Efficiency gains in least squares estimation: A new approach

Abstract

In pursuit of efficiency, we propose a new way to construct least squares estimators, as the minimizers of an augmented objective function that takes explicitly into account the variability of the error term and the resulting uncertainty, as well as the possible existence of heteroskedasticity. We initially derive an infeasible estimator which we then approximate using Ordinary Least Squares (OLS) residuals from a first-step regression to obtain the feasible “HOLS” estimator. This estimator has negligible bias, is consistent and outperforms OLS in terms of finite-sample Mean Squared Error, but also in terms of asymptotic efficiency, under all skedastic scenarios, including homoskedasticity. Analogous efficiency gains are obtained for the case of Instrumental Variables estimation. Theoretical results are accompanied by simulations that support them.

Keywords:

• Heteroskedasticity
• least squares
• efficiency
• excess kurtosis
• instrumental variables

JEL CLASSIFICATION:

• C13
• C21
• C36

Disclosure statement

There was no funding for this work. The authors declare no conflict of interest.

Correction Statement

This article has been republished with minor changes. These changes do not impact the academic content of the article.

Notes

1 See e.g. Greene (2012, p.105).

2 The truncated version of the flagship of distributions without moments, Cauchy, has been studied in Nadarajah and Kotz (2006).

3 In fact we implemented this approach, and found that, although the estimator obtained thus is asymptotically equivalent to HOLS, it performed visibly worse in small samples.

4 The concept is here defined as having an asymptotic distribution with zero mean, rather than in relation to the convergence of the moment sequence, see Lehmann and Casella (1998, p.438).

5 The case of exact identification can be worked out along the same lines.