Journal of Applied Mathematics and Physics, 2014, 2, 214-218
Published Online April 2014 in SciRes.
How to cite this paper: Das, P. (2014) Is There Chaos in Inflation Data? Journal of Applied Mathematics and Physics, 2,
Is There Chaos in Inflation Data?
Pritha Das
Department of Mathematics, Bengal Engineering and Science University, Shibpur, Howrah, India
Received January 2014
Economic indicators are snippets of financial and economic data published regularly by govern-
mental agencies and the private sector. An exchange rate represents the value of one currency in
another and it fluctuates over time. ForEx rates are affected by many highly correlated economic,
political and even psychological factors. It was observed that changes in the exchange rate are re-
lated to news in the fundamentals which cover Inflation for the country concerned. In a series of
work, we investigated and confirmed the chaotic property of ForEx Rates of several countries. In
this perspective, we concentrate on nonlinear data analysis of inflation data of nine countries. We
find existence of chaos in inflation data for some countries.
Inflation, Surrogate Method, Lyapunov Exponent, C h aos
1. Introduction
An exchange rate represents the value of one currency in another. An exchange rate between two currencies
fluctuates over time. Foreign exchange (ForEx) rates are amongst the most important economic indices in the
international monetary markets. It was observed that changes in the exchange rate are related to news in the
fundamentals. Set of fundamentals covers: inflation for the country concerned, money supply for the country
under scrutiny, Money Market Rate etc. In a series of work (2007, 2012, 2013), we investigated and confirmed
the chaotic property of Foreign Exchange Rates of several countries [1]-[4]. Chaotic processes are characterized
by positive Lyapunov Exponent (LE)s and we calculated LEs from ForEx data.
Economic indicators are snippets of financial and economic data published regularly by governmental agen-
cies and the private sector. Here we introduce CPI as defined below for measuring inflation.
Consumer Price Index (CPI): Measures the average price level paid by urban consumers (80% of the popula-
tion in major currency countries) for a fixed basket of goods and services. It reports price changes in over 200
categories. The CPI also includes various user fees and taxes directly associated with the prices of specific
goods and services [5].
In this work we have collected data of some of these indicators from year 2000 to 2013 of the above few im-
portant indicators. Figures plotted from this data indicate that ForEx is closely related to other indicators. Our
main focus is to explore the nature of dependence of CPI on ForEx rate. We like to address the specific question:
Is CPI data also chaotic as with other indicator, it influences ForEx whose fluctuation is already found to be
P. Das
Little work was attempted in this direction. Guastello (1995) showed low-dimensional chaos for US inflation
rates during the 1948-1995 era with Lyapunov dimensionality 1.5. He also confirmed that although there was
short time linear prediction of inflation rates, the global picture was, nonetheless, chaotic [6]. In another study
(2001) he stressed the presence of chaotic attractors for inflation rate in the US [7]. Results from linear and
nonlinear analyses provide overwhelming evidence in support of the nonstationarity of the inflation rate in
Africa [8]. We like to study this situation for some EU countries, as well as some other countries having its own
ForEx rate. Obviously, we are not attempting any analysis of justification of Euro. Our analysis will be confined
to understand the economic indicators, particularly inflation data in relation to ForEx rate through data analysis.
2. Data Collection
Detailed Inflation or CPI data: The inflation rate is based upon the consumer price index (CPI). The CPI
inflation rates used are on a yearly basis (compared to the same month the year before). For example, inflation
for January 2013 is difference over that in January 2012 expressed as per cent. [9] maintains
historical data for many countries which have been used in this paper. We have collected data on monthly basis
from January 2000 to September 2013 for the following countries: France, Italy, Germany, Spain, Greece, India
and UK. So each country has a dataset consisting 165 dataone for each month. For Sri Lanka, data from
January, 2001 to April, 2008 are available, so data points are 88 in number from Department of Census and
Statistics. Government of Sri Lanka [10]. Singapore CPI data was taken from ‘Time Series on Monthly CPI
(2009 = 100) And Percentage Change Over Corresponding Period Of Previous Year’, Government of Singapore
3. Nonlinear Analysis of Inflation Data
Here we shall concentrate on detailed nonlinear data analysis of inflation data collected to get more insight of it.
The basic point we like to investigate is if CPI data analysis show chaos or not. For characterizing chaos both
qualitatively and quantitatively, we have to find Largest Lyapunov Exponent (LLE).
3.1. Test for Nonlinearity Using Surrogate Data Method
We follow the approach of Theiler et al. (1992) [12]. The surrogate signal is produced by phase-randomizing the
given data. It has spectral properties similar to the given data, that is, the surrogate data sequence has the same
mean, the same variance, the same autocorrelation function, and therefore the same power spectrum as the
original sequence, but (nonlinear) phase relations are destroyed. Details of the method for the countries con-
sidered have been given in the previous work [1] or as used with additional noise reduction (Çoban et al., 2012)
[3]. We used the TSTOOL package by Parlitz et al. (1998) [13], under MATLAB (2008) [14] software to create
surro gat e data for a scalar time series. From this analysis, we got some idea about the degree of nonlinearity
associated with the time series of foreign exchange data up to year 2008. We are not repeating the same analysis
because we are considering the same countries and compared to our previous data, we now have 450 more
points, which is only 5% of total only (from January 2008 to October 2009). But we certainly have to use the
result s.
3.2. Finding Lyapunov Exponent Using TSTOOL Package
Chaotic processes are characterized by positive Lyapunov Exponent (LE)s calculated following the approach of
Wolf et al. [15], as explained in previous works [1] [2]. Again, we used the TSTOOL to find the LLE. The
function used is largelyap which is an algorithm based on work by Wolf (1985), it computes the average
exponential growth of the distance of neighboring orbits via the prediction error. The increase of the prediction
error versus the prediction time allows an estimation of the LLE [7]. In the particular MATLAB code, largelyap,
the average exponential growth of the distance of neighboring orbits is studied in a logarithmic scale, this time
via prediction error p(k). Dependence of p(k) on the number of time steps may be divided into three phases.
Phase I is the transient where the neighboring orbits converges to the direction corresponding to the λ the LLE.
During phase II, the distance grows exponentially with exp (λ tk) until it exceeds the range of validity of the
linear approximation of the flow. Then phase III begins where the distance increases slower than exponentially
until it decreases again due to folding in the state space. If the phase II is sufficiently long, a linear segment with
P. Das
slope λ appears in the p(k) versus k diagram [13]. While calculating the LLE, we have obtained the prediction
error p(k) versus k diagrams as output and are given.
4. Results and Discussion
To ascertain the nonlinearity of data, we have applied surrogate methods as described in Section 3.1, we have
used Theiler algorithm [12] to produce three surrogate datasets of each series. We have calculated LLE from
original as well as three surrogate sets and compared the values to see how much they change in per cent. This is
given in Table 1. For a sufficiently chaotic dataset, LL for a surrogate set would differ considerably for reasons
described in Section 3.1. In this paradigm, percent deviation of LLE values in corresponding surrogate sets will
indicate chaotic nature of the original data. From the results given in the Table 1, we find:
For Inflation data, chaotic natures of data are the following:
Low (change < 20%): France, Germany, Spain.
High (change < 40%): Greece, India.
Very high ((change > 40%): Italy, Singapore, Sri Lanka and UK.
From above results, we can draw following conclusions:
We can say that from above calculation, LLE values indicate that inflation data shows chaos. They are high-
ly chaotic for at least four countries listed above.
Inflation data points for very low in number. Consumer Price Index is calculated monthly, so during the pe-
riod under present study (year 2000 to 2013), we have about 160 points only. But there is evidence, for many
countries (say, for example India, Sri Lanka) prices fluctuate daily like in ForEx markets, but only monthly
values are recorded.
Anothe r important aspect of inflation data is that, as this factor immediately affects people’s life, they are
more closely monitored by respective Governments. Even the intervention during present global recession is
so acute that some reverse phenomena like deflation has taken place in many European countries. Deflation
refers to fall in money prices of commodities also constitutes a serious threat to the monitory system.
Inflation falling low, as seen in Figure 1 for countries (say, for example Greece) threatens Europe’ [16]. In
the present study, we are interested in fluctuations of the CPI, not its absolute values. So, whether inflation is
high or low is not of direct importance here, but such interventions some times in unspoken terms, makes
data analysis more difficult. Particularly, these become important when we try to see CPI in relation to other
economic indicators mentioned in this paper.
5. Discussion
We found chaotic nature of the inflation rate. But as pointed out earlier, experimentations with higher amount of
Table 1. Calculation of Largest Lyapunov Exponent (LLE) using TSTOOL in Mat Lab: Inflation
(Monthly data, January 2000 to September 2013).
Country Original
data surrogate
% change in
surrogate over
% change in
surrogate over
% change in
surrogate over
France 1.8 2.1 16.67 1.9 5.56 2.2 22.22
Germany 1.7 2.1 23.53 1.9 11.76 2 17.65
Greece 1.7 2.2 29.41 1.7 0 2.2 29.41
India 1.1 1.4 27.27 1.3 18.18 1.5 36.36
Italy 1.2 1.9 58.33 1.6 33.33 1.7 41.67
Singapore 0.7 1.2 71.43 1.2 71.43 0.8 14.29
Spain 1.7 1.8 5.88 1.7 0 1.9 11.76
Sri Lanka 0.6 0.8 33.33 0.4 -33.33 0.85 41.67
UK 1.3 1.9 46.15 1.5 15.38 1.4 7.69
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Figure 1. Plot of LLE from inflation data. Dashed line for original data and other three represents surrogate counter parts.
data have to be made. For that reason, more rigorous data collection practice has to be adopted. If we can
establish at least some empirical mathematical relation of indicators towards setting up a model of a system
consisting of these indicators, we can investigate important parameters controlling the system. These issues
deserve more thoughtful studies which will have far reaching effect to handle the system for the benefit of
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P. Das
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