Hypothesis testing of varying coefficients for regional quantiles

Testing the behavior of varying coefficients (VC) over a range of quantiles is important in the field of regression analysis. This study tests whether coefficient functions in varying quantile regression share common structural information across a certain range of quantile levels, even when linear combinations of covariates are unspecified in the null hypothesis. Our approach allows varying the coefficients, β(τ,t), as a function of the quantile level, τ∈Δ, and a random variable, t∈T, where Δ is the quantile region of interest and T is a domain of t. By incorporating an interval of quantiles in the inference, the proposed method can test whether the model can be justified by using a VC quantile model against other important reduced models, such as the linear quantile model, varying coefficient linear model, or linear model with homogeneous errors. We use bivariate B-splines to approximate the varying quantile functions, β(τ,t), and utilize composite quantile regression to estimate the parameters. Furthermore, we develop constrained composite quantile regression to provide a more efficient estimate in case the null hypothesis is not rejected. We show that the proposed test admits normal approximations. Using simulation and real data analysis, we demonstrate the superiority of the proposed test over other tests designed for a finite number of quantile levels.