# The chaotic behavior of foreign exchange rates.

1. Introduction

In the early 1970s, many economists believed that the floating currency exchange rates that were to characterize the post-Bretton Woods period could be well explained by the purchasing power parity theory [see Bilson and Marston, 1984]. As empirical data soon demonstrated, however, the theory was not sufficient to explain the large fluctuations in exchange rates, and the latter

part of the decade saw the development of several new economic theories of exchange rate determination. Among these are the theories of Dornbusch (1976), Mussa (1976), and Frenkel (1976). The exchange rate was no longer viewed merely as an equalizer of relative inflation rates as suggested by the PPP theory. Rather, the new theories suggested that a country's exchange rate was the market price of local money in the world market. Among the determinants of this price a key element was deemed to be the supply of and the demand for the local currency. The theories also accounted for the effects of other economic variables, and rational expectations. An example of an economic model based on this approach to exchange rate determination is Woo (1985). Similar paper used vector autoregression (VAR), a time series technique that analyzes the evolution of exchange rates in conjunction with other economic variables [see, for example, Branson, 1984]. The technique, which is equivalent to simultaneous equations [Zellner and Palm, 1974], requires the specification of a particular model for exchange rates. Consistent with the view of the exchange rate as the market price of an asset traded in an efficient market, the VAR approach has not met with much success in explaining and predicting exchange rate movements [see Diebold, 1988].

In this paper, we follow a new approach to exchange rates. We believe that one explanation for the persistency of large unpredictable fluctuations in exchange rates is that the mechanism determining the exchange rates may be chaotic. We use a technique developed by Grassberger and Proaccai (1983) for measuring the amount of chaos inherent in a time series. To do so, we compute an estimate of the correlation dimension of a series, and use the bootstrap [Efron (1979) to determine the estimator standard error. The estimated correlation dimension of various exchange rates then provides us with information about the behavior of exchange rates, without having to assume an econometric model.

We use the correlation dimension for two purposes. First, we estimate this parameter for series of daily (U.S. Dollar) exchange rates of five currencies: Sterling, the French franc, the Italian lira, the German mark, and the Singapore dollar. We show how the correlation dimension reflects the degree to which chaotic behavior characterizes the fluctuations in the exchange rate, and how this conforms with ideas about exchange rate management. Second, we use the correlation dimension and its estimated standard error in carrying out an intervention analysis. The analysis is aimed at determining whether the exchange rates of the five currencies were affected by the stock market crash of October 19, 1987. Generally, an intervention analysis would have required building separate time series models for each currency [Box and Tiao (1975)], or a single VAR model for all the currencies [Abraham (1980)]. In our case, neither is necessary and the correlation dimension leads to clear results.

2. The Correlation Dimension

Since Lorenz (1963) first showed that relatively simple mathematical systems can lead to unpredictable behavior, much work has gone into trying to understand the complex behavior characterizing chaos. The developed theory is linked to fractals in the sense of Mandelbrot (1983) because truly chaotic motion often possesses the intricate geometry of the fractal. Pure chaos is deterministic in nature, yet impossible to predict. Physicists studying turbulence observe chaotic systems, and so do workers in biology, ecology, and other areas. Only recently did the concept begin to emerge in the economics literature [see Scheinkman and LeBaron (19890 and Hsieh (1989)]. This is surprising given the economic behavior is often erratic, and that small initial fluctuations in prices or quantities of traded assets may lead to large, unpredictable deviations as the system evolves. This is exactly the characteristic of chaos. In fact, in his leading paper on chaos, Ruelle (1980) notes: |One imagines that strange attractors may play a role in economics . . . suppose that the macroeconomic evolution equations contain a parameter m describing, say, the level of technological development. By analogy with hydrodynamics we would guess that for small m the economy is in a steady state and that, as m increases, periodic or quasiperiodic cycles may develop. For high m chaotic behavior with sensitive dependence on initial condition would be present.'

Ways of quantifying the degree to which a system is chaotic again relate to fractals. The Hausdorff dimension is a measure of how intricately the fractal fills the space in which it exists. A related measure, a lower bound for the Hausdorff dimension useful for time series data, is the correlation dimension developed by Grassberger and Procaccia (1983). The measure is defined as d, where: (1) [Mathematical Expression Omitted]

The correlation integral C(e) measures the fraction of the total number of pairs of points such that the distance between them is at most [Epsilon]. As an example, we consider the logistic equation: (2) [x.sub.t+1] = [ax.sub.t] (1 - [x.sub.t])

Figure 1 shows iterates of Equation 2 in two different conditions: a stable attractor (a = 3.0) and chaos (a = 3.95). Figure 2 illustrates the corresponding correlation dimension calculations. Note that the correlation dimension d changes from 0.4 to 0.8 as the system becomes chaotic. When data are distributed approximately on a line, their dimension (the limit in Equation (1)) is approximately the dimension of a line, 1. When data are distributed uniformly on a plane, their dimension is that of plane, 2. Data distributed more discretely have a dimension less than that of a line, 0<D<1, and data distributed on a dense curve that covers some of the plane have dimension between that of a line and that of a plane, 1<D<2. This definition is very close to the definition of the dimension of a fractal, given by Mandelbrot (1983). The more chaotic the mechanism generating the data, the more densely the data fill the n-dimensional space in which they exist, and the general the correlation dimension D. Hence, for single-variable data, we would except D to range between 0 and 2. A series with a large D is more chaotic than a similar series with a lower value for the correlation dimension.

3. Analysis of Daily Foreign Exchange Rates

Daily foreign exchange data for the five currencies, British Sterling, Deutchmark, Italian lira, French franc, and Singapore dollar, from 3/31/89 to 9/12/89 (115 observations) were collected. Figure 3 is a superposition of the five currencies normalized to their maximum values. As can be seen from the figure, movement in the European currencies seem to be highly correlated with each other, as expected, while movements of the Singapore dollar are less variable and apparently controlled to within a particular range. This accords with information about the |managed float' regime followed by the Monetary Authority of Singapore [see MAS Report (1989), and Lee (1984)].

Figure 4 shows a phase diagram for each of the five currencies. The diagram describes the state of the system for each of the currencies in the sense that it depicts each currency value on the horizontal axis and the difference in currency values on the vertical axis. Phase diagrams are useful for exploring the stability of a system and possibly identifying chaotic behavior. Long-term patterns of behavior form an attractor: a set of points on the diagram to which the system gravitates. Attractors of chaotic systems are called strange attractors [Ruelle (1980)].

4. Exchange Rate Correlation Dimension

As a first step in the analysis, we performed an ARIMA identification procedure on each of the five currencies as a single series. The European currencies all appeared as random walks, while Singapore identified as an autoregressive process of order 1. In the second step, we computed the estimated correlation dimension of each series using Equation (1). Again, Singapore yielded a lower parameter estimate, reflecting less chaotic behavior as may be expected from the tight management policy followed by the monetary authority. The results of the analysis are shown in Table (1). the standard errors of the estimated correlation dimension for each currency were obtained using the bootstrap method. [Tabular Data 1 Omitted]

Singapore has a significant difference in the correlation dimension from the four European currencies, all of which do not seem different from each other. Figure 5 shows the numerator and the denominator of Equation (1) necessary for estimating the correlation dimension.

5. Intervention Analysis

We used the correlation dimension in testing for the effect of the U.S. stock market crash of October 19, 1987 on our five currencies. For each currency, daily data were collected for 35 observations before and after the intervention on October 19. The data are shown in Figure 6. For each currency and time span (before, after), we computed the estimated correlation dimension and the bootstrapped standard error. The results are shown in Table 2.

As can be seen from Table 2, there is a statistically significantly difference in the correlation dimension for each one of the European currencies studied, but no significant change in the correlation dimension for Singapore. We conclude that the four European currencies were affected by the stock market crash in terms of the degree of chaos in the system, but that no such change occurred in the more controlled Singapore currency. In addition, the decrease in correlation dimension may be interpreted as the result of significant intervention by the monetary authorities of the four European countries. [Tabular Data 2 Omitted]

6. Conclusion

In this paper, we showed how the correlation dimension, a concept used by physicists to describe the degree of chaos in a system, may be used to characterize the behavior of foreign exchange rate time series. We estimated the correlation dimension of five currencies and showed that the degree of chaos in the European currencies was higher than that of Singapore, a managed-float currency. As a quantifier of chaos, the correlation dimension may also be used by traders and international investors in assessing the risk associated with a foreign currency.

We used the estimated correlation dimension in performing an intervention analysis to determine which currencies, if any, were affected by the stock market crash of October 19, 1987. We concluded that the chaotic behavior of the four European currencies we studied was indeed changed (at least temporarily) by the event, while the Singapore currency was not significantly affected. We noted that in an intervention analysis, the correlation dimension is useful as a detector of change in a time series process without requiring the use of any econometric model. Since often no adequate econometric model exists, the use of the correlation dimension holds great promise as quickly-computed indicator of fundamental change.

References

Abraham, B.,1980, Intervention Analysis and Multiple Time Series, Biometrika, 67, 1, pp. 73-8. Bilson, J. F. O., and R. C. Marston, 1984, Introduction to Exchange Rate Theory and Practice, Chicago, Ill.: The University of Chicago Press, pp.1-10. Box, G. E. P., and G. Tiao, 1975, Intervention Analysis with Applications to Economic and Environmental Problems, Journal of the American Statistical Association, 70, pp. 70-9. Branson, W. H., 1984, Exchange Rate Policy after a Decade of Floating, in Exchange Rate Theory and Practice, J. Bilson and R. Marston, eds., Chicago, Ill.: University of Chicago Press. Diebold, F. X., 1988, Empirical Modeling of Exchange Rate Dynamics, New York: Springer-Verlag. Dornbusch, R., 1976, Expectations and Exchange Rate Dynamics, Journal of Political Economy, 84, December, pp. 1161-76. Efron, B., 1979 Bootstrap methods: Another look at the Jackknife, Annals of Statistics, 7. pp. 1-26. Frenkel, J. A., 1976, A Monetary Approach to the Exchange Rate: Doctrinal Aspects and Empirical Evidence, Scandinavian Journal of Economics, 78, May, pp. 200-24. Hsieh, D. A., 1989, Testing for Nonlinear Dependence in Daily Foreign Exchange Rates, Journal of Business, Vol. 62, no. 3, pp. 339-368. Lee, S. Y., 1984, Some Aspects of Foreign Exchange Management, in Singapore, Asia Pacific Journal of Management, May pp. 207-217 Lorenz, E. N., 1963, Deterministic Nonperiodic Flow, Journal of Atmospheric Science, 20, 130. Mandelbrot, B. B., 1983, The Fractal Geometry of Nature, New York: W. H. Freeman. M.A.S. Annual Report, 1989, Singapore: The Monetary Authority of Singapore. Mussa, M. L., 1976, Exchange Rate, The Balance of Payments, and Monetary and Fiscal Policy under a Regime of Controlled Floating, Scandinavian Journal of Economics, 78 May, pp. 229-48. Ruelle, D., 1980, Strange Attractors, La Recherche, No. 108, February. Scheinkman, J. A., and B. LeBaron, 1989, Nonlinear dynamics and stock returns, Journal of Business, vol. 62, no. 3, pp. 311-337. Woo, W. T., 1985, The Monetary Approach to Exchange Rate Determination, Journal of International Economics, 18, pp.1-16. Zellner, A., and F. Palm, 1974, Time Series Analysis and Simultaneous Equations Econometric Models, Journal of Econometrics, 2, pp. 17-54. Amir D. Aczel and Norman H. Josephy Associate Professors of Mathematical Science, Bentley College.

In the early 1970s, many economists believed that the floating currency exchange rates that were to characterize the post-Bretton Woods period could be well explained by the purchasing power parity theory [see Bilson and Marston, 1984]. As empirical data soon demonstrated, however, the theory was not sufficient to explain the large fluctuations in exchange rates, and the latter

part of the decade saw the development of several new economic theories of exchange rate determination. Among these are the theories of Dornbusch (1976), Mussa (1976), and Frenkel (1976). The exchange rate was no longer viewed merely as an equalizer of relative inflation rates as suggested by the PPP theory. Rather, the new theories suggested that a country's exchange rate was the market price of local money in the world market. Among the determinants of this price a key element was deemed to be the supply of and the demand for the local currency. The theories also accounted for the effects of other economic variables, and rational expectations. An example of an economic model based on this approach to exchange rate determination is Woo (1985). Similar paper used vector autoregression (VAR), a time series technique that analyzes the evolution of exchange rates in conjunction with other economic variables [see, for example, Branson, 1984]. The technique, which is equivalent to simultaneous equations [Zellner and Palm, 1974], requires the specification of a particular model for exchange rates. Consistent with the view of the exchange rate as the market price of an asset traded in an efficient market, the VAR approach has not met with much success in explaining and predicting exchange rate movements [see Diebold, 1988].

In this paper, we follow a new approach to exchange rates. We believe that one explanation for the persistency of large unpredictable fluctuations in exchange rates is that the mechanism determining the exchange rates may be chaotic. We use a technique developed by Grassberger and Proaccai (1983) for measuring the amount of chaos inherent in a time series. To do so, we compute an estimate of the correlation dimension of a series, and use the bootstrap [Efron (1979) to determine the estimator standard error. The estimated correlation dimension of various exchange rates then provides us with information about the behavior of exchange rates, without having to assume an econometric model.

We use the correlation dimension for two purposes. First, we estimate this parameter for series of daily (U.S. Dollar) exchange rates of five currencies: Sterling, the French franc, the Italian lira, the German mark, and the Singapore dollar. We show how the correlation dimension reflects the degree to which chaotic behavior characterizes the fluctuations in the exchange rate, and how this conforms with ideas about exchange rate management. Second, we use the correlation dimension and its estimated standard error in carrying out an intervention analysis. The analysis is aimed at determining whether the exchange rates of the five currencies were affected by the stock market crash of October 19, 1987. Generally, an intervention analysis would have required building separate time series models for each currency [Box and Tiao (1975)], or a single VAR model for all the currencies [Abraham (1980)]. In our case, neither is necessary and the correlation dimension leads to clear results.

2. The Correlation Dimension

Since Lorenz (1963) first showed that relatively simple mathematical systems can lead to unpredictable behavior, much work has gone into trying to understand the complex behavior characterizing chaos. The developed theory is linked to fractals in the sense of Mandelbrot (1983) because truly chaotic motion often possesses the intricate geometry of the fractal. Pure chaos is deterministic in nature, yet impossible to predict. Physicists studying turbulence observe chaotic systems, and so do workers in biology, ecology, and other areas. Only recently did the concept begin to emerge in the economics literature [see Scheinkman and LeBaron (19890 and Hsieh (1989)]. This is surprising given the economic behavior is often erratic, and that small initial fluctuations in prices or quantities of traded assets may lead to large, unpredictable deviations as the system evolves. This is exactly the characteristic of chaos. In fact, in his leading paper on chaos, Ruelle (1980) notes: |One imagines that strange attractors may play a role in economics . . . suppose that the macroeconomic evolution equations contain a parameter m describing, say, the level of technological development. By analogy with hydrodynamics we would guess that for small m the economy is in a steady state and that, as m increases, periodic or quasiperiodic cycles may develop. For high m chaotic behavior with sensitive dependence on initial condition would be present.'

Ways of quantifying the degree to which a system is chaotic again relate to fractals. The Hausdorff dimension is a measure of how intricately the fractal fills the space in which it exists. A related measure, a lower bound for the Hausdorff dimension useful for time series data, is the correlation dimension developed by Grassberger and Procaccia (1983). The measure is defined as d, where: (1) [Mathematical Expression Omitted]

The correlation integral C(e) measures the fraction of the total number of pairs of points such that the distance between them is at most [Epsilon]. As an example, we consider the logistic equation: (2) [x.sub.t+1] = [ax.sub.t] (1 - [x.sub.t])

Figure 1 shows iterates of Equation 2 in two different conditions: a stable attractor (a = 3.0) and chaos (a = 3.95). Figure 2 illustrates the corresponding correlation dimension calculations. Note that the correlation dimension d changes from 0.4 to 0.8 as the system becomes chaotic. When data are distributed approximately on a line, their dimension (the limit in Equation (1)) is approximately the dimension of a line, 1. When data are distributed uniformly on a plane, their dimension is that of plane, 2. Data distributed more discretely have a dimension less than that of a line, 0<D<1, and data distributed on a dense curve that covers some of the plane have dimension between that of a line and that of a plane, 1<D<2. This definition is very close to the definition of the dimension of a fractal, given by Mandelbrot (1983). The more chaotic the mechanism generating the data, the more densely the data fill the n-dimensional space in which they exist, and the general the correlation dimension D. Hence, for single-variable data, we would except D to range between 0 and 2. A series with a large D is more chaotic than a similar series with a lower value for the correlation dimension.

3. Analysis of Daily Foreign Exchange Rates

Daily foreign exchange data for the five currencies, British Sterling, Deutchmark, Italian lira, French franc, and Singapore dollar, from 3/31/89 to 9/12/89 (115 observations) were collected. Figure 3 is a superposition of the five currencies normalized to their maximum values. As can be seen from the figure, movement in the European currencies seem to be highly correlated with each other, as expected, while movements of the Singapore dollar are less variable and apparently controlled to within a particular range. This accords with information about the |managed float' regime followed by the Monetary Authority of Singapore [see MAS Report (1989), and Lee (1984)].

Figure 4 shows a phase diagram for each of the five currencies. The diagram describes the state of the system for each of the currencies in the sense that it depicts each currency value on the horizontal axis and the difference in currency values on the vertical axis. Phase diagrams are useful for exploring the stability of a system and possibly identifying chaotic behavior. Long-term patterns of behavior form an attractor: a set of points on the diagram to which the system gravitates. Attractors of chaotic systems are called strange attractors [Ruelle (1980)].

4. Exchange Rate Correlation Dimension

As a first step in the analysis, we performed an ARIMA identification procedure on each of the five currencies as a single series. The European currencies all appeared as random walks, while Singapore identified as an autoregressive process of order 1. In the second step, we computed the estimated correlation dimension of each series using Equation (1). Again, Singapore yielded a lower parameter estimate, reflecting less chaotic behavior as may be expected from the tight management policy followed by the monetary authority. The results of the analysis are shown in Table (1). the standard errors of the estimated correlation dimension for each currency were obtained using the bootstrap method. [Tabular Data 1 Omitted]

Singapore has a significant difference in the correlation dimension from the four European currencies, all of which do not seem different from each other. Figure 5 shows the numerator and the denominator of Equation (1) necessary for estimating the correlation dimension.

5. Intervention Analysis

We used the correlation dimension in testing for the effect of the U.S. stock market crash of October 19, 1987 on our five currencies. For each currency, daily data were collected for 35 observations before and after the intervention on October 19. The data are shown in Figure 6. For each currency and time span (before, after), we computed the estimated correlation dimension and the bootstrapped standard error. The results are shown in Table 2.

As can be seen from Table 2, there is a statistically significantly difference in the correlation dimension for each one of the European currencies studied, but no significant change in the correlation dimension for Singapore. We conclude that the four European currencies were affected by the stock market crash in terms of the degree of chaos in the system, but that no such change occurred in the more controlled Singapore currency. In addition, the decrease in correlation dimension may be interpreted as the result of significant intervention by the monetary authorities of the four European countries. [Tabular Data 2 Omitted]

6. Conclusion

In this paper, we showed how the correlation dimension, a concept used by physicists to describe the degree of chaos in a system, may be used to characterize the behavior of foreign exchange rate time series. We estimated the correlation dimension of five currencies and showed that the degree of chaos in the European currencies was higher than that of Singapore, a managed-float currency. As a quantifier of chaos, the correlation dimension may also be used by traders and international investors in assessing the risk associated with a foreign currency.

We used the estimated correlation dimension in performing an intervention analysis to determine which currencies, if any, were affected by the stock market crash of October 19, 1987. We concluded that the chaotic behavior of the four European currencies we studied was indeed changed (at least temporarily) by the event, while the Singapore currency was not significantly affected. We noted that in an intervention analysis, the correlation dimension is useful as a detector of change in a time series process without requiring the use of any econometric model. Since often no adequate econometric model exists, the use of the correlation dimension holds great promise as quickly-computed indicator of fundamental change.

References

Abraham, B.,1980, Intervention Analysis and Multiple Time Series, Biometrika, 67, 1, pp. 73-8. Bilson, J. F. O., and R. C. Marston, 1984, Introduction to Exchange Rate Theory and Practice, Chicago, Ill.: The University of Chicago Press, pp.1-10. Box, G. E. P., and G. Tiao, 1975, Intervention Analysis with Applications to Economic and Environmental Problems, Journal of the American Statistical Association, 70, pp. 70-9. Branson, W. H., 1984, Exchange Rate Policy after a Decade of Floating, in Exchange Rate Theory and Practice, J. Bilson and R. Marston, eds., Chicago, Ill.: University of Chicago Press. Diebold, F. X., 1988, Empirical Modeling of Exchange Rate Dynamics, New York: Springer-Verlag. Dornbusch, R., 1976, Expectations and Exchange Rate Dynamics, Journal of Political Economy, 84, December, pp. 1161-76. Efron, B., 1979 Bootstrap methods: Another look at the Jackknife, Annals of Statistics, 7. pp. 1-26. Frenkel, J. A., 1976, A Monetary Approach to the Exchange Rate: Doctrinal Aspects and Empirical Evidence, Scandinavian Journal of Economics, 78, May, pp. 200-24. Hsieh, D. A., 1989, Testing for Nonlinear Dependence in Daily Foreign Exchange Rates, Journal of Business, Vol. 62, no. 3, pp. 339-368. Lee, S. Y., 1984, Some Aspects of Foreign Exchange Management, in Singapore, Asia Pacific Journal of Management, May pp. 207-217 Lorenz, E. N., 1963, Deterministic Nonperiodic Flow, Journal of Atmospheric Science, 20, 130. Mandelbrot, B. B., 1983, The Fractal Geometry of Nature, New York: W. H. Freeman. M.A.S. Annual Report, 1989, Singapore: The Monetary Authority of Singapore. Mussa, M. L., 1976, Exchange Rate, The Balance of Payments, and Monetary and Fiscal Policy under a Regime of Controlled Floating, Scandinavian Journal of Economics, 78 May, pp. 229-48. Ruelle, D., 1980, Strange Attractors, La Recherche, No. 108, February. Scheinkman, J. A., and B. LeBaron, 1989, Nonlinear dynamics and stock returns, Journal of Business, vol. 62, no. 3, pp. 311-337. Woo, W. T., 1985, The Monetary Approach to Exchange Rate Determination, Journal of International Economics, 18, pp.1-16. Zellner, A., and F. Palm, 1974, Time Series Analysis and Simultaneous Equations Econometric Models, Journal of Econometrics, 2, pp. 17-54. Amir D. Aczel and Norman H. Josephy Associate Professors of Mathematical Science, Bentley College.

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Author: | Aczel, Amir D.; Josephy, Norman. |
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Publication: | American Economist |

Date: | Sep 22, 1991 |

Words: | 2193 |

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