Segmentation analysis on a multivariate time series of the foreign exchange rates

This study considers the multivariate segmentation procedure under the assumption of the multivariate Gaussian mixture. Jensen-Shannon divergence between two multivariate Gaussian distributions is employed as a discriminator and a recursive segmentation procedure is proposed. The daily log-return time series for 30 currency pairs consisting of 12 currencies for the last decade (January 3, 2001 to December 30, 2011) are analyzed using the proposed method. The proposed method can detect several important periods related to the significant affairs of the international economy. Introduction Over the last two decades, the statistical properties of asset price returns have been successively studied in the literature of econophysics [1, 2]. One important property is that the probability distribution of returns exhibits a fat-tailed distribution [3, 4]. In this study, I hope to provide some insights on the problem of finding transition points in the global economy. In the context of economics and finance, there are various methods that can be used to segment highly nonstationary financial time series into stationary segments called regimes or trends. Following the pioneering works of Goldfeld and Quandt [5], there is much literature on detecting structural breaks or change points separating stationary segments. Recently, a recursive entropic scheme to separate financial time series was proposed [6]. The multivariate time series can be modeled by using multivariate Gaussian distribution. However, in the case of financial time series, we normally observe the multivariate time series as a mixture of multivariate Gaussian distributions with a different variance-covariance matrix and mean due to the nonstationarity of variance-covariance matrix and mean values. Therefore, we obtain the variance-covariance matrix of an unconditional distribution when we compute an empirical variance-covariance matrix from all the data. Furthermore, the mixture of multivariate Gaussian distributions is normally a non-Gaussian distribution. In this study, we consider a segmentation procedure for multivariate time series under the assumption of local stationarity. We assume that the multivariate time series are generated from different multivariate Gaussian distributions. The proposed procedure is applied to segmenting multiple daily log-return time series for 30 selected currency pairs from the period of January 4, 2001 to December 30, 2011. This article is organized as follows. In Sec. 2, the recursive segmentation procedure is briefly explained. In Sec. 3 the proposed method is applicable to segmenting a mixture of multivariate Gaussian samples with the given variance-covariance matrix. In Sec. 4, an empirical analysis with daily log-returns for the last 10 years is conducted. Sec. 5 is devoted to conclusions. Segmentation procedure Let ) (t ri (i = 1,..., M; t = 1,...,T) be M-dimensional multiple log-return time series, defined as ) ( ) 1 ( ) ( t R t R t r i i i    , where ) (t Ri (t = 1,...,T + 1) is the daily exchange rate of i-th currency pair at day t. From successive works on financial markets, the log-return time series for foreign exchange rates are modeled by q-Gaussian distributions [2] and/or Lévy distributions [1,3]. Both the q-Gaussian and Lévy distributions are given by an infinite mixture of Gaussian distribution with Gamma distribution multipliers[2]. In the context of finance, the q-Gaussian distributions are referred as to Student-t distributions. Let us assume that this multivariate time series consist of n sequences sampled from n different multivariate Gaussian distributions. I further assume that the log-return movements in segment k follow multivariate Gaussian distributions with variance-covariance matrix ) (k C . To determine the n stationary segments from the given multiple time series ) (t ri , I employ the recursive segmentation procedure introduced by Cheong et al. [6]. In this segmentation procedure, we check whether the likelihood value is suitable in order to separate the multiple time series into two segments at the point t. To do so, we denote the likelihood as 1 2 log ) ( log ) ( L t L t    , (1) where                . ) ( ) ( 2 1 exp 2 1 , ; , , , , ); ( , ), ( , ); ( , ), ( ) ( , , ); ( , ), ( 1 1 1 2 / 1 2 / 1 1 ) ( ) ( 1 1 ) ( ) ( 1 2 1 1 1            