Internet Appendix to Are Stocks Really Less Volatile in the Long Run?

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1 Internet Appendix to Are Stocks Really Less Volatile in the Long Run? by Ľuboš Pástor and Robert F. Stambaugh January 1, 211

2 B1. Roadmap This Appendix is organized as follows. A simple illustration demonstrating the effects of parameter uncertainty on long-horizon predictive variance is provided in Section B2. The Bayesian empirical methodology for System 2 is presented in Section B3. Additional empirical evidence that complements the evidence presented in the paper is reported in Section B4. That section first presents the estimates of predictive regressions for both annual and quarterly data, followed by various robustness results regarding the long-horizon predictive variance of stock returns. B2. Parameter uncertainty: A simple illustration In the paper, we compute Var(r T,T+k D T ) and its components empirically, incorporating parameter uncertainty via Bayesian posterior distributions. Here we use a simpler setting to illustrate the effects of parameter uncertainty on multiperiod return variance. All the notation that is not defined here is defined in Section 2 of the paper. The elements of the parameter vector φ are viewed as random, given that they are unknown to an investor. For the purpose of this illustration, we make several assumptions related to φ with the objective of simplifying the variance ratio calculations in the presence of parameter uncertainty. Our first assumption is that, given data D T, the elements of φ are distributed independently of each other. We also make two assumptions about the deviations of the conditional mean from the unconditional mean: given D T, µ T E r is uncorrelated with E r, and the quantity z T b T E r is fixed across φ. Next, let ρ µb denote the true unconditional correlation between µ T and b T, ρ µb Corr(µ T, b T φ) (this correlation is true in that it depends on the true parameter values φ, and it is unconditional in that it does not condition on D T ). If the observed predictors capture µ T perfectly, then ρ µb = 1; otherwise ρ µb < 1. We make the homoskedasticity assumption that q T is constant across D T, which implies that q T = (1 ρ 2 µb)σ 2 µ = (1 ρ 2 µb)r 2 σ 2 r, (B1) where σµ 2 and σ2 r are the true unconditional variances of µ t and r t+1, respectively (i.e., σµ 2 Var(µ t φ) and σr 2 Var(r t+1 φ)). The parameter vector is φ = [β, R 2, ρ uw, E r, σ r, ρ µb ]. Finally, we specify τ such that Var(E r D T ) = τe(σr 2 D T), (B2) so that the uncertainty about the unconditional mean return E r is as large as the imprecision in a sample mean computed over a sample of length 1/τ. 1

3 None of the above simplifying assumptions hold in the Bayesian empirical framework in the paper; they are made for the purpose of this simple illustration only. Given these assumptions, we are able to characterize the variance ratio in a parsimonious fashion that does not depend on the unconditional variance of single-period returns. The variance of the k-period return can be decomposed as follows: Var(r T,T+k D T ) = E {Var(r T,T+k φ, µ T, D T ) D T } + Var {E(r T,T+k φ, µ T, D T ) D T } = E {Var(r T,T+k φ, µ T ) D T } + Var {E(r T,T+k φ, µ T ) D T } = E { kσr(1 2 R 2 )[1 + 2 dρ uw A(k) + d } 2 B(k)] D T { +Var ke r + 1 } βk 1 β (µ T E r ) D T = ke(σr 2 D T)E { (1 R 2 )[1 + 2 dρ uw A(k) + d 2 B(k)] D T } { } 1 β +k 2 k Var(E r D T ) + Var 1 β (µ T E r ) D T = ke(σ 2 r D T )E { (1 R 2 )[1 + 2 dρ uw A(k) + d 2 B(k)] D T } +k 2 τe(σ 2 r D T) + Var { 1 β k 1 β (µ T E r ) D T }. (B3) The next to last equality uses the property that E r is uncorrelated with µ T E r, and the last equality uses (B2). Next observe that { } 1 β k Var 1 β (µ T E r ) D T { [ ] } 1 β k = E Var 1 β (µ T E r ) φ, D T D T { [ ] } 1 β k +Var E 1 β (µ T E r ) φ, D T D T ( ) 1 β k 2 = E Var(µ T φ, D T ) D T 1 β { } 1 β k +Var 1 β E(µ T E r φ, D T ) D T ( ) 1 β k 2 = E q T D T 1 β { } 1 β k +Var 1 β (b T E r ) D T ( ) 1 β k 2 = E σ 2 1 β rr 2 (1 ρ 2 µb) D T { } 1 β k +Var 1 β z T D T 2

4 ( ) 1 β k 2 = E D T 1 β E(σ2 r D T)E { R 2 (1 ρ 2 µb ) D } T { } 1 β +zt 2 k Var 1 β D T. (B4) Substituting the right-hand side of (B4) for the last term in (B3) then gives Var(r T,T+k D T ) = ke(σr D 2 T ))E { (1 R 2 )[1 + 2 dρ uw A(k) + d } 2 B(k)] D T ( ) 1 β +k 2 τe(σr D 2 T ) + E(σr D 2 k 2 T )E D T 1 β E { R 2 (1 ρ 2 µb) D T } { } 1 β +zt 2 k Var 1 β D T. (B5) When k = 1, equation (B5) simplifies to Var(r T,T+1 D T ) = E(σ 2 r D T ) [ 1 + τ E(R 2 D T )E(ρ 2 µb D T ) ]. (B6) Observe that the k-period variance in (B5) depends on the value of zt 2, which enters the last term multiplied by the variance of (1 β k )/(1 β). This dependence makes sense: when µ T is estimated to be farther from the unconditional mean E r, so that the absolute value of z T is large, uncertainty about the speed with which µ T reverts to E r is more important. To achieve a further algebraic simplification, we evaluate the variance in (B5) by setting zt 2 equal to the posterior mean of the true unconditional mean of zt 2 : z 2 T = E[E(z 2 T φ) D T ]. (B7) To evaluate the right-hand side of (B7), first note that Var(z T φ) = Var(b T φ), since z T = b T E r and E r is in φ. Also observe that E(z T φ) =, since across samples (D T s), the expected conditional mean b T is the unconditional mean E r. Therefore, E(zT 2 φ) = Var(z T φ) = Var(b T φ). We thus have E[E(zT 2 φ) D T] = E[Var(b T φ) D T ] = E[ρ 2 µbvar(µ T φ) D T ] = E[ρ 2 µbσrr 2 2 D T ] = E(ρ 2 µb D T )E(σr D 2 T )E(R 2 D T ). (B8) We compute the k-period variance ratio, V (k) = Var(r T,T+k D T ) k Var(r T,T+1 D T ), 3 (B9)

5 where Var(r T,T+1 D T ) is given in equation (B6) and Var(r T,T+k D T ) denotes the value of (B5) obtained by substituting the right-hand side of (B8) for zt 2. Note that the variance ratio in equation (B9) does not depend on E(σr 2 D T). We set τ = 1/2 in equation (B2), so that the uncertainty about the unconditional mean return E r corresponds to the imprecision in a 2-year sample mean. To specify the uncertainty for the remaining parameters, we choose the probability densities displayed in Figure A1, whose medians are.86 for β,.12 for R 2, -.66 for ρ uw, and.7 for ρ µb. Table A1 displays the 2-year variance ratio, V (2), under different specifications of uncertainty about the parameters. In the first row, β, R 2, ρ uw, and E r are held fixed, by setting the first three parameters equal to their medians and by setting τ = in (B2). Successive rows then specify one or more of those parameters as uncertain, by drawing from the densities in Figure A1 (for β, R 2, and ρ uw ) or setting τ = 1/2 (for E r ). For each row, ρ µb is either fixed at one of the values,.7 (its median), and 1, or it is drawn from its density in Figure A1. Note that the return variances are unconditional when ρ µb = and conditional on full knowledge of µ T when ρ µb = 1. Table A1 shows that when all parameters are fixed, V (2) < 1 at all levels of conditioning (all values of ρ µb ). That is, in the absence of parameter uncertainty, the values in the first row range from.95 at the unconditional level to.77 when µ T is fully known. Thus, this fixed-parameter specification is consistent with mean reversion playing a dominant role, causing the return variance (per period) to be lower at the long horizon. Rows 2 through 5 specify one of the parameters β, R 2, ρ uw, and E r as uncertain. Uncertainty about β exerts the strongest effect, raising V (2) by 17% to 26% (depending on ρ µb ), but uncertainty about any one of these parameters raises V (2). In the last row of Table A1, all parameters are uncertain, and the values of V (2) substantially exceed 1, ranging from 1.17 (when ρ µb = 1) to 1.45 (when ρ µb = ). Even though the density for ρ uw in Figure A1 has almost all of its mass below, so that mean reversion is almost certainly present, parameter uncertainty causes the long-run variance to exceed the short-run variance. As discussed in the paper, uncertainty about E r implies V (k) as k. We can see from Table A1 that uncertainty about E r contributes nontrivially to V (2), but somewhat less than uncertainty about β or R 2 and only slightly more than uncertainty about ρ uw. With uncertainty about only the latter three parameters, V (2) is still well above 1, especially when ρ µb < 1. Thus, although uncertainty about E r must eventually dominate variance at sufficiently long horizons, it does not do so here at the 2-year horizon. The variance ratios in Table A1 increase as ρ µb decreases. In other words, less knowledge about µ T makes long-run variance greater relative to short-run variance. We also see that drawing ρ µb 4

6 from its density in Figure A1 produces the same values of V (2) as fixing ρ µb at its median. B3. Predictive System 2 System 2 is implemented in the paper for one asset, but it can be specified for n assets, in which case r t+1 and π t+1 are n 1 vectors. We begin the analysis with that more general form and then later restrict it to the single-asset setting. With multiple assets, System 2 is given by r t+1 x t+1 π t+1 = a θ + A 12 I A 22 A 33 r t x t π t + u t+1 v t+1 η t+1, (B1) which can also be written as r t+1 E r x t+1 E x π t+1 E π = A 12 I A 22 A 33 r t E r x t E x π t E π + u t+1 v t+1 η t+1, (B11) where E x = (I A 22 ) 1 θ, E r = a + A 12 E x, and without loss of generality E π =. We begin working with multiple assets, so that not only x t but also r t and π t are vectors. We assume the errors in (B11) are i.i.d. across t = 1,..., T : u t v t η t N, Σ uu Σ uv Σ uη Σ vu Σ vv Σ vη Σ ηu Σ ηv Σ ηη. (B12) We assume the eigenvalues of both A 22 and A 33 lie inside the unit circle. As the elements of Σ ηη approach zero, the above model approaches the perfect-predictor specification, [ rt+1 E r x t+1 E x ] = [ A12 A 22 ] [ rt E r x t E x ] + [ ut+1 v t+1 ]. (B13) Let Ā denote the entire coefficient matrix in (B11), and let Σ denote the entire covariance matrix in (B12). Define the vector ζ t = r t x t π t, and let V ζζ denote its unconditional covariance matrix. Then V ζζ = V rr V rx V rπ V xr V xx V xπ V πr V πx V ππ 5 = ĀV ζζā + Σ, (B14) (B15)

7 which can be solved as vec (V ζζ ) = [I (Ā Ā)] 1 vec(σ), (B16) using the well known identity vec (DFG) = (G D)vec (F). Let z t denote the vector of the observed data at time t, ] z t = [ rt Denote the data we observe through time t as D t = (z 1,..., z t ), and note that our complete data consist of D T. Also define [ ] [ ] [ ] Er Vrr V E z =, V E zz = rx Vrπ, V x V xr V zπ =. (B17) xx V xπ x t. Let φ denote the full set of parameters in the model, φ (Ā, Σ, E z), and let π denote the full time series of π t, t = 1,..., T. To obtain the joint posterior distribution of φ and π, denoted by p(φ, π D T ), we use an MCMC procedure in which we alternate between drawing π from the conditional posterior p(π φ, D T ) and drawing φ from the conditional posterior p(φ π, D T ). The procedure for drawing π from p(π φ, D T ) is described in Section B3.1. The procedure for drawing φ from p(φ π, D T ) p(φ)p(d T, π φ) is described in Section B3.2. B3.1. Drawing π given the parameters To draw the time series of the unobservable values of π t conditional on the current parameter draws, we apply the forward filtering, backward sampling (FFBS) approach developed by Carter and Kohn (1994) and Frühwirth-Schnatter (1994). See also West and Harrison (1997, chapter 15). B Filtering The first stage follows the standard methodology of Kalman filtering. Define a t = E(π t D t 1 ) b t = E(π t D t ) e t = E(z t π t, D t 1 ) (B18) f t = E(z t D t 1 ) P t = Var(π t D t 1 ) Q t = Var(π t D t ) (B19) R t = Var(z t π t, D t 1 ) S t = Var(z t D t 1 ) G t = Cov(z t, π t D t 1) (B2) Conditioning on φ is assumed throughout this section but suppressed in the notation for convenience. First observe that π 1 D N(a 1, P 1 ), (B21) 6

8 where D denotes the null information set, so that the unconditional moments of π are given by a 1 = E π = and P 1 = V ππ. Also, z 1 D N(f 1, S 1 ), (B22) where f 1 = E z and S 1 = V zz. Note that and that where G 1 = V zπ z 1 π 1, D N(e 1, R 1 ), (B23) (B24) e 1 R 1 Combining this density with equation (B21) using Bayes rule gives = f 1 + G 1 P 1 1 (π 1 a 1 ) (B25) = S 1 G 1 P 1 1 G 1. (B26) π 1 D 1 N(b 1, Q 1 ), (B27) where b 1 Q 1 = a 1 + P 1 (P 1 + G 1R 1 1 G 1 ) 1 G 1R 1 1 (z 1 f 1 ) (B28) = P 1 (P 1 + G 1R 1 1 G 1 ) 1 P 1. (B29) Continuing in this fashion, we find that all conditional densities are normally distributed, and we obtain all the required moments for t = 2,..., T : a t = A 33 b t 1 (B3) f t = S t = G t = [ Er + A 12 (x t 1 E x ) + b t 1 E x + A 22 (x t 1 E x ) [ Qt 1 [ Qt 1 A 33 ] ] + + [ Σuu Σ uv Σ vu [ Σuη Σ vη ] Σ vv ] ] (B31) (B32) (B33) P t = A 33 Q t 1 A 33 + Σ ηη (B34) e t R t b t Q t = f t + G t P 1 t (π t a t ) (B35) = S t G t P 1 t G t (B36) = a t + P t (P t + G tr 1 t G t ) 1 G tr 1 t (z t f t ) (B37) = a t + G ts 1 t (z t f t ) (B38) = P t (P t + G t R 1 t G t ) 1 P t. (B39) 7

9 The values of {a t, b t, Q t, S t, G t, P t } for t = 1,..., T are retained for the next stage. Equations (B32) through (B34) are derived as [ St ] G t G t P t = Var(ζ t D t 1 ) = ĀVar(ζ t 1 D t 1 )Ā + Σ = Ā Ā + Σ Q t 1 = Q t 1 Q t 1 A 33 A 33 Q t 1 A 33 Q t 1 A 33 + Σ uu Σ uv Σ uη Σ vu Σ vv Σ vη Σ ηu Σ ηv Σ ηη. B Sampling drawing π We wish to draw (π 1,..., π T ) conditional on D T (and the parameters, φ). The backwardsampling approach relies on the Markov property of the evolution of ζ t and the resulting identity, p(ζ 1,..., ζ T D T ) = p(ζ T D T )p(ζ T 1 ζ T, D T 1 ) p(ζ 1 ζ 2, D 1 ). (B4) We first sample π T from p(π T D T ), the normal density obtained in the last step of the filtering. Then, for t = T 1, T 2,..., 1, we sample π t from the conditional density p(ζ t ζ t+1, D t ). (Note that the first two subvectors of ζ t are already observed and thus need not be sampled.) To obtain that conditional density, first note that ([ ] [ ]) ft+1 St+1 G ζ t+1 D t N, t+1 a t+1 G, (B41) t+1 P t+1 and ζ t D t N r t x t b t, Cov(ζ t, ζ t+1 D t ) = Var(ζ t D t )Ā = = Q t Q t Q t Q t A 33 8, A 12 A 22 I A 33. (B42) (B43)

10 Therefore, where and ζ t ζ t+1, D t N(h t, H t ), h t = E(ζ t D t ) + [ Cov(ζ t, ζ t+1 D t ) ] [Var(ζ t+1 D t )] 1 [ζ t+1 E(ζ t+1 D t )] = r t x t b t + Q t Q t A 33 [ St+1 G t+1 G t+1 P t+1 ] 1 [ zt+1 f t+1 π t+1 a t+1 ] (B44) H t = Var(ζ t D t ) [ Cov(ζ t, ζ t+1 D t ) ] [Var(ζ t+1 D t )] 1 [ Cov(ζ t, ζ t+1 D t ) ] = Q t Q t Q t A 33 [ St+1 G t+1 G t+1 P t+1 ] 1 Q t A 33 Q t The mean and covariance matrix of π t are taken as the relevant elements of h t and H t. In the rest of the Appendix, we discuss the special case (implemented in the paper) in which r t and π t are scalars. The dimensions of Ā and Σ are then (K +2) (K +2), where K is the number of predictors in the vector x t. In this case, A 12 is a K 1 vector, which we denote as b (distinct from b t, defined previously), and A 33 is a scalar that we denote as δ. We also denote A 22 as simply A (distinct from Ā, defined previously). B3.2. B Drawing the parameters given π Prior distributions With r t and π t being scalars, Σ = σu 2 σ uv σ uη σ vu Σ vv σ vη σ ηu σ ηv ση 2 = [ Σξξ σ ξη σ ξη ση 2 ], where ξ t [u t v t]. We wish to be informative about ση, 2 to varying degrees, while being noninformative about Σ ξξ and σ ξη. This objective is accomplished by specifying a prior for Σ equal to the posterior that obtains when a diffuse prior for Σ is combined with a hypothetical sample of T observations of η t and S observations of ξ t, where S T and the shorter period is a subset of the longer. This posterior is given by Stambaugh (1997), except that we impose the additional restriction that the population means of η t and ξ t are zero. The posterior in Stambaugh (1997) 9

11 relies on a change of variables, which we adopt here as well. Specifically, c = (1/ση)σ 2 ξη, and Ω = Σ ξξ σηcc 2. The hypothetical T observations of η t produce the sample variance ση,. 2 The S observations of ξ t and η t produce the sample covariance matrix ˆΣ, giving ˆσ η, 2, ĉ and ˆΩ. (All hypothetical second moments are non-central; equivalently the hypothetical sample means are zero.) With the change in variables from Σ to (ση, 2 c, Ω), the latter quantities have the posterior, taken here as the prior, given by p(ση, 2 c, Ω) = p(ση)p(c Ω)p(Ω), 2 where 1 ση 2 T ση, 2 (B45) χ 2 T K 1 ( ) 1 c Ω N ĉ, Ω (B46) S ˆσ η, 2 Ω IW(S ˆΩ, S K). (B47) Be specifying different values of σ 2 η, and T, we vary the degree to which the prior imposes a belief that σ 2 η is close to zero, where the limiting case of σ 2 η = corresponds to perfect predictors. The larger we make T, the greater is the prior information we supply about σ 2 η. We set S = K+1, where K is the number of predictors, which makes the prior on c and Ω essentially noninformative (as informative as a sample of only K + 1 observations, where K = 1 or 3). The specification of ˆΣ is thus inconsequential, and we simply set it to a scalar times the identity matrix. For the cases with imperfect predictors, we set T = K + 2 and then set σ η, equal to either.5 or.1 when using annual data and either.1 or.3 when using quarterly data. We also wish to be informative about δ, the autocorrelation of π t. Here we specify priors for δ identical to those specified for the autocorrelation (β) of the conditional mean µ t in the predictive system analyzed previously. Specifically, the prior for δ is a truncated Normal distribution, with the mean and standard deviation of the corresponding non-truncated distribution denoted as δ and 1 The densities are given as follows: p(ση 2 ) σ (T K+1) η exp { T σ 2 η, 2σ 2 η } ( p(σ η ) ση (T K) exp ( 1 p(c Ω) Ω (K+1) 2 exp 1 2 (c ĉ ) 1 Ω) S ˆσ η, 2 (c ĉ ) { 12 } tr(s ˆΩ )Ω 1. p(ω) Ω (S +2) 2 exp { T σ 2 η, 2σ 2 η }) 1

12 σ δ. The distribution is then truncated to satisfy the stationarity requirement δ < 1. We set σ δ to.25 with annual data and to.15 with quarterly data, and we set δ to.99 in both cases. The priors on the remaining parameters are non-informative, except for the condition required for stationary of x t that ρ(a), the spectral radius of A, be less than 1. Specifically, define Then the prior for ψ is given by [ ] a θ ψ = vec b A. δ (B48) { p(ψ) exp 1 2 (ψ ψ) } Vψ 1 (ψ ψ) 1 S, (B49) where ψ = [ δ ] [ σ 2, V ψ = ψ I (K+1) 2 σδ 2 ], (B5) σ 2 ψ is set to a large number, 1 S equals 1 under the conditions that ρ(a) < 1 and δ < 1, and 1 S equals otherwise. We also assume that the priors for Σ and ψ are independent. B Posterior distributions Define y t+1 = r t+1 π t x t+1 π t+1 y =, Y t = y 2. y T [ IK+1 [1 x t], Y = π t Y 1. Y T 1 ], υ = The sample representation of the predictive system in (B1) is then y = Y ψ + υ,, υ t+1 = υ 2. υ T. u t+1 v t+1 η t+1 for ψ defined in (B48). For tractability we employ the conditional likelihood, which treats values in period t = 1 as non-stochastic. We can then apply Gibbs sampling to draw ψ and Σ. This simplification seems reasonable, given that T = 26 for our main results. 2 2 An alternative would be to employ the exact likelihood that includes the density of the values at t = 1 and then use the Metropolis-Hastings algorithm, with the conditional posteriors given here used as proposal densities., 11

13 B Drawing ψ Given the prior for ψ in (B49), we can apply standard results from the multivariate regression model (e.g., Zellner, 1971) to obtain the full conditional posterior for ψ as ψ N( ψ, Ṽψ) 1 S (B51) where and. ψ = Ṽ ψ [ V 1 ψ ψ + Y (I T 1 Σ 1 )y ] Ṽ ψ = [ V 1 ψ + Y (I T 1 Σ 1 )Y ] 1 B Drawing Σ Define ˆΣ = 1 [ ] T 1 ˆΣξξ (y t+1 Y t ψ)(y t+1 Y t ψ) ˆσ = ξη T 1 ˆσ t=1 ξη ˆσ η 2 [ ] Σ Σ = ξξ σ ξη = 1 S ( SˆΣ + (T 1)ˆΣ ), S = S + (T 1) σ ξη σ 2 η c = (1/ σ 2 η) σ ξη, Ω = Σ ξξ σ 2 η c c, σ 2 η = 1 T ( T σ 2 η, + (T 1)ˆσ 2 η ), T = T + (T 1). A draw of Σ is constructed by drawing (ση 2, c, Ω). The joint full conditional of the latter is given by p(σ 2 η, c, Ω ) = p(σ 2 η )p(c Ω, )p(ω ), where the posteriors on the right-hand side are of the same form as (B45) through (B47): σ 2 η T σ 2 η χ 2 T K 1 ( ) (B52) 1 c Ω, N c, Ω (B53) S σ η 2 Ω IW( S Ω, S K). (B54) The draw of Σ is then obtained by computing Σ ξξ = Ω + σ 2 ηcc and σ ξη = σ 2 ηc. Our results based on 25, draws from the posterior distribution. First, we generate a sequence of 76, draws. We discard the first 1, draws as a burn-in and take every third draw from the rest to obtain a series of 25, draws that exhibit little serial correlation. 12

14 B3.3. Predictive variance In addition to the notation from equations (B11), (B12), and (B14), define also E ζ = E r E x, ɛ t = u t v t η t, (B55) and Equation (B11) can then be written as ζ e t = ζ t E ζ. ζ e t+1 = Āζe t + ɛ t+1. (B56) (B57) For i > 1, successive substitution using (B57) gives ζ e t+i = Āi ζ e t + Āi 1 ɛ t+1 + Āi 2 ɛ t ɛ t+i. (B58) Define ζ T,T+k = ζt,t+k e = Summing (B57) over k periods then gives k ζ T+i, i=1 k ζt+i e = ζ T,T+k ke ζ. i=1 ζ e T,T+k = ( k Ā )ζ i t e + ( I + Ā + + Āk 1) ɛ t+1 i=1 + ( I + Ā + + Āk 2) ɛ t ɛ t+k = ( Λ k+1 I)ζ e t + Λ k ɛ t+1 + Λ k 1 ɛ t ɛ t+k, (B59) where Λ i = I + Ā + + Āi 1 = (I Ā) 1 (I Āi ). (B6) It then follows that E ( ζ e T,T+k D T, φ, π T ) = ( Λk+1 I)ζ e T, E (ζ T,T+k D T, φ, π T ) = ( Λ k+1 I)ζ e T + ke ζ, (B61) 13

15 and Var (ζ T,T+k D T, φ, π T ) = Var ( ζ e T,T+k D T, φ, π T ) = k Λ i Σ Λ i. The first and second moments of (B61) given D T and φ are given by r T E r E(ζ T,T+k D T, φ) = ( Λ k+1 I) x T E x + ke ζ b T i=1 (B62) (B63) and Var [E (ζ T,T+k D T, φ, π T ) D T, φ] = ( Λ k+1 I) Q T ( Λ k+1 I). (B64) Combining (B62) and (B64) gives Var (ζ T,T+k D T, φ) = E [Var (ζ T,T+k D T, φ, π T ) D T, φ] + Var [E(ζ T,T+k D T, φ, π T ) D T, φ] = k Λ i Σ Λ i + ( Λ k+1 I) i=1 Q T ( Λ k+1 I). (B65) By evaluating (B63) and (B65) for repeated draws of φ from its posterior, the predictive variance of ζ T,T+k can be computed using the decomposition, Var(ζ T,T+k D T ) = E {Var(ζ T,T+k φ, D T ) D T } + Var {E(ζ T,T+k φ, D T ) D T }. (B66) Finally, the predictive variance of r T,T+k is the (1,1) element of Var(ζ T,T+k D T ). B3.4. Perfect predictors If the predictors are perfect, then π t is absent from the first equation in (B1) and the third equation simply drops out. In that case the model consists of the two equations, r t+1 = a + b x t + u t+1 (B67) x t+1 = θ + Ax t + v t+1, (B68) combined with the distributional assumption on the residuals, [ ] ([ ] [ ut σ 2 N, u σ uv v t σ vu Σ vv ]). (B69) 14

16 This perfect-predictor model obtains as the limiting case of the above imperfect-predictor setting as ση 2. Our perfect-predictor results can be computed in that manner, by setting T to a large number and σ η, to a small number. For completeness, however, we also provide here the simplified calculations that arise in that special case. In this case φ denotes the full set of parameters in equations (B75), (B68), and (B69), and Σ denotes the covariance matrix in (B69). Let B denote the matrix of coefficients in (B75) and (B68), [ ] a θ B = b A, and let ψ = vec(b). B Posterior distributions under perfect predictors We specify the prior distribution on φ as p(φ) = p(ψ)p(σ). The prior on ψ is p(ψ) 1 S, where 1 S equals 1 if ρ(a) < 1 and equals otherwise. The prior on Σ is p(σ) Σ (K+2)/2. Define the following notation: r = [r 2 r 3 r T ], Q + = [x 2 x 3 x T ], Q = [x 1 x 2 x T 1 ], X = [ι T 1 Q], where ι T 1 denotes a (T 1) 1 vector of ones, Y = [r Q + ], ˆB = (X X) 1 X Y, and S = (Y X ˆB) (Y X ˆB). We first draw Σ 1 from a Wishart distribution with T K 2 degrees of freedom and parameter matrix S 1. Given that draw of Σ 1, we then draw ψ from a normal distribution with mean ˆψ = vec ( ˆB) and covariance matrix Σ (X X) 1. That draw of φ is retained as a draw from p(φ D T ) if ρ(a) < 1. B Predictive variance under perfect predictors The conditional moments of the k-period return r T,T+k are given by E(r T,T+k D T, φ) = ka + b Ψ k 1 θ + b Λ k x T (B7) ( k 1 ) Var(r T,T+k D T, φ) = kσu 2 + 2b Ψ k 1 σ vu + b Λ i Σ vv Λ i b, (B71) where i=1 Λ i = I + A + + A i 1 = (I A) 1 (I A i ) (B72) Ψ k 1 = Λ 1 + Λ Λ k 1 = (I A) 1 [ki (I A) 1 (I A k )]. (B73) The first term in (B71) reflects i.i.d. uncertainty. The second term reflects correlation between unexpected returns and innovations in future x T+i s, which deliver innovations in future µ T+i s. 15

17 That term can be positive or negative and captures any mean reversion. The third term, always positive, reflects uncertainty about future x T+i s, and thus uncertainty about future µ T+i s. This third term, which contains a summation, can also be written without the summation as ( k 1 b i=1 Λ i Σ vv Λ i ) b = (b b ) [ (I A) 1 (I A) 1] [ ki Λ k I I Λ k +(I A A) 1 (I (A A) k ) ] vec (Σ vv ). Applying the standard variance decomposition Var(r T,T+k D T ) = E{Var(r T,T+k D T, φ) D T } + Var{E(r T,T+k D T, φ) D T }, (B74) the predictive variance Var(r T,T+k D T ) can be computed as the sum of the posterior mean of the right-hand side of equation (B71) and the posterior variance of the right-hand side of equation (B7). These posterior moments are computed from the posterior draws of φ, which are described in Section B B4. Additional empirical results B4.1. Predictive regressions This section reports the results from standard predictive regressions, r t+1 = a + b x t + e t+1, (B75) for various combinations of the three predictors used in the paper. The results, obtained by OLS, are reported in Table A2. Panel A reports the results based on annual data; the results based on quarterly data are reported in Panel B. In both panels, the first three regressions contain just one predictor, while the fourth regression contains all three. The table reports the estimated coefficients as well as the t-statistics, along with the bootstrapped p-values associated with these t-statistics as well as with the R In the bootstrap, we repeat the following procedure 2, times: (i) Resample T pairs of (ˆv t, ê t ), with replacement, from the set of OLS residuals from regressions (B68) and (B75); (ii) Build up the time series of x t, starting from the unconditional mean and iterating forward on equation (B68), using the OLS estimates (ˆθ, Â) and the resampled values of ˆv t ; (iii) Construct the time series of returns, r t, by adding the resampled values of ê t to the sample mean (i.e., under the null that returns are not predictable); (iv) Use the resulting series of x t and r t to estimate regressions (B68) and (B75) by OLS. The bootstrapped p-value associated with the reported t-statistic (or R 2 ) is the relative frequency with which the reported quantity is smaller than its 2, counterparts bootstrapped under the null of no predictability. 16

18 Table A2 shows that all the predictors exhibit significant ability to predict returns, especially in the multivariate regressions that involve all three predictors. In those regressions, the estimated correlation between e t+1 and the estimated innovation in expected return, b v t+1, is negative. Pástor and Stambaugh (29) suggest this correlation as a diagnostic in predictive regressions, with a negative value being what one would hope to see for predictors able to deliver a reasonable proxy for expected return. B4.2. Long-horizon predictive variance This section provides detailed robustness results on the long-horizon predictive variance, expanding the evidence reported in the paper for both predictive systems. We examine the following cases: Annual data: Baseline case reported in the paper System 1: Table A3; Figures A2, A3 System 2: Figure A18, left column Annual data: First subperiod System 1: Table A4; Figures A4, A5 System 2: Figure A19, left column Annual data: Second subperiod System 1: Table A5; Figures A6, A7 System 2: Figure A19, right column Annual data: One instead of three predictors System 1: Table A6; Figures A8, A9 System 2: Figure A2, left column Annual data: Excess instead of real returns System 1: Table A7; Figures A1, A11 System 2: Figure A21, left column Quarterly data: Baseline case reported in the paper System 1: Table A8; Figures A12, A13 System 2: Figure A18, right column Quarterly data: One instead of three predictors System 1: Table A9; Figures A14, A15 System 2: Figure A2, right column Quarterly data: Excess instead of real returns System 1: Table A1; Figures A16, A17 17

19 System 2: Figure A21, right column We do not report subperiod results for quarterly data because the (post-war) quarterly sample is already rather short compared to the 26-year annual sample. Parameter uncertainty generally plays a larger role in shorter samples. 18

20 Table A1 Effects of Parameter Uncertainty on 2-Year Variance Ratio The table displays the ratio (1/2)Var(r T,T+2 D T )/Var(r T+1 D T ), where D T is information used by an investor at time T. The value of the ratio is computed under various parametric scenarios for β (autocorrelation of the conditional expected return µ t ), R 2 (fraction of variance in r t+1 explained by µ t ), ρ uw (correlation between unexpected returns and innovations in expected returns), ρ µb (correlation between µ T and its best available estimate given D T ), and E r (the unconditional mean return). For β, R 2, ρ uw, and ρ µb, each parameter is either drawn from its density in Figure A1 when uncertain or set to a fixed value. The parameters β, R 2, and ρ uw are set to their medians when held fixed, while ρ µb is fixed at its median as well as and 1. The medians are.86 for β,.12 for R 2, -.66 for ρ uw, and.7 for ρ µb. The variance of E r given D T is either (when fixed) or 1/2 times the expected variance of one-year returns (when uncertain). fixed (F) or uncertain (U) ρ µb fixed at ρ µb β R 2 ρ uw E r.7 1 uncertain F F F F U F F F F U F F F F U F F F F U U U U F U U U U

21 Table A2 Predictive Regressions This table summarizes the results from predictive regressions r t = a + b x t 1 + e t, where r t denotes real log stock market return and x t 1 contains the predictors (listed in the column headings) lagged by one year. Innovations in expected returns are constructed as b v t, where v t contains the disturbances estimated in a vector autoregression for the predictors, x t = θ +Ax t 1 +v t. The table reports the estimated slope coefficients ˆb, the correlation Corr(e t, b v t ) between unexpected returns and innovations in expected returns, and the (unadjusted) R 2 from the predictive regression. The independent variables are rescaled to have unit variance. The correlations and R 2 s are reported in percent (i.e., 1). The OLS t-statistics are given in parentheses ( ). The t-statistic of Corr(e t, b v t ) is computed as the t-statistic of the slope from the regression of the sample residuals ê t on ˆbˆv t. The p-values associated with all t-statistics and R 2 s are computed by bootstrapping and reported in brackets [ ]. Each p-value is the relative frequency with which the reported quantity is smaller than its 2, counterparts bootstrapped under the null of no predictability. See footnote 3 of this document for more details on the bootstrapping procedure. Panel A. Annual data (182 27) Dividend Yield Term Spread Bond Yield Corr(e t, b v t ) R (1.891) (-9.88) [.7] [.57] [1.] (.69) (3.298) [.498] [.236] [.] (2.129) (-2.86) [.34] [.18] [.997] (2.383) (2.137) (2.373) (-1.988) [.13] [.21] [.17] [.1] [.973] Panel B. Quarterly data (1952Q1 26Q4) Bond Yield Dividend Yield CAY Corr(e t, b v t ) R (3.299) (3.466) [.2] [.1] [.] (1.951) ( ) [.95] [.91] [1.] (3.866) (-8.885) [.] [.] [1.] (3.145) (1.876) (3.11) (-4.236) [.] [.2] [.67] [.4] [1.] 2

22 Table A3 (same as Table 1 in the paper) Variance Ratios and Components of Long-Horizon Variance The first row of each panel reports the ratio (1/k)Var(r T,T+k D T )/Var(r T+1 D T ), where Var(r T,T+k D T ) is the predictive variance of the k-year return based on 26 years of annual data for real equity returns and the three predictors over the period. The second row reports Var(r T,T+k D T ), multiplied by 1. The remaining rows report the five components of Var(r T,T+k D T ), also multiplied by 1 (they add up to total variance). Panel A contains results for k = 25 years, and Panel B contains results for k = 5 years. Results are reported under each of three priors for ρ uw, R 2, and β. As the prior for one of the parameters departs from the benchmark, the priors on the other two parameters are held at the benchmark priors. The tight priors, as compared to the benchmarks, are more concentrated towards 1 for ρ uw, for R 2, and 1 for β; the loose priors are less concentrated in those directions. ρ uw R 2 β Prior Tight Bench Loose Tight Bench Loose Tight Bench Loose Panel A. Investment Horizon k = 25 years Variance Ratio Predictive Variance IID Component Mean Reversion Uncertain Future µ Uncertain Current µ Estimation Risk Panel B. Investment Horizon k = 5 years Variance Ratio Predictive Variance IID Component Mean Reversion Uncertain Future µ Uncertain Current µ Estimation Risk

23 Table A4 Variance Ratios and Components of Long-Horizon Variance First subperiod ( ) Counterpart of Table A3 ρ uw R 2 β Prior Tight Bench Loose Tight Bench Loose Tight Bench Loose Panel A. Investment Horizon k = 25 years Variance Ratio Predictive Variance IID Component Mean Reversion Uncertain Future µ Uncertain Current µ Estimation Risk Panel B. Investment Horizon k = 5 years Variance Ratio Predictive Variance IID Component Mean Reversion Uncertain Future µ Uncertain Current µ Estimation Risk

24 Table A5 Variance Ratios and Components of Long-Horizon Variance Second subperiod (195 27) Counterpart of Table A3 ρ uw R 2 β Prior Tight Bench Loose Tight Bench Loose Tight Bench Loose Panel A. Investment Horizon k = 25 years Variance Ratio Predictive Variance IID Component Mean Reversion Uncertain Future µ Uncertain Current µ Estimation Risk Panel B. Investment Horizon k = 5 years Variance Ratio Predictive Variance IID Component Mean Reversion Uncertain Future µ Uncertain Current µ Estimation Risk

25 Table A6 Variance Ratios and Components of Long-Horizon Variance One instead of three predictors (Dividend yield) Counterpart of Table A3 ρ uw R 2 β Prior Tight Bench Loose Tight Bench Loose Tight Bench Loose Panel A. Investment Horizon k = 25 years Variance Ratio Predictive Variance IID Component Mean Reversion Uncertain Future µ Uncertain Current µ Estimation Risk Panel B. Investment Horizon k = 5 years Variance Ratio Predictive Variance IID Component Mean Reversion Uncertain Future µ Uncertain Current µ Estimation Risk

26 Table A7 Variance Ratios and Components of Long-Horizon Variance Excess instead of real returns Counterpart of Table A3 ρ uw R 2 β Prior Tight Bench Loose Tight Bench Loose Tight Bench Loose Panel A. Investment Horizon k = 25 years Variance Ratio Predictive Variance IID Component Mean Reversion Uncertain Future µ Uncertain Current µ Estimation Risk Panel B. Investment Horizon k = 5 years Variance Ratio Predictive Variance IID Component Mean Reversion Uncertain Future µ Uncertain Current µ Estimation Risk

27 Table A8 Variance Ratios and Components of Long-Horizon Variance Quarterly data (1952Q1 26Q4) Counterpart of Table A3 ρ uw R 2 β Prior Tight Bench Loose Tight Bench Loose Tight Bench Loose Panel A. Investment Horizon k = 25 years Variance Ratio Predictive Variance IID Component Mean Reversion Uncertain Future µ Uncertain Current µ Estimation Risk Panel B. Investment Horizon k = 5 years Variance Ratio Predictive Variance IID Component Mean Reversion Uncertain Future µ Uncertain Current µ Estimation Risk

28 Table A9 Variance Ratios and Components of Long-Horizon Variance Quarterly data (1952Q1 26Q4) One predictor instead of three (dividend yield) Counterpart of Table A3 ρ uw R 2 β Prior Tight Bench Loose Tight Bench Loose Tight Bench Loose Panel A. Investment Horizon k = 25 years Variance Ratio Predictive Variance IID Component Mean Reversion Uncertain Future µ Uncertain Current µ Estimation Risk Panel B. Investment Horizon k = 5 years Variance Ratio Predictive Variance IID Component Mean Reversion Uncertain Future µ Uncertain Current µ Estimation Risk

29 Table A1 Variance Ratios and Components of Long-Horizon Variance Quarterly data (1952Q1 26Q4) Excess instead of real returns Counterpart of Table A3 ρ uw R 2 β Prior Tight Bench Loose Tight Bench Loose Tight Bench Loose Panel A. Investment Horizon k = 25 years Variance Ratio Predictive Variance IID Component Mean Reversion Uncertain Future µ Uncertain Current µ Estimation Risk Panel B. Investment Horizon k = 5 years Variance Ratio Predictive Variance IID Component Mean Reversion Uncertain Future µ Uncertain Current µ Estimation Risk

30 β R ρ uw ρ µ b Figure A1. Distributions for uncertain parameters The plots display the probability densities used to illustrate the effects of parameter uncertainty on long-run variance. In the R 2 panel, the solid line plots the density of the true R 2 (predictability given µ T ), and the dashed line plots the implied density of the R-squared in a regression of returns on b T. The dashed line incorporates the uncertainty about ρ µb. 29

31 .55 Panel A. Predictive Variance of Stock Returns.5 Variance (per year) Predictive Horizon (years).6 Panel B. Components of Predictive Variance.4 Variance (per year) IID component Mean reversion Uncertain future µ Uncertain current µ Estimation risk Predictive Horizon (years) Figure A2 (same as Figure 6 in the paper). Predictive variance of multiperiod return and its components. Panel A plots the variance of the predictive distribution of long-horizon returns, Var(r T,T+k D T ). Panel B plots the five components of the predictive variance. All quantities are divided by k, the number of periods in the return horizon. The results are obtained by estimating the predictive system on annual real U.S. stock market returns in Three predictors are used: the dividend yield, the bond yield, and the term spread. 3

32 .6 Predictive Variance of Stock Returns: Different Priors.55.5 Variance (per year) Predictive Horizon (years) Figure A3. Predictive variance for seven different priors. The figure plots the variance of the predictive distribution of long-horizon returns, Var(r T,T+k D T ), as a function of the investment horison k. The variance is divided by k, the number of periods in the return horizon. The results are obtained by estimating the predictive system on annual real U.S. stock market returns in Three predictors are used: the dividend yield, the bond yield, and the term spread. 31

33 .4 Panel A. Predictive Variance of Stock Returns Variance (per year) Predictive Horizon (years).3 Panel B. Components of Predictive Variance.2 Variance (per year).1.1 IID component Mean reversion Uncertain future µ Uncertain current µ Estimation risk Predictive Horizon (years) Figure A4. Predictive variance of multiperiod return and its components. First subperiod ( ). Counterpart of Figure A2. 32

34 .5 Predictive Variance of Stock Returns: Different Priors.45.4 Variance (per year) Predictive Horizon (years) Figure A5. Predictive variance for seven different priors. First subperiod ( ). Counterpart of Figure A3. 33

35 .55 Panel A. Predictive Variance of Stock Returns.5 Variance (per year) Predictive Horizon (years).6 Panel B. Components of Predictive Variance.4 Variance (per year).2.2 IID component Mean reversion Uncertain future µ Uncertain current µ Estimation risk Predictive Horizon (years) Figure A6. Predictive variance of multiperiod return and its components. Second subperiod (195 27). Counterpart of Figure A2. 34

36 .75 Predictive Variance of Stock Returns: Different Priors Variance (per year) Predictive Horizon (years) Figure A7. Predictive variance for seven different priors. Second subperiod (195 27). Counterpart of Figure A3. 35

37 .4 Panel A. Predictive Variance of Stock Returns.38 Variance (per year) Predictive Horizon (years).4 Panel B. Components of Predictive Variance Variance (per year) IID component Mean reversion Uncertain future µ Uncertain current µ Estimation risk Predictive Horizon (years) Figure A8. Predictive variance of multiperiod return and its components. One instead of three predictors (dividend yield). Counterpart of Figure A2. 36

38 .44 Predictive Variance of Stock Returns: Different Priors Variance (per year) Predictive Horizon (years) Figure A9. Predictive variance for seven different priors. One instead of three predictors (dividend yield). Counterpart of Figure A3. 37

39 .38 Panel A. Predictive Variance of Stock Returns.36 Variance (per year) Predictive Horizon (years).3 Panel B. Components of Predictive Variance.2 Variance (per year).1.1 IID component Mean reversion Uncertain future µ Uncertain current µ Estimation risk Predictive Horizon (years) Figure A1. Predictive variance of multiperiod return and its components. Excess instead of real returns. Counterpart of Figure A2. 38

40 .44 Predictive Variance of Stock Returns: Different Priors Variance (per year) Predictive Horizon (years) Figure A11. Predictive variance for seven different priors. Excess instead of real returns. Counterpart of Figure A3. 39

41 .15 Panel A. Predictive Variance of Stock Returns Variance (per quarter) Predictive Horizon (quarters).15 Panel B. Components of Predictive Variance Variance (per quarter) IID component Mean reversion Uncertain future µ Uncertain current µ Estimation risk Predictive Horizon (quarters) Figure A12. Predictive variance of multiperiod return and its components. Quarterly data (1952Q1 26Q4). Counterpart of Figure A2. 4

42 .24 Predictive Variance of Stock Returns: Different Priors Variance (per quarter) Predictive Horizon (quarters) Figure A13. Predictive variance for seven different priors. Quarterly data (1952Q1 26Q4). Counterpart of Figure A3. 41

43 .2 Panel A. Predictive Variance of Stock Returns.18 Variance (per quarter) Predictive Horizon (quarters).15 Panel B. Components of Predictive Variance Variance (per quarter) IID component Mean reversion Uncertain future µ Uncertain current µ Estimation risk Predictive Horizon (quarters) Figure A14. Predictive variance of multiperiod return and its components. Quarterly data (1952Q1 26Q4). One instead of three predictors (dividend yield). Counterpart of Figure A2. 42

44 .55 Predictive Variance of Stock Returns: Different Priors Variance (per quarter) Predictive Horizon (quarters) Figure A15. Predictive variance for seven different priors. Quarterly data (1952Q1 26Q4). One instead of three predictors (dividend yield). Counterpart of Figure A3. 43

45 .15 Panel A. Predictive Variance of Stock Returns Variance (per quarter) Predictive Horizon (quarters).15 Panel B. Components of Predictive Variance Variance (per quarter) IID component Mean reversion Uncertain future µ Uncertain current µ Estimation risk Predictive Horizon (quarters) Figure A16. Predictive variance of multiperiod return and its components. Quarterly data (1952Q1 26Q4). Excess instead of real returns. Counterpart of Figure A2. 44

46 .22 Predictive Variance of Stock Returns: Different Priors Variance (per quarter) Predictive Horizon (quarters) Figure A17. Predictive variance for seven different priors. Quarterly data (1952Q1 26Q4). Excess instead of real returns. Counterpart of Figure A3. 45

47 1 A. Prior for Standard Deviation of π 1 B. Prior for Standard Deviation of π Density (rescaled) Density (rescaled) σ π σ π 1 C. Posterior for Increase in R 2 1 D. Posterior for Increase in R 2 Density (rescaled) Density (rescaled) Variance (per year) R 2 E. Predictive Variance of Returns Horizon (years) Variance (per quarter) R 2 F. Predictive Variance of Returns less imperfect more imperfect perfect (σ π =) Horizon (quarters) Figure A18 (same as Figure 7 in the paper). Predictive variance and predictor imperfection. The plots display results under the predictive system (System 2) in which expected return depends on a vector of observable predictors, x t, as well as a missing predictor, π t, that obeys an AR(1) process. The top panels display prior distributions for σ π, the standard deviation of π t, under different degrees of predictor imperfection. The middle panels display the corresponding posteriors of R 2, the true R 2 for one-period returns minus the observed R 2 when conditioning only on x t. The bottom panels display the predictive variances for the two imperfect-predictor cases as well for the case of perfect predictors (σ π = R 2 = ). The left-hand panels are based on annual data from for real U.S. stock returns and three predictors: the dividend yield, the bond yield, and the term spread. The right-hand panels are based on quarterly data from 1952Q1 26Q4 for real returns and three predictors: the dividend yield, CAY, and the bond yield. 46