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As stated in the question.

In particular, how does a researcher know when to apply which estimation method and are there any examples that can show when one case is more appropriate than the other?

Maybe not just in terms of practical usage as in here, but also in terms of the philosophical basis and assumptions being made.

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  • $\begingroup$ I do not understand how the choice of a statistical estimation technique relates to Philosophy... $\endgroup$
    – Xi'an
    Mar 4, 2015 at 13:59
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    $\begingroup$ The origin of science is historically a subset of philosophy. The historical etymology of the Greek word 'philosophy' is philo ("love") plus sophia ("wisdom, A.K.A. knowledge"). The proper application of statistics to data is the extraction of order in chaotic systems and very much a search for knowledge. $\endgroup$
    – Carl
    Aug 11, 2016 at 18:47
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    $\begingroup$ stats.stackexchange.com/questions/112451/… $\endgroup$ May 1, 2017 at 20:14

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Main idea in GMM is to have a bit relaxed view concerning data generation process since one does not need to specify full parametric density as in maximum likelihood estimation.

And there are possibilities to formulate theories which can be tested if they specify certain moment conditions.

Here is some discussion by LP Hansen who introduced this method to econometrics.

http://www.larspeterhansen.org/documents/2007_GMM_BookPDE_Generalized_Method_of_Moments_Estimation.pdf

This GMM actually looks quite a bit like M/Z - estimators in statistics.

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  • $\begingroup$ GMM estimators are in fact a special case of M-estimators where the quantity being minimized is a quadratic form of the moment conditions. $\endgroup$
    – jayk
    Jun 15, 2015 at 0:12

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