Analysis of Financial Time Series, Third Edition

11.6 Forecasting

Suppose that the forecast origin is t and we are interested in predicting yt+j for j = 1, … , h, where h > 0. Also, we adopt the minimum mean-squared error forecasts. Similar to the ARMA models, the j-step-ahead forecast yt(j) turns out to be the expected value of yt(j) given Ft and the model. That is, yt(j) = E(Tt+jIFt). In what follows, we show that these forecasts and the covariance matrices of the associated forecast errors can be obtained via the Kalman filter in Eq. (11.64) by treating yt+1, …, yt+h as missing values, that is, the first case in Section 11.5.

Consider the 1-step-ahead forecast. From Eq. (11.27),

where st+1It is available via the Kalman filter at the forecast origin t. The associated forecast error is

Therefore, the covariance matrix of the 1-step-ahead forecast error is

This is precisely the covariance matrix Vt+1 of the Kalman filter in Eq. (11.64). Thus, we have showed the case for h = 1.

Now, for h > 1, we consider 1-step- to h-step-ahead forecasts sequentially. From Eq. (11.27), the j-step-ahead forecast is

11.83

and the associated forecast ...