Abstract
Multivariate GARCH models constitute the workhorse of empiricalapplications in several fields, a notable example being financialeconometrics. Unfortunately, ML (or quasi-ML) estimation of such models,although relatively straightforward in theory, is often made difficult bythe fact that available software relies on numerical methods for computingthe first derivatives of the log-likelihood; the fact that these modelsoften include a large number of parameters makes it impractical toestimate even medium-sized models. In this paper, closed-form expressionsfor the score of the BEKK model of Engle and Kroner (1995) are obtained,and strategies for efficient computation are discussed.
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References
Andersen, T. G., Chung, H.-J. and Sørensen, B. E. (1999). Efficient method of moments estimation of a stochastic volatility model: A Monte Carlo study. Journal of Econometrics, 91, 61–87.
Bollerslev, T. (1986). Generalized autoregressive conditional heteroscedasticity. Journal of Econometrics, 31, 307–327.
Bollerslev, T. (1990). Modelling the coherence in short-run nominal exchange rates: A multivariate generalized ARCH model. Review of Economics and Statistics, 72, 498–505.
Bollerslev, T., Engle, R. F. and Nelson, D. (1994). ARCH models. In R. F. Engle and D. McFadden (eds.): Handbook of Econometrics, Vol. 4. North-Holland, Chapt. 49, pp. 2959–3040.
Bollerslev, T., Engle, R. F. and Wooldridge, J. M. (1988). A capital asset pricing model with time varying covariances. Journal of Political Economy, 96, 116–131.
Diebold, F. X. and Nerlove, M. (1989). The dynamics of exchange rate volatility: A multivariate latent factor ARCH model. Journal of Applied Econometrics, 4, 1–21.
Doornik, J. A.: 1998, Object-Oriented Matrix Programming Using Ox 2.0. Timberlake Consultants, 2nd Edition.
Engle, R. F. (1982). Autoregressive conditional heteroskedasticity with estimates of the variance of United Kingdom inflation. Econometrica, 50, 987–1008.
Engle, R. F. and Kroner, K. F. (1995). Multivariate simultaneous generalized ARCH. Econometric Theory, 11, 122–150.
Engle, R. F., Ng, V. K. and Rothschild, M. (1990). Asset pricing with a FACTOR-ARCH covariance structure: Empirical estimates for treasury bills. Journal of Econometrics, 45, 213–237.
Fiorentini, G., Calzolari, G. and Panattoni, L. (1996). Analytic derivatives and the computation of GARCH estimates. Journal of Applied Econometrics, 11, 399–417.
Giannini, C. and Rossi, E. (1999). A principal components multivariate GARCH technique for medium size portfolio management. Collana Studi del Credito Italiano, 6/99.
Gourieroux, C. and Monfort, A. (1996). Simulation-Based Econometric Methods. Oxford University Press.
Jeantheau, T. (1998). Strong consistency of estimators for multivariate ARCH models. Econometric Theory, 14, 70–86.
Lütkepohl, H. (1996). Handbook of Matrices. John Wiley & Sons.
Magnus, J. (1988). Linear Structures. Charles Griffin & Co.
Magnus, J. and Neudecker, H. (1988). Matrix Differential Calculus. John Wiley & Sons.
Pollock, D. (1979). The Algebra of Econometrics. John Wiley & Sons.
Quandt, R. E. (1983). Computational problems and methods. In Z. Griliches and M. D. Intriligator (eds.), Handbook of Econometrics, No. 1, North-Holland, Chapt. 12, pp. 699–764.
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Lucchetti, R. Analytical Score for Multivariate GARCH Models. Computational Economics 19, 133–143 (2002). https://doi.org/10.1023/A:1015001204774
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DOI: https://doi.org/10.1023/A:1015001204774