Abstract
Quasi maximum likelihood estimation and inference in multivariate volatility models remains a challenging computational task if, for example, the dimension of the parameter space is high. One of the reasons is that typically numerical procedures are used to compute the score and the Hessian, and often they are numerically unstable. We provide analytical formulae for the score and the Hessian for a variety of multivariate GARCH models including the Vec and BEKK specifications as well as the recent dynamic conditional correlation model. By means of a Monte Carlo investigation of the BEKK–GARCH model we illustrate that employing analytical derivatives for inference is clearly preferable to numerical methods.
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References
Alexander CO (2001) Orthogonal GARCH. In: Alexander CO (ed) Mastering risk, vol II. Prentice Hall, New York, pp 21–38
Bauwens L, Laurent S, Rombouts JVK (2006) Multivariate GARCH models: a survey. J Appl Econom 21:79–109
Bollerslev T (1986) Generalized autoregressive conditional heteroskedasticity. J Econom 31:307–327
Bollerslev T (1990) Modelling the coherence in short-run nominal exchange rates: a multivariate generalized ARCH approach. Rev Econ Stat 72:498–505
Bollerslev T, Wooldridge JM (1992) Quasi-maximum likelihood estimation and inference in dynamic models with time-varying covariances. Econom Rev 11:143–172
Bollerslev T, Engle RF, Wooldridge JM (1988) A capital asset pricing model with time varying covariances. J Polit Econ 96:116–131
Comte F, Lieberman O (2003) Asymptotic theory for multivariate GARCH processes. J Multivariate Anal 84:61–84
Engle RF, Kroner KF (1995) Multivariate simultaneous generalized ARCH. Econom Theory 11:122–150
Engle RF, Ng VM, Rothschild M (1990) Asset pricing with a factor-ARCH structure: empirical estimates for treasury bills. J Econom 45:213–237
Fiorentini G, Sentana E, Calzolari G (2003) Maximum likelihood estimation and inference in multivariate conditionally heteroskedastic dynamic regression models with student t innovations. J Bus Econ Stat 21:532–546
Hafner CM, Franses PH (2003) A generalized dynamic conditional correlation model for many assets, Econometric Institute Report 18, Erasmus University Rotterdam
Hafner CM, Herwartz H (2000) Testing for linear autoregressive dynamics under heteroskedasticity. Econom J 3:177–197
Hafner CM, Herwartz H (2006) Volatility impulse responses for multivariate GARCH models: an exchange rate illustration. J Int Money Financ 25:719–740
Herwartz H (2004) Conditional heteroskedasticity. In: Lütkepohl H., Krätzig M (eds). Applied time series econometrics. Cambridge University Press, Cambridge, pp 197–221
Herwartz H, Neumann MH (2005) Bootstrap inference in systems of single equation error correction models. J Econom 128:165–193
Jeantheau T (1998) Strong consistency of estimators for multivariate ARCH models. Econom Theory 14:70–86
Ling S, McAleer M (2003) Asymptotic theory for a vector ARMA-GARCH model. Econom Theory 19:280–309
Lucchetti R (2001) Analytical score for multivariate GARCH models. Comput Econ 19:133–143
Lütkepohl H (1996) Handbook of matrices. Wiley, New York
Magnus JR, Neudecker H (1988) Matrix differential calculus with applications in statistics and econometrics. Wiley, New York
van der Weide R (2002) GO-GARCH: a multivariate generalized orthogonal GARCH model. J Appl Econom 17:549–564
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Hafner, C.M., Herwartz, H. Analytical quasi maximum likelihood inference in multivariate volatility models. Metrika 67, 219–239 (2008). https://doi.org/10.1007/s00184-007-0130-y
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DOI: https://doi.org/10.1007/s00184-007-0130-y