Building Econometric Models

BEKK and Model Estimation for M-GARCH

• ‘Introductory Econometrics for Finance’ © Chris Brooks 2013

• Neither the VECH nor the diagonal VECH ensure a positive definite variance-covariance matrix.
• An alternative approach is the BEKK model (Engle & Kroner, 1995).
• The BEKK Model uses a Quadratic form for the parameter matrices to ensure a positive definite variance / covariance matrix Ht.
• In matrix form, the BEKK model is
• Model estimation for all classes of multivariate GARCH model is again performed using maximum likelihood with the following LLF:
• where N is the number of variables in the system (assumed 2 above),  is a vector containing all of the parameters, and T is the number of obs.

Correlation Models and the CCC

• ‘Introductory Econometrics for Finance’ © Chris Brooks 2013

• The correlations between a pair of series at each point in time can be constructed by dividing the conditional covariances by the product of the conditional standard deviations from a VECH or BEKK model
• A subtly different approach would be to model the dynamics for the correlations directly
• In the constant conditional correlation (CCC) model, the correlations between the disturbances to be fixed through time
• Thus, although the conditional covariances are not fixed, they are tied to the variances
• The conditional variances in the fixed correlation model are identical to those of a set of univariate GARCH specifications (although they are estimated jointly):

More on the CCC

• ‘Introductory Econometrics for Finance’ © Chris Brooks 2013

• The off-diagonal elements of Ht, hij,t (i  j), are defined indirectly via the correlations, denoted ρij:
• Is it empirically plausible to assume that the correlations are constant through time?
• Several tests of this assumption have been developed, including a test based on the information matrix due and a Lagrange Multiplier test
• There is evidence against constant correlations, particularly in the context of stock returns.