Journal of Forecasting
J. Forecast. (2011)
Published online in Wiley Online Library
(wileyonlinelibrary.com) DOI: 10.1002/for.1231
Price–Dividend Ratios and Stock
Price Predictability
JYH-LIN WU1,2* AND YU-HAU HU3
1 Institute of Economics, National Sun Yat-Sen University,
Kaohsiung, Taiwan
2 Department of Economics, National Chung-Cheng University,
Chia-Yi, Taiwan
3 Department of International Trade, Cheng-Shiu Technological
University, Kaohsiung, Taiwan
ABSTRACT
A long-standing puzzle to financial economists is the difficulty of outperforming
the benchmark random walk model in out-of-sample contests. Using data from
the USA over the period of 1872–2007, this paper re-examines the out-ofsample predictability of real stock prices based on price–dividend (PD) ratios.
The current research focuses on the significance of the time-varying mean and
nonlinear dynamics of PD ratios in the empirical analysis. Empirical results
support the proposed nonlinear model of the PD ratio and the stationarity of the
trend-adjusted PD ratio. Furthermore, this paper rejects the non-predictability
hypothesis of stock prices statistically based on in- and out-of-sample tests
and economically based on the criteria of expected real return per unit of risk.
Copyright © 2011 John Wiley & Sons, Ltd.
KEY WORDS
price–dividend ratios; time-varying means; out-of-sample
predictability; random walks; long-horizon regression tests,
economic significance
INTRODUCTION
Since the seminal papers by Campbell and Shiller (1988) and Fama and French (1988), the predictability of stock returns has become a central research topic in empirical financial economics over the
past two decades. The predictability of stock returns is typically analyzed based on a long-horizon
predictive regression equation with a price–dividend (PD) ratio as a regressor.1 This is because a
stationary PD ratio suggests predictable movements of stock prices. Almost all of the empirical
evidence related to return or stock price predictability has relied on long-horizon regression equations, and the evidence with regard to predictability is interpreted based on the significance of t- or
F -statistics and high R2 values. Based on in-sample tests, several researchers find that the PD ratio
* Correspondence to: Jyh-Lin Wu, Institute of Economics, National Sun Yat-Sen University, Kaohsiung, Taiwan.
E-mail: [email protected]
1
Useful survey papers offered by Campbell (2000) and Campbell et al. (1997) provide detailed discussions.
Copyright © 2011 John Wiley & Sons, Ltd.
J.-L. Wu and Y.-H. Hu
has little forecasting power for stock price changes when the forecast horizon is 1 year, but its forecasting power increases significantly when forecast horizons are long (Campbell and Shiller, 1998,
2001).2 However, Campbell and Yogo (2006) and Torous et al. (2004) point out that the conventional
t-test is incorrect when regressors in the prediction equation are nearly integrated. After adopting
the framework of local-to-unity, they find evidence of in-sample return predictability. On the other
hand, there is no convincing evidence for out-of-sample predictability of stock price movement in the
literature (Goyal and Welch, 2003; Lettau and Van Nieuwerburgh, 2008; Welch and Goyal, 2008).
Avramov (2002) find that Bayesian methodology reveals the existence of in- and out-of-sample predictability of stock returns; however, their results also indicate that dividend yields have relatively
small posterior probabilities of being correlated with future returns. The above results give rise to the
puzzle of why PD ratios do not provide a strong and stable predictive power for future movements of
stock prices in out-of-sample prediction tests.
A key assumption embedded in conventional long-horizon regressions is the stability of the PD
ratio, which implies that the means of the ratio are relatively unchanged over time. Figure 1 plots the
US historical PD ratio over the relevant sample period, and reveals that in this 136-year period PD
ratios fluctuate around 2.97 before 1955. There is an unprecedented rise in the PD ratio from 1955 to
1982 and another rise from 1982 all the way until 2000, after which the PD ratio goes down slightly.
The average PD ratio is 2.97 from 1872 to 1954, 3.32 from 1955 to 1981, and 3.71 from 1982 to
2007. The mean of the PD ratio varies in the above three periods and there appears to have been an
upward trend in the mean.
Several papers find significant structural changes in PD ratios and predictive regression models
(Rapach and Wohar, 2005b; Coakley and Fuertes, 2006; Paye and Timmermann, 2006; Lettau and
Van Nieuwerburgh, 2008). In such a case, out-of-sample prediction of stock price changes is difficult
for investors since they need to estimate a break date and the new mean of the PD ratio if a new break
is detected (Lettau and Van Nieuwerburgh, 2008).
Instead of focusing on discrete structural changes, this paper argues that the long-run equilibrium
mean of the ratio is time varying and hence the PD ratio reverts to its time-varying mean. Moreover, this work focuses on the nonlinear adjustments of the ratio toward its time-varying mean. The
current research adopts an exponential smooth transition autoregressive (ESTAR) model to describe
the nonlinear dynamics of PD ratios and a nonlinear deterministic trend function to model the timevarying mean of the ratio. The above specification on the dynamics of PD ratios is motivated by the
view that both fundamental and non-fundamental components affect stock prices. The existence of a
time-varying mean of PD ratios renders a long-horizon prediction equation unstable if such behavior of the mean is not taken into account and could be one of the reasons for finding poor in- and
out-of-sample predictability in stock prices. If our argument of a time-varying mean is correct then
we expect to find evidence for stock price predictability at different forecast horizons based on bootstrapped critical values constructed from an appropriate representation of data. This paper therefore
performs in- and out-of-sample tests to investigate the predictability of stock returns statistically and
economically.
Based on data from the USA over the period of 1872–2007, this paper obtained several results of
interest. First, an ESTAR model with a time-varying mean well describes the dynamics of the PD ratio
and the PD ratio could be a unit root process whereas the trend-adjusted PD ratio is stationary. The
2
The weakness of the PD ratio as a predictor may be due to its exclusion of non-dividend cash flows. Robertson and Wright
(2006) show that a new cash-flow yield including both dividend and non-dividend cash flows provides a strong and stable
in-sample predictive power for aggregate returns.
Copyright © 2011 John Wiley & Sons, Ltd.
J. Forecast. (2011)
DOI: 10.1002/for
Stock Price Predictability
4.5
4.0
3.5
3.0
2.5
2.0
1880 1890 1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000
Figure 1. PD ratios (—) and simulated PD ratios (---) generated based on a non-parametric bootstrap
above results indicate that modeling the impacts of fundamental and non-fundamental components in
stock prices is crucial in empirical analysis. Second, assuming a constant mean in the conventional
linear dynamics of PD ratios provides no evidence of stock price predictability based on in- and outof-sample bootstrap tests. Furthermore, the local-to-unity framework of Campbell and Yogo (2006)
fails to reject the hypothesis of no in-sample predictability of stock returns, which is consistent with
our results from the bootstrap t-test. Third, imposing the time-varying mean in ESTAR dynamics of
the PD ratio, our bootstrap tests provide strong in- and out-of-sample evidence of stock price predictability. The above results of beating the random walk forecast model are crucial since Lettau and
Van Nieuwerburgh (2008), Welch and Goyal (2008) and Goyal and Welch (2003) fail to beat the random walk model in their out-of-sample contests. Our findings reconcile inconsistent results of in- and
out-of-sample predictive tests in extant literature. Fourth, applying the criteria of expected return per
unit of risk, the economic significance of the proposed nonlinear model is supported. Fifth, our bootstrap tests achieve reasonable power, but suffer a slight size distortion for in-sample statistics when
forecast horizons are less than 3 years. Nevertheless, our conclusions on stock price predictability are
not affected by the slight size distortion.
The next section provides a nonlinear ESTAR model with a time-varying mean to describe the
dynamics of PD ratios and discuss its properties. The third section discusses in- and out-of-sample
bootstrap tests of a competitive model and the random walk with drift, respectively. The fourth section reports empirical results and the fifth section examines the economic significance of models
under consideration. The sixth section examines the power and size of the bootstrap tests. Finally, the
seventh section concludes our discussion.
THE NONLINEAR ESTAR MODEL OF PD RATIOS
This paper argues that there exist both short-run nonlinear fluctuations and long-run persistent movements in PD ratios. Theoretically, the long-run value of the PD ratio (the mean of the PD ratio) is
Copyright © 2011 John Wiley & Sons, Ltd.
J. Forecast. (2011)
DOI: 10.1002/for
J.-L. Wu and Y.-H. Hu
affected by the mean growth rate of real dividends and the mean real discount rate as discussed in
Poterba and Summers (1986). Facing permanent regime switches in stock markets, investors will
continuously revise their expectations of the mean of the previous two fundamental factors resulting
in persistent movements in the long-run PD ratio.3
Several permanent fundamental changes are usually offered as explanations for the upward movement of the mean growth rate of real dividends or the downward movement of the mean real discount
rate. These include increased expected future economic growth brought about by the information
revolution (Hobijn and Jovanovic, 2001); institutional changes such as shifting payout policies away
from paying dividends to share repurchases (Cole et al., 1996; Liang and Sharpe, 1999; Siegel, 2002);
a decrease in the equity premium due to lower transaction costs resulting from the advance of information technology (Siegel, 1999); broader participation in financial markets and greater diversification
in portfolios (Vissing-Jorgenson, 1998; Heaton and Lucas, 1999); demographic effects of the baby
boomers (Siegel, 2002); and a decline in inflation (Ritter and Warr, 2002; Sharpe, 2002). In other
words, the assumption of the stability of the PD ratio is challenged when stock markets witness consecutive permanent regime switches. Fama and French (2002) also find that the downward trend in
dividend yields is permanent.
Our arguments are consistent with the fact that financial markets experience consecutive permanent
regime switches even though those changes are small. We set up an ESTAR model with a time-varying
mean to describe the dynamics of the PD ratio:
t D t C .t 1 t 1 /F Œt 1 , , t 1 C ut
(1)
F Œt 1 , , t 1 D expŒ.t 1 t 1 /2 ,
ut i.i.d..0, 2 /,
< 0,
t D h.t/
where t is the PD ratio, defined as the deviation of the log of real S&P 500 price index, pt , from
the log of real dividend, dt , and hence t D pt dt . The term t is the time-varying mean of the PD
ratio, and its trend specification, h.t/, is data determined and hence will vary with sample periods.
The trend specification of t in the PD ratio can be justified by the ‘new era’ explanation in the
literature. The development of new technology results in investors expecting higher future earning
and dividends that shift the growth rate of real dividends upward. The increased ease of trade results
in investors reducing the real discount rate. In other words, consecutive regime switches in financial
markets shift the mean growth rate of real dividends upward or the mean real discount rate downward
which in turn results in an upward trend movement of the mean of the PD ratio (Balke and Wohor,
2001, 2002; Caporale and Gil-Alana, 2004; Psaradakis, 2004). In addition, the trend specification of
t in equation (1) is also consistent with our finding below (‘The stationarity of the PD ratio’) that the
trend-adjusted PD ratio is stationary.
Based upon the lengthy period from 1872 to 2007, the quadratic trend is identified among four different trend specifications: constant, linear, quadratic and cubic formulations. The time-varying mean,
t , in equation (1) is therefore specified as t D a + bt + ct2 . It is worth noting that the specification
of t is re-identified when more data are available. As pointed out by Lettau and Van Nieuwerburgh
(2008), the uncertainty of estimating the time-varying mean of PD ratios is responsible for the difficulty of forecasting stock returns in real time. If our trend measure of the time-varying mean of the
PD ratio is at variance with the true mean, then short-run deviations of the PD ratio in equation (1)
3
The property of time-varying PD ratio is discussed in Poterba and Summers (1986) in which the driving forces of the model
are the growth rate of real dividends and the real discount rate. The derivation of the model is not reported here but is available
upon request from the authors.
Copyright © 2011 John Wiley & Sons, Ltd.
J. Forecast. (2011)
DOI: 10.1002/for
Stock Price Predictability
should also be incorrectly measured and will result in the failure of beating the benchmark random
walk model in out-of-sample contests. Therefore, results from out-of-sample prediction tests allow
us to answer whether our trend specification of t provides a reasonable measure of the time-varying
mean of the PD ratio.
The transition function, F Œt 1 , , t 1 , captures the nonlinearity of the model, which is a function of the lagged PD ratio and its time-varying mean. The parameter (<0) is the parameter that
determines the extent of nonlinearities. It also denotes the speed of transition between regimes. The
model in equation (1) degenerates into an ESTAR model with a constant mean if t D a. The value of
the transition function is bounded between zero and one. In the case where the deviation of the lagged
PD ratio from its long-run equilibrium is small then the ratio is highly persistent since the value of
F is close to one. As the ratio moves further away from equilibrium, the transition function F moves
toward zero and hence the PD ratio follows a stationary process. The PD ratio, as equation (1) indicates, is stationary overall although it is highly persistent when the ratio is very close to its mean at
time t. This could be the reason for failing to reject the unit root hypothesis for the PD ratio in the literature. In addition, if the parameter determining the extent of nonlinearities, , is zero then the model
degenerates to a unit root process. Otherwise it is a nonlinear stationary process. Therefore, testing
the hypothesis of D 0 is the same as testing a linear model against a nonlinear ESTAR model.4
There are two theoretical justifications for a smooth transition autoregressive specification for
short-run deviations of the PD ratio. The first one is the significance of noise trading on asset price
dynamics as suggested by Poterba and Summers (1986) and Shleifer and Summers (1990). Given
the actions of noise traders, the perceived deviations of asset prices from their fundamental value
represent risky arbitrage opportunities. Small deviations from fundamentals may not be arbitraged
since the perceived gains may be too small to outweigh this risk. However, arbitrage increases as the
degree of deviations from the fundamentals increases. The risk arbitrage hypothesis implies nonlinear
dynamics of asset prices (Gallagher and Taylor, 2001; McMillan, 2007).
The second one is the existence of transaction costs in financial markets. Facing investment
uncertainties, the existence of transaction costs leads investors to wait for large arbitrage opportunities before entering the market. Theoretical justifications for the argument of transaction costs are
provided by Dumas (1992) and Sercu et al. (1995).
EMPIRICAL METHODOLOGY
In-sample tests
The long-horizon predictive regression model is typically applied in examining the predictability of
stock prices, which is given as follows:
k pt D ˛k C ˇk .t t / C "t Ck ,
k D 1, 2, 3, : : : , 10
(2)
where t t is the deviation of the PD ratio from its long-run equilibrium value, and k pt pt Ck pt . Equation (2) differs from a conventional long-horizon regression equation in that
t t , instead of t , is used to serve as an error correction component. It is t t that reflects
deviations from long-run equilibrium given that the PD ratio mean reverts to its time-varying
mean.
4
The critical values of testing the hypothesis of D 0 is constructed through bootstrap.
Copyright © 2011 John Wiley & Sons, Ltd.
J. Forecast. (2011)
DOI: 10.1002/for
J.-L. Wu and Y.-H. Hu
Based on equation (2), the non-predictability of stock price changes can be examined by testing the
hypothesis of ˇk D 0 versus ˇk < 0 for a given forecast horizon of k. We adopt the Newey and West
(1987) method to construct a consistent estimate of the covariance matrix when the forecast horizon
is greater than one. The lag order in constructing the Newey–West covariance matrix is determined
based upon Andrews’ (1991) procedure. We also apply a joint test to examine whether the smallest
t ratio among the 10 horizons, tmin D min ft.k/ W k D 1, 2, 3, : : : , 10}, is significant.
It is well known that the estimate of ˇk in equation (2) is subject to the problem of finite-sample
bias since the innovations of PD ratios and price changes may be correlated (Keim and Stambaugh,
1986; Richardson and Stock, 1989). Therefore the inference of ˇk based on a conventional t-test
may not be appropriate (Campbell and Yogo, 2006). The inference of ˇk in the paper is conducted
based on a bootstrapped t-test in which the critical values of the tˇ statistic are constructed through
bootstrap (Kilian and Taylor, 2003; Rapach and Wohar, 2005a). In addition, this paper investigates
the inference of ˇk based on a local-to-unity framework below (‘Empirical Investigation’), which
assumes that PD ratios follow a process with a root near unity.
k
Out-of-sample tests
To evaluate the out-of-sample predictability of the long-horizon predictive equation relative to the
random walk with drift, this work constructs a sequence of recursive forecasts from the above two
models, respectively. The first 30-year observations are reserved for estimation and hence the out-ofsample forecast period starts from 1902. This paper re-identifies the trend function and re-estimates
the long-horizon regression equation each time we add one observation to our sample and thus use
only data available up to the date of forecast.
Conventionally, researchers compare the value of root mean square errors (RMSE) from different models to conclude for the superiority of a model in out-of-sample contests if the model has the
smallest RMSE. Simply comparing the values of RMSE does not give any idea of the significance
of differences. In our case, the null model is a random walk and it is nested with the alternative of
a long-horizon regression equation. This paper therefore adopts the CW test provided by Clark and
West (2006, 2007) to examine the predictability of stock price movements. The Newey–West method
is applied to construct an autocorrelation consistent standard error. If the p-value of the CW statistic
is less than 0.1 then this paper asserts that the long-horizon predictive equation outperforms random
walk with drift in out-of-sample contests. A joint test is also applied to examine whether the largest
CW statistic among the 10 horizons, CWmax D max fCW.k)I k D 1, 2, 3, 4, : : : , 10g, is significant. The
critical values of the CW statistic are constructed through a non-parametric bootstrap. Results from
out-of-sample tests allow us to examine the significance of the assumptions of time-varying mean and
ESTAR dynamics of PD ratios.
EMPIRICAL INVESTIGATION
Estimation results of the ESTAR model
Annual data for the US consumer price index (CPI), nominal stock prices and dividends for S&P 500
from 1872 to 2004 are obtained from Robert Shiller’s web page, and are updated to 2007. The CPI
is used to convert the series of nominal stock prices into real terms. The PD ratio is defined as the
log of real S&P 500 price index minus the log of real dividend paid on the portfolio of the previous
year.
This paper applies an ESTAR(1) model to describe short-run deviations of PD ratios. The estimation results, reported in Table I, reveal that all estimates are significant at the 5% level. As for
Copyright © 2011 John Wiley & Sons, Ltd.
J. Forecast. (2011)
DOI: 10.1002/for
Stock Price Predictability
Table I. Estimation of the ESTAR model
t D t C fexpŒ1.567
.t1 t1 /2 g.t1 t1 / C "t
.3.596/
Œ0.005
t D 3.151
0.013
t C 0.0002
t2
.27.64/
.3.325/
.5.195/
Q(1) = 3 105 [0.995], LM(1) = 1.5 105 [0.997], Q2 (1) = 2.851 [0.091], ARCH(1) = 2.798 [0.097],
Reset(1) = 0.013 [0.909] ; ET = 0.242 [ 0.623 ]
Note: t is the PD ratio. A number in parentheses under an estimate is its t statistic, and a number in square brackets is
a p-value. LM(p/ and Q(p/ are respectively the Lagrange multiplier and the Ljung–Box autocorrelation tests for up to a
pth-order autocorrelation. They are 2 distributions with p degrees of freedom. Q2 .p/ is the Ljung–Box autocorrelation test
for squared residuals for up to a pth-order autocorrelation, which has a 2 distribution with p degrees of freedom. ARCH(p/
is a test statistic for up to a pth-order autoregressive conditional heteroscedasticity. It has 2 distribution with p degrees of
freedom. RESET is the Ramsey regression specification error test, which has an F distribution. ET is the statistic for firstorder residual autocorrelation in a nonlinear model provided by Eitrheim and Teräsvirta (1996). The p-value of the estimated
transition parameter, , is constructed based on a non-parametric bootstrap.
residual diagnostic tests, the hypothesis of no serial correlation of residuals is examined using Q,
LM and ET statistics.5 The Q statistic on squared residuals, Q2 , and the ARCH statistic of Engle
(1982) are applied to examine the autoregressive conditional heteroscedasticity of residuals. Finally,
the RESET statistic by Ramsey (1969) is applied to examine the misspecification of the model. Based
on these statistics, we find that estimated residuals are well behaved since there is no evidence of serial
correlation or conditional heteroscedasticity as indicated by Q, LM, Q2 , and ARCH statistics, respectively. In addition, there is also no evidence of model misspecification as indicated by the RESET
statistic.
The current research simulates PD ratios based on a non-parametric bootstrap and plots the simulated and realized ratios in Figure 1. Simulated PD ratios fit realized ratios well and the simulated
ratio also captures the unprecedented rise of the PD ratio during the period of 1955-1970 and that
of post 1982. This further strengthens the appropriateness of the ESTAR model in equation (1) to
describe the dynamics of PD ratios.
The stationarity of the PD ratio
Based on equation (2), the stationarity of the trend-adjusted PD ratio is crucial since it makes little
sense to predict price changes with I (1) variables. In addition, neglecting the unit-root problem
of regressors leads to spurious results of the predictability of stock returns (Lanne, 2002). This
paper applies two different linear unit root tests provided by Dicky and Fuller (1979, ADF) and
Elliott et al. (1996, DF-GLS), respectively, to examine the stationarity of PD ratios. In addition
to linear unit root tests, this work also applies the nonlinear unit root test provided by Kapetanios
et al. (2003, KSS). The KSS test examines the unit-root hypothesis against the hypothesis of a stationary nonlinear smooth transition autoregressive process. The model includes a constant but no
trend, and the lag order of the model is set to one in the KSS test based on the Akaike information
criteria (AIC) and it is selected based on the modified AIC rule in the ADF and DF-GLS tests as
suggested by Ng and Perron (2001). It is worth noting that applying the critical values of the abovementioned unit root tests will bias test results toward rejecting the unit root hypothesis since the
5
The Q and LM statistics are Ljung–Box and Breusch–Goldfrey statistics for residual autocorrelation. The ET statistic,
provided by Eitrheim and Teräsvirta (1996), tests for first-order residual autocorrelation in a nonlinear model.
Copyright © 2011 John Wiley & Sons, Ltd.
J. Forecast. (2011)
DOI: 10.1002/for
J.-L. Wu and Y.-H. Hu
trend-adjusted PD ratio is an estimated series rather than an observed series. To correct the involved
estimation uncertainty in our case, we construct the finite sample distribution of unit root tests using a
non-parametric bootstrap.
Results from Table II indicate that the unit root hypothesis of PD ratios is not rejected at the 5%
level by all three statistics. However, the unit root hypothesis of the trend-adjusted ratio (t t /
is rejected at the 10% level by the ADF statistic and is rejected at the 5% level by DF-GLS and
KSS statistics. In addition, results from Table I indicate that PD ratios exhibit significant nonlinear
trends. Results from Tables I and II imply that the PD ratio is stationary and has a time-varying
mean.
Our results are crucial for several reasons. First, there are several articles that support the unit
root hypothesis of the PD ratio and hence reject the present value model (Froot and Obstfeld, 1991;
Lamont, 1998; Balke and Wohar, 2002). Several articles in the literature explain the apparent failure of the present value model through the presence of non-fundamental components. A typical
non-fundamental component has been ascribed to the presence of a rational bubble, but there are serious theoretical doubts regarding its existence (Campbell et al., 1997, pp. 259–260). An interesting
alternative is to consider a nonlinear modeling approach arising from noise traders or the existence
of transaction costs (Shleifer, 2000; McMillan, 2007; Gallagher and Taylor, 2001). Following this
alternative approach, we model the PD ratio by an ESTAR model with a trend, and find that the
trend-adjusted PD ratio is stationary. In contrast to existing findings, our results support that stock
prices include both fundamental and non-fundamental components. It also indicates that capturing
the impact of non-fundamental components is crucial in empirical analysis. Second, Campbell and
Shiller (2001) argue that it is unwise to reject the mean reversion hypothesis of the PD ratio since no
pure empirical evidence can indicate that the mean of the ratio has changed. Results of rejecting the
unit root hypothesis for the trend-adjusted PD ratio indicate that the long-run PD ratio is time varying,
and hence it could be misleading to treat it as a constant. Third, our results point out that conventional
long-horizon regression equations using the level of PD ratio, t , as an error correction component
are not justified empirically.
Stock price predictability under the ESTAR dynamics of PD ratios
Given that the ESTAR model with a time-varying mean fits the data well, it is interesting to question
the significance of nonlinearities in a data-generating process in improving the out-of-sample predictability of stock prices. This paper estimates predictive regressions at different forecast horizons,
Table II. Unit root tests of the PD ratio
ADF
t
Statistics
CV 5%
CV 10%
1.577
2.883
2.578
DF-GLS
t t
3.458
3.691
3.400
t
1.203
1.943
1.615
KSS
t t
2.672
2.659
2.353
t
t t
1.250
2.930
2.660
4.173
4.055
3.774
Note: t and t t are the PD ratio and trend-adjusted PD ratio, respectively. CV 5% and CV 10% are 5% and 10% critical
values. Asterisks (*) and (**) indicate significance at the 10% and 5% level, respectively. ADF is the augmented Dickey–
Fuller test, DF-GLS is the unit root test provided by Elliott et al. (1996) and KSS is the nonlinear stationarity test provided by
Kapetanios et al. (2003). The lag order of the KSS test is determined based on the Akaiki information rule but the lag order
of ADF and DF-GLS tests is selected based on the MAIC rule as suggested by Ng and Perron (2001). Critical values of ADF,
DF-GLS and KSS tests for trend-adjusted PD ratio are constructed using a non-parametric bootstrap.
Copyright © 2011 John Wiley & Sons, Ltd.
J. Forecast. (2011)
DOI: 10.1002/for
Stock Price Predictability
as in Campbell and Shiller (1988, 1998), and then examine the hypothesis of no return predictability
under the ESTAR dynamics of PD ratios.
This work first examines stock price predictability over the period 1872–2007. Table III reports
the marginal significance level (p-value) of the t-statistic at different forecast horizons under the
hypothesis that stock prices follow the process of random walk with drift. The finite sample
distribution of the t-statistic under different forecast horizons is obtained through a bootstrap procedure in which the DGP of the PD ratio is given by equation (1) with a quadratic function,
t D aCbt Cct 2 . Results from the last column of Table III indicate that the bootstrap t-test rejects, at
the 5% level, the no-predictability hypothesis of stock price changes for all forecast horizons. Next,
the current research examines the out-of-sample predictability of stock prices. Based on the longhorizon regression equation and the model of the random walk with drift, this work computes the
ratio of root mean square error (RMSE) from the above two models. The bootstrap CW test is then
applied to investigate the hypothesis of no significant difference in the accuracy of two competing
forecasts.
Results found in the last two columns on the nonlinear DGP panel of Table IV indicate that the
ratios of RMSEs are less than one and CW statistics reject the no-predictability hypothesis at the 10%
level when the forecast horizon is greater than 1 year. In addition, the joint statistic is also significant
at the 5% level. These results are interesting since they reveal both in- and out-of-sample evidence
of predictability of stock prices even when the forecast horizon is short. Although RMSE ratios are
less than but close to one when the forecast horizon is greater than 1 year under the assumptions
of time-varying means and ESTAR nonlinearities of PD ratios, these ratios are significant statistically. These results reveal statistical significance of the above two assumptions in predicting US
stock prices.
Table III. In-sample predictability test of stock price changes
pt ptk D ˛k C ˇk .tk tk / C "t
k
(year)
1
2
3
4
5
6
7
8
9
10
JS
Linear DGP
1872–1997
Nonlinear DGP
1872–2007
1872–1997
1872–2007
CM
TVM
CM
TVM
CM
TVM
CM
TVM
0.338
0.182
0.234
0.178
0.080
0.078
0.060
0.014
0.020
0.036
0.036
0.146
0.040
0.098
0.050
0.024
0.026
0.016
0.004
0.008
0.018
0.018
0.480
0.330
0.358
0.292
0.266
0.320
0.372
0.404
0.458
0.490
0.416
0.084
0.024
0.028
0.012
0.010
0.010
0.010
0.002
0.010
0.010
0.018
0.332
0.172
0.244
0.134
0.062
0.048
0.038
0.012
0.026
0.042
0.028
0.112
0.026
0.066
0.032
0.014
0.010
0.010
0.004
0.004
0.008
0.012
0.516
0.330
0.372
0.316
0.284
0.334
0.374
0.402
0.454
0.474
0.430
0.046
0.010
0.018
0.006
0.004
0.004
0.002
0.002
0.002
0.006
0.002
Note: The number in the table is the bootstrapping P -value of in-sample t -statistic, in which the finite sample distribution of
the statistic is constructed through bootstrap. The t statistics are constructed based on serial correlation robust standard errors.
If a number in the table is less than 0.1 then we claim that the in-sample predictability of stock price changes is rejected at
the 10% level of significance. DGP, TVM and CM are a data-generating process, a time-varying mean and a constant mean,
respectively. k is the forecast horizon and JS indicates a joint statistic. Numbers in bold indicate significance at the 10% level.
Copyright © 2011 John Wiley & Sons, Ltd.
J. Forecast. (2011)
DOI: 10.1002/for
J.-L. Wu and Y.-H. Hu
Table IV. Out-of-sample predictability tests of stock prices
k
1872–1997
(year)
CM
RMSE
1872–2007
TVM
CM
TVM
CW(k/
RMSE
CW(k )
RMSE
CW(k )
RMSE
CW(k )
Nonlinear DGP
1
1.0044
2
0.9967
3
1.0005
4
0.9951
5
0.9825
6
0.9773
7
0.9716
8
0.9446
9
0.9351
10
0.9309
JS
–
0.408
0.142
0.196
0.130
0.096
0.104
0.104
0.046
0.056
0.074
0.100
1.0010
0.9882
0.9931
0.9862
0.9784
0.9698
0.9680
0.9439
0.9334
0.9347
–
0.284
0.054
0.009
0.068
0.048
0.058
0.056
0.018
0.014
0.030
0.036
1.0083
1.0156
1.0238
1.0292
1.0311
1.0326
1.0368
1.0342
1.0285
1.0215
–
0.540
0.302
0.370
0.272
0.208
0.248
0.276
0.260
0.314
0.332
0.388
1.0025
0.9964
0.9987
0.9949
0.9920
0.9856
0.9844
0.9688
0.9579
0.9552
–
0.266
0.072
0.104
0.076
0.054
0.062
0.052
0.024
0.024
0.042
0.046
Linear DGP
1
1.0044
2
0.9967
3
1.0005
4
0.9951
5
0.9825
6
0.9773
7
0.9716
8
0.9446
9
0.9351
10
0.9309
JS
–
0.416
0.152
0.214
0.146
0.104
0.126
0.090
0.058
0.060
0.066
0.126
1.0048
0.9957
1.0028
1.0009
0.9952
0.9857
0.9857
0.9637
0.9530
0.9536
–
0.528
0.092
0.208
0.152
0.102
0.074
0.074
0.028
0.020
0.028
0.052
1.0083
1.0156
1.0238
1.0292
1.0311
1.0326
1.0368
1.0342
1.0285
1.0215
–
0.506
0.294
0.352
0.246
0.208
0.264
0.288
0.268
0.298
0.316
0.358
1.0059
1.0027
1.0071
1.0086
1.0088
1.0029
1.0048
0.9925
0.9823
0.9797
–
0.482
0.120
0.196
0.130
0.084
0.088
0.092
0.054
0.058
0.068
0.128
Note: DGP, TVM and CM are a data-generating process, a time-varying mean and a constant mean, respectively. k is the
forecast horizon and JS indicates a joint statistic. The numbers under the ‘RMSE’ column are the ratio of the root mean square
error from the predictive equation to that from the random walk with drift under different forecast horizons. If the value of
RMSE ratios is less than one then we claim that the long-horizon predictive equation provides more accurate forecasts than
random walks. CW(k/ is the statistic provided by Clark and West (2007) under the forecast horizon k in which the truncated
lag in constructing the Newey–West covariance matrix is determined based on Andrews’ (1991) procedure. The number under
the ‘CW(k/’ column is the bootstrapping p-value of the CW statistic, in which its finite sample distribution is constructed
through a bootstrap method. If the value of CW(k/ is less than 0.1 then we claim that the long horizon predictive equation
outperforms random walks in out-of-sample contests. ‘–’ indicates that a statistic is not available. Numbers in bold indicate
significance at the 10% level.
Under the assumption of a constant mean, the trend function in the dynamics of PD ratio, equation
(1) is assumed to be a constant, t D a. Results from the eighth column of Table III indicate that the
hypothesis of no predictability of stock returns is not rejected by the bootstrap t-test when a constant
mean of PD ratio is assumed. Under the assumption of stable PD ratios, the sixth and seventh columns
on the nonlinear DGP panel of Table IV report results of out-of-sample tests. These findings reveal
that RMSE ratios are greater than one and CW statistics are insignificant at all forecast horizons.
In short, there is no evidence of stock price predictability based on both in- and out-of-sample tests
under the assumption of stable PD ratios. These results challenge the appropriateness of the stability
assumption of PD ratios.
Copyright © 2011 John Wiley & Sons, Ltd.
J. Forecast. (2011)
DOI: 10.1002/for
Stock Price Predictability
Shiller (2005) points out that the period after 1997 is full of institutional changes in stock markets.
Top capital gains tax rate was cut from 28% to 20% at 1997 and a large number of users did not
discover the Web until 1997. In addition, PD ratios reveal an enormous spike at 2000. The 10-year
average of PD ratios (without taking logs) over 1998–2007 is 67, which is about two to three times
the historical average by this measure.6 This paper therefore excluded the period 1998–2007 and then
applied this sub-period to check the robustness of our results. Results from the seventh column of
Table III reject the no-predictability hypothesis of stock returns as long as the forecast horizon is
greater than 1 year when the time-varying mean of PD ratios is allowed. RMSE ratios and CW statistics found in the fourth and fifth columns on the nonlinear DGP panel of Table IV indicate evidence of
out-of-sample predictability when forecast horizons are greater than 1 year. In addition, results found
in the sixth column of Table III and the second and third columns on the nonlinear DGP panel of
Table IV support the evidence of stock price predictability under the stability hypothesis of PD ratios
only when the forecast horizon is long.
Based on the nonlinear DGP panel of Tables III and IV, this paper finds two interesting results.
First, the evidence of stock price predictability is sensitive to sample periods under a constant mean
assumption of PD ratios yet it is robust to sample periods when the assumption of time-varying means
is made. Second, the assumption on the time-varying mean of PD ratio is helpful to strengthen the inand out-of-sample evidence of return predictability.
Lettau and Van Nieuwerburgh (2008) challenge the assumption of a fixed long-run value of the PD
ratio. Allowing for structural changes in the predictive equation, they find support for the in-sample
predictability of stock returns, but fail to find evidence of beating the benchmark random walk model
in out-of-sample contests. Results from the nonlinear DGP panel of Tables III and IV indicate both
in- and out-of-sample predictability of US stock prices regardless of sample periods when the timevarying mean and nonlinearity of PD ratios are taken into account. The result of beating the random
walk model is crucial since a general consensus in extant literature indicates the failure of outperforming the random walk model in out-of-sample tests (Goyal and Welch, 2003; Lettau and Van
Nieuwerburgh, 2008; Welch and Goyal, 2008). A last question further arises: Why is it that outof-sample evidence of stock price predictability is much weaker than in-sample evidence in the
literature? Our results show that these inconsistent results can be reconciled once the time-varying
mean and nonlinearities of PD ratios are controlled.
Stock price predictability under a linear AR model for PD ratios
For the purpose of comparison, this paper estimates predictive regressions at different forecast horizons and then examines the hypothesis of no return predictability under a linear AR model of PD
ratios. The linear dynamic of PD ratios is described as follows:
t D t C
m
P
ut i.i.d..0, 2 /
i .t i t i / C ut ,
i D1
(3)
t D a C bt C ct
2
where m is set to one based on the Akaike information rule. This research first examines stock price
predictability over the period 1872–2007. Results from the fifth column of Table III indicate that
the hypothesis of no predictability of stock prices is rejected at the 10% level by the bootstrap t-test
6
In addition, the average of the 10-year average of PD ratios (without taking logs) over 1872–1997 is 23.
Copyright © 2011 John Wiley & Sons, Ltd.
J. Forecast. (2011)
DOI: 10.1002/for
J.-L. Wu and Y.-H. Hu
regardless of forecast horizons. As for out-of-sample tests, results found in the last two columns on
the linear DGP panel of Table IV indicate that RMSE ratios are less than one when the forecast horizon is greater than 7 years. The CW statistic rejects the no-predictability hypothesis of stock prices, at
the 10% level, when the forecast horizon is greater than 4 years. However, the joint CW statistic has a
P -value of 0.128, which fails to reject the no-predictability hypothesis. It is worth noting that RMSE
ratios are greater than 1.0 but the P -values of CW statistic are less than 0.1 when the forecast horizon
is 5, 6 and 7 years. This is possible since the CW statistic uses the MSPE adjusted series, instead of
MSPE series, to examine the no-predictability hypothesis of stock prices. Based on CW statistics and
RMSE ratios, this research concludes that the no-predictability hypothesis of stock prices is rejected,
at the 10% level, when the forecast horizon is greater than 7 years. Our results also reveal that it might
be misleading to reject the no-predictability hypothesis solely based on the CW statistic.
Under the assumption of a constant mean, the t-statistics from column four of Table III indicate no
evidence of in-sample return predictability, at the 10% level, regardless of forecast horizons. Results
found in the six and seventh columns on the linear DGP panel of Table IV indicate that RMSE ratios
are greater than one and that CW statistics fail to reject the no-predictability hypothesis of stock
prices, at the 10% level, regardless of forecast horizons.
Are the foregoing results sensitive to sample periods? To check the robustness of our findings to
the sample periods, this paper repeats the experiment of stock price predictability over the period
1872–1997. Results from second and third columns of Table III indicate that the hypothesis of no
predictability of stock price changes is rejected by the bootstrap t-test for the forecast horizon being
greater than 1 year when the time-varying mean of the PD ratio is assumed. However, the above
hypothesis is rejected for the forecast horizon being greater than 4 years under the assumption of
stable PD ratio.
Under the time-varying mean assumption of PD ratios, results found in columns four and five
on the linear DGP panel of Table IV indicate that RMSE ratios are less than one when the forecast
horizon is greater than 4 years. In addition, CW statistics are significant when the forecast horizon is
greater than 5 years and the joint CW statistic has a P -value of 0.052, rejecting the no-predictability
hypothesis. Under the assumption of a constant mean, results found in column two on the linear DGP
panel of Table IV indicate that RMSE ratios are less than one when the forecast horizon is greater
than 3 years. Results found in the third column on the linear DGP panel of Table IV reveal that CW
statistics are significant when the forecast horizon is greater than 6 years. In other words, under the
assumption of a linear AR model for PD ratios, the evidence of out-of-sample predictability on stock
prices is observed only when the forecast horizon is long.
Overall, results from Tables III and IV are interesting. First, the evidence of stock price predictability is strengthened once the nonlinearity and the time-varying mean of the PD ratio are taken into
account and the evidence is robust to sample periods. Second, results of stock price predictability are
sensitive to sample periods when the mean of PD ratios is assumed to be a constant even though the
nonlinearity of the ratio has been taken into account. Why are the results of stock price predictability
sensitive to sample periods under the assumption of stable PD ratios? Our empirical investigation
points out that the reason could be structural changes in the predictive equation when the mean of PD
ratios is time varying but is assumed to be a constant.
Return predictability within a local-to-unity framework
The inference in predictive regressions depends critically on the assumed stochastic properties of
the PD ratio. Although this paper fails to reject the unit root hypothesis of PD ratios as found in
Table II, it is very likely that we are not able to distinguish whether the ratio has a unit root. To
Copyright © 2011 John Wiley & Sons, Ltd.
J. Forecast. (2011)
DOI: 10.1002/for
Stock Price Predictability
take this uncertainty into account, several recent articles re-examine the evidence of in-sample return
predictability using tests that are available within a local-to-unity framework (Cavanagh et al., 1995;
Torous et al., 2004; Campbell and Yogo, 2005, 2006). Local-to-unity asymptotics provide an accurate
approximation to the finite-sample distribution of test statistics when the predictor variables are highly
persistent. This paper applies a new Bonferroni procedure, based on the DF-GLS test and the Q test,
provided by Campbell and Yogo (2006), to re-examine the in-sample predictability of stock price
changes.
Table V reports the correlation coefficients (ı/ between the innovation of the PD ratio and price
changes, the 95% confidence interval of the autoregressive root (
/ for the PD ratio, the estimate of
the slope coefficient (ˇk / in the predictive equation, the t-statistic of ˇk constructed based on a serial
correlation robust standard error, and the 90% confidence interval of ˇk based on DF-GLS and Q
statistics. Based on the period 1872–2007, results found in column two on the 1872–2007 panel of
Table V indicate that the correlation coefficients are positive and large when the forecast horizon is
short, and they decrease with the forecast horizon. This is because movements in stock price changes
and PD ratios mostly come from the movement in stock prices when the forecast horizon is short.
Table V. Tests of return predictability within the local-to-unity framework
pt ptk D ˛k C ˇk tk C "t
t D ! C t1 C ut
D 1 C c=T with c a fixed constant and T is the number of observations
k
(year)
ı
ˇk
t statistic
95% CI: 90% CI ˇk : Q-test
1872–2007
1
2
3
4
5
6
7
8
9
10
0.831
0.645
0.446
0.457
0.384
0.283
0.285
0.314
0.240
0.201
0.028
0.065
0.070
0.104
0.128
0.124
0.121
0.143
0.150
0.154
0.610
1.208
1.138
1.472
1.620
1.395
1.177
1.204
1.206
1.240
[0.887,1.012]
[0.889,1.013]
[0.889,1.014]
[0.869,1.006]
[0.894,1.016]
[0.902,1.019]
[0.900,1.018]
[0.873,1.010]
[0.832,0.993]
[0.795,0.977]
[0.063,0.089]
[0.124,0.058]
[0.152,0.057]
[0.198,0.038]
[0.235,0.025]
[0.252,0.042]
[0.269,0.069]
[0.313,0.077]
[0.335,0.077]
[0.343,0.070]
1872–1997
1
2
3
4
5
6
7
8
9
10
0.828
0.647
0.428
0.420
0.332
0.229
0.227
0.251
0.162
0.140
0.063
0.137
0.154
0.216
0.262
0.252
0.255
0.302
0.311
0.318
1.087
1.857
1.739
2.352
3.175
3.252
3.684
5.367
5.004
4.330
[0.766, 0.964]
[0.910, 1.025]
[0.910, 1.025]
[0.898, 1.022]
[0.895, 1.022]
[0.882, 1.017]
[0.885, 1.019]
[0.883, 1.019]
[0.878, 1.017]
[0.876, 1.017]
[0.093, 0.119]
[0.128, 0.102]
[0.216, 0.073]
[0.285, 0.016]
[0.332, 0.058]
[0.334, 0.075]
[0.325, 0.092]
[0.345, 0.152]
[0.381, 0.172]
[0.411, 0.166]
Note: ı indicates the correlation between the innovations of PD ratio and price changes. CI indicates a confidence interval.
t .k/ is the t statistic of ˇk under the forecast horizon k and it is constructed based on serial correlation-robust standard
errors. A bold confidence interval of ˇk indicates rejection of the hypothesis of no-return predictability at the 10% level
of significance.
Copyright © 2011 John Wiley & Sons, Ltd.
J. Forecast. (2011)
DOI: 10.1002/for
J.-L. Wu and Y.-H. Hu
The confidence interval of does contain a unit root except when the forecast horizon is greater
than 8 years. Results found in the fourth column of the 1872–2007 panel of Table V indicate that the
no-predictability hypothesis is not rejected by t statistics at all forecast horizons. The last column in
the 1872–2007 panel of Table V indicates that a 90% Bonferroni confidence interval for ˇk , based
on a Q test, does not lie strictly below zero for all forecast horizons. Hence the current research fails
to reject the hypothesis of no in-sample predictability of stock price changes at the 10% level regardless of forecast horizons. These results are consistent with those from the bootstrap t-test under the
constant mean assumption of PD ratios.
With the period 1872–1997, results found in the 1872–1997 panel of Table V indicate that the contemporaneous correlation of innovations is high with a short horizon and decreases with the length
of the forecast horizon. The confidence interval of does contain a unit root except for the forecast
horizon of 1 year. The t statistics reject the no-predictability hypothesis at the 10% level when the
forecast horizon is greater than 1 year. However, a 90% Bonferroni confidence interval for ˇk based
on a Q test lies strictly below zero when the forecast horizon is greater than 4 years, revealing evidence of long-horizon predictability of stock price changes. Again, this paper finds that results from
a linear model within the local-to-unity framework are sensitive to sample periods.
Campbell and Yogo (2006) examine one-step-ahead return predictability based on several financial ratios within the framework of nearly integrated regressors. They fail to find evidence of return
predictability based on PD ratios. Our results are consistent with theirs.7 It is interesting to find that
results from the last column of Table V are consistent with those from the second and fourth columns
of Table III. This further confirms the appropriateness of applying bootstrap tests to examine the
predictability of stock returns.
ECONOMIC SIGNIFICANCE
Given the statistical evidence of rejecting the no-predictability hypothesis of stock prices, it is interesting to ask if a long-horizon predictive equation is helpful for investors to make a profit in stock
markets. This paper investigates the economic significance of the following four models: the random
walk with drift (RW), a linear long horizon predictive equation with a constant mean (MD1) and a
linear long horizon predictive equation with a time-varying mean in the linear autoregressive process
of PD ratios (MD2) and the ESTAR process of PD ratios (MD3), respectively. The economic significance of a model is evaluated based on the following two different criteria: expected real return per
unit of risk and ex post real profit.
The annualized expected real return per unit of risk is obtained by dividing the conditional
p real return (Et .pt Ch pt //h/ by the conditional standard deviation of the return at time
t
vart .pt Ch pt /= h . To construct ex post real returns, this paper sets up the following trading
rule. Given the information set at time t, an investor predicts h-period-ahead log real stock price
(pOt Ch / based on the long horizon prediction equation in (1). If an investor predicts an increase of the
real stock price (pOt Ch > pt /, then the investor will buy one unit of portfolio at time t and then sell
the unit at t C h. The annualized ex post real return after trading at t C h is .pt Ch pt /= h. Otherwise,
the investor will sell a unit of portfolio at time t and then buy back a unit of portfolio at time t C h. In
such a case, the ex post real return at t C h is .pt pt Ch /= h. For each criterion, this paper computes
the expected real return and the ex post real return for each out-of-sample period and then reports their
7
It is worth noting that the definitions of return and sample periods in Campbell and Yogo (2006) are different from ours.
Copyright © 2011 John Wiley & Sons, Ltd.
J. Forecast. (2011)
DOI: 10.1002/for
Stock Price Predictability
means over the out-of-sample period under a specific forecast horizon in Table VI. The significance
of these means is examined using a simple t-test.
Results from the top panel of Table VI report the economic significance of models based on the
expected real return per unit of risk. The means of expected real return per unit of risk under different forecast horizons are positive for all models under consideration. The proposed nonlinear model
(MD3) results in the highest mean of expected real return per unit of risk regardless of forecast horizons in the sub period (1872–1997), which is also true when the forecast horizon is greater than 4
years throughout the entire period (1872–2007). Results from the bottom panel of Table VI indicate
that the means of ex post real return from different forecast horizons based on the constant mean
model (MD1) are positive for 9 out of 10 horizons in the sub-period, but the means are all negative in
the whole period. The random walk model reveals the highest mean of ex post real return among the
four models for most horizons regardless of sample periods. Although the means of ex post real return
based on MD2 and MD3 are all positive, the value of significance based on MD3 is much larger than
that from MD2 regardless of sample periods.
Table VI. Economic significance
RW: k pt D ˛k C "tCk ,
MD1: k pt D ˛k C ˇk .t c/ C "tCk I k D 1, 2, 3, : : : , 10
MD2: k pt D ˛k C ˇk .t t / C "tCk I t D t C 1 .t1 t1 / C ut , t D h.t/, k D 1, 2, 3, : : : , 10
MD3: k pt D ˛k C ˇk .t t / C "tCk I t D t C .t1 t1 / expŒ.t1 t1 /2 C ut I t D h.t/,
k D 1, 2, 3, : : : , 10
k
1872–1997
RW
MD1
1872–2007
RW
MD1
MD2
MD3
1
2
3
4
5
6
7
8
9
10
Annualized expected real return per unit of risk
0.093
0.058
0.098
0.101#
0.094
0.044
0.108
0.113#
0.107
0.061
0.115
0.118#
0.111
0.057
0.133
0.136#
0.118
0.069
0.155
0.160#
0.137
0.098
0.170
0.173#
0.148
0.115
0.184
0.187#
0.150
0.123
0.191
0.194#
0.152
0.128
0.186
0.187#
0.155
0.134
0.186
0.187#
0.096
0.096
0.108
0.112
0.117
0.135
0.145
0.146
0.147
0.149
0.053
0.025
0.045
0.031
0.034
0.061
0.079
0.084
0.094
0.103
0.084
0.083
0.092
0.104
0.121
0.138
0.156
0.165
0.164
0.168
0.089
0.091
0.098
0.110
0.128#
0.144#
0.160#
0.169#
0.168#
0.172#
1
2
3
4
5
6
7
8
9
10
Annualized ex post real returns
0.039
0.020
0.005
0.036
0.007
0.030
0.033
0.001
0.007
0.031
0.009
0.008
0.029
0.011
0.010
0.028
0.013
0.012
0.027
0.014
0.011
0.026
0.023
0.013
0.025
0.023
0.018
0.024
0.024
0.019
0.057
0.055
0.055
0.054
0.051
0.052
0.055
0.058
0.060
0.059
0.062
0.017
0.034
0.018
0.015
0.015
0.018
0.014
0.016
0.016
0.023
0.025
0.009
0.013
0.005
0.001
0.028
0.004
0.038
0.039
MD2
MD3
0.014
0.048#
0.022
0.029
0.027
0.031#
0.016
0.023
0.024
0.022
0.032
0.044
0.029
0.033
0.032
0.041
0.032
0.038
0.044
0.047
Note: k indicates the forecast horizon. Numbers in the table are means of expected real return per unit of risk and ex post real
return under different models and forecast horizons. Numbers in bold indicate significance at the 10% level. ‘#’ indicates that
the number is greater than that from the random walk.
Copyright © 2011 John Wiley & Sons, Ltd.
J. Forecast. (2011)
DOI: 10.1002/for
J.-L. Wu and Y.-H. Hu
Overall, results from Table VI indicate that the expected real return per unit of risk from the proposed nonlinear model is higher than that from the random walk and other competitive models for
most forecast horizons. Ex post real returns from the proposed nonlinear model are significantly positive if the forecast horizon is greater than 1 year. The above results are robust to sample periods.
These results echo our findings from Tables III and IV indicating the significance of the time-varying
mean and nonlinear dynamics of PD ratios in the empirical analysis.
SIZE AND POWER OF BOOTSTRAP TESTS
Given the rejection of the hypothesis of no stock price predictability as indicated in Tables III and IV,
it is interesting to inquire whether our findings result from the problem of size distortion or from a reasonable power of bootstrap tests. To address this question, we adopt a three-step strategy to evaluate
the power and size of bootstrap tests, which is available upon request from the authors.
Results presented in Table VII indicate that the effective size of out-of-sample tests at different
forecast horizons is close to 0.10. However, the effective size of the in-sample test is about 0.13 when
the forecast horizon is less than 3 years, and is close to 0.10 for the rest of the horizons. Given that the
P -values of the bootstrap t statistic in the last column of Table III are fairly low (less than 0.05), the
slight size distortion will not affect our conclusion about in-sample predictability. Overall, this paper
concludes that the rejection of the no-predictability hypothesis of stock prices found in the last column
of Table III and the nonlinear DGP panel of Table IV is not likely due to the problem of size distortion.
Following the three-step bootstrapping strategy, we report our results of power analysis in
Table VII. The power of the in-sample bootstrap test is in general high and varies around 0.90. As for
the power of the out-of-sample bootstrap test, it varies in the range of 0.50–0.67. Based on the size
and power analysis, we conclude that the major reason for rejecting the non-predictability hypothesis
of stock prices is the reasonable power of bootstrap tests.
Results shown in the sixth and seventh columns of the nonlinear DGP panel of Table IV, respectively, indicate no evidence of stock price predictability when the assumption of a constant mean of
PD ratios is imposed. Could these results be due to the fact of neglecting a time-varying mean of PD
ratios in empirical analysis? If our conjecture is correct, then we should find low powers of bootstrap
tests when the existing time-varying mean of PD ratios is neglected in empirical analysis. To verify
this conjecture, we investigate the power of bootstrap tests where a time-varying mean of PD ratio
exists in data but is treated as a constant in long-horizon prediction analysis and in bootstrapping
the finite sample distribution of various statistics. The power and size of bootstrap tests for this case
Table VII. Size and power of bootstrap tests
k
(year)
Size
t.k/
CW(k/
1
2
3
4
5
6
7
8
9
10
JS
0.132
0.122
0.138
0.116
0.130
0.122
0.114
0.118
0.122
0.108
0.112
0.114
0.106
0.112
0.112
0.110
0.106
0.102
0.110
0.108
0.108
0.110
Power
t.k/
CW(k/
0.916
0.566
0.938
0.622
0.938
0.652
0.938
0.668
0.930
0.650
0.920
0.640
0.914
0.604
0.904
0.570
0.886
0.528
0.866
0.492
0.904
0.562
Note: k is the forecast horizon. t .k/ and CW(k/ are t and CW statistics under the forecast horizon k. The data-generating
processes for size and power analysis are available upon request from the authors.
Copyright © 2011 John Wiley & Sons, Ltd.
J. Forecast. (2011)
DOI: 10.1002/for
Stock Price Predictability
Table VIII. Size and power of bootstrap tests: without correcting the time-varying mean in the long-horizon
regression equation
k
(year)
Size
t.k/
CW(k/
1
2
3
4
5
6
7
8
9
10
JS
0.080
0.084
0.070
0.070
0.074
0.076
0.076
0.072
0.076
0.076
0.072
0.068
0.070
0.066
0.072
0.066
0.072
0.066
0.072
0.064
0.064
0.060
Power
t.k/
CW(k/
0.238
0.334
0.242
0.354
0.266
0.374
0.242
0.382
0.234
0.380
0.214
0.342
0.200
0.308
0.190
0.282
0.176
0.268
0.160
0.232
0.166
0.216
Note: k is the forecast horizon. t .k/ and CW(k/ are t and CW statistics under the forecast horizon k. Detailed simulation
procedures are available upon request from the authors.
are reported in Table VIII. The reported results show that the power of in-sample (out-of-sample)
bootstrap tests varies from 0.16 to 0.27 (0.22 to 0.38).8 The effective size of in- and out-of-sample
bootstrap tests varies from 0.06 to 0.08, given the nominal size of 10%, indicating a medium size
distortion. In sum, the incorrect assumption of a constant mean in the PD ratio results in low power of
bootstrap tests. This could be a major reason for failing to find out-of-sample predictability of stock
prices in Table IV and existing literature.
CONCLUSIONS
There has never been convincing evidence that PD ratios were ever useful for out-of-sample predictability of stock returns (Goyal and Welch, 2003). Using US data from 1872 to 2007, this paper
reinvestigates the predictability of stock prices based on PD ratios. In contrast to the conventional stability assumption of the PD ratio, we stress the significance of its time-varying mean and its ESTAR
dynamics. Several interesting results are obtained from our empirical investigation.
First, we find that the dynamics of the PD ratio can be well described by an ESTAR model with a
time-varying mean and that the PD ratio could be a unit root process, whereas the trend-adjusted PD
ratio is stationary. These results support the premise that stock prices include both fundamental and
non-fundamental components and that modeling the impact of both components is crucial in empirical
analysis. Second, our bootstrap tests, based on a nonlinear data-generating process with time-varying
means, provide significant in- and out-of-sample evidence to reject the no-predictability hypothesis
of stock prices. The hypothesis is not rejected when the stability of the PD ratio is assumed in the
data-generating process. Third, the expected real return per unit of risk from the proposed nonlinear
model is larger than that from the random walk and other competitive models for most forecast horizons. Ex post real profits from the proposed nonlinear model rather than the constant mean model are
significantly positive if the forecast horizon is greater than 1 year. However, ex post real profits from
the proposed nonlinear mode are less than those from the random walk in general. Fourth, the above
results are robust to a subsample of 1872–1997. Fifth, our bootstrap tests have good size and achieve
reasonable power. These results indicate that allowing for a time-varying mean in the PD ratio is crucial for predicting stock price movements and hence support our assertion that the mean of the US PD
ratio is time varying. Our results also indicate that inconsistent results for in- and out-of-sample tests
8
It is meaningless to compare the power of tests in Table VII with that of Table VIII since their DGPs are different.
Copyright © 2011 John Wiley & Sons, Ltd.
J. Forecast. (2011)
DOI: 10.1002/for
J.-L. Wu and Y.-H. Hu
in the extant literature are reconciled once the time-varying mean and nonlinearity of the PD ratio are
taken into account in regressions.
ACKNOWLEDGEMENTS
For helpful comments we are grateful to an anonymous referee, Alan Taylor, as well as participants
in seminars at UC Davis and National Taiwan University. Part of the paper was finished while Wu
was visiting UC Davis, where the Department of Economics graciously provided him with superior
research facilities. Wu is grateful for financial support from the National Science Council in Taiwan.
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Authors’ biographies:
Jyh-Lin Wu graduated from Tamkang University in 1980 with a B.A. in international trade, and earned a PhD
in economics from the Ohio State University in 1990. In 1990 he began to work at National Chung-cheng
University until 2003. He worked at the Sun Yat-sen university since 2004. His research interests include empirical
international finance and empirical macroeconomics.
Yu-Hau Hu graduated from National Cheng-chi University in 1994 with a B.A. in economics, and earned a PhD
in International Economics from National Chung-cheng University in 2007. In 1998 he began to work at the
Cheng-Shiu Technological University. His research interests include empirical international finance and empirical
macroeconomics.
Authors’ addresses:
Jyh-Lin Wu, Institute of Economics, National Sun Yat-Sen University, Kaohsiung, Taiwan; and Department of
Economics, National Chung-Cheng University, Chia-Yi, Taiwan.
Yu-Hau Hu, Department of International Trade, Cheng-Shiu Technological University, Kaohsiung, Taiwan.
Copyright © 2011 John Wiley & Sons, Ltd.
J. Forecast. (2011)
DOI: 10.1002/for