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I was reading Chapter 22 of Hansen's Econometrics on M-estimators and he lists OLS, Non-linear LS and Maximum Likelihood as examples of M estimators. After a few paragraphs, he also writes "[...] not all non-linear estimators are m-estimators. Examples include method of moments, GMM, and minimum distance"

My questions are:

  1. What does he mean when he says non-linear estimators?
  2. What is the difference between M-estimators and minimum distance estimators? Is OLS both an m-estimator and a minimum distance? If so, then why is GMM not considered an m-estimator?
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OLS is "the" linear estimator. When I hear "nonlinear estimator" in the context of econometrics, I'm thinking "anything except OLS". This isn't entirely true, as GLS is a linear estimator, but FGLS is not, and FGLS is what is used in practice.

An extremum estimator is an estimator that maximizes (or minimizes) an objective function.

An m-estimator is an extremum estimator in which the objective function is a sample average.

Minimum distance is on section 8.5 of Bruce Hansen's textbook. I think it'd be simplest to just reference that rather than repeat his text here.

EDIT: I neglected earlier to mention that 2SLS is also a linear estimator.

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