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LOCAL WHITTLE ESTIMATION OF FRACTIONAL INTEGRATION FOR NONLINEAR PROCESSES

Published online by Cambridge University Press:  14 May 2007

Xiaofeng Shao  and
Wei Biao Wu
Xiaofeng Shao
Affiliation:
University of Illinois at Urbana-Champaign
Wei Biao Wu
Affiliation:
University of Chicago
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Abstract

We study asymptotic properties of the local Whittle estimator of the long memory parameter for a wide class of fractionally integrated nonlinear time series models. In particular, we solve the conjecture posed by Phillips and Shimotsu (2004, Annals of Statistics 32, 656–692) for Type I processes under our framework, which requires a global smoothness condition on the spectral density of the short memory component. The formulation allows the widely used fractional autoregressive integrated moving average (FARIMA) models with generalized autoregressive conditionally heteroskedastic (GARCH) innovations of various forms, and our asymptotic results provide a theoretical justification of the findings in simulations that the local Whittle estimator is robust to conditional heteroskedasticity. Additionally, our conditions are easily verifiable and are satisfied for many nonlinear time series models.We thank Liudas Giraitis for providing the manuscript by Dalla, Giraitis, and Hidalgo (2006). We are grateful to the two referees and the editor for their detailed comments, which led to substantial improvements. We also thank Michael Stein for helpful comments on an earlier version. The work is supported in part by NSF grant DMS-0478704.

Type
Research Article
Information
Econometric Theory , Volume 23 , Issue 5 , October 2007 , pp. 899 - 929
Copyright
© 2007 Cambridge University Press

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References

REFERENCES

Baillie, R.T., C.F. Chung, & M.A. Tieslau (1996) Analyzing inflation by the fractionally integrated ARFIMA-GARCH model. Journal of Applied Econometrics 11, 23–40.Google Scholar
Bollerslev, T. (1986) Generalized autoregressive conditional heteroscedasticity. Journal of Econometrics 31, 307–327.Google Scholar
Brillinger, D.R. (1975) Time Series: Data Analysis and Theory. Holden-Day.
Brockwell, P.J. & R.A. Davis (1991) Time Series: Theory and Methods. 2nd ed. Springer-Verlag.
Dalla, V., L. Giraitis, & J. Hidalgo (2006) Consistent estimation of the memory parameter for nonlinear time series. Journal of Time Series Analysis 27, 211–251.Google Scholar
Ding, Z., C. Granger, & R. Engle (1993) A long memory property of stock market returns and a new model. Journal of Empirical Finance 1, 83–106.Google Scholar
Geweke, J. & S. Porter-Hudak (1983) The estimation and application of long memory time series models. Journal of Time Series Analysis 4, 221–238.Google Scholar
Giraitis, L., P. Kokoszka, & R. Leipus (2000) Stationary ARCH models: Dependent structure and central limit theorem. Econometric Theory 16, 3–22.Google Scholar
Giraitis, L., P.M. Robinson, & D. Surgailis (2000) A model for long memory conditional heteroscedasticity. Annals of Applied Probability 10, 1002–1024.Google Scholar
Gray, H.L., N.-F. Zhang, & W.A. Woodward (1989) On generalized fractional processes. Journal of Time Series Analysis 10, 233–257.Google Scholar
Hall, P. & C.C. Heyde (1980) Martingale Limit Theory and Its Applications. Academic Press.
Hannan, E.J. (1973) Central limit theorems for time series regression. Zeitschrift für Wahrscheinlichskeitstheorie und Verwandte Gebiete 26, 157–170.Google Scholar
Hauser, M.A. & R.M. Kunst (1998a) Fractionally integrated models with ARCH errors: With an application to the Swiss 1-month euromarket interest rate. Review of Quantitative Finance and Accounting 10, 95–113.Google Scholar
Hauser, M.A. & R.M. Kunst (1998b) Forecasting high-frequency financial data with the ARFIMA-ARCH model. Journal of Forecasting 20, 501–518.Google Scholar
Henry, M. (2001) Robust automatic bandwidth for long memory. Journal of Time Series Analysis 22, 293–316.Google Scholar
Henry, M. & P.M. Robinson (1996) Bandwidth choice in Gaussian semiparametric estimation of long range dependence. In P.M. Robinson & M. Rosenblatt (eds.), Athens Conference on Applied Probability and Time Series Analysis, vol. II: Time Series Analysis, In Memory of E.J. Hannan, Lecture Notes in Statistics, 115, 220–232. Springer-Verlag.
Hurvich, C.M., E. Moulines, & P. Soulier (2005) Estimating long memory in volatility. Econometrica 73, 1283–1328.Google Scholar
Kallianpur, G. (1981) Some ramifications of Wiener's ideas on nonlinear prediction. In N. Wiener & P. Masani (eds.), Norbert Wiener, Collected Works with Commentaries, pp. 402–424. MIT Press.
Kim, C.S. & P.C.B. Phillips (1999) Log Periodogram Regression: The Nonstationary Case. Technical report, Yale University.
Künsch, H. (1987) Statistical aspects of self-similar processes. In Yu.A. Prohorov & V.V. Sazonov (eds.), Proceedings of the 1st World Congress of the Bernoulli Society, vol. 1, pp. 67–74. Science Press.
Li, W.K., S. Ling, & M. McAleer (2002) A survey of recent theoretical results for time series models with GARCH errors. Journal of Economic Survey 16, 245–269.Google Scholar
Lien, D. & Y.K. Tse (1999) Forecasting the Nikkei spot index with fractional cointegration. Journal of Forecasting 18, 259–273.Google Scholar
Ling, S. & W.K. Li (1997) On fractionally integrated autoregressive moving-average time series models with conditional heteroscedasticity. Journal of the American Statistical Association 92, 1184–1194.Google Scholar
Ling, S. & M. McAleer (2002a) Necessary and sufficient moment conditions for the GARCH(p,q) and asymmetric power GARCH(p,q) models. Econometric Theory 18, 722–729.Google Scholar
Ling, S. & M. McAleer (2002b) Stationarity and the existence of moments of a family of GARCH processes. Journal of Econometrics 106, 109–117.Google Scholar
Marinucci, D. & P.M. Robinson (1999a) Weak convergence of multivariate fractional processes. Stochastic Processes and Their Applications 86, 103–120.Google Scholar
Marinucci, D. & P.M. Robinson (1999b) Alternative forms of fractional Brownian motion. Journal of Statistical Planning and Inference 80, 111–122.Google Scholar
Min, W. (2004) Inference on time series driven by dependent innovations. Ph.D. Dissertation, University of Chicago.
Moulines, E. & P. Soulier (2003) Semiparametric spectral estimation for fractional processes. In P. Doukhan, G. Oppenheim, & M. Taqqu (eds.), Theory and Applications of Long-Range Dependence, pp. 252–301. Springer Verlag.
Nelson, D.B. (1991) Conditional heteroskedasticity in asset returns: A new approach. Econometrica 59, 347–370.Google Scholar
Nielsen, M.O. & P.H. Frederiksen (2005) Finite sample comparison of parametric, semiparametric, and wavelet estimators of fractional integration. Econometric Reviews 24, 405–443.Google Scholar
Phillips, P.C.B. (1999a) Discrete Fourier Transforms of Fractional Processes. Technical report, Yale University.
Phillips, P.C.B. (1999b) Unit Root Log Periodogram Regression. Technical report, Yale University.
Phillips, P.C.B. & K. Shimotsu (2004) Local Whittle estimation in nonstationary and unit root cases. Annals of Statistics 32, 656–692.Google Scholar
Phillips, P.C.B. & V. Solo (1992) Asymptotics for linear processes. Annals of Statistics 20, 971–1001.Google Scholar
Priestley, M.B. (1988) Nonlinear and Nonstationary Time Series Analysis. Academic Press.
Robinson, P.M. (1991) Testing for strong serial correlation and dynamic conditional heteroskedasticity in multiple regression. Journal of Econometrics 47, 67–84.Google Scholar
Robinson, P.M. (1995a) Log-periodogram regression of time series with long range dependence. Annals of Statistics 23, 1048–1072.Google Scholar
Robinson, P.M. (1995b) Gaussian semiparametric estimation of long range dependence. Annals of Statistics 23, 1630–1661.Google Scholar
Robinson, P.M. (2005) The distance between rival nonstationary fractional processes. Journal of Econometrics 128, 283–300.Google Scholar
Robinson, P.M. & M. Henry (1999) Long and short memory conditional heteroskedasticity in estimating the memory parameter of levels. Econometric Theory 15, 299–336.Google Scholar
Rosenblatt, M. (1959) Stationary processes as shifts of functions of independent random variables. Journal of Mathematical Mechanics 8, 665–681.Google Scholar
Rosenblatt, M. (1971) Markov Processes. Structure and Asymptotic Behavior. Springer-Verlag.
Rosenblatt, M. (1985) Stationary Sequences and Random Fields. Birkhäuser.
Shao, X. & W.B. Wu (2005) Local Whittle Estimation of Fractional Integration for Nonlinear Processes. Technical report 560, Department of Statistics, University of Chicago. Available at http://galton.uchicago.edu/techreports/tr560r1.pdf.
Shao, X. & W.B. Wu (2007) Asymptotic spectral theory for nonlinear time series. Annals of Statistics, forthcoming.Google Scholar
Shimotsu, K. & P.C.B. Phillips (2000) Modified Local Whittle Estimation of the Memory Parameter in the Nonstationary Case. Technical report, Yale University.
Shimotsu, K. & P.C.B. Phillips (2005) Exact local Whittle estimation of fractional integration. Annals of Statistics 33, 1890–1933.Google Scholar
Shimotsu, K. & P.C.B. Phillips (2006) Local Whittle estimation of fractional integration and some of its variants. Journal of Econometrics 103, 209–233.Google Scholar
Subba Rao, T. & M.M. Gabr (1984) An Introduction to Bispectral Analysis and Bilinear Time Series Models. Lecture Notes in Statistics 24. Springer-Verlag.
Tong, H. (1990) Non-linear Time Series: A Dynamical System Approach. Oxford University Press.
Velasco, C. (1999a) Non-stationary log-periodogram regression. Journal of Econometrics 91, 325–371.Google Scholar
Velasco, C. (1999b) Gaussian semiparametric estimation of non-stationary time series. Journal of Time Series Analysis 20, 87–127.Google Scholar
Velasco, C. (2000) Non-Gaussian log-periodogram regression. Econometric Theory 16, 44–79.Google Scholar
Wiener, N. (1958) Nonlinear Problems in Random Theory. MIT Press.
Wu, W.B. (2005) Nonlinear system theory: Another look at dependence. Proceedings of the National Academy of Sciences 102, 14150–14154.Google Scholar
Wu, W.B. (2007) Strong invariance principles for dependent random variables. Annals of Probability, forthcoming.Google Scholar
Wu, W.B. & W. Min (2005) On linear processes with dependent innovations. Stochastic Processes and Their Applications 115, 939–958.Google Scholar
Wu, W.B. & X. Shao (2004) Limit theorems for iterated random functions. Journal of Applied Probability 41, 425–436.Google Scholar
Wu, W.B. & X. Shao (2006) Invariance principles for fractionally integrated nonlinear processes. In J. Sun, A. DasGupta, V. Melfi, & C. Page (eds.), Recent Developments in Nonparametric Inference and Probability. IMS Lecture Notes—Monograph Series 50, 20–30. Institute of Mathematical Statistics.
Wu, W.B. & X. Shao (2007) A limit theorem for quadratic forms and its applications. Econometric Theory 23 (this issue).
Yao, J.-F. (2004) Stationarity, Tails and Moments of Univariate GARCH Processes. Preprint, University of Rennes, France.
Zaffaroni, P. (2004) Stationarity and memory of ARCH(∞) models. Econometric Theory 20, 147–160.Google Scholar