Nuances of chaos in foreign exchange markets.

Pandey, Vivek K. ; Kohers, Theodor ; Kohers, Gerald 等

ABSTRACT

Is the economy an evolutionary process? Recently, scientists have begun to think that the economic dynamics of free-market societies can be explained by evolutionary dynamics. If so, on the aggregate level then, foreign exchange markets may be driven by a collective "image of the future" that societies are driven by. When economies are viewed as evolutionary processes, it is just possible that on the aggregate, but a subconscious level, competitive forces in foreign exchange markets become endogenous in a system that drives exchange rates towards a collective futuristic image. Moreover, such a system could be deterministic. This paper investigates such possibility in the daily dollar price movements of five major trading currencies and three less actively traded currencies over a 25-year time span beginning with the inception of the floating exchange rate system in 1973. The results of this study suggest that none of the examined currencies are influenced by low-dimensional chaotic determinism. Although three of the examined exchange rates do exhibit signs of being driven by higher-dimensional chaos, this finding does not significantly favor the possibility of predicting these currency movements. As such, very little evidence of a deterministic driving force behind foreign exchange rates is uncovered in this study.

INTRODUCTION

Of late, there has been some deliberation about viewing economies as evolutionary processes. In such a case, it is just possible that on the aggregate, but a subconscious level, competitive forces in foreign exchange markets become endogenous in a system that drives exchange rates towards a collective "image of the future". Grabbe [1996] presents the possibility of self-organization of human societies, and thus by implication of the economy, with a shared image or a vision of the future. At the singular level, this vision might be subconscious or nonexistent, but at the aggregate level such a vision might be discernible. In the foreign exchange markets, most of the trading occurs while traders are marketmakers or speculators. They may not afford the luxury of acting late on any relevant news. Very often, the trader must anticipate other traders' moves and try to preempt such moves. As such, each trader must not just act on his or her expectations but rather act on anticipation of other traders moves who themselves are trying to anticipate the first's and everyone else's moves and so on. Evolutionary dynamics provide a solution in the form of spontaneous order involving dynamic feedback at a higher, or aggregate, level. In the foreign exchange markets context, what appears to be competition amongst traders and central banks at the lower level, where expectations are generated, functions as co-ordination at the higher (global) level (Grabbe [1996]). Recent research in behavioral economics has also yielded explanations of chaotic influences in economic and financial data series based on equilibrium solutions under conditions of imperfect foresight (Sorger [1996]).

If such is the case, foreign exchange rates may be driven by nonlinear deterministic systems. Recent advances in the study of nonlinear dynamics and chaotic processes have yielded tools that can distinguish stochastic variables from seemingly random data that are, in fact, generated by low-complexity nonlinear deterministic processes. Tests for informational efficiency in foreign exchange markets can now be strengthened by employing these tests for chaotic dynamics among time series of security returns. Since some forms of chaotic determinism can generate seemingly random variates, it is imperative that tests for nonlinear dependencies become an integral part of market efficiency tests.

In examining the pricing efficiency of foreign exchange markets, the vast majority of research has relied on linear modeling techniques, which have serious limitations in detecting multidimensional patterns. This study intends to broaden the scope of previous research on the subject. This approach employs a battery of tests designed to avoid the problem of nonstationarity often associated with financial data and proceeds in its attempts to detect any driving influence of low-dimensional deterministic chaos in exchange rate movements of five major trading currencies and three less actively traded currencies. More specifically, this study examines the spot markets for the exchange of dollars for the following currencies: the Canadian dollar, the German mark, the Japanese yen, the Swiss franc, the British pound, the Australian dollar, the Malaysian ringgit, and the Spanish peseta. Foreign exchange rate movements exhibiting low-dimensional deterministic chaos may contain some informational inefficiency (in the weak form sense); thus, it may be possible to use nonlinear dynamics to predict future currency exchange rates.

Several recent studies on foreign exchange rate movements have ascertained that these rate changes depict a high level of non-normality. Aggarwal [1990] detected a high degree of skewness and kurtosis in the forward price movements of foreign currencies. He also observed that significant non-normality plagued the spot rates for foreign exchange.

Upon reviewing some prior literature on the statistical properties of foreign exchange rate movements, Hsieh [1991b] concludes that daily changes in foreign exchange rates are not identically and independently distributed. Furthermore, he ascertains that higher order moments characterize the distribution of foreign exchange rates.

The observations made in the above mentioned studies suggest the possibility that nonlinear dynamics may be driving foreign exchange price changes. The fact that statistical properties characterizing the distribution of foreign exchange rate movements display significant third and fourth order moments makes the possibility of mean or variance nonlinearities influencing exchange rates a plausible issue. Although international stock markets have undergone intense scrutiny regarding nonlinear properties and chaotic behavior inherent in their return movements (e.g., Pandey et al. [1998, 1999], Sewell et al. [1997]), very few studies have investigated the possibility of the existence of nonlinear dynamics in foreign exchange rate movements. Hsieh [1989] investigated the movements of the dollar exchange rates for the British pound, the Canadian dollar, the German mark, the Japanese yen, and the Swiss franc over the ten year period spanning 1974 through 1983. His conclusions confirm the existence of variance nonlinearities (significant GARCH effects) in the examined foreign exchange rate movements. However, the above study was unable to confirm the presence of chaotic dynamics in any of the examined currencies.

Some recent studies have successfully attempted to forecast foreign-exchange rate movements using nonlinear deterministic models. Fernandez-Rodriguez et al. [1999] use a simultaneous nearest neighbors (SNN) predictor to daily exchange rate data for nine emerging market currencies during the period 1978-1994. They report that the nonlinear SNN predictors outperform the traditional (linear) ARIMA predictors by a significant margin. Lisi and Schiavio [1999] use monthly exchange rates for four major European countries from 1973-1997 and apply a vector valued local linear approximation (VLLR) method to mimic chaotic predictions. They conclude that the performance of this chaotic prediction technique was inferior to that of a neural network trained on the same data. It must be noted that while the above studies have examined the performance of specific chaotic models or approximation techniques, the results cannot be generalized to the whole universe of chaotic processes.

This study undertakes the task of examining the daily exchange rate movements of eight foreign currencies for almost a twenty-five year period spanning from March 1973 through December 1997. The focus of this research is restricted to identifying the presence of chaotic dynamics in the examined series. Nonlinearities of variance are not investigated in this study as their detection does not have any significant implications on the pricing efficiency of the examined foreign exchange markets.

The remainder of this paper is organized as follows. The methodology, including a brief description of the nonlinear dynamics approach, time frame, and data selection for the eight foreign currencies is explained in Section II. The findings are discussed in Section III. The paper concludes with a summary in Section IV.

DATA AND METHODOLOGY

This study examines five major trading currencies and three less actively traded currencies for the existence of low-dimensional deterministic chaos. Specifically, the following spot markets are analyzed for the exchange of U.S. dollars for these currencies: the Canadian dollar, the German mark, the Japanese yen, the Swiss franc, the British pound, the Australian dollar, the Malaysian ringgit, and the Spanish peseta. The prices for these currencies are noon buying rates in the interbank market for cable transfers of foreign exchange, certified for customs purposes by the Federal Reserve Bank of New York.

The period examined in this study extends from March 1973, the beginning of the floating exchange rate system, through December 1997. The prices analyzed are on a daily basis. To avoid biases arising from possible structural shifts from regime changes and other shifts in market dynamics, the overall time frame is also subdivided into four subperiods, that is March 1, 1973 February 6, 1979, February 7, 1979-August 30, 1984, August 31, 1984-December 28, 1990, and December 31, 1990-December 31, 1997.

Testing for Nonlinear Dynamics:

In examining the efficiency of foreign exchange markets, the first step lies in testing for the randomness of security or portfolio returns. Such an approach was adopted in earlier studies of market efficiency using linear statistical theory and very general nonparametric procedures. Examinations of chaotic dynamics have revealed that deterministic processes of a nonlinear nature can generate variates that appear random and remain undetected by linear statistics. Hence, this study employs tests that have recently evolved from statistical advances in chaotic dynamics. One of the more popular statistical procedures that has evolved from recent progress in nonlinear dynamics is the BDS statistic, developed by Brock et al. [1987], which tests whether a data series is independently and identically distributed (IID).

The BDS statistic, which can be denoted as Wm,T(e) is given by

(1) [W.sub.m, T] (epsilon) = [square of T] [C.sub.m, T] ([epsilon]) - [C.sub.1, T] [([epsilon]).sup.m]]/ [[sigma].sub.m, T] ([epsilon]) (1)

where:

T = the number of observations,

e = a distance measure,

m = the number of embedding dimensions,

C = the Grassberger and Procaccia correlation integral, and

s = a standard deviation estimate of C.

For more details about the development of the BDS statistic, see Brock et al. [1987]. Simulations in Brock et al. demonstrate that the BDS statistic has a limiting normal distribution under the null hypothesis of independent and identical distribution (IID) when the data series consists of more than five hundred observations. The use of the BDS statistic to test for independent and identical distribution of pre-whitened data has become a widely used and recognized process (e.g., Brorsen and Yang [1994], Hsieh [1991a,b, 1993], Philippatos et al. [1993], Sewell et al. [1992]). After data has been pre-whitened and nonstationarity is ruled out, the rejection of the null of IID by the BDS statistic points towards the existence of some form of nonlinear dynamics.

Rejection of the null hypothesis of IID by the BDS is not conclusive evidence of the presence of nonlinear dynamics. Structural shifts in the data series can be a significant contributor to the rejection of the null. To avoid biases arising from structural shifts from regime changes and other shifts in market dynamics, the overall sample period is also subdivided into four subperiods of equal length, which are examined individually for the violation of the IID assumption.

Furthermore, in order to ascertain whether the data series are, indeed, a result of chaotic processes, two other tests are performed. The Rescaled Range (R/S) analysis is a powerful indicator of the persistence of a series where the influence of a set of past price changes on a set of future price changes is effectively captured. In addition, the three moments test effectively distinguishes deterministic (mean) nonlinearities from nonlinearities of variance, the latter of which could result from a stochastic rather than deterministic influence.

The R/S statistic, which was developed by Hurst (1951), has been used in several studies for the purpose of detecting long term dependencies in time series data, (see, for example, Greene and Fielitz (1977), Booth and Kaen (1979), Booth, Kaen, and Koveos (1982), Helms, Kaen, and Rosenman (1984), and Lo and MacKinlay (1988, 1990)). Over the years, a number of modifications and refinements have been made to the classical R/S statistic (e.g., see Mandelbrot (1972, 1975), Mandelbrot and Taqqu (1979), and Lo (1991)). According to Lo (1991), one of the drawbacks of the classical R/S statistic is that it detects short range as well as long range dependency, but does so without distinguishing between them. Thus, if a time series were to have strong short range dependencies, the R/S statistic may be biased towards an indication that long range dependence also exists.

Both the R/S statistic and the Modified R/S statistic have been utilized by researchers as a measurement for detecting long range dependencies in time series data. In these studies, the classical R/S statistic is computed along with the Modified R/S for comparison purposes (for example, see Lo (1991) and Pan et al. (1996)).

The classical R/S statistic is defined as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

where:

[S.sub.n] [equivalent to] [[1/n [summation over j] [([X.sub.j] - [[bar.X].sub.n).sub.2]].sub.1/2] (3)

The Modified R/S statistic differs from the classical R/S in that, in addition to the variance, it incorporates the autocovariance of X in the denominator (see equations below). (For a more detailed description of the Modified R/S statistic, see Lo (1991).)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

where

[[omega].sub.j] (q) [equivalent to] 1 - j/q + 1, q < n (5)

The adjusted variance/covariance function, which is used to scale the ranges for the Modified R/S statistic may then be denoted as:

[[??].sup.2.sub.n](Q) [approximately equal to] [[??].sup.2.sub.x] + 2 [q.summation over (j=1)] [[omega].sub.j](q)[[??].sub.j]] (6)

When properly normalized, as the sample size n increases without bound, the rescaled range converges in distribution to a well-defined random variable V, that is,

1/[square root of n] [[??].sub.n] [??] V (7)

The cumulative distribution function of V takes on the following form:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

Fractiles of this cumulative distribution function are used as benchmarks for the evaluation of R/S statistics presented in Table 7.

Hsieh [1989, 1991a] and Brock et al. [1991] developed the three moments test to specifically capture mean nonlinearity in a given series. Briefly stated, this test uses the concept that mean nonlinearity implies additive autoregressive dependence, whereas variance nonlinearity implies multiplicative autoregressive dependence. Using this notion and exploiting its implications, Hsieh [1989, 1991a] constructed a test that examines the third order moments of a given series. Additive dependencies will lead to some of these third order moments being correlated. By its construction, this test will not detect variance nonlinearities.

The third order sample correlation coefficients are computed as:

[r.sub.(xxx)] (i, j) = [1/T] [summation] [x.sub.t][x.sub.t-i][x.sub.t-j] / [[1/T][summation][[x.sup.2.sub.t]].sup.1.5] (9)

where: [r.sub.(xxx) (i, j) = the third order sample correlation coefficient of [x.sub.t] with [x.sub.t-1] and [x.sub.t-j'] and T = the length of the data series being examined.

Hsieh [1991] developed the estimates of the asymptotic variance and covariance for the combined effect of these third order sample correlation coefficients which can be used to construct a c2 statistic to test for the significance of the joint influence of the r(xxx) (i,j)'s for specific values of j, such that 1 [pounds sterling] i [pounds sterling] j. If the c2 statistics for relatively low values of j are significant, this outcome would be a strong indicator of the presence of mean nonlinearity in the examined series. As chaotic determinism is a form of mean-nonlinearity, the three moments test provides strong evidence of the presence of chaos.

Hence, first the examined index returns are filtered for linear influences using autoregressive filters. The lag lengths for these AR processes are determined by using the Akaike Information Criteria (AIC) (see Akaike [1974]). Next, each of these series is tested for the null of IID using the BDS statistic. The above process is then repeated for four shorter time periods, which subdivides the sample period into four subperiods of equal lengths, to rule out the possibility that any observed rejection of the IID assumption in the previous step was due to nonstationarity of the data series. Further, an examination of the rescaled ranges, traditional and modified R/S statistics, will reveal if the rejection of IID, in case it is observed, results from nonlinear dynamics. Finally, the three moments test is used to determine if the observed nonlinearities are, indeed, mean (deterministic) nonlinearities to provide conclusive evidence of the existence of chaotic determinism.

Strengths and Limitations of the Tests Used in this Study

All of the statistical procedures employed in this study have originated from studies of physical and natural phenomena. Due to the fact that, unlike data in the physical sciences, financial data are severely restricted in size and consistency across time, these tests have been modified to make them more applicable to financial data. However, all limitations of data sets analyzed are not overcome by the modified tests employed in this study. Ramsey et al. [1990] observe that with the techniques used to date, no evidence of the presence of simple chaotic attractors exist for economic time series. The BDS statistic, which is an adaptation of the correlation integral (standardized with an asymptotic variance estimate) has good power against the null of independence, but does not point towards the existence of a chaotic attractor. In this analysis, the BDS statistic is only used to reject the null of independence, a task at which it has demonstrated robustness.

Further, Ramsey et al. [1990] assert that bearing the limitations of small data sets in mind, "Finding a nonlinear evolutionary dynamical path, possibly subject to sporadic shocks, is a much more feasible task." This paper assumes precisely such a task in its search for chaotic deterministic influences in the examined data sets. Since chaotic systems are highly dependent on initial conditions, such a system, if discovered, could only be employed for short-term predictions. Shocks to such systems could make its path highly deviant in a short span. Given that such sporadic shocks are not uncommon in foreign exchange markets, a grandiose aspiration of finding a long-term steady-state dynamical path by locating a chaotic attractor would be impractical. This paper makes no attempt or pretense of locating a chaotic attractor.

Although Ramsey and Yuan [1989] and Ramsey et al. [1990] do not examine the R/S analysis, (perhaps because the R/S analysis was not popular in the study of economic data sets before Peters [1991]), it does not claim to locate a chaotic attractor either. However, the R/S analysis is a good indicator of temporal dependency in a time series. Lo [1991] points out that the R/S analysis is not robust to short-term dependencies, and hence, cannot be employed to study longterm dependencies. He further suggests a Modified R/S analysis which addresses this weakness. This paper addresses these shortcomings of the Classical R/S statistics by employing Lo's [1991] Modified R/S statistics.

Hsieh's three moments test is one of few existing direct test for nonlinear deterministic influence in noisy data. The attractiveness of this test lies in its ability to distinguish deterministic influences from stochastic influences. No published work appears to exist that brings out any significant weaknesses in the application of this test to data sets with more than 1,000 observations. The biggest limitation of the three moments test lies in its inability to capture all forms of chaotic influences (Hsieh [1991a]). Yet, once a data set is identified as having significant third-order correlations in low-order embedding dimensions, then one can be fairly certain that it is driven by a chaotic process.

RESULTS

The BDS statistics, by its very design, is very sensitive to linear as well as nonlinear processes. Since this study concerns itself with the detection of a specific type of nonlinear dynamics, all linear autocorrelations are filtered out using suitable autoregressive models. The appropriate lag lengths used for linear filtration were determined using the Akaike Information Criterion (AIC). Table 1 shows the lag lengths used to filter each currency's exchange rate movements.

Table 2 lists the BDS statistics for the overall period (March 1973--December 1997) for each of the eight exchange rates. The BDS statistics for each examined series is computed using dimensions m = 2, ... , 10 and the distance measure e = 0.5s, 0.75s, and 1.00s. There is an intuitive explanation for the BDS statistic. For example, a positive BDS statistic indicates that the probability of any two m histories (xt, xt-1, ... , xt-m+1) and (xs, xs-1, ... , xs-m+1) being close together is higher than in truly random data. In other words, some clustering is occurring too frequently in an m-dimensional space. Thus, some patterns of exchange rate movements are taking place more frequently than is possible with truly random data.

An inspection of Table 2 reveals that all BDS statistics are significantly positive. For each exchange rate examined, the BDS statistics are computed for e = 0.5s, 0.75s, and 1.00s. A lower e value represents a more stringent criteria since points in the m-dimensional space must be clustered closer together to qualify as being "close" in terms of the BDS statistic. Hence, e = 0.50s reflects the most stringent test while e = 1.00s is the most relaxed criterion used in this analysis.

In this study, the values of m examined go only as high as 10. Two reasons dictate the choice of 10 as the highest dimension analyzed. First, with m = 10, only 621 non-overlapping m-history points exist in each series. Examining a higher dimensionality will restrict the confidence one may place in the computed BDS statistic. Second, and more importantly, the interest of this study lies only in detecting low dimensional deterministic chaos. High dimensional chaos is, for all practical purposes, as good as randomness (see, e.g., Brock et al. [1987]).

One must remember, however, that the BDS statistic only reveals whether or not an examined data series is different from an identically and independently distributed (IID) series. The results in Table 2 present a summary rejection of the null hypothesis of IID for all the foreign exchange movement series examined. However, it is possible that nonstationarity of the examined data series is the root cause of this rejection of the null. Exogenous influences such as regime changes or regulatory reforms, among others, could impact the exchange rate series in such a way that they give the appearance of not being random (although they are truly random in stable times) over the 25 year period under investigation in this study.

Tables 3 through 6 provide the BDS statistics for the same foreign exchange rate movements for the four subperiods. Each subperiod covers approximately six years of exchange rate movements. As is evident from a quick review of these tables, a summary rejection of the null is provided for each examined currency over each one of the four subperiods. Consistent rejection of the null of IID over each of the four subperiods and across the overall period rules out the possibility that such rejection of the null is entirely an artifact of latent nonstationarity in these currencies.

The next step in this examination is to attempt to determine if the observed rejection of the IID hypothesis is indeed a result of latent memory effects, more specifically, chaotic dynamics present in these foreign exchange rate movements. The two tools employed for further evaluation are the rescaled range analysis and the three moments test.

Table 7 reports the results of the rescaled range analysis for each examined foreign exchange rate movement series. The table contains the Classical R/S ([n.sub.n]) and the Modified R/S ([n.sub.n](q)) statistics. The four Modified R/S statistics are computed with q-values of 90, 180, 270, and 360 days. In each case, the "%-Bias" is also reported. The "%-Bias" is the estimated bias of the Classical R/S statistic and is computed as [(nn)/(nn(q) - 1] '100. This bias term indicates an inclination of the Classical R/S statistic to be influenced by short-term dependence in its search for long-term dependence.

From the reported results in Table 7, one may conclude that some evidence of temporal dependence exists for the exchange rate movements of the Australian dollar, the German mark, the Japanese yen, the Spanish peseta, the Swiss franc, and the British pound. The Australian dollar exhibits evidence of long-term dependence only for the first subperiod examined. The later subperiods do not show signs of any temporal dependence. The temporal dependence observed for the German mark is of a short-term nature (less than 270 days) and exists only for the second subperiod. The Japanese yen appears to be influenced by temporal dependence during subperiods 2 and 3 of the sample period. This nonlinear influence appears to have been of a long-term nature in more recent times (subperiod 3). The Spanish peseta seems to be driven by a short-term (less than 180 days) nonlinear influence that is mostly present in the second subperiod examined. The exchange rate movements of the Swiss franc display long-term temporal dependence for the last subperiod examined. For subperiods 1 and 2, evidence of short-term dependence (less than 180 days) can be gleaned for this currency. The British pound exhibits strong evidence of short-term dependence (less than 270 days) only for subperiods 1 and 2 of the sample period.

In summary, the evidence of temporal dependence obtained from the R/S analysis is sporadic and mainly of a short-term nature. Only in the case of the Japanese yen, the Spanish peseta, and the Swiss franc, may one conclude that temporal dependence might exist during the most recent subperiod analyzed.

The results presented in this study so far suggest the presence of nonlinear dynamics in all the examined foreign exchange rate movements. In order to ascertain whether this observed nonlinearity is deterministic in nature, the results of the three moments test are presented in Table 8. This table shows the c2 statistics for a combined test of significance of all examined three moment correlations r(xxx)(i,j) up to a certain lag length. Where 1=i=j=5, the c2 statistic has 15 degrees of freedom. On the other hand, when three moment correlations up to a lag length of 10 are examined (i.e., 1=i=j=10), the c2 statistic has 55 degrees of freedom. As one may observe from Table 7, none of the c2(15) statistics are significant for any of the examined exchange rate movements. Hence, one is unable to find evidence of low-dimensional chaos in any of these currencies while 1=i=j=5.

The c2(55) statistics reveal that the exchange rate movements of the Canadian dollar, the Japanese yen, and the British pound are, in fact, influenced by chaotic dynamics. However, this detected deterministic nonlinearity is only high dimensional in nature. Consequently, even though the driving influence of deterministic chaos is detected in the above three currencies, predictability of these exchange rates is highly improbable as this deterministic influence is high-dimensional.

To sum up the findings of this study, the exchange rate movements of all examined currencies exhibit non-IID behavior. None of this rejection of the null of IID is purely an artifact of nonstationary data. The rescaled range analysis suggested the existence of some form of nonlinear dynamics in the exchange rate movements of the Australian dollar, the German mark, the Japanese yen, the Spanish peseta, the Swiss franc, and the British pound. Moreover, most of these observed dependencies are of a short-term nature. Furthermore, only the Japanese yen, the Spanish peseta, and the Swiss franc exhibit nonlinear dependence during more recent times. Finally, the three moments test revealed that none of the examined currency price changes appear to be driven by low dimensional chaos. The exchange rate movements of the Canadian dollar, the Japanese yen, and the British pound are, however, influenced by high dimensional chaos.

SUMMARY AND CONCLUSIONS

This study examined the daily exchange rate movements of the following eight foreign currencies: the Canadian dollar, the German mark, the Japanese yen, the Swiss franc, the British pound, the Australian dollar, the Malaysian ringgit, and the Spanish peseta over the floating exchange rate period from March 1973 through December 1997. The results provide evidence to suggest that some form of nonlinear dynamics appears to be influencing some of the examined foreign exchange rate movements. However, further examination reveals that none of the observed nonlinear dynamics manifest themselves in the form of low-dimensional chaos. High dimensional chaos is detected in the exchange rate movements of the German mark, the Japanese yen, and the British pound. However, this observation is not encouraging to the technical analyst as it renders the probability of being able to predict these movements as being very slim.

The above observations have important implications concerning the efficiency of the eight examined foreign exchange markets. Although evidence of nonlinear dynamics abounds in these markets, none of the evidence appears to be in the form of low-dimensional chaos. As such, the possibility of predictability of foreign exchange price changes appears to be an improbable task with currently available technology. Moreover, the examined foreign exchange markets appear to be at least weak-form efficient. The significance of this observation lies in the breadth of the foreign exchange rates examined in this study, which include major currencies as well as less actively traded currencies.

It may be important to recall that the three moments test does not detect all forms of deterministic influences. In light of this observation, one cannot rule out the possibility that a chaotic deterministic mechanism lies latent in the examined foreign exchange returns, awaiting discovery by more powerful tests at a later date.

Some recent studies such as Fernandez-Rodriguez et al. [1999] and Lisi and Schiavio [1999] have successfully attempted to forecast foreign-exchange rate movements using nonlinear deterministic models. It must be noted that while the above studies have examined the performance of specific chaotic models or chaotic approximation techniques, the results cannot be generalized to the whole universe of chaotic processes. The tests employed in this study investigate the generalized possibility of using certain classes of chaotic models to predict the exchange rate movement series examined. Although three of the examined exchange rates do exhibit signs of being driven by higher-dimensional chaos, this finding does not significantly favor the possibility of predicting these currency movements, for reasons mentioned earlier. As such, very little evidence of a deterministic driving force behind foreign exchange rates is uncovered in this study.

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Vivek K. Pandey, The University of Texas at Tyler

Theodor Kohers, Mississippi State University

Gerald Kohers, Sam Houston State University

Table 1
Autoregression Lags Used to Filter Movements in Foreign
Exchange Rates Analyzed

Country Stock Market Index: Autoregressive Model Used to Filter:

Canadian Dollar AR(5)
German Mark AR(1)
Japanese Yen AR(9)
Swiss Franc AR(3)
British Pound AR(10)
Australian Dollar AR(2)
Malaysian Ringgit AR(7)
Spanish Peseta AR(1)

NOTE: AR(x) = Autoregressive model with (x) lags. Lags are determined
via the Akaike Information Criterion.

Table 2: BDS Statistics for Filtered Movements in Foreign Exchange
Rates March 1, 1973-December 31, 1997

e/s m: Canadian German Japanese Swiss
 Dollar Mark Yen Franc

0.50 2 17.55 14.18 19.99 25.85
0.50 3 22.43 21.66 30.31 29.01
0.50 4 28.72 29.47 44.79 32.99
0.50 5 36.37 40.46 68.19 38.27
0.50 6 48.39 57.88 108.88 46.07
0.50 7 65.26 86.70 181.87 56.82
0.50 8 89.95 134.27 318.02 72.38
0.50 9 127.23 217.54 588.52 95.19
0.50 10 186.29 369.48 1124.5 129.79
0.75 2 17.48 13.29 16.77 25.89
0.75 3 21.59 18.83 22.72 27.79
0.75 4 26.22 23.81 29.80 30.02
0.75 5 31.23 29.60 38.40 32.62
0.75 6 38.07 37.17 50.25 35.85
0.75 7 46.57 47.57 66.81 39.57
0.75 8 57.68 61.78 90.10 44.14
0.75 9 72.04 81.43 125.29 49.89
0.75 10 91.04 109.11 178.38 57.44
1.00 2 17.49 12.60 14.98 25.69
1.00 3 21.02 16.89 19.24 27.05
1.00 4 24.57 20.63 23.66 28.37
1.00 5 27.97 24.43 27.96 29.68
1.00 6 32.33 28.83 33.01 31.24
1.00 7 37.21 34.08 38.97 32.82
1.00 8 42.94 40.33 46.10 34.62
1.00 9 49.64 47.89 55.88 36.76
1.00 10 57.46 57.21 68.38 39.37

e/s British Austral. Malaysian Spanish
 Pound Dollar Ringgit Peseta

0.50 19.45 27.62 24.80 19.45
0.50 28.82 37.18 32.04 28.82
0.50 41.23 46.96 38.95 41.23
0.50 60.72 60.46 47.02 60.72
0.50 94.47 79.04 57.45 94.47
0.50 152.19 105.76 71.61 152.19
0.50 262.36 145.57 91.74 262.36
0.50 477.17 203.89 120.77 477.17
0.50 903.92 291.67 162.61 903.92
0.75 17.58 25.71 24.32 17.58
0.75 23.61 32.39 29.69 23.61
0.75 29.96 37.96 33.81 29.96
0.75 38.09 44.25 38.37 38.09
0.75 49.76 51.67 43.71 49.76
0.75 65.23 60.91 49.97 65.23
0.75 86.87 72.49 57.84 86.87
0.75 117.96 87.46 67.93 117.96
0.75 165.61 106.88 80.92 165.61
1.00 15.92 23.82 23.63 15.92
1.00 20.59 28.68 27.30 20.59
1.00 24.64 32.43 29.71 24.64
1.00 29.19 35.78 32.28 29.19
1.00 35.18 39.31 35.22 35.18
1.00 41.93 43.22 38.35 41.93
1.00 49.86 47.67 41.93 49.86
1.00 59.27 53.00 46.10 59.27
1.00 71.71 59.24 51.14 71.71

Note: m = embedding dimensions; Number of observations = 6,214 per
currency; All BDS statistics are significant at the .01 level.

Table 3: BDS Statistics for Filtered Movements in Foreign Exchange
Rates March 1, 1973-February 6, 1979

e/s m: Canadian German Japanese Swiss
 Dollar Mark Yen Franc

0.50 2 12.00 12.08 17.02 16.79
0.50 3 16.11 17.60 23.43 20.53
0.50 4 19.09 24.25 30.47 26.15
0.50 5 22.67 33.65 39.45 34.76
0.50 6 28.44 48.27 51.92 48.23
0.50 7 35.41 70.62 69.08 68.70
0.50 8 44.80 106.07 93.67 99.71
0.50 9 59.49 163.02 131.67 147.32
0.50 10 79.77 254.98 188.51 223.29
0.75 2 11.92 10.97 13.48 16.09
0.75 3 15.12 14.93 17.85 18.80
0.75 4 17.16 19.00 21.92 22.19
0.75 5 19.29 23.50 26.15 26.69
0.75 6 22.41 29.31 30.92 32.64
0.75 7 26.08 36.70 36.17 40.46
0.75 8 30.59 46.40 42.63 50.96
0.75 9 36.76 59.30 51.28 65.08
0.75 10 45.83 76.53 62.01 84.81
1.00 2 11.62 10.16 11.48 15.71
1.00 3 14.45 13.24 14.97 18.07
1.00 4 15.94 15.90 17.86 20.45
1.00 5 17.42 18.55 20.42 23.25
1.00 6 19.21 21.40 23.04 26.46
1.00 7 21.27 24.67 25.76 30.08
1.00 8 23.57 28.47 28.74 34.51
1.00 9 26.29 33.03 32.40 40.22
1.00 10 29.74 38.32 36.57 47.61

e/s British Austral. Malaysian Spanish
 Pound Dollar Ringgit Peseta

0.50 14.89 8.40 12.58 11.47
0.50 20.52 9.38 16.03 14.13
0.50 26.31 10.25 20.44 15.81
0.50 35.29 11.33 25.40 17.02
0.50 49.53 11.83 31.20 18.31
0.50 72.09 12.26 39.21 19.73
0.50 109.43 12.71 51.04 21.17
0.50 170.95 13.03 68.42 22.64
0.50 273.43 13.31 92.58 24.19
0.75 13.10 7.55 12.62 9.59
0.75 16.58 9.18 15.21 11.67
0.75 19.23 9.95 17.88 12.91
0.75 22.41 10.53 20.79 13.64
0.75 26.52 10.70 23.84 14.15
0.75 31.78 10.66 27.61 14.53
0.75 39.10 10.69 32.57 14.93
0.75 49.37 10.62 38.85 15.23
0.75 62.92 10.55 46.85 15.57
1.00 12.06 5.01 12.07 9.04
1.00 14.23 6.49 13.99 9.57
1.00 15.50 7.06 15.66 10.20
1.00 16.81 7.70 17.20 10.89
1.00 18.48 8.09 18.91 11.31
1.00 20.30 8.18 20.73 11.49
1.00 22.43 8.30 23.02 11.64
1.00 25.33 8.28 25.70 11.68
1.00 28.79 8.20 28.93 11.79

Note: m = embedding dimensions; Number of observations = 1,491 per
currency; All BDS statistics are significant at the .01 level.

Table 4: BDS Statistics for Filtered Movements in Foreign Exchange
Rates February 7, 1979-August 30, 1984

e/s m: Canadian German Japanese Swiss
 Dollar Mark Yen Franc

0.50 2 8.05 4.07 5.32 11.80
0.50 3 9.68 7.53 7.17 13.08
0.50 4 11.82 10.88 8.94 15.25
0.50 5 14.56 15.64 10.28 18.23
0.50 6 19.56 22.49 11.86 22.92
0.50 7 26.34 30.54 14.49 29.74
0.50 8 37.62 42.21 19.41 38.23
0.50 9 54.39 62.40 27.05 54.36
0.50 10 81.14 93.94 40.36 84.30
0.75 2 8.37 4.31 5.64 12.03
0.75 3 10.04 7.42 7.45 13.26
0.75 4 11.44 10.27 9.11 15.00
0.75 5 13.24 13.49 10.46 16.93
0.75 6 16.04 17.69 11.91 19.68
0.75 7 19.20 22.36 13.68 22.85
0.75 8 23.47 28.34 15.43 26.75
0.75 9 28.54 36.96 18.13 32.71
0.75 10 34.88 48.61 21.88 41.56
1.00 2 8.98 4.43 5.95 12.10
1.00 3 10.59 6.93 7.73 13.13
1.00 4 11.68 9.29 9.10 14.54
1.00 5 12.89 11.57 10.14 15.89
1.00 6 14.61 14.19 11.24 17.56
1.00 7 16.29 16.84 12.41 19.35
1.00 8 18.36 20.04 13.57 21.52
1.00 9 20.59 24.16 15.27 24.33
1.00 10 22.95 29.03 17.21 28.08

e/s British Austral. Malaysian Spanish
 Pound Dollar Ringgit Peseta

0.50 3.63 10.80 11.80 13.56
0.50 5.23 12.89 15.87 22.00
0.50 6.92 14.16 19.99 35.95
0.50 8.47 16.15 25.89 62.75
0.50 11.16 18.82 34.55 110.22
0.50 13.65 21.70 46.51 198.08
0.50 17.51 25.23 65.67 377.32
0.50 23.24 29.42 96.53 745.07
0.50 36.40 35.44 147.97 1524.00
0.75 3.77 11.96 10.69 10.15
0.75 5.10 13.72 13.27 14.83
0.75 6.52 14.90 15.06 21.27
0.75 7.98 16.60 17.45 31.00
0.75 10.40 18.49 20.72 43.74
0.75 12.61 20.54 24.33 62.45
0.75 15.12 22.72 29.11 90.87
0.75 18.17 25.22 35.31 134.74
0.75 22.37 28.63 43.99 206.01
1.00 3.72 12.68 10.14 8.49
1.00 4.81 14.27 11.64 11.51
1.00 5.78 15.13 12.37 15.37
1.00 6.66 16.35 13.60 20.34
1.00 8.20 17.44 15.13 25.53
1.00 9.48 18.61 16.65 32.07
1.00 10.75 19.71 18.50 40.03
1.00 12.17 20.99 20.50 50.16
1.00 14.06 22.68 23.15 64.35

Note: m = embedding dimensions; Number of observations = 1,491 per
currency; All BDS statistics are significant at the .01 level.

Table 5
BDS Statistics for Filtered Movements in Foreign Exchange Rates
August 31, 1984-December 28, 1990

e/s m: Canadian German Japanese Swiss
 Dollar Mark Yen Franc

0.50 2 6.86 3.31 5.01 12.35
0.50 3 8.75 4.78 6.64 12.30
0.50 4 10.97 5.81 8.47 12.48
0.50 5 13.48 6.35 10.09 13.47
0.50 6 16.56 7.54 12.15 14.85
0.50 7 21.05 8.70 15.73 16.43
0.50 8 26.69 10.55 21.49 19.63
0.50 9 35.32 13.54 27.75 19.06
0.50 10 49.88 16.87 37.01 21.73
0.75 2 7.36 2.76 4.44 12.07
0.75 3 9.01 4.00 5.59 11.98
0.75 4 10.82 5.01 7.14 12.30
0.75 5 12.54 5.28 8.21 12.84
0.75 6 14.56 5.91 9.50 13.77
0.75 7 16.96 6.49 11.45 14.54
0.75 8 19.57 7.37 13.80 15.96
0.75 9 23.20 8.18 16.93 17.12
0.75 10 28.21 8.81 20.92 18.82
1.00 2 7.78 2.53 4.23 11.92
1.00 3 9.34 3.60 4.95 11.68
1.00 4 10.76 4.61 6.16 11.91
1.00 5 12.06 5.00 6.88 12.31
1.00 6 13.37 5.79 7.62 13.12
1.00 7 14.73 6.52 8.52 13.74
1.00 8 16.10 7.31 9.52 14.76
1.00 9 17.62 8.08 11.03 15.89
1.00 10 19.45 8.75 12.86 17.23

e/s British Austral. Malaysian Spanish
 Pound Dollar Ringgit Peseta

0.50 5.24 9.08 11.83 5.71
0.50 6.07 10.75 15.48 6.73
0.50 6.87 12.70 18.43 7.84
0.50 7.63 13.87 22.23 8.54
0.50 9.03 14.78 27.66 9.59
0.50 10.75 15.91 35.11 10.30
0.50 12.52 17.46 45.07 11.68
0.50 14.05 20.69 58.04 11.81
0.50 11.78 25.20 76.27 12.87
0.75 5.37 9.88 11.81 5.96
0.75 6.21 11.63 14.85 7.16
0.75 7.03 13.37 16.90 8.44
0.75 7.89 14.43 19.31 9.31
0.75 9.30 15.29 22.43 10.38
0.75 10.65 16.20 26.13 11.21
0.75 11.53 17.16 30.52 12.30
0.75 12.44 19.00 36.37 13.41
0.75 13.50 21.59 44.00 14.34
1.00 5.45 10.91 11.54 6.00
1.00 6.46 12.66 13.73 7.04
1.00 7.40 14.07 15.10 8.23
1.00 8.33 14.97 16.58 9.16
1.00 9.68 15.60 18.51 10.20
1.00 10.90 16.19 20.55 10.99
1.00 11.84 16.75 22.74 11.88
1.00 12.65 17.97 25.53 12.76
1.00 13.59 19.57 28.93 13.62

Note: m = embedding dimensions; Number of observations = 1,490
per currency; All BDS statistics are significant at the .01 level.

Table 6
BDS Statistics for Filtered Movements in Foreign Exchange Rates
December 31, 1990-December 31, 1997

e/s m: Canadian German Japanese Swiss
 Dollar Mark Yen Franc

0.50 2 6.20 7.34 5.52 12.06
0.50 3 6.99 9.01 5.84 11.72
0.50 4 9.58 9.75 6.13 11.53
0.50 5 12.18 11.35 7.19 11.60
0.50 6 16.57 12.74 8.80 11.71
0.50 7 23.33 15.94 9.28 12.05
0.50 8 34.35 19.45 10.05 12.44
0.50 9 50.33 20.51 11.79 12.94
0.50 10 73.52 19.95 13.68 13.76
0.75 2 6.20 7.18 4.88 13.89
0.75 3 6.80 8.43 5.15 12.99
0.75 4 8.84 8.92 5.65 12.50
0.75 5 10.48 10.35 6.69 12.12
0.75 6 13.29 11.85 7.85 11.86
0.75 7 17.02 14.45 8.61 11.74
0.75 8 22.29 17.57 9.39 11.61
0.75 9 29.01 20.49 10.10 11.64
0.75 10 37.64 24.92 10.78 11.83
1.00 2 6.15 7.55 4.59 16.44
1.00 3 6.59 8.53 4.74 14.90
1.00 4 8.14 8.93 5.21 13.95
1.00 5 9.12 10.01 6.04 13.03
1.00 6 10.96 11.15 6.80 12.39
1.00 7 13.20 12.81 7.32 12.02
1.00 8 16.04 14.55 7.70 11.74
1.00 9 19.26 16.24 8.01 11.53
1.00 10 22.89 18.55 8.25 11.42

e/s British Austral. Malaysian Spanish
 Pound Dollar Ringgit Peseta

0.50 8.75 4.39 10.17 9.02
0.50 11.46 5.15 12.65 11.17
0.50 13.81 5.29 14.38 12.54
0.50 16.98 5.56 15.83 14.27
0.50 21.85 6.64 17.49 16.34
0.50 27.67 7.78 20.08 19.35
0.50 37.78 9.03 23.50 22.71
0.50 50.47 10.98 27.76 25.56
0.50 70.28 14.20 33.09 29.67
0.75 8.26 4.17 11.42 9.46
0.75 10.85 5.01 13.13 11.05
0.75 12.65 5.32 13.90 12.02
0.75 15.06 5.57 14.50 13.40
0.75 18.43 6.00 15.25 15.04
0.75 22.61 6.59 16.35 17.27
0.75 28.50 7.53 17.93 20.01
0.75 35.19 8.89 19.81 22.82
0.75 44.26 10.35 22.10 26.95
1.00 7.55 3.70 12.15 9.66
1.00 9.90 4.73 13.17 10.85
1.00 11.21 5.04 13.42 11.70
1.00 12.96 5.28 13.63 12.86
1.00 15.03 5.73 13.85 14.13
1.00 17.40 6.25 14.26 15.62
1.00 20.29 6.90 14.85 17.38
1.00 23.34 7.68 15.56 19.31
1.00 27.19 8.51 16.41 21.91

Note: m = embedding dimensions; Number of observation = 1,641
per currency; All BDS statistics are significant at the .01 level.

Table 7
Classical R/S and Modified R/S (q=90, 180, 270, and 360) Statistics
with %-Bias ((Vn/Vn(q)-1) x 100)

Currency: Vn Vn(90) %-Bias Vn(180) %-Bias

Australia
All (3/73-12/97) 1.37 1.34 0.02 1.45 -5.7%
Subperiod 1 1.35 1.60 -0.16 ** 1.78 -24.4%
Subperiod 2 1.63 1.42 0.15 1.50 8.6%
Subperiod 3 1.16 1.27 -0.09 1.44 -20.0%

Canada

All (3/73-12/97) 1.56 1.62 -0.04 1.60 -2.8%
Subperiod 1 1.69 1.43 0.18 1.34 26.3%
Subperiod 2 1.48 1.73 -0.14 1.70 -12.9%
Subperiod 3 1.53 1.66 -0.07 ** 1.83 -16.3%

Germany

All (3/73-12/97) 1.58 1.41 0.12 1.43 10.3%
Subperiod 1 1.26 1.02 0.23 1.20 5.3%
Subperiod 2 0 0 0.11 ** 1.79 20.2%
Subperiod 3 0.97 0.94 0.03 1.07 -10.0%

Japan

All (3/73-12/97) 1.49 1.33 0.12 1.29 14.7%
Subperiod 1 1.65 1.47 0.12 1.34 22.9%
Subperiod 2 ** 1.80 1.59 0.13 1.49 20.5%
Subperiod 3 ** 1.84 1.72 0.07 * 1.87 -1.8%

Malaysia

All (3/73-12/97) 1.13 1.03 0.10 1.12 0.8%
Subperiod 1 1.01 0.88 0.14 1.00 1.2%
Subperiod 2 0.87 0.92 -0.06 1.01 -13.5%
Subperiod 3 1.16 1.05 0.11 1.17 -0.6%

Spain

All (3/73-12/97) 0 1.71 0.13 1.58 22.4%
Subperiod 1 1.47 1.30 0.13 1.23 19.9%
Subperiod 2 0 ?? 0.11 1.71 28.9%
Subperiod 3 1.46 1.29 0.13 1.31 11.7%

Switzerland

All (3/73-12/97) 1.45 1.34 0.08 1.35 7.4%
Subperiod 1 0 1.43 0.60 1.50 51.5%
Subperiod 2 0 ** 1.83 0.63 1.73 72.5%
Subperiod 3 1.01 1.42 -0.29 1.59 -36.5%

U. K.

All (3/73-12/97) 1.45 1.32 0.10 1.31 10.8%
Subperiod 1 0 ** 1.82 0.16 1.72 22.3%
Subperiod 2 ?? ** 1.81 0.05 ** 1.78 7.1%
Subperiod 3 1.09 0.99 0.10 1.06 2.1%

Currency: Vn(270) %-Bias Vn(360) %-Bias

Australia
All (3/73-12/97) 1.43 -0.04 1.46 -5.7%
Subperiod 1 ** 1.79 -0.25 * 1.88 -28.2%
Subperiod 2 1.46 0.12 1.46 11.5%
Subperiod 3 1.39 -0.17 1.44 -19.9%

Canada

All (3/73-12/97) 1.47 0.06 1.42 9.8%
Subperiod 1 1.33 0.27 1.40 20.7%
Subperiod 2 1.53 -0.03 1.42 4.3%
Subperiod 3 1.62 -0.06 1.58 -3.0%

Germany

All (3/73-12/97) 1.37 0.15 1.34 17.9%
Subperiod 1 1.16 0.09 1.19 5.6%
Subperiod 2 1.68 0.29 1.57 37.6%
Subperiod 3 1.11 -0.13 1.22 -20.8%

Japan

All (3/73-12/97) 1.25 0.19 1.23 21.1%
Subperiod 1 1.28 0.28 1.31 25.9%
Subperiod 2 1.41 0.27 1.37 31.0%
Subperiod 3 * 1.87 -0.02 ** 1.81 1.8%

Malaysia

All (3/73-12/97) 1.25 -0.10 1.31 13.7%
Subperiod 1 1.07 -0.06 1.13 10.8%
Subperiod 2 1.26 -0.31 1.40 37.5%
Subperiod 3 1.39 -0.16 1.48 21.5%

Spain

All (3/73-12/97) 1.46 0.32 1.41 37.0%
Subperiod 1 1.16 0.27 1.18 24.5%
Subperiod 2 1.53 0.43 1.40 56.8%
Subperiod 3 1.28 0.14 1.36 7.6%

Switzerland

All (3/73-12/97) 1.32 0.10 1.31 10.8%
Subperiod 1 1.47 0.55 1.48 53.5%
Subperiod 2 1.66 0.80 1.58 88.5%
Subperiod 3 1.61 -0.37 ** 1.76 -42.5%

U. K.

All (3/73-12/97) 1.28 0.13 1.28 12.6%
Subperiod 1 1.57 0.34 1.49 41.1%
Subperiod 2 1.71 0.12 1.66 15.0%
Subperiod 3 1.16 -0.06 1.35 -19.4%

* Significant at the .05 level; ** Significant at the .10 level.

Fractiles of the Distribution [F.sub.v] (v)

P(V<v) .005 .025 .050 .100 .200

v 0.721 0.809 0.861 0.927 1.018

P(V<v) .800 .900 .950 .975 .995

v 1.473 1.620 1.747 1.862 2.098

Table 8
Chi-Square Statistics for the Joint Influence of Three Moment
Correlations for the Filtered Foreign Exchange Rate Movements

Foreign Exchange: Lags/Statistic: Lags/Statistic:

Currency: 1 [pounds sterling] 1 [pounds sterling]
 i [pounds sterling] i [pounds sterling]
 j [pounds sterling] 5 j [pounds sterling]10
 c2(15) c2(55)

Canadian Dollar 11.73 257.94 *
German Mark 3.63 8.96
Japanese Yen 3.12 121.91 *
Swiss Franc 4.99 54.02
British Pound 2.59 130.77 *
Australian Dollar 3.83 10.48
Malaysian Ringgit 4.38 5.82
Spanish Peseta 8.88 19.33

* Significant at the 1% level for a right-tailed test.