X.Zhang et al.Physica A 415 (2014)43-53 野 2.2.Granger causality analysis To investigate the dynamic systemic risk,we must measure both the degree of interconnectedness between the subsectors of the shipping industry and the direction of these relationships[35-37].To this end,using Granger causality analysis we propose a statistical definition of causality based on the relative forecasting power of two series.Specifically, let R and R be two stationary time series,and for simplicity we assume they both have zero mean.We can represent their linear inter-relationships using the model [38,39] Ri1=aR:+bR+e+1. (3) Ri+1=R:+bR:+e+1 (4) where e ande are two uncorrelated white noise processes.The definition of causality implies that Rcauses R when b is statistically significant from zero.Likewise,R causes R whenb is statistically significant from zero.When both b and bi are statistically significant from zero,there is a feedback relationship between the two time series.In practice,the causality is based on the F-test where the null hypothesis is defined such that coefficients a'and a are equal to zero. We analyze the pairwise Granger causality between the t and t+1 monthly returns of the shipping physical market and the shipping stock market.We follow the definition of the dynamic causality index(DCI)[40]series. number of causal relationships over a given period Lpa(t)= (5) total possible number of causal relationships 2.3.Brownian distance Distance correlation is a new approach proposed by Szekely and Rizzo to measure statistical interdependence between two random vectors of arbitrary,not necessarily equal dimension41.Brownian distance covariance captures the non-linear dependence,which make up deficiency of the classical measure of dependence,such as the Pearson correlation coefficient, that is mainly sensitive to a linear relationship between two variables [41l. According to the basic definition of distance correlation,Brownian covariance(v(X,Y))between fxfy and fx.y is obtained as the square root of v2(X,Y)=llfx.y(t,s)-x(t)fy(s)2where ll l is the joint characteristic function of X and Y.Brownian covariance is based on Brownian motion or Wiener process with an important property that v(X,Y)=0 if and only if X and Y are independent [42].The Brownian covariance is equal to the distance covariance.The distance correlation R(X,Y) can be defined from the following expression: v2(X,Y) u2(X)u2(Y)>0. √u2X)u2(Y) (6) 0 v2(X)u2(Y)=0. In this paper we utilize Brownian distance correlation between current value of time series Y:and I lagged value of another time series vector X-exploring then the non-linear causality effect.In general,if R(X-1.Y)0 and I>0.then X-leads the series Yr.Additionally,if R(X-1,Yt)0.R(X,Y-1)0,and I>0,there is a unidirectional relationship between X and Y. 3.Data We investigate two datasets.Dataset I comprises the prices of the real shipping market.Dataset Il comprises the stock prices of publicly-listed shipping companies.For the shipping market we select 45 monthly price indicator series for the time period from January 2003 to June 2013,provided by world leading shipping database Clarksons.The dataset includes three shipping markets,the new ship market,the second-hand ship sale and purchase market,and the world-wide chartering market.For each market we use price indicators according to ship type,oil tankers,container carriers,and bulk carriers.The New-building ship market price indicators investigated are shown in Table 1.The Secondhand ship market price indicators are shown in Table 2.The freight rates are shown in Table 3. We also select 40 publicly-listed shipping companies to represent the shipping financial market.The sample includes the oil tanker,container carrier,and bulk carrier industry as well as the ship-building industry.The monthly closing price of each stock is recorded from January 2003 to June 2013,provided by Yahoo Finance. 4.Real shipping market hierarchical structure In this section,using 45 physical shipping market price indicators,we present the MSTs and the HTs,and investigate the topology and structure of the correlation networks in the shipping market.We find that MSTs and HTs both showX. Zhang et al. / Physica A 415 (2014) 43–53 45 2.2. Granger causality analysis To investigate the dynamic systemic risk, we must measure both the degree of interconnectedness between the subsectors of the shipping industry and the direction of these relationships [35–37]. To this end, using Granger causality analysis we propose a statistical definition of causality based on the relative forecasting power of two series. Specifically, let R i t and R j t be two stationary time series, and for simplicity we assume they both have zero mean. We can represent their linear inter-relationships using the model [38,39] R i t+1 = a i R i t + b ijR j t + e i t+1 , (3) R j t+1 = a j R j t + b jiR i t + e j t+1 , (4) where e i t+1 and e j t+1 are two uncorrelated white noise processes. The definition of causality implies that R j t causes R i t+1 when b ij is statistically significant from zero. Likewise, R i t causes R j t+1 when b ji is statistically significant from zero. When both b ij and b ji are statistically significant from zero, there is a feedback relationship between the two time series. In practice, the causality is based on the F -test where the null hypothesis is defined such that coefficients a i and a j are equal to zero. We analyze the pairwise Granger causality between the t and t +1 monthly returns of the shipping physical market and the shipping stock market. We follow the definition of the dynamic causality index (DCI) [40] series, LDCI(t) = number of causal relationships over a given period total possible number of causal relationships . (5) 2.3. Brownian distance Distance correlation is a new approach proposed by Székely and Rizzo to measure statistical interdependence between two random vectors of arbitrary, not necessarily equal dimension [41]. Brownian distance covariance captures the non-linear dependence, which make up deficiency of the classical measure of dependence, such as the Pearson correlation coefficient, that is mainly sensitive to a linear relationship between two variables [41]. According to the basic definition of distance correlation, Brownian covariance (v(X, Y)) between fX fY and fX,Y is obtained as the square root of v 2 (X, Y) = ∥fX,Y (t, s)−fX (t)fY (s)∥ 2 where ∥·∥ is the joint characteristic function of X and Y. Brownian covariance is based on Brownian motion or Wiener process with an important property that v(X, Y) = 0 if and only if X and Y are independent [42]. The Brownian covariance is equal to the distance covariance. The distance correlation R(X, Y) can be defined from the following expression: R 2 =    v 2 (X, Y)  v 2 (X)v2 (Y) , v2 (X)v2 (Y) > 0. 0, v2 (X)v2 (Y) = 0. (6) In this paper we utilize Brownian distance correlation between current value of time series Yt and l lagged value of another time series vector Xt−l exploring then the non-linear causality effect. In general, if R(Xt−l, Yt) ̸= 0 and l > 0, then Xt−l leads the series Yt . Additionally, if R(Xt−l, Yt) ̸= 0, R(Xt, Yt−l) ̸= 0, and l > 0, there is a unidirectional relationship between X and Y. 3. Data We investigate two datasets. Dataset I comprises the prices of the real shipping market. Dataset II comprises the stock prices of publicly-listed shipping companies. For the shipping market we select 45 monthly price indicator series for the time period from January 2003 to June 2013, provided by world leading shipping database Clarksons. The dataset includes three shipping markets, the new ship market, the second-hand ship sale and purchase market, and the world-wide chartering market. For each market we use price indicators according to ship type, oil tankers, container carriers, and bulk carriers. The New-building ship market price indicators investigated are shown in Table 1. The Secondhand ship market price indicators are shown in Table 2. The freight rates are shown in Table 3. We also select 40 publicly-listed shipping companies to represent the shipping financial market. The sample includes the oil tanker, container carrier, and bulk carrier industry as well as the ship-building industry. The monthly closing price of each stock is recorded from January 2003 to June 2013, provided by Yahoo Finance. 4. Real shipping market hierarchical structure In this section, using 45 physical shipping market price indicators, we present the MSTs and the HTs, and investigate the topology and structure of the correlation networks in the shipping market. We find that MSTs and HTs both show