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. Inflation.eu [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 data—one 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
[11].
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