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

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1 Technical Appendix to Are Stocks Really Less Volatile in the Long Run? by Ľuboš Pástor and Robert F. Stambaugh February 14, 9

2 This Technical Appendix describes how the predictive variance of multiperiod returns is computed in two alternative frameworks that are different from the basic framework used in the paper. Both alternative frameworks are analyzed for the purpose of assessing the robustness of the paper s results, and both feature noninformative prior beliefs. The first framework is the general VAR form of the predictive system, whose results are reported in Section 4. in the paper. This framework is discussed in Section B1 below. The second framework is the perfect-predictor model, whose results are reported in Section 5 in the paper. That framework is discussed in Section B below. B1. Predictive system: General VAR We begin working with multiple assets, so that not only x t but also r t and t are vectors. The predictive system in its most general form is a VAR for r t, x t, and t, with coefficients restricted so that t is the conditional mean of r tc1. We assume that x t and t are stationary with means E x and E r. We work with the first-order VAR: 1 r tc1 E r I 4 x tc1 E x 5 4 A 1 A A 5 4 tc1 E r A 1 A A We assume the errors in (B1) are i.i.d. across t 1; : : : ; T : u t 4 v t w t ; 4 r t x t t E r E x E r uu uv uw vu vv vw wu wv ww 5 C 4 1 5A : u tc1 v tc1 w tc1 5 : (B1) Let N A denote the entire coefficient matrix in (B1), and let denote the entire covariance matrix in (B). efine the vector t 4 and let V denote its unconditional covariance matrix. Then V rr V rx V r V 4 V xr V xx V x 5 AV N AN C ; V r V x V r t x t t (B) 5 ; (B) 1 The basic framework used in the paper is a special case of (B1) in which the coefficient matrix is restricted as I I 4 A 1 A A 5 4 A 5 : A 1 A A B The predictive system in (B1) can also be viewed alternatively as an unrestricted VAR for returns and predictors when some predictors are unobserved. See the Technical Appendix to Pástor and Stambaugh (9). (B4) 1

3 which can be solved as vec.v / ŒI. N A N A/ 1 vec. /; (B5) using the well known identity vec.fg/.g /vec.f/. Let z t denote the vector of the observed data at time t, rt z t : x t enote the data we observe through time t as t.z 1 ; : : : ; z t /, and note that our complete data consist of T. Also define E z Er E x ; V zz Vrr V xr V rx ; V V z xx Vr V x : (B6) Let denote the full set of parameters in the model,. N A; ; E z /, and let denote the full time series of t, t 1; : : : ; T. To obtain the joint posterior distribution of and, denoted by p.; j T /, we use an MCMC procedure in which we alternate between drawing from the conditional posterior p.j; T / and drawing from the conditional posterior p.j; T /. The procedure for drawing from p.j; T / is described in Section B1.1. The procedure for drawing from p.j; T / / p./p. T ; j/ is described in Section B1.. B1.1. rawing the time series of t To draw the time series of the unobservable values of 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. efine a t E. t j t 1 / b t E. t j t / e t E.z t j t ; t 1 / (B7) f t E.z t j t 1 / P t Var. t j t 1 / Q t Var. t j t / (B8) R t Var.z t j t ; t 1 / S t Var.z t j t 1 / G t Cov.z t ; t j t 1/ (B9) Conditioning on the (unknown) parameters of the model is assumed throughout but suppressed in the notation for convenience. First observe that j N.b ; Q /; (B1)

4 where denotes the null information set, so that the unconditional moments of are given by b E r and Q V. Also, 1 j N.a 1 ; P 1 /; (B11) where a 1 E r and P 1 V, and z 1 j N.f 1 ; S 1 /; (B1) where f 1 E z and S 1 V zz. Note that G 1 V z (B1) and that where z 1 j 1 ; N.e 1 ; R 1 /; (B14) e 1 f 1 C G 1 P a 1 / (B15) R 1 S 1 G 1 P 1 1 G 1 : (B16) Combining this density with equation (B11) using Bayes rule gives 1 j 1 N.b 1 ; Q 1 /; (B17) where b 1 a 1 C P 1.P 1 C G 1 R 1 1 G 1/ 1 G 1 R 1 1.z 1 f 1 / (B18) Q 1 P 1.P 1 C G 1 R 1 1 G 1/ 1 P 1 : (B19) Continuing in this fashion, we find that all conditional densities are normally distributed, and we obtain all the required moments for t ; : : : ; T : a t.i A 1 A /E r A E x C A 1 r t 1 C A x t 1 C A b t 1 (B) f t b t 1.I A /E x.a 1 C A /E r C A 1 r t 1 C A x t 1 C A b t 1 (B1) S t Q t 1 Q t 1 A A Q t 1 A Q t 1 A C uu vu uv vv (B) G t Q t 1 A A Q t 1 A C uw vw (B) P t A Q t 1 A C ww (B4)

5 e t f t C G t P 1 t. t a t / (B5) R t S t G t P 1 t G t (B6) b t a t C P t.p t C G t R 1 t G t / 1 G t R 1 t.z t f t / (B7) a t C G t S 1 t.z t f t / (B8) Q t P t.p t C G t R 1 t G t / 1 P t : (B9) The values of fa t ; b t ; Q t ; S t ; G t ; P t g for t 1; : : : ; T are retained for the next stage. Equations (B) through (B4) are derived as St G t Var. P t j t 1 / t G t AVar. N t 1 j t 1 / AN C AN 4 5 AN C Q t 1 4 Q t 1 Q t 1 A Q t 1 A A Q t 1 A Q t 1 A A Q t 1 A A Q t 1 A Q t 1 A A Q t 1 A 5 C 4 uu uv uw vu vv vw wu wv ww 5 : B1.1.. Sampling We wish to draw. ; 1 ; : : : ; T / conditional on T. The backward-sampling approach relies on the Markov property of the evolution of t and the resulting identity, p. ; 1 ; : : : ; T j T / p. T j T /p. T 1 j T ; T 1 / p. 1 j ; 1 /p. j 1 ; /: (B) We first sample T from p. T j T /, the normal density obtained in the last step of the filtering. Then, for t T 1; T ; : : : ; 1;, we sample t from the conditional density p. t j tc1 ; t /. (Note that the first two subvectors of t are already observed and thus need not be sampled.) To obtain that conditional density, first note that tc1 j t N ftc1 a tc1 StC1 ; G tc1 G tc1 P tc1 ; (B1) t j t 4 r t x t b t 5 ; 4 Q t 1 5A ; (B) and Cov. t ; tc1 j t/ Var. t j t / AN 4

6 4 4 Q t 5 4 Q t Q t A Q t A A 1 A 1 A A I A A 5 5 : (B) Therefore, where t j tc1 ; t N.h t ; H t /; (B4) h t E. t j t / C Cov. t ; tc1 j t/ ŒVar. tc1 j t / 1 Œ tc1 E. tc1 j t / r t 1 4 x t 5 C 4 5 StC1 G tc1 ztc1 f tc1 b t Q t Q t A Q t A G tc1 P tc1 tc1 a tc1 and H t Var. t j t / 4 Q t Cov.t ; tc1 j t/ 1 ŒVar. tc1 j t / Cov. t ; tc1 j t/ 1 Q StC1 G t tc1 4 Q t Q t A Q ta G A tc1 P Q t tc1 A Q t 5 The mean and covariance matrix of 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 t are scalars. The dimensions of N A and are then.k C /.K C /, where K is the number of predictors in the vector x t. B1.. rawing the parameters This section describes how we obtain the posterior draws of all parameters conditional on the current draw of the time series of t. The parameters are. A, N, E xr ), where E xr.ex E r/. B1..1. Prior distributions We specify noninformative priors for all parameters. The prior on AN is flat, except for the stationarity restriction that all eigenvalues of AN must lie inside the unit circle. The prior on E xr is normal, E xr N.E xr ; V xr /. The prior covariance matrix V xr has very large elements on the main diagonal and zeros off the diagonal, so the choice of E xr is unimportant. The prior on is inverted Wishart with a small number of degrees of freedom: IW. ; /. We 5

7 choose K C 4, so that 7 in our basic specification with three predictors. The choice of, the prior mean of, is unimportant because the prior standard deviation of is very large, corresponding to the posterior from a hypothetical sample of only seven observations. All parameters are independent a priori. B1... Posterior distributions rawing given.e xr ; N A/ Given the normal likelihood and the inverted Wishart prior for, the posterior of is also inverted Wishart, j IW.S C ;.T 1/ C /, where S.Y XB/.Y XB/ and r E r x E x K B 1 E r C : : : A r T E r xt E x K T E r 1 r 1 E r x1 E x K 1 E r B C : : : A r T 1 E r xt 1 E x K T 1 E r B N A : Above, K denotes a K 1 vector of ones. The dimensions of both X and Y are.t 1/.K C/. rawing E xr given. ; N A/ The conditional posterior of E xr is normal, E xr j N. QE xr ; QV xr /, where QV xr V 1 xr C.T 1/Q 1Q 1 # QE xr QV xr "V 1 xr E xr C Q TX 1 1. tc1 A t / Q 4 t1 I K A.A 1 C A / A 1 A 1 A (B5) (B6) 5 : (B7) Above, Q is.k C /.K C 1/, I K is a K K identity matrix, and t is defined in equation (B). rawing N A given.e xr ; / In order to draw A, N we need to draw the.k C /.K C 1/ matrix A 1 A 1 A 4 A A 5 : (B8) A A 6

8 First, we establish some notation. enote Z I KC1 X, where I KC1 is a.k C 1/.K C 1/ identity matrix, so that Z is Œ.T NY 1/.K C 1/ Œ.K C /.K C 1/. Let x E x K 1 E : : : : : : A xt E x K T E r and let z vec. NY /, so that z is Œ.T u v u 4 : 5 ; V 4 : u T vt 1/.K C 1/ 1. enote w r ; w 4 : 5 ; r 4 : w T r T ;.l/ 4 1 : T ; (B9) where u, w, r, and.l/ are.t 1/ 1 vectors and V is a.t 1/ K matrix. Let v vec.v /. efine 1, 1, and so that That is, 1 Œ uv uw and uu 1 vv wv 1 vw ww With the above notation, we can write the system (B1) without its first equation as v z Zb C ; w where b vec.a/. Conditional on u, the error terms Œv w are normally distributed with the mean of 1 1 uu u and the covariance matrix of. 1 1 uu 1/ I T 1. Recall that the prior distribution on b is given by 1 bs, which is equal to one when the stationarity restriction on AN is satisfied and zero otherwise. Given the normal likelihood, the full conditional posterior distribution of b is then given by bj N Qb; QV b 1 bs ; where : : (B4) Qb.Z Z/ 1 Z z 1 1 uu.r.l// (B41) QV b. 1 1 uu 1/.X X/ 1 : (B4) We obtain the posterior draws of b by making draws from N Qb; QV b and retaining only draws that satisfy b S. The posterior draws of A are constructed from the posterior draws of b vec.a/. 7

9 B1.. Predictive variance for general VAR In addition to the notation from equations (B1), (B), and (B), define also E r E 4 E x E r 5 ; u t t 4 v t w t 5 ; (B4) and Equation (B1) can then be written as e t t E : (B44) e tc1 N A e t C tc1: (B45) For i > 1, successive substitution using (B45) gives e tci AN i e t C AN i 1 tc1 C N A i tc C C tci : (B46) efine T;T CK Summing (B45) over K periods then gives KX T Ci ; i1 K e T;T CK X e T Ci T;T CK KE : i1 K! e T;T CK X AN i e t C I C AN C C N i1 A K 1 tc1 C I C AN C C AN K tc C C tck. N KC1 I/ e t C N K tc1 C N K 1 tc C C tck ; (B47) where N i I C AN C C AN i 1.I A/ N 1.I AN i /: (B48) It then follows that E e T;T CK j T ; ; T. N KC1 I/ e T ; E. T;T CK j T ; ; T /. N KC1 I/ e T C KE ; (B49) 8

10 and Var. T;T CK j T ; ; T / Var e T;T CK j T ; ; T KX N i N i : (B5) The first and second moments of (B49) given T and are given by i1 E. T;T CK j T ; /. N KC1 I/ 4 r T x T b T E r E x E r 5 C KE (B51) and Var ŒE. T;T CK j T ; ; T / j T ;. N KC1 I/ 4 Combining (B5) and (B5) gives Q T 5. N KC1 I/ : (B5) Var. T;T CK j T ; / E ŒVar. T;T CK j T ; ; T / j T ; C Var ŒE. T;T CK j T ; ; T / j T ; KX i1 N i N i C. N KC1 I/ 4 Q T 5. N KC1 I/ : (B5) By evaluating (B51) and (B5) for repeated draws of from its posterior, the predictive variance of T;T CK can be computed using the decomposition, Var. T;T CK j T / E fvar. T;T CK j; T /j T g C Var fe. T;T CK j; T /j T g : (B54) Finally, the predictive variance of r T;T CK is the (1,1) element of Var. T;T CK j T /. B. Perfect predictors Note: The notation in this section is distinct from the notation in the previous section. The paper focuses on the realistic scenario in which the observable predictors x t are imperfect, in that they do not perfectly capture the conditional expected return t. In this section, we discuss the calculation of predictive variance in an alternative framework with perfect predictors, for which t a C b x t. In that case, the predictive system is replaced by a model consisting of equations r tc1 a C b x t C e tc1 (B55) x tc1 C Ax t C v tc1 ; (B56) 9

11 combined with the following distributional assumption on the residuals: et N ; e ev : (B57) v t Let ı denote the full set of parameters in equations (B55), (B56), and (B57). Let denote the covariance matrix in (B57), let B denote the matrix of the slope coefficients in (B55) and (B56), a B b A ; ve vv and let c vec.b/. Note that ı consists of the elements of c and. B.1. Posterior distributions under perfect predictors We specify the prior distribution on ı as p.ı/ p.c/p. /. The priors on c and are noninformative, except for the restriction that the spectral radius of A,.A/, is less than 1. The prior on c is p.c/ / IŒ.A/ < 1, where IŒ denotes the indicator function. The prior on is p. / / j j.mc1/=, where m is the number of rows in (i.e., x t is.m 1/ 1). To obtain the posterior distribution of ı, the prior p.ı/ is combined with the normal likelihood function p. T jı/ implied by equation (B57). The posterior draws of ı can be obtained by applying standard results from the multivariate regression model (e.g., Zellner, 1971). efine the following notation: r Œr 1 r r T, Q C Œx 1 x x T, Q Œx x 1 x T 1, X Œ T Q, where T denotes a T 1 vector of ones, Y r Q C, OB.X X/ 1 X Y, and S.Y X OB/.Y X OB/. We first draw 1 from a Wishart distribution with T m degrees of freedom and parameter matrix S 1. Given that draw of 1, we then draw c from a normal distribution with mean Oc vec. OB/ and covariance matrix.x X/ 1. That draw of ı is retained as a draw from p.ıj T / if.a/ < 1. B.. Predictive variance under perfect predictors The conditional moments of the k-period return r T;T Ck are given by E.r T;T Ck j T ; ı/ ka C b k 1 C b k x T (B58)! Var.r T;T Ck j T ; ı/ k e C Xk 1 b k 1 ve C b i vv i b; (B59) i1 1

12 where i I C A C C A i 1.I A/ 1.I A i / (B6) k 1 1 C C C k 1.I A/ 1 ŒkI.I A/ 1.I A k / : (B61) The first term in (B59) reflects i.i.d. uncertainty. The second term reflects correlation between unexpected returns and innovations in future x T Ci s, which deliver innovations in future T Ci s. That term can be positive or negative and captures any mean reversion. The third term, always positive, reflects uncertainty about future x T Ci s, and thus uncertainty about future T Ci s. This third term, which contains a summation, can also be written without the summation as! Xk 1 b i vv i b b b.i A/ 1.I A/ 1 h ki k I I k i1 i C.I A A/ 1.I.A A/ k / vec. vv / : Applying the standard variance decomposition Var.r T;T Ck j T / EfVar.r T;T Ck j T ; ı/j T g C VarfE.r T;T Ck j T ; ı/j T g; (B6) the predictive variance Var.r T;T Ck j T / can be computed as the sum of the posterior mean of the right-hand side of equation (B59) and the posterior variance of the right-hand side of equation (B58). These posterior moments are computed from the posterior draws of ı, which are described in Section B.1. References Carter, Chris K., and Robert Kohn, 1994, On Gibbs sampling for state space models, Biometrika 81, Frühwirth-Schnatter, Sylvia, 1994, ata augmentation and dynamic linear models, Journal of Time Series Analysis 15, 18. Pástor, Ľuboš, and Robert F. Stambaugh, 9, Predictive systems: Living with imperfect predictors, Journal of Finance, forthcoming. West, Mike, and Jeff Harrison, 1997, Bayesian Forecasting and ynamic Models (Springer-Verlag, New York, NY). 11