19972-1 - The Journal of Financial Research 0 Vol XXIII...

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Unformatted text preview: The Journal of Financial Research 0 Vol. XXIII. No.4 0 Pages 523-544 0 Winter 2000 A STATE-SPACE MODEL OF SHORT- AND LONG-HORIZON STOCK RETURNS Chunsheng Zhou University of California, Riverside Chang Qing Inner Mongolia Polytechnic University, China Abstract In this article we propose a new parsimonious state-space 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 state-space 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 short-horizon returns and stronger negative autocorrelations of long-horizon returns; and (d) the forecasts of asset returns obtained with the slate-space 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 state-space 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 ex-ante 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 state-space model does not require an a priori specification of the predictive variables. Conrad and Kaul (1988) use a state-space approach to study time variation of expected returns. They focus on the predictability of high-fiequency weekly returns and use a less flexible model than that proposed here. As a result, their model does not fit low-fi'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 long-horizon stock returns, but it does not explain the positive autocorrelation of short-horizon stock returns. A parsimonious state-space 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 long-term 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 long-term returns data to estimate the autocorrelation of long-term stock returns. However, there are not many nonoverlapping observations of multiyear stock returns. My state-space model infers the behavior of long-term stock returns fiom the history of short-term stock returns. As a result, a much bigger nonoverlapping data sample is available to estimate the state-space 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 State-Space 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 long-horizon returns, we Copyright © 2001 All Rights Reserved A State-Space 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 = qt-l + 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 one-period stock return hour the end of t to the end of t+l, and denote w, = E,[r,+,] as an expected one-period return at the end of t.l It follows that: r at =7 “1 -zt+ e h! = W: + eta-I + "(+1 + Ybfi-l! 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 state-space model (1) in terms of stock returns, we obtain: rt = z: ' zt-l + at" (4) z! 4) Y sz-l 11: = + . (s) x: o A xI-l 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 random-walk and transitory components are independent, we have Pa“) 5 Convmk’ rt-lu) 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 first-order autocorrelation of k-period stock return r,” ' is the same as the first-order autocorrelation of k-period changes in the transitory price component 2,. The first-order autocorrelation of k-period 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 State-Space 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" Zk-l + Y 22 Aid>k-l-I _ 2 “bu—H 0:. (14) 1 ‘ M) H) (:0 ‘ The second term on the right-hand 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 Short-Horizon Stock Returns _ In considering the autocorrelation of short-horizon stock returns, we look at the autocorrelation of one-period stock returns. It follows from equation (4) and equation (5) that the one-period 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 one-period stock returns are positively autocorrelated as long as inequality (16) holds, [W)J>[fl)[yzé+ 2¢72°i+oi]. (16) 1-2.4) " 1+4: ‘ 1-14) " Because 0 < A < 1 and 0 < d) < 1, both sides on equation (16) are positive. However, if 4) goes to one, the right-hand 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 one-period 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 mean-reverting process. One-period stock returns are therefore negatively autocorrelated. Intuitively, equation (17) shows that the state variable x can lead to a positive autocorrlation of short-term 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 State-Space Model 529 Properties of Long-Hon'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 mean-reverting 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 two-variable state-space model of stock prices is parsimonious, it offers rich dynamics at both short and long horizons. Figures I and II illustrate several possible first-order 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 —=¢=0-9,7=5; --= ¢=137=1;----= ¢=0-9a‘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 State-Space Model At first glance, the state-space 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 positive-feedback trading strategies may contribute largely to the positive autocorrelation of short-horizon asset returns. Feedback speculators buy assets after Copyright © 2001 All Rights Reserved A State-Space 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 short-run 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, short-horizon stock returns are positively correlated but long-horizon 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 positive-feedback 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 time-varying risk premiums. As argued by Poterba and Summers (1988), transitory components in stock prices always imply variation in ex-ante returns. Variation in ex-ante 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 habit-formation model of Constantinides (1990)) that affects investors’ preferences at time t. Ill. Empirical Results for CRSP Indexes [rt-Sample Forecasts We now evaluate the state-space 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 State-Space 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) 1946-94 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 first-order 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 dividend-price 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 first-order autocorrelation of stock returns over various horizons in Table 1. In the table, krepresents the period length (in months) and rw denotes the k-period compounded return (in logarithm) from t to H-k. 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 long-horizon returns in the post-war period is typically not as strong as that of returns in the pre-war period and full sample period, but the hypothesis that the autocorrelation of the pro-war stock returns and the post-war 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 post-war period, 1946—94. Because we do not find significant differences between the results for the post-war period and the results for the full sample period, the following discussion focuses on the results for the full sample period. 2Both value-weighted 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 value-weighted returns. Copyright © 2001 All Rights Reserved 534 The Joumal of Financial Research TABLE 2. Parameter Estimates of the State-Space Model. rr=zr'zr-l +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 state-space 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 state-space 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 state-space 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 goodness-of-fit; 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 short-horizon 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 State-Space 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 1946-94 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. Auto-correlatioan —peflodrehrme Worms) -—-: Autocorrelations Implied by Model; — —: Sample Autocorrelations; - - —-: 95% Confidence Interval of Sample Autocorrelations; Note: The figure plots moor-relation 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‘ Ir-Perlod 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 k-Period 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) p-value 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) p-value 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 k-month 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 Newey-West (1987). The p-value 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 long-horizon returns are high. For example, over a five-year horizon, the R2 of the regression can reach 40 percent or higher. Also, a 4 percent R2 for one-month returns is higher than the typical R2 values obtained by vector auto-regression (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 p-values 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 long-horizon 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 State-Space 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) p-value 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) p-value 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 k-month 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 Newey-West (1987). The p-value 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 long-horizon stock returns. If dividend yields contain additional information on fiiture stock returns beyond the expected returns extracted from the state-space 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., dividend-price 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. Out-of-Sample Forecasts The results in the previous subsection offer strong evidence that the two- state-variable model characterizes the patterns of short- and long-horizon stock returns. To check the stability of these results, further testing is in order. One approach is to use the model to forecast out-of-sample 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 goodness-of-fit of the model. An initial thirty-year 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 thirty-year period and for all the observations that immediately precede the period being estimated. To be more precise, the out-of-sample 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 out-of-sample forecasts in“. We use the following regression, rm]: = bo + blfmk + em” (23) to test the goodness-of-fit of out-of-sarnple forecasts. This regression is similar to equation (21), which regresses the ex-post returns on the in-sample forecasts of stock returns. Table 6 reports the regression results. These results are as strong as the in-sarnple results reported in Table 4. The Rz’s are high, the t-ratios 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 out-of-sample forecasts support the earlier finding that the two-state-variable model captures important patterns of stocks returns. “The data of stock returns used in out-of-sample forecasts are not demeaned because the mean of the whole sample (1927—94) is not observed for the out-of-sample 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 State-Space Model 539 TABLE 6. Estimates “Regressions of Realized k-Perlod Equally Weighted Nominal CRSP Returns on Corresponding Out-of-Sample Forecasts of the State-Space 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) p-value 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 k-month retum from the end of month! to the end of month 1+]: and Fm, is the corresponding out-of-sample 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 Newey-West (1987). The p-value 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 ofOut-oflSample Forecasting Ability The previous discussion shows the strong ability of the state-space model (1) in matching the autocorrelations of historical stock returns and making both in-sample and out-of-sample predictions of stock returns. As mentioned in the introduction, some VAR models are also successfirl in generating both short-term positive and long-term negative autocorrelations of historical stock returns. However, the out-of-sample forecasting ability of VAR models is not well studied. For the sake of comparison, we briefly discuss this issue. We consider a first-order VAR model with three popular predicting variables that is similar to Campbell (1991). The first variable is the one-month stock return r,, the second is an interest rate variable (one-month T-bill rate) i,, and the third is the dividend-price ratio dp,. We follow the strategy used in the previous subsection to construct the out-of-sample 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 out-of-sample forecasts. The estimated coefficient is close to one only for one-month returns, for which the regression R2 is low. For long-term returns, the estimated coefficient bl has the wrong predicting sign. A comparison between Table 7 and Table 6 suggests the state-space model (1) has superior out-of-sample 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 k-Perlod Equally Weighted Nominal CRSP 4 Returns on Corresponding Out-of-Sample 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) p-value 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 k-month return from the end of month t to the end of month 1+]: and 5-9,, is the conesponding out-of-sample 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 Newey-West (1987). The p-value 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 state-space 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. One-month 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 State-Space 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. 1946-94 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. 1945-94 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 k-Period Returns on Corresponding Forecasted Returns: . rw=bo+blnfl+uw _____________—_——— 1: (Months) 1 12 24 36 48 60 ____________—______—————— Pancl A. Decile 1, 1927-94 —________—_—__——-—————————— 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) p-value 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, 1927-94 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) p-value 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) p-value 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) p-value 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 State-Space Model 543 TABLE 10. Continued. 1: (Months) 1 12 24 36 48 60 Panel F. Decile 10, 1946-94 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) p-value 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 k-month return and 7:“ is the corresponding expected return extracted from the model. The numbers in parentheses are standard errors. The p-value 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 state-space model, which captures the autocorrelations of returns over both short and long horizons. Although the forecasts obtained with the state-space model are based solely on past returns, they subsume the information in other potential predictor variables such as dividend yields. The state-space 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 Economic Studies 60, 1—34. Conrad, J. and G. Kaul, 1988. Time-variation in expected returns, Journal of Business 61, 409—25. Constantinides, G. M., 1990, Habit formation: A resolution of the equity premium puzzle, Journal of Political Economy 98, 519—43. Culter, D. M., J. M. Poterba, and L. H. Summers, 1990, Speculative dynamics and the role of feedback traders, American Economic Review, Papers and Proceedings 80, 63—68. De Long, J. B., A. Shleifer, L. H. Summers, and R. J. Waldrnann, 1990, Positive feedback investment strategies and destabilizing rational speculation, Journal of Finance 45, 379—95. Fama, E. F. and K. R. French, 1988a, Permanent and temporary components of stock prices, Journal of Political Economy 96, 246—73. , 1988b, Dividend yields and expected stock returns, Journal of Financial Economics 22, 3—25. Ferson, W. E. and R. A. Korajczyk, 1995, Do arbitrage pricing models explain the predictability of stock returns?, Journal of Business 68, 309-49. Hansen, L. P. andR. J. Hodrick, 1980, Forward exchange rates as optimal predictors of future spotratcs: An econometric analysis, Journal of Political Economy 88, 829-53. Harvey, A. C., 1989, Forecasting, Structural Time Series Models and the Kalman Filter (Cambridge University Press). Hodrick, R. J., 1992, Dividend yields and expected stock returns: Alternative procedures for inference and measurement, Review of Financial Studies 5, 357—86. Newey, W. K. and K. D. West, 1987, A simple, positive definite, heteroscedasticity and autocorrelation consistent covariance matrix, Econometrica 55, 703—8. Poterba, J. M. and L. H. Summers, 1988, Mean reversion in stock prices: Evidence and implications, Journal of Financial Economics 22, 27—59. Richardson, M., 1993, Temporary components of stock prices: A skeptic’s view, Journal of Business and Economic Statistics 11, 199—207. Richardson, M. and J. H. Stock, 1989, Drawing inferences fi‘orn statistics based on multiyear asset returns, Journal of Financial Economics 25, 323-48. White, H. 1980, A heteroscedasticity-consistent covariance matrix estimator and a direct test for heteroscedasticity, Econometrica 48, 817—38. Copyright © 2001 All Rights Reserved ...
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