Notes On Method-of-Moments Estimation

for some r-dimensional vector of functions m(zi, θ) of the observable data vector zi and possible parameter value θ in some parameter space Θ. Assuming that θ0 is the unique solution of this population moment equation (equivalent to identification when only (*) is imposed), a method-of-moments estimator θ̂ is defined as a solution (or near-solution) of a sample analogue to (*), replacing the population expectation by a sample average. Generally, for θ0 to uniquely solve (*), the number of components r of the moment function m(·) must be at least as large as the number of components p in θ — that is, r ≥ p, known as the “order condition” for identification. When θ0 is identified and r = p — termed “just identification” — a natural analogue of the population moment equation for θ0 defines the method-of-moment estimator as the solution to the p-dimensional sample moment equation