All of Linear Regression

Understand

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.