  In this paper we propose a new test for the hypothesis of a constant
coefficient of variation in the common nonparametric regression model. The test
is based on an estimate of the $L^2$-distance between the square of the
regression function and variance function. We prove asymptotic normality of a
standardized estimate of this distance under the null hypothesis and fixed
alternatives and the finite sample properties of a corresponding bootstrap test
are investigated by means of a simulation study. The results are applicable to
stationary processes with the common mixing conditions and are used to
construct tests for ARCH assumptions in financial time series.
