Standard BEKK parameters

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?

1 Answer

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.
• Not sure whether this applies to the particular form of BEKK studied here, but McAleer "What they did not tell you about algebraic (non-) existence, mathematical (ir-) regularity and (non-) asymptotic properties of the full BEKK dynamic conditional covariance model" (2019) shows that BEKK might not even exist except under restrictive conditions, pulling the rug from under 4500+ papers citing BEKK. – Richard Hardy Jun 11 '19 at 9:19
• @Duffau a great answer but have you got any ideas on what the gap between A and B should be? – Francis Origi Jun 11 '19 at 9:20
• Thanks @FrancisOrigi! So remember that A and B are matrices so there is no clear notion of "gap". In dynamical systems where the process is defined by matrices, often some kind of eigenvalue determines the stability of the system. Like for the BEKK the stability (stationarity and weak dependence) is governed by the eigenvalues of the transformed matrices I described above. If you want to learn more I would look into linear Vector Autoregressions, they are the simplest type with multivariate dynamics. They are the equivalent to AR-models in the univariate world. – Duffau Jun 11 '19 at 11:45