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
and the associated forecast error is
Recall that st+jIt and ∑t+jIt are, respectively, the conditional mean and covariance matrix of st+j given Ft. The prior equation says that
11.84 
Furthermore, from Eq. (11.26),
which in turn implies that
Consequently,
Note that Var [et(j)] = Vt+j and Eqs. (11.83) and (11.85) are the recursion of the Kalman filter in Eq. (11.64) for t + j with j = 1, … , h when vt+j = 0 and Kt+j = 0. Thus, the forecast yt(j) and the covariance matrix of its forecast error et(j) can be obtained via the Kalman filter with missing values.
Finally, the prediction error series {vt} can be used to evaluate the likelihood function for estimation and the standardized prediction errors 
can be used for model checking, where D = diag{Vt(1, 1), …, Vt, (k, k)} with Vt(i, i) being the (i, i)th element of Vt.