Multidimensional specification test based on non-stationary time series

In the literature, most works of the specification tests focus on the problem with one-dimensional response or fixed multidimensional responses. In this paper, we develop a new specification test for the parametric models with non-stationary regressor under multidimensional setup, where the dimension of responses may tend to infinity, which fills a gap in the literature. The theoretical results about the asymptotic properties of the proposed test are studied and the optimal rate of the local departure under the alternative hypothesis is also given which ensures the models underpinning by the null and alternative hypotheses can be differentiated. Some simulation studies are done to evaluate the performance of the proposed test with the finite sample. Besides, a real data example based on the US aggregate consumers’ consumption data is employed to illustrate the performance. The results of simulation studies and real data analysis both demonstrate the efficiency of our proposed method.