Forecasting :: SAS/ETS(R) 13.1 User's Guide

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

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

The STATESPACE Procedure

Forecasting