Journal of Banking & Finance 31 (2007) 455–475
www.elsevier.com/locate/jbf
Stock returns, dividend yield,
and book-to-market ratio
Xiaoquan Jiang
a
a,*
, Bong-Soo Lee
b,c
Department of Finance, College of Business, University of Northern Iowa, Cedar Falls, IA 50614, United States
b
Department of Finance, College of Business, Florida State University, Tallahassee, FL 32306, United States
c
KAIST Graduate School of Finance, Korea Advanced Institute of Science and Technology, Seoul, Korea
Received 17 February 2006; accepted 12 July 2006
Available online 16 October 2006
Abstract
A dividend yield model has been widely used in previous research that relates stock market valuations to cash flow fundamentals. Given controversies about using dividends as a proxy for cash
flows, a loglinear book-to-market model has recently been proposed. However, these models rely
on the assumption that dividend yield and book-to-market ratio are both stationary, and empirical
evidence for this is, at best, mixed. We develop a new model, the loglinear cointegration model, that
explains future profitability and excess stock returns in terms of a linear combination of log book-tomarket ratio and log dividend yield. The loglinear cointegration model performs better than the log
dividend yield model and the log book-to-market model in terms of cross-equation restriction tests
and forecasting performance comparisons. The superior performance of the loglinear cointegration
model suggests that the linear combination may be a better indicator of intrinsic fundamentals than
the dividend yield or the book-to-market ratio separately.
Published by Elsevier B.V.
JEL classification: C52; G12
Keywords: Present value model; Dividend yield; Book-to-market ratio; Cointegration
*
Corresponding author.
E-mail addresses: xq.jiang@uni.edu (X. Jiang), blee2@cob.fsu.edu (B.-S. Lee).
0378-4266/$ - see front matter Published by Elsevier B.V.
doi:10.1016/j.jbankfin.2006.07.012
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X. Jiang, B.-S. Lee / Journal of Banking & Finance 31 (2007) 455–475
1. Introduction
In explaining fluctuations in stock market valuation levels, Campbell and Shiller’s
(1988) dividend yield model has been widely used. The Campbell–Shiller model relates
the dividend–price ratio to a present value of expected future returns and future dividend
growth rates: high prices should eventually be followed by high future dividends, low
future returns, or some combination of the two. This model is useful and convenient
for empirical implementation but relies on the stability of corporate dividend policy, which
is often suspected for various reasons. In particular, many firms, especially those that are
high-tech and high-growth, do not pay regular cash dividends until later in their life cycle.1
Instead, share repurchases have recently become very popular. Therefore, the dividend
yield model which uses regular cash dividends may be less attractive.2
Vuolteenaho (2000, 2002) developed an alternative, loglinear book-to-market model.
To replace dividends in the loglinear dividend yield model, he introduces the clean surplus
accounting relation: Book value this year equals book value last year plus earnings less
dividends. His model relates the current book-to-market ratio to expected future profitability, interest rates, and excess stock returns. The model implies that the book-to-market
ratio can be (temporarily) low if the future cash flows are high and/or the future excess
stock returns are low.
Both models provide a very useful framework in understanding stock price fluctuation
in terms of cash flow fundamentals or profitability. In particular, Vuolteenaho’s log bookto-market model is attractive in that it does not rely on possibly unstable corporate dividend policy. However, the loglinear book-to-market model relies on the assumption that
the difference of log book value and log market value is stationary, even though both series
are non-stationary. That is, log book value and log market value are assumed to be cointegrated with a cointegrating vector [1, 1]. However, empirical evidence on this property
of the variables is very weak.3
In this paper, we propose a new model, called a loglinear cointegration (LLCI) model,
that explains future profitability and excess stock returns in terms of a linear combination
of (or spread between) log book-to-market ratio and log dividend yield. The LLCI model
shows that a linear combination of the log book-to-market ratio and log dividend yield
can be written as a present value of all expected future returns and returns on equity
(accounting returns or profitability). Furthermore, we show that the LLCI model performs better than either the log dividend yield model or the log book-to-market model
in terms of cross-equation restriction tests and various forecasting performance
comparisons.
The intuition behind the LLCI model is simple and straightforward. Previous studies
find that both dividend yields and book-to-market ratios have some predictive power
for stock returns (e.g., Fama and French, 1988, 1989, 1993; Campbell and Shiller, 1988;
1
Fama and French (2001) document that the percent of firms paying cash dividends among NYSE, AMEX,
and NASDAQ non-financial, non-utility firms fell from 66.5 in 1978 to 20.8 in 1999.
2
Given the recent tax law change in 2003 in favor of dividends, the recent trend that share repurchases are
increasing relative to dividends may reverse itself.
3
Even Vuolteenaho (2000) acknowledges that there is marginal evidence against the presence of a unit root in
the book-to-market ratio. See Table A.3 in Vuolteenaho (2000).
X. Jiang, B.-S. Lee / Journal of Banking & Finance 31 (2007) 455–475
457
Hodrick, 1992; Pontiff and Schall, 1998; Vuolteenaho, 2000, 2002; Ali et al., 2003a). However, the assumption of stationarity of these two variables is often suspected. If so, they
may share a common trend, and a linear combination of these variables may yield a better
predictive power for stock returns. Since book value is closely related to earnings, the cointegration between log book-to-market and log dividend yield seems consistent with the
comovements (or cointegration) of earnings, dividends, and stock prices (e.g., Lee,
1996, 1998).
The LLCI model can be thought of as an extension of the models in Campbell and
Shiller (1988) and Vuolteenaho (2000, 2002). Therefore, the LLCI model shares all the
benefits of their loglinear properties and has additional interesting features. First, given
mixed evidence on the previous loglinear models’ assumptions (about the stationarity of
log dividend yield and log book-to-market), the LLCI model exploits possible cointegration between log dividend yield and log book-to-market variables in an explicit manner.
Second, loglinear models’ dynamic implications can be summarized by cross-equation
restrictions on vector autoregression (VAR) coefficients. As a result of taking into account
the cointegration relation, the LLCI model tends to perform better in the cross-equation
restriction tests than the other two loglinear models. In addition, the LLCI model tends to
outperform the other two loglinear models in in-sample fit of excess returns and in out-ofsample forecast performance tests.
Third, the LLCI model incorporates both dividend yield and book-to-market ratio into
a closed form present value relation that explains expected future profitability and stock
returns. For stock return forecasts, some studies find that dividend yields have predictive
power while others find that book-to-market ratio is informative. The former is related to
a finance approach based on the conventional dividend discount model, while the latter is
related to an accounting approach based on the accounting clean surplus relation. As
such, the LLCI approach provides an integration of the two approaches.
The paper is organized as follows. In Section 2, we briefly introduce the loglinear dividend yield model and the loglinear book-to-market model and propose a loglinear cointegration model. Section 3 describes data and reports the results of various unit root tests
for variables in the three loglinear models. In Section 4, as a means of testing implications
of the three loglinear models, we implement cross-equation restrictions tests. Section 5
presents the results of estimation of returns based on each model, and Section 6 reports
out-of-sample forecast performance. Section 7 presents forecast performance based on a
bootstrapping method. Section 8 examines cointegration among dividends, book value,
and market value, and Section 9 concludes the paper.
2. Loglinear models
2.1. Model 1: Loglinear dividend yield model
The realized log gross return on a portfolio, held from the beginning of time t to the
beginning of time t + 1, can be written as
logð1 þ Rt Þ logðP t þ Dt Þ logðP t1 Þ;
ð1Þ
where Rt is the realized return during the period t, Pt is the real price of a stock or stock
portfolio measured at the end of time period t, and Dt is the real dividend paid on the portfolio during period t.
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X. Jiang, B.-S. Lee / Journal of Banking & Finance 31 (2007) 455–475
Campbell and Shiller (1988) assume that the ratio of the price to the sum of price and
dividend is approximately constant through time at the level q. That is, q = P/(P + D),
where P and D are the mean values of stock price and dividend, respectively. By using
a Taylor approximation, they derive the following equation:
rt ¼ qpt þ ð1 qÞd t pt1 þ k;
ð2Þ
where the lowercase letters represent logs of the corresponding uppercase letters (e.g.,
rt = log(1 + Rt)). The parameter q is slightly smaller than 1, and k is a constant term. They
rewrite Eq. (2) in terms of the dividend–price ratio dt = dt pt and the dividend growth
rate Ddt:
rt ¼ k þ dt1 qdt þ Dd t :
ð3Þ
Solving forward by imposing a transversality condition, ignoring a constant term, and
taking the conditional expectation, they obtain
"
#
1
X
dt ¼ E t
qj ½rtþjþ1 Dd tþjþ1 :
ð4Þ
j¼0
Eq. (4) states that the spread, the log dividend–price ratio, is an expected discounted value
of all future returns less dividend growth rate discounted at the discount rate q. In other
words, the log dividend–price ratio is a present value of all expected future one-period
‘growth-adjusted discount rates’, rt+j Ddt+j. Therefore, the log dividend–price ratio provides the optimal forecast of the present value of all expected future returns less future dividend growth rates.
2.2. Model 2: Loglinear book-to-market model
In accounting literature, an alternative valuation model, the residual income model
(RIM), has become popular recently primarily due to its formalization by Ohlson
(1990, 1991, 1995) and Feltham and Ohlson (1995) (see also Ohlson, 2005).4 The RIM
maintains that the current stock price equals the current book value of equity plus the
present value of expected future residual income (or abnormal earnings), which is defined
as the difference between accounting earnings and the previous period book value multiplied by the cost of equity. Jiang and Lee (2005) examine the empirical validity of the dividend discount model and the RIM, and find that the RIM performs better in the variance
bounds test and the VAR-based cross-equation restrictions test (see also Ali et al., 2003b).
In finance literature, the book-to-market equity ratio has been widely used as a risk factor since Fama and French (1992, 1993, 1995, 1996) carefully reexamine the book-to-market effect.5 They show that book-to-market ratio is related to relative distress.6 However,
4
Frankel and Lee (1998) find that fundamental value (based on a residual income model)-to-price ratio is a
good predictor of long-term cross-sectional returns. Ali et al. (2003b) find that the predictive power of this ratio
for future returns is more consistent with a mispricing explanation than a risk-proxy explanation. Dechow et al.
(1999) provide evidence to support information dynamics of residual income model.
5
Examples include Fama and French (1992, 1993, 1995, 1996), Pontiff and Schall (1998), Kothari et al. (1995),
Breen and Korajczyk (1993), Ali et al. (2003a) and Vuolteenaho (2000, 2002).
6
In contrast to a popular interpretation that book-to-market is a proxy for a state variable associated with
relative financial distress, in an attempt to explain the value effect, Zhang (2005) shows that the value anomaly
arises naturally in the neoclassical framework with rational expectations based on costly reversibility and
countercyclical price of risk.
X. Jiang, B.-S. Lee / Journal of Banking & Finance 31 (2007) 455–475
459
Ali et al. (2003a) provide a market mispricing explanation for the book-to-market effect.7
Kothari and Shanken (1997) find evidence that both book-to-market and dividend yield
track the variation in expected stock returns over time. However, Kothari et al. (1995)
and Breen and Korajczyk (1993) argue that there is a survivorship bias in the data used
to test these new asset pricing specifications.8
Vuolteenaho (2000, 2002) proposes an alternative, accounting-based, approximate present value model.9 He derives the loglinear book-to-market-ratio model by using a log-linearized RIM which is based on the clean surplus relation, allowing for time-varying
discount rates. In deriving the model, Vuolteenaho assumes that the difference of log book
value (bvt) and log market value (mvt) is stationary even though both series are non-stationary. That is, bvt and mvt are cointegrated with a cointegrating vector [1, 1], and the
log book-to-market ratio, ht, is stationary. He also assumes that the log dividend–price
ratio is stationary.
Let BVt, MVt, Xt and Dt be the book value of equity, market value of equity, earnings,
and dividends, respectively. Then, the book-to-market ratio can be written as:
BVt
ð1 þ X t =BVt1 ÞBVt1 Dt
¼
MVt ð1 þ ðDMVt þ Dt Þ=MVt1 ÞMVt1 Dt
¼
ð1 þ X t =BVt1 Dt =BVt1 Þ
BVt1
:
ðð1 þ DMVt þ Dt Þ=MVt1 Dt =MVt1 Þ MVt1
ð5Þ
Using first-order Taylor series approximations, solving forward a difference equation,
and taking the expectations, Vuolteenaho approximates this nonlinear relation of the
log book-to-market ratio, ht = log (BVt/MVt), as a linear model:
ht1 ¼
N
X
j¼0
qj Et ðrtþj ftþj Þ N
X
qj Et ðartþj ftþj Þ þ k t ;
ð6Þ
j¼0
where q is a parameter, q < 1, art is the log ROE (i.e., art = log(1 + Xt/BVt1), ft is the log
one plus the interest rate, (rt ft) is the excess log stock return, and kt is the one-period
approximation error. Eq. (6) states that the log book-to-market ratio is an infinite discounted sum of expected future excess stock returns less profitability (art+ j ft+j). The
book-to-market ratio can be (temporarily) low if future cash flows are high and/or future
excess stock returns are low.
7
Ali et al. (2003a) find, among other things, that the book-to-market effect is greater for stocks with higher
idiosyncratic volatility. Griffin and Lemmon (2002) also document that high book-to-market cannot be explained
as a risk factor, instead it is related to mispricing. Bali and Wu (2005) find that the loading of book-to-market is
not significant.
8
Lo and Mackinlay (1990) raised the issue of data snooping in general. However, Kim (1997) finds that bookto-market equity still has predictive power after carefully considering the potential issue of data snooping:
selection bias and errors-in-variables bias. Ferson and Harvey (1999) emphasize the importance of conditioning
information in testing these new multifactor asset pricing models, while Harvey and Siddique (2000) propose an
asset pricing model that incorporates conditional skewness.
9
Campbell and Vuolteenaho (2004) propose a two-beta model that captures a stock’s risk in two risk loadings,
cash-flow beta and discount-rate beta. The return on the market portfolio can be split into two components, one
reflecting news about the market’s future cash flows and another reflecting news about the market’s discount
rates.
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X. Jiang, B.-S. Lee / Journal of Banking & Finance 31 (2007) 455–475
2.3. Model 3: Loglinear cointegration model
In the spirit of Campbell and Shiller (1988) and Vuolteenaho (2000, 2002), we begin
with the definitions of market and accounting returns (i.e., ROE):
rt log½ðP t þ Dt Þ=P t1 ¼ logðP t þ Dt Þ lnðP t1 Þ;
ð7Þ
art logð1 þ X t =Bt1 Þ ¼ log½ðBt þ Dt Þ=Bt1 ¼ logðBt þ Dt Þ lnðBt1 Þ;
ð8Þ
where rt is the log of one plus the real return on a stock held from time t 1 to time t, art is
the log of one plus the return on equity or accounting return, and Bt is book value. Using a
Taylor expansion and ignoring a constant term, we obtain
rt ¼ qpt þ ð1 qÞd t pt1 ; and
art ¼ q1 bt þ ð1 qÞd t bt1 ;
ð9Þ
ð10Þ
The parameters q = P/(P + D) and q1 = B/(B + D) are constants where P, D, and B are
the mean values of stock price, dividend, and book value, respectively.
Solving forward, taking the conditional expectation, ignoring a constant term and
imposing the transversality condition, we obtain the log book-to-market, bpt, as
"
#
1
1
1
X
X
1 X
j
j
j
q Et rtþj q Et artþj þ ðq q1 Þ
q Et dbtþj ;
bpt ¼
ð11Þ
q j¼1
j¼1
j¼1
where dbt denotes the log dividend-to-book value ratio. In Vuolteenaho (2000, 2002), the
last term in Eq. (11) is ignored. However, Lamont (1998) finds that the dividend payout
ratio contains primary information about short run variations in stock returns. Now,
we approximate the log dividend-to-book value ratio, dbt, by using an AR(1) process,
dbt ¼ udbt1 þ et ;
ð12Þ
where E(et) = 0 for all t. It follows that
Et ðdbtþj Þ ¼ uj dbt :
By substituting Eq. (13) into (11), we obtain a loglinear cointegration model:
1
1X
qj Et ðrtþj artþj Þ;
st ¼
q j¼1
ð13Þ
ð14Þ
where st is a spread between (or a linear combination of) bpt and dpt: st =
q((1 + k)bpt kdpt), with k = (q q1)u/(1 qu).10 Eq. (14) implies that the linear combination of the book-to-market ratio and dividend yield is high if future returns are high,
and/or if the future profitability (or cash flows) is low. It explains the log spread, st, as a
present value of all expected future ‘‘fundamental adjusted discount rates,’’ rt+j art+j.11
This represents the combined effect of expected future discount rates and accounting
10
A more detailed derivation of Eq. (14) is available from the authors upon request.
Comparison of (14) with the log book-to-market ratio model of Vuolteenaho (6) indicates that the
approximation error in (6) amounts to the last term on the right-hand-side of (11). As such, the approximation
error in (6) may be safely ignored when the mean values of book equity and market equity are equal (i.e., q = q1).
In Vuolteenaho’s model, the error term can be safely ignored for the purpose of variance decompositions.
However, the error term may matter in deriving cross-equation restrictions and in-sample and out-of-sample
forecasts based on the model.
11
X. Jiang, B.-S. Lee / Journal of Banking & Finance 31 (2007) 455–475
461
returns on the spread. Thus, the LLCI model combines and extends the residual income
model and dividend discount model by taking into account the possible cointegration.
Whether this loglinear cointegration model turns out to be a useful extension remains
an empirical issue to which we now turn our attention.
3. Data and preliminary empirical results
3.1. Data
For empirical estimation and tests of the three loglinear models, we employ the
annual S&P industrial index for the sample period of 1946–2004, which is obtained from
Standard & Poor’s Analysts’ Handbook 2005. We choose the S&P Industrials Index,
instead of the S&P 500 Index, in part because the former is available for a longer period.
The price index is the end of calendar year price. Dividend is the total amount of cash dividends for both common and preferred stocks. Earnings are basic earnings per share
adjusted to remove the defect of all special items from the calculation; they reflect earnings
per share which exclude the effect of all non-recurring events. Book value represents the
common and preferred shareholder’s interests. It includes capital surplus, common stock,
non-redeemable preferred stock, redeemable preferred stock, retained earnings, and treasury stock. All variables are deflated by the consumer price index.12
3.2. Tests for cointegration
One of the major assumptions in the loglinear book-to-market model of Vuolteenaho
(2000, 2002) is that the difference of log book value (bvt) and log market value (mvt) is stationary even though both series are non-stationary. That is, bvt and mvt are cointegrated
with a cointegrating vector [1, 1], and the log book-to-market ratio, ht, is stationary.
He also assumes that the log dividend–price ratio is stationary. As a means of evaluating
the empirical validity of these assumptions, we implement the augmented Dickey–Fuller
(ADF) and Phillips and Perron (PP) tests of unit root for the variables in the loglinear models. While these two procedures test for the null hypothesis of a unit root in a variable, we
also implement the KPSS (Kwiatkowski et al., 1992) tests for the null of stationarity.
Although we fully recognize the problems associated with the power of various unit
root tests, we include these tests for two reasons. First, the standard theory of inference
in regressions with stochastic regressors requires that all variables be stationary. If we
regress expected excess returns on variables with a unit root, the conventional standard
errors may be misleading. Second, the estimation and test results are likely to be sensitive
to the stationarity of variables because all loglinear models either assume (Campbell and
Shiller, and Vuolteenaho) or imply (loglinear cointegration model) the stationary of the
variables on the left-hand-side. When the right-hand-side variables are stationary, having
a non-stationary variable on the left-hand-side would be inconsistent.
We present the unit root test results in Table 1. First, we test for a unit root in the righthand-side variables in the three loglinear models: dividend growth-adjusted return
12
For a robustness check, we have used the DJIA data for the sample period of 1920–2004, and replicated
Tables 1–5. The results are very similar to those using the S&P index. To save space, we do not report the results.
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X. Jiang, B.-S. Lee / Journal of Banking & Finance 31 (2007) 455–475
Table 1
Unit root and stationarity tests for loglinear models
Variables
q
ADF
PP
KPSS (mu)
KPSS (tau)
rt Ddt
1
2
3
4
6.395***
4.806***
3.965***
2.968**
7.603***
7.674***
7.747***
7.716***
0.080
0.087
0.090
0.088
0.055
0.059
0.061
0.060
rt Dbdt
1
2
3
4
6.232***
6.494***
6.259***
3.150**
7.186***
7.243***
7.542***
8.104***
0.061
0.078
0.114
0.174
0.021
0.028
0.041
0.064
rt art
1
2
3
4
5.253***
3.592***
3.067**
2.815*
6.439***
6.428***
6.427***
6.427***
0.104
0.111
0.112
0.110
0.094
0.101
0.103
0.101
dpt
1
2
3
4
0.870
0.738
0.965
1.483
0.712
0.665
0.666
0.718
1.854***
1.283***
0.994***
0.823***
0.343***
0.246***
0.195**
0.164**
bdpt
1
2
3
4
3.020*
2.176
1.440
0.850
3.049**
2.953**
2.808*
2.747*
0.470**
0.374*
0.320
0.279
0.365***
0.294***
0.255***
0.224***
bpt
1
2
3
4
1.394
1.098
1.461
1.443
1.147
1.069
1.067
1.089
1.858***
1.283***
0.993***
0.820***
0.330***
0.237***
0.188**
0.159**
st
1
2
3
4
3.229**
2.344
2.943**
2.753*
3.128**
3.016**
3.056**
3.101**
0.620**
0.473**
0.395*
0.351*
0.139*
0.109
0.094
0.086
We present the results of the unit root tests and the stationarity tests using the annual S&P industrial index data
(from 1946 to 2004). dt, bdt, rt, and art are regular dividend, broad dividend, market return, and accounting return,
respectively. rt Ddt, rt Dbdt, and rt art are regular dividend adjusted return, broad dividend adjusted return,
and accounting return adjusted return, respectively. dpt, bdpt, bpt, and st are log dividend yield, broad dividend
yield, book-to-market ratio, and loglinear cointegration model’s spread, respectively. All variables are in real values.
We report the augmented Dickey–Fuller (ADF), Phillips–Perron (PP), and KPSS test statistics. We consider the lag
lengths (q) of one to four for each variable for robustness checks. The Schwarz Bayesian criterion (SBC) chooses lag
3 for bdp and s, and lag 1 for all others. For the ADF and PP unit root tests of the spreads S, critical values with 100
(200) observations are 10%, 3.03 (3.02); 5%, 3.37(3.37); and 1%, 4.07(4.00), respectively (see Engle and
Yoo, 1987, Table, p. 157). For KPSS tests, critical values are 10%, 0.347 (0.119); 5%, 0.463 (0.146); and 1%, 0.739
(0.216), for mu (tau), respectively. *, **, and *** represent significance at 10%, 5% and 1% levels, respectively.
(rt Ddt), broad dividend growth-adjusted return (rt Dbdt, where bdt is the broadly
defined dividend), and accounting return adjusted return (rt art). The null hypothesis
of a unit root in each variable is rejected by both ADF and PP tests, in particular when
the lag length in the test is chosen by Schwarz Bayesian information criterion (SBC). Consistent with this result, the null hypothesis of the stationarity of these variables is not
rejected by the KPSS test for any lags chosen. This implies that the right-hand-side variables in all three loglinear models are stationary.
X. Jiang, B.-S. Lee / Journal of Banking & Finance 31 (2007) 455–475
463
Now we test for a unit root in the left-hand-side variables in the three loglinear models.
The null hypothesis of a unit root in the log dividend yield (dpt, bdpt) and log book-tomarket (bpt) series is not rejected by either ADF or PP tests for any lag lengths considered.
In addition, the null of stationarity in dividend yield and book-to-market is rejected by the
KPSS test for any lags considered. These indicate that the log dividend yield and the log
book-to-market series are non-stationary. This result is consistent with the findings of
Campbell and Shiller (1987).
However, the null of a unit root in the spread, st, in the LLCI model is rejected by both
ADF and PP tests for any lag length except for the ADF test with two lags. Similarly, the
null of stationarity of the spread is not rejected by the KPSS test for any lags considered.
This suggests that the spread in the LLCI model is stationary, and thus log dividend yield
and log book-to-market are cointegrated.13 In sum, the results in Table 1 are generally
supportive of the assertion that log book-to-market and log dividend yield are cointegrated as implied by the LLCI model. Thus, the LLCI model seems more consistent with
the data than the loglinear dividend yield model and the loglinear book-to-market model.
4. Cross-equation restriction tests
Now we turn to the test of the implications of the three loglinear models. It is noted that
all three models are in the dynamic expectations framework and linear in the log. Therefore, we can summarize the models’ implications by cross-equation restrictions on a VAR
system of the relevant variables. For the loglinear dividend yield model, consider a bivariate vector autoregressive (BVAR) representation of dpt (=dt) and rt Ddt,
aðLÞ bðLÞ
dpt1
u1t
dpt
¼
þ
;
ð15Þ
cðLÞ dðLÞ rt1 Dd t1
rt Dd t
u2t
where the variables in the vector are demeaned, and a(L),
hP b(L), c(L),
i and d(L) are the kthk
j1
order polynomials in the lag operator (e.g., aðLÞ ¼
, with L being the lag
j¼1 aj L
operator, LkXt = Xtk), u1t = dpt E[dptjdpts, rts Ddts,] for s = 1, 2, . . . , k, and
u2t = (rt Ddt) E[(rts Ddts)jdpts, rts Ddts], for s = 1, 2, . . . , k. This BVAR can
be stacked into a first-order VAR system as
32
3 2
3 2 3
2
dpt1
a1 ak b1 bk
u1t
dpt
7
7
7
7
6
6
6
6
76
7 61
7 607
6
76
7 6
7 6 7
6
76
7 6
7 6 7
6
0 0
76
7 6
7 6 7
6
7
7
7 6 7
6
6
6
dptkþ1
1
76 dptk 7 6 0 7
7 6
6
ð16Þ
76
7¼6
7þ6 7
6
7 6 c1 ck d 1 d k 76 rt1 Dd t1 7 6 u2t 7
6
rt Dd t
76
7 6
7 6 7
6
76
7 6
7 607
6
1
76
7 6
7 6 7
6
76
7 6
7 6 7
6
54
5 4
5 4 5
4
0
0
rtkþ1 Dd tkþ1
1
rtk Dd tk
0
13
We employ a dynamic least squares (DLS) technique proposed by Stock and Watson (1993) to generate the
optimal estimates of the cointegrating parameters in the spread, s, since the spread itself is endogenously
determined.
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X. Jiang, B.-S. Lee / Journal of Banking & Finance 31 (2007) 455–475
or
zt ¼ Azt1 þ ut :
The first-order VAR representation is useful because we can obtain forecasts of future
zts as
E½ztþk jH t ¼ Ak zt ;
ð17Þ
where Ht includes current and past values of zt (i.e., dptj and rtj Ddtj for all j P 0).
Define g1 0 and g2 0 as row vectors with 2k elements, all of which are zero except for the
first element of g1 0 and the (k + 1)st element of g2 0 , being unity. Then, these vectors can
select dpt and rt Ddt as
dpt ¼ g10 zt ;
and
rt Dd t ¼ g20 zt :
Thus, by projecting the loglinear dividend yield model (4) onto the information set Ht, we
can characterize the loglinear dividend yield model as the following restriction:
1
X
dpt ¼ g10 zt ¼
qj g20 Ajþ1 zt :
ð18Þ
j¼0
Assuming a non-singular variance–covariance matrix of u1t and u2t, we can rewrite Eq.
(18) as
1
g10 ¼ g20 AðI qAÞ ;
g10 ¼ ðqg10 þ g20 ÞA:
or
ð19Þ
Specifically, the constraints imposed by Eq. (19) are
qai þ ci ¼ 1;
qai þ ci ¼ 0;
for i ¼ 1;
for i ¼ 2; 3; :::k;
qbi þ d i ¼ 0;
for all i ¼ 1; 2; 3; . . . ; k:
and
ð20Þ
Thus, under the null hypothesis that the loglinear dividend yield model holds, the restrictions in (19) (or (20)) should hold.
For the loglinear book-to-market model, we consider a BVAR representation of the log
book-to-market ht and (rt ft) (art ft). For the loglinear cointegration model, we consider a BVAR representation of the spread st and rt art. Then, by following a similar
procedure, we can summarize each model by a set of cross-equation restrictions on the
BVAR coefficients.
In Table 2, we provide results of the cross-equation restriction tests for the loglinear
dividend yield model (model 1), the loglinear book-to-market model (model 2), and the
LLCI model (model 3). For the convenience of comparison and to better illustrate the nature of the loglinear cointegration model, we use both narrow dividends (cash dividends) in
model 1a, and broad dividends (generated by the clean surplus relation) in model 1b for
model 1.
We observe in Table 2 that the restrictions derived from model 1 (including 1a and 1b)
and model 2 are strongly rejected by the data summarized by a VAR for any lags considered. In contrast, the restrictions from model 3 are not rejected by the data for any lags
considered. While the p-values for models 1 (including 1a and 1b) and 2 for any four lags
are substantially less than 1%, the p-values for model 3 with lags one to four are 0.525,
X. Jiang, B.-S. Lee / Journal of Banking & Finance 31 (2007) 455–475
465
Table 2
Cross-equation restriction tests for loglinear models
q
1
2
3
4
Model 1a
Model 1b
Model 2
Model 3
v2
p-value
v2
p-value
v2
p-value
v2
p-value
312.217
591.033
589.996
598.750
0.000
0.000
0.000
0.000
26.852
37.947
48.061
56.055
0.000
0.000
0.000
0.000
16.430
19.271
22.784
26.167
0.000
0.001
0.001
0.001
1.289
3.936
5.879
14.289
0.525
0.415
0.437
0.075
We present the test results of cross-equation restrictions derived from loglinear models using the annual S&P
industrial index data (from 1946 to 2004). The term ‘q’ represents the number of lag length in the VAR. Models
1a, 1b, 2, and 3 represent (narrow) dividend yield model, (broad) dividend yield model, book-to-market model,
and loglinear cointegration model, respectively.
0.415, 0.437, and 0.075, respectively. In summary, the VAR-based cross-equation restriction tests show that model 3, the LLCI model, fits the data substantially better than
models 1 and 2.
5. Forecasting expected returns
Empirical research finds that expected excess return has a positive relation with dividend yield and book-to-market ratio in both cross-section and time-series relations. For
example, Litzenberger and Ramaswamy (1979), Kothari and Shanken (1992), and Brennan et al. (1998) find that dividend yield has some predictive power for cross-section excess
stock return, while Shiller (1984), Fama and French (1988, 1989), Campbell and Shiller
(1988, 2001), Kothari and Shanken (1997), and Lamont (1998) find that dividend yield
has predictive power for time-series excess returns. Book-to-market ratio has also been
found to have predictive power in both cross-section and time-series excess returns (e.g.,
Fama and French, 1992, 1993; Kothari and Shanken, 1997; Lewellen, 1999). Guo
(2006) provides evidence of out-of-sample forecast of stock returns. Similarly, we explore
the predictive ability of the loglinear spread in model 3 and compare the result with that of
the log book-to-market ratio in model 2 and the dividend yield in model 1. Based on the
above theoretical analysis and empirical tests, we conjecture the predictive ability of the
spread to perform at least as well as that of the log book-to-market ratio and the dividend
yield.
Panel A of Table 3 presents the results of regressing annual log real return, rt, on the
lagged log dividend yield (models 1a and 1b), lagged log book-to-market (model 2), and
lagged log spread of the LLCI model (model 3), respectively. The model forecasts real
returns in 2004, using 2003 values of the regressors. Estimation error is the standard error
of the point estimate based on sampling error in the coefficients. Total forecast error
includes both sampling error and residual error. The t-statistics computed using
Newey–West heteroskedastic-robust standard errors with two lags are in parentheses.
The term R2 in the table is the adjusted R2.
The slope coefficients on dividend yield using both narrow and broad dividend, on log
book-to-market, and on LLCI model’s spread are 0.080, 0.042, 0.057 and 0.191, respectively. While the coefficients on dividend yield and book-to-market are not significantly
different from zero, the coefficient on LLCI model’s spread is significantly different from
zero. The adjusted R2 for the four models are 0.039, 0.016, 0.007 and 0.091, respectively.
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X. Jiang, B.-S. Lee / Journal of Banking & Finance 31 (2007) 455–475
Table 3
OLS and VAR estimates
a
Panel A: OLS
Model 1a
Model 1b
Model 2
Model 3
R2
b
rt+1 = a + bdpt + ut+1
0.024
0.080
(0.323)
(1.480)
[0.435]
[1.776]
rt+1 = a + bbdpt + ut+1
0.195**
0.042
(2.202)
(1.287)
[2.139]
[1.411]
rt+1 = a + bbpt + ut+1
0.107***
0.057
(2.903)
(0.959)
[2.834]
[1.200]
rt+1 = a + bst + ut+1
0.070***
0.191***
(3.409)
(2.648)
[3.466]
[2.602]
Forecast return
0.039
0.000
0.011
0.168
Forecast
Estimate error
Total forecast error
0.016
0.057
0.003
0.161
Forecast
Estimate error
Total forecast error
0.007
0.036
0.007
0.166
Forecast
Estimate error
Total forecast error
0.091
0.070
0.001
0.154
Forecast
Estimate error
Total forecast error
Panel B: Vector autoregressive (VAR) model estimates
Model 1a Constant
rt
Ddt
dpt
R2
Forecast
0.079
(1.624)
0.046
0.031
0.001
0.162
Forecast
Estimate error
Total forecast error
bdpt
R2
0.040
(1.071)
0.015
0.061
0.001
0.161
Forecast
Estimate error
Total forecast error
Dbt
bpt
R2
0.218
(0.819)
0.071
(1.407)
0.019
0.039
0.001
0.161
Forecast
Estimate error
Total forecast error
rt
Dbpt
Ddpt
st
R2
0.092
(0.331)
0.346
(1.168)
0.064
(1.350)
0.229***
(2.904)
0.141
0.031
0.001
0.148
Forecast
Estimate error
Total forecast error
0.030
(0.435)
Model 1b
Constant
0.137
(1.241)
rt
0.178*
(1.779)
Model 2
Constant
Model 3
Constant
0.078***
(3.583)
Dbdt
0.188
(1.333)
rt
0.114***
(3.828)
0.178
(0.683)
0.197
(1.606)
0.042
(0.728)
We report regressions of stock returns using the S&P industrial index data of 1946–2004. The dependent variable
rt is log return. dpt, bdpt, bpt, and st are log (narrow) dividend yield, log broad dividend yield, log book-tomarket, and loglinear cointegration model’s spread, respectively. All variables are in real values. Models 1a, 1b, 2,
and 3 represent (narrow) dividend yield model, broad dividend yield model, book-to-market model, and loglinear
cointegration model, respectively. Estimation error is the standard error of the point estimate, based on sampling
error in the coefficients. Total forecast error includes both sampling error and residual error. We compute the tstatistics in parentheses and brackets using the Newey–West heteroskedastic-robust standard errors with two lags
and using bootstrapped standard errors, respectively. R2 is the adjusted R2. *, **, and *** represent significance at
10%, 5% and 1% levels, respectively.
In addition, the total forecast errors (estimation error) for the four models are 0.168
(0.011), 0.161 (0.003), 0.166 (0.007) and 0.154 (0.001), respectively. This result shows that
X. Jiang, B.-S. Lee / Journal of Banking & Finance 31 (2007) 455–475
467
the predictive ability of the LLCI model’s spread is better than that of either the dividend
yield (narrow or broad) or the log book-to-market from the perspective of tracking stock
return, goodness of fit and accuracy.
Panel B of Table 3 reports the results of the VAR estimation and forecast based on the
three loglinear models. To save space, we only report the return forecast regression. The
standard errors are corrected for heteroskedasticity (Newey and West, 1987). Consistent
with the OLS forecast, the LLCI model’s spread is the only regressor which can forecast
the stock returns. The coefficient on the spread is 0.229 and the t-statistic computed using
Newey–West heteroskedastic-robust standard errors with two lags is 2.904. The R2 statistics of expected real return equations in the loglinear dividend yield model (narrow and
broad), the loglinear book-to-market model, and the LLCI model are 0.046, 0.015,
0.019, and 0.141 respectively. The total forecast errors in the four models are 0.162,
0.161, 0.161 and 0.148 respectively. The total forecast error in the LLCI model is the
smallest. In summary, as in the OLS forecast, the VAR forecast shows that the LLCI
spread is the best forecast variable among those considered.
6. Out-of-sample forecast
We now consider out-of-sample forecasts of each model to examine whether the above
results are materially affected by small-sample biases. We compare the root-meansquared errors (RMSE) from a series of one-year-ahead out-of-sample forecasts obtained
from the LLCI model to that of the loglinear book-to-market model and the loglinear
dividend yield model (see Lettau and Ludvigson, 2001a,b). One possible concern may
be the potential for look-ahead bias because the spread st is estimated using the full sample. To address this concern, we use recursive regressions, re-estimating both the spread
st and the forecast model for each period using only data available at the time of the
forecast, adding one year observations at a time and calculating a series of one-stepahead forecasts.
Since our forecast comparison is not purely regression-based but model-based, we
implement the Diebold and Mariano (1995) non-nested test. For the model-based comparison, the stock returns are adjusted by the fundamentals according to the models’
specifications. The ROE is used as a fundamental in both Vuolteenaho’s model and
the LLCI model while the dividend growth rate is used as a fundamental in the Campbell–Shiller model. To address whether the alternative models encompass the LLCI
model, we conduct the Diebold and Mariano (DM) test. The DM test provides a formal
hypothesis testing procedure for the analysis of competing forecasts from non-nested
models. The null hypothesis of the DM test is that the competitor models – the loglinear
dividend yield model (narrow or broad dividends) and the loglinear book-to-market
model – and the preferred model, the LLCI model, have equal forecast accuracy. The
alternative hypothesis is that the preferred model provides superior forecasts to any of
the competitor models.
In Table 4, we present the results using the RMSE of model 3 to that of model 1 (or 2),
RMSE3/RMSE1,2, and the DM test. If the preferred model (e.g., the LLCI model) performs better than the competitor models (e.g., the loglinear dividend yield model and
the loglinear book-to-market model), the RMSE3/RMSE1,2, should be less than one.
We observe from rows 1, 2, and 3 that the root-mean-squared error ratios, RMSE3/
RMSE1,2, are 0.829, 0.910, and 0.863, respectively. They are all less than one and are sig-
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X. Jiang, B.-S. Lee / Journal of Banking & Finance 31 (2007) 455–475
Table 4
Out-of-sample forecast
Row
Model comparison
RMSE3/RMSE1,2
1
2
3
st vs. dpt
st vs. bdpt
st vs. bpt
0.829
0.910
0.863
4
5
6
7
dpt vs. constant
bdpt vs. constant
bpt vs. constant
st vs. constant
1.273
1.060
1.176
0.875
DM statistic
2.917***
1.194
2.526***
p-value
0.002
0.116
0.006
RMSE1,2,3/RMSR0
1.225
0.690
1.630
0.810
0.890
0.755
0.949
0.209
We report the results of the one-year-ahead, model-based out-of-sample forecast evaluation using the S&P
Industrial (1946–2004) index. dpt, bdpt, bpt, and st are log dividend yield, log broad dividend yield, log book-tomarket, and loglinear cointegration model’s spread, respectively. All variables are in real values. In each row, two
models are compared. Models 1a, 1b, 2, and 3 represent (narrow) dividend yield model, (broad) dividend yield
model, book-to-market model, and loglinear cointegration model, respectively. The left-hand-side is the real
return. Model 3 uses lagged s as a predictive variable, while models 1a, 1b and 2 use lagged dpt, bdpt and bpt as
predictive variables, respectively. The column labeled RMSE3/RMSE1,2 reports the ratio of the RMSE of model 3
to that of model 1a, 1b, and 2. The rows 4, 5, 6, and 7 provide comparison of each model to a random walk
model. RMSE1,2,3/RMSR0 reports the ratio of the RMSE of model 1a, 1b, 2, and 3 to that of the random walk
model. The column labeled ‘‘DM Statistic’’ gives the Diebold and Mariano (1995) test statistic. In rows 1, 2, and
3, the null hypothesis is that the model 1a, 1b, 2, and 3 have equal forecast accuracy. In rows 4, 5, 6, and 7, the
null hypothesis is that the model 1a, 1b, 2, 3, and the random walk model have equal forecast accuracy The initial
estimation period begins in 1946 and ends in 1994. *, **, and *** represent significance at 10%, 5%, and 1% levels,
respectively.
nificant in rows 1 and 3. Thus, the null hypothesis of equal forecast accuracy is rejected in
favor of the LLCI model.
It is interesting to compare each model with a constant expected return model (model
0). We report the comparison results in rows 4–7 of Table 4. It is not surprising to find that
none of the models considered outperforms the constant expected return model in the outof-sample forecast.14 However, it is noted that the RMSE1,2,3/RMSR0 is greater than one
for models 1a, 1b, and 2, while it is less than one for model 3. This is consistent with the
results in rows 1–3 that model 3 provides a better out-of-sample forecast than models 1a,
1b, and 2.
In summary, the results in Table 4 show that the LLCI spread st has statistically significant out-of-sample predictive power for the real returns and contains information that is
not included in either log book-to-market or dividend yield. The DM test results suggest
that the LLCI model outperforms both the loglinear dividend model and the loglinear
book-to-market model. The superior performance may also suggest that the LLCI spread
is a better indicator for intrinsic fundamentals than the dividend yield or the book-to-market ratio separately.
14
Our finding that dividend yield and book-to-market do not outperform the constant model in the out-ofsample forecast is consistent with previous finding; see Bossaerts and Pierre (1999) and Goyal and Welch
(forthcoming).
X. Jiang, B.-S. Lee / Journal of Banking & Finance 31 (2007) 455–475
469
7. Bootstrapping forecast
Stambaugh (1986, 1999) shows that the OLS estimator of the regressor coefficient may
be biased in small samples if the regressor is highly persistent. Consider the model of
returns analyzed by Stambaugh (1986, 1999), Mankiw and Shapiro (1986), and Nelson
and Kim (1993):
rt ¼ a þ bX t1 þ ut ;
ð21Þ
X t ¼ d þ cX t1 þ vt ;
ð22Þ
where rt is the stock return, Xt1 is a candidate forecastor variable, and {(ut, vt) 0 } is an
independently and identically distributed vector sequence. Eq. (21) is the forecast equation
and Eq. (22) specifies the evolution of the forecaster. Since Xt follows an AR(1) process,
the residuals ut and vt are correlated. The OLS estimator of c in a sample of T observations
is biased toward zero with the bias given by
1 þ 3c
Eð^c cÞ :
ð23Þ
T
Stambaugh (1986, 1999) shows that the size of bias in the OLS of b in the forecast equation is proportional to the bias of c in AR(1) process:
^ bÞ ¼ ruv Eð^c cÞ:
Eðb
ð24Þ
r2v
It is noted from Eq. (23) that the bias in estimates c and b can be large for small T. Several recent studies discuss alternative econometric methods for correcting the Stambaugh
bias and conducting valid inference (Cavangh et al., 1995; Ang and Bekaert, 2003; Jansson
and Moreira, 2003; Polk et al., 2006; Lewellen, 2004; Torous et al., 2004; Campbell and
Yogo, 2006).
To deal with the small sample issue, we evaluate the forecasting models by using a bootstrap. In this bootstrap, the observed distribution of the random variables is the best estimate of the actual distribution. We implement the bootstrap-based statistical inference as
follows:
1. Estimate each forecasting model and calculate residuals for each model.
2. Generate 1000 bootstrap error samples of size 59 from each forecasting model.
3. Use the bootstrap errors to compute 1000 series of bootstrap returns for each
model.
4. Run a forecasting regression for each model to obtain the root-mean-squared error
ratio (RMSE3/RSME1,2) for in-sample and out-of-sample cases.
We choose the RMSE ratio for three reasons. First, it is simple. Second, RMSE is arguably the most commonly used measure of forecasting ability (West, 2005). Third, we can
compare the results for in-sample and out-of-sample.
Table 5 presents the bootstrap forecasting results. We observe that the RMSE3/
RMSE1,2 for the in-sample (out-of-sample) forecasts are 0.861 (0.898), 0.860 (0.952),
and 0.856 (0.914), respectively. They are all less than one. The significance level is always
less than 10%. The bootstrap results show that the LLCI outperforms other logliner
models in forecasting returns, which is consistent with the forecasting results in the
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X. Jiang, B.-S. Lee / Journal of Banking & Finance 31 (2007) 455–475
Table 5
Bootstrapping for in-sample forecast and out-of-sample forecast evaluation
Row
1
2
3
Model comparison
st vs. dpt
st vs. bdpt
st vs. bpt
In sample forecast
Out-of-sample forecast
RMSE3/RMSE1,2
p-value
RMSE3/RMSE1,2
p-value
0.861
0.860
0.856
0.034**
0.029**
0.031**
0.898
0.952
0.914
0.030**
0.053*
0.036**
We report ratios of root-mean-squared errors for model 3 to models 1a, 1b, and 2 for both in sample and out-ofsample forecasts using the S&P Industrial (1946–2004) index. dpt, bdpt, bpt, and st are log dividend yield, log
broad dividend yield, log book-to-market, and loglinear cointegration model’s spread, respectively. All variables
are in real values. The rows 1, 2, and 3 provide the model-based forecast comparisons. The column labeled
RMSE3/RMSE1,2 reports the ratio of bootstrapping mean of the root-mean-squared forecasting error of model 3
to that of models 1a, 1b, and 2. Models 1a, 1b, 2, and 3 represent (narrow) dividend yield model, (broad) dividend
yield model, book-to-market model, and loglinear cointegration model, respectively. P-value is the bootstrapping
significance level. *, **, and *** represent significance at 10%, 5% and 1% levels, respectively.
previous section. They imply that the superior performance of the LLCI model is not due
to a small sample bias.
8. Cointegration among dividends, book value, and market value
Our finding of the bivariate cointegrations between log dividend yield and log book-tomarket ratio suggests that there may be a trivariate cointegration relation among log dividends (dt), log book values (bt), and log market values (pt). Thus, it is worth examining the
dynamic relations among these variables in a unified framework.
For this purpose, we test for possible cointegration among the three variables by using
the procedure of Johansen (1988, 1991). In Panel A of Table 6, we present the results of
three variable cointegration tests based on the maximal eigenvalue test and the trace test
of Johansen. The term r denotes the number of linearly independent cointegrating vectors. The null of zero cointegration vector (r = 0) is rejected, whereas the null of either
less than one cointegration vector (r 6 1) or less than two cointegration vectors (r 6 2)
is not rejected at the conventional significance level of 10%. To confirm this, we follow
Johansen and Juselius (1992) and further examine the determination of the number of
cointegrating vectors based on a formal testing. Using three eigenvalues, we compute
three possible cointegration terms: S1, S2, and S3. Unit root tests and stationary tests
in Panel B of Table 6 show that S1 is stationary, but S2 and S3 are non-stationary. Overall, the tests indicate that there is one cointegration vector. Further unit root tests
for possibly as many as three cointegration terms confirm that there is indeed one cointegration term, which we call S1. This indicates that dividends, book values, and market
values tend to commove over time sharing a common stochastic trend.15 We report
the estimation result of the trivariate VECM in Panel C of Table 6. We find that the
cointegration term is significant in the dividend change and the book value change
equations.
15
We thank a referee for suggesting cointegration test for dividends, book value, and market value.
X. Jiang, B.-S. Lee / Journal of Banking & Finance 31 (2007) 455–475
471
Table 6
Trivariate cointegration test
Trace
H0: r
pr
k-max90
Trace90
Panel A: Johansen cointegration test
0.344
24.00
36.24
0.159
9.85
12.24
0.041
2.39
2.39
r=0
r61
r6 2
3
2
1
13.39
10.60
2.71
26.70
13.31
2.71
Eigenvalue
Variables
k-max
Lag q
ADF
PP
KPSS (mu)
KPSS (tau)
4.492***
4.488***
4.498***
0.661**
0.505**
0.431*
0.371***
0.285***
0.245***
2.263
1.710
2.042
2.195
2.128
2.150
0.485**
0.347**
0.277
0.319***
0.230***
0.185**
0.530
0.718
0.749
Dpt1
0.319
0.346
0.345
1.606***
1.121***
0.878***
S1t1
0.285***
0.202**
0.160**
R2
Panel B: unit root and KPSS tests
S1
1
3.940***
2
3.632***
3
3.372***
S2
S3
1
2
3
1
2
3
Constant
Ddt1
Dbt1
0.310
1.076
0.338**
2.420
0.244**
2.494
0.273
0.861
0.168
0.904
0.075
0.574
Panel C: Vector error correction model estimates
Dpt
t-stat
Ddt
t-stat
Dbt
t-stat
0.017
0.357
0.118***
5.196
0.072***
4.038
0.122
1.218
0.089**
2.185
0.063
1.587
0.017
1.319
0.033***
5.198
0.017***
2.605
0.012
0.274
0.088
We report the results of the Johansen cointegration tests for dt (log dividend), bt (log book value), and pt (log
price) using the S&P Industrial (1946–2004) index. We also report the cointegration terms’ unit root tests using
the augmented Dickey–Fuller (ADF), Phillips–Perron (PP), and KPSS test. We consider the lag lengths (q) of one
to three for each variable for robustness checks. In panel A, r is the number of linearly independent cointegrating
P
vectors. Tracestatistic ¼ T ni¼rþ1 lnð1 ki Þ; k-max statistic = T ln(1 ki), where T is the number of observations, n is the dimension of the vector (here n = 3), and ki is the ith smallest squared canonical correlations in
Johansen (1988, 1991) or Johansen and Juselius (1992). Various spreads Si for i = 1, 2, and 3 are calculated as
follows. s1t = 10.083 * dt + 1.967 * bt + 2.120 * pt, s2t = 3.911 * dt 8.538 * bt + 2.496 * pt, and s3t = 4.182 *
dt 4.135 * bt 1.243 * pt.
For the ADF and PP unit root tests of the spreads Si for i = 1, 2, and 3, critical values with 100 (200) observations
are 10%, 3.03(3.02); 5%, 3.37(3.37); and 1%, 4.07(4.00), respectively (see Engle and Yoo, 1987, Table,
p. 157). For KPSS tests, critical values are 10%, 0.347 (0.119); 5%, 0.463 (0.146); and 1%, 0.739 (0.216), for mu
(tau), respectively. *, **, and *** represent 10%, 5% and 1% significant levels, respectively.
Campbell and Shiller (1987) show that a present value model implies a cointegration
between two variables in the model. Based on the dividend discount model, they show that
dividends and stock prices are cointegrated, although the cointegration vector is not
[1, 1]. That is, dividends and stock prices tend to move together over time. Similarly,
the RIM (residual income model), which is a present value model, suggests that there is
a cointegration relation among market value, book value, and earnings. Some dividend
models (e.g., the permanent earning hypothesis and the partial adjustment hypothesis),
which are represented as present value models, suggest that earnings and dividends
are cointegrated. Considered together, it is quite possible that dividends, book value,
472
X. Jiang, B.-S. Lee / Journal of Banking & Finance 31 (2007) 455–475
and market value are cointegrated, sharing a common trend that may reflect a permanent
component of fundamentals.16 Intuitively, market value and book value tend to move
together over time around their fundamental variables such as earnings and dividends.
9. Concluding remarks
Campbell–Shiller’s dividend yield model has been widely used in previous research that
relates stock price to cash flow fundamentals. Given unstable corporate dividend policy, a
loglinear book-to-market model has been proposed recently by Vuolteenaho based on the
accounting clean surplus relation. However, these models rely on the assumption that dividend yield and book-to-market ratio are stationary. Empirical evidence for this is, at best,
mixed.
By extending Campbell and Shiller’s and Vuolteenaho’s models, we have proposed a
loglinear cointegration model, which accounts for potential cointegration and explains
future profitability and excess stock returns in terms of a linear combination of log
book-to-market and log dividend yield. The loglinear cointegration model takes into
account possible non-stationarity of these variables. We have shown that the loglinear
cointegration model performs better than either the log dividend yield model or the log
book-to-market model in terms of cross-equation restrictions tests, excess return forecasting performance, and out-of-sample forecasting performance comparisons. The superior
performance of the loglinear cointegration model suggests that the spread may be a better
indicator for intrinsic fundamentals than dividend yield or book-to-market ratio. Since the
loglinear cointegration model can be viewed as a combination of the log dividend yield
model and the log book-to-market model, its superior performance also suggests that
the spread may contain useful information that is not contained in either the dividend
yield or the book-to-market ratio separately.
Acknowledgements
We would like to thank seminar participants at the University of Northern Iowa, 2003
Midwest Finance Association, St. Louis, MO and 2004 Financial Management Association, New Orleans, LA for their useful comments. We especially thank Giorgio Szego
(the Editor), and two anonymous referees for detailed and valuable comments.
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