# When should one consider using GMM?

One of the things which makes econometrics unique is the use of the Generalized Method of Moments technique.

What types of problems make GMM more appropriate than other estimation techniques? What does using GMM buy you in terms of efficiency or reduced bias or more specific parameter estimation?

Conversely, what do you lose by using GMM over MLE, etc.?

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GMM is a semi-parametric method; it is also a partial information method, as compared to (full-information) MLE. –  Dimitris Nov 29 '11 at 16:34
GMM techniques are not unique to econometrics - although other flavors of statistician tend to have other names for the same ideas. They are popular anywhere you want to do statistical inference but can't justify a full modeling approach (or don't want to) - see applications in biostatistics, survey research, social science, and probably much else. –  guest May 4 '12 at 5:40
Note, the tag [gmm] is applied to this thread, & should remain on this thread only so that it will not disappear. The tag itself is ambiguous & should not be used in general; instead the specific tags [generalized-moments], [gaussian-mixture-model], or [growth-mixture-model] should be used for future threads. –  gung Oct 25 '13 at 16:55
If you want to fold TSLS under GMM, then you may as well say the same for OLS, so saying that GMM is TSLS and GMM and TSLS help get rid of endogeneity kind of misses the point. The point here is "why would you want to go to the extra trouble of some specialized GMM model?" That may be valid and deep question, especially if it is difficult to test the strength or validity of any instruments you might be attempting to use to purge endogeneity. –  user36939 Jan 6 at 20:53
Why should we use GMM? Why should you migrate from other models to GMM? –  user43003 Apr 3 at 4:27

GMM is practically the only estimation method which you can use, when you run into endogeneity problems. Since these are more or less unique to econometrics, this explains GMM atraction. Note that this applies if you subsume IV methods into GMM, which is perfectly sensible thing to do.

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Well you can estimate IV lots of ways right? TSLS, etc.... But GMM is probably the most flexible. –  Ari B. Friedman Dec 13 '11 at 17:38
TSLS is GMM with special weighting matrix. –  mpiktas Dec 13 '11 at 17:47
This may be nitpicky semantics, but I'd view TSLS as its own procedure, that can be viewed as a special case of GMM. Just because you can run OLS in a GLM doesn't make OLS:=GLM.... –  Ari B. Friedman Dec 14 '11 at 0:14
Historically yes. But treating TSLS as GMM procedure is very natural. See Wooldridge's Econometric Analysis of Cross Section and Panel Data, chapter 8, for example. I do not know for sure, but I think GMM was thought as a generalisation of TSLS, so including it into GMM would seem to be prudent. –  mpiktas Dec 14 '11 at 7:14
Like I said...semantics. :-) But +1 for a good answer. –  Ari B. Friedman Dec 14 '11 at 12:17

The implications of economic theories are often naturally formulated in terms of conditional moment restrictions (see e.g. the original asset pricing application of LP Hansen) which nest a variety of unconditional restrictions thus leading to overidentification. Rather than arbitrarily picking "which squares to minimize" to satisfy a subset of those restriction exactly using whatever-LS, GMM provides a way of efficiently combining all of them.

MLE requires a complete specification - all of the moments of all the random variables included in the model should be matched. If those additional restrictions are satisfied in the population, you are naturally getting a more efficient estimator, perhaps, with a better behaving objective function to be optimized.

In the context of simulation estimation, however, nonlinearity of likelihood functions introduces an additional source of bias, complicating the comparison with SMM.

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One partial answer seems to be that:

"In models for which there are more moment conditions than model parameters, GMM estimation provides a straightforward way to test the specification of the proposed model. This is an important feature that is unique to GMM estimation."

This seems like it would be important but insufficient to wholly explain the popularity of GMM in metrics.

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That's exactly right; I don't know why you think this is a partial answer. To supplement: suppose that 1 moment condition would be enough for identification of parameters, but theory provides a set of moment conditions, all of which are equally valid. In that case, rather than choosing one moment condition at random, it is intuitively more appealing to minimize some weighted average of deviations from each of the moment conditions. This is, roughly speaking, what the GMM estimator does. –  Zermelo Nov 30 '11 at 19:28
Ah, I just noticed that your question asks for more than just why GMM is used. –  Zermelo Nov 30 '11 at 19:29
@Zermelo: Precisely ;-) –  Ari B. Friedman Nov 30 '11 at 19:44

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