Standard BEKK parameters

Question

I am looking at a BEKK Multivariate GARCH model.

In a standard GARCH model, we generally expect,

$$h_t=\omega+\alpha u_{t-1}^2 +\beta\sigma_{t-1}^2$$

The alpha ($\alpha$) coefficient to be considerably smaller than the beta ($\beta$), see for example Verbeeks 'Guide to modern econometrics chapter on GARCH', with around 0.1 alpha and 0.8 beta.

I am now moving into a multivariate setting, to a BEKK(1),

$$\left[\begin{matrix}h_{11,t}&h_{12,t}\\h_{21,t}&h_{22,t}\\\end{matrix}\right]=\left[\begin{matrix}k_{11}&k_{12}\\k_{21}&k_{22}\\\end{matrix}\right]+\left[\begin{matrix}a_{11}&a_{12}\\a_{21}&a_{22}\\\end{matrix}\right]\left[\begin{matrix}e_{1,t-1}\\e_{2,t-1}\\\end{matrix}\right]\left[\begin{matrix}e_{1,t-1}\\e_{2,t-1}\\\end{matrix}\right]^\prime\left[\begin{matrix}a_{11}&a_{12}\\a_{21}&a_{22}\\\end{matrix}\right]^\prime$$

i.e. an MV-ARCH(1),

Would anyone know suitable parameters for the $A_{ij}$ matrix, with a reference? And also the BEKK(1,1) with the GARCH term,

$$H_t=C^\ast{C^\ast}^\prime+A_{11}\varepsilon_{t-1}\varepsilon_{t-1}^\prime A_{11}^\prime+B_{11}H_{t-1}B_{11}^\prime$$

I need suitable parameter values (as in what we would expect) for A and B. I understand this will change considerably between datasets etc. But in general any values we might expect?

Answer 1

Unfortunately, there are no straight forward checks on the $a_{ij}$'s and $b_{ij}$'s coefficients in the BEKK case, like $\alpha+\beta<1$ ensure stationarity and weak time dependence in the GARCH(1,1) case. The conditions are a bit more convoluted in the BEKK-case.

The process is stationary and weakly time dependent (in the sense that it's a geometrically ergodic Harris recurrent Markov chain), if all the eigenvalues of the $k^2 \times k^2$ matrix $A_{11} \otimes A_{11} + B_{11} \otimes B_{11}$ are less than 1 and $C^{\ast}C^{\ast\prime}$ is positive definite, but that will always be the case with $C^{\ast}C^{\ast\prime}$, since it's positive definite by construction. The $\otimes$ denotes the Kronecker product.

Theorem 2 in Comte and Lieberman (2003) says that this condition ensures that the maximum likelihood estimator is consistent, and if we further assume that the process has finite sixth order moment , that is $E \left\|X^6\right\| < \infty$ , then Theorem 3 in Hafner and Preminger (2009) establishes asymptotic normality of the MLE.

To my knowledge the literature gives no straight forward parameter restrictions, which ensures finite sixth order moments of the BEKK-process. Theorem C.1 in the appendix of Pedersen and Rahbek (2014) provide sufficient conditions for the ARCH-version of the Gaussian BEKK-process ($B_{11} = 0$), to have $E \left\|X^6\right\| < \infty$. This conditions is that all eigenvalues of $A_{11} ⊗ A_{11}$ should be less than $15^{−1/3} \approx 0.4055$.

• F. Comte and O. Lieberman. Asymptotic theory for multivariate GARCH processes. Journal of Multivariate Analysis, 84(1):61 – 84, 2003.
• C. M. Hafner and A. Preminger. On asymptotic theory for multivariate GARCH models. Journal of Multivariate Analysis, 100(9):2044 – 2054, 2009.
• R. S. Pedersen and A. Rahbek. Multivariate variance targeting in the bekk -garch model. The Econometrics Journal, 17(1):24–55, 2014.