Appendix A: Posterior Properties of the Random Walk Model

The purpose of this Appendix is to provide technical details on the predictive density of the random walk model with a standard diffuse prior on the residual covariance matrix. An analytical expression of the predictive density is derived and its mean vector and covariance matrix are also determined. To these ends, let yt = yt−1 + εt, t = 1,..., T, (A.1) where the residuals εt are assumed to be i.i.d. N(0,Ω) with Ω positive definite and y0 is fixed. The diffuse prior is given by p(Ω) ∝ ∣∣Ω∣∣−(n+1)/2. (A.2) Stacking the model in (A.1) into n × T matrices y = [y1 · · · yT] and ε = [ε1 · · · εT], with the realized values, for convenience, being denoted the same way, the posterior distribution is proportional to the prior times the likelihood, which in natural logarithms can be expressed as log p y ∣∣y0,Ω) + log p(Ω) = −nT2 log(2π) − T + n + 1 2 lo