Semiparametric models and inference for the effect of a treatment when the outcome is nonnegative with clumping at zero

The outcome in a randomized experiment is sometimes nonnegative with a clump of observations at zero and continuously distributed positive values. One widely used model for a nonnegative outcome with a clump at zero is the Tobit model, which assumes that the treatment has a shift effect on the distribution of a normally distributed latent variable and the observed outcome is the maximum of the latent variable and zero. We develop a class of semiparametric models and inference procedures that extend the Tobit model in two useful directions. First, we consider more flexible models for the treatment effect than the shift effect of the Tobit model; for example, our models allow for the treatment to have a larger in magnitude effect for upper quantiles. Second, we make semiparametric inferences using empirical likelihood that allow the underlying latent variable to have any distribution, unlike the original Tobit model that assumes the latent variable is normally distributed. We apply our approach to data from the RAND Health Insurance Experiment. We also extend our approach to observational studies in which treatment assignment is strongly ignorable.