[Paper Review] Dealing with Logs and Zeros in Regression Models

David Benatia, Christophe Bellégo+1 more|arXiv (Cornell University)|Mar 22, 2022

Statistical Methods and Inference46 citations

TL;DR

The paper introduces a new family of estimators, iOLS (and i2SLS for endogenous settings), to handle zeros in dependent variables in log-linear models, unifying log(Y+1) and Poisson approaches, with an accompanying model selection test.

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.

Motivation & Objective

• Clarify the log of zero problem in log-linear and log-log regressions and review existing approaches and their limitations.
• Propose a flexible, iterative OLS-based estimator family (iOLS) that nests log-linear and Poisson models.
• Develop an endogenous-extension (i2SLS) and provide a model selection procedure to choose among moment conditions.
• Show theoretical properties (consistency, asymptotic normality) and practical performance via simulations and replications.

Proposed method

• Introduce a continuum of models via a hyper-parameter delta that weights Y and exp(X'beta) inside a log transform.
• Define a transformed dependent variable tilde{Y}_i(beta, delta) = log(Y_i + delta exp(X_i'beta)) - c(beta, delta) and estimate beta via a fixed-point iteration (iOLS).
• Iteratively update beta by solving beta_{t+1} = [X'X]^{-1} X' tilde{Y}(beta_t), starting from an initial beta_0.
• Show that special cases converge to log-linear (delta -> 0) and Poisson (delta -> infinity) models.
• Provide identification under exogeneity assumptions and derive the asymptotic distribution with robust variance.

Experimental results

Research questions

• RQ1 How should zeros in the dependent variable be treated in log-linear/log-log models to obtain consistent estimates?
• RQ2 Can we create a unified, computation-friendly estimator that nests log-linear and Poisson models and handles many fixed effects?
• RQ3 How can we perform model selection among different moment conditions induced by the delta parameter?
• RQ4 What are the asymptotic properties (consistency, normality) of the iOLS estimators in both exogenous and endogenous settings?

Key findings

• The iOLS family provides a consistent, asymptotically normal estimator that nests log-linear and Poisson models as special cases.
• The delta parameter creates a continuum of models, with delta -> 0 recovering log-linear and delta -> infinity yielding a multiplicative Poisson framework.
• The estimator is computationally fast, only requiring a single X'X inversion and iterative updates of beta.
• An endogenous extension (i2SLS) is developed to handle IV contexts with many fixed effects.
• Specification tests are proposed to assess external validity of models against observed zeros.