The VARMAX Procedure

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 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.

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

and hence

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

Thus, the covariance matrix of the -step-ahead prediction error is given as

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

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

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.

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

where and .

The -step-ahead prediction of is

The -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