Inference on Functionals Under First Order Degeneracy

Abstract

This paper presents a unified framework for 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 known with null first order derivative at $\theta_0$. We show the ``standard'' bootstrap is consistent if and only if the second order derivative $\phi_{\theta_0}''=0$ 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 heterskedasticity features that improves upon Dovonon and Renault (2013).

Keywords: First order degeneracy, Second order Delta method, Bootstrap consistency, Babu correction, Common CH features, J-test