The Frisch--Waugh--Lovell Theorem for Standard Errors

TL;DR

This paper extends the Frisch--Waugh--Lovell Theorem to standard errors, demonstrating their equivalence in regression analysis and highlighting differences in standard error estimates in stratified experiments.

Contribution

It introduces a new result showing the equivalence of various standard errors, enhancing understanding of error estimation in regression models.

Findings

Standard errors are equivalent under certain conditions.

Discrepancies exist between model-based and design-based standard errors in stratified experiments.

Theoretical extension of the Frisch--Waugh--Lovell Theorem to standard errors.

Abstract

The Frisch--Waugh--Lovell Theorem states the equivalence of the coefficients from the full and partial regressions. I further show the equivalence between various standard errors. Applying the new result to stratified experiments reveals the discrepancy between model-based and design-based standard errors.