Bayesian prediction mean squared error for state space models with estimated parameters (Q2703256)

The authors consider the random walk plus noise model \( y_t=\mu_0+\sum_{i=1}^t\eta_t+\varepsilon_t,\) where \(\varepsilon_t\sim N(0,\sigma^2)\) and \(\eta_t\sim N(0,\sigma^2q)\) are uncorrelated. The first-order parameters are \(\mu_0\) (fixed effect) and \(\alpha_t=\sum_{i=1}^t\eta_t\) (random effects). The second order parameters are \(\sigma^2\) and \(q\). An empirical Bayes estimator for the first-order parameters is described. The naive estimators for the second-order parameters are defined by sums of squares of residuals and estimates for \(\eta_t\). It is noted that use of these estimates leads to underestimation. Plug-in (delta) and Monte-Carlo methods of correction for this problem are considered. Simulation results are presented. A case study of a series `Purse Snatching in Chicago' is discussed.