The STATESPACE Procedure

Forecasting

Given estimates of ${\mb{F}}$, ${\mb{G}}$, and ${\bSigma _{\mb{ee}}}$, forecasts of ${\mb{x}_{t}}$ are computed from the conditional expectation of ${\mb{z}_{t}}$.

In forecasting, the parameters F, G, and ${\bSigma _{\mb{ee}}}$ are replaced with the estimates or by values specified in the RESTRICT statement. One-step-ahead forecasting is performed for the observation ${\mb{x}_{t}}$, where ${t{\leq }n-b}$. Here ${n}$ is the number of observations and b is the value of the BACK= option. For the observation ${\mb{x}_{t}}$, where ${t > n-b}$, m-step-ahead forecasting is performed for ${m = t-n + b}$. The forecasts are generated recursively with the initial condition ${\mb{z}_{0} = 0}$.

The m-step-ahead forecast of ${\mb{z}_{t+m}}$ is ${\mb{z}_{t+m|t}}$, where ${\mb{z}_{t+m|t}}$ denotes the conditional expectation of ${\mb{z}_{t+m}}$ given the information available at time t. The m-step-ahead forecast of ${\mb{x}_{t+m}}$ is ${\mb{x}_{t+m|t} = \mb{H} \mb{z}_{t+m|t}}$, where the matrix ${\mb{H} = [\mb{I}_{r} \mb{0} ]}$.

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

The state vector ${\mb{z}_{t+m}}$ can be represented as

\[ \mb{z}_{t+m} = \mb{F} ^{m} \mb{z}_{t} + \sum _{i=0}^{m-1}{\bPsi _{i} \mb{e}_{t+m-i}} \]

Since ${\mb{e}_{t+i|t} = \mb{0}}$ for ${i>0}$, the m-step-ahead forecast ${\mb{z}_{t+m|t}}$ is

\[ \mb{z}_{t+m|t} = \mb{F} ^{m} \mb{z}_{t} = \mb{F} \mb{z}_{t+m-1|t} \]

Therefore, the m-step-ahead forecast of ${\mb{x}_{t+m}}$ is

\[ \mb{x}_{t+m|t} = \mb{H} \mb{z}_{t+m|t} \]

The m-step-ahead forecast error is

\[ \mb{z}_{t+m}-\mb{z}_{t+m|t} = \sum _{i=0}^{m-1}{\bPsi _{i} \mb{e}_{t+m-i}} \]

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

\[ \mb{V}_{z,m} = \sum _{i=0}^{m-1}{\bPsi _{i} \bSigma _{\mb{ee}} {\Psi }_{i}’} \]

Letting ${\mb{V}_{z,0} = \mb{0}}$, the variance of the m-step-ahead forecast error of ${\mb{z}_{t+m}}$, ${\mb{V}_{z,m}}$, can be computed recursively as follows:

\[ \mb{V}_{z,m} = \mb{V}_{z,m-1} + \bPsi _{m-1} \bSigma _{\mb{ee}} \bPsi ^{'}_{m-1} \]

The variance of the m-step-ahead forecast error of ${\mb{x}_{t+m}}$ is the ${r \times r}$ left upper submatrix of ${\mb{V}_{z,m}}$; that is,

\[ \mb{V}_{x,m} = \mb{H} \mb{V}_{z,m}\mb{H} ’ \]

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 ${\mb{y}_{t}}$ 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 ${\bDelta ({B})}$ be the ${s \times s}$ matrix polynomial in the backshift operator that corresponds to the differencing specified by the VAR statement. The off-diagonal elements of ${\bDelta _{i}}$ are 0. Note that ${\bDelta _{0} = \mb{I}_{s}}$, where ${\mb{I}_{s}}$ is the ${s \times s}$ identity matrix. Then ${\mb{z}_{t} = \bDelta ({B})\mb{y}_{t}}$.

This gives the relationship

\[ \mb{y}_{t} = \bDelta ^{-1}(B) \mb{z}_{t} = \sum _{i=0}^{{\infty }}{\bLambda _{i}\mb{z}_{t-i}} \]

where ${\bDelta ^{-1}(B) =\sum _{i=0}^{{\infty }}{\bLambda _{i} B^{i}}}$ and ${\bLambda _{0} = \mb{I}_{s}}$.

The m-step-ahead forecast of ${\mb{y}_{t+m}}$ is

\[ \mb{y}_{t+m|t} = \sum _{i=0}^{m-1}{\bLambda _{i} \mb{z}_{t+m-i|t}} + \sum _{i=m}^{{\infty }}{\bLambda _{i} \mb{z}_{t+m-i}} \]

The m-step-ahead forecast error of ${\mb{y}_{t+m}}$ is

\begin{eqnarray*} \sum _{i=0}^{m-1}{\bLambda _{i} \left(\Strong{z}_{t+m-i} - \Strong{z}_{t+m-i|t}\right)} = \sum _{i=0}^{m-1} \left(\sum _{u=0}^{i}{\bLambda _{u} \bPsi _{i-u}}\right) \Strong{e}_{t+m-i} \nonumber \end{eqnarray*}

Letting ${\mb{V}_{y,0} = \mb{0}}$, the variance of the m-step-ahead forecast error of ${\mb{y}_{t+m}}$, ${\mb{V}_{y,m}}$, is

\begin{eqnarray*} \Strong{V}_{y,m} & =& \sum _{i=0}^{m-1}{\left(\sum _{u=0}^{i}{\bLambda _{u} \bPsi _{i-u}}\right) \bSigma _{\mb{ee}} \left(\sum _{u=0}^{i}{\bLambda _{u} \bPsi _{i-u}}\right)’} \\ & =& \Strong{V}_{y,m-1} + \left(\sum _{u=0}^{m-1}{\bLambda _{u} \bPsi _{m-1-u}}\right) \bSigma _{\mb{ee}} \left(\sum _{u=0}^{m-1}{\bLambda _{u} \bPsi _{m-1-u}}\right)’ \nonumber \end{eqnarray*}