In the past, I encountered problems related to the BEKK model when studying volatility models. Recently, I encountered similar problems again. Let me summarize what I have learned.
I used “Introductory Econometrics For Finance” by Chris Brooks for my introduction. The order of my study was ARCH model → GARCH model → asymmetric GARCH model → multivariate GARCH model, and the order of studying the multivariate GARCH model was VECH model → BEKK model → CCC model → DCC model. After learning the basic model, I read a paper that was specifically mentioned in the book and learned about the improved version of the model that encompasses all of these, called the ADC model.
BEKK-MGARCH is actually the basic GARCH model, which extends one dimension to multiple dimensions. Therefore, the conditional variance involved becomes a matrix containing conditional variance and conditional covariance, and the residual square term becomes the product of the residual vector containing all the sequence residuals at a certain moment.
First, we generalize this idea, and we will get the VECH model. In order to solve the problem that the conditional variance matrix is often non-positive, we need to ensure that the right side of the equation is positive as much as possible, so the BEKK model, which is in the form of square terms, came into being.
Kroner and Ng expanded BEKK in the paper “Modeling Asymmetric Comovements of Asset Returns”, which I think is also a very interesting form. They combined the constant correlation of CCC-GARCH with BEKK and proposed the GDC model:
It is not difficult to see that the first half of GDC is taken from the CCC model, and the second half comes from the BEKK model.
Correspondingly, in order to reflect asymmetric shocks, they referred to the GJR model and improved the GDC model to obtain the ADC model:
It can be seen that the nij term added on the basis of GDC is similar to the GJR model, and the change from taking 1 when the residual is negative in GJR to taking negative further reflects the impact of the shock size on the model. The ADC model can be said to be the culmination of all the volatility models I have come across so far. It combines multiple CCC and BEKK, and also takes into account asymmetric shocks.
If you have any questions or suggestions about this, please feel free to come and discuss with me in depth!