Statistica Sinica 7(1997), 973-991

MIXED MODEL REPRESENTATION OF STATE SPACE

MODELS: NEW SMOOTHING RESULTS AND THEIR

APPLICATION TO REML ESTIMATION

John V. Tsimikas and Johannes Ledolter

University of Massachusetts and University of Iowa

Abstract: Known results for the general linear mixed model and its special case, the variance components model, are applied to inference in state space models. New state and disturbance smoothing algorithms that accomodate fixed effects and diffuse initial conditions are developed. The algorithms are based on an augmented Kalman filter, and they avoid the backward recursions of standard smoothing algorithms.

The disturbance smoother is used to develop an EM algorithm for REML estimation of variance components in state space models. The EM algorithm for the structural time series model with polynomial trend and additive seasonality is illustrated in detail.

Key words and phrases: Diffuse prior distribution, EM algorithm, Kalman filter, mixed-model prediction, restricted maximum likelihood, state space model, variance components.