The generalized method of moments (GMM) estimates parameters by forcing selected sample moments to be as close as possible to their theoretical counterparts.
Core Mechanics
Let (\theta) be parameters and (g(z_t, \theta)) be moment conditions such that:
[ E[g(z_t, \theta_0)] = 0 ]
GMM chooses (\hat\theta) to minimize:
[ J(\theta) = \bar g(\theta)’ W \bar g(\theta) ]
where (\bar g(\theta)) is the sample mean of moments and (W) is a weighting matrix.
Why Economists Use GMM
- Works with heteroskedasticity under robust weighting.
- Handles over-identified models with more moments than parameters.
- Fits naturally in dynamic macro and finance models where full likelihood is hard to write.
Practical Interpretation
The over-identification test (often based on the minimized (J)-statistic) checks whether the full set of instruments and restrictions is jointly plausible.
Weak or invalid instruments can still bias results, so model design matters as much as computation.