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), $\operatorname{arcsinh}(Y)$). 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 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.

中文翻译：

---

带零的日志？一些问题及解决方案


当研究弱正但可能为零的结果 Y（例如收入）时，研究人员经常估计“类对数”转换的平均治疗效果 (ATE)，该转换的行为类似于大 Y 的 log (Y)，但定义为零（例如 log (1 + Y), $\operatorname{arcsinh}(Y)$）。我们认为，类对数转换的 ATE 不应被解释为近似百分比效应，因为与百分比不同，它们取决于结果的单位。事实上，我们表明，如果处理影响扩展边缘，只需在进行类似对数变换之前重新调整 Y 的单位即可获得任何大小的处理效果。出现这种任意单位依赖性的原因是，对于接受治疗时结果从零变为非零的个体来说，个体水平的百分比效应没有明确定义，并且结果的单位隐含地决定了类似对数转换的 ATE 的权重位于广泛边缘上。我们进一步建立了一个三难困境：当结果可以为零时，不存在作为个体水平治疗效果、单位不变性和识别点的平均值的治疗效果参数。我们讨边际，以及 (iii) 估计两个边际的单独影响（例如使用 Lee 界限）。我们通过三个实证应用来说明这些方法。