# Inference on Functionals Under First Order Degeneracy

75 Pages Posted: 21 Sep 2015 Last revised: 25 Jan 2019

See all articles by Qihui Chen

## Qihui Chen

The Chinese University of Hong Kong, Shenzhen

## Zheng Fang

Texas A&M University - Department of Economics

Date Written: January 9, 2018

### 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

Suggested Citation

Chen, Qihui and Fang, Zheng, Inference on Functionals Under First Order Degeneracy (January 9, 2018). Available at SSRN: https://ssrn.com/abstract=2662758 or http://dx.doi.org/10.2139/ssrn.2662758