A note on centering in subsample selection for linear regression

TL;DR

This paper investigates the implications of centering in subsample selection for linear regression, demonstrating that fitting models without intercept on un-centered subsamples remains valid and can improve estimation efficiency.

Contribution

It proves the unbiasedness and variance reduction of slope estimators from un-centered subsamples and suggests using full data weighted means to enhance efficiency in weighted subsampling.

Findings

Un-centered subsamples yield unbiased slope estimates without intercept.

Variance of slope estimator is smaller when fitting without intercept.

Using full data weighted means improves efficiency in weighted subsampling.

Abstract

Centering is a commonly used technique in linear regression analysis. With centered data on both the responses and covariates, the ordinary least squares estimator of the slope parameter can be calculated from a model without the intercept. If a subsample is selected from a centered full data, the subsample is typically un-centered. In this case, is it still appropriate to fit a model without the intercept? The answer is yes, and we show that the least squares estimator on the slope parameter obtained from a model without the intercept is unbiased and it has a smaller variance covariance matrix in the Loewner order than that obtained from a model with the intercept. We further show that for noninformative weighted subsampling when a weighted least squares estimator is used, using the full data weighted means to relocate the subsample improves the estimation efficiency.