Analysis of Financial Time Series, Third Edition

11.4 Kalman Filter and Smoothing

In this section, we study the Kalman filter and various smoothing methods for the general state-space model in Eqs. (11.26) and (11.27). The derivation follows closely the steps taken in Section 11.1. For readers interested in applications, this section can be skipped at the first read. A good reference for this section is Durbin and Koopman (2001, Chapter 4).

11.4.1 Kalman Filter

Recall that the aim of the Kalman filter is to obtain recursively the conditional distribution of st+1 given the data Ft = {yt, …, yt} and the model. Since the conditional distribution involved is normal, it suffices to study the conditional mean and covariance matrix. Let st+1 and ∑jIi be the conditional mean and covariance matrix of st given Fi, that is, sj I Fi ~ Ni(sjIi, ∑jIi) From Eq. (11.26),

11.55

11.56

Similarly to that of Section 11.1, let ytIt-1 be the conditional mean of yt given Ft−1. From Eq. (11.27),

Let

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be the 1-step-ahead forecast error of yt given Ft−1. It is easy to see that (a) E(vtIFt-1 = 0; (b) ut is independent of Ft−1, that is, Cov(vtyt) = 0 for ...