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Inference on Functionals under First Order Degeneracy

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Document pages: 75 pages

Abstract: This paper presents a unified second order asymptotic framework forconducting inference on parameters of the form $ phi( theta 0)$, where$ theta 0$ is unknown but can be estimated by $ hat theta n$, and $ phi$ is aknown map that admits null first order derivative at $ theta 0$. For a largenumber of examples in the literature, the second order Delta method reveals anondegenerate weak limit for the plug-in estimator $ phi( hat theta n)$. Weshow, however, that the `standard bootstrap is consistent if and only if thesecond order derivative $ phi { theta 0} =0$ under regularity conditions,i.e., the standard bootstrap is inconsistent if $ phi { theta 0} neq 0$, andprovides degenerate limits unhelpful for inference otherwise. We thus identifya source of bootstrap failures distinct from that in Fang and Santos (2018)because the problem (of consistently bootstrapping a textit{nondegenerate}limit) persists even if $ phi$ is differentiable. We show that the correctionprocedure in Babu (1984) can be extended to our general setup. Alternatively, amodified bootstrap is proposed when the map is textit{in addition} secondorder nondifferentiable. Both are shown to provide local size control undersome conditions. As an illustration, we develop a test of common conditionalheteroskedastic (CH) features, a setting with both degeneracy andnondifferentiability -- the latter is because the Jacobian matrix is degenerateat zero and we allow the existence of multiple common CH features.

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