Maritime
Businesses: Volatile Stock Prices and
Market Valuation Inefficiencies
Robert F. Mulligan, Ph.D.
Department of Business Computer Information
Systems & Economics
Phone: 8282273329
Fax: 8282277414
Email: mulligan@wcu.edu
Gary A. Lombardo, Ph.D.
Professor of Maritime Business
Department of Marine Transportation
Phone: 5167735066
Email: lombardog@usmma.edu
Acknowledgements
Robert F. Mulligan is associate professor of economics in
the Department of
Abstract
This
paper examines twelve maritime equity price series for behavioural
stability and efficient market pricing for the 19892002 period. Five selfaffine fractal analysis techniques
for estimating the
Introduction
Twelve maritime
businesses are examined in terms of their stock prices and market
valuations. Statistical tests focusing
on five alternative methods for estimating
The paper is organized as follows. A literature review is provided in the second section. The data are documented in the third section. Methodology and results are presented in the fourth and fifth sections. The conclusions are provided in the sixth section. Additionally, the glossary and appendix have been prepared to assist the reader in understanding the uniqueness of the specialized statistical language used in this paper.
Mandelbrot's
(1972a, 1975, 1977) and Mandelbrot and Wallis's (1969) R/S or rescaled range
analysis characterizes time series as one of four types: 1.) dependent or autocorrelated series, 2.) persistent, trendreinforcing
series, also called biased random walks, random walks with drift, or fractional
Brownian motion, 3.) random walks, or
4.) antipersistent, ergodic, or meanreverting
series. MandelbrotLévy
distributions are a general class of probability distributions derived from the
generalized central limit theorem, and include the normal or Gaussian and
Cauchy as limiting cases (Lévy 1925; Gnedenko and Kolmolgorov
1954). They are also referred to as
stable, Lévystable, Lstable, stableParetian, and ParetoLévy. Samuelson (1982) popularized the term
MandelbrotLévy, but Mandelbrot avoids this
expression, perhaps out of modesty, and the other terms remain current. The reciprocal of the MandelbrotLévy
characteristic exponent alpha is the
Mandelbrot (1972b, 1974) and Mandelbrot, Fisher, and Calvet (1997) have developed the multifractal model of asset returns (MMAR), which shares the longmemory feature of the Fractional Brownian Motion (FBM) model introduced by Mandelbrot and van Ness (1968). Fractal structure in equity prices indicates traditional statistical and econometric methods are inadequate for analyzing security markets. The statistical theory necessary to identify empirical regularities and local scaling properties of MMAR processes with local Hölder exponents is developed by Calvet, Fisher, and Mandelbrot (1997) and applied by Fisher, Calvet, and Mandelbrot (1997).
<<Table
1 about here>>
Literature review
The search for long memory in capital markets has been a fixture in the literature applying fractal geometry and chaos theory to economics since Mandelbrot (1963b) shifted his attention from income distribution to speculative prices. Fractal analysis has been applied extensively to equities (Green and Fielitz 1977; Lo 1991; Barkoulas and Baum 1996; Peters 1994, 1996; Koppl et al 1997; Kraemer and Runde1997; Barkoulas and Travlos 1998; Koppl and Nardone 2001; and Mulligan 2003) interest rates (Duan and Jacobs 1996; and Barkoulas and Baum 1997a, 1997b), commodities (Barkoulas, Baum, and Ogutz 1998), exchange rates (Cheung 1993; Byers and Peel 1996; Koppl and Yeager 1996; Barkoulas and Baum 1997c; Chou and Shih 1997; Andersen and Bollerslev 1997; Koppl and Broussard 1999, and Mulligan 2000a), and derivatives (Fang, Lai, and Lai 1994; Barcoulas, Labys, and Onochie 1997; and Corazza, Malliaris, and Nardelli 1997).
Cheung and Lai (1993) suggest Heiner's (1980) and Kaen and Rosenman's (1986) competencedifficulty (CD) gap hypothesis as a potential source of long memory in asset prices, offering a theoretical expectation of long memory. The CD gap is a discrepancy between investors' competence to make optimal decisions and the complexity of exogenous risk, to which the maritime industry is particularly vulnerable. A wide CD gap leads to investor dependency on deterministic rules, which can lead to persistent price movements in one direction – either a crash or speculative bubble. Due to the irregular arrival of new information, Kaen and Rosenman argue persistent price movements may suddenly reverse direction, leading to nonperiodic cycles.
Mussa (1984) introduced a disequilibriumovershooting model for exchange rate determination. Disequilibriumovershooting, where market participants overcorrect when adjusting prices and quantities toward equilibrium, would be supported for equity markets by a finding of antipersistence or ergodicity, that is, of consistent price overadjustment, often followed by overcorrection in the opposite direction.
<<Table
2 about here>>
Data
The data are daily closing prices reported by the exchanges for each traded equity. A sample of twelve maritime firms was selected. Table 3 lists the twelve maritime firms examined, giving the sample range, number of daily price observations, and the standard beta measure of relative volatility for the last year in the sample.
The maximum sample
period is
The data are adjusted for stock splits. Failure to adjust raw price series for splits would have introduced discontinuities unrelated to any fundamentals. Such shifts in the mean can introduce biases toward finding fractal dynamics (e.g., Barkoulas, Baum, and Oguz 1998; Granger and Hyung 1999; Diebold and Inoue 2000). It is essential to test for fractal behavior using splitadjusted prices to reflect the actual return on the equities and ensure the findings are not a statistical artifact due to changes in the mean of the process, but instead reflect genuine features of the underlying data generating process. Data not adjusted for stock splits would introduce nonstationarities, the property that the mean of the data changes over time.
Methodology
Long memory series exhibit nonperiodic long cycles, or persistent dependence between observations far apart in time; i.e., observable patterns which tend to repeat. Long memory or persistent series tend to reverse themselves less often than a purely random series. Thus, they display a trend, and are also called black noise, in contrast to purely random white noise. Persistent series have long memory in that events are correlated over long time periods. In contrast, shortterm dependent time series include standard autoregressive moving average and Markov processes, and have the property that observations far apart exhibit little or no statistical dependence. R/S or rescaled range analysis distinguishes random from nonrandom or deterministic series. The rescaled range is the range divided (rescaled) by the standard deviation. Seemingly random time series may be deterministic chaos, fractional Brownian motion (FBM), or a mixture of random and nonrandom components.
Conventional statistical techniques lack power to distinguish unpredictable random components from highly predictable deterministic components. R/S analysis evolved to address this difficulty. R/S analysis exploits the structure of dependence in time series irrespective of their marginal distributions, statistically identifying nonperiodic cyclic long run dependence as distinguished from short dependence or Markov character and periodic variation (Mandelbot 1972a: 259260). The difference between longmemory processes, also called nonperiodic long cycles, and shortterm dependence, is that each observation in long memory processes has a persistent effect, on average, on all subsequent observations, up to some horizon after which memory is lost, whereas in contrast, shortterm dependent processes display little or no memory of the past, and what shortterm dependence can be observed often diminishes with the square of the time elapsed. For equity prices, long memory can be observed when a stock follows a trend or repeats a cyclical movement, even though the cycles can have timevarying frequencies. Shortterm dependence is indicated when there are no observable trends or patterns beyond a very short time span, and the impact of any outliers or extreme values diminishes rapidly over time.
Mandelbrot (1963a, 1963b) demonstrated all
speculative prices can be categorized in accordance with their
Results
Many maritime security
prices are antipersistent
or ergodic,
meanreverting,
or pink noise
processes with (0.00 < H < 0.50), indicating they are more
volatile than a random walk. Pink noise
processes are used to model dynamic turbulence.
B&H Ocean Shippers was the only firm that did not provide evidence
of ergodicity for at least part of the period
19892002. Ergodic
or antipersistent processes reverse themselves more
often than purely random series. Ergodicity, that is, H significantly below 0.50, strongly
disconfirms the efficient
market hypothesis, indicating market participants persistently
overreact to new information, imposing more stock price volatility than would
be consistent with market efficiency, and participants never learn not to
overreact. This observed phenomenon is
directly analogous to Mussa's (1984) disequilibrium
overshooting, in which the market process of adjustment toward final equilbrium prices is unstable, and never quiets down. Hs significantly above 0.50 demonstrate stock
prices are not random walks, shedding some doubt on weak market efficiency and
indicating technical analysis can provide systematic returns. Any findings of nonnormality or nonGaussian
character have serious implications for pricing financial derivatives. Because the BlackScholes
(1972; 1973) option pricing model assumes normallydistributed prices for
underlying securities, financial derivatives based on nonnormal securities
prices cannot be priced efficiently.
<<Table 3 about here>>
This section
discusses and interprets the results of five alternative fractal analysis
methods for measuring the
Five techniques
for estimating the
1.) Rescaledrange or R/S analysis: R/S analysis is the traditional technique introduced by Mandelbrot (1972a). Hs estimated by this method are mostly around 0.50, superficially suggesting Gaussian processes and supporting the efficient market hypothesis. However, the difference between estimated Hs and 0.50, though generally not great, is statistically significant over the whole sample range and both subsamples for Alexander & Baldwin and Maritrans. For Seacor Smit, Kirby, CSX, and OMI, H is significantly different from 0.50 for the earlier subsample, and for MC Shipping, H is significantly different from 0.50 for the later subsample. These results formally support the multifractal model of asset returns, which is more general, by disconfirming the weak form of the efficient market hypothesis for the maritime firms in those periods. Significant Hs are below 0.50, indicating ergodicity or antipersistence (e.g., negative serial correlation meaning the market price persistently overcorrects) for Alexander & Baldwin (all ranges), CSX (early range), Kirby (early range), MC Shipping (late range), OMI (early range), and Maritrans (all ranges). These results provide further support for the multifractal model of asset returns and further difficulty for weak form efficiency, the least restrictive form of the efficient market hypothesis. This measurable antipersistence or ergodicity demonstrates market participants habitually overreact to new information, and never learn not to.
Hs different from 0.50 demonstrate the return series have not been random walks, shedding significant doubt on weak market efficiency and indicating technical analysis could have provided systematic returns. Nevertheless, this finding may be due to shortterm dependence still present after taking AR1 residuals, or systematic bias due to information asymmetries, or both.
2.) Power spectral density analysis: Hs estimated by this technique in the persistent range (H > 0.50) are OMI (all ranges), B&H and Kirby (early and late ranges), Seacor Smit, International Shipholding, MC Shipping, and Overseas Shipping Group (early range), Tidewater (whole and early ranges), and Mexican Maritime Transportation (late range). Note these results often flatly contradict those provided by other techniques. Spectral density does not provide a standard error for H, and thus cannot be used for formal hypothesis testing. Powerspectral density analysis indicates Alexander & Baldwin, CSX, and Maritrans to be consistently ergodic over all samples, providing further evidence of price overcorrection and market inefficiency.
3.) Roughnesslength relationship method: This method provides the most consistent support for the multifractal model of asset returns. Formal hypothesis tests reject the Gaussian null for all series and all sample ranges, except B&H, International Shipholding, and Mexican Marine Transportation (whole and late ranges), though the null of normality is rejected for the early subsample for these three firms. One difficulty in applying the roughnesslength method is that the standard errors are always so low the null hypothesis of H = 0.50 is nearly always rejected no matter how nearly normal the asset returns. The seemingly unambiguous rejection of weak market efficiency provided by this technique is best viewed cautiously. The rule of thumb adopted here is to look for rejection of the Gaussian null by at least one other technique.
4.) Variogram analysis: Variogram analysis supports the multifractal model of asset returns and disconfirms weak market efficiency for only two equities out of the sample of twelve: Alexander & Baldwin (late range) and Mexican Marine Transportation (early range). Viewed in isolation, the interpretation would be that most maritime equities are valued efficiently, but clearly not all.
5.) Wavelet analysis: This method was developed by Daubechies (1990), Beylkin (1992), and Coifman et al (1992). Wavelet H estimates indicate antipersistence or ergodicity (H < 0.50) for Alexander & Baldwin (all ranges), MC Shipping (late range), OMI (early range), Tidewater (late range), and Maritrans (whole range), indicating persistence (H > 0.50) elsewhere.
<<Table
4 about here>>
Hypothesis tests are constructed to test for 1.) the Gaussian character or normality of the underlying time series, 2.) Cauchycharacter, and 3.) changes in price behavior between 19891994 and 19952002:
1.) Tests of Gaussian character or normality: Table 4 presents tstatistics for tests of
the null hypothesis H = 0.50, along with twotail probability levels. Tstatistics are computed as the
Gaussian character is a sufficient condition for weak market efficiency, but not a necessary condition. This result is generally interpreted as support for the more general multifractal model of asset returns, and disconfirmation of the weakform efficient market hypothesis; however, it really does not prove that the market is inefficient. More importantly, findings of H < 1 strongly reject weak market efficiency because they demonstrate ergodicity of antipersistence. These findings are absolutely fatal to the BlackScholes [5051] option pricing model and its underlying assumption of normallydistributed prices for the underlying securities. Financial derivatives based on nonnormal securities prices cannot be priced efficiently. Thus, even if the equity markets for maritime stocks are efficient, in spite of the substantial empirical evidence against market efficiency, the derivatives markets clearly are not efficient.
2.) Tests of Cauchy character: the MandelbrotLévy characteristic exponent test: Various statistics are available to test
the null hypothesis of normality, but not for the Cauchy distribution, the
other extreme. The MandelbrotLévy characteristic exponent alpha is computed as the
reciprocal of the
<<Table 5 about here>>
3.) Tests of structural change: Table 5 presents tstatistics testing for significant differences among Hs estimated over the whole sample range and the two subsamples, referred to a ranges A, B, and C. The first null hypothesis tested for each equity is that the Hs for the two subsamples are equal (B=C), with degrees of freedom equal to the sum of sample average R/Ss in the two regressions estimating H for each subsample. The second and third null hypotheses are that the H for each subsample is equal to the H estimated over the whole sample (A=B and A=C) with degrees of freedom equal to the number of R/Ss in the wholesample regression, because the standard error of the whole sample H is treated as the pooled standard error.
With time series that may be fractional Gaussian noise, that is, apparently random combinations of otherwise statistically wellbehaved processes scrambled together with periodicallychanging parameters and characteristics, it is not strictly correct to infer structural change in the conventional sense. For example, a random scrambling of several different finitevariance processes can result in an infinitevariance process over a larger sample range. The multifractal model of asset returns achieves extreme generality by incorporating the possibility of such finitevariance processes. The finding that H is not constant over two subsamples and the whole sample range is wholly consistent with a stable fractal process, but more importantly, it points to some difference in fundamentals, or at least in behavior of the variable studied, from one period to the other.
Conclusion
Equities traded in
efficient markets should have
All the maritime securities in this study yield strong evidence of antipersistence, ergodicity, or negative serial correlation, except B&H Ocean Carriers. This means the markets for these securities must be described by the multifractal model of asset returns and cannot be described by the efficient market hypothesis, not even in its least restrictive, weak form. Formal hypothesis tests indicate H is significantly less than 0.50, though often not by a great magnitude, for eleven of the twelve securities sampled (B&H being the exception), by at least one technique (usually roughnesslength, but occasionally by R/S, variogram, or some combination). The conclusion suggested is that market participants are incapable of efficiently valuing some maritime equities and that they persistently overreact to the arrival of new information, and never learn not to overreact.
A possible scenario that renders this finding more intuitive is that information relevant to the valuation of a given maritime firm arrives frequently and seemingly at random. Market participants habitually ignore the vast majority of this information, until it accumulates a critical mass they must finally recognize. Then, perceiving they have ignored a body of relevant information which they have allowed to accumulate, they attempt to compensate for their history of informational sloth by overreacting. Furthermore, most investors are laypeople not well suited to evaluate relevant market information pertaining to a technically specialized industry like maritime shipping. Confirmation of the more general multifractal model of asset returns is an important implication of findings of ergodicity which disconfirm the weak form of the efficient market hypothesis. Disconfirmation of the efficient market hypothesis in its weak form suggests possibilities for constructing nonlinear econometric models for improved price forecasting and option valuation.
Any findings of nonnormality or nonGaussian character have serious implications for pricing financial derivatives. Because the BlackScholes (1972; 1973) option pricing model assumes normallydistributed prices for underlying securities, financial derivatives based on nonnormal securities prices cannot be priced efficiently. This finding of inefficient derivatives markets is completely independent of, and robust to, inefficient markets for equities which underlie the derivative assets.
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Glossary of
Fractal Analysis Terms 
Antipersistence –
a series that reverses itself more often than a purely random series, also
called pink noise, ergodicity, 1/f noise, or
negative serial correlation. (Peters 1994: 306). 
Black noise – a series that reverses itself less
often than a purely random series, displaying trends or repetitive patterns
over time, also called persistence, positive serial correlation, or
autocorrelation. (Peters 1994: 183187). 
Brown noise – the cumulative sum of a
normallydistributed random variable, also called Brownian motion. (Peters
1994: 183185; Osborne 1959). 
Efficient Market Hypothesis – the proposition that market prices fully and correctly reflect all relevant information. A market is described as efficient with respect to an information set if prices do not change when the information is revealed to all market participants. There are three levels of market efficiency: weak, semistrong, and strong. (Fama et al 1969; Malkiel 1987). 
Long memory – the property that any value a series takes on generally has a long and persistent effect, e.g., extreme values that repeat at fairly regular intervals. (Peters 1994: 274). 
Multifractal Model of Asset Returns (MMAR) – a very general model of asset pricing behavior allowing for longmemory and fattailed distributions. Instead of infinitevariance distributions such as the MandelbrotLévy and Cauchy distributions, the MMAR relies on fractional combinations of random variables with nonconstant mean and variance, providing many of the properties of infinitevariance distributions. (Mandelbrot, Fisher, and Calvet 1997). 
Nonperiodic long cycles  a characteristic of longmemory processes, i.e., of statistical processes where each value has a long and persistent impact on values that follow it, that identifiable patterns tend to repeat over similar, though irregular, cycles (nonperiodic cycles.) Also called the Joseph effect. (Peters 1994: 266). 
Nonstationarity – the property that a series has a systematically varying mean and variance. Any series with a trend, e.g., U.S. GDP, has a growing mean and therefore is nonstationary. Brownnoise processes are nonstationary, but whitenoise processes are stationary. (Granger 1989: 58). 
Persistence or persistent dependence – a series that reverses itself less often than a purely random series, and thus tends to display a trend, also called black noise. Persistent series have long memory in that events are correlated over long time periods, and thus display nonperiodic long cycles. (Peters 1994: 310). 
Semistrongform Market Efficiency – the
intermediate form of the efficient market hypothesis, asserting that market
prices incorporate all publicly available information, including both
historical data on the prices in question, and any other relevant,
publiclyavailable data, and thus it is impossible for any market participant
to gain advantage and earn excess profits, in the absence of inside
information. (Peters 1994: 308). 
Shortterm dependence  the property that any value
a series takes on generally has a transient effect, e.g., extreme values bias
the series for a certain number of observations that follow. Eventually, however, all memory of the
extreme event is lost, in contrast to longmemory or the Joseph effect. Special cases include Markov processes and
serial correlation. (Peters 1994:
274). 
Spectral density or Powerspectral Density Analysis – a fractal analysis based on the power spectra calculated through the Fourier transform of a series. (Peters 1994: 170171). 
Stationarity – the property that a series has a constant mean and variance. White, pink, and blacknoise processes are all stationary. Because it is the cumulative sum of a whitenoise process, a brownnoise process is nonstationary. (Granger 1989: 58). 
Strongform Market Efficiency – the most restrictive version of the efficient market hypothesis, asserting that all information known to any one market participant is fully reflected in the price, and thus insider information provides no speculative advantage and cannot offer above average returns. (Malkiel 1987: 120). 
Weakform Market Efficiency – the least restrictive version of the efficient market hypothesis, asserting that current prices fully reflect the historical sequence of past prices. One implication is that investors cannot obtain aboveaverage returns through analyzing patterns in historical data, i.e., through technical analysis. Also referred to as the Random Walk Hypothesis. One common way of testing for weakform efficiency is to test price series for normality, however, normality is a sufficient rather than a necessary condition. (Malkiel 1987: 120). 
White noise – a perfectly random process. Normallydistributed processes are a special case. (Peters 1994: 312). 
Appendix
Statistical methodology
Rescaledrange or R/S
analysis: R/S analysis is
the conventional method introduced by Mandelbrot (1972a). Time series are classified according to the
estimated value of the
R/S
= an^{H}
where R is the average range of all subsamples of size n, S is the average standard deviation for all samples of size n, a is a scaling variable, and n is the size of the subsamples, which is allowed to range from an arbitrarily small value to the largest subsample the data will allow. Putting this expression in logarithms yields
log(R/S) = log(a) + H log(n)
which is used to estimate H as a regression slope. Standard errors are given in parentheses. H ranges from 1.00 to 0.50 for persistent series, is exactly equal to 0.50 for random walks, ranges from zero to 0.50 for antipersistent series, and is greater than one for a persistent or autocorrelated series. Mandelbrot, Fisher, and Calvet (1997) refer to H as the selfaffinity index or scaling exponent.
Power spectral density analysis: This method uses the properties of power spectra of selfaffine traces, calculating the power spectrum P(k) where k = 2p/l is the wavenumber, and l is the wavelength, and plotting the logarithm of P(k) versus log(k), after applying a symmetric taper function which transforms the data smoothly to zero at both ends. If the series is selfaffine, this plot follows a straight line with a negative slope –b, which is estimated by regression and reported as beta, along with its standard error. This coefficient is related to the fractal dimension by: D = (5  beta)/2. H and alpha are computed as H = 2 – D, and alpha = 1/H. Power spectral density is the most common technique used to obtain the fractal dimension in the literature, although it is also highly problematic due to spectral leakage.
Roughnesslength relationship method: This method is similar to R/S, substituting the rootmeansquare (RMS) roughness s(w) and window size w for the standard deviation and range. Then H is computed by regression from a logarithmic form of the relationship s(w) = w^{H}. As noted above, the roughnesslength method provides standard errors so low the null hypothesis of H = 0.500 is nearly always rejected no matter how nearly normal the asset returns.
Variogram
analysis: The variogram, also known as variance of the increments, or
structure function, is defined as the expected value of the squared difference
between two y values in a series separated by a distance w. In other words, the sample variogram V(w) of a series y(x) is
measured as: V(w) = [y(x) – y(x+w)]^{2}, thus V(w) is the average value of the
squared difference between pairs of points at distance w . The distance of
separation w is also referred to as the lag.
The
Wavelet analysis: Wavelet analysis exploits localized variations in power by decomposing a series into time frequency space to determine both the dominant modes of variability and how those modes vary in time. This method is appropriate for analysis of nonstationary traces such as asset prices, i.e. where the variance does not remain constant with increasing length of the data set. Fractal properties are present where the wavelet power spectrum is a power law function of frequency. The wavelet method is based on the property that wavelet transforms of the selfaffine traces also have selfaffine properties.
Consider n wavelet
transforms each with a different scaling coefficient a_{i},
where S_{1}, S_{2},..., S_{n} are
the standard deviations from zero of the scaling coefficients a_{i}. Then
define the ratio of the standard deviations G_{1}, G_{2}, ..., G_{n1} as: G_{1} = S_{1}/S_{2}, G_{2} = S_{2}/S_{3},
..., G_{n1} = S_{n1}/S_{n}. Then the average value of G_{i}
is estimated as G_{avg} = (G_{i})/(n – 1). The estimated
MandelbrotLévy characteristic exponent test: The
MandelbrotLévy distributions are a family of
infinitevariance distributions without explicit analytical expressions, except
for special cases. Limiting
distributions include the normal, with finite variance, and the Cauchy, with
the most extreme platykurtosis or fat tails. Paul Lévy (1925) developed
the theory of these distributions. The
log
f(t) = i(delta)t – (gamma)t^{alpha}[1 + i(beta)(sign(t)(tan[(alpha)(pi/2)])],
where delta is the expectation or mean of t if alpha > 1; gamma is a scale parameter; alpha is the characteristic exponent; and i is the square root of 1. Gnedenko and Kolmogorov (1954) showed the sum of n independent and identically distributed MandelbrotLévy variables is:
n
log f(t) = in(delta)t – n(gamma)t^{alpha}[1 + i(beta)(sign(t)(tan[(alpha)(pi/2)])],
and thus the distributions exhibit stability under addition. Many applications of the central limit theorem only demonstrate MandelbrotLévy character. The result of normality generally depends on an unjustified assumption of finite variance. Mandelbrot (1972a) introduced a technique for estimating alpha by regression, further refined by Lo (1991). Mulligan (2000b) estimates the distribution of alpha for Cauchydistributed random variables. This distribution is used to test estimated alphas for technology equities against the Cauchy null.
Table 1 Fractal Taxonomy of Time Series 

Term 
'Colour' 

Fractal dimension 
Characteristic exponent 
Antipersistent, Ergodic, Meanreverting, Negative serial correlation, 1/f noise 
Pink noise 
0 ≤ H < ½ 
1.50 < D ≤ 2.00 
2.00 < a ≤ ∞ 
Gaussian process, Normal distribution 
White noise 
H º ½ 
D º 1.50 
a º 2.00 
Brownian motion, Wiener process 
Brown noise 
H º ½ 
D º 1.50 
a º 2.00 
Persistent, Trendreinforcing, 
Black noise 
½ < H < 1 
1.00 < D < 1.50 
1 < a < 2.00 
Cauchy process, Cauchy distribution 
Cauchy noise 
H º 1 
D º 1 
a º 1 
Note: Brown noise or Brownian motion is the cumulative sum of a whitenoise process, including a normallydistributed process. The changes in, or returns on, a Brownian motion, are white noise. The fractal statistics are the same for Brown and white noise because the brownnoise process should be differenced as part of the estimation process, yielding white noise. 
Table 2 Maritime Equity Price Series 

Firm 
Ticker 
Exchange(s) 
Dates 
N 
beta 
Alexander & Baldwin, Inc. 
ALEX 
NASD 
12/30/198808/23/2002 
3446 (1517+1929) 
0.32 
B&H Ocean Carriers, Ltd. 
BHO 
AMEX 
12/30/198808/23/2002 
3447 (1517+1930) 
0.91 
Seacor Smit, Inc. 
CKH 
NYSE 
12/21/199208/23/2002 
2440 (511+1929) 
1.14 
CSX Corporation 
CSX 
NYSE 
12/30/198808/23/2002 
3447 (1517+1930) 
0.41 
International Shipholding Corporation 
ISH 
NYSE 
12/30/198808/27/2002 
3449 (1517+1932) 
0.14 
Kirby Corporation 
KEX 
NYSE 
12/30/198808/23/2002 
3447 (1517+1930) 
0.44 
MC Shipping, Inc. 
MCX 
AMEX 
05/25/198908/23/2002 
3347 (1417+1930) 
0.14 
OMI Corporation 
OMM 
NYSE 
12/30/198808/23/2002 
3448 (1517+1931) 
0.60 
Overseas Shipping Group, Inc. 
OSG 
NYSE 
12/30/198808/23/2002 
3447 (1517+1930) 
0.73 
Tidewater, Inc. 
TDW 
NYSE 
12/30/198808/23/2002 
3446 (1516+1930) 
1.25 
Grupo TMM (Transportación Marítime Mexicana) 
TMM 
NYSE 
06/10/199208/23/2002 
2577 (647+1930) 

Maritrans, Inc. 
TUG 
NYSE 
12/30/198808/23/2002 
3447 (1517+1930) 
0.34 
Note: All raw price series are adjusted for stock splits and converted to logarithmic returns. 
Table 3 Fractal Analyses of Maritime Securities Estimated (Standard Errors in Parentheses) 

Firm 
Range 
R/S 
Power Spectrum 
Roughnesslength 
Variogram 
Wavelet 
ALEX 
12/30/198808/23/2002 
0.431 (0.0282) 
0.396 
0.397 (0.0028) 
0.382 (0.2215) 
0.479 
12/30/198812/30/1994 
0.451 (0.0116) 
0.465 
0.390 (0.0023) 
0.436 (0.0533) 
0.460 

01/03/199508/23/2002 
0.383 (0.0317) 
0.478 
0.333 (0.0056) 
0.376 (0.0665) 
0.468 

BHO 
12/30/198808/23/2002 
0.495 (0.0659) 
0.448 
0.474 (0.0178) 
0.526 (0.3473) 
0.644 
12/30/198812/30/1994 
0.492 (0.0322) 
0.538 
0.535 (0.0056) 
0.477 (0.0800) 
0.632 

01/03/199508/23/2002 
0.508 (0.0232) 
0.552 
0.505 (0.0053) 
0.534 (0.1104) 
0.562 

CKH 
12/21/199208/23/2002 
0.517 (0.0302) 
0.479 
0.480 (0.0039) 
0.405 (0.2034) 
0.617 
12/21/199212/30/1994 
0.553 (0.0167) 
0.587 
0.527 (0.0033) 
0.458 (0.0422) 
0.686 

01/03/199508/23/2002 
0.513 (0.0192) 
0.482 
0.476 (0.0068) 
0.404 (0.1802) 
0.565 

CSX 
12/30/198808/23/2002 
0.459 (0.0307) 
0.496 
0.424 (0.0072) 
0.491 (0.1707) 
0.551 
12/30/198812/30/1994 
0.427 (0.0348) 
0.479 
0.350 (0.0080) 
0.490 (0.1429) 
0.568 

01/03/199508/23/2002 
0.451 (0.0424) 
0.481 
0.428 (0.0054) 
0.496 (0.1029) 
0.574 

ISH 
12/30/198808/27/2002 
0.476 (0.1182) 
0.461 
0.504 (0.0064) 
0.350 (0.7979) 
0.607 
12/30/198812/30/1994 
0.525 (0.0605) 
0.551 
0.468 (0.0031) 
0.413 (0.1895) 
0.530 

01/03/199508/27/2002 
0.476 (0.0565) 
0.448 
0.506 (0.0086) 
0.435 (0.2343) 
0.633 

KEX 
12/30/198808/23/2002 
0.463 (0.0651) 
0.466 
0.438 (0.0069) 
0.397 (0.2080) 
0.545 
12/30/198812/30/1994 
0.452 (0.0218) 
0.513 
0.412 (0.0045) 
0.458 (0.0590) 
0.567 

01/03/199508/23/2002 
0.473 (0.0369) 
0.548 
0.447 (0.0034) 
0.384 (0.4151) 
0.546 

MCX 
12/30/198808/23/2002 
0.498 (0.0244) 
0.412 
0.465 (0.0068) 
0.557 (0.3552) 
0.603 
12/30/198812/30/1994 
0.501 (0.0109) 
0.535 
0.461 (0.0019) 
0.579 (0.2239) 
0.625 

01/03/199508/23/2002 
0.437 (0.0250) 
0.496 
0.383 (0.0030) 
0.460 (0.3809) 
0.497 

Table 3continued Fractal Analyses of Maritime Securities Estimated (Standard Errors in Parentheses) 

Firm 
Range 
R/S 
Power Spectrum 
Roughnesslength 
Variogram 
Wavelet 
OMM 
12/30/198808/23/2002 
0.482 (0.0493) 
0.505 
0.477 (0.0063) 
0.487 (0.1216) 
0.539 
12/30/198812/30/1994 
0.446 (0.0250) 
0.521 
0.401 (0.0034) 
0.506 (0.2125) 
0.493 

01/03/199508/23/2002 
0.503 (0.0254) 
0.584 
0.515 (0.0077) 
0.485 (0.2333) 
0.522 

OSG 
12/30/198808/23/2002 
0.504 (0.0381) 
0.465 
0.462 (0.0063) 
0.439 (0.1682) 
0.558 
12/30/198812/30/1994 
0.503 (0.0163) 
0.521 
0.435 (0.0045) 
0.492 (0.0775) 
0.594 

01/03/199508/23/2002 
0.463 (0.0635) 
0.465 
0.448 (0.0033) 
0.456 (0.0467) 
0.520 

TDW 
12/30/198808/23/2002 
0.489 (0.0299) 
0.515 
0.431 (0.0066) 
0.464 (0.0666) 
0.566 
12/30/198812/30/1994 
0.459 (0.0404) 
0.422 
0.446 (0.0045) 
0.365 (0.2412) 
0.617 

01/03/199508/23/2002 
0.476 (0.0184) 
0.521 
0.425 (0.0031) 
0.490 (0.0297) 
0.488 

TMM 
06/10/199208/23/2002 
0.491 (0.1180) 
0.480 
0.509 (0.0108) 
0.391 (1.2583) 
0.599 
06/10/199212/30/1994 
0.463 (0.0231) 
0.496 
0.439 (0.0033) 
0.431 (0.0175) 
0.543 

01/03/199508/23/2002 
0.519 (0.0647) 
0.529 
0.509 (0.0100) 
0.437 (0.8114) 
0.653 

TUG 
12/30/198808/23/2002 
0.451 (0.0261) 
0.464 
0.390 (0.0097) 
0.539 (0.4529) 
0.487 
12/30/198812/30/1994 
0.432 (0.0222) 
0.440 
0.387 (0.0031) 
0.469 (0.1096) 
0.520 

01/03/199508/23/2002 
0.436 (0.0249) 
0.495 
0.398 (0.0039) 
0.528 (0.2402) 
0.569 

Note: The
MandelbrotLévy characteristic exponent alpha is
the reciprocal of the 
Table 4 Hypothesis Tests for Normality or Gaussian Character (H = 0.50) 

Firm 
d.f. 
R/S 
Roughnesslength 
Variogram 

t(R/S) 
p 
t(R) 
p 
t(V) 
p 

ALEX 
38 
2.447 ** 
0.019 
36.786 *** 
0.000 
0.533 
0.597 
26 
4.224 *** 
0.000 
47.826 *** 
0.000 
1.201 
0.241 

30 
3.691 ** 
0.001 
29.821 *** 
0.000 
1.865 * 
0.072 

BHO 
38 
0.076 
0.940 
1.461 
0.152 
0.075 
0.941 
26 
0.248 
0.806 
6.250 *** 
0.000 
0.288 
0.776 

30 
0.345 
0.733 
0.943 
0.353 
0.308 
0.760 

CKH 
34 
0.563 
0.578 
5.128 *** 
0.000 
0.467 
0.644 
30 
3.174 * 
0.007 
8.182 *** 
0.000 
0.995 
0.337 

30 
0.677 
0.506 
3.529 *** 
0.002 
0.533 
0.600 

CSX 
30 
1.336 
0.190 
10.556 *** 
0.000 
0.053 
0.958 
14 
2.098 ** 
0.046 
18.750 *** 
0.000 
0.070 
0.945 

20 
1.156 
0.257 
13.333 *** 
0.000 
0.039 
0.969 

ISH 
38 
0.203 
0.840 
0.625 
0.536 
0.188 
0.852 
26 
0.413 
0.683 
10.323 *** 
0.000 
0.459 
0.650 

30 
0.425 
0.674 
0.698 
0.491 
0.277 
0.783 

KEX 
38 
0.568 
0.573 
8.986 *** 
0.000 
0.495 
0.623 
26 
2.202 ** 
0.037 
19.556 *** 
0.000 
0.712 
0.483 

30 
0.732 
0.470 
15.588 *** 
0.000 
0.279 
0.782 

MCX 
32 
0.082 
0.935 
5.147 *** 
0.000 
0.160 
0.874 
26 
0.092 
0.928 
20.526 *** 
0.000 
0.353 
0.727 

30 
2.520 ** 
0.017 
39.000 *** 
0.000 
0.105 
0.917 

OMM 
38 
0.365 
0.717 
3.651 *** 
0.001 
0.107 
0.915 
26 
2.160 ** 
0.040 
29.118 *** 
0.000 
0.028 
0.978 

30 
0.118 
0.907 
1.948 * 
0.061 
0.064 
0.949 

OSG 
38 
0.105 
0.917 
6.032 *** 
0.000 
0.363 
0.719 
26 
0.184 
0.855 
14.444 *** 
0.000 
0.103 
0.919 

30 
0.583 
0.564 
15.758 *** 
0.000 
0.942 
0.354 

TDW 
38 
0.368 
0.715 
10.455 *** 
0.000 
0.541 
0.592 
26 
1.015 
0.320 
12.000 *** 
0.000 
0.560 
0.580 

30 
1.304 
0.202 
24.194 *** 
0.000 
0.337 
0.739 

TMM 
37 
0.076 
0.940 
0.833 
0.410 
0.087 
0.931 
15 
1.602 
0.130 
18.485 *** 
0.000 
3.943 *** 
0.001 

30 
0.294 
0.771 
0.900 
0.375 
0.078 
0.939 

TUG 
38 
1.877 * 
0.068 
11.340 *** 
0.000 
0.086 
0.932 
26 
3.063 *** 
0.005 
36.452 *** 
0.000 
0.283 
0.780 

30 
2.570 ** 
0.015 
26.154 *** 
0.000 
0.117 
0.908 

Note: Hs
computed by R/S, Roughnesslength, and Variogram
methods are used for conventional hypothesis tests where the null hypothesis
is H = 0.500, (i.e., equivalently alpha = 2, D = 1.500, or normality of the
asset returns). Three independent
hypothesis tests are performed for each time series. The 
Table 5 Hypothesis Tests for Structural Stability 

Firm 
Null 
d.f. 
R/S 
Roughnesslength 
Variogram 

t 
p (t) 
T 
p (t) 
t 
p (t) 

ALEX 
B=C 
56 
2.411 ** 
0.019 
20.357 * 
0.000 
0.271 
0.787 
A=B 
38 
0.709 
0.483 
2.500 ** 
0.017 
0.244 
0.809 

A=C 
38 
1.702 * 
0.097 
22.857*** 
0.000 
0.027 
0.979 

BHO 
B=C 
56 
0.243 
0.809 
1.685 * 
0.097 
0.164 
0.870 
A=B 
38 
0.046 
0.964 
3.427 *** 
0.001 
0.141 
0.889 

A=C 
38 
0.197 
0.845 
1.742 * 
0.090 
0.023 
0.982 

CKH 
B=C 
34 
1.325 
0.194 
13.077 
0.000 
0.265 
0.792 
A=B 
30 
1.192 
0.243 
12.051*** 
0.000 
0.261 
0.796 

A=C 
30 
0.132 
0.896 
1.026 
0.313 
0.005 
0.996 

CSX 
B=C 
56 
0.782 
0.438 
10.833*** 
0.000 
0.035 
0.972 
A=B 
38 
1.042 
0.304 
10.278*** 
0.000 
0.006 
0.995 

A=C 
38 
0.261 
0.796 
0.556 
0.582 
0.029 
0.977 

ISH 
B=C 
56 
0.415 
0.680 
5.938 *** 
0.000 
0.028 
0.978 
A=B 
38 
0.415 
0.681 
5.625 *** 
0.000 
0.079 
0.937 

A=C 
38 
0.000 
1.000 
0.313 
0.756 
0.107 
0.916 

KEX 
B=C 
56 
0.323 
0.748 
5.072 *** 
0.000 
0.356 
0.723 
A=B 
38 
0.169 
0.867 
3.768 *** 
0.001 
0.293 
0.771 

A=C 
38 
0.154 
0.879 
1.304 
0.200 
0.063 
0.950 

MCX 
B=C 
56 
2.623 ** 
0.011 
11.471*** 
0.000 
0.335 
0.739 
A=B 
32 
0.123 
0.903 
0.588 
0.561 
0.062 
0.951 

A=C 
32 
2.500 ** 
0.018 
12.059*** 
0.000 
0.273 
0.787 

OMM 
B=C 
56 
1.156 
0.252 
18.095*** 
0.000 
0.173 
0.863 
A=B 
38 
0.730 
0.470 
12.063*** 
0.000 
0.156 
0.877 

A=C 
38 
0.426 
0.673 
6.032 *** 
0.000 
0.016 
0.987 

OSG 
B=C 
56 
1.050 
0.298 
2.063 ** 
0.044 
0.214 
0.831 
A=B 
38 
0.026 
0.979 
4.286 *** 
0.000 
0.315 
0.754 

A=C 
38 
1.076 
0.289 
2.222 ** 
0.032 
0.101 
0.920 

TDW 
B=C 
56 
0.569 
0.572 
3.182 *** 
0.002 
1.877 * 
0.066 
A=B 
38 
1.003 
0.322 
2.273 ** 
0.029 
1.486 
0.145 

A=C 
38 
0.435 
0.666 
0.909 
0.369 
0.390 
0.698 

TMM 
B=C 
45 
0.475 
0.637 
6.481 *** 
0.000 
0.005 
0.996 
A=B 
37 
0.237 
0.814 
6.481 *** 
0.000 
0.032 
0.975 

A=C 
37 
0.237 
0.814 
0.000 
1.000 
0.037 
0.971 

TUG 
B=C 
56 
0.153 
0.879 
1.134 
0.261 
0.130 
0.897 
A=B 
38 
0.728 
0.471 
0.309 
0.759 
0.155 
0.878 

A=C 
38 
0.575 
0.569 
0.825 
0.415 
0.024 
0.981 

Note: Three
independent hypothesis tests are performed for each time series. The 