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Unformatted text preview: The Journal of Financial Research 0 Vol. XXIII. No.4 0 Pages 523544 0 Winter 2000 A STATESPACE MODEL OF SHORT AND LONGHORIZON
STOCK RETURNS Chunsheng Zhou
University of California, Riverside Chang Qing
Inner Mongolia Polytechnic University, China Abstract In this article we propose a new parsimonious statespace model in
which state variables characterize the stochastic movements of stock returns.
Using the equally weighted and decile monthly stock returns, we show that (a) a
parsimonious statespace model characterizes the variation in expected returns at
any horizon; (b) the extracted expected returns explain a substantial proportion of
the variance in realized returns, and the magnitude of this proportion increases
signiﬁcantly with the horizon of returns; (c) the model successfully captures the
empirical fact that returns of smaller ﬁrms have both stronger positive
autocorrelations of shorthorizon returns and stronger negative autocorrelations
of longhorizon returns; and (d) the forecasts of asset returns obtained with the
slatespace model subsume the information in other potential predictor variables
such as dividend yields. JEL classiﬁcation: G10, G12. I. Introduction . Considerable evidence shows that stock returns are predictable and that
stock prices do not follow random walks. For example, the sample autocorrelation
structure of asset returns impliesipositive serial correlation at short horizons and
negative serial correlation at long horizons. In this article we propose a ﬂexible
statespace model in which two state variables characterize the stochastic behavior
of stock returns. Although the model is highly parsimonious, it ﬁts stock returns
over both short and long horizons well. Many researchers focus on multivariate models for exante stock returns
(e.g., Patna and French (1988b), Campbell (1991), Hodrick (1992)). They use
dividend yields, interest rates, realized past returns, and other variables to forecast We would like to thank our colleagues and an anonymous referee for helpful comments. 523 Copyright © 2001 All Rights Reserved 524 The Journal of Financial Research stock returns at various horizons. Even though many multivariate models carefully
choose predicting variables based on certain economic theories and successfully
capture the autocorrelation structure of stock returns, many economists still worry
about whether multivariate models omit some important predicting variable (e.g.,
Campbell (1991)). In comparison with the multivariate regression models, the current
statespace model does not require an a priori speciﬁcation of the predictive
variables. Conrad and Kaul (1988) use a statespace approach to study time variation
of expected returns. They focus on the predictability of highﬁequency weekly
returns and use a less ﬂexible model than that proposed here. As a result, their
model does not ﬁt lowﬁ'equency data well and does not generate the typical pattern
of negative autocorrelation of stock returns over long horizons. Fama and French
(1988a) and Poterba and Summers (198 8) propose a model in which the stock price
is the sum of a permanent component plus a temporary component. Their model captures the negative autocorrelation of longhorizon stock returns, but it does not
explain the positive autocorrelation of shorthorizon stock returns. A parsimonious statespace model like the one presented here also has a
technical advantage over the traditional univariate regression approach used by Fama and French (1988a) and Poterba and Summers (1988) to analyze the behavior
of longterm stock returns. A practical problem with the traditional univariate
regression approach is that it often delivers weak evidence against the hypothesis
that all autocorrelations are zero (e.g., Richardson (1993), Richardson and Stock
(1989)). This is because the univariate regression approach needs to use longterm
returns data to estimate the autocorrelation of longterm stock returns. However,
there are not many nonoverlapping observations of multiyear stock returns. My
statespace model infers the behavior of longterm stock returns ﬁom the history of
shortterm stock returns. As a result, a much bigger nonoverlapping data sample is
available to estimate the statespace model. The contribution of this article is twofold. First, it sets up a parsimonious
model to characterize the variation in expected returns without resorting to any
predictive variables. The model matches various features of historical stock returns,
namely: (a) the autocorrelation ﬁmction of stock returns over both short and long
horizons, (b) the relation between autocorrlation function and the size of ﬁrms, and
(c) the predictability of stock returns. Second, the modeling of expected stock
returns in this article is linked to economic theories such as noise or feedback
trading. II. The StateSpace Model An important stylized fact of the securities market is that asset returns are
positively autocorrelated over short horizons but negatively autocorrelated over long
horizons. To capture the negative autocorrelation of longhorizon returns, we Copyright © 2001 All Rights Reserved A StateSpace Model 525 assume the stock price has two components: a permanent component q, and a
transitory component 2,, as suggested by Fama and French (1988a) and Poterba and
Summers (1988). Furthermore, we assume the transitory component 2, depends on
some other autoregressive state variable y, so that stock returns can be positively
autocorrelated over short horizons. Formally, the model can be written as: p: = 9. + z"
qt = qtl + et’
2: = ¢ZH + yyt + nt’ y: = Ayg—l + 6p where p, is the logarithm of the detrended asset price at t, q, and z, are two
components of p,, y, is a state variable, 8, ~ iid N(0, of , 1], ~ iid MD, of), 6, ~ iid N(0,
0:),andd>andlareconstantssuchthat0<¢<land0<A<1.Foreaseof
estimation, we assume the noise terms 6, c, and n are mutually independent, which
signiﬁcantly reduces the number of parameters to be estimated. Some authors, such as Campbell (1993), rely on a negative correlation
between the errors in the equations of actual and expected returns to yield greater
power of the predictability of stock returns over long horizons. Note that the mutual
independence among 6, e, and n is not contradictory to the negative correlation in
the error terms of actual and expected returns. To see this, denote rM as a oneperiod
stock return hour the end of t to the end of t+l, and denote w, = E,[r,+,] as an
expected oneperiod return at the end of t.l It follows that: r at =7 “1 zt+ e h! = W: + etaI + "(+1 + Ybﬁl! WM = Er+,[rt+2] = Ell": ‘ 2m + 8n = (¢  1)¢Z, + (<l> + l  DYly, + (¢  I)?!“ + (¢ + A  1)Y5..1. (3) where n,1 is the actual stock return realized from t to 1+1. Because ((1)  1) is
negative (d) < 1), it is possible the error terms in actual returns and expected returns
are negatively correlated. 'In Campbell (1993), E,[r,,,] is expressed as 1:, Copyright © 2001 All Rights Reserved 526 The Journal of Financial Research Let xH =y, and EH = 6,. Rewriting the above statespace model (1) in terms
of stock returns, we obtain: rt = z: ' ztl + at" (4)
z! 4) Y szl 11: = + . (s)
x: o A xIl E: This model is estimated with stock market data in section HI. Implications of State Variables z and x (or y) Because p, is the logarithm of the stock price at time t, the continuously
compounded return from t to t+k is: rtﬁk = phk — pt = [an _ qt] + [21¢  23] The state variable x, has an effect on stock returns because of its effect on the
transitory price component 2,. Because the changes in the randomwalk and
transitory components are independent, we have Pa“) 5 Convmk’ rtlu) sznl:  21) :0 z —z,z “Z ——_—.
01T( 1+1; 1 t N) Var(zt+k  2,) + Var(q,+i ' qt) (7) This means the sign of the ﬁrstorder autocorrelation of kperiod stock return r,”
' is the same as the ﬁrstorder autocorrelation of kperiod changes in the transitory
price component 2,. The ﬁrstorder autocorrelation of kperiod changes in 2,, or the
slope in regression of z,“ — 2, on 2, — 2H, is Cov(z,+k  z,,z,  zH) p’a" = we” — 2,) (8) The numerator covariance is C°v(zt+k " zt’zt — 2H) = ‘0: + 2C°V(z:+k'zt) " C°V(zt+2bz¢)9 (9) Copyright © 2001 All Rights Reserved A StateSpace Model 527 and the variance in the denominator is Var(z,,,,  2,) = 20:  2Cov(z,+k,z,). (10)
It is straightforward to show:
I: 2 A‘12 H I k—l—i 2
Cov(z,,,,,z,) = «9 a2 + l _ M) Mb 0,, (11)
where
2
2 0:
ox = Var(x,) = 1 AZ, (12)
and
2
__ _ l 2 2 2 72°: 2
o,Var(z,)l—¢2[y x+l_M)+ n] (13) Substituting equation (1 1) into equation (9) yields C°v(zt+k " zt’zt " 21) = '(l — dﬂzoi A 2 k" Zkl
+ Y 22 Aid>klI _ 2 “bu—H 0:. (14)
1 ‘ M) H) (:0 ‘ The second term on the righthand side of equation (14) is attributed to the
variation in the state variable x,. It is easy to verify that this term is nonnegative if
0 < A < 1 and 0 < d) < 1. As a result, the autocorrelation of stock returns could have
a positive sign if this term is large enough. Without this term, stock returns would
be always negatively autocorrelated. Properties of ShortHorizon Stock Returns _ In considering the autocorrelation of shorthorizon stock returns, we look
at the autocorrelation of oneperiod stock returns. It follows from equation (4) and
equation (5) that the oneperiod stock return r,+1 = rml satisﬁes rm = zhl — z: + 8M = '(l ' ¢)zt + 71x: + nu»! + YEM' Copyright © 2001 All Rights Reserved 528 The Journal of Financial Research From section II, we have l =___—.._.__
p,() Vail“ _ I)
M1 ¢)+(1 4)) ,2 _ 1¢ 7202,, 2¢vzoi +02
= 1—1:!) " 1+¢ ‘ 14¢ ". (15) sznl — r) This implies p,(l) > 0 or the oneperiod stock returns are positively autocorrelated
as long as inequality (16) holds, [W)J>[ﬂ)[yzé+ 2¢72°i+oi]. (16)
12.4) " 1+4: ‘ 114) " Because 0 < A < 1 and 0 < d) < 1, both sides on equation (16) are positive.
However, if 4) goes to one, the righthand side must go to zero. As a result, the
inequality in (16) will hold when d), the mean reversion speed of the transitory price
component 2,, is sufﬁciently close to one. On the other hand, if either A or y is close
to zero, the inequality in (16) will be reversed and the oneperiod stock returns will
be negatively autocorrelated. Because the autocorrlation of stock returns is determined by changes in the
temporary price component 2,, the following expression may offer some insights into
the sign of the autocorrlation of stock returns: 21+] — z: = _(l  4))2: + Yx: + 11m = (1 _ (by; + 1 + 111+] + anl' If 4) is close to one, we have the approximation zM  z, = yx, + 1],”. The sign of the
autocorrelation of 2M — 2, will be the same as that of x,, which is positive (A > 0). If
y or A are close to zero, however, we will have the approximation zM = (b2, + n,”
+ y€,. That is, 2, can be approximately expressed as a univariate meanreverting
process. Oneperiod stock returns are therefore negatively autocorrelated.
Intuitively, equation (17) shows that the state variable x can lead to a positive
autocorrlation of shortterm stock returns because a high .acH implies not only a high
r,, but also a possible high x, (because of the positive correlation between xH and x,)
and, as a result, a possible high rm. Copyright © 2001 All Rights Reserved A StateSpace Model 529 Properties of LongHon'zon Stock Returns Because 0 < A <1 and 0 < d) < 1, equation (1 1) implies that as k ° 41»,
Cov(z,,,, 2,) ~ 0. In other words, because 2, is mean reverting, the current innovation
in 2, does not have a persistent effect on itself or on future stock prices. It follows immediately from (9) and (10) that . 2
hm Cove”  2,,2,  z”) = —02 k  a
and that
Em Var(2",,  ,) = 203.
As a result
Hm plat) = m Cov(z”  z,,z,  2”)
kn: kn» Var(z,,,r — 2,)
= 0.5. (19)
Using equation (7), we have
Var(z +  2
pa) = p,(k) ”‘ ’ va:(zt+k — Z) + quuk  qt) 203  2Cov(z,,,,,z,) = 9:00 20, — 2Cov(z,,,,,z,) + Itoe (20) If a: = 0, then limb, p(k) = 0.5; otherwise, lirnk... p(k) = 0. The asymptotic properties of p,(k) and p(k) in the current model are the
same as those in the model of Fama and French (1988a). The intuition is
straightforward: because both 2, and x, are meanreverting processes, state variable
x, does not have a permanent effect on stock prices. The effect of x, on the
autocorrelation structure of asset returns fades away as the return horizons become
long. These results suggest that although the twovariable statespace model of
stock prices is parsimonious, it offers rich dynamics at both short and long horizons.
Figures I and II illustrate several possible ﬁrstorder autocorrelation curves implied
by the model. These ﬁgures show the model can capture a wide range of autocorrelation patterns. Copyright © 2001 All Rights Reserved 530 The Journal of Financial Research ' o 20 40 so so 100 120 140 160 —=¢=09,7=5; = ¢=137=1;= ¢=09a‘7=0 Note: The ﬁgure plots autocorrelation curve p(k) = Comm“, rm) based on the following parameter values:
oE= l,on=2,o,=0,l=0.7. Figure 1. Asset Return Autocorrelation Without a Permanent Price Component. Economic Intuition of the StateSpace Model At ﬁrst glance, the statespace model (1) is a simple variation of Fama and
French (1988a). If y = 0, the model will be reduced to Fama and French’s model.
The inclusion of the additional state variable x, (i.e., y a 0) provides the model with
additional ﬂexibility to ﬁt stock returns over both long and short horizons. Like the
model of Fama and French or Poterba and Summers (1988), the model in (1) seems
ad hoc in the sense that it is originally motivated by the observations of historical
data. However, the model has economic intuition. For example, the state variable y, in model (1) may be related to transitory
market factors such as noise (e.g., Campbell and Kyle (1993)) and feedback trading
(Culter, Poterba, and Summers (1990), De Long et al. (1990)). Campbell and Kyle
(1993) show that under certain assumptions on dividend processes and information
variables, stock returns are positively autocorrelated over short horizons. This
positive autocorrelation is strengthened by noise. Culter, Poterba, and Summers (1990) and De Long et al. (1990) argue that
positivefeedback trading strategies may contribute largely to the positive
autocorrelation of shorthorizon asset returns. Feedback speculators buy assets after Copyright © 2001 All Rights Reserved A StateSpace Model 531
0.1
.
0.1 0.2 M) o.3 o.4 .0‘50 20 40 so a: 100 120 140 160 '
—:¢=0.9,'y=5; —: ¢=.9,'y=0 Note: The ﬁgure plots autocorrelation curve 90:) = Coer rm) based on the following parameter values:
o¢= 1, a‘=2, a,= l,A=0.7. Figure II. Asset Return Autoeorrelation with a Permanent Price Component. price increases and sell after price declines. This moves the price even further in the same direction. However, because feedback trading is a shortrun behavior, it does
not have a permanent effect on stock prices. In other words, the stock price should go back to its fundamental value over long horizons. As a result, shorthorizon stock returns are positively correlated but longhorizon stock returns are negatively
correlated. De Long et al. (1990) present a feedback model that consists of four periods
and three types of investors: feedback traders, informed rational speculators, and
passive investors. The feedback traders’ demand is proportional to price change in
the previous period. If the price has risen, they buy; if the price has fallen, they sell.
Feedback trading tends to move the stock price in the same direction. Passive
investors’ demand for assets follows a simple rule: if the price is higher than the
investors’ perceived asset fundamental value, they buy; otherwise, they sell. Passive
investors are so named because they do not incorporate expected future movement
in the stock price in their portfolio decisions. This strategy drags the stock price to
its fundamental value. Informed rational speculators, who determine their demand
based on their information and trading strategies of other investors, are utility Rational speculators can either stabilize asset prices or destabilize asset Copyright © 2001 All Rights Reserved 532 The Journal of Financial Research prices, depending on the model parameters. The trading strategies of informed
speculators and feedback traders are listed by period as follows. Period 0 is a reference period. No signals are received by investors and no
trading takes place. As a result, the stock price is set at its initial fundamental value,
assumed to be zero in the model. In period 1, informed rational speculators receive a (noisy) signal about
period 2 fundamental news. Suppose the signal suggests a high fundamental value
for the stock. Then, speculators bet on the fundamental being high in period 2 and
drive the period 1 price above zero; this in turn raises positive feedback traders’
demand for the asset in period 2 no matter what happens in period 2. In period 2, the price rises ﬁrrther because of positive feedback demand. The
rational speculators unload their positions or even sell the stock short as positive
feedback demand keeps the price higher than its fundamental value. In period 3, the total payoff to the stock is publicly known, and investors receive their payoffs. Because the payoff is known, rational investors pin the stock
price to its fundamental value, which is the ﬁnal payoff of the stock in this period. This analysis demonstrates that the feedback trading generates a positively
correlated component in the stock price, as captured by y, in expression (1). The
price effect of passive investors and informed rational speculators can be
summarized as the component 2, in expression (1). In that expression, 2, is also a
function of y,. This is because the current size of feedback trading provides a signal
on future demand of feedback traders as suggested by the AR(1) process of y, In the
presence of positivefeedback investors, it might be rational for speculators to jump
on the bandwagon and not buck the trend. Rational speculators who expect some
future buying by noise traders buy today in the hope of selling at a higher price
tomorrow. In model (1), qt can be interpreted as the stock’s fundamental value. Model (1) may also be interpreted in terms of timevarying risk premiums.
As argued by Poterba and Summers (1988), transitory components in stock prices
always imply variation in exante returns. Variation in exante returns could be
explained by changes in interest rates and risk premiums. If one interprets 2, as a
map of risk premium at time t, then y, can be naturally interpreted as a state variable
(say consumption habit of investors as in the habitformation model of
Constantinides (1990)) that affects investors’ preferences at time t. Ill. Empirical Results for CRSP Indexes [rtSample Forecasts We now evaluate the statespace model empirically. For the sake of
comparability with the previous literature, we use a standard data set: the Center for Copyright © 2001 All Rights Reserved A StateSpace Model 533 TABLE 1. Summary Statistics of Equally Weighted Monthly CRSP Nominal Stock Returns. k (Months)
1 6 12 24 36 48 60 1927—94 Autocorrelation 0.164 0.030 0.030 0.172 0.290 o.465 0.482 (Std. error) (.035) (.070) (.103) (.133) (.123) (.113) (.121)
194694 Autocorrelation 0.166 —0.047 o.100 0.208 0.053 0.197 o.411 (Std. error) (.041) (.084) (.116) (.152) (.190) (.239) (.240)
_______________ Note: The table reports the ﬁrstorder autocorrelations of stock returns at various time horizons. The standard
errors of the autocorrelations are adjusted for the overlap of observations on longer horizon returns following
the method of Hansen and Hodrick (1980). Research in Securities Prices (CRSP) equally weighted monthly returns.2 The data
set used here runs ﬁ'om 1926 to 1994, but we reserve the ﬁrst year to construct the
dividendprice ratios so that our sample period is from January 1927 to December
1994. All returns are transformed into logarithms and are expressed in monthly
percentage points. Because the CRSP data set is well documented, we do not discuss this data
set in detail. For the sake of comparison, we report some relevant summary statistics
of the ﬁrstorder autocorrelation of stock returns over various horizons in Table 1.
In the table, krepresents the period length (in months) and rw denotes the kperiod
compounded return (in logarithm) from t to Hk. Both series of stock returns in Table 1 show the typical characteristics of
autocorrelation mentioned in the introduction: positive over short periods but negative over longer horizons. Similar to what is reported in Fama and French
(1988a), the negative autocorrelation of longhorizon returns in the postwar period is typically not as strong as that of returns in the prewar period and full sample
period, but the hypothesis that the autocorrelation of the prowar stock returns and
the postwar returns are equal cannot be rejected (Fama and French (1988a, p. 25 7)). To evaluate the model more carefully, we apply the model not only to the
data in the full sample period, 1927—94, but also to the data in the postwar period,
1946—94. Because we do not ﬁnd signiﬁcant differences between the results for the
postwar period and the results for the full sample period, the following discussion
focuses on the results for the full sample period. 2Both valueweighted and equally weighted US. stock returns are positively autocorrelated over short
investment horizons but negatively autocorrelated over longer investment horizons. We use the equally weighted returns because these returns show a stronger pattern of autocorrelations. Similar results are
obtained if the model is applied to the valueweighted returns. Copyright © 2001 All Rights Reserved 534 The Joumal of Financial Research TABLE 2. Parameter Estimates of the StateSpace Model. rr=zr'zrl +8: [::]=(‘::][::]+[2:}' d) y A a, 0,.
1927—94 0.186 7.205 0.973 0.801 0.302
(.039) (.274) (.012) (1.83) (.980)
1946—94 0.181 5.145 0.974 0.011 0.053
(.041) (.150) (.010) (.028) (.030) _______—__—————————— Note: The parameters reported in this table are estimated with equally weighted monthly CRSP nominal
stock returns. Here, r, is the stock returnrealized from time t—l to time t, and (x, 2,) are two state variables
that characterize the movement of the temporary component of the stock price. Standard errors are in
parentheses. The standard deviation of £, 05, is normalized to 1. If x, is an unobservable state variable, it cannot be identiﬁed in the
statespace model without a normalization. For this reason, the state variable x, is
normalized so that 0% = 1. The normalization is a change in scale and does not have
any effect on the dynamics of stock prices.3 The statespace model is estimated by the Kah‘nan ﬁlter approach using
monthly observations of US. stock returns. A general discussion of this approach
can be found in Harvey (1989). The estimated parameters of the model are reported
in Table 2. The large estimated values of y and A signal that state variable x, plays
an important role in characterizing the stochastic movements in stock prices. To assess the ability of the statespace model to ﬁt the historical data, we
use the following criteria: (a) forecasts that ﬁt the important properties of the
observed data such as the autocorrelation of stock returns; (b) conditional forecast
unbiasedness at various horizons, that is, an intercept close to zero and a slope
coefﬁcient close to one for the regression of returns on the corresponding forecasted
returns; (c) high goodnessofﬁt; and (d) high informational efﬁciency, that is, the
forecasts incorporate important information available to investors such as dividend
yields. Table 3 presents the implied autocorrelation of the model. The implied
autocorrelation matches the sample autocorrelation in Table l for both shorthorizon
returns and longshorizon returns. To illustrate this point more clearly and more
intuitively, Figure HI exhibits the implied autocorrelation and the corresponding ’The same dynamics of stock prices are obtained if one chooses another normalization, say a: = 2, or
5, or any other positive number. Copyright © 2001 All Rights Reserved A StateSpace Model 535 TABLE 3. Implied Autocorrelaﬂons of Equally Weighted Nominal Stock Returns. 1: (Months) 1 6 12 24 36 43 so
1927—94 0.164 o.o43 o.123 o.2so o.3o4 o.3se o.393
194694 0.166 —o.041 o.120 0.226 o.3oo o.354 o.393 Note: The implied statistics are functions of the estimated parameters of the model. Autocorrelatioan —peﬂodrehrme Worms) —: Autocorrelations Implied by Model; — —: Sample Autocorrelations;
  —: 95% Conﬁdence Interval of Sample Autocorrelations; Note: The ﬁgure plots moorrelation curve p(k) = Corr(r,,,,, r,_,) based on the model and CRSP equally
weighted nominal returns (1927—94). Figure 111. Implied and Sample First—Order Autocorrelations ol‘ IrPerlod Stock Returns. sample autoeorrelation for equally weighted CRSP monthly returns. The implied
autocorrelatipn curve and the sample autocorrelation curve are close to each other,
and the implied autocorrelation always lies in the 95 percent conﬁdence intervals of
the sample autocorrelation. Expression (21) investigates the relation between realized return r,M and the
corresponding extracted expected return rIH: rmk = bo + 1:17;}, + amt. (21) Copyright © 2001 All Rights Reserved 536 The Journal of Financial Research TABLE 4. Estimates of Regressions of Realized kPeriod Returns on Corresponding Forecasted
Returns: 0
r,M=bo+blr,M+uw 11: (Months) 1 12 24 36 48 60 Panel A. Equally Weighted Nominal Returns, 1927—94 b, 0.154 0.165 0.141 0.114 0.115 0.124
(.257) (.272) (.230) (.201) (.177) (.146)
b. 1.010 1.040 1.119 1.131 1.221 1.240
(.184) (.452) (.242) (.176) (.169) (.152)
pvalue 0.359 0.760 0.800 0.680 0.330 0.130
R’ 0.036 0.092 0.167 0.221 0.288 0.348 Panel B. Equally Weighted Nominal Returns, 1946—94 b.) 0.095 0.115 0.126 0.122 0.127 0.130
(.214) (.194) (.168) (.149) (.135) (.121)
bI 1.018 1.033 1.038 1.038 1.122 1.350
(.213) (.355) (.240) (.180) (. 169) (.230)
pvalue 0.819 0.840 0.760 0.710 0.470 0.080
R2 0.038 0.103 0.182 0.244 0.331 0.455 Note: r“... is the realized kmonth return ﬁom the end of month t to the end of month 1+]: and ﬂ” is the
corresponding expected return extracted ﬁom the model. The numbers in parentheses are standard errors
adjusted for the residual autocorrelation due to overlap of observations and for heteroskedasticity following
the methods of Hansen and Hodrick (1980), White (1980), and NeweyWest (1987). The pvalue is
associated with the joint null hypothesis b0 = 0 and bl = 1. If ﬂ” is an unbiased forecast of rm,” one would expect that b0 = 0 and bl = l in
equation (21). The regression results of equation (21) are presented in Table 4. The Rz’s
for longhorizon returns are high. For example, over a ﬁveyear horizon, the R2 of
the regression can reach 40 percent or higher. Also, a 4 percent R2 for onemonth
returns is higher than the typical R2 values obtained by vector autoregression
(VAR) models with several variables. More important, Table 4 shows that the
constant intercept terms are close to zero and that the slopes of the expected returns
in the regression are close to one. The joint hypothesis that bo = 0 and bl = 1 cannot
be rejected at any conventional signiﬁcance level according to the pvalues reported
in Table 4. This is what is expected with a good model. Table 4 shows that the regression R2 increases monotonically with the time
horizon. This result is consistent with the typical evidence about the predictability
of stock returns (e.g., see Fama and French (1988a, b) and Ferson and Korajczyk
(1995)): the longhorizon stock returns are more predictable. But the implication of
this result is different ﬁ'om that of the VAR model proposed in Campbell (1991). Copyright © 2001 All Rights Reserved A StateSpace Model 537 TABLE 5. The Relation Between Forecast Errors and Dividend Yields. 1: (Months)
12 24 36 48 60
Panel A. Equally Weighted Nominal Returns, 1927—94 p. 0.0251 0.0008 0.0055 0.0336 0.0035
(.2269) (.2213) (0.2005) (0.1743) (0.1442) 0, 0.0086 0.0003 0.0013 0.0013 0.0011
(.0049) (.0031) (.0021) (.0020) (.0007) pvalue 02100 0.9900 0.6200 0.8100 0.1600 R’ 0.0070 0.0000 0.0010 0.0010 0.0010
Panel B. Equally Weighted Nominal Returns, 1946—94 0., 0.0171 0.0008 0.0016 0.0013 0.0028
(.1857) (.1681) (0.1350) (0.1486) (0.1258)
0, 0.0088 0.0004 0.0008 0.0007 0.0015
(.0061) (.0037) (.0024) (.0020) (0.0014)
pvalue 0.3400 0.9900 0.9400 0.9300 0.5000
R1 0.0080 0.0000 0.0000 0.0000 0.0020 Note: The table presents regression results of forecast errors in stockremrns (uw= rm,  5'“) on dividend
Yields (4P1): "m " pa + Bldpn + “we where rwis the realized kmonth return and rip, is the corresponding expected return extracted from the
model. This regression examines if dividend yield, apopularpredictor for stock returns, has additional power
to forecast stock returns. The numbers in parentheses are standard errors adjusted for the residual
autocorrelation due to overlap ofobservations and for heteroskedasticity following the methods of Hansen
and Hodrick (1980), White (1980), and NeweyWest (1987). The pvalue is associated with the joint null
hypothesis [3,  0 and B. = 0. The implied R2 statistics of the VAR model in Campbell is hump shaped, which
peaks around three years and then declines steadily. Since Fama and French (1988b), dividend yields have been widely used in
stock return forecasts. Most researchers ﬁnd that dividend yields have good forecast
power for longhorizon stock returns. If dividend yields contain additional
information on ﬁiture stock returns beyond the expected returns extracted from the
statespace model, the forecast error of the model should be correlated with dividend
yields. Table 5 reports the relation between the forecast error of the model, u,M =
r,“  ﬂ“, and dividend yields (i.e., dividendprice ratios), (11),, based on the
following set of regressions: um; = $0 + Mp. + row. (22) where r1“, represents the expected return and forecastied return extracted from the
model. The table does not show any evidence of a signiﬁcant relation between Copyright © 2001 All Rights Reserved 538 The Journal of Financial Research forecast errors and dividend yields. The high p—values for the joint null hypothesis
I30 = 0 and B1 = 0 suggest the null hypothesis cannot be rejected. Altogether, the
evidence shows that the forecasts based on the Kalman ﬁlter model subsume the
information in dividend yields, the most popular predictor for stock returns. OutofSample Forecasts The results in the previous subsection offer strong evidence that the two
statevariable model characterizes the patterns of short and longhorizon stock
returns. To check the stability of these results, further testing is in order. One
approach is to use the model to forecast outofsample returns.4 This approach takes
a subsample of all available observations to generate a series of return estimates for
the next period. The forecasted estimates are then compared in a regression with the realized observations to test the goodnessofﬁt of the model. An initial thirtyyear
period of observations from 1927.1 to 1956.12 is reserved to generate return estimates starting from 1957.1. Each estimated forecast is made conditioning on all
the data for the thirtyyear period and for all the observations that immediately
precede the period being estimated. To be more precise, the outofsample forecast
of the stock return rm,” denoted as h”, is estimated with all available data up to date
t. For example, to forecast the return in 1960.01, we use coefﬁcients and state
variables estimated with monthly returns for 1927.01 to 1959.12. By this method,
we can obtain a series of outofsample forecasts in“.
We use the following regression, rm]: = bo + blfmk + em” (23) to test the goodnessofﬁt of outofsarnple forecasts. This regression is similar to
equation (21), which regresses the expost returns on the insample forecasts of
stock returns. Table 6 reports the regression results. These results are as strong as the
insarnple results reported in Table 4. The Rz’s are high, the tratios of intercepts b0
are low, the slope coefﬁcients bl are close to l, and the p—values for the joint null
hypothesis bo = 0 and bl = 1 are high. In summary, the outofsample forecasts
support the earlier ﬁnding that the twostatevariable model captures important
patterns of stocks returns. “The data of stock returns used in outofsample forecasts are not demeaned because the mean of the
whole sample (1927—94) is not observed for the outofsample study. As an alternative, we allow for a drift
din stockremrns: r,=d+z, z,_l +cr ’I‘hevariabledisestimatedinthe samewayasotherparameters, such
as o and 7. Copyright © 2001 All Rights Reserved A StateSpace Model 539 TABLE 6. Estimates “Regressions of Realized kPerlod Equally Weighted Nominal CRSP
Returns on Corresponding OutofSample Forecasts of the StateSpace Model: rw= b, + bi,” + cw.
1: (Months)
1 12 24 36 48 60 b" 0.033 0.199 0.208 0.134 —0.025 —0.359 (.380) (.511) (.378) (.294) (.271) (0.305)
bI 1.016 0.812 0.806 0.857 1.000 1.304 (.246) (.494) (.367) (.302) (.281) (.261)
pvalue 1.000 0.930 0.860 0.890 0.990 0.460
R2 0.036 0.064 0.106 0.152 0.213 0.328 Note: rm.Mt is the realized kmonth retum from the end of month! to the end of month 1+]: and Fm, is the
corresponding outofsample forecast. The numbers in parentheses are standard errors adjusted for the
residual antocorrelation due to overlap of observations and for heteroskedasticity following the methods of
Hansen and Hodrick (1980), White (1980), and NeweyWest (1987). The pvalue is associated with the joint
null hypothesis bo = 0 and bI = 1. Sample period is 1927.01—1994.12. Comparing with VAR Models in Terms ofOutoflSample Forecasting Ability The previous discussion shows the strong ability of the statespace model
(1) in matching the autocorrelations of historical stock returns and making both
insample and outofsample predictions of stock returns. As mentioned in the
introduction, some VAR models are also successﬁrl in generating both shortterm
positive and longterm negative autocorrelations of historical stock returns.
However, the outofsample forecasting ability of VAR models is not well studied.
For the sake of comparison, we brieﬂy discuss this issue. We consider a ﬁrstorder VAR model with three popular predicting
variables that is similar to Campbell (1991). The ﬁrst variable is the onemonth stock return r,, the second is an interest rate variable (onemonth Tbill rate) i,, and
the third is the dividendprice ratio dp,. We follow the strategy used in the previous subsection to construct the
outofsample forecasts i'm, with the VAR model. We then regress the observed
stock returns on the forecasted stock returns using the same regression equation as
(23). The regressed result is reported in Table 7. As mentioned earlier, the basic requirement for a good forecasting model
is that bo = 0 and bl = l in regression (23). Based on Table 7, the VAR model seems
to have limited ability for outofsample forecasts. The estimated coefﬁcient is close
to one only for onemonth returns, for which the regression R2 is low. For longterm
returns, the estimated coefﬁcient bl has the wrong predicting sign. A comparison between Table 7 and Table 6 suggests the statespace model
(1) has superior outofsample forecasting ability. This may further justify adding
such a model to the existing literature. Copyright © 2001 All Rights Reserved 540 The Joumal of Financial Research TABLE 7. Estimates of Regressions of Realized kPerlod Equally Weighted Nominal CRSP
4 Returns on Corresponding OutofSample Forecasts of the VAR Model: rW= b°+ b,i,,,,+e,,,, 1: (Months)
1 12 24 36 48 60 b0 0.325 1.591 1.783 1.570 1.427 1.423 (.748) (.629) (0.387) (.294) (.307) (0.268)
bl 1.126 0.861 1.l83 —0.896 —0.735 —0.780 (.898) (.790) (0.407) (.203) (.170) (.137)
pvalue 0.330 0.040 0.000 0.000 0.000 0.000
R’ 0.008 0.035 0.155 0.164 0.165 0.231 Note: rw, is the realized kmonth return from the end of month t to the end of month 1+]: and 59,, is the
conesponding outofsample forecast based on the vector autoregression (VAR) model. The numbers in
parentheses are standard errors adjusted for the residual autocorrelation due to overlap of observations and
for heteroskedasticity following the methods ofHansen andHodn'ck(l980), White (1980), and NeweyWest
(1987). The pvalue is associated with the joint null hypothesis b, = 0 and bI = 1. IV. Applying the Model to Other Returns The preceding section shows that the statespace model does well when
applied to the returns on the CRSP market index. We now apply the model to the
returns of different size portfolios. Following Fama and French (1988a) and Conrad
and Kaul (1988), we group stocks in the CRSP data set into ten (decile) portfolios.
At the end of each year, stocks are sorted into these deciles based on size (shares
outstanding times price per share), with the smallest capitalizations placed in decile
1 and so on. Onemonth portfolio returns, with equal weighting of securities, are
calculated and transformed into continuously compounded returns, expressed in
monthly percentage points. As shown in Table 8, the autocorrelation structure of decile portfolios
displays a consistent pattern when moving from the smallest portfolio (decile l) to
the largest portfolio (decile 10): the magnitude of autocorrlations at both short and
long horizons declines monotonically. That is, the smaller portfolio returns not only
have larger positive autocorrelations over short horizons, but also have higher
negative autocorrelations over long horizons. This pattern is consistent with the
results of Fama and French (1988a) and Conrad and Kaul (1988). This pattern may
also explain why the equally weighted market ‘ portfolio displays higher
autocorrelations than does the value—weighted portfolio (Fama and French (1988a),
Poterba and Summers (1988)): the former is tilted more toward small stocks. Table 9 reports the implied autocorrelations of decile returns. Generally, the
implied autocorrelations ﬁt the observed autocorrelations of decile portfolios. In
particular, the implied autocorrelations display the pattern displayed in observed Copyright © 2001 All Rights Reserved A StateSpace Model 541 TABLE 8. Summary Statistics of Monthly CRSP Nominal Deeile Portfolio Returns. 1: (Months)
1 6 12 24 36 48 60 Panel A. 1927—94 Decile 1 0.187 0.006 0.034 0.141 0.317 0.565 0.589
(Standard error) (.035) (.060) (.084) (.150) (.116) (.1 19) (.1 19)
Decile 6 0.147 0.045 0.024 0.126 0.219 .0352 0.355
(Standard error) (.035) (.078) (.131) (.124) (.100) (.084) (.090)
Decile 10 0.103 0.080 0.045 0. 188 0.175 —0. 161 0.082
(Standard error) (.035) (.099) (.154) (.093) (.082) (.104) (.090)
Panel 8. 194694 Decile 1 0.202 0.054 0.017 —0.155 0.l90 —0.464 —0.647
(Standard error) (.040) (.086) (.095) (.140) (.152) (.155) (.179)
Decile 6 0.151 0.036 0.102 0.164 0.001 0.112 0.310
(Standard error) (.041) (.081) (.114) (.121) (.231) (.207) (.172)
Decile 10 0.042 —0.038 0.185 0.259 0.137 0.138 0.216
(Standard error) (.041) (.080) (.101) (.063) (.120) (.171) (.150) TABLE 9. Implied Autoeorrelations of CRSP Declle Portfolio Returns.
——————_____—_—__—__ 1: (Months)
1 12 24 36 4s 60 Panel A. 1927—94 Decile 1 0.186 0.113 0.217 0201 o.345 —o.3s4
Decilc6 0.145 —o.oso 0.163 o.227 —0.278 0.320
Decile 10 0.103 0.o72 —o.144 —0.2o2 o.2s1 —0.291
Panel 13. 194594 Decilc 1 0.203 0.113 0.219 0.293 —0.348 —0.387
Decile6 0.150 0.107 0.205 0.276 —0.329 .0370
Decile 10 0.042 0.064 o.124 0.176 —o.210 0.239 autocorrelations: the magnitude of autocorrlations at both short and long horizons
declines monotonically with size. Table 10 presents the regression result of realized returns on the
corresponding forecasts extracted from the model. The result is similar to that for
the equally weighted index reported in Table 3. The extracted forecasts explain a
substantial portion of variance in realized portfolio returns, especially in smaller Copyright © 2001 All Rights Reserved 542 The Jouma! of Financial Research TABLE 10. Estimates of Regressions of Realized kPeriod Returns on Corresponding Forecasted
Returns: .
rw=bo+blnﬂ+uw _____________—_———
1: (Months) 1 12 24 36 48 60 ____________—______——————
Pancl A. Decile 1, 192794 —________—_—__———————————— bo 0.028 0.041 0.021 0.01 1 0.032 0.062
(.371) (.349) (.330) (.302) (.272) (.238)
bl 0.997 0.981 1.121 1.163 1.286 1.300
(.164) (.352) (.307) (.261) (.302) (.325)
p—value 0.994 0.990 0.920 0.820 0.640 0.640
R2 0.043 0.090 0.176 0.245 0.325 0.374 ________________.___—_——————
Panel B. Decile 6, 1927—94 bo 0.098 0.099 0.075 0.053 0.055 0.065
(.254) (.254) (.230) (.213) (.135) (.172)
bl 1.004 1.004 1.020 1.023 1.073 1.071
(.212) (.456) (.289) (.266) (.258) (.245)
pvalue 0.863 0.890 0.950 0.970 0.920 0.870
R2 0.027 0.056 0.098 0.135 0.175 0.211 Panel C; Decile 10, 192794 b.) —0.051 0.062 0.073 0.088 0.089 0.079
(.195) (.167) (.171) (.166) (.162) (.151)
bl 0.994 0.961 0.962 0.948 0.955 0.978
(.286) (.395) (.333) (.392) (.405) (.369)
pvalue 0.932 0.930 0.860 0.730 0.700 0.750
R2 0.015 0.041 0.071 0.096 0.1 19 0.145 ___________________————
Panel D. Decile 1, 1945—94 _________.__——_———_——— 110 0.043 —0.050 0.041 0.046 0.044 o.050
(.307) (.349) (.330) (.242) (.218) (.238)
b, 0.999 0.959 1.115 1.181 1.270 1.371
(.181) (.313) (.278) (.296) (319) (.355)
pvalue 0.980 0.970 0.910 0.820 0.690 0.580
11' 0.049 0.087 0.174 0.249 5 0.335 0.411 Panel B. Decile 6, 1946—94 b, 0.065 0.084 0.099 0.099 0.108 0.110
(.219) (.198) (.175) (.162) (.152) (.140)
b, 1.018 1.023 1.012 1.035 1.123 1352
(.233) (.394) (.290) (.223) (.191) (.210)
pvalue 0.910 0.910 0.840 0.830 0.680 0.110
R’ 0.031 0.090 0.158 0220 0.301 0.417
(Continued) Copyright © 2001 All Rights Reserved A StateSpace Model 543 TABLE 10. Continued. 1: (Months)
1 12 24 36 48 60 Panel F. Decile 10, 194694
6, o.060 0.050 0.016 0.000 0.010 —0.004 (.175) (.140) (.114) (.105) (.101) (.119)
b, 1.096 1.071 0.841 0.705 0.660 0.891 (.575) (.490) (.326) (329) (.369) (.391)
pvalue 0.880 0.920 0.890 0.660 0.630 0.891
R’ 0.006 0.045 0.060 0.066 0.076 0.132 Note: rw is the realized kmonth return and 7:“ is the corresponding expected return extracted from the
model. The numbers in parentheses are standard errors. The pvalue is associated with the joint null
hypothesis b. = 0 and b, = l. capitalizations, which are usually more predictable (Fama and French (1988a),
Conrad and Kaul (1988)). V. Concluding Remarks We propose a parsimonious model with two state variables to characterize
the stochastic behavior of asset returns. We ﬁnd that the time variation of expected returns can be characterized by the structural statespace model, which captures the
autocorrelations of returns over both short and long horizons. Although the forecasts obtained with the statespace model are based solely on past returns, they subsume
the information in other potential predictor variables such as dividend yields. The
statespace model not only predicts the short horizon returns well, but it also predicts
longer horizon returns successfully. Moreover, the extracted expected returns can
explain a substantial proportion of the variation in realized returns. At a horizon of
two to three years, this proportion reaches about 20 percent to 25 percent; at a
horizon of ﬁve years, the proportion can reach 40 percent or higher. The model
successﬁilly captures the fact that returns of smaller ﬁrms have higher positive
autocorrelations at short horizons and stronger negative autocorrelations at long
horizons. The ﬁndings in this article provide implications for asset allocation and
option pricing because they depend on expected returns and risks. A better
understanding of the behavior of stock returns (and therefore related risks) at long
and short horizons should help in the evaluation of these models. Future research is
needed to explore these implications. Copyright © 2001 All Rights Reserved 544 The Journal of Financial Research References Campbell, 1. Y., 1991, A variance decomposition for stock returns, Economic Journal 101, 157—79. , 1993, Why long horizon? A study of power against persistent alternatives, NBER Technical Working Paper, No. 142. Campbell, I. Y. and A. S. Kyle, 1993, Smart money, noise trading and stock price behavior, Review of
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