Calzolari, Giorgio and Fiorentini, Gabriele and Panattoni, Lorenzo (1993): Alternative estimators of the covariance matrix in GARCH models. Published in: Universita' di Messina, Istituto di Economia, Statistica e Analisi del Territorio No. Quaderno No. 11 (1993): pp. 133.

PDF
MPRA_paper_24433.pdf Download (665Kb)  Preview 
Abstract
With most of the available software packages, estimates of the parameter covariance matrix in a GARCH model are usually obtained from the outer products of the first derivatives of the loglikelihoods (BHHH estimator). However, other estimators could be defined and used, analogous to the covariance matrix estimators in maximum likelihood studies described in the literature for other types of models (linear regression model, linear and nonlinear simultaneous equations, Probit and Tobit models). These alternative estimators can be derived from: (1) the Hessian (observed information), (2) the estimated information (expected Hessian), (3) a mixture of Hessian and outer products matrix (White's QML covarjance matrix). Signifacant differences among these estimates can be interpreted as an indication of misspecification, or can be due to systematic inequalities between alternative estimators in small samples. Unlike other types of models, from our Monte Carlo study we do not encounter very large differences, presumably because GARCH estimation is usually applied when the sample size is rather large. However, analogously to otber types of models we find in this Monte Carlo study that, even in absence of misspecification, the sign of the differences between some estimators is almost systematic. This suggests that, as for other types of models, the choice of the covariance estimator is not neutral, but the results of hypotheses testing are not strongly affected by such a choice.
Item Type:  MPRA Paper 

Original Title:  Alternative estimators of the covariance matrix in GARCH models 
Language:  English 
Keywords:  GARCH model; Hessian matrix; outer products; maximum likelihood 
Subjects:  C  Mathematical and Quantitative Methods > C6  Mathematical Methods; Programming Models; Mathematical and Simulation Modeling > C63  Computational Techniques; Simulation Modeling C  Mathematical and Quantitative Methods > C2  Single Equation Models; Single Variables > C22  TimeSeries Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models 
Item ID:  24433 
Depositing User:  Giorgio Calzolari 
Date Deposited:  16. Aug 2010 11:51 
Last Modified:  12. Feb 2013 20:39 
References:  Baillie, R. T., and T. Bollerslev (1989): "The Message in Daily Exchange Rates: A ConditionalVariance Tale", Journal of Business and Economic Statistics 7, 297305. Baillie, R. T., and R. P. De Gennaro (1990): "Stock Returns and Volatility", Journal of Financial and Quantitative Analysis 25, 203214. Baillie, R. T., and R. J. Myers (1991): "Bivariate GARCH Estimation of the Optimal Commodity Futures Hedge", Journal of Applied Econometrics 6, 109124. Berndt, E. K., B. H. Hall, R. E. all, and J. A. Hausman (1974): "Estimation and Inference in Nonlinear Structural Models", Annals of Economic and Social Measurement 3, 653665. Bianchi, C., G. Calzolari, and F. P. Sterbenz (1991): "Simulation of Interest Rate Options Using ARCH". Universita' di Messina, discussion paper presented at the European Meeting of the Econometric Society, Cambridge, U.K., September 26. Bollerslev, T. (1986): "Generalized Autoregressive Conditional Heteroskedasticity", Journal of Econometrics 31, 307327. Bollerslev, T. (1 987): "A Conditional Heteroskedastic Time Series Model for Speculative Prices and Rates of Return", The Review of Economics and Statistics 69, 542547. Bollerslev, T., R. F. Engle, and J. M. Wooldridge (1988): "A Capital Asset Pricing Model with Timevarying Covariances", Journal of Political Economy 96, 116131. Buzzigoli, L. (1992): "Modelli di Analisi delle Serie Temporali Finanziarie: Rassegna e Applicazioni di Modelli di Tipo ARCH". Universita' di Padova: Dipartimento di Scienze Statistiche, PhD dissertation. Calzolari, G., and G. Fiorentini (1993, forthcoming): "Alternative Covariance Estimators of the Standard Tobit Model", Economics Letters. Calzolari, G., and G. Fiorentini (1992): "Alternative Methods for GARCH Estimation", Universita' di Firenze: Dipartimento Statistico, Working Paper No. 44, presented at the European Meeting of the Econometric Society, Bruxelles, August 2428. Calzolari, G., and L. Panattoni (1988a): "Alternative Estimators of FIML Covariance Matrix: A Monte Carlo Study", Econometrica 56, 701714. Calzolari, G., and L. Panattoni (1988b): "Finite Sample Performance of the Robust Wald Test in Simultaneous Equation Systems", in Advances in Econometrics, Vol. 7, ed. by G. F. Rhodes, Jr., and T. B. Fomby. Greenwich, CO: JAI Press Inc., 163191. Chesher, A. (1989): "Hajek Inequalities, Measures of Leverage and the Size of Heteroskedasticity Robust Wald Tests", Econometrica 57, 971977. Chou, R. Y. (1988): "Volatility Persistence and Stock Valuations: Some Empirical Evidence using GARCH", Journal of Applied Econometrics 3, 279294. Demos, A. and E. Sentana, (1991): "Testing for GARCH effects: A Onesided Approach", discussion paper presented at the European Meeting of the Econometric Society, Cambridge, U.K., September 26. De Santis, G., and A. M. Sbordone (1990): "A CAPM with a Multivariate Generalized ARCH Process: An Empirical Analysis of the Italian Financial Market". University of Chicago: Graduate School of Business, discussion paper. Engle, R. F. (1982): "Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation", Econometrica 50, 9871007. Engle, R. F., and T. Bollerslev (1986): "Modelling the Persistence of Conditional Variances", Econometric Reviews 5, 150. Engle, R. F., D. F. Hendry, and J. F. Richard (1983): "Exogeneity", Econometrica 51, 277304. Engle, R. F., D. M. Lilien, and R. P. Robins (1987): "Estimating Time Varying Risk Premia in the Term Structure: The ARCHM Model", Econometrica 55, 391407. Engle, R. F., and M. Rothschild, eds. (1992): "ARCH Models in Finance", Journal of Econometrics 52, 1311. Gourieroux, C., A. Monfort, and A. Trognon (1984): "Pseudo Maximum Likelihood Methods: Theory", Econometrica 52, 681700. Griffiths, W. E., R. C. Hill, and P. J. Pope (1987): "Small Samples Properties of Probit Model Estimators", Journal of the American Statistical Association 82, 929937. Lamoureaux, C. G., and W. D. Lastrapes (1990): "Persistence in Variance, Structural Change, and the GARCH Model", Journal of Business and Economic Statistics 8, 225234. MacKinnon, J. G., and H. White (1985): "Some Heteroskedasticity Consistent Covariance Matrix Estimators with Improved Finite Sample Properties", Journal of Econometrics 29, 305325. Nelson, D. B. (1991): "Conditional Heteroskedasticity in Asset Returns: A New Approach", Econometrica 59, 347370. Parks, R W. and N. E. Savin (1990): "The Choice of Coefficients Covariance Matrix Estimator: Outer Product versus Hessian". The University of Iowa, Department of Economics, working paper presented at the 6th World Congress of the Econometric Society, Barcelona, August 2228. Schweppe, F. (1965): "Evaluation of Likelihood Functions for Gaussian Signals", IEEE Transactions on Information 11, 6170. Thomas, A. (1991): "Implied Nonlinear ARCH Models in Computing Optimal Auction Bids". Toulouse: GREMAQ, discussion paper. White, H. (1982): "Maximum Likelihood Estimation of Misspecified Models", Econometrica 50, 125. White, H. (1983): "Corrigendum", Econometrica 51, 513. 
URI:  http://mpra.ub.unimuenchen.de/id/eprint/24433 