Nonlinear Log-Periodogram Regression for Perturbed Fractional Processes

This paper studies fractional processes that may be perturbed by weakly dependent time series. The model for a perturbed fractional process has a components framework in which there may be components of both long and short memory. All commonly used estimates of the long memory parameter (such as log periodogram (LP) regression) may be used in a components model where the data are affected by weakly dependent perturbations, but these estimates can suffer from serious downward bias. To circumvent this problem, the present paper proposes a new procedure that allows for the possible presence of additive perturbations in the data. The new estimator resembles the LP regression estimator but involves an additional (nonlinear) term in the regression that takes account of possible perturbation effects in the data. Under some smoothness assumptions at the origin, the bias of the new estimator is shown to disappear at a faster rate than that of the LP estimator, while its asymptotic variance is inflated only by a multiplicative constant. In consequence, the optimal rate of convergence to zero of the asymptotic MSE of the new estimator is faster than that of the LP estimator. Some simulation results demonstrate the viability and the bias-reducing feature of the new estimator relative to the LP estimator in finite samples. A test for the presence of perturbations in the data is given.

[1]  É. Moulines,et al.  Broadband log-periodogram regression of time series with long-range dependence , 1999 .

[2]  Donald W. K. Andrews,et al.  A BIAS-REDUCED LOG-PERIODOGRAM REGRESSION ESTIMATOR FOR THE LONG-MEMORY PARAMETER , 2003 .

[3]  P. Robinson.  Gaussian Semiparametric Estimation of Long Range Dependence , 1995 .

[4]  Marc Henry,et al.  Bandwidth Choice in Gaussian Semiparametric Estimation of Long Range Dependence , 1996 .

[5]  Clive W. J. Granger,et al.  The Correlogram of a Long Memory Process Plus a Simple Noise , 1997 .

[6]  H. Künsch.  Discrimination between monotonic trends and long-range dependence , 1986 .

[7]  C. Hurvich,et al.  Plug‐in Selection of the Number of Frequencies in Regression Estimates of the Memory Parameter of a Long‐memory Time Series , 1998 .

[8]  Marc Henry,et al.  Higher-Order Kernel Semiparametric M-Estimation of Long Memory , 2002 .

[9]  Marc Henry,et al.  Robust Automatic Bandwidth for Long Memory , 2001 .

[10]  D. Andrews,et al.  Local Polynomial Whittle Estimation of Long-range Dependence , 2001 .

[11]  F. Breidt,et al.  The detection and estimation of long memory in stochastic volatility , 1998 .

[12]  P. Phillips.  Discrete Fourier Transforms of Fractional Processes , 1999 .

[13]  Peter C. B. Phillips,et al.  Exact Local Whittle Estimation of Fractional Integration , 2002 .

[14]  Clifford M. Hurvich,et al.  Broadband Semiparametric Estimation of the Memory Parameter of a Long‐Memory Time Series Using Fractional Exponential Models , 1998 .

[15]  J. Geweke,et al.  THE ESTIMATION AND APPLICATION OF LONG MEMORY TIME SERIES MODELS , 1983 .

[16]  Pooled Log Periodogram Regression , 2000 .

[17]  Yixiao Sun,et al.  Adaptive Local Polynomial Whittle Estimation of Long-Range Dependence , 2002 .

[18]  C. Hurvich,et al.  ON THE LOG PERIODOGRAM REGRESSION ESTIMATOR OF THE MEMORY PARAMETER IN LONG MEMORY STOCHASTIC VOLATILITY MODELS , 2001, Econometric Theory.

[19]  Rohit S. Deo,et al.  The mean squared error of Geweke and Porter‐Hudak's estimator of the memory parameter of a long‐memory time series , 1998 .

[20]  O. Linton,et al.  Local nonlinear least squares: Using parametric information in nonparametric regression , 2000 .

[21]  P. Phillips,et al.  Published online in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/jae.760 UNDERSTANDING THE FISHER EQUATION , 2022 .

[22]  É. Moulines,et al.  Log-Periodogram Regression Of Time Series With Long Range Dependence , 1999 .

[23]  C. Velasco,et al.  NON-GAUSSIAN LOG-PERIODOGRAM REGRESSION , 2000, Econometric Theory.