Working Papers
• Inference under First-Order Degeneracy
(with Xinyue Bei).
Abstract
We study inference in models where a transformation of parameters exhibits first-order degeneracy — that is, its gradient is zero or close to zero, making the standard delta method invalid. A leading example is causal mediation analysis, where the indirect effect is a product of coefficients and the gradient degenerates near the origin. In these local regions of degeneracy the limiting behaviors of plug-in estimators depend on nuisance parameters that are not consistently estimable. We show that this failure is intrinsic — around points of degeneracy, both regular and quantile-unbiased estimation are impossible. Despite these restrictions, we develop minimum-distance methods that deliver uniformly valid confidence intervals. We establish sufficient conditions under which standard chi-square critical values remain valid, and propose a simple bootstrap procedure when they are not. We demonstrate favorable power in simulations and in an empirical application linking teacher gender attitudes to student outcomes.
• An Identification and Dimensionality Robust Test for Instrumental Variables Models. Revise and Resubmit, Journal of Econometrics.
Abstract
Using modifications of Lindeberg’s interpolation technique, I propose a new identification-robust test for the structural parameter in a heteroskedastic instrumental variables model. While my analysis allows the number of instruments to be much larger than the sample size, it does not require many instruments, making my test applicable in settings that have not been well studied. Instead, the proposed test statistic has a limiting chi-squared distribution so long as an auxiliary parameter can be consistently estimated. This is possible using machine learning methods even when the number of instruments is much larger than the sample size. To improve power, a simple combination with the sup-score statistic of Belloni et al. (2012) is proposed. I point out that first-stage F-statistics calculated on LASSO selected variables may be misleading indicators of identification strength and demonstrate favorable performance of my proposed methods in both empirical data and simulation study.
• Identification and Estimation in a Class of Potential Outcomes Models
(with Rodrigo Pinto and Andres Santos). Revise and Resubmit, Econometrica.
Abstract
This paper develops a class of potential outcomes models characterized by three main features: (i) Unobserved heterogeneity can be represented by a vector of potential outcomes and a “type” describing the manner in which an instrument determines the choice of treatment; (ii) The availability of an instrumental variable that is conditionally independent of unobserved heterogeneity; and (iii) The imposition of convex restrictions on the distribution of unobserved heterogeneity. The proposed class of models encompasses multiple classical and novel research designs, yet possesses a common structure that permits a unifying analysis of identification and estimation. In particular, we establish that these models share a common necessary and sufficient condition for identifying certain causal parameters. Our identification results are constructive in that they yield estimating moment conditions for the parameters of interest. Focusing on a leading special case of our framework, we further show how these estimating moment conditions may be modified to be doubly robust. The corresponding double robust estimators are shown to be asymptotically normally distributed, bootstrap based inference is shown to be asymptotically valid, and the semi-parametric efficiency bound is derived for those parameters that are root-n estimable. We illustrate the usefulness of our results for developing, identifying, and estimating causal models through an empirical evaluation of the role of mental health as a mediating variable in the Moving To Opportunity experiment.