Multivariate GARCH Modeling :: SAS/ETS(R) 12.3 User's Guide

BEKK Representation

Engle and Kroner (1995) propose a general multivariate GARCH model and call it a BEKK representation. Let $\mathcal{F}(t-1)$ be the sigma field generated by the past values of $\bepsilon _ t$ , and let be the conditional covariance matrix of the -dimensional random vector $\bepsilon _ t$ . Let be measurable with respect to $\mathcal{F}(t-1)$ ; then the multivariate GARCH model can be written as

$\displaystyle \bepsilon _ t\|\mathcal{F}(t-1)$	$\displaystyle \sim$	$\displaystyle N(0,H_ t)$
$\displaystyle H_ t$	$\displaystyle =$	$\displaystyle C + \sum _{i=1}^ q A_{i}’ \bepsilon _{t-i} \bepsilon _{t-i}’A_{i} + \sum _{i=1}^ p G_{i}’H_{t-i}G_{i}$

where , $A_{i}$ and $G_{i}$ are $k\times k$ parameter matrices.

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

$\displaystyle H_ t$	$\displaystyle =$	$\displaystyle \left[ \begin{array}{cc} c_{11} & c_{12} \\ c_{12} & c_{22} \end{array} \right] +\left[ \begin{array}{cc} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array} \right]’ \left[ \begin{array}{cc}\epsilon ^2_{1,t-1} & \epsilon _{1,t-1}\epsilon _{2,t-1} \\ \epsilon _{2,t-1}\epsilon _{1,t-1} & \epsilon ^2_{2,t-1} \end{array} \right] \left[ \begin{array}{cc} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array} \right]$
$\displaystyle$	$\displaystyle$	$\displaystyle + \left[ \begin{array}{cc} g_{11} & g_{12} \\ g_{21} & g_{22} \end{array} \right]’ H_{t-1}\left[ \begin{array}{cc} g_{11} & g_{12} \\ g_{21} & g_{22} \end{array} \right]$

or, representing the univariate model,

$\displaystyle h_{11,t}$	$\displaystyle =$	$\displaystyle c_{11} + a_{11}^2\epsilon ^2_{1,t-1} + 2a_{11}a_{21}\epsilon _{1,t-1}\epsilon _{2,t-1} + a_{21}^2\epsilon ^2_{2,t-1}$
$\displaystyle$	$\displaystyle$	$\displaystyle + g_{11}^2 h_{11,t-1} + 2g_{11}g_{21}h_{12,t-1} + g_{21}^2h_{22,t-1}$
$\displaystyle h_{12,t}$	$\displaystyle =$	$\displaystyle c_{12} + a_{11}a_{12}\epsilon ^2_{1,t-1} + (a_{21}a_{12}+a_{11}a_{22}) \epsilon _{1,t-1}\epsilon _{2,t-1} + a_{21}a_{22}\epsilon ^2_{2,t-1}$
$\displaystyle$	$\displaystyle$	$\displaystyle + g_{11}g_{12} h_{11,t-1} + (g_{21}g_{12}+g_{11}g_{22})h_{12,t-1} + g_{21}g_{22}h_{22,t-1}$
$\displaystyle h_{22,t}$	$\displaystyle =$	$\displaystyle c_{22} + a_{12}^2\epsilon ^2_{1,t-1} + 2a_{12}a_{22}\epsilon _{1,t-1}\epsilon _{2,t-1} + a_{22}^2\epsilon ^2_{2,t-1}$
$\displaystyle$	$\displaystyle$	$\displaystyle + g_{12}^2 h_{11,t-1} + 2g_{12}g_{22}h_{12,t-1} + g_{22}^2h_{22,t-1}$

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

$\displaystyle H_ t = D_ t \Gamma D_ t$

where is a $k \times k$ stochastic diagonal matrix with element $\sigma _{it}$ and $\Gamma$ is a $k \times k$ time-invariant matrix with the typical element $\rho _{ij}$ .

The elements of are

$\displaystyle h_{ii,t}$	$\displaystyle =$	$\displaystyle c_ i + \sum _{l=1}^ q a_{ii,l} \bepsilon ^2_{i,t-l} + \sum _{l=1}^ p g_{ii,l} h_{ii,t-l} ~ ~ i,j=1,\ldots k$
$\displaystyle h_{ij,t}$	$\displaystyle =$	$\displaystyle \rho _{ij}(h_{ii,t}h_{jj,t})^{1/2} ~ ~ i\neq j$

Estimation of GARCH Model

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

$\displaystyle \ell = - \frac{1}{2}\sum _{t=1}^{T} [ \log |H_ t| + \bepsilon _{t}’H_ t^{-1}\bepsilon _{t} ]$

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 $\bepsilon _{t}$ is used as the starting values for the GARCH constant parameters, and the starting value used for the other GARCH parameters is either $10^{-6}$ or $10^{-3}$ 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

$\mb {h} _ t = \sum _{i=1}^\infty G(B)^{i-1}[\mb {c} +A(B)\bm {\eta }_ t]$

where $\mb {h} _ t = \mbox{vec}(H_{t})$ , $\mb {c} = \mbox{vec}(C_0)$ , and $\bm {\eta }_ t = \mbox{vec}(\bepsilon _{t}\bepsilon _{t}’)$ . This representation is equivalent to a GARCH() model by the following algebra:

$\displaystyle \mb {h}_ t$	$\displaystyle =$	$\displaystyle \mb {c} + A(B)\bm {\eta }_ t + \sum _{i=2}^\infty G(B)^{i-1}[\mb {c} +A(B)\bm {\eta }_ t]$
$\displaystyle$	$\displaystyle =$	$\displaystyle \mb {c} + A(B)\bm {\eta }_ t + G(B)\sum _{i=1}^\infty G(B)^{i-1}[tmb{c} +A(B)\bm {\eta }_ t]$
$\displaystyle$	$\displaystyle =$	$\displaystyle \mb {c} + A(B)\bm {\eta }_ t + G(B) \mb {h}_ t$

Defining $A(B)= \sum _{i=1}^ q (A_ i \otimes A_ i)’B^ i$ and $G(B)= \sum _{i=1}^ p (G_ i \otimes G_ i)’B^ i$ 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 36.61 through Figure 36.65 show the details of this example. Figure 36.61 shows the initial values of parameters.

Figure 36.61: 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 36.62 shows the final parameter estimates.

Figure 36.62: 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 36.63 shows the conditional variance using the BEKK representation of the ARCH(1) model. The ARCH parameters are estimated by the vectorized parameter matrices.

$\displaystyle \bepsilon _ t\|\mathcal{F}(t-1)$	$\displaystyle \sim$	$\displaystyle N(0,H_ t)$
$\displaystyle H_ t$	$\displaystyle =$	$\displaystyle \left[ \begin{matrix} 8.35905 & -0.18250 \\ -0.18250 & 1.60275 \\ \end{matrix} \right]$
$\displaystyle$	$\displaystyle +$	$\displaystyle \left[ \begin{matrix} 0.37757 & 0.05649 \\ 0.03216 & 0.71002 \\ \end{matrix} \right]’ \bepsilon _{t-1}\bepsilon _{t-1}’ \left[ \begin{matrix} 0.37757 & 0.05649 \\ 0.03216 & 0.71002 \\ \end{matrix} \right]$

Figure 36.63: ARCH(1) Parameter Estimates for the VAR(1)–ARCH(1) Model

The VARMAX Procedure

Type of Model	VAR(1)-ARCH(1)
Estimation Method	Maximum Likelihood Estimation
Representation Type	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 36.64 shows the AR parameter estimates and their significance.

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

$\mb {y} _ t = \left[ \begin{matrix} 1.94399 \\ 4.07390 \\ \end{matrix} \right] + \left[ \begin{matrix} 1.22094 & -0.52712 \\ 0.60826 & 0.30301 \\ \end{matrix} \right] \mb {y} _{t-1} + \bepsilon _ t$

Figure 36.64: 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 36.65 shows the roots of the AR and ARCH characteristic polynomials. The eigenvalues have a modulus less than one.

Figure 36.65: 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