On portmanteau-type tests for nonlinear multivariate time series

Abstract

A general framework to devise portmanteau-type test statistics for a general class of multivariate nonlinear time series models with vector martingale difference errors is formulated. Based on this framework a suite of individual and mixed multivariate test statistics is considered. Two applications are developed: single- and multiple-lag test statistics. In each case, the resulting portmanteau test statistic is based on multivariate residuals and multivariate squared residuals. Moreover, single- and multiple-lag mixed multivariate portmanteau-type tests are introduced. These test statistics are designed to detect different forms of inadequacies in the model residuals jointly. All proposed tests take uncertainty due to model estimation properly into account. The asymptotic null distribution of each test statistic follows from the asymptotic distribution of the general portmanteau-type test statistic in a natural way. Some considerations are given to the empirical size and power of six test statistics via a simulation study. All tests have satisfactory size and power properties in finite samples. To demonstrate their practical use, the proposed test statistics are applied to the residuals of a vector bivariate nonlinear threshold model fitted to U.S. interest rates.