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