Before revision: In this paper , we suggest and analyze a new class of specification tests for random coefficient models. These tests allow to assess the validity of central structural features of the model , in particular linearity in coefficients , generalizations of this notion like a known nonlinear functional relationship, or degeneracy of the distribution of a random coefficient, i. e. , whether a coefficient is fixed or random, including whether an associated variable can be omitted altogether.
After revision: This paper provides a new methodology to analyze unobserved heterogeneity when observed characteristics are modeled nonlinearly. The proposed model builds on varying random coefficients (VRC) that are determined by nonlinear functions of observed regressors and additively separable unobservables. This paper proposes a novel estimator of the VRC density based on weighted sieve minimum distance. The main example of sieve bases are Hermite functions which yield a numerically stable estimation procedure. This paper shows inference results that go beyond what has been shown in ordinary RC models. We provide in each case rates of convergence and also establish pointwise limit theory of linear functionals, where a prominent example is the density of potential outcomes. In addition, a random coefficient, i. e. , whether a coefficient is fixed or random, including whether an associated variable can be omitted altogether.