 is both a statistically signi?cant and economically important predictor of next
year's dividend growth, ?dt. This ?nding?in conjunction with the evidence that the coin-
tegrating residual c t 1 is stationary but persistent?implies that although there is some
short-run predictability in the growth of consumption and labor income (as exhibited by the
dependence of these variables on lagged growth rates), it is dividend growth that exhibits
error-correction behavior and therefore predictability over long horizons. Indeed, regressions
of long-horizon growth rates on lagged growth rates and c t 1 (not shown) demonstrate that
dividend growth becomes substantially more forecastable than consumption or labor income
growth as the horizon over which these variables are measured increases. These ?ndings im-
ply that when log dividends deviate from their habitual ratio with log labor income and log
consumption, it is dividends, rather than consumption or labor income, that is forecast to
slowly adjust until the cointegrating equilibrium is restored. As for asset wealth, dividends
are mean reverting and adapt over long-horizons to match the smoothness in consumption
and labor income.
Figure 1 plots the demeaned values of c t and dt over the period spanning 1948 to
2001. The ?gure shows that the two present-value relations tend to move together over time,
19
although there are some notable episodes in which they diverge. One such episode is the year
2000, when an extraordinary 30% decline in recorded dividends (an extreme outlier in our
sample) pushed c t well above its historical mean. According to the results just presented,
the current levels of c t are forecasting a increase in future dividend growth rates. By
contrast, dt had been forecasting a sharp decrease in stock market returns as of 1999, but
has now been largely restored to its long-run mean after the broad market declines of 2000
and 2001.
A more direct way to understand mean reversion in dividend growth is to investigate
regressions of long-horizon dividend growth onto the cointegrating residual c t 1: The the-
ory behind (3) and (4) makes clear that both the log consumption-wealth ratio and the
log consumption-dividend ratio should track longer-term tendencies in asset markets rather
than provide accurate short-term forecasts of booms or crashes.18 We focus in this paper on
explaining the historical behavior of excess stock market returns and dividend growth. Table
4 presents the results of univariate regressions of the return on the CRSP value-weighted
stock market index in excess of the three-month Treasury bill rate, at horizons ranging from
one to 6 years. In each regression, the dependent variable is the H-period log excess re-
turn, rt+1   rf;t+1 + ::: + rt+H   rf;t+H, where rf;t is used to denote the Treasury bill rate,
or ?risk-free? rate. The independent variable is either dt   pt, dt, or c t. The table re-
ports the estimated regression coe?cient, the adjusted R2 statistic in square brackets, and
a heteroskedasticity and autocorrelation-consistent t-statistic for the hypothesis that the
regression coe?cient is zero in parentheses. The table also reports, in curly brackets, the
rescaled t-statistic recommended by Valkanov (2001) for the hypothesis that the regression
coe?cient is zero. We discuss this rescaled statistic below. Table 5 presents the same output
for predicting long-horizon CRSP dividend growth, ?dt+1 + ::: + ?dt+H.
The ?rst row of Table 4 shows that the log dividend-price ratio has little power for forecast
aggregate stock market returns from one to 6 years in this sample, according to conventional
18The forecasting tests in this paper are intentionally in-sample tests. This makes our results directly
comparable with existing literature which has investigated the predictability of long-horizon dividend growth
using in-sample regressions. In addition, both the theoretical framework presented above, and the hypothesis
of cointegration imply long-horizon predictability, but not stable predictive power over short subsamples of
a ?nite data set. Inoue and Kilian (2002) show that the subsample analysis inherent in out-of-sample
forecasting tests causes them to be signi?cantly less powerful than in-sample forecasting tests, a pattern
that would be exacerbated in an investigation of long-horizon forecasting power. See Lettau and Ludvigson
(2001b) for further discussion on the relative merits of in-sample and out-of-sample forecasting tests using
consumption-based present value relations.
20
statistical analysis. These results di?er from those reported elsewhere primarily because we
have included the last few years of stock market data in the sample. The extraordinary
increase in stock prices in the late 1990s substantially weakens the statistical evidence for
predictability by dt   pt that had been a feature of previous samples. If we end the sample
in 1998, the log dividend price ratio displays forecasting power for excess returns, but its
strongest forecasting power is exhibited over horizons that are far longer than that exhib-
ited by the consumption-wealth ratio proxy, dt (see Lettau and Ludvigson (2001a)).19 It
should be noted, however, that although recent data has weakened the statistical evidence
in favor of predictability by the dividend-price ratio according to conventional statistical
analyses, Lewellen (2001) ?nds that the dividend-price ratio remains a strong predictor of
excess stock returns even in samples that include recent data. Noting that the dividend-
price ratio is very persistent, Lewellen incorporates the information conveyed by the sample
autocorrelation of the dividend-price ratio when assessing its predictive power. Under the
assumption that the dividend-price ratio is stationary, episodes in which the dividend yield
remains deviated from its long-run mean for an extended period of time do not necessarily
constitute evidence against predictability over very long-horizons when the forecasting vari-
able is extremely persistent. Similar results are reported in recent work by Campbell and
Goto (2002), who ?nd evidence of return predictability by ?nancial ratios if one is willing to
rule out an explosive root in the ratios. These results corroborate previous ?ndings that the
dividend yield captures very slow-moving forecasts of excess returns, but has considerably
more di?culty predicting medium-horizon returns, over business cycle frequencies.
The second row of Table 4 shows that dt has statistically signi?cant forecasting power
for future excess returns at horizons ranging from one to six years. This evidence is consistent
with that reported in Lettau and Ludvigson (2001a) using quarterly data. Using this single
variable alone achieves an R2 of 0.49 for excess returns over a two year horizon, and 0.52 for
19Campbell (2001) reports that dt   pt explains about 40 percent of the variation in excess stock returns
at a 16 quarter horizon in data from 1947 to 1998. His results di?er from those reported in Table 4 for two
reasons. First, the unusual movements in the dividend-price ratio over just the last three years of our sample
have had a dramatic impact on the statistical evidence on predictability by the log dividend-price ratio.
Second, Campbell performs long-horizon regressions using quarterly data, rather than annual as we use here.
Quarterly and annual regressions of long-horizon returns di?er in the number of overlapping residuals relative
to the number of observations. Valkanov (2001) shows that the ?nite-sample distributions of R2 statistics
in long-horizon regressions do not converge to their population values when there are overlapping residuals,
with the degree of divergence dependent on the amount of overlap. This ?nding implies that di?erent data
sets (e.g., quarterly versus annual) are likely to generate di?erent distributions for this statistic. We address
this di?culty below by using a vector autoregressive approach to impute long-horizon R2 statistics.
21
excess returns over a six year horizon.
The remaining row of Table 4 gives an indication of the forecasting power of the estimated
present value relation, c t, for long-horizon excess returns. At a one year horizon, c t, dis-
plays little statistical forecasting power for future returns in this sample. For returns over all
longer horizons, however, the present-value relation for dividend growth displays forecasting
power for future returns. In addition, the coe?cients from these predictive regressions are
positive, indicating that a high c t forecasts high excess returns just as a high dt forecasts
high excess returns. The t-statistics are above four for all horizons in excess of one year, and
the R2 statistic rises from .20 at a three year horizon to .32 at a six year horizon. Thus, the
results suggest that both dt and c t capture some component of time-varying expected
returns.
Table 5 displays results from the same forecasting exercise for long horizon dividend
growth. In this sample, which includes data in the last half of the 1990s, the log dividend-
price ratio displays some forecasting power for future dividend growth (row 1), but has
the wrong sign (positive), consistent with evidence in Campbell (2001) who also uses data
that include the second half of the 1990s. Rows 2 and 3 present the results of predictive
regressions using dt and c t. Previous research found that dt had predictive power
for future returns; Row 2 shows that it also has statistically signi?cant predictive power
for dividend growth rates in our sample, with high values of dt predicting high dividend
growth rates. The forecasting power of dt is, however, weaker than that displayed by c t
at every horizon in excess of one year (row 3). For example, at a four year horizon, c t
explains about 20 percent of the variation in dividend growth, while dt explains 9 percent.
At a ?ve year horizon, c t explains about 28 percent of the variation in dividend growth,
while dt explains 10 percent. Still, just as for excess returns, the results suggest that both
dt and c t capture some component of time-varying expected dividend growth.
The results in Tables 5 and 6 suggest that there is common variation in expected returns
and expected dividend growth. The consumption-wealth ratio proxy, dt; which is a strong
predictor of excess stock market returns, is also a predictor of stock market dividend growth.
Conversely, c t, a strong predictor of stock market dividend growth, is also a predictor of
excess stock market returns. Does either variable have independent predictive power for
excess returns and dividend growth? To address this question, Table 6 presents the results
of multivariate regressions of long-horizon returns and dividend growth using both dt and
ccdyt as regressors. The ?rst panel of Table 6 shows that, in forecasting long horizon excess
returns, dt drives out c t, implying that the latter contains no information about future
22
returns that is independent of that contained in dt. dt does contain some information
for future returns that is independent of that contained in c t, suggesting the existence a
component of time-varying expected returns that is not associated with rational forecasts of
dividend growth.
The second panel of Table 6 shows that the opposite pattern is borne out in long-horizon
forecasting regressions for dividend growth: c t drives out dt in forecasting future dividend
growth at all forecasting horizons greater than three years. But for horizons between 2 and
3 years, the information contained in dt and c t is su?ciently similar that the regression
has di?culty distinguishing their independent e?ects (although c t is statistically signi?cant
at the 6 percent level). The two variables are strongly jointly signi?cant as a predictor of
long-horizon dividend growth (the p-value for the F-test is less than 0.000001). This ?nding
that is consistent with results discussed below which suggest that much of the variation in
expected dividend growth is common to variation in expected returns. In addition, for every
forecasting horizon in excess of one year, the regression results imply that dt contains no
information about future dividend growth that is independent of that already contained in
ccdyt.
Comparing adjusted R2 statistics in rows 2 and 3 of Table 4, it is evident that ?uctuations
in c t account for about half of the variation in future excess returns that is captured by
dt at a three year horizon, and more than half at a six year horizon. Similarly, ?uctuations
in dt explain about half of variation in future dividend growth that is captured by c t
at four and ?ve year horizons. Thus, these statistics are also consistent with the hypothesis
that expected dividend growth and expected returns are positively correlated, and that a
signi?cant fraction of their movement is common. Section 5, below, pursues two ways of
quantifying this common movement.
In summary, the evidence presented above suggests that there is important predictability
of dividend growth over long horizons in direct long-horizon regressions, and that predictable
variation in dividend growth is correlated with that in excess returns. This evidence is
largely new. Other researchers, cited in the introduction, have found that dividend growth
predictability?if evident at all in these regressions?occurs at relatively short horizons and is
not highly correlated with predictable variation in excess returns. More recently, Ang (2002)
performs direct long-horizon regressions using annual data from 1927-2000 and ?nds some
evidence that the dividend-price ratio predicts dividend growth at a one-year horizon, but
no evidence that it does so at longer horizons. These ?ndings are consistent with those of
the earlier papers cited in the introduction. One recent study that does ?nd predictability
23
in dividend growth is Ang and Bekaert (2001), who report results based on observations
from 1952:Q4 to 1999:Q4 on the S&P 500 stock market index. Although they con?rm earlier
?ndings that dividend growth is not predictable by the dividend-price ratio in univariate
forecasting regressions, they ?nd that the dividend-price ratio has marginal predictive power
once the earnings yield is included as a predictive variable. (The earnings yield also has
marginal predictive power.) There are two main di?erences between our predictability results
and those in Ang and Bekaert. First, the joint forecasting power of the dividend yield and
the earnings yield for dividend growth is concentrated at shorter horizons than in regressions
using c t and dt. Second, the R-squares for the regressions using the former variables are
substantially lower than those using the latter. For example, in the sample used in Ang and
Bekaert (2001), the dividend yield and the earnings yield jointly explain about 21 percent of
dividend growth one year ahead, and about 13 percent a ?ve year horizon. The comparable
numbers using c t alone as a predictive variable are 31 percent and 34 percent.20
5.1 Additional Statistical Tests
There are at least two potential econometric hazards with interpreting the long-horizon re-
gression results just presented. One is that the use of overlapping data in long-horizon
regressions can skew statistical inference in ?nite samples. Valkanov (2001) shows that, in
?nite samples where the forecasting horizon is a nontrivial fraction of the sample size, (i)
the t-statistics of long-horizon regression coe?cients do not converge to a well de?ned distri-
bution, and (ii) the ?nite-sample distributions of R2 statistics in long-horizon regressions do
not converge to their population values. A second possible econometric hazard with inter-
preting the long-horizon regression results presented in the previous section occurs because
(like most long-horizon forecasting variables) c t and d are persistent variables, which,
although predetermined, are not exogenous. This lack of exogeneity can create a small sam-
ple bias in the regression coe?cient that works in the direction of indicating predictability
even where none is present (Nelson and Kim (1993) and Stambaugh (1999)).
To address these potential inference problems, we perform three robustness checks. The
?rst is to compute the rescaled t=pT statistic (where T is the sample size), recommended
by Valkanov (2001). In contrast to the standard t-statistic, this rescaled t has a well de?ned
distribution in ?nite samples that is straightforward to simulate, once two nuisance param-
eters have been estimated. Second, we use vector autoregessions to impute long-horizon
20These numbers are higher than those reported in Table 4 because we use the slightly shorter sample
employed by Ang and Bekaert (2001) in order to make the results directly comparable.
24
R2 statistics rather than estimating them directly from long-horizon regressions. Third, we
perform bootstrapped estimates of the empirical distribution of the predictive regression co-
e?cients and adjusted R2 statistics under the null of no predictability; these bootstrapped
estimates are reported in Table 8 and discussed below. The Valkanov rescaled t-statistic and
the bootstrapping methodology provide distinct approximations to the true ?nite sample
distribution of the relevant statistics. Since the two approximations are unlikely to be the
identical in any given sample, we look at the complete picture and perform both exercises.
The rescaled t-statistics are reported in curly brackets in Table 4, for univariate predictive
regressions of excess returns on dt and c t, and in Table 5, for univariate predictive
regressions of dividend growth on dt and c t. After obtaining estimates of the nuisance
parameters for our application, we compute the rescaled t-statistic and use Valkanov's critical
values to determine statistical signi?cance; the table reports the rescaled t-statistic and
whether its value implies that the predictive coe?cient in each regression is statistically
signi?cant at the 5, 2.5 and 1 percent levels. According to this rescaled t-statistic, dt
is a powerful forecaster of excess returns (statistically signi?cant at the 1% level) at every
horizon ranging from one to six years, as is c t at all but the one-year horizon (Table 4). For
future dividend growth (Table 5), the rescaled t-statistic implies that c t is a statistically
signi?cant predictor at the 1% percent level at every horizon from one to six years, whereas
dt is a statistically signi?cant predictor of dividend growth at the 1% level at every horizon
ranging from one to four years. The analysis suggests that the forecasting power of dt and
ccdyt for excess stock market returns and stock market dividend growth cannot be attributed
to biases arising from the use of overlapping data in ?nite samples.
Finite sample problems with overlapping data in long-horizon regressions may also be
avoided by using vector autoregregressions to impute implied long-horizon R2 statistics for
univariate forecasting regressions, rather than estimating them directly from long-horizon
returns. The methodology for measuring long-horizon statistics by estimating a VAR has
been covered by Campbell (1991), Hodrick (1992), and Kandel and Stambaugh (1989), and
we refer the reader to those articles for further details. Hodrick (1992) also compares the
asymptotic distributions of these statistics to their empirical distributions under the null
and under alternatives, and ?nds that they have good ?nite sample properties. We present
the results from using this methodology in Table 7, which investigates the long-horizon
predictive power of both dt and c t for future returns and future dividend growth using
bivariate, ?rst-order VARs. For each forecasting horizon we consider, we calculate an implied
R2 statistic using the coe?cient estimates of the VAR and the estimated covariance matrix
25
of the VAR residuals.
Table 7 shows that the pattern of the implied R2 statistics from the vector autoregressions
is very similar to those from the produced from the single equation long-horizon regressions.
The implied R2 statistics for forecasting dividend growth with c t (row 3) peaks around
three years, according to this metric, equal to 0.20. In addition, the implied R2 statistics
are considerably larger using lagged c t as a predictive variable for both excess returns and
dividend growth (rows 3 and 6) than when a simple autoregressive speci?cation for returns
or dividend growth is employed (rows 1 and 4). A similar pattern holds for the implied R2
statistics for forecasting with dt: the implied R2 statistic for forecasting excess returns with
dt is as high as 49% at a three year horizon; for forecasting dividend growth with dt; it
reaches 24% at a three year horizon. Thus, the evidence favoring predictability of dividend
growth and excess stock returns using c t and dt is robust to the VAR methodology,
implying that the size of the long-horizon R2 statistics cannot be readily attributed to
inference problems with the use of overlapping data in ?nite samples.
An alternate method for addressing potential ?nite sample biases is to estimate the
empirical distribution of regression coe?cients and adjusted R2 statistics from predictive
regressions in which dt and c t are used as forecasting variables. We present the results
of doing so in Table 8. The methodology is based on bootstrap simulation under the null
hypothesis that expected excess returns and dividend growth are constant (i.e., the coe?cient
on dt and c t are restricted to be zero), and a ?rst-order autoregressive model for the
predictive variables dt and c t.21 Arti?cial sequences of excess returns and dividend
growth are generated by drawing randomly (with replacement) from the sample residual
pairs.22 The simulations were repeated 10,000 times. To avoid di?culties caused by the use
of overlapping data, we focus here on the one-year ahead regressions presented in Tables 5
and 6.
Table 8 summarizes the estimated sampling distribution for the slope coe?cient and the
R2 statistic in univariate regressions of annual excess returns for the CRSP index on lagged
dt and c t (top panel), as well as univariate regressions of annual dividend growth for the
21It is known that the standard bootstrap is not consistent if the data series have a near-unit root. Although
dt and dt are relatively persistent variables (see Table 1), they do not appear well-characterized as near-
unit root processes, since?unlike the log dividend-price ratio?standard cointegration tests strongly reject
the hypothesis that they are I (1) random variables.
22Nelson and Kim (1993) also perform randomization, which di?ers from bootstrapping only in that
sampling is without replacement. We also performed the simulations using randomization and found that
the results were not a?ected by this change.
26
CRSP index on lagged dt and c t (bottom panel). In almost every case, the estimated
regression coe?cient and R2 statistic lies outside of the 95 percent con?dence interval based
on the empirical distribution under the null of no predictability. In most cases they lie outside
of the 99 percent con?dence interval. The one exception is for the case in which excess returns
are regressed on the one-year lagged value of c ; in this case, we cannot reject the hypothesis
that one-step ahead forecasting power of c t is not statistically indistinguishable from zero.
This is not surprising, since even the standard asymptotic statistics suggest that c t only
has signi?cant predictive power for returns at horizons longer than one year. For all of the
other regressions and forecasting horizons, however, we ?nd that the slope coe?cients and R2
statistics are large relative to their sampling distributions. For both the dividend growth and
excess return forecasting regressions, the estimated regression coe?cients and R2 statistics
are large compared with their empirical values generated under the null of no predictability.
These results, like those discussed above using the rescaled t-statistic and VAR-imputed R2
statistics, imply that the predictability of excess returns and dividend growth documented
here is unlikely to be attributable to small sample biases in the regression coe?cients or R2
statistics.
5.2 Interpretation
The empirical results presented above indicate that ?uctuations in the log consumption-
aggregate dividend ratio are informative about the future path of dividend growth. One
theoretical framework that is consistent with these ?ndings is the rational expectations
framework presented above and in Lettau and Ludvigson (2001a). If investors want rela-
tively ?at consumption paths over time, forward looking agents will attempt to smooth out
transitory ?uctuations in both wealth and total income from aggregate wealth, or aggregate
dividends. The present-value relation cayt will be high when agents are expecting high future
returns, and low when they are expecting low future returns. Similarly, the present value
relation cdyt will be high when agents are expecting high future dividend growth, and low
when agents are expecting low dividend growth.
Of course, the ?nding that returns and dividend growth are forecastable does not mean
that agents have perfect foresight, or even that investors can foresee a large fraction of the
variation in future returns and future dividend growth. But they are consistent with some
ability on the part of stockholders to gauge the future path of returns and dividend growth.
They suggest a somewhat di?erent perspective on investor behavior than that found in recent
27
theoretical treatments which explain the variability in the dividend-price ratio by assuming
that investors make systematic errors in forecasting dividend growth (e.g., Barsky and De
Long (1993)).
Although rational, forward looking behavior is consistent with the ?ndings presented
above, explanations for these phenomena which involve some form of irrational behavior
or irrational expectations may also be consistent. One possibility, raised by the work of
Campbell and Mankiw (1989), is that stockholders follow a simple rule-of-thumb in which
they consume their total income every period. If income itself is close to a random walk,
such a rule could imply that consumption and income could de?ne the trend in wealth
(implying that the consumption-wealth ratio forecasts returns) without investors possessing
any real foresight. The evidence presented here, however, exposes the potential limitations
of such an explanation for stockholder behavior. Although labor income, the dividend to
human capital, may be close to a random walk, dividends from stock market wealth have an
important transitory component, yet consumption tracks just the long-term components in
equity dividends. Any explanation of stockholders' behavior based on a simple rule of thumb
of this type could only be consistent with these ?ndings by sheer coincidence, since such a
rule must consistently make no important systematic errors in forecasting dividend growth.
This observation does not, of course, rule out an explanation based on heterogeneity in stock
market participation, a possibility raised by the work of Mankiw and Zeldes (1991), Vissing-
J?rgensen (1999), Vissing-J?rgensen (2002), and others. For example, nonstockholders may
follow a simple Campbell-Mankiw rule of thumb while stockholders are forward-looking and
make an optimal consumption choice. Such an explanation is in fact very close in spirit
to the original Campbell and Mankiw (1989) framework. Lacking a long time-series on the
consumption of stockholders, however, such a hypothesis may be di?cult to test empirically.
5.3 Including Share Repurchases
So far we have focused on measuring dividends as the actual cash paid to shareholders of the
CRSP value-weighted index. We do this in order to make our results directly comparable
with the existing literature which has focused on forecasting the growth rate in this particular
measure of dividends. This measure is of interest because it represents the predominant form
of payout to shareholders over much of the post-war period. Moreover, as noted by Campbell
and Shiller (2001), traditional dividends are an appealing indicator of fundamental value for
long-term shareholders, because the end-of-period share price becomes trivially small when
28
discounted from the end to the beginning of a long holding period.
Nonetheless, there is a growing view that changing corporate ?nance policy has led
many ?rms, in recent years, to compensate shareholders through repurchase programs rather
than through dividends (Fama and French (2001); Grullon and Michaely (2002)), even if
large ?rms with high earnings have continued to increase traditional dividend payouts over
time (DeAngelo, DeAngelo, and Skinner (2002)). In this section we show that our main
conclusions are not altered by adjusting dividends to account for share repurchase activity.
One way to adjust dividends for such shifts in corporate ?nancial policy is to add re-
purchases (dollars spent on repurchases) to dividends.23 We do so here by using aggregate
share repurchase expenditures for the Industrial Compustat ?rms reported in Grullon and
Michaely (2002). These data cover the period 1972 to 2000 and are added to the CRSP
dividends after being converted to match the same scale as our dividend series. As Grullon
and Michaely (2002) note, repurchases activity prior to 1972 represented a tiny fraction of
shareholder compensation for U.S. corporations; thus the traditional dividend series should
provide an excellent approximation of actual payouts in data prior to 1972.
Table 9 presents the results of univariate long horizon forecasting regressions for the
growth in dividends plus repurchase activity, using dt and c t as forecasting variables in
separate regressions.24 The results should be compared with those in Table 5, which presents
the analogous ?ndings using CRSP value-weighted dividends. Comparing the output from
the two tables, it is immediately evident that the inclusion of share repurchases delivers
?ndings that are very similar to those excluding repurchases. dt and c are both strong
predictors of the long-horizon growth rates in this series, with t-statistics that begin above 4
for horizons at one year and increase, and R-squared statistics that are in line with those in
Table 4. We conclude that adjusting dividends for repurchases does not alter the main ?nding
in this paper, namely that the growth in compensation to shareholders is forecastable in post-
war data, and over horizons previously associated exclusively with return forecastability.
23We add gross repurchases rather than net repurchases (dollars spent on repurchases less dollars spent on
reissues) to dividends, since dividends are also measured on a gross basis. This procedure is conservative for
our purpose, since it insures that share repurchase activity is maximally represented relative to traditional
dividend payments.
24The altered dividend series (including repurchases) is used both to construct the cointegrating relation
dcdyt, and in the long horizon growth rate to be forecast.
29
6 The Comovement in Expected Returns and Expected
Dividend Growth
This section pursues two ways of quantifying the comovement between expected excess re-
turns and expected dividend growth, as measured by the state variables, or driving processes,
cayt and cdyt, respectively. We begin by investigating a simple principal components frame-
work for the bivariate system (dt; c t)0. We use a standard eigenvalue decomposition to
identify principal components in this bivariate system (see Theil (1971)). Figure 2 plots the
?rst principal component, along with our estimate of cdyt. The ?rst principal component
is almost identical to c t itself. The behavior of these two variables is well-described by
the ?rst principal component, which explains 80.2 percent of the variation in the system
(dt; c t)0. Thus, a large fraction of the variation in expected returns and expected divi-
dend growth is indeed common, and the common component is very well captured by c t.
Moreover, the ?nding that the ?rst principal component is almost identical to c t suggests
that virtually all of the variation in expected dividend growth is common to variation in
expected returns. The ?rst principal component is less highly correlated with dt, sug-
gesting that some variation in expected returns is independent of that in expected dividend
growth. In short, most of the variation in rational forecasts of long-horizon dividend growth
is common to variation in rational forecasts of long-horizon returns, but not the other way
around. Given that common variation in expected returns and expected dividend growth
has o?setting e?ects on the log dividend price ratio, the ?ndings are consistent with the
well-documented result that the log dividend-price ratio contains signi?cantly more infor-
mation about future long-horizon returns than it does about long horizon dividend growth.
In summary, the common components analysis suggests that variation in expected dividend
growth is mostly common to variation in expected returns, while expected returns have a
component that is independent of expected dividend growth.
The second way we quantify the common variation in dt and c t is to employ the
frequency domain measure of comovement developed by Croux, Forni, and Reichlin (2002).
Croux, Forni and Reichlin call this measure of comovement ?dynamic correlation?when ap-
plied to two series. We refer the reader to that paper for details an merely outline the
approach here. Dynamic correlation measures how important are cycles of di?erent frequen-
cies in accounting for the comovement between series. Here we use dynamic correlation to
quantify the comovement between our empirical state variables for expected returns and
30
expected dividend growth. The dynamic correlation statistic is de?ned
?xy(?) = Cxy(?)S
x(?)Sy(?)
;
where ? is the frequency  ? ? ? ? ?, Cxy is the co-spectrum, and Sx and Sy are the spectral
density functions of x and y respectively.25
Figure 2 displays four plots. The ?rst three are the individual spectra for the dividend-
price ratio, dt and c t. Recall that a white noise series has a ?at spectrum. Thus the very
steep spectrum for the dividend-price ratio indicates that the low frequency components of
dt pt are by far the most important determinants of its sample variance, so much so that the
variable appears nonstationary in this sample. By contrast, the low frequency components
of dt and c t are far less important; the spectra for these two variables are much ?atter
and they take on much lower values at low frequencies than does the spectrum for the log
dividend-price ratio. These results are not surprising given the autocorrelation coe?cients
reported in Table 2.
The fourth panel of Figure 2 displays the dynamic correlations between each series. The
panel shows that dt and c t are most highly correlated at periods ranging from 2 to 4
years; they have a correlation of almost 70 percent at frequencies corresponding to 2 years
and a correlation in excess of 50 percent at horizons less than 4 years. By contrast, dt is
most highly correlated with dt   pt at low frequencies, when it is least correlated with c t.
In addition, although c t is not highly correlated with dt   pt at any frequency, it is least
highly correlated with dt   pt at low frequencies, where most of the sampling variation in
the dividend-price ratio occurs.
Taken together, these results imply that the low-frequency variation in dt pt that so dom-
inates its sampling variation is better described as capturing slow-moving forecasts of excess
returns than time-varying expected dividend growth. Variation in expected dividend growth,
as captured by c t, occurs at higher frequencies than those over which dt  pt predominately
varies. Thus the common component in expected dividend growth and expected returns ap-
pears to vary at business cycle frequencies, those corresponding to periods of a few years.
By contrast, the dividend-price ratio is most highly correlated with the consumption-wealth
ratio at low frequencies, when the comovement between expected returns and expected divi-
dend growth is least pronounced. It follows that the common component in dt and dt   pt
appears to be a low frequency component that is primarily associated with time variation in
expected returns.
25The co-spectrum is the real part of the cross-spectral density between two series.
31
7 Implications for the Equity Premium Puzzle
There has been a growing interest in equilibrium asset pricing models that explicitly connect
consumption to dividends.26 One reason for this interest is the recognition that autonomous
variation in dividends and consumption o?er one way of rationalizing the observed equity
premium, since those models can in principal generate volatile stock returns without substan-
tial variation in consumption or real interest rates. Bansal and Yaron (2000) and Campbell
(2001) consider models of this form. Neither of these models, however, allow aggregate con-
sumption and dividends to be cointegrated. In this section we consider how implications for
the equity premium in models with autonomous dividend variability are a?ected by allowing
consumption and measures of aggregate dividends to be cointegrated.
The framework above implies that aggregate consumption can be thought of as the divi-
dend paid on all invested wealth, dw;t: Many authors have assumed that the aggregate stock
market is a good proxy for aggregate wealth and thus set consumption equal to the stock
market dividend, dt (e.g., Grossman and Shiller (1981), Lucas (1978); Mehra and Prescott
(1985)). Alternatively, Campbell (1986) and Abel (1999) assume that dividends on equity
equal aggregate consumption raised to a power ?, so that, in logs, dt = ?ct: Abel (1999)
shows that ? can be thought of as a measure of leverage, with levered equity represented
by values of ? ? 1. Values for ? greater than one make dividend growth more volatile than
consumption growth and therefore help resolve the equity premium puzzle. We follow Abel
(1999) and assume that ? ? 1. Campbell (2001) generalizes this set-up to allow for separate
variation in dividends and consumption by specifying a simple bivariate model taking the
form
?ct+1 = ?t+1
?dt+1 = ??t+1 + ?t+1; (12)
where ?t+1 is an autonomous component in dividends, assumed to be uncorrelated with ?t+1.
In this model, there is no cointegration between ct and dt.
The purpose of the calculations in this section is to examine the role of cointegration in
a?ecting the equity premium. Such a speci?cation makes sense if there is a saving technology
so that agents are not forced to consume their endowment (dividend) every period. In this
case, consumption and aggregate dividends will be cointegrated and move together over the
long term, but will not necessarily be equal in every period.
26See, for example, Abel (1999), Bansal and Yaron (2000), Bansal, Dittmar, and Lundblad (2001), Camp-
bell (1986), and Campbell (2001).
32
To build intuition, it is useful to begin by considering a simple example in which stock
market wealth is equal to aggregate wealth, implying wt = pt, and aggregate dividends, dw;t,
are equal to stock market dividends, dt. In this case, the share of stock market wealth in
aggregate wealth is unity, implying that cayt in (3) collapses to ct   wt = ct   pt, and cdyt
in (4) collapses to ct   dt. Below we relax this simplifying assumption and consider a case in
which ct is cointegrated with dt and yt, consistent with empirical evidence reported above.
We may allow for cointegration between ct and dt by modifying (12) so that consumption
and stock market dividends have the following error-correction representation
?ct+1 = ?t+1 (13)
?dt+1 = ?(ct   dt) + ??t+1 + ?t+1:
Equation (13) generalizes equation (12). The last two terms in (13) are the same as in
(12) and account for the sensitivity of dividends to innovations in consumption, allowing
for some independent variation between consumption and dividends. The ?rst term arises
because consumption and dividends are cointegrated; ?, which we will refer to as the ad-
justment parameter, measures the e?ect on dividend growth of last period's cointegrating
error, (ct   dt). Notice that the budget constraint analysis discussed above implies dividends
can no longer be a levered version of the level of consumption, dt = ?ct, ? > 1, otherwise
consumption would eventually become a negligible fraction of dividends. Instead, dividends
may respond to a consumption innovations, ?t+1, in a levered fashion, but cointegration re-
quires the volatility of dividend growth to be identical to that of consumption growth when
measured over su?ciently long horizons.
In this simple bivariate model, ? tells us how quickly dividends adjust to restore the
consumption-aggregate dividend ratio to its long-run mean, subsequent to an equilibrium-
distorting shock, or ?cointegration gap.? Higher values of ? correspond to swifter adjust-
ments and less persistent variation in (ct   dt) : Equation (13) collapses to (12) in the special
case where dividends and consumption are not cointegrated, which in this simple exam-
ple occurs as ? ! 0: This example assumes that consumption growth is unforecastable, so
that deviations from the common trend in consumption and dividends are eventually elimi-
nated by a subsequent movement in dividend growth, consistent with the empirical ?ndings
presented above.
How does cointegration between consumption and dividends a?ect risk premia in this
simple example? To address this question, we follow classic papers on the equity risk premium
and assume there is a representative agent that maximizes a time-separable constant relative
33
risk aversion (CRRA) utility function de?ned over aggregate consumption
U (Ct) = C
1  
t   1
1     ;
where   is the coe?cient of relative risk aversion. For expositional convenience, we assume
that expected asset returns are constant and aggregate consumption and asset returns are
jointly lognormal and homoskedastic. The expected log equity risk premium (adjusted for a
Jensen's inequality term) is equal to
Etrt+1   rf;t+1 + ?2r=2 =   Cov(rt+1;?ct+1): (14)
In addition, Campbell (1991) shows that asset return surprises are given (up to a ?rst-order
approximation) by the following expression
rt+1   Etrt+1 = (Et+1   Et)
1X
j=0
?j?dt+1+j   (Et+1   Et)
1X
j=1
?jrt+1+j
If expected returns are constant, return surprises are entirely attributable to revisions in ex-
pectations about future dividend growth. In this case, the risk-premium of the consumption
claim is
Etrt+1   rf;t+1 + ?2r=2 =  ?2?: (15)
To derive the risk-premium of the dividend claim, use the cointegrated system (13). The
unexpected return of the dividend claim reduces to
rt+1   Etrt+1 = ?(1   ?) + ??1   ? + ?? ?t+1 + 1   ?1   ? + ???t+1:
De?ning ? ? ?(1 ?)+??1 ?+?? ; equation (14) implies that the risk premium on the dividend claim
is
Etrt+1   rf;t+1 + ?2r=2 =  ??2?: (16)
We may use (16) to illustrate how the equity premium is a?ected by cointegration. Notice
that ? is increasing in the leverage parameter, ?, but decreasing in the adjustment parameter
?. In the model of equation (13), only dividends participate in the adjustment needed to
restore the consumption-dividend ratio to its long-run mean, subsequent to an innovation in
ct   dt. The higher is ?, the more quickly this adjustment takes place and the less persistent
is ct   dt. If dividends are not cointegrated with consumption, the adjustment never takes
place, and ? = 0. In this case, ? = ? and the model collapses to the simple levered models of
the type considered by Campbell (1986) and Abel (1999). By contrast, if consumption and
34
dividends are cointegrated and dividends fully adjust in the period immediately following an
innovation in the consumption-dividend ratio, ? = 1 and ? = ?(1   ?) + ? ? 1, since ? is
close to one. In this case the equity risk-premium of the dividend claim is almost identical to
that of the consumption claim, and the presence of leverage does not help resolve the equity
premium puzzle. In summary, relative to the risk premium of the consumption claim, the
risk premium of the dividend claim is decreasing in ?; and is therefore largest when there is
no cointegration (? = 0).
To get a feel for the magnitude of this e?ect on the equity premium, consider the following
parameterization:   = 50; ?? = 1:1% (from Table 1), ? = 10 and ? = 0:95. The risk premium
of a dividend claim that is not cointegrated with consumption (? = 0) is 6.05%, in line with
post-war U.S. data. If dividends adjust very quickly to close the cointegration gap, the risk
premium on the dividend claim decreases dramatically. With ? = 1 the risk premium is
0.605. For more interesting intermediate values, the e?ect of cointegration is still important
even if the adjustment parameter is substantially smaller; for example, if ? = 0:5, the risk
premium is only 1.2%.
The value for risk-aversion required to match the empirical equity premium in this pa-
rameterization is considerably larger than that required in Abel (1999) to match the equity
premium, as is the number for ?, which he estimates is consistent with the relative volatilities
of consumption and dividends when it takes the value ? = 2:74. The reason for these dif-
ferences is that Abel calibrates his model to data used in Mehra and Prescott (1985), which
includes the signi?cantly more volatile pre-war consumption data and smaller equity risk pre-
mium. Those values, as reported by Mehra and Prescott (1985) in data from 1889-1978, are
?? = 3:6% and Etrt+1  rf;t+1 = 4:48%. Using this ?gure for ??, Abel obtains a risk-premium
of 3.9% for the log-linear approximate version of his model by setting ? = 2:74, and relative
risk aversion equal to 11.048. The same value for the equity risk-premium (3.92%) may be
obtained in the model of this section by setting ? = 0 (so that the framework collapses to the
simple levered case without cointegration), and by plugging the values ? = 2:74,   = 11:048,
and ?? = 3:6% into (16).
So far we have considered a simple example in which consumption is cointegrated with
stock market dividends. The analysis above, however, suggests not that consumption and
stock market dividends are cointegrated, but that consumption and measures of aggregate
dividends, or ct, dt, and labor income yt, are cointegrated. It is straightforward to generalize
the analysis further to allow for cointegration among consumption, dividends and labor
income, as in (4). For example, we might modify (13) so that ct, at, and yt have the
35
following error-correction speci?cation
?ct+1 = ?t+1 (17)
?yt+1 = ??t+1 + ut+1
?dt+1 = ?dt + ??t+1 + ?ut+1 + ?t+1:
The set of equations in (17) generalizes those in (13) and is directly analogous. Requiring
? > 1 is analogous to requiring ? > 1, so that ? can be thought of as a ?leverage? parameter
for the dividend to human capital, yt. Similarly, the autonomous innovation ut+1 is analogous
to the autonomous innovation ?t+1. The cointegrating residual is now dt ? ct  bddt  byyt;
rather than ct dt. The parameter ? tells us how quickly dividends adjust to restore dt to its
long-run mean; higher values of ? again correspond to swifter adjustments and less persistent
variation in dt: Moreover, this case is comparable to the previous example in that stock
market dividends and human capital dividends may behave in a levered fashion in response
to innovations in consumption growth, but the relative volatility of the growth rates in ct,
dt, and yt is governed in the long-run by their common trends, which insure that the precise
linear combination of variables given by dt is stationary. Finally, this example models log
consumption and log labor income as random walks, thereby capturing the empirical ?nding
documented above that only stock market dividends, ?dt+1, are signi?cantly related to dt:
Although the data suggest that log consumption and log labor income are not exactly random
walks, the empirical ?ndings presented above suggest that (17) is likely to be a reasonable
approximation of the vector time-series properties for these variables.
Under the assumptions about preferences and the stochastic properties of returns and
consumption made above, the risk-premium on the consumption claim is again given by (15).
To derive the risk-premium of the dividend claim, use the cointegrated system (17). In this
case, the unexpected return of the dividend claim reduces to
rt+1   Etrt+1 = ?(1   ?) + ??(1  
b?y?)
1   ? + ?bd? ?t+1 +
?(1   ?)   ???y
1   ? + ?bd? ut+1 +
1   ?
1   ? + ?bd??t+1:
De?ning ? ? ?(1 ?)+??(1 b?y?)1 ?+?b?
d?
; equation (14) implies that the risk premium on the dividend
claim is
Etrt+1   rf;t+1 + ?2r=2 =  ??2?:
The parameter ? collapses to ? when the cointegrating coe?cient on labor income, by; is
zero, and the cointegrating coe?cient on stock market dividends, bd, is one.
36
We may again gain an understanding of the magnitude of this e?ect on the equity pre-
mium by considering the parameterization used in Abel (1999):   = 11:048; ?? = 3:6%;
? = 2:74 and ? = 0:95. We also set by = 0:68 and bd = 0:13 to match the empirical
estimates reported above. The parameter ? may be calibrated by matching the covariance
between ?ct and ?yt implied by (17) with that from the US data displayed in Table 1. The
covariance implied by (17) is
Cov (?ct;?yt) = ??2?;
where ?2? is the variance of consumption growth. Given the estimates of the standard devi-
ations and correlation of ?ct and ?yt, a value of ? = 1:26 is obtained.
Under these parameter values, the risk-premium of the consumption claim is 1.43%.
When there is no cointegration (? ! 0), the risk-premium of the dividend claim again
collapses to that implied by Abel's speci?cation and the presence of leverage generates the
much larger 3.92% premium. By contrast, when cointegration among ct, dt, and yt is present
and ? = 2:4 (the estimated value reported in Table 3), the risk-premium of the dividend
claim is only slightly larger than that of the consumption claim, equal to 1.91%. This
occurs because an innovation in consumption growth causes dt to deviate from its mean,
and therefore necessitates a subsequent adjustment in dividends in order to restore dt to
its mean. Such an adjustment reduces the covariance between news about future dividend
growth and today's consumption innovation, thereby reducing the equity premium. To
obtain an equity premium of 3.92% in this setting (when ? = 2:4), an unrealistically large
value for the stock market leverage parameter is required, equal to ? = 13.
Of course, the model above could account for a higher equity risk-premium without
leverage if it were modi?ed to allow for catching up with the Joneses preferences, as in Abel
(1999). Abel emphasizes, however, that one cannot rely on catching up with the Joneses
preferences alone to match both the equity premium and the volatility of the short-term
riskless rate. To rationalize both phenomena, leverage must be su?ciently high, since only
leverage increases the equity premium implied by the model without increasing the volatility
of the short-term riskless rate. The calculations in this section suggest that the presence of
cointegration among consumption, stock market dividends and labor income, once calibrated
to match the estimates of the cointegrating relation we document here, can signi?cantly limit
the scope for leverage to increase the equity risk-premium implied by the model. Even if
deviations from the common trend in consumption, dividends and labor income are quite
persistent and dividends are considerably more volatile than consumption in the short-run,
the dividend stream may not be signi?cantly more risky than the consumption stream if all
37
three variables are tied together in the long-run. In the simple asset pricing model studied
above, even large values of leverage are insu?cient to explain the equity risk premium when
dividends, consumption and labor income are cointegrated.
8 Why Might Expected Dividend Growth Covary with
Expected Returns?
Why might expected dividend growth be positively correlated with expected Returns? Some
authors have noted that the present value model for dividends and prices, (5), together
with the hypothesis that managers smooth dividends, can be used to interpret the empirical
behavior of the dividend-price ratio. Such explanations imply that expected dividend growth
is roughly constant. For example, Cochrane (1994) argues that if managers perfectly smooth
dividends by setting them equal to the discounted value of future earnings (discounted at the
risk-free rate), dividends will follow a random walk, explaining why dividend growth is close
to unforecastable by the log dividend-price ratio. The investigation of this paper ?nds that
dividend growth is not unforecastable, but is instead predictable over horizons of several
years, horizons shorter than those corresponding to frequencies over which the persistent
dividend-price ratio primarily varies. Of course, dividends are smoother than earnings, and
the dividend payout ratio has some forecasting power for future dividends (Lamont (1998)),
consistent with the hypothesis that managers do some dividend smoothing. But the results
presented suggest that dividend smoothing by managers may be imperfect.
If investors themselves desire smooth consumption paths, why don't managers perfectly
smooth dividend payments? One possibility is that although dividend-smoothing may
be possible over long horizons (implying that dividend growth may be unforecastable by
variables such as the dividend-price ratio whose sampling variation is dominated by low-
frequency ?uctuation), it may be more di?cult over horizons corresponding to business
cycle ?uctuations. Several researchers have presented evidence that is suggestive of this
hypothesis. Gertler and Hubbard (1993) study ?rm-level data from Compustat and ?nd
that ?rm dividend payouts vary with macroeconomic conditions: they are lower during a
slow-down in economic growth and higher during periods of economic expansion. Others
have documented that the relative cost of obtaining external ?nance rises during an eco-
nomic slow-down, suggesting that managers who need to ?nance long-term projects have
a greater need to retain earnings in recessions than in expansions. Bernanke and Gertler
38
(1989) and Bernanke, Gertler, and Gilchrist (1996) present theoretical and empirical evi-
dence of countercyclical variation in the external ?nance premium. The equity risk-premium
also appears counter-cyclical: it rises during an economic slow-down and falls during peri-
ods of economic growth (Fama and French (1989); Ferson and Harvey (1991); Lettau and
Ludvigson (2001a)). Taken together, these ?ndings suggest that high risk-premia occur in
periods of economic recession and coincide with a temporarily low stock price, temporarily
low earnings and temporarily low dividends. According to this hypothesis, consumers may
be better able to smooth transitory ?uctuations in earnings than managers, implying that
earnings growth should be predictably higher when, according to c t, dividend growth and
excess stock returns are predictably higher.
Table 10 presents some evidence that is supportive of this hypothesis using earnings data
for NYSE ?rms. The earnings data are from Lewellen (2001) and are operating earnings
before depreciation to market value. Unfortunately, only a short sample is available that
is limited by when Compustat data are available: 1964-2000.27 Table 10 presents long-
horizon forecasts of earnings growth using c t as a predictive variable. Earnings growth is
predictably higher when predictable dividend growth, as captured by c t, is higher. The
present-value relation, c t, is strongly statistically signi?cant as a predictor of earnings
growth, with t-statistics in excess of four for one to three year forecasting horizons, and in
excess of three for a four year horizon. The univariate forecasting regression explains about
14 percent of the variation in earnings growth 4 years hence. The results suggest that when
consumption is high relative to its common trend with stock market dividends and labor
income, both dividends and earnings are temporarily low and forecast to grow more quickly
in the future. These ?ndings are consistent with the hypothesis that consumers are better
able to smooth consumption than managers are able to smooth dividends through business
cycles.
9 Conclusion
This paper presents evidence that changing forecasts of dividend growth make an important
contribution to ?uctuations in the U.S. stock market, despite the failure of the dividend-price
ratio to uncover such variation. We ?nd that aggregate consumption, stock market dividends
and labor income are cointegrated, and that deviations from their common trend are typi-
27We use Lewellen's data and not earnings per share since that measure is contaminated by variability in
share issuance.
39
cally restored by a forecastable movement in stock market dividend growth. Although these
?ndings contradict the common conclusion that expected dividend growth is roughly con-
stant, they reinforce the textbook conclusion that expected returns are time-varying and
make an important contribution to aggregate stock market ?uctuations. Dividend forecasts
covary with changing forecasts of excess stock returns, and are positively correlated with
business cycle variation in expected returns. The ?ndings provide at least a partial explana-
tion for why the consumption-wealth ratio has been found superior to the log dividend-price
ratio as a predictor of excess stock market returns over medium-term horizons.
The ?ndings have several implications for asset pricing. First, they imply that the log
dividend-price ratio will have di?culty predicting both dividend growth and excess returns
over anything but extremely long horizons precisely because expected excess returns ?uctuate
and covary with expected dividend growth at those horizons. Thus, the results suggest
that an important component of time-varying expected returns and time-varying expected
dividend growth is not captured by the log dividend-price ratio, or likely by other aggregate
?nancial ratios. This stacks the deck against such ?nancial ratios in statistical tests of return
or dividend growth predictability. Second, the ?ndings imply that time-varying investment
opportunities will be poorly captured by variation in the log dividend-price ratio, because
it fails to reveal signi?cant movements in the investment opportunity set that occur over
business cycle horizons. Third, autonomous variation in dividends as a levered version of
consumption is unlikely to provide a complete resolution of the equity premium puzzle. If
consumption, stock market dividends and labor income are cointegrated, as the evidence
presented here suggests, the dividend claim may not be signi?cantly more risky than the
consumption claim even for large values of leverage.
We caution that the ?ndings presented here provide but one piece of a larger puzzle
concerning the behavior of the dividend-price ratio, especially that more recently. There is
a growing view that shifts in corporate ?nancial policy may have created persistent changes
in dividend growth rates. For example, ?rms have been distributing an increasing fraction
of total cash paid to shareholders in the form of stock repurchases (e.g., Fama and French
(2001)). It is too soon to tell whether such shifts in corporate ?nancial policy will be
sustained. At the same time, however, stock prices relative to earnings and other measures
of economic fundamentals have followed patterns similar to that of the dividend-price ratio
(Campbell and Shiller (2001)), while the consumption-based valuation ratio for dividend
growth studied here has been less a?ected. These observations suggest that factors other than
changes in corporate payout policy may be partly responsible for the behavior of aggregate
40
?nancial ratios in recent data. Whatever the reason for these changes, the results presented
here suggest that some of the di?erences between the log dividend-price ratio and the log
consumption-wealth ratio have been attributable historically to changing forecasts of long-
horizon dividend growth.
41
Appendix A: Derivation of Consumption-Based Loglinear Present
Value Expressions
This appendix derives the consumption-based present value expression for dividend growth,
cdyt. First, note that cayt is derived by expressing the market value of human capital in terms
of future expected returns to human wealth and future dividend ?ows from human wealth, namely
labor income (Lettau and Ludvigson (2001a)). This allows a reformulation of the bivariate cointe-
grating relation between ct and wt as a trivariate cointegrating relation involving three observable
variables, namely ct, at, and yt;where yt denotes log labor income. Denote the net return to non-
human capital Ra;t and the net return to human capital Rh;t, and assume that human capital, Ht;
takes the form, Ht = Et P1j=0 Qji=0(1 + Rh;t+i) iYt+j, so that labor income is the dividend to
human wealth. A log-linear approximation of Ht yields
ht = ?H + yt + Et
1X
j=1
?jh(?yt+j   rh;t+j) (18)
where ?H is a constant and ?h ? 1=(1 + exp(y   h). We follow Campbell and Mankiw (1989)
and assume that ?h = ?w ? 1=(1 + exp(c   w). (The equations below can easily be extended to
relax this assumption but nothing substantive is gained by doing so since the generalization would
merely yield more complicated of future expected returns and future expected income ?ows from
wealth.) An approximation for log aggregate wealth, Wt, as a function of its component elements,
At and Ht is given by
wt t !at + (1   !) ht: (19)
In addition, we follow ?? and linearize the relationship
Rw;t+1 = !tRa;t+1 + (1   !t)Rh;t+1 (20)
around the sample means of !t, ra;t and rh;t assuming that the latter two are equal. Recall that
Rw;t+1 denotes the net return, while rw;t denotes the log return. The result is
rw;t+1 ? !ra;t+1 + (1   !) rh;t+1: (21)
Combining (2), (18), (19) and (21) furnishes an approximate equation for the log consumption-
aggregate wealth ratio given in equation (3). Notice that terms involving the unobservable return
to human capital drop out because they appear on both the left-hand and right-hand sides of (2).
To derive cdyt; we express the market value of stock market wealth in terms of future expected
returns to stock market wealth and future dividend ?ows from stock market wealth. We do the
same for all nonstock wealth. The present value relation for stock market wealth, St;
St = Et
1X
j=0
jY
i=0
(1 + Rt+i) iDt+j (22)
is analogous to the more familiar present value relation for prices and dividends:
Ps;t = Et
1X
j=1
jY
i=1
(1 + Rs;t+i) iDs;t+j; (23)
where Ps;t and Ds;t in (23) denote price and dividends per share, respectively. However, there are
two di?erences. The ?rst is that (22) uses total market value, St; and total dividends, rather price
per share, Pt; and dividends per share, as in (23). The second is that we use a di?erent timing
convention in order to be consistent with the timing convention for aggregate wealth in (1). Since
wealth is measured at the beginning of the period, asset values are cum-dividend values, rather than
the typically assumed ex-dividend value associated with measuring price at the end of the period.
To obtain a dividend-based consumption ratio, we suppose total wealth consists of Nt shares, and
let Pt ? Ps;tNt be the ex-dividend value of an asset making a dividend payment Dt:
St = (Pt + Dt)
= Nt(Ps;t + Ds;t):
Then the return on wealth can be written
Rt+1 = (Pt+1 + Dt+1) =Pt;
and St = Pt + Dt is the cum-dividend stock market value at time t. Thus the expression for the
return can be written in terms of cum-dividend total value:
St+1 = Rt+1 (St   Dt) ;
which is in the same form as (1) and can be linearized in the same way. We obtain a loglinear
approximate expression
st = ?s + dt + Et
1X
j=1
?jw(?dt+j   rt+j); (24)
where and ?s is a constant. The same derivation can be applied to all nonstock forms of aggregate
wealth (primarily human capital) denoted ft, where ft includes human capital and nonstock
forms of net worth. In the text we conserve on notation by using Yt to denote both the income
?ows from human capital and the income ?ows from all nonstock forms of net worth, since the
latter is primarily human capital. To be precise, we now use et+1 for the latter, implying that
fWt+1 = eRt+1 ?fWt   Yt?;
and a loglinear approximate expression for ft takes the form
et = ?ew + et + Et
1X
j=1
?jw(?et+j   et+j); (25)
where et is the log return to ft. Implicitly we are assuming that the mean dividend-price ratio for
each form of aggregate wealth equals the mean consumption-wealth ratio, since the same parameter
?w appears in (18), (24) and (25). An approximation for log aggregate wealth, Wt, as a function
of its component elements, St and ft is given by
wt t ?st + (1   ?) et; (26)
where ? is the average share of stock market wealth in aggregate wealth. Combining (25), (24),
(26) and (2), we obtain (4).
Appendix B: Data Description
The sources and description of each data series we use are listed below.
CONSUMPTION
Consumption is measured as either total personal consumption expenditure or expenditure on
nondurables and services, excluding shoes and clothing. The quarterly data are seasonally adjusted
at annual rates, in billions of chain- weighted 1996 dollars. The components are chain-weighted
together, and this series is scaled up so that the sample mean matches the sample mean of total
personal consumption expenditures. Our source is the U.S. Department of Commerce, Bureau of
Economic Analysis.
AFTER-TAX LABOR INCOME
Labor income is de?ned as wages and salaries + transfer payments + other labor income -
personal contributions for social insurance - taxes. Taxes are de?ned as [wages and salaries/(wages
and salaries + proprietors' income with IVA and Ccadj + rental income + personal dividends
+ personal interest income)] times personal tax and nontax payments, where IVA is inventory
valuation and Ccadj is capital consumption adjustments. The annual data are in current dollars.
Our source is the Bureau of Economic Analysis.
WEALTH
Total wealth is household net wealth in billions of current dollars, measured at the end of the
period. Stock market wealth includes direct household holdings, mutual fund holdings, holdings
of private and public pension plans, personal trusts, and insurance companies. Nonstock wealth is
the residual of total wealth minus stock market wealth, and includes ownership of privately traded
companies in noncorporate equity. Our source is the Board of Governors of the Federal Reserve
System.
DIVIDENDS
Dividends are constructed from the CRSP index returns. The CRSP dividends, Dc;t, are scaled
by the average ratio of stock market wealth, St to the price of the value-weighted CRSP index, Pc;t
to re?ect dollar values, i.e., Dt ? E(St=Pc;t)Dc;t.
POPULATION
A measure of population is created by dividing real total disposable income by real per capita
disposable income. Consumption, wealth, labor income, and dividends are in per capita terms.
Our source is the Bureau of Economic Analysis.
PRICE DEFLATOR
The nominal after-tax labor income, stock market dividend and wealth data are de?ated by the
personal consumption expenditure chain-type de?ator (1996=100), seasonally adjusted. (Source:
Bureau of Economic Analysis.)
Appendix C: Cointegration Tests
This appendix presents the results of cointegration tests. Dickey-Fuller tests for the presence
of a unit root in c, y, a, d, and p (not reported) are consistent with the hypothesis of a unit root
in those series.
Table C-I reports test statistics corresponding to two cointegration tests. Reported in the far
right column are Phillips and Ouliaris (1990) residual based cointegration test statistics. The table
shows both the Dickey-Fuller t-statistic and the relevant ?ve and 10 percent critical values. The test
is carried out without a deterministic trend in the static regression. We applied the data dependent
procedure suggested in Campbell and Perron (1991) for choosing the appropriate lag length in an
augmented Dickey-Fuller test. This procedure suggested that the appropriate lag length was one
for both the (c;a;y)0 system and the (c;d;y)0 system. The tests reject the null of no cointegration
both systems at the ?ve percent level. The persistent dividend-price ratio displays no evidence
favoring cointegration in this sample.
Table C-I also reports the outcome of testing procedures suggested by Johansen (1988) and Jo-
hansen (1991) that allow the researcher to estimate the number of cointegrating relationships. This
procedure presumes a p-dimensional vector autoregressive model with k lags, where p corresponds
to the number of stochastic variables among which the investigator wishes to test for cointegration.
For our application, p = 3. The Johansen procedure provides two tests for cointegration: under the
null hypothesis, H0, that there are exactly r cointegrating relations, the `Trace' statistic supplies
a likelihood ratio test of H0 against the alternative, HA, that there are p cointegrating relations,
where p is the total number of variables in the model. A second approach uses the `L-max' statistic
to test the null hypothesis of r cointegrating relations against the alternative of r + 1 cointegrating
relations.
The critical values obtained using the Johansen approach also depend on the trend character-
istics of the data. We present results allowing for linear trends in data, but assuming that the
cointegrating relation has only a constant. See the articles by Johansen for a more detailed dis-
cussion of these trend assumptions. In choosing the appropriate trend model for our data, we are
guided by both theoretical considerations and statistical criteria. Theoretical considerations imply
that the long-run equilibrium relationship between consumption, labor income and wealth do not
have deterministic trends, although each individual data series may have deterministic trends. The
Table also reports the 90 percent critical values for these statistics.
Both the L-max and Trace test results establish evidence of a single cointegrating relation among
log consumption, log labor income, and the log of household wealth, and among log consumption,
log dividends and the log of labor income. Table C-I shows that we may reject the null of no
cointegration against the alternative of one cointegrating vector. In addition, we cannot reject the
null hypothesis of one cointegrating relationship against the alternative of two or three.
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Table 1: Summary Statistics
?ct ?yt ?dt ?pt ?at
Univariate Summary Statistics
Mean (in %) 2.01 2.30 4.01 6.88 2.45
Standard Deviation (in %) 1.14 1.83 12.24 16.13 4.05
Correlation Matrix
?ct 1.00 0.78 -0.13 -0.00 0.32
?yt 1.00 -0.10 -0.10 0.18
?dt 1.00 0.64 0.52
?pt 1.00 0.83
?at 1.00
Notes: This table reports summary statistics for annual growth of real per capita consumption
?ct, labor income ?yt, CRSP-VW dividends ?dt, CRSP-VW price ?pt and asset wealth ?at.
The sample spans the period 1948 to 2001.
Table 2: Autocorrelations of Ratios
Ratio ?1 ?2 ?3 ?4
d   p 0.875 0.724 0.596 0.473
c   0:29 a   0:60 y 0.551 0.130 0.085 0.051
c   0:13 d   0:68 y 0.475 0.217 0.258 0.171
Notes: This table reports autocorrelations of ratios involving consumption ct, labor income yt,
CRSP-VW dividends dt, CRSP-VW price pt and asset wealth at. ?i denotes the autocorrelation of
order i (in years). The cointegrating coe?cients in the last two rows are estimates using dynamic
least squares with 2 leads and lags. The sample is annual and spans the period 1948 to 2001.
Table 3: Estimates From Cointegrated VARs
Panel A: (d;p)
Equation
Dependent Variable ?dt ?pt
?dt 1 -0.194 0.364
(t-stat) (-1.059) (1.352)
?pt 1 -0.192 -0.210
(t-stat) (-1.441) (-1.079)
dt 1   pt 1 0.103 0.070
(t-stat) (2.205) (1.021)
?R2 0.183 0.046
Panel B: (c;a;y)
Equation
Dependent variable ?ct ?yt ?at
?ct 1 0.267 0.449 -0.523
(t-stat) (1.279) (1.220) (-0.696)
?yt 1 -0.039 -0.148 0.433
(t-stat) (-0.294) (-0.641) (0.916)
?at 1 0.112 0.128 0.392
(t-stat) (2.777) (1.794) (2.702)
dt 1 -0.007 0.102 1.726
(t-stat) (-0.053) (0.457) (3.803)
?R2 0.199 0.050 0.207
Panel C: (c;d;y)
Equation
Dependent variable ?ct ?yt ?dt
?ct 1 0.469 0.652 -0.136
(t-stat) (2.284) (1.869) (-0.060)
?yt 1 -0.074 -0.156 -0.252
(t-stat) (-0.572) (-0.709) (-0.176)
?dt 1 0.029 0.052 -0.129
(t-stat) (2.311) (2.389) (-0.917)
dcdyt 1 -0.038 0.219 2.400
(t-stat) (-0.408) (1.377) (2.314)
?R2 0.179 0.098 0.104
Notes: The table reports estimated coe?cients from cointegrated ?rst-order vector autoregres-
sions of the column variable on the row variable; ct is log consumption, yt is log labor income, at is
log asset wealth (net worth), dt is log stock market dividends, and pt is the log CRSP value-weighted
price index. t-statistics are reported in parentheses. Estimated coe?cients that are signi?cant at
the 5% level are highlighted in bold face. The sample is annual and spans the period 1948 to 2001.
Table 4: Univariate Long-horizon Regressions ? Excess Stock Returns
h-period regression: Phi=1(rt+i   rf;t+i) = k +   zt + ?t;t+h
Horizon h (in years)
zt = 1 2 3 4 5 6
dt   pt 0.14 0.24 0.27 0.34 0.52 0.73
(1.90) (1.40) (1.21) (0.73) (0.84) (1.12)
[0.08] [0.10] [0.10] [0.10] [0.16] [0.23]
dt 6.48 11.78 13.23 13.62 16.81 21.94
(4.19) (5.42) (5.42) (5.27) (7.07) (5.46)
{0:57???} {0:74???} {0:74???} {0:72???} {0:96???} {0:74???}
[0.27] [0.49] [0.46] [0.37] [0.39] [0.52]
dcdyt 1.32 5.21 6.11 6.77 18.09 11.40
(1.47) (7.38) (4.13) (4.28) (4.92) (4.45)
{0:20} {1:00???} {0:56???} {0:58???} {0:67???} {0:61???}
[0.01] [0.16] [0.20] [0.20] [0.20] [0.32]
Notes: This tables reports the results of h-period regressions of CRSP-VW returns in excess of a
3-month Treasury-bill rate, rr;t, on the variable listed in the ?rst column: Phi=1(rt+i  rf;t+i) = k+
  zt+?t;t+h, where zt are the cointegration residuals listed in the ?rst column. ct is log consumption,
yt is log labor income, at is log asset wealth (net worth), dt is log stock market dividends, and pt
is the log CRSP value-weighted price index. dt and dt are estimated cointegrating residuals
for the systems (ct;at;yt)0 and (ct;dt;yt)0, respectively. For each regression, the table reports OLS
estimates of the regressors, Newey and West (1987) corrected t-statistics (in parentheses), the
t=pT test suggested in Valkanov (2001) in curly brackets and adjusted R2 statistics in square
brackets. Signi?cant coe?cients using the standard t-test at the 5% level are highlighted in bold
face. Signi?cance at the 5%, 2.5% and 1% level of the t=pT test using Valkanov's (2001) critical
values is indicated by ?;?? and ? ? ?, respectively. The sample is annual and spans the period 1948
to 2001.
Table 5: Univariate Long-horizon Regressions ? Dividend Growth
h-period regression: dt+h   dt = k +   zt + ?t;t+h
Horizon h (in years)
zt = 1 2 3 4 5 6
dt   pt 0.09 0.18 0.19 0.23 0.29 0.34
(2.94) (2.11) (2.70) (2.27) (2.70) (2.41)
[0.07] [0.15] [0.13] [0.14] [0.15] [0.19]
dt 4.74 5.89 4.90 4.30 5.13 5.72
(6.26) (4.86) (3.33) (2.80) (2.17) (1.50)
{0:85???} {0:66???} {0:45???} {0:38???} {0:30?} {0:20}
[0.29] [0.30] [0.16] [0.09] [0.10] [0.12]
dcdyt 2.74 3.95 3.65 3.99 5.24 6.13
(4.06) (5.84) (4.13) (3.60) (5.38) (3.65)
{0:55???} {0:79???} {0:56???} {0:49???} {0:73???} {0:50???}
[0.20] [0.24] [0.20] [0.20] [0.28] [0.37]
Notes: This tables reports results from h-period regression of CRSP-VW dividend growth: dt+h  
dt = k +   zt + ?t;t+h, where zt are the cointegration residuals listed in the ?rst column. ct
is log consumption, yt is log labor income, at is log asset wealth (net worth), dt is log stock
market dividends, and pt is the log CRSP value-weighted price index. dt and dt are estimated
cointegrating residuals for the systems (ct;at;yt)0 and (ct;dt;yt)0, respectively. For each regression,
the table reports OLS estimates of the regressors, Newey and West (1987) corrected t-statistics
(in parentheses), the t=pT test suggested in Valkanov (2001) in curly brackets and adjusted R2
statistics in square brackets. Signi?cant coe?cients using the standard t-test at the 5% level are
highlighted in bold face. Signi?cance at the 5%, 2.5% and 1% level of the t=pT test using Valkanov's
(2001) critical values is indicated by ?;?? and ? ? ?, respectively. The sample is annual and spans
the period 1948 to 2001.
Table 6: Multivariate Long-horizon Regressions
Horizon h (in years)
Variables 1 2 3 4 5 6
h-period regression: excess stock returns
dt 6.94 11.15 12.22 11.95 15.33 18.47
(3.27) (3.73) (3.82) (3.43) (3.86) (4.12)
dcdyt -0.74 0.89 1.14 1.70 1.46 3.44
(-0.81) (0.69) (0.80) (1.02) (0.76) (1.81)
[0.27] [0.48] [0.45] [0.36] [0.38] [0.53]
h-period regression: dividend growth
dt 3.71 4.27 2.62 0.62 -0.30 -0.82
(3.08) (1.95) (1.09) (0.31) (-0.14) (-0.24)
dcdyt 1.64 2.29 2.58 3.72 5.37 6.48
(2.57) (1.86) (1.87) (2.74) (4.78) (3.32)
[0.34] [0.35] [0.21] [0.18] [0.26] [0.36]
Notes: This tables reports results from h-period regression of CRSP-VW returns in excess of a 3-
month Treasury-bill rate (top panel), and dividend growth (bottom panel). ct is log consumption,
yt is log labor income, at is log asset wealth (net worth), dt is log stock market dividends, and pt
is the log CRSP value-weighted price index. dt and dt are estimated cointegrating residuals
for the systems (ct;at;yt)0 and (ct;dt;yt)0, respectively. For each regression, the table reports OLS
estimates of the regressors, Newey and West (1987) corrected t-statistics (in parentheses) and
adjusted R2 statistics in square brackets. Signi?cant coe?cients at the 5% level are highlighted in
bold face. The sample is annual and spans the period 1948 to 2001.
Table 7: Implied Long-Horizon R2 from VARs
row Variables Implied R2 for Forecast Horizon H
1 2 3 4 5 6
1 ?dt 0.12 0.09 0.05 0.05 0.04 0.03
2 ?dt; dt 0.34 0.31 0.24 0.20 0.17 0.14
3 ?dt; dt 0.17 0.19 0.20 0.19 0.19 0.19
4 rt 0.08 0.09 0.05 0.03 0.03 0.03
5 rt; dt 0.36 0.52 0.49 0.42 0.39 0.36
6 rt; dt 0.20 0.26 0.26 0.28 0.31 0.32
Note: The table reports implied R2 statistics for H-year dividend growth and excess
returns obtained from second-order vector autoregressions. The column denoted ?Vari-
ables? lists the variables included in the VAR. The implied (unadjusted) R2 statistics
for dividend growth in rows 1, 2 and 3 and excess returns in rows 4, 5 and 6 for horizon
H are calculated from the estimated parameters of the VAR and the estimated co-
variance matrix of VAR residuals. Row 1 gives the implied R2 statistic for forecasting
dividend growth with lagged dividend growth, row 2 with lagged dt and row 3 with
lagged dt. Row 4 gives the implied R2 statistic for forecasting excess stock market
returns with lagged returns, row 5 with lagged dt, and row 6 with lagged dt. The
sample is annual and spans the period 1948 to 2001.
Table 8: Bootstrap Con?dence Intervals of Slope and R2
xt b 95% CI 99% CI R2 95% CI 99% CI
rt+1   rft+1 = ? + ?xt + ut+1; xt+1 = ? + ?xt 1 + vt+1
dt 6.48 (-2.78, 3.86) (-4.13, 5.44) 0.27 (-0.01, 0.06) (-0.02, 0.11)
dcdyt 1.32 (-2.09, 2.96) (-3.10, 4.15) 0.01 (-0.01, 0.07) (-0.02, 0.10)
?dt+1 = ? + ?xt + ut+1; xt+1 = ? + ?xt 1 + vt+1
dt 4.74 (-2.12, 2.61) (-3.10, 3.78) 0.29 (-0.02, 0.06) (-0.02, 0.11)
dcdyt 2.74 (-1.32, 2.19) (-1.94, 3.09) 0.20 (-0.02, 0.06) (-0.02, 0.11)
Notes: This tables reports con?dence intervals from a bootstrap procedure. 10,000 arti?cial time
series are generated under the null hypothesis ? = 0 by drawing (with replacement) from the
residuals of the system estimated under the null hypothesis. The columns denoted b and R2 report
data estimates using annual data from 1948 to 2001.
Table 9: Univariate Long-horizon Regressions ? Including Share Repurchases
h-period regression: dt+h   dt = k +   zt + ?t;t+h
Horizon h (in years)
zt = 1 2 3 4 5 6
dt   pt 0.09 0.10 0.10 0.11 0.15 0.19
(1.76) (1.12) (0.81) (0.70) (0.81) (0.88)
[0.01] [0.01] [0.00] [0.00] [0.01] [0.02]
dt 4.66 6.36 6.52 6.51 7.97 9.24
(4.58) (4.52) (3.44) (2.97) (4.15) (3.95)
[0.24] [0.25] [0.19] [0.15] [0.18] [0.22]
dcdyt 4.28 5.10 4.59 4.77 6.47 8.29
(5.67) (5.05) (2.91) (2.32) (3.23) (3.98)
[0.20] [0.19] [0.12] [0.10] [0.16] [0.24]
Notes: This tables reports results from h-period regression of CRSP-VW dividend growth: dt+h  
dt = k+  zt +?t;t+h, where dividends are adjusted to include share repurchases using the estimates
in Grullon and Michaely (2002). ct is log consumption, yt is log labor income, at is log asset
wealth (net worth), dt is log stock market dividends plus repurchases, and pt is the log CRSP
value-weighted price index. dt and dt are estimated cointegrating residuals for the systems
(ct;at;yt)0 and (ct;dt;yt)0, respectively. For each regression, the table reports OLS estimates of the
regressors, Newey and West (1987) corrected t-statistics (in parentheses) and adjusted R2 statistics
in square brackets. Signi?cant coe?cients using the standard t-test at the 5% level are highlighted
in bold face. The sample is annual and spans the period 1948 to 2000, since the repurchases data
from Grullon and Michaely are only available through 2000.
Table 10: Long-horizon Regression ? Earnings Growth
Horizon h (in years)
Variables 1 2 3 4 5 6
dcdyt 2.16 3.46 4.73 6.68 6.75 7.05
(4.88) (6.50) (4.51) (3.56) (2.20) (2.01)
[0.07] [0.06] [0.07] [0.14] [0.13] [0.13]
Notes: This tables reports results from h-period regression of earnings growth: et+h   et = k +
? dt + ?t;t+h. The earnings data are from Lewellen (2001). For each regression, the table reports
OLS estimates of the regressors, Newey and West (1987) corrected t-statistics (in parentheses) and
adjusted R2 statistics in square brackets. Signi?cant coe?cients at the 5% level are highlighted in
bold face. The sample is annual and spans the period 1964 to 2000.
Table C-I: Cointegration Tests
L-max Test Trace Test t-Test
Variables H0 : r = 0 1 2 H0 : r = 0 1 2 H0 : no cointegration
10% Critical Values 10.60 2.71 13.31 2.71 -2.60
5% Critical Values 14.07 3.76 15.41 3.76 -2.93
d;p 6.06 4.56 10.62 4.56 -0.47
10% Critical Values 13.39 10.60 2.71 26.70 13.31 2.71 -3.52
5% Critical Values 20.97 14.07 3.76 29.68 15.41 3.76 -3.80
c;a;y 25.34 6.57 0.07 31.98 6.64 0.07 -4.13
c;d;y 27.58 5.36 1.08 34.01 6.43 1.08 -3.77
Notes: The ?rst two columns report the L-max and Trace test statistics described in Johansen
(1988) and Johansen (1991). The former tests the null hypothesis that there are r cointegrating
relations against the alternative of r+1; the latter tests the null of r cointegrating relations against
the alternative of p, where p is the number of variables in the cointegrated system. The last
column reports the Dickey-Fuller test for dt   pt and the Phillips-Ouliaris (1990) cointegration test
for (c;a;y) and (c;d;y). The critical values for the Phillips-Ouliaris tests allow for trends in the
data while the Dickey-Fuller regression does not include a trend, since according to the theory,
there should be no trend in d-p. One lag was used for all tests. The variables are consumption
ct, labor income yt, CRSP-VW dividends dt, CRSP-VW price pt and asset wealth at. The null
hypothesis is no cointegration; signi?cant statistics at the 10% level are highlighted in bold face.
The sample is annual and spans the period 1948 to 2001.
Figure 1
Figure 2
Figure 3