Asymptotic Distribution of Least Squares Estimators for Linear Models with Dependent Errors: Regular Designs

Abstract

In this paper, we consider the usual linear regression model in the case where the error process is assumed strictly stationary. We use a result from Hannan, who proved a Central Limit Theorem for the usual least squares estimator under general conditions on the design and the error process. We show that for a large class of designs, the asymptotic covariance matrix is as simple as the independent and identically distributed case. We then estimate the covariance matrix using an estimator of the spectral density whose consistency is proved under very mild conditions. 1 independent and identically distributed. 1 (ǫ i ) is satisfied, we propose a consistent estimator of the spectral density of (ǫ i ) (as a byproduct, we get an estimator of the covariance series).Wu and Liu [14] considered the problem of estimating the spectral density for a large class of short-range dependent processes. They proposed a consistent estimator for the spectral density, and gave some conditions under which the centered estimator satisfies a Central Limit Theorem. These results are based on the asymptotic theory of stationary processes developed by Wu [23]. This framework enables to deal with most of the statistical procedures from time series, including the estimation of the spectral density. However the class of processes satisfying the L 2 "physical dependence measure" introduced by Wu is included in the class of processes satisfying Hannan's condition. In this paper, we prove the consistency of an estimator of the spectral density of the error process under Hannan's condition. Compared to Wu's precise results on the estimation of the spectral density (Central Limit Theorem, rates of convergence, deviation inequalities), our result is only a consistency result, but it holds under Hannan's condition, that is for most of short-range dependent processes.The paper is organized as follows. In Section 2, we recall Hannan's Central Limit Theorem for the least squares estimator, and we define the class of « regular designs » (we also give many examples of such designs). In Section 3, we focus on the estimation of the spectral density of the error process under Hannan's condition. Finally, some examples of stationary processes satisfying Hannan's condition are presented in Section 4.Theorem 2.1 is very general because it includes a very large class of designs. In this paper, we will focus on the case where the design is regular in the following sense: Definition 2.3.1 (Regular design). A fixed design X is called regular if, for any j, l in {1, . . . , p}, the coefficients ρ j,l (k) do not depend on k.A large class of regular designs is the one for which the columns are regularly varying sequences. Let us recall the definition of regularly varying sequences: Definition 2.3.2 (Regularly varying sequence [21]). A sequence S(·) is regularly varying if and only if it can be written as:where −∞ < α < ∞ and L(·) is a slowly varying sequence. 2 The transpose of a matrix X is denoted by X t .

Self-normalized Cramér type moderate deviations for stationary sequences and applications

“…Accordingly, a confidential interval for linear regression can be obtained via Hannan's theorem (cf. Caron and Dede [4]), but with larger risk probability; the risk probability can be significantly improved, using Cramér type moderate deviations of Wu and Zhao [38] and Cuny and Merlevède [9] on stationary sequences. Notice that the results of [38] and [9] also hold when X i has finite pth moments with p > 2.…”

Asymptotic distribution of least square estimators for linear models with dependent errors

“…However, Hannan's condition is satisfied for most short-range dependent stationary processes. The reader can see the paper [7] where some examples checking Hannan's condition are developed.…”

“…and the uniform distribution on [0, 1] is the unique invariant distribution by K. Hence, the chain (Z i ) i≥1 is strictly stationary. Furthermore, it is not α-mixing in the sense of Rosenblatt [5], but it isφ-dependent in the sense of Dedecker-Prieur [10] (see also Caron-Dede [7], Section 4.4). Indeed, one can prove that the coefficientsφ(k) of the chain (Z i ) i≥1 decrease geometrically [10]:φ(k) ≤ 2 −k .…”

“…Let us quote that most of short-range dependent processes satisfies Hannan's conditions on the error process, for instance the class of linear processes with summable coefficients and square integrable innovations, a large class of functions of linear processes, many processes under various mixing conditions and the 2-strong stable processes introduced by Wu [15]. We refer to our previous paper [7], which presents many classes of short range processes satisfying Hannan's condition.…”

“…In this paper, we present an estimator of the asymptotic covariance matrix of the normalized least square estimators of the parameters. This estimator is derived from the estimator of the spectral density of the error process introduced in [7]. Once the asymptotic covariance matrix is consistently estimated, it is then possible to obtain confidence regions and test procedures for the unknown parameter β.…”