Lag selection and model specification testing in nonparametric autoregressive conditional heteroscedastic models

•A lag dependence test is developed using high-dimensional one-way ANOVA.•The asymptotic distribution of the test statistic is derived under the null and local alternatives.•A test for the adequacy of the functional form of a pre-specified model is developed.•The lag selection method is based on PACF and false discovery rate.

The aim of this paper is two-fold: for a nonparametric autoregressive conditional heteroscedastic model, to propose a lag selection procedure; and given a parametric form specified a priori, to develop a model specification test. Both methods are based on a new powerful test statistic developed to check the influence of a lag on the conditional mean function. The asymptotic properties of this test statistic are investigated under the null and local alternative hypotheses. Extensive simulation studies suggest that the proposed lag selection method and the model specification test outperform existing methods for nonlinear models.