Abstract
The dynamic conditional correlation (DCC) model has been widely used for modeling the conditional correlation of multivariate time series by Engle (2002). However, the stationarity conditions have been established only recently and the asymptotic theory of parameter estimation for the DCC model has not yet to be fully discussed. In this paper, we propose an alternative model, namely the scalar dynamic conditional correlation (SDCC) model. Sufficient and easily-checked conditions for stationarity, geometric ergodicity, and β-mixing with exponential-decay rates are provided. We then show the strong consistency and asymptotic normality of the quasi-maximum-likelihood estimator (QMLE) of the model parameters under regular conditions. The asymptotic results are illustrated by Monte Carlo experiments. As a real-data example, the proposed SDCC model is applied to analyzing the daily returns of the FSTE (financial times and stock exchange) 100 index and FSTE 100 futures. Our model improves the performance of the DCC model in the sense that the Li-McLeod statistic of the SDCC model is much smaller and the hedging efficiency is higher.
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References
Aielli G P. Dynamic conditional correlations: On properties and estimation. J Bus Econom Statist, 2013, 31: 282–299
Aielli G P, Caporin M. Variance clustering improved dynamic conditional correlation MGARCH estimators. Comput Statist Data Anal, 2014, 76: 556–576
Bauwens L, Laurent S, Rombouts J V K. Multivariate GARCH models: A survey. J Appl Econometrics, 2006, 21: 79–109
Berkes I, Horváth L, Kokoszka P. GARCH processes: Structure and estimation. Bernoulli, 2003, 9: 201–227
Bollerlsev T. Modelling the coherence in short run nominal exchange rates: A multivariate generalized arch model. Rev Econ Statist, 1990, 72: 498–505
Bollerlsev T, Engle R, Wooldridge J. A capital asset pricing model with time varying covariances. J Political Economy, 1988, 96: 116–131
Boussama F, Fuchs F, Stelzer R. Stationarity and geometric ergodicity of BEKK multivariate GARCH models. Stochastic Process Appl, 2011, 121: 2331–2360
Brownlees C, Engle R F. SRISK: A conditional capital shortfall measure of systemic risk. Rev Financ Stud, 2017, 30: 48–79
Cappiello L, Engle R F, Sheppard K. Dynamics in the correlations of global equity and bond returns. J Financ Econometrics, 2006, 4: 537–572
Comte F, Lieberman O. Asymptotic theory for multivariate GARCH processes. J Multivariate Anal, 2003, 84: 61–84
Dennis J G, Hansen E, Rahbek A. ARCH innovations and their impact on cointegration rank testing. Working paper no. 22. Aarhus: University of Aarhus, 2002
Diebold F X, Nerlove M. The dynamics of exchange rate volatility: A multivariate latent factor ARCH model. J Appl Econometrics, 1989, 4: 1–21
Doz C, Renault E. Conditionally heteroskedastic factor models: Identification and instrumental variables estimation. CIRANO working paper. https://doi.org/www.cirano.qc.ca/files/publications/2004s-37.pdf, 2004
Engle R F. Dynamic conditional correlation: A simple class of multivariate generalized autoregressive conditional heteroskedasticity models. J Bus Econom Statist, 2002, 20: 339–350
Engle R F, Kroner K. Multivariate simultaneous generalized ARCH. Econometric Theory, 1995, 11: 122–150
Engle R F, Sheppard K. Theoretical and empirical properties of dynamic conditional correlation multivariate GARCH. https://doi.org/www.nber.org/papers/w8554, 2001
Fan J, Yao Q. Nonlinear Time Series: Nonparametric and Parametric Methods. New York: Springer-Verlag, 2003
Fernamian J D, Malongo H. On the stationary of dynamic conditional correlation models. Econometric Theory, 2017, 33: 636–663
Fiorentini G, Sentana E. Identification, estimation and testing of conditionally heteroskedastic factor models. J Econometrics, 2001, 102: 143–164
Francq C, Zakoïan J M. Maximum likelihood estimation of pure GARCH and ARMA-GARCH processes. Bernoulli, 2004, 10: 605–637
Francq C, Zakoïan J M. GARCH Models: Structure, Statistical Inference and Financial Applications. New York: John Wiley & Sons, 2010
Francq C, Zakoïan J M. QML estimation of a class of multivariate asymmetric GARCH models. Econometric Theory, 2011, 27: 1–28
Francq C, Zakoïan J M. Estimating multivariate volatility models equation by equation. J R Stat Soc Ser B Stat Methodol, 2016, 78: 613–635
Franses P H, Hafner C M. A generalized dynamic conditional correlation model for many asset returns. Econometric Rev, 2009, 28: 612–631
Grossman S J, Shiller R J. The determinants of the variability of stock market prices. Amer Econ Rev, 1981, 71: 222–227
Hafner C M. Fourth moment structure of multivariate GARCH models. J Financ Econometrics, 2003, 1: 26–54
Hafner C M, Preminger A. On asymptotic theory for multivariate GARCH models. J Multivariate Anal, 2009, 100: 2044–2054
He C, Teräsvirta T. An extended constant conditional correlation GARCH model and its fourth-moment structure. Econometric Theory, 2004, 20: 904–926
Jeantheau T. Strong consistency of estimators for multivariate ARCH models. Econometric Theory, 1998, 14: 70–86
Kristensen D. Uniform ergodicity of a class of Markov chains with applications to time series models. Working paper. New York: Columbia University, 2007
Lien D, Yang L. Asymmetric effect of basis on dynamic futures hedging: Empirical evidence from commodity markets. J Banking Finance, 2008, 32: 187–198
Ling S, McAleer M. Asymptotic theory for a vector ARMA-GARCH model. Econometric Theory, 2003, 19: 280–310
Magnus J R, Neudecker H. Matrix Differential Calculus with Applications in Statistics and Econometrics. New York: Wiley, 1988
McAleer M, Hoti S, Chan F. Structure and asymptotic theory for multivariate asymmetric conditional volity. Econometric Rev, 2009, 28: 422–440
Sherrer W, Ribarits E. On the parametrization of multivariate GARCH models. Econometric Theory, 2007, 23: 464–484
Silvennoinen A, Teräsvirta T. Multivariate GARCH Models. Handbook of Financial Time Series. New York: Springer, 2009
Tse Y K, Tsui K C. A multivariate generalized autoregressive conditional heteroskedasticity model with time-varying correlation. J Bus Econom Statist, 2002, 20: 315–362
Tweedie R L. Invariant measure for Markov chains with no irreductibility assumptions. J Appl Probab, 1988, 25: 275–285
Zakoïan J M. A class of DCC asymmetric GARCH models driven by exogenous variables. Rapport de Recherche, RR-FiME-10-12. https://doi.org/www.fime-lab.org/fr/rapports-de-recherche, 2010
Acknowledgements
This work was supported by National Natural Science Foundation of China (Grant No. 71771224), National Social Science Foundation of China (Grant Nos. 14ZDA044 and 15BGJ037), the Program for National Statistics Science Research Plan (Grant No. 2016LD02) and the Program for Innovation Research in Central University of Finance and Economics.
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Wang, H., Pan, J. A scalar dynamic conditional correlation model: Structure and estimation. Sci. China Math. 61, 1881–1906 (2018). https://doi.org/10.1007/s11425-017-9273-x
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DOI: https://doi.org/10.1007/s11425-017-9273-x