We consider varying coefficient models which are an extension of the clas-
sical linear regression models in the sense that the regression coefficients are replaced
by functions in certain variables (often time). Varying coefficient models have been
popular in longitudinal data and panel data studies, and have been applied in fields,
such as finance and health sciences. We estimate the coefficient functions by splines.
An important question in a varying coefficient model is whether a coefficient function
is monotone or convex. We develop consistent testing procedures for monotonicity
and convexity. Moreover, we provide procedures to test simultaneously the shapes of
certain coefficient functions in a varying coefficient model. The tests use constrained
and unconstrained regression splines. The performances of the proposed tests are
illustrated on simulated data. We also give a real data application.

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• Shape testing in varying coefficient models