A Brief History of Maritime Econometrics, 1934-2012

This article reviews the seminal econometric1 models published 
by maritime economists between 1934 and 2012, indicating the main mistakes 
committed during this period, so that future research can avoid them. The 
errors were spurious regression, identification and the false assumption that 
maritime markets are sufficiently efficient, if co-integrated. Three 
further mistakes are noted: the belief that a shipping firm is its vessel; the 
assumption that a random walk is appropriate for modeling tanker markets, and 
the assumption that shipping markets are random and linear. Koopmans was 
correct (in 1939) in stating that the discrepancy (surplus/deficit) between supply and demand for ship space determines freight rates, something 
that passed unnoticed until recently. The papers reviewed cast in four centers 
-on the basis of the academic domicile of their authors: 1) the pioneering 
Dutch Center, 1934-1939, 2) the Zannetos 
Model, 1966, 3) the Norwegian Center, 1976-2012, and 4) the 
Beenstock-Vergottis Model, 1985-1993 and English Center, 1987- 2002. Unfortunately, 
each new shipping model rejected almost all previous ones and consequently the 
research did not build a clear picture. Moreover, maritime markets are not perfectly competitive, and most 
maritime economic concepts need a re-definition, including: shortrun, shipowners expectations and marginal cost.


Introduction
Models are important because they attempt to represent reality. A model is a simplified representation of actual phenomena, and has three specific purposes: to explain, to predict and to control reality. In econometric language, this triptych entails three targets: 1) a structural 2 analysis, 2) a forecasting and 3) policy evaluation.
There are four main types of model ( Figure 1). The first type was used by Adam Smith (1776) [2] and involved the invisible hand and the division of labor. Physical models represent a real-world system with a physical entity. Geometrical models use diagrams, few variables, including supply and demand. Such models appeared in shipping economics, because the price or freight rate is theoretically determined by supply and demand for hiring ship space. Algebraic models provide systems of equations, needing econometric methods, which can be linear or non-linear (Intriligator, (1978) [1]).
Historically, freight markets were a popular subject of shipping economics and gained priority in shipping econometrics (e.g. Koopmans, (1939) [3]; Zannetos, (1966) [4];   [5]; Kavussanos, (1996) [6]. Moreover, there were often cycles of freight rates. In this industry, supply and demand drive each of its own way independently and supply must adapt to (unforeseen) changes in demand, which in turn depends on global seaborne trade. Demand cannot adapt to supply, unless the freight rate is so high so that to affect the final price of the goods in the market.
The paper surveys of the econometric models applied to shipping economics, between 1934 and 2012, without repeating material from our two other papers which covered the period 1996 to 2016 [16] [17], unless this is absolutely necessary.
The paper is organized in seven parts. Part 2, describes the evolution of econometrics; Part 3 presents the models of the Dutch Center, 1934-1965. Part 4, 3 Shipbuilding, (as well Cruising), left behind! Shipbuilding belongs also to manufacturing. 4 A branch of maritime economics dealing with all available means of transport in a combination strategy. ML aims at the minimum cost of containers' transport from point of production to point of consumption, including information (GPS). ML emerged when sea transport wanted to provide door-to-door and just-in-time services. It combines: ports, sea, air, land, rivers; storage depots and assembly centers. It seeks to minimize transport time. It deals also with such issues like the management of empty containers.

The Evolution of Econometrics
The most popular econometric tool is "linear regression", invented by Francis Galton in 1885 [18], who is also the father of the concept of correlation 1889.
In rigorous terms, a linear regression is denoted as: where the error term u t , is assumed to be identically and independently distributed round its mean, with 0 mean and variance equal to σ 2 . The exact form of any regression, (i.e. estimating the parameters α and β) is determined mathematically by the method of least squares, devised by both Legendre in 1805 and Gauss in 1809 5 .
The work done by the Cowles Commission was an intellectual success, (but an) empirical failure (Heckman, (2000) [25]; Qin, (2013) [22]). The work was closely associated with the structural approach pioneered by Frisch (1938) [19], who suggested models with many variables, attempting to explain changes in a variable by current or past values of other (explanatory) variables (Brooks, (2014) [26]).
In 1958, econometrics abandoned structural models, i.e. y x α β = + (2) with exogenous 6 x and explanatory variable(s) fixed over repeated samples (Brooks, (2014) [26]). If the dependent variable is linearly determined by k independent variables: 1 2 , , , k x x x , and there is one endogenous 7 variable, then Equation (2) can be written: where 1, , t n = (i.e. a linear static function applicable to time series).
More specifically, in "structural equation modelling" (SEM), each endogenous variable has its own equation. Econometricians considered Equation (3) insufficient, and added a (stochastic) error term 8 , u t , turning it into a statistical 5 This method caused a debate between the two authors on the paternity of the method. 6 An exogenous variable is the one whose values are determined outside the equation(s), un-correlated with the error term.

model.
A causal relationship was first considered as coming from x' variables to y. But, causality 9 can also be from y to x'. The variables may thus be interrelated in a system of linear simultaneous equations. An endogenous variable can also appear on the right hand side of an equation creating the simultaneity problem.
This means that if a (stochastic) price and quantity appear in two equations, it is impossible to estimate them validly using ordinary least squares, and maintaining that x and u are independent, and errors uncorrelated with explanatory variables [26]. SEM, however, was popular in published papers, and books, until 1970, accounting for 20% in 1951, rising to 48% in 1959-1966 and falling back a little to 33% in 1968-1970 [22].
There is also a feedback mechanism operating between many economic variables. Economic data should then be described as a system of (simultaneous) relations. A SEM is then properly used to describe relationships among random economic variables.

What Is Random 10 ?
The property of randomness in a variable, or time series, should be tested (e.g. by the Jarque-Bera test), and when tested the results should be taken into account. A diagrammatic test of randomness is presented in Figure 2, using MATLAB and "Rescaled Range Analysis 11 ". It shows the first logarithmic differences between a random variable (e.g. Baltic Panamax Index-BPI) (red line) and its actual representation (blue line).
As shown, the supposedly random curve of daily BPI for 1999-2012 (blue line) departs from random (red line).  The classical linear regression model assumed that the explanatory variables are non-stochastic. But this assumption can be violated. There are also errors-in-variables and the use of proxies. 10 A common term is random walk. This is used when the current value of a series is the previous value perturbed by a white noise error term. 11 A method developed by Hurst H E (in 1951) to determine long-memory effects and fractional Brownian motion. A measurement of how distance covered by a particle increases over longer and longer time scales. For Brownian motion the distance covered increases with the square root of time. Haavelmo (1943) [20] drew attention to the identification problem connected with SEM and the associated estimation problem (i.e. simultaneity bias) of OLS estimators. Marschak (1946) [27] tried to combine theoretical and empirical research. Econometric steps were then formalized: 1) a model specification, 2) identification and 3) estimation. Finally, SEM is the most general form of a structural model, having the conceptual adequacy of the Walrasian general equilibrium system. SEM was then transformed into a reduced form, which is more useful for forecasting. The main method adopted was the maximum likelihood, which became the limited information maximum likelihood model, with associated and appropriate computing methods (Epstein, (1987), [28], (1989) [29]). This is a method for estimating the parameters of nonlinear models, based on the construction and maximization of a likelihood function.
Rational expectations are based on the principle that people make, on average, correct guesses about future, using all available and relevant information. In order to have a perfect foresight, information must be complete, and the uncertainty zero… Models that are related to VAR are: ARMA 13 , VECM 14 and VMA 15 .
Dynamic models include lagged or differenced terms of the dependent or independent variables. A new methods to estimate these models was the two stages least squares developed by Theil (1953) [32] and Basmann (1957) [33].
The method was used to estimate parameters called instrumental variables, developed by Sargan (1958) [34], 1959 [35]); also the SUR method emerged developed by Zellner (1962) [36] 16 . IV are correlated with the variables they replace, but not with error term. SUR applied to shipping by Kavussanos M. SUR is suitable for models with movements of several highly related dependent variables, using a time series regression, and at the same time, allowing correlation in error terms. 12 A time series specification with many variables, where the values of lagged variables of all variables appear on RHS in all equations of the unrestricted model. 13 A VAR model with lagged values of error term in each equation. 14 Embedded into a VAR framework for relationships (short-and long-term) between a set of variables to be modelled simultaneously. 15 A multivariate time series model, where series is expressed as a combination of lagged values of a vector of white noise processes. 16 Instrumental variables are used when we replace endogenous variables on the RHS of a regression.

Stationary Maritime Time Series
Shipping economists used non-stationary variables in their models. Using such methods one can easily fit regressions using levels of co-trending, but unrelated variables, and find reasonable goodness of fit, and heavy serial correlation, even though this correlation is spurious (Phillips, 1986) [37])! A regression involving two or more independent non-stationary variables, where the estimates of the slopes appear highly significant in standard statistical tests, can have highly significant t-ratios, even though in reality there is no relationship. This restriction to the method was first considered in the 1970s, and appeared in shipping models 10 years later. Because of this phenomenon, the conclusions of any paper published before 1990 must be considered invalid. The time series involved did not have constant means, and included a variance and an auto-covariance.

Co-Integration: The Great Breakthrough!
After 1986-1987, econometricians used methods to analyze time series with many co-integrated variables/equations, (i.e. variables showing a fixed relationship in the long-run), with error corrected (i.e. variables that were stationary/1st differenced-which were combined with a term capturing movements back to their long-run equilibrium) (Engle & Granger, (1987), [38]).
Maritime economists adopted these innovative approaches roughly six years after their first appearance. Many maritime economists incorporated assumptions into their models, including the idea that markets were efficient and that people (traders) have rational expectations. In combination, these suggest that asset prices, (or their natural logarithms), follow a random walk (with or without drift) (with unpredictable differences).
Kavussanos and Alizadeh-M (2002) [39] [40] rejected previous shipping models, which used co-integration, (e.g. Hale and Vanags' (1992) [41]), and argued that co-integration is a sufficient, but not necessary 17 , condition for an efficient market. They used a generalized conditional heteroscedastic model (GARCH-M), previously developed by Campbell and Shiller, (1987) [42]). Two dry cargo markets (new-buildings and second hand) were found not to be efficient (due to stable risk premiums) using data from 1980 to 1997. They assumed that time charters (of 12 months) should reflect the weighted average of the expected monthly (spot) freight contracts over the next 12 months, if they were efficient. They applied the expectations hypothesis of term structure (EHTS). Their results did not support the hypothesis because returns were excessively volatile. This implies that shipowners prefer to be in less volatile markets, but between 2003 and 2008 shipowners preferred to be in the spot market with very high profits, extreme volatility, and more risk.
Kavussanos and Alizadeh-M [40] also showed that the dry bulk shipping freight rate market is not efficient and does not price risk as expected, because of 17 This confirmed by e-mail to author by Prof. Kavussanos. the way that shipowners perceive risk; they are prepared to suffer a loss to secure a contract for a longer duration. Between 2003 and 2008, shipowners found it very rewarding to be in the spot market, but after the end of 2008, they regretted it. It can be seen that the level of net profit is decisive, and its variability is a secondary matter. The level of profit seems to overshadow risk. Shipowners prefer to accept a lower time charter rate-lower than spot-if they anticipate a falling spot rate in the future.
Certain models presuppose that investors should be compensated for taking more risk by getting a higher return. We argue that in shipping when seeking profit risk is ignored. Engle et al. (1987) [43] suggested applying ARCH-M model. However, GARCH-M 18 was more popular, and replaced If d > 0 (then a risk premium exists), and if it is statistically significant, then increased risk leads to a rise in the mean return.
Goulielmos (2009) [44] argued that the Aframax markets showed different levels of long term volatility, and risk, with the second hand ship market (alpha = 1.32) (shorter) safer (less volatile) followed by the new-buildings market (longer) (alpha = 1.30). This means that in shipping shorter-run investors run lower risks than longer-term ones, though the above alphas are close. Alternatively, in practice charterers avoid long-term time charters out of fear that a marine accident may happen. This is a matter of concern where overage tankers (say more than ten years of age) are involved. This suggests a testable hypothesis that has not been adequately examined: "Will shipowners prefer to be in the spot market when profits are very high, even if the market is then more volatile and of a higher risk?" In econometrics, the sequence of the events was: 1) the "autoregressive model" (AR 19 ), which replaced the multivariate and inappropriate structural model.

The Dutch Center, 1934-1965
The earliest shipping models produced by Dutch Center were due to Tinbergen 21 and Koopmans 22 , who also played an important role in the development of Eco- 18 GARCH-M is a dynamic model for volatility, where standard deviation-σ enters into the generating process; σ, however, does not reflect risk as good as alpha coefficient-α. 19 AR is a time series model where current value of a series is fitted with its previous values. 20 ARIMA is a univariate time series model derived from a generalization of ARMA for a stationary-made process after differenced it d times. 21 Mathematician and physicist, with numerous publications (1935)(1936)(1937)(1938)(1939)(1940)(1941)(1942)(1943)(1944)(1945)(1946)(1947)(1948)(1949)(1950)(1951)(1952); 50% of them dealt with business cycles (1935)(1936)(1937)(1938)(1939); one with econometrics (1951) and one with the theory of economic policy (1952)-considered influential-1939-1975. He made significant contributions to the concept of "shadow prices". He was also known for his macro-dynamic model, and especially for that of US economy (1939), which is the first major econometric effort to model "business cycle". A subsequent methodological debate over his models with Keynes, played a crucial role for him and for the wider formalization of econometrics (Qin, (2013) [22]). 22 He was a student of Tinbergen and physicist. nometrics (Qin, (2013) [22]). They are considered fundamental contributors to maritime economics (Veenstra and De La Fosse, (2006) [7]). Tinbergen (1934; [15]) related freight rates to changes in demand, (stated in ton miles), which he considered to be perfectly inelastic. Using data from 1880 to 1911, and to changes in supply (also in ton miles) as well as laid-up tonnage, he showed that

Tinbergen's 1934 Model
where P = the price of fuel (coal) -other costs assumed constant-and K = the tonnage level. The equilibrium condition is that demand equals supply.
Solving (5) at equilibrium for freight rate gives a linear regression: where e 1 = 1/g, e 2 = a/g and e 3 = b/g. This states that the freight rate is determined, at equilibrium, by demand, D for tonnage, where K = level of existing tonnage (i.e. supply) and P = cost of fuel. Equation (6) [48], argued that equilibrium means a "balance of forces". Economists adopted the term without providing a strict definition, and shipping economists did the same. Koopmans (1939) [3] modeled, at equilibrium, the demand for tanker services as equal to supply (in ton miles), i.e.

Koopmans' Model
( ) 23 If the system was up in the previous period, it is more likely to be down in next period (Peters, (1996) [46])? where P = costs (all costs). Supply is proportional to fleet size (K) and to the ratio of freight rate to costs. Coefficient g was found to be 0.15, indicating a strong influence of laid-up tonnage on elasticity of supply. Koopmans also showed the effect of laid-up tonnage on freight rates accounting for short-run ups and downs. He also, related ship values to freight rates for tankers and linked the freight market with shipbuilding and scrapping. Koopmans' model is presented diagrammatically in Figure 3. Figure 3 presents a diagrammatic determination of short-run tanker freight rates, F 1-3 . The supply curve, (expressed in ton miles), is initially (on left) perfectly elastic (part sf), because the available tonnage is not hired and some ships are laid-up. If the demand curve intersects this part of the supply curve, the impact on equilibrium freight rates will be negligible.
If demand increases, as shown by the shift of demand curves to the right, and intersects the right hand part of the supply curve, the fleet is fully employed. Increases in demand gradually draw into use all sea-worthy tonnage. The full impact of increased demand, in the shortrun, passes entirely to freight rates, no more tonnage is available (not even from shipbuilding). Certain factors in the ceteris paribus cupboard, like: average speed of ships, ships lost, ships scrapped and ships converted, are assumed to have exerted all their influence on supply and to remain constant at equilibrium.
The construction of additional ships requires time, which, by definition, is beyond the shortrun. Equilibrium is delayed and demand outstrips supply until supply responds. The time taken to reach equilibrium varies. There are waiting and construction times for new buildings, also variable, depending on whether a boom or a depression is on the way. The time span may be from a few months to a few years.
The various factors mentioned above show that a simple two-variable model-like those of Koopmans and Tinbergen-left out a number of factors. Tinbergen and Koopmans indirectly incorporated sea distances, scrapping, and laid-up tonnage. Sea distances and lay-up tonnage play an important role, but the impact the speed of ships in low steaming is completely ignored. Figure 3. Short-run equilibrium freight rate determined by supply and demand (tankers). Source: inspired by Koopmans (1939) [3]. and supply (a two dimensional space). This is a market photograph, valid only at the specific moment when it was taken. To approach reality, time must be introduced into the models.
In Figure 3, equal shifts in demand, AB = BC, correspond to very unequal increases in freight rates, because demand is absolutely inelastic at this stage: F 1 F 2 < F 2 F 3 . This explains why there are large fluctuations in freight rates (Evans (1994) [49]). There is a saying that "shipping creates great fortunes, but also great misfortunes". For example, Onassis gained a freight rate six times the value of his ship for a single trip during the Korean War (1951). Another shipowner, from Sweden, gained a freight rate equal to $1m for one trip using a super tank-er… A valid hypothesis made by Koopmans [3] was not tested, due to lack of suitable data. He argued that expectations about the equilibrium of supply and demand for ship services, for a few years ahead, exert the most decisive influence on tanker buildings. Goulielmos (2014) [50] tested this, using figures 24 for the surplus of supply over demand. The correlation coefficient was high (=0. 84) between surpluses in demand and supply, and time charters (1976)(1977)(1978)(1979)(1980)(1981)(1982)(1983)(1984)(1985)(1986)(1987)(1988)).

Zannetos' Model, 1966
Zannetos' model is presented in  Zannetos was dissatisfied with the traditional static economic analysis, which, as Hicks noted, ignored time, and was unable to explain how tanker freight rates are formed. He was puzzled because freight rates and demand for tanker services were cyclical, while demand for oil was not...
Zannetos found his theoretical framework in Hicks' theory of expectations.
His model is a product of his PhD thesis (1956)(1957)(1958)(1959) at Massachusetts Institute of Technology. His book, based on it, was a continuation of the work of Koopmans (Zannetos, (1987) [57]). Hicks (1946) [58] argued that when elasticity of expectations is greater than unity, a change in current prices makes people feel that they can recognize a trend to extrapolate. Zannetos argued that a substantial rate movement away from equilibrium creates expectations. More recent theory provides not one, but three main theories for expectations: adaptive, rational and behavioral. Because people anticipate the future and act on those assumptions, future freight rates will be proportionally greater or less than the current level. If this assumption is valid-which should be tested econometrically, using data provided by shipowners-then almost all Zannetos' theory follows. This could explain the cyclical behavior of markets, as expectations work as a stop-go mechanism. The elasticity of expectations is unity, and it is this elasticity that creates market instability. Moreover, new-building prices also move the system into market instability (Zannetos [4]; Dikos, (2004) [59]). Georgantzi (2005) [60] found that the majority of people she interviewed in her MSc thesis considered that the parties involved in the chartering market for dry cargoes, are affected from 80% to 100%, by forecasts future freight rates. This is of great importance, and should also be tested econometrically. There is a need for accurate forecasting in maritime businesses-an area where many maritime economists failed as have many ship owners.
Zannetos also recognized the term structure of freight rates in the long-run (Veenstra and De La Fosse, (2006) [7]). Short-run freight rates are formed by demand, influenced by freight rate expectations, and static supply. The longer run freight rate, he argued, is closer to equilibrium. As charter duration rises, freight rates approach long term equilibrium freight rate from above (in a boom) or from below (in a depression). Figure 5 shows the impact of price elastic expectations, away from six equilibria, denoted by R 1 , R 2 , R 3 , R 4 , R 5 and R s , between short and long run rates, whenever supply equals demand and when slopes are right for stability. Partial equilibria are also possible (Marshall, (1920) [61] in regions, R 2 (unstable) and R 3 (stable from below). R 4 is unstable, and R 5 is stable from above. Zannetos studied situations away 25 from equilibrium, where there is a dynamic process moving towards equilibrium.
Zannetos argued that freight rates create a non-symmetrical behavior between buyers and sellers of ship space that is regular, non-uniform, and creates 25 Prigogine (2003) [62] argued that non-equilibrium Physics gave him a better understanding of the mechanism of the emergence of the events.  ( ) where F stands for spot rates. Freight rates have long troughs and brief sudden peaks that can be simulated by an adapted cobweb model (Tvedt, (2003) [63]; Adland and Cullinane, forthcoming, [64] We believe that the oil majors needed a specific percentage of ownership of the tanker fleet to control the market in an oligo-psonistic manner .
Also, oil-majors induced independent owners to build new ships by negotiating with them long term (15 to 20 years) contracts at freight rates lower than the cost of their own ships (Goulielmos, (2013) [65], 2018 [52]). Independents were more competitive than oil companies, exploiting faster economies of scale and taking quicker decisions.
This policy, however, to increase existing supply (and removing a potential demand from the market), and in the end to lowering the freight rates to be paid by the oil majors, was brilliant. Oil majors have bureaucratic boards of directors.
Some argue that the spot freight rate was the yardstick in negotiations. We believe that the oil majors' operating costs were the yardstick. Independents could not easily refuse a long term contract provided by a first class charterer that could be used in a bank to obtain finance for between 60% and 80% to build the ship involved. Independents relied on profit from their lower operating costs and economies of scale building larger ships than had previously in the market (e.g. Onassis, Niarchos).
Strandenes (2012) [66] argued that Zannetos conceived the shape of the supply curve empirically, and added expectations to explain cycles in freight rates. For Zannetos cycles is inherent in shipping. Koopmans [3] explained, wrongly, that the shipping cycle is due to the replacement of ships, which did not satisfy Zannetos-quite rightly. Greek shipping companies, when there are rock bottom ship prices, buy ships, adding to their fleets larger and younger ships (a strategy = of firm growth). Also, they buy used ships that are about five years old, of new types (diversification; product carriers; VLCC). Glen and Martin (1998) [67] argued that Zannetos developed a formal explanation forthe spot freight rate to converge (from above or below) to the long run marginal cost of the vessel, using a "bareboat" 26 charter as a proxy.

The Norwegian Center, 1976-1985
Eriksen and Norman (1976) [68] related ship values to tanker freight rates. Eriksen (1977) [69] was the first to deal with the estimation of demand (dry cargo T F − = (9) where T is world's bulk trade, and e is assumed positive; d is roughly unity.
Wergeland used the methods: two stages least squares, full information maximum likelihood and a log-linear approach. He proved that the existence of laid-up tonnage determines elasticity of supply, something argued also by 26 The ship is capable, but not ready, to transport cargo. It is a bare… ship. Koopmans [3]. Norman and Wergeland (1981) [71] optimized vessels' speed to find freight rates, using a solvable (analytical) model. Strandenes (1981) [72]  This means that in order for a market to be efficient, all relevant information has to have been included in the price. But the hypothesis that the market is efficient has been disputed for some time, especially after 1987 (Black Monday). 27 Transporting more than one type of cargos. Strandenes (1990) [79] examined the effects of the problem of capacity utilization shipbuilding and shipping in 1990s.

The "Beenstock-Vergottis" Model and the English Center, 1986-1995
This period is characterized by the work of Beenstock and Vergottis (1989 a, b) [80] [81], 1993 [5] and 1995 [82]). Hawdon (1978) [83] modeled supply and demand, the fleet, fuel price, and: the price of new-buildings, wages of seamen, average size of ships, freight rates of dry cargo ships (in grain transport) and some dummies variables. He found the latter not to be statistically significant for freight rates estimation, using a reduced form regression in long and shortrun.
He also estimated the cross effects of spot freight rates between tankers and dry cargoes . He found no evidence for the impact of combined carriers 28 .
Maritime economists at this time were heavily preoccupied with the cross effects of spot freight rates between tankers and dry bulks, due to the appearance of combination carriers. These ships, initially called OBO (oil-bulk-ore), of a total of 49m dwt globally, had handsome profits between 1967 and 1970 and in 1973. Stopford (2009) [14] million mentioned a $2000 to $3000 revenue premium per day. Initially, maritime economists used short-run models. Charemza and Gronicki (1981) [84] argued that demand is completely insensitive to freight rates 29 . They created a linear bulk market model with freight rates in disequilibrium, supply and demand taken at their minima, and freight rates changed proportionally to excess demand. Their econometric models were separate for dry bulk and tankers, where ship values made it possible to react to changes in freight rates. Beenstock (1985) [75] proposed a dynamic general equilibrium model to determine ship prices given current and expected freight rates, assuming market efficiency, rational expectations and an index for global economic activity. He used a present value model. Hale and Vanags (1989) [85] examined spot and period bulk freight rates modeling risk aversion, or liquidity preference (i.e. to be in the spot market).
They performed several tests for market efficiency of shipping freight rates using single equation regression. They found no evidence of term structure between freight rates. Glen (1990) [86] dealt with the relationship between risk and ship size (annually). He found that volatility, i.e. the variation of estimated gross profit margins, systematically increased with size (except on the Caribbean-US Gulf route) 30 . Hale and Vanags (1992) [41] examined the efficiency, (in its weak form in the sense of 1970 Fama, so that historical share prices cannot be used, of the second 28 A new type of ships emerged at the beginning of 1960s. 29 If the % of F in CIF price is low; if F is substantial, it affects CIF price and thus demand! This has been proved when the oil of price increased substantially, the demand of oil fell. Economists committed the erroneous assumption of a very inelastic demand for oil! 30 He admitted that Glen and Martin's (1998) [67] model, used short data of only 9 years was also naive. b s α = (13) where b = bunker (fuel) consumption in tons and α is a technology constant coefficient, assumed to be greater than unity (=decreasing returns to scale). A more interesting case that has been totally neglected by economists is where there are increasing returns to scale , [89]).
They argued [5] that the assumption of α > 1 is theoretically necessary for maximization.

Fs Pb OC
where P is the price of fuel, OC fixed costs (operating). Now To maximize profits, speed must vary with the ratio of freight rate to fuel price.
Equation (17) implies that maximum profit is related to freight rates and fuel prices. Following certain further steps-omitted here-they can determine total market supply. Next, demand is assumed exogenous and completely inelastic. For market equilibrium, where K = carrying capacity in dwt of all ships. No data could be found on profits or expected profits. They used 2SLS (equivalent to 3SLS), OLS, instrumental variables in place of 2SLS (where the full economic structural system of equations could be used), simulation, with an emphasis on rational expectations, 3SLS (better is the generalized method of moments-GMM, 1982, also in 3 stages), using optimization (maximizing profits), and perfect competition (in a structural model).
Expectations are formed in the most effective way through an information mechanism so that forecasting error is random and unforeseen. Rational expectations are the result of rational maximizing behavior in acquiring and processing information to form a view about the future. They also modeled the interdependencies between dry cargo and tanker markets and the market for ships and rational expectations.
Papadopoulos and Tamvakis (1994) [90] used probability theory in investment decision-making and simulation, with a Monte Carlo method overcoming a number of problems. Forecasting using simulation is usually inaccurate. The χ 2 method, however, was found to be more appropriate. Evans (1994) [49] modeled revenue per ship mile: where s is the optimum speed and s * o is the voyage speed, k is a constant, d is distance, p is the price of fuel, and t is voyage time. If t = 0, revenue per ship mile = 3pks 2 . Glen (1991) [91] reviewed Evans and Marlow's (1990) book [92], and argued that the latter ignored the steady increase in the number of shipping econometric models over the period from 1980 to 1990, while they used difference equations and modeled expectations.

A Critique of Models Presented
Most of maritime papers reviewed above reached contradictory findings and ul- In Tinbergen's model [15] no degree of goodness of fit was mentioned (e.g. R 2 with no intercept). No error term appeared; his equation was exact. Koopmans (1939) [3] argued that expectations about future returns do not influence investment decisions, contrary to the opinion of Keynes and almost all economists.
In addition, there are many definitions of maritime marginal cost, indicating an apparent weakness of theory in providing a unique definition. Zannetos (1966) [4] implicitly recognized the long-range dependence in his time series for rates (1949)(1950)(1951)(1952)(1953)(1954)(1955)(1956)(1957)(1958), where the past continued to influence random fluctuations in the present (Mandelbrot and Hudson, (2006/8) [24], in another context). He argued that, "The quantitative exactness of regression models is not always reliable" (p. 232). The spot freight rate is not the only factor that influences investing in a new building (Goulielmos, (2014) [50] Owners order ships if trade increases, freight rates rise, and prices (of new buildings) fall. Also, the price of steel can be inserted into the regression of freight rates (both affecting new building prices). Orders affect deliveries (positively). Scrapping depends on expected future profits of old ships and on the size of the fleet (supply). Freight rates and demand affect (negatively) scrapping (if high). The fleet size is the supply, and the fleet grows when there are new deliveries and shrinks when vessels are scrapped. Wergeland (1981) [70] failed to estimate correctly the fleet size coefficient in the supply function, because his demand data was incompatible with that of the fleet. He adjusted a coefficient, in a simulation model, arbitrarily, to unity 1, and he assumed the elasticity of demand with respect to freight rates to be −0.4, in order to get better results. Supply was influenced by lay-up. Strandenes (1984) [74] argued that prices are more affected by changes in the long-term equilibrium profits than changes in current profits. This conclusion was different from her previous findings, but was similar in that of Keynes in his General Theory etc. book. Hale and Vanags (1992) [41] rejected the hypothgesis that market is efficient, contrary to the findings of Beenstock (1985) [75] and Strandenes (1986) [77]. This is also a serious blow to perfect competition. The data in Beenstock and Vergottis' work (1993) [5] was found to have certain problems, and their book lacked further editing [96]. Eales (1995 book review) [96] suggested the use of neural networks. Such a model appeared by Li and Parsons two years later (1997) [97].

Suggestions for Further Shipping Econometric Research
Three essential claims about market behavior derive from the above assumptions, and these need to be carefully evaluated: 1) the best estimate of tomorrow's freight rate is today's; 2) tomorrow's freight rate is independent of past freight rates; 3) changes in freight rate vary in accordance with a normal distribution. The assumption of efficient markets implies that asset prices will rapidly reflect all relevant and available information though the exact speed required has not been specified.

Unit of Analysis: Vessel or Firm
A mistake committed by Thorburn (1960;p. 11) [51]) was to argue that the vessel was the unit on which his investigation should be based, and that in tramp 31 shipping, the vessel forms the economic unit of analysis (i.e. the shipping company), while the shipping company per se plays a subordinate role. This approach, though wrong especially after 1960, was also adopted by Zannetos [4] (for a detailed analysis about this serious mistake see Goulielmos, (2018) [52]. Metaxas (1971) [53]  cost is better thought of as the addition to total cost of the vessel for a one way voyage. Metaxas however, was known for his contribution to social costs in shipping and his accurate knowledge of operating costs.

Other Important Maritime Economists
Professor Heaver (1993) [54]; 2012) [55]) considered the books of Zannetos (1966) [4] and Shimojo (1979) [56] to be the best two analytic books. Shimojo presented the way freight rates are determined in tramp shipping, showing time in a third axis. Goss was the founder of maritime economics in UK   [52]). Svendsen was the first to write a good mathematical shipping economics textbook in 1958. Georgandopoulos published a shipping textbook in 1953 (in Greek) and he organized the first international conference of maritime economists in 1976 in Piraeus [55].

Further Questions
It is important to know whether elastic expectations really apply to shipping, and whether perfect competition occurs in the bulk and tanker industries (Goulielmos, (2013) [65], (2018) [52]. There is also the problem as to whether (and when) spot or time charter freight rate are more decisive in a variety of ship owners' decisions.
Moreover, free-exit from the market is not as free as certain maritime economists believe (the Hajin case). It is also worth re-examining the concept of the shortrun in shipping. A shipping company almost always makes decisions in the shortrun as most decisions of a shipping firm to change the level of the firm's capital, or number of ships need no more than one to two months.
The most crucial trigger for econometrics was the widespread failure of macro-econometric models to predict global recession after the 1973 oil crisis. Macro-econometric models also failed to predict the depression at the end of-2008 (Lucas and Sargent, (1979) [101]; Lucas, (1976) [102]).
A new feature has now arisen with the appearance of futures markets. It needs to be investigated whether the parties involved in a chartering market for dry cargos and tankers are substantially affected by futures forecasts. Above all it is important to understand the influence of future freight rates agreements (FFAs).
The concept of the shortrun in shipping needs to be re-examined, as mentioned. For the industry, shortrun is when new ships cannot be built. This time can be from one to four years. For the firm, shortrun is when additional ships cannot be bought. This time is from one to two months.

Conflicts of Interest
The author declares no conflicts of interest regarding the publication of this paper.