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RANK-BASED ESTIMATION FOR GARCH PROCESSES

Published online by Cambridge University Press:  27 April 2012

Beth Andrews
Beth Andrews*
Affiliation:
Northwestern University
*
*Address correspondence to Beth Andrews, Department of Statistics, Northwestern University, Evanston, IL 60208, USA; e-mail: [email protected].
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Abstract

We consider a rank-based technique for estimating generalized autoregressive conditionally heteroskedastic (GARCH) model parameters, some of which are scale transformations of conventional GARCH parameters. The estimators are obtained by minimizing a rank-based residual dispersion function similar to the one given in Jaeckel (1972, Annals of Mathematical Statistics43, 1449–1458). They are useful for GARCH order selection and preliminary estimation. We give a limiting distribution for the rank estimators that holds when the true parameter vector is in the interior of its parameter space and when some GARCH parameters are zero. The limiting theory is used to show that the rank estimators are robust, can have the same asymptotic efficiency as maximum likelihood estimators, and are relatively efficient compared to traditional Gaussian and Laplace quasi-maximum likelihood estimators. The behavior of the estimators for finite samples is studied via simulation, and we use rank estimation to fit a GARCH model to exchange rate log-returns.

Type
Articles
Information
Econometric Theory , Volume 28 , Issue 5 , October 2012 , pp. 1037 - 1064
Copyright
Copyright © Cambridge University Press 2012

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References

REFERENCES

Abramson, A. & Cohen, I. (2008) Single-sensor audio source separation using classification and estimation approach and GARCH modeling. IEEE Transactions on Audio, Speech and Language Processing 16, 1528–1540.Google Scholar
Andrews, B. (2008) Rank-based estimation for autoregressive moving average time series models. Journal of Time Series Analysis 29, 51–73.Google Scholar
Andrews, B., Davis, R.A., & Breidt, F.J. (2007) Rank-based estimation for all-pass time series models. Annals of Statistics 35, 844–869.Google Scholar
Andrews, D.W.K. (1999) Estimation when a parameter is on a boundary. Econometrica 67, 1341–1383.Google Scholar
Andrews, M.E. (2003) Parameter estimation for all-pass time series models. Ph.D. Dissertation, Colorado State University.Google Scholar
Berkes, I. & Horváth, L. (2004) The efficiency of the estimators of the parameters in GARCH processes. Annals of Statistics 32, 633–655.Google Scholar
Berkes, I., Horváth, L., & Kokoszka, P. (2003) GARCH processes: Structure and estimation. Bernoulli 9, 201–227.Google Scholar
Billingsley, P. (1961) The Lindeberg-Lévy theorem for martingales. Proceedings of the American Mathematical Society 12, 788–792.Google Scholar
Billingsley, P. (1999) Convergence of Probability Measures, 2nd ed.Wiley.Google Scholar
Bollerslev, T. (1986) Generalized autoregressive conditional heteroskedasticity. Journal of Econometrics 31, 307–327.Google Scholar
Bollerslev, T. (1987) A conditionally heteroskedastic time series model for speculative prices and rates of return. Review of Economics and Statistics 69, 542–547.Google Scholar
Bougerol, P. & Picard, N. (1992) Stationarity of GARCH processes and of some nonnegative time series. Journal of Econometrics 52, 115–127.Google Scholar
Brockwell, P.J. & Davis, R.A. (1991) Time Series: Theory and Methods, 2nd ed.Springer-Verlag.CrossRefGoogle Scholar
Campbell, S.D. & Diebold, F.X. (2005) Weather forecasting for weather derivatives. Journal of the American Statistical Association 100, 6–16.Google Scholar
Chernoff, H. & Savage, I.R. (1958) Asymptotic normality and efficiency of certain nonparametric test statistics. Annals of Mathematical Statistics 29, 972–994.Google Scholar
Davis, R.A., Knight, K., & Liu, J. (1992) M-estimation for autoregressions with infinite variance. Stochastic Processes and Their Applications 40, 145–180.Google Scholar
Engle, R.F. (1982) Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation. Econometrica 50, 987–1008.Google Scholar
Engle, R.F. & González-Rivera, G. (1991) Semiparametric ARCH models. Journal of Business & Economic Statistics 9, 345–359.Google Scholar
Ewing, B.T., Kruse, J.B., & Thompson, M.A. (2008) Analysis of time-varying turbulence in geographically-dispersed wind energy markets. Energy Sources, Part B: Economics, Planning, and Policy 3, 340–347.Google Scholar
Fan, J. & Yao, Q. (2003) Nonlinear Time Series: Nonparametric and Parametric Methods. Springer-Verlag.Google Scholar
Francq, C. & Zakoïan, J.M. (2004) Maximum likelihood estimation of pure GARCH and ARMA-GARCH processes. Bernoulli 10, 605–637.Google Scholar
Francq, C. & Zakoïan, J.M. (2007) Quasi-maximum likelihood estimation in GARCH processes when some coefficients are equal to zero. Stochastic Processes and Their Applications 117, 1265–1284.Google Scholar
Francq, C. & Zakoïan, J.M. (2009) Testing the nullity of GARCH coefficients: Correction of the standard test and relative efficiency comparisons. Journal of the American Statistical Association 104, 313–324.Google Scholar
Gastwirth, J.L. & Wolff, S.S. (1968) An elementary method for obtaining lower bounds on the asymptotic power of rank tests. Annals of Mathematical Statistics 39, 2128–2130.Google Scholar
Hall, P. & Yao, Q. (2003) Inference in ARCH and GARCH models with heavy-tailed errors. Econometrica 71, 285–317.Google Scholar
Hettmansperger, T.P. & McKean, J.W. (1998) Robust Nonparametric Statistical Methods. Arnold.Google Scholar
Hoti, S., McAleer, M., & Chan, F. (2005) Modelling the spillover effects in the volatility of atmospheric carbon dioxide concentrations. Mathematics and Computers in Simulation 69, 46–56.Google Scholar
Huang, H.-H., Shiu, Y.-M., & Lin, P.-S. (2008) HDD and CDD option pricing with market price of weather risk for Taiwan. Journal of Futures Markets 28, 790–814.Google Scholar
Jaeckel, L.A. (1972) Estimating regression coefficients by minimizing the dispersion of the residuals. Annals of Mathematical Statistics 43, 1449–1458.Google Scholar
Jurečková, J. & Sen, P.K. (1996) Robust Statistical Procedures: Asymptotics and Interrelations. Wiley.Google Scholar
Koul, H.L. (2002) Weighted Empirical Processes in Dynamic Nonlinear Models, 2nd ed.Springer-Verlag.Google Scholar
Koul, H.L. & Ling, S. (2006) Fitting an error distribution in some heteroscedastic time series models. Annals of Statistics 34, 994–1012.Google Scholar
Mukherjee, K. (2007) Generalized R-estimators under conditional heteroscedasticity. Journal of Econometrics 141, 383–415.Google Scholar
Mukherjee, K. (2008) M-estimation in GARCH models. Econometric Theory 24, 1530–1553.Google Scholar
Muler, N. & Yohai, V.J. (2008) Robust estimates for GARCH models. Journal of Statistical Planning and Inference 138, 2918–2940.Google Scholar
Nelder, J.A. & Mead, R. (1965) A simplex method for function minimization. Computer Journal 7, 308–313.CrossRefGoogle Scholar
Nelson, D. (1990) Stationarity and persistence in the GARCH(1,1) model. Econometric Theory 6, 318–334.Google Scholar
Peng, L. & Yao, Q. (2003) Least absolute deviations estimation for ARCH and GARCH models. Biometrika 90, 967–975.Google Scholar
Rao, C.R. (1973) Linear Statistical Inference and Its Applications, 2nd ed.Wiley.Google Scholar
Romilly, P. (2005) Time series modelling of global mean temperature for managerial decision-making. Journal of Environmental Management 76, 61–70.Google Scholar
Shephard, N. (1996) Statistical aspects of ARCH and stochastic volatility. In Cox, D.R., Hinkley, D.V., & Barndorff-Nielsen, O.E. (eds.), Time Series Models In Econometrics, Finance, and Other Fields, pp. 1–67. Chapman & Hall.Google Scholar
Silverman, B.W. (1978) Weak and strong uniform consistency of the kernel estimate of a density and its derivatives. Annals of Statistics 6, 177–184.Google Scholar
Silverman, B.W. (1986) Density Estimation for Statistics and Data Analysis. Chapman & Hall.Google Scholar
Straumann, D. (2005) Estimation in Conditionally Heteroscedastic Time Series Models. Springer-Verlag.Google Scholar