Predictability of Stock Market Returns

Transcription

1 Predictability of Stock Market Returns May 3, 23 Present Value Models and Forecasting Regressions for Stock market Returns Forecasting regressions for stock market returns can be interpreted in the framework of the dynamic dividend growth model. The dynamic dividend growth model of Campbell and Shiller(988) uses a loglinear approximation to the de nition of returns on the stock market to express the log of the price-dividend ratio p t d t as a linear function of the future discounted dividend growth,d t+j ; and of future returns,h s t+j : (p t d t ) = m + X j d t+j rt+j s : This basic relation allows to classify virtually all forecasting regression of stock market returns in terms of di erent approaches to proxying the future expected variables included in the linearized relations. The classical Gordon-growth model posits constant dividend growth E t d t+j = g and constant returns E t h s t+j = r:in this case we have p t = k + d t + g g r r The Lander et al.(997) model also known as the FED model can be understood by substituting out the no-arbitrage restrictions in (??) E t h s t+j = E t r t+j + s t+j and then by assuming constant dividend growth and a close relation between the risk premium on long-term bonds and the risk premium on stocks in this case we have: p t = k + d t + g g R t where R t is th yield to maturity on long-term bonds

2 Asness(23) considers the assumption of proportionality between the stock market risk premium and the bond market risk premium as problematic and corrects the FED model with the following speci cation: p t = k + d t + g g R t 2 s t B t where s t B t is the ratio between the historical volatility of stock and bonds. Lamont(24) argues that the log dividend payout ratio (d t e t ) is the most appropriate proxy for future stock market returns to consider the following model: p t = k + d t + g g + (d t e t ) Ribeiro(25) higlight the importance of labour income in predicting future dividends and posits VECM error correction model for dividend growth and future returns with two cointegrating vectors de ned as (d t y t ) and (d t p t ), hence the iplicit equilibrium stock market price is: p t = k + d t + g g + (d t y t ) The empirical investigation of the dynamic dividend growth model has established a few empirical results: (i) dp t is a very persistent time-series and forecasts stock market returns and excess returns over horizons of many years (Fama and French (988), Campbell and Shiller (988), Cochrane (25, 27). (ii) dp t does not have important long-horizon forecasting power for future discounted dividend-growth (Campbell (99), Campbell, Lo and McKinlay (997) and Cochrane (2)). (iii) the very high persistence of dp t has led some researchers to question the evidence of its forecasting power for returns, especially at short-horizons. Careful statistical analysis that takes full account of the persistence in dp t provides little evidence in favour of the stock-market return predictability based on this nancial ratio ( Nelson and Kim (993); Stambaugh (999); Ang and Bekaert (27); Valkanov (23); Goyal and Welch (23) and Goyal and Welch (28)). Structural breaks have also been found in the relation between dp t and future returns (Neely and Weller (2), Paye and Timmermann (26) and Rapach and Wohar (26)). (iv) More recently, Lettau and Ludvigson (2, 25) have found that dividend growth and stock returns are predictable by long-run equilibrium relationships derived from a linearized version of the consumer s intertemporal budget constraint. The excess consumption with respect to its long run equilibrium value is de ned by the authors alternatively as a linear combination of labour 2

3 income and nancial wealth, cay t ; or as a linear combination of aggregate dividend payments on human and non-human wealth, cdy. cay t and cdy t are much less persistent than dp t ; they are predictors of stock market returns and dividend-growth, and, when included in a predictive regression relating stock market returns to dp t ; they swamp the signi cance of this variable. Lettau and Ludvigson (25) interpret this evidence in the light of the presence of a common component in dividend growth and stock market returns. This component cancels out from (??), cay t and cdy t are instead able to capture it as the linearized intertemporal consumer budget constraint delivers a relationship between excess consumption and expected dividend growth or future stock market returns that is independent from their di erence. 2 The Dog that did not bark Consider the following DGP: r s t+ = r + b r dp t + " ;t+ d t+ = d + b d dp t + " 2;t+ dp t+ = dp + 'dp t + " 3;t+ given that we have r s t+ = d t+ dp t+ + dp t b r = ' + b d " ;t+ = " 2;t+ " 3;t+ = In particular, as long as ' is nonexplosive, ' < /.4, we cannot choose anull in which both dividend growth and returns are unforecastable, i.e. in which both b r = and b d =. To generate a coherent null with br =, we must assume an equally large bd of theopposite sign, and then we must address the failure of this dividend growth forecastability in the data. 3 Time varying mean for dp A recent strand of the empirical literature has related the contradictory evidence on the dynamic dividend growth model to the potential weakness of its fundamental hypothesis that the log dividend-price ratio is a stationary process (Lettau and Van Nieuwerburgh (28), LVN henceforth). LVN use a century 3

4 of US data to show evidence on the breaks in the constant mean dp. We report the time series of US data on dp t over the last century in Figure. As a matter of fact, the evidence from univariate test for non-stationarity and bivariate cointegration tests does not lead to the rejection of the null of the presence of a unit-root in dpso far lineariztion of the dividend price ratio has been implemented around a constant. An interesting possibility is to allow for a time-varying mean in dp : Time Series of Log Dividend-Price Ratio d t+ = " ;t+ () dp t+ = ' 22 dp t + ' 23 X t+ + " 2;t+ (2) rt+ s = d t+ dp t dp t+ + dpt dp t + "3;t+ (3) 4 " ;t " 2;t 5 s " 3;t 4@ A ; 5 Where X t+ is a a time-varying variable that determines the mean of dp t+ :In Appendix B of their paper, Lettau&VanNieuwerburgh (28) derive the following log-linear approximation of returns : r s t+ r t+ = dt+ d t+ rt+ s = ln Pt+ +D t+ ; P t dp t = ln Dt P t t dpt+ dp t+ + dpt dp t + dpt+ (4) 4

5 We obtain our equation (??) from (4) by assuming that of the three processes for returns, dividend growth and the dividend-price ratio only the last one is persistent r t = d t = const, that the time-varying mean of the dividend-price ratio is very slowly evolving, i.e. dp t+ and that the linearization parameter is constant, t =. 2 We introduce an error term " 3;t+ to capture the e ect of our approximation 3. The speed of mean reversion towards a constant mean of the dividend-price ratio is very di erent from that of annual real returns and annual real dividend growth. We model this feature of the data by introducing a time-varying mean for the dividend-price ratio, driven by X t+. In practice, without this step it would be very hard to reconcile the time-series properties of dp t with those of rt s and d t+j : By solving eq. (??) forward we obtain: mx j rt+j s = X m dp t dp t + j (d t+j ) m dp t+m dp t+m (5) mx + j (" ;t+j + " 3;t+j ) Eq. (5) clearly shows that deviations of the dividend/price ratio from its equilibrium value at time t have a predictive power for m-period ahead stock market returns (and/or dividend growth) that increases with the horizon, as the larger is m the smaller is the e ect of future noise in the dividend-price ratio dp t+m dp t+m : However, this term cannot be ignored in the computation of the term structure of stock market risk that considers typically horizons from -year onwards. To bring (5) to the data, an observable counterpart of the time varying linearization value for the dividend-price must be considered. Consistently with (??), we assume that the relevant linearization value for computing returns from time t to time t + m is the conditional expectation of the dividend-yield for time t + m, given the information available at time t. We then have 2 Rytchkov (28) estimates a system of equation similar to ours and study how sensitive ML parameters are to variation in this parameter. He concludes that there is almost no sensitivity to the choice of (see Table in his paper). 3 The validity of these assumptions will be subject to a test in the empirical section where we shall evaluate the "restricted" model generated by all our assumptions against an unrestricted one. 5

6 2 mx j rt+j s mx = dpt 4' m 22dp t + u t+m = = ( ' m 22) dp t m X ' j 22 ' 23X t+m+ j ' j 3 22 ' 23X t+m+ j + u t+m mx X m j (" ;t+j + " 3;t+j ) m 5 + u t+m (6) ' j 22 " 2;t+m+ j Note that the speci cation of the model requires that future values of X are used to predict return. One alternative, followed by Lettau and Van Niuewenburgh is to allow for shift in the mean, another possibility, followed by Favero et al., is to make X function of demographic variables. 6

7 Table 3: System Estimation (9-28) dp t+ = ' 2 + ' 22 dp t + ' 23 MY t+ + " 2t+ X m UM: p m X m RM: p m rt+j s = m + m p m dp t + 2m m mx ' j 22 MY t+j rt+j s = m + p m ( ' m ' 22) dp 23 t mx A + u t+m m = ; :::; ' j 22 MY t+j A + u t+m horizon m in years UM m (t stat) 2m (t stat) ' 22 (t stat) ' 23 (t stat) RM ' 22 (t stat) ' 23 (t stat) :52 (:7) :95 ( 6:28) :77 (2:94) :5 (4:2) 2 3 7:28 (:9) :49 (:93) :96 (7:9) :65 (7:39) :86 (7:62) 2 2 6:44 (:) :7 (2:5) :83 (7:54) :76 (24:89) :83 (7:2) :77 (24:24) :83 (7:) :77 (28:2) :84 (6:83) :77 (27:75) :86 (6:77) :78 (26:87) :9 (6:85) :79 (24:96) :94 (6:9) :78 (8:8) :98 (6:95) DepV ar :98 :2 :9 :9 :88 :8 :8 :78 :74 :7 ut+m UM ut+m RM adjr 2 UM : :2 :34 :43 :47 :5 :56 :58 :59 adjr 2 RM :7 :9 :25 :34 :42 :44 :49 :54 :55 :55 Table : The restricted system is estimated using GMM with optimal Newey- West bandwidth selection to compute GMM standard errors. Ann:Unc: is the annualized unconditional standard deviation. Ann: r (m) is the annualized conditional standard deviation of the compounded (over m periods) returns, i.e. our measure of stock market risk. The e ective sample period is

8 Information Middle (4 to 49) to Young(2 to 29) ratio (left scale) 2 Year annualized real US Stock Market Returns (right scale) Figure.2: 2-year real US stock market returns and demographic trends 8

9 3. Spurious Regressions and the predictability of returns at di erent frequencies The evidence for cointegration between price and dividends is not so clear-cut. In fact, the log of the price-dividends ratio is a very persistent time series and the possibility that it contains a unit root cannot be rule-out a priori. As a matter of fact we have used in our empirical analysis so far the UK dividend price ratio but the evidence from US data speaks less favorably in favour of a mean-reverting (log) of dividend price ratio. A widespread use empirical evidence in favour of the dynamic dividend growth model, that supports the stationarity of the log dividend yield, is the one based on multi-period predictive regressions for stock market returns. The performance of the log dividend yield as a predictor of stock market returns improves as the length of the horizon at which returns are de ned increases. Table 3.4 illustrates this evidence by reporting the performance of predictive regression for stock UK -quarter, -year, 2-year and 3-year stock market returns based on the dividend yield. Table 3.4. Forecasting UK Stock-Market Returns at di erent horizons Dependent variable (h s t+j ), regression by OLS, 973:-2:4 horizon k k R 2 S.E -quarter :34 : (:93) (:28) -year :8 (:6) 2-year :85 (:98) 3-year 2:49 (:2) :3 (:5) :52 (:6) :7 (:6) (h s;uk t+j ) = k k (p t d t ) + " t;t+j k = ; 4; 8; 2 h s t are log total annualized real UK stock market returns. The evidence that long-horizon variables seem to nd signi cant results where short-term approaches have failed, has been questioned. Valkanov(23) argues that long-horizon regressions will always produce signi cant results, whether or not there is a structural relation between the underlying variables. This result depend on the fact that a rolling summation of series integrated of order zero behaves asymptotically as a series integrated of order one and, whenever the regressor is persistent, the well-know occurrence of spurious regression between I() variables emerges. Having established that estimation and testing using long-horizon variables cannot be carried out using the usual regression methods, Valkanov(23) provides a simple guide on how to conduct estimation and inference using long-horizon regressions. The author proposes propose a 9

10 rescaled t-statistic, t/ p T ; for testing long-horizon regressions. the asymptotic distribution of this statistic, although non-normal, is easy to simulate and the results are applicable to a general class of long-horizon regressions. In deriving his correction Valkanov also illustrates that the problem related to spurious regression goes beyond the inadequacy of statistical asymptotic approximation when using overlapping variables. In fact he shows that, even after correcting for serially correlated errors, using Hansen and Hodrick (98) or Newey-West (987) standard errors, the small-sample distribution of the estimators and the t-statistics are very di erent from the asymptotic normal distribution. To illustrate the Valkanov rescaling procedure consider the following DGP: r t+ = + lpd t + t, (7) ( + L) lpd t = + 2t : t 2t = + c T N:I:D:, where the parameter c measures deviations from the unit root in a decreasing (at rate T) neighborhood of. The unit-root case corresponds to c =. The long-horizon variables are r k t+ = r t+ The regression at di erent horizon is run by projecting Zt k on lpd t :The simulation of the relevant distribution requires an estimate of the nuisance parameter c: To this end long-run restrictions implied by the dynamic dividend growth model can be used. As shown in the introductory chapter, the model implies that one-period total return can be approximated as follows: r t+ = + lpd t+ + d t+ lpd t (8) assuming that the log-dividends follows an autoregressive process: by substituting from (9) into (8) we have that lpd t+ = lpd t + u t : (9) r t+ = lpd t + " t+ () " t+ = d t+ + u t : = ( ) where " t+ is a stationary variable and therefore the E t r k t+ = lpd t : The k-period horizon return can then be written as follows:

11 rt+ k k ~ k x t + ~ t+ () " # k = ( ) i i=: Now, we can write k = " ( ) k # (2) If = then: k = k (3) Now, remember that = + c T and we can express k in terms of the total length of the available sample as,k = bt c, from which T k. Then: k = + c k = T lim + c k = e c k! k lim k = lim bt c = e c k! T! + c k (4) k Since we can estimate k consistently, we can also nd a consistent estimate of c by using the transformation: c CONSIST ENT = log ( k) Given the knowledge of c and, the model can be simulated under the null to obtain the critical values of the Valkanov t-statistics. Note that the empirical literature on predictability also cast doubts on the validity of the cointegrating relationships between dividend and prices and different models have been proposed based on alternative cointegrating relationships (see, for example the FED model by Lander et al.(997) or the cay model by Lettau and Ludgvison(24)). The instability of parameter estimates in econometric models has generated alternative approaches based on stationary representations of the return dynamics ( Ferreira and Santa Clara(2)).

12 4 Risk, Returns and Portfolio Allocation with Cointegrated VARs Consider the continuously compounded stock market return from time t to time t +, r t+. De ne t ; the conditional expected log return given information up to time t; as follows: r t+ = t + u t+ where u t+ is the unexpected log return. De ne the k-period cumulative return from period t + through period t + k; as follows: r t;t+k = The term structure of risk is de ned as the conditional variance of cumulative returns, given the investor s information set, scaled by the investment horizon i= r t+i r (k) k V ar(r t;t+k j D t ) (5) where D t fz k : k tg consists of the full histories of returns as well as predictors that investors use in forecasting returns. 4. Inspecting the mechanism: a bivariate case Consider the continuously compounded stock market return from time t to time t +, r t+. De ne t ; the conditional expected log return given information up to time t; as follows: r t+ = t + u t+ where u t+ is the unexpected log return. De ne the k-period cumulative return from period t + through period t + k; as follows: r t;t+k = The term structure of risk is de ned as the conditional variance of cumulative returns, given the investor s information set, scaled by the investment horizon i= r t+i r (k) k V ar(r t;t+k j D t ) (6) where D t fz k : k tg consists of the full histories of returns as well as predictors that investors use in forecasting returns. We illustrate the econometrics of the term structure of stock market risk by considering a simple bi-variate rst-order VAR for continuously compounded total stock market returns, r s t ; and the log dividend price,dp t : 2

13 (z t E z ) = (z t E z ) + t t N (; ) where v;t v 2;t z t = = s r s t dp t ';2 ' 2;2 ; ; E z = Er s E d p The bivariate model for returns and the predictor features a restricted dynamics such that only the lagged predictor is signi cant to determine current returns ' ; = and the predictor is itself a strongly exogenous variable ' 2; =. Given the VAR representation and the assumption of constant V ar t [(z t+ + ::: + z t+k ) j D t ] = + (I + ) (I + ) + (7) from which we can derive: (I ) (I ) + ::: +(I + + ::: + k ) (I + + ::: + k ) r (k) = D i Di k i= D i = I + i i > i = i + i i > D I; I Note that, under the chosen speci cation of the matrix we can write the generic term D i Di, as follows: D i Di M M = 2 M2 (8) M 22 M = ; + ;2 (22) i ;2 + ;2 (22) i ;2 + ;2 (22) i 2;2 (22) i ;2 M 2 = (22) i ;2 + (22) i 2;2 (22) i ;2 M 22 = (22) i 2;2 (22) i 3

14 and where we have used the fact that i = ix j= i B X i = X i j= j 2;2 j= j 2;2 D i = I + i X i B I ;2 X i j= j 2;2 j= j 2;2 Eq. (8) implies that, in our simple bivariate example, the term structure of stock market risk takes the form where 2 r(k) = 2 + 2' ;2 ;2 (k) + ' 2 ;2 2 2;2 2 (k) (9) (k) = k 2(k) = k 2 l= i= 2 l= lx ' i 2;2 k > C A C A 2 lx ' 2;2! i k > i= () = 2 () = The total stock market risk can be decomposed in three components: i.i.d uncertainty, 2, mean reversion, 2' ;2 ;2 (k), and uncertainty about future predictors, ' 2 ;2 2 2;2 2 (k). Without predictability ' ;2 = the entire term structure is at at the level 2. This is the classical situation where portfolio choice is independent of the investment horizon. The possible downward slope of the term structure of risk depends on the second term, and it is therefore crucially a ected by predictability and a negative correlation between the innovations in dividend price ratio and in stock market returns ( ;2 ) ; the third term is always positive and increasing with the horizon when the autoregressive coe cient in the dividend yield process is positive. 4