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Alternative estimators of the covariance matrix in GARCH models

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  • Calzolari, Giorgio
  • Fiorentini, Gabriele
  • Panattoni, Lorenzo

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 log-likelihoods (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.

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Paper provided by University Library of Munich, Germany in its series MPRA Paper with number 24433.

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Date of creation: 1993
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Handle: RePEc:pra:mprapa:24433

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Keywords: GARCH model; Hessian matrix; outer products; maximum likelihood;

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References

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  1. Lamoureux, Christopher G & Lastrapes, William D, 1990. "Persistence in Variance, Structural Change, and the GARCH Model," Journal of Business & Economic Statistics, American Statistical Association, vol. 8(2), pages 225-34, April.
  2. E.K. Berndt & B.H. Hall & R.E. Hall, 1974. "Estimation and Inference in Nonlinear Structural Models," NBER Chapters, in: Annals of Economic and Social Measurement, Volume 3, number 4, pages 103-116 National Bureau of Economic Research, Inc.
  3. Bollerslev, Tim, 1986. "Generalized autoregressive conditional heteroskedasticity," Journal of Econometrics, Elsevier, vol. 31(3), pages 307-327, April.
  4. MacKinnon, James G. & White, Halbert, 1985. "Some heteroskedasticity-consistent covariance matrix estimators with improved finite sample properties," Journal of Econometrics, Elsevier, vol. 29(3), pages 305-325, September.
  5. Nelson, Daniel B, 1991. "Conditional Heteroskedasticity in Asset Returns: A New Approach," Econometrica, Econometric Society, vol. 59(2), pages 347-70, March.
  6. Baillie, Richard T & Myers, Robert J, 1991. "Bivariate GARCH Estimation of the Optimal Commodity Futures Hedge," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 6(2), pages 109-24, April-Jun.
  7. Chou, Ray Yeutien, 1988. "Volatility Persistence and Stock Valuations: Some Empirical Evidence Using Garch," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 3(4), pages 279-94, October-D.
  8. Bianchi, Carlo & Calzolari, Giorgio & Sterbenz, Frederic P., 1991. "Simulation of interest rate options using ARCH," MPRA Paper 24844, University Library of Munich, Germany.
  9. Calzolari, Giorgio & Panattoni, Lorenzo, 1988. "Alternative Estimators of FIML Covariance Matrix: A Monte Carlo Stud y," Econometrica, Econometric Society, vol. 56(3), pages 701-14, May.
  10. Baillie, Richard T. & DeGennaro, Ramon P., 1990. "Stock Returns and Volatility," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 25(02), pages 203-214, June.
  11. Engle, Robert F & Lilien, David M & Robins, Russell P, 1987. "Estimating Time Varying Risk Premia in the Term Structure: The Arch-M Model," Econometrica, Econometric Society, vol. 55(2), pages 391-407, March.
  12. Calzolari, Giorgio & Fiorentini, Gabriele, 1993. "Alternative covariance estimators of the standard Tobit model," Economics Letters, Elsevier, vol. 42(1), pages 5-13.
  13. Gourieroux, Christian & Monfort, Alain & Trognon, Alain, 1984. "Pseudo Maximum Likelihood Methods: Theory," Econometrica, Econometric Society, vol. 52(3), pages 681-700, May.
  14. Chesher, Andrew, 1989. "Hajek Inequalities, Measures of Leverage and the Size of Heteroskedasticity Robust Wald Tests," Econometrica, Econometric Society, vol. 57(4), pages 971-77, July.
  15. White, Halbert, 1982. "Maximum Likelihood Estimation of Misspecified Models," Econometrica, Econometric Society, vol. 50(1), pages 1-25, January.
  16. 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.
  17. Demos, Antonis & Sentana, Enrique, 1998. "Testing for GARCH effects: a one-sided approach," Journal of Econometrics, Elsevier, vol. 86(1), pages 97-127, June.
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