Seland [5] argued that ancient Greeks used oracles⁵; Romans had also their own; medieval times had their fortune-tellers. Today, we have economists. The esteem in which economists are held is rather varied: often, fairly high, by those who read prophesies, and generally rather low, by those who remember them...He advised his students to forecast for 10 years―at least―ahead…then they will have the option to say: “As I wrote ten years ago…” or remain silent…

Hearth [6] argued that the overall forecast of demand for oil was right: a 7% - 9% compound increase between 1965 and 1980. Forecasters failed in anticipating ‘economies of scale’! Tankers above 100,000 dwt forecast to be 4.5m dwt by 1980 -since 1965―but at May 1970 51.2 m dwt already were on order and construction, of 150,000 dwt size and over each…Onassis by instinct believed in both―an increasing demand and economies of scale.

Stopford [7] reviewed 3 main segments of shipping for their likely development by 2003. He said that the Greek (he was Latin) poet Horace (65-68 B.C.) wrote: “adversity reveals the genius, prosperity conceals them”. Recessions and shipping go together, Stopford wrote. The year 2000 was as good as 1973 (rather 1974). The “Clarksea index” peaked at $25,575/day in end-2000. The bulk carrier market paid a heavy price for its investment boom in 1999; so the market remained weak and moved into recession; the “order book” forecast to wind down over 2002, and bulk trade to decline. Was he right? No, the 2000 dry cargo index (Stopford, 2009 [3] ; and Figure 1 here) showed for 2000 164 units compared with 135 in 1973 (and 168 in 1974), 155 in 2002 and 253 in 2003. Market improved due to “China effect”.

Lorange (2009) [8] argued that forecasting comes first from consulting firms on forecasting (like Marsoft; Jefferies & Co Inc.; Maritime strategies int.―MSI; China ship economy research center; Drewry and others); research units within traditional shipbrokers (like Clarkson, Platou, Fearnleys); specialized ship consulting firms (like Carl Bro, Vik & Sanbdvik, Skipsteknisk) after ship designs. He admitted: good forecasting is the key, and its role is important⁶.

Eslami et al. [9] argued that their model―a hybrid one based on “Artificial Neural Network” ―ANN, and an “adaptive generic algorithm”―was superior of regression, moving averages and pre-existing ANN studies. The RMSE (“root mean squared error”) was for their model 11.2WS units (worldscale⁷) vis-à-vis regression of 21.6WS and moving averages of 17.5WS and other ANN studies from 14.6 to 15.8WS (1983-2003).

Looking at their predictions, (their Figure 6), over 30 or so months, however, the difference between actual and their model was 30WS units at the ~27th or so month, while regression model was worse by 42WS, but moving averages were

closer, by 20WS…To understand what 1WS means, we may say that 50WS means for a charterer to pay $8.65 per ton of oil from Jubail to Rotterdam (authors used Ras Tanura⁸ to Rotterdam); in other words, a tanker of 250,000 dwt would expect to earn $4.375 m per voyage according to authors’ forecast (~100WS) and in fact she will get 30% (30WS) less…A prediction model is not enough to be better than others, but also closer to reality to be useful to shipowners. This is another maritime forecasting failure.

Makridakis [10] argued that many people view forecasting as a crystal-ball attempt with little, or no, scientific or rational basis. Such a view was reinforced by prominent errors in forecasting events that did not materialize―like the forthcoming widespread use of nuclear energy and the appearance and impact of computers and information revolution. He suggested the meta-forecasting, based on rational principles arriving at future predictions scientifically sound -using feedback and learning.

Faggini and Parziale (2012) [11] stated their dissatisfaction for the inability of economics to foresee and proposed to resort to “Chaos Theory⁹”.

Weatherall (2013) [12] ) doubted the idea that markets are efficient (p. 158)! This (“efficient market hypothesis¹⁰”―EMH) is the great postulate of the mainstream economic theory, representing one of the 3 sacred cows of modern economics. Mandelbrot and Hudson [2] argued¹¹ that EMH holds in an ideal market in which “all relevant information is already priced into a security (today)”. Weatherall (2013) [12] argued that “wild randomness¹²” can be a sign of underlying…chaos…and there is often enough structure to make useful predictions…He left us a hope for predict-ability.

Buchanan (2013) [13] argued that economists, almost all, have virtually ignored the most profound scientific discoveries of the past few decades―chaos theory―where the ongoing fluctuations in economic and financial systems are quite normal. The automatic efficiency of the “invisible hand” of Adam Smith is a chimeric bygone dream. Buchanan [13] (in jacket) argued that a smarter economics accepts that markets act much like the weather…while centuries of classical financial thought trained us to understand “the market” as a system that always returns to equilibrium¹³.

Wolf¹⁴ (2014) [14] argued that he personally lacked the imagination to anticipate (p. xvi); he was guilty of working with a mental model of the economy that did not allow him to foresee the crisis in the world’s most advanced economies (italics added). The crisis was the outcome of…a “rational carelessness” [14] , he wrote...

In summary, we believe that the idea of the “invisible hand”, which no doubt is present in the celestial system and Nature, when applied to economy―as Adam Smith did (1723-1790)―automatically rejects Man’s free will. But if a man is free to choose, as he/she is, forecasting is even more difficult (Priesmeyer, 1992) [15] . Priesmeyer (1992) [15] (and after) argued that managers do not need forecasting as they are that kind of men/women to “make things happen”…

All maritime economists say: shipping markets are cyclical; i.e. they encounter periods of high rates (prices) and periods of low ones. A good question is: are the periods of good times equal to those of bad times? Unfortunately, not.

Research, originally, ignored cycles as it thought to be phenomena that “come and go”, like the weather. Then noticed that cycles “come, go, but return”...Today, cycles are tidily classified in short, medium, long and secular (=near a century), (Goulielmos, 2017 [24] , in this journal, discovered shipping’s “over-secular” cycles, i.e. > 100 years), and are explained―after they happened- how and why occurred...What surely is lacking is: “when cycles come, when cycles go, when cycles come back and when cycles turn”...This means that forecasting business cycles’ timing, and their turning points, are missing, but at the same time these are very important.

Shipping history has taught us that between 1741 and 2015 (274 years), a shipowner―if lived so long- had to face ~27 cycles, 1 every ~10 years on average (Figure 1).

Does this 10-year average cycle mean that shipowners have to amortize a new ship within her first 10 years in anticipation of a cycle? Ships, as we know, are built to live for as long as 30+ years. Should this change? The car industry―we know this by experience―produces less durable cars as time goes-by.

More important is what Keynes wrote²⁴ [16] ―in relation to durability of assets: “…we have a substantial reason (to believe) that the duration of (a) slump (has) a definite relationship to…the durability of assets”. This idea is based on the assumption that the scarcity²⁵ of durable goods raises MEC, and creates needs for new investment. It is here that technical change comes-in if it makes existing capital goods obsolete. However, the myths in connection with business cycles will occupy us also in the next section.

Persistent time series are found by calculating their “exponent H” (defined below). Letter H²⁹ is given to this exponent―after Hurst’s name³⁰―by Mandelbrot. Hurst used letter K. Now, an H larger than 0.50 ≤ 1 indicates that data are “persistent”³¹. Also, its statistical test can be done through the (non-parametric³²) “rescaled range analysis”, or the range divided by local “standard deviation”.

The case where H is = to 1/2, proved mathematically by Einstein³³ [30] . He proved that the distance: D, which a random particle³⁴ covers, increases by the square root of time (used in measuring it): so, D = Time 1 / 2 = t = t 0.50 {1}, where “time”―t―stands for the time index³⁵.

Hurst³⁶ [31] showed that ( R / S ) n = c ∗ n H {2}, where R/S is the ‘rescaled range’, (with zero mean and a local standard deviation S), divided by S; the (adjusted; nonnegative) range R_n now is = to max ( Y 1 , ⋯ , Y n ) − min ( Y 1 , ⋯ , Y n ) {3}, which gives the distance that a system travels for time index n; where c is a constant; H is the Hurst exponent and n refer to the R/S values for ( X 1 , ⋯ , X n ) time series [32] . Taking logs on {2}: log ( R / S n ) = log ( c ) + H ∗ log ( n ) {4}. {4} is used to estimate H.

To test the above theory, we used 256 (265-1 for 1^st log differences―8 yearly figures were lost due to war: 1939-1946) yearly observations (1741-2015), already presented in Figure 1, and we have applied above regression {4} (using least squares), and found³⁷: H_n≥10 = 0.685823 = 0.69 (rounded) > 0.50 ≤ 1, indicating persistent maritime time series.

As argued by Peters [28] , in Chaos theory, non-periodic cycles exist with an average duration. R/S can detect periodic cycles too. This is so for periodic cycles the range R―defined above―could never grow beyond the amplitude of the relevant wave, and so R/S would reach a maximum after one cycle. The first cycle in our data is 84 years as shown below (Figure 4).

Figure 5 gives a better estimate of the cycle length by presenting the results from equation {5}. Equation 5 originally was used by Hurst to test for stability,

and it works particularly well in the presence of noise. The plot of V versus log n (Figure 5) is flat if the process is random. This is not true for the whole range of the maritime time series used here (1741-2015).

An alternative picture is given by Figure 6 presenting coefficient α. If α = 2, time series are random. As shown in Figure 6, the curve deviates from alpha = 2 at various times; the deviation is stronger from 223^rd year till end-2002, where α → 1.46 (rounded). This points towards (1741 + 223 years + 8 years for which data is lost) approximately years 1971-1972. During these years and after, randomness disappeared and risk increased in shipping…(Figure 6). Shipowners in age round their 50s today had to be more careful, as dry cargo markets became very tough since 1971-1972. After 1971-1972, indeed many (five) peaks occurred: 1973-1974; 1979-1981; 1988-1997; 2003-2007; 2008; out of 9 since 1947.

The last myth is that the probability of an event―occurring in a given population ―fulfilling some basic conditions―is given only by bell curve. Let x be a variable’s level that we want to study; let the Greek letter μ stands for the average value of all x’s in the population; let σ stands for the standard deviation -indicating the degree of how broadly x’s, around μ, are scattered; then the bell curve is: f ( x ) = 1 / σ 2 π ∗ e − ( ( x − μ ) / σ ) 2 / 2 (known also as normal distribution)³⁹.

Now, if x is close to μ, the probability for this to appear is high. The opposite will hold if x is found in tails. Standard deviation, σ, determines if the curve will

be squat and low, or narrow and tall. The myth that the bell curve is the unique one for time series is false. In fact there are another two interesting ones (Figure 7).

As shown, the more flat curve―the bell―does not go beyond the 2 points indicated by the 2 arrows on the right and left tail; or in statistical language, the curve does not go beyond ±3σ from the mean μ. The two other curves do. This means that the probability of x to get extreme values -like over ±3σs from the mean μ―is only ±0.25%. It is thus very important to know what type of curve time series under study follows.

The “Dow Jones Ind. Index” had many changes beyond 5σ, 10σ (1930-1935) (2000 and after), and even 22σ (1987; black Monday) (Mandelbrot and Hudson, 2004, [2] p. 93-94). This happened also in shipping industry 9 times since 1914: in 1914: 4.57σ; in 1920: 3.45σ; in 1950: 3.06σ; in 1970: 3.28σ; in 1972: 4.9σ; in 1973: 7.04σ; in 1974: 3.65σ; in 2004: 5.23σ; and the highest 9.56σ, in October, 2008 meltdown [37] . Moreover, the freight rates obtained by a vessel between 1989 and 2008 (1043 weeks) [37] showed a right fat tail going up to +4.65σ.

Alpha appears in the formula of the broader L-stable characteristic function (οr L-stable probability distribution): log f ( t ) = i δ t − γ | t | α [ 1 + i β ( t / | t | ) tan ( α π / 2 ) ] , [written in Mandelbrot and Hudson [2] . This curve has 4 key parameters that determine its shape. In fact these parameters determine whether we have a bell curve, a Pareto-style one, etc.: delta-δ―is the location parameter; gama-γ―is the scale parameter, determining the magnitude of probabilities overall; beta-β- determines skewness: if 0, the curve is symmetrical like bell; alpha determines the amount of “fat” in the tails. If x lays 22σ from the mean (as in 1987), the fat is quite significant.

In Figure 6 we showed that maritime alpha⁴⁰ tended to 1.46 (rounded). This is best presented by the in-between curve, (Figure 7), not by the bell! Mandelbrot and Hudson [2] found alpha = 1.70 for cotton prices, and characterized these as subject to “strong variation”. Maritime prices, however, were wilder after⁴¹ 1971-1972, than cotton prices, as shipping alpha → 1.46 < 1.70…

First myth is that the exclusive “appropriate” time series models⁴² are linear⁴³ (autoregressive-ΑR model; GARCH). Very important is that the models {1 & 2}, shown in footnote 42, are unable to explain a number of important features shown by shipping and finance time series such as: leptokurtosis (fat tails; excess peak); volatility clustering or pooling (where volatility appears in bunches); leverage effects (where volatility rises more in large price falls) [40] .

Dealing with this matter for a long time, we found, in many shipping papers, that shipping time series behave as those in finance. Jing et al. [41] argued apropos that the daily returns of BCI, BPI and JEHSI, indices for Cape, Panama and Handy-sized ships―though found non-normal by the Bera and Jarque [42] test―have the typical features of stock returns, i.e. fat tails and spiked peaks (for 01/03/1999-23/12/2005)!

As argued by Peters [28] and by K Syriopoulos [43] , a very important characteristic of finance―and shipping, we add―time series, is their “nonlinear dependence”, measured by H. This leads the classical linear analysis of time series in relative ineffectiveness. If the linear dependence, or zero hypotheses, is rejected, then time series can be non-stationary, obey a low-dimensional chaos and as well having other properties. The existence of a low dimensional chaos, i.e. < 10 variables, permits forecasting…

A test of nonlinearity―as concluded from the above―would apparently be useful. One of the widely used tests⁴⁴ is the BDS one [44] . This last test assumes that data are completely random, or depart to deterministic chaos etc. Goulielmos and Psifia [37] applied this to shipping time series and found them non-IID⁴⁵ with confidence level 99.5% for embedding dimensions from 2 to 5 and radius values from 1/2σ to 2σ [45] .

Whoever can forecast demand (= seaborne trade) is King in shipping. But, tiny deviations of seaborne trade, however, give huge amounts of tons required to be transported… due to the different scales of demand and supply. Stopford (2009; [3] p. 718 and after) forecast seaborne trade over 1996-2005, fitting a linear regression model on data regressing seaborne trade on global GDP: ST t = a + b GDP {1}; he found ST t = − 26 . 289 + 3 0. 9GDP t {2}, which indicates that for 1 point increase in global GDP, seaborne trade rises by ~31 million tons, with a correlation coefficient 0.99.

The above regression used by Stopford (2009 [3] ) to predict seaborne trade from 1995 to 2005. The 2005 prediction gave 6.79 billion tons, where the actual was 6.78b tons. The difference was 10 m tons. This thus underestimated demand by 200 ships of 50,000 net dwt each (200 * 50,000 = 10m dwt). This, however, cannot be considered as a successful forecast as considered by Stopford [3] .

In shipping, it matters the size of what is predicted as a tiny deviation of only ~0.15%―as in this case on 6.79 billion tons sea trade―means a ~10 million tons demand for ships… Predictions by Stopford (2009 [3] ; p. 719) for “oil trade” and “dry cargo one” deviated more than in the case of total seaborne trade, i.e. by −400 m tons and +300 m tons respectively for 2005! The ships required in this instance for dry cargoes were 6000 additional ones (of 50,000 net dwt each) and for oil 8,000 ships were not needed…

For shipping time series, our interest is on “nonlinear dynamics⁴⁶”. First concern―as mentioned―is if the “Random Walk” hypothesis is valid, i.e. if X t + 1 = X t + U t [weak form], where X_t stands for logX at time t and U_t stands for a white noise process [46] . Using Bera and Jarque [42] test it would be easily to show that almost all shipping (and finance) time series do not pass normality test [41] .

The methods that can be used for nonlinear forecasting are five⁴⁷. Eight applications from 1990 to 1999 (not mentioned here, but in NLTSA 2000 [35] ) showed that forecasting a time series “looking as if it was random” (=chaotic), gave better⁴⁸ results than by using an autoregressive model―AR(1).

Here we will use a different approach: using the time series up to 2008, presented already in Figure 1, we will forecast inside the sample the last 7 years (2009-2015) (n = 260 − 1) with all five methods (Table 1). Since we know the actual figures for 2009-2015, we will find out which method gives the closer predictions. Then we will use this method to forecast next 5 years (2016-2020) outside the sample.

As shown, KDE gives predictions closer to actual, except for 2010, where PCR is better (the closer predictions are highlighted). In all 5 methods of Table 1, we assumed embedding dimension 5; time delay 1 and the number of nearest neighbors equal to 16. These values were selected based on our previous experience [47] and many tests carried out inside the sample by trying different coefficients; except for time delay = 1, which is recommended by NLTSA [35] computer program.

KDE stands for the Kernel density estimation. Forecasts with this method are

given by X ( N + 1 ) prediction = ∑ j = 1 k W j X j + p + ( m − 1 ) T {1}, where―if possible― ∑ W i = 1 . W i = f ( R i ) / ∑ f ( R j ) {2}, where j = 1 to k. R_j are the distances to which we attach a weight f ( R i ) = exp { − R i ( square ) / c 2 } {3}, where c is the

mean of R_i. For the last point X_T (bold stands for a vector), we select the k nearest neighbors X_j, where j = 1(1)k. m stands for the embedding dimension, the time delay is T, k is the number of the nearest neighbors and p = T − k.

KDE (n = 259 years) gave the following forecasting outside the sample for 2016-2020 (Table 1; Figure 8): 2016: 175 (improved 192); 2017: 172 (190); 2018: 173 (188); 2019: 170.5 (179) and 2020: 157 (167.4). The improved forecasts mentioned in parenthesis (and closer to real than the previous ones shown in Figure 8)

(*) Variance = 80%; (**) b = 1.5; e d = 5; t d = 1; k = 16; n = 259; H = 0.70 (rounded).

obtained after changing the embedding dimension⁴⁹ to 8, from 5, and k to 20 from 16. The new predictions and forecasts are shown in Table 1 marked “new”. As shown 5 yearly predictions and forecasts have improved and two: *** and **** were worse.

In more detail, the new prediction inside the sample for 2009 are even closer to actual (+1.9% greater); for 2015 the difference ?as expected- is greater, i.e. -~10%. The method KDE forecast, however, the falling trend for 2009-2020. We can use here our dictum: “an up to 90% - 98% accurate forecasting in shipping is better than no forecasting at all”. Shipping is considered unpredictable and a scientific forecasting even from 2% to 10% away from reality for certain years―as above―is of substantial help, we believe. Moreover, the above predictions and forecasts do not indicate a cycle, or any turning points, but only a falling trend. This is also very important information.

Paper’s contributions are: 1) to show that many time series are persistent, and a test exists for that, and these time series are prone to cycles following trends; time series “travel” in different speeds, where Einstein’s one travels at the square root of time. Have time series a memory? Do they have Long term dependence? Are they chaotic? 2) the torturing issue for me as a maritime economist for many decades is to forecast shipping cycles and their turning points; our hopes ?since most of the linear models have failed as these have been tested in reality for the second time in meltdown and since 1918. As a result we have resorted to alternative models like those of nonlinear chaotic nature.