Quasi-maximum likelihood estimation of the Component-GARCH model using the stochastic approximation algorithm with application to the S&P 500

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 $N\geq 1$. In this paper, a CGARCH parameter space constructed from the GARCH(1,1) component parameter spaces is provided a priori to identifying its GARCH form.  Such a space fulfils the relaxed constraints of the CGARCH conditional variance non-negativity to be pre-estimated ensuring the existence of a QML estimation in the sense of the stochastic approximation algorithm.  Simulation experiment as well as empirical application to the S&P500 index are presented and both show the performance of the proposed method.