The purpose of this paper is to present a theoretical methodology for estimation of a multivariate constant conditional correlation hyperbolic GARCH (CCC-HGARCH) model. The proposed model captures correlations between several time series exhibiting long memory property in volatility and finite variance observed on the same time period. This model extends the class of hyperbolic GARCH processes with finite variance to a multivariate framework. The goal of this work is to model volatility of multivariate hyperbolic GARCH processes by conditional covariance matrix based on a time invariant correlation matrix. We first provide a sufficient condition for the existence of strictly stationary solution of the CCC-HGARCH model. Secondly, the asymptotic properties of the quasi-maximum likelihood estimator are established.

HGARCH model, multivariate long memory models, strong consistency, asymptotic normality

Received: August 9, 2024; Accepted: October 21, 2024; Published: February 17, 2025

How to cite this article: Lanciné BAMBA, Brahima SORO, Konan Jean Geoffroy KOUAKOU and Ouagnina HILI, On QML-estimation of multivariate constant conditional correlation hyperbolic GARCH models, Far East Journal of Theoretical Statistics 69(1) (2025), 81‑108. https://doi.org/10.17654/0972086325004.

This Open Access Article is Licensed under Creative Commons Attribution 4.0 International License

References:

[1] T. Amemiya, Advanced Econometrics, Harvard University Press, Cambridge, Massachusetts, 1985.
[2] R. T. Baillie, T. Bollerslev and H. O. Mikkelsen, Fractionally integrated generalized autoregressive conditional heteroscedasticity, J. Econometrics 74 (1996), 3-30.
[3] L. Bamba, O. Hili, A. K. Diongue and A. N’Guessan, M-estimate for the stationary hyperbolic GARCH models, Metron 79 (2021), 303-351.
[4] A. Bibi, Asymptotic properties of QML estimation of multivariate periodic CCC-GARCH models, Math. Methods Statist. 27 (2018), 184-204.
[5] P. Billingsley, Convergence of Probability Measures, 2nd ed., John Wiley & Sons, Inc., 1999.
[6] C. Brunetti and C. L. Gilbert, Bivariate FIGARCH and fractional cointegration, Journal of Empirical Finance 7 (2000), 509-530.
[7] J. Davidson, Moment and memory properties of linear conditional heteroscedasticity models, and a new model, J. Bus. Econom. Statist. 22 (2004), 16-29.
[8] Z. Ding, C. W. Granger and R. F. Engle, A long memory property of stock market returns and a new model, Journal of Empirical Finance 1 (1993), 83-106.
[9] P. Doukhan, Stochastic Models for Time Series, Springer International Publishing AG, 2018.
[10] C. Francq and J.-M. Zakoïan, QML estimation of a class of multivariate asymmetric GARCH models, Econometric Theory 28 (2011), 179-206.
[11] M. Li, W. K. Li and G. Li, A new hyperbolic GARCH model, J. Econometrics 189 (2015), 428-436.
[12] W. F. Stout, Almost Sure Convergence, Academic Press, Inc., 1974.
[13] D. Straumann, Estimation in Conditionally Heteroscedastic Time Series Models, Berlin Heidelberg Springer, 2005.
[14] G. Teyssière, Modelling exchange rates volatility with multivariate long-memory arch processes, Humboldt-Universität zu Berlin, Wirtschaftswissenschaftliche Fakultät, 1998.