Exact Initial Kalman Filtering and Smoothing for Nonstationary Time Series Models
Siem Jan Koopman · 2005
Abstract Assume that a vector of observations yt is generated by the Gaussian state-space model where p 1 vector αtis the state vector. The system matrices, Zt , Tt , Gt, and Ht, for t 1,..., n, are assumed to be fixed and known. The former equation of (1) is referred to as the observation equation, whereas the latter equation is called the transition equation. A normally distributed random vector x with mean µ and variance matrix A is denoted by x N(µ, A). The disturbance vector εtis serially uncorrelated. The appearance of εtin both equations is general rather than restrictive. The special case HtG± t 0, for t 1,…, n, implies mutual independence between the two sets of disturbances. The mean vector and variance matrix P of the initial state vector are assumed to be known. The state-space model (1) is often used as a framework for representing linear time series models such as autoregressive integrated moving average (ARIMA)