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Conditional independence graph for nonlinear time series and its application to international financial markets Wei Gao a,∗ , Hongxia Zhao b
School of Statistics, Xi’an University of Finance and Economics, Xi’an Shaanxi 710061, China School of College English, Xi’an University of Finance and Economics, Xi’an Shaanxi 710061, China
Conditional independence graphs are proposed for describing the dependence structure of multivariate nonlinear time series, which extend the graphical modeling approach based on partial correlation. The vertexes represent the components of a multivariate time series and edges denote direct dependence between corresponding series. The conditional independence relations between component series are tested efficiently and consistently using conditional mutual information statistics and a bootstrap procedure. Furthermore, a method combining information theory with surrogate data is applied to test the linearity of the conditional dependence. The efficiency of the methods is approved through simulation time series with different linear and nonlinear dependence relations. Finally, we show how the method can be applied to international financial markets to investigate the nonlinear independence structure. © 2012 Elsevier B.V. All rights reserved.
Article history: Available online 14 March 2013 Keywords: Nonlinear time series Conditional independence graphs Conditional mutual information Financial markets
1. Introduction In recent years increasing research studies involved the interaction structure between national stock markets and many empirical works in financial data used graphical models, e.g. Refs. [1,2]. Allali et al.  applied a partial correlation graph to study the interaction structure between international markets and investigated the conditional interaction structure between the most important international financial returns. They suggested using graphical interaction models to analyze the partial associations of stock markets. However, nonlinearity is a typical feature of financial data. Thus graphical models which can present nonlinear relations should be applied to analyze financial time series. Graphical models have become an important tool for analyzing multivariate data. Through merging the probabilistic concept of conditional independence with graph theory by representing possible dependence among the variables of a multivariate distribution in a graph. Graphical models led to simple graphical criteria for identifying and visualizing the conditional independence relations that are implied by a model associated with a given graph. For an introduction to graphical models we refer to the monographs by Whittaker , Edwards , and Cox and Wermuth ; a mathematically more rigorous treatment can be found in Ref. . From the study in Refs. [8,9] there has been an increasing interest in the use of graphical modeling techniques to analyze multivariate time series (e.g., Refs. [10–14]). However, most of these works have been restricted to the analysis of linear interdependence among the variables whereas the recent trend in time series analysis has shifted towards non-linear parametric and non-parametric models (e.g., Ref. ). Eichler  introduced a graphical time series model for the analysis of dynamic relationships among variables in multivariate time series. The modeling approach is based on the notion of strong Granger causality and can be applied to time series with non-linear
Corresponding author. E-mail address: firstname.lastname@example.org (W. Gao).
0378-4371/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2012.07.002
W. Gao, H. Zhao / Physica A 392 (2013) 2460–2469
dependence. Chu and Glymour  proposed an approach to learn a...