Weak and strong consistency of the least squares estimators in regression models

Abstract

This paper is a continuation of a previous paper [2] on the consistency of the least squares estimator ^ -+ aN-X Iv YN of 6 in the sequence of regression-models yu= X N (~ + u N obtained after N observations have been made. It is the purpose of this paper to show that the consistency-condition 2,~i,(X'NXN)--*oO can be obtained without any nornaality assumptions. This result was conjectured by Eicker (Ann. Math. Statist., 34, 447-456, (1963)). Moreover, it is shown, that under rather weak assumptions the sequence {X~ YN} possesses the remarkable property that lira X~v yN=Z in the sense of L2-convergence, whatever the sequence {XN} N~o~ may be.If the components of u N are supposed to be stochastically independent and there is only one regressor necessary and sufficient conditions for strong consistency of a N can be obtained. This result is then applied to obtain sufficient conditions for the strong consistency of a N in the general case. Moreover, it is shown that these conditions are met if N-iX~v XN--> Z, Z > 0, the usual assumption made in econometric textbooks to establish weak consistency. * This work was done when the author was a research worker at "Sonderforschungsbereich Okonometrie und Unternehmensforschung der Universit~it Bonn".

Consistent Estimation of Structural Parameters in Regression Models with Adaptive Learning

“…In the Gauss-Markov model, condition (4.7) is also necessary for weak convergence (cf. Eicker (1963) and Drygas (1976)). Needless to say that (4.7) is much weaker than the classical textbook condition…”

Strong Consistency of the Least-Squares Estimator in Simple Regression Models with Stochastic Regressors

“…convergence θ n → θ in probability. Actually, as shown by Eicker (1963) (for normal disturbances) and Drygas (1976), it is also necessary for weak consistency. Needless to say that (2.1) is much weaker than the classical textbook condition…”

A Note on an Estimation Problem in Models with Adaptive Learning

“…convergence with probability one) of the OLS estimator, cf. Anderson & Taylor (1976), Drygas (1976), Lai & Robbins (1977) and Lai, Robbins & Wei (1978, 1979, providing both sufficient and necessary conditions. For stochastic regressors, the situation is more delicate.…”