Inference on Directionally Differentiable Functions

This article studies an asymptotic framework for conducting inference on parameters of the form $\phi(\theta_0)$, where $\phi$ is a known directionally differentiable function and $\theta_0$ is estimated by $\hat \theta_n$. In these settings, the asymptotic distribution of the plug-in estimator $\phi(\hat \theta_n)$ can be derived employing existing extensions to the Delta method. We show, however, that (full) differentiability of $\phi$ is a necessary and sufficient condition for bootstrap consistency whenever the limiting distribution of $\hat \theta_n$ 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 $\phi$. 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.