## Minimum Distance Estimation

Probably the best source on “asymptotic theory” of extremum estimators is this survey by Newey and McFadden. I, for one, have learnt much of what I know from this source. However, there are a couple of things in this paper that I don’t get. Maybe one of readers can enlighten me.

Let me start with a couple of points needed for the discussion: First, 4 assumptions are often required for consistency of an extremum estimator: compactness, continuity, identification, and uniform convergence. Second, there is a general estimation method called the minimum distance (MD) which has two special cases: the generalised method of moments (GMM) and the classical minimum distance (CMD). According to the paper, “This framework (i.e. MD) is useful for analyzing asymptotic normality of GMM and CMD …” but apparently not for consistency because “the first-order condition can have multiple roots even when the objective function has a unique maximum”. No consistency theorem is provided for the general MD.

What I don’t understand is the difference between MD and GMM in terms of identification and the consistency proof. If one assumes identification for MD [i.e. $plim \ g_N(\theta)=0$ has a unique solution $\theta_0$] as the paper make such an assumption for GMM, then one should be able to use a proof very similar to Lemma 2.3 to show that limiting MD quadratic objective function has a unique minimum at $\theta_0$ and the consistency should follow under the other assumptions. Am I missing something here?

The second thing I don’t get is that they provide conditions and proofs for GMM in details but leave everything for CMD as an exercise for the reader. I think CMD is important enough to deserve statement of assumptions and proofs especially because there is no other modern source providing them. There are important things requiring clarification e.g.  I think for CMD, the often difficult to prove uniform convergence, is automatically satisfied but I would have liked to actually see this (if true) in a major reference.

In general, why don’t econometric books start with the more general minimum distance estimation and then go to the important special case of GMM?