Inference on Directionally Differentiable Functions

By Zheng Fang, Andrés Santos. The Review of Economic Studies, 2014

210 citations recorded by the source corpus.

This paper studies an asymptotic framework for conducting inference on parameters of the form ( 0), where is a known directionally dierentiable function and 0 is estimated by ^ n. In these settings, the asymptotic distribution of the plug-in estimator ( ^ n) can be readily derived employing existing extensions to the Delta method. We show, however, that the \standard" bootstrap is only consistent under overly stringent conditions { in particular we establish that dierentiability of is a necessary and sucient condition for bootstrap consistency whenever the limiting distribution of ^ n is Gaussian. An alternative resampling scheme is proposed which remains consistent when the bootstrap fails, and is shown to provide local size control under restrictions on the directional derivative of . We illustrate the utility of our results by developing a test of whether a Hilbert space valued parameter belongs to a convex set { a setting that includes moment inequality problems and certain tests of shape restrictions as special cases.

Subfields: Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/f6347d39ab2814da711d0fb71863add2520027a7

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Citations in the corpus

Statistical Inference for Average Treatment Effects Estimated by Synthetic Control Methods (2019, methodology): “We use projection theory (Zarantonello, 1971; Fang and Santos, 2014) to derive the asymptotic distributions of the standard and the modified synthetic control ATE estimators with long panels.”
Regression Kink With an Unknown Threshold (2017, background): “Important contributions to this literature include Hinkley (1969, 1971) and Feder (1975).”
Delta-method inference for a class of set-identified SVARs (2018, methodology): “Fang and Santos (2015) show that the standard bootstrap is not consistent when applied to parameters of the form v(μ), where v is a directionally differentiable function.”
Using Instrumental Variables for Inference About Policy Relevant Treatment Effects (2018, unknown): “In particular, a naive “plug-in” bootstrap approach fails due to β? not being (fully) differentiable in the constraints; see Shapiro (1989) for the lack of differentiability and Fang and Santos (2014) for the implication of bootstrap failure.”
Inference on breakdown frontiers (2017, methodology): “Hence we can use the delta method for Hadamard directionally differentiable functions (see Theorem 2.1 in Fang and Santos (2019)) to find that[√ N ( φ1(θ̂)−φ1(θ0) )] (x w) [ φ′1 θ0(Z1) ] (x w) ≡ Z̃2(x w) This result holds uniformly over any finite grid of values for y ∈ R and c ∈ C by considering…”