  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. $\log(1+Y)$, $\mathrm{arcsinh}(Y)$). 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 re-scaling 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 non-zero 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.
