986 JJ.Xu et aL/Transportation Research Part E 47(2011)983-991 (Kavussanos,1997).forward freight agreement(FFA)price(Batchelor et al.,2005).The GARCH model has been widely used to examine the time-varying volatilities of shipping related time series.The ARCH model considers the variance of the cur- rent error term to be a function of the variances of the previous time period's error terms.ARCH relates the error variance to the square of a previous period's error.As the name suggests,the model has the following properties: (1)Autoregression-Uses previous estimates of volatility to calculate subsequent(future)values.Hence volatility values are closely related. (2)Heteroskedasticity-The probability distributions of the volatility varies with the current value. In this paper,we apply AR-GARCH(p,g)to model the conditional volatility of freight rate,since it has been proved that a GARCH model adequately fits many economic time-series(Bollerslev.1987). rt bo+birt-1+b2rr-2+...+bmrt-m Et,t~iid(0,ht) (3) P h=o+-+Bh-i (4) i1 i=1 where r is the natural logarithm of the monthly freight rate change evaluated by first difference of monthly freight rate r=AFR.h is the conditional variance.&r is the error term that follows a normal distribution with mean zero and time-vary- ing variance he p is the order of the GARCH terms hr and q is the order of the ARCH terms 8?. Using the freight rate volatility(h)derived from AR-GARCH model to represent freight market risk,we then analyze the relationship between freight market risk and the change of fleet size.h is regressed against the change in fleet size by In FS and In FS,the change in freight level by In FR,the change in demand for shipping services by In IP.and the change in trans- portation costs by In BP.as shown in Eq.(2).We first employed Ordinary Least Squares(OLS)to test the result.However,OLS or Generalized Least Squares(GLS)often lead to inconsistent estimation.The coefficient value or significant level may be seriously upward biased due to failures of some assumptions,such as collinearity,autocorrelation,and heteroscedasticity. which imply inefficient standard errors. Thus,we consider estimating the model using the generalized method of moments(GMM)approach.GMM is a very gen- eral statistical method for obtaining estimates of parameters of statistical models.In the 20 years since it was first intro- duced by Hansen(1982)of the method of moments.GMM has become a very popular tool among empirical researchers. It is also a very useful heuristic tool.Many standard estimators,including instrument variable(IV)and Ordinary Least Squares (OLS).can be seen as special cases of GMM estimators. GMM is a good estimator for dealing with autocorrelation and heterogeneity issues.The GMM approach allows an instru- ment to be used,thereby avoiding any simultaneity bias.It also brings the advantage of consistent estimation in the presence of heteroscedasticity and autocorrelation(Newey and West,1987).Baum et al.(2003)also mentioned that GMM makes use of the orthogonality conditions to allow for efficient estimation in the presence of heteroskedasticity of unknown form. 4.Data description and empirical results 4.1.Data description In the analysis,the data sets consist of monthly freight rate,fleet size (FS).industrial production (IP)and bunker price(BP). The freight rate is specified into Panamax and Capesize spot rate (SPR)and 1-year time-charter rate (TCR)in the dry bulk shipping industry while the fleet size is also divided into Panamax and Capesize bulk carriers (FS p,FS_c)as two types of dry bulk supply.The samples for Panamax spot rate (SPR_p)and Capesize spot rate (SPR_p)cover the period from January 1973 to October 2010,the sample for Panamax time-charter rate(TCR_p)covers the period from January 1976 to October 2010 and Capesize time-charter rate(TCR c)from January 1977 to October 2010.All freight rates,fleet size and bunker price data are collected from Clarkson Research Services Ltd.,while the industrial production indices are from OECD Statistics.The time series are transformed into natural logarithmic form. Descriptive statistics of logarithmic freight rates and fleet size are presented in Table 1.The J-B statistic rejects the hypotheses of normality for freight rates and fleet size in both ship types.The Ljung-Box Q-statistics are for auto-correlation test and the test results indicate that the p-value of the first 12 lags of the raw series and of the squared series is 0,which demonstrates significant auto-correlation.The Augmented Dickey-Fuller(ADF)unit root test on the monthly log first-differ- ence freight rate and fleet size series is applied to examine whether the series are stationary.The results indicate that for both ship types the log first-difference of freight rate and fleet size series are stationary. 4.2.Empirical results To analyze the relationship between freight market risk and the change in fleet size,one-step ahead conditional volatility estimates (h)of freight rates are constructed through the AR-GARCH model.We first choose the best auto-regression(AR) model for the four freight rate series(SPR_p,TCR_p,SPR_c and TCR_c).determined by Schwartz Information Criterion (SIC).(Kavussanos, 1997), forward freight agreement (FFA) price (Batchelor et al., 2005). The GARCH model has been widely used to examine the time-varying volatilities of shipping related time series. The ARCH model considers the variance of the cur￾rent error term to be a function of the variances of the previous time period’s error terms. ARCH relates the error variance to the square of a previous period’s error. As the name suggests, the model has the following properties: (1) Autoregression – Uses previous estimates of volatility to calculate subsequent (future) values. Hence volatility values are closely related. (2) Heteroskedasticity – The probability distributions of the volatility varies with the current value. In this paper, we apply AR-GARCH (p, q) to model the conditional volatility of freight rate, since it has been proved that a GARCH model adequately fits many economic time-series (Bollerslev, 1987). rt ¼ b0 þ b1rt1 þ b2rt2 þþ bmrtm þ et; et  iidð0; htÞ ð3Þ ht ¼ x þXq i¼1 aie2 ti þXp i¼1 bihti ð4Þ where rt is the natural logarithm of the monthly freight rate change evaluated by first difference of monthly freight rate rt = DFRt. ht is the conditional variance. et is the error term that follows a normal distribution with mean zero and time-vary￾ing variance ht. p is the order of the GARCH terms ht and q is the order of the ARCH terms e2 t . Using the freight rate volatility (ht) derived from AR-GARCH model to represent freight market risk, we then analyze the relationship between freight market risk and the change of fleet size. ht is regressed against the change in fleet size by ln FSt and ln FS2 t , the change in freight level by ln FRt, the change in demand for shipping services by ln IPt, and the change in trans￾portation costs by ln BPt, as shown in Eq. (2). We first employed Ordinary Least Squares (OLS) to test the result. However, OLS or Generalized Least Squares (GLS) often lead to inconsistent estimation. The coefficient value or significant level may be seriously upward biased due to failures of some assumptions, such as collinearity, autocorrelation, and heteroscedasticity, which imply inefficient standard errors. Thus, we consider estimating the model using the generalized method of moments (GMM) approach. GMM is a very gen￾eral statistical method for obtaining estimates of parameters of statistical models. In the 20 years since it was first intro￾duced by Hansen (1982) of the method of moments, GMM has become a very popular tool among empirical researchers. It is also a very useful heuristic tool. Many standard estimators, including instrument variable (IV) and Ordinary Least Squares (OLS), can be seen as special cases of GMM estimators. GMM is a good estimator for dealing with autocorrelation and heterogeneity issues. The GMM approach allows an instru￾ment to be used, thereby avoiding any simultaneity bias. It also brings the advantage of consistent estimation in the presence of heteroscedasticity and autocorrelation (Newey and West, 1987). Baum et al. (2003) also mentioned that GMM makes use of the orthogonality conditions to allow for efficient estimation in the presence of heteroskedasticity of unknown form. 4. Data description and empirical results 4.1. Data description In the analysis, the data sets consist of monthly freight rate, fleet size (FS), industrial production (IP) and bunker price (BP). The freight rate is specified into Panamax and Capesize spot rate (SPR) and 1-year time-charter rate (TCR) in the dry bulk shipping industry while the fleet size is also divided into Panamax and Capesize bulk carriers (FS_p, FS_c) as two types of dry bulk supply. The samples for Panamax spot rate (SPR_p) and Capesize spot rate (SPR_p) cover the period from January 1973 to October 2010, the sample for Panamax time-charter rate (TCR_p) covers the period from January 1976 to October 2010 and Capesize time-charter rate (TCR_c) from January 1977 to October 2010. All freight rates, fleet size and bunker price data are collected from Clarkson Research Services Ltd., while the industrial production indices are from OECD Statistics. The time series are transformed into natural logarithmic form. Descriptive statistics of logarithmic freight rates and fleet size are presented in Table 1. The J–B statistic rejects the hypotheses of normality for freight rates and fleet size in both ship types. The Ljung–Box Q-statistics are for auto-correlation test and the test results indicate that the p-value of the first 12 lags of the raw series and of the squared series is 0, which demonstrates significant auto-correlation. The Augmented Dickey–Fuller (ADF) unit root test on the monthly log first-differ￾ence freight rate and fleet size series is applied to examine whether the series are stationary. The results indicate that for both ship types the log first-difference of freight rate and fleet size series are stationary. 4.2. Empirical results To analyze the relationship between freight market risk and the change in fleet size, one-step ahead conditional volatility estimates (ht) of freight rates are constructed through the AR-GARCH model. We first choose the best auto-regression (AR) model for the four freight rate series (SPR_p, TCR_p, SPR_c and TCR_c), determined by Schwartz Information Criterion (SIC). 986 J.J. Xu et al. / Transportation Research Part E 47 (2011) 983–991