On consistent testing for serial correlation of unknown form in vector time series models

Multivariate autoregressive models with exogenous variables (VARX) are often used in econometric applications. Many properties of the basic statistics for this class of models rely on the assumption of independent errors. Using results of Hong (Econometrica 64 (1996) 837), we propose a new test statistic for checking the hypothesis of non-correlation or independence in the Gaussian case. The test statistic is obtained by comparing the spectral density of the errors under the null hypothesis of independence with a kernel-based spectral density estimator. The asymptotic distribution of the statistic is derived under the null hypothesis. This test generalizes the portmanteau test of Hosking (J. Amer. Statist. Assoc. 75 (1980) 602). The consistency of the test is established for a general class of static regression models with autocorrelated errors. Its asymptotic slope is derived and the asymptotic relative efficiency within the class of possible kernels is also investigated. Finally, the level and power of the resulting tests are also studied by simulation.