Preminger, Arie and Storti, Giuseppe (2014): Least squares estimation for GARCH (1,1) model with heavy tailed errors.

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Abstract
GARCH (1,1) models are widely used for modelling processes with time varying volatility. These include financial time series, which can be particularly heavy tailed. In this paper, we propose a logtransformbased least squares estimator (LSE) for the GARCH (1,1) model. The asymptotic properties of the LSE are studied under very mild moment conditions for the errors. We establish the consistency, asymptotic normality at the standard convergence rate of square rootofn for our estimator. The finite sample properties are assessed by means of an extensive simulation study. Our results show that LSE is more accurate than the quasimaximum likelihood estimator (QMLE) for heavy tailed errors. Finally, we provide some empirical evidence on two financial time series considering daily and high frequency returns. The results of the empirical analysis suggest that in some settings, depending on the specific measure of volatility adopted, the LSE can allow for more accurate predictions of volatility than the usual Gaussian QMLE.
Item Type:  MPRA Paper 

Original Title:  Least squares estimation for GARCH (1,1) model with heavy tailed errors 
English Title:  Least squares estimation for GARCH (1,1) model with heavy tailed errors 
Language:  English 
Keywords:  GARCH (1,1), least squares estimation, consistency, asymptotic normality. 
Subjects:  C  Mathematical and Quantitative Methods > C1  Econometric and Statistical Methods and Methodology: General > C13  Estimation: General C  Mathematical and Quantitative Methods > C1  Econometric and Statistical Methods and Methodology: General > C15  Statistical Simulation Methods: General C  Mathematical and Quantitative Methods > C2  Single Equation Models ; Single Variables > C22  TimeSeries Models ; Dynamic Quantile Regressions ; Dynamic Treatment Effect Models ; Diffusion Processes 
Item ID:  59082 
Depositing User:  Prof. Giuseppe Storti 
Date Deposited:  04 Oct 2014 21:54 
Last Modified:  05 Apr 2017 15:34 
References:  [1] Andrews B. (2012). Rank based estimation for GARCH processes. Econometric Theory, 28(5), 10371064. [2] Andrews D.W.K.(2001). Testing when a parameter is on the boundary of the maintained hypothesis. Econometrica, 69, 683734. [3] Berkes I. Horvath L. and P.S. Kokoszka (2003). GARCH processes: structure and estimation. Bernoulli, 9, 201227. [4] Berkes I. and L. Horvath (2003). The rate of consistency of the quasimaximum likelihood estimator. Statistics and Probability Letters, 61, 133143. [5] Billingsley P. (1968) Convergence of Probability Measures. New York, John Wiley. [6] Billingsley P. (1995). Probability and Measure. New York, John Wiley. [7] Bollerslev T. (1986). Generalized autoregressive conditional heteroskedasticity. Journal of Econometrics, 31, 307327. [8] Bollerslev T. and J.M. Wooldridge (1992). Quasimaximum likelihood estimation and inference in dynamic models with timevarying covariances. Econometric Reviews, 11, 143172. [9] Boussama F. (2000). Normalité asymptotique de l'estimateur du pesudomaximum de vraisemblance d'un mode'le GARCH. C.R.Acad Sci. Paris, 331, 8184. [10] Dacorogna M.M., Gencay R., Muller U., Olsen R.B. and O.V. Pictet (2001). An Introduction to High Frequency Finance. San Diego, CA: Academic Press. [11] Ding, Z. and C. W. J. Granger (1996). Modelling volatility persistence of speculative returns: a new approach, Journal of Econometrics, 73, 185215. [12] Francq C. and JM. Zakoian (2007). Quasilikelihood inference in GARCH processes when some coefficients are equal to zero. Stochastic Processes and their Applications, 117(9), 12651284. [13] Francq C. and JM. Zakoian (2009). A Tour in the asymptotic theory of GARCH estimation, Handbook of Financial Time Series, 85111, Berlin, SpringerVerlag. [14] Francq C. and JM. Zakoian (2013). Estimating the marginal law of a time series with Applications to heavytailed distributions, Journal of Business and Economic Statistics, 31(4), 412425. [15] Hall P. and Q. Yao (2003). Inference in ARCH and GARCH models with heavytailed errors. Econometrica, 71, 285317. [16] Hansen P.R and A.Lunde (2005). A forecast comparison of volatility models: does anything beat a GARCH(1,1)? Journal of Applied Econometrics, 20(7), 873889. [17] Harvey A.C., Ruiz E. and N.G. Shephard (1994). Multivariate stochastic variance models. Review of Economic Studies, 61, 247264. [18] Huang D., Wang H. and Q. Yao (2008). Estimating GARCH models: when to use what? Econometrics Journal, 11, pp. 27{38 [19] Lee S.W. and B.E. Hansen (1994). Asymptotic theory for the GARCH(1,1) quasi maximum likelihood estimator. Econometric Theory, 10, 2952. [20] Linton O, Pan J. and H Wang (2010). Estimation for A nonstationary semistrong GARCH(1,1) models with heavytailed errors. Econometric theory, 26(1), 128. [21] Loeve M. (1977). Probability Theory 1, New York, Springer. [22] Lumsdaine, R.L. (1996). Consistency and asymptotic normality of the quasimaximum likelihood estimator in IGARCH(1,1) and covariance stationary GARCH (1,1) models. Econometrica, 64, 575596. [23] Mittnik S. and S.T. Rachev (2000) Stable Paretian Models in Finance. New York, JohnWiley. [24] Nelson D.B. (1990). Stationarity and persistence in the GARCH(1,1) model. Econometric Theory, 6(3), 318334. [25] Mukherjee K. (2008). Mestimation in GARCH models. Econometric Theory, 24(6), 15301553. [26] Patton A. (2011) Volatility forecast evaluation and comparison using imperfect volatility proxies. Journal of Econometrics, 160, 246{256. [27] Peng, L. and Q. Yao (2003). Least absolute deviation estimation for ARCH and GARCH models. Biometrika, 90, 967975. [28] Rekkasa M. and A. Wong (2008). Implementing likelihoodbased inference for fattailed distributions. Finance Research Letters, 5(1), 3246. [29] Robinson P.M. and P. Zaffaroni (2006). Pseudomaximum likelihood estimation of ARCH(1) models. Annals of Statistics, 34, 1049{1074. [30] Ruiz E. (1994). Quasimaximum likelihood estimation of stochastic volatility models. Journal of Econometrics, 63, 289{306. [31] Sakata S. and H. White (2001). Sestimation of nonlinear regression models with dependent and heterogeneous observations. Journal of Econometrics, 103, 572. [32] Stinchcombe M. B. and H. White (1992). Some measurability results for extrema of random functions over random sets. Review of Economic Studies, 59 (3), 495514. [33] Straumann D. (2005) Estimation in Conditionally Heteroscedastic Time Series Models, Lecture Notes in Statistics, Springer. [34] White H. (1994). Estimation, Inference and Specification Analysis. New York, Cambridge University Press. 
URI:  https://mpra.ub.unimuenchen.de/id/eprint/59082 