Shape testing in quantile varying coefficient models with heteroscedastic error

The interest is in regression quantiles in varying coefficient models for analysing longitudinal data. The coefficients are allowed to vary with time, and the error variance (the variability function) varies with the covariates to allow for heteroscedasticity. The functional coefficients are estimated using penalized splines (P-splines), not requiring specification of the error distribution. A likelihood-ratio-type test is considered to test the shape (constancy, monotonicity and/or convexity) of the functional coefficients. Further, testing procedures based on $ L_1 $ L1-norm, $ L_2 $ L2-norm and $ L_\infty $ L∞-norm of the differences of the P-splines coefficients are considered to test for constant functional coefficients. These norm-based tests perform better than the likelihood-ratio-type test in our simulation study. An extreme value test for testing monotonicity or convexity also performs better than the likelihood-ratio-type test. The likelihood-ratio-type test is, however, useful when testing the shape of the coefficients in signal and in variability function simultaneously. A real-data example demonstrates the testing procedures.