Robust inference in nonlinear models with mixed identification strength

The paper studies inference in regression models composed of nonlinear functions with unknown transformation parameters and loading coefficients that measure the importance of each component. In these models, non-identification and weak identification present in multiple parts of the parameter space, resulting in mixed identification strength for different unknown parameters. This paper proposes robust tests and confidence intervals for sub-vectors and linear functions of the unknown parameters. In particular, the results cover applications where some nuisance parameters are non-identified under the null (Davies (1977, 1987)) and some nuisance parameters are subject to a full range of identification strength. To construct this robust inference procedure, we develop a local limit theory that models mixed identification strength. The asymptotic results involve both inconsistent estimators that depend on a localization parameter and consistent estimators with different rates of convergence. A sequential argument is used to peel the criterion function based on identification strength of the parameters.