Hostname: page-component-586b7cd67f-2plfb Total loading time: 0 Render date: 2024-11-22T04:54:49.081Z Has data issue: false hasContentIssue false

NONSTATIONARITY-EXTENDED WHITTLE ESTIMATION

Published online by Cambridge University Press:  04 November 2009

Xiaofeng Shao*
Affiliation:
University of Illinois at Urbana-Champaign
*
*Address correspondence to Xiaofeng Shao, Department of Statistics, University of Illinois at Urbana-Champaign, 725 South Wright St, Champaign, IL, 61820 USA; e-mail: [email protected]

Abstract

For long memory time series models with uncorrelated but dependent errors, we establish the asymptotic normality of the Whittle estimator under mild conditions. Our framework includes the widely used fractional autoregressive integrated moving average models with generalized autoregressive conditional heteroskedastic-type innovations. To cover nonstationary fractionally integrated processes, we extend the idea of Abadir, Distaso, and Giraitis (2007, Journal of Econometrics 141, 1353–1384) and develop the nonstationarity-extended Whittle estimation. The resulting estimator is shown to be asymptotically normal and is more efficient than the tapered Whittle estimator. Finally, the results from a small simulation study are presented to corroborate our theoretical findings.

Type
ARTICLES
Copyright
Copyright © Cambridge University Press 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Abadir, K.M., Distaso, W., & Giraitis, L. (2007) Nonstationarity-extended local Whittle estimation. Journal of Econometrics 141, 13531384.CrossRefGoogle Scholar
Baillie, R.T., Chung, C.F., & Tieslau, M.A. (1996) Analyzing inflation by the fractionally integrated ARFIMA-GARCH model. Journal of Applied Econometrics 11, 2340.3.0.CO;2-M>CrossRefGoogle Scholar
Beran, J. (1995) Maximum likelihood estimation of the differencing parameter for invertible short and long memory autoregressive integrated moving average models. Journal of the Royal Statistical Society, Series B 57, 659672.Google Scholar
Bollerslev, T. (1986) Generalized autoregressive conditional heteroscedasticity. Journal of Econometrics 31, 307327.CrossRefGoogle Scholar
Chiu, S.T. (1988) Weighted least squares estimators on the frequency domain for the parameters of a time series. Annals of Statistics 16, 13151326.CrossRefGoogle Scholar
Dahlhaus, R. (1989) Efficient parameter estimation for self-similar processes. Annals of Statistics 17, 17491766.CrossRefGoogle Scholar
Davidson, J. (2004) Moment and memory properties of linear conditional heteroscedasticity models, and a new model. Journal of Business & Economic Statistics 22, 1629.CrossRefGoogle Scholar
Ding, Z., Granger, C., & Engle, R. (1993) A long memory property of stock market returns and a new model. Journal of Empirical Finance 1, 83106.CrossRefGoogle Scholar
Doukhan, P., Oppenheim, G. & Taqqu, M.S. (2003) Theory and Applications of Long-Range Dependence. Birkhaüser.Google Scholar
Elek, P. & Márkus, L. (2004) A long range dependent model with nonlinear innovations for simulating daily river flows. Natural Hazards and Earth System Sciences 4, 277283.CrossRefGoogle Scholar
Fan, J. & Yao, Q. (2003) Nonlinear Time Series: Nonparametric and Parametric Methods. Springer-Verlag.CrossRefGoogle Scholar
Fox, R. & Taqqu, M.S. (1986) Large-sample properties of parameter estimates for strongly dependent stationary Gaussian time series. Annals of Statistics 14, 517532.CrossRefGoogle Scholar
Giraitis, L. & Robinson, P.M. (2001) Whittle estimation of ARCH models. Econometric Theory 17, 608631.CrossRefGoogle Scholar
Giraitis, L. & Surgailis, D. (1990) A central limit theorem for quadratic forms in strongly dependent random variables and its application to asymptotic normality of Whittle’s estimate. Probability Theory and Related Fields 86, 87104.CrossRefGoogle Scholar
Giraitis, L. & Surgailis, D. (2002) ARCH-type bilinear models with double long memory. Stochastic Processes and Their Applications 100, 275300.CrossRefGoogle Scholar
Giraitis, L. & Taqqu, M.S. (1999) Whittle estimator for non-Gaussian long-memory time series. Annals of Statistics 27, 178203.CrossRefGoogle Scholar
Gray, H.L., Zhang, N.-F., & Woodward, W.A. (1989) On generalized fractional processes. Journal of Time Series Analysis 10, 233257.CrossRefGoogle Scholar
Hannan, E.J. (1973a) Central limit theorems for time series regression. Zeitschrift Für Wahrscheinlichskeitstheorie und Verwandte Gebiete 26, 157170.CrossRefGoogle Scholar
Hannan, E.J. (1973b) The asymptotic theory of linear time series models. Journal of Applied Probability 10, 130145.CrossRefGoogle Scholar
Hauser, M.A. & Kunst, R.M. (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, 95113.CrossRefGoogle Scholar
Hauser, M.A. & Kunst, R.M. (1998b) Forecasting high-frequency financial data with the ARFIMA-ARCH model. Journal of Forecasting 20, 501518.CrossRefGoogle Scholar
Hosoya, Y. (1997) A limit theory for long-range dependence and statistical inference on related models. Annals of Statistics 25, 105137.CrossRefGoogle Scholar
Keenan, D.M. (1987) Limiting behavior of functionals of higher-order sample cumulant spectra. Annals of Statistics 15, 134151.CrossRefGoogle Scholar
Koopman, S.J., Oohs, M., & Carnero, M.A. (2007) Periodic seasonal Reg-ARFIMA-GARCH models for daily electricity spot prices. Journal of the American Statistical Association 102, 1627.CrossRefGoogle Scholar
Lien, D. & Tse, Y.K. (1999) Forecasting the Nikkei spot index with fractional cointegration. Journal of Forecasting 18, 259273.3.0.CO;2-7>CrossRefGoogle Scholar
Ling, S. & Li, W.K. (1997) On fractionally integrated autoregressive moving-average time series models with conditional heteroscedasticity. Journal of the American Statistical Association 92, 11841194.CrossRefGoogle Scholar
Ling, S. & McAleer, M. (2002) Necessary and sufficient moment conditions for the GARCH(r, s) and asymmetric power GARCH(r, s) models. Econometric Theory 18, 722729.CrossRefGoogle Scholar
Marinucci, D. & Robinson, P.M. (1999) Alternative forms of fractional Brownian motion. Journal of Statistical Planning and Inference 80, 111122.CrossRefGoogle Scholar
Mayoral, L. (2007) Minimum distance estimation of stationary and non-stationary ARFIMA processes. Econometrics Journal 10, 124148.CrossRefGoogle Scholar
Nelson, D.B. (1991) Conditional heteroskedasticity in asset returns: A new approach. Econometrica 59, 347370.CrossRefGoogle Scholar
Phillips, P.C.B. (1999) Discrete Fourier Transforms of Fractional Processes. Technical Report, Yale University.Google Scholar
Phillips, P.C.B. & Shimotsu, K. (2004) Local Whittle estimation in nonstationary and unit root cases. Annals of Statistics 32, 656692.CrossRefGoogle Scholar
Robinson, P.M. (1995a) Log-periodogram regression of time series with long range dependence. Annals of Statistics 23, 10481072.CrossRefGoogle Scholar
Robinson, P.M. (1995b) Gaussian semiparametric estimation of long range dependence. Annals of Statistics 23, 16301661.CrossRefGoogle Scholar
Robinson, P.M. (2003) Time Series with Long Memory. Oxford University Press.CrossRefGoogle Scholar
Robinson, P.M. (2005) The distance between rival nonstationary fractional processes. Journal of Econometrics 128, 283300.CrossRefGoogle Scholar
Shao, X. & Wu, W.B. (2007a) Asymptotic spectral theory for nonlinear time series. Annals of Statistics 35, 17731801.CrossRefGoogle Scholar
Shao, X. & Wu, W.B. (2007b) Local Whittle estimation of fractional integration for nonlinear processes. Econometric Theory 23, 899929.CrossRefGoogle Scholar
Shimotsu, K. & Phillips, P.C.B. (2006) Local Whittle estimation of fractional integration and some of its variants. Journal of Econometrics 103, 209233.CrossRefGoogle Scholar
Subba Rao, T. & Gabr, M.M. (1984) An Introduction to Bispectral Analysis and Bilinear Time Series Models, Lecture Notes in Statistics, 24. Springer-Verlag.Google Scholar
Taniguchi, M. (1982) On estimation of the integrals of the 4th order cumulant spectral density. Biometrika 69, 117122.CrossRefGoogle Scholar
Tong, H. (1990) Non-linear Time Series: A Dynamical System Approach. Oxford University Press.CrossRefGoogle Scholar
Velasco, C. & Robinson, P.M. (2000) Whittle pseudo-maximum likelihood estimation for nonstationary time series. Journal of the American Statistical Association 95, 12291243.CrossRefGoogle Scholar
Walker, A.M. (1964) Asymptotic properties of least-squares estimates of parameters of the spectrum of a stationary non-deterministic time-series. Journal of the Australian Mathematical Society 4, 363384.CrossRefGoogle Scholar
Wu, W.B. (2005) Nonlinear system theory: Another look at dependence. Proceedings of the National Academy of Science, USA 102, 1415014154.CrossRefGoogle Scholar
Wu, W.B. & Min, M. (2005) On linear processes with dependent innovations. Stochastic Processes and Their Applications 115, 939958.Google Scholar
Wu, W.B. & Shao, X. (2004) Limit theorems for iterated random functions. Journal of Applied Probability 41, 425436.CrossRefGoogle Scholar
Zaffaroni, P. & d’Italia, B. (2003) Gaussian inference on certain long-range dependent volatility models. Journal of Econometrics 115, 199258.CrossRefGoogle Scholar