Abstract
In this paper, we explore some probabilistic and statistical properties of constant conditional correlation (CCC) multivariate periodic GARCH models (CCC − PGARCH for short). These models which encompass some interesting classes having (locally) long memory property, play an outstanding role in modelling multivariate financial time series exhibiting certain heteroskedasticity. So, we give in the first part some basic structural properties of such models as conditions ensuring the existence of the strict stationary and geometric ergodic solution (in periodic sense). As a result, it is shown that the moments of some positive order for strictly stationary solution of CCC − PGARCH models are finite.Upon this finding, we focus in the second part on the quasi-maximum likelihood (QML) estimator for estimating the unknown parameters involved in the models. So we establish strong consistency and asymptotic normality (CAN) of CCC − PGARCH models.
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A. Aknouche and A. Bibi, “Quasi-Maximum Likelihood Estimation of Periodic GARCH s and Periodic ARMA − GARCH s Processes”, J. Time Series Analysis 30, 19–46 (2008).
A. Aue, S. Hörmann, L. Horváth, and M. Reimherr “Break Detection in the Covariance Structure of Multivariate Time Series Models”, Ann. Statist. 37, 4046–4087 (2009).
I. Berkes, L. Horvath, and P. Kokoskza, “GARCH Processes: Structure and Estimation”, Bernoulli 9, 201–207 (2003).
L. Bauwens, S. Laurent, and J. V. K. Rombouts, “Multivariate GARCH Models: A Survey”, J. Appl. Econometrics 21, 79–109 (2006).
A. Bibi and A. Aknouche, “On Periodic GARCH Processes: Stationarity, Existence of Moments and Geometric Ergodicity”, Math. Methods Statist. 4, 305–316 (2007).
P. Billingsley, Probability and Measure, 3rd ed. (Wiley, New York, 1995).
T. Bollerslev, “Generalized Autoregressive Conditional Heteroskedasticity”, J. Econometrics 31, 307–327 (1986).
T. Bollerslev, “Modelling the Coherence in Short-Run Nominal Exchange Rates: A Multivariate Generalized ARCH Model”, Rev. Eco. Statist. 72, 498–505 (1990).
T. Bollerslev and J. M. Wooldridge, “Quasi-Maximum Likelihood Estimation and Inference in Dynamic Models with Time-Varying Covariances”, Econometric Reviews 11, 143–172 (1992).
P. Bougerol and N. Picard, “Stationarity of GARCH Processes and of Some Nonnegative Time Series”, J. Econometrics 52, 115–127 (1992).
N. H. Chan and C. T. Ng, “Statistical Inference for Nonstationary GARCH(p, q) Models”, Elect. J. Statist. 956–992 (2009).
F. Comte and O. Lieberman, “Asymptotic Theory for Multivariate GARCH Processes”, J. Multivariate Anal. 84, 61–84 (2003).
R. F. Engle, “Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflations”, Econometrica 50, 987–1008 (1982).
R. F. Engle and K. F. Kroner, “Multivariate Simultaneous Generalized ARCH”, Econometric Theory 11, 122–150 (1995).
R. F. Engle, Anticipating Correlations. A New Paradigm for Risk Management (Princeton Univ. Press, 2009).
C. Francq and J.-M. Zakoïan, “QML Estimation of a Class of Multivariate GARCH Models without Moment Conditions on the Observed Process”, Unpublished working paper (2010).
C. Francq and J.-M. Zakoïan, “QML Estimation of a Class of Multivariate Asymmetric GARCH Models”, Econometric Theory 28, 179–206 (2012).
C. Francq and J.-M. Zakoïan, “Estimating Multivariate Volatility Models Equation by Equation”, J. Roy. Statist. Soc., Ser. B, 78, 613–635 (2016).
L. Giraitis and P. M. Robinson, “Whittle Estimation of ARCH Models”, Econometrics Theory 17, 608–631 (2001).
C. M. Hafner and A. Preminger, “Asymptotic Theory for a Factor GARCH Model”, Econometric Theory 25, 336–363 (2009).
C. Hafner and H. Herwartz, “Analytical Quasi Maximum Likelihood Inference in Multivariate Volatility Models”, Metrika 67, 219–239 (2008).
M. Haas and J.-C. Liu, “Theory for a Multivariate Markov-switching GARCH Model with an Application to Stock Markets”, Beitrage zur Jahrestagung des Vereins fur Socialpolitik: Ökonomische Entwicklung–Theorie und Politik, Session: Financial Econometrics, No. B22-V2 (2015).
C. He and T. Teräsvirta, “An Extended Constant Conditional Correlation GARCH Model and Its Fourth-Moment Structure”, Econometric Theory 20, 904–926 (2004).
R. Horn and C. Johnsen, Matrix Analysis (Cambridge Univ. Press. New York, 1985).
T. Jeantheau, “Strong Consistency of Estimations of Multivariate ARCH Model”, Econometric Theory 14, 70–86 (1998).
S. T. Jensen and A. Rahbek, “Asymptotic Inference for Nonstationary GARCH”, Econometric Theory 20, 1203–1226 (2004a).
S. T. Jensen and A. Rahbek, “Asymptotic Normality of the QMLE Estimator of ARCH in the Nonstationary Case”, Econometrica 72, 641–646 (2004b).
S. Ling and M. McAleer, “Asymptotic Theory for a Vector ARMA − GARCH Model”, Econometric Theory 19, 280–310 (2003).
T. Mikosch and D. Straumann, “Whittle Estimation in Heavy-Tailed GARCH(1, 1) Model”, Stoch. Proc. Appl. 100, 187–222 (2002).
D. R. Osborn, C. S. Sawa, and L. Gill, “Periodic Dynamic Conditional Correlations between Stock Markets in Europe and the US”, J. Financial Econometrics 6 (3), 307–325 (2008).
A. Silvennoinen and T. Teräsvirta, Multivariate GARCH Models. Handbook of Financial Time Series, Ed. by T. G. Andersen, R. A. Davis, J-P. Kreiss, and T. Mikosch (Springer, New York, 2009).
D. Straumann, Estimation in Conditionally Heteroscedastic Time Series Models, in Lecture Notes in Statist. (Springer, Berlin–Heidelberg, 2005).
R. L. Tweedie, “Drift Conditions and Invariant Measures for Markov Chains”, Stoch. Proc. Appl. 92, 345–354 (2001).
P. Zaffaroni, “Whittle Estimation of EGARCH and Other Exponential Volatility Models”, J. Econometrics 151 (2), 190–200 (2009).
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Bibi, A. Asymptotic Properties of QML Estimation of Multivariate Periodic CCC − GARCH Models. Math. Meth. Stat. 27, 184–204 (2018). https://doi.org/10.3103/S106653071803002X
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DOI: https://doi.org/10.3103/S106653071803002X
Keywords
- multivariate periodic GARCH models
- strict periodic stationarity
- geometric ergodicity
- strong consistency
- asymptotic normality