Abstract
When modeling conditionally heteroscedastic processes, it is usually the case that exact likelihood is not known to researchers or it is too complicated for practical purposes. Consequently, a quasi-maximum likelihood (QML) is frequently employed rather than the exact maximum likelihood (cf. Gourieroux (1997, Ch. 4), Straumann (2005, Ch. 5)). When a GARCH process is partially specified only through the first and second order conditional moments, a systematic and unified approach for inference is via the so called quasilikelihood (QL, see Godambe (1985)). This article aims to discriminate between the QL and QML for general GARCH-type processes.Itisverified that the QL differs with the QML in terms of conditional skewness and kurtosis. Asymptotics for the QL and QML are obtained and then compared with each other. A class of skewed t-distributions (cf. Fernandez and Steel (1998)) is considered to illustrate the difference between the QL and QML.
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References
Andersen, T. G., Davis, R. A., Kreiss, J.-P., & Mikosch, T. (2009). Handbook of financial time series. Berlin: Springer.
Baek, J. S., Park, J. A., & Hwang, S. Y. (2012). Preliminary test of fit in a general class of conditionally heteroscedastic nonlinear time series. Journal of Statistical Computation and Simulation, 82, 82–763.
Choi, M. S., Park, J. A., & Hwang, S. Y. (2012). Asymmetric GARCH processes featuring both threshold effect and bilinear structure. Statistics & Probability Letters, 82, 82–419.
Fahrmeir, L., & Kaufmann, H. (1985). Consistency and asymptotic normality of maximum likelihood estimator in generalized linear models. Annals of Statistics, 13, 13–342.
Fernandez, C., & Steel, M. F. J. (1998). On Bayesian modeling of fat tails and skewness. Journal of the American Statistical Association, 93, 93–359.
Francq, C., & Zakoian, J. M. (2009). In T. G. Andersen, R. A. Davis, J.-P. Kreiss, & T. Mikosch (Eds.), Handbook of financial time series, A tour in the asymptotic theory of GARCH estimation (pp. 85–111). Berlin: Springer.
Francq, C., & Zakoian, J. M. (2010). GARCH models: structure, statistical inference and financial applications. Wiley: Chichester.
Francq, C., & Zakoian, J. M. (2013). Optimal predictions of powers of conditionally heteroscedastic processes. Journal of the Royal Statistical Society, B, 75, 75–345.
Godambe, V. P. (1985). The foundation of finite sample estimation in stochastic processes. Biometrika, 72, 72–419.
Gourieroux, C. (1997). ARCH models and financial applications. New York: Springer-Verlag.
Heyde, C. C. (1997). Quasi-likelihood and its application. New York: Springer.
Hwang, S. Y., & Basawa, I. V. (2011a). Asymptotic optimal inference for multivariate branching-Markov processes via martingale estimating functions and mixed normality. Journal of Multivariate Analysis, 102, 102–1018.
Hwang, S. Y., & Basawa, I. V. (2011b). Godambe estimating functions and asymptotic optimal inference. Statistics & Probability Letters, 81, 81–1121.
Hwang, S. Y., & Kim, T. Y. (2004). Power transformation and threshold modeling for ARCH innovations with applications to tests for ARCH structure. Stochastic Processes and their Applications, 110, 110–295.
Hwang, S. Y., Basawa, I. V., Choi, M. S., & Lee, S. D. (2013). Non-ergodic martingale estimating functions and related asymptotics. Statistics, 1–21. http://dx.doi.org/10.1080/02331888.2012.748772. online published.
Nicholls, D. F., & Quinn, B. G. (1982). LNS: Vol. 11. Random coefficient autoregressive models: an introduction. New York: Springer.
Straumann, D. (2005). LNS: Vol. 181. Estimation in conditionally heteroscedastic time series models. Berlin: Springer.
Straumann, D., & Mikosch, T. (2006). Quasi-maximum-likelihood estimationin conditionally heteroscedastic time series: astochastic recurrence equations approach. Annals of Statistics, 34, 34–2449.
Tsay, R. S. (2010). Analysis of financial time series, (3rd ed.). New York: Wiley.
Tong, H. (1990). Nonlinear time series, Oxford: Oxford University Press.
Zivot, E. (2009). Practical issues in the analysis of univariate GARCH models. In T. G. Andersen, R. A. Davis, J.-P. Kreiss, & T. Mikosch (Eds.), Handbook of financial time series, (pp. 113–155). Berlin: Springer.
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Hwang, S.Y., Choi, M.S. & Yeo, I.K. Quasilikelihood and quasi-maximum likelihood for GARCH-type processes: Estimating function approach. J. Korean Stat. Soc. 43, 631–641 (2014). https://doi.org/10.1016/j.jkss.2014.01.005
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DOI: https://doi.org/10.1016/j.jkss.2014.01.005