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# Optimal ship speed and the cubic law revisited:
Empirical evidence from an oil tanker fleet

Roar Adland, Pierre Cariou, Francois-Charles Wolff

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Roar Adland, Pierre Cariou, Francois-Charles Wolff. Optimal ship speed and the cubic law revisited: Empirical evidence from an oil tanker fleet: .. Transportation Research Part E: Logistics and Transportation Review, 2020, 140, pp.101972. 10.1016/j.tre.2020.101972. hal-03422276

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# Optimal ship speed and the cubic law revisited: Empirical evidence from an oil tanker fleet

<figure><img src="image_1.png" /></figure>

Roar Adland<sup>a, *</sup>, Pierre Cariou<sup>b</sup>, Francois-Charles Wolff<sup>c</sup>

## ARTICLE INFO

### Keywords:

Fuel consumption

Cubic law

Optimal speed

Noon reports

## ABSTRACT

We develop a flexible framework for the estimation of the fuel consumption-speed curve for ships which allows for speed-dependent elasticity with endogenous thresholds. Using a large dataset of noon reports for 16 crude oil tankers, we estimate the corresponding elasticities net of weather effects. Our empirical findings confirm that the classical cubic law for fuel consumption is valid only near the design speed but also that the sensitivity with regards to sailing speed can be substantially lower at the sailing speeds actually observed. Our results can be used to question the economic and environmental benefits of slow-steaming and fuel levies.

## 1. Introduction

The move towards shipping decarbonization (OECD, 2009) initiated by the International Maritime Organization (IMO, 2014) has put renewed focus on the relationship between vessel sailing speeds and carbon emissions, particularly as speed reduction is often cited as the best short-term mitigation measure to curb emissions (Cariou, 2011; Lindstad et al., 2011; Faber et al., 2012, 2017; Comer et al., 2018; Leaper, 2019). Given the one-to-one relationship between fuel consumption and emissions, the optimal policy direction is crucially dependent on the correct modelling of vessels' fuel consumption as a function of sailing speed. Similarly, at the micro-economic level, the assessment of the economically optimal sailing speed for a given voyage is highly dependent on the true elasticity of fuel consumption to vessel speed.

For a given vessel design, cargo intake and voyage parameters, the daily fuel consumption of a ship (C) is proportional to its speed (S) to the power of $\beta$, $C \sim S^\beta$. The parameter $\beta$ measures the responsiveness of fuel consumption to changes in speed and is a ship-specific elasticity (Wang and Meng, 2012). Despite its importance and an extensive literature (see, for instance, Wang and Meng, 2012; Yao et al., 2012; Psaraftis and Kontovas, 2013, 2014; Xia et al., 2015; Meng et al., 2016), there is no clear consensus on the empirical characteristics of the consumption-speed function.

The predominant view has been that the daily bunker fuel consumption, and therefore carbon emissions, can be well approximated by a cubic function of speed ($\beta = 3$) (see, for instance, Ryder and Chappell, 1980; Ronen, 1982, 2011; Wang and Meng, 2012) and that this 'cubic law' should be used when there is not enough historical consumption data available (Wang and Meng, 2012). However, this approximation is also known to hold when vessels are sailing at a speed close to design speed (IMO, 2014), implying that the elasticity might be better modelled as a function of vessel speed itself.

The typical 'cubic law' assumption is therefore potentially problematic when market conditions are such that slow-steaming is the

<footnote><sup>a</sup> Norwegian School of Economics (NHH), Helleveien 30, 5045 Bergen, Norway</footnote>

<footnote><sup>b</sup> Kedge Business School, 680 Cours de la Libération, 33405, Talence, France</footnote>

<footnote><sup>c</sup> LEMNA, University of Nantes, BP 52231 Chemin de la Censive du Tertre, 44322 Nantes Cedex 3, France</footnote>

<footnote>* Corresponding author.
E-mail addresses: roar.adland@nhh.no (R. Adland), pierre.cariou@kedgebs.com (P. Cariou), francois.wolff@univ-nantes.fr (F.-C. Wolff).</footnote>
# Optimal ship speed and the cubic law revisited: Empirical evidence from an oil tanker fleet

<figure><img src="image_3.png" /></figure>

Roar Adland<sup>a, *</sup>, Pierre Cariou<sup>b</sup>, Francois-Charles Wolff<sup>c</sup>

## ARTICLE INFO

### Keywords:

Fuel consumption

Cubic law

Optimal speed

Noon reports

## ABSTRACT

We develop a flexible framework for the estimation of the fuel consumption-speed curve for ships which allows for speed-dependent elasticity with endogenous thresholds. Using a large dataset of noon reports for 16 crude oil tankers, we estimate the corresponding elasticities net of weather effects. Our empirical findings confirm that the classical cubic law for fuel consumption is valid only near the design speed but also that the sensitivity with regards to sailing speed can be substantially lower at the sailing speeds actually observed. Our results can be used to question the economic and environmental benefits of slow-steaming and fuel levies.

## 1. Introduction

The move towards shipping decarbonization (OECD, 2009) initiated by the International Maritime Organization (IMO, 2014) has put renewed focus on the relationship between vessel sailing speeds and carbon emissions, particularly as speed reduction is often cited as the best short-term mitigation measure to curb emissions (Cariou, 2011; Lindstad et al., 2011; Faber et al., 2012, 2017; Comer et al., 2018; Leaper, 2019). Given the one-to-one relationship between fuel consumption and emissions, the optimal policy direction is crucially dependent on the correct modelling of vessels' fuel consumption as a function of sailing speed. Similarly, at the micro-economic level, the assessment of the economically optimal sailing speed for a given voyage is highly dependent on the true elasticity of fuel consumption to vessel speed.

For a given vessel design, cargo intake and voyage parameters, the daily fuel consumption of a ship (C) is proportional to its speed (S) to the power of $\beta$, $C \sim S^\beta$. The parameter $\beta$ measures the responsiveness of fuel consumption to changes in speed and is a ship-specific elasticity (Wang and Meng, 2012). Despite its importance and an extensive literature (see, for instance, Wang and Meng, 2012; Yao et al., 2012; Psaraftis and Kontovas, 2013, 2014; Xia et al., 2015; Meng et al., 2016), there is no clear consensus on the empirical characteristics of the consumption-speed function.

The predominant view has been that the daily bunker fuel consumption, and therefore carbon emissions, can be well approximated by a cubic function of speed ($\beta = 3$) (see, for instance, Ryder and Chappell, 1980; Ronen, 1982, 2011; Wang and Meng, 2012) and that this 'cubic law' should be used when there is not enough historical consumption data available (Wang and Meng, 2012). However, this approximation is also known to hold when vessels are sailing at a speed close to design speed (IMO, 2014), implying that the elasticity might be better modelled as a function of vessel speed itself.

The typical 'cubic law' assumption is therefore potentially problematic when market conditions are such that slow-steaming is the

<footnote><sup>a</sup> Norwegian School of Economics (NHH), Helleveien 30, 5045 Bergen, Norway</footnote>

<footnote><sup>b</sup> Kedge Business School, 680 Cours de la Libération, 33405, Talence, France</footnote>

<footnote><sup>c</sup> LEMNA, University of Nantes, BP 52231 Chemin de la Censive du Tertre, 44322 Nantes Cedex 3, France</footnote>

<footnote>* Corresponding author.
E-mail addresses: roar.adland@nhh.no (R. Adland), pierre.cariou@kedgebs.com (P. Cariou), francois.wolff@univ-nantes.fr (F.-C. Wolff).</footnote>
prevailing mode of operation, as it has been the case in shipping since 2008 (IMO, 2014; Cariou et al., 2019). For instance, Clarkson Research (October 2019) reported reductions in average fleet speeds of 16% for bulkers, 18% for tankers and 24% for containership compared to 2008. Under such operational conditions, the general assumption that fuel consumption is proportional to sailing speed to a power of three needs to be reevaluated (Psaraftis and Kontovas, 2013; Wang and Meng, 2012; Kristensen, 2019).

The main methodological innovation in our paper is that we allow the speed elasticity β to be a function of vessel speed and estimate its functional form using real data from vessel operations. Improved access to high-frequency vessel performance and weather data now enable researchers to estimate the empirical speed-consumption relationship ‘ex-post’ for existing vessels. We follow this line of research in the current paper. For instance, the impact of external factors such as weather attributes (in particular wind and waves) has been reexamined using artificial neural network techniques (ANN) that are able to deal with multi-dimensional weather data (Pedersen and Larsen, 2009; Petersen et al., 2012; Beşikçi et al., 2016; Jeon et al., 2018; Wang et al., 2018, 2019; Du et al., 2019). Similarly, empirical speed and draft observations from the Automated Identification System (AIS) can be used to estimate aggregate emissions from the world fleet (IMO, 2014), despite limitations mostly related to human error in the data input (Aldous et al., 2013; Meng et al., 2016).

The objective of this paper is to investigate empirically how the speed-consumption relationship changes across the full observed speed range. Our empirical analysis is based on a unique dataset of more than 11,000 noon reports for a fleet of 6 Suezmax and 10 Aframax oil tankers deployed in global trade.<fnref n="1" /> Noon reports record the daily fuel consumption, voyage data and prevailing weather conditions at noon of every day that the vessel is sailing. While it is the most common way to transmit such data from the vessel to the shore office, data recorded in noon reports are mainly based on human input. As a consequence, they may suffer from inaccuracies.

Our contributions to the literature are threefold. First, we present a new econometric framework for the data-driven estimation of the speed-to-consumption elasticity without relying on naval architecture principles. Aside from speed, our estimates account for the role of important explanatory variables like draft, trim, sea current, waves and wind (and its interaction with the ship’s heading). Second, we propose a new fuel consumption model that allows for speed-dependent elasticities across speed intervals. We develop an econometric methodology based on the estimation of a piecewise linear function for determining these intervals endogeneously. Third, we illustrate the difference between constant and speed-varying fuel consumption elasticities in the determination of the economically optimal speed of vessels.

Our empirical results show that the elasticity of fuel consumption with regards to speed is dependent on the vessel speed itself. While the “cubic law” is found to hold true near the design speed of vessels, the elasticity of fuel consumption with regards to vessel speed is substantially lower in the speed range where ships mostly operate. This is important for economic models of ship fuel consumption and has potentially profound effects for maritime environmental policy. In particular, our findings can explain why the observed adjustment of vessel speed to freight market conditions and fuel prices is limited (Adland and Jia, 2016, 2018). Our findings can also be used to challenge the general political view that a large reduction in CO2 emissions can be achieved through further speed reduction (OECD, 2009), and that an increase in fuel price (through for instance the implementation of a bunker levy) would force ship operators to reduce speed and therefore CO2 emissions. At the microeconomic level, improved evaluation of fuel consumption would enable shipowners and operators to make more accurate voyage calculations and improve chartering decisions. It would also be a key input in calculations of the tradeoff between fleet size and optimal vessel speed in liner shipping, as considered in Ng (2019).

The remainder of the paper is organized as follows. In Section 2, we outline the fundamental relationship between ship fuel consumption and speed and report estimates of the speed-consumption elasticity in the literature. In Section 3, we provide some descriptive statistics for our sample of noon reports. In Section 4, we present the econometric framework when using a constant power function for speed to fuel consumption and show that the assumption of a speed elasticity equal to three is rejected. Then, we estimate a more general specification with speed-varying elasticities showing that elasticities are much lower than 3 in the lower part of the distribution of speed values. The implications of our empirical findings on profit maximization and CO2 emissions reduction are discussed in Section 5. Finally, Section 6 concludes.

## 2. Fundamentals of the speed-consumption relationship

The theoretical foundation of ship fuel consumption (Telfer, 1926; Todd, 1967), often termed as ‘resistance modelling’, aims at estimating the ship’s total resistance (or, equivalently, power requirement) as a function of speed and external factors such as wind and waves. Formally, if the effective horse power for propulsion at the propeller for a given speed is $P_s$ (Lewis, 1988; Lloyd, 1989; Lindstad et al., 2013, 2015; Meng et al., 2016) and the additional power requirement to account for the impact of waves and wind is $P_w$ and $P_a$, respectively, then fuel consumption FC may be expressed as:

$$FC = \frac{P_s + P_w + P_a}{\eta} \qquad (1)$$

where η is defined as the propulsion efficiency. Propulsion efficiency is a product of the efficiency of the main engine, the propeller efficiency, and their interaction at various engine loads and for different external operating environments. Importantly for our purposes, propulsion efficiency is a function of engine load, and therefore implicitly vessel speed itself, through the Specific Fuel

<footnote n="1">¹ While the ability to generalize our results cannot be confirmed in the absence of much larger datasets, our sample remain substantially larger than those in the literature, for instance, the four container services investigated in Meng et al. (2016). We also note that the global tanker fleet is rather homogeneous within defined size brackets (VLCC, Suezmax, Aframax etc.) due to highly standardized oil cargo sizes.</footnote>
prevailing mode of operation, as it has been the case in shipping since 2008 (IMO, 2014; Cariou et al., 2019). For instance, Clarkson Research (October 2019) reported reductions in average fleet speeds of 16% for bulkers, 18% for tankers and 24% for containership compared to 2008. Under such operational conditions, the general assumption that fuel consumption is proportional to sailing speed to a power of three needs to be reevaluated (Psaraftis and Kontovas, 2013; Wang and Meng, 2012; Kristensen, 2019).

The main methodological innovation in our paper is that we allow the speed elasticity β to be a function of vessel speed and estimate its functional form using real data from vessel operations. Improved access to high-frequency vessel performance and weather data now enable researchers to estimate the empirical speed-consumption relationship ‘ex-post’ for existing vessels. We follow this line of research in the current paper. For instance, the impact of external factors such as weather attributes (in particular wind and waves) has been reexamined using artificial neural network techniques (ANN) that are able to deal with multi-dimensional weather data (Pedersen and Larsen, 2009; Petersen et al., 2012; Beşikçi et al., 2016; Jeon et al., 2018; Wang et al., 2018, 2019; Du et al., 2019). Similarly, empirical speed and draft observations from the Automated Identification System (AIS) can be used to estimate aggregate emissions from the world fleet (IMO, 2014), despite limitations mostly related to human error in the data input (Aldous et al., 2013; Meng et al., 2016).

The objective of this paper is to investigate empirically how the speed-consumption relationship changes across the full observed speed range. Our empirical analysis is based on a unique dataset of more than 11,000 noon reports for a fleet of 6 Suezmax and 10 Aframax oil tankers deployed in global trade.<fnref n="1" /> Noon reports record the daily fuel consumption, voyage data and prevailing weather conditions at noon of every day that the vessel is sailing. While it is the most common way to transmit such data from the vessel to the shore office, data recorded in noon reports are mainly based on human input. As a consequence, they may suffer from inaccuracies.

Our contributions to the literature are threefold. First, we present a new econometric framework for the data-driven estimation of the speed-to-consumption elasticity without relying on naval architecture principles. Aside from speed, our estimates account for the role of important explanatory variables like draft, trim, sea current, waves and wind (and its interaction with the ship’s heading). Second, we propose a new fuel consumption model that allows for speed-dependent elasticities across speed intervals. We develop an econometric methodology based on the estimation of a piecewise linear function for determining these intervals endogeneously. Third, we illustrate the difference between constant and speed-varying fuel consumption elasticities in the determination of the economically optimal speed of vessels.

Our empirical results show that the elasticity of fuel consumption with regards to speed is dependent on the vessel speed itself. While the “cubic law” is found to hold true near the design speed of vessels, the elasticity of fuel consumption with regards to vessel speed is substantially lower in the speed range where ships mostly operate. This is important for economic models of ship fuel consumption and has potentially profound effects for maritime environmental policy. In particular, our findings can explain why the observed adjustment of vessel speed to freight market conditions and fuel prices is limited (Adland and Jia, 2016, 2018). Our findings can also be used to challenge the general political view that a large reduction in CO2 emissions can be achieved through further speed reduction (OECD, 2009), and that an increase in fuel price (through for instance the implementation of a bunker levy) would force ship operators to reduce speed and therefore CO2 emissions. At the microeconomic level, improved evaluation of fuel consumption would enable shipowners and operators to make more accurate voyage calculations and improve chartering decisions. It would also be a key input in calculations of the tradeoff between fleet size and optimal vessel speed in liner shipping, as considered in Ng (2019).

The remainder of the paper is organized as follows. In Section 2, we outline the fundamental relationship between ship fuel consumption and speed and report estimates of the speed-consumption elasticity in the literature. In Section 3, we provide some descriptive statistics for our sample of noon reports. In Section 4, we present the econometric framework when using a constant power function for speed to fuel consumption and show that the assumption of a speed elasticity equal to three is rejected. Then, we estimate a more general specification with speed-varying elasticities showing that elasticities are much lower than 3 in the lower part of the distribution of speed values. The implications of our empirical findings on profit maximization and CO2 emissions reduction are discussed in Section 5. Finally, Section 6 concludes.

## 2. Fundamentals of the speed-consumption relationship

The theoretical foundation of ship fuel consumption (Telfer, 1926; Todd, 1967), often termed as ‘resistance modelling’, aims at estimating the ship’s total resistance (or, equivalently, power requirement) as a function of speed and external factors such as wind and waves. Formally, if the effective horse power for propulsion at the propeller for a given speed is $P_s$ (Lewis, 1988; Lloyd, 1989; Lindstad et al., 2013, 2015; Meng et al., 2016) and the additional power requirement to account for the impact of waves and wind is $P_w$ and $P_a$, respectively, then fuel consumption FC may be expressed as:

$$FC = \frac{P_s + P_w + P_a}{\eta} \qquad (1)$$

where η is defined as the propulsion efficiency. Propulsion efficiency is a product of the efficiency of the main engine, the propeller efficiency, and their interaction at various engine loads and for different external operating environments. Importantly for our purposes, propulsion efficiency is a function of engine load, and therefore implicitly vessel speed itself, through the Specific Fuel

<footnote n="1">¹ While the ability to generalize our results cannot be confirmed in the absence of much larger datasets, our sample remain substantially larger than those in the literature, for instance, the four container services investigated in Meng et al. (2016). We also note that the global tanker fleet is rather homogeneous within defined size brackets (VLCC, Suezmax, Aframax etc.) due to highly standardized oil cargo sizes.</footnote>
Consumption function (SFC). SFC refers to the fuel required per produced kWh [g/kWh] and is a U-shaped non-linear function of engine load<fnref n="2" />. Vessels are designed to sail at speeds equivalent to 70–80% engine load, which is where the SFC function reaches its minimum (MAN Diesel & Turbo, 2004). Additionally, the ship resistance introduced by the wave-making of the hull is speed-dependent, creating non-linearities in the total power requirement as a function of vessel speed. Both sources of non-linearity (engine efficiency and wave-making resistance) implies that the elasticity of fuel consumption to speed cannot be constant. In particular, we expect a priori that the elasticity is a decreasing function of speed itself.

In reality, traditional resistance modeling, as represented by Eq. (1), is too simplified to be able to account for the multitude of determinants of fuel consumption and their complex, stochastic, potentially non-linear interactions. It suffices to mention that aside from vessel speed, the fuel consumption for a specific ship is known to be influenced by its draught, trim, hull fouling condition, wave height and direction, wind force and direction (Adland et al., 2018), water salinity, water depth, and water temperature, to mention only some. Accordingly, many fuel consumption functions have been proposed in the literature (see, for instance, Wang and Meng, 2012; Yao et al., 2012; Psaraftis and Kontovas, 2013; Xia et al., 2015; Meng et al., 2016; He et al., 2017; Wang et al., 2019). All of these recognize vessel speed as the principal determinant of a ship's fuel consumption, through its direct impact on the required horsepower and the SFC (Psaraftis and Kontovas, 2013), even though its relative importance changes with the residual resistance from weather conditions (Lo and McCord, 1995; Meng et al., 2016; Adland et al., 2018; Du et al., 2019).

The proposed models differ primarily in their complexity and whether researchers take a ‘model-free’, purely data-driven approach or base their models, at least in part, on naval architecture principles. Pedersen and Larsen (2009) were the first to argue in favor of the former, using artificial neural networks to predict propulsion power from the variables influencing ship resistance, such as ship speed, relative wind speed and direction, air temperature and sea water temperature. More recent empirical studies that were also based on ship log data from noon reports include Wang and Meng (2012), Du et al. (2019) and Adland et al. (2018). Conversely, Kristensen (2019) provides the most comprehensive study that relies on traditional resistance modelling methods for estimating the speed-fuel elasticities under different weather conditions and for three vessels types (seven tankers, six bulk carriers and five containerships).

Other studies combine elements from naval architecture principles with “black box” machine-learning models built on empirical data to produce hybrid “gray-box models” for the optimization of trim (Coraddu et al., 2018) or the estimation of the fuel consumption function (Leifsson et al., 2008; Yang et al., 2019). Lepore et al. (2017) compare several regression techniques, including variable selection methods, penalized regression methods, latent variable methods and tree-based ensemble methods, to develop an empirical prediction model for CO2 emissions for a passenger ferry.

We note here that vessel performance data are typically considered to be sensitive and are not currently available from any third-party provider or on a large scale. As a consequence, the empirical studies are necessarily limited in scope to small fleets (typically, between 1 and 3 vessels in the above papers) that are operated by a single shipping company. Theoretical or numerical studies based purely on traditional naval architecture principles do not have such limitations. In this context, our sample consisting of operational data from a total of 16 vessels is a significant improvement.

Importantly, with the exception of Kristensen (2019), all models in the literature assume a constant speed-consumption elasticity across the assumed range of sailing speeds for a particular vessel type, size and loading condition. Table 1 reports some examples from the existing literature on the estimated speed-consumption elasticity.

The wide range of elasticity estimates in Table 1 – admittedly across a wide variety of vessel types and varying assumptions regarding weather effects – suggests that the assumption of constant elasticity needs to be re-evaluated. In this paper, we propose a new model framework for fuel consumption that allows for speed-dependent elasticity and estimate the corresponding ranges en-dogeneously.

### 3. Data and descriptive statistics

For our empirical analysis, we consider a dataset on a fleet of ships belonging to one (anonymous) international shipping company. The fleet under consideration consists of 10 Aframax product tankers with an average carrying capacity of around 120,000 DWT and 6 Suezmax vessels with an average carrying capacity of around 160,000 DWT. Within each fleet category, all vessels are sister ships.

Information on noon reports was collected on a daily basis for each vessel between January 2013 and December 2016 (see Aldous et al., 2013, for a detailed discussion of noon report data and quality). The noon report is manually compiled by the crew. It includes details on the vessel's position and weather conditions as snapshots at noon, along with the cumulative sailed distance and fuel consumption since the previous report. The database includes the following covariates: a vessel identifier, the date (year, month, day), time since last report (in hours), port of departure and port of destination, draft forward and draft aft (in meters), whether the vessel is in ballast, two measures of daily average speed expressed in knots (GPS speed corresponding to speed over ground and LOG speed corresponding to speed through water), fuel consumption expressed in tonnes per day, relative wind direction, wind type, sea state, swell direction and swell type.

Fig. 1 presents the distribution of vessels' speed values over the entire period and for each type of vessels, when the vessel is in ballast and when laden<fnref n="3" />. A first result is a general bimodal distribution with a mass point at zero that corresponds to days when a

<footnote n="2">² The IMO (2014) estimate of SFC as a function of engine load is $SFC = SFC_{base} * (0.455.Load^2 - 0.71.Load + 1.28)$.</footnote>

<footnote n="3">³ Results reported for LOG speed are similar than when using GPS speed.</footnote>
Consumption function (SFC). SFC refers to the fuel required per produced kWh [g/kWh] and is a U-shaped non-linear function of engine load<fnref n="2" />. Vessels are designed to sail at speeds equivalent to 70–80% engine load, which is where the SFC function reaches its minimum (MAN Diesel & Turbo, 2004). Additionally, the ship resistance introduced by the wave-making of the hull is speed-dependent, creating non-linearities in the total power requirement as a function of vessel speed. Both sources of non-linearity (engine efficiency and wave-making resistance) implies that the elasticity of fuel consumption to speed cannot be constant. In particular, we expect a priori that the elasticity is a decreasing function of speed itself.

In reality, traditional resistance modeling, as represented by Eq. (1), is too simplified to be able to account for the multitude of determinants of fuel consumption and their complex, stochastic, potentially non-linear interactions. It suffices to mention that aside from vessel speed, the fuel consumption for a specific ship is known to be influenced by its draught, trim, hull fouling condition, wave height and direction, wind force and direction (Adland et al., 2018), water salinity, water depth, and water temperature, to mention only some. Accordingly, many fuel consumption functions have been proposed in the literature (see, for instance, Wang and Meng, 2012; Yao et al., 2012; Psaraftis and Kontovas, 2013; Xia et al., 2015; Meng et al., 2016; He et al., 2017; Wang et al., 2019). All of these recognize vessel speed as the principal determinant of a ship's fuel consumption, through its direct impact on the required horsepower and the SFC (Psaraftis and Kontovas, 2013), even though its relative importance changes with the residual resistance from weather conditions (Lo and McCord, 1995; Meng et al., 2016; Adland et al., 2018; Du et al., 2019).

The proposed models differ primarily in their complexity and whether researchers take a ‘model-free’, purely data-driven approach or base their models, at least in part, on naval architecture principles. Pedersen and Larsen (2009) were the first to argue in favor of the former, using artificial neural networks to predict propulsion power from the variables influencing ship resistance, such as ship speed, relative wind speed and direction, air temperature and sea water temperature. More recent empirical studies that were also based on ship log data from noon reports include Wang and Meng (2012), Du et al. (2019) and Adland et al. (2018). Conversely, Kristensen (2019) provides the most comprehensive study that relies on traditional resistance modelling methods for estimating the speed-fuel elasticities under different weather conditions and for three vessels types (seven tankers, six bulk carriers and five containerships).

Other studies combine elements from naval architecture principles with “black box” machine-learning models built on empirical data to produce hybrid “gray-box models” for the optimization of trim (Coraddu et al., 2018) or the estimation of the fuel consumption function (Leifsson et al., 2008; Yang et al., 2019). Lepore et al. (2017) compare several regression techniques, including variable selection methods, penalized regression methods, latent variable methods and tree-based ensemble methods, to develop an empirical prediction model for CO2 emissions for a passenger ferry.

We note here that vessel performance data are typically considered to be sensitive and are not currently available from any third-party provider or on a large scale. As a consequence, the empirical studies are necessarily limited in scope to small fleets (typically, between 1 and 3 vessels in the above papers) that are operated by a single shipping company. Theoretical or numerical studies based purely on traditional naval architecture principles do not have such limitations. In this context, our sample consisting of operational data from a total of 16 vessels is a significant improvement.

Importantly, with the exception of Kristensen (2019), all models in the literature assume a constant speed-consumption elasticity across the assumed range of sailing speeds for a particular vessel type, size and loading condition. Table 1 reports some examples from the existing literature on the estimated speed-consumption elasticity.

The wide range of elasticity estimates in Table 1 – admittedly across a wide variety of vessel types and varying assumptions regarding weather effects – suggests that the assumption of constant elasticity needs to be re-evaluated. In this paper, we propose a new model framework for fuel consumption that allows for speed-dependent elasticity and estimate the corresponding ranges en-dogeneously.

### 3. Data and descriptive statistics

For our empirical analysis, we consider a dataset on a fleet of ships belonging to one (anonymous) international shipping company. The fleet under consideration consists of 10 Aframax product tankers with an average carrying capacity of around 120,000 DWT and 6 Suezmax vessels with an average carrying capacity of around 160,000 DWT. Within each fleet category, all vessels are sister ships.

Information on noon reports was collected on a daily basis for each vessel between January 2013 and December 2016 (see Aldous et al., 2013, for a detailed discussion of noon report data and quality). The noon report is manually compiled by the crew. It includes details on the vessel's position and weather conditions as snapshots at noon, along with the cumulative sailed distance and fuel consumption since the previous report. The database includes the following covariates: a vessel identifier, the date (year, month, day), time since last report (in hours), port of departure and port of destination, draft forward and draft aft (in meters), whether the vessel is in ballast, two measures of daily average speed expressed in knots (GPS speed corresponding to speed over ground and LOG speed corresponding to speed through water), fuel consumption expressed in tonnes per day, relative wind direction, wind type, sea state, swell direction and swell type.

Fig. 1 presents the distribution of vessels' speed values over the entire period and for each type of vessels, when the vessel is in ballast and when laden<fnref n="3" />. A first result is a general bimodal distribution with a mass point at zero that corresponds to days when a

<footnote n="2">² The IMO (2014) estimate of SFC as a function of engine load is $SFC = SFC_{base} * (0.455.Load^2 - 0.71.Load + 1.28)$.</footnote>

<footnote n="3">³ Results reported for LOG speed are similar than when using GPS speed.</footnote>
<table><caption>**Table 1**
Examples of fuel consumption-speed elasticity ($\beta$) estimates.</caption><thead><tr><th>Study</th><th>Vessel type</th><th>β</th></tr></thead><tbody><tr><td>MAN Diesel & Turbo (2004)</td><td>Feeder containership</td><td>3.5</td></tr><tr><td></td><td>Large containership</td><td>4</td></tr><tr><td>Psaraftis and Kontovas (2010)</td><td>Oil Tanker</td><td>3-4</td></tr><tr><td>Notteboom and Cariou (2009)</td><td>Containership</td><td>3.3</td></tr><tr><td>Wang and Meng (2012)</td><td>Containerships (&lt; 20 kt)</td><td>2.7-3.3</td></tr><tr><td>Adland, Cariou and Wolff (2018)</td><td>Oil Tanker (Aframax/Suezmax)</td><td>1.5-2.1</td></tr><tr><td>Kristensen (2019)</td><td>Oil Tanker (4.5-15 kt)<br>(15,000-110,000) & (110 000 + ) dwt</td><td>(1.6-4.8) & (1.7-3.7)</td></tr><tr><td></td><td>Bulk Carrier (4.1-15 kt)<br>(15 000-75 000) & (175 000+) dwt</td><td>(1.6-4.3) & (1.6-3.7)</td></tr><tr><td></td><td>Containership (5.6 to 23.5 kt)<br>(1500 TEU) & (5000 + TEU)</td><td>(1.9-4.4) & (1.8-3.5)</td></tr></tbody></table>

<figure><img src="image_2.png" /><figcaption>**Fig. 1. Distribution of vessel speed.** Source: data from a tanker company, authors' calculations.</figcaption></figure>

vessel is at anchorage or in port. Vessels are usually sailing at a higher speed when laden rather than in ballast. This difference is explained by contractual obligations when ships carry cargoes and by a lower fuel consumption when a vessel is in ballast.<fnref n="4" /> A second result is that over this period of time (2013–2016), vessels were sailing at a speed which is far below design speed. For the Aframax

<footnote n="4">⁴ From a legal point of view, the shipowner undertakes that the ship shall proceed on the carrying (laden) voyage with “utmost despatch”, i.e. without delay, due to speed reduction or otherwise.</footnote>
<table><caption>**Table 1**
Examples of fuel consumption-speed elasticity ($\beta$) estimates.</caption><thead><tr><th>Study</th><th>Vessel type</th><th>β</th></tr></thead><tbody><tr><td>MAN Diesel & Turbo (2004)</td><td>Feeder containership</td><td>3.5</td></tr><tr><td></td><td>Large containership</td><td>4</td></tr><tr><td>Psaraftis and Kontovas (2010)</td><td>Oil Tanker</td><td>3-4</td></tr><tr><td>Notteboom and Cariou (2009)</td><td>Containership</td><td>3.3</td></tr><tr><td>Wang and Meng (2012)</td><td>Containerships (&lt; 20 kt)</td><td>2.7-3.3</td></tr><tr><td>Adland, Cariou and Wolff (2018)</td><td>Oil Tanker (Aframax/Suezmax)</td><td>1.5-2.1</td></tr><tr><td>Kristensen (2019)</td><td>Oil Tanker (4.5-15 kt)<br>(15,000-110,000) & (110 000 + ) dwt</td><td>(1.6-4.8) & (1.7-3.7)</td></tr><tr><td></td><td>Bulk Carrier (4.1-15 kt)<br>(15 000-75 000) & (175 000+) dwt</td><td>(1.6-4.3) & (1.6-3.7)</td></tr><tr><td></td><td>Containership (5.6 to 23.5 kt)<br>(1500 TEU) & (5000 + TEU)</td><td>(1.9-4.4) & (1.8-3.5)</td></tr></tbody></table>

<figure><img src="image_4.png" /><figcaption>**Fig. 1. Distribution of vessel speed.** Source: data from a tanker company, authors' calculations.</figcaption></figure>

vessel is at anchorage or in port. Vessels are usually sailing at a higher speed when laden rather than in ballast. This difference is explained by contractual obligations when ships carry cargoes and by a lower fuel consumption when a vessel is in ballast.<fnref n="4" /> A second result is that over this period of time (2013–2016), vessels were sailing at a speed which is far below design speed. For the Aframax

<footnote n="4">⁴ From a legal point of view, the shipowner undertakes that the ship shall proceed on the carrying (laden) voyage with “utmost despatch”, i.e. without delay, due to speed reduction or otherwise.</footnote>
<table><caption>**Table 2**

Descriptive statistics for fuel consumption and speed.</caption><thead><tr><th rowspan="2">Vessel</th><th colspan="3">Fuel consumption (in tonnes per day)</th><th colspan="3">Speed (in knots)</th></tr><tr><th>All</th><th>Ballast</th><th>Laden</th><th>All</th><th>Ballast</th><th>Laden</th></tr></thead><tbody><tr><td colspan="7"><strong>Panel A. Aframax</strong></td></tr><tr><td>V1N = 785</td><td>32.7</td><td>24.8</td><td>36.8</td><td>12.3</td><td>11.5</td><td>12.7</td></tr><tr><td>V2N = 802</td><td>31.6</td><td>24.6</td><td>35.4</td><td>12.2</td><td>11.5</td><td>12.6</td></tr><tr><td>V3N = 779</td><td>36.3</td><td>26.2</td><td>42.0</td><td>12.1</td><td>11.0</td><td>12.6</td></tr><tr><td>V4N = 630</td><td>32.2</td><td>27.3</td><td>34.9</td><td>11.5</td><td>11.1</td><td>11.7</td></tr><tr><td>V5N = 668</td><td>32.7</td><td>23.5</td><td>37.8</td><td>11.8</td><td>10.8</td><td>12.4</td></tr><tr><td>V6N = 784</td><td>33.5</td><td>25.0</td><td>38.4</td><td>12.1</td><td>11.3</td><td>12.6</td></tr><tr><td>V7N = 780</td><td>34.2</td><td>22.8</td><td>39.0</td><td>11.7</td><td>10.5</td><td>12.3</td></tr><tr><td>V8N = 727</td><td>33.6</td><td>25.5</td><td>36.7</td><td>12.3</td><td>11.6</td><td>12.5</td></tr><tr><td>V9N = 815</td><td>31.1</td><td>22.8</td><td>34.3</td><td>12.2</td><td>11.3</td><td>12.5</td></tr><tr><td>V10N = 830</td><td>33.4</td><td>26.6</td><td>38.0</td><td>12.2</td><td>11.5</td><td>12.7</td></tr><tr><td colspan="7"><strong>Panel B. Suezmax</strong></td></tr><tr><td>V11N = 562</td><td>37.3</td><td>30.4</td><td>45.1</td><td>11.7</td><td>11.0</td><td>12.4</td></tr><tr><td>V12N = 857</td><td>39.3</td><td>30.9</td><td>46.9</td><td>12.2</td><td>11.7</td><td>12.8</td></tr><tr><td>V13N = 696</td><td>34.8</td><td>25.8</td><td>44.6</td><td>11.8</td><td>11.0</td><td>12.5</td></tr><tr><td>V14N = 643</td><td>36.6</td><td>32.3</td><td>41.0</td><td>11.4</td><td>11.2</td><td>11.5</td></tr><tr><td>V15N = 840</td><td>40.6</td><td>34.1</td><td>45.3</td><td>12.4</td><td>12.1</td><td>12.6</td></tr><tr><td>V16N = 727</td><td>38.6</td><td>31.8</td><td>43.8</td><td>12.0</td><td>11.6</td><td>12.4</td></tr><tr><td>AllN = 11925</td><td>34.9</td><td>27.6</td><td>39.6</td><td>12.0</td><td>11.3</td><td>12.5</td></tr></tbody></table>

Source: Data from a tanker company, authors' calculations.

Note: The sample is restricted to daily speeds between 6 and 16 knots.

tankers, for instance, around 93% of observations are below 14 knots and when in ballast condition Aframax vessels sail at less than 10 knots as much as 40.7% of the time.

The maximum observed speed is naturally constrained by the design speeds of the vessels (15.2 knots for the Aframax tankers and 15.5 knots for the Suezmax tankers, according to the Clarksons World Fleet Register database). Conversely, extremely low daily average speeds (below 6 knots) are rare and represent special circumstances such as arrivals or departures from port, navigation in constrained waters (such as canals) and vessels adrift. In our empirical estimates, we therefore restrict our attention to the speed interval between 6 and 16 knots, where vessels are sailing in the open sea. The remaining sample comprises 11,925 daily noon reports.

Table 2 presents some descriptive statistics on the average speed and main engine fuel consumption. There is limited variation across vessels in terms of their average speed, ranging from 11.5 to 12.4 knots: the average speed is 12.1 knots for Aframax and 12.0 knots for Suezmax. For comparison, according to IMO (2014), the speed ranges for tankers were 9.1–12.5 knots in 2012 and 10.3–14.6 knots in 2007. Suezmax are larger vessels and are burning more fuel than Aframax (38 tonnes per day against 33.1). In line with Psaraftis and Kontovas (2013), fuel consumption is higher when vessels are laden (39.6 tonnes per day) than in ballast (27.6) due to the lower speed and displacement in the latter loading condition.

Although sailing at similar average speed, differences in fuel consumption remain across vessels. This may be explained for instance by differences in trim, sailing condition (weather) or hull condition (Adland et al., 2018). Fig. 2 illustrates this heterogeneity in reporting the sailing speed fuel consumption curves obtained from regression models using nonlinear least squares. The figures suggest that speed remains the main determinant of fuel consumption, but other factors like weather or operating conditions need to be considered in our estimations.

Using information available in the noon reports, we consider a large set of control variables when explaining fuel consumption. First, we account for sea current (defined as the difference between GPS and LOG speed), average draft (the sum of draft forward and draft aft divided by two) and trim measured as the difference between draft forward and draft aft. We have also detailed information on the weather condition: wind direction, wind strength, swell type and wave height. We consider a flexible specification for wind and define 25 dummy variables corresponding to interactions between wind direction (head, astern, side) and wind force (gentle breeze or less, moderate breeze, fresh breeze, strong breeze, near gale). We include six categories for swell type (low or less, light, moderate, moderate rough, rough or high) and swell height (< 0.5 m, 0.5–1.25 m, 1.25–2.5 m, 2.5–4 m, 4–6 m, 6–9 m). Table A in Appendix A presents summary statistics of these variables for Aframax and Suezmax, respectively.

# 4. Estimates of the speed elasticity

## 4.1. The case of constant elasticity

We turn to an econometric framework to assess the magnitude of the consumption-speed elasticity in real sea conditions. This setting allows us to estimate parameters of the fuel consumption function within a parametric framework (power function) and account for speed-varying elasticities. However, such approach has been challenged in recent years as data mining techniques have
<table><caption>**Table 2**

Descriptive statistics for fuel consumption and speed.</caption><thead><tr><th rowspan="2">Vessel</th><th colspan="3">Fuel consumption (in tonnes per day)</th><th colspan="3">Speed (in knots)</th></tr><tr><th>All</th><th>Ballast</th><th>Laden</th><th>All</th><th>Ballast</th><th>Laden</th></tr></thead><tbody><tr><td colspan="7"><strong>Panel A. Aframax</strong></td></tr><tr><td>V1N = 785</td><td>32.7</td><td>24.8</td><td>36.8</td><td>12.3</td><td>11.5</td><td>12.7</td></tr><tr><td>V2N = 802</td><td>31.6</td><td>24.6</td><td>35.4</td><td>12.2</td><td>11.5</td><td>12.6</td></tr><tr><td>V3N = 779</td><td>36.3</td><td>26.2</td><td>42.0</td><td>12.1</td><td>11.0</td><td>12.6</td></tr><tr><td>V4N = 630</td><td>32.2</td><td>27.3</td><td>34.9</td><td>11.5</td><td>11.1</td><td>11.7</td></tr><tr><td>V5N = 668</td><td>32.7</td><td>23.5</td><td>37.8</td><td>11.8</td><td>10.8</td><td>12.4</td></tr><tr><td>V6N = 784</td><td>33.5</td><td>25.0</td><td>38.4</td><td>12.1</td><td>11.3</td><td>12.6</td></tr><tr><td>V7N = 780</td><td>34.2</td><td>22.8</td><td>39.0</td><td>11.7</td><td>10.5</td><td>12.3</td></tr><tr><td>V8N = 727</td><td>33.6</td><td>25.5</td><td>36.7</td><td>12.3</td><td>11.6</td><td>12.5</td></tr><tr><td>V9N = 815</td><td>31.1</td><td>22.8</td><td>34.3</td><td>12.2</td><td>11.3</td><td>12.5</td></tr><tr><td>V10N = 830</td><td>33.4</td><td>26.6</td><td>38.0</td><td>12.2</td><td>11.5</td><td>12.7</td></tr><tr><td colspan="7"><strong>Panel B. Suezmax</strong></td></tr><tr><td>V11N = 562</td><td>37.3</td><td>30.4</td><td>45.1</td><td>11.7</td><td>11.0</td><td>12.4</td></tr><tr><td>V12N = 857</td><td>39.3</td><td>30.9</td><td>46.9</td><td>12.2</td><td>11.7</td><td>12.8</td></tr><tr><td>V13N = 696</td><td>34.8</td><td>25.8</td><td>44.6</td><td>11.8</td><td>11.0</td><td>12.5</td></tr><tr><td>V14N = 643</td><td>36.6</td><td>32.3</td><td>41.0</td><td>11.4</td><td>11.2</td><td>11.5</td></tr><tr><td>V15N = 840</td><td>40.6</td><td>34.1</td><td>45.3</td><td>12.4</td><td>12.1</td><td>12.6</td></tr><tr><td>V16N = 727</td><td>38.6</td><td>31.8</td><td>43.8</td><td>12.0</td><td>11.6</td><td>12.4</td></tr><tr><td>AllN = 11925</td><td>34.9</td><td>27.6</td><td>39.6</td><td>12.0</td><td>11.3</td><td>12.5</td></tr></tbody></table>

Source: Data from a tanker company, authors' calculations.

Note: The sample is restricted to daily speeds between 6 and 16 knots.

tankers, for instance, around 93% of observations are below 14 knots and when in ballast condition Aframax vessels sail at less than 10 knots as much as 40.7% of the time.

The maximum observed speed is naturally constrained by the design speeds of the vessels (15.2 knots for the Aframax tankers and 15.5 knots for the Suezmax tankers, according to the Clarksons World Fleet Register database). Conversely, extremely low daily average speeds (below 6 knots) are rare and represent special circumstances such as arrivals or departures from port, navigation in constrained waters (such as canals) and vessels adrift. In our empirical estimates, we therefore restrict our attention to the speed interval between 6 and 16 knots, where vessels are sailing in the open sea. The remaining sample comprises 11,925 daily noon reports.

Table 2 presents some descriptive statistics on the average speed and main engine fuel consumption. There is limited variation across vessels in terms of their average speed, ranging from 11.5 to 12.4 knots: the average speed is 12.1 knots for Aframax and 12.0 knots for Suezmax. For comparison, according to IMO (2014), the speed ranges for tankers were 9.1–12.5 knots in 2012 and 10.3–14.6 knots in 2007. Suezmax are larger vessels and are burning more fuel than Aframax (38 tonnes per day against 33.1). In line with Psaraftis and Kontovas (2013), fuel consumption is higher when vessels are laden (39.6 tonnes per day) than in ballast (27.6) due to the lower speed and displacement in the latter loading condition.

Although sailing at similar average speed, differences in fuel consumption remain across vessels. This may be explained for instance by differences in trim, sailing condition (weather) or hull condition (Adland et al., 2018). Fig. 2 illustrates this heterogeneity in reporting the sailing speed fuel consumption curves obtained from regression models using nonlinear least squares. The figures suggest that speed remains the main determinant of fuel consumption, but other factors like weather or operating conditions need to be considered in our estimations.

Using information available in the noon reports, we consider a large set of control variables when explaining fuel consumption. First, we account for sea current (defined as the difference between GPS and LOG speed), average draft (the sum of draft forward and draft aft divided by two) and trim measured as the difference between draft forward and draft aft. We have also detailed information on the weather condition: wind direction, wind strength, swell type and wave height. We consider a flexible specification for wind and define 25 dummy variables corresponding to interactions between wind direction (head, astern, side) and wind force (gentle breeze or less, moderate breeze, fresh breeze, strong breeze, near gale). We include six categories for swell type (low or less, light, moderate, moderate rough, rough or high) and swell height (< 0.5 m, 0.5–1.25 m, 1.25–2.5 m, 2.5–4 m, 4–6 m, 6–9 m). Table A in Appendix A presents summary statistics of these variables for Aframax and Suezmax, respectively.

# 4. Estimates of the speed elasticity

## 4.1. The case of constant elasticity

We turn to an econometric framework to assess the magnitude of the consumption-speed elasticity in real sea conditions. This setting allows us to estimate parameters of the fuel consumption function within a parametric framework (power function) and account for speed-varying elasticities. However, such approach has been challenged in recent years as data mining techniques have
<figure><img src="image_3.png" /><figcaption>Fig. 2. Correlation between fuel consumption and speed. Source: Data from a tanker company, authors’ calculations. Note: The solid curves plot estimates from nonlinear least squares regressions explaining fuel consumption as a power function of speed. The sample is restricted to average daily speeds between 6 and 16 knots.</figcaption></figure>

also been considered to predict ship fuel consumption under different conditions. Although our focus is on the elasticity of fuel consumption with respect to speed (net of other factors) rather than on finding the best predictive specification with all available regressors, we provide in Appendix B a brief comparison in terms of model fit between our regression approach and results from different data-mining techniques.

In terms of notation, let $C_{it}$ be the fuel consumption of vessel $i$ observed at date $t$ (with $t$ a given day). We start from a power function for the theoretical consumption function such that $C_{it} = \alpha S_{it}^{\beta}$. Here, $S$ stands for the average speed for vessel $i$ in last 24 h $t$, $\alpha$ is a scale parameter and $\beta$ is the speed-fuel consumption elasticity. With real data, we turn to a log-log specification and estimate the following model:

$$ \ln C_{it} = \ln\alpha + \beta * \ln S_{it} + X_{it}\delta + \varepsilon_{it} \quad (2) $$

where $X_{it}$ is a set of observable factors influencing consumption, $\delta$ the vector of associated coefficients, and $\varepsilon_{it}$ a residual perturbation such that $E(\varepsilon_{it}) = 0$ and $V(\varepsilon_{it}) = \sigma^2$. Specification (2) corresponds to the standard framework with a constant speed elasticity $\beta$. Estimates for various linear regressions with robust standard errors are presented in Table 4.

In columns (1A) and (2A) and for Aframax and Suezmax, respectively, we consider a first set of control variables that include, in addition to speed, average draft, trim and sea current. From a theoretical point of view, a vessel that is deeper in water is subject to higher frictional resistance from water and waves but to lower resistance from air (Meng et al., 2016), and the former effect is generally higher (MAN Diesel & Turbo, 2004). In columns (1B) and (2B), the list of covariates additionally includes weather conditions. In Eq. (2), the residual $\varepsilon_{it}$ picks up the influence of all unobservable factors. Since we have longitudinal data, we can further decompose $\varepsilon_{it}$ in the following way:

$$ \varepsilon_{it} = \Theta_{i} + \lambda_{t(i)} + \xi_{it} \quad (3) $$
<figure><img src="image_5.png" /><figcaption>Fig. 2. Correlation between fuel consumption and speed. Source: Data from a tanker company, authors’ calculations. Note: The solid curves plot estimates from nonlinear least squares regressions explaining fuel consumption as a power function of speed. The sample is restricted to average daily speeds between 6 and 16 knots.</figcaption></figure>

also been considered to predict ship fuel consumption under different conditions. Although our focus is on the elasticity of fuel consumption with respect to speed (net of other factors) rather than on finding the best predictive specification with all available regressors, we provide in Appendix B a brief comparison in terms of model fit between our regression approach and results from different data-mining techniques.

In terms of notation, let $C_{it}$ be the fuel consumption of vessel $i$ observed at date $t$ (with $t$ a given day). We start from a power function for the theoretical consumption function such that $C_{it} = \alpha S_{it}^{\beta}$. Here, $S$ stands for the average speed for vessel $i$ in last 24 h $t$, $\alpha$ is a scale parameter and $\beta$ is the speed-fuel consumption elasticity. With real data, we turn to a log-log specification and estimate the following model:

$$ \ln C_{it} = \ln\alpha + \beta * \ln S_{it} + X_{it}\delta + \varepsilon_{it} \quad (2) $$

where $X_{it}$ is a set of observable factors influencing consumption, $\delta$ the vector of associated coefficients, and $\varepsilon_{it}$ a residual perturbation such that $E(\varepsilon_{it}) = 0$ and $V(\varepsilon_{it}) = \sigma^2$. Specification (2) corresponds to the standard framework with a constant speed elasticity $\beta$. Estimates for various linear regressions with robust standard errors are presented in Table 4.

In columns (1A) and (2A) and for Aframax and Suezmax, respectively, we consider a first set of control variables that include, in addition to speed, average draft, trim and sea current. From a theoretical point of view, a vessel that is deeper in water is subject to higher frictional resistance from water and waves but to lower resistance from air (Meng et al., 2016), and the former effect is generally higher (MAN Diesel & Turbo, 2004). In columns (1B) and (2B), the list of covariates additionally includes weather conditions. In Eq. (2), the residual $\varepsilon_{it}$ picks up the influence of all unobservable factors. Since we have longitudinal data, we can further decompose $\varepsilon_{it}$ in the following way:

$$ \varepsilon_{it} = \Theta_{i} + \lambda_{t(i)} + \xi_{it} \quad (3) $$
where $\theta_i$ is a vessel-specific fixed effect, $\lambda_{it(i)}$ is a time trend and $\xi_{it}$ is the residual term. The fixed effect $\theta_i$ captures the unobserved differences in fuel consumption, such as, for instance, differences in sea temperatures if vessels operate in different parts of the world. The influence (if any) of the crew is also controlled by this fixed effect, although the magnitude of the vessel-specific fixed effect is expected to be limited as we consider sister ships. The second term $\lambda_{it(i)}$ is a vessel-specific time trend. We construct the time-trend for each vessel to control for the influence of hull condition.<fnref n="5" /> Specifically, we used quarter dummies for each vessel in order to avoid an excessive number of time dummies.

The corresponding estimates of the fixed effect regressions defined by (2) and (3) are reported in columns (1C) and (2C). In Table 3, we only report the speed coefficient as it is the focus of our research<fnref n="6" />. Nonetheless, we will briefly discuss hereafter the influence of the selected control variables. We estimate separate regressions for Aframax (columns 1A, 1B and 1C) and Suezmax (columns 2A, 2B and 2C) and separate regressions for the whole sample (panel A) and when vessels are either in ballast (panel B) or laden (panel C) conditions.

According to Table 3, our estimates explain up to 82.4% of variations in fuel consumption for Aframax and 87.3% for Suezmax (columns 1C and 2C). Whatever the configuration, the fuel consumption to speed elasticity is far below 3. Indeed, according to our results, the estimated constant elasticity is approximatively 1.9 for Aframax (column 1C) and around 2 for Suezmax (column 2C). Elasticities for ballast or laden do not display significant differences. These results are still lower than those reported by Kristensen (2019) for Aframax and Suezmax vessel and for a speed of 12 knots, with a speed elasticity estimated at 2.4–2.5. Given that the ship and propulsion configuration is not optimized for the observed average sailing speeds, resulting in higher specific fuel consumption per energy unit input, it is reasonable to postulate that this deviation from the cubic rule is speed related. This motivates a closer investigation of the possible speed-dependency of the elasticity itself.

In Fig. 3, we investigate potential heterogeneous effects by estimating the constant fuel consumption-speed elasticity for each vessel. Substantial differences remain across vessels, with an elasticity ranging from 1.5 to 2.2 for ballast and laden conditions. An explanation could be that all vessels have different average speed values (see Table 3). At the same time, there may be other factors not included in our covariates, such as water temperature or salinity.

The influence of the other control variables are in line with expectations. We find a positive correlation between fuel consumption and average draft (vessels that are lower in the water consume more fuel) and with trim, while sea current does not have any influence. Fuel consumption is naturally very sensitive to the various combinations of wind direction and wind force. On the one hand, fuel consumption increases with headwind and especially strong wind (above 22 knots). On the other hand, fuel consumption is reduced with wind from astern, with the reduction being an increasing function of wind strength. Finally, we find a positive correlation between fuel consumption and both swell and wave height.

## 4.2. The case of speed-dependent elasticities

As emphasized in the section on fuel consumption fundamentals, the value of the speed to fuel consumption elasticity $\beta$ is likely to change within specific speed intervals, an effect which is not accounted for in Eq. (2). Hence, when the operating speed is low, the assumption of a power of three is expected to under-estimate fuel consumption. The issue is then to investigate how elasticities changes from one speed interval to another.

To account for these effects, we turn to a more flexible specification. Let $\phi$ be an integer with $\phi \in \{1, \dots, G\}$ such that $\phi$ characterizes the interval of speed values $[\ln S^{\phi}; \ln S^{\phi+1}]$. The speed elasticity is now allowed to depend on $\phi$ and is denoted by $\beta^{\phi}$. Let $\ln S^{\phi}$ the variable such that:

$$ \ln S^{\phi} = \begin{cases} 0 & \text{if } S \le S^{\phi} \\ \ln S - \ln S^{\phi} & \text{if } S^{\phi} \le S < S^{\phi+1} \\ \ln S^{\phi+1} - \ln S^{\phi} & \text{if } S^{\phi+1} \le S \end{cases} \quad (4) $$

The specification with speed-dependent elasticities may be expressed as:<fnref n="7" />

$$ \ln C_{it} = \ln \alpha + \sum \beta^{\phi} * \ln S_{it}^{\phi} + X_{it} \delta + \varepsilon_{it} \quad (5) $$

We expect the estimated values of $\beta^{\phi}$ to increase as $S$ is near the vessel’s design speed and to be close to 3. A key issue is the determination of speed intervals. In a preliminary step, we choose to rely on an ad hoc procedure based on exogenous thresholds with four speed intervals: < 10 knots, 10–12 knots, from 12 to 14 knots, > 14 knots. Results are reported in Fig. 4.

For Aframax tankers (either in ballast or laden condition), the elasticity is extremely low when speed is less than 10 knots, at around 0.4 (with t-value = 4.99). Then, it increases to 2–2.5 when speed is from 10 to 14 knots. Above 14 knots and closer to the design speed of 15+ knots, the elasticity is equal to 3.02 (t = 25.96) with a confidence interval ranging between 2.79 and 3.25. For high speed values, the estimated elasticity is 3.2 in ballast (t = 13.98) and 2.70 when in the laden (t = 19.57) condition. For Suezmax, the estimated elasticity is at 1.3 when speed is below 10 knots (t = 12.35), at around 2.3 when speed is between 10 and 14

<footnote n="5">⁵ With a pure time-fixed effect $\lambda_t$ (instead of $\lambda_{it(i)}$), we would only account for seasonality. Such effect is likely to be misleading because seasons are reversed in the northern and southern hemispheres.</footnote>

<footnote n="6">⁶ The detailed estimates, not reported, are available upon request.</footnote>

<footnote n="7">⁷ Compared to (2), this specification includes a piecewise linear function for the logarithm of speed.</footnote>
where $\theta_i$ is a vessel-specific fixed effect, $\lambda_{t(i)}$ is a time trend and $\xi_{it}$ is the residual term. The fixed effect $\theta_i$ captures the unobserved differences in fuel consumption, such as, for instance, differences in sea temperatures if vessels operate in different parts of the world. The influence (if any) of the crew is also controlled by this fixed effect, although the magnitude of the vessel-specific fixed effect is expected to be limited as we consider sister ships. The second term $\lambda_{t(i)}$ is a vessel-specific time trend. We construct the time-trend for each vessel to control for the influence of hull condition.<fnref n="5" /> Specifically, we used quarter dummies for each vessel in order to avoid an excessive number of time dummies.

The corresponding estimates of the fixed effect regressions defined by (2) and (3) are reported in columns (1C) and (2C). In Table 3, we only report the speed coefficient as it is the focus of our research<fnref n="6" />. Nonetheless, we will briefly discuss hereafter the influence of the selected control variables. We estimate separate regressions for Aframax (columns 1A, 1B and 1C) and Suezmax (columns 2A, 2B and 2C) and separate regressions for the whole sample (panel A) and when vessels are either in ballast (panel B) or laden (panel C) conditions.

According to Table 3, our estimates explain up to 82.4% of variations in fuel consumption for Aframax and 87.3% for Suezmax (columns 1C and 2C). Whatever the configuration, the fuel consumption to speed elasticity is far below 3. Indeed, according to our results, the estimated constant elasticity is approximatively 1.9 for Aframax (column 1C) and around 2 for Suezmax (column 2C). Elasticities for ballast or laden do not display significant differences. These results are still lower than those reported by Kristensen (2019) for Aframax and Suezmax vessel and for a speed of 12 knots, with a speed elasticity estimated at 2.4–2.5. Given that the ship and propulsion configuration is not optimized for the observed average sailing speeds, resulting in higher specific fuel consumption per energy unit input, it is reasonable to postulate that this deviation from the cubic rule is speed related. This motivates a closer investigation of the possible speed-dependency of the elasticity itself.

In Fig. 3, we investigate potential heterogeneous effects by estimating the constant fuel consumption-speed elasticity for each vessel. Substantial differences remain across vessels, with an elasticity ranging from 1.5 to 2.2 for ballast and laden conditions. An explanation could be that all vessels have different average speed values (see Table 3). At the same time, there may be other factors not included in our covariates, such as water temperature or salinity.

The influence of the other control variables are in line with expectations. We find a positive correlation between fuel consumption and average draft (vessels that are lower in the water consume more fuel) and with trim, while sea current does not have any influence. Fuel consumption is naturally very sensitive to the various combinations of wind direction and wind force. On the one hand, fuel consumption increases with headwind and especially strong wind (above 22 knots). On the other hand, fuel consumption is reduced with wind from astern, with the reduction being an increasing function of wind strength. Finally, we find a positive correlation between fuel consumption and both swell and wave height.

## 4.2. The case of speed-dependent elasticities

As emphasized in the section on fuel consumption fundamentals, the value of the speed to fuel consumption elasticity $\beta$ is likely to change within specific speed intervals, an effect which is not accounted for in Eq. (2). Hence, when the operating speed is low, the assumption of a power of three is expected to under-estimate fuel consumption. The issue is then to investigate how elasticities changes from one speed interval to another.

To account for these effects, we turn to a more flexible specification. Let $\phi$ be an integer with $\phi \in \{1, \dots, G\}$ such that $\phi$ characterizes the interval of speed values $[\ln S^{\phi}; \ln S^{\phi+1}]$. The speed elasticity is now allowed to depend on $\phi$ and is denoted by $\beta^{\phi}$. Let $\ln S^{\phi}$ the variable such that:

$$ \ln S^{\phi} = \begin{cases} 0 & \text{if } S \le S^{\phi} \\ \ln S - \ln S^{\phi} & \text{if } S^{\phi} \le S < S^{\phi+1} \\ \ln S^{\phi+1} - \ln S^{\phi} & \text{if } S^{\phi+1} \le S \end{cases} \quad (4) $$

The specification with speed-dependent elasticities may be expressed as:<fnref n="7" />

$$ \ln C_{it} = \ln \alpha + \sum \beta^{\phi} * \ln S_{it}^{\phi} + X_{it} \delta + \varepsilon_{it} \quad (5) $$

We expect the estimated values of $\beta^{\phi}$ to increase as $S$ is near the vessel's design speed and to be close to 3. A key issue is the determination of speed intervals. In a preliminary step, we choose to rely on an ad hoc procedure based on exogenous thresholds with four speed intervals: < 10 knots, 10–12 knots, from 12 to 14 knots, > 14 knots. Results are reported in Fig. 4.

For Aframax tankers (either in ballast or laden condition), the elasticity is extremely low when speed is less than 10 knots, at around 0.4 (with t-value = 4.99). Then, it increases to 2–2.5 when speed is from 10 to 14 knots. Above 14 knots and closer to the design speed of 15+ knots, the elasticity is equal to 3.02 (t = 25.96) with a confidence interval ranging between 2.79 and 3.25. For high speed values, the estimated elasticity is 3.2 in ballast (t = 13.98) and 2.70 when in the laden (t = 19.57) condition. For Suezmax, the estimated elasticity is at 1.3 when speed is below 10 knots (t = 12.35), at around 2.3 when speed is between 10 and 14

<footnote n="5">⁵ With a pure time-fixed effect $\lambda_t$ (instead of $\lambda_{t(i)}$), we would only account for seasonality. Such effect is likely to be misleading because seasons are reversed in the northern and southern hemispheres.</footnote>

<footnote n="6">⁶ The detailed estimates, not reported, are available upon request.</footnote>

<footnote n="7">⁷ Compared to (2), this specification includes a piecewise linear function for the logarithm of speed.</footnote>
<table><caption>**Table 3**
Estimates of the logarithm of fuel consumption.</caption><thead><tr><th rowspan="2">Variables</th><th colspan="3">Aframax</th><th colspan="3">Suezmax</th></tr><tr><th>(1A)</th><th>(1B)</th><th>(1C)</th><th>(2A)</th><th>(2B)</th><th>(2C)</th></tr></thead><tbody><tr><td colspan="7"><strong>Panel A. All</strong></td></tr><tr><td>LOG speed (log)</td><td>1.488***<br>(74.32)</td><td>1.783***<br>(100.97)</td><td>1.871***<br>(114.01)</td><td>1.989***<br>(94.05)</td><td>1.997***<br>(107.02)</td><td>1.978***<br>(115.19)</td></tr><tr><td>Average draft, trim and sea current</td><td>YES</td><td>YES</td><td>YES</td><td>YES</td><td>YES</td><td>YES</td></tr><tr><td>Weather</td><td>NO</td><td>YES</td><td>YES</td><td>NO</td><td>YES</td><td>YES</td></tr><tr><td>Vessel fixed effects</td><td>NO</td><td>NO</td><td>YES</td><td>NO</td><td>NO</td><td>YES</td></tr><tr><td>Vessel-specific time trend</td><td>NO</td><td>NO</td><td>YES</td><td>NO</td><td>NO</td><td>YES</td></tr><tr><td>Observations</td><td>7600</td><td>7600</td><td>7600</td><td>4325</td><td>4325</td><td>4325</td></tr><tr><td>R<sup>2</sup></td><td>0.662</td><td>0.769</td><td>0.824</td><td>0.763</td><td>0.825</td><td>0.873</td></tr><tr><td colspan="7"><strong>Panel B. Ballast</strong></td></tr><tr><td>LOG speed (log)</td><td>1.869***<br>(55.31)</td><td>1.999***<br>(69.39)</td><td>2.013***<br>(72.18)</td><td>2.006***<br>(66.71)</td><td>1.986***<br>(73.63)</td><td>1.918***<br>(74.10)</td></tr><tr><td>Average draft, trim and sea current</td><td>YES</td><td>YES</td><td>YES</td><td>YES</td><td>YES</td><td>YES</td></tr><tr><td>Weather</td><td>NO</td><td>YES</td><td>YES</td><td>NO</td><td>YES</td><td>YES</td></tr><tr><td>Vessel fixed effects</td><td>NO</td><td>NO</td><td>YES</td><td>NO</td><td>NO</td><td>YES</td></tr><tr><td>Vessel-specific time trend</td><td>NO</td><td>NO</td><td>YES</td><td>NO</td><td>NO</td><td>YES</td></tr><tr><td>Observations</td><td>2575</td><td>2575</td><td>2575</td><td>2068</td><td>2068</td><td>2068</td></tr><tr><td>R<sup>2</sup></td><td>0.615</td><td>0.738</td><td>0.797</td><td>0.694</td><td>0.777</td><td>0.839</td></tr><tr><td colspan="7"><strong>Panel C. Laden</strong></td></tr><tr><td>LOG speed (log)</td><td>1.112***<br>(45.81)</td><td>1.518***<br>(67.30)</td><td>1.621***<br>(75.82)</td><td>1.966***<br>(65.12)</td><td>2.034***<br>(79.42)</td><td>1.966***<br>(84.79)</td></tr><tr><td>Average draft, trim and sea current</td><td>YES</td><td>YES</td><td>YES</td><td>YES</td><td>YES</td><td>YES</td></tr><tr><td>Weather</td><td>NO</td><td>YES</td><td>YES</td><td>NO</td><td>YES</td><td>YES</td></tr><tr><td>Vessel fixed effects</td><td>NO</td><td>NO</td><td>YES</td><td>NO</td><td>NO</td><td>YES</td></tr><tr><td>Vessel-specific quartic time trend</td><td>NO</td><td>NO</td><td>YES</td><td>NO</td><td>NO</td><td>YES</td></tr><tr><td>Observations</td><td>5025</td><td>5025</td><td>5025</td><td>2257</td><td>2257</td><td>2257</td></tr><tr><td>R<sup>2</sup></td><td>0.350</td><td>0.557</td><td>0.688</td><td>0.664</td><td>0.772</td><td>0.862</td></tr></tbody></table>

Source: data from a tanker company, authors' calculations.

Note: the reported estimates are obtained from linear regressions with robust standard errors. Significance levels are 1% (***), 5% (**) and 10% (\*\*\*). The sample is restricted to average daily speeds comprised between 6 and 16 knots. Additional controls include average draft, trim, sea current, wind direction crossed with wind force, swell and wave height.

knots, and at 2.84 when above 14 knots (t = 22.98). For this last interval, the upper limit of the confidence interval is 3.08 so that we cannot reject the null assumption $H_0: \beta = 3$ in this interval. Overall, our findings with exogenous threshold values confirm that the speed-consumption elasticity is close to 3 only at speeds above 14 knots.

These results emphasize that the selection of speed intervals is critical for fuel consumption estimation. Rather than relying on predetermined exogenous thresholds, we next propose a more general specification corresponding to a piecewise linear log-log function with endogenous thresholds. Let us consider the case of the existence of one threshold, meaning that $\phi$ is either $\phi = 1$ or $\phi = 2$. Speed values range in the interval $[S; \bar{S}]$ and $S^1$ is the threshold. The model to estimate is:

$$ \ln C_{it} = \ln \alpha + \beta^1 * \ln S_{it}^1 + \beta^2 * \ln S_{it}^2 + X_{it} \delta + \varepsilon_{it} \quad (6) $$

As long as the threshold $S^1$ is unknown (endogeneity assumption), this means that it has to be estimated along with the other parameters of the regression, provided that $\ln S_{it}^1$ and $\ln S_{it}^2$ are themselves conditional on the value of $S^1$. The specification (6) defines a complex non-linear function. While estimation through non-linear least squares is a possibility, its implementation remains difficult due to the presence of the two fixed effects $\theta_i$ and $\lambda_{it}$. Following Cox et al. (2004), we consider instead a conditional least squares approach with the objective to minimize the sum of squared errors (SSE) criterion. Consider first the case of one endogenous threshold that is obtained using the two following steps:

- step 1: let $S^k$ be any value within interval $[S; \bar{S}]$; for any value of $S^k$, estimate the model (6) by a least squares method and calculate the score $SSE(S^k)$, which is a function of $S^k$;

- step 2: the endogenous estimate of $S^1$ is found in searching over the various $S^k$ the threshold for which $S^1 = \min SSE(S^k)$.

To avoid irrelevant estimators, we require a minimal number of 150 observations within each speed intervals $[S; S^1]$ and $[S^1; \bar{S}]$. The procedure can be easily extended to several endogenous thresholds. If there are for instance three thresholds $S^1, S^2$ and $S^3$ (corresponding to four speed intervals), then the search is for all combinations $(S^1, S^2, S^3)$ such that $S^1 < S^2$ and $S^2 < S^3$ which minimize the function $SSE(S^1, S^2, S^3)$. Again, a minimum of 150 observations is required within each endogenous speed interval.

The various results are reported in Table 4. In panel A, estimates are similar to Table 4 corresponding to the assumption of a
<table><caption>**Table 3**
Estimates of the logarithm of fuel consumption.</caption><thead><tr><th rowspan="2">Variables</th><th colspan="3">Aframax</th><th colspan="3">Suezmax</th></tr><tr><th>(1A)</th><th>(1B)</th><th>(1C)</th><th>(2A)</th><th>(2B)</th><th>(2C)</th></tr></thead><tbody><tr><td colspan="7"><strong>Panel A. All</strong></td></tr><tr><td>LOG speed (log)</td><td>1.488***<br>(74.32)</td><td>1.783***<br>(100.97)</td><td>1.871***<br>(114.01)</td><td>1.989***<br>(94.05)</td><td>1.997***<br>(107.02)</td><td>1.978***<br>(115.19)</td></tr><tr><td>Average draft, trim and sea current</td><td>YES</td><td>YES</td><td>YES</td><td>YES</td><td>YES</td><td>YES</td></tr><tr><td>Weather</td><td>NO</td><td>YES</td><td>YES</td><td>NO</td><td>YES</td><td>YES</td></tr><tr><td>Vessel fixed effects</td><td>NO</td><td>NO</td><td>YES</td><td>NO</td><td>NO</td><td>YES</td></tr><tr><td>Vessel-specific time trend</td><td>NO</td><td>NO</td><td>YES</td><td>NO</td><td>NO</td><td>YES</td></tr><tr><td>Observations</td><td>7600</td><td>7600</td><td>7600</td><td>4325</td><td>4325</td><td>4325</td></tr><tr><td>R<sup>2</sup></td><td>0.662</td><td>0.769</td><td>0.824</td><td>0.763</td><td>0.825</td><td>0.873</td></tr><tr><td colspan="7"><strong>Panel B. Ballast</strong></td></tr><tr><td>LOG speed (log)</td><td>1.869***<br>(55.31)</td><td>1.999***<br>(69.39)</td><td>2.013***<br>(72.18)</td><td>2.006***<br>(66.71)</td><td>1.986***<br>(73.63)</td><td>1.918***<br>(74.10)</td></tr><tr><td>Average draft, trim and sea current</td><td>YES</td><td>YES</td><td>YES</td><td>YES</td><td>YES</td><td>YES</td></tr><tr><td>Weather</td><td>NO</td><td>YES</td><td>YES</td><td>NO</td><td>YES</td><td>YES</td></tr><tr><td>Vessel fixed effects</td><td>NO</td><td>NO</td><td>YES</td><td>NO</td><td>NO</td><td>YES</td></tr><tr><td>Vessel-specific time trend</td><td>NO</td><td>NO</td><td>YES</td><td>NO</td><td>NO</td><td>YES</td></tr><tr><td>Observations</td><td>2575</td><td>2575</td><td>2575</td><td>2068</td><td>2068</td><td>2068</td></tr><tr><td>R<sup>2</sup></td><td>0.615</td><td>0.738</td><td>0.797</td><td>0.694</td><td>0.777</td><td>0.839</td></tr><tr><td colspan="7"><strong>Panel C. Laden</strong></td></tr><tr><td>LOG speed (log)</td><td>1.112***<br>(45.81)</td><td>1.518***<br>(67.30)</td><td>1.621***<br>(75.82)</td><td>1.966***<br>(65.12)</td><td>2.034***<br>(79.42)</td><td>1.966***<br>(84.79)</td></tr><tr><td>Average draft, trim and sea current</td><td>YES</td><td>YES</td><td>YES</td><td>YES</td><td>YES</td><td>YES</td></tr><tr><td>Weather</td><td>NO</td><td>YES</td><td>YES</td><td>NO</td><td>YES</td><td>YES</td></tr><tr><td>Vessel fixed effects</td><td>NO</td><td>NO</td><td>YES</td><td>NO</td><td>NO</td><td>YES</td></tr><tr><td>Vessel-specific quartic time trend</td><td>NO</td><td>NO</td><td>YES</td><td>NO</td><td>NO</td><td>YES</td></tr><tr><td>Observations</td><td>5025</td><td>5025</td><td>5025</td><td>2257</td><td>2257</td><td>2257</td></tr><tr><td>R<sup>2</sup></td><td>0.350</td><td>0.557</td><td>0.688</td><td>0.664</td><td>0.772</td><td>0.862</td></tr></tbody></table>

Source: data from a tanker company, authors' calculations.

Note: the reported estimates are obtained from linear regressions with robust standard errors. Significance levels are 1% (***), 5% (**) and 10% (\*\*\*). The sample is restricted to average daily speeds comprised between 6 and 16 knots. Additional controls include average draft, trim, sea current, wind direction crossed with wind force, swell and wave height.

knots, and at 2.84 when above 14 knots ($t = 22.98$). For this last interval, the upper limit of the confidence interval is 3.08 so that we cannot reject the null assumption $H_0: \beta = 3$ in this interval. Overall, our findings with exogenous threshold values confirm that the speed-consumption elasticity is close to 3 only at speeds above 14 knots.

These results emphasize that the selection of speed intervals is critical for fuel consumption estimation. Rather than relying on predetermined exogenous thresholds, we next propose a more general specification corresponding to a piecewise linear log-log function with endogenous thresholds. Let us consider the case of the existence of one threshold, meaning that $\phi$ is either $\phi = 1$ or $\phi = 2$. Speed values range in the interval $[S; \bar{S}]$ and $S^1$ is the threshold. The model to estimate is:

$$ \ln C_{it} = \ln \alpha + \beta^1 * \ln S_{it}^1 + \beta^2 * \ln S_{it}^2 + X_{it} \delta + \varepsilon_{it} \quad (6) $$

As long as the threshold $S^1$ is unknown (endogeneity assumption), this means that it has to be estimated along with the other parameters of the regression, provided that $\ln S_{it}^1$ and $\ln S_{it}^2$ are themselves conditional on the value of $S^1$. The specification (6) defines a complex non-linear function. While estimation through non-linear least squares is a possibility, its implementation remains difficult due to the presence of the two fixed effects $\theta_i$ and $\lambda_{it}$. Following Cox et al. (2004), we consider instead a conditional least squares approach with the objective to minimize the sum of squared errors (SSE) criterion. Consider first the case of one endogenous threshold that is obtained using the two following steps:

- step 1: let $S^k$ be any value within interval $[S; \bar{S}]$; for any value of $S^k$, estimate the model (6) by a least squares method and calculate the score $SSE(S^k)$, which is a function of $S^k$;

- step 2: the endogenous estimate of $S^1$ is found in searching over the various $S^k$ the threshold for which $S^1 = \min SSE(S^k)$.

To avoid irrelevant estimators, we require a minimal number of 150 observations within each speed intervals $[S; S^1]$ and $[S^1; \bar{S}]$. The procedure can be easily extended to several endogenous thresholds. If there are for instance three thresholds $S^1, S^2$ and $S^3$ (corresponding to four speed intervals), then the search is for all combinations $(S^1, S^2, S^3)$ such that $S^1 < S^2$ and $S^2 < S^3$ which minimize the function $SSE(S^1, S^2, S^3)$. Again, a minimum of 150 observations is required within each endogenous speed interval.

The various results are reported in Table 4. In panel A, estimates are similar to Table 4 corresponding to the assumption of a
<figure><img src="image_4.png" /><figcaption>**Fig. 3.** Fuel Consumption-speed elasticity by vessel, Source: Data from a company, authors' calculations. Note: Each elasticity is obtained from a vessel-specific linear regression explaining fuel consumption as a function of speed, average draft, trim, sea current, wind direction crossed with wind force, swell, wave height and time trend. The sample is restricted to average daily speeds between 6 and 16 knots.</figcaption></figure>

constant elasticity over all speed values. In panel B, we consider the possibility of one threshold, meaning two speed intervals with specific elasticity, while panels C and D are for two and three thresholds respectively.

When considering panel B with both ballast and laden conditions, the threshold is 9.79 knots for Aframax and 10.26 knots for Suezmax. The elasticity is low below the threshold (less than 0.5 for Aframax and less than 1.4 for Suezmax) which implies that at these levels, the impact of speed level on fuel consumption is very limited. Above the threshold, the elasticity increases to around 2.3 for both vessels. When considering panels C and D, the significant and varying impact of speed on fuel consumption is even clearer. Panel D defines four speed intervals which are, for instance, for Aframax equal to less than 8.4, (8.4–9.8), (9.9–14.7) and above 14.8 knots with a corresponding elasticity being equal to 0.1, 0.7, 2.3 and 3.7.

At the global level, our results show that considering one or several thresholds (panels B, C and D) makes a difference compare to the assumption of a constant elasticity (panel A), but the assumption of one or many endogenous thresholds is not leading to significant changes in R². The next section shows that these new specifications have, however, some significant implications for regulators and practitioners. We note that the concept of piecewise linearization has also been used in Wang et al. (2019), albeit as a linearization of the fuel consumption function itself for the purposes of solving the speed optimization problem. Here, we estimate endogeneously the thresholds for a piecewise linear elasticity function as shown in Fig. 4.

**5. Impact on optimal speed**

To illustrate the implications of using a constant or a speed-dependent elasticity of fuel consumption with respect to speed, we focus on ship-operators' practices when determining the optimal speed and in particular how they adjust to fuel price. The optimal speed of a vessel (Strandenes, 1981; Evans and Marlow, 1990; Ronen, 2011; Gkonis and Psaraftis, 2012; Psaraftis and Kontovas, 2013; Adland and Jia, 2018; Adland et al., 2018) is, in its simplest form,<fnref n="8" /> the speed that maximizes the Time Charter Equivalent per day under a voyage charter, given by:<fnref n="9" />

<footnote n="8">⁸ In our setting, we do not consider fuel consumption in port for the auxiliary engines and potential payments related to demurrage.</footnote>

<footnote n="9">⁹ Time Charter Equivalent is a shipping industry measure which is used to calculate the average earnings of a vessel net of voyage-related costs.</footnote>
<figure><img src="image_6.png" /><figcaption>**Fig. 3.** Fuel Consumption-speed elasticity by vessel, Source: Data from a company, authors' calculations. Note: Each elasticity is obtained from a vessel-specific linear regression explaining fuel consumption as a function of speed, average draft, trim, sea current, wind direction crossed with wind force, swell, wave height and time trend. The sample is restricted to average daily speeds between 6 and 16 knots.</figcaption></figure>

constant elasticity over all speed values. In panel B, we consider the possibility of one threshold, meaning two speed intervals with specific elasticity, while panels C and D are for two and three thresholds respectively.

When considering panel B with both ballast and laden conditions, the threshold is 9.79 knots for Aframax and 10.26 knots for Suezmax. The elasticity is low below the threshold (less than 0.5 for Aframax and less than 1.4 for Suezmax) which implies that at these levels, the impact of speed level on fuel consumption is very limited. Above the threshold, the elasticity increases to around 2.3 for both vessels. When considering panels C and D, the significant and varying impact of speed on fuel consumption is even clearer. Panel D defines four speed intervals which are, for instance, for Aframax equal to less than 8.4, (8.4–9.8), (9.9–14.7) and above 14.8 knots with a corresponding elasticity being equal to 0.1, 0.7, 2.3 and 3.7.

At the global level, our results show that considering one or several thresholds (panels B, C and D) makes a difference compare to the assumption of a constant elasticity (panel A), but the assumption of one or many endogenous thresholds is not leading to significant changes in R². The next section shows that these new specifications have, however, some significant implications for regulators and practitioners. We note that the concept of piecewise linearization has also been used in Wang et al. (2019), albeit as a linearization of the fuel consumption function itself for the purposes of solving the speed optimization problem. Here, we estimate endogeneously the thresholds for a piecewise linear elasticity function as shown in Fig. 4.

**5. Impact on optimal speed**

To illustrate the implications of using a constant or a speed-dependent elasticity of fuel consumption with respect to speed, we focus on ship-operators' practices when determining the optimal speed and in particular how they adjust to fuel price. The optimal speed of a vessel (Strandenes, 1981; Evans and Marlow, 1990; Ronen, 2011; Gkonis and Psaraftis, 2012; Psaraftis and Kontovas, 2013; Adland and Jia, 2018; Adland et al., 2018) is, in its simplest form,<fnref n="8" /> the speed that maximizes the Time Charter Equivalent per day under a voyage charter, given by:<fnref n="9" />

<footnote n="8">⁸ In our setting, we do not consider fuel consumption in port for the auxiliary engines and potential payments related to demurrage.</footnote>

<footnote n="9">⁹ Time Charter Equivalent is a shipping industry measure which is used to calculate the average earnings of a vessel net of voyage-related costs.</footnote>
<figure><img src="image_5.png" /><figcaption>**Fig. 4.** Linear spline speed-fuel consumption elasticity, with three exogenous thresholds, Source: Data from a company, authors' calculations. Note: The curves are obtained from piecewise linear regressions (with thresholds at 10, 12 and 14 knots) explaining fuel consumption as a function of speed, sea current, average draft, trim, wind direction crossed with wind force, swell, wave height, vessel fixed effects and vessel-specific time trend. The sample is restricted to average daily speeds between 6 and 16 knots.</figcaption></figure>

$$TCE = \frac{R * DWT}{(PT + \frac{d}{24 * s})} - p * FC \quad (7)$$

where TCE is the Daily Time Charter Equivalent (in USD/day), R the freight rate (in USD/tonne of cargo), DWT the deadweight available for cargo (in tonnes), d the distance travelled (in nautical miles), PT the Port time, s the speed (in knots, i.e. nautical miles/hour), p the price of fuel (in USD per tonne) and FC the fuel consumption (in tonnes per day). Under the assumption of a constant speed elasticity to fuel consumption (FC = α * sβ), the optimal speed s* which maximizes the TCE obtained from the first-order condition $\frac{dTCE}{ds} = 0$ is:

$$s^* = \left( \frac{24 * R * DW}{\beta * p * \alpha * d} \right)^{\frac{1}{\beta-1}} \quad (8)$$

For a given voyage where the deadweight available for cargo DWT and the distance to travel d are fixed, the optimal speed depends on the ratio of freight rate to fuel price $\frac{R}{p}$ and on the values of the parameters α and β. Under the assumption of a speed-dependant elasticity, the optimal speed can be estimated inside each speed range using estimates of β from Table 5. The derivation of optimal vessel speed in our tramp shipping case is very different from that of liner shipping, where optimal speed will also depend on the number of port calls, the size of ships, and the number of ships deployed within a service (see, Wu (2020) for a recent optimization example).

For illustration, we use the characteristics of a representative contract for an Aframax vessel carrying Jet A1 Fuel from Ulsan (South Korea) to Botany Bay (Australia). The selected values for the different parameters are reported in Table 5.

Fig. 5 shows how fuel consumption varies according to speed when considering three distinct scenarios: first a unique elasticity equal to 3, second a unique elasticity equal to 1.87 (as in Table 5, panel A, column 1A) and then a case with two thresholds (Table 5, panel C, column 1A) such that the elasticity is 0.372 for speed values between 6 and 9.79 knots, 2.327 for speed values between 9.8 and 14.76 knots, and 3.776 for speed values above 14.76 knots.

Using the traditional assumption of an elasticity equal to 3, such as in Faber et al (2017), and the average speed of 12 knots for an Aframax tanker, the reduction in fuel consumption compared to a design speed of 15 knots (per day and per transport work) is 48%
<figure><img src="image_7.png" /><figcaption>Fig. 4. Linear spline speed-fuel consumption elasticity, with three exogenous thresholds, Source: Data from a company, authors' calculations. Note: The curves are obtained from piecewise linear regressions (with thresholds at 10, 12 and 14 knots) explaining fuel consumption as a function of speed, sea current, average draft, trim, wind direction crossed with wind force, swell, wave height, vessel fixed effects and vessel-specific time trend. The sample is restricted to average daily speeds between 6 and 16 knots.</figcaption></figure>

$$TCE = \frac{R * DWT}{(PT + \frac{d}{24 * s})} - p * FC \quad (7)$$

where TCE is the Daily Time Charter Equivalent (in USD/day), R the freight rate (in USD/tonne of cargo), DWT the deadweight available for cargo (in tonnes), d the distance travelled (in nautical miles), PT the Port time, s the speed (in knots, i.e. nautical miles/hour), p the price of fuel (in USD per tonne) and FC the fuel consumption (in tonnes per day). Under the assumption of a constant speed elasticity to fuel consumption ($FC = \alpha * s^\beta$), the optimal speed $s^*$ which maximizes the TCE obtained from the first-order condition $\frac{dTCE}{ds} = 0$ is:

$$s^* = \left( \frac{24 * R * DW}{\beta * p * \alpha * d} \right)^{\frac{1}{\beta-1}} \quad (8)$$

For a given voyage where the deadweight available for cargo DWT and the distance to travel d are fixed, the optimal speed depends on the ratio of freight rate to fuel price $\frac{R}{p}$ and on the values of the parameters $\alpha$ and $\beta$. Under the assumption of a speed-dependant elasticity, the optimal speed can be estimated inside each speed range using estimates of $\beta$ from Table 5. The derivation of optimal vessel speed in our tramp shipping case is very different from that of liner shipping, where optimal speed will also depend on the number of port calls, the size of ships, and the number of ships deployed within a service (see, Wu (2020) for a recent optimization example).

For illustration, we use the characteristics of a representative contract for an Aframax vessel carrying Jet A1 Fuel from Ulsan (South Korea) to Botany Bay (Australia). The selected values for the different parameters are reported in Table 5.

Fig. 5 shows how fuel consumption varies according to speed when considering three distinct scenarios: first a unique elasticity equal to 3, second a unique elasticity equal to 1.87 (as in Table 5, panel A, column 1A) and then a case with two thresholds (Table 5, panel C, column 1A) such that the elasticity is 0.372 for speed values between 6 and 9.79 knots, 2.327 for speed values between 9.8 and 14.76 knots, and 3.776 for speed values above 14.76 knots.

Using the traditional assumption of an elasticity equal to 3, such as in Faber et al (2017), and the average speed of 12 knots for an Aframax tanker, the reduction in fuel consumption compared to a design speed of 15 knots (per day and per transport work) is 48%
<table><caption>**Table 4**
Estimates of the logarithm of fuel consumption with endogenous thresholds.</caption><thead><tr><th rowspan="2">Variables</th><th colspan="3">Aframax</th><th colspan="3">Suezmax</th></tr><tr><th>(1A)<br>All</th><th>(1B)<br>Ballast</th><th>(1C)<br>Laden</th><th>(2A)<br>All</th><th>(2B)<br>Ballast</th><th>(2C)<br>Laden</th></tr></thead><tbody><tr><td colspan="7"><strong>Panel A. Constant elasticity</strong></td></tr><tr><td>LOG speed (log)</td><td>1.871***<br>(114.01)</td><td>2.013***<br>(72.18)</td><td>1.621***<br>(75.82)</td><td>1.978***<br>(115.19)</td><td>1.918***<br>(74.10)</td><td>1.966***<br>(84.79)</td></tr><tr><td>Extended controls</td><td>YES</td><td>YES</td><td>YES</td><td>YES</td><td>YES</td><td>YES</td></tr><tr><td>Observations</td><td>7600</td><td>2575</td><td>5025</td><td>4325</td><td>2068</td><td>2257</td></tr><tr><td>R<sup>2</sup></td><td>0.824</td><td>0.797</td><td>0.688</td><td>0.873</td><td>0.839</td><td>0.862</td></tr><tr><td colspan="7"><strong>Panel B. One endogenous threshold (K)</strong></td></tr><tr><td>Estimated<br>K<sub>1</sub></td><td>9.79</td><td>9.89</td><td>10.50</td><td>10.26</td><td>10.16</td><td>10.68</td></tr><tr><td>LOG speed (log)</td><td>0.360***<br>(8.78)</td><td>0.588***<br>(9.80)</td><td>0.400***<br>(9.05)</td><td>1.339***<br>(36.77)</td><td>1.248***<br>(24.42)</td><td>1.735***<br>(33.42)</td></tr><tr><td>LOG speed (log)</td><td>2.351***<br>(122.19)</td><td>2.642***<br>(76.54)</td><td>2.143***<br>(82.88)</td><td>2.388***<br>(89.97)</td><td>2.405***<br>(58.97)</td><td>2.125***<br>(54.05)</td></tr><tr><td>Test of equal elasticity</td><td>1564.04</td><td>680.86</td><td>954.20</td><td>387.41</td><td>223.70</td><td>24.88</td></tr><tr><td>F-test</td><td>0.000</td><td>0.000</td><td>0.000</td><td>0.000</td><td>0.000</td><td>0.000</td></tr><tr><td>Prob-value</td><td>YES</td><td>YES</td><td>YES</td><td>YES</td><td>YES</td><td>YES</td></tr><tr><td>Extended controls</td><td>YES</td><td>YES</td><td>YES</td><td>YES</td><td>YES</td><td>YES</td></tr><tr><td>Observations</td><td>7600</td><td>2575</td><td>5025</td><td>4325</td><td>2068</td><td>2257</td></tr><tr><td>R<sup>2</sup></td><td>0.854</td><td>0.841</td><td>0.738</td><td>0.884</td><td>0.855</td><td>0.863</td></tr><tr><td colspan="7"><strong>Panel C. Two endogenous thresholds (K)</strong></td></tr><tr><td>Estimated<br>K<sub>1</sub></td><td>9.79</td><td>8.77</td><td>10.50</td><td>8.15</td><td>9.76</td><td>10.16</td></tr><tr><td>Estimated<br>K<sub>2</sub></td><td>14.76</td><td>10.09</td><td>14.32</td><td>10.79</td><td>11.92</td><td>12.53</td></tr><tr><td>LOG speed (log)</td><td>0.372***<br>(9.07)</td><td>0.176<br>(1.50)</td><td>0.415***<br>(9.38)</td><td>0.764***<br>(7.01)</td><td>1.223***<br>(20.06)</td><td>1.714***<br>(26.53)</td></tr><tr><td>LOG speed (log)</td><td>2.327***<br>(115.93)</td><td>1.274***<br>(10.86)</td><td>2.099***<br>(75.09)</td><td>1.719***<br>(34.76)</td><td>2.060***<br>(30.12)</td><td>2.030***<br>(38.36)</td></tr><tr><td>LOG speed (log)</td><td>3.776***<br>(10.95)</td><td>2.638***<br>(70.49)</td><td>3.036***<br>(14.11)</td><td>2.424***<br>(76.70)</td><td>2.655***<br>(36.18)</td><td>2.196***<br>(31.57)</td></tr><tr><td>Test of equal elasticity</td><td>792.27</td><td>350.01</td><td>487.42</td><td>209.34</td><td>122.47</td><td>13.49</td></tr><tr><td>F-test</td><td>0.000</td><td>0.000</td><td>0.000</td><td>0.000</td><td>0.000</td><td>0.000</td></tr><tr><td>Prob-value</td><td>YES</td><td>YES</td><td>YES</td><td>YES</td><td>YES</td><td>YES</td></tr><tr><td>Extended controls</td><td>YES</td><td>YES</td><td>YES</td><td>YES</td><td>YES</td><td>YES</td></tr><tr><td>Observations</td><td>7600</td><td>2575</td><td>5025</td><td>4325</td><td>2068</td><td>2257</td></tr><tr><td>R<sup>2</sup></td><td>0.854</td><td>0.842</td><td>0.739</td><td>0.885</td><td>0.856</td><td>0.863</td></tr><tr><td colspan="7"><strong>Panel D. Three endogenous thresholds (K)</strong></td></tr><tr><td>Estimated<br>K<sub>1</sub></td><td>8.43</td><td>8.77</td><td>9.60</td><td>8.15</td><td>9.76</td><td>10.26</td></tr><tr><td>Estimated<br>K<sub>2</sub></td><td>9.89</td><td>9.89</td><td>10.50</td><td>10.68</td><td>11.92</td><td>12.53</td></tr><tr><td>Estimated<br>K<sub>3</sub></td><td>14.76</td><td>10.19</td><td>14.32</td><td>14.71</td><td>14.56</td><td>13.71</td></tr><tr><td>LOG speed (log)</td><td>0.114<br>(1.19)</td><td>0.185<br>(1.52)</td><td>0.360***<br>(4.95)</td><td>0.760***<br>(6.92)</td><td>1.218***<br>(19.98)</td><td>1.719***<br>(27.45)</td></tr><tr><td>LOG speed (log)</td><td>0.694***<br>(8.85)</td><td>1.233***<br>(6.77)</td><td>0.556***<br>(3.58)</td><td>1.720***<br>(32.75)</td><td>2.088***<br>(29.83)</td><td>2.046***<br>(36.14)</td></tr><tr><td>LOG speed (log)</td><td>2.325***<br>(111.30)</td><td>1.826***<br>(3.75)</td><td>2.091***<br>(71.86)</td><td>2.363***<br>(70.52)</td><td>2.540***<br>(26.40)</td><td>2.143***<br>(19.92)</td></tr><tr><td>LOG speed (log)</td><td>3.783***<br>(10.96)</td><td>2.639***<br>(64.57)</td><td>3.048***<br>(14.14)</td><td>3.667***<br>(10.24)</td><td>3.430***<br>(8.06)</td><td>2.306***<br>(12.06)</td></tr><tr><td>Test of equal elasticity</td><td>532.60</td><td>233.37</td><td>325.24</td><td>144.02</td><td>82.88</td><td>9.12</td></tr><tr><td>F-test</td><td>0.000</td><td>0.000</td><td>0.000</td><td>0.000</td><td>0.000</td><td>0.000</td></tr><tr><td>Prob-value</td><td>YES</td><td>YES</td><td>YES</td><td>YES</td><td>YES</td><td>YES</td></tr><tr><td>Extended controls</td><td>YES</td><td>YES</td><td>YES</td><td>YES</td><td>YES</td><td>YES</td></tr><tr><td>Observations</td><td>7600</td><td>2575</td><td>5025</td><td>4325</td><td>2068</td><td>2257</td></tr><tr><td>R<sup>2</sup></td><td>0.855</td><td>0.842</td><td>0.739</td><td>0.885</td><td>0.856</td><td>0.863</td></tr></tbody></table>

Source: data from a tanker company, authors' calculations.

Note: the reported estimates are obtained from linear regressions with robust standard errors. Significance levels are 1% (***), 5% (**) and 10% (***). The sample is restricted to average daily speeds comprised between 6 and 16 knots. Additional controls include average draft, trim, sea current, wind direction crossed with wind force, swell, wave height, vessel fixed effects and vessel-specific time trend.

using the cubic law, 34% using an elasticity of 1.87 and 41% using speed-dependent elasticity.<fnref n="10" /> This result implies that most studies (see Psaraftis, 2019) that assess the impact of slow steaming on CO₂ emissions on the basis of the cubic rule over-estimate the impact of such operational measures – even before the adverse impact of weather is considered. Fig. 6a and b report our estimated TCE as a function of sailing speed for an Aframax vessel for a fuel price at 400 USD/tonne of fuel and then 600 USD/tonne respectively.

Using our estimates, an increase in fuel price from 400 to 600 USD/tonne, for example as a result of a 200 USD/tonne bunker levy,

<footnote n="10">¹⁰ To account for constant transport work, the reduction in consumption/emission from a speed S<sub>ds</sub> to S<sub>1</sub> is (S<sub>ds</sub>/S<sub>1</sub>) x reduction per transport work.</footnote>
<table><caption>**Table 4**
Estimates of the logarithm of fuel consumption with endogenous thresholds.</caption><thead><tr><th rowspan="2">Variables</th><th colspan="3">Aframax</th><th colspan="3">Suezmax</th></tr><tr><th>(1A)<br>All</th><th>(1B)<br>Ballast</th><th>(1C)<br>Laden</th><th>(2A)<br>All</th><th>(2B)<br>Ballast</th><th>(2C)<br>Laden</th></tr></thead><tbody><tr><td colspan="7"><strong>Panel A. Constant elasticity</strong></td></tr><tr><td>LOG speed (log)</td><td>1.871***<br>(114.01)</td><td>2.013***<br>(72.18)</td><td>1.621***<br>(75.82)</td><td>1.978***<br>(115.19)</td><td>1.918***<br>(74.10)</td><td>1.966***<br>(84.79)</td></tr><tr><td>Extended controls</td><td>YES</td><td>YES</td><td>YES</td><td>YES</td><td>YES</td><td>YES</td></tr><tr><td>Observations</td><td>7600</td><td>2575</td><td>5025</td><td>4325</td><td>2068</td><td>2257</td></tr><tr><td>R<sup>2</sup></td><td>0.824</td><td>0.797</td><td>0.688</td><td>0.873</td><td>0.839</td><td>0.862</td></tr><tr><td colspan="7"><strong>Panel B. One endogenous threshold (K)</strong></td></tr><tr><td>Estimated<br>K<sub>1</sub></td><td>9.79</td><td>9.89</td><td>10.50</td><td>10.26</td><td>10.16</td><td>10.68</td></tr><tr><td>LOG speed (log)</td><td>0.360***<br>(8.78)</td><td>0.588***<br>(9.80)</td><td>0.400***<br>(9.05)</td><td>1.339***<br>(36.77)</td><td>1.248***<br>(24.42)</td><td>1.735***<br>(33.42)</td></tr><tr><td>LOG speed (log)</td><td>2.351***<br>(122.19)</td><td>2.642***<br>(76.54)</td><td>2.143***<br>(82.88)</td><td>2.388***<br>(89.97)</td><td>2.405***<br>(58.97)</td><td>2.125***<br>(54.05)</td></tr><tr><td>Test of equal elasticity</td><td>1564.04</td><td>680.86</td><td>954.20</td><td>387.41</td><td>223.70</td><td>24.88</td></tr><tr><td>F-test</td><td>0.000</td><td>0.000</td><td>0.000</td><td>0.000</td><td>0.000</td><td>0.000</td></tr><tr><td>Prob-value</td><td>YES</td><td>YES</td><td>YES</td><td>YES</td><td>YES</td><td>YES</td></tr><tr><td>Extended controls</td><td>YES</td><td>YES</td><td>YES</td><td>YES</td><td>YES</td><td>YES</td></tr><tr><td>Observations</td><td>7600</td><td>2575</td><td>5025</td><td>4325</td><td>2068</td><td>2257</td></tr><tr><td>R<sup>2</sup></td><td>0.854</td><td>0.841</td><td>0.738</td><td>0.884</td><td>0.855</td><td>0.863</td></tr><tr><td colspan="7"><strong>Panel C. Two endogenous thresholds (K)</strong></td></tr><tr><td>Estimated<br>K<sub>1</sub></td><td>9.79</td><td>8.77</td><td>10.50</td><td>8.15</td><td>9.76</td><td>10.16</td></tr><tr><td>Estimated<br>K<sub>2</sub></td><td>14.76</td><td>10.09</td><td>14.32</td><td>10.79</td><td>11.92</td><td>12.53</td></tr><tr><td>LOG speed (log)</td><td>0.372***<br>(9.07)</td><td>0.176<br>(1.50)</td><td>0.415***<br>(9.38)</td><td>0.764***<br>(7.01)</td><td>1.223***<br>(20.06)</td><td>1.714***<br>(26.53)</td></tr><tr><td>LOG speed (log)</td><td>2.327***<br>(115.93)</td><td>1.274***<br>(10.86)</td><td>2.099***<br>(75.09)</td><td>1.719***<br>(34.76)</td><td>2.060***<br>(30.12)</td><td>2.030***<br>(38.36)</td></tr><tr><td>LOG speed (log)</td><td>3.776***<br>(10.95)</td><td>2.638***<br>(70.49)</td><td>3.036***<br>(14.11)</td><td>2.424***<br>(76.70)</td><td>2.655***<br>(36.18)</td><td>2.196***<br>(31.57)</td></tr><tr><td>Test of equal elasticity</td><td>792.27</td><td>350.01</td><td>487.42</td><td>209.34</td><td>122.47</td><td>13.49</td></tr><tr><td>F-test</td><td>0.000</td><td>0.000</td><td>0.000</td><td>0.000</td><td>0.000</td><td>0.000</td></tr><tr><td>Prob-value</td><td>YES</td><td>YES</td><td>YES</td><td>YES</td><td>YES</td><td>YES</td></tr><tr><td>Extended controls</td><td>YES</td><td>YES</td><td>YES</td><td>YES</td><td>YES</td><td>YES</td></tr><tr><td>Observations</td><td>7600</td><td>2575</td><td>5025</td><td>4325</td><td>2068</td><td>2257</td></tr><tr><td>R<sup>2</sup></td><td>0.854</td><td>0.842</td><td>0.739</td><td>0.885</td><td>0.856</td><td>0.863</td></tr><tr><td colspan="7"><strong>Panel D. Three endogenous thresholds (K)</strong></td></tr><tr><td>Estimated<br>K<sub>1</sub></td><td>8.43</td><td>8.77</td><td>9.60</td><td>8.15</td><td>9.76</td><td>10.26</td></tr><tr><td>Estimated<br>K<sub>2</sub></td><td>9.89</td><td>9.89</td><td>10.50</td><td>10.68</td><td>11.92</td><td>12.53</td></tr><tr><td>Estimated<br>K<sub>3</sub></td><td>14.76</td><td>10.19</td><td>14.32</td><td>14.71</td><td>14.56</td><td>13.71</td></tr><tr><td>LOG speed (log)</td><td>0.114<br>(1.19)</td><td>0.185<br>(1.52)</td><td>0.360***<br>(4.95)</td><td>0.760***<br>(6.92)</td><td>1.218***<br>(19.98)</td><td>1.719***<br>(27.45)</td></tr><tr><td>LOG speed (log)</td><td>0.694***<br>(8.85)</td><td>1.233***<br>(6.77)</td><td>0.556***<br>(3.58)</td><td>1.720***<br>(32.75)</td><td>2.088***<br>(29.83)</td><td>2.046***<br>(36.14)</td></tr><tr><td>LOG speed (log)</td><td>2.325***<br>(111.30)</td><td>1.826***<br>(3.75)</td><td>2.091***<br>(71.86)</td><td>2.363***<br>(70.52)</td><td>2.540***<br>(26.40)</td><td>2.143***<br>(19.92)</td></tr><tr><td>LOG speed (log)</td><td>3.783***<br>(10.96)</td><td>2.639***<br>(64.57)</td><td>3.048***<br>(14.14)</td><td>3.667***<br>(10.24)</td><td>3.430***<br>(8.06)</td><td>2.306***<br>(12.06)</td></tr><tr><td>Test of equal elasticity</td><td>532.60</td><td>233.37</td><td>325.24</td><td>144.02</td><td>82.88</td><td>9.12</td></tr><tr><td>F-test</td><td>0.000</td><td>0.000</td><td>0.000</td><td>0.000</td><td>0.000</td><td>0.000</td></tr><tr><td>Prob-value</td><td>YES</td><td>YES</td><td>YES</td><td>YES</td><td>YES</td><td>YES</td></tr><tr><td>Extended controls</td><td>YES</td><td>YES</td><td>YES</td><td>YES</td><td>YES</td><td>YES</td></tr><tr><td>Observations</td><td>7600</td><td>2575</td><td>5025</td><td>4325</td><td>2068</td><td>2257</td></tr><tr><td>R<sup>2</sup></td><td>0.855</td><td>0.842</td><td>0.739</td><td>0.885</td><td>0.856</td><td>0.863</td></tr></tbody></table>

Source: data from a tanker company, authors' calculations.

Note: the reported estimates are obtained from linear regressions with robust standard errors. Significance levels are 1% (***), 5% (**) and 10% (***). The sample is restricted to average daily speeds comprised between 6 and 16 knots. Additional controls include average draft, trim, sea current, wind direction crossed with wind force, swell, wave height, vessel fixed effects and vessel-specific time trend.

using the cubic law, 34% using an elasticity of 1.87 and 41% using speed-dependent elasticity.<fnref n="10" /> This result implies that most studies (see Psaraftis, 2019) that assess the impact of slow steaming on CO₂ emissions on the basis of the cubic rule over-estimate the impact of such operational measures – even before the adverse impact of weather is considered. Fig. 6a and b report our estimated TCE as a function of sailing speed for an Aframax vessel for a fuel price at 400 USD/tonne of fuel and then 600 USD/tonne respectively.

Using our estimates, an increase in fuel price from 400 to 600 USD/tonne, for example as a result of a 200 USD/tonne bunker levy,

<footnote n="10">¹⁰ To account for constant transport work, the reduction in consumption/emission from a speed S<sub>ds</sub> to S<sub>1</sub> is (S<sub>ds</sub>/S<sub>1</sub>) x reduction per transport work.</footnote>
<table><caption>**Table 5**
Selected values for parameters to determine optimal speed.</caption><tr><td>Spot freight rate (R)</td><td>13 USD/tonne</td></tr><tr><td>Distance (d)</td><td>4670 Nautical miles</td></tr><tr><td>Payload (DW)</td><td>76 000 tonnes</td></tr><tr><td>Fuel consumption design speed (15 kt)</td><td>55 tonnes/day</td></tr><tr><td>Port time (PT)</td><td>10 days</td></tr><tr><td>Fuel price</td><td>400–600 USD/tonne</td></tr></table>

<figure><img src="image_6.png" /><figcaption>**Fig. 5.** Fuel consumption with constant and speed-dependent elasticities, Source: authors’ calculations, using estimates and parameters from Tables 5 and 6.</figcaption></figure>

will have a different impact depending on the selected scenario as shown in Table 6. For instance, with an elasticity of 3, the speed would reduce from 10.4 to 8.9 knots, and the fuel consumption per day would decrease by 36%. For an elasticity of 1.87, the speed would decrease from 10.8 to 8.3 knots, and emissions by 39%. Finally, with speed-dependent elasticity, the speed reduces from 10.5 to 9.8 knots, and emissions by 15%. This result confirms that accurate estimates of the speed-fuel consumption elasticity have a large impact on operators’ decision on the optimal speed (Adland and Jia, 2016), but also on the likely impact of a bunker levy.

## 6. Conclusion

In this paper, we have proposed a new and flexible methodology that allows for the empirical estimation of a variable elasticity of fuel consumption with regards to speed using real-life operational data. We show that once weather effects are accounted for, the real elasticity of fuel consumption with regards to speed is dependent on vessel speed itself.

While the “cubic law” holds true near the design speed of vessels, the elasticity of fuel consumption with regards to vessel speed can be substantially lower in the speed range where ships mostly operate. We also show that the differences in optimal speed can be substantial between the classical models that assume constant elasticity and our more flexible functional form. Clearly, accurate data-driven estimates of the speed-consumption relationship therefore becomes extremely important for the optimization of ship operations as well as the evaluation of the impact of potential upcoming environmental regulations such as speed limits or a CO2 bunker levy.

We acknowledge that our study has certain weaknesses. First, we have only considered one type of vessels, i.e. medium-size crude oil tankers. While the results for bulk carriers can be expected to be similar, containerships are different in terms of both design and
<table><caption>**Table 5**
Selected values for parameters to determine optimal speed.</caption><tr><td>Spot freight rate (R)</td><td>13 USD/tonne</td></tr><tr><td>Distance (d)</td><td>4670 Nautical miles</td></tr><tr><td>Payload (DW)</td><td>76 000 tonnes</td></tr><tr><td>Fuel consumption design speed (15 kt)</td><td>55 tonnes/day</td></tr><tr><td>Port time (PT)</td><td>10 days</td></tr><tr><td>Fuel price</td><td>400–600 USD/tonne</td></tr></table>

<figure><img src="image_8.png" /><figcaption>**Fig. 5.** Fuel consumption with constant and speed-dependent elasticities, Source: authors' calculations, using estimates and parameters from Tables 5 and 6.</figcaption></figure>

will have a different impact depending on the selected scenario as shown in Table 6. For instance, with an elasticity of 3, the speed would reduce from 10.4 to 8.9 knots, and the fuel consumption per day would decrease by 36%. For an elasticity of 1.87, the speed would decrease from 10.8 to 8.3 knots, and emissions by 39%. Finally, with speed-dependent elasticity, the speed reduces from 10.5 to 9.8 knots, and emissions by 15%. This result confirms that accurate estimates of the speed-fuel consumption elasticity have a large impact on operators' decision on the optimal speed (Adland and Jia, 2016), but also on the likely impact of a bunker levy.

## 6. Conclusion

In this paper, we have proposed a new and flexible methodology that allows for the empirical estimation of a variable elasticity of fuel consumption with regards to speed using real-life operational data. We show that once weather effects are accounted for, the real elasticity of fuel consumption with regards to speed is dependent on vessel speed itself.

While the “cubic law” holds true near the design speed of vessels, the elasticity of fuel consumption with regards to vessel speed can be substantially lower in the speed range where ships mostly operate. We also show that the differences in optimal speed can be substantial between the classical models that assume constant elasticity and our more flexible functional form. Clearly, accurate data-driven estimates of the speed-consumption relationship therefore becomes extremely important for the optimization of ship operations as well as the evaluation of the impact of potential upcoming environmental regulations such as speed limits or a CO2 bunker levy.

We acknowledge that our study has certain weaknesses. First, we have only considered one type of vessels, i.e. medium-size crude oil tankers. While the results for bulk carriers can be expected to be similar, containerships are different in terms of both design and
<figure><img src="image_7.png" /><figcaption>Fig. 6. Optimal speed and profit function, Source: authors' calculations, using estimates and parameters from Tables 5 and 6.</figcaption></figure>

<table><caption>**Table 6**

Impact of a bunker levy of 200 USD/tonne of fuel as a function of elasticity (β).</caption><thead><tr><th rowspan="2">Outcomes</th><th colspan="3">Bunker price 400 USD/tonne</th><th colspan="3">Bunker price 600 USD/tonne</th></tr><tr><th>β = 3</th><th>β = 1.87</th><th>β variable</th><th>β = 3</th><th>β = 1.87</th><th>β variable</th></tr></thead><tbody><tr><td>Optimal speed S* (kt)</td><td>10.4</td><td>10.8</td><td>10.5</td><td>8.9</td><td>8.3</td><td>9.8</td></tr><tr><td>Profit/day (USD)</td><td>26,267</td><td>22,902</td><td>24,704</td><td>23,707</td><td>18,247</td><td>20,561</td></tr><tr><td>Emissions/day (tonne)</td><td>18.3</td><td>30.0</td><td>23.9</td><td>11.7</td><td>18.2</td><td>20.15</td></tr></tbody></table>

Source: authors' calculations, using estimates and parameters from Tables 5 and 6.

operation (higher average sailing speeds) and could therefore exhibit differing results (Cariou, 2011; Cariou et al., 2019).

Second, we rely on the use of noon reports which, although commonly used in the empirical literature, are likely to be less accurate than high-frequency sensor data (Aldous et al., 2013). In particular, the state of the weather conditions in which the vessel operates is recorded manually by the crew at noon every day and this snapshot may not give an accurate picture of the average conditions in the preceding 24 h over which the daily fuel consumption is measured. Nevertheless, the fact that our models have such high explanatory power suggests that this is not a major concern.

Third, we have considered regression models to explain fuel consumption as a function of speed. Recently, this approach has been complemented by data mining and machine learning techniques. Although parametric, the methodology described in this paper provides a very flexible tool to account for non-linearities in the speed-consumption relationship. The main shortcoming of the regression framework is that it is not designed to find the best fit of the data. However, when applied to the noon report data, we find that our linear regressions perform reasonably well, with R² values exceeding 0.8 both for Aframax and Suezmax. As shown in Appendix B, the use of neural networks provides a better fit of the data, but the improvement remains moderate.

As we have only investigated the relationship for a limited fleet of sister vessels in the Aframax and Suezmax tanker segment, we cannot be certain of the extent to which our results can be generalized to other ship designs. However, our results provide robust evidence that the consumption-to-speed elasticity for our chosen vessel type is speed dependent that the simplified cubic law only applies near the vessels' design speed.

Future research in this area should expand to other vessel types and sizes. It would also be important to include weather effects in
<figure><img src="image_9.png" /><figcaption>Fig. 6. Optimal speed and profit function, Source: authors' calculations, using estimates and parameters from Tables 5 and 6.</figcaption></figure>

<table><caption>**Table 6**

Impact of a bunker levy of 200 USD/tonne of fuel as a function of elasticity (β).</caption><thead><tr><th rowspan="2">Outcomes</th><th colspan="3">Bunker price 400 USD/tonne</th><th colspan="3">Bunker price 600 USD/tonne</th></tr><tr><th>β = 3</th><th>β = 1.87</th><th>β variable</th><th>β = 3</th><th>β = 1.87</th><th>β variable</th></tr></thead><tbody><tr><td>Optimal speed S* (kt)</td><td>10.4</td><td>10.8</td><td>10.5</td><td>8.9</td><td>8.3</td><td>9.8</td></tr><tr><td>Profit/day (USD)</td><td>26,267</td><td>22,902</td><td>24,704</td><td>23,707</td><td>18,247</td><td>20,561</td></tr><tr><td>Emissions/day (tonne)</td><td>18.3</td><td>30.0</td><td>23.9</td><td>11.7</td><td>18.2</td><td>20.15</td></tr></tbody></table>

Source: authors' calculations, using estimates and parameters from Tables 5 and 6.

operation (higher average sailing speeds) and could therefore exhibit differing results (Cariou, 2011; Cariou et al., 2019).

Second, we rely on the use of noon reports which, although commonly used in the empirical literature, are likely to be less accurate than high-frequency sensor data (Aldous et al., 2013). In particular, the state of the weather conditions in which the vessel operates is recorded manually by the crew at noon every day and this snapshot may not give an accurate picture of the average conditions in the preceding 24 h over which the daily fuel consumption is measured. Nevertheless, the fact that our models have such high explanatory power suggests that this is not a major concern.

Third, we have considered regression models to explain fuel consumption as a function of speed. Recently, this approach has been complemented by data mining and machine learning techniques. Although parametric, the methodology described in this paper provides a very flexible tool to account for non-linearities in the speed-consumption relationship. The main shortcoming of the regression framework is that it is not designed to find the best fit of the data. However, when applied to the noon report data, we find that our linear regressions perform reasonably well, with R² values exceeding 0.8 both for Aframax and Suezmax. As shown in Appendix B, the use of neural networks provides a better fit of the data, but the improvement remains moderate.

As we have only investigated the relationship for a limited fleet of sister vessels in the Aframax and Suezmax tanker segment, we cannot be certain of the extent to which our results can be generalized to other ship designs. However, our results provide robust evidence that the consumption-to-speed elasticity for our chosen vessel type is speed dependent that the simplified cubic law only applies near the vessels' design speed.

Future research in this area should expand to other vessel types and sizes. It would also be important to include weather effects in
the calculations of optimal speed (our selected elasticities are net of weather and wave conditions) along with an evaluation of the impact of parameter uncertainty. As a related example, Ng (2019) uses stochastic programming to account for the uncertainty in the cubic rule. It would also be beneficial to include higher-resolution weather and wave data as well as new covariates such as sea water temperature, salinity, water depth, and other variables known to affect the power requirement for oceangoing vessels.

### CRediT authorship contribution statement

**Roar Adland:** Conceptualization, Writing - original draft, Funding acquisition, Writing - review & editing. **Pierre Cariou:** Conceptualization, Writing - original draft, Writing - review & editing. **Francois-Charles Wolff:** Conceptualization, Data curation, Methodology, Formal analysis, Visualization, Writing - original draft, Writing - review & editing.

### Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

### Acknowledgements

We are indebted to three anonymous reviewers for their helpful comments and suggestions on a previous draft. We also would like to thank conference participants at the 2019 IAME Conference (Athens). This research was partly funded by the Research Council of Norway under the project 'Real Energy Efficiency and Emissions in the Seaway', project no. 255672/O80.

## Appendix A. Descriptive statistics

See Table A.

<table><caption>**Table A**

Descriptive statistics of the sample.</caption><thead><tr><th rowspan="2">Variables</th><th colspan="3">Aframax</th><th colspan="3">Suezmax</th></tr><tr><th>All</th><th>Ballast</th><th>Laden</th><th>All</th><th>Ballast</th><th>Laden</th></tr></thead><tbody><tr><td>Fuel consumption (log)</td><td>3,438</td><td>3,134</td><td>3,594</td><td>3,558</td><td>3,335</td><td>3,762</td></tr><tr><td>LOG Speed (log)</td><td>2,481</td><td>2,407</td><td>2,518</td><td>2,470</td><td>2,423</td><td>2,512</td></tr><tr><td>Sea current (difference in speed)</td><td>-0,012</td><td>-0,031</td><td>-0,002</td><td>0,007</td><td>0,000</td><td>0,014</td></tr><tr><td>Average draft</td><td>11,475</td><td>8,166</td><td>13,170</td><td>11,795</td><td>8,007</td><td>15,266</td></tr><tr><td>Trim</td><td>-0,608</td><td>-1,681</td><td>-0,059</td><td>-1,377</td><td>-2,845</td><td>-0,032</td></tr><tr><td>Headwind &lt; 10 k</td><td>0,079</td><td>0,101</td><td>0,068</td><td>0,033</td><td>0,025</td><td>0,039</td></tr><tr><td>Headwind 11-16 k</td><td>0,064</td><td>0,064</td><td>0,064</td><td>0,042</td><td>0,035</td><td>0,048</td></tr><tr><td>Headwind 17-21 k</td><td>0,042</td><td>0,036</td><td>0,045</td><td>0,033</td><td>0,019</td><td>0,046</td></tr><tr><td>Headwind 22-27 k</td><td>0,030</td><td>0,016</td><td>0,037</td><td>0,037</td><td>0,023</td><td>0,051</td></tr><tr><td>Headwind &gt; 27 k</td><td>0,014</td><td>0,004</td><td>0,019</td><td>0,021</td><td>0,009</td><td>0,031</td></tr><tr><td>Headwind 45° &lt; 10 k</td><td>0,100</td><td>0,122</td><td>0,089</td><td>0,078</td><td>0,086</td><td>0,072</td></tr><tr><td>Headwind 45° 11-16 k</td><td>0,097</td><td>0,108</td><td>0,091</td><td>0,104</td><td>0,098</td><td>0,110</td></tr><tr><td>Headwind 45° 17-21 k</td><td>0,054</td><td>0,060</td><td>0,051</td><td>0,080</td><td>0,065</td><td>0,093</td></tr><tr><td>Headwind 45° 22-27 k</td><td>0,031</td><td>0,028</td><td>0,032</td><td>0,046</td><td>0,033</td><td>0,058</td></tr><tr><td>Headwind 45° &gt; 27 k</td><td>0,012</td><td>0,006</td><td>0,015</td><td>0,025</td><td>0,021</td><td>0,029</td></tr><tr><td>Side &lt; 10 k</td><td>0,043</td><td>0,047</td><td>0,041</td><td>0,061</td><td>0,073</td><td>0,051</td></tr><tr><td>Side 11-16 k</td><td>0,045</td><td>0,050</td><td>0,043</td><td>0,066</td><td>0,067</td><td>0,065</td></tr><tr><td>Side 17-21 k</td><td>0,029</td><td>0,033</td><td>0,027</td><td>0,045</td><td>0,045</td><td>0,045</td></tr><tr><td>Side 22-27 k</td><td>0,014</td><td>0,016</td><td>0,013</td><td>0,016</td><td>0,021</td><td>0,012</td></tr><tr><td>Side &gt; 27 k</td><td>0,005</td><td>0,004</td><td>0,006</td><td>0,006</td><td>0,007</td><td>0,005</td></tr><tr><td>Wind astern 45° &lt; 10 k</td><td>0,064</td><td>0,066</td><td>0,063</td><td>0,064</td><td>0,077</td><td>0,051</td></tr><tr><td>Wind astern 45° 11-16 k</td><td>0,073</td><td>0,066</td><td>0,076</td><td>0,087</td><td>0,094</td><td>0,081</td></tr><tr><td>Wind astern 45° 17-21 k</td><td>0,047</td><td>0,044</td><td>0,048</td><td>0,050</td><td>0,060</td><td>0,041</td></tr><tr><td>Wind astern 45° 22-27 k</td><td>0,026</td><td>0,025</td><td>0,026</td><td>0,019</td><td>0,026</td><td>0,013</td></tr><tr><td>Wind astern 45° &gt; 27 k</td><td>0,010</td><td>0,008</td><td>0,011</td><td>0,006</td><td>0,005</td><td>0,006</td></tr><tr><td>Wind astern &lt; 10 k</td><td>0,037</td><td>0,035</td><td>0,038</td><td>0,024</td><td>0,031</td><td>0,016</td></tr><tr><td>Wind astern 11-16 k</td><td>0,040</td><td>0,026</td><td>0,047</td><td>0,029</td><td>0,036</td><td>0,022</td></tr><tr><td>Wind astern 17-21 k</td><td>0,027</td><td>0,019</td><td>0,030</td><td>0,019</td><td>0,028</td><td>0,011</td></tr><tr><td>Wind astern 22-27 k</td><td>0,013</td><td>0,010</td><td>0,014</td><td>0,008</td><td>0,014</td><td>0,003</td></tr><tr><td>Wind astern &gt; 27 k</td><td>0,004</td><td>0,006</td><td>0,004</td><td>0,001</td><td>0,002</td><td>0,000</td></tr><tr><td>Swell low or less</td><td>0,353</td><td>0,386</td><td>0,337</td><td>0,324</td><td>0,358</td><td>0,293</td></tr><tr><td>Swell light</td><td>0,275</td><td>0,283</td><td>0,271</td><td>0,213</td><td>0,232</td><td>0,197</td></tr></tbody></table>

(continued on next page)
the calculations of optimal speed (our selected elasticities are net of weather and wave conditions) along with an evaluation of the impact of parameter uncertainty. As a related example, Ng (2019) uses stochastic programming to account for the uncertainty in the cubic rule. It would also be beneficial to include higher-resolution weather and wave data as well as new covariates such as sea water temperature, salinity, water depth, and other variables known to affect the power requirement for oceangoing vessels.

### CRediT authorship contribution statement

**Roar Adland:** Conceptualization, Writing - original draft, Funding acquisition, Writing - review & editing. **Pierre Cariou:** Conceptualization, Writing - original draft, Writing - review & editing. **Francois-Charles Wolff:** Conceptualization, Data curation, Methodology, Formal analysis, Visualization, Writing - original draft, Writing - review & editing.

### Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

### Acknowledgements

We are indebted to three anonymous reviewers for their helpful comments and suggestions on a previous draft. We also would like to thank conference participants at the 2019 IAME Conference (Athens). This research was partly funded by the Research Council of Norway under the project 'Real Energy Efficiency and Emissions in the Seaway', project no. 255672/O80.

## Appendix A. Descriptive statistics

See Table A.

<table><caption>**Table A**

Descriptive statistics of the sample.</caption><thead><tr><th rowspan="2">Variables</th><th colspan="3">Aframax</th><th colspan="3">Suezmax</th></tr><tr><th>All</th><th>Ballast</th><th>Laden</th><th>All</th><th>Ballast</th><th>Laden</th></tr></thead><tbody><tr><td>Fuel consumption (log)</td><td>3,438</td><td>3,134</td><td>3,594</td><td>3,558</td><td>3,335</td><td>3,762</td></tr><tr><td>LOG Speed (log)</td><td>2,481</td><td>2,407</td><td>2,518</td><td>2,470</td><td>2,423</td><td>2,512</td></tr><tr><td>Sea current (difference in speed)</td><td>-0,012</td><td>-0,031</td><td>-0,002</td><td>0,007</td><td>0,000</td><td>0,014</td></tr><tr><td>Average draft</td><td>11,475</td><td>8,166</td><td>13,170</td><td>11,795</td><td>8,007</td><td>15,266</td></tr><tr><td>Trim</td><td>-0,608</td><td>-1,681</td><td>-0,059</td><td>-1,377</td><td>-2,845</td><td>-0,032</td></tr><tr><td>Headwind &lt; 10 k</td><td>0,079</td><td>0,101</td><td>0,068</td><td>0,033</td><td>0,025</td><td>0,039</td></tr><tr><td>Headwind 11-16 k</td><td>0,064</td><td>0,064</td><td>0,064</td><td>0,042</td><td>0,035</td><td>0,048</td></tr><tr><td>Headwind 17-21 k</td><td>0,042</td><td>0,036</td><td>0,045</td><td>0,033</td><td>0,019</td><td>0,046</td></tr><tr><td>Headwind 22-27 k</td><td>0,030</td><td>0,016</td><td>0,037</td><td>0,037</td><td>0,023</td><td>0,051</td></tr><tr><td>Headwind &gt; 27 k</td><td>0,014</td><td>0,004</td><td>0,019</td><td>0,021</td><td>0,009</td><td>0,031</td></tr><tr><td>Headwind 45° &lt; 10 k</td><td>0,100</td><td>0,122</td><td>0,089</td><td>0,078</td><td>0,086</td><td>0,072</td></tr><tr><td>Headwind 45° 11-16 k</td><td>0,097</td><td>0,108</td><td>0,091</td><td>0,104</td><td>0,098</td><td>0,110</td></tr><tr><td>Headwind 45° 17-21 k</td><td>0,054</td><td>0,060</td><td>0,051</td><td>0,080</td><td>0,065</td><td>0,093</td></tr><tr><td>Headwind 45° 22-27 k</td><td>0,031</td><td>0,028</td><td>0,032</td><td>0,046</td><td>0,033</td><td>0,058</td></tr><tr><td>Headwind 45° &gt; 27 k</td><td>0,012</td><td>0,006</td><td>0,015</td><td>0,025</td><td>0,021</td><td>0,029</td></tr><tr><td>Side &lt; 10 k</td><td>0,043</td><td>0,047</td><td>0,041</td><td>0,061</td><td>0,073</td><td>0,051</td></tr><tr><td>Side 11-16 k</td><td>0,045</td><td>0,050</td><td>0,043</td><td>0,066</td><td>0,067</td><td>0,065</td></tr><tr><td>Side 17-21 k</td><td>0,029</td><td>0,033</td><td>0,027</td><td>0,045</td><td>0,045</td><td>0,045</td></tr><tr><td>Side 22-27 k</td><td>0,014</td><td>0,016</td><td>0,013</td><td>0,016</td><td>0,021</td><td>0,012</td></tr><tr><td>Side &gt; 27 k</td><td>0,005</td><td>0,004</td><td>0,006</td><td>0,006</td><td>0,007</td><td>0,005</td></tr><tr><td>Wind astern 45° &lt; 10 k</td><td>0,064</td><td>0,066</td><td>0,063</td><td>0,064</td><td>0,077</td><td>0,051</td></tr><tr><td>Wind astern 45° 11-16 k</td><td>0,073</td><td>0,066</td><td>0,076</td><td>0,087</td><td>0,094</td><td>0,081</td></tr><tr><td>Wind astern 45° 17-21 k</td><td>0,047</td><td>0,044</td><td>0,048</td><td>0,050</td><td>0,060</td><td>0,041</td></tr><tr><td>Wind astern 45° 22-27 k</td><td>0,026</td><td>0,025</td><td>0,026</td><td>0,019</td><td>0,026</td><td>0,013</td></tr><tr><td>Wind astern 45° &gt; 27 k</td><td>0,010</td><td>0,008</td><td>0,011</td><td>0,006</td><td>0,005</td><td>0,006</td></tr><tr><td>Wind astern &lt; 10 k</td><td>0,037</td><td>0,035</td><td>0,038</td><td>0,024</td><td>0,031</td><td>0,016</td></tr><tr><td>Wind astern 11-16 k</td><td>0,040</td><td>0,026</td><td>0,047</td><td>0,029</td><td>0,036</td><td>0,022</td></tr><tr><td>Wind astern 17-21 k</td><td>0,027</td><td>0,019</td><td>0,030</td><td>0,019</td><td>0,028</td><td>0,011</td></tr><tr><td>Wind astern 22-27 k</td><td>0,013</td><td>0,010</td><td>0,014</td><td>0,008</td><td>0,014</td><td>0,003</td></tr><tr><td>Wind astern &gt; 27 k</td><td>0,004</td><td>0,006</td><td>0,004</td><td>0,001</td><td>0,002</td><td>0,000</td></tr><tr><td>Swell low or less</td><td>0,353</td><td>0,386</td><td>0,337</td><td>0,324</td><td>0,358</td><td>0,293</td></tr><tr><td>Swell light</td><td>0,275</td><td>0,283</td><td>0,271</td><td>0,213</td><td>0,232</td><td>0,197</td></tr></tbody></table>

(continued on next page)
<table><caption>**Table A (continued)**</caption><thead><tr><th rowspan="2">Variables</th><th colspan="3">Aframax</th><th colspan="3">Suezmax</th></tr><tr><th>All</th><th>Ballast</th><th>Laden</th><th>All</th><th>Ballast</th><th>Laden</th></tr></thead><tbody><tr><td>Swell moderate</td><td>0,226</td><td>0,216</td><td>0,231</td><td>0,334</td><td>0,317</td><td>0,350</td></tr><tr><td>Swell moderate rough</td><td>0,084</td><td>0,073</td><td>0,090</td><td>0,080</td><td>0,059</td><td>0,100</td></tr><tr><td>Swell rough</td><td>0,048</td><td>0,037</td><td>0,054</td><td>0,036</td><td>0,024</td><td>0,047</td></tr><tr><td>Swell high</td><td>0,013</td><td>0,005</td><td>0,017</td><td>0,012</td><td>0,010</td><td>0,013</td></tr><tr><td>Wave height &lt; 0.5 m</td><td>0,171</td><td>0,203</td><td>0,154</td><td>0,160</td><td>0,188</td><td>0,134</td></tr><tr><td>Wave height 0.5–1.25 m</td><td>0,373</td><td>0,389</td><td>0,364</td><td>0,345</td><td>0,362</td><td>0,328</td></tr><tr><td>Wave height 1.25–2.5 m</td><td>0,278</td><td>0,264</td><td>0,285</td><td>0,329</td><td>0,326</td><td>0,332</td></tr><tr><td>Wave height 2.5–4 m</td><td>0,134</td><td>0,114</td><td>0,143</td><td>0,124</td><td>0,096</td><td>0,150</td></tr><tr><td>Wave height 4–6 m</td><td>0,039</td><td>0,029</td><td>0,044</td><td>0,036</td><td>0,022</td><td>0,048</td></tr><tr><td>Wave height 6–9 m</td><td>0,007</td><td>0,002</td><td>0,009</td><td>0,007</td><td>0,006</td><td>0,008</td></tr><tr><td>Number of observations</td><td>7600</td><td>2575</td><td>5025</td><td>4325</td><td>2068</td><td>2257</td></tr></tbody></table>

Source: data from a tanker company, authors' calculations.

<caption>**Table B**</caption>

Sensitivity of model fit to estimation technique.

<table><thead><tr><th rowspan="2">Estimation technique</th><th rowspan="2"></th><th colspan="3">Aframax</th><th colspan="3">Suezmax</th></tr><tr><th>(1A)<br>All</th><th>(1B)<br>Ballast</th><th>(1C)<br>Laden</th><th>(2A)<br>All</th><th>(2B)<br>Ballast</th><th>(2C)<br>Laden</th></tr></thead><tbody><tr><td colspan="8"><strong>A. Linear regression</strong></td></tr><tr><td rowspan="3">OLS</td><td>R<sup>2</sup></td><td>0.824</td><td>0.797</td><td>0.688</td><td>0.873</td><td>0.839</td><td>0.862</td></tr><tr><td>RMSE</td><td>0.156</td><td>0.183</td><td>0.129</td><td>0.151</td><td>0.175</td><td>0.107</td></tr><tr><td>MAE</td><td>0.110</td><td>0.130</td><td>0.092</td><td>0.104</td><td>0.122</td><td>0.077</td></tr><tr><td colspan="8"><strong>B. Boosting</strong></td></tr><tr><td rowspan="3">One interaction</td><td>R<sup>2</sup></td><td>0.763</td><td>0.724</td><td>0.487</td><td>0.767</td><td>0.747</td><td>0.574</td></tr><tr><td>RMSE</td><td>0.169</td><td>0.191</td><td>0.149</td><td>0.179</td><td>0.202</td><td>0.161</td></tr><tr><td>MAE</td><td>0.128</td><td>0.142</td><td>0.113</td><td>0.131</td><td>0.145</td><td>0.122</td></tr><tr><td rowspan="3">Two interactions</td><td>R<sup>2</sup></td><td>0.763</td><td>0.726</td><td>0.517</td><td>0.772</td><td>0.764</td><td>0.579</td></tr><tr><td>RMSE</td><td>0.160</td><td>0.184</td><td>0.138</td><td>0.167</td><td>0.184</td><td>0.151</td></tr><tr><td>MAE</td><td>0.118</td><td>0.135</td><td>0.104</td><td>0.122</td><td>0.132</td><td>0.113</td></tr><tr><td rowspan="3">Three interactions</td><td>R<sup>2</sup></td><td>0.759</td><td>0.721</td><td>0.522</td><td>0.769</td><td>0.775</td><td>0.584</td></tr><tr><td>RMSE</td><td>0.161</td><td>0.179</td><td>0.136</td><td>0.165</td><td>0.174</td><td>0.143</td></tr><tr><td>MAE</td><td>0.118</td><td>0.130</td><td>0.103</td><td>0.120</td><td>0.124</td><td>0.103</td></tr><tr><td rowspan="3">Four interactions</td><td>R<sup>2</sup></td><td>0.760</td><td>0.714</td><td>0.520</td><td>0.770</td><td>0.778</td><td>0.583</td></tr><tr><td>RMSE</td><td>0.159</td><td>0.182</td><td>0.133</td><td>0.167</td><td>0.168</td><td>0.141</td></tr><tr><td>MAE</td><td>0.117</td><td>0.132</td><td>0.100</td><td>0.122</td><td>0.119</td><td>0.100</td></tr><tr><td rowspan="3">Five interactions</td><td>R<sup>2</sup></td><td>0.758</td><td>0.710</td><td>0.521</td><td>0.773</td><td>0.779</td><td>0.586</td></tr><tr><td>RMSE</td><td>0.159</td><td>0.179</td><td>0.127</td><td>0.166</td><td>0.165</td><td>0.140</td></tr><tr><td>MAE</td><td>0.117</td><td>0.129</td><td>0.094</td><td>0.122</td><td>0.115</td><td>0.099</td></tr><tr><td colspan="8"><strong>C. Backpropagation network</strong></td></tr><tr><td rowspan="3">One hidden layer</td><td>R<sup>2</sup></td><td>0.832</td><td>0.824</td><td>0.666</td><td>0.864</td><td>0.826</td><td>0.809</td></tr><tr><td>RMSE</td><td>0.153</td><td>0.170</td><td>0.133</td><td>0.156</td><td>0.182</td><td>0.125</td></tr><tr><td>MAE</td><td>0.109</td><td>0.124</td><td>0.097</td><td>0.112</td><td>0.130</td><td>0.096</td></tr><tr><td rowspan="3">Two hidden layers</td><td>R<sup>2</sup></td><td>0.863</td><td>0.846</td><td>0.744</td><td>0.879</td><td>0.861</td><td>0.874</td></tr><tr><td>RMSE</td><td>0.138</td><td>0.159</td><td>0.117</td><td>0.147</td><td>0.163</td><td>0.102</td></tr><tr><td>MAE</td><td>0.098</td><td>0.116</td><td>0.085</td><td>0.107</td><td>0.115</td><td>0.077</td></tr><tr><td rowspan="3">Three hidden layers</td><td>R<sup>2</sup></td><td>0.869</td><td>0.859</td><td>0.772</td><td>0.901</td><td>0.875</td><td>0.885</td></tr><tr><td>RMSE</td><td>0.135</td><td>0.152</td><td>0.110</td><td>0.132</td><td>0.154</td><td>0.097</td></tr><tr><td>MAE</td><td>0.098</td><td>0.112</td><td>0.080</td><td>0.093</td><td>0.109</td><td>0.073</td></tr><tr><td rowspan="3">Four hidden layers</td><td>R<sup>2</sup></td><td>0.872</td><td>0.873</td><td>0.759</td><td>0.893</td><td>0.884</td><td>0.883</td></tr><tr><td>RMSE</td><td>0.133</td><td>0.145</td><td>0.113</td><td>0.139</td><td>0.148</td><td>0.098</td></tr><tr><td>MAE</td><td>0.095</td><td>0.105</td><td>0.084</td><td>0.098</td><td>0.105</td><td>0.075</td></tr><tr><td rowspan="3">Five hidden layers</td><td>R<sup>2</sup></td><td>0.854</td><td>0.879</td><td>0.780</td><td>0.907</td><td>0.890</td><td>0.903</td></tr><tr><td>RMSE</td><td>0.142</td><td>0.141</td><td>0.108</td><td>0.129</td><td>0.144</td><td>0.089</td></tr><tr><td>MAE</td><td>0.109</td><td>0.103</td><td>0.081</td><td>0.090</td><td>0.103</td><td>0.068</td></tr></tbody></table>

Source: data from a tanker company, authors' calculations.

Note: The sample is restricted to average daily speeds comprised between 6 and 16 knots. Controls include speed, average draft, trim, sea current, wind direction crossed with wind force, swell, wave height, vessel fixed effects and vessel-specific time trend.
<table><caption>**Table A (continued)**</caption><thead><tr><th rowspan="2">Variables</th><th colspan="3">Aframax</th><th colspan="3">Suezmax</th></tr><tr><th>All</th><th>Ballast</th><th>Laden</th><th>All</th><th>Ballast</th><th>Laden</th></tr></thead><tbody><tr><td>Swell moderate</td><td>0,226</td><td>0,216</td><td>0,231</td><td>0,334</td><td>0,317</td><td>0,350</td></tr><tr><td>Swell moderate rough</td><td>0,084</td><td>0,073</td><td>0,090</td><td>0,080</td><td>0,059</td><td>0,100</td></tr><tr><td>Swell rough</td><td>0,048</td><td>0,037</td><td>0,054</td><td>0,036</td><td>0,024</td><td>0,047</td></tr><tr><td>Swell high</td><td>0,013</td><td>0,005</td><td>0,017</td><td>0,012</td><td>0,010</td><td>0,013</td></tr><tr><td>Wave height &lt; 0.5 m</td><td>0,171</td><td>0,203</td><td>0,154</td><td>0,160</td><td>0,188</td><td>0,134</td></tr><tr><td>Wave height 0.5–1.25 m</td><td>0,373</td><td>0,389</td><td>0,364</td><td>0,345</td><td>0,362</td><td>0,328</td></tr><tr><td>Wave height 1.25–2.5 m</td><td>0,278</td><td>0,264</td><td>0,285</td><td>0,329</td><td>0,326</td><td>0,332</td></tr><tr><td>Wave height 2.5–4 m</td><td>0,134</td><td>0,114</td><td>0,143</td><td>0,124</td><td>0,096</td><td>0,150</td></tr><tr><td>Wave height 4–6 m</td><td>0,039</td><td>0,029</td><td>0,044</td><td>0,036</td><td>0,022</td><td>0,048</td></tr><tr><td>Wave height 6–9 m</td><td>0,007</td><td>0,002</td><td>0,009</td><td>0,007</td><td>0,006</td><td>0,008</td></tr><tr><td>Number of observations</td><td>7600</td><td>2575</td><td>5025</td><td>4325</td><td>2068</td><td>2257</td></tr></tbody></table>

Source: data from a tanker company, authors' calculations.

**Table B**

Sensitivity of model fit to estimation technique.

<table><thead><tr><th rowspan="2">Estimation technique</th><th rowspan="2"></th><th colspan="3">Aframax</th><th colspan="3">Suezmax</th></tr><tr><th>(1A)<br>All</th><th>(1B)<br>Ballast</th><th>(1C)<br>Laden</th><th>(2A)<br>All</th><th>(2B)<br>Ballast</th><th>(2C)<br>Laden</th></tr></thead><tbody><tr><td colspan="8"><strong>A. Linear regression</strong></td></tr><tr><td rowspan="3">OLS</td><td>R<sup>2</sup></td><td>0.824</td><td>0.797</td><td>0.688</td><td>0.873</td><td>0.839</td><td>0.862</td></tr><tr><td>RMSE</td><td>0.156</td><td>0.183</td><td>0.129</td><td>0.151</td><td>0.175</td><td>0.107</td></tr><tr><td>MAE</td><td>0.110</td><td>0.130</td><td>0.092</td><td>0.104</td><td>0.122</td><td>0.077</td></tr><tr><td colspan="8"><strong>B. Boosting</strong></td></tr><tr><td rowspan="3">One interaction</td><td>R<sup>2</sup></td><td>0.763</td><td>0.724</td><td>0.487</td><td>0.767</td><td>0.747</td><td>0.574</td></tr><tr><td>RMSE</td><td>0.169</td><td>0.191</td><td>0.149</td><td>0.179</td><td>0.202</td><td>0.161</td></tr><tr><td>MAE</td><td>0.128</td><td>0.142</td><td>0.113</td><td>0.131</td><td>0.145</td><td>0.122</td></tr><tr><td rowspan="3">Two interactions</td><td>R<sup>2</sup></td><td>0.763</td><td>0.726</td><td>0.517</td><td>0.772</td><td>0.764</td><td>0.579</td></tr><tr><td>RMSE</td><td>0.160</td><td>0.184</td><td>0.138</td><td>0.167</td><td>0.184</td><td>0.151</td></tr><tr><td>MAE</td><td>0.118</td><td>0.135</td><td>0.104</td><td>0.122</td><td>0.132</td><td>0.113</td></tr><tr><td rowspan="3">Three interactions</td><td>R<sup>2</sup></td><td>0.759</td><td>0.721</td><td>0.522</td><td>0.769</td><td>0.775</td><td>0.584</td></tr><tr><td>RMSE</td><td>0.161</td><td>0.179</td><td>0.136</td><td>0.165</td><td>0.174</td><td>0.143</td></tr><tr><td>MAE</td><td>0.118</td><td>0.130</td><td>0.103</td><td>0.120</td><td>0.124</td><td>0.103</td></tr><tr><td rowspan="3">Four interactions</td><td>R<sup>2</sup></td><td>0.760</td><td>0.714</td><td>0.520</td><td>0.770</td><td>0.778</td><td>0.583</td></tr><tr><td>RMSE</td><td>0.159</td><td>0.182</td><td>0.133</td><td>0.167</td><td>0.168</td><td>0.141</td></tr><tr><td>MAE</td><td>0.117</td><td>0.132</td><td>0.100</td><td>0.122</td><td>0.119</td><td>0.100</td></tr><tr><td rowspan="3">Five interactions</td><td>R<sup>2</sup></td><td>0.758</td><td>0.710</td><td>0.521</td><td>0.773</td><td>0.779</td><td>0.586</td></tr><tr><td>RMSE</td><td>0.159</td><td>0.179</td><td>0.127</td><td>0.166</td><td>0.165</td><td>0.140</td></tr><tr><td>MAE</td><td>0.117</td><td>0.129</td><td>0.094</td><td>0.122</td><td>0.115</td><td>0.099</td></tr><tr><td colspan="8"><strong>C. Backpropagation network</strong></td></tr><tr><td rowspan="3">One hidden layer</td><td>R<sup>2</sup></td><td>0.832</td><td>0.824</td><td>0.666</td><td>0.864</td><td>0.826</td><td>0.809</td></tr><tr><td>RMSE</td><td>0.153</td><td>0.170</td><td>0.133</td><td>0.156</td><td>0.182</td><td>0.125</td></tr><tr><td>MAE</td><td>0.109</td><td>0.124</td><td>0.097</td><td>0.112</td><td>0.130</td><td>0.096</td></tr><tr><td rowspan="3">Two hidden layers</td><td>R<sup>2</sup></td><td>0.863</td><td>0.846</td><td>0.744</td><td>0.879</td><td>0.861</td><td>0.874</td></tr><tr><td>RMSE</td><td>0.138</td><td>0.159</td><td>0.117</td><td>0.147</td><td>0.163</td><td>0.102</td></tr><tr><td>MAE</td><td>0.098</td><td>0.116</td><td>0.085</td><td>0.107</td><td>0.115</td><td>0.077</td></tr><tr><td rowspan="3">Three hidden layers</td><td>R<sup>2</sup></td><td>0.869</td><td>0.859</td><td>0.772</td><td>0.901</td><td>0.875</td><td>0.885</td></tr><tr><td>RMSE</td><td>0.135</td><td>0.152</td><td>0.110</td><td>0.132</td><td>0.154</td><td>0.097</td></tr><tr><td>MAE</td><td>0.098</td><td>0.112</td><td>0.080</td><td>0.093</td><td>0.109</td><td>0.073</td></tr><tr><td rowspan="3">Four hidden layers</td><td>R<sup>2</sup></td><td>0.872</td><td>0.873</td><td>0.759</td><td>0.893</td><td>0.884</td><td>0.883</td></tr><tr><td>RMSE</td><td>0.133</td><td>0.145</td><td>0.113</td><td>0.139</td><td>0.148</td><td>0.098</td></tr><tr><td>MAE</td><td>0.095</td><td>0.105</td><td>0.084</td><td>0.098</td><td>0.105</td><td>0.075</td></tr><tr><td rowspan="3">Five hidden layers</td><td>R<sup>2</sup></td><td>0.854</td><td>0.879</td><td>0.780</td><td>0.907</td><td>0.890</td><td>0.903</td></tr><tr><td>RMSE</td><td>0.142</td><td>0.141</td><td>0.108</td><td>0.129</td><td>0.144</td><td>0.089</td></tr><tr><td>MAE</td><td>0.109</td><td>0.103</td><td>0.081</td><td>0.090</td><td>0.103</td><td>0.068</td></tr></tbody></table>

Source: data from a tanker company, authors' calculations.

Note: The sample is restricted to average daily speeds comprised between 6 and 16 knots. Controls include speed, average draft, trim, sea current, wind direction crossed with wind force, swell, wave height, vessel fixed effects and vessel-specific time trend.
## Appendix B. Regression versus machine learning

In recent years, other estimation techniques have been adopted to predict ship fuel consumption under different sailing speed and sea and weather conditions. In our framework, the parametric specification estimated using linear regressions is motivated by the underlying cubic function and weather conditions are used as additional controls without attempting to find necessarily the best fit of the data. By contrast, data mining techniques focus on prediction and provide more complicated models that capture additional details in the relationship between the dependent variable and the explanatory variables. In this appendix, we compare the fit of three different estimation tools applied to our dataset: regression models, boosting, and neural networks.

We consider two different machine learning techniques. The first one is boosting which may be viewed as an ensemble method (Freund and Schapire, 1997; Friedman, 2001; Hastie et al., 2001). Boosting applies to any generalized linear models and the different covariates are included without specifying the functional relationship to the dependent variable. The boosting algorithm has been proved to have a very high predictive accuracy (Friedman et al., 2000). In boosting, each term of the model consists of a tree and the number of splits used for fitting each regression tree has to be chosen. The second is based on artificial neural networks which correspond to a collection of connected units (artificial neurons). Each connection (edge) is able to transmit a signal to the other neurons and neurons are aggregated in subgroups called layers. A backpropagation algorithm, which computes the gradient of the loss function, is used in the training of the network (Rumelhart et al., 1986). The introduction of hidden layers, which lie between input and output layers, helps the network to exhibit non-linear behavior (Banati et al., 2018).

Table B1 reports the coefficient of determination ($R^2$), the root mean square error (RMSE) and the mean absolute error (MAE) obtained from our different estimation techniques for various samples. Panel A corresponds to the OLS specification shown in column (1C) and (2C) of Table 4. Panel B presents estimates from boosting with up to five-way interactions. Panel C reports estimates from a backpropagation network with up to five hidden layers.

Both for Aframax and Suezmax, more than 80% of variation in fuel consumption is explained by the linear regression (panel A). Interestingly, the fit of the boosting model is slightly lower than that of OLS models: the $R^2$ is up to 0.763 for instance for Aframax (panel B). As the boosting $R^2$ is computed on a test dataset, we recalculate the OLS $R^2$ using the same test dataset only, but reach very similar conclusion compared to using the whole dataset (with an $R^2$ of 0.8 for Aframax). Finally, as shown in panel C, the best fit of

<figure><img src="image_8.png" /><figcaption>**Fig. B.** Calibration plots for linear regressions, boosting and backpropagation network (Aframax), Source: Data from a tanker company, authors' calculations. Note: Comparison of actual and fitted values from linear regression, boosting (five interactions) and backpropagation network (five hidden layers). The sample is restricted to Aframax with average daily speeds between 6 and 16 knots. Controls include speed, average draft, trim, sea current, wind direction crossed with wind force, swell, wave height, vessel fixed effects and vessel-specific time trend.</figcaption></figure>
## Appendix B. Regression versus machine learning

In recent years, other estimation techniques have been adopted to predict ship fuel consumption under different sailing speed and sea and weather conditions. In our framework, the parametric specification estimated using linear regressions is motivated by the underlying cubic function and weather conditions are used as additional controls without attempting to find necessarily the best fit of the data. By contrast, data mining techniques focus on prediction and provide more complicated models that capture additional details in the relationship between the dependent variable and the explanatory variables. In this appendix, we compare the fit of three different estimation tools applied to our dataset: regression models, boosting, and neural networks.

We consider two different machine learning techniques. The first one is boosting which may be viewed as an ensemble method (Freund and Schapire, 1997; Friedman, 2001; Hastie et al., 2001). Boosting applies to any generalized linear models and the different covariates are included without specifying the functional relationship to the dependent variable. The boosting algorithm has been proved to have a very high predictive accuracy (Friedman et al., 2000). In boosting, each term of the model consists of a tree and the number of splits used for fitting each regression tree has to be chosen. The second is based on artificial neural networks which correspond to a collection of connected units (artificial neurons). Each connection (edge) is able to transmit a signal to the other neurons and neurons are aggregated in subgroups called layers. A backpropagation algorithm, which computes the gradient of the loss function, is used in the training of the network (Rumelhart et al., 1986). The introduction of hidden layers, which lie between input and output layers, helps the network to exhibit non-linear behavior (Banati et al., 2018).

Table B1 reports the coefficient of determination ($R^2$), the root mean square error (RMSE) and the mean absolute error (MAE) obtained from our different estimation techniques for various samples. Panel A corresponds to the OLS specification shown in column (1C) and (2C) of Table 4. Panel B presents estimates from boosting with up to five-way interactions. Panel C reports estimates from a backpropagation network with up to five hidden layers.

Both for Aframax and Suezmax, more than 80% of variation in fuel consumption is explained by the linear regression (panel A). Interestingly, the fit of the boosting model is slightly lower than that of OLS models: the $R^2$ is up to 0.763 for instance for Aframax (panel B). As the boosting $R^2$ is computed on a test dataset, we recalculate the OLS $R^2$ using the same test dataset only, but reach very similar conclusion compared to using the whole dataset (with an $R^2$ of 0.8 for Aframax). Finally, as shown in panel C, the best fit of

<figure><img src="image_10.png" /><figcaption>**Fig. B.** Calibration plots for linear regressions, boosting and backpropagation network (Aframax), Source: Data from a tanker company, authors' calculations. Note: Comparison of actual and fitted values from linear regression, boosting (five interactions) and backpropagation network (five hidden layers). The sample is restricted to Aframax with average daily speeds between 6 and 16 knots. Controls include speed, average draft, trim, sea current, wind direction crossed with wind force, swell, wave height, vessel fixed effects and vessel-specific time trend.</figcaption></figure>
the data is obtained using the backpropagation algorithm. For Aframax, the improvement in the $R^2$ is up to 5.8% with four hidden layers (0.872/0.824 = 1.058). For Suezmax, the improvement is slightly lower, equal to 3.9% with five hidden layers.

As the $R^2$ may be more suitable to evaluate the fitting performance in linear regression models, it is important to further compare the models with respect to RMSE and MAE. When considering Aframax (column 1), the RMSE is equal to 0.156 with OLS. Similar values are obtained with boosting, but they are lower with the backpropagation algorithm since the decrease is up to 15% with four hidden layers (0.133/0.156 = 0.853). For Suezmax, the RMSE is 0.151 with OLS, slightly higher with boosting and around 17% lower when using the backpropagation algorithm with five hidden layers (0.129/0/0156 = 0.827). When considering MAE, we also find a better fit with the backpropagation algorithm. The improvement is around 14% for Aframax and 12% for Suezmax.

Overall, this comparison suggests that our estimation strategy based on linear regressions perform reasonably well compared to machine learning techniques. Finally, to assess whether any other models is better calibrated, we present in Fig. B1 plots showing fitted versus actual values for both linear regressions, boosting and backpropagation network. Compared to machine learning techniques, we note that there is more distance between actual and fitted values for a few observations located both in the lower and upper part of the distribution of observed fuel consumption values, but the different calibration plots remain nonetheless very similar.

## Appendix C. Supplementary material

Supplementary data to this article can be found online at https://doi.org/10.1016/j.tre.2020.101972.

## References

Adland, R., Cariou, P., Wolff, F.-C., 2018. The energy efficiency effects of periodic ship hull cleaning. J. Cleaner Prod. 178, 1–13.

Adland, R., Jia, H., 2018. Dynamic speed choice in bulk shipping. Maritime Econ. Logist. 20 (2), 253–266.

Adland, R., Jia, H., 2016. Vessel speed analytics using satellite-based ship position data. In: Proceedings of the 2016 IEEE IEM Conference, Bali, pp. 1299–1303. https://doi.org/10.1109/IEEM.2016.7798088.

Aldous, L., Smith, T., Buchnall, R., 2013. Noon report data uncertainty. In: Low Carbon Shipping Conference, London, September.

Banati, H., Metha, S., Kaur, P., 2018. Nature-inspired Algorithms for Big Data Frameworks. IGI Global.

Besikçi, E.B., Arslan, O., Turan, O., Ölçer, A.I., 2016. An artificial neural network based decision support system for energy efficient ship operations. Comput. Oper. Res. 66, 393–401.

Cariou, P., 2011. Is slow steaming a sustainable means of reducing CO2 emissions from container shipping? Transp. Res. Part D 16, 260–264.

Cariou, P., Parola, F., Notteboom, T., 2019. Towards low carbon global supply chains: a multi-trade analysis of CO2 emission reductions in container shipping. Int. J. Prod. Econ. 208, 17–28.

Comer, B., Chen, C., Rutherford, D., 2018. Relating short-term measures to IMO's minimum 2050 emissions reduction target. In: Appendix to paper MEPC 73/INF.27 presented to IMO Marine Environment Protection Committee. 73rd session, London.

Cox, D., Hansen, B.E., Jimenez, E., 2004. How responsive are private transfers to income? Evidence from a laissez-faire economy. J. Public Econ. 88, 2193–2219.

Coraddu, A., Oneto, L., Baldi, F., Anguita, D., 2018. Vessels fuel consumption: a data analytics perspective to sustainability. In: Cruz Corona C. (Ed.), Soft Computing for Sustainability Science. Studies in Fuzziness and Soft Computing, vol. 358. Springer, Cham.

Du, Y., Meng, Q., Wang, S., Kuang, H., 2019. Two-phase optimal solutions for ship speed and trim optimization over a voyage using voyage report data. Transp. Res. Part B 122, 88–114.

Evans, J., Marlow, P., 1990. Quantitative Methods in Maritime Economics. Fairplays Publication.

Faber, J., Huigen, T., Nelissen, D., 2017. Regulating Speed: A Short- term Measure to Reduce Maritime GHG Emissions. CE Delft Publication, Netherlands.

Faber, J., Nelissen, D., Hon, G., Wang, H., Tsimplis, M., 2012. Regulated Slow Steaming in Maritime Transport: An Assessment of Options, Costs and Benefits. CE Delft, Delft.

Freund, Y., Schapire, R.E., 1997. A decision-theoretic generalization of online learning and an application to boosting. J. Comput. Syst. Sci. 55, 119–139.

Friedman, R., 2001. Greedy function approximation: a gradient boosting machine. Ann. Stat. 29, 1189–1232.

Friedman, J., Hastie, T., Tibshirani, R., 2000. Additive logistic regression: a statistical view of boosting. Ann. Stat. 28, 337–407.

Gkonis, K.G., Psaraftis, H.N., 2012. Modelling tankers' optimal speed and emissions. In: SNAME 2012 Transactions 120. pp. 90–115.

Hastie, T., Tibshirani, R., Friedman, J., 2001. The Elements of Statistical Learning. Springer, New York.

He, Q., Zhang, X., Nip, K., 2017. Speed optimization over a path with heterogeneous arc costs. Transp. Res. Part B 104, 198–214.

IMO, 2014. Third IMO Greenhouse Gas Study. International Maritime Organization, London.

Jeon, M., Noh, Y., Shin, Y., Lim, O.-K., Cho, D., 2018. Prediction of ship fuel consumption by using an artificial neural network. J. Mech. Sci. Technol. 32 (12), 5785–5796.

Kristensen, H.O., 2019. Evaluation of Different Measures for Reduction of Green-house gas (GHG) Emissions for Different Ship Types. HOK Manrinconsult, Danish Technical University. https://gitlab.gbar.dtu.dk/oceanwave3d/Ship-Desmo.

Leaper, R., 2019. The role of slower vessel speeds in reducing greenhouse gas emissions, underwater noise and collision risk to whales. Front. Mar. Sci. 6, 505.

Leifsson, L.P., Sævarsdóttir, H., Sigurðsson, S.Þ., Vésteinsson, A., 2008. Grey-box modeling of an ocean vessel for operational optimization. Simul. Model. Pract. Theory 16 (8), 923–932.

Lepore, A., dos Reis, M.S., Palumbo, B., Rendall, R., Capezza, C., 2017. A comparison of advanced regression techniques for predicting ship CO2 emissions. Qual. Reliab. Eng. Int. 33 (6), 1281–1292.

Lewis, E.D., 1988. Principles of Naval Architecture, vol. II The Society of Naval Architects and Marine Engineers.

Lindstad, H., Eskeland, G.S., Psaraftis, H., Sandaas, I., Strømman, A.H., 2015. Maritime shipping and emissions: a three-layered, damage-based approach. Ocean Eng. 110, 94–101.

Lindstad, H., Asbjørnslett, B., Strømman, A., 2011. Reductions in greenhouse gas emissions and cost by shipping at lower speeds. Energy Policy 39, 3456–3464.

Lindstad, H., Asbjørnslett, B.E., Jullumstrø, E., 2013. Assessment of profit, cost and emissions by varying speed as a function of sea conditions and freight market. Transp. Res. Part D 19, 5–12.

Lloyd, A.R.J.M, 1989. Seakeeping, Ship Behaviour in Rough Weather. Ellis Horwood.

Lo, H.K., McCord, M.R., 1995. Routing through dynamic ocean currents: general heuristics and empirical results in the Gulf Stream region. Transp. Res. Part B 29 (2), 109–124.

MAN Diesel & Turbo, 2004. Basic principles of ship propulsion. http://www.mandieselturbo.com/files/news/filesof5405/5510_004_02%20low.pdf.

Meng, Q., Du, Y., Wang, Y., 2016. Shipping log data based container ship fuel efficiency modeling. Transp. Res. Part B 83, 207–229.

Ng, M., 2019. Vessel speed optimisation in container shipping: A new look. J. Oper. Res. Soc. 70 (4), 541–547.

Notteboom, T., Cariou, P., 2009. Fuel surcharge practices of container shipping lines: is it about cost recovery or revenue making? The 2009 International Association
the data is obtained using the backpropagation algorithm. For Aframax, the improvement in the $R^2$ is up to 5.8% with four hidden layers (0.872/0.824 = 1.058). For Suezmax, the improvement is slightly lower, equal to 3.9% with five hidden layers.

As the $R^2$ may be more suitable to evaluate the fitting performance in linear regression models, it is important to further compare the models with respect to RMSE and MAE. When considering Aframax (column 1), the RMSE is equal to 0.156 with OLS. Similar values are obtained with boosting, but they are lower with the backpropagation algorithm since the decrease is up to 15% with four hidden layers (0.133/0.156 = 0.853). For Suezmax, the RMSE is 0.151 with OLS, slightly higher with boosting and around 17% lower when using the backpropagation algorithm with five hidden layers (0.129/0/0156 = 0.827). When considering MAE, we also find a better fit with the backpropagation algorithm. The improvement is around 14% for Aframax and 12% for Suezmax.

Overall, this comparison suggests that our estimation strategy based on linear regressions perform reasonably well compared to machine learning techniques. Finally, to assess whether any other models is better calibrated, we present in Fig. B1 plots showing fitted versus actual values for both linear regressions, boosting and backpropagation network. Compared to machine learning techniques, we note that there is more distance between actual and fitted values for a few observations located both in the lower and upper part of the distribution of observed fuel consumption values, but the different calibration plots remain nonetheless very similar.

## Appendix C. Supplementary material

Supplementary data to this article can be found online at https://doi.org/10.1016/j.tre.2020.101972.

## References

Adland, R., Cariou, P., Wolff, F.-C., 2018. The energy efficiency effects of periodic ship hull cleaning. J. Cleaner Prod. 178, 1–13.

Adland, R., Jia, H., 2018. Dynamic speed choice in bulk shipping. Maritime Econ. Logist. 20 (2), 253–266.

Adland, R., Jia, H., 2016. Vessel speed analytics using satellite-based ship position data. In: Proceedings of the 2016 IEEE IEM Conference, Bali, pp. 1299–1303. https://doi.org/10.1109/IEEM.2016.7798088.

Aldous, L., Smith, T., Buchnall, R., 2013. Noon report data uncertainty. In: Low Carbon Shipping Conference, London, September.

Banati, H., Metha, S., Kaur, P., 2018. Nature-inspired Algorithms for Big Data Frameworks. IGI Global.

Besikçi, E.B., Arslan, O., Turan, O., Ölçer, A.I., 2016. An artificial neural network based decision support system for energy efficient ship operations. Comput. Oper. Res. 66, 393–401.

Cariou, P., 2011. Is slow steaming a sustainable means of reducing CO2 emissions from container shipping? Transp. Res. Part D 16, 260–264.

Cariou, P., Parola, F., Notteboom, T., 2019. Towards low carbon global supply chains: a multi-trade analysis of CO₂ emission reductions in container shipping. Int. J. Prod. Econ. 208, 17–28.

Comer, B., Chen, C., Rutherford, D., 2018. Relating short-term measures to IMO's minimum 2050 emissions reduction target. In: Appendix to paper MEPC 73/INF.27 presented to IMO Marine Environment Protection Committee. 73rd session, London.

Cox, D., Hansen, B.E., Jimenez, E., 2004. How responsive are private transfers to income? Evidence from a laissez-faire economy. J. Public Econ. 88, 2193–2219.

Coraddu, A., Oneto, L., Baldi, F., Anguita, D., 2018. Vessels fuel consumption: a data analytics perspective to sustainability. In: Cruz Corona C. (Ed.), Soft Computing for Sustainability Science. Studies in Fuzziness and Soft Computing, vol. 358. Springer, Cham.

Du, Y., Meng, Q., Wang, S., Kuang, H., 2019. Two-phase optimal solutions for ship speed and trim optimization over a voyage using voyage report data. Transp. Res. Part B 122, 88–114.

Evans, J., Marlow, P., 1990. Quantitative Methods in Maritime Economics. Fairplays Publication.

Faber, J., Huigen, T., Nelissen, D., 2017. Regulating Speed: A Short-term Measure to Reduce Maritime GHG Emissions. CE Delft Publication, Netherlands.

Faber, J., Nelissen, D., Hon, G., Wang, H., Tsimplis, M., 2012. Regulated Slow Steaming in Maritime Transport: An Assessment of Options, Costs and Benefits. CE Delft, Delft.

Freund, Y., Schapire, R.E., 1997. A decision-theoretic generalization of online learning and an application to boosting. J. Comput. Syst. Sci. 55, 119–139.

Friedman, R., 2001. Greedy function approximation: a gradient boosting machine. Ann. Stat. 29, 1189–1232.

Friedman, J., Hastie, T., Tibshirani, R., 2000. Additive logistic regression: a statistical view of boosting. Ann. Stat. 28, 337–407.

Gkonis, K.G., Psaraftis, H.N., 2012. Modelling tankers' optimal speed and emissions. In: SNAME 2012 Transactions 120. pp. 90–115.

Hastie, T., Tibshirani, R., Friedman, J., 2001. The Elements of Statistical Learning. Springer, New York.

He, Q., Zhang, X., Nip, K., 2017. Speed optimization over a path with heterogeneous arc costs. Transp. Res. Part B 104, 198–214.

IMO, 2014. Third IMO Greenhouse Gas Study. International Maritime Organization, London.

Jeon, M., Noh, Y., Shin, Y., Lim, O.-K., Cho, D., 2018. Prediction of ship fuel consumption by using an artificial neural network. J. Mech. Sci. Technol. 32 (12), 5785–5796.

Kristensen, H.O., 2019. Evaluation of Different Measures for Reduction of Green-house gas (GHG) Emissions for Different Ship Types. HOK Manrinconsult, Danish Technical University. https://gitlab.gbar.dtu.dk/oceanwave3d/Ship-Desmo.

Leaper, R., 2019. The role of slower vessel speeds in reducing greenhouse gas emissions, underwater noise and collision risk to whales. Front. Mar. Sci. 6, 505.

Leifsson, L.P., Sævarsdóttir, H., Sigurðsson, S.P., Vésteinsson, A., 2008. Grey-box modeling of an ocean vessel for operational optimization. Simul. Model. Pract. Theory 16 (8), 923–932.

Lepore, A., dos Reis, M.S., Palumbo, B., Rendall, R., Capezza, C., 2017. A comparison of advanced regression techniques for predicting ship CO2 emissions. Qual. Reliab. Eng. Int. 33 (6), 1281–1292.

Lewis, E.D., 1988. Principles of Naval Architecture, vol. II The Society of Naval Architects and Marine Engineers.

Lindstad, H., Eskeland, G.S., Psaraftis, H., Sandaas, I., Strømman, A.H., 2015. Maritime shipping and emissions: a three-layered, damage-based approach. Ocean Eng. 110, 94–101.

Lindstad, H., Asbjørnslett, B., Strømman, A., 2011. Reductions in greenhouse gas emissions and cost by shipping at lower speeds. Energy Policy 39, 3456–3464.

Lindstad, H., Asbjørnslett, B.E., Jullumstrø, E., 2013. Assessment of profit, cost and emissions by varying speed as a function of sea conditions and freight market. Transp. Res. Part D 19, 5–12.

Lloyd, A.R.J.M, 1989. Seakeeping, Ship Behaviour in Rough Weather. Ellis Horwood.

Lo, H.K., McCord, M.R., 1995. Routing through dynamic ocean currents: general heuristics and empirical results in the Gulf Stream region. Transp. Res. Part B 29 (2), 109–124.

MAN Diesel & Turbo, 2004. Basic principles of ship propulsion. http://www.mandieselturbo.com/files/news/filesof5405/5510_004_02%20low.pdf.

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Petersen, J.P., Winther, O., Jacobsen, D.J., 2012. A machine-learning approach to predict Main energy consumption under realistic operational conditions. Ship Technol. Res. 59 (1), 64–72.

Psaraftis, H.N., 2019. Speed optimization vs speed reduction: the choice between speed limits and a bunker levy. Sustainability 11 (8), 22–49.

Psaraftis, H.N., Kontovas, C.A., 2010. Balancing the economic and environmental performance of maritime transportation. Transp. Res. Part D: Transp. Environ. 15 (8), 458–462.

Psaraftis, H.N., Kontovas, C.A., 2013. Speed models for energy-efficient maritime transportation: A taxonomy and survey. Transp. Res. Part C 26, 331–351.

Psaraftis, H.N., Kontovas, C.A., 2014. Ship speed optimization: Concepts, models and combined speed-routing scenarios. Transp. Res. Part C 44, 52–69.

Ronen, D., 2011. The effect of oil price on containership speed and fleet size. J. Oper. Res. Soc. 62 (1), 211–216.

Ronen, D., 1982. The effect of oil price on the optimal speed of ships. J. Oper. Res. Soc. 33 (11), 1035–1040.

Rumelhart, D., Hinton, G., Williams, R., 1986. Learning representations by back-propagating errors. Nature 323, 533–536.

Ryder, S., Chappell, D., 1980. Optimal speed and ship size for the liner trades. Maritime Policy Manage. 7 (1), 55–57.

Strandenes, S.P., 1981. Demand substitution between tankers of different sizes. Norwegian Maritime Res. 11, 27–36.

Telfer, E.V., 1926. The Practical Analysis of Merchant Ship Trials and Service Performance. North East Coast Institution of Engineers and Shipbuilders.

Todd, F.H., 1967. Principles of naval architecture. Chapter VIII, Resistance and Propulsio. Society of Naval Architecture and Marine Engineers.

Wang, S., Meng, Q., 2012. Sailing speed optimization for container ships in a liner shipping network. Transp. Res. Part E 48, 701–714.

Wang, S., Gao, S., Tan, T., Yang, W., 2019. Bunker fuel cost and freight revenue optimization for a single liner shipping service. Comput. Oper. Res. 111, 67–83.

Wang, S., Ji, B., Zhao, J., Liu, W., Xu, T., 2018. Predicting ship fuel consumption based on LASSO regression. Transp. Res. Part D 65, 817–824.

Xia, J., Li, K.X., Ma, H., Xu, Z., 2015. Joint planning of fleet deployment, speed optimization, and cargo allocation for liner shipping. Transp. Sci. 49 (4), 922–938.

Yao, Z., Ng, S.H., Lee, L.-H., 2012. A study on bunker fuel management for the shipping liner services. Comput. Oper. Res. 39 (5), 1160–1172.

Yang, L., Chen, G., Rytter, N.G.M., et al., 2019. A genetic algorithm-based grey-box model for ship fuel consumption prediction towards sustainable shipping. Ann. Oper. Res. 1–27.

Wu, W.-M., 2020. The optimal speed in container shipping: Theory and empirical evidence. Transp. Res. Part E 136, 1–21.