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

This Appendix is organized as follows. A simple illustration demonstrating the effects of pa-rameter 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(rT,T+k|DT) 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 DT, 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 DT, µT − Er is uncorrelated with Er, and the quantity zT ≡ bT − Er is fixed across φ. Next, let ρµb denote the true unconditional correlation between µT and bT, ρµb ≡ Corr(µT, bT |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 DT). If the observed predictors capture µT perfectly, then ρµb = 1; otherwise ρµb < 1. We make the “homoskedasticity ” assumption that qT is constant across DT, which implies that qT = (1 − ρ 2 µb)