Seasonality patterns in tanker spot freight rate markets

Economic Modelling 19 Ž2002. 747᎐782

Seasonality patterns in tanker spot freight rate markets Manolis G. Kavussanos a,b,U , Amir H. Alizadeh-M b a

Department of Accounting and Finance, Athens Uni¨ ersity of Economics and Business, 76 Patission St, Athens, TK 104 34, Greece b Department of Shipping, Trade and Finance, City Uni¨ ersity Business School, Frobisher Crescent, Barbican Centre, London EC2Y 8HB, UK

Accepted 20 April 2001

Abstract The aim of this paper is to investigate the existence and nature of seasonality Ždeterministic or stochastic . in tanker freight markets and measure and compare it across sub-sectors and under different market conditions Žexpansionary and contractionary. for the period January 1978 to December 1996. The existence of stochastic seasonality is rejected for all freight series while results on deterministic seasonality indicate increases in rates in November and December and decreases in rates from January to April. Seasonality is found to be varying across markets depending on vessel size and market condition. Seasonality comparisons under different market conditions, an issue investigated for the first time in the econometrics literature using Markov Switching models, reveal that seasonal rate movements are more pronounced when the market is recovering compared to smaller changes when the market is falling. This is well in line with the low and high elasticity of supply expected in expansionary and contractionary periods of shipping markets. The results have implications for tactical shipping operations such as budget planning, timing of dry-docking, vessel speed adjustments and repositioning. As expected, the out-of-sample forecasting performance of these Markov Regime Switching models is lacking somewhat, a result which is thought to be a consequence of having to predict ‘states’ simultaneously with mean values. 䊚 2002 Elsevier Science B.V. All rights reserved. JEL classifications: L91; L92; C22; C51; C53 Keywords: Shipping; Seasonality; Tanker freight rates; Markov switching; Market conditions; Forecasting U

Corresponding author. Tel.: q44-20-7040-8681; fax: q44-20-7040-8853. E-mail address: [email protected], [email protected] ŽM.G. Kavussanos..

0264-9993r02r$ - see front matter 䊚 2002 Elsevier Science B.V. All rights reserved. PII: S 0 2 6 4 - 9 9 9 3 Ž 0 1 . 0 0 0 7 8 - 5

748

M.G. Ka¨ ussanos, A.H. Alizadeh-M r Economic Modelling 19 (2002) 747᎐782

1. Introduction Freight rates and prices in the tanker shipping industry fluctuate considerably in short periods of time. While part of these fluctuations are due to the general world economic activity as well as the state of the tanker shipping market, a large proportion of the within-the-year fluctuations is attributed to seasonal factors. Seasonal behaviour of freight rates is an important factor in the formation of transportation policy and affects the shipowners’ cash flow and charterers’ costs. Yet, to our knowledge, there has been no systematic attempt in the past to understand the nature and measure the magnitude of seasonal movements in tanker shipping freight rates.1 The aim of this paper is to fill this gap in the literature by concentrating on the movement of tanker freight rates within the year and compare them across sub-sectors of the industry as well as over different market conditions. The out-of-sample forecasting performance of different models for seasonality is also investigated. Like any other market, tanker freight rate markets are characterised by the interaction of supply and demand for tanker shipping services. The demand for tanker services is a derived demand which depends on the economics of the oil markets and trade, world economic activity and the related macroeconomic variables of major economies, such as imports and consumption of energy commodities Žsee Stopford, 1997 p. 238.. Such macroeconomic variables have been shown elsewhere to be non-stationary with deterministic seasonal components in most cases Žsee Osborn, 1990; Beaulieu and Miron, 1992; Canova and Hansen, 1995.. The same is also true for trade in oil and oil derivatives; for instance Moosa and Al-Loughani Ž1994. found that there are seasonal elements in the petroleum and petroleum products trades, Moosa Ž1995. found that there are elements of seasonality in oil import figures of Group 7 countries; Girma and Paulson Ž1998. also found seasonal patterns in the petroleum products futures markets. It is possible then that, the seasonal patterns evident in the variables that generate demand for shipping services are likely to make their way into tanker freight rates. For analysis, the tanker market is disaggregated by vessel size into different sub-markets, since different vessel sizes are involved in different trading routes and regions of the world.2 Such sub-markets are thought to be distinct in terms of their risk᎐return characteristics Žsee for example Kavussanos 1996a,b, 1998a,b.. Yet there has been no systematic attempt in the past to understand the stochastic behaviour of rates in each of these sub-markets. The issue ties in with the investigation of seasonality since when, for instance, monthly data are used for modelling, stochastic seasonal roots may appear in the data. This has important implications when modelling the series. As Wallis Ž1974. notes, it is important to study the seasonal behaviour of the data because using seasonally adjusted data may distort the dynamics of the constructed models and result in biased estimates.

1 2

Seasonality, though, has been investigated in dry bulk markets by Kavussanos and Alizadeh Ž2001.. Four sub-sectors may be distinguished in the tanker market, details of which are in the data section.

M.G. Ka¨ ussanos, A.H. Alizadeh-M r Economic Modelling 19 (2002) 747᎐782

749

At the same time, knowing the seasonal behaviour of the data allows better model specification, which in turn can improve the reliability of forecasts. This paper contributes to the literature in a number of ways. First, it investigates systematically the univariate properties of monthly tanker freight rate series in terms of stationarity and seasonal unit roots. As a result, correct decisions can be made about econometric modelling issues. Second, as a consequence of this investigation decisions can also be made about the existence and nature of seasonality in tanker freight markets. Third, seasonality is measured and compared across tanker shipping freight markets and different market conditions. For this purpose, Markov Regime Switching Seasonal ŽMRSS. models are used, for the first time in the econometrics literature, to compare seasonal fluctuations in freight rates between periods of market expansion and contraction. Finally, the out-ofsample forecasting performance from these models is investigated with some interesting results. Examining the form and magnitude of seasonal fluctuations in tanker rates can be beneficial to shipowners and charterers in their decisions regarding shipping investments and operations. From the shipowners’ point of view such decisions include: budget planning according to annual cash flow; as well as short vs. long haul contracts in order to reduce back haul during high seasons; timing of dry docking; speed adjustments; and vessel repositioning. Charterers also may use the information on seasonal fluctuations in tanker freight rates to optimise their cost by adjusting their inventory in order to reduce chartering requirements during high seasons by building up sufficient inventory before the high season. Overall, additional quantitative information about likely market conditions Že.g. seasonality measurements. can improve operational decisions and strategic business plans. This paper has four more sections. Section 2 reviews briefly the different types of seasonality and various methods used to test and detect them. Section 3 describes the tanker freight rate data, and their sources. Section 4 presents empirical results, which identify the nature of seasonality, and measures and compares seasonal patterns over sub-sectors of the industry and different market conditions. Section 5 discusses the forecasting performance of seasonal models and the final section concludes.

2. Seasonality A time series, measured more than once a year Že.g. at monthly, quarterly or semi-annual intervals., is said to contain seasonal components when there are systematic patterns in the series at the measured points Žseasons. within the year. This may be due to changes in the weather, the calendar, or the behaviour of agents involved in decision making. These systematic changes may or may not be regular due to different circumstances in which factors such as technology, politics, etc., can be influential. In shipping, freight rate seasonality may arise because of factors that influence the demand for shipping services; that is, the demand for international commodity transport. These are thought to be primarily the weather

M.G. Ka¨ ussanos, A.H. Alizadeh-M r Economic Modelling 19 (2002) 747᎐782

750

conditions and calendar effects, such as the increase in heating oil consumption during the winter, and increased demand for dry bulk commodities by Japan before the change of financial year every March. Seasonal behaviour of economic time series can take three forms: stochastic; deterministic; or a combination of the two. A series with deterministic seasonality has the same seasonal behaviour Žpeaks and troughs. every year; for instance seasonal consumption of heating oil in the Northern Hemisphere, increases every winter and declines every summer. In contrast a series with stochastic seasonality follows a seasonal pattern, which changes over time; for example, summer peaks in the series shift to winter and winter trough moves to summer.3 In addition, series with stochastic seasonality retain the shocks for a long period, unlike deterministic seasonal series in which shocks diminish relatively quickly. It is important to distinguish between different types of seasonality in time series analysis both from the econometric and the economic point of view. Failing to recognise the existence of stochastic seasonality in time series may lead to spurious regression results Žsee for example, Hylleberg et al., 1990., while taking account of deterministic seasonality of time series may improve the explanatory power of econometric models and result in better forecasts. From the economic point of view, distinguishing between stochastic and deterministic seasonality would improve the effectiveness of decisions and policies based on the seasonal behaviour of the series. This is because if the pattern of seasonality changes over time Ži.e. if it is stochastic ., then policies should be revised periodically. 2.1. Deterministic seasonality The deterministic seasonal variations in a series can be investigated by regressing the growth rate of the variable, ⌬ X t Žwhere X t is the natural logarithm of the series., against a constant, ␤ 0 , and a set of seasonal dummy variables, as in Eq. Ž1., 12

⌬ Xt s ␤0 q

Ý ␤i Qi ,t q ␧ t

Ž1.

is2

Where Q it , i s 2,...,12, are relative seasonal dummies,4 12 is the number of periods the variable is measured over the year Ž12 is used here because of our 3

Hylleberg et al. Ž1990. document such behaviour in UK total and non-durables consumption as well as public investment series. This is thought to be a result of changes in consumers’ tastes and spending behaviour. 4 Relative seasonal dummies are constructed as Q i,t s Di,t y D 1,t , i s 2,...,12 where D 1,t ,..., D 12,t are 0, 1 monthly dummies. In this case, the coefficient for the base month, January, can be 12

calculated as ␤ 1 s y from

the

is2

12

Var Ž ␤i . q 2

The standard error of the January coefficient can be calculated

is2

estimated

12

½Ý

Ý ␤i .

variance ᎐ covariance

12

Ý Ý CovŽ ␤i , ␤ j . is2- js2

1r2

5

matrix

of

the

coefficients

as

se Ž ␤ 1 . s

. See also Suits Ž1984. and Greene and Seaks Ž1991. for an

alternative restricted least squares procedure.

M.G. Ka¨ ussanos, A.H. Alizadeh-M r Economic Modelling 19 (2002) 747᎐782

751

monthly data series, otherwise i s 2,3,4 for quarterly series., ␤i are the parameters of interest and ␧ t is a white noise error term.5 The significance of each seasonal dummy indicates the existence of deterministic seasonality in the respective period; that is, that there is a significant change in the dependent variable compared to its long-run mean, ␤ 0 . 2.2. Alternati¨ e models of deterministic seasonality under different market conditions Canova and Ghysels Ž1994. argue that the magnitude of seasonality in a series might not be constant over time. They find that the deterministic seasonal coefficients in many macroeconomic variables depend on the prevailing market conditions; that is, on the business cycle phase. This is an important issue in the cyclical shipping freight markets since the elasticity of supply is thought to be high during troughs and low in peaks of the shipping business cycle, as indicated by the shape of the freight service supply curve in Fig. 1 Žsee Stopford, 1997; McConville, 1999, p. 109.. This is because at low freight rates the least efficient ships are laid up and the fleet at sea is slow steaming. As market conditions improve, and freight rates rise, the speed of the fleet increases and more and more ships are gradually taken out of lay up. At very high levels of freight rates all vessels are active and the fleet operates at full speed, creating a vertical supply curve. The demand for tanker freight services is shown to be relatively inelastic as the proportion of the transportation cost to the final value of the good is minimal ŽMcConville, 1999, p. 104.. As a result, changes in demand during the recovery period of the cycle produce stronger reactions in rates compared to market downturns.

Fig. 1. The shipping freight market.

5 Alternatively, one can regress the growth rate of the series, ⌬ X t , on 12 seasonal dummies, Di,t where i s 1,...,12. In such a case, the significance of a dummy coefficient indicates the average change in the series in that particular month compared to the previous month.

752

M.G. Ka¨ ussanos, A.H. Alizadeh-M r Economic Modelling 19 (2002) 747᎐782

To investigate empirically whether seasonal effects vary under different market conditions, Eq. Ž1. is extended to the following two-state Markov Regime Switching Seasonal regression ŽMRSS. model, which allows structural shifts in the behaviour of the time series over the estimation period Žsee, e.g. Hamilton, 1989, 1994.. 12

⌬ X t s ␤ 0,S t q

Ý ␤i ,S Qi ,t q ␧ S ,t , t

t

St s 1,2

Ž2.

is2

where St is an unobserved state variable, which determines the state of the market; that is, expansion or contraction. Therefore, seasonal parameters in Eq. Ž2. depend on the state of the market, St . The variable St follows a two-state first-order Markovian process with fixed transition probabilities, see Appendix A for more details.6 A comparison between the maximum likelihood estimates ŽMLE. of parameters in Eq. Ž2., ␤i,1 and ␤i,2 , i s 0, . . . ,12, can give an indication of differences in seasonal behaviour of freight rate series under different market conditions. A two-state MRSS model can be assumed as a combination of two linear models, which describe changes in freight rates under two different market conditions.7 These two linear models are related through the regime probabilities, which can be estimated by filtering the sample. The first model explains the seasonal behaviour of freight rates when the market is in the contraction phase Žpoint A to B in Fig. 1. while the second model explains the seasonal behaviour of freight rates when the market is in the expansion phase Žpoint B to C in Fig. 1.. We also estimate and compare two restricted versions of the MRSS model of Eq. Ž2..8 The first one in Eq. Ž3., known as the restricted MRSS model ŽRMRSS., assumes that seasonal changes under different market conditions Žexpansion or contraction. follow the same pattern and differ only in magnitude; i.e. the same seasonal coefficients are estimated under both market expansion and contraction. In this case a state-dependent coefficient, ␥ S t , which takes a value of one when the market is in expansion phase, is used to determine the magnitude of the reduction in seasonal fluctuations of freight rates under poor market conditions. 12

⌬ X t s ␤ 0,S t q ␥ S t

Ý ␤i Qi ,t q ␧S ,t , t

St s 1,2

Ž3.

is2

6 More recent studies such as those of Diebold et al. Ž1994., Filardo Ž1994. relax the assumption of fixed transition probabilities, modelling them as logistic functions or through other deterministic variables. We do not pursue these models here as the large number of seasonal parameters would make them difficult to converge. 7 Three and four-state MRSS models are also estimated to allow for more than two regimes in the freight market. Results, which are not reported here, indicate that in the case of three- and four-state MRSS models only two regimes are the prominent ones and other regimes are not significant. In addition, since the number of parameters in the model increases with the number of regimes, parameter estimates render inefficient and unreliable results. 8 We would like to thank the referee for suggesting this alternative MRS seasonal model.

M.G. Ka¨ ussanos, A.H. Alizadeh-M r Economic Modelling 19 (2002) 747᎐782

753

The next model, which is the most restricted form of the MRSS specification, is in fact a Segmented Random Walk ŽSRW. model. Mathematically ⌬ X t s ␤ 0,S t q ␧ S t ,t ,

S t s 1,2

Ž4.

This model assumes that there is no seasonal fluctuation in the series, but the trend in the series depends on the prevailing market condition. Akaike, Hannan and Quinn, and Schwarz Bayesian Information Criteria ŽAIC, HQC and SBIC. as well as R 2 are used to select the best model in the sample period.9 2.3. Stochastic seasonality An additional problem when testing for deterministic seasonality is that if the X t series is Žseasonally. stochastic, then inferences are invalidated, see Franses et al. Ž1995.. This suggests determining the stochastic properties of the series before considering the issue of deterministic seasonality. Several procedures have been proposed in the literature for testing the stochastic properties of seasonal Žperiodic. series; see for example, Dickey et al. Ž1984., Osborn et al. Ž1988., Hylleberg et al. Ž1990.; Franses Ž1991.; Franses Ž1994.. The so-called HEGY ŽHylleberg et al., 1990. approach seems to be the most promising. HEGY Ž1990. recognise that unit roots in a periodic series may exist at more than one frequency.10 They propose a procedure to test for the existence of unit roots at all possible frequencies, seasonal and non-seasonal, for quarterly series.11 The intuition behind the HEGY test is to filter the series from all possible unit roots except one and test for the significance of that unit root. Then use another filter to separate a different set of unit roots except one and test for the latter. This procedure is continued until the existence 9 Akaike Ž1969., Hannan and Quinn Ž1979., Schwarz Ž1978. information criteria are log-likelihood based model selection criteria, defined as: AIC s LL y k, HQC s LL y lnŽlnŽ n..U k and SBIC s LL y 0.5U kU lnŽ n.. Where LL, k and n represent the value of the log-likelihood, number of regressors and number of observations, respectively. The choice between alternative models with any of these criteria is based on selecting the model with the largest value of the criterion in question. These criteria are designed to penalise the log-likelihood function for extra coefficients included in the model, and in that respect take into account the desired property of parsimony Žin conjunction with goodness of fit. in estimated models. 10 For time series that can be observed more than once a year Že.g. at weekly, monthly or quarterly. the number of observations or data points within a year is called the periodicity of the data, denoted as s Že.g. s s 12 for monthly data.. Correspondingly, a series with a periodicity equal to s may contain Ž s y 1. seasonal cycles. Each cycle is associated with a seasonal frequency which can be denoted as ␣ s 2␲ jrs, j s 1,...,s y 1, and a zero frequency which is associated with no cycle. 11 A major problem with the other methods of testing seasonal unit roots, Dickey et al. Ž1984., Osborn et al. Ž1988., is that they do not recognise the possibility that unit roots may exist at different frequencies Žpossibly more than one.. In modelling, this leads to over-differencing of the series, since these test procedures require the series to be differenced twice at the zero frequency. Abeysinghe Ž1994. outlines the problems of over-differencing and under-differencing. He argues that over-differencing can result in loss of important information regarding the relation among the variables, while under differencing can lead to spurious regression results.

754

M.G. Ka¨ ussanos, A.H. Alizadeh-M r Economic Modelling 19 (2002) 747᎐782

of all possible unit roots is tested for. Beaulieu and Miron Ž1993. extend the HEGY Ž1990. method to monthly series, testing through the following equation. 12

Ž 1 y B 12 . X t s ⌬12 X t s ␣ 0 q ␤ 0 t q

12

Ý ␤i Qi ,t q Ý ␲ j Yj,ty1 is2

js1

p

q

Ý ␥ k ⌬12 X tyk q ␧ t

Ž5.

ks1

where Yj,ty1 are different seasonal filters in the form of back-shift polynomials defined in Appendix B, and ␲ j are the seasonal and non-seasonal unit root coefficients. In Eq. Ž5., tests for the significance of ␲ j , j s 1,...,12, as proposed by Beaulieu and Miron Ž1993., are equivalent to testing for seasonal unit roots at the associated frequencies. Critical values are in Beaulieu and Miron Ž1993.. The null hypothesis of the existence of a unit root at each frequency is: Ho : ␲ j s 0, j s 1,...,12 for monthly data. The alternative of stationarity is H1: ␲ j - 0 for j s 1, 2; that is, zero and one cycle per year frequencies. The condition for a unit root to exist for all other frequencies is, ␲ j s 0, for j s 2 and a joint F-test of ␲ jy1 s ␲ j s 0, for j s 4,6,8,10,12 . . . . It is possible to reject the existence of a unit root at all frequencies other that zero, if ␲ j / 0, for j s 2. The joint F-test is used because the pairs of complex roots cannot be distinguished and they always operate together. Beaulieu and Miron Ž1993. produced and tabulated the critical values for testing the significance of the parameters of interest for monthly data with different combinations of intercept, trend and seasonal dummies.12

3. Tanker markets and freight rates For analysis, the tanker sector is divided into four markets. The types of vessels operating in these markets are: very large crude carriers ŽVLCC, 160 000 dwt and over.; Suezmax Ž80 000᎐160 000 dwt.; Aframax Ž40 000᎐80 000 dwt.; and Handysize Ž20 000᎐40 000 dwt.. The two larger size tankers are involved in crude oil transportation. Aframax vessels are also involved in transportation of crude oil, however, they contribute to oil product transportation to some extent. Handysize tankers are mainly engaged in transportation of oil products,13 although they can be employed in short haul crude oil transportation at times. The employment of 12

Other methods exist to test for monthly seasonal unit roots. Franses Ž1991. uses a similar approach to Beaulieu and Miron Ž1993., with results which are equivalent to Beaulieu and Miron Ž1993.. In another approach, proposed by Franses Ž1994. the seasonal series Žmonthly or quarterly. can be decomposed into s Ž12 or 4. different annual series. Johansen’s multivariate approach is used then to determine the existence of cointegrating vectors among the annual series. A problem with this approach is that when the sample period is not long it is difficult to apply the test. 13 Handysize tankers involved in transportation of dirty petroleum products and small shipments of crude oil are examined in this study.

M.G. Ka¨ ussanos, A.H. Alizadeh-M r Economic Modelling 19 (2002) 747᎐782

755

Handysize tankers in clean product transportation is mainly due to the small parcel size of petroleum products, which rarely exceed 60 000 tons. Due to the limited number of petroleum export and import areas around the world as well as draught and capacity restrictions in certain oil terminals, ports and canals, operation of larger tankers is restricted to certain routes. For instance, the maximum size of a loaded tanker that can go through the Suez Canal is approximately 160 000 dwt. The three major routes for VLCCs are from the Persian Gulf to Ži. the Far East, Žii. North America and Žiii. north-west Europe via the Cape. Suezmax tankers are mainly operating between the Persian Gulf and north-west Europe through the Suez Canal as well as west Africa to the US Gulf and the US east coast. The major routes for Aframax tankers are from west Africa and North Sea to the US east coast, from north Africa to the Mediterranean and north Europe, and from the Persian Gulf to the Far East. Handysize tankers are mainly employed in regional ŽUS Gulf᎐US east coast, Persian Gulf, west Europe and the Far East. dirty product transportation, however, they occasionally serve long haul routes such as Middle East to Far East and Europe. In the tanker sector, the size of the vessel determines the flexibility of operation in terms of serving different routes, size cargoes and ports. Therefore, one would expect smaller size tankers to be more flexible compared to larger ones in terms of their commercial operation Žsee, for example, Kavussanos 1996a,b, 1998a,b.. Monthly spot freight rate indices for the four categories of tankers Žon worldscale 14 basis. are obtained from the Institute of Shipping Economics and Logistics ŽBremen. for the period January 1978᎐December 1996. These are constructed from the Lloyds Ship Manager database. They are plotted in Fig. 2. It can be seen that, while there are co-movements between the series in the long run, they behave quite differently in the short run. The co-movement of the series in the long run is because rates are driven by the aggregate demand for international oil transport. Differences between the behaviour of freight rates in the short term emanate from the specific factors affecting the trade in routes these different size tankers are serving. Table 1 presents descriptive statistics of the monthly series. Significantly negative coefficients of excess kurtosis for VLCC and Suezmax spot rates indicate that the distributions of these series have fat tails. The distributions of freight rates of smaller vessels are mesokurtic though. Coefficients of skewness show VLCC and 14

Tanker voyage rates are based and reported on the Worldscale index. The index, shows the break-even rate for a certain size tanker Žstandard vessel. in a particular route, normally a major route, considering all the voyage costs and expenses. The rates for all other sizes are negotiated between the brokers and charterers as a percentage of this break-even rate. The elements entering in the calculation of the break-even rate such as costs, bunker prices, port dues, etc., are revised every year. Differential tables also provide the differences to be charged for vessels employed in carrying cargoes between the ports for which the WrS is not calculated. For example, if a 250 000 dwt tanker is hired to carry a cargo from Kharg Island to a port in Malaysia, the WrS route, Kharg Island to Singapore, which is the standard route will be charges plus the differential from Singapore to the Malaysian port. The offered rate for the 250 000 dwt tanker is a percentage of the break-even rate for the loaded standard vessel, which carry the cargo from Kharg Island to the Malaysian port.

M.G. Ka¨ ussanos, A.H. Alizadeh-M r Economic Modelling 19 (2002) 747᎐782

Fig. 2. Monthly spot freight rates for different size tankers.

756

M.G. Ka¨ ussanos, A.H. Alizadeh-M r Economic Modelling 19 (2002) 747᎐782

757

Table 1 Summary statistics of logarithmic tanker freight rates, sample: 1978:1᎐1996:12

Mean S.D. CV Skewness Kurtosis ARCHŽ12. L᎐BŽ12. J᎐B

VLCC

Suezmax

Aframax

Handysize

3.69 0.35 9.55 0.00 w0.986x y0.66 w0.044x 50.34 w0.000x 975.66 w0.000x 4.18 w0.123x

4.35 0.32 7.32 y0.15 w0.369x y1.09 w0.000x 110.33 w0.000x 1365.62 w0.000x 12.05 w0.000x

4.76 0.31 6.48 0.27 w0.096x y0.38 w0.245x 97.90 w0.000x 1259.40 w0.000x 4.19 w0.123x

5.12 0.29 5.69 0.79 w0.000x y0.40 w0.225x 108.78 w0.000x 1216.77 w0.000x 24.53 w0.000x

Figures in w x are P-values.Coefficient of variation ŽCV. is a relative measure of risk, defined as the standard deviation of a series over its mean value. ARCHŽ12. is the F-test for 12th order autoregressive conditional heteroscedasticity. L᎐BŽ12. is the Ljung᎐Box Ž1978. test for 12th order autocorrelation. The 5% critical value for this statistic is 21.03.J᎐B is the Jarque᎐Bera normality test. The 5% critical value for this statistic is 5.99.

Suezmax rates to be symmetrically distributed around their means. However, significantly positive coefficients of skewness for Aframax and Handysize tankers indicate that distributions for these series are skewed to the right. There seems to be a positive relation between coefficients of variation and size; that is, freight rates for larger tankers show higher variations than for smaller size ones.15 Statistics for ARCH, autocorrelation and normality tests for tanker freight rate series, reported in the last three rows of the table, indicate that the log-levels of all time series are heteroskedastic, autocorrelated and non-normal Žexcept for VLCC and Aframax. at any conventional significance levels.

4. Estimation results Following the Franses et al. Ž1995. suggestion on testing deterministic and stochastic seasonality in a simultaneous framework, logarithmically transformed data are used to estimate three different specifications of Eq. Ž5.: Ži. with intercept only; Žii. with intercept and trend; Žiii. with intercept, trend and seasonal dummies. The final results for the selected model on tanker freight rate series are in Table 2. The lag structure and deterministic components of each equation is determined using AIC, HQC and SBIC, while ensuring that there is no autocorrelation left in the residuals. ARCH effects are not a problem as long as the ARCH coefficients show stationarity Žsee Greene, 1997, p. 570.. Since deterministic terms are found significant in most regression equations, inferences are based on regressions with all the deterministic components present; that is, a constant, trend and seasonal 15

See Kavussanos Ž1996b, 1998a,b. for a formal analysis of time varying freight rate volatilities in the tanker sector.

758

M.G. Ka¨ ussanos, A.H. Alizadeh-M r Economic Modelling 19 (2002) 747᎐782

Table 2 Seasonal unit roots test results for monthly tanker freight rates, sample: 1978:1᎐1996:12 12

Ž1 y B12 . Xt s ⌬12 Xt s ␣0 q ␤ 0 t q

is2

Frequency

0 ␲ "␲r2 "2␲r3 "␲r3 "5␲r6 "␲r6

␣0 2.68 ␤0 ␲1 s 0 ␲2 s 0 F3,4 ␲3 ,␲4 s 0 F5,6 ␲5 ,␲6 s 0 F7,8 ␲7 ,␲8 s 0 F9,10 ␲9 ,␲10 s 0 F11,12 ␲11 ,␲12 s 0

Lags of ⌬12 Xt R2 DW L᎐BŽ12. ARCHŽ12. White J᎐B test LL AIC HQC SBIC

p

12

Ý ␤i Qi,t q Ý ␲j Yj,ty1 q Ý ␥k ⌬12 Xtyk q ␧t js1

Eq. Ž5.

ks1

VLCC

Suezmax

Aframax

Handysize

2.49 2.01 y2.72 y4.11 20.18 20.24 16.54 32.73 21.61

2.45 1.91 y2.57 y4.83 20.70 22.26 18.72 20.34 30.45

2.42 1.50 y2.56 y3.97 23.12 17.09 20.21 28.69 26.12

0.85 y2.50 y4.60 23.98 14.46 22.30 34.49 22.69

0 0.86 1.98 4.76 w0.965x 1.66 w0.077x 0.532 w0.465x 7.60 w0.022x 162.932 137.632 120.586 95.441

0 0.87 2.00 5.19 w0.951x 5.18 w0.000x 8.21 w0.004x 4.80 w0.090x 155.605 130.605 113.559 88.413

0 0.87 2.00 5.50 w0.938x 0.78 w0.666x 10.3 w0.001x 2.65 w0.266x 171.636 146.636 129.591 104.445

0 0.77 1.98 3.94 w0.995x 1.69 w0.071x 0.00 w0.987x 23.95 w0.000x 76.086 51.086 33.872 8.440

The regression includes a constant, a trend and seasonal dummies.LL, AIC, HQC and SBIC represent log-likelihood, Akaike Ž1969., Hannan and Quinn Ž1979., Schwarz Ž1978. information criteria, respectively, which are used to select the number of lagged dependent variables in each equation. AIC s LL y k, HQC s LL y lnŽlnŽ n..U k, SBIC s LL y 0.5kU lnŽ n., where k and n represent the number of regressors and observations, respectively. The model with the highest AIC, HQC and SBIC is the most parsimonious model. Figures in w.x are P-values. DW is the Durbin᎐Watson test for first order serial correlation. L᎐BŽ12. is the Ljung᎐Box test for 12th order serial correlation in the residuals. ARCHŽ12. is the F-test for 12th order ARCH effects. White is the White Ž1980. test for heteroscedasticity. J᎐B is the Jarque and Bera Ž1980. test for normality. The 5% critical value for this statistic is ␹ 2 Ž2. s 5.99.1, 2.5 and 5% critical values for the seasonal unit root test statistics are wSource: Beaulieu and Miron Ž1993.x:1%: t Ž␲ 1. s y3.83, t Ž␲ 2. s y3.31, F s 5.25;2.5%: t Ž␲ 1. s y3.54, t Ž␲ 2. s y3.02, F s 7.14;5%: t Ž␲ 1. s y3.28, t Ž␲ 2. s y2.75, F s 6.23.

dummies. Having obtained well-specified equations, seasonal unit root tests are performed next as described earlier. 4.1. Stochastic seasonality In seasonal unit root test equations, the F-statistics for ␲ i,iq1 s 0 Ž i s 3,5,7,9,11., testing for the joint existence of complex seasonal unit roots at the corresponding frequencies Ž"␲r2, "2␲r3, "␲r3, "5␲r6, "␲r6., are rejected at the 5% level

M.G. Ka¨ ussanos, A.H. Alizadeh-M r Economic Modelling 19 (2002) 747᎐782

759

of significance. The null hypothesis of unit root at the ␲ frequency, or six cycles per year, is also rejected for all freight rate series. Contrary to the above, tests for the existence of a unit root at zero Žlong run. frequency, i.e. ␲ 1 s 0, point to there being a unit root at the zero frequency for all the series examined.16 Therefore, it can be argued that the existence of stochastic seasonality at all the seasonal frequencies is rejected for all sub-markets in the tanker sector for the period 1978᎐1996. However, the existence of a unit root at zero frequency could not be rejected for all the series indicating that freight rate series are non-stationary of order one,17 I Ž1.. The existence of unit roots at zero frequency in tanker freight rate series suggests that these series are serially correlated and consequently have a long memory, which in turn implies that the effect of shocks to these series persist. As a result, when modelling tanker freight rates in a univariate framework, ARIMA models are appropriate, while modelling of rates in a multivariate framework requires application of vector autoregression ŽVAR. techniques, which incorporates the degree of integration of these series. 4.2. Deterministic seasonality Testing the existence of deterministic seasonality when stochastic seasonality is not present through Eq. Ž5. will reduce the power of the test. This is due to the loss of degrees of freedom in estimating the parameters of the seasonal unit roots. Moreover, excluding seasonal filters from Eq. Ž5. in order to test only the deterministic seasonality is not appropriate, since the dependent variable is the 12th difference of the series rather than the monthly growth rate of the series. Therefore, the existence of deterministic seasonality in tanker sub-markets is investigated through Eq. Ž1. for the period January 1978 to December 1996. The results of the most parsimonious model in each sub-market of the tanker industry are in Table 3. Diagnostic tests reveal that some equations have non-spherical disturbances, in which case, the variance᎐covariance matrices are corrected for heteroscedasticity andror serial correlation using White Ž1980. or Newey and West Ž1987. estimates. In equations with significant ARCH effects it is found that these are stationary. As a result Žsee Greene, 1997, p. 570. the unconditional variance of the residuals is constant and OLS yield the BLUE. Significance of a t statistic for ␤i , i s 1,2, . . . ,12, parameters Žmonths. is an indication of a significant increase or decrease in monthly freight rate growth at a 16

It should be noted that seasonal unit root tests Žas well as DF and ADF tests. for monthly shipping freight rates depend on the sample period examined. For example, using a short sample period may reveal non-stationary series, however, testing over a longer sample which includes the whole business cycle may reverse the result. 17 The results are also confirmed by the Franses and Hobijn Ž1997. seasonal unit root test. This test uses a similar approach as HEGY Ž1990. and only differs from HEGY Ž1990. in the method used for linearising the seasonal back shift polynomial. Also ordinary unit root tests such as DF, ADF and Philips᎐Perron tests confirm that series are non-stationary in the levels, but are stationary in logarithmic first differences. The results are available from the authors on request.

760

M.G. Ka¨ ussanos, A.H. Alizadeh-M r Economic Modelling 19 (2002) 747᎐782

Table 3 Deterministic seasonality in tanker freight rate series; Sample: 1978:1᎐1996:12 12

⌬ Xt s ␤0 q ⌺ ␤i Qi,t q ␧ t Eq. Ž1. is2

Month

Coef

Const.

␤0

0.004 Ž0.350.

0.004 Ž0.443.

0.003 Ž0.347.

Jan. Feb. Mar. Apr. May June July Aug. Sept. Oct. Nov. Dec.

␤1 ␤2 ␤3 ␤4 ␤5 ␤6 ␤7 ␤8 ␤9 ␤10 ␤11 ␤12

I0.110 (I2.933) I0.067 (I1.695)

I0.049 (I2.034)

I0.030 (I1.900)

I0.048 (I2.056)

I0.041 (I2.662)

R2 L᎐BŽ1. L᎐BŽ12. ARCHŽ12. White J᎐B LL AIC HQC SBIC

VLCC

Suezmax

Aframax

Handysize 0.002 Ž0.239.

I0.059 (I2.701)

0.105 (2.951) I0.052 (I3.068)

0.066 (1.847)

0.105 (4.321)

0.110 (4.725) 0.033 (1.832)

0.077 (4.894)

0.059 1.464 w0.226x 24.44 w0.017x 1.039 w0.413x 0.463 w0.496x 20.58 w0.000x 53.461 56.870 53.461 48.432

0.074 0.915 w0.339x 10.43 w0.578x 1.774 w0.054x 0.121 w0.728x 18.00 w0.000x 152.513 155.240 152.513 148.489

0.084 1.702 w0.192x 17.60 w0.128x 1.857 w0.042x 0.008 w0.929x 46.54 w0.000x 142.009 146.100 142.009 135.974

0.048 2.516 w0.113x 16.82 w0.156x 1.264 w0.242x 0.812 w0.367x 74.59 w0.000x 160.615 162.660 160.615 157.597

See notes in Table 2. t-statistics in brackets are corrected for heteroscedasticity and serial correlation using the Newey᎐West method where appropriate.The coefficient for January dummies, ␤ 1 , are calculated as ␤ 1 s yŽ ␤ 2 q ...q ␤ 12 .. Standard errors for January dummies are calculated from the variance᎐covariance matrix of the coefficients, see footnote 4 in the text for details.Figures in bold represent significant coefficients.

particular month compared to the average over the sample period. The results indicate that freight rates in all size tankers experience a significant increase in November compared to the average monthly growth rate of zero Ž ␤ 0 s 0. over the period. This increase is 6.6% for VLCCs, 10.5% for Suezmax, 11.0% for Aframax, 7.7% for Handysize rates. The increase in freight rates in early winter ŽNovember. is due to the increase in demand for oil by oil-importing countries Žcompanies., which are in the process of building up sufficient inventory levels Žcrude oil. for winter.18 Aframax spot rates continue to rise in December by 3.3% due to the 18 The coldest winter months are usually January and February; so excess demand for crude oil is generated sometime in November to give enough lead time for crude to be transported and refined Žfrom the long haul, Middle East routes., which can take 6᎐8 weeks.

M.G. Ka¨ ussanos, A.H. Alizadeh-M r Economic Modelling 19 (2002) 747᎐782

761

increase in demand for small shipments, usually from short haul routes Žsuch as from Venezuela and Mexico to the US., to satisfy any remaining inventory requirements for the winter months. In January, VLCC rates decrease significantly by 11.0% because of the decline in the need for inventory building using large vessels after the cold season in the Northern Hemisphere. This decline in demand continues and results in a further 6.7% drop in VLCC rates in February. The February dummy shows a significant 4.9% drop in both Suezmax and 3.0% in Aframax rates. During April, Suezmax, Aframax and Handysize rates decline even further by 4.8, 4.1 and 5.9%, respectively; this can be linked to the decline in the level of petroleum imports and trade activities during the spring months, and the routine maintenance program of refineries around the US Gulf and the Far East which takes place during this period. VLCC rates show a significant rise of 10.5% in June. This is due to inventory building that takes place after the end of routine maintenance programs of refineries and terminals during April and May, increase in Japanese imports, and to stock up for the US driving season taking place from mid-July to the end of August. A decline of 5.2% in Aframax freight rates is observed in July. This is thought to be a result of the decline in demand for those vessels operating in the Mediterranean as well as annual maintenance in the North Sea terminals, which use primarily Aframax tankers. The overall variations of tanker freight rates explained purely by seasonal factors are measured by the adjusted coefficients of determination, R 2 . These are 5.9, 7.4, 8.4 and 4.8% for VLCC, Suezmax, Aframax and Handysize, respectively. 4.3. Seasonality patterns under different market conditions To allow for different seasonality effects under expansionary and contractionary periods in freight markets, MRSS models corresponding to Eqs. Ž2. ᎐ Ž4. are estimated and presented in Tables 4᎐6. For example, estimation results from the most parsimonious MRSS model of Eq. Ž2. are presented in Table 4. Under each size category, two columns are designated as ‘Expansion’ and ‘Contraction’, distinguishing between expansion and contraction periods, respectively. Constant terms in the regressions represent the average monthly growth or decline in rates over the estimation period, under different market conditions. Comparison of the results of other coefficients shows that seasonal fluctuations in rates during market recoveries, when supply is inelastic, are stronger compared to market downturns when the supply schedule is thought to be elastic. This can be seen, for instance, in the case of Handysize rates; they show significant declines of 6.5 and 12.7% in February and April, respectively, in expansionary markets, but no significant change in weak markets. A similar situation is observed when demand rises seasonally in November and December; that is, rates rise by 13.8 and 6.8% in these 2 months, when the market is strong, but show no significant rise when markets are weak. Differences in seasonal coefficients under different market conditions are also evident in the Aframax sector. The pre-winter rise in Aframax rates is 9.4, 31.3 and

is 2

Ž expansion . Eq. Ž2 . 2 Ž contraction . Pr ŽSt s 2
Pr ŽSt s 1
St s

½

VLCC Expansion Ž ␤ i,1 . Const.

0.028 ( 2.605 )

Jan. Feb.

I0.069 ( I2.221 )

1

Suezmax Contraction Ž ␤ i,2 . y0.012 Žy0.530 . I0.201 ( I3.956 )

Expansion Ž ␤ i,1 . 0.012 Ž0.696 .

I0.121 ( I2.381 )

Mar. I0.148 (I2.698 ) 0.138 ( 2.543 )

Apr. May June July Aug.

I0.069 ( I2.697 ) 0.154 ( 5.407 ) 0.116 ( 4.192 )

Aframax

Handysize

Contraction Ž ␤ i,2 .

Expansion Ž ␤ i,1 .

Contraction Ž ␤ i,2 .

0.0005 Ž0.070 .

0.005 Ž0.340 . I0.260 ( I4.523 ) I0.075 ( I2.191 )

0.003 Ž0.570 . 0.070 ( 3.134 )

I0.047 ( I2.588 )

I0.149 (I3.421 ) 0.104 ( 1.783 )

I0.062 ( I3.552 )

Expansion Ž ␤ i,1 . 0.005 Ž0.450 .

I0.065 ( I1.681 )

I0.127 (I5.759 )

0.201 ( 3.956 ) I0.053 ( I1.716 )

I0.112 ( I3.338 )

I0.040 ( I2.250 )

0.218 ( 5.427 )

0.094 ( 2.346 ) 0.313 ( 6.590 )

0.032 ( 2.158 )

Sept. Oct. No¨ .

I0.106 ( I3.759 ) 0.041 ( 2.344 )

0.138 ( 2.732 )

Contraction Ž ␤ i,2 . y0.0001 Žy0.032 .

M.G. Ka¨ ussanos, A.H. Alizadeh-M r Economic Modelling 19 (2002) 747᎐782

12

⌬ X t s ␤ 0,S t q ⌺ ␤ i,S t Q i,t q ␧ S t,t ,

762

Table 4 Estimates of Markov Regime Switching Seasonal Ž MRSS. model for variations in spot rates under different market conditions Žexpansion and contraction ., Sample: 1978:1 to 1996:12

Table 4

Ž Continued . 12

is 2

Pr ŽSt s 1
St s

½

VLCC Expansion Ž ␤ i,1 .

1

Suezmax Contraction Ž ␤ i,2 .

Expansion Ž ␤ i,1 .

Aframax Contraction Ž ␤ i,2 .

Dec.

P 12 P 21 ␴1 ␴2 R2 L ᎐B Ž1 . L ᎐B Ž12 . ARCH Ž12 . White J᎐B LL AIC HQC SBIC

Expansion Ž ␤ i,1 .

Handysize Contraction Ž ␤ i,2 .

0.084 ( 2.208 ) 0.149 0.201 0.070 0.220 0.214 2.988 18.98 1.041 0.024 63.29 90.581 77.581 68.717 55.642

Ž2.224 . Ž2.549 . Ž7.245 . Ž14.88 .

w0.084 x w0.089 x w0.413 x w0.877 x w0.000 x

0.041 0.086 0.069 0.158 0.241 1.155 9.749 1.590 1.105 43.93 193.07 180.070 171.206 158.131

Ž2.465 . Ž2.196 . Ž13.43 . Ž13.03 .

w0.282 x w0.638 x w0.097 x w0.293 x w0.000 x

0.026 0.080 0.087 1.140 0.413 2.070 16.28 1.484 3.646 1.445 189.01 171.010 158.737 140.632

Expansion Ž ␤ i,1 .

Contraction Ž ␤ i,2 .

0.068 ( 2.063 ) Ž1.403 . Ž1.531 . Ž17.14 . Ž16.87 .

w0.100 x w0.178 x w0.132 x w0.056 x w0.486 x

0.016 0.020 0.076 0.137 0.147 5.254 21.15 1.855 1.600 59.96 184.18 174.180 167.362 157.304

Ž1.068 . Ž1.743 . Ž12.74 . Ž17.82 .

w0.022 x w0.048 x w0.042 x w0.206 x w0.000 x

See notes in Tables 2 and 3. t-statistics in brackets are corrected for heteroscedasticity and serial correlation using the Newey᎐West method where appropriate.The coefficient for January dummies, ␤ 1, are calculated as ␤ 1 s y Ž ␤ 2 q ...q ␤ 12 .. Standard errors for January dummies are calculated from the variance᎐covariance matrix of the coefficients, see footnote 4 in the text for details.Figures in bold represent significant coefficients. ␤ i,1 and ␤ i,2 refer to coefficient estimates under expansionary Žstate 1 . and contractionary Žstate 2 . periods, respectively. P12 , P 21 are regime switching probabilities, and ␴ 1, ␴ 2 are standard errors of regression for each regime.

M.G. Ka¨ ussanos, A.H. Alizadeh-M r Economic Modelling 19 (2002) 747᎐782

Ž expansion . Eq. Ž2 . 2 Ž contraction . Pr ŽSt s 2
⌬ X t s ␤ 0,S t q ⌺ ␤ i,S t Q i,t q ␧ S t,t ,

763

764

M.G. Ka¨ ussanos, A.H. Alizadeh-M r Economic Modelling 19 (2002) 747᎐782

8.4% in October, November and December during market expansions as opposed to a 3.2 and 7.0% rise in November and January in weak markets. The after winter decline in Aframax rates during market recoveries is estimated to be 26.0, 7.5 and 14.9% in January, February and April, respectively. The decline in Aframax rates in the post winter periods is reduced to 6.2% and takes place only in March during market downturns. There is also a 10.4% rise in Aframax rates in May and an 11.2% decline in July, before and after the US driving season. The July decline in Aframax rates reduces to 4.0% in the same month in market downturns.19 Suezmax seasonal movements of freight rates are very similar to those of Aframax tankers. This is expected since these two size tankers are close substitutes in many routes and trades Že.g. West Africa and North Sea to US Gulf and US east coast.. During market expansions, Suezmax rates show a pre-winter rise of 21.8% in November as opposed to a rise of only 4.1% in the same month when markets are weak. The after-winter drop in Suezmax rates is found to be 12.1 and 14.8% in February and April, respectively, when the market is in recovery. During market recessions this decline is reduced to 4.7% in March. Similar to Aframax, Suezmax rates also show a rise of 13.8% in May and a drop of 5.3% in July during a recovery, but no change in market troughs. In contrast to results for smaller size tankers, the MRSS model results for VLCCs are not as consistent. This may be due to the high volatility of VLCC rates compared to smaller size tankers, which when coupled with the large number of parameters in the seasonal Markov Switching model makes convergence to the global maximum difficult. Thus, we do not get the expected pre-winter rise in rates, as in other sizes, and the results are peculiar. However, adjusted R 2 values for all MRSS models show higher values compared to the deterministic seasonal models of Table 3. These are 0.214, 0.241, 0.413 and 0.147 for VLCC, Suezmax, Aframax and Handysize regressions, respectively. These are increases of three to four times in comparison to Eq. Ž1. estimates. Results of the most parsimonious RMRSS models of Eq. Ž3. are in Table 5. Coefficients of ␥ S t in each equation represent the amount by which seasonal coefficients should be multiplied with to yield the seasonal effect of the respective month under a market downturn. In that respect seasonal coefficients are not allowed to change in magnitude separately for each month between market conditions, i.e. the pattern of seasonality remains the same between market conditions. This is more restrictive in comparison to the MRSS of model 4. ␥ S t is set equal to one across models, when the market is in the expansionary phase. The fact that these coefficients are insignificant implies that there is no seasonal variation in tanker freight rates during market troughs, while there are sharp fluctuations in freight rates during periods of market recovery. The patterns of seasonality for each size during the market recovery are similar to those from the

19

As mentioned earlier, changes in demand for freight services takes place 4᎐6 weeks prior to the actual increase or decrease in demand for petrol due to the time needed for transportation and refining.

12

⌬ Xt s ␤0,S t q ␥St ⌺ ␤i Qi,t q ␧S t,t , is2

PrŽSt s 1
½

␥St s 1

Eq. Ž3. ␥St s ␥St Ž contraction . PrŽSt s 2
VLCC

␥St Cons. Jan. Feb. Mar. Apr. May. June July Aug. Sept. Oct. No¨ . Dec. P12 P21 ␴1 ␴2

Ž expansion .

y0.254 Žy0.719. 0.012 Ž1.317. I0.143 (3.130) I0.118 (I3.882)

Suezmax 0.104 Ž1.391. 0.011 Ž0.643.

Aframax

Handysize

I0.119 (I2.969)

0.032 Ž0.587. y0.001 Žy0.075. I0.165 (I2.058) I0.094 (I1.943)

0.046 Ž0.322. 0.005 Ž0.407.

I0.128 (I2.543) 0.104 (1.712)

I0.180 (I2.639) 0.130 (2.276)

0.091 (2.158) 0.120 (5.351)

I0.063 (I1.830)

I0.151 (I3.643)

0.050 (1.722)

0.235 (3.850)

0.116 (3.668) 0.345 (6.668) 0.119 (3.727)

0.138 (3.849) 0.070 (2.528)

0.102 Ž2.527. 0.259 Ž3.293. 0.116 Ž11.05. 0.282 Ž9.563.

0.048 Ž2.062. 0.088 Ž2.145. 0.071 Ž14.68. 0.158 Ž11.14.

0.015 Ž1.279. 0.065 Ž1.384. 0.095 Ž17.85. 0.140 Ž9.040.

0.016 Ž0.996. 0.021 Ž1.405. 0.076 Ž11.96. 0.137 Ž17.82.

I0.064 (I2.112) I0.122 (I4.228)

M.G. Ka¨ ussanos, A.H. Alizadeh-M r Economic Modelling 19 (2002) 747᎐782

Table 5 Estimates of the Restricted Markov Regime Switching model for seasonality ŽRMRSS. Sample: 1978:1 to 1996:12

765

766

12

is2

PrŽSt s 1
R2 L᎐B Ž1. L᎐B Ž12. ARCH Ž12. WIHTE J᎐B LL AIC HQC SBIC

␥St s 1

Ž expansion . Eq. Ž3. ␥St s ␥St Ž contraction . PrŽSt s 2
⌬ Xt s ␤0,S t q ␥St ⌺ ␤i Qi,t q ␧S t,t ,

½

VLCC

Suezmax

Aframax

Handysize

0.132 2.502 w0.114x 23.87 w0.021x 1.193 w0.290x 2.087 w0.149x 74.66 w0.000x 81.63 70.630 63.130 52.066

0.217 1.701 w0.192x 9.503 w0.659x 1.562 w0.105x 1.603 w0.205x 37.07 w0.000x 189.62 178.620 171.120 160.056

0.369 2.926 w0.087x 14.05 w0.297x 0.958 w0.489x 2.502 w0.114x 0.157 w0.924x 181.19 167.190 157.645 143.563

0.145 5.217 w0.022x 21.03 w0.050x 1.839 w0.044x 1.302 w0.253x 59.62 w0.000x 184.23 174.230 167.412 157.354

See notes in Tables 2 and 3. t-statistics in brackets are corrected for heteroscedasticity and serial correlation using the Newey᎐West method where appropriate. The coefficient for January dummies, ␤ 1 , are calculated as ␤ 1 s yŽ ␤ 2 q ...q ␤ 12 .. Standard errors for January dummies are calculated from the variance᎐covariance matrix of the coefficients, see footnote 4 in the text for details. Figures in bold represent significant coefficients.

M.G. Ka¨ ussanos, A.H. Alizadeh-M r Economic Modelling 19 (2002) 747᎐782

Table 5 Ž Continued.

Table 6 Estimates of the Markov Regime Switching Segmented Random Walk model ŽSRW . Sample: 1978:1᎐1996:12

⌬ X t s ␤ 0,S t q ␧ St,t ,

S t s 1 Ž expansion . S t s 2 Ž contraction .

Pr ŽSt s 1
␤ 0,1 ␤ 0,2 P 12 P 21 ␴1 ␴2 R2 L ᎐B Ž1 . L ᎐B Ž12 . ARCH Ž12 . White J᎐B LL AIC HQC SBIC

Eq. Ž4 .

Pr ŽSt s 2
Suezmax

Aframax

Handysize

0.019 Ž1.558 . y0.014 Žy0.450 .

0.011 Ž0.479 . y0.0006 Žy0.079 .

0.004 Ž0.551 . y0.002 Žy0.063 .

0.004 Ž0.273 . y0.0004 Žy0.047 .

0.113 0.155 0.115 0.257

Ž1.395 . Ž2.067 . Ž9.618 . Ž9.243 .

Na 2.502 w0.114 x 23.87 w0.021 x 1.193 w0.290 x 2.087 w0.149 x 74.66 w0.000 x 68.21 62.210 58.119 52.084

0.049 0.087 0.072 0.181

Ž1.847 . Ž1.943 . Ž12.63 . Ž9.890 .

Na 0.382 w0.535 x 13.19 w0.355 x 2.161 w0.015 x 93.27 w0.000 x 20.32 w0.000 x 176.85 170.850 166.759 160.724

0.008 0.042 0.099 0.216

Ž0.595 . Ž0.846 . Ž14.48 . Ž9.387 .

Na 0.316 w0.574 x 20.19 w0.063 x 4.229 w0.000 x 53.94 w0.000 x 71.45 w0.000 x 163.57 157.570 153.479 147.444

0.018 0.021 0.075 0.149

Ž0.879 . Ž1.294 . Ž11.21 . Ž16.45 .

Na 1.200 w0.273 x 19.24 w0.083 x 1.495 w0.128 x 23.27 w0.000 x 139.1 w0.000 x 174.20 168.200 164.109 158.074

767

The regression includes a constant, a trend and seasonal dummies.LL, AIC, HQC and SBIC represent log-likelihood, Akaike Ž1969 ., Hannan and Quinn Ž1979 ., Schwarz Ž1978 . information criteria, respectively, which are used to select the number of lagged dependent variable in each equation. AIC s LL y k, HQC s LL y ln Žln Ž n ..U k, SBIC s LL y 0.5k U ln Ž n ., where k and n represent number of regressors and observations, respectively. The model with the highest AIC, HQC and SBIC is the most parsimonious model. Figures in w.x are P-values.DW is the Durbin ᎐Watson test for first order serial correlation.L ᎐B Ž12 . is the Ljung᎐Box test for 12th order serial correlation in the residuals. ARCH Ž12 . is the F-test for 12th order ARCH effects. White is the White Ž1980 . test for heteroscedasticity.J ᎐B is the Jarque and Bera Ž1980 . test for normality. The 5% critical value for this statistic is ␹ 2 Ž2 . s 5.99.1, 2.5 and 5% critical values for the seasonal unit root test statistics are wSource: Beaulieu and Miron Ž1993 .x :1%: t Ž␲ 1 . s y3.83, t Ž␲ 2 . s y3.31, F s 5.25;2.5%: t Ž␲ 1 . s y3.54, t Ž␲ 2 . s y3.02, F s 7.14;5%: t Ž␲ 1 . s y3.28, t Ž␲ 2 . s y2.75, F s 6.23. t-statistics in brackets are corrected for heteroscedasticity and serial correlation using the Newey᎐West method where appropriate. The coefficient for January dummies, ␤ 1, are calculated as ␤ 1 s y Ž ␤ 2 q ...q ␤ 12 .. Standard errors for January dummies are calculated from the variance᎐covariance matrix of the coefficients, see footnote 4 in the text for details.Figures in bold represent significant coefficients.

M.G. Ka¨ ussanos, A.H. Alizadeh-M r Economic Modelling 19 (2002) 747᎐782

½

768

M.G. Ka¨ ussanos, A.H. Alizadeh-M r Economic Modelling 19 (2002) 747᎐782

unrestricted MRSS model; that is, freight rates increase during the winter months and decline during the spring and summer months. For example, significant coefficients of January and February dummies in the VLCC model indicate 14.3 and 11.8% drop in VLCC rates in respective months when the market is in the recovery stage. Also, positive and significant July, August and November coefficients show 9.1, 12 and 5% rise in VLCC rates, respectively, under a market recovery. Investigating the seasonal behaviour of Suezmax and Aframax freight rates using the RMRSS model reveals similar seasonal patterns as in the case of the unrestricted MRSS models when the market is in the expansionary phase. Results for Handysize rates are almost identical to those of the MRSS model. This is because in both models it is found that there is no seasonality for Handysize vessels during recession periods. This makes sense, given that these smaller size vessels can switch between routes and trades in bad market conditions, when freight rates are low. However, during tight market situations, the rates of these vessels are also affected, as they are called to service to fill the seasonal capacity gap that riddles across segments of the tanker industry. Table 6 reports the results from an even more restricted Markov Regime Switching model, which contains no seasonal elements. In this Segmented Random Walk model ŽSRW. of Eq. Ž4. the growth rate of the series is allowed to be different under different market conditions. It is used as a benchmark model to compare the rest. Results indicate that the standard deviation of freight rates under tight market conditions, ␴ 2 , is larger than the standard deviation of the rates when weak market conditions prevail, ␴ 1 , in line with evidence on freight markets Žsee Kavussanos, 1996b.. However, there seems to be no significant difference between the mean growth rates of freight rates under different market conditions. Comparison of AIC, HQC and R 2 across different models; i.e. SSDV, MRSS, RMRSS and SRW models, for each size vessel show that the unrestricted MRSS model fits the data better in every case, where models with higher values of AIC and HQC are preferable. The results from using the SBIC are clear in rejecting SSDV in every case, but not as clear in their choice of the best model; MRSS is selected for VLCC, SRW for Suezmax and Aframax, and RMRSS for Handysize. However, based on the AIC and the HQC, the R 2 and also the additional economic information observed from the varying estimated coefficients for each season by market state, the MRSS model seems to perform best.20 That is, based on statistical criteria the MRSS model is preferable, which when coupled with our economic knowledge of freight markets Žsee p. 6 and Fig. 1. reinforce our choice of MRSS as the best model. A little further discussion is due on the results of the selected MRSS models of Eq. Ž2.. Estimated P12 and P21 values reported for each model, in Table 5, show 20

In addition, since the likelihood function in MRS models has a mixture distribution, and the fact that the estimated log-likelihood for models is not the global maximum log-likelihood Žsee Hamilton, 1994, p. 689., we believe that applying maximum penalty to the estimated log-likelihood for inclusion of regressors, as in the case of SBIC, might not be appropriate.

M.G. Ka¨ ussanos, A.H. Alizadeh-M r Economic Modelling 19 (2002) 747᎐782

769

the probability of change in the state of the market from expansionary to contractionary and vice-versa, respectively. The duration of a market expansion or recession for each tanker sector can be calculated as 1rP12 and 1rP21. Results indicate that in the Suezmax sector the average duration of expansion and recession periods in this market are 24 and 12 months, respectively. In the Aframax market, the average duration of expansion and recession periods are 38 and 13 months, respectively. The average duration of expansion and recession periods in the Handysize market are 62 and 50 months, respectively. The transition probability values for VLCC rates suggest an expansionary period of 7 months and a recessionary period of approximately 5 months, which indicate that the whole cycle in this market may last 1 year. The fact that the cycles in the VLCC market are shorter compared to the other tanker sub-markets may be attributed to the domination of seasonal cycles in this market over the cyclical variations and the higher volatility levels in this market compared to markets for smaller tankers. The combination of these two effects makes it more difficult to disentangle seasonal and cyclical variations in the market for VLCCs.21 In general, the results suggest that; first, expansionary periods in the tanker freight markets last longer compared to contractionary periods. This may be because additions to the stock of the fleet, through new buildings in the expansionary periods, take longer in comparison to deletions, which take place through the much faster process of scrapping during contractionary periods. Second, freight rates for larger tankers show more cyclical changes Žboth expansionary and contractionary. compared to smaller tankers; a result which is in line with the argument of positive relation between vessel size and freight rate volatilities in tanker freight markets Žsee Kavussanos 1996b, 1998a,b.. Fig. 3 is used to illustrate how the Markov Regime Switching model is applied to derive results under different market conditions for Handysize rates. Panel A plots the log of the Handysize spot rate. It forms the basis upon which to distinguish between expansionary and contractionary conditions in the freight market. Such regime probabilities over the estimation period are shown in panel B. When the market is in expansionary phases regime Žstate. probabilities are close to zero, while when the market is in contractionary phases state probabilities tend to one.22 It can be seen that periods of market expansion are identified correctly by the model to be 1978:6᎐1981:6 and 1986:2᎐1992:6, while recession periods are 1981:6᎐1986:1 and 1992:6᎐1994:12. Panel C compares the actual charges in freight rates with the fitted seasonal changes, the latter tracking well the former ones. Panel D shows the seasonal fluctuations in freight rates based on the MRSS model, utilising the regime switching probabilities of panel B; clearly, the magnitude of seasonal changes in freight vary according to the state of the market, confirming visually the numerical results of Table 5. In general, the results for the tanker sector suggest that, seasonality in freight 21

We would like to thank an anonymous referee for pointing this out. Details of how the Markov Regime Switching model estimates the regime switching probabilities are in Appendix A. 22

770 M.G. Ka¨ ussanos, A.H. Alizadeh-M r Economic Modelling 19 (2002) 747᎐782 Fig. 3. Markov Regime Switching Model for Handysize spot rates. Panels: Ža. log-level of Handysize spot rates; Žb. regime probabilities determining market being in a particular state; Žc. seasonal changes of freight rates under different market conditions; and Žd. fitted and actual changes in Handysize freight rates.

M.G. Ka¨ ussanos, A.H. Alizadeh-M r Economic Modelling 19 (2002) 747᎐782

Fig. 3. Ž Continued..

771

772

M.G. Ka¨ ussanos, A.H. Alizadeh-M r Economic Modelling 19 (2002) 747᎐782

rates is deterministic rather than stochastic and is quite different across the size of vessels and under different market conditions. More specifically: Ž1. the levels of freight rates for different size tankers increase during November and December and drop after January; Ž2. precise seasonal patterns are related to size; Ž3. seasonal movements are found to be asymmetric under different market conditions; that is, seasonal variations are more pronounced during market expansions compared to market contractions, which is in line with the expected low and high elasticities of the supply for freight services under the respective market conditions.

5. Forecasting performance The aim of the analysis so far has been to understand and measure seasonality patterns in different segments of the tanker sector. A useful exercise for business decisions is to consider the possibility of forecasting seasonality using a particular model. This may not be the same model as the one used so far to understand seasonality patterns. This section considers the out-of-sample forecasting performance of competing models ŽSSDV, SRW, MRSS and RMRSS. for each sub-sector by using the most parsimonious specification of each model to produce recursive 23 multiple step ahead forecasts over the period January 1997᎐December 1998. Results of R.M.S.E. values for forecasts for 1᎐6, 9 and 12 months ahead are reported in Table 7, and the Diebold and Mariano Ž1995. test is used to compare the forecasting performance of models. While forecasting changes in tanker freight rates at time t q m using the standard seasonal model of Eq. Ž1. is straightforward, forecasting with MRS models of Eqs. Ž2. ᎐ Ž4. requires determination of regime probabilities at time t q m Žsee for example, Engle, 1994 and Marsh, 2000. prior to forecasting changes in freight rates at t q m. Therefore, in order to use the MRS models to forecast ˆ changes in tanker freight rates at time t q m, estimates of the transition matrix, P, and the filtered regime probabilities at time t, ˆ pŽ st s 1. and ˆ pŽ st s 2., are used to predict regime probabilities; i.e. probabilities of the state variable, Stqm at time t q m; that is, ˆ pŽ stqm s 1. and ˆ pŽ stqm s 2..

ˆp11 Žˆ p Ž stqm s 1 . ˆ p Ž stqm s 2 .. s Ž ˆ p Ž st s 1. ˆ p Ž st s 2 .. ˆp 21

ž

ˆp12 ˆp 22

m

/

Ž6.

Once regime probabilities at time t q m are determined, forecasts of changes in 23 Recursive forecasts are made by first estimating each model over the sample period ŽJanuary 1978 to December 1996. and forecasting the series 1᎐24 steps ahead; that is, January 1997 to December 1998. Next, the estimation period is increased by one observation to cover the period January 1978 to January 1997 and forecasting the series 1᎐23 steps ahead; that is, February 1997 to December 1998. This process is repeated until the estimation period covered the period January 1978 to November 1998 and 1 step ahead forecast for December 1998 is obtained. In this way, 24 observations for 1-step ahead forecasts, 23 observations for 2-step ahead forecasts, and so on, are obtained which are then used to calculate the R.M.S.E. values for 1᎐12 steps ahead forecasts.

Table 7 Comparison of forecasting performance of different seasonal models in forecasting changes in freight rates for different size tankers Handysize

2-Month ahead

3-Month ahead

4-Month ahead

5-Month ahead

6-Month ahead

9-Month ahead

SSDV

SRW

SSDV SRW RMRSS MRSS

24

0.0793 1.0573U 0.9923 1.0148

0.0838 0.9386UU 0.9598UU

SSDV SRW RMRSS MRSS

23

SSDV SRW RMRSS MRSS

22

SSDV SRW RMRSS MRSS

21

SSDV SRW RMRSS MRSS

20

SSDV SRW RMRSS MRSS

19

SSDV SRW RMRSS MRSS

16

SSDV SRW RMRSS MRSS

13

0.0824 1.0420U 1.0237 1.0389 0.0835 1.0425U 0.9968 1.0074 0.0851 1.0459U 0.9951 1.0051 0.0872 1.0435 1.0053 1.0174 0.0880 1.0442 1.0013 1.0104

0.0858 0.9825 0.9971 0.0871 0.9562UU 0.9663UU 0.0891 0.9515UU 0.9610UU 0.0909 0.9633U 0.9750 0.0918 0.9589U 0.9676

0.0877 1.0505 0.9885 0.9924

0.0921 0.9410 0.9447

0.0913 1.0856UU 1.0071 1.0177

0.0991 0.9277 0.9375

RMRSS

0.0787 1.0226UU

0.0843 1.0149U

0.0832 1.0106

0.0847 1.0100

0.0876 1.0121UU

0.0881 1.0090UU

0.0866 1.0040

0.0919 1.0106U

MRSS

SSDV

SRW

RMRSS

MRSS

0.0805

0.1116 0.9107UU 0.9166UU 0.8773UU

0.1016 1.0064 0.9634

0.1023 0.9572

0.0979

0.0856

0.1106 0.9197UU 0.9212UU 0.9166UU

0.1018 1.0016 0.9966

0.1019 0.9950

0.1014

0.0841

0.1131 0.9182UU 0.9222 UU 0.9225UU

0.1038 1.0043 1.0047

0.1043 1.0004

0.1043

0.0856

0.1151 0.9166UU 0.9211UU 0.9326UU

0.1055 1.0050 1.0175

0.1060 1.0125

0.1074

0.0887

0.0980 0.9436 0.9373 0.9526

0.0925 0.9933 1.0095

0.0919 1.0163

0.0934

0.0889

0.0977 0.9384 0.9345 0.9480

0.0917 0.9959 1.0103

0.0913 1.0145

0.0926

0.0870

0.1046 0.9255 0.9226 0.9631

0.0968 0.9969 1.0406

0.0965 1.0439

0.1007

0.0929

0.1135 0.9019 0.9047 0.9782

0.1023 1.0031 1.0847

0.1026 1.0813

0.1110

773

12-Month ahead

N

M.G. Ka¨ ussanos, A.H. Alizadeh-M r Economic Modelling 19 (2002) 747᎐782

1-Month ahead

Aframax

774

Table 7

Ž Continued .

1-Month ahead

2-Month ahead

3-Month ahead

4-Month ahead

5-Month ahead

6-Month ahead

VLCC

N

SSDV

SRW

SSDV SRW RMRSS MRSS

24

0.0803 0.8822UU 0.9454UU 0.8924UU

0.0709 1.0716UU 1.0115

SSDV SRW RMRSS MRSS

23

SSDV SRW RMRSS MRSS

22

SSDV SRW RMRSS MRSS

21

SSDV SRW RMRSS MRSS

20

SSDV SRW RMRSS MRSS

19

0.0836 0.8695UU 0.9483UU 0.9075UU

0.0727 1.0907UU 1.0438UU

0.0822 0.8898UU 0.9743 0.9423UU

0.0731 1.0951UU 1.0590UU

0.0834 0.8913UU 0.9844 0.9636U

0.0743 1.1045UU 1.0812UU

0.0797 0.9206UU 1.0065 0.9935

0.0734 1.0933UU 1.0792UU

0.0794 0.9165UU 0.9913 0.9814

0.0728 1.0816U 1.0708U

RMRSS

0.0760 0.9440UU

0.0793 0.9569UU

0.0801 0.9671UU

0.0821 0.9789

0.0803 0.9871

0.0787 0.9900

MRSS

SSDV

SRW

RMRSS

MRSS

0.0717

0.1220 1.1508UU 1.0714UU 1.0578UU

0.1405 0.9310 0.9192UU

0.1308 0.9874

0.1291

0.0759

0.1256 1.1404UU 1.0764UU 1.0306U

0.1432 0.9439 0.9037UU

0.1352 0.9575UU

0.1294

0.0774

0.1272 1.1498UU 1.0720UU 1.0241

0.1462 0.9323U 0.8906UU

0.1363 0.9552UU

0.1302

0.0804

0.1301 1.1481UU 1.0741UU 1.0161

0.1494 0.9355 0.8850UU

0.1397 0.9460UU

0.1322

0.0792

0.1327 1.1491UU 1.0764UU 1.0088

0.1525 0.9367 0.8779UU

0.1428 0.9373UU

0.1339

0.0779

0.1299 1.1617 UU 1.0822UU 1.0059

0.1510 0.9316 0.8659UU

0.1406 0.9295UU

0.1307

M.G. Ka¨ ussanos, A.H. Alizadeh-M r Economic Modelling 19 (2002) 747᎐782

Suezmax

Ž Continued . Suezmax

9-Month ahead

12-Month ahead

VLCC

N

SSDV

SRW

SSDV SRW RMRSS MRSS

16

0.0846 0.9113UU 1.0195 1.0183

0.0771 1.1187U 1.1173U

SSDV SRW RMRSS MRSS

13

0.0859 0.9470 1.0550 1.0507

0.0813 1.1140 1.1095

RMRSS

0.0863 0.9987

0.0906 0.9959

MRSS

SSDV

SRW

RMRSS

MRSS

0.0862

0.1383 1.1264U 1.0525UU 0.9997

0.1557 0.9344 0.8875U

0.1455 0.9498UU

0.1382

0.0902

0.1417 1.1222 1.0535UU 0.9803

0.1590 0.9388 0.8736

0.1492 0.9306UU

0.1389

SSDV, SRW, RMRSS and MRSS represent the Standard Seasonal Dummy Variable model wEq. Ž1.x , the Segmented Random Walk model wEq. Ž4.x , the Restricted Markov Regime Switching Seasonal model wEq. Ž3 .x and the unrestricted Markov Regime Switching Seasonal model wEq. Ž2 .x, respectively. N is the number of forecasts. Figures in the table are rounded up to four digits and numbers in bold indicate the lowest R.M.S.E. between the models. Numbers on the principal diagonal are the R.M.S.E. of forecasts from each model and the off diagonal numbers are the ratios of the R.M.S.E. of the model on the corresponding column to the R.M.S.E. of the model on the corresponding row. However, significance between R.M.S.E. values is shown by the Diebold᎐Mariano test. The Diebold᎐Mariano pairwise test of the hypothesis that the R.M.S.E. values from two competing models are equal is estimated using a Newey᎐West covariance estimator with a truncation lag equal to 1r3 of the sample Žnumber of forecasts .. U and UU indicate significance at the 10 and 5% level, respectively.

M.G. Ka¨ ussanos, A.H. Alizadeh-M r Economic Modelling 19 (2002) 747᎐782

Table 7

775

776

M.G. Ka¨ ussanos, A.H. Alizadeh-M r Economic Modelling 19 (2002) 747᎐782

freight rates are obtained by multiplying the regime probabilities by their respective mean values; i.e. by the mean of the model under each market condition, and adding them up Žsee Hamilton, 1994 for more details.. 12

⌬ X tqm s Ž ˆ p Ž stqm s 1 . ˆ p Ž stqm s 2 ..



␤ˆ1,0 q

Ý ␤ˆ1,i Qi ,t is1 12

␤ˆ2,0 q

Ý ␤ˆ2,i Qi ,t is1

0

Ž7.

In Table 7 R.M.S.E. values for the four competing models, SSDV, SRW, RMRSS and the unrestricted MRSS are shown on the main diagonal, for each forecast horizon. The off-diagonal numbers are the ratio of the R.M.S.E. of the model on the row to the R.M.S.E. values of the model on the column. When this number is greater Žlower. than unity the R.M.S.E. of the model on that row Žcolumn. is greater than the R.M.S.E. of the model on that column, which indicate that the model on the column Žrow. provides relatively more accurate forecasts compared to the model on the row Žcolumn.. In order to test whether the forecasting accuracy of one model significantly outperforms that of the other, we employ Diebold and Mariano Ž1995. tests, the significance of which are indicated next to the R.M.S.E. ratios using asterisks. In the case of Handysize rates, the SSDV model has the lowest R.M.S.E. for 2, 5, 6 and 12-step ahead forecasts. The RMRSS model has the lowest R.M.S.E. values for 1, 3, 4 and 9-step ahead forecasts among all models. The Diebold and Mariano Ž1995. test for the equality of R.M.S.E. values show that both the RMRSS and the unrestricted MRSS models perform better than the SRW model for 1, 3 and 4 months ahead, and the RMRSS model performs significantly better than the MRSS model for 1, 2, 5, 6 and 12-step ahead forecasts. Turning next to the Aframax market results on the same table, the unrestricted MRSS model performs better than other competing models for up to 2-step ahead forecasts. From 3-step to 12-step ahead forecasts, R.M.S.E. values for both the SRW and the RMRSS models are very close and they outperform the other models. However, results of the Diebold and Mariano Ž1995. test indicate that all models perform better than the SSDV model up to 4-step ahead and thereafter there is no significant difference between their forecasting performances. Diebold and Mariano’s tests on equality of R.M.S.E. values of both RMRSS and unrestricted MRSS models also indicate that there is no significant difference between predictions from these models. Turning to comparison of forecasting performance of seasonal models for Suezmax freight rates, it can be seen that the SRW model outperforms other seasonal models according to R.M.S.E. values. Results of the Diebold and Mariano Ž1995. tests indicate that the SRW model produces significantly more accurate forecasts than the RMRSS and the unrestricted MRSS models up to 9-step ahead, while there is no significant difference between the forecasting performance of SSDV and RMRSS as well as MRSS models for 5, 6, 9 and 12 months ahead

M.G. Ka¨ ussanos, A.H. Alizadeh-M r Economic Modelling 19 (2002) 747᎐782

777

forecasts. There is also no significant difference between the performance of models for 12-month ahead forecasts in the Suezmax market. Comparison of forecasting performance between seasonal models reveals that the unrestricted MRSS model outperforms other seasonal models significantly up to 6-month ahead, while according to the Diebold᎐Mariano test, there is no difference between forecasts of seasonal models beyond 4-step ahead forecasts. Finally, the results for VLCC freight rates indicate that the SSDV model outperforms the other models significantly up to 6-months ahead, with the exception that there is no significant difference between performance of the SSDV model and the unrestricted MRSS model for seasonality beyond 3-step ahead forecasts. The unrestricted MRSS model also outperforms the SRW and RMRSS models in terms of R.M.S.E. up to 12-months ahead, with the exception that there is no significant difference between the performance of MRSS and SRW models for 12-month ahead forecasts and MRSS and RMRSS models for 1-month ahead forecasts. According to the Diebold᎐Mariano’s test results, forecasts from the unrestricted MRSS model are significantly better than from the SRW model and the RMRSS model in most cases.

6. Conclusion The existence and type of seasonality present in tanker freight rates as well as the stochastic properties of these series are examined in this paper. Tanker freight rates seem to have a unit root at zero frequency, but not at seasonal frequencies for the period examined. This by itself suggests that ARIMA and VAR models are appropriate when modelling the series in univariate or multivariate frameworks, respectively. However, deterministic seasonal patterns are present in freight rates, and indicate seasonal increases in November and December to stock up for the winter and decline in rates from January to April. Similarities in seasonal rate patterns amongst segments of the tanker industry are attributed to the nature and pattern of trade in commodities transported by these ships, while differences emanate from factors that sub-divide tanker shipping and oil markets such as ship size, flexibility, route and commodity parcel size. It is also found that seasonal movements are higher during expansionary periods and lower during contractionary periods, which is in line with the theory of freight rate formation. Results suggest that shipowners may be able to use the information on the seasonal movements of freight markets in order to make business decisions such as; budget planning, dry-docking of vessels when freight rates are expected to drop Že.g. February to April., adjusting vessel speeds to increase productivity and ship repositioning to loading areas during peak seasons. Charterers also can use the information derived here to optimise their transportation costs by timing, for instance, their inventory build up outside peak seasons. Economically, it seems that Markov Regime Switching Seasonal models are promising in modelling markets where business cycles are prominent. Seasonality

778

M.G. Ka¨ ussanos, A.H. Alizadeh-M r Economic Modelling 19 (2002) 747᎐782

patterns also do vary according to the phase of the cycle the industry is in, and this feature can be picked up nicely by these kinds of model. However, forecasting with these models may not be as successful since they require the simultaneous forecasting of the state of the market and the mean of variables. Acknowledgements The authors would like to thank participants’ comments on an earlier version of this paper presented at the International Conference on World Shipping Markets Facing the 21st Century, 12᎐15 October 1998, Shenzhen, China. Also Eric Shawyer of Gibson Shipbrokers, Andrew Hamilton of Clarkson Research Studies, Ian Marsh of City University Business School, Keith Cuthbertson of Imperial College and an anonymous referee have provided insightful comments. Naturally, all remaining errors are ours. Appendix A: Markov Regime Switching Seasonal (MRSS) Model Hamilton Ž1989. introduced the Markov Regime Switching model, which allows for structural shifts in the behaviour of the time series over the estimation period through a state variable, St , as in Eq. Ž2.. This follows a two-state first-order Markovian process with the following transition probabilities; Pr Ž St s 1 < Sty1 s 2 . s P21

,

Pr Ž St s 2 < Sty1 s 2 . s P22 s Ž 1 y P21 . Ž A.1.

Pr Ž St s 2 < Sty1 s 1 . s P12

,

Pr Ž St s 1 < S ty1 s 1 . s P11 s Ž 1 y P12 .

where transition probability P12 gives the probability that state 1 will be followed by state 2, and transition probability P21 gives the probability that state 2 will be followed by state 1. Transition probabilities P11 and P22 give probabilities that the current state will not change in the next period. Thus, the unconditional probability that the process will be in any state at a point is given by Pr Ž St s 1 . s P1 s

1 y P22 2 y P11 y P22

,

Pr Ž St s 2 . s P2 s

1 y P11 2 y P11 y P22 Ž A.2.

Assuming normality, the likelihood function for each regime Žstate of the market. can be written as follows

f Ž ⌬ X t < St s j,Q i ,t ;␤ . s

1

'2␲␴

j s 1, 2

¡y ž ⌬ X y ␤ y exp~ 2␴ ¢ t

2 j

0, j

2

j

12

⌺ ␤i , j Q i ,t

is2

/ ¦¥ 2

§

,

Ž A.3.

M.G. Ka¨ ussanos, A.H. Alizadeh-M r Economic Modelling 19 (2002) 747᎐782

779

where, ␤ s Ž ␤ 0, j , . . . , ␤ 12, j , ␴j ., j s 1, 2, is the vector of parameters. The likelihood function for the entire sample is then formed by a mixture of the probability distribution of the state variable and the density function for each regime. 2

f Ž ⌬ X t ;␤ . s ⌺ P Ž ⌬ X t ,St s j;␤ . js1

s

q

P1

t

'2␲␴

2 1

P2

'2␲␴

¡y ž ⌬ X y ␤ y ~ exp 2␴ ¢

⌺ ␤i ,1 Q i ,t

is2

2 1

¡y ž ⌬ X y ␤ y ~ exp 2␴ ¢ t

2 2

0,1

12

0,2

12

⌺ ␤i ,2 Q i ,t

is2

2 2

/ ¦¥ 2

§

/ ¦¥ 2

§

Ž A.4.

where ␤ s ( ␤ 0, S t , . . . , ␤ 12,S t , ␴S t, PS t ., St s 1, 2 and P1 , P2 are the probabilities of the regime being in state 1 or 2, respectively. Therefore, the log-likelihood function is T

LŽ␤ . s

Ý log f Ž ⌬ X t ;␤ .

Ž A.5.

ts1

which can be maximised using numerical optimisation methods, subject to the constraint that P1 q P2 s 1 and 0 F P1 , P2 F 1. Appendix B: Definition of the Yi t variables in Eq. (5) For monthly time series, Yit variables in the Beaulieu and Miron Ž1993. test are derived from the following trigonometric Fourier transformations: 12

Y1,t s

Ý cos Ž0 i␲ . B iy1 Ž X t . s X t q X ty1 q X ty2 q X ty3 q X ty4 q X ty5 q X ty6 is1

q X ty7 q X ty8 q X ty9 q X ty10 q X ty11 12

Y2,t s

Ý cos Ž i␲ . B iy1 Ž X t . s yXt q X ty1 y X ty2 q X ty3 y X ty4 q X ty5 is1

y X ty6 q X ty7 y X ty8 q X ty9 y X ty10 q X ty11

M.G. Ka¨ ussanos, A.H. Alizadeh-M r Economic Modelling 19 (2002) 747᎐782

780

12

Ý cos Ž i␲r2. B iy1 Ž X t . s yXty1 q X ty3 y X ty5 q X ty7 y X ty9 q X ty11

Y3,t s

is1

12

Y4,t s y Ý sin Ž i␲r2. B iy1 Ž X t . s yX t q X ty2 y X ty4 q X ty6 y X ty8 q X ty10 is1

12

1

Ý cos Ž2 i␲r3. B iy1 Ž X t . s y 2 Ž X t q X ty1 y 2 X ty2 q X ty3 q X ty4

Y5,t s

is1

y2 X ty5 q X ty6 q X ty7 y 2 X ty8 q X ty9 q X ty10 y 2 X ty11 . 12

Ý sin Ž2 i␲r3. B iy1 Ž X t . s

Y6 ,t s

'3

Ž X t y X ty1 q X ty3 y X ty4

2

is1

qX ty6 y X ty7 q X ty9 y X ty10 . 12

1

Ý cos Ž i␲r3. B iy1 Ž X t . s y 2 Ž X t y X ty1 y 2 X ty2 y X ty3 q X ty4

Y7,t s

is1

q2 X ty5 q X ty6 y X ty7 y 2 X ty8 y X ty9 q X ty10 q 2 X ty11 . 12

Ý sin Ž i␲r3. B iy1 Ž X t . s y

Y8,t s

'3

is1

2

Ž X t q X ty1 y X ty3 y X ty4

qX ty6 q X ty7 y X ty9 y X ty10 . 12

1

Ý cos Ž5i␲r6. B iy1 Ž X t . s y 2 Ž '3 X t y X ty1 q X ty3 y '3 X ty4

Y9,t s

is1

q2 X ty5 y '3 X ty6 q X ty7 y X ty9 q '3 X ty10 y 2 X ty11 . 12

Y10 ,t s

1

Ý sin Ž5i␲r6. B iy1 Ž X t . s 2 Ž X t y '3 X ty1 q 2 X ty2 y '3 X ty3

is1

qX ty4 y X ty6 q '3 X ty7 y 2 X ty8 q '3 X ty9 y X ty10 . 12

Y11 ,t s

1

Ý cos Ž i␲r6. B iy1 Ž X t . s y 2 Ž '3 X t q X ty1 y X ty3 y '3 X ty4

is1

y2 X ty5 y '3 X ty6 y X ty7 q X ty9 q '3 X ty10 q 2 X ty11 .

M.G. Ka¨ ussanos, A.H. Alizadeh-M r Economic Modelling 19 (2002) 747᎐782 12

Y12 ,t s y

781

1

Ý cos Ž i␲r6. B iy1 Ž X t . s y 2 Ž X t q '3 X ty1 q 2 X ty2 q '3 X ty3

is1

qX ty4 y X ty6 y '3 X ty7 y 2 X ty8 q '3 X ty9 y X ty10 . Where B iy1 is the lag operator and X t is the monthly series. References Abeysinghe, T., 1994. Deterministic seasonal models and spurious regressions. J. Econom. 61, 259᎐272. Akaike, H., 1969. Fitting autoregressive models for predictions. Ann. Institute Stat. Math. 21, 243᎐247. Beaulieu, J.J., Miron, J.A., 1992. A cross-country comparison of seasonal cycles and business cycles. Econ. J. 102, 772᎐788. Beaulieu, J.J, Miron, J.A., 1993. Seasonal unit roots in aggregate US data. J. Econom. 55, 305᎐328. Canova, F., Ghysels, E., 1994. Changes in seasonal patterns. J. Econ. Dyn. Control 18, 1143᎐1171. Canova, F., Hansen, B.E., 1995. Are seasonal patterns constant over time? A test for seasonal stability. J. Bus. Econ. Stat. 13, 237᎐252. Dickey, D.A., Hasza, D.P., Fuller, W.A., 1984. Testing for unit roots in seasonal time series. J. Am. Stat. Assoc. 79, 355᎐376. Diebold, F.X., Mariano, R.S., 1995. Comparing predictive accuracy. J. Bus. Econ. Stat. 13 Ž3., 252᎐263. Diebold, F.X, Lee, J.H, Weinback, G.C., 1994. Regime switching with time-varying transition probabilities. In: Hargreaves, C. ŽEd.., Nonstationary Time Series Analysis and Cointegration. Oxford University Press. Engle, C., 1994. Can the Markov Switching Model forecast exchange rates? J. Int. Econ. 36, 151᎐165. Filardo, A.J., 1994. Business cycle phases and their transition dynamics. J. Bus. Econ. Stat. 12, 299᎐308. Franses, P.H., 1994. A multivariate approach to modelling univariate seasonal time series. J. Econom. 63, 133᎐151. Franses, P.H., Hobijn, B., 1997. Critical values for unit root tests in seasonal time series. J. Appl. Stat. 24 Ž1., 25᎐47. Franses, P.H., Hylleberg, S., Lee, H.S., 1995. Spurious deterministic regression. Econ. Lett. 48, 249᎐256. Girma, P.B., Paulson, A.S., 1998. Seasonality in petroleum future spreads. J. Futures Mark. 18 Ž5., 581᎐598. Greene, W.H., 1997. Econometric Analysis, 3rd ed. Prentice Hall, Inc. Hamilton, J.D., 1989. A new approach to the economic analysis of nonstationary time series and business cycle. Econometrica 57, 357᎐384. Hamilton, J.D., 1994. Time Series Analysis. Princeton University Press, Princeton, NJ. Hannan, E.J., Quinn, B.G., 1979. The determination of the order on an autoregression. J. R. Stat. Soc. B 41, 190᎐195. Hylleberg, S., Engle, R.F., Granger, C.W.J, Yoo, B.S., 1990. Seasonal integration and cointegration. J. Econom. 44, 215᎐238. Jarque, C.M, Bera, A.K., 1980. Efficient test for normality, homoscedasticity and serial dependence of regression residuals. Econ. Lett. 6, 255᎐259. Kavussanos, M.G., 1996a. Price risk modelling of different size vessels in tanker industry using Autoregressive Conditional Heteroscedasticity ŽARCH. models. Logistics and Transp. Rev. 32 Ž2., 161᎐176. Kavussanos, M.G., 1996b. Measuring risk differences among segments of the tanker freight markets, Discussion Paper No. 18, Dept. of Shipping, Trade and Finance, City University Business School. Also presented at the IAME conference, Vancouver, Canada, July 26᎐28 1996 Kavussanos, M.G., 1998a. Freight risks in the tanker sector. Lloyds Shipping Economist, 6᎐9, June. Kavussanos, M.G., 1998b. Freight risks in the tanker sector. Lloyds Shipping Economist, p. 9, July. Kavussanos, M.G. and Alizadeh, A.H. Ž2001., Seasonality patterns in dry bulk shipping spot and time-charter freight rates, Transportation Research ᎏ Part E; Logistics Transp. Rev., forthcoming.

782

M.G. Ka¨ ussanos, A.H. Alizadeh-M r Economic Modelling 19 (2002) 747᎐782

Ljung, G.M., Box, G.E.P.E., 1978. On a measure of lack of fit in time series models. Biometrika 65, 297᎐303. Marsh, I.W., 2000. High-frequency Markov Switching Models in the foreign exchange market. J. Forecasting 19, 123᎐134. McConville, J., 1999. Economics of Maritime Transport. Witherby Publishers, London. Moosa, I.A., 1995. Deterministic and stochastic seasonality in the oil imports of the G7 countries. OPEC Review, 181᎐201, Autumn. Moosa, I.A., Al-Loughani, N.E., 1994. Some time-series properties of Japanese oil imports. J. Int. Econ. Stud. 8, 109᎐122. Newey, W., West, K., 1987. A simple positive semi-definite, heteroscedasticity and autocorrelation consistent covariance matrix. Econometrica 55, 703᎐708. Osborn, D.R., 1990. A survey of seasonality in UK macroeconomic variables. Int. J. Forecasting 6, 327᎐336. Osborn, D.R., Chui, A.P.L., Smith, J.P., Birchenhall, C.R., 1988. Seasonality and order of integration for consumption. Oxf. Bull. Econ. Stat. 50, 361᎐377. Schwarz, G., 1978. Estimating the dimension of a model. Ann. Stat. 6, 461᎐464. Stopford, M., 1997. Maritime Economics. Unwin Hyman, London. Wallis, K.F., 1974. Seasonal adjustment and relations between variables. J. Am. Stat. Assoc. 69, 18᎐31. White, H., 1980. A heteroscedasticity-consistent covariance matrix estimator and a direct test for heteroscedasticity. Econometrica 48, 817᎐838.

Further Reading Franses, P.H., 1991. Seasonality, nonstationarity and the forecasting of time series. Int. J. Forecast. 7, 199᎐208. Greene, W.H., Seaks, T.G., 1991. The restricted least square estimator: A pedagogical note. Rev. Econ. Stat. 73, 563᎐567. Suits, D.B., 1984. Dummy variables: mechanics vs. interpretation. Rev. Econ. Stat. 66, 177᎐180.