Information Theoretic Approaches to Inference in Moment Condition Models

Gel Criteria for Moment Condition Models

“…Moreover, the ET estimator is identical to that based on the criterion − log ³ T −1 P T t=1 exp(kλ 0 g tT (β))´since it is a monotonic transformation of the ET criterion. More generally, members of the Cressie-Read (1984) power divergence family of discrepancies discussed by Imbens, Spady, and Johnson (1998) are included in the GEL class with ρ(v) = −γ 2 (1 + v) (γ+1)/γ /(γ + 1). In this case, the GEL optimisation problem is a dual of that arising from the Cressie-Read (1984) family.…”

“…The EL estimator is a GEL estimator with ρ(v) = log(1 − v), see Imbens (1997), Qin andLawless (1994) andSmith (2000), and the ET estimator of Imbens, Spady, and Johnson (1998) is also GEL with ρ(v) = − exp(v). The CUE of Hansen, Heaton and Yaron (1996) is obtained if ρ(·) is quadratic; see Theorem 2.1, p.223, of NS.…”

“…Hansen, Heaton, and Yaron (1996) suggested the continuous updating estimator (CUE). Other estimators include empirical likelihood (EL) [Imbens (1997), Qin and Lawless (1994)], and exponential tilting (ET) [Imbens, Spady and Johnson (1998), 1 Let T denote the sample size. The relevant optimality concept for estimation throughout this paper is that of minimum asymptotic variance among root-T consistent and asymptotic normal GMM estimators based on a given set of unconditional moment restrictions.…”

Efficient information theoretic inference for conditional moment restrictions

“…Imbens (1997) and Qin and Lawless (1994), and a local exponential tilting (ET) criterion P n i=1 P n j=1 π ij log(π ij /w ij ) obtains if γ = 0, cf. Imbens, Spady and Johnson (1998) and Kitamura and Stutzer (1997). Similarly to the unconditional context, in comparison to EL, ET substitutes the weight π ij for the unrestricted weight w ij .…”

Gel Methods for Nonsmooth Moment Indicators

“…The GEL estimator is then defined aŝ β = arg min β∈B sup λ∈Λn(β)P n (β, λ) , (2.4) whereΛ n (β) = {λ : λ 0 g i (β) ∈ V, i = 1, ..., n}; see NS andSmith (1997, 2001). EL and ET estimators are obtained with ρ(v) = log(1 − v) and V = (−∞, 1) [Qin and Lawless (1994), Smith (1997)] and ρ(·) = − exp(v) [Kitamura and Stutzer, 1997, Imbens et al, 1998, Smith, 1997 whereas CUEβ CUE = arg min β∈Bĝ (β) 0Ω (β) −1ĝ (β) [Pakes andPollard, 1989, Hansen et al, 1996] is a GEL estimator when ρ(·) is quadratic [NS,Theorem 2.1,p.223]. Moreover, MD estimators [Corcoran, 1998] are GEL if the discrepancy function belongs to the Cressie and Read (1984) family [NS,Theorem 2.2,p.224].…”