In parametric models a suﬀicient condition for local identiﬁcation is that the vector of moment conditions is diﬀerentiable at the true parameter with full rank derivative matrix. We show that there are corresponding suﬀicient conditions for nonparametric models. A nonparametric rank condition and diﬀerentiability of the moment conditions with respect to a certain norm imply local identiﬁcation. It turns out these conditions are slightly stronger than needed and are hard to check, so we provide weaker and more primitive conditions. We extend the results to semiparametric models. We illustrate the suﬀicient conditions with endogenous quantile and single index examples. We also consider a semiparametric habit-based, consumption capital asset pricing model. There we ﬁnd the rank condition is implied by an integral equation of the second kind having a one-dimensional null space.
Chen, Xiaohong; Chernozhukov, Victor; Lee, Sokbae; and Newey, Whitney, "Local Identification of Nonparametric and Semiparametric Models" (2011). Cowles Foundation Discussion Papers. 2138.