Why Transform Y? A Critical Assessment of Dependent-Variable Transformations in Regression Models for Skewed and Sometimes-Zero Outcomes

Dependent variables that are non-negative, follow right-skewed distributions, and have large probability mass at zero arise often in empirical economics. Two classes of models that transform the dependent variable y — the natural logarithm of y plus a constant and the inverse hyperbolic sine — have been widely used in empirical work. We show that these two classes of models share several features that raise concerns about their application. The concerns are particularly prominent when dependent variables are frequently observed at zero, which in many instances is the main motivation for using them in the first place. The crux of the concern is that these models have an extra parameter that is generally not determined by theory but whose values have enormous consequences for point estimates. As these parameters go to extreme values estimated marginal effects on outcomes' natural scales approach those of either an untransformed linear regression or a normed linear probability model. Across a wide variety of simulated data, two-part models yield correct marginal effects, as do OLS on the untransformed y and Poisson regression. If researchers care about estimating marginal effects, we recommend using these simpler models that do not rely on transformations.