Before revision:  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. Our tests are nonparametric in nature, and use sieve estimators of the characteristic function. We provide formal power analysis against global as well as against local alternatives. Moreover, we perform a Monte Carlo simulation study , and apply the tests to analyze the degree of nonlinearity in a heterogeneous random coefficients demand model. While we find some evidence against the popular QUAIDS specification with random coefficients, it is not strong enough to reject the specification at the conventional significance level .
After revision:  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 multiplier bootstrap procedure is proposed to construct uniform confidence bands. A Monte Carlo study examines finite sample properties of the estimator and shows that it performs well even when the regressors associated to RC are far from being heavy tailed. Finally, the methodology is applied to analyze heterogeneity in income elasticity of demand for housing .