On Optimal Estimate Functions for Asymmetric GARCH Models.
Abstract
High frequency data exhibit non-constant variance. This paper models the exhibited fluctuations via asymmetric GARCH models. The Maximum Likelihood Estimation (MLE) and Estimating Functions (EF) are used in the estimation of the asymmetric GARCH family models. This EF approach utilizes the third and fourth moments which are common features in financial time series data analysis and does not rely on distributional assumptions of the data. Optimal estimating functions have been constructed as a combination of linear and quadratic estimating functions. The results show that estimates from the estimating functions approach are better than those of the traditional estimation methods such as the MLE especially in cases where distributional assumptions on the data are seriously violated. The implementation of the EF approach to asymmetric GARCH models assuming a generalized student-t distribution innovation reveals the efficiency benefits of the EF approach over the MLE method in parameter estimation especially for non-normal cases.
References
Achumba, I. C., Ighomereho, O. S. & Akpor-Robaro, M. O. M. Security Challenges in Nigeria and the Implications for Business Activities and Sustainable Development. Journal of Economics and Sustainable Development, l4(2), 79-99 (2013).
Mukolu, M. O & Ogodor, B. N. Insurgency and Implication on Nigeria economic growth. International Journal of Development and Sustainability, 7(2), 492-501 (2018).
Onyeka-Ubaka, J. N. A BL-GARCH Model for Distribution with Heavy Tails. A Ph.D. Thesis, Department of Mathematics, University of Lagos, Nigeria (2013).
Onyeka-Ubaka, J. N., Abass, O. & Okafor, R. O. Conditional Variance Parameters in Symmetric Models. International Journal of Probability and Statistics, 3(1), 1-7 (2014).
Ogbogbo, C. P. Modeling crude oil spot price as an Ornstein- Uhlenbeck Process. International Journal of Mathematical Analysis and Optimization: Theory and Applications, 2018, 261-275 (2018).
Storti, G. & Vitale, C. BL-GARCH models and asymmetries in volatility. Statistical Methods and Applications, 12, 19-40 (2003).
Black, F. & Scholes, M. The Pricing of Options and Corporate Liabilities. Journal of Political Economics, 81, 637-54 (1973).
Engle, R. F. Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation. Econometrica, 50(4), 987-1007 (1982).
Bollerslev, T. Generalized autoregressive conditional heteroscedasticity. Journal of Econometrics, 31, 307-328 (1986).
Nelson, D.B. Conditional heteroskedasticity in Asset returns: A new Approach. Econometrica, 59, 347- 370 (1991).
Glosten, L. R., Jagannathan, R. & Runkle, D. On the relation between the expected value and the volatility of nominal excess returns on stocks. Journal of Finance, 48: 1779-1801 (1993).
Engle, R. F. & Ng, V. K. Measuring and testing the impact of news on volatility. Journal of Finance, 48(5), 1749-1778 (1993).
Zakoian, J.M. Threshold Heteroskedasticity Models. Journal of Economic Dynamics and Control, 19, 931-944 (1994).
Sentana, E. Quadratic ARCH Models. Review of Economic Studies, 62, 639-661 (1995).
Bolleslev, T., Engle, R. F. and Nelson, D. B. ARCH Models in Handbook of Econometrics, Vol. IV, R. F. Engle and D. L. McFadden, eds. Amsterdan: North Holland, 2959-3038 (1994).
Rossi, E. Lecture Notes on GARCH Models. University of Pavia, 1, 33-45 (2004).
Nelson, D. B. & Cao, C. Q. Inequality constraints in the univariate GARCH model. Journal of Business & Economic Statistics,10, 229-235 (1992).
Engle, R. F. & Bollerslev T. Modelling the persistence of conditional variances, Econometric Reviews, 5, 1-50 (1986).
Engle, R. F. Stock Volatility and the Crash of '87. Review of Financial Studies, 3(1), 103-106 (1990).
Engle R. F., Lilien, D. M. & Robins, R. P. Estimating Time Varying Risk Premia in the Term Structure: The ARCH-M Model. Econometrica, 55(2), 391-407 (1987).
Diongue, A. K., Guegan, D. & Wolff, R. C. Exact Maximum likelihood Estimator for the BL-GARCH model under elliptical distributed innovations. Journal of Statistical Computation and Simulation, 7, 4-10 (2010).
Godambe, V.P. The foundations of finite sample estimation in stochastic processes, Biometrika, 72, 319-328 (1985).
Onyeka-Ubaka, J. N., Abass, O. & Okafor, R. O. A Generalized t-Distribution based Filter for Stochastic Volatility Models, AMSE JOURNALS-2016-Series:Advances D, 21(1), 19-37 (2016).
Campbell, J. Y. Lo, A. W. & Mackinlay, A. C. The Econometrics of Financial Markets. Princeton, New Jersey, Princeton University Press. Journal of Finance. 25, 383-417 (1997).
Pagan, A. R. and Schwert, G. W. (1990). Alternative Models for Conditional Stock Volatility, Journal of Economics, 45, 267-290.
Locke, P.R. & Sayers, C.L. Intra-Day Futures Prices Volatility: Information Effects and Variance Persistence. Journal of Applied Econometrics, 8, 15-30 (1993).
Muller, N. & Yohai, V. Robust Estimates for ARCH Models Time Series Analysis, 23, 341-375 (2002).
Eraker, B., Johannes, M. & Polson, N. The Impact of Jumps in Volatility and Returns. Journal of Finance. 58, 1269 & 1300 (2003).
Mutunga, T. N., Islam, A. S. & Orawo, L. A. Estimation of Asymmetric GARCH Models: The Estimating Functions Approach. International Journal of Applied Science and Technology 4(5), 198 - 209 (2014).
Godambe, V.P. & Thompson, M.E. An extension of the Quasi-likelihood Estimation, Journal of Statistical Planning and Inference, 22, 137-172 (1989).
United Nations. Global Economic Outlook 2006. Global Economic and Monitoring Unit, Development Policy and Analysis Division, Department of Economic and Social Affairs, United Nations (2006).
Pitt, M. K. & Shephard, N. Filtering via Simulation: Auxiliary Particle Filters. Journal of the American Statistical Association, 94, 590-599 (1999).
Storvik, G. Particle Filters for State-Space Models with the Presence of Unknown Static Parameters. Institute of Electrical and Electronics Engineers (IEEE) Transactions on Signal Processing, 50, 281-289 (2002).
Johannes, M., Polson, N. & Stroud, J. Sequential Parameter Estimation in Stochastic Volatility Models with Jumps. Working paper, Graduate School of Business, University of Chicago
(2006).
Raggi, D. & Bordignon, S. Sequential Monte Carlo Methods for Stochastic Volatility Methods with Jumps. Lecture Notes, University of Bologna/Padova, Italy, October 13, 2006, (2006).
Zivot. E.V. Practical Issues in the Analysis of Univariate GARCH Models, Handbook of Financial Time Series, Springer, New York (2008).
Mandelbrot, B. The Variation of Certain Speculative Prices. Journal of Business, 36, 394-419 (1963).
This is an Open Access article distributed under the terms of the Creative Commons Attribution 4.0 International License, which permits unrestricted use, distribution, adaptation, and reproduction in any medium, provided that the original work is properly cited.
