Chaotic analysis of the foreign exchange rates
Introduction
Changes in the exchange rate are related to news in the fundamentals. Set of fundamentals covers: (i) the inflation for the country concerned, (ii) the money supply 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 proxies of the long-term interest rate. The latter was however only available for the low inflation countries, (v) industrial production and (vi) the trade balance relative to the GDP. Characterizing the nature of the relationship between exchange rate changes and the news in its underlying fundamentals has long been an objective of empirical international macroeconomics. [8]
After intensive empirical testing it was concluded that the relation is not linear at least for the exchange rates of countries experiencing relatively low levels of inflation. Studies of the early 1980s documented that there is no stable relationship between exchange rate movements and the news in the fundamental variables. The exchange rate appears to be disconnected from its underlying fundamentals most of the time [3], [13] while Faust et al. [7] found that most of the changes in the exchange rates occur when there is no observable news in the fundamental economic variables. This finding contradicted the theoretical models, which imply that the exchange rate can only move when there is news in the fundamentals. While there is dramatic increase in the volatility of the exchange rate, no such increase could be detected in the volatility of the underlying economic variables [10].
Naturally, the possible nonlinear nature of the relationship between exchange rate changes and the news in the underlying fundamentals were explored. The first non-linearity stressed by Obstfeld and Rogoff [15] was transaction costs. A second non-linear feature was due to the various heterogeneous agents who use different information sets [11].
The regime confronted by the exchange rates of the major industrialized countries comes close to the regime identified to be the one producing complexity, speculative noise, and structural breaks between exchange rates and underlying fundamentals. The relation between the exchange rate and the fundamentals of low inflation countries is characterized by frequent regimes shifts. It was found that the coefficients of these fundamentals change over time quite often from significant values to insignificant ones, and vice versa. This feature is absent in the exchange rate equations of high inflation countries. In those countries we find that the coefficients of the fundamentals are quite stable. The movements of the exchange rates of these countries can be explained much better by movements in underlying fundamentals (e.g., inflation differentials) [11]. In this work, we attempt to relate the exchange rate with another fundamental, namely, balance of payment (=Import − Export) with the US.
The foreign exchange market is a 24-hour financial market. The trading in the foreign exchange markets generally involves the US dollar. In the present globalized economy, most countries accept pegging their currencies to the US dollar. There are several reasons for this creeping return to pegged exchange rates. Most of the countries are buying the US dollar in order to curb the appreciation of their currencies [9].
In this work, we investigate chaotic property of Foreign Exchange Rates of several countries. Some of the related earlier works found evidence of chaotic structures in foreign exchange rates (for example, in case of the Canadian and Australian dollars over their floating rate periods), some studies found little evidence of chaos, however, many of them showed evidence of nonlinear structure. Earlier studies found little evidence of chaos, however, many of them showed evidence of nonlinear structure [12]. Bask [1], [2] considered Swedish Kroner versus Deutche Mark, ECU, US $ and Yen in his study, using data from daily observation from January 1986 to August 1995 (2409 points). By measuring the largest Lyapunov exponent, he found indication of deterministic chaos in all exchange rate series. De Grauwe and Vansteenkiste [8] found that most of the time the hypothesis of the existence of chaotic dynamics in the foreign exchange markets must be rejected. There are only a few episodes where chaotic dynamics can be detected. They stress that it is generally difficult to conclusively find evidence for the existence of chaotic dynamics because the available techniques do not allow to separate the exogenous noise from chaos. This lack of strong evidence for the existence of chaos has been confirmed by other researchers (see [9] and references therein).
This type of conflicting claims are common in nonlinear analyses of financial data, as shown in our earlier work [5]. The purpose of the present work is to test foreign exchange data for the nonlinearity and chaos. For this, daily data were collected for twelve countries, over the span of nearly 36 years. We have thus a time series for each country as described Section 2. We test the nonlinearity in the data by surrogate method as given in Section 3. In Section 4, by calculating the largest Lyapunov exponent (LLE), we found indication of deterministic chaos in all exchange rate series and discuss its implication. Finally, in Section 6, we discuss how LLEs calculated from foreign exchange data fail to reflect real economic system using data from balance of payment with the US. We added few comments in Section 6.
Section snippets
Data collection
Board of Governors of the Federal Reserve System has Foreign Exchange Rates for different countries [6], based on noon buying rates in New York City for cable transfers payable in foreign currencies. Exchange Rates data are provided by Economic Research, Federal Reserve Bank of St. Louis and are freely downloadable for research purpose. The data are available in ASCII text as well as XLS format. We collected data for twelve countries Australia, Canada, China, India, Japan, Malaysia, Singapore,
Test for nonlinearity using surrogate data method
We follow the approach of Theiler et al. [19]. 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. In the case of data shuffling, the histograms of the surrogate sequence and the reference sequence are
Finding Lyapunov exponent using TSTOOL package
Chaotic processes are characterized by positive Lyapunov Exponent (LE)s calculated following approach of Wolf et al. [20]. For details, please refer to our earlier work [4]. For fixed evolution time (FET) program: For Given the time series x(t) for m dimensional phase space with delay coordinate τ that is a point on the attractor is given byWe locate nearest neighbor to initial pointAnd denote the distance between these two points as L(t0
What does LLE represent
As indicated in the Introduction that exchange rate and hence its fluctuation depends on different type of news and agents. We have attempted to quantify nonlinearity caused by different factors discussed earlier. Finally we have the LLE values to characterize chaos in the data sets.
We now attempt to find in reality, what does LLE represent? To check for the relation of one of the fundamental news to foreign exchange rate, we present in Table 2 the balance of payment of the select countries
Conclusion
We have found that nonlinearity of varying degree exists in Foreign Exchange market. Evidence of chaos for different countries was also detected in terms of positive LE. As indicated in the theory that various news as well as agents control the exchange rate, but there is nonlinear relationship among them. This is reinforced in our finding. But in calculating the LLE to establish chaos, much information about the news and its relation to the exchange rates is probably lost. We illustrated this
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