# What does “special case of GMM” mean?

I was researching purchasing this text book: http://www.amazon.com/dp/0691010188/ref=wl_it_dp_o_pd_nS_ttl?_encoding=UTF8&colid=2QTISO1Y8TYVW&coliid=I3FUEFWL47AC4L

In its description it talks about how the book teaches estimation techniques as special cases of GMM. This is not the first time I have heard that phrase before. What exactly does it mean?

For example, what does OLS as a special case of GMM mean? I was under the impression GMM was its own estimate technique.

For a weighting matrix $\hat W$, regressor matrix $Z$ (following the somewhat unusual notation used in the book by Hayashi you are referring to) and instrument matrix $X$, the GMM estimator to estimate $\delta$ in a linear model $y=Z\delta+\epsilon$ can be written as $$\widehat{\delta}(\widehat{W})=(Z'X\widehat{W}X'Z)^{-1}Z'X\widehat{W}X'y.$$ If you assume your regressors to be "valid" (aka exogenous, uncorrelated with the error term), the $Z$ may be a subset of the set of instruments $X$, or they may even coincide with $X$, $Z=X$.

Consider the latter case first. GMM becomes (notice the matrices in the inverse then are square so that the inverse can be written as the product of the inverses, in reverse order) \begin{align*} \widehat{\delta}(\widehat{W})&=(Z'Z\widehat{W}Z'Z)^{-1}Z'Z\widehat{W}Z'y\\ &=(Z'Z)^{-1}\widehat{W}^{-1}(Z'Z)^{-1}Z'Z\widehat{W}Z'y\\ &=(Z'Z)^{-1}Z'y, \end{align*} which is the OLS estimator.

For the first case, we get equality for the particular choice of weighting matrix (optimal under conditional homoskedasticity) $\hat W=(X'X)^{-1}$:

GMM then becomes \begin{align*} \widehat{\delta}((X'X)^{-1})&=(Z'X(X'X)^{-1}X'Z)^{-1}Z'X(X'X)^{-1}X'y\\ &:=(Z'P_XZ)^{-1}Z'P_Xy, \end{align*} known as two-stage least squares. Now, if $Z\subset X$, projecting $Z$ on $X$ via $P_XZ$ will simply give $Z$ again (you explain something by itself and other instruments, and the best way to explain a variable is by itself - you will get perfect fit). Hence, GMM will again reduce to OLS.

It's generalized method of moments, for which Hansen got Economic Nobel a couple of years ago. It's very popular in econometrics.

You can show that OLS estimates are the same as GMM estimates under certain conditions. It's similar to MLE and OLS being the same under certain conditions (normal errors).

• I edited my post to be more clear. I understand what GMM is but I am not sure what OLS as a special case of GMM means. – Michael May 18 '15 at 23:23
• @Michael, GMM will produce the same estimate as OLS – Aksakal May 18 '15 at 23:47