Uniform Post Selection Inference for Z-estimation problems

In this talk we will consider inference with high dimensional data. We propose new methods for estimating and constructing confidence regions for a regression parameter of primary interest alpha_0, a parameter in front of the regressor of interest, such as the treatment variable or a policy variable. We show how to apply these methods to Z-estimators (for example, logistic regression and quantile regression). These methods allow to estimate alpha_0 at the root-n rate when the total number p of other regressors, called controls, exceed the sample size n, using the sparsity assumptions. The sparsity assumption means that only s unknown controls are needed to accurately approximate the nuisance part of the regression function, where s is smaller than n. Importantly, the estimators and these resulting confidence regions are “honest” in the formal sense that their properties hold uniformly over s-sparse models. Moreover, these procedures do not rely on traditional “consistent model selection” arguments for their validity; in fact, they are robust with respect to “moderate” model selection mistakes in variable selection steps. Furthermore, the estimators are semi-parametrically efficient in the sense of attaining the semi-parametric efficiency bounds for the class of models in this paper.