A Comparison of Maximum Likelihood Versus Blue Estimators

Abstract

T HE most frequently used estimating technique for applied economic research has been ordinary least squares (OLS). There are two theoretical justifications for its use. First, the Gauss-Markov theorem suggests that OLS estimators will outperform all other techniques in the class of linear unbiased estimators.1 Secondly, under the assumption that the error terms are normally distributed, OLS estimators can be derived as the maximum likelihood estimates. Frequently when OLS is introduced in elementary texts, the method of minimizing the sum of absolute deviations (MAD) is presented for comparison purposes, but rarely is it given serious consideration for applied uses.) Certainly one reason for such behavior stems from previous evaluations of OLS versus MAD estimators. Asher and Wallace (1963) found that the use of MAD meant one should be prepared to give up considerable efficiency 3 More recently Glahe and Hunt (1970), suggested several estimators derived under the general minimization criterion. In contrast to the Asher-Wallace study, the Glahe-Hunt model was a two-equation linear simultaneous system. Their results show that neither form of absolute deviation estimator outperformed either OLS or two-stage least squares.4 The purpose of this paper is to suggest that in at least one aspect these comparisons have given OLS a differential advantage. That is, both of these studies employed errors for their hypothesized models which were drawn from normal distributions. Consequently the OLS estimators are both maximum likelihood and best linear unbiased estimators (BLUE) under such circumstances. Since recently published works by Zeckhauser and Thompson (1970), Fama (1965) and others have called into question the assumption of normally distributed error terms, attention has begun to shift to other alternatives.5 Blattberg and Sargent (1971) have examined three techniques including both OLS and MAD when the errors are drawn from a stable Paretian distribution. Their findings indicate that the MAD estimator . performs sufficiently well that it deserves further study and elaboration. G Accordingly we have chosen to explore the relative merits of OLS and MAD for a single equation model whose errors are drawn from a double exponential parent distribution. This distribution was chosen because both estimators will exhibit theoretically desirable properties. OLS remains the BLUE estimator, while MAD is the maximum likelihood estimator. Furthermore, this distribution is one member of the power distribution suggested by Zeckhauser and Thompson as an alternative to the normal. This paper is divided into, three sections. The first describes the design of the experiments. Section II presents the empirical results and the last summarizes the primary findings of the paper.