Approaches to Estimation: A Synthesis - Sem. Bilias 20/5 Cassino

[Giovanni Porzio <porzio@eco.unicas.it>]

Giovedì 20 maggio alle ore 14.30 presso l'aula del dottorato della Facoltà
di Economia (3° piano) via Mazzaroppi snc, 03043 Cassino

si terrà il seminario

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"THE METHOD OF MOMENTS, MAXIMUM ENTROPY, MAXIMUM LIKELIHOOD,
EMPIRICAL LIKELIHOOD, ESTIMATING FUNCTION AND GENERALIZED METHOD OF MOMENTS.
Approaches to Estimation: A Synthesis"

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Prof. Yannis Bilias

Department of Economics
University of Cyprus
Nicosia CY 1678
CYPRUS

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Abstract

The 20th century began on an auspicious statistical note with the publication 
of Karl Pearson's (1900) goodness-of-test, which is regarded as one of the 
most important scientic breakthroughs. The basic motivation behind this test 
was to see whether an assumed probability model adequately described the data 
at hand. Pearson (1894) also introduced a formal approach to statistical 
estimation through his method of moments (MM) estimation.
Ronald A. Fisher, while he was a third year undergraduate at the Gonville and 
Caius College, Cambridge, suggested the maximum likelihood estimation (MLE) 
procedure as an alternative to Pearson's MM approach. In 1922 Fisher published 
a monumental paper that introduced such basic concepts as consistency, 
efficiency, sufficiency (and even the term "parameter" with its present 
meaning). Fisher (1922) provided the analytical foundation of MLE and studied 
its efficiency relative to the MM estimator. Fisher (1924a) established
the asymptotic equivalence of minimum Chi^2 and ML estimators and wrote in 
favor of using minimum Chi^2 method rather than Pearson's MM approach. 
Recently, econometricians have found working under assumed likelihood 
functions restrictive, and have suggested using a generalized version of 
Pearson's MM approach, commonly known as the GMM estimation procedure as 
advocated in Hansen (1982). Earlier, Godambe (1960) and Durbin (1960) 
developed the estimating function (EF) approach to estimation that has been 
proven very useful for many statistical models. A fundamental result is that 
score is the optimum EF.
Ferguson (1958) considered an approach very similar to GMM and showed that 
estimation based on the Pearson chi-squared statistic is equivalent to 
efficient GMM. Golan, Judge and Miller (1996) developed entropy-based 
formulation that allowed them to solve a wide range of estimation and 
inference problems in econometrics. More recently, Imbens, Spady
and Johnson (1998), Kitamura and Stutzer (1997) and Mittelhammer, Judge and 
Miller (2000) put GMM within the framework of empirical likelihood (EL) and 
maximum entropy (ME) estimation. It can be shown that many of these estimation 
techniques can be obtained as special cases of minimizing Cressie and Read 
(1984) power divergence criterion that comes directly from the Pearson (1900) 
chi-squared statistic. In this way we are able to assimilate a number of 
seemingly unrelated estimation techniques into a unified framework.

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