PROC STATESPACE: Forecasting :: SAS/ETS(R) 9.2 User's Guide

The STATESPACE Procedure

Given estimates of $\text{[math]}$ , $\text{[math]}$ , and $\text{[math]}$ , forecasts of $\text{[math]}$ are computed from the conditional expectation of $\text{[math]}$ .

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

The m-step-ahead forecast of $\text{[math]}$ is $\text{[math]}$ , where $\text{[math]}$ denotes the conditional expectation of $\text{[math]}$ given the information available at time t. The m-step-ahead forecast of $\text{[math]}$ is $\text{[math]}$ , where the matrix $\text{[math]}$ .

Let $\text{[math]}$ . Note that the last $\text{[math]}$ elements of $\text{[math]}$ consist of the elements of $\text{[math]}$ for $\text{[math]}$ .

The state vector $\text{[math]}$ can be represented as

$\text{[math]}$

Since $\text{[math]}$ for $\text{[math]}$ , the m-step-ahead forecast $\text{[math]}$ is

$\text{[math]}$

Therefore, the m-step-ahead forecast of $\text{[math]}$ is

$\text{[math]}$

The m-step-ahead forecast error is

$\text{[math]}$

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

$\text{[math]}$

Letting $\text{[math]}$ , the variance of the m-step-ahead forecast error of $\text{[math]}$ , $\text{[math]}$ , can be computed recursively as follows:

$\text{[math]}$

The variance of the m-step-ahead forecast error of $\text{[math]}$ is the $\text{[math]}$ left upper submatrix of $\text{[math]}$ ; that is,

$\text{[math]}$

Unless the NOCENTER option is specified, the sample mean vector is added to the forecast. When differencing is specified, the forecasts x $\text{[math]}$ plus the sample mean vector are integrated back to produce forecasts for the original series.

Let $\text{[math]}$ 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 $\text{[math]}$ be the $\text{[math]}$ matrix polynomial in the backshift operator that corresponds to the differencing specified by the VAR statement. The off-diagonal elements of $\text{[math]}$ are 0. Note that $\text{[math]}$ , where $\text{[math]}$ is the $\text{[math]}$ identity matrix. Then $\text{[math]}$ .