BEKK Representation

Engle and Kroner (1995) propose a general multivariate GARCH model and call it a BEKK representation. Let be the sigma field generated by the past values of , and let be the conditional covariance matrix of the -dimensional random vector . Let be measurable with respect to ; then the multivariate GARCH model can be written as

where , and are parameter matrices.

Consider a bivariate GARCH(1,1) model as follows:

or, representing the univariate model,

For the BEKK representation of the bivariate GARCH(1,1) model, the SAS statements are:

   model y1 y2;
   garch q=1 p=1 form=bekk;

CCC Representation

Bollerslev (1990) propose a multivariate GARCH model with time-varying conditional variances and covariances but constant conditional correlations.

The conditional covariance matrix consists of

where is a stochastic diagonal matrix with element and is a time-invariant matrix with the typical element .

The elements of are

Estimation of GARCH Model

The log-likelihood function of the multivariate GARCH model is written without a constant term

The log-likelihood function is maximized by an iterative numerical method such as quasi-Newton optimization. The starting values for the regression parameters are obtained from the least squares estimates. The covariance of is used as the starting values for the GARCH constant parameters, and the starting value used for the other GARCH parameters is either or depending on the GARCH models representation. For the identification of the parameters of a BEKK representation GARCH model, the diagonal elements of the GARCH constant, the ARCH, and the GARCH parameters are restricted to be positive.

Covariance Stationarity

Define the multivariate GARCH process as

where , , and . This representation is equivalent to a GARCH() model by the following algebra:

Defining and gives a BEKK representation.

The necessary and sufficient conditions for covariance stationarity of the multivariate GARCH process is that all the eigenvalues of are less than one in modulus.

An Example of a VAR(1)–ARCH(1) Model

The following DATA step simulates a bivariate vector time series to provide test data for the multivariate GARCH model:

   data garch;
      retain seed 16587;
      esq1 = 0; esq2 = 0;
      ly1 = 0;  ly2 = 0;
      do i = 1 to 1000;
         ht = 6.25 + 0.5*esq1;
         call rannor(seed,ehat);
         e1 = sqrt(ht)*ehat;
         ht = 1.25 + 0.7*esq2;
         call rannor(seed,ehat);
         e2 = sqrt(ht)*ehat;
         y1 = 2 + 1.2*ly1 - 0.5*ly2 + e1;
         y2 = 4 + 0.6*ly1 + 0.3*ly2 + e2;
         if i>500 then output;
         esq1 = e1*e1; esq2 = e2*e2;
         ly1 = y1;  ly2 = y2;
      end;
      keep y1 y2;
   run;

The following statements fit a VAR(1)–ARCH(1) model to the data. For a VAR-ARCH model, you specify the order of the autoregressive model with the P=1 option in the MODEL statement and the Q=1 option in the GARCH statement. In order to produce the initial and final values of parameters, the TECH=QN option is specified in the NLOPTIONS statement.

   proc varmax data=garch;
      model y1 y2 / p=1
            print=(roots estimates diagnose);
      garch q=1;
      nloptions tech=qn;
   run;

Figure 30.61 through Figure 30.65 show the details of this example. Figure 30.61 shows the initial values of parameters.