Inference on Directionally Differentiable Functions

This paper 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 θˆn. In these settings, the asymptotic distribution of the plug-in estimator φ(θˆn) can be derived employing existing extensions to the Delta method. We show, however, that (full) differentiability of φ is a necessary and suf- ficient condition for bootstrap consistency whenever the limiting distribution of θˆn is Gaussian. An alternative resampling scheme is proposed that remains consistent when the bootstrap fails, and is shown to provide local size control under restric- tions on the directional derivative of φ. These results enable us to reduce potentially challenging statistical problems to simple analytical calculations – a feature we il- lustrate 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 United States.

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