of Extremum Estimators
EXTREMUM ESTIMATORS ARE estimators obtained by either maximizing or minimizing a criterion function over the admissible parameter space. In this chapter we introduce more formally the concept of an extremum estimator and discuss the large-sample properties of these estimators.1 After briefly setting up notation and describing the probability environment within which we discuss estimation, we describe regularity conditions under which an estimator converges almost surely to its population counterpart.
We then turn to the large-sample distributions of extremum estimators. Throughout this discussion we maintain the assumption that θT is a consistent estimator of θ0 and focus on properties of the distribution of θT as T gets large. Whereas discussions of consistency are often criterion-function specific, the large-sample analyses of most of the extremum estimators we will use subsequently can be treated concurrently. We formally define a family of estimators that encompasses the first-order conditions of the ML, standard GMM, and LLS estimators as special cases. Then, after we present a quite general central limit theorem, we establish the asymptotic normality of these estimators. Finally, we examine the relative asymptotic efficiencies of the GMM, LSS, and ML estimators and interpret their asymptotic efficiencies in terms of the restrictions on the joint distribution of the data used in estimation.
Notationally, we let Ω denote the sample space, F the set of events about which we want to make probability statements (a “σ-algebra” of events), and
1 The perspective on the large-sample properties of extremum estimators taken in this chapter has been shaped by my discussions and collaborations with Lars Hansen over the past 25 years. In particular, the approach to establishing consistency and asymptotic normality in Sections 3.2–3.4 follows that of Hansen (1982b, 2005).