Dealing with Logs and Zeros in Regression Models

TL;DR

This paper introduces the iterated OLS (iOLS) estimator to effectively handle zeros in log transformations within regression models, maintaining interpretability and addressing fundamental limitations of traditional methods.

Contribution

The paper proposes iOLS, a new estimator that consistently handles zeros in log transformations, extending to endogenous regressors and improving upon existing ad-hoc solutions.

Findings

iOLS provides consistent, asymptotically normal estimates.

The method is computationally efficient and globally convergent.

It extends naturally to endogenous regressors using iterated 2SLS.

Abstract

The log transformation is widely used in linear regression, mainly because coefficients are interpretable as proportional effects. Yet this practice has fundamental limitations, most notably that the log is undefined at zero, creating an identification problem. We propose a new estimator, iterated OLS (iOLS), which targets the normalized average treatment effect, preserving the percentage-change interpretation while addressing these limitations. Our procedure is the theoretically justified analogue of the ad-hoc log(1+Y) transformation and delivers a consistent and asymptotically normal estimator of the parameters of the exponential conditional mean model. iOLS is computationally efficient, globally convergent, and free of the incidental-parameter bias, while extending naturally to endogenous regressors through iterated 2SLS. We illustrate the methods with simulations and revisit three influential publications.