Time Series Analysis: The Kalman Filter

CHAPTER ONE

INTRODUCTION

Time Series Analysis is a statistical technique used mainly to infer properties of a system by the analysis of its data measured in time. This is done by fitting a typical model to the data with the aim of discovering the underlying structure as closely as possible. Traditional time series analysis is based on assumptions of linearity and stationarity. However, many real world problems do not satisfy the assumptions of linearity and/or stationarity. This gave rise to increased interest in studying nonlinear and nonstationary time series models in many practical problems. For example, the financial markets are one of the areas where there is a greater need to explain behaviours that are far from being even approximately

Kalman filter was applied to navigation for the Apollo Project, which required estimates of the trajectories of manned spacecraft going to the Moon and back. With the lives of the astronauts at stake, it was essential that the Kalman filter be proven effective and reliable before it could be used. There are several state-space models which were proposed by several authors, and each consists of two equations for a process {Yt}. Suppose that an observed vector series {Yt} can be written in terms of an observed state vector {Xt} (of dimension v). This first equation is known as the observation equation and the second equation is known as state equation. The state equation determines the evolution of the state Xt at time ‘t’ in terms of the previous state Xt−1 and a noise term Vt. This method is reviewed in the Next Chapter where it is shown that the Kalman filter is a linear, discrete time, finite dimensional time-varying system that evaluates the state estimate that minimizes the mean-square

When either the system state dynamics or the observation dynamics is nonlinear, the conditional probability density functions that provide the minimum mean-square estimate are no longer Gaussian. The optimal non-linear filter transmits these non-Gaussian functions and evaluate their mean, which represents a high computational burden. A non-optimal approach to solve the problem, in the frame of linear filters, is the Extended Kalman filter (EKF) or the Unscented Kalman filter (UKF). The EKF or UKF implements a Kalman filter for a system dynamics that results from the linearization of the original nonlinear filter dynamics around the previous state estimates.

This research develops an algorithm for the application of the Smooth Transition Autoregressive (STAR) methodology by Terasvirta (1994) to the estimation of the state equation of the Kalman filtering technique.

Smooth Transition Autoregressive (STAR)