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| The VARMAX Procedure |
| Forecasting |
The optimal (minimum MSE) 
-step-ahead forecast of 
is
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with 
and 
for 
. For the forecasts 
, see the section State-Space Representation.
Covariance Matrices of Prediction Errors without Exogenous (Independent) Variables
Under the stationarity assumption, the optimal (minimum MSE) -step-ahead forecast of has an infinite moving-average form, 
. The prediction error of the optimal -step-ahead forecast is 
, with zero mean and covariance matrix,
 |
where 
with a lower triangular matrix 
such that 
. Under the assumption of normality of the 
, the -step-ahead prediction error 
is also normally distributed as multivariate 
. Hence, it follows that the diagonal elements 
of 
can be used, together with the point forecasts 
, to construct -step-ahead prediction intervals of the future values of the component series, 
.
The following statements use the COVPE option to compute the covariance matrices of the prediction errors for a VAR(1) model. The parts of the VARMAX procedure output are shown in Figure 32.36 and Figure 32.37.
proc varmax data=simul1;
model y1 y2 / p=1 noint lagmax=5
printform=both
print=(decompose(5) impulse=(all) covpe(5));
run;
Figure 32.36 is the output in a matrix format associated with the COVPE option for the prediction error covariance matrices.
| Prediction Error Covariances | |||
|---|---|---|---|
| Lead | Variable | y1 | y2 |
| 1 | y1 | 1.28875 | 0.39751 |
| y2 | 0.39751 | 1.41839 | |
| 2 | y1 | 2.92119 | 1.00189 |
| y2 | 1.00189 | 2.18051 | |
| 3 | y1 | 4.59984 | 1.98771 |
| y2 | 1.98771 | 3.03498 | |
| 4 | y1 | 5.91299 | 3.04856 |
| y2 | 3.04856 | 4.07738 | |
| 5 | y1 | 6.69463 | 3.85346 |
| y2 | 3.85346 | 5.07010 | |
Figure 32.37 is the output in a univariate format associated with the COVPE option for the prediction error covariances. This printing format more easily explains the prediction error covariances of each variable.
Covariance Matrices of Prediction Errors in the Presence of Exogenous (Independent) Variables
Exogenous variables can be both stochastic and nonstochastic (deterministic) variables. Considering the forecasts in the VARMAX(
,
,
) model, there are two cases.
As defined in the section State-Space Representation, 
has the representation
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and hence
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Therefore, the covariance matrix of the -step-ahead prediction error is given as
 |
where 
is the covariance of the white noise series 
, and is the white noise series for the VARMA(,) model of exogenous (independent) variables, which is assumed not to be correlated with 
or its lags.
The optimal forecast of 
conditioned on the past information and also on known future values 
can be represented as
 |
and the forecast error is
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Thus, the covariance matrix of the -step-ahead prediction error is given as
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Decomposition of Prediction Error Covariances
In the relation 
, the diagonal elements can be interpreted as providing a decomposition of the -step-ahead prediction error covariance 
for each component series 
into contributions from the components of the standardized innovations .
If you denote the (
)th element of 
by 
, the MSE of 
is
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Note that 
is interpreted as the contribution of innovations in variable 
to the prediction error covariance of the -step-ahead forecast of variable 
.
The proportion, 
, of the -step-ahead forecast error covariance of variable accounting for the innovations in variable is
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The following statements use the DECOMPOSE option to compute the decomposition of prediction error covariances and their proportions for a VAR(1) model:
proc varmax data=simul1;
model y1 y2 / p=1 noint print=(decompose(15))
printform=univariate;
run;
The proportions of decomposition of prediction error covariances of two variables are given in Figure 32.38. The output explains that about 91.356% of the one-step-ahead prediction error covariances of the variable 
is accounted for by its own innovations and about 8.644% is accounted for by 
innovations.
Forecasting of the Centered Series
If the CENTER option is specified, the sample mean vector is added to the forecast.
Forecasting of the Differenced Series
If dependent (endogenous) variables are differenced, the final forecasts and their prediction error covariances are produced by integrating those of the differenced series. However, if the PRIOR option is specified, the forecasts and their prediction error variances of the differenced series are produced.
Let 
be the original series with some appended zero values that correspond to the unobserved past observations. Let 
be the 
matrix polynomial in the backshift operator that corresponds to the differencing specified by the MODEL statement. The off-diagonal elements of 
are zero, and the diagonal elements can be different. Then 
.
This gives the relationship
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where 
and 
.
The -step-ahead prediction of 
is
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The -step-ahead prediction error of is
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Letting 
, the covariance matrix of the l-step-ahead prediction error of , 
, is
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If there are stochastic exogenous (independent) variables, the covariance matrix of the l-step-ahead prediction error of , , is
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