International Journal of Forecasting 15 (1999) 421–430 www.elsevier.com / locate / ijforecast Nonlinear deterministic forecasting of daily dollar exchange rates a b, * Liangyue Cao , Abdol S. Soofi a Department of Mathematics, University of Western Australia, Nedlands, WA 6907, Australia b Department of Economics, University of Wisconsin –Platteville, and Visiting Scholar, School of Business Administration, University of Wisconsin –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 techniques and a local linear predictor. We compared our predictions with those by a mean value predictor. Some of our predictions of the exchange rate returns outperform the predictions of the same series by the mean value predictor. However, these improvements were not statistically significant. Another interesting result in this paper which was obtained by using a recently developed technique of nonlinear dynamics is that all exchange rate return series we tested have a very high embedding dimension. Additionally, evidence indicates that these series are likely generated by high dimensional systems with measurement noise or by high dimensional nonlinear stochastic systems, that is, nonlinear deterministic systems with dynamic 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 embedding showed 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’s econometric models of the exchange rates, has raised dynamics through observed time series. Several the possibility that some of the series are generated related algorithms, such as calculation of correlation by stochastic rather than deterministic processes. dimension (Grassberger & Procaccia, 1983) and Determining 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 make was hampered by the inadequacy of the mathemati- characterizing dynamical behavior from time series cal theory in the past. Recent developments in time data possible. These algorithms have had a large series techniques from chaos theory (see e.g., Ott et number of applications in detecting nonlinear de- terminism from observed time series, e.g., economic and 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 sample 229-6957. E-mail address: asoofi@uwm.edu (A.S. Soofi) sizes, misleading results could still occur. 0169-2070 / 99 / $ – see front matter  1999 Elsevier Science B.V. All rights reserved. PII: S0169-2070(99)00024-2