Irregular Identification, Support Conditions, and Inverse Weight Estimation
揭示了加权矩条件模型中识别与可估性之间的微妙联系，指出若识别需要权重取任意大值，则参数虽可识别但无法以常规参数速率估计，收敛速度可能慢至n^{-1/4}，并分析了二元响应模型和处理效应模型中的具体表现。
In weighted moment condition models, we show a subtle link between identification and estimability that limits the practical usefulness of estimators based on these models. In particular, if it is necessary for (point) identification that the weights take arbitrarily large values, then the parameter of interest, though point identified, cannot be estimated at the regular (parametric) rate and is said to be irregularly identified. This rate depends on relative tail conditions and can be as slow in some examples as n−1/4. This nonstandard rate of convergence can lead to numerical instability and/or large standard errors. We examine two weighted model examples: (i) the binary response model under mean restriction introduced by Lewbel (1997) and further generalized to cover endogeneity and selection, where the estimator in this class of models is weighted by the density of a special regressor, and (ii) the treatment effect model under exogenous selection (Rosenbaum and Rubin (1983)), where the resulting estimator of the average treatment effect is one that is weighted by a variant of the propensity score. Without strong relative support conditions, these models, similar to well known "identified at infinity" models, lead to estimators that converge at slower than parametric rate, since essentially, to ensure point identification, one requires some variables to take values on sets with arbitrarily small probabilities, or thin sets. For the two models above, we derive some rates of convergence and propose that one conducts inference using rate adaptive procedures that are analogous to Andrews and Schafgans (1998) for the sample selection model.