Cassim, Lucius (2018): A semiparametric GARCH (1, 1) estimator under serially dependent innovations.

PDF
MPRA_paper_86572.pdf Download (1MB)  Preview 
Abstract
The main objective of this study is to derive semi parametric GARCH (1, 1) estimator under serially dependent innovations. The specific objectives are to show that the derived estimator is not only consistent but also asymptotically normal. Normally, the GARCH (1, 1) estimator is derived through quasimaximum likelihood estimation technique and then consistency and asymptotic normality are proved using the weak law of large numbers and Lindeberg central limit theorem respectively. In this study, we apply the quasimaximum likelihood estimation technique to derive the GARCH (1, 1) estimator under the assumption that the innovations are serially dependent. Allowing serial dependence of the innovations has however brought problems in terms of methodology. Firstly, we cannot split the joint probability distribution into a product of marginal distributions as is normally done. Rather, the study splits the joint distribution into a product of conditional densities to get around this problem. Secondly, we cannot use the weak laws of large numbers or/and the Lindeberg central limit theorem. We therefore employ the martingale techniques to achieve the specific objectives. Having derived the semi parametric GARCH (1, 1) estimator, we have therefore shown that the derived estimator not only converges almost surely to the true population parameter but also converges in distribution to the normal distribution with the highest possible convergence rate similar to that of parametric estimators
Item Type:  MPRA Paper 

Original Title:  A semiparametric GARCH (1, 1) estimator under serially dependent innovations 
English Title:  A semiparametric GARCH (1, 1) estimator under serially dependent innovations 
Language:  English 
Keywords:  GARCH(1,1), semi parametric , Quasi Maximum Likelihood Estimation, Martingale 
Subjects:  C  Mathematical and Quantitative Methods > C4  Econometric and Statistical Methods: Special Topics C  Mathematical and Quantitative Methods > C5  Econometric Modeling > C58  Financial Econometrics 
Item ID:  86572 
Depositing User:  Mr Lucius Cassim 
Date Deposited:  10 May 2018 04:20 
Last Modified:  27 Sep 2019 15:12 
References:  Amemiya, T. (1985). Advanced Econometrics. Cambridge: Havard University Press. Andersen, T. G. (1996). GMM estimation of stochastic volatility models:A monte Carlo Study. Journal of Business and Economic Statistics,48, 328352. Avran, F. (1988). Weak Convergence of the variations,iterated integrals and DoleansDade exponentials of sequences of semimartingales. Anals of probability 16, 246250. Baillie, R. T., & Bollerslev, T. (1987). The message in Daily Exchange Rates: A conditional Variance Tale. Econometrica. Bollerslev. (1987). A conditional Heteroskedastic time series model for speculative prices and rates of return. Review of Economics and Statistics,69, 542547. Bollerslev, T. (1986). Generalized Autoregressive Heteroskedasticity. Journal of Econometrics, 307327. Bollersslev, J., & Woodridge, J. (1992). Quasi maximum likelihood estimation and inference in dynamic models with time varying covariances. Econometric Reviews,11(2), 143172. Buhlman, P., & McNeil, A. J. (2000). Nonparametric GARCH Models. Zurich, Switzerland: Seminar Fur Statistik:CH8092. Cameron, C. A., & Trivedi, P. (2005). Microeconometrics:Methods and Applications. New York,USA: Cambrdge University Press. Choi, E. J. (2004). Estimation of Stochastic Volatility Models by Simulated Maximum Likelihood Method. University of Waterloo. Chung, S. S. (2012). A class of nonparametric volatility models:Application to financial time series. Journal of Econometrics. Dahl, C. M., & Levine, M. (2010). Nonparametric estimation of volatility models under serially dependent innovations. Econometrica. Davidson, J. (2000). Econometric Theory. Blackwel: Oxford University Press. Davidson, R., & Mackinnon, J. (1993). Estimation and inference in Econometrics. London: Oxford University Press. Drost, F. C., & Klasssen, C. (1996). Efficient estimation in Semiparametric GARCH Models. Discussion paper;vol(199638),Tilburg. Duan, J. (1997). Augmented GARCH(p,q) process and its diffusion limit. Journal of Econometrics,79(1), 97127. Engle, R. (1982). Autoregressive Conditional Heteroskedasticity with estimation of the variance of U.K inflation. Econometrica, 9871008. Engle, R. F., & GonzaleRivera, G. (October,1991). Semiparametric ARCH Models. Journal of Business and Economic Statistics, 9(4), 345359. Engle, R. F., & Ng, V. (1993). Measuring and testing the impact of news on volatility. Journal of finance,48, 17471778. Fan, & Gijbels, I. (1995). Datadriven bandwidth selection in local polynomial fitting:variable bandwidth and spatial adaptation. Journal of Royal Statistical Society,B,57, 371394. Fan, J., & Yao, Q. (1998). Efficient estimation of conditional variance functions in stochastic regression. Biometrika,85, 645660. Gallant, A. R., & Hsieh, D. (1989). Fitting a Recalcitrant series:The Pound/Dollar Exchange Rate,197483. Econometrica. Geweke, J. (1986). Modelling Persistence in Conditional Variances: A Comment. Econometric Review,5, 5661. Glosten, L. R., & Runkle, D. (1993). On the relation between the expected value and volatility of the nominal excess return of stocks. Journal of Finance,48, 17791801. Gourieroux, C., & Trognon, A. (1984). Pseudo Maximum Likelihood Methods:Theory,52(1). Econometrica, 681700. Haafner, C. M. (2003). Analytical quasi maximum likelihood inference in BEKKGARCH models. Econometric Institution,Erasmus University,Rotterdam. Hafner, C. M., & Rombonts, J. (2002). Semiparametric multivariate GARCH models . Discussion paper,2002/XX,CORE. Hansen, B. C. (2006). Econometrics. New York: Cambridge University Press. Hansen, P., & Heyde, C. (1980). Martingale limit theory and its applications. New York: Academic Press. Hansen, R. P., & Lunde, A. (2001). A comparison of volatility models:Does anything beat GARCH(1,1)? Centre for analytica finance:University of AARHUS. Hentshel, L. (1995). All in the family: Nesting Symetric and Asymetric GARCH Models. Journal of Financial Economics,39, 71104. Herwartz, H. (2004). ConditionalHeteroskedasticity.In H. Lutkepoh, & M. Kratzig (Eds.) Themes in Modern Econometrics, pp. 197220. Higging, M. L., & Bera, A. (1992). A Class of nonlinear ARCH Models. International Economic Review. Holly. (2009). Modelling Risk using fourth order Pseudo Maximum Likelihood Methods. University of Lausanne, Institute of Healthy Economics. Holly, A., & Montifort, A. (2010). Fourth Order Pseudo Maximum Likelihood Methods. Econometrica. Holly, A., & Pentsak, Y. (2004). Maximum Likelihhod Estimation of the Conditional mean E(YX) for Skewed Dependent variables in FourthParameter families of Distributions. Technical Report, University of Lausanne, Institute of Healthy Economics and Management. Hood, W., & Koopman, T. (1953). The estimation of simultaneous linear economic relationships. Econometric Methods. Ibragimov, R., & Philips, P. (2010). Regression asymptotics using martingale convergence. Yale University press. Kouassi, E. (2015). Consistency of Pseudomaximum likelihood estimation in ARCH(1) under dependent innovations. Working paper. Linton, O., & Mammen, E. (May, 2003). Estimating Semiparametric ARCH(∞) nodels by kernel smoothing methods. Discussion paper,No:EM/03/453. McCullagh, P. (1994). Exponential mixtures and quadratic exponential families. Biometrika, 81(4), 721729. Nadaraya, E. (1964). "On Estimating Regression".The Theory of Probability and its Applications. Econometrica, 1412. Nielsen, B. (1978). Information and exponential families in statiatical theory. New York: Wiley. Pantula, S. G. (1986). Modelling Persistence in Conditional Variances: A comment. Econometric Review,5, 7174. Posedel, P. (2005). Properties and estimation of GARCH(1,1) model. Metodoloski Zvezki, 2, 243257. Rao, C. R. (1973). Linear statistical inference and its applications. New York: John Willey & Sons. Rossi, E. (2004, march). A note on GARCH models. Working paper. Sentana, C. (1995). Quadratic ARCH Models. Review of Economic Studies,62(4), 639661. Sherphard, N. (2008). Statistical aspects of ARCH and Stochastic Volatility. In D. R. Cox, D. Hinkly, & O. Barndorff (Eds) Time series models in Econometrics, Finance and other fields.Monographs on statistics and Applied probability,65, pp. 165. Sousi, P. (2013). Advanced Probability. New York: Cambridge University Press. Stout, W. F. (1974). Almost Sure Convergence. New York: Academic Press. Su, L., Ullah, A., & Mashra, S. (2011). Nonparametric and semiparametric volatility models:specification,estimation and testing. Econometrica. Tapia, R. A., & Thompson, J. (n.d.). Nonparametric Probability Density Estimation. Taylor, S. (1986). Modelling Financial Time Series. John Wiley & Sons. Tsay, R. S. (2010). Analysis of time series. Willey. Tschiernig, R. (2004). Nonparametric Econometrics. In H. Lutkepoh, & M. Kratzig (Eds) Themes in Econometrics, pp. 243289. Watson, G. (1964). Smooth Regression Analysis. The Indian Journal of Statistics, 359372. Weiss, A. A. (1986). Asymptotic Theory for ARCH Models: Estimation and Testing. Econometric Theory,2, 107131. Williams, D. (1991). Probability with martingales. New York: Cambridge University Press. Yang, L., & Song, Q. (2012). Efficient Semiparametric GARCH Modelling of Financial Volatility. Statistica Sinica, 22, 249270. Zakoian, M. J. (1994). Threshold heteroskedastic models. Journal of Economic Dynamics and Control,18, 931955. 
URI:  https://mpra.ub.unimuenchen.de/id/eprint/86572 