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 35.11.
/*--- Bayesian Vector Autoregressive Process ---*/ proc varmax data=simul1; model y1 y2 / p=1 noint prior=(lambda=0.9 theta=0.1); run;
The output in Figure 35.11 shows that parameter estimates are slightly different from those in Figure 35.3. 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 35.11: Parameter Estimates for the BVAR(1) Model
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 Error |
t Value | Pr > |t| | Variable |
y1 | AR1_1_1 | 1.05623 | 0.04999 | 21.13 | 0.0001 | y1(t-1) |
AR1_1_2 | -0.34707 | 0.04807 | -7.22 | 0.0001 | y2(t-1) | |
y2 | AR1_2_1 | 0.40068 | 0.04867 | 8.23 | 0.0001 | y1(t-1) |
AR1_2_2 | 0.48728 | 0.05670 | 8.59 | 0.0001 | y2(t-1) |
Covariances of Innovations | ||
---|---|---|
Variable | y1 | y2 |
y1 | 1.36278 | 0.45343 |
y2 | 0.45343 | 1.48077 |