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What They Did Not Tell You About Algebraic (Non-)Existence, Mathematical (IR-)Regularity and (Non-)Asymptotic Properties of the Full BEKK Dynamic Conditional Covariance Model

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2019
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Facultad de CC Económicas y Empresariales. Instituto Complutense de Análisis Económico (ICAE)
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Persistently high negative covariances between risky assets and hedging instruments are intended to mitigate against risk and subsequent financial losses. In the event of having more than one hedging instrument, multivariate covariances need to be calculated. Optimal hedge ratios are unlikely to remain constant using high frequency data, so it is essential to specify dynamic covariance models. These values can either be determined analytically or numerically on the basis of highly advanced computer simulations. Analytical developments are occasionally promulgated for multivariate conditional volatility models. The primary purpose of the paper is to analyse purported analytical developments for the most widely-used multivariate dynamic conditional covariance model to have been developed to date, namely the Full BEKK model of Baba et al. (1985), which was published as Engle and Kroner (1995). Dynamic models are not straightforward (or even possible) to translate in terms of the algebraic existence, underlying stochastic processes, specification, mathematical regularity conditions, and asymptotic properties of consistency and asymptotic normality, or the lack thereof. The paper presents a critical analysis, discussion, evaluation and presentation of caveats relating to the Full BEKK model, and an emphasis on the numerous dos and don’ts in implementing Full BEKK in practice.
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Baba, Y., R. Engle, D. Kraft and K. Kroner (1985), Multivariate simultaneous generalized ARCH, Unpublished Paper, University of California, San Diego.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)]. Bollerslev, T. (1986), Generalized autoregressive conditional heteroskedasticity, Journal of Econometrics, 31, 307-327. Chang, C.-L. and M. McAleer (2019), The fiction of full BEKK: Pricing fossil fuels and carbon emissions, Finance Research Letters, 28, 11-19. 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. and K.F. Kroner (1995), Multivariate simultaneous generalized ARCH, Econometric Theory, 11(1), 122-150. Hafner, C.M. and A. Preminger (2009), On asymptotic theory for multivariate GARCH models, Journal of Multivariate Analysis, 100, 2044-2054. McAleer, M. (2014), Asymmetry and leverage in conditional volatility models, Econometrics, 2(3), 145-150. McAleer, M. (2019), What they did not tell you about algebraic (non-)existence, mathematical (ir-)regularity and (non-)asymptotic properties of the dynamic conditional correlation (DCC) model, unpublished paper, Department of Finance, Asia University, Taiwan. McAleer, M., F. Chan, S. Hoti and O. Lieberman (2008), Generalized autoregressive conditional correlation, Econometric Theory, 24(6), 1554-1583. Tsay, R.S. (1987), Conditional heteroscedastic time series models, Journal of the American Statistical Association, 82, 590-604.