Asymptotic Theory for Rotated Multivariate GARCH Models

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Facultad de Ciencias Económicas y Empresariales. Instituto Complutense de Análisis Económico (ICAE)
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In this paper, we derive the statistical properties of a two step approach to estimating multivariate GARCH rotated BEKK (RBEKK) models. By the definition of rotated BEKK, we estimate the unconditional covariance matrix in the first step in order to rotate observed variables to have the identity matrix for its sample covariance matrix. In the second step, we estimate the remaining parameters via maximizing the quasi-likelihood function. For this two step quasi-maximum likelihood (2sQML) estimator, we show consistency and asymptotic normality under weak conditions. While second-order moments are needed for consistency of the estimated unconditional covariance matrix, the existence of finite sixthorder moments are required for convergence of the second-order derivatives of the quasilog-likelihood function. We also show the relationship of the asymptotic distributions of the 2sQML estimator for the RBEKK model and the variance targeting (VT) QML estimator for the VT-BEKK model. Monte Carlo experiments show that the bias of the 2sQML estimator is negligible, and that the appropriateness of the diagonal specification depends on the closeness to either of the Diagonal BEKK and the Diagonal RBEKK models.
The authors are most grateful to Yoshi Baba for very helpful comments and suggestions. The first author acknowledges the financial support of the Japan Ministry of Education, Culture, Sports, Science and Technology, Japan Society for the Promotion of Science, and the Australian Academy of Science. The second author thanks the Ministry of Science and Technology (MOST) for financial support. The third author is most grateful for the financial support of the Australian Research Council, Ministry of Science and Technology (MOST), Taiwan, and the Japan Society for the Promotion of Science.
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Avarucci, M., E. Beutner, and P. Zaffaroni (2013), “On Moment Conditions for Quasi-Maximum Likelihood Estimation of Multivariate ARCH models”, Econometric Theory, 29, 545–566. Baba, Y., R. Engle, D. Kraft and K. Kroner (1985), “Multivariate Simultaneous Generalized ARCH”, Unpublished Paper, University of California, San Diego. [Published as Engle and Kroner (1995)]. Bauwens L., S. Laurent, and J. K. V. Rombouts (2006), “Multivariate GARCH Models: A Survey”, Journal of Applied Econometrics, 21, 79–109. Bollerslev, T. (1986), “Generalized Autoregressive Conditional Heteroskedasticity”, Journal of Econometrics, 31, 307–327. Boussama, F., F. Fuchs, and R. Stelzer (2011), “Stationarity and Geometric Ergodicity of BEKK Multivariate GARCH Models”, Stochastic Processes and their Applications, 121, 2331–2360. Chang, C.-L. and M. McAleer (2018), “The Fiction of Full BEKK: Pricing Fossil Fuels and Carbon Emissions”, to appear in Finance Research Letters. Comte, F. and O. Lieberman (2003), “Asymptotic Theory for Multivariate GARCH Processes”, Journal of Multivariate Analysis, 84, 61–84. Engle, R. F. (1982), “Autoregressive Conditional Heteroskedasticity with Estimates of the Variance of United Kingdom Inflation”, Econornetrica, 50, 987–1007. Engle, R. F. (2002), “Dynamic Conditional Correlation: A Simple Class of Multivariate Generalized Autoregressive Conditional Heteroskedasticity Models”, Journal of Business & Economic Statistics, 20, 339–350. Engle, R.F. and K.F. Kroner (1995), “Multivariate Simultaneous Generalized ARCH”, Econometric Theory, 11, 122–150. Francq, C., L. Horv´ath, and J.-M. Zako¨ıan (2011), “Merits and Drawbacks of Variance Targeting in GARCH Models”, Journal of Financial Econometrics, 9, 619–656. Hafner, C. M. and A. Preminger (2009), “On Asymptotic Theory for Multivariate GARCH Models”, Journal of Multivariate Analysis, 100, 2044–2054. Laurent, S., J. V. K. Rombouts, and F. Violante (2012), “On the Forecasting Accuracy of Multivariate GARCH Models”, Journal of Applied Econometrics, 27, 934–955. L¨utkephol, H. (1996), Handbook of Matrices, New York: John Wiley & Sons. McAleer, M. (2005), “Automated Inference and Learning in Modeling Financial Volatility”, Econometric Theory, 21, 232–261. McAleer, M. (2018), “Stationarity and Invertibility of a Dynamic Correlation Matrix”, Kybernetika, 54(2), 363–374. McAleer, M., F. Chan, S. Hoti and O. Lieberman (2008), “Generalized Autoregressive Conditional Correlation”, Econometric Theory, 24, 1554–1583. Newey,W. K. and D. McFadden (1994), “Large Sample Estimation and Hypothesis Testing”, In R. F. Engle and D. McFadden (Eds.), Handbook of Econometrics, Volume 4, 2111-2245. Amsterdam: Elsevier. Noureldin, D., N. Shephard, and K. Sheppard (2014), “Multivariate Rotated ARCH Models”, Journal of Econometrics, 179, 16–30. Pedersen, R. S. and A. Rahbek (2014), “Multivariate Variance Targeting in the BEKK-GARCH Model”, Econometrics Journal, 17, 24–55. Silvennoinen, A., and T. Ter¨asvirta (2009), “Multivariate GARCH Models”, In T. G. Andersen, R.A. Davis, J.-P. Kreiss, and T. Mikosch (eds.), Handbook of Financial Time Series, 201–229, New York: Springer. Tse, Y. K. and A. K. C. Tsui (2002), “A Multivariate Generalized Autoregressive Conditional Heteroscedasticity Model with Time-Varying Correlations”, Journal of Business & Economic Statistics, 20, 351–361.