In this paper, we are devoted to the problem of testing nonparametric hypotheses for locally stationary diffusion models. We consider the question whether the time-varying drift function has the same shape over a given time interval. The weighted L₂-test statistic is proposed, and the asymptotic distribution is obtained under both the null hypothesis and the alternatives.

diffusion model, local stationarity, drift function, nonparametric testing, asymptotic normality.