The VARMAX Procedure

Bayesian Vector Autoregressive Process

The Bayesian vector autoregressive (BVAR) model is used to avoid problems of collinearity and over-parameterization that often occur with the use of VAR models. BVAR models do this by imposing priors on the AR parameters.

The following statements fit a BVAR(1) model to the simulated data. You specify the PRIOR= option with the hyperparameters. The LAMBDA=0.9 and THETA=0.1 options are hyperparameters controlling the prior covariance. Part of the VARMAX procedure output is shown in Figure 36.11.

/*--- Bayesian Vector Autoregressive Process ---*/

proc varmax data=simul1;
   model y1 y2 / p=1 noint
                 prior=(lambda=0.9 theta=0.1);

The output in Figure 36.11 shows that parameter estimates are slightly different from those in Figure 36.3. By choosing the appropriate priors, you might be able to get more accurate forecasts by using a BVAR model rather than by using an unconstrained VAR model. See the section Bayesian VAR and VARX Modeling for details.

Figure 36.11: Parameter Estimates for the BVAR(1) Model

The VARMAX Procedure

Type of Model BVAR(1)
Estimation Method Maximum Likelihood Estimation
Prior Lambda 0.9
Prior Theta 0.1

Model Parameter Estimates
Equation Parameter Estimate Standard
t Value Pr > |t| Variable
y1 AR1_1_1 1.05623 0.05050 20.92 0.0001 y1(t-1)
  AR1_1_2 -0.34707 0.04824 -7.19 0.0001 y2(t-1)
y2 AR1_2_1 0.40068 0.04889 8.20 0.0001 y1(t-1)
  AR1_2_2 0.48728 0.05740 8.49 0.0001 y2(t-1)

Covariances of Innovations
Variable y1 y2
y1 1.36278 0.45343
y2 0.45343 1.48077