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.
a b s t r a c tIn the functional regression model where the responses are curves, new tests for the functional form of the regression and the variance function are proposed, which are based on a stochastic process estimating L 2 -distances. Our approach avoids the explicit estimation of the functional regression and it is shown that normalized versions of the proposed test statistics converge weakly. The finite sample properties of the tests are illustrated by means of a small simulation study. It is also demonstrated that for small samples, bootstrap versions of the tests improve the quality of the approximation of the nominal level.
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