  In this paper an autoregressive time series model with conditional
heteroscedasticity is considered, where both conditional mean and conditional
variance function are modeled nonparametrically. A test for the model
assumption of independence of innovations from past time series values is
suggested. The test is based on an weighted $L^2$-distance of empirical
characteristic functions. The asymptotic distribution under the null hypothesis
of independence is derived and consistency against fixed alternatives is shown.
A smooth autoregressive residual bootstrap procedure is suggested and its
performance is shown in a simulation study.
