 Logs with Zeros? Some Problems and Solutions*Jiafeng Chen and Jonathan Roth The Quarterly Journal of Economics, 2024, vol. 139, issue 2, 891-936 Abstract: When studying an outcome Y that is weakly positive but can equal zero (e.g., earnings), researchers frequently estimate an average treatment effect (ATE) for a “log-like” transformation that behaves like log (Y) for large Y but is defined at zero (e.g., log (1 + Y), ). We argue that ATEs for log-like transformations should not be interpreted as approximating percentage effects, since unlike a percentage, they depend on the units of the outcome. In fact, we show that if the treatment affects the extensive margin, one can obtain a treatment effect of any magnitude simply by rescaling the units of Y before taking the log-like transformation. This arbitrary unit dependence arises because an individual-level percentage effect is not well-defined for individuals whose outcome changes from zero to nonzero when receiving treatment, and the units of the outcome implicitly determine how much weight the ATE for a log-like transformation places on the extensive margin. We further establish a trilemma: when the outcome can equal zero, there is no treatment effect parameter that is an average of individual-level treatment effects, unit invariant, and point identified. We discuss several alternative approaches that may be sensible in settings with an intensive and extensive margin, including (i) expressing the ATE in levels as a percentage (e.g., using Poisson regression), (ii) explicitly calibrating the value placed on the intensive and extensive margins, and (iii) estimating separate effects for the two margins (e.g., using Lee bounds). We illustrate these approaches in three empirical applications. Date: 2024 Downloads: (external link) Related works: Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text Persistent link: https://EconPapers.repec.org/RePEc:oup:qjecon:v:139:y:2024:i:2:p:891-936. Ordering information: This journal article can be ordered from Access Statistics for this article The Quarterly Journal of Economics is currently edited by Robert J. Barro, Lawrence F. Katz, Nathan Nunn, Andrei Shleifer and Stefanie Stantcheva More articles in The Quarterly Journal of Economics from President and Fellows of Harvard College |
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