Valid Post-Selection and Post-Regularization Inference: An Elementary, General Approach

Here we present an expository, general analysis of valid post-selection or post-regularization inference about a low-dimensional target parameter, α, in the presence of a very high-dimensional nuisance parameter, η, which is estimated using modern selection or regularization methods. Our analysis relies on high-level, easy-to-interpret conditions that allow one to clearly see the structures needed for achieving valid post-regularization inference. Simple, readily verifiable sufficient conditions are provided for a class of affine-quadratic models. We rely on asymptotic statements which dramatically simplifies theoretical statements and helps highlight the structure of the problem. We focus our discussion on estimation and inference procedures based on using the empirical analog of theoretical equations