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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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  • Michael McAleer

    ( Department of Quantitative Finance National Tsing Hua University, Taiwan and Econometric Institute Erasmus School of Economics Erasmus University Rotterdam, The Netherlands and Department of Quantitative Economics Complutense University of Madrid, Spain And Institute of Advanced Sciences Yokohama National University, Japan.)

Abstract

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.

Suggested Citation

  • Michael McAleer, 2019. "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," Documentos de Trabajo del ICAE 2019-18, Universidad Complutense de Madrid, Facultad de Ciencias Económicas y Empresariales, Instituto Complutense de Análisis Económico.
    • Handle: RePEc:ucm:doicae:1918
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    References listed on IDEAS

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    1. Engle, Robert F. & Kroner, Kenneth F., 1995. "Multivariate Simultaneous Generalized ARCH," Econometric Theory, Cambridge University Press, vol. 11(1), pages 122-150, February.
    2. Michael McAleer, 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," JRFM, MDPI, vol. 12(2), pages 1-9, April.
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    4. Bollerslev, Tim, 1986. "Generalized autoregressive conditional heteroskedasticity," Journal of Econometrics, Elsevier, vol. 31(3), pages 307-327, April.
    5. Chang, Chia-Lin & McAleer, Michael, 2019. "The fiction of full BEKK: Pricing fossil fuels and carbon emissions," Finance Research Letters, Elsevier, vol. 28(C), pages 11-19.
    6. Michael McAleer, 2019. "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," JRFM, MDPI, vol. 12(2), pages 1-7, April.
    7. Michael McAleer, 2014. "Asymmetry and Leverage in Conditional Volatility Models," Econometrics, MDPI, vol. 2(3), pages 1-6, September.
    8. McAleer, Michael & Chan, Felix & Hoti, Suhejla & Lieberman, Offer, 2008. "Generalized Autoregressive Conditional Correlation," Econometric Theory, Cambridge University Press, vol. 24(6), pages 1554-1583, December.
    9. Engle, Robert F, 1982. "Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation," Econometrica, Econometric Society, vol. 50(4), pages 987-1007, July.
    10. Hafner, Christian M. & Preminger, Arie, 2009. "On asymptotic theory for multivariate GARCH models," Journal of Multivariate Analysis, Elsevier, vol. 100(9), pages 2044-2054, October.
    11. Nicholls, D. F. & Quinn, B. G., 1981. "The estimation of multivariate random coefficient autoregressive models," Journal of Multivariate Analysis, Elsevier, vol. 11(4), pages 544-555, December.
    12. Comte, F. & Lieberman, O., 2003. "Asymptotic theory for multivariate GARCH processes," Journal of Multivariate Analysis, Elsevier, vol. 84(1), pages 61-84, January.
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    Cited by:

    1. Asai, Manabu & Gupta, Rangan & McAleer, Michael, 2020. "Forecasting volatility and co-volatility of crude oil and gold futures: Effects of leverage, jumps, spillovers, and geopolitical risks," International Journal of Forecasting, Elsevier, vol. 36(3), pages 933-948.
    2. Michael McAleer, 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," JRFM, MDPI, vol. 12(2), pages 1-9, April.
    3. Das, Sonali & Demirer, Riza & Gupta, Rangan & Mangisa, Siphumlile, 2019. "The effect of global crises on stock market correlations: Evidence from scalar regressions via functional data analysis," Structural Change and Economic Dynamics, Elsevier, vol. 50(C), pages 132-147.
    4. Chia-Lin Chang & Michael McAleer & Jiarong Tian, 2019. "Modeling and Testing Volatility Spillovers in Oil and Financial Markets for the USA, the UK, and China," Energies, MDPI, vol. 12(8), pages 1-24, April.
    5. Hou, Yang (Greg) & Li, Steven, 2020. "Volatility and skewness spillover between stock index and stock index futures markets during a crash period: New evidence from China," International Review of Economics & Finance, Elsevier, vol. 66(C), pages 166-188.
    6. Semei Coronado & Rangan Gupta & Besma Hkiri & Omar Rojas, 2020. "Time-Varying Spillovers between Currency and Stock Markets in the USA: Historical Evidence From More than Two Centuries," Advances in Decision Sciences, Asia University, Taiwan, vol. 24(4), pages 44-76, December.
    7. Katsiampa, Paraskevi & Yarovaya, Larisa & Zięba, Damian, 2022. "High-frequency connectedness between Bitcoin and other top-traded crypto assets during the COVID-19 crisis," Journal of International Financial Markets, Institutions and Money, Elsevier, vol. 79(C).
    8. Mai, Te-Ke & Foley, Aoife M. & McAleer, Michael & Chang, Chia-Lin, 2022. "Impact of COVID-19 on returns-volatility spillovers in national and regional carbon markets in China," Renewable and Sustainable Energy Reviews, Elsevier, vol. 169(C).
    9. Hafner, Christian M. & Herwartz, Helmut & Maxand, Simone, 2022. "Identification of structural multivariate GARCH models," Journal of Econometrics, Elsevier, vol. 227(1), pages 212-227.
    10. Asai, Manabu & Chang, Chia-Lin & McAleer, Michael, 2022. "Realized matrix-exponential stochastic volatility with asymmetry, long memory and higher-moment spillovers," Journal of Econometrics, Elsevier, vol. 227(1), pages 285-304.
    11. Hsu, Shu-Han & Sheu, Chwen & Yoon, Jiho, 2021. "Risk spillovers between cryptocurrencies and traditional currencies and gold under different global economic conditions," The North American Journal of Economics and Finance, Elsevier, vol. 57(C).
    12. Aloui, Chaker & Hamida, Hela ben & Yarovaya, Larisa, 2021. "Are Islamic gold-backed cryptocurrencies different?," Finance Research Letters, Elsevier, vol. 39(C).
    13. Elie Bouri & Konstantinos Gkillas & Rangan Gupta, 2020. "Trade uncertainties and the hedging abilities of Bitcoin," Economic Notes, Banca Monte dei Paschi di Siena SpA, vol. 49(3), September.
    14. Zolfaghari, Mehdi & Ghoddusi, Hamed & Faghihian, Fatemeh, 2020. "Volatility spillovers for energy prices: A diagonal BEKK approach," Energy Economics, Elsevier, vol. 92(C).
    15. Michael McAleer, 2019. "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," JRFM, MDPI, vol. 12(2), pages 1-7, April.
    16. Semei Coronado & Rangan Gupta & Besma Hkiri & Omar Rojas, 2020. "Time-Varying Spillovers between Currency and Stock Markets in the USA: Historical Evidence From More than Two Centuries," Advances in Decision Sciences, Asia University, Taiwan, vol. 24(4), pages 44-76, December.
    17. Konstantinos Gkillas & Christoforos Konstantatos & Costas Siriopoulos, 2021. "Uncertainty Due to Infectious Diseases and Stock–Bond Correlation," Econometrics, MDPI, vol. 9(2), pages 1-18, April.

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    More about this item

    Keywords

    Hedging; Covariances; Correlations; Existence; Mathematical regularity; Invertibility; Likelihood function; Statistical asymptotic properties; Caveats; Practical implementation.;
    All these keywords.

    JEL classification:

    • C22 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes
    • C32 - Mathematical and Quantitative Methods - - Multiple or Simultaneous Equation Models; Multiple Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes; State Space Models
    • C51 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Model Construction and Estimation
    • C52 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Model Evaluation, Validation, and Selection
    • C58 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Financial Econometrics
    • C62 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Existence and Stability Conditions of Equilibrium
    • G32 - Financial Economics - - Corporate Finance and Governance - - - Financing Policy; Financial Risk and Risk Management; Capital and Ownership Structure; Value of Firms; Goodwill

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