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 The STATESPACE Procedure

## Forecasting

Given estimates of F, G, and ,forecasts of xt are computed from the conditional expectation of zt.

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 xt, where .Here n is the number of observations and b is the value of the BACK= option. For the observation xt, where t > n-b, m-step-ahead forecasting is performed for m = t-n + b. The forecasts are generated recursively with the initial condition z0 = 0.

The m-step-ahead forecast of zt+m is ,where denotes the conditional expectation of zt+m given the information available at time t. The m-step-ahead forecast of xt+m is ,where the matrix H = [Ir 0].

Let .Note that the last s-r elements of zt consist of the elements of for u>t.

The state vector zt+m can be represented as

Since for i>0, the m-step-ahead forecast is

Therefore, the m-step-ahead forecast of xt+m is

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

Letting Vz,0 = 0, the variance of the m-step-ahead forecast error of zt+m, Vz,m, can be computed recursively as follows:

The variance of the m-step-ahead forecast error of xt+m is the r ×r left upper submatrix of Vz,m; that is,

Vx,m = HVz,mH'

Unless the NOCENTER option is specified, the sample mean vector is added to the forecast. When differencing is specified, the forecasts xt+m|t plus the sample mean vector are integrated back to produce forecasts for the original series.

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

This gives the relationship

where and .

The m-step-ahead forecast of yt+m is

The m-step-ahead forecast error of yt+m is

Letting Vy,0 = 0, the variance of the m-step-ahead forecast error of yt+m, Vy,m, is

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