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Studying the history of “economic turbulence”, we ended up with the certainty that the statistical methods used so far, by economists, are unsuitable to model it. We have mainly the methods of Normal distribution and Random Walk in mind. Moreover, the picture of reality we get from time series depends on the time-frame of the data used… Different time-frame data, different reality… In addition, econometricians provided a whole family of econometric models to approach reality, starting in early 1980s, with “autoregressive models—AR” combined or not with MA (moving averages). But even its 1986 flagship, the GARCH, with its many variations, cannot cope with a number of characteristics, one of which is leptokurtosis (small alpha, higher peaks and long tails), though some argue that it can cater for outliers. Economic turbulence, low or high—despite its characterization by Science as rare—became frequent since 1987 (Black Monday)... In late 1990s e.g. the global financial system underwent 6 crises—which have been called “near turbulences”—over a number of countries, including Russia in 1998. The next turbulence will not be one generation apart—we reckon. This paper is an attempt to invite writers to write a “theory of economic turbulence”. Turbulence is a nightmare, which wakes people up suddenly, and unexpectedly, but it is something people wish to forget… till it strikes again: turbulence stroke in 1929 on (Black) Tuesday, then in 1987 on (Black) Monday and in end-2008, the Great Recession—on 29th September. In Black Monday stock markets around the world crashed losing a huge value in a matter of very short time (Hong Kong, Europe, and USA). The Dow fell ~23%. At that time OPEC collapsed in 1986 and the price of oil doubled… The dry cargo shipping sector entered a turbulent situation since 1989, which has been deteriorated since 2015 reaching finally an alpha equal to ~1.43 < 1.70 by 2035…

Economies, and industrial sectors, suffered suddenly in the past from economic turbulences from time to time. In particular in the shipping sector, there were periods when ship owners made great fortunes and others when they became bankrupt [

The frequency of turbulences is related to the ability, which managers have now to take faster decisions, using more rapid computers and multi-service mobile phones. In addition, millions of people work now in the financial sector. After all, we all live in interconnected and globalized world-being citizens of one and the same village: the Globe.

In order not to be misled, computer models, are not yet in a position to cope with the rapidity that real economic life changes, as manifested in the situations when stock exchanges collapsed faster than computers, contrary to what was supposed by theory.

The “Economics of Turbulence” is now at the agenda given the meltdown^{1} in end-2008, which reminded us of both the 1929 “Great Depression^{2}” and the 1987 Black Monday, just ~21 years ago. USA’s annual real GDP during 1929-1934 fell from $1000 b (constant 2005$) to $700 b (−30%) and recovered by 1938-1939 to $1000 b again (1910 = $550 b). Additional depressions have been recorded in 1819, in 1873-1896, in 1907, and in 1910-1911. Another Black Monday was the 09/05/1873 for Vienna stock exchange, and the “Long Depression” that followed.

Should the models based on normal distribution forecast turbulence? Normal distribution―as shown in

As shown, 68.2% of all outcomes are within ±1σ from mean μ, 95.4% are within ±2σ and 99.6% are within ±3σ. Beyond ±3σ, the probability of an outcome to be there is between ±0.2% and ±0.4%: points at which distribution curve touches zero.

In addition to the reliance on normal distribution, science has relied on models of 1) “classical linear regression”, 2) “ARMA”(autoregressive moving average models) and 3) “VAR” (multivariate time series with lagged values on right hand side), which are linear in nature, i.e. linear in parameters: Y = Xβ + u_{t}, where u_{t}~ N(0, σ^{2}), where “errors” follow normal distribution with mean 0 and variance σ^{2} [

Many relationships in finance, and shipping, are intrinsically non-linear [

This paper aims at looking on “economic turbulence”, which shipping and financial time series exhibit at occasions. Also, a method to forecast turbulence in the shipping sector for 20 years to come is used, outside the sample, i.e. for 2016-2035 using the nonlinear method―the “Kernel density estimation”.

The paper is organized as follows: next, is a literature review, followed by methodology. Then, the concepts of volatility, risk and uncertainty are presented. Next, the more suitable method to model economic turbulences is indicated. The “maritime economics freight index” is then presented. Next, the phenomenon of power-laws in stock exchanges is analyzed, and the meaning of coefficient alpha is given. Finally, we forecast the turbulences that we expect to appear in shipping sector till 2035, and finally conclude.

Ruelle and Takens [

Feynman [

Mello [^{3} processes” [

The scientific community, Mello [

Anonymous [

As shown, the peak of the distribution (alpha = 1/2) is, at f(x) = 0.65, at a higher height than if alpha was = 1, 1.5 or 2. This is the impact of alpha on the peak of the distribution, producing also fat tails, or long tails, beyond ±5σ.

Mandelbrot and Hudson [

events that common economic wisdom says should not happen and calls them “statistical outliers”. The assumptions underlying models are wrong. Standard financial models focus on a typical market behavior, i.e. modest price changes, real or seeming trends, a risky, but ultimately closely manageable world. Risk = ±3σ. For shipping this means ±308 units of the shipping index (

Trevethan and Chanson [

Juarez [^{4} model, applied in “weather forecasting” by the meteorologist Lorenz [

When Juarez used conventional methods the explained variance was 6%, but after certain dynamic transformations he made, he succeeded to increase that to 73%, fitting a linear regression (

As shown, the dependent variable is the “profit & loss X”, and the independent variables are: “cash flow Y” and “assets Z”. This model resulted in a better prediction, as argued.

Zou et al. [^{5} (02/01/2002-13/02/2009)― deviated from “normal” levels (kurtosis = 3; skewness 0). This indicates that the market exhibits significant abnormal return changes. The null hypothesis of “Jarque-Bera” “test of normality” and “BDS” “test of independence”, were both

rejected. This means that market returns contain unknown nonlinear dynamics, not easily captured by traditional linear models (italics added).

In summary, we showed, among other things, that turbulences are considered in a probabilistic manner, as rare economic phenomena. This, however, we doubt; looking at Black Monday, 1987, and at the Great Recession, in end-2008, there were only 21 years apart... Turbulences are also always chaotic and are expressed exclusively by the “Levy stable distributions”. Alpha < 2 creates fat tails and high peaks, a serious number of outliers and thus unexpected outcomes (turbulences).

First, we distinguish volatility/turbulence in 3 states: 1) when volatility is mild, and modelled by Normal distribution, and σ ≤ ±3. 2) When turbulence is modelled by an “in -between” distribution, and turbulence = alpha = 1.5 ≤ 1.70 and σ > ±3 ≤ ±12.99. This is a representation of the shipping sector. 3) When turbulence is modelled by a Cauchy distribution, being wild, and alpha = 1 and σ ≥ ±13.

We assume also that shipping time series behave like financial ones [

The relationship between Hurst exponent and alpha coefficient: Let R_{n} be the sum of a stable variable in a particular interval n, and R_{1} be its initial value, then [

R n = R 1 ∗ n 1 / alpha (1)

means that the sum of n values scales by n^{1/alpha} times their initial value. Taking logs:

alpha = log ( n ) / log ( R n ) − log ( R 1 ) = log ( n ) / o g ( R / S ) l = 1 / H (2)

where

H = log ( c ) + log ( R / S n ) / log n (3)

where log(c) = 0 being constant.

logR_{n} - R_{1} ~ R/S and thus alpha = 1/H (4)

Equation (3) has been developed by Hurst [_{n}); also

(R/S)_{n} = c × n^{H} (5)

(Rescaled) range is a method dealing with volatility rescaled to make it independent of time. It is a generalized method to measure the speed of a time series including the case of Einstein 1905 [

Distance = n (6)

n stands for the number of observations or of time steps. With H = 0.69 (rounded) there is “black noise” in shipping markets, which is characterized by catastrophes and discontinuities, which are abrupt up and down, indicating high peaks at the mean and fat tails…

In forecasting, we used the “Kernel Density Estimation” method―KDE. It is slightly simpler than the other local linear forecasting methods, being in fact a weighted average. We used this method successfully in other occasions [

X ( N + 1 ) prediction = ∑ i = 1 k W j X j + p + ( m − 1 ) T (7)

where―if possible―

∑W_{i} = 1. W_{i} = f(R_{i})/∑f(R_{j}) (8)

where j = 1 to k. R_{j} are the distances to which a weight f(R_{i}) is

attached = exp {−R^{2}_{i}/c^{2}} (9)

where c is the mean of R_{i}. For the last point X_{T} (bolds stand for vectors), we select the k nearest neighbors X_{j}, where

j = 1(1)k (10)

k. m stands for embedding dimension, the time delay is T, k is the number of the nearest neighbors and p = T − k. This method is due to Sugihara and May [_{τ} are included in x_{j}. Forecasting is based on the mean of the relocated points weighted, as far as the initial distance is concerned.

Business life―blackboard theorizing, and uncountable conferences―became familiar with the concepts of risk, volatility and uncertainty. Companies need help and effective ways of their protection from these three enemies (i.e. use of hedging). As a result, managers may easily become victims, where the “Sirens” are in the form of all kinds of derivatives.

Is volatility = turbulence?

The σ^{2} measures the spread of data round mean:

S ( n ) = { 1 / n ∑ n = 0 n - 1 ( u n - ( u ) n ) 2 } 1 / 2 , (11)

where u_{t} stands for discrete time series, (u)_{n} stands for the mean over time lag n; σ 2 = σ [^{6} 1993 etc. Volatility (mild) in addition is not disastrous. e.g. an earthquake at a low point on Richter’s climax causes volatility, but not devastation. Tsunamis, hurricanes, the Katrina, the Caroline, are different. Volatility^{7} is like a severe fever… to put it simply, and a broader concept of turbulence.

For econometricians, volatility is the degree to which a time series varies over time. It is usually measured by variance^{8}, denoted by the Greek letter σ^{2}. In finance, σ^{2} is often also used as a measure of risk^{9}. Risk^{10} is very important for companies; firms wish to manage, avoid or profit from it. Moreover, σ^{2} appears in the equation of normal distribution, and the probability of the risk of a decision is locked in ±3σ from mean… maximum, as mentioned.

Let us present volatility in shipping markets (

As shown (left), all 3 main ship sizes and markets had strong variations in their earnings since 2011; par excellence Capes^{11} (blue line) had for the first time in 2012 hires (=time charters), which fell from >$42,350 per day to $2,350 (!), as well in 2016. The “BDI-Baltic Dry Index” (right) fell from a high of ~11,800 units in 2008 to ~400 by 2015. Capes were heavily involved in China’s imports in serious volumes.

Shipping and Finance cannot be modeled by Normal distribution... due to the alpha they have [

Normal distribution

The equation of normal^{12} distribution (

f ( x ) = 1 / σ 2 π ∗ e − ( ( x − μ ) / σ ) 2 / 2 (12)

where x is the level of a specific variable under study, μ is the average value of all x’s round average, σ stands for standard deviation―and e is a constant^{13}.

As shown, the tails approach zero at ±3.5σ from mean^{14} (X = 0).

Normal distribution is the statistical expression of “human justice”, which indicates that there is an “equal” return for everyone (egalitarianism). “Normal” distribution describes a world of mild circumstances. Normal markets are fair... Normality also means that all price changes taken together from small to large vary in accordance with the mild, bell-curve, distribution. What about if data

show that the magnitude of price changes depends on those of the past? Can markets exhibit dependence without correlation? Can large price changes tend to be followed by more large changes, + or ?, and vice versa? Is volatility clustering?

For some time, and especially since 1987, evidence showed that normal distribution is not adequate at times to describe market returns, and thus a need arises to

replace it. We know that there are cases where amplifications occur at extreme values, and often we have a long-tailed distribution.

As shown (

The long-tailed distributions led Levy (1937) to formulate a “generalized density function”, where “Normal” (and “Cauchy”) distribution are special cases, using a generalized “Central Limit Theorem^{16}”. These distributions―called “stable Levy distributions―SLDs”―are useful in describing the statistical properties of turbulent flows.

One characteristic function of an SLD is:

log f ( t ) = i δ t − γ | t | α [ 1 + i β ( t / | t | ) tan ( α π / 2 ) ] (13)

[^{17} = 0, these stand for Cauchy distribution, and if alpha = 1.50 and β ≠ 0, these stand for “shipping and finance distributions”, as well as Pareto’s distribution. Alpha takes values from 0 to 2, with variance 2 × γ^{2}, and for shipping and finance alpha takes values from 1 to 1.7.

A proper model for turbulence is the one which allows: wild price fluctuations-big jumps, fat tails (alpha < 2), volatility clustering―here and there―periods of big price changes, grouping together, interspersed by intervals of more calm variation―long memory and persistence (H > 0.5 ≤ 1), and scaling of price series.

The time series of “maritime economics freight index”, 1741-2007 (extended to 2015) [

As shown, the shipping distribution―fitted to 266 years―exhibits a positive skew, and is not normal. The skew is positive and greater than zero^{18}. More important is that the fitted distribution indicates a long tail on the right, and peak at ~144 units, which are measured by alpha―mentioned below―equal to 1.46 < 2. The shipping standard deviation went away from its mean by 6.52σ in 2008…

Turbulences/volatility in shipping historically is as follows: In class I, recorded none. In class II recorded none too. In class III were recorded 14 cases: in 1916 3.8; 1917 7.1; 1918 7.7σ; 1919 5; 1920 3.8σ; 2004 3.4σ; 2007 5.7; 2008 6.5σ; 2009 3.6; 2010 4.5; 2011 3.9; 2012 3.4; 2013 3.3 and in 2014σ = 3.1.

The dry cargo market held up till 1974 and for small bulk carriers into 1975. There was stock building in the world economy due to commodity price inflation and the heavy congestion in Middle East and Nigeria due to increased oil revenue [^{nd} half in 1981 till 1^{st} half in 1987. Stock markets are, however, wilder.

so years (~45 years) showed turbulences from 9% to 14% and the rest 15 years (12.5%), showed turbulences from 15% to ~25%.

Turbulence, though studied for more than 100 years, is only partly understood by theory [^{19}, as called by economists and econometricians. Also, the size of firms and industry’s concentration influence profits, and in turn profits determine stock prices.

The key traits of turbulence are: “scaling”-γ and “long-term dependence”―or a Hurst exponent H > 0.5 ≤ 1. Also, “self-similarity” is a property of invariance against changes in scale or size. Small parts of an object are qualitatively the same―or similar―to the whole [

Economic history is also full of “near”^{20}―turbulent events―like that of October 27^{th}, 1997. The Dow then lost a 7.2%; cascades of selling occurred across exchange―forcing authorities to halt trading in 2 times [

The place where chaos^{21} reigns over charts is Wall Street. In stock and commodity exchanges, “self-similarity” weighs in on many scales. The paradox here is that the chart of minute-by-minute stock averages looks much alike the daily averages and so on in weekly, and in monthly prices… There is an uncharacteristic jump in the data―as in October 1987 as shown in

The “linear” (normality) assumption is that in stock averages the actual prices are generated by independent increments. This has a power spectrum that is proportional to the inverse square of the frequency, called “brown noise”, as an allusion to “Brownian motion”. The name of this motion is after the Scottish botanist Robert Brown (1773-1858), who in 1827 saw floating dust specks to “move” in a microscope. The innovation process consists of the independent “kicks” given to the suspended particles by the molecules of the liquid (in which they float/hover).

Pareto’s Power LawPareto (1848-1923) was an industrialist, economist and sociologist. The field of economics evolved by his efforts from a branch of social philosophy―as practiced by Adam Smith (1723-1790)―into a data―intensive field of scientific research and mathematical equations. He attracted by the way “power” and “wealth” is distributed. He gathered extensive data from 1471 to 1512, and plotted them (in 1909) on a graph (

As shown, the curve is not symmetrical. The slope of the Pareto curve― alpha―is equal to −3/2 = −1.5 (power law)… Money begets money, power creates power. “The first 1/2 a million $ is difficult, while the $1 m is easier thereafter” or “increasing returns to scale”, which economists ignored^{22}... [

P(u) = (u/m)^{−}^{α} (14)

where P is the proportion of people earning more than some level of income u, and m is the minimum income^{23}.

Alpha is a characteristic exponent restricted to values [0-2] or rather [1-2]― where 0 < alpha ≤ 2―shapes the distribution it belongs to. The smaller alpha, the heavier (fatter) the tails of the (stable density) distribution are. Several outliers (extreme outcomes) will be observed.

As shown, alpha wanders round 2, meaning: a normal distribution since 1750 and till 1975 (225^{th} year). From 1975 and till 2015, however, alpha gradually fell from 2 to 1.4561976 = 1.46 (rounded) for n = 10 on 266 yearly observations. As argued by Peters [^{24} is related to the velocity of a fluid and not to fluid’s movement, entailing 0 ≤ H < 0.50. As alpha decreases, both the occurrence rate and the strength of outliers increase, resulting to very impulsive processes. In

is plotted against time (years) since 1741 and till 2015^{25,26}.

Financial (and shipping) time series are known to be non-stationary, where the statistical calculations for σ etc., are valid only for stationary series. Here we have applied 1^{st} logarithmic differences to obtain stationarity. This transformation is due to Box and Jenkins [

The tendency towards catastrophes (turbulences) has been called by Mandelbrot [

Equation: H = log ( c ) + log ( R / S n ) / log n (15)

will be used, where c is a constant, R is the range, n is a time index and it equals the number of observations, S is the local standard deviation and H is the Hurst exponent. This is based on Hurst’s equation

log ( R / σ ) = K log ( N / 2 ) (16)

or R = σ ( Ν / 2 ) Κ (17)

where K = H. Hurst [

It has the advantage to be non-parametric―ignoring the organization of the original data [

The “fractional Brownian motion”, as a generalization to processes which grow at different rates

t^{H} is: ([X_{H}(t) - X_{H}(0)]^{2})^{1/2} with a linear | t | H (18)

where 0 < H < 1 is the Hurst exponent. Einstein [

R = kT1/2 (19)

where R is the distance covered, k a constant and T the time index or n.

This, however, can be generalized to:

R/S = k T^{H} (20)

where T = the time index or n, or N. R/S has no dimension; the power value is equal to H. There is no matter if periods are separated by many years; and there is no need for time series to have a characteristic scale. Equation (6) has a characteristic of fractal geometry, as it is scaling in accordance with a power law― the power of H. As a result a time series may move covering time at the square root of time-this characterized the time series which are random and H = 1/2 = 0.50 (normality).

The most interesting case is the one where time series move at H > 1/2 ≤ 1, called black noise, persistent, as well possessing the Joseph and the Noah effects [^{27} as mentioned―are faster, and cover more distance than random walk-the system increasing in one period is more likely to keep increasing in the immediately following period. More important is that these time series has the potential of sudden catastrophes (the Noah effect) or of turbulences! This also allows for short term turbulences, say of 7 years down (according to Joseph effect)…

Forecasting turbulences outside the sample

For forecasting turbulences, we will use 1/H = alpha for 2016-2035 using NLTSA (2000) computer program due to Syriopoulos and Leontitsis [

As a result, the forecasts indicate that turbulences in shipping dry cargo market will continue and will be intensified by and including 2035; we have 18 years still to wait to see reality if still alive…

Turbulence is an extreme economic phenomenon caused by forces, which should have a certain degree of intensity/energy, exhibiting always chaotic behavior. More important, turbulences cannot be captured by a probability density distribution like the Gaussian, because by definition it does not allow for long tails and high peaks―fact indicated by alpha < 2. Alpha < 2 ≥ 1 shows turbulences in the market to which time series belongs to.

The problem of turbulence is one of methodology: markets behave in one of three states: slow, mild and wild. Economists created only one (convenient) method to model all three: “normal distribution”. Moreover, the great majority of econometricians do not admit that normal distribution is unable to model “wild” circumstances, with ∞ variance or undefined mean, proposing^{28} linear, models like GARCH.

GARCH in particular, when volatility jumps, it plugs-in new parameters to make the bell curve grow... and vice versa. The bell distribution is thus treated like a balloon to cope with actual volatility. Moreover, shipping sector and finance are special cases, because their time series―returns, indices or time series―do not follow either Normal distribution or Cauchy distribution…

Very astonishing is the fact that shipping sector since 1741 had time series of dry cargo freight rates, distributed following normal distribution till 1975, as indicated by the prevailing alphas. Turbulences in the sector occurred indeed after 1975: one depression appeared in 1981 till 1987, one in 2009―due to the end-2008 meltdown, and the recent depression―after China’s slowdown―in 2011-2018_{. }

Economists indeed accommodated 3 important concepts: “volatility, uncertainty and risk” into one, i.e. the σ of the bell curve, without reference to the distribution it belongs to. For Cauchy distribution―even if we wanted to define σ―we cannot do it… because variance is ∞ or undefined...

In addition, normality is a phenomenon of…the long run, it appears when recessions, depressions and turbulences, calm/boil down or even disappears from time series! This denies what Keynes wrote in 1936 that ‘in the long run, we are all dead’; because in the long run everything in the economy is more peaceful, as shown by normal distribution… At short time-frames prices vary wildly, and at longer time frames they settle down.

Despite the above, the yearly observations we used for shipping, detected wilderness in the last 13 years (2003-2015) pushing alpha from 2 in the interval [1.70, 1.43]. Shipping companies (and maritime economists) thus may be misled about reality according to the time-frame of data they look at… Minute by minute, hour by hour, day by day, week by week are time-frames that reveal the true wild face of markets; it is then also that decisions are taken… long runs are suitable for plans, but days are only suitable for decisions. Decisions transformed in actions are the blood and the energy for life in the economic organisms like markets. It is this energy that causes turbulences according to its energy, power and intensity.

Goulielmos, A.M. (2018) The Nature of Economic Turbulence: The Power Destructing Economies, with Application to Shipping. Modern Economy, 9, 1023-1044. https://doi.org/10.4236/me.2018.95066