An Efficient Nonparametric Estimator for Models with Nonlinear Dependence

We provide a convenient econometric framework for the analysis of nonlinear dependence in financial applications. We introduce models with constrained nonparametric dependence, which specify the conditional distribution or the copula in terms of a one-dimensional functional parameter. Our approach is intermediate between standard parametric specifications (which are in general too restrictive) and the fully unrestricted approach (which suffers from the curse of dimensionality). We introduce a nonparametric estimator defined by minimizing a chi-square distance between the constrained densities in the family and an unconstrained kernel estimator of the density. We derive the nonparametric efficiency bound for linear forms and show that the minimum chi-square estimator is nonparametrically efficient for linear forms.