Exact Initial Kalman Filtering and Smoothing for Nonstationary Time Series Models

Abstract

This article presents a new exact solution for the initialization of the Kalman filter for state space models with diffuse initial conditions. For example, the regression model with stochastic trend, seasonal and other nonstationary autoregressive integrated moving average components requires a (partially) diffuse initial state vector. The proposed analytical solution is easy to implement and computationally efficient. The exact solution for smoothing is also given. Missing observations are handled in a straightforward manner. All proofs rely on elementary results.

Key Words:

• Autoregressive integrated moving average component models
• Diffuse initial conditions
• Likelihood function and score vector
• Missing observations
• State space.