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The STATESPACE Procedure |
Given estimates of F, G, and ,forecasts of x_{t} are computed from the conditional expectation of z_{t}.
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 x_{t}, where .Here n is the number of observations and b is the value of the BACK= option. For the observation x_{t}, where t > n-b, m-step-ahead forecasting is performed for m = t-n + b. The forecasts are generated recursively with the initial condition z_{0} = 0.
The m-step-ahead forecast of z_{t+m} is ,where denotes the conditional expectation of z_{t+m} given the information available at time t. The m-step-ahead forecast of x_{t+m} is ,where the matrix H = [I_{r} 0].
Let .Note that the last s-r elements of z_{t} consist of the elements of for u>t.
The state vector z_{t+m} can be represented as
Since for i>0, the m-step-ahead forecast is
Therefore, the m-step-ahead forecast of x_{t+m} is
The m-step-ahead forecast error is
The variance of the m-step-ahead forecast error is
Letting V_{z,0} = 0, the variance of the m-step-ahead forecast error of z_{t+m}, V_{z,m}, can be computed recursively as follows:
The variance of the m-step-ahead forecast error of x_{t+m} is the r ×r left upper submatrix of V_{z,m}; that is,
Unless the NOCENTER option is specified, the sample mean vector is added to the forecast. When differencing is specified, the forecasts x_{t+m|t} plus the sample mean vector are integrated back to produce forecasts for the original series.
Let y_{t} 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 I_{s} is the s ×s identity matrix. Then .
This gives the relationship
where and .
The m-step-ahead forecast of y_{t+m} is
The m-step-ahead forecast error of y_{t+m} is
Letting V_{y,0} = 0, the variance of the m-step-ahead forecast error of y_{t+m}, V_{y,m}, is
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