  We propose a general framework for the specification testing of continuous
treatment effect models. We assume a general residual function, which includes
the average and quantile treatment effect models as special cases. The null
models are identified under the confoundedness condition and contain a
nonparametric weighting function. We propose a test statistic for the null
model in which the weighting function is estimated by solving an expanding set
of moment equations. We establish the asymptotic distributions of our test
statistic under the null hypothesis and under fixed and local alternatives. The
proposed test statistic is shown to be more efficient than that constructed
from the true weighting function and can detect local alternatives deviated
from the null models at the rate of $O_{P}(N^{-1/2})$. A simulation method is
provided to approximate the null distribution of the test statistic.
Monte-Carlo simulations show that our test exhibits a satisfactory
finite-sample performance, and an application shows its practical value.
