<p>OLS is &quot;the&quot; linear estimator. When I hear &quot;nonlinear estimator&quot; in the context of econometrics, I'm thinking &quot;anything except OLS&quot;. This isn't entirely true, as GLS is a linear estimator, but FGLS is not, and FGLS is what is used in practice.</p>
<p>An extremum estimator is an estimator that maximizes (or minimizes) an objective function.</p>
<p>An m-estimator is an extremum estimator in which the objective function is a sample average.</p>
<p>Minimum distance is on section 8.5 of <a href="https://www.ssc.wisc.edu/%7Ebhansen/econometrics/Econometrics.pdf" rel="nofollow noreferrer">Bruce Hansen's textbook</a>. I think it'd be simplest to just reference that rather than repeat his text here.</p>
<p>EDIT: I neglected earlier to mention that 2SLS is also a linear estimator.</p>

CC BY-SA 4.0
2022-04-21T07:39:08.467Z
2022-04-27T11:04:08.547Z
2022-04-27T11:04:08.547Z
Michael Gmeiner
<p>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 &quot;[...] not all non-linear estimators are m-estimators. Examples include method of moments, GMM, and minimum distance&quot;</p>
<p>My questions are:</p>
<ol>
<li>What does he mean when he says non-linear estimators?</li>
<li>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?</li>
</ol>

<econometrics><applied-econometrics><statistics>
Difference between M-estimators and Minimum Distance estimators
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# Difference between M-estimators and Minimum Distance estimators

(Asked by: username_0 on 2022-04-20)

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?

**Tags:** econometrics, applied-econometrics, statistics


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## 1 answers

### Answer by username_1 on 2022-04-21. Score: 2

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](https://www.ssc.wisc.edu/%7Ebhansen/econometrics/Econometrics.pdf). 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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