A Note on Feasible Gls Estimation with Heteroskedasticity of Unspecified Form

We give conditions for consistency and asymptotic normality of a FGLS estimator under conditions similar to the ones in [White, 1980a]. This is motivated by the estimator suggested in [Wansbeek, 2004]. In particular, we show that this estimator is not always consistent (even when the OLS estimator is) and is generally inefficient. 1. Motivation and Problem Formulation Consider the linear regression model defined by Yi = Xiβ+εi for i = 1, . . . , n. In matrix form the model is customarily written as Y = Xβ+ ε. Let σ2 i ! E ( εi |Xi ) and Ω ! diag ( σ2 i ) . It is well known that the (unfeasible) GLS estimator βn = ( X′Ω−1X )−1 X′Ω−1Y is, under reasonable hypotheses, consistent, asymptotically normal and, by Aitken’s Theorem ([Davidson, 2000], p. 24), efficient in the class of linear unbiased estimators. If a consistent estimator of Ω exists, the feasible estimator (FGLS) obtained replacing Ω with its estimator enjoys similar properties (see e.g. [White, 1980b], and in particular p. 732 for a discussion on efficiency, and [Mandy and Martins-Filho, 1994, Mandy and Martins-Filho, 1997] for general results). We are interested in what happens when Ω in βn is replaced with the nonconsistent estimator Ω̂ = diag ( σ̂2 i ) , where σ̂2 i = ( Yi − Xiβ̂n )2 and β̂n = (X′X) −1 X′Y is the OLS estimator. Therefore, consider the following estimator β̃n: β̃n = ( X′Ω̂ −1 X )−1 X′Ω̂ −1 Y = β0 + ( X′Ω̂ −1 X )−1 X′Ω̂ −1 ε, (1.1) where β0 is the true value of the parameter (see Assumption 1). The study of the properties of this estimator has been suggested in [Wansbeek, 2004]. We show that the properties of βn are not necessarily shared by the feasible estimator β̃n. In particular, β̃n is not necessarily consistent and is generally inefficient. This is in contrast with what happens in a related but different situation. White [White, 1980a] considers the estimation of the variance of β̂n under heteroskedasticity. In particular, in the stochastic regressor case, he shows that the unfeasible variance is n · (X′X) (X′ΩX) (X′X) and that n · (X′X) ( X′Ω̂X ) (X′X) is a consistent estimator of the variance. The reason is explained in [White, 1980a] (p. 820; see also [Spanos, 1987], p. 465) and amounts to saying that, even if σ̂ i is not consistent for σ2 i , ∑n k=1 XkiXkj σ̂ 2 k and ∑n k=1 XkiXkjσ 2 k have the same limit. This is due to the fact that the estimated variances enter linearly in the sum, and therefore this reasoning does not extend to the present case. Date: February 15, 2008. 1 A NOTE ON FEASIBLE GLS ESTIMATION 2 2. Assumptions We consider the following Assumptions. In particular, Assumption 1 (largely inspired by [White, 1980a], p. 818) defines the model under scrutiny. Assumption 1: The model is known to be Yi = Xiβ0 + εi i = 1, . . . , n where (Yi,Xi)i=1,...,n is a sequence of n independent not (necessarily) identically distributed (inid) random vectors, such that Xi (a (K × 1)−vector) and εi (a scalar) satisfy E (Xiεi) = 0. εi is unobservable while Yi and Xi are observable. β0 is a finite unknown parameter (K × 1)−vector to be estimated. Define the (n × K)−matrix X and the (n × 1)−vectors Y and ε as usual. We will also need to introduce the scalar Zij ! Xij εi , the (K × 1)−vector Zi ! [Zij ]j=1,...,K and the related (n × K)−matrix Z ! [Zij ]i=1,...,n;j=1,...,K . Assumption 2: (i) There exist δ > 0 and ∆ > 0 such that, for any i, k and $, E |εi| < ∆ and E |XikXi#| < ∆. (ii) The matrix 1 n · ∑n i=1 E [XiXi] is positive definite for any sufficiently large n, with a determinant that is uniformly in n bounded away from 0. Remark that under Assumptions 1 and 2, Lemma 1 in [White, 1980a] holds and the estimator β̂n converges to β0 in probability. Assumption 3: There exist δ > 0 and ∆ > 0 such that E |ZikZi#| < ∆ for any i, k and $. Remark 2.1. If the regressors X and the errors ε are independent, Assumption 3 requires the existence of moments of negative order of the variables |εi|. Conditions for the existence of negative moments are given in [Chao and Strawderman, 1972, Cressie et al., 1981, Piegorsch and Casella, 1985]. The last paper gives some results that can be used in our case. Indeed, if the distribution of ε is absolutely continuous with respect to the Lebesgue measure with density f , the moment of order −2 · (1 + δ) can be rewritten as: E |ε| = ∫ +∞ −∞ |u| f (u) du = ∫ +∞ 0 u−(2+2δ) [f (u) + f (−u)] du