Asymptotic normality of regression estimators with long memory errors

doi:10.1016/0167-7152(95)00188-3

Statistics & Probability Letters

Volume 29, Issue 4, 16 September 1996, Pages 317-335

https://doi.org/10.1016/0167-7152(95)00188-3 Get rights and content

Abstract

This paper discusses asymptotic normality of certain classes of M- and R-estimators of the slope parameter vector in linear regression models with long memory moving average errors, extending recent results of Koul (1992) and Koul and Mukherjee (1993). Like in the case of the long memory Gaussian errors, it is observed that all these estimators are asymptotically equivalent to the least squares estimator, a fact that is in sharp contrast with the i.i.d. errors case.

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There are more references available in the full text version of this article.

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An M-estimator for the long-memory parameter
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This paper proposes an $M$ -estimator for the fractional parameter of stationary long-range dependent processes as an alternative to the classical GPH (Geweke and Porter-Hudak, 1983) method. Under very general assumptions on the long-range dependent process the consistency and the asymptotic normal distribution are established for the proposed method. One of the main results is that the convergence rate of the $M$ -estimator is $N^{β / 2}$ , for some positive $β$ , which is the same rate as the standard GPH estimator. The asymptotic properties of the $M$ -estimation method is investigated through Monte-Carlo simulations under the scenarios of ARFIMA models using contaminated with additive outliers and outlier-free data. The GPH approach is also considered in the study for comparison purposes, since this method is widely used in the literature of long-memory time series. The empirical investigation shows that $M$ and GPH-estimator methods display standardized densities fairly close to the standard Gaussian density in the context of non-contaminated data. On the other hand, in the presence of additive outliers, the $M$ -estimator remains unaffected with the presence of additive outliers while the GPH is totally corrupted, which was an expected performance of this estimator. Therefore, the $M$ -estimator here proposed becomes an alternative method to estimate the long-memory parameter when dealing with long-memory time series with and without outliers.
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Citation Excerpt :
This led to further developments in this direction. We refer to Giraitis et al. (1996), Koul and Surgailis (1997, 2000, 2002) for original results in such framework, as well as to a recent monograph (Giraitis et al., 2012). Introduction to general theory of weighted empirical processes can be found in Koul (1992, 2002).
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A limit theory is developed for mildly explosive autoregression under both weakly and strongly dependent innovation errors. The asymptotic behaviour of the sample moments is affected by the memory of the innovation process both in the form of the limiting distribution and, in the case of long range dependence, in the rate of convergence. However, this effect is not present in least squares regression theory as it is cancelled out by the interaction between the sample moments. As a result, the Cauchy regression theory of Phillips and Magdalinos (2007a) is invariant to the dependence structure of the innovation sequence.
Asymptotic expansion for nonparametric M-estimator in a nonlinear regression model with long-memory errors
2011, Journal of Statistical Planning and Inference
We consider asymptotic expansion of the nonparametric M-estimator in a fixed-design nonlinear regression model when the errors are generated by long-memory linear processes. Under mild conditions, we show that the nonparametric M-estimator is first-order equivalent to the Nadaraya–Watson (NW) estimator, which implies that the nonparametric M-estimator has the same asymptotic distribution as that of the NW estimator. Furthermore, we study the second-order asymptotic expansion of the nonparametric M-estimator and show that the difference between the nonparametric M-estimator and the NW estimator has a limiting distribution after suitable standardization. The nature of the limiting distribution depends on the range of long-memory parameter $α$ . We also compare the finite sample behavior of the two estimators through a numerical example when the errors are long-memory.

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¹: Research supported by the Alexander von Humboldt-Foundation.

²: Research partly supported by the NSF Grant DMS-94 02904.

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Asymptotic normality of regression estimators with long memory errors

Abstract

Statist. Probab. Lett.

Generalized powers of strongly dependent random variables

Ann. Probab.

Convergence of Probability Measures

M-estimators for location for Gaussian and related processes with slowly decaying serial correlations

J. Amer. Statist. Assoc.

Statistical methods for data with long range dependence

Statist. Sci.

The empirical processes of some long range dependent sequences with application to U-statistics

Ann. Statist.

Multivariate Appell polynomials and the central limit theorem

A limit theorem for polynomials of linear process with long range dependence

Lithuanian J. Math.