  We propose a bivariate quantile regression method for the bivariate varying
coefficient model through a directional approach. The varying coefficients are
approximated by the B-spline basis and an $L_{2}$ type penalty is imposed to
achieve desired smoothness. We develop a multistage estimation procedure based
the Propagation-Separation~(PS) approach to borrow information from nearby
directions. The PS method is capable of handling the computational complexity
raised by simultaneously considering multiple directions to efficiently
estimate varying coefficients while guaranteeing certain smoothness along
directions. We reformulate the optimization problem and solve it by the
Alternating Direction Method of Multipliers~(ADMM), which is implemented using
R while the core is written in C to speed it up. Simulation studies are
conducted to confirm the finite sample performance of our proposed method. A
real data on Diffusion Tensor Imaging~(DTI) properties from a clinical study on
neurodevelopment is analyzed.
