A fractal analysis of US industrial sector stocks

Taro Ikeda

November 2016
Discussion Paper No.1643

GRADUATE SCHOOL OF ECONOMICS
KOBE UNIVERSITYROKKO, KOBE, JAPANA fractal analysis of US industrial sector stocks
Taro Ikeda∗
November 19, 2016

Abstract
This paper found that most of US industrial sectors are fractal, and therefore
have a long autocorrelated dependence.
Keywords: Fractal geometry, Hurst exponent, random walk
JEL classification: C18, E39, F31

1Introduction

The fractal geometry has a concern with a long autocorrelated dependence of time-series
by a power law distributions. In this methodology, we can consider the wider degreeof autocorrelations than the unitroot process (random walk). In this regard, the fractal
geometry is a method to detect generalized autocorrelations more than the unitroot
process.
The generalization of the random walk model with the fractals suggests that theeﬃcient market hypothesis does not valid anymore if the autocorrelation is higher thanthe one suggested in the unitroot case, because an independence among sequence of pastshocks on asset prices is destroyed by the higher order autocorrelations of the fractals.
We believe that the fractal geometry provides new perspectives on the economics,
however, the formal application of the fractal geometry to the economics are not enoughexcept for Ikeda (2016), which found that the whole US stock market is fractal rather
than the random walk with Hurst’s R/S analysis.1
This paper aims to provide a formal fractal analysis of the US industrial sector stock
prices. To the best of our knowledge, there is no investigation to provide a fractal analysis
for the stock prices by industrial sectors. We distinguish the long-run dependence for theUS industrial sector stocks.
This paper proceeds as follows. Section 2 introduces a fractal Brownian motionand R/S analysis to determine the long dependence of stock prices. Section 3 presents
empirical results. Section 4 concludes.
∗Research Fellow, Graduate School of Economics, Kobe University, 2-1 Rokko-dai, Nada, Kobe,Hyogo, Japan(E-mail : young.tubo315o@gmail.com)
1Also, see Booth et al. (1982) for the case of exchange rates of developed European countries.

12Methodology

This section introduces a fractional Brownian motion which provides mathematical backgrounds of the fractal geometry, and then the Hurst’s R/S analysis to determine thegeneralized autocorrelation of stock prices.
A fractional Brownian motion
A fractional Brownian motion (fBm) of the fractal geometry is a key concept to generalize
the ordinary Brownian motion (random walk) in the unit root analysis. To reveal therelationship between the two, we denote the diﬀerence of the fractional Brownian motion
BH (t) for any two times t and t0 as |BH (t) − BH (t0 )|. Therefore, the second moment of
the diﬀerence can be calculated as:⟨|BH (t) − BH (t0 )|2 ⟩ ∝ |t − t0 |2H ,(1)where ⟨⟩ is an ensemble average and H is the Hurst exponent which takes a value 1/2
for the random walk case.2 Accordingly, it is relevant that the variance of the ordinary
Brownian motion is the special case of the fBm’s variance equation.
Also, note that the variance of the fBm goes to infinity with time increasing, and therate is faster than the one of the random walk in the case 1/2 < H < 1.
The Hurst’s R/S analysis
The method, so-called R/S analysis, has been proposed by Hurst (1955) in order todetermine the long-run persistencein time-series.
We describe the increments of time()
series in logarithms as x(t) ∈ x(1), ..., x(T ) , and calculate the ensemble average overany window-size τ as:1⟨xt |τ ⟩ =ττl∑x(t),(2)τ (l−1)+1for l = 1 : ceil(T /τ ).
Then, we calculate the accumulated sum of dispersion of x(t) from the average asfollows:x(t, τ ) =T∑()
x(t) − ⟨x(t)|τ ⟩ .

(3)

t=1

By the subtraction of the local averages, constant trends in the sample are excluded.
In addition, we calculate R(τ ) of the max-min range over the window-size τ and S(τ )of the standard deviations in each segment as:R(τ ) =max
τ (l−1)+1≤t≤τ l2x(t, τ ) −minx(t, τ ),

τ (l−1)+1≤t≤τ lThe variance equation of the random walk takes well-known formulation as:⟨|BH (t) − BH (t0 )|2 ⟩ ∝ |t − t0 |.

2(4)andvuu1S(τ ) = tττl∑()2x(t) − ⟨x(t)⟩ .

(5)τ (l−1)+1

Clearly, the range R is an increasing function for τ .
Finally, the R/S statistics is defined as a fraction of (4) and (5), which obeys thepower law distribution as:
R(τ )/S(τ ) ∝ τ H .

(6)where, H is the Hurst exponent as already introduced in equation (1).
Using Hurst’s H, we can categorize the autocorrelation nature of the time-series
into three cases. First, the case 0 < H < 1/2 corresponds to antipersistence or shortcorrelation, where the time-series increasing today must decrease tomorrow. Second, thecase H = 1/2 implies the random walk as frequently used assumption in financial timeseries analysis, which implies the shock terms of the series are not interdependent overtime-diﬀerences. Third, 1/2 < H < 1 denotes long dependence where the past shocks
do not disappear and do have an eternal eﬀect on the series. In the long dependence,increasing trends of the variable can turn suddenly to decreasing trends.3
Accordingly, it is apparent that the fractal geometry develops the concept of autocorrelation in traditional time-series analyses.

3

Empirical results

This section exhibits the statistical Hurst exponents for the 15 industrial sector stockindexes. To this end, we utilize the sector stock indexes categorized by Dow Jones.45
Here, we implement 5555 times bootstrapping to construct confidence intervals for
estimated H. This is because the variance for H goes to infinity in the case 1/2 < H < 1.
The nonparametric bootstrapping is appropriate in such situation.

3.1

Benchmark results

Table 1 presents the statistical Hurst exponents for the industrial sector stocks. The longautocorrelated dependence is significant in 7/15 industries: Auto and Parts, Consumer
Goods, Consumer Services, Media, Real Estate, Technology, Telecommunications. The
lower confidence intervals for the Hurst exponents of these stocks take values more than
0.50.6
Figure 1 confirms these regressions. In almost all figures, the fitted lines replicatedby 5555 times bootstrapping contain the realized R/S values, which grantees the appropriateness for our regressions. However, there are only two exceptions: Farming and3See Feder (1988, Chap. 9) for this classifications.
All data are weekly-frequency and obtained from Thomson Reuters Datastream.
5
All results in this paper are very similar to daily observations.
6
The result of Real Estate is consistent with Ikeda (2016), who suggested the fractality of the US
stock market.
43

Fishing and Oil and Gas. These realized values violate our bootstrapped lines, which implies the large structural changes for H. This problem deeply corresponds to a crossovers
phenomenon, discussed in next section.

3.2

Time-dependent crossovers for the Hurst exponents

Following Peng et al. (1994), time-series has crossovers in which estimated H changes
largely.78 This section examines whether the time-dependent crossovers appears on thestock price indexes by sequential estimations.
Figure 2 suggests the crossovers for all sectors. The results are amazing in the sensethat the most of the sectors (12/15 sectors) has long-run dependence. This fact suggeststhat the crossovers are critical, also in economic time-series.

4Conclusion

This paper found that the most of the US industrial sectors has a long-run dependence
when we consider the crossovers for the statistical Hurst exponent. The exact specification for the crossovers by multiracial is challenges of the future.

References[1] Booth, Geoﬀrey G., Kaen, Fred R., Koveos, Peter E., (1982) “R/S analysis of foreignexchange rates under two international monetary regimes.” Journal of Monetary
Economics, 10, 407–415.
[2] Feder, Jens, (1988) Fractals, Plenum Press, New York.
[3] Hurst, Edwin H., (1955) “Methods of using long-term storage in reservoirs.” Proceedings of the Institution of Civil Engineers, part 1, 519–577.
[4] Ikeda, Taro (2016) “Relume: a fractal analysis for the US stock market.” Discussion
Papers, 1637, Kobe University.
[5] Peng, C.K., Buldyrev, S.V., Havlin, S., Simons, M., Stanley, H.E., Goldberger, A.L.,(1994) “Mosaic organization of DNA nucleotides.” Physical Review E, 49, 1685–1689.

78

They stressed that the crossovers should be considered in DNA sequences.
The crossovers suggest changes of the shape of the power law distributions for the variable.

4Figures and Tables
Table 1: Hurst exponents by sector
No.
123456789101112131415

Name
Hurst exponent
The Americas Auto and Parts
0.64570
The Americas Basic Resources
0.50475
The Americas Consumer Goods
0.60409
The Americas Consumer Services
0.62969
The Americas Farming and Fishing
0.49060
The Americas Financial Services
0.56600
The Americas Food and Beverages
0.56925
The Americas Media
0.68863
The Americas Oil and Gas
0.49178
The Americas Real Estate
0.57230
The Americas Retail0.57423
The Americas Technology
0.68715The Americas Telecommunications
0.65490
The Americas Travel and Leisure0.52539
The Americas Utilities
0.52157

Lower
0.54456
0.465360.53906
0.58543
0.390370.47879
0.482160.553700.41657
0.535310.47975
0.574310.525440.418850.45394

Note: ‘Lower’ and ‘Upper’ denote bootstrapped 95% confidence intervals.

Upper0.75352
0.604800.651100.71856
0.53957
0.632850.606220.831860.509220.67868
0.59372
0.755180.74653
0.58782
0.62740Figure 1: Bootstrapped regressions
Note: Gray lines are generated by regressions using 5555 bootstrapping.

Note: Gray lines are generated by regressions using 5555 bootstrapping.

Figure 2: Sequential estimation of H
Note: Shaded area denotes 95% confidence interval for H.

Note: Shaded area denotes 95% confidence interval for H.