Two moment conditions, one parameter

Question

For a Poisson(\lambda), both the mean and the variance equal \lambda. That gives you two moment conditions but only one parameter to estimate. Explain what "more conditions than parameters" means, why it is useful rather than a problem, and how the generalized method of moments resolves the surplus.

Accepted Answer

Overidentification. When the number of moment conditions exceeds the number of parameters, the model is called overidentified. Here two facts, "the mean is \lambda" and "the variance is \lambda", both constrain a single unknown. In a finite sample the sample mean and the sample variance will not be exactly equal, so the two conditions cannot both be satisfied at once. You have a surplus of information and no way to honor all of it perfectly. Why a surplus is good news, not bad. Extra moment conditions are extra evidence. Rather than forcing you to pick one, they let you do two valuable things…

Two moment conditions, one parameter

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