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.
- Engle R. F. Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation. Econometrica: Journal of the econometric society. 50 (4), 987–1007 (1982).
- Bollerslev T. Generalized autoregressive conditional heteroskedasticity. Journal of econometrics. 31 (3), 307–327 (1986).
- Bollerslev T. On the correlation structure for the generalized autoregressive conditional heteroskedastic process. Journal of Time Series Analysis. 9, 121–131 (1988).
- Ding Z., Granger C. W. Modelling volatility persistence of speculative returns: a new approach. Journal of econometrics. 73 (1), 185–215 (1996).
- Ding Z., Granger C. W., Engle R. F. A long memory property of stock market returns and a new model. Journal of empirical finance. 1 (1), 83–106 (1993).
- Andersen T. G., Bollerslev T. Heterogeneous information arrivals and return volatility dynamics: Uncovering the long\-run in high frequency returns. The journal of Finance. 52, 975–1005 (1997).
- Andersen T. G., Bollerslev T., Diebold F. X. The distribution of realized exchange rate volatility. Journal of the American statistical association. 96 (453), 42–55 (2001).
- Bollerslev T., Wright J. H. Semiparametric estimation of long-memory volatility dependencies: The role of high\-frequency data. Journal of econometrics. 98 (1), 81–106 (2000).
- Karanasos M. The second moment and the autocovariance function of the squared errors of the GARCH model. Journal of Econometrics. 90 (1), 63–76 (1999).
- Maheu J. Can GARCH models capture long-range dependence? Studies in Nonlinear Dynamics & Econometrics. 9 (4) (2005).
- Engle R. F., Lee G. A long-run and short-run component model of stock return volatility. Cointegration, Causality, and Forecasting: A Festschrift in Honour of Clive WJ Granger. 475 (1999).
- Settar A., Idrissi N. F., Badaoui M. New approach in dealing with the non-negativity of the conditional variance in the estimation of GARCH model. Central European Journal of Economic Modelling and Econometrics. 13, 55 (2021).
- Allal J., Benmoumen M. Parameter Estimation for GARCH(1,1) Models Based on Kalman Filter. Advances and Applications in Statistics. 25, 15 (2011).
- Spall J. C. Implementation of the simultaneous perturbation algorithm for stochastic optimization. IEEE Transactions on aerospace and electronic systems. 34 (3), 817–823 (1998).
- Spall J. C. Multivariate stochastic approximation using a simultaneous perturbation gradient approximation. IEEE transactions on automatic control. 37 (3), 332–341 (1992).
- Bhatnagar S., Prashanth H. L., Prashanth L. A. Lecture Notes in Control and Information Sciences. Series Advisory. 434 (2013).
- Francq C., Zakoian J. M. GARCH models: structure, statistical inference and financial applications. John Wiley & Sons (2019).