Local Whittle estimation in nonstationary and unit root cases. (English) Zbl 1091.62084

Summary: Asymptotic properties of the local Whittle estimator in the nonstationary case \((d>1/2)\) are explored. For \(1/2 < d \leq 1\), the estimator is shown to be consistent, and its limit distribution and the rate of convergence depend on the value of \(d\). For \(d=1\), the limit distribution is mixed normal. For \(d>1\) and when the process has a polynomial trend of order \(a>1/2\), the estimator is shown to be inconsistent and to converge in probability to unity.


62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
62E20 Asymptotic distribution theory in statistics
62G20 Asymptotic properties of nonparametric inference
Full Text: DOI arXiv


[1] Andrews, D. W. K. and Sun, Y. (2001). Local polynomial Whittle estimation of long-range dependence. Discussion Paper 1293, Cowles Foundation, Yale Univ. Available at cowles.econ.yale.edu. · Zbl 1131.62317
[2] Corbae, D., Ouliaris, S. and Phillips, P. C. B. (2002). Band spectral regression with trending data. Econometrica 70 1067–1109. · Zbl 1121.62556
[3] Erdélyi, A., ed. (1953). Higher Transcendental Functions 1 . Krieger, Malabar, FL. · Zbl 0051.30303
[4] Gray, H. L., Zhang, N.-F. and Woodward, W. A. (1989). On generalized fractional processes. J. Time Ser. Anal. 10 233–257. · Zbl 0685.62075
[5] Hall, P. and Heyde, C. C. (1980). Martingale Limit Theory and Its Application . Academic Press, New York. · Zbl 0462.60045
[6] Hurvich, C. M. and Chen, W. W. (2000). An efficient taper for potentially overdifferenced long-memory time series. J. Time Ser. Anal. 21 155–180. · Zbl 0958.62085
[7] Kim, C. S. and Phillips, P. C. B. (1999). Log periodogram regression: The nonstationary case. Mimeograph, Cowles Foundation, Yale Univ.
[8] Künsch, H. (1987). Statistical aspects of self-similar processes. In Proc. First World Congress of the Bernoulli Society (Yu. Prokhorov and V. V. Sazanov, eds.) 1 67–74. VNU Science Press, Utrecht. · Zbl 0673.62073
[9] Marinucci, D. and Robinson, P. M. (2000). Weak convergence of multivariate fractional processes. Stochastic Process. Appl. 86 103–120. · Zbl 1028.60030
[10] Nelson, C. R. and Plosser, C. I. (1982). Trends and random walks in macroeconomic time series: Some evidence and implications. J. Monetary Economics 10 139–162.
[11] Phillips, P. C. B. (1999a). Discrete Fourier transforms of fractional processes. Discussion Paper 1243, Cowles Foundation, Yale Univ. Available at cowles.econ.yale.edu.
[12] Phillips, P. C. B. (1999b). Unit root log periodogram regression. Discussion Paper 1244, Cowles Foundation, Yale University. Available at cowles.econ.yale.edu.
[13] Robinson, P. M. (1995). Gaussian semiparametric estimation of long range dependence. Ann. Statist. 23 1630–1661. JSTOR: · Zbl 0843.62092
[14] Robinson, P. M. and Marinucci, D. (2001). Narrow-band analysis of nonstationary processes. Ann. Statist. 29 947–986. · Zbl 1012.62100
[15] Schotman, P. and van Dijk, H. K. (1991). On Bayesian routes to unit roots. J. Appl. Econometrics 6 387–401.
[16] Shimotsu, K. and Phillips, P. C. B. (2002). Exact local Whittle estimation of fractional integration. Discussion Paper 1367, Cowles Foundation, Yale Univ. Available at cowles.econ.yale.edu. · Zbl 1081.62069
[17] Velasco, C. (1999). Gaussian semiparametric estimation of non-stationary time series. J. Time Ser. Anal. 20 87–127. · Zbl 0922.62093
[18] Zygmund, A. (1959). Trigonometric Series , 2nd ed. Cambridge Univ. Press. · Zbl 0085.05601
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.