The optimal (minimum MSE) stepahead forecast of is






with and for . For the forecasts , see the section StateSpace Representation.
Under the stationarity assumption, the optimal (minimum MSE) stepahead forecast of has an infinite movingaverage form, . The prediction error of the optimal stepahead forecast is , with zero mean and covariance matrix,

where with a lower triangular matrix such that . Under the assumption of normality of the , the stepahead 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 stepahead 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 36.36 and Figure 36.37.
proc varmax data=simul1; model y1 y2 / p=1 noint lagmax=5 printform=both print=(decompose(5) impulse=(all) covpe(5)); run;
Figure 36.36 is the output in a matrix format associated with the COVPE option for the prediction error covariance matrices.
Figure 36.36: Covariances of Prediction Errors (COVPE Option)
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 36.37 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.
Figure 36.37: Covariances of Prediction Errors
Prediction Error Covariances by Variable  

Variable  Lead  y1  y2 
y1  1  1.28875  0.39751 
2  2.92119  1.00189  
3  4.59984  1.98771  
4  5.91299  3.04856  
5  6.69463  3.85346  
y2  1  0.39751  1.41839 
2  1.00189  2.18051  
3  1.98771  3.03498  
4  3.04856  4.07738  
5  3.85346  5.07010 
Exogenous variables can be both stochastic and nonstochastic (deterministic) variables. Considering the forecasts in the VARMAX(,,) model, there are two cases.
When exogenous (independent) variables are stochastic (future values not specified):
As defined in the section StateSpace Representation, has the representation

and hence

Therefore, the covariance matrix of the stepahead 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.
When future exogenous (independent) variables are specified:
The optimal forecast of conditioned on the past information and also on known future values can be represented as

and the forecast error is

Thus, the covariance matrix of the stepahead prediction error is given as

In the relation , the diagonal elements can be interpreted as providing a decomposition of the stepahead 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

Note that is interpreted as the contribution of innovations in variable to the prediction error covariance of the stepahead forecast of variable .
The proportion, , of the stepahead forecast error covariance of variable accounting for the innovations in variable is

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 36.38. The output explains that about 91.356% of the onestepahead prediction error covariances of the variable is accounted for by its own innovations and about 8.644% is accounted for by innovations.
Figure 36.38: Decomposition of Prediction Error Covariances (DECOMPOSE Option)
Proportions of Prediction Error Covariances by Variable 


Variable  Lead  y1  y2 
y1  1  1.00000  0.00000 
2  0.88436  0.11564  
3  0.75132  0.24868  
4  0.64897  0.35103  
5  0.58460  0.41540  
y2  1  0.08644  0.91356 
2  0.31767  0.68233  
3  0.50247  0.49753  
4  0.55607  0.44393  
5  0.53549  0.46451 
If the CENTER option is specified, the sample mean vector is added to the forecast.
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 offdiagonal elements of are zero, and the diagonal elements can be different. Then .
This gives the relationship

where and .
The stepahead prediction of is

The stepahead prediction error of is

Letting , the covariance matrix of the lstepahead prediction error of , , is






If there are stochastic exogenous (independent) variables, the covariance matrix of the lstepahead prediction error of , , is





