The optimal (minimum MSE) l-step-ahead forecast of is
with and for . For the forecasts , see the section State-Space Representation.
Under the stationarity assumption, the optimal (minimum MSE) l-step-ahead forecast of has an infinite moving-average form, . The prediction error of the optimal l-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 l-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 l-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 35.36 and Figure 35.37.
proc varmax data=simul1; model y1 y2 / p=1 noint lagmax=5 printform=both print=(decompose(5) impulse=(all) covpe(5)); run;
Figure 35.36 is the output in a matrix format associated with the COVPE option for the prediction error covariance matrices.
Figure 35.37 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.
Exogenous variables can be both stochastic and nonstochastic (deterministic) variables. Considering the forecasts in the VARMAX(p,q,s) model, there are two cases.
When exogenous (independent) variables are stochastic (future values not specified):
As defined in the section State-Space Representation, has the representation
and hence
Therefore, the covariance matrix of the l-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(p,q) 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 l-step-ahead prediction error is given as
In the relation , the diagonal elements can be interpreted as providing a decomposition of the l-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
Note that is interpreted as the contribution of innovations in variable n to the prediction error covariance of the l-step-ahead forecast of variable i.
The proportion, , of the l-step-ahead forecast error covariance of variable i accounting for the innovations in variable n 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 35.38. 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.
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 off-diagonal elements of are zero, and the diagonal elements can be different. Then .
This gives the relationship
where and .
The l-step-ahead prediction of is
The l-step-ahead prediction error of is
Letting , the covariance matrix of the l-step-ahead prediction error of , , is
If there are stochastic exogenous (independent) variables, the covariance matrix of the l-step-ahead prediction error of , , is