Inference on Directionally Differentiable Functions

This article studies an asymptotic framework for conducting inference on parameters of the form φ(θ₀), where φ is a known directionally differentiable function and θ₀ is estimated by θₙ. In these settings, the asymptotic distribution of the plug-in estimator φ( θₙ) can be derived employing existing extensions to the Delta method. We show, however, that (full) differentiability of φ is a necessary and sufficient condition for bootstrap consistency whenever the limiting distribution of θₙ is Gaussian. An alternative resampling scheme is proposed that remains consistent when the bootstrap fails, and is shown to provide local size control under restrictions on the directional derivative of φ. These results enable us to reduce potentially challenging statistical problems to simple analytical calculations—a feature we illustrate by developing a test of whether an identified parameter belongs to a convex set. We highlight the empirical relevance of our results by conducting inference on the qualitative features of trends in (residual) wage inequality in the U.S.