Testing linearity of regression models with dependent errors by kernel based methods

In a recent paper Gonz alez Manteiga and Vilar Fern andez (1995) considered the problem of testing linearity of a regression under MA(1) structure of the errors using a weighted L2-distance between a parametric and a nonparametric t. They established asymptotic normality of the corresponding test statistic under the hypothesis and under local alternatives. In the present paper we extend these results and establish asymptotic normality of the statistic under xed alternatives. This result is then used to prove that the optimal (with respect to uniform maximization of power) weight function in the test of Gonz alez Manteiga and Vilar Fern andez (1995) is given by the Lebesgue measure independently of the design density. The paper also discusses several extensions of tests proposed by Azzalini and Bowman (1993), Zheng (1996) and Dette (1999) to the case of non-independent errors and compares these methods with the method of Gonz alez Manteiga and Vilar Fern andez (1995). It is demonstrated that among the kernel based methods the approach of the latter authors is the most e cient from an asymptotic point of view.