Research Article Investigating the Existence of Chaos in Inflation Data in relation to Chaotic Foreign Exchange Rate

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1 Economics Research International, Article ID 78, 8 ages htt://dx.doi.org/.//78 Research Article Investigating the Existence of Chaos in Inflation Data in relation to Chaotic Foreign Exchange Rate Pritha Das and Atin Das Deartment of Mathematics, Indian Institute of Engineering Science & Technology, Shibur, Howrah 7, India N H School, /7 Naktala, Kolkata 7 7, India Corresondence should be addressed to Atin Das; dasatin@yahoo.co.in Received May ; Revised Setember ; Acceted 8 Setember ; Published October Academic Editor: Laura Gardini Coyright P. Das and A. Das. This is an oen access article distributed under the Creative Commons Attribution License, which ermits unrestricted use, distribution, and reroduction in any medium, rovided the original work is roerly cited. Foreign exchange (ForEx) rates are amongst the most imortant economic indices in the international monetary markets. ForEx rate reresents the value of one currency in another and it fluctuates over time. It is related to indicators like inflation, interest rate, gross domestic roduct, and so forth. In a series of works, we investigated and confirmed the chaotic roerty of ForEx rates by finding ositive largest Lyaunov exonent (LLE). As inflation influences ForEx, in this work we would like to address the secific question, Is inflation data also chaotic? We collected data for time eriod of to and tested for nonlinearity in data by surrogate method. Calculating LLE, we find existence of chaos in inflation data for some countries.. Introduction An exchange rate reresents the value of one currency in another. An exchange rate between two currencies fluctuates over time. The foreign exchange market is the largest and most liquid of the financial markets. Foreign exchange rates are amongst the most imortant economic indices in the international monetary markets. As floating exchange rate system was imlemented by industrialized countries in 97, they are no more determined almost solely by balance of ayments as in the fixed rate regime. Foreign exchange rates are affected by many highly correlated economic, olitical, and even sychological factors. The interaction of these factorsisinaverycomlexfashion[]. Ever since the breakdown of (linear) structural exchange model, the dominant view of exchange rate dynamics has been used on the news model. It was observed that changes intheexchangeratearerelatedtonewsinthefundamentals. Set of fundamentals covers (i) the inflation for the country concerned, (ii) the money suly for the country under scrutiny, (iii) the Money Market Rate, which is used as a measure of the short term interest rate, (iv) the lending rate and the long-term government bond yield which are both roxies of the long-term interest rate, and the latter was, however, only available for the low inflation countries, (v) industrial roduction, and (vi) the trade balance relative to the GDP. Characterizing the nature of the relationshi betweenexchangeratechangesandthenewsinitsunderlying fundamentals has long been an objective of emirical international macroeconomics []. In an analysis of high frequency exchange data, Goodhart [] documents that very often the exchange rate does not resond to observable news and that many exchange rate movements cannot be associated with the news. This emirical research suggests that, in addition to random shocks, there are other driving forces in the exchange market that are imortant to understanding its dynamics []. Most of the countries are buying the US dollar in order to curb the areciation of their currencies []. There is no clear relationshi among the factors as well as their collective relationshi to determine ForEx rate of a country. From the comlex viewoint of its deendencies on different factors and adding the most difficult factor of seculation in this huge market, ForEx is almost imossible to redict. Nonlinearity of seculative dynamics leads to chaotic motion of the exchange rate as mentioned before. Bask considered Swedish Kroner versus Deutsche Mark, ECU, US$,andYeninhisstudyusingdataofdailyobservationand found indication of deterministic chaos in all exchange rate

2 Economics Research International series [, 7]. In a series of works (7,, and ), we investigated and confirmed the chaotic roerty of foreign exchange rates of several countries [8 ]. Chaotic rocesses are characterized by ositive Lyaunov Exonents (LEs) and we calculated LEs from ForEx data. Economic indicators are sniets of financial and economicdataublishedregularlybygovernmentalagenciesand the rivate sector. Traders can measure the economic health of a given country (and its currency) through its economic indicators. These statistics hel market observers monitor the economy s ulse which are followed in the financial markets. We briefly introduce working definitions of them below. In this work, we collected data of gross domestic roduct (GDP), Consumer Price Index (CPI) and inflation, and ForEx rate of different countries. In our earlier work [] we investigated balance of trade (BoT) with resect to US for some countries in relation to their ForEx rate []. As discussed earlier, the relationshis of these and other indicators in defining ForEx are yet to be established. In this work we have collected data of some of these indicators from year to of the above few imortant indicators. Figures lotted from this data indicate that ForEx is closely related to other indicators. Our main focus is to exlore the nature of deendence of Consumer Price Index (CPI) on ForEx rate. We would like to address the secific question, Is CPI data is also chaotic as along with other indicators, it influences ForEx whose fluctuation is already found to be chaotic?. This question has become more ertinent in resent years because, in Euroean Union (EU) countries, the common currency Euro have a common exchange rate to US Dollar for member countries who have different economic conditions and different CPI. If domestic economic indicators define artly or wholly ForEx rate, what haens for these EU countries? Needless to say roblems some of the Eurozone countries are facing are hard to control robably for this reason. As Gilles [] ut it, over the last decade, countries in theeurozonehavequietlybutstubbornlydivergedinterms of inflation, growth, fiscal erformance, and cometitiveness. Some have had % inflation on average, others have %, some have built u trade surluses, others are increasingly indebted with resect to the rest of the world, some have ket their government budget in check, and others have let debt grow. This divergence comes from different olicy choices, different institutions, and different cultures. But the common currency has in fact added to the divergence. Now what wouldhavehaenedtosuchacountryifithadnotbeen a member of the Eurozone? Perhas its central bank would have been worried about inflation and would have raised the interest rate. This would have cooled the economy down andeseciallyhittheoversizedconstructionindustryand erhas even deflated the housing bubble, or foreign investors would have taken into account the critical develoments of the external accounts and attacked the currency. Little work was attemted in this direction. Guastello (99) showed low-dimensional chaos for US inflation rates during the era with Lyaunov dimensionality.. He also confirmed that although there was short time linear rediction of inflation rates, the global icture was, nonetheless, chaotic []. In another study () he stressed the resence of chaotic attractors for inflation rate in the US []. Results from linear and nonlinear analyses rovide overwhelming evidence in suort of the nonstationarity of the inflation rate in Africa [].Wewouldliketostudy this situation for some EU countries, as well as some other countries having their own ForEx rate. Obviously, we are not attemting any analysis of justification of Euro. Our analysis will be confined to understanding the economic indicators, articularly inflation data in relation to ForEx rate through data analysis. Details of data collection are given in Section, while in Section we estimate qualitatively the erformance of three economic indicators, as well as ForEx data. In Section, we briefly exlain nonlinear analysis tools alied here, that is, surrogate test and Lyaunov exonent. Results of analysis are given in Section which are discussed in Section as conclusion. We make some remarks under Discussion in Section7.. Data Collection and Descrition To get an idea of gross domestic roduct (GDP) annual growthrate(exressedasercentchangeoverreviousyear), we have collected data for the following countries: France, Germany, Greece, Italy, India, Malaysia, Sain, Singaore, Sri Lanka, United Kingdom, and average for Euro area. We have also collected the data of same countries on annual basis of inflation, consumer rices (annual %), obtained from World Databank, maintained by the World Bank for the eriod to on annual basis []. Detailed inflation or CPI data are as follows: the inflation rateisbaseduontheconsumerpriceindex(cpi).thecpi inflation rates used are on a yearly basis (comared to the same month the year before). For examle, CPI for January is difference in inflation over that in January exressed as er cent. Inflation.eu [7] maintains historical dataformanycountrieswhichhavebeenusedinthisaer. WehavecollecteddataonmonthlybasisfromJanuary to Setember for the following countries: France, Italy, Germany, Sain, Greece, India, Singaore, Sri Lanka, and UK. So each country has a dataset consisting of data, one for each month. For Sri Lanka, data from January to Aril 8 are available, so dataoints are 88 in number from Deartment of Census and Statistics, government of Sri Lanka [8]. Singaore CPI data was taken from data maintained by the government of Singaore [9]. Foreign Exchange data are taken from OANDA which maintains historical data, from January to November. So each set has 9 oints [].. Performance of Indicators Here we shall reort the analysis of data collected for reviously defined indicators. Earlier we have shown () that GDP has overall imortant role in influencing ForEx rate goinguordowninthelonghand.herethegdpconcerns the nominal value, that is, without inflation adjustment over

3 Economics Research International Table : Showing qualitatively erformance of indicators for different countries. Country ForEx rate (to US$) GDP growth annual % Inflation annual % France.8 to. Decreasing Stable around Germany Same Less than - value Stable around Greece Same Negative since 9 Decreasing since Italy Same Decreasing since Increasing Sain Same Decreasing, negative High, about Australia Stable around for last years Stable around Decreasing India Increased from to 9 Decreasing, less than value Very high Malaysia Decreased from.8 to. Same as 8 Higher than Singaore Decreased from.7 to. Lower than 8 value Higher than 8 value UK Same, near - value Lower than 8 value High, more than 7 value eriod of study. Over time, the growth in GDP causes inflation, and inflation begets hyerinflation. Once this rocess is in lace, it can quickly become a self-reinforcing feedback loo. This is because, in a world where inflation is increasing, eole will send more money because they know that it will be less valuable in the future. This causes further increases in GDP in the short term, bringing about further rice increases. Here, in Figure, we have lotted the GDP data for ten countries and we observe the following. () For all countries without any excetion, GDP growth rate is negative in year 9. There is a clear di in the curve. This is clearly due to fall in GDP value due to global recession. () In subsequent year, GDP growth has increased for many countries. But for last two years ( and ), GDP is again sliding down. This is true for all three categories. For Greece, GDP growth rate never turned back to ositive values, almost the same for Sain. () During -, GDP growth rate was sharly down for almost every country though in lesser scale comared to 9 fall. We have also lotted in Figure inflation of consumer rices as annual er cent change and ForEx rate for these countries against time eriod of to. At the to of each block of Figure, we lot ForEx rate daily data. For EU countries, this is not alicable as they have common ForEx rate of Euro given in searate block. From the lots of Figure the indicators changed as shown in Table.. Nonlinear Analysis of Inflation Data Here we shall concentrate on detailed nonlinear data analysis of inflation data collected to get more insight into it. The basic ointweliketoinvestigateiswhethercpidataanalysisshows chaos or not. First we check the nonlinearity of data using surrogate method. For characterizing chaos both qualitatively and quantitatively, we find LLE using TSTOOL ackage. Due toscarcityofdata,weuseanotheralgorithmtocheckthe results... Test for Nonlinearity Using Surrogate Data Method. We follow the aroach of Theiler et al.[]. The surrogate signal is roduced by hase-randomizing the given data. It has sectral roerties 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 ower sectrum as the original sequence, but (nonlinear) hase relations are destroyed. Details of the method for the countries considered have been given in the revious work [8] or as used with additionalnoisereduction []. We used the TSTOOL ackage by Parlitz et al. [], under MATLAB [] software to create surrogate data for a scalar time series. From this analysis, we got some idea aboutthedegreeofnonlinearityassociatedwiththetime series of foreign exchange data u to year 8. We are not reeating the same analysis because we are considering the same countries and comared to our revious data, we now have more oints, which are only % of total (from January8toOctober9).Butwecertainlyhavetouse the results... Finding Lyaunov Exonent Using TSTOOL Package. Chaotic rocesses are characterized by ositive Lyaunov exonent (LE) calculated following the aroach of Wolf et al. [] as detailed in Das et al. [8, 9]. Again, we used the TSTOOL [] tofindthelle.thefunctionusedis largelya which is an algorithm based on work by Wolf et al.; it comutes the average exonential growth of the distance of neighboring orbits via the rediction error. The increase of the rediction error versus the rediction time allows an estimation of the LLE. In the articular MATLAB code, largelya, the average exonential growth of the distance of neighboring orbits is studied in a logarithmic scale, this time via redictionerror (k). Deendence of (k) on the number of time stes may be divided into three hases. Phase I is the transient where the neighboring orbits converge to the direction corresonding to the LLE (λ). During hase II, the distance grows exonentially with ex (λt k ) until it exceeds the range of validity of the linear aroximation of the flow. Then hase III begins where the distance increases slower than exonentially until it decreases again due to folding in

4 Economics Research International Euro-USD exchange rate.8 France Germany Greece Italy Australia Sain India UK Malaysia Singaore Inflation, consumer rices (annual %) GDP growth (annual %) Official exchange rate (er US$) Figure : Plot of ForEx daily rate (solid line), GDP annual growth rate (dotted and marked), and monthly inflation (dashed line) for ten countries and EU (names given in each block). The horizontal axis is time eriod of year to (corresonds to to in horizontal axis of each block). Note that EU countries (here, first six) have common currency and hence the same ForEx rate. the state sace. If hase II is sufficiently long, a linear segment with sloe λ aears in the (k) versus k diagram []. While calculatingthelle,wehaveobtainedtheredictionerror (k) versus k diagramsasoututandaregiveninfigure... Finding Lyaunov Exonent Using Rosenstein Algorithm. The algorithm is also attractive from a ractical standoint because it does not require large datasets []. The first ste of the aroach involves reconstructing the attractor dynamics from a single time series. The reconstruction deends on the lag or reconstruction delay and the embedding dimension []. This method works well when dataoints are small, to for the systems examined. The greatest difficulty occurs with the Rossler attractor where a % negative bias in the results for even number of oints is equal to. In our case, inflation dataoints are small (nearly ). For comlex data like inflation, the embedding dimension is high, but the above method fails with higher dimension but smaller datasets. We used MATLAB rogram []tofind LLE for some countries, keeing in mind the error range. We find LLE from inflation data of France as., India as., Sain as., and Germany as.. The outut of the rogram is given in Figure. Itistobenotedthatabsolute value of LLE varies deending on the algorithm used, but it s sign should be the same as ositive LLE indicates chaos. In our case, the results obtained using both TSTOOL and Rosenstein algorithm indicate chaos.

5 Economics Research International.... France Germany Greece India Italy Singaore Sain..... Sri Lanka... UK Figure : Showing LLE values of inflation data calculated using TSTOOL; solid line indicates original data and the other three (dot, dash, and dot-dash) indicate three surrogate sets. Name of concerned country is given in each block.. Results To ascertain the nonlinearity of data, we have alied surrogate methods as described in Section.. We have used Theiler algorithm [] to roduce three surrogate datasets of each series. We have calculated LLE from original and three surrogate sets and comared the values to see how much they change in er cent. This is given in Table.For a sufficiently chaotic dataset, LLE for a surrogate set would differ considerably for reasons described in Section..In this aradigm, er cent deviation of LLE values in corresonding surrogatesetswillindicatechaoticnatureoftheoriginaldata. Fromtheresultsgivenin Table,wefindthefollowing. (A) For inflation data, chaotic nature of data is as follows: low (change < %): France, Germany, and Sain; high (change < %): Greece and India; very high ((change > %):Italy,Singaore,Sri Lanka, and UK. (B) For ForEx data, chaotic nature of data is as follows: low: EU and Australia; high: Malaysia; very high: Singaore, Sri Lanka, and India.. Conclusion From above results, we can draw the following conclusions. () We can say that, from above calculation, LLE values indicate that inflation data exhibits chaos. They are highly chaotic for at least four countries listed above. () Inflation data oints are very low in number. Consumer Price Index is calculated monthly, so during the eriod under resent study (year to ), we have about oints only. But there is evidence that for many countries (say, e.g., India and Sri Lanka) rices fluctuate daily like in ForEx markets, but only monthly values are recorded. Considering algorithm to calculate LLE, such small number of dataoints is not sufficient. Comared to this, each ForEx dataset contains nearly oints. ()Anotherimortantasectofinflationdataisthat as this factor immediately affects eole s life they aremorecloselymonitoredbyresectivegovernments. Even the intervention during resent global recession is so acute that some reverse henomena likedeflationhavetakenlaceinmanyeuroean countries. Deflation which refers to fall in money rices of commodities also constitutes a serious threat to the monitory system. Inflation falling low as seen

6 Economics Research International Lyaunov exonent Lyaunov exonent France Germany Lyaunov exonent Lyaunov exonent India Sain Figure : Showing LLE values of inflation data calculated using Rosenstein algorithm. Country Original data Surrogate set Table : Calculation of largest Lyaunov exonent (LLE) using TSTOOL in MATLAB. %changein surrogate over original Surrogate set %changein surrogate over original Surrogate set %changein surrogate over original Inflation (monthly data, January to Setember ) France Germany Greece Italy Sain UK Sri Lanka India Singaore Foreign exchange rate (daily data, January to October ) EU Australia India Singaore Malaysia Sri Lanka

7 Economics Research International 7 in Figure for countries (say, e.g., Greece) threatens Euroe [7].In theresentstudy,weareinterested in fluctuations of the CPI, not its absolute values. So, whether inflation is high or low is not of direct imortance here, but such interventions sometimes inunsokentermsmakedataanalysismoredifficult. Particularly, these become imortant when we try to see CPI in relation to other economic indicators mentioned in this aer. () If we comare the LLE values of inflation data and ForEx rate data, we find ForEx rate to be more chaotic in nature. For India, Singaore, and Sri Lanka we have calculated LLEs from both inflation and ForEx data. We find for these countries, LLE of ForEx data is more than double of LLE of inflation data. So, even if inflation is chaotic, it is much lesser chaotic than ForEx rate. () In our earlier analysis (7), we showed ForEx market to exhibit chaos [8]. For some countries (like India, China, and Sri Lanka) chaos is much higher comared to countries like Jaan, Singaore, and Sweden when countries like Malaysia and Thailand standinbetween.theresultswerecalculatedfor eriod (year 97 to ) before recession started. Here we find that the same results are reflected for data during year to. Chaotic nature of ForEx countries is retained even during the global recession as Singaore, Sri Lanka, and India show high chaotic ForEx market. () Interestingly, we find countries with chaotic ForEx market show higher chaotic inflation rate (Singaore, Sri Lanka, and India). So there may be good relation between the two indicators. But countries within EU show different levels of chaos in inflation data (France and Germany low but Italy high). Historically, to articiate in the currency, member states are meant to meet strict criteria, such as a budget deficit of less than three er cent of their GDP, a debt ratio of less than sixty er cent of GDP (both of which were ultimately widely flouted after introduction), low inflation, and interest rates close to the EU average [8]. But current analysis shows different icture in this regard. So the relationshi is more comlex in nature. The imortant oint that arises is how and why countries with same ForEx (that is Euro) have different inflation rate. So robably, in relating inflation and ForEx rates, other domestic and international factors have to be accounted for. 7. Discussion We found chaotic nature of the inflation rate. But as ointed out earlier, exerimentation with higher amount of data has to be made. For that reason, more rigorous data collection racticehastobeadoted.ifwecanestablishatleastsome emirical mathematical relation of indicators towards setting u a model of a system consisting of these indicators, we can investigate imortant arameters controlling the system. These issues deserve more thoughtful studies which will have far reaching effect on handling the system for the benefit of citizens. Conflict of Interests The authors declare that there is no conflict of interests regarding the ublication of this aer. References [] M. A. Torkamani, S. Mahmoodzadeh, S. Pourroostaei, and C. Lucas, Chaos theory and alication in foreign exchange rates vs. IRR (Iranian rial), International Human and Social Sciences,vol.,article7,7. [] P. Grauwe de and I. Vansteenkiste, Exchange rates and fundamentals A nonlinear relationshi? Working Paer 77, Center for Economic Studies & Ifo Institute for Economic Research,, htt:// [] C. Goodhart, News and the foreign exchange market, LSE Financial Market Grou Discussion Paer 7, 99. []P.deGrauweandH.Dewachter, Achaoticmodelofthe exchange rate: the role of fundamentalists and chartists, Oen Economies Review,vol.,no.,. 79,99. [] P. de Grauwe and M. Grimaldi, Intervention in the Foreign Exchange Market in a Model with Noise Traders, University of Leuven, Leuven, Belgium,. [] M. Bask, Dimensions and Lyaunov exonents from exchange rate series, Chaos, Solitons and Fractals,vol.7,no.,.99, 99. [7] M. Bask, A ositive Lyaunov exonent in Swedish exchange rates? Chaos, Solitons and Fractals,vol.,no.8,.9,. [8] A. Das and P. Das, Chaotic analysis of the foreign exchange rates, Alied Mathematics and Comutation,vol.8,no.,. 88 9, 7. [9] G. Çoban, A. H. Büyüklü, and A. Das, A linearization based non-iterative aroach to measure the gaussian noise level for chaotic time series, Chaos, Solitons and Fractals,vol.,no.,. 78,. [] A. Das, P. Das, and G. 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8 8 Economics Research International [7] Inflation.eu worldwide inflation data, December, htt:// [8] Movements of Sri Lanka CCPI, Deartment of Census and Statistics, Government of Sri Lanka,, htt:// [9] Singaore inflation data, Government of Singaore,, htt://data.gov.sg/. [] OANDA a trusted source for currency data, November, htt:// [] J.Theiler,S.Eubank,A.Longtin,B.Galdrikian,andJ.Doyne Farmer, Testing for nonlinearity in time series: the method of surrogate data, Physica D: Nonlinear Phenomena, vol. 8, no.,. 77 9, 99. [] U. Parlitz, I. Wedekind, W. Lauterborn, and C. Merkwirth, TSTOOL & User Manual, Ver.., DPI Göttingen,, htt:// [] MATLAB Release, The MathWorks, Natick, Mass, USA,. [] A. Wolf, J. B. Swift, H. L. Swinney, and J. A. Vastano, Determining Lyaunov exonents from a time series, Physica D. Nonlinear Phenomena,vol.,no.,.8 7,98. [] M. T. Rosenstein, J. J. Collins, and C. J. de Luca, A ractical method for calculating largest Lyaunov exonents from small data sets, PhysicaD:NonlinearPhenomena,vol.,no.-,. 7, 99. [] S. Mohammadi, Lyarosen.m A Matlab Imlementation of Rosenstein Algorithm, University of Tehran, 9. [7] Wall Street J, November, htt://online.wsj.com/homeage. [8] Wikiedia, EURO,, htt://en.wikiedia.org/wiki/euro.

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