## 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 $N\geq 1$.