Post-Selection Inference for Generalized Linear Models with Many Controls

Abstract

This paper considers generalized linear models in the presence of many controls. We lay out a general methodology to estimate an effect of interest based on the construction of an instrument that immunize against model selection mistakes and apply it to the case of logistic binary choice model. More specifically 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. These methods allow to estimate ${\alpha }_0$ at the root-$n$ rate when the total number $p$ of other regressors, called controls, potentially exceed the sample size $n$ using sparsity assumptions. The sparsity assumption means that there is a subset of $s<n$ controls which suffices to accurately approximate the nuisance part of the regression function. Importantly, the estimators and these resulting confidence regions are valid uniformly over $s$-sparse models satisfying $s^2{\log }^{2}p=o(n)$ and other technical conditions. 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. Under suitable conditions, the estimators are semi-parametrically efficient in the sense of attaining the semi-parametric efficiency bounds for the class of models in this paper.

References in corpus (3)

• High-dimensional Ising model selection using ${\ell }_1$-regularized logistic regression
• Sparse Estimators and the Oracle Property, or the Return of Hodges' Estimator
• On the Distribution of Penalized Maximum Likelihood Estimators: The LASSO, SCAD, and Thresholding

Cited by in corpus (9)

• Robust Inference on Average Treatment Effects with Possibly More Covariates than Observations
• Valid Post-Selection and Post-Regularization Inference: An Elementary, General Approach
• High Dimensional Expectation-Maximization Algorithm: Statistical Optimization and Asymptotic Normality
• A General Theory of Hypothesis Tests and Confidence Regions for Sparse High Dimensional Models
• A General Framework for Robust Testing and Confidence Regions in High-Dimensional Quantile Regression
• On Semiparametric Exponential Family Graphical Models
• Uniform Inference for High-dimensional Quantile Regression: Linear Functionals and Regression Rank Scores
• A Likelihood Ratio Framework for High Dimensional Semiparametric Regression
• Supplementary Appendix for "Inference on Treatment Effects After Selection Amongst High-Dimensional Controls"