What does it mean that GMM is robust to distributional assumptions?
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It is a fancy, confusing and indirect way to say that GMM, being a Method-of-Moments estimator, does not require distributional assumptions in order to estimate regression coefficients, just like Ordinary Least Squares, and in contrast to maximum likelihood that needs to make distributional assumptions.
The wording is confusing, because "robust to distributional assumptions" means that we get consistent estimates even if we have made the wrong distributional assumptions.