All of Linear Regression

TL;DR

This paper provides a unified, finite-sample analysis of ordinary least squares (OLS) regression, addressing convergence, dependence, variable selection, and inference in high-dimensional and potentially dependent data settings.

Contribution

It introduces a simple deterministic inequality that underpins finite-sample results for OLS under various conditions, including high-dimensional and dependent data, and studies inference after variable selection.

Findings

Finite-sample bounds for OLS convergence.

Analysis of inference after variable selection with increasing covariates.

First study of inference post-variable selection in high-dimensional settings.

Abstract

Least squares linear regression is one of the oldest and widely used data analysis tools. Although the theoretical analysis of the ordinary least squares (OLS) estimator is as old, several fundamental questions are yet to be answered. Suppose regression observations $(X_1,Y_1),\ldots,(X_n,Y_n)\in\mathbb{R}^d\times\mathbb{R}$ (not necessarily independent) are available. Some of the questions we deal with are as follows: under what conditions, does the OLS estimator converge and what is the limit? What happens if the dimension is allowed to grow with $n$? What happens if the observations are dependent with dependence possibly strengthening with $n$? How to do statistical inference under these kinds of misspecification? What happens to the OLS estimator under variable selection? How to do inference under misspecification and variable selection? We answer all the questions raised above with one simple deterministic inequality which holds for any set of observations and any sample size. This implies that all our results are a finite sample (non-asymptotic) in nature. In the end, one only needs to bound certain random quantities under specific settings of interest to get concrete rates and we derive these bounds for the case of independent observations. In particular, the problem of inference after variable selection is studied, for the first time, when $d$, the number of covariates increases (almost exponentially) with sample size $n$. We provide comments on the ``right'' statistic to consider for inference under variable selection and efficient computation of quantiles.