A Generalized Portmanteau Test for Independence of Two Infinite Order Vector Autoregressive Series

In many situations, we want to verify the existence of a relationship between multivariate time series. Here, we propose a semiparametric approach for testing the independence between two infinite order vector autoregressive (VAR()) series which is an extension of Hong's (1996a) univariate results. We first filter each series by a finite-order autoregression and the test statistic is a standardized version of a weighted sum of quadratic form in residual cross-correlation at all possible lags. The weights depend on a kernel function and on a truncation parameter. Using a result of Lewis and Reinsel (1985), the asymptotic distribution of the statistic test is derived under the null hypothesis and its consistency is also established for a fixed alternative of serial cross-correlation of unknown form. Apart from standardization factors, the multivariate portmanteau statistic proposed by Bouhaddioui and Roy (2003) that takes into account a fixed number of lags can be viewed as a special case by using the truncated uniform kernel. However, many kernels lead to a greater power, as shown in an asymptotic power analysis and by a small simulation study in finite samples. A numerical example with real data is also presented.
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