  The objective of the present paper is to develop a minimax theory for the
varying coefficient model in a non-asymptotic setting.
  We consider a high-dimensional sparse varying coefficient model where only
few of the covariates are present and only some of those covariates are time
dependent. Our analysis allows the time dependent covariates to have different
degrees of smoothness and to be spatially inhomogeneous. We develop the minimax
lower bounds for the quadratic risk and construct an adaptive estimator which
attains those lower bounds within a constant (if all time-dependent covariates
are spatially homogeneous) or logarithmic factor of the number of observations.
