ap-5-11.dvi


Acta Polytechnica Vol. 51 No. 5/2011

Chaos in GDP

R. Kř́ıž

Abstract

This paper presents an analysis of GDP and finds chaos in GDP. I tried to find a nonlinear lower-dimensional discrete
dynamicmacroeconomicmodel thatwould characterizeGDP.Thismodel is represented bya set of differential equations.
I have used the Mathematica and MS Excel programs for the analysis.

Keywords: macroeconomic modeling, gross domestic product, chaos theory.

1 Introduction
Humanity has always been concerned with the ques-
tion whether the processes in the real world are of a
stochastic or deterministic nature. Answers are ex-
plored by theologians, philosophers and scientists in
various fields. I take the view that real processes are
more deterministic in nature. An interesting case of
determinism is deterministic chaos. The only purely
stochastic process is amathematicalmodel described
by mathematical statistics. The statistical model of-
tenworksand is the onlypossible description ifwedo
not know the system. This also applies to economic
quantities, including forecasts for GDP. In this pa-
per I have tried to grasp the hidden essence of the
problem in order to formulate a predictionmore eas-
ily. The basic question is therefore the existence of
chaotic behavior. If the system behaves chaotically,
we are forced to accept only limited predictions. In
this paper I will try to show the chaotic behavior of
GDP and then propose a simple lower-dimensional
system under which the system evolves.

2 Methodology for the
analysis

I will briefly state the basic definitions and describe
the basic methods for examining the input data.

2.1 Gross domestic product

Gross domestic product (GDP) is a major
macroeconomic indicator. It measures the overall
production performance of the economy. It is the
totalmarket value of all final goods and services pro-
duced within a country during some period, usually
one year, expressed in monetary units [8]. It is of-
ten considered an indicator of a country’s standard
of living. GDP can be determined in three ways, all
of which should, in principle, give the same result.
They are the product (or output) approach, the in-

come approach, and the expenditure approach.

Y = C + I + G +(X − M) (1)

• C (consumption) is normally the largestGDP
component in the economy, consisting of pri-
vate (household final consumption expenditure)
in the economy.

• I (investment) includes business investment in
equipment, for example, and does not include
exchanges of existing assets.

• G (government spending) is the sum of gov-
ernment expenditures on final goods and ser-
vices.

• X − M (net exports) represents gross exports
X — gross imports.

2.2 Hurst exponent

The Hurst exponent is widely used to character-
ize some processes. The Hurst exponent is a mea-
sure that has been widely used to evaluate the self-
similarity and correlation properties of fractional
Brownian noise, the time-series produced by a frac-
tional (fractal) Gaussian process.
As originallydefinedbyMandelbrot [5], theHurst

exponent H describes (among other things) the scal-
ing of the variance of a stochastic process y(t),

σ2 =
∫ +∞
−∞

y2f(y, t)dy = ct2H (2)

where c is constant.
The Hurst exponent is used to evaluate the pres-

ence or absence of long-range dependence and its de-
gree in a time-series.
The Hurst exponent (H) is defined in terms of

the asymptotic behavior of the rescaled range as a
function of the time span of a time series, as follows

E

[
R(n)
S(n)

]
= CnH as n → ∞, (3)

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Acta Polytechnica Vol. 51 No. 5/2011

where [R(n)/S(n)] is the rescaled range; E[y] is ex-
pected value; n is number of data points in a time
series, C is a constant.
An algorithm for calculation is used from

Wikipedia [12]. To calculate the Hurst exponent,
one must estimate the dependence of the rescaled
range on the time span n of observation. The av-
erage rescaled range is then calculated for each value
of n. For a (partial) time series of length n, Y =
Y1, Y2, . . . , Yn, the rescaled range is calculated as fol-
lows:
1. Create a mean-adjusted series

Ut = Yt −
1
n

n∑
i=1

Yi for t =1,2, . . . , n (4)

2. Calculate the cumulative deviate series V ;

Vt =
n∑

i=1

Ui for t =1,2, . . . , n (5)

3. Compute the range R;

R(n)=max(V1, V2, . . . , Vn)−min(V1, V2, . . . , Vn)
(6)

4. Compute the standard deviation S

S(n)=

√√√√1
n

n∑
i=1

(Yi − Ȳ )2 (7)

5. Calculate the rescaled range and average over
all the partial time series of length n. The Hurst
exponent is estimated by fitting the power law, ac-
cording to the definition.
The values of the Hurst exponent vary between

0 and 1, with higher values indicating a smoother
trend, less volatility, and less roughness. Random
walk has a Hurst exponent of 0.5.

2.3 Fractal

The term “fractal” was first introduced by Mandel-
brot [4]. A fractal is a complicated geometric figure
that, unlike a conventional complicated figure, does
not simplifywhen it ismagnified. In thewaythatEu-
clideangeometryhas servedas adescriptive language
for classicalmechanics ofmotion, fractal geometry is
being used for the patterns produced by chaos [9].
The fractal dimension, D, is a statistical quan-

tity that gives an indication of howcompletely a frac-
tal appears to fill space, as one zooms down to finer
and finer scales. There are many specific definitions
of fractal dimension.
Hurst exponent H is directly related to fractal di-

mension D, because the maximum fractal dimension
for a planar tracing is 2:

D + H =2 (8)

2.4 Correlation dimension

Correlationdimension (DC) describes thedimension-
ality of the underlying process in relation to its ge-
ometrical reconstruction in phase space. The corre-
lation dimension is calculated using the fundamental
definition. Define the correlation integral for set of
data M:

C(r) =
1

M(M −1)

M∑
i, j =1

i �= j

H (r − ‖yi − yj‖) (9)

where H is the Heaviside step function.

H(x)=

⎧⎪⎪⎨
⎪⎪⎩
0 y < 0
1
2

y =0

1 y > 0

(10)

A Euclidean metric is used for all calculations in
this paper. When a lower limit exists, the correlation
dimension is then defined as

DC = lim
r → 0
M → ∞

ln(C(r))
ln(r)

(11)

2.5 Lyapunov exponents

The Lyapunov exponent or the Lyapunov character-
istic exponent of a dynamical system is a quantity
that characterizes the rate of separation of infinitesi-
mally close trajectories. Quantitatively, two trajecto-
ries in phase spacewith initial separation δZ0 diverge
(provided that the divergence can be treated within
the linearized approximation)

δZ(t) ≈ eλt |δZ0| (12)

where λ is the Lyapunov exponent.
The maximal Lyapunov exponent can be defined

as follows:

λ = lim
δZ0 → 0
t → ∞

1
t
ln

|δZ(t)|
|δZ0|

(13)

The limit δZ0 → 0 ensures the validity of the
linear approximation at any time.
MaximalLyapunov exponentdetermines a notion

of predictability for a dynamical system. A positive
Maximal Lyapunov exponent is usually taken as an
indication that the system is chaotic (provided some

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Acta Polytechnica Vol. 51 No. 5/2011

other conditions are met, e.g., phase space compact-
ness) [13].

3 Introduction to nonlinear
dynamics

If the world is not linear (and there is no qualitative
reason to assume that it is linear), it should be nat-
ural to model dynamical economic phenomena non-
linearly [2]. If it is present in the general system of
nonlinear dynamics, a deterministic system can gen-
erate random-looking results, but may include hid-
den order. In this paper I have studied only a low-
dimensional discrete dynamical system.

3.1 Discrete dynamical system

Definition from Goldsmith [6]: A discrete-time dy-
namical system on a set Y is just the function

Φ:Y → Y (14)

This function, often called a map, may describe
the deterministic evolution of some physical system:
if the system is in state y at time t, then it will be in
state Φ(y) at time t +1. The study of discrete-time
dynamical systems is concerned with iterates of the
map: the sequence

y,Φ(y),Φ2(y), . . . (15)

is called the trajectory of y and the set of its points is
called the orbit of y. These two terms are often used
interchangeably, although we will remain consistent
in their usage. The set Y in which the states of the
system exist is referred to as the phase space or state
space. We will restrict our attention to maps Φ(y)
such that Y is a subset of Rd.

3.2 One-dimensional discrete
dynamical system

The one-dimensional discrete system is

yt+1 = f(yt) yt ∈ � (16)

One-dimensional, discrete dynamical systems in
economics are surely suited for demonstrating the
relative easy with which complex behavior can be
modeled. The mathematical properties of one-
dimensional dynamical systems seemtobebetter un-
derstood than higher-dimensional systems [2].
The logistic equation is a typical example of easy

one-dimensional discrete dynamical systems which
can be chaotic.

yt+1 = μyt(1− yt) (17)

The logistic equation has been described inmany
books andpapers e.g. [1,2,6,9,10]. The logistic equa-
tion is chaotic, when control parameter μ is between
3.5699 . . . and 4.

4 Analysis of GDP

4.1 Input data

The gross domestic product by type of expenditure
in current prices is used in this paper. I have used
data (quarterly, without seasonal adjustment) from
the Czech Statistical Office. I analyze data from the
Czech Republic between 1995 and 2010 [11].

Fig. 1: GDP with linear trend of GDP

4.2 Calculation of H and Dc

The main problem in analyzing GDP is the lack of
data. Therefore, all results are only estimates.
I have computed the Hurst exponent for GDP

H = 0.75 according to the algorithm in chapter 2.2.
This value is in accordance with expectations. We
know that the value of H is between 0 and 1, whilst
real time series are usually higher than 0.5. If the
exponent value is close to 0 or 1, it means that the
time-series has long-range dependence. Value 0.75
is directly between the stochastic and deterministic
process. I think that 0.75 value is a sufficient value
for credible prediction. Nowwe also know the fractal
dimension 2−0.75=1.25.

Fig. 2: Correlation integral, value with a linear trend

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Acta Polytechnica Vol. 51 No. 5/2011

The correlation dimension is calculated using the
fundamental definition in Section 2.4. Wehavehad a
problemwith lack of data for this computing. I have
put the calculated data into a graph in logarithmic
coordinates, and I have made a linear interpolation.
(cf. Figure 2). On this basis, the correlation dimen-
sion for the small value of r can be estimated.
The estimate of the correlation dimension is a

value lower than 2. If the correlation dimension is
low, the Lyapunov exponent is positive and the Kol-
mogorov entropy has a finite positive value, chaos is
probably present. From estimates H and Dc it can
be concluded that GDP is a deterministic chaos.

4.3 Analyzing in phase space

In the previous section (Section 4.2) we verified the
presence of chaos in GDP.
Data with a trend can cause problems for future

analysis. The trend is removed by subtracting the
linear interpolation. Denote GDP without trend as
Y (t).

Fig. 3: GDP without trend

Fig. 4: GDP in phase space

A phase portrait 2D of GDP is constructed so
that each ordered pair of {Yt;Yt−1, t = 2, . . . , N} is

displayed in theplanewhere the x-axis represents the
values of Yt and y-axis value Yt−1 (cf. Figure 4). The
individual points {Yt;Yt−1} of phase space are con-
nected by a smooth curve. This curve looks like a
chaotic attractor. The points are located mainly in
the first and third quadrant.
The phase portrait 3D of GDP is constructed

so that each ordered trio of {Yt;Yt−1, Yt−2, t =
3, . . . , N} is displayed in the space. The individual
points {Yt;Yt−1, Yt−2} of 3D phase space are con-
nected by a line (cf. Figure 5) or coveredby a surface
(cf. Figure 6). Figure 6 looks very strange.

Fig. 5: GDP in 3D phase space (Yt, Yt−1, Yt−2)

Fig. 6: GDP in 3D phase space (Yt, Yt−1, Yt−2)

5 GDP as a dynamical system

I have tried to find a system that will be similar to
GDP, based on the previous analysis.

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Acta Polytechnica Vol. 51 No. 5/2011

5.1 Comparison of GDP with an
one-dimensional discrete
dynamical system

Chaotic course Yt can be immediately simulated by
a logistic equation, but a progression in the phase
space is completely different. A logistic equation can
be appropriate for simulation, but not for forecast-
ing. If we consider a function, it should be an odd
function.
I have studied several one-dimensional systems

with a cubic function. It is reasonable to assume that
that function has one root 0. The desired equation
can be written in the form:

yt+1 = yt(αy
2
t − β) (18)

Fig. 7: Cubic function: Plot [y^3-2.8y,y,-2,2]

Fig. 8: Bifurcation diagramof function y3−by, {b,2.25,3}

Fig. 9: Chaos in cubic function Yt+1 = Y
3

t − 2.8Yt

6 Conclusion
I have shown in this paper thatGDP can be chaotic.
I found a very simple nonlinear differential equa-
tionwhich properly captures GDP. According to the
phase portrait, it seems that the function should be
odd. The most appropriate function is the cubic
equationwith a single zero root. It is a simple system
that is easy to interpret. It should be noted that this
system is not perfect and it would be useful to find
a set of differential equations of higher order. The
biggest problem in finding a suitable system is the
lack of data. This paper analyzes GDP as such. It
would be more sophisticated to create a theoretical
model and calibrate it.

Acknowledgement

The research presented in the paper was supervised
by doc. Ing. Helena Fialová, CSc., FEE CTU in
Prague and supported by the Czech Grant Agency
under grant No. 402/09/H045 “Nonlinear Dynamics
in Monetary and Financial Economics. Theory and
Empirical Models.”

References

[1] Allen, R. G. D.: Matematická ekonomie.
Academia, 1971.

[2] Lorenz, H.-W.: Nonlinear Dynamical Eco-
nomics and Chaotic Motion. Springer-Verlag,
1989.

[3] Flaschel,P., Franke,R.,Demmler,W.: Dynamic
Macroeconomics. The MIT Press, 1997.

[4] Mandelbrot,B.B.: TheFractalGeometry ofNa-
ture. W. H. Freeman and Co., 1983.

[5] Mandelbrot, B. B., Ness Van, J. W.: Fractional
Brownian motions, fractional noises and appli-
cations. SIAM Rev. 10 (1968), pp. 422.

[6] Goldsmith, M.: The Maximal Lyapunov Expo-
nent of a Time Series. Thesis, 2009.

[7] Grassberg, P., Procaccia, I.: Characterization
of strange attractors,Phys. Rev. Lett.50 (1983)
346.

[8] Fialová, H., Fiala, J.: Ekonomický slovńık. A
plus, 2009.

[9] Alligood, T. K., Sauer, D. T., Yorke, J. A.:
CHAOS an introduction to dynamical systems.
Springer, 2000.

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[10] Chiarella, C.: The Elements of a Nonlinear
Theory ofEconomicDynamics.Springer-Verlag,
1990.

[11] http://www.czso.cz/eng/redakce.nsf/i/time series

[12] http://en.wikipedia.org/wiki/Hurst exponent

[13] http://en.wikipedia.org/wiki/Lyapunov exponent

About the author

Radko Kř́ıž,MSc. wasborn inBroumov. Hegrad-
uated at the Faculty of Electrical Engineering of the
CzechTechnicalUniversity in Prague, specializing in

Economics and Management in Electrical Engineer-
ing, and he is currently aPhD student at theFaculty
of Electrical Engineering of theCzechTechnicalUni-
versity in Prague. His major fields of specialization
are Macroeconomics, Energetics, Nonlinear Dynam-
ics in Economics and Chaos Theory.

Radko Kř́ıž
E-mail: krizradk@fel.cvut.cz
Dept. of Economics
Management and Humanities
Faculty of Electrical Engineering
Czech Technical University
Technická 2, 166 27 Praha, Czech Republic

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