GARCH Models, 2nd Edition

Appendix D The Kalman Filter

The state‐space representation is a useful tool for analysing many dynamic models. When this representation exists, the Kalman filter can be applied to estimate the model parameters, to compute predictions or to smooth the series. This technique was introduced by Kalman (1960) in the field of engineering but has been used in various domains, in particular in economics.

We first introduce state‐space representations and useful notations. Let (y _t ) denote a ℝ ^N ‐valued observable process, and let (α _t ) a ℝ ^m ‐valued latent (in general non‐observable, or only partially observable) process. A state‐space model is defined by

D.1

where M _t , d _t , T _t , c _t and R _t are deterministic matrices of appropriate dimensions, (u _t ) and ( ) are white noises, respectively, valued in ℝ ^N and ℝ ^m . The vector α _t is called space vector. The first equation is called measurement equation, and the second equation transition equation.

The Kalman filter is an algorithm for

1. (i) predicting the period‐ t space vector from observations of y up to time t − 1;
2. (ii) filtering, that is predicting the period‐ t space vector from observations of y up to time t ;
3. (iii) smoothing, that is estimating the value of α _t from observations of ...