Abstract
This paper considers a class of multivariate ARCH models with scalar weights. A new specification with hyperbolic weighted moving average (HWMA) is proposed as an analogue of the EWMA model. Despite the restrictive dynamics of a scalar weight model, the proposed model has a number of advantages that can deal with the curse of dimensionality. The empirical application illustrates that the (pseudo) out-of-sample multistep forecasts can be surprisingly more accurate than those from the DCC model.
References
Aielli, G. P. 2013. “Dynamic Conditional Correlation: On Properties and Estimation.” Journal of Business & Economic Statistics 31: 282–99. https://doi.org/10.1080/07350015.2013.771027.Search in Google Scholar
Baillie, R. T., T. Bollerslev, and H. O. Mikkelsen. 1996. “Fractionally Integrated Generalized Autoregressive Conditional Heteroskedasticity.” Journal of Econometrics 74: 3–30. https://doi.org/10.1016/s0304-4076(95)01749-6.Search in Google Scholar
Bollerslev, T., and H. O. Mikkelsen. 1996. “Modeling and Pricing Long Memory in Stock Market Volatility.” Journal of Econometrics 73: 151–84. https://doi.org/10.1016/0304-4076(95)01736-4.Search in Google Scholar
Boudt, K., A. Galanos, S. Payseur, and E. Zivot. 2019. “Multivariate GARCH Models for Large-Scale Applications: A Survey.” In Handbook of Statistics, vol. 41, Chap. 7, 193–242. Elsevier.10.1016/bs.host.2019.01.001Search in Google Scholar
Boussama, F. 2006. “Ergodicity of Markov Chains in an Algebraic Manifold: Application to Multivariate GARCH Models.” Comptes Rendus Mathematique 343: 275–8. https://doi.org/10.1016/j.crma.2006.06.027.Search in Google Scholar
Chiriac, R., and V. Voev. 2011. “Modelling and Forecasting Multivariate Realized Volatility.” Journal of Applied Econometrics 26: 922–47. https://doi.org/10.1002/jae.1152.Search in Google Scholar
Comte, F., and O. Lieberman. 2003. “Asymptotic Theory for Multivariate GARCH Processes.” Journal of Multivariate Analysis 84: 61–84. https://doi.org/10.1016/s0047-259x(02)00009-x.Search in Google Scholar
Conrad, C., and B. R. Haag. 2006. “Inequality Constraints in the Fractionally Integrated GARCH Model.” Journal of Financial Econometrics 4: 413–49. https://doi.org/10.1093/jjfinec/nbj015.Search in Google Scholar
Corsi, F. 2009. “A Simple Approximate Long-Memory Model of Realized Volatility.” Journal of Financial Econometrics 7: 174–96. https://doi.org/10.1093/jjfinec/nbp001.Search in Google Scholar
Creal, D., S. J. Koopman, and A. Lucas. 2011. “A Dynamic Multivariate Heavy-Tailed Model for Time-Varying Volatilities and Correlations.” Journal of Business & Economic Statistics 29: 552–63. https://doi.org/10.1198/jbes.2011.10070.Search in Google Scholar
De Nard, G., O. Ledoit, and M. Wolf. 2020. “Factor Models for Portfolio Selection in Large Dimensions: The Good, the Better and the Ugly.” Journal of Financial Econometrics, forthcoming.10.1093/jjfinec/nby033Search in Google Scholar
Ding, J., and A. Zhou. 2007. “Eigenvalues of Rank-One Updated Matrices with Some Applications.” Applied Mathematics Letters 20: 1223–6. https://doi.org/10.1093/jjfinec/nby033.Search in Google Scholar
Ding, Z., C. W. J. Granger, and R. F. Engle. 1993. “A Long Memory Property of Stock Market Returns and a New Model.” Journal of Empirical Finance 1: 83–106. https://doi.org/10.1016/0927-5398(93)90006-d.Search in Google Scholar
Engle, R. F. 1982. “Autoregressive Conditional Heteroskedasticity with Estimates of the Variance of U.K. Inflation.” Econometrica 50: 987–1008. https://doi.org/10.2307/1912773.Search in Google Scholar
Engle, R. F. 2002. “Dynamic Conditional Correlation: A Simple Class of Multivariate Generalized Autoregressive Conditional Heteroskedasticity Models.” Journal of Business & Economic Statistics 20: 339–50. https://doi.org/10.1198/073500102288618487.Search in Google Scholar
Engle, R. F., and K. F. Kroner. 1995. “Multivariate Simultaneous Generalized ARCH.” Econometric Theory 11: 122–50. https://doi.org/10.1017/s0266466600009063.Search in Google Scholar
Engle, R. F., O. Ledoit, and M. Wolf. 2019. “Large Dynamic Covariance Matrices.” Journal of Business & Economic Statistics 37: 363–75. https://doi.org/10.1080/07350015.2017.1345683.Search in Google Scholar
Engle, R. F., and J. Mezrich. 1996. “GARCH for Groups.” Risk Magazine: 36–40.Search in Google Scholar
Engle, R. F., and K. Sheppard. 2001. Theoretical and Empirical Properties of Dynamic Conditional Correlation Multivariate GARCH. Unpublished.10.3386/w8554Search in Google Scholar
Francq, C., and J.-M. Zakoïan. 2010. GARCH Models. Wiley.10.1002/9780470670057Search in Google Scholar
Francq, C., and J.-M. Zakoïan. 2016. “Estimating Multivariate Volatility Models Equation by Equation.” Journal of Royal Statistical Society: Series B 78: 613–35. https://doi.org/10.1111/rssb.12126.Search in Google Scholar
Hafner, C. .. 2003. “Fourth Moment Structure of Multivariate GARCH Models.” Journal of Financial Econometrics 1: 26–54. https://doi.org/10.1093/jjfinec/nbg001.Search in Google Scholar
Hafner, C. M., and H. Herwartz. 2006. “Volatility Impulse Responses for Multivariate GARCH Models: An Exchange Rate Illustration.” Journal of International Money and Finance 25: 719–40. https://doi.org/10.1016/j.jimonfin.2006.04.006.Search in Google Scholar
Hafner, C. M., and O. Reznikova. 2012. “On the Estimation of Dynamic Conditional Correlation Models.” Computational Statistics & Data Analysis 56: 3533–45. https://doi.org/10.1016/j.csda.2010.09.022.Search in Google Scholar
Hager, W. W. 1989. “Updating the Inverse of a Matrix.” SIAM Review 31: 221–39. https://doi.org/10.1137/1031049.Search in Google Scholar
Hansen, P. R., A. Lunde, and J. M. Nason. 2011. “The Model Confidence Set.” Econometrica 79: 453–97. https://doi.org/10.3982/ECTA5771.Search in Google Scholar
Jeantheau, T. 1998. “Strong Consistency of Estimators for Multivariate ARCH Models.” Econometric Theory 14: 70–86. https://doi.org/10.1017/s0266466698141038.Search in Google Scholar
Jensen, A. N., and M. Ø. Nielsen. 2014. “A Fast Fractional Difference Algorithm.” Journal of Time Series Analysis 35: 428–36. https://doi.org/10.1111/jtsa.12074.Search in Google Scholar
Klein, T., and T. Walther. 2017. “Fast Fractional Differencing in Modeling Long Memory of Conditional Variance for High-Frequency Data.” Finance Research Letters 22: 274–9. https://doi.org/10.1016/j.frl.2016.12.020.Search in Google Scholar
Laurent, S., J. V. K. Rombouts, and F. Violante. 2013. “On Loss Functions and Ranking Forecasting Performances of Multivariate Volatility Models.” Journal of Econometrics 173: 1–10. https://doi.org/10.1214/19-AOS1921.Search in Google Scholar
Ledoit, O., and M. Wolf. 2020. “Analytical Nonlinear Shrinkage of Large-Dimensional Covariance Matrices.” Annals of Statistics, forthcoming.10.1214/19-AOS1921Search in Google Scholar
Müller, U. A., M. M. Dacorogna, R. D. Davé, R. B. Olsen, O. V. Pictet, and J. E. von Weizsäcker. 1997. “Volatilities of Different Time Resolutions—Analyzing the Dynamics of Market Components.” Journal of Empirical Finance 4: 213–39. https://doi.org/10.1016/s0927-5398(97)00007-8.Search in Google Scholar
Pakel, C., N. Shephard, K. Sheppard, and R. F. Engle. 2020. “Fitting Vast Dimensional Time-Varying Covariance Models.” Journal of Business & Economic Statistics. https://doi.org/10.1080/07350015.2020.1713795.Search in Google Scholar
Patton, A. J. 2011. “Volatility Forecast Comparison Using Imperfect Volatility Proxies.” Journal of Econometrics 160: 246–56. https://doi.org/10.1016/j.jeconom.2010.03.034.Search in Google Scholar
Pedersen, R. S., and A. Rahbek. 2014. “Multivariate Variance Targeting in the BEKK-GARCH Model.” Econometrics Journal 17: 24–55. https://doi.org/10.1111/ectj.12019.Search in Google Scholar
Supplementary Material
The online version of this article offers supplementary material (https://doi.org/10.1515/jem-2020-0004).
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