# Inference on Directionally Differentiable Functions

Zheng Fang and Andres Santos

Review of Economic Studies, 2019, vol. 86, issue 1, 377-412

Abstract: 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.

Keywords: Delta method; Bootstrap consistency; Directional differentiability; Shape restrictions; Residual wage inequality; C1; C12; C15; J31 (search for similar items in EconPapers)
Date: 2019
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http://hdl.handle.net/10.1093/restud/rdy049 (application/pdf)

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