Forecasting spot and forward prices in t (1)

International Journal of Forecasting 23 (2007) 101 – 114
www.elsevier.com/locate/ijforecast

Forecasting spot and forward prices in the
international freight market
Roy Batchelor a,⁎, Amir Alizadeh a,1 , Ilias Visvikis b,2
b

a
Cass Business School, City University, 106 Bunhill Row, London EC1Y 8TZ, UK
Athens Laboratory of Business Administration (ALBA), Athinas Ave and 2A Areos Str, 166 71 Vouliagmeni, Athens, Greece

Abstract
This paper tests the performance of popular time series models in predicting spot and forward rates on major seaborne
freight routes. Shipping is a nonstorable service, so the forward price is not tied to the spot by any arbitrage relationship. The
developing forward market is dominated by hedgers, and it is an empirical question whether forward rates contain information
about future spot rates. We find that vector equilibrium correction (VECM) models give the best in-sample fit, but implausibly
suggest that forward rates converge strongly on spot rates. In out-of-sample forecasting all models easily outperform a random
walk benchmark. Forward rates do help to forecast spot rates, suggesting some degree of speculative efficiency. However, in
predicting forward rates, the VECM is unhelpful, and ARIMA or VAR models forecast better. The exercise illustrates the
dangers of forecasting with equilibrium correction models when the underlying market structure is evolving, and coefficient

estimates conflict with sensible priors.
© 2006 International Institute of Forecasters. Published by Elsevier B.V. All rights reserved.
JEL classification: G13; G14
Keywords: Forecasting; Freight market; Commodity market: Vector equilibrium correction model; ARIMA model

1. Introduction
In this paper we investigate the performance of
alternative univariate and bivariate linear time-series
models in generating short-term forecasts of spot
⁎ Corresponding author. Tel.: +44 207 040 8733; fax: +44 207 040
8881.
E-mail addresses: r.a.batchelor@city.ac.uk (R. Batchelor),
a.alizadeh@city.ac.uk (A. Alizadeh), ivisviki@alba.edu.gr
(I. Visvikis).
1
Tel.: +44 207 040 0199; fax: +44 207 040 8881.
2
Tel.: +30 210 896 4531 8; fax: +30 210 896 4737.

freight rates in the international dry bulk shipping

market, and corresponding rates fixed in the Forward
Freight Agreement (FFA) market. The FFA market is
interesting for several reasons. First, it is relatively new
and under-researched, and our findings come from a
unique and specially constructed database of forward
freight rates. There is practical value to users of the
market – ship owners and shipping agents – in knowing whether and how forward rates can best be used to
predict spot rates. Second, the underlying asset traded
in the FFA market is a service rather than a storable
commodity. This means that arbitrage between spot

0169-2070/$ - see front matter © 2006 International Institute of Forecasters. Published by Elsevier B.V. All rights reserved.
doi:10.1016/j.ijforecast.2006.07.004

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R. Batchelor et al. / International Journal of Forecasting 23 (2007) 101–114

and forward markets is not possible, so spot and
forward prices are not linked by the rigid cost-of-carry

relationships observed in most commodity markets.
Third, unlike the large established markets in commodities and financial futures, the FFA market is small
and dominated by the activities of hedgers rather than
speculators. It cannot therefore be taken for granted
that all information relevant to future freight rates is
automatically incorporated into the forward price.
In these circumstances we should expect to observe
certain characteristics in the time series of spot and
forward freight rates. In speculatively efficient markets
for nonstorable commodities, forward prices are unbiased forecasts of future spot prices, and changes in
forward prices for fixed target dates are close to random, reflecting the arrival of news. The thinness of the
FFA market and the absence of a strong speculative
interest mean that forward freight rates may exhibit
neither of these properties. Similarly, in arbitragedominated markets, forward rates are tied to spot rates,
and both tend to move to ensure convergence at the
expiry of contracts, as the cost of carry falls. The
absence of arbitrage in the FFA market means that the
spot rate may converge on the forward rate, provided
that the forward rate embodies some expectations
about future spot rates. However, there is no reason

why the forward rate should converge on the spot rate.
The most general model we use is a vector errorcorrection (VECM) model linking spot and forward
rates for four major shipping routes. This model is used
to make inferences about the efficiency and usefulness
of FFA rates. For example, if forward rates are expectations of spot rates we would expect (a) there to be a
cointegrating vector linking spot and forward rates,
and (b) the cointegrating vector to be the basis (that is,
spot rate − forward rate = 0), and (c) this equilibrium to
be established by spot rates converging on forward
rates, but not vice versa.
The validity of the VECM model is tested by
benchmarking forecasts from it against forecasts from a
number of alternative linear time series models, and
against the random walk. Even if two price series are
cointegrated, incorporating the information contained
in the cointegrating relationship in the model is not
guaranteed to improve predictability. Moreover, as
discussed by Clements and Hendry (1995, 1998,
2001), the VECM is not robust to structural change.
The equilibrium correction term forces variables to

their average historical relationship, so long-term forecasts in particular may be inaccurate if the underlying
relationship has shifted. Stock and Watson (1996)
show that most mainstream macroeconomic variables
have been subject to significant structural change in
recent decades. As a consequence, Allen and Fildes
(2001) find that in practice, VECM models have a
mixed track record in forecasting such time series. In
commodity markets, the balance of evidence seems to
favour the VECM approach. For example, Zeng and
Swanson (1998) estimate VECM and other models for
spot and futures prices of the S&P500 index, the US
30-year T-bond, gold and oil. They find that the VECM
predicts better than all simpler models, and also the
random walk. However, this is not surprising since in
their chosen markets the possibility of arbitrage
ensures that the basis (futures price−spot price) is
equal to the cost of carry (borrowing cost less own rate
of return on the spot asset), and their best-performing
models use this as the cointegrating vector. Cash-andcarry arbitrage (borrow funds, buy spot, sell forward) is

not feasible in the freight market since the underlying
asset cannot be stored. Cullinane (1992) and Cullinane,
Mason, and Cape (1999) report success in forecasting
spot freight rates using simpler univariate ARIMA
models. Kavussanos and Nomikos (2003) compare
joint VECM forecasts of spot freight rates and the now
defunct exchange traded BIFFEX futures freight rates
with forecasts from ARIMA, VAR and Random Walk
models. They find that the VECM generates the most
accurate forecasts of spot prices but not of futures
prices. Their tests use overlapping forecast intervals,
which Tashman (2000) argues may bias forecast
evaluation, and we have been careful to design our
forecast evaluation procedures to avoid this problem.
Alizadeh and Nomikos (2003) examine the directional
forecast accuracy of FFAs and Freight Futures
contracts (BIFFEX) in four routes and concluded that
FFAs do not seem to be very accurate in revealing the
direction of future freight rates. They report that the
directional accuracy of FFAs in forecasting freight

rates varies between 46% and 74%, and, in general,
forecasting accuracy declines as maturity increases.
The paper is organised as follows. Section 2 describes the data and the models that are used to generate
the forecasts. Sections 3 and 4 present the different
models used to generate forecasts and discuss in-sample
estimation results. Section 5 evaluates the forecasting

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R. Batchelor et al. / International Journal of Forecasting 23 (2007) 101–114

performance of the alternative models. Finally, Section 6
summarises our conclusions.

Table 1
Baltic Panamax Index (BPI)—route definitions
Routes Descriptions

Size of
Weight

vessel
in index
(dwt, tonnes) (%)

1

55,000

10

70,000

20

54,000

12.5

70,000

12.5

54,000

10

70,000

20

70,000

15

2. Spot and forward freight rate data
The aim of the creation of the Freight Forward
Market in 1992 was to provide a mechanism for
hedging freight rate risk in the dry-bulk and wet-bulk
sectors of the shipping industry. FFAs are principal-toprincipal contracts between a seller and a buyer to settle
a freight or hire rate for a specified quantity of cargo or

type of vessel, for one (usually) or more major trade
routes. One party – the charterer – is concerned that in
some future month the spot price of a standard freight
may be higher than expected, and buys FFA contracts.
The other party – the shipowner – is concerned about
rates falling, takes the opposite position, and sells FFA
contracts. Settlement is made on the difference between
the contracted forward price and the average price for
the route selected in the index over the last seven
working days of the settlement month. The market is
intermediated by a small number of specialist brokers.
The standard routes underlying the FFA contracts
have changed since its inception. Currently, FFA contracts are written on freight rates in routes of the Baltic
Panamax Index (BPI), the Baltic Handymax Index
(BHMI), the Baltic Capesize Index (BCI), the Baltic
Dirty Tanker Index (BDTI), and Baltic Clean Tanker
Index (BCTI). The most liquid routes, those in the BPI,
are described in Table 1. Our data set consists of daily
spot and FFA prices in Panamax Atlantic routes 1 and
1A from 16 January 1997 to 31 July 2000, and daily

spot and FFA prices in Panamax Pacific routes 2 and
2A from 16 January 1997 to 30 April 2001. The
difference in the sample periods between the Atlantic
and Pacific routes arises because the Atlantic routes are
characterised by modest FFA trading and FFA brokers
stopped publishing FFA quotes for those routes after
July 2000. FFA trading is concentrated mostly on the
Pacific routes 2 and 2A, since most market agents
operate their Panamax vessels in the Pacific region.
Spot price data are from the Baltic Exchange. FFA
rates are not available from any data vendor. Our data
on daily bid and offer quotes for every trading route of
the BPI, and for the nearby FFA contracts for the four
Panamax routes, have been collected manually from
the records of Clarkson's Securities Ltd., a leading
broker in the FFA market. FFA prices are always those

1A

2

2A

3
3A

4

1–2 safe berths/anchorages
Mississippi River not above
Baton Rouge/Antwerp,
Rotterdam, Amsterdam
Transatlantic (including ESCA)
round of 45/60 days on the basis
of delivery and redelivery
Skaw–Gibraltar range
1–2 safe berths/anchorages
Mississippi River not above
Baton Rouge/1 no combo port
South Japan
Basis delivery Skaw–Gibraltar
range, for a trip via Gulf to the
Far East, redelivery Taiwan–
Japan range, duration 50/
60 days
1 port US North Pacific/1 no
combo port South Japan
Transpacific round of 35/50
days either via Australia or
Pacific (but not including short
rounds such as Vostochy/Japan),
delivery and redelivery Japan/
South Korea range
Delivery Japan/South Korea
range for a trip via US West
Coast–British Columbia range,
redelivery Skaw–Gibraltar
range, duration 50/60 days

Source: Baltic Exchange.

of the nearby contract because it is highly liquid and is
the most active contract. However, to avoid thin
market and expiration effects (when futures and
forward contracts approach their settlement day, the
trading volume decreases sharply), we roll over to the
next nearest contract 1 week before the nearby contract
expires. In 1999, the total FFA volume was about 1200
contracts, a figure similar to 1998. According to
Clarkson's Securities Ltd., in 2000 the total FFA
volume was about 2000 contracts, while more than
8000 contracts were traded in 2004 and about 8500 in
2005, with a market value of around US$29 billion. In
practice, most of the trading concentrates in the nearby
(one month) and first-out (2 months) contracts, with
the second-out (3 months) contracts characterised by

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R. Batchelor et al. / International Journal of Forecasting 23 (2007) 101–114

Fig. 1. FFA and Spot Prices in Route 1 (16/01/97–31/07/00).

very low volume figures. More recently, quarter and
calendar contracts have also been traded in certain
routes and on time-charter rates or daily earnings of
different types of vessels.
Combining information from FFA contracts with
different times to maturity may create breaks in the
series at the date of the forward rollover, since FFA
returns for that day are calculated between the price of
the expiring contract and the price of the next nearest
contract. To address this issue, we experimented with a
series of synthetic prices for a “perpetual” 22-day
ahead FFA contract. The prices are calculated as a
weighted average of near and distant FFA contracts,
weighted according to their respective number of days
from maturity. This procedure generates a series of
FFA prices with constant maturity and avoids the
problem of price-jumps caused by the expiration of a
particular FFA contract. However, use of these data
yields empirical results which are qualitatively the

same as those reported below, so there is no evidence
that the FFA contract rollover biases our findings.
Figs. 1–4 present co-movements and fluctuating
patterns of spot and FFA rates for routes 1, 1A, 2 and
2A, respectively. Freight rates weakened after the
economic crises in Russia and the Far East in 1997–
1998, but recovered through 1999 to their starting
levels by 2000–2001. Since then, as world trade – and
especially trade with China – has accelerated, freight
rates have risen very sharply, and in 2004–2005 were
some three to four times higher than in 2001. Forward
rates are typically close to spot rates, but occasionally
(for example, in 1999–2000) drift away. Kavussanos
and Visvikis (2004) investigate the lead–lag relationships between forward and spot markets, and find
bidirectional causality in price movements in all routes,
but find less clear evidence on the direction of volatility
spillovers between spot and forward prices across
different routes. In another study, Kavussanos,

Fig. 2. FFA and Spot Prices in Route 1A (16/01/97–31/07/00).

R. Batchelor et al. / International Journal of Forecasting 23 (2007) 101–114

105

Fig. 3. FFA and Spot Prices in Route 2 (16/01/97–10/08/01).

Visvikis, and Menachof (2004) show that FFA prices 1
and 2 months prior to maturity are generally unbiased
predictors of the realised spot prices, while Kavussanos, Visvikis, and Batchelor (2004) argue that the onset
of FFA trading did have a general stabilising influence
on the spot price volatility.
For the purpose of analysis, all prices are
transformed to natural logarithms. Summary statistics
of logarithmic first differences (“log returns”) of daily
spot and FFA prices are presented in Table 2 for the
whole period, in the four Panamax routes. The results
indicate excess kurtosis in all series, and excess
skewness in most, and the Jarque–Bera tests indicate
departures from normality for both spot and FFA
prices in all routes. The Ljung–Box Q(36) and Q2(36)
statistics on the first 36 lags of the sample autocorrelation function of the log-level series and of the logsquared series indicate significant serial correlation
and existence of heteroskedasticity, respectively. In
contrast to speculative storable commodities, such as

stock prices and exchange rates, there is no reason to
expect changes in spot freight rates to be serially
uncorrelated. Demand and supply for freight services
are determined by the needs of trade. Scope for
intertemporal substitution and substitution across
routes and vessel types is severely limited. Serial
correlation in spot prices may also be exaggerated by
the way shipbroking companies calculate freight rates.
These rates are mostly based on actual fixtures, but in
the absence of an actual fixture they depend on the
shipbroker's estimate of what the rate would be if there
was a fixture; discussions with brokers suggest that
their estimates are typically a mark-up over the
previous day's rate.
Augmented Dickey Fuller (ADF) and Phillips–Perron
(PP) unit root tests on the log levels and log first
differences of the daily spot and FFA price series indicate
that all variables are log-first difference stationary, all
having a unit root on the log levels representation. ADF
and PP tests are sometimes criticised for their lack of

Fig. 4. FFA and Spot Prices in Route 2A (16/01/97–10/08/01).

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R. Batchelor et al. / International Journal of Forecasting 23 (2007) 101–114

Table 2
Descriptive statistics for spot and FFA prices
Statistics

N
Skew
Kurtosis
Jarque–Bera
Q(36)
QSQ(36)
ADF (levels)
PP (levels)
ADF (differences)
PP (differences)
KPSS

Route 1

Route 1A

Route 2

Route 2A

Spot

FFA

Spot

FFA

Spot

FFA

Spot

FFA

891
− 0.17
13.35
6609.83
216.43
264.77
− 1.53
− 1.32
− 9.17
− 15.75
0.93

891
− 0.15
5.43
1096.71
304.47
283.47
− 1.65
− 1.57
−31.72
−32.07
0.86

891
1.81
30.78
35,637.27
186.29
290.45
− 1.89
− 1.67
− 10.34
− 14.05
0.74

891
− 0.04
4.71
822.28
258.59
221.35
− 1.61
− 1.78
− 29.55
− 29.71
0.75

1078
0.63
522.25
12,271.08
507.73
215.81
− 1.85
− 1.83
− 14.00
− 32.77
0.83

1078
0.29
5.04
1158.85
285.55
276.36
−1.52
−1.63
−30.42
−30.46
0.86

1078
− 1.28
31.35
44,392.48
157.80
246.01
− 2.12
− 1.89
− 12.48
− 15.22
0.74

1078
0.10
6.23
1747.41
292.49
404.80
−1.89
−1.95
−30.01
−30.03
0.75

The table shows descriptive statistics for the log differences of spot and FFA rates on four routes. Data are daily in the period 16 January 1997 to
31 July 2000 (routes 1 and 1A) and 16 January 1997 to 30 April 2001 (routes 2 and 2A). N shows the number of daily observations. Skewness and
kurtosis are estimated centralised third and fourth moments of the data. The Bera and Jarque (1980) test for normality is distributed as χ2(2), with
a critical 5% value of 5.99. Q(36) and QSQ(36) are the Ljung and Box (1978) Q statistics on the first 36 lags of the sample autocorrelation
function of the raw series and of the squared series. Both statistics are distributed as χ2(36) under the nulls of no serial correlation in returns and no
serial correlation in squared returns (no heteroskedasticity), with a critical 5% value of 51.48. ADF is the Augmented Dickey and Fuller (1981)
test. The ADF regressions include an intercept term; the lag length of the ADF test is determined by minimising the SBIC. PP is the Phillips and
Perron (1988) test, and KPSS the Kwiatkowski, Phillips, Schmidt, and Shin (1992). The 5% critical value for the ADF and PP tests is −2.89, and
the 5% critical value for the KPSS test is 0.146.

power in rejecting the null hypothesis of a unit root when
it is false. This lack of power is addressed by the KPSS
test proposed by Kwiatkowski et al. (1992), which has
stationarity as the null hypothesis, and the results of the
KPSS tests confirm our inferences.

used to evaluate independent out-of-sample N-days
ahead forecasts.
The ARIMA(p,1,q) model for spot and forward
freight rates is

3. Forecasting models

DSt ¼ a10 þ

To identify the model that provides the most accurate
short-term forecasts of spot and FFA prices in the market,
four time series models are considered. These are the Box
and Jenkins (1970) ARIMA model, the Vector Autoregression (VAR) model of Sims (1980) on price
changes, a general Vector Equilibrium Correction
model (VECM) suggested for cointegrated variables by
Engle and Granger (1987), and a restricted VECM
model. The VAR model here can be considered as a
restricted version of the VECM in which the equilibrium
correction term is dropped. For evaluation purposes, the
data are split into an in-sample estimation set and an outof-sample forecast set. The various time-series models are
initially estimated over the period 16 January 1997 to 30
June 1998 for all routes. The period from 1 July 1998 to
31 July 2000 for the Atlantic routes and the period from 1
July 1998 to 30 April 2001 for the Pacific routes are then

p
X

a1i DSt−i þ

i−1

þ e1t ;

DFt ¼ a20 þ

e1t fiid

p
X

þ e2t ;

e2t fiid

b1i e1t−j

j−1
N ð0; r 21 Þ

a2i DFt−i þ

i−1

q
X

q
X

b2i e2t−j

j−1
N ð0; r 22 Þ

ð1Þ

where ΔFt and ΔSt are changes in log futures and spot
prices, respectively, and the εt are random error terms.
The corresponding bivariate VAR(p) model is:
DSt ¼ a10 þ

p
X

DFt ¼ a20 þ

i−1
p
X
i−1

a1i DSt−1 þ

p
X

b1i Ft−j þ e1t

a2i DSt−i þ

j−1
p
X

b2i Ft−j þ e2t :

j−1

ð2Þ

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R. Batchelor et al. / International Journal of Forecasting 23 (2007) 101–114

The potential advantage of the bivariate VAR model over
the univariate ARIMA model is that it takes into account
the information content in the spot price movement in
determining the forward price and vice versa.
Our bivariate VECM (p) models are of the form
DSt ¼ a10 þ

p
X

a1i DSt−i þ

p
X

priate than the univariate ARIMA and bivariate VAR
models in modelling the spot and forward prices as it
takes into account both the short-run dynamics and the
long-run relationship between variables. In Eq. (3) the
coefficients γ1 and γ2 measure the speed of adjustment of spot and forward prices to their long-run
equilibrium.
The VAR model is itself a restricted version of the
VECM, where the two equilibrium correction terms
are omitted (γ1 = γ2 = 0). The VAR model therefore
may require a larger number of parameters than the
VECM to capture the dynamic behaviour of the variables, and this lack of parsimony may cause problems
when the model is used for forecasting. One problem is
that the collinearity between the different lagged
variables may lead to imprecise coefficient estimates.
Another problem is that the large number of parameters may lead us to model some sample-specific
noise in finite samples, and this overfitting can give the

b1i Ft−j

j−1

i−1

þ g1 ðSt−1 −d0 −d1 Ft−1 Þ þ e1t

DFt ¼ a20 þ

p
X

a2i DSt−1 þ

p
X

b2i Ft−j

j−1

i−1

þ g2 ðSt−1 −d0 −d1 Ft−1 Þ þ e2t :

ð3Þ

The term in brackets represents the cointegrating
(long-run) relationship between the spot and forward
prices. The VECM model is argued to be more appro-

Table 3.1
Estimates of the forecasting models for route 1
Regressor

ARIMA
ΔSt

VAR
ΔFt

ΔSt

ΔFt

ΔSt

ΔFt

–

0.601⁎
(3.213)
− 0.128
(0.669)
0.167
(0.877)
− 0.223
(1.302)
− 0.090⁎⁎
(1.651)
− 0.076
(1.299)
0.055
(0.944)
− 0.087
(1.510)
–

0.585⁎
(3.152)
− 0.142
(0.749)
0.153
(0.811)
− 0.231
(1.358)
− 0.054
(0.968)
− 0.046
(0.763)
0.081
(1.374)
− 0.065
(1.117)
0.056⁎
(2.85)
0.0385
10.048
[0.526]

0.360⁎
(7.579)
–

0.412⁎
(3.039)
–

0.244
(5.823)
–

–

0.0258
12.099
[0.356]

0.350⁎
(6.385)
0.001
(0.002)
0.234⁎
(4.203)
− 0.010
(0.209)
0.102⁎
(6.193)
0.035⁎
(1.989)
0.031⁎⁎
(1.773)
0.000
(0.004)
− 0.007
(1.016)
0.4216
2.74
[0.994]

ΔSt−4
ΔFt−1

–

ΔFt−2

–

ΔFt−3

–

ΔFt−4

–

− 0.052
(0.981)
0.003
(0.062)
0.109⁎⁎
(1.931)
–

zt−1

–

–

0.348⁎
(6.354)
− 0.002
(0.032)
0.233⁎
(4.174)
− 0.012
(0.229)
0.107⁎
(6.704)
0.039⁎
(2.261)
0.034⁎
(1.995)
0.003
(0.161)
–

−2
R
Q(12)

0.3536
2.245
[0.997]

0.0047
9.956
[0.534]

0.4215
2.705
[0.994]

ΔSt−2
ΔSt−3

S-VECM

ΔSt

0.451⁎
(8.68)
0.041
(0.709)
0.212⁎
(4.074)
–

ΔSt−1

VECM

ΔFt

–
–
–

–

0.104⁎
(6.623)
0.033⁎
(2.039)
–

–

–

–

− 0.008
(1.266)
0.4219
3.117
[0.989]

0.061⁎
(2.832)
0.0355
19.536
[0.052]

–
–

Estimated coefficients of ARIMA, VAR, VECM and restricted VECM (S-VECM) estimated using daily data on spot and FFA freight rates over
the period 16/01/97 to 30/06/98. The cointegrating vector is restricted to be the lagged basis St−1 − Ft−1 in route 1. Figures in parentheses under
estimated coefficients are t-statistics, adjusted using the White (1980) heteroskedasticity consistent variance–covariance matrix. ⁎ and ⁎⁎ denote
significance at the 5% and 10% levels, respectively. Q(12) is the Ljung and Box (1978) Q-statistic testing up to 12th-order serial correlation in the
residuals, and the figures in square brackets show exact significance levels for these tests.

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R. Batchelor et al. / International Journal of Forecasting 23 (2007) 101–114

Table 3.1A
Estimates of the forecasting models for route 1A
Regressor

ARIMA
ΔSt

VAR
ΔSt

ΔFt

ΔSt

ΔFt

ΔSt

ΔFt

–

0.481⁎
(2.23)
0.219
(1.048)
− 0.089
(1.648)
− 0.135⁎
(2.498)
–

0.510⁎
(2.392)
0.221
(1.071)
− 0.033
(0.586)
− 0.091
(1.635)
0.093⁎
(3.051)
0.0641
10.21
[0.512]

0.602⁎
(11.586)
0.104⁎
(2.116)
0.027⁎
(1.976)
0.031⁎
(2.371)
− 0.013⁎⁎
(1.660)
0.5574
14.27
[0.218]

0.596
(4.223)
–

0.0425
14.797
[0.192]

0.596⁎
(11.114)
0.121⁎
(2.321)
0.024⁎⁎
(1.708)
0.025⁎⁎
(1.78)
−0.014⁎⁎
(1.783)
0.5578
14.217
[0.221]

ΔFt−1
ΔFt−2

–

0.138⁎
(4.625)
–

zt−1

–

–

0.601⁎
(11.171)
0.121⁎
(2.320)
0.033⁎
(2.410)
0.031⁎
(2.324)
–

−
R2
Q(12)

0.545
15.663
[0.154]

0.0096
10.079
[0.523]

0.5551
15.119
[0.177]

ΔSt−2

S-VECM

ΔFt

0.643⁎
(12.275)
0.126⁎
(2.399)
–

ΔSt−1

VECM

–

–
–
0.108
(3.857)
0.0626
12.874
[0.302]

See notes to Table 3.1.
For route 1A, the cointegrating vector is St−1 − 1.0143 ⁎ Ft−1 + 0.1585.

appearance of a good within-sample fit, but poor outof-sample forecasts.
We estimate both an unrestricted VECM and a
restricted version which is simply the parsimonious
version of Eq. (3) derived by successively eliminating statistically insignificant coefficients. In general
this model has different regressors in the two
equations, and is therefore estimated as a system
of Seemingly Unrelated Regression Equations
(SURE), since this method yields more efficient
estimates than Ordinary Least Squares in these
circumstances (Zellner, 1962). The restricted model
is denoted S-VECM.

4. Estimation results
As noted above, the results of the unit root tests on the
log levels and log first differences of the daily spot and
FFA price series indicate that all variables are log-first
difference stationary, all having a unit root on the loglevels representation. This means that the first differences
of spot and forward series should be used in the ARMA
and VAR models, while cointegration tests should be
performed to ascertain the long-run relationship between
the series if the VECM model is going to be used.
Johansen's (1988) multivariate cointegration test results
indicate that spot and FFA prices are cointegrated in all

Table 3.2
Estimates of the forecasting models for route 2
Regressor

ARIMA

VAR

ΔSt

ΔFt

ΔSt

ΔFt

ΔSt

ΔFt

ΔSt

ΔFt

ΔSt−1

–

ΔFt−1

0.668⁎
(17.012)
–
–

0.434⁎
(4.107)
− 0.03
(0.531)
–

−
R2
Q(12)

0.443
8.717
[0.648]

0.0161
6.843
[0.812]

0.4592
12.239
[0.346]

0.0417
7.553
[0.723]

0.556⁎
(12.911)
0.047⁎
(2.012)
− 0.052⁎
(4.544)
0.4872
9.747
[0.553]

0.464⁎
(4.219)
− 0.012
(0.209)
0.029
(0.986)
0.0417
6.761
[0.818]

0.555⁎
(13.057)
0.049⁎
(2.343)
− 0.052⁎
(4.574)
0.4871
9.841
[0.545]

0.457⁎
(4.392)
–

zt−1

0.161⁎
(4.443)
–

0.611⁎
(14.376)
0.079⁎
(3.452)
–

See notes to Table 3.1.
For route 2, the cointegrating vector is St−1 − 1.0067 ⁎ Ft−1 + 0.0324.

VECM

S-VECM

0.031
(1.105)
0.0442
6.601
[0.830]

109

R. Batchelor et al. / International Journal of Forecasting 23 (2007) 101–114

are higher than those of forward rates, indicating
potentially higher predictability in spot rates than
forward rates.
The next set of columns in Tables 3.1–3.2A present
the estimates of coefficients of the VAR models. These
are similar to the ARIMA model in terms of the
appropriate number of lag lengths used and the
diagnostic tests. As expected, the adjusted coefficients
of determination for the VAR models are slightly higher
than those of the ARIMA model due to the use of extra
information, namely, the lagged forward rates in the spot
equation and vice versa. The final columns of Tables
3.1–3.2A show the estimation results for the unrestricted VECM and restricted S-VECM models. These fit
the data a little better than the VAR models, and the SVECM is preferred by the Schwartz Bayesian criterion
in all cases. “Granger Causality”, as measured by the
significance of lagged forward rates in the spot equation
and lagged spot rates in the forward equation, seems to
run both ways. In all cases, the one-period lagged
change in forward rates is significant in the spot rate
equation, and the one-period lagged change in spot rates
is significant in the forward equation. There are also

routes. The cointegrating vector zt−1 = (St−1 +δ1Ft−1 +δ0)
is restricted to be the lagged basis (St−1 −Ft−1) in routes 1
and 2A; that is, δ1 = −1 and δ0 = 0. The other cointegrating vectors are zt−1 = St−1 − 1.0143 ⁎ Ft−1 + 0.1585 in
route 1A and zt−1 =St−1 − 1.0067 ⁎ Ft−1 + 0.0324 in route
2. The results of the likelihood ratio tests for the overidentifying restrictions applied to the cointegrating
vectors are 4.901 [0.086] for route 1; 8.011 [0.018] for
route 1A; 8.481 [0.014] for route 2; and 3.581 [0.167] for
route 2A, where figures in square brackets are p-values.
This discrepancy in the results might be result of the
different economic and trading conditions that prevailed
in each trading route in our sample period.
The results of ARIMA models for spot and forward
rates for the four routes are presented in the first
columns of Tables 3.1, 3.1A, 3.2 and 3.2A. The lag
length for the autoregressive and moving average parts
are chosen to minimise the Schwartz Bayesian
Criterion (SBC; Schwartz, 1978). All ARIMA models
seem to be well specified, as is indicated by relevant
diagnostic tests for autocorrelation and heteroskedasticity. It can be noted that across all routes, the adjusted
coefficient of determination for changes in spot rates
Table 3.2A
Estimates of the forecasting models for route 2A
Regressor

ARIMA

VAR

ΔSt

ΔFt

ΔSt

ΔFt

ΔSt

ΔFt

ΔSt

ΔFt

ΔSt−1

0.739⁎
(20.621)

–

0.416
(2.571)
0.034
(0.185)
0.108
(0.593)
−0.046
(0.306)
0.106
(1.889)
−0.109
(1.879)
−0.061
(1.042)
−0.037
(0.629)
–

0.416⁎
(2.59)
0.009
(0.051)
0.085
(0.468)
−0.065
(0.438)
0.167⁎
(2.723)
−0.054
(0.873)
−0.018
(0.301)
−0.001
(0.009)
0.088⁎
(2.362)
0.0588
9.977
[0.532]

0.570⁎
(13.289)
–

0.401⁎
(3.783)
–

–

–

–

–

0.066⁎
(3.181)
0.046⁎
(2.38)
0.038⁎
(2.063)
0.056⁎
(3.071)
− 0.030⁎
(2.444)
0.5892
17.396
[0.097]

0.174⁎
(3.026)
–

0.0467
12.469
[0.329]

0.558⁎
(10.141)
0.052
(0.827)
− 0.044
(0.710)
0.013
(0.255)
0.065⁎
(3.09)
0.036⁎⁎
(1.71)
0.037⁎⁎
(1.799)
0.055⁎
(2.677)
− 0.032⁎
(2.479)
0.5868
16.952
[0.109]

ΔSt−4

–

–

ΔFt−1

–

0.169⁎
(3.263)

ΔFt−2

–

ΔFt−3

–

ΔFt−4

–

–

zt−1

–

–

0.558
(10.072)
0.043
(0.681)
− 0.053
(0.839)
0.006
(0.118)
0.087
(4.537)
0.056
(2.819)
0.053
(2.635)
0.068
(3.399)
–

−
R2
Q(12)

0.5396
14.047
[0.230]

0.0259
10.209
[0.512]

0.5808
17.264
[0.100]

ΔSt−2

–

ΔSt−3

–

See notes to Table 3.1.
For route 2A, the cointegrating vector is the lagged basis St−1 − Ft−1.

VECM

S-VECM

–
–
0.101⁎
(3.145)
0.0715
10.433
[0.492]

110

R. Batchelor et al. / International Journal of Forecasting 23 (2007) 101–114

lagged equilibrium correction terms show that rates do
converge, and the relative importance of movements in
spot rates and movements in forward rates differs across
routes. In Route 1 only the forward rate adjusts to correct
disequilibrium, but in Route 2 only the spot rate moves.
In Routes 1A and 2A, adjustment coefficients on both
spot and forward rates are significant. In routes 1 and 1A
the adjustment coefficient on the forward rate is some
eight times higher than that on the spot rate, and in route

effects from longer lags of the forward rate on the spot
price in all models and all routes, but not vice versa,
which suggests that forward rates lead spot rates at
longer horizons.
However, the size and significance of the equilibrium
correction coefficients in the VECM models are not
consistent with our prior that in an efficient market for a
nonstorable service, the spot rate should converge on the
forward, but not vice versa. The coefficients of the

Table 4
Forecast error reduction for spot freight rate models
Route

1

1A

2

2A

Horizon

1
2
3
4
5
10
15
20
1
2
3
4
5
10
15
20
1
2
3
4
5
10
15
20
1
2
3
4
5
10
15
20

Number
of
Forecasts

Benchmark
RW

Incremental % reduction in RMSE of
ARIMA

VAR

VECM

S-VECM

%
reduction
from
S-VECM

520
260
173
130
104
52
34
26
520
260
173
130
104
52
34
26
707
353
235
176
141
70
47
35
707
353
235
176
141
70
47
35

1.64
0.72
0.92
0.99
1.16
0.87
1.54
1.06
1.31
1.45
1.75
1.67
2.30
1.92
2.57
1.68
0.79
0.99
1.05
1.28
1.18
0.92
1.22
0.92
1.34
1.76
1.65
2.35
1.94
1.34
2.39
1.75

12
−2
− 17
− 23
− 30
− 44
− 42
− 47
0
− 13
− 23
− 28
− 37
− 42
− 38
− 61
− 10
− 28
− 37
− 42
− 43
− 45
− 46
− 36
−11
− 36
− 38
− 48
− 39
− 47
− 45
− 55

−20
−19
−18
−18
−9
−1
− 11
−2
−14
−18
−18
−19
−4
−1
−6
−1
−3
−4
−1
−3
−3
−3
−1
−2
−7
−5
−2
−5
−2
0
−3
−1

0
0
1
1
0
−2
1
−1
−1
−1
0
0
0
0
0
−1
−2
−2
−2
−1
−1
−3
−1
−5
0
−1
0
−1
−1
−2
0
−2

−1
−1
−1
−1
−1
0
−1
0
0
0
0
0
0
0
0
−1
0
0
0
0
0
0
0
0
−1
0
−1
0
−1
−1
−1
−1

−9
− 22
− 36
− 41
− 40
− 47
− 54
− 51
− 14
− 32
−41
− 47
− 42
− 43
− 43
− 64
− 15
− 34
− 39
− 47
− 47
− 51
− 48
− 44
− 19
− 42
− 41
− 53
− 42
− 50
− 49
− 60

For horizons of 1, 2, 3, 4, 5, 10, 15, and 20 days, the table shows the number of non-overlapping forecasts in the period 1 July 1998 to 31 July 2000
(routes 1 and 1A) and 1 July 1998 to 30 April 2001 (routes 2 and 2A). The next column shows 100× the root mean square error of the benchmark nochange random walk forecast of the natural logarithm of the spot freight rate. The remaining columns show the percentage reduction of this
root mean square error achieved by using the next model in the sequence of progressively more statistically preferred models ARIMA, VAR, VECM
and S-VECM.

111

R. Batchelor et al. / International Journal of Forecasting 23 (2007) 101–114

prices up to 20-steps ahead. Then as recommended in
Tashman (2000), independent out-of-sample N-period
ahead forecasts are generated over the forecast period;
that is, from 1 July 1998 to 31 July 2000 for the Atlantic
routes and the period from 1 July 1998 to 30 April 2001
for the Pacific routes. In order to avoid the bias induced
by serially correlated overlapping forecast errors, we
recursively augment our estimation period by N-periods
ahead every time (where N corresponds to the number of
steps ahead). For example, in order to compute 5 steps-

2A some three times higher. So in three out of the four
routes, it is the forward rate rather than the spot rate that
bears the burden of adjustment.
5. Forecasting performance of the time-series
models
These alternative univariate and multivariate models,
estimated over the initial estimation period, are used to
generate independent forecasts of the spot and FFA
Table 5
Forecast error reduction for forward freight agreement models
Route

1

1A

2

2A

Horizon

1
2
3
4
5
10
15
20
1
2
3
4
5
10
15
20
1
2
3
4
5
10
15
20
1
2
3
4
5
10
15
20

Number
of
Forecasts

Benchmark
RW

Incremental % reduction in RMSE of
ARIMA

VAR

VECM

S-VECM

%
reduction
from
S-VECM

520
260
173
130
104
52
34
26
520
260
173
130
104
52
34
26
707
353
235
176
141
70
47
35
707
353
235
176
141
70
47
35

3.13
3.18
2.60
3.22
2.41
1.47
0.81
1.35
3.83
4.11
4.18
4.74
3.62
3.38
4.46
2.73
2.28
2.35
2.55
2.58
1.97
2.03
2.09
2.37
3.88
3.39
4.62
3.86
4.06
4.23
5.73
4.32

−33
−30
−31
−29
−30
−34
−37
−34
−29
−33
−28
−35
−27
−29
−31
−31
−27
−35
−28
−32
−28
−31
−27
−38
−28
−31
−29
−34
−20
−34
−31
−38

−1
−2
0
−1
−1
1
12
7
−2
−1
−2
−3
−2
−2
−4
−2
−1
0
−2
−4
0
0
−2
−2
−2
−2
−2
−2
−4
−3
−2
−3

0
0
−1
0
1
4
−3
−1
0
0
0
−1
0
2
1
1
0
0
−1
−1
0
1
0
0
0
0
0
−1
1
2
2
0

0
−1
0
0
−1
−3
7
1
−1
0
−1
0
−1
−2
−1
0
0
0
0
0
0
0
0
0
−1
0
−1
0
−1
−1
−1
0

−34
−32
−31
−31
−31
−31
−20
−26
−31
−34
−31
−39
−30
−31
−34
−33
−28
−36
−32
−36
−28
−30
−30
−40
−30
−33
−31
−37
−24
−37
−33
−41

For horizons of 1, 2, 3, 4, 5, 10, 15, and 20 days, the table shows the number of non-overlapping forecasts in the period 1 July 1998 to 31 July
2000 (routes 1 and 1A) and 1 July 1998 to 30 April 2001 (routes 2 and 2A). The next column shows 100× the root mean square error of the
benchmark no-change random walk forecast of the natural logarithm of the forward freight agreement rate. The remaining columns show the
percentage reduction of this root mean square error achieved by using the next model in the sequence of progressively more statistically preferred
models ARIMA, VAR, VECM and S-VECM.

112

R. Batchelor et al. / International Journal of Forecasting 23 (2007) 101–114

ahead forecasts, we augment our estimation period by
N = 5 observations each time. This method yields 104
and 141 independent non-overlapping forecasts in the
Atlantic and Pacific routes, respectively. Similarly, in
order to compute 10 steps-ahead forecasts, the method
yields 52 and 70 independent non-overlapping forecasts
in the Atlantic and Pacific routes. This methodology
provides two desirable characteristics of an out-ofsample test; adequacy (enough forecasts at each
forecasting horizon) and diversity (desensitising forecast error measures to special events).
The forecasting performance of each model for spot
prices across the different forecast horizons is presented in matrix form in Tables 4 and 5 for spot rate
forecasts and forward rate forecasts, respectively. The
forecast accuracy of each model is assessed using the
conventional root mean square error metric (RMSE).
The RMSE is considered a fairly accurate forecast
accuracy metric. It assumes a symmetric loss function
for forecast users, which seems reasonable given the
monetary amount of FFA contracts which can be
anything over $0.5 m for a monthly contract and up to
several million dollars in the case of a calendar
contract. We benchmark the models against each
other, and also against random walk, no-change,
forecasts for spot and forward rates. To highlight the
relative performance of the models, we report the
RMSE of the random walk model, and the incremental
percentage reduction in this RMSE achieved by
moving to progressively more complicated (and
statistically preferred) models. Consider the first row
of Table 4. The RMSE of the 1-day ahead forecast of
the (log) spot rate from the random walk model is
0.00642, or approximately 0.64% of the spot price. The
corresponding RMSEs for the ARIMA, VAR, VECM
and S-VECM models are 0.00716, 0.00590, 0.00592,
and 0.00585, respectively. The ARIMA model is less
accurate than the random walk and increases the RMSE
by 100 × (.00642 − .00716)/.00642 = 12%, and this is
reported in Table 4. The VAR model is much more
accurate than the ARIMA model, and improves the
RMSE by 100 × (0.00590 − 0.00716)/0.00642 = − 20%.
This represents an error reduction of about 12 − 20 =
− 8% of the RMSE of the random walk forecast, and so
on. Positive entries in Tables 4 and 5 indicate that the
model has failed to improve on the previous model.
Negative entries indicate an error reduction. The size of
the entry indicates the gain from using that model rather

than the previous model, expressed as a percentage of
the RMSE of the random walk model. To test the
significance of any outperformance, we also apply the
Diebold and Mariano (1995) test of the hypothesis that
the RMSEs from two competing models are equal,
exhaustively to all pairs of models. Results indicate that
the RMSEs of the S-VECM and VECM forecasts are,
in general, significantly lower than other competing
models, especially for shorter maturities. Full tables of
these model comparisons are not shown here but are
available from the authors.
The final column of the table shows the total
percentage reduction in RMSE achieved by using the
S-VECM model, the sum of the incremental gains
from all the models. In the case of 1-day ahead spot
rate forecasts for route 1, the use of the S-VECM
model reduces the RMSE from the random walk
forecast by 9% (as it happens, an atypically small
improvement).
Several regularities stand out from the tables. First,
as expected, the forward rates are much harder to
forecast than the spot rates. The higher volatility of
the forwards is evidenced by RMSEs from the nochange random walk model that are some three to
four times higher than the corresponding spot rate
RMSEs.
Second, all the S-VECM models comfortably
outperform the random walk benchmark. On average
over all routes and horizons they achieve a reduction of
about 40% for the spot rate RMSE and 30% for the
forward rate RMSE. For spot rate forecasts the
improvement is less marked at very short horizons
(about 15–20% for 1–4 days ahead), but about 50%
for longer horizons. For the forward rate forecasts, the
advantage of the model over the random walk is
uniform across horizons. Not surprisingly, in all cases,
the Diebold–Mariano statistic rejects the hypothesis of
equal RMSE between the random walk and S-VECM
models. These results are comparable to the findings of
Ghosh (1993), who reports reductions in the RMSE for
1-day ahead spot forecasts ranging from 15% to 34%
for the S&P500 and Commodity Research Bureau
(CRB) spot indices respectively. Corresponding reductions for the 1-day-ahead futures forecasts range from
24% to 39%. Of course, the greater volatility of
forward rates means that the RMSEs from the forward
rate forecasting models are still two to three times
higher than forecasts from the spot rate models.

R. Batchelor et al. / International Journal of Forecasting 23 (2007) 101–114

Third, the models with equilibrium correction features (VECM and S-VECM) perform better than VAR
models for forecasts of spot rates, but not for forecasts
of forward rates. Looking at the spot rate forecasts in
Table 4, in the liquid Pacific routes 2 and 2A, the use of
the VECM in place of the VAR reduces the benchmark
RMSE by up to 5%, and the use of the restricted SVECM reduces it by a further 1%. In the less liquid
Atlantic routes 1 and 1A, the improvements in accuracy are less marked. Most improvements in forecast
accuracy can be achieved by a VAR in place of the
univariate ARIMA model, and the incremental gains
from using a VECM are small. But at no horizon does
the use of the VECM for spot rate forecasting reduce
the forecast accuracy relative to the VAR. In contrast,
the statistics in Table 5 show that for forward rate
forecasting most improvements in accuracy relative to
the random walk can be achieved by a simple ARIMA
model. Use of the VAR actually reduces accuracy in
longer term forecasts in route 1 and reduces the RMSE
by up to 4% in other routes, and the use of the VECM
and S-VECM models in place of the VAR leads to very
small and inconsistent changes in accuracy in forecasting forward rates.
6. Conclusion
The forward freight market is relatively new,
relatively illiquid and relatively under-researched. In
this paper, we have explored the value of various
standard linear time series models as forecasting tools
to jointly predict the spot and forward freight rates. If
the forward market is liquid enough to embody some
information about futures spot rates into the prices, we
should observe cointegration between spot and forward
rates, and convergence of spot towards forward rates,
rather than vice versa. Spot and forward freight rates are
indeed cointegrated. But the models in Tables 3.1–3.2A
suggest that, contrary to our expectations, forward rates
adjust more strongly than spot rates to close the gap
between spot and forward rates. However, out-ofsample forecasting with these equilibrium correction
models show that they are not helpful in predicting
forward rate behaviour, but do help predict spot rates, a
finding more consistent with market efficiency.
For shipowners and charterers, the findings of our
study are encouraging, in the sense that they suggest
that spot freight rates are forecastable, and the rates

113

offered by Forward Freight Agreements to some extent
help anticipate spot freight rates. For analysts of
commodity markets the message is more cautionary,
an illustration of the dangers of forecasting with an
equilibrium correction model when the underlying
market is evolving, and the parameter estimates
conflict with sensible theoretical priors.
Acknowledgements
We thank Professor Michael Clements and an
anonymous referee for their extremely helpful comments. This paper has also largely benefited from the
comments of participants at the 2003 International
Association of Maritime Economists Conference,
Busan, Korea, and the 2003 International Symposium
on Forecasting, Merida, Mexico.
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