Interdependence between Exchange Rates: Evidence from Multivariate Fractional Cointegration

I. INTRODUCTION The interdependence between the foreign exchange markets has been largely examined in the literature and numerous empirical studies have focused on investigating the relationship between the exchange rate series. The existence of such relationship is important and implies that the price of the current market is related to the price of the foreign market in the sense that we could be able to outperform the market using additional information of the others markets. Consequently, we reject the weak efficient market hypothesis which stipulates that current prices fully reflect past prices and information available. Earlier studies consider that the exchange rate series are non stationary and thus employ the conventional cointegration tests developed by Engle and Granger (1987) and Johansen (1988) to check for a possible equilibrium relationship between the variables (Baillie and Bollerslev (1989), Diebold et al. (1994)). Others studies show that the conventional cointegration tests may fail to find evidence of cointegration when the variables and particularly their equilibrium relationships exhibit characteristics that are more consistent with long memory (Cheung and Lai (1993), Andersson and Gredenhoff (1999), Gonzalo and Lee (2000), Smallwood and Norrbin (2002)). For that, they consider the fractional cointegration approach developed by Granger (1986). This approach consists on testing if the estimated residuals from the linear cointegration relationship are I(1) against distributed as a fractional process. Although this approach permits to test the presence of fractional cointegration, it supposes that the series are stationnary and is limited to univariate framework. Recently, Davidson (2002) develops an alternative approach for testing fractional cointegration in multivariate framework. In particular, this approach considers that the processes are fractionally integrated and is powerful enough to distinguish between linear cointegration and fractional cointegration, and thus provide more robust results than conventional cointegration tests. The aim of this paper is to examine the empirical relationship between the exchange rates using multivariate fractional cointegration approach (Davidson (2002)) and to analyse the causal links between the variables (Granger (1988)). In order to determine the cointegrating rank, we apply in addition to the minimal algorithm of Davidson (1998a,b) the procedure of Nielson and Shimotsu (2007) based the exact local Whittle analysis of Shimotsu (2005). This paper is structured as follows. Section II describes the fractional cointegration model as advanced by Davidson (2002) and describes the procedures of determining the cointegrating rank. Section III presents the data and reports the empirical results and Section IV concludes the paper. II. FRACTIONAL COINTEGRATION The concept of fractional cointegration is based on the basic principle that a set of k fractionally integrated variables [x.sub.jt] ~ I([d.sub.j]), j = l, ..., k, there exist r independent linear combinations that are integrated to lower orders. This concept is introduced by Granger (1986) and developed by Davidson (2002, 2005) and Davidson et al. (2006). The general framework of the fractional vector error correction model (Granger (1986)) is given by: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1) where [X.sub.t](k x 1) is a vector of observed variables, [D.sub.t](s x 1) is a vector of exogenous variables, typically dummies, [PHI](k x s) is a matrix of coefficients. [[epsilon].sub.t] (k x 1) is a vector of error terms [[epsilon].sub.t] ~ i.i.d.(0, [SIGMA]), C(L)(k x k) is a finite-order matrix polynomials in the lag operator with all roots outside the unit circle to represent short run effects. a and [beta] are constant matrices of dimension (k x r), having rank r which represent, respectively, the error correction and the cointegration coefficients. …