Logs with zeros? Some problems and solutions

Many economic settings involve an outcome $Y$ that is weakly positive but can equal zero (e.g. earnings). In such settings, it is common to estimate an average treatment effect (ATE) for a transformation of the outcome that behaves like $\log(Y)$ when $Y$ is large but is defined at zero (e.g. $\log(1+Y)$, $\mathrm{arcsinh}(Y)$). This paper argues that ATEs for such log-like transformations should not be interpreted as approximating a percentage effect, since unlike a percentage, they depend arbitrarily on the units of the outcome when the treatment affects the extensive margin. Intuitively, this dependence arises because an individual-level percentage effect is not well-defined for individuals whose outcome changes from zero to non-zero when receiving treatment, and the units of the outcome implicitly determine how much weight the ATE places on such extensive margin changes. We further establish that 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 a variety of 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.