Testing Serial Independence via Density-Based Measures of Divergence

Abstract

This article reviews the nonparametric serial independence tests based on measures of divergence between densities. Among others, the well-known Kullback-Leibler, Hellinger and Tsallis divergences are analyzed. Moreover, the copulabased version of the considered divergence functionals is defined and taken into account. In order to implement serial independence tests based on these divergence functionals, it is necessary to choose a density estimation technique, a way to compute p-values and other settings. Via a wide simulation study, the performance of the serial independence tests arising from the adoption of the divergence functionals with different implementation is compared. Both single-lag and multiple-lag test procedures are investigated in order to find the best solutions in terms of size and power.

An Updated Literature Review of Distance Correlation and Its Applications to Time Series

“…/ is the indicator function, we form the corresponding empirical estimates. Similar distance measures are obtained by employing density functions instead of distribution functions (see Tjøstheim, 1996, andBagnato et al, 2014, for an overview). Skaug and Tjøstheim (1993) extended the work by Blum et al (1961) and considered the asymptotic behaviour of the Cramer-von Mises-type statistic (27) at lag j , under ergodicity of ¹X t º.…”

“…Replacing the theoretical distribution functions with their empirical analogues where is the indicator function, we form the corresponding empirical estimates. Similar distance measures are obtained by employing density functions instead of distribution functions (see Skaug & Tjøstheim, , and Bagnato et al ., , for an overview).…”

SDD: AnRPackage for Serial Dependence Diagrams

“…As observed by Anderson, Hall, and Titterington (1994), in testing procedures a relative oversmoothing may be appropriate for some dependence functionals (test statistics). Nevertheless, the simulation results of Bagnato et al (2013b) highlight that when the GK density estimator is adopted to define the estimator of ∆ (r) , the use of h LCV is appropriate and, then, no oversmoothing is applied in SDD.…”

“…Among them, the ∆ 1 -ADF proposed in Bagnato et al (2013a) is considered as default. This choice follows from the results of a wide simulation study on several data generating processes presented by Bagnato et al (2013b) which shows that ∆ (r) 1 seems to be, among the eight aforementioned functionals, the best performer. Several methodologies are proposed in the literature to implement ∆ (r) ; the differences among the various approaches stem from the way: (i) the densities f r and g are estimated, (ii) the integral in (13) is computed, (iii) the p values q r , r = 1, .…”

“…, l, are obtained. Among the alternatives considered by Bagnato et al (2013b), the Gaussian kernel density estimator, an approximated numerical integration (similarly to Granger, Maasoumi, and Racine 2004), and the permutation approach, appear to be the best solutions for (i), (ii) and (iii), respectively. Details on these settings are given below.…”