On Testing for Independence between Several Time Series

This paper aims at developing a robust and omnibus procedure for checking the independence of two time series. proposed a robustified version of classic portmanteau statistic which is based on a fixed number of lagged residual cross-correlations. In order to obtain a consistent test for independence against an alternative of serial crosscorrelation of an arbitrary form between the two series, Hong's (1996a) introduced a class of statistics that take into account all possible lags. The test statistic is a weighted sum of residual cross-correlations and the weighting is determined by a kernel function. With the truncated uniform kernel, we retrieve a normalized version of Haugh's statistic. However, several kernels lead to a greater power. Here, we introduce a robustified version of Hong's statistic. We suppose that for each series, the true ARMA model is estimated by a n 1/2 -consistent robust method and the robust cross-correlation is so obtained. Under the null hypothesis of independence, we show that the robust statistic asymptotically follows a N (0, 1) distribution. Using a result of Li and Hui, we also propose a robust procedure for checking independence at individual lags and a descriptive causality analysis in the Granger's sense is discussed. The level and power of the robust version of Hong's statistic are studied by simulation in finite samples. Finally, the proposed robust procedures are applied to a set of financial data.

Canadian Journal of Statistics, 2012

Test statistics for checking the independence between the innovations of several time series are developed. The time series models considered allow for general specifications for the conditional mean and variance functions that could depend on common explanatory variables. In testing for independence between more than two time series, checking pairwise independence does not lead to consistent procedures. Thus a finite family of empirical processes relying on multivariate lagged residuals are constructed, and we derive their asymptotic distributions. In order to obtain simple asymptotic covariance structures, Möbius transformations of the empirical processes are studied, and simplifications occur. Under the null hypothesis of independence, we show that these transformed processes are asymptotically Gaussian, independent, and with tractable covariance functions not depending on the estimated parameters. Various procedures are discussed, including Cramér-von Mises test statistics and tests based on non-parametric measures. The ranks of the residuals are considered in the new methods, giving test statistics which are asymptotically margin-free. Generalized cross-correlations are introduced, extending the concept of cross-correlation to an arbitrarily number of time series; portmanteau procedures based on them are discussed. In order to detect the dependence visually, graphical devices are proposed. Simulations are conducted to explore the finite sample properties of the methodology, which is found to be powerful against various types of alternatives when the independence is tested between two and three time series. An application is considered, using the daily log-returns of Apple, Intel and Hewlett-Packard traded on the Nasdaq financial market. The Canadian Journal of Statistics xx: 1-32; 2012 c ⃝ 2012 Statistical Society of Canada Résumé: Des statistiques de test pour vérifier l'indépendance entre les innovations de plusieurs séries chronologiques sont développées. Les modèles de séries chronologiques considérés permettent des spécifications générales pour les fonctions de moyenne et de variance conditionnelle qui pourraient dépendre de variables explicatives communes. Pour tester l'indépendance entre plus de deux séries chronologiques, vérifier l'indépendance deuxà deux ne conduit pasà des procédures convergeantes. Ainsi, une famille finie de processus empiriques reposant sur plusieurs résidus décalés sont construits, et nous dérivons leurs distributions asymptotiques. Afin d'obtenir des structures simples de covariances asymptotiques, des transformations de Möbius des processus empiriques sontétudiées, et des simplifications se produisent. Sous l'hypothèse nulle d'indépendance, nous montrons que ces processus transformés sont asymptotiquement gaussiens, indépendants, et avec des fonctions de covariance commodes qui ne sont pas fonction de l'estimation des paramètres. Différentes procédures sont discutées, incluant les statistiques de test de Cramér-von Mises et les tests basés sur des mesures non paramétriques. Les rangs des résidus sont considérés dans les nouvelles méthodes, donnant des statistiques de test qui ne sont asymptotiquement pas fonction des marges. Des corrélations croisées généralisées sont introduites, fournissant une extension du concept de corrélation croiséeà un nombre arbitraire de séries temporelles; des procédures portemanteaux basées sur elles sont discutées. Afin de détecter la dépendance visuellement, des méthodes graphiques sont proposées. Des simulations sont réalisées afin d'étudier les propriétés pour deséchantillons finis de la c ⃝ 2012 Statistical Society of Canada / Société statistique du Canada CJS ??? DUCHESNE, GHOUDI AND RÉMILLARD Vol. xx, No. yy méthodologie, qui se trouveêtre puissante contre divers types de contre-hypothèses lorsque l'indépendance est testée entre deux et trois séries chronologiques. Une application est considérée, en utilisant les logrendements quotidiens d'Apple, d'Intel et de Hewlett-Packard négociées sur le marché financier Nasdaq. La revue canadienne de statistique xx: 1-32; 2012 c ⃝ 2012 Société statistique du Canada

Statistica Sinica

This paper proposes some novel one-sided omnibus tests for independence between two multivariate stationary time series. These new tests apply the Hilbert-Schmidt independence criterion (HSIC) to test the independence between the innovations of both time series. Under regular conditions, the limiting null distributions of our HSICbased tests are established. Next, our HSIC-based tests are shown to be consistent. Moreover, a residual bootstrap method is used to obtain the critical values for our HSIC-based tests, and its validity is justified. Compared with the existing cross-correlation-based tests for linear dependence, our tests examine the general (including both linear and non-linear) dependence to give investigators more complete information on the causal relationship between two multivariate time series. The merits of our tests are illustrated by some simulation results and a real example. 1. Introduction. Before applying any sophisticated method to describe relationships between two time series, it is important to check whether they are independent or not. If they are dependent, causal analysis techniques, such as copula and multivariate modeling, can be used to investigate the relationship between them, and this may lead to interesting insights or effective predictive models; otherwise, one should

1998

When autocorrelation is small, existing statistical techniques may not be powerful enough to reject the hypothesis that a series is free of autocorrelation. We propose two new and simple statistical tests (RHO and PHI) based on the unweighted sum of autocorrelati on and partial autocorrelation coefficients. We analyze a set of simulated data to show the higher power of RHO and PHI in comparison to conventional tests for autocorrelation, especially in the presence of small but persistent autocorrelation. We show an application of our tests to data on currency futures to demonstrate their practical use. Finally, we indicate how our methodology could be used for a new class of time series models (the Generalized Autoregressive, or GAR models) that take into account the presence of small but persistent autocorrelation.

Communications in Statistics - Theory and Methods, 2012

This short note suggests a heuristic method for detecting the dependence of random time series that can be used in the case when this dependence is relatively weak and such that the traditional methods are not effective. The method requires to compare some special functionals on the sample characteristic functions with the same functionals computed for the benchmark time series with a known degree of correlation. Some experiments for financial time series are presented. This short note presents some statistical experiments with the purpose to estimate the dependence for time series. We suggest to compare historical time series with a series with given and known correlation using a functional formed from empirical characteristic functions defined similarly to . It gives a simple empirical method that allows to estimate the dependence by comparing the values of this functional for two time series.

SSRN Electronic Journal, 2021

We propose consistent nonparametric tests of conditional independence for time series data. Our methods are motivated from the difference between joint conditional cumulative distribution function (CDF) and the product of conditional CDFs. The difference is transformed into a proper conditional moment restriction (CMR), which forms the basis for our testing procedure. Our test statistics are then constructed using the integrated moment restrictions that are equivalent to the CMR. We establish the asymptotic behavior of the test statistics under the null, the alternative, and the sequence of local alternatives converging to conditional independence at the parametric rate. Our tests are implemented with the assistance of a multiplier bootstrap. Monte Carlo simulations are conducted to evaluate the finite sample performance of the proposed tests. We apply our tests to examine the predictability of equity risk premium using variance risk premium for different horizons and find that there exist various degrees of nonlinear predictability at mid-run and long-run horizons.

Journal of Statistical Planning and Inference, 1999

We propose a nonparametric test of independence of two autoregressive time series. The test statistic is based on lagged cross-correlation coe cients computed from autoregression rank scores, and extends the traditional correlogram-based method of Haugh (1976). It is easily computable, asymptotically distribution-free, and, contrary to its traditional parametric competitor, it does not require any estimation of the unknown autoregression parameters. The test is applied in a study of the relations between outdoor temperature and the daily mortality related to cardiovascular problems in Brussels, during the period 1980-1989.

Journal of Multivariate Analysis, 2001

This paper presents nonparametric tests of independence that can be used to test the independence of p random variables, serial independence for time series, or residuals data. These tests are shown to generalize the classical portmanteau statistics. Applications to both time series and regression residuals are discussed.

Econometric Theory, 2002

Annals of Economics and Finance, 2001

This paper proposes an asymptotic one-sided N (0, 1) test for independence between two stationary time series using the empirical characteristic function. Unlike the tests based on the cross-correlation function (e.g. Haugh, 1976; Hong, 1996; Koch & Yang 1986), the proposed test has power against all pairwise cross-dependencies, including those with zero cross-correlation. By differentiating the empirical characteristic function at the origin, the present approach yields a modified version of Hong's (1996) test, which in turn generalizes Haugh's (1976) test. Other new tests can be derived by further differentiating the empirical characteristic function properly. A simulation study compares the new test with those of Haugh (1976), Hong (1996) and Koch & Yang (1986) in finite samples; the results show that the new test has reasonable sizes and good powers against linear and nonlinear cross-dependencies.