Least squares estimation for GARCH (1,1) model with heavy tailed errors

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

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 log-transform-based 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 root-of-n 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 quasi-maximum 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
• 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 - Time-Series Models ; Dynamic Quantile Regressions ; Dynamic Treatment Effect Models ; Diffusion Processes

References

[1] Andrews B. (2012). Rank based estimation for GARCH processes. Econometric Theory, 28(5), 1037-1064.

[2] Andrews D.W.K.(2001). Testing when a parameter is on the boundary of the maintained hypothesis. Econometrica, 69, 683-734.

[3] Berkes I. Horvath L. and P.S. Kokoszka (2003). GARCH processes: structure and estimation. Bernoulli, 9, 201-227.

[4] Berkes I. and L. Horvath (2003). The rate of consistency of the quasimaximum likelihood estimator. Statistics and Probability Letters, 61, 133-143.

[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, 307-327.

[8] Bollerslev T. and J.M. Wooldridge (1992). Quasi-maximum likelihood estimation and inference in dynamic models with time-varying covariances. Econometric Reviews, 11, 143-172.

[9] Boussama F. (2000). Normalité asymptotique de l'estimateur du pesudomaximum de vraisemblance d'un mode'le GARCH. C.R.Acad Sci. Paris, 331, 81-84.

[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, 185-215.

[12] Francq C. and J-M. Zakoian (2007). Quasi-likelihood inference in GARCH processes when some coefficients are equal to zero. Stochastic Processes and their Applications, 117(9), 1265-1284.

[13] Francq C. and J-M. Zakoian (2009). A Tour in the asymptotic theory of GARCH estimation, Handbook of Financial Time Series, 85-111, Berlin, Springer-Verlag.

[14] Francq C. and J-M. Zakoian (2013). Estimating the marginal law of a time series with Applications to heavy-tailed distributions, Journal of Business and Economic Statistics, 31(4), 412-425.

[15] Hall P. and Q. Yao (2003). Inference in ARCH and GARCH models with heavy-tailed errors. Econometrica, 71, 285-317.

[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), 873-889.

[17] Harvey A.C., Ruiz E. and N.G. Shephard (1994). Multivariate stochastic variance models. Review of Economic Studies, 61, 247-264.

[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, 29-52.

[20] Linton O, Pan J. and H Wang (2010). Estimation for A non-stationary semi-strong GARCH(1,1) models with heavy-tailed errors. Econometric theory, 26(1), 1-28.

[21] Loeve M. (1977). Probability Theory 1, New York, Springer.

[22] Lumsdaine, R.L. (1996). Consistency and asymptotic normality of the quasi-maximum likelihood estimator in IGARCH(1,1) and covariance stationary GARCH (1,1) models. Econometrica, 64, 575-596.

[23] Mittnik S. and S.T. Rachev (2000) Stable Paretian Models in Finance. New York, John-Wiley.

[24] Nelson D.B. (1990). Stationarity and persistence in the GARCH(1,1) model. Econometric Theory, 6(3), 318-334.

[25] Mukherjee K. (2008). M-estimation in GARCH models. Econometric Theory, 24(6), 1530-1553.

[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, 967-975.

[28] Rekkasa M. and A. Wong (2008). Implementing likelihood-based inference for fat-tailed distributions. Finance Research Letters, 5(1), 32-46.

[29] Robinson P.M. and P. Zaffaroni (2006). Pseudo-maximum likelihood estimation of ARCH(1) models. Annals of Statistics, 34, 1049{1074.

[30] Ruiz E. (1994). Quasi-maximum likelihood estimation of stochastic volatility models. Journal of Econometrics, 63, 289{306.

[31] Sakata S. and H. White (2001). S-estimation of nonlinear regression models with dependent and heterogeneous observations. Journal of Econometrics, 103, 5-72.

[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), 495-514.

[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.