Unified equations for the slope, intercept, and standard errors of the best straight line

TLDR

Least‑squares and maximum‑likelihood estimation give the same slope and intercept for linear regression with normally distributed errors. The authors aim to derive all known regression solutions from the York equations and provide a compact set of equations for slope, intercept, and standard errors. They present a compact set of equations for slope, intercept, and unified standard errors. They show that least‑squares and maximum‑likelihood estimation yield identical standard errors for slope and intercept when evaluated at the least‑squares‑adjusted points, and this holds even when x and y errors are correlated and vary across points.

Abstract

It has long been recognized that the least-squares estimation method of fitting the best straight line to data points having normally distributed errors yields identical results for the slope and intercept of the line as does the method of maximum likelihood estimation. We show that, contrary to previous understanding, these two methods also give identical results for the standard errors in slope and intercept, provided that the least-squares estimation expressions are evaluated at the least-squares-adjusted points rather than at the observed points as has been done traditionally. This unification of standard errors holds when both x and y observations are subject to correlated errors that vary from point to point. All known correct regression solutions in the literature, including various special cases, can be derived from the original York equations. We present a compact set of equations for the slope, intercept, and newly unified standard errors.