The VARMAX Procedure

Bayesian Vector Error Correction Model

Bayesian inference on a cointegrated system begins by using the priors of $\bbeta$ obtained from the VECM( $p$ ) form. Bayesian vector error correction models can improve forecast accuracy for cointegrated processes.

The following statements fit a BVECM(2) form to the simulated data. You specify both the PRIOR= and ECM= options for the Bayesian vector error correction model. The VARMAX procedure output is shown in Figure 35.17.

/*--- Bayesian Vector Error-Correction Model ---*/

proc varmax data=simul2;
   model y1 y2 / p=2 noint
                 prior=( lambda=0.5 theta=0.2 )
                 ecm=( rank=1 normalize=y1 )
                 print=(estimates);
run;

Figure 35.17 shows the model type fitted to the data, the estimates of the adjustment coefficient ( $\balpha$ ), the parameter estimates in terms of lag one coefficients ( $\mb {y} _{t-1}$ ), and lag one first differenced coefficients ( $\Delta \mb {y} _{t-1}$ ).

Figure 35.17: Parameter Estimates for the BVECM(2) Form

The VARMAX Procedure

Type of Model	BVECM(2)
Estimation Method	Maximum Likelihood Estimation
Cointegrated Rank	1
Prior Lambda	0.5
Prior Theta	0.2

Alpha
Variable	1
y1	-0.34392
y2	0.16659

Parameter Alpha * Beta' Estimates
Variable	y1	y2
y1	-0.34392	0.67262
y2	0.16659	-0.32581

AR Coefficients of Differenced Lag
DIF Lag	Variable	y1	y2
1	y1	-0.80070	-0.59320
	y2	0.33417	-0.53480