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The Fiction of Full BEKK: Pricing Fossil Fuels and Carbon Emissions

Author

Listed:
  • Chia-Lin Chang

    ( Department of Applied Economics and Department of Finance National Chung Hsing University, Taiwan.)

  • 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

The purpose of the paper is to (i) show that univariate GARCH is not a special case of multivariate GARCH, specifically the Full BEKK model, except under parametric restrictions on the off-diagonal elements of the random coefficient autoregressive coefficient matrix, that are not consistent with Full BEKK, and (ii) provide the regularity conditions that arise from the underlying random coefficient autoregressive process, for which the (quasi-) maximum likelihood estimates (QMLE) have valid asymptotic properties under the appropriate parametric restrictions. The paper provides a discussion of the stochastic processes that lead to the alternative specifications, regularity conditions, and asymptotic properties of the univariate and multivariate GARCH models. It is shown that the Full BEKK model, which in empirical practice is estimated almost exclusively compared with Diagonal BEKK (DBEKK), has no underlying stochastic process that leads to its specification, regularity conditions, or asymptotic properties, as compared with DBEKK. An empirical illustration shows the differences in the QMLE of the parameters of the conditional means and conditional variances for the univariate, DEBEKK and Full BEKK specifications.

Suggested Citation

  • Chia-Lin Chang & Michael McAleer, 2018. "The Fiction of Full BEKK: Pricing Fossil Fuels and Carbon Emissions," Documentos de Trabajo del ICAE 2018-08, Universidad Complutense de Madrid, Facultad de Ciencias Económicas y Empresariales, Instituto Complutense de Análisis Económico.
  • Handle: RePEc:ucm:doicae:1808
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    References listed on IDEAS

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    1. Nelson, Daniel B., 1990. "ARCH models as diffusion approximations," Journal of Econometrics, Elsevier, vol. 45(1-2), pages 7-38.
    2. Ling, Shiqing & McAleer, Michael, 2003. "Asymptotic Theory For A Vector Arma-Garch Model," Econometric Theory, Cambridge University Press, vol. 19(02), pages 280-310, April.
    3. Engle, Robert F. & Kroner, Kenneth F., 1995. "Multivariate Simultaneous Generalized ARCH," Econometric Theory, Cambridge University Press, vol. 11(01), pages 122-150, February.
    4. Nelson, Daniel B, 1991. "Conditional Heteroskedasticity in Asset Returns: A New Approach," Econometrica, Econometric Society, vol. 59(2), pages 347-370, March.
    5. Kwiatkowski, Denis & Phillips, Peter C. B. & Schmidt, Peter & Shin, Yongcheol, 1992. "Testing the null hypothesis of stationarity against the alternative of a unit root : How sure are we that economic time series have a unit root?," Journal of Econometrics, Elsevier, vol. 54(1-3), pages 159-178.
    6. Dickey, David A & Fuller, Wayne A, 1981. "Likelihood Ratio Statistics for Autoregressive Time Series with a Unit Root," Econometrica, Econometric Society, vol. 49(4), pages 1057-1072, June.
    7. Michael McAleer & Christian M. Hafner, 2014. "A One Line Derivation of EGARCH," Econometrics, MDPI, Open Access Journal, vol. 2(2), pages 1-6, June.
    8. Jeantheau, Thierry, 1998. "Strong Consistency Of Estimators For Multivariate Arch Models," Econometric Theory, Cambridge University Press, vol. 14(01), pages 70-86, February.
    9. Chia-Lin Chang & Michael McAleer & Guangdong Zuo, 2017. "Volatility Spillovers and Causality of Carbon Emissions, Oil and Coal Spot and Futures for the EU and USA," Sustainability, MDPI, Open Access Journal, vol. 9(10), pages 1-22, October.
    10. Perron, Pierre & Rodriguez, Gabriel, 2003. "GLS detrending, efficient unit root tests and structural change," Journal of Econometrics, Elsevier, vol. 115(1), pages 1-27, July.
    11. repec:eee:rensus:v:81:y:2018:i:p1:p:1002-1018 is not listed on IDEAS
    12. Glosten, Lawrence R & Jagannathan, Ravi & Runkle, David E, 1993. " On the Relation between the Expected Value and the Volatility of the Nominal Excess Return on Stocks," Journal of Finance, American Finance Association, vol. 48(5), pages 1779-1801, December.
    13. Bollerslev, Tim, 1986. "Generalized autoregressive conditional heteroskedasticity," Journal of Econometrics, Elsevier, vol. 31(3), pages 307-327, April.
    14. Chang, Chia-Lin & McAleer, Michael & Wang, Yu-Ann, 2018. "Modelling volatility spillovers for bio-ethanol, sugarcane and corn spot and futures prices," Renewable and Sustainable Energy Reviews, Elsevier, vol. 81(P1), pages 1002-1018.
    15. Michael McAleer, 2014. "Asymmetry and Leverage in Conditional Volatility Models," Econometrics, MDPI, Open Access Journal, vol. 2(3), pages 1-6, September.
    16. Perron, Pierre & Rodriguez, Gabriel, 2003. "GLS detrending, efficient unit root tests and structural change," Journal of Econometrics, Elsevier, vol. 115(1), pages 1-27, July.
    17. Chang, C-L. & McAleer, M.J. & Wang, Y-A., 2016. "Modelling Volatility Spillovers for Bio-ethanol, Sugarcane and Corn," Econometric Institute Research Papers EI2016-15, Erasmus University Rotterdam, Erasmus School of Economics (ESE), Econometric Institute.
    18. McAleer, Michael & Chan, Felix & Hoti, Suhejla & Lieberman, Offer, 2008. "Generalized Autoregressive Conditional Correlation," Econometric Theory, Cambridge University Press, vol. 24(06), pages 1554-1583, December.
    19. 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.
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    Citations

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    Cited by:

    1. Chia-Lin Chang & Michael McAleer & Guangdong Zuo, 2017. "Volatility Spillovers and Causality of Carbon Emissions, Oil and Coal Spot and Futures for the EU and USA," Sustainability, MDPI, Open Access Journal, vol. 9(10), pages 1-22, October.
    2. Fuentes, Fernanda & Herrera, Rodrigo & Clements, Adam, 2018. "Modeling extreme risks in commodities and commodity currencies," Pacific-Basin Finance Journal, Elsevier, vol. 51(C), pages 108-120.
    3. repec:gam:jrisks:v:6:y:2018:i:4:p:120-:d:175263 is not listed on IDEAS
    4. repec:gam:jeners:v:11:y:2018:i:6:p:1595-:d:153161 is not listed on IDEAS
    5. Chia-Lin Chang & Yiying Li & Michael McAleer, 2018. "Volatility Spillovers between Energy and Agricultural Markets: A Critical Appraisal of Theory and Practice," Energies, MDPI, Open Access Journal, vol. 11(6), pages 1-19, June.
    6. Chang, Chia-Lin & Mai, Te-Ke & McAleer, Michael, 2019. "Establishing national carbon emission prices for China," Renewable and Sustainable Energy Reviews, Elsevier, vol. 106(C), pages 1-16.
    7. 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," Journal of Risk and Financial Management, MDPI, Open Access Journal, vol. 12(2), pages 1-7, April.
    8. Chia-Lin Chang & Shu-Han Hsu & Michael McAleer, 2018. "Risk Spillovers in Returns for Chinese and International Tourists to Taiwan," Tinbergen Institute Discussion Papers 18-031/III, Tinbergen Institute.
    9. Asai, M. & Chang, C-L. & McAleer, M.J. & Pauwels, L., 2018. "Asymptotic Theory for Rotated Multivariate GARCH Models," Econometric Institute Research Papers EI2018-38, Erasmus University Rotterdam, Erasmus School of Economics (ESE), Econometric Institute.
    10. 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, Open Access Journal, vol. 12(8), pages 1-24, April.
    11. David E. Allen & Michael McAleer, 2018. "Theoretical and Empirical Differences between Diagonal and Full BEKK for Risk Management," Energies, MDPI, Open Access Journal, vol. 11(7), pages 1-19, June.
    12. repec:gam:jjrfmx:v:11:y:2018:i:4:p:58-:d:172906 is not listed on IDEAS
    13. Chia-Lin Chang & Tai-Lin Hsieh & Michael McAleer, 2018. "Connecting VIX and Stock Index ETF with VAR and Diagonal BEKK," Journal of Risk and Financial Management, MDPI, Open Access Journal, vol. 11(4), pages 1-25, September.

    More about this item

    Keywords

    Random coefficient stochastic process; Off-diagonal parametric restrictions; Diagonal BEKK; Full BEKK; Regularity conditions; Asymptotic properties; Conditional volatility; Univariate and multivariate models; Fossil fuels and carbon emissions.;

    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
    • C52 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Model Evaluation, Validation, and Selection
    • C58 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Financial Econometrics

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