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. Onestepahead 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 , mstepahead forecasting is performed for . The forecasts are generated recursively with the initial condition .
The mstepahead forecast of is , where denotes the conditional expectation of given the information available at time t. The mstepahead 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 mstepahead forecast is

Therefore, the mstepahead forecast of is

The mstepahead forecast error is

The variance of the mstepahead forecast error is

Letting , the variance of the mstepahead forecast error of , , can be computed recursively as follows:

The variance of the mstepahead 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 offdiagonal elements of are 0. Note that , where is the identity matrix. Then .
This gives the relationship

where and .
The mstepahead forecast of is

The mstepahead forecast error of is

Letting , the variance of the mstepahead forecast error of , , is





