Abstract
This paper discusses how conditional heteroskedasticity models can be estimated efficiently without imposing strong distributional assumptions such as normality. Using the generalized method of moments (GMM) principle, we show that for a class of models with a symmetric conditional distribution, the GMM estimates obtained from the joint estimating equations corresponding to the conditional mean and variance of the model are efficient when the instruments are chosen optimally. A simple ARCH(1) model is used to illustrate the feasibility of the proposed estimation procedure.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bollerslev, T. P., “Generalized Autoregressive Conditional Heteroskedasticity.” Journal of Econometrics 31, 309–328, 1986.
Bollerslev, T. P., “A Conditionally Heteroskedasticity Time Series Model for Security Prices and Rates of Return Data.” Review of Economics and Statistics 69, 542–547, 1987.
Bollerslev, T. P., R. Y. Chou and K. F. Kroner, “ARCH Modelling in Finance: A Review of the Theory and Empirical Evidence,” Journal of Econometrics 52, 5–69, 1990.
Bollerslev, T. P., R. F. Engle and D. B. Nelson, “ARCH Models.” in R. F. Engle and D. McMadden eds. The Handbook of Econometrics, Vol 4. Elsevier: Nort Holland, 1994.
Bollerslev, T. P. and J. M. Wooldridge, “Quasi-Maximum Likelihood Estimation of Dynamic Models With Time Varying Covariances.” Discussion Paper 505, Masachussetts Institute of Technology, Dept. of Economics, 1988.
Chamberlain, G., “Asymptotic Efficiency in Estimation with Conditional Moment Restriction.” Journal of Econometrics 34, 305–334, 1987.
Davidson, R. and J. G. MacKinnon, “Specification Tests Based on Artificial Regression.” International Economic Review 25, 485–502, 1984.
Engle, R. F., “Autoregressive Conditional Heteroskedasticity With Estimates of the Variance of United Kingdom Inflation.” Econometrica 50, 987–1007, 1982.
Engle, R. F. and T. P. Bollerslev, “Modelling the Persistence of Conditional Variances.” Econometrics Review 5, 1–50, 1986.
Engle, R. F. and G. Gonzales-Rivera, “Semiparametric ARCH Models.” Journal of Business and Economic Statistics 9, 345–360, 1991.
Engle, R. F., D. F. Hendry and J. F. Richard, “Exogeneity.” Econometrica 51, 277–304, 1983.
Engle, R. F., D. M. Lillien and R. P. Robins, “Estimating Time Varying Risk Premia in the Term Structure.” Econometrica 55, 391–407, 1987.
French, K. R., G. W. Schwert and E. F. Stambaugh, “Expected Stock Returns and Volatility.” Journal of Financial Economics 19, 3–29, 1987.
Fuhrer, J., G. Moore and S. Schuh, “Estimating the Linear-Quadratic Inventory Model: Maximum Likelihood Versus Generalized Method of Moments.” Finance and Economics Discussion Series 93-11, Federal Reserve Board, Wasihington, D. C., (April 1993).
Gallant, A. R., D. Hsieh and G. Tauchen, “On Fitting a Recalcitrant Series: The Pound/Dollar Exchange Rate. 1974–83.” in W. A. Barnett, J. Powell and G. E. Tauchen eds. Nonparametric and Semiparametric Methods in Econometrics and Statistics. Cambridge: Cambridge University Press, 1991.
Hansen, L. P., “Large Sample Properties of Generalized Method of Moments Estimator.” Econometrica 50, 1029–1054, (July 1982).
Hong, C., “Options, Volatilities and the Hedge Strategy.” unpublished Ph.D. Dissertation. University of California at San Diego, Department of Economics, 1988.
Li, David X. and T. S. Wirjanto, “Estimating Functions and Conditional Heteroskedasticity Models, unpublished manuscript, University of Waterloo, 1996.
Newey, W. K., “Efficient Instrumental Variables Estimation of Non-linear Moels.” Econometrica 58, 809–837, 1990.
Pagan, A. R and Y. Jung, “Understanding the Failure of Some Instrumental Variables Estimators.” Mimeo, Australian National University, (June 1993).
Rich, R., J. Raymond and J. S. Butler, “Generalized Instrumental Variables Estimation of Autoregressive Conditional Heteroskedastic Models.” Economics Letters 35, 179–185, 1991.
Robinson, P. M. “Asymptotically Efficient Estimation in the Presence of Heteroskedasticity of Unknown Form.” Econometrica 55, 875–891, 1987.
Singleton, J. C. and J. Wigender, “Skewness Persistence in Common Stock Returns.” Journal of Financial and Quantitative Analysis 21, 335–341, 1986.
Staiger, D. and J. H. Stock, “Instrumental Variables Regressions With Weak Instruments.” National Bureau of Economic Research Technical Working Paper No. 151, (January 1994).
Weiss, A. A., “ARMA Models with ARCH Errors.” Journal of Time Series Analysis 15, 129–143, 1984.
Weiss, A. A., “Asymptotic Theory for ARCH Models: Estimation and Testing.” Econometric Theory 2, 107–131, 1986.
White, H., “Maximum Likelihood Estimation of Misspecified Models.” Econometrica 50, 1–26, 1982.
Wirjanto, T. S., “On Asymptotically Efficient Estimation of a Class of Conditionally Heteroskedastic Nonlinear Regression Models.” Waterloo Economic Series. Working Paper No. 9115, University of Waterloo, (August 1991).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Lee, T.Y., Wirjanto, T. On the Efficiency of Conditional Heteroskedasticity Models. Review of Quantitative Finance and Accounting 10, 21–37 (1998). https://doi.org/10.1023/A:1008244029475
Issue Date:
DOI: https://doi.org/10.1023/A:1008244029475