## Abstract

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 [Biometrika (1996c) vol. 83, 615-625] 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 forms in the residual cross-correlation matrices at all possible lags. The weights depend on a kernel function and on a truncation parameter. Using a result of Lewis and Reinsel [Journal of Multivariate Analysis (1985) Vol. 16, pp. 393-411], the asymptotic distribution of the test statistic 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 [Statistics and Probability Letters (2006) vol. 76, pp. 58-68] 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.

Original language | English |
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Pages (from-to) | 505-544 |

Number of pages | 40 |

Journal | Journal of Time Series Analysis |

Volume | 27 |

Issue number | 4 |

DOIs | |

Publication status | Published - Jul 2006 |

Externally published | Yes |

## Keywords

- Asymptotic power
- Consistency
- Independence
- Infinite-order vector autoregressive process
- Kernel function
- Portmanteau statistic
- Residual crosscorrelation

## ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty
- Applied Mathematics