Inference for multi-valued heterogeneous treatment effects when the number of treated units is small

Abstract: We propose a method for conducting asymptotically valid inference for treatment effects in a multi-valued treatment framework where the number of units in the treatment arms can be small and do not grow with the sample size. We accomplish this by casting the model as a semi-/non-parametric conditional quantile model and using known finite sample results about the law of the indicator function that defines the conditional quantile. Our framework allows for structural functions that are non-additively separable, with flexible functional forms and heteroskedasticy in the residuals, and it also encompasses commonly used designs like difference in difference. We study the finite sample behavior of our test in a Monte Carlo study and we also apply our results to assessing the effect of weather events on GDP growth.