Goodness-of-fit tests for multiplicative models with dependent data

Several classical time series models can be written as a regression model of the form Yt = m(Xt) + σ(Xt)εt, where (Xt, Yt), t = 0,±1,±2, . . ., is a bivariate strictly stationary process. Some of those models, such as ARCH or GARCH models, share the property of proportionality of the regression function, m, and the scale function, σ. In this article, we present a procedure to test for this feature in a nonparametric context, which is a preliminary step to identify certain time series models. The test is based on the difference between two nonparametric estimators of the distribution of the regression error. Asymptotic results are proved and some simulations are shown in the paper in order to illustrate the finite sample properties of the procedure.