Francq, Christian and Meintanis, Simos (2012): Fourier--type estimation of the power garch model with stable--paretian innovations.
Download (447Kb) | Preview
We consider estimation for general power GARCH models under stable--Paretian innovations. Exploiting the simple structure of the conditional characteristic function of the observations driven by these models we propose minimum distance estimation based on the empirical characteristic function of corresponding residuals. Consistency of the estimators is proved, and we obtain a singular asymptotic distribution which is concentrated on a hyperplane. Efficiency issues are explored and finite--sample results are presented as well as applications of the proposed procedures to real data from the financial markets. A multivariate extension is also considered.
|Item Type:||MPRA Paper|
|Original Title:||Fourier--type estimation of the power garch model with stable--paretian innovations|
|Keywords:||GARCH model; Minimum distance estimation; Heavy--tailed distribution; Empirical characteristic function|
|Subjects:||C - Mathematical and Quantitative Methods > C3 - Multiple or Simultaneous Equation Models; Multiple Variables > C32 - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models
C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General > C13 - Estimation: General
C - Mathematical and Quantitative Methods > C2 - Single Equation Models; Single Variables > C22 - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models
|Depositing User:||Christian Francq|
|Date Deposited:||01. Oct 2012 18:24|
|Last Modified:||13. Feb 2013 15:00|
Adler, R.J., Feldman, R.E., and Taqqu, M.S. (Eds.) A Practical Guide to Heavy Tails. Statistical Techniques and Applications. Birkhäuser, Boston, 1998.
Akgül, I., and Sayyan, H. 2008 Modelling and forecasting long memory in exchange rate volatility vs. stable and integrated GARCH models. Appl. Financ. Econom., 18, 463-482.
Billingsley, P. Probability and Measure. Wiley and Sons, New York, 1995.
Bryant, J.L. and Paulson, A.S. 1979 Some comments on characteristic function based estimators. Sankhy$\bar\rma$, 41, 109-116.
Bonato, M. 2009 Modeling fat tails in stock returns: a multivariate stable-GARCH approach. Electronic copy available at http://ssrn.com/abstract=1015477.
Bougerol, P., and Picard, N. 1992 Stationarity of GARCH processes and of some nonnegative time series. J. Econometr., 52, 115-127.
Carrasco, M., Chernov, M., Florens, J.P., and Ghysels, E. 2007 Efficient estimation of general dynamic models with a continuum of moment conditions. J. Econometr., 140, 529-573.
Curto, J.D., Pinto, J.C., and Tavares, G.N. 2009 Modelling stock markets' volatility using GARCH models with Normal, Student's $t$ and stable Paretian distribution. Statist. Papers, 50, 311-321.
DuMouchel, W.H. 1983 Estimating the stable index $\alpha$ in order to measure tail thickness: A critique. Ann. Statist., 11, 1019-1031.
Fama, E. 1965 The behavior of stock market prices. J. Business, 38, 34-105.
Feuerverger, A. 1990
Feuerverger, A. and McDunnough, P. 1981a On the efficiency of empirical characteristic function procedures. J. Roy. Statist. Soc. Ser. B , 38, 20-27.
Feuerverger, A. and McDunnough, P. 1981b On some Fourier methods for inference. J. Amer. Statist. Assoc., 76, 379-387.
Hamadeh, T., and Zakoïan, J.M. 2011 Asymptotic properties of LS and QML estimators for a class of nonlinear GARCH processes. J. Statist. Plann. Inference, 141, 488-507.
Hansen, L.P. 1982 Large sample properties of generalized method of moments estimators. Econometrica, 50, 1029-1054.
Heathcote, C.R. 1977 The integrated squared error estimation of parameters. Biometrika, 64, 255-264.
Kotchoni, R. 2012 Applications of the characteristic function-based continuum GMM in finance. Comput. Statist. Dat. Anal. , 56, 3599-3622.
Koutrouvelis, I.A., and Meintanis, S.G. 1999 Testing for stability based on the empirical characteristic function with applications to financial data. J. Statist. Comput. Simul., 64, 275-300.
Liu, S.M., and Brorsen, B.W. 1995a GARCH-stable as a model for future price movements. Rev. Quantitat. Financ. Account., 5, 155-167.
Liu, S.M., and Brorsen, B.W. 1995b Maximum likelihood estimation of a GARCH-stable model. J. Appl. Econometr., 10, 273-285.
Mandelbrot, B. 1963 The variation of certain speculative prices. J. Business, 36, 394-419.
Mittnik, S., and Rachev, S.T. 1993 Modelling asset returns with alternative stable models. Econometr. Rev., 12, 261-330.
Mittnik, S., Rachev, S.T., Doganoglu, T, and Chenyao, D. 1999 Maximum likelihood estimation of stable Paretian models. Math. Comput. Model., 29, 275-293.
Newey, W.K., and McFadden, D. 1994 \newblockLarge sample estimation and hypothesis testing. In: Engle and McFadden (Eds.) \em Handbook of Econometrics, Vol. IV, 2111-2245, Elsevier.
Nolan, J.P. Stable Distributions - Models for Heavy Tailed Data. Birkhauser, Boston, 2012.
Paolella, M.S. 2001 Testing the stable Paretian assumption. Math. Comput. Model. , 34, 1095-1112.
Rachev, S. (Ed.) Handbook of Heavy Tailed Distributions in Finance. Elsevier/North-Holland, Amsterdam, 2003. Rachev, S., and Mittnik, S. Stable Paretian Models in Finance. Wiley, New York, 2000.
Samorodnitsky, G., and Taqqu, M.S. Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance. Chapman and Hall, New York, 1994.
Tauchen, G.E. 1985 Diagnostic testing and evaluation of maximum likelihood models . J. Econometr., 30, 415-443.
Taufer, E. and Leonenko, N. 2009 Characteristic function estimation of non-Gaussian Ornstein-Uhlenbeck processes. J. Statist. Plann. Infer. , 139, 3050-3063.
Tavares, A.B., Curto, J.D., and Tavares, G.N. 2008 Modelling heavy tails and asymmetry using ARCH-type models with stable Paretian distribution. Nonlinear Dyn., 51, 231-243.
Thornton, J.C. and Paulson, A.S. 1977 Asymptotic distribution of characteristic function-based estimators for the stable laws. Sankhy$\bar\rma$, 39, 341-354.
Tsionas, E. 2002 Likelihood-based comparison of stable Paretian and competing models: Evidence from daily exchange rates. J. Statist. Comput. Simul. , 72, 341-353.
Xu, W., Wu, C., Dong, Y., and Xiao, W. 2011 Modeling Chinese stock returns with stable distribution. Math. Comput. Model., 54, 610-617.