On Least Squares Linear Regression Without Second Moment

TL;DR

This paper demonstrates that least squares linear regression parameters can be identified without requiring the second moment of the response variable, under certain conditions involving conditional expectation and covariance.

Contribution

It establishes that regression parameters are recoverable without finite second moments, extending classical results to broader settings.

Findings

Regression parameters equal their classical estimates without second moments

Conditional expectation being affine implies mean independence

Zero covariance leads to mean independence

Abstract

If X and Y are real valued random variables such that the first moments of X, Y, and XY exist and the conditional expectation of Y given X is an affine function of X, then the intercept and slope of the conditional expectation equal the intercept and slope of the least squares linear regression function, even though Y may not have a finite second moment. As a consequence, the affine in X form of the conditional expectation and zero covariance imply mean independence.