quasi-maximum likelihood

To guarantee the non-negativity of the conditional variance of the GARCH process, it is sufficient to assume the non-negativity of its parameters.  This condition was empirically violated besides rendering the GARCH model more restrictive.  It was subsequently  relaxed for some GARCH orders by necessary and sufficient constraints.  In this paper, we generalized an approach for the QML estimation of the GARCH$(p,q)$ parameters for all orders $p\geq 1$ and $q\ge 1$ using a constrained Kalman filter.  Such an approach allows a relaxed QML estimation of the GARCH without th

The component GARCH (CGARCH) is suitable to better capture the short and long term of the volatility dynamic.  Nevertheless, the parameter space constituted by the constraints of the non-negativity of the conditional variance, stationary and existence of moments, is only ex-post defined via the GARCH representation of the CGARCH.  This is due to the lack of a general method to determine a priori the relaxed constraints of non-negativity of the CGARCH($N$) conditional variance for any $N\geq 1$.