Log-like? ATEs defined with zero outcomes are (arbitrarily) scale-dependent

Researchers frequently estimate the average treatment effect (ATE) in logs, which has the desirable property that its units approximate percentages. When the outcome takes on zero values, researchers often use alternative transformations (e.g., $\log(1+Y)$, $\mathrm{arcsinh}(Y)$) that behave like $\log(Y)$ for large values of $Y$, and interpret the units as percentages. In this paper, we show that ATEs for transformations other than $\log(Y)$ cannot be interpreted as percentages, at least if one imposes the seemingly reasonable requirement that a percentage does not depend on the original scaling of the outcome (e.g. dollars versus cents). We first show that if $m(y)$ is a function that behaves like $\log(y)$ for large values of $y$ and the treatment affects the probability that $Y=0$, then the ATE for $m(Y)$ can be made arbitrarily large or small in magnitude by re-scaling the units of $Y$. Moreover, we show that any parameter of the form $\theta_g = E[ g(Y(1),Y(0)) ]$ is necessarily scale dependent if it is point-identified and defined with zero-valued outcomes. We conclude by outlining a variety of options available to empirical researchers dealing with zero-valued outcomes, including (i) estimating ATEs for normalized outcomes, (ii) explicitly calibrating the value placed on the extensive versus intensive margins, or (iii) estimating separate effects for the intensive and extensive margins.