Analysis of Financial Time Series, Third Edition

11.5 Missing Values

For the general state-space model in Eqs. (11.26) and (11.27), we consider two cases of missing values. First, suppose that similar to the local trend model in Section 11.1 the observations yt at t = ℓ + 1, … , ℓ + h are missing. In this case, there is no new information available at these time points and we set

The Kalman filter in Eq. (11.64) can then proceed as usual. That is,

for t = ℓ + 1, … , ℓ + h. Similarly, the smoothed state vectors can be computed as usual via Eq. (11.74) with

for t = ℓ + 1, … , ℓ + h.

In the second case, some components of yt are missing. Let be the vector of observed data at time t, where J is an indicator matrix identifying the observed data. More specifically, rows of J are a subset of the rows of the k × k identity matrix. In this case, the observation equation (11.27) of the model can be transformed as

where , , and = Jet with covariance ...