Forecasting

The STATESPACE Procedure

Forecasting

Given estimates of F, G, and ${{\Sigma}_{ee}}$ ,forecasts of x_t are computed from the conditional expectation of z_t.

In forecasting, the parameters F, G, and ${{\Sigma}_{ee}}$ 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 ${t{\leq}n-b}$ .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 z₀ = 0.

The m-step-ahead forecast of z_t+m is ${z_{t+m| t}}$ ,where ${z_{t+m| t}}$ 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 ${x_{t+m| t} = H{z}_{t+m| t}}$ ,where the matrix H = [I_r 0].

Let ${{\Psi}_{i} = F^i G}$ .Note that the last s-r elements of z_t consist of the elements of ${x_{u| t}}$ for u>t.

The state vector z_t+m can be represented as

$z_{t+m} = F^m z_{t} + \sum_{i=0}^{m-1}{{\Psi}_{i} e_{t+m-i}}$

Since ${e_{t+i| t} = 0}$ for i>0, the m-step-ahead forecast ${z_{t+m| t}}$ is

$z_{t+m| t} = F^m z_{t} = F z_{t+m-1| t}$

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

$x_{t+m| t} = H{z}_{t+m| t}$

The m-step-ahead forecast error is

$z_{t+m}-z_{t+m| t} = \sum_{i=0}^{m-1}{{\Psi}_{i} e_{t+m-i}}$

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

$V_{z,m} = \sum_{i=0}^{m-1}{{\Psi}_{i} {\Sigma}_{ee} {\Psi}_{i}'}$

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:

$V_{z,m} = V_{z,m-1} + {\Psi}_{m-1} {\Sigma}_{ee} {\Psi}^{'}_{m-1}$

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,

V_x,m = HV_z,mH'

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 ${{\Delta}(B)}$ be the s ×s matrix polynomial in the backshift operator corresponding to the differencing specified by the VAR statement. The off-diagonal elements of ${{\Delta}_{i}}$ are 0. Note that ${{\Delta}_{0} = I_{s}}$ , where I_s is the s ×s identity matrix. Then ${z_{t} = {\Delta}(B)y_{t}}$ .

This gives the relationship

$y_{t} = {\Delta}^{-1}(B) z_{t} = \sum_{i=0}^{{\infty}}{{\Lambda}_{i}z_{t-i}}$

where ${{\Delta}^{-1}(B) =\sum_{i=0}^{{\infty}}{{\Lambda}_{i} B^i}}$ and ${{\Lambda}_{0} = I_{s}}$ .

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

$y_{t+m| t} = \sum_{i=0}^{m-1}{{\Lambda}_{i} z_{t+m-i| t}} + \sum_{i=m}^{{\infty}}{{\Lambda}_{i} z_{t+m-i}}$

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

$\sum_{i=0}^{m-1}{{\Lambda}_{i} (z_{t+m-i} - z_{t+m-i| t})} = \sum_{i=0}^{m-1} (\sum_{u=0}^i{{\Lambda}_{u} {\Psi}_{i-u}}) e_{t+m-i}$

Letting V_y,0 = 0, the variance of the m-step-ahead forecast error of y_t+m, V_y,m, is

$V_{y,m} &=& \sum_{i=0}^{m-1}{(\sum_{u=0}^i{{\Lambda}_{u} {\Psi}_{i-u}}) {\Sigma... ...Psi}_{m-1-u}}) {\Sigma}_{ee} (\sum_{u=0}^{m-1}{{\Lambda}_{u} {\Psi}_{m-1-u}})'$

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