An Automatic MDDM-Based Test for Martingale Difference Hypothesis

Checking whether the error term is a marginal difference sequence (MDS) in the multivariate time series model with a parametric conditional mean is a crucial problem.Tests based on the martingale difference divergence matrix (MDDM) are an effective statistical method for testing MDS in the residuals of multivariate time series models.However, MDDM-based tests require specifying the lag order.To solve this problem, we propose a data-driven MDDM-based test that automatically selects the lag order.This method has three main advantages: first, researchers do not need to specify the lag order while the test automatically selects it from the data; second, under the null hypothesis, the lag order is one; third, the proposed automatic tests have good performance in detecting model inadequacy caused by highorder dependence.In theory, we prove the asymptotical property of the proposed method.Furthermore, we demonstrate the effectiveness of this method through simulations and real data analysis.