Generalized method of moments

Generalized method of moments (GMM) is an estimation framework that generalizes the classical method of moments. A moment condition is an expectation that equals zero at the true parameter values. Under certain theoretical moment conditions, parameters can be estimated by forcing their sample analogues as close to zero as possible. While GMM yields consistent, asymptotically normal estimates, it is flexible as it does not rely on a specific probability distribution for the data, contrary to other estimation methods such as maximum likelihood.

Relation to OLS and IV

Ordinary least square (OLS) and instrumental variable (IV) estimators can be seen as special cases of GMM. In an OLS regression $y_i=\beta x_i+{ϵ}_i$, the assumption $\mathbb{𝔼}(x_i{ϵ}_i)=0$ is a moment condition. The OLS estimator can be derived by setting the sample analogue:

$\frac{1}{N}\sum _i{x}_i(y_i-\hat{\beta }{x}_i)=0$