GMM with Many Weak Moment Conditions and Nuisance Parameters: General Theory and Applications

Music City Center 

Weak identification is a common issue for many statistical problems. In instrumental variable literature, weak instruments problem arises when the instruments are weakly correlated with treatment; in proximal causal inference literature, weak proxies issue can also arise when the proxies are weakly correlated with unmeasured confounders. Under weak-identification, standard estimation methods, such as Generalized method of moments (GMM), have poor finite sample performance. In this paper, we studied the estimation and inference problem with many weak moment conditions with Neyman orthogonality. We developed a two-step continuous updating estimator which allows first step possibly non-parametric estimation for nuisance parameter, and proved its consistency and asymptotic normality. Our theory is applicable even when there are high-dimensional moment conditions and high-dimensional nuisance parameters. We applied our general theory to study estimation and inference for weak instruments and weak proxies problems.

Causal Inferece

Instrumental variables

proximal causal inference

Generalized method of moments

weak identification 

Section on Nonparametric Statistics