Dealing with Logs and Zeros in Regression Models

Abstract

Log-linear models are prevalent in empirical research. Yet, how to handle zeros in the dependent variable remains an unsettled issue. This article clarifies it and addresses the ``log of zero'' by developing a new family of estimators called iterated Ordinary Least Squares (iOLS). This family nests standard approaches such as log-linear and Poisson regressions, offers several computational advantages, and corresponds to the correct way to perform the popular log(Y+1) transformation. We extend it to the endogenous regressor setting (i2SLS) and overcome other common issues with Poisson models, such as controlling for many fixed-effects. We also develop specification tests to help researchers select between alternative estimators. Finally, our methods are illustrated through numerical simulations and replications of landmark publications.

Keywords: Log(0), Log of zero, Log-log, Bias, Elasticity, PPML, Contraction mapping, Gravity model, Iterative estimator, Log-linear, Selection bias.

JEL Classification: C26, C52, C55