Given estimates of , , and , forecasts of are computed from the conditional expectation of .

In forecasting, the parameters **F**, **G**, and are replaced with the estimates or by values specified in the RESTRICT statement. One-step-ahead forecasting is performed
for the observation , where . Here is the number of observations and *b* is the value of the BACK= option. For the observation , where , *m*-step-ahead forecasting is performed for . The forecasts are generated recursively with the initial condition .

The *m*-step-ahead forecast of is , where denotes the conditional expectation of given the information available at time *t*. The *m*-step-ahead forecast of is , where the matrix .

Let . Note that the last elements of consist of the elements of for .

The state vector can be represented as

Since for , the *m*-step-ahead forecast is

Therefore, the *m*-step-ahead forecast of is

The *m*-step-ahead forecast error is

The variance of the *m*-step-ahead forecast error is

Letting , the variance of the *m*-step-ahead forecast error of , , can be computed recursively as follows:

The variance of the *m*-step-ahead forecast error of is the left upper submatrix of ; that is,

Unless the NOCENTER option is specified, the sample mean vector is added to the forecast. When differencing is specified,
the forecasts **x** plus the sample mean vector are integrated back to produce forecasts for the original series.

Let be the original series specified by the VAR statement, with some 0 values appended that correspond to the unobserved past observations. Let B be the backshift operator, and let be the matrix polynomial in the backshift operator that corresponds to the differencing specified by the VAR statement. The off-diagonal elements of are 0. Note that , where is the identity matrix. Then .

This gives the relationship

where and .

The *m*-step-ahead forecast of is

The *m*-step-ahead forecast error of is

Letting , the variance of the *m*-step-ahead forecast error of , , is