The VARMAX Procedure

Stochastic volatility modeling is important in many areas, particularly in finance. To study the volatility of time series, GARCH models are widely used because they provide a good approach to conditional variance modeling.

### 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.

Figure 30.61 Start Parameter Estimates for the VAR(1)–ARCH(1) Model
The VARMAX Procedure

Optimization Start
Parameter Estimates
N Parameter Estimate Gradient
Objective
Function
1 CONST1 2.249575 5.787988
2 CONST2 3.902673 -4.856056
3 AR1_1_1 1.231775 -17.155796
4 AR1_2_1 0.576890 23.991176
5 AR1_1_2 -0.528405 14.656979
6 AR1_2_2 0.343714 -12.763695
7 GCHC1_1 9.929763 -0.111361
8 GCHC1_2 0.193163 -0.684986
9 GCHC2_2 4.063245 0.139403
10 ACH1_1_1 0.001000 -0.668058
11 ACH1_2_1 0 -0.068657
12 ACH1_1_2 0 -0.735896
13 ACH1_2_2 0.001000 -3.126628

Figure 30.62 shows the final parameter estimates.

Figure 30.62 Results of Parameter Estimates for the VAR(1)–ARCH(1) Model
The VARMAX Procedure

Optimization Results
Parameter Estimates
N Parameter Estimate
1 CONST1 1.943991
2 CONST2 4.073898
3 AR1_1_1 1.220945
4 AR1_2_1 0.608263
5 AR1_1_2 -0.527121
6 AR1_2_2 0.303012
7 GCHC1_1 8.359045
8 GCHC1_2 -0.182483
9 GCHC2_2 1.602739
10 ACH1_1_1 0.377569
11 ACH1_2_1 0.032158
12 ACH1_1_2 0.056491
13 ACH1_2_2 0.710023

Figure 30.63 shows the conditional variance using the BEKK representation of the ARCH(1) model. The ARCH parameters are estimated by the vectorized parameter matrices.

Figure 30.63 ARCH(1) Parameter Estimates for the VAR(1)–ARCH(1) Model
The VARMAX Procedure

Type of Model VAR(1)-ARCH(1) Maximum Likelihood Estimation BEKK

GARCH Model Parameter Estimates
Parameter Estimate Standard
Error
t Value Pr > |t|
GCHC1_1 8.35905 0.73116 11.43 0.0001
GCHC1_2 -0.18248 0.21706 -0.84 0.4009
GCHC2_2 1.60274 0.19398 8.26 0.0001
ACH1_1_1 0.37757 0.07470 5.05 0.0001
ACH1_2_1 0.03216 0.06971 0.46 0.6448
ACH1_1_2 0.05649 0.02622 2.15 0.0317
ACH1_2_2 0.71002 0.06844 10.37 0.0001

Figure 30.64 shows the AR parameter estimates and their significance.

The fitted VAR(1) model with the previous conditional covariance ARCH model is written as follows:

Figure 30.64 VAR(1) Parameter Estimates for the VAR(1)–ARCH(1) Model
Model Parameter Estimates
Equation Parameter Estimate Standard
Error
t Value Pr > |t| Variable
y1 CONST1 1.94399 0.21017 9.25 0.0001 1
AR1_1_1 1.22095 0.02564 47.63 0.0001 y1(t-1)
AR1_1_2 -0.52712 0.02836 -18.59 0.0001 y2(t-1)
y2 CONST2 4.07390 0.10574 38.53 0.0001 1
AR1_2_1 0.60826 0.01231 49.42 0.0001 y1(t-1)
AR1_2_2 0.30301 0.01498 20.23 0.0001 y2(t-1)

Figure 30.65 shows the roots of the AR and ARCH characteristic polynomials. The eigenvalues have a modulus less than one.

Figure 30.65 Roots for the VAR(1)–ARCH(1) Model
Roots of AR Characteristic Polynomial
Index Real Imaginary Modulus Radian Degree
1 0.76198 0.33163 0.8310 0.4105 23.5197
2 0.76198 -0.33163 0.8310 -0.4105 -23.5197

Roots of GARCH Characteristic Polynomial
Index Real Imaginary Modulus Radian Degree
1 0.51180 0.00000 0.5118 0.0000 0.0000
2 0.26627 0.00000 0.2663 0.0000 0.0000
3 0.26627 0.00000 0.2663 0.0000 0.0000
4 0.13853 0.00000 0.1385 0.0000 0.0000

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