Inference on functionals under first order degeneracy

This paper presents a unified framework for inference on parameters of the form ϕ(θ0), where θ0 is unknown but can be estimated by θˆn, and ϕ is known with null first order derivative at θ0. We show the “standard” bootstrap is consistent if and only if the second order derivative ϕθ0′′=0 under regularity conditions, thereby identifying a source of bootstrap failures distinct from that in Fang and Santos (2018). Two consistent bootstrap schemes are proposed: one based on Babu (1984) that applies to differentiable maps and the other one based on Fang and Santos (2018) that applies to (second order) nondifferentiable maps. As an illustration, we develop a test of existence of (potentially multiple) common conditional heteroskedasticity features that improves upon Dovonon and Renault (2013).