A Generalized Portmanteau Test For Independence Of Two Infinite‐Order Vector Autoregressive Series

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.

A Nonparametric Test for Granger Causality in Distribution With Application to Financial Contagion

“…Moreover, as discussed by Hong, Liu, and Wang (2009), the choice of kernel is not important because they lead to comparable powers except for the uniform kernel, which does not discount higher-order lags. See Bouhaddioui and Roy (2006) for the same conclusion in a multivariate setting.…”

“…Following Bouhaddioui and Roy (2006), our test statistic is a centered and scaled version of the quadratic form in (28), that is,…”

“…The proof of Lemma 3 proceeds by combining elements in the proofs of Proposition 3 in Bouhaddioui andRoy (2006), Theorems A.1. andA.3.…”

“…To summarize, our testing procedure consists of dividing the distribution support of each series into a multivariate process of dynamic interquantile event variables. The test for causality in the distribution between the two series is enabled by an analysis of the cross-correlation structures of the multivariate processes and relies on the generalized portmanteau test for independence between multivariate processes developed by Bouhaddioui and Roy (2006).…”

An Improved Measure for Lack of Fit in Time Series Models

“…One could take the results of Section 4.1 in Robbins and Fisher (2015) and construct a statistic for gauging the cross-correlation between two series using a statistic such asQ m herein. Following Hong (1996b); Bouhaddioui and Roy (2006); Robbins and Fisher (2015), a weighted variant can further improve power. The results of Peña and Rodríguez (2002) and Mahdi and McLeod (2012) are based on large Toeplitz matrices with the kth off-diagonal populated with anR k term -one could develop an analogous matrix-based test using the proposed measure of correlation.…”