nonlinear deterministic forecasting of daily dollar exchange rates

International Journal of Forecasting 15 (1999) 421–430www.elsevier.com/ locate / ijforecast

Nonlinear deterministic forecasting of daily dollar exchange rates

a b ,*Liangyue Cao , Abdol S. SoofiaDepartment of Mathematics, University of Western Australia, Nedlands, WA 6907, Australia

bDepartment of Economics, University of Wisconsin–Platteville, and Visiting Scholar, School of Business Administration, University ofWisconsin–Milwaukee, Business Administration Building, P.O. Box 742, Milwaukee, WI 53201, USA

Abstract

We perform out-of-sample predictions on several dollar exchange rate returns by using time-delay embedding techniquesand a local linear predictor. We compared our predictions with those by a mean value predictor. Some of our predictions ofthe exchange rate returns outperform the predictions of the same series by the mean value predictor. However, theseimprovements were not statistically significant. Another interesting result in this paper which was obtained by using arecently developed technique of nonlinear dynamics is that all exchange rate return series we tested have a very highembedding dimension. Additionally, evidence indicates that these series are likely generated by high dimensional systemswith measurement noise or by high dimensional nonlinear stochastic systems, that is, nonlinear deterministic systems withdynamic noise. 1999 Elsevier Science B.V. All rights reserved.

Keywords: Exchange rates; Time series; Embedding dimension; Nonlinear forecasting

1. Introduction al., 1994) have brought great progress in distinguish-ing deterministic chaos from randomness.

Publication of Meese and Rogoff (1983) which The mathematical theory of time-delay embeddingshowed that a random walk model outperforms out- by Takens (1981) and later by Sauer et al. (1991)of-sample forecasts of both structural and time series has provided a technique to view the system’seconometric models of the exchange rates, has raised dynamics through observed time series. Severalthe possibility that some of the series are generated related algorithms, such as calculation of correlationby stochastic rather than deterministic processes. dimension (Grassberger & Procaccia, 1983) andDetermining the dynamics of the data generating calculation of Lyapunov exponents (e.g., Wolf et al.,process (DGP) of the observed time series, however, 1985) have thereafter been developed, which makewas hampered by the inadequacy of the mathemati- characterizing dynamical behavior from time seriescal theory in the past. Recent developments in time data possible. These algorithms have had a largeseries techniques from chaos theory (see e.g., Ott et number of applications in detecting nonlinear de-

terminism from observed time series, e.g., economicand financial time series. These algorithms, however,often need a large number of observations for*Corresponding author. Tel.: 11-414-229-4235; fax: 11-414-reliable computations, and even with large sample229-6957.

E-mail address: [email protected] (A.S. Soofi) sizes, misleading results could still occur.

0169-2070/99/$ – see front matter 1999 Elsevier Science B.V. All rights reserved.PI I : S0169-2070( 99 )00024-2

422 L. Cao, A.S. Soofi / International Journal of Forecasting 15 (1999) 421 –430

In applications of nonlinear dynamics to exchange method of minimizing prediction error (by localrate economics, Bajo-Rubio et al. (1992) tested for linear predictor) in choosing the embedding dimen-the presence of deterministic chaos on daily Spanish sion.peseta–US dollar rates (spot and forwards rates) and In our work we use a zero-order approximationreported presence of chaotic behavior for the time model to determine the embedding dimension. Thisseries. Furthermore, they tested prediction using has two advantages: (1) it is extremely fast inseveral local predictors which outperformed the out- computation as the zero-order approximation modelof-sample forecast by a random walk model. Soofi has no free-parameters to be fitted, and (2) it mayand Cao (1999) applied a simple nonlinear deter- avoid overfitting because the zero-order model doesministic technique of a local linear predictor to the not need any fitting at all. In contrast to Lisi andsame data set used by Bajo-Rubio et al., and showed Medio’s results, the embedding dimensions by ourthe results outperform the prediction results of a method are much higher.mean value predictor in two out of three sample In this paper we use a new method of optimalperiods. Larsen and Lam (1992) computed the determination of embedding dimension (Cao et al.,correlation dimension for daily dollar rates for Swiss 1998) in testing for presence of determinism infranc, Japanese yen, and French franc and reported several dollar exchange rate returns. We further buildthat these time series are not purely stochastic and predictive models for out-of-sample forecasting exer-found evidence of some deterministic behavior. De cises and compare the results with those by a meanGrauwe et al. (1993) developed a nonlinear model of value predictor. We have found that (1) the exchangeexchange rate based on behaviors of two classes of rate returns time series we tested have very highspeculators: fundamentalists and chartists and embedding dimension, (2) our prediction methodshowed that the process is capable of chaotic be- outperforms a mean value predictor on some of thehaviors. Cecen and Erkal (1996) tested high-fre- exchange rate returns, and does not outperform onquency (hourly) daily dollar spot rates for the British other exchange rate returns, and (3) the results wepound, the Deutschmark, the yen, and the Swiss obtained here are not statistically significant.franc and reported that the exchange rates returnsexhibit nonlinear stochastic dependence. Lisi andMedio (1997) reconstructed the phase spaces for a

2. Methodologynumber of monthly dollar exchange rates and usedlocal linear approximation in out-of-sample forecast-

2.1. Choosing optimal model dimensioning of the exchange rates and showed that theirmethod outperforms a random walk model. Diebold

Suppose we have a scalar time series, x ,x , ? ? ?1 2and Nason (1990) using a nonparametric prediction,x . We make a time-delay reconstruction of theNtechnique studied 10 OECD currencies and con-phase-space with the reconstructed vectors:cluded that the exchange rates do not contain non-

linearities that are exploitable for enhanced pointV 5 (x ,x , ? ? ? ,x ), (1)n n n2t n2(d21)tprediction. Furthermore, their method’s performance

`vis-a-vis a random walk model was mixed. Mizrach(1993) used a multivariate nonlinear modeling of the where t is time-delay, d is embedding dimension,exchange rates and obtained prediction results which and n 5 (d 2 1)t 1 1, ? ? ? ,N. d represents the dimen-outperformed a random walk model but the improve- sion of the state space in which to view the dynamicsment was not statistically significant. of the underlying system. The time-delay (time lag),

Due to similarity of work between the present t, represents the time interval between the succes-study and that by Lisi and Medio (1997), we make sively sampled observations used in constructing thethe following observations. d-dimensional embedding vectors. For details, see an

First, our data sets are daily exchange rates which excellent introductory book by Hilborn (1994).are generally more noisy than the monthly exchange If the time series is generated by a deterministicrates they tested. Second, Lisi and Medio used the system, then by the embedding theorems (Takens,

L. Cao, A.S. Soofi / International Journal of Forecasting 15 (1999) 421 –430 423

1981; Sauer et al., 1991), there generically exists a peared in the literature. Cao (1997) modified thed dfunction F: R ∞R such that method of false neighbors and developed a method

of the averaged false neighbors which does notV 5 F(V ), (2)n11 n contain any subjective parameter provided the time-if the observation function of the time series is delay has been chosen. A more general methodsmooth, and d is sufficiently large. And this mapping based on zero-order approximations has been de-has the same dynamic behavior as that of the original veloped by Cao et al. (1998), which can be used tounknown system in the sense of topological equiva- determine the embedding dimension from any di-lent. mensional time series including scalar and multi-

In practical applications, we usually use a scalar variate time series. This method is very simple,mapping rather than the mapping in (2), that is, requires fewer computational resources as compared

to the other methods, performs well, and does notx 5 f(V ), (3)n11 n require subjective parameters. In this paper we usewhich is equivalent to (2). this method to choose the embedding dimension.

Then the remaining problem is how to choose the Below we briefly describe the method.t and d, i.e. time-delay and embedding dimension, For a given dimension d, we can get a series ofsuch that the above mapping exists. From the delay vectors V defined in (1), (note that we haven

Takens’ theorem, it does not matter what time delay already chosen the time delay t). For each V we findn

is selected in a ‘generic’ sense. But in practice, its nearest neighbor V , i.e.,h(n)

because we have only a finite number of data pointsV 5 argminhiV 2V i: javailable with finite measurement precision, a good h(n) n j

choice of t is deemed to be important in phase space 5 (d 2 1)t 1 1, ? ? ? ,N, j ± nj, (4)reconstructions. In addition, determining a good

or in other words, h(n) is an integer such thatembedding dimension d depends on a judiciouschoice of t. For more discussions on this topic, see

iV 2V i 5 minhiV 2V i: jh(n) n n je.g., Ott et al. (1994).There are several methods to choose a time delay t 5 (d 2 1)t 1 1, ? ? ? ,N, j ± nj

from a scalar time series, such as mutual informationwhere the norm(Fraser & Swinney, 1986) and autocorrelation func-

tion methods. The importance of choosing a good iV 2V i 5 i(x ,x , ? ? ? ,x )n j n n2t n2(d21)ttime-delay is that it could make the necessary

2 (x ,x , ? ? ? ,x )iembedding dimension lower. For the time series we j j2t j2(d21)t

tested in this paper, we used the method of mutual d21 1 / 22information to choose the time delay, and we found 5 O (x 2 x ) .F Gn2it j2it

i50it is equal to 1. Therefore we fix the time-delay t 5 1in the rest of this paper. Then we define:

The more interesting issue is the choice of theN211embedding dimension from a time series. Generally ]]E(d) 5 O ux 2 x u, Jn11 h(n)11 0N 2 Jthere are three basic methods used in the published 0 n5J0

literature, which include computing some invariant5 (d 2 1)t 1 1. (5)

(e.g., correlation dimension, Lyapunov exponents) onthe attractor (e.g., Grassberger & Procaccia, 1983), E depends on the dimension d. Actually E(d) is thesingular value decomposition (Broomhead & King, average absolute prediction error using a zero-order1986; Vautard & Ghil, 1989), and the method of false approximation predictor for a given d. Note that a

ˆ ˆneighborhoods (Kennel et al., 1992). All these zero order predictor f is x 5 f(V ) and x 5n11 n n11

methods contain some subjective parameters or need x , where h(n) is an integer such that V is theh(n)11 h(n)

subjective judgment to choose the embedding dimen- nearest neighbor of V . Furthermore, note that the Nn

sion. Many comments on these methods have ap- in (5) represents only the number of available data


points for fitting, which does not include the datapoints for out-of-sample forecasting.

To choose the embedding dimension d , wee

simply minimize the E, i.e.,

d 5 argminhE(d): d [ Z and d $ 1j. (6)e

The embedding dimension d we choose gives thee

minimum prediction error if we use a zero-orderapproximation predictor. It is reasonable to infer thatthis d will also give good predictions if we use ae

high-order (e.g. local-linear) approximation predic-tor, since a high-order predictor is better than azero-order predictor when making out-of-samplepredictions.

In practical computations, it is certainly impos-sible to minimize the E over all positive integers. Soin real calculations we replace (6) with

d 5 argminhE(d): 1 # d # D j, (7)e max

where D is the maximum dimension with whichmax

one would like to search the minimum value of E(d).In this paper we let D 5 60.max

In summary, the above method is to find theembedding dimension by minimizing the one-stepprediction errors using a zero-order approximationpredictive model. The method is very fast in compu-tations. We have tested a number of time series Fig. 1. Theoretical pattern of E(d)’s behavior for time series

generated from different systems: (a) low-dimensional and noiseincluding artificial and experimental time series; infree systems; (b) high-dimensional and noise free systems; (c)all cases the method worked well.low-dimensional and noisy systems or low-dimensional stochasticShown in Fig. 1a–e is the typical behavior of E(d)systems; (d) high-dimensional and noisy systems or high-dimen-

for low-dimensional and noise free, high-dimension- sional stochastic systems; and (e) purely random systems.al and noise free, low-dimensional and noisy (andlow-dimensional stochastic), high-dimensional andnoisy (and high-dimensional stochastic), pure ran- mation, which runs very fast in computers and isdom time series data, respectively. For details about widely used in the area of chaotic time seriesthis method, see Cao et al. (1998). prediction. For the daily dollar exchange rates data,

we found that the embedding dimension is very high,e.g., up to 56. Using such a high embedding dimen-2.2. Local-linear predictionsion, the sophisticated approximation methods suchas neural networks and wavelet decomposition willHaving solved the problem of choosing embed-run extremely slow in numerical simulations, whileding dimension and time-delay for the vectors Vn

the local linear approximation will still run fast. Thisdefined in (1) we now use model (3) for prediction.is also the reason that we choose local linear methodThe remaining problem is how to approximate thein this paper. We need to mention that we use thefunction f. Certainly we can use any approximationzeroth order approximation to determine the embed-techniques, such as local-linear approximation, poly-ding dimension as described in the above sectionnomial approximation, neural networks, radial basisprior to making prediction using local-linear approxi-function, and wavelet decomposition. Here wemation described below.choose the simplest method: local-linear approxi-


Suppose we have N samples of time series data There are two reasons why we use a zero-orderf

available for fitting the function, i.e., we have x ,x , approximation to minimize the E(d) and a linear1 2

? ? ? ,x . Therefore we have time-delay vectors V , approximation to predict. Firstly, initially we did useN nf

a linear approximation to minimize the E(d), but wen 5 J ,J 1 1, ? ? ? ,N and J 5 (d 2 1)t 1 1. We0 0 f 0

did not find that the results of embedding dimensionwant to predict x . The local-linear approximationN 11f

are significantly different, while using the zero-orderprocedures follow.approximation method requires much less computertime than the linear approximation method. Second-1. Impose a metric on the delay-vector space, de-ly, our experiences show that better out-of-samplenoted by i i. Here we use the root-square norm,

d 2 1 / 2 predictions are obtained by using a linear approxi-i.e. iai 5 i(a ,a , ? ? ? ,a )i 5 (o a ) .1 2 d i51 i

mation than by using a zero-order approximation.2. Find the l nearest neighbors of V , denote themNf

by V ,V , ? ? ? ,V , J # j , N , (k 5 1,2, ? ? ? ,l),j j j 0 k f1 2 l

then for any k 5 1,2, ? ? ? ,l, iV 2V i # iV 2j N n 2.3. Evaluation of prediction accuracyk f

V i (J # n , N and n ± j for any k 5 1, ? ? ? ,l).N 0 f kf

3. Construct a local-linear predictor, regarding each Because we test our predictions on the exchangeneighbor V as a point in the domain and x asj j 11 rate returns, i.e. the differenced exchange rates, wek k

the corresponding point in the range. That is, compare our predictions with those by a mean valuefitting a linear function to the l pairs (V , x )j j 11 predictor rather than a random walk predictor. Thek k

(k 5 1,2, ? ? ? ,l). We use the least-squares method reason we make this comparison follows. Accordingˆto fit this linear function. Denote it by F, then we to a random walk model without a drift (x 2(t )ˆhave o ux 2 F(V )u minimized.k j 11 j x 5 e ), the exchange rate returns may comek k (t21) tˆ4. The predicted value of x is F(V ), i.e.,N 11 N from a white noise process. For a white noisef f

ˆx̂ 5 F(V ). process, the best possible prediction of the futureN 11 Nf f

behavior would be the mean value of the process.The number of neighbors, i.e., l, is usually chosen In addition, we do statistical test on significance of

larger than or equal to the number of parameters in differences between two predictions (i.e. the predic-the linear approximation, that is, l $ d 1 1. This will tions by our method and by the mean value predic-ensure the stability of the solution (Farmer & tor) using a modified Diebold–Mariano test statisticSidorowich, 1987). And l should be smaller than the (Harvey et al., 1997).total number of available delay-vectors, i.e., N 2 J . ˆ ¯ ¯A mean value predictor is, x 5 x, where x is af 0 i11

Following the above procedures, one can then constant: the mean of the time series hx j, i.e. takeipredict x if the new data point x at the time the mean of the time series as the prediction of theN 12 N 11f f

N 1 1 becomes available, where one can search the l future value. In comparing the out-of-sample fore-f

nearest neighbors either up to N 1 1 (i.e., in the casting results of the proposed method in this studyf

procedures step 2: J # j , N 1 1) or still up to N . with those of the mean value predictor, we use the0 f f

In the former case, we keep the most up-to-date following statistics:available data in the fitting set. In this paper wealways use the most up-to-date data in the prediction. 12

]ˆO (x 2 x ) 2i iF GIn the same way one can predict x , and so on ifN 13 i[Pf]]]]]RMSE 5 , (8)new data points become available. This is one-step 12

]¯O (x 2 x ) 2iF Gprediction.i[P

One can also predict x , x , and so on byN 12 N 13f f

iteration or free-run if the new data points are not where P represents the samples for out-of-sampleavailable. That is, once the predicted value of xN 11 ˆprediction, x is the predicted value of x , RMSE is af i iis at hand, it is used as a known value to predict root-mean-square error comparing with the predic-x . One can also directly predict x by fitting a ˆ ¯tion by the predictor x 5 x. Given a time series hx j,N 12 N 1T i if f

Nˆ ¯local linear predictor x 5 F(V ). In this paper we x 5 o x /N, where N is the number of availablen1T n i51 i

only consider one-step prediction. data points. If RMSE50, the predictions are perfect;


RMSE $ 1 indicates that the performance is no better exchange rate return time series has an obviousthan the mean value predictor. decreasing trend as the dimension d goes from 1 to

higher values. This may be evidence that the ex-change rate data we tested have some deterministicdynamics. In fact, from the theoretical patterns of

3. Results embedding dimensions for different systems, whichare shown in Fig. 1, it is very unlikely that the above

We test five daily dollar exchange rates time exchange rate return data are generated by purelyseries: Canadian dollar (Ca$), British pound, Ger- random processes. They may be generated by highman mark, Japanese yen, and French franc. The time dimensional systems contaminated by (measurement)period of the data we test is from October 1, 1993 to noise or nonlinear deterministic systems with sto-October 3, 1997. The total number of data points chastic driving forces (i.e. dynamic noise) (these

1recorded is 1009. Following a standard approach in systems are usually called nonlinear stochastic sys-dealing with nonstationary exchange rates time tems). For comparison with those patterns of embed-series, we test differenced-log time series: ln (x ) 2 ding dimension shown in Fig. 2, we furthermoret

ln (x ), i.e., exchange rate returns, instead of the present some embedding dimension results computedt21

original time series. directly from artificial time series data generated byWe divide each time series into three parts. The (a) a high-dimensional chaotic system, i.e. Mackey–

first part, with 1st–948th observations, is used to fit Glass delay-differential equation with the delaythe predictive model; the second part, with 949th– being 60 (Mackey & Glass, 1977) with 10% uni-978th sample data, is referred to as the internal formly distributed measurement noise; (b) a high-validation part which is used to search for the dimensional nonlinear stochastic system, i.e. Mac-optimal value of model parameter (i.e. the number of key–Glass delay-differential equation with the delayneighborhoods, see Section 2.2) using a local linear being 60 with uniformly distributed stochastic driv-predictor by minimizing the sum squared prediction ing force; (c) a low-dimensional nonlinear stochasticerrors on this part of data, and the third part, with system, i.e. chaotic Ikeda map (Ikeda, 1979) with979th–1008th observations, is used to test real or uniformly distributed stochastic driving force; andout-of-sample predictions using the fitted model. We (d) a uniformly distributed white noise. All theseonly test one-step prediction based on the most artificially generated time series have the sameup-to-date information available at the time of a length as the exchange rate time series we tested.given forecast. The results are shown in Fig. 3. This figure may

The first and the second parts of data, that is, further support that the exchange rate return time1st–978th sample data, are also used to determine series tested above are likely generated by highthe embedding dimensions for the local linear dimensional systems with measurement noise or bymodels. The values of E(d) for different embedding high dimensional nonlinear stochastic systems.dimensions for the five differenced-log (i.e. ex- Next we turn to test out-of-sample prediction ofchange rate return) time series are shown in Fig. the above five exchange rate return time series using2a–e. These values were computed by the method the local linear method. We evaluate our predictionproposed in Section 2.1 with the time-delay t being by calculating the root-mean-square error RMSE1. Then by minimizing E(d), we get the optimal which is defined in Section 2.3. The results areembedding dimension for each time series. We list summarized in Table 1; where one can see that ourthe embedding dimensions in Table 1 (see the row of predictions outperform the mean value predictor for‘Embed. Dim.’), observing that the embedding di- the pound/dollar and the yen/dollar rate returns, butmensions are very high. not for the other three exchange rate returns.

From Fig. 2a–e, it is clear that the E(d) for each To test statistical significance of the difference (orequality of prediction) between our predictions and

1 those by the mean value predictor, we calculate theWe used more data points, but they did not significantly affectour prediction outcomes. modified Diebold–Mariano test statistic defined in


Fig. 2. The quantity E(d) versus d for the differenced-log exchange rates time series. The horizontal axis is the dimension d and the verticalaxis is the quantity E(d). (a) Canadian dollar /US dollar. (b) British pound/US dollar. (c) German mark/US dollar. (d) Japanese yen/USdollar. (e) French franc /US dollar.

Table 1RMSE of predictions

Name Ca$/$ Pound/$ Mark/$ Yen/$ Franc /$

Embed. Dim. 44 39 27 56 51aN. Neigh. 896 459 1 880 308

RMSE 1.02 0.91 1.08 0.93 1.10a N. Neigh. stands for the number of neighborhoods used for the local linear approximation.

Harvey et al. (1997). The critical values are used test results are shown in Table 2. For the fivefrom the Student’s t distribution with (n 2 1) degrees exchange rate return time series we tested, the null ofof freedom (n is the number of data points predicted, equality of prediction errors by our method and thei.e. n 5 30 in this paper) at a nominal 10% level. The mean value predictor are all accepted.


Fig. 3. The quantity E(d) versus d for four artificial time series: (a) Mackey–Glass time series with 10% uniformly distributed measurementnoise; (b) Mackey–Glass time series generated by Mackey–Glass equation with uniformly distributed stochastic driving force; (c) Ikeda timeseries generated by Ikeda map with uniformly distributed stochastic driving force; and (d) time series generated by uniformly distributedwhite noise.

Table 2aStatistical test of equality of prediction errors

Name Ca$/$ Pound/$ Mark/$ Yen/$ Franc /$

Value of test statistic 0.45 21.48 1.04 20.96 1.58Rejection of equality No No No No No

a The test statistic is the modified Diebold–Mariano statistic (Harvey et al., 1997); the rejection of equality is based on the critical value ofStudent’s t distribution with (n21) degrees of freedom at a nominal 10% level, (i.e. uxu$1.699), and n is the number of data pointspredicted, i.e. n530 in our cases.

4. Summary and comments od, we found that these series have very highembedding dimensions. The reason for having so

We have tested five exchange rate return time high embedding dimensions may be explained by theseries. By applying the time-delay embedding meth- fact that exchange rates are determined by a large


10number of real and nominal variables such as interest 10, 10 data points are needed, which makes itrates, price level, national income, money supply, nearly impossible to run in the present generation ofetc. Also, it is possible that the exchange rates may computers.come from a delay-differential equation, which We believe that the out-of-sample predictions atmathematically represents a system with an infinite best are sensitive to the number of neighbours used.dimension. A good choice of the number of neighbours may be

Our results on the estimation of embedding dimen- determined by (1) the number of data points in thesion may have provided evidence that the exchange part of time series which is used for the internalrate returns we tested are likely generated from validation of the model, and (2) by the similarhigh-dimensional systems with noise or with stochas- dynamics in the part of the data used in validationtic driving force (the latter one is usually called and the data used in out-of-sample prediction. Thenonlinear stochastic systems). The high dimensional finite number of observations, the high level of noise,systems may be indistinguishable from stochastic and nonstationarity of the data, create difficulties insystems in practice. Because of this, we cannot having a good choice of the number of neighboursexclude the possibility that the exchange rate returns needed for the local-linear predictor. However, thewe tested are generated from nonlinear stochastic use of internal validation in choosing the number ofsystems. But we could still claim it is unlikely that neighbours is a common practice. Moreover, it is notthe exchange rate returns are generated by purely clear why the number of neighbours is either ex-random processes. tremely large in most cases, and extremely small in

We would like to mention that differencing a time one of the cases presented in this paper. Neverthe-series may amplify the noise effect in the time series. less, we would like to point out that noise is always aTherefore, differencing may not be a good way to factor playing an important role in the above radical-process a nonstationary time series if the time series ly different results. Further investigation on this issueis contaminated by noise. We will be investigating is needed and the results may be provided in ourother ways to make a nonstationary series stationary future work.without amplifying the noise effect. In that case wemay obtain better predictions than we obtained inthis paper. This work will be presented in the future. Acknowledgements

The techniques proposed in this paper are ratheruseful in understanding complex dynamics of ex- We are deeply indebted to the anonymous refereeschange rates, especially the method of searching for for their highly valuable and detailed comments andembedding dimension. The high embedding dimen- criticisms which, we hope, have enhanced the claritysions we found in this paper imply very complex of this paper.dynamics in the exchange rates. In contrast, manypublished results use fewer (usually less than 10)embedding dimensions in prediction of exchange Referencesrates. The low embedding dimensions may haveundesirable distorting effects on observing the com- Bajo-Rubio, O., Fernandez-Rodriguez, F., & Sosvilla-Rivero, S.

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Introduction For Scientists and Engineers, Oxford Universitymathematics at the Institute of Systems Science, Chinese

Press, Oxford.Academy of Sciences, with a thesis on Chaos and Reconstruction

Ikeda, K. (1979). Multiple-valued stationary state and its instabili-of Dynamics from Experimental Signals. Dr. Cao’s research

ty of the transmitted light by a ring cavity system. Opticinterests include chaotic data analysis, time series modelling and

Communications 30, 257.prediction, and their application in economics and finance as well

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geometrical construction. Physical Review A 45, 3403–3411.Mathematics, University of Western Australia.

Larsen, C., & Lam, L. (1992). Chaos and the foreign exchangemarket. In: Lam, L., & Naroditsky, V. (Eds.), Modeling

Abdol SOOFI received a doctorate in economics from theComplex Phenomena, Springer, New York.University of California, Riverside. Dr. Soofi’s primary researchLisi, F., & Medio, A. (1997). Is a random walk the best exchangeinterest is exchange-rate dynamics. In addition to his work onrate predictor? International Journal of Forecasting 13, 255–nonlinear modeling and forecasting of exchange rate, he has used267.the autoregressive fractional integrated moving average (AR-Mackey, M., & Glass, L. (1977). Oscillation and chaos inFIMA) model in testing for long memory in several exchange ratephysiological control systems. Science 197, 287.time series. His works also involve developing a method forMeese, R., & Rogoff, K. (1983). Empirical exchange rate modelsestimating the parameters of the ARFIMA model with Bayesianof the seventies: do they fit out-of-sample? Journal of Interna-global optimization method. Dr. Soofi is a Professor of Economicstional Economics 14, 3–24.at the University of Wisconsin–Platteville, and is currently aMizrach, B. (1993). Multivariate nearest-neighbor forecasts ofVisiting Scholar at the School of Business Administration, theEMS exchange rates. In: Pesaran, M. H., & Potter, S. M.University of Wisconsin–Milwaukee.(Eds.), Nonlinear Dynamics, Chaos and Econometrics, John

Wiley, New York.Ott, E., Sauer, T., & Yorke, J. A. (1994). Copying With Chaos:

Analysis of Chaotic Data and the Exploitation of ChaoticSystems, John Wiley, New York.

nonlinear deterministic forecasting of daily dollar exchange rates

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