Improving prediction of exchange rates using Differential EMD

https://doi.org/10.1016/j.eswa.2012.07.048Get rights and content

Abstract

Volatility is a key parameter when measuring the size of errors made in modelling returns and other financial variables such as exchanged rates. The autoregressive moving-average (ARMA) model is a linear process in time series; whilst in the nonlinear system, the generalised autoregressive conditional heteroskedasticity (GARCH) and Markov switching GARCH (MS-GARCH) have been widely applied. In statistical learning theory, support vector regression (SVR) plays an important role in predicting nonlinear and nonstationary time series variables. In this paper, we propose a new algorithm, differential Empirical Mode Decomposition (EMD) for improving prediction of exchange rates under support vector regression (SVR). The new algorithm of Differential EMD has the capability of smoothing and reducing the noise, whereas the SVR model with the filtered dataset improves predicting the exchange rates. Simulations results consisting of the Differential EMD and SVR model show that our model outperforms simulations by a state-of-the-art MS-GARCH and Markov switching regression (MSR) models.

Highlights

► The algorithm of Differential EMD has the capability of smoothing and reducing the noise. ► The SVR model can predict exchange rates. ► In prediction, the Differential EMD and SVR model outperforms the MS-GARCH model.

Introduction

With an average daily turnover of $3981 billion, foreign exchange instruments which comprise spot transactions, outright forwards, foreign exchange swap options, and other derivative products, are extremely active. The central banks, investment banks, financial institutions, multinational companies, and stock markets all buy and sell deposits denominated in foreign currencies. US dollars and the euro make up a large part of the trading volume, with market share of 42.45% and 19.55% respectively.1 Since the onset of the last global financial crisis and market crash in 2008, the financial markets have experienced periodic extreme volatility and turmoil, due largely to the occurrence of black swan and heavy tail events. Increasingly, governments, economists and the financial communities are now showing renewed interest in financial econometric modelling using time-varying dynamic macroeconomic parameters to forecast economic growth, market crashes and to mitigate systemic financial risks.

In terms of the classification and prediction of exchange rates, there are few models that can handle nonlinear, high frequency and high volatility time- series variables in applications such as bioinformatics, oceanography, oil and gas explorations, finance, and so on. These models migrated from different origins, including; (i) the Markov switching generalised auto regressive conditional heteroskedasticity (MS-GARCH) model which is suitable for econometric applications, (ii) the Empirical Mode Decomposition (EMD) de-noising model, which emerged in the signal processing area, and (iii) the support vector machine (SVM) model which originated from statistical learning theory.

The following sections provide background of EMD, support vector egression (SVR), and finally MS-GARCH model and Markov switching regression (MSR).

The analysis of nonlinear and nonstationary data becomes relatively important in many applications such as bioinformatics (Shi, Chen, & Li, 2008); signal processing (Huang and Shen, 2005, Huang et al., 1996, Huang et al., 1998); geophysics, (Datig & Schlurmann, 2004); and finance (Guhathakurta et al., 2008, Huang and Shen, 2005). In 1996, Huang et al., cited by Huang and Attoh-Okine (2005), invented an a posteriori algorithm with adaptive control over a separate data structure. The invention was later called the Hilbert–Huang transform (HHT) (Huang et al., 1998). This new transform enhances the limitation of the Hilbert transform, which is only suitable for a narrow band-passed signal (Huang et al., 1996). The key part of the HHT algorithm is the EMD, in which any complicated dataset can be decomposed into a finite and often small number of intrinsic mode functions (IMF) that admit a well-behaved Hilbert transform. Since the decomposition is based on the local characteristic time scale of the data, it is applicable to nonlinear and nonstationary processes (Huang & Attoh-Okine, 2005). In signal processing, high frequency noise from input data may be considered as different simple intrinsic mode oscillations (Huang et al., 1998). The EMD algorithm, which is fundamental to the HHT, can reduce a high frequency noise from input data such as noise from retail trades in the stock exchange.

Fisher (1936) invented the first algorithm for pattern recognition. A few decades later, Vapnik and Lerner (1963) introduced the generalized portrait algorithm which has since been the template for support vector machines. Later, Duda et al. (1973) discussed large margin hyperplanes in the input space. However, the field ‘statistical learning theory’ was already active under the introduction of Vapnik and Chervonenkis (1994). At the Conference on Learning Theory (COLT), Boser, Bernhard, Guyon, and Vapnik (1992) introduced SVMs by combining the Generalized Portrait algorithm comprising a large margin hyperplane and kernel functions. Nowadays, SVM has empirically shown a better performance than other machine learning methods, including artificial neural networks (ANN). This is because of the employment of structural risk minimisation, in which they can avoid multiple local minima. Moreover, the computational complexity of SVM does not depend on the dimensionality of the input space. This is what makes an SVM so flexible when selecting large controlling parameters. Therefore, the SVM model is suitable for handling datasets that are complex in dimensionality, such as exchange rates.

In terms of regression n analysis, the SVM model was first compared against the benchmark of time series prediction tests with the Boston housing problem, ANN, and (using artificial data) the PET operator inversion problem (Burges, 1998). Technically, SVMs consist of a set of related supervised learning methods. The algorithm sets a hyperplane characterising a functional margin that holds all possible data points situated in a finite dimensional nonlinear space. A kernel function, K(x, y), then defines the cross-products that have been separated by the hyperplane. Each data point demonstrates its vector potential depending on its distance from the hyperplane. In the peak computational mode, the SVM model may have difficulty in finding the best-fit hyperplane. As a result it, can present overflow memory errors (Burges, 1998, Kim, 2003, Rychetsky, 2001).

In econometrics, volatility is a key parameter when measuring the size of errors made in modelling returns and other financial variables. The ARMA model is a linear time series, in which the mean remains conditionally changed, but the variance is constant. Subsequently, an error can occur when handling nonlinear nonstationary for time series systems.

Engle et al., cited by Diebold (2004) introduced autoregressive conditional heteroskedasticity (ARCH), a breakthrough in econometrics modelling that uses a stochastic process to predict an average size of the error terms. Bollerslev (2010) introduced the Generalised GARCH model, which is well suited to handle nonstationary random variables, in which while variance of the error is independent. It is known that The GARCH is an advance on the ARCH model and uses a weighted average of past squared residuals.

There are a number of studies that introduce conditional variance with regression analysis. Quandt (1958) estimated that parameters of a linear regression system obey two separate regimes. Later, in 1973, Goldfeld–Quandt, cited by Rana, Midi, and Rahmatullah Imon (2008) introduced parametric test checks for homoscedasticity in regression analyses. Following this, they introduced a simple intuitive diagnostic for heteroskedasticity errors for both univariate and multivariate regression models. Bollerslev et al., 1992, Bollerslev et al., 1994 introduced models that can forecast conditional variances. Those models have now been employed in various branches of econometrics, particularly in financial analysis of time-varying volatility. In summary, the ARCH and GARCH models used autoregression (AR) technique to estimate coefficient of an error, which is conditional, uncertain, and fluctuating.

The Markov chain random process has currently been popular in dynamic macroeconomics and financial econometrics for modelling asset prices and market instability. In a Markov switching model, the latent state variables controlling regime shifts follows the Markov chain principle. Hamilton (1989) first introduced regime switching to describe an autoregressive regime switching process and extended the Markov switching models for dependent data. In particular, he used autoregression to model switching between periods of high volatility and low volatility of asset returns. The rationale behind the switching framework is to estimate markets that change periodically between a stable low-volatility state and a more unstable highly volatile regime. Periods of high volatility may arise, for example, because of short-term political or economic uncertainties. Further, Hamilton and Susmel (1994) analysed the ARCH in several regime switching models, varying the number of regimes and the form of the model within regimes. Their objective was to model various weekly econometric series; and for these, they applied the more complicated ARCH models within regimes, and called ‘Regime switching GARCH (RS-GARCH) (Diebold, 2004, Bauwens et al., 2010, Klaassen, 2002). The Markov chain algorithm has later been applied to the RS-GARCH model. It is then known as MS-GARCH (Hass, Mittnik, & Paollela, 2004), and is used in the financial industry to analyse volatility in the stock and foreign exchange markets.

In terms of regression analysis, the GARCH model cannot accommodate any independent variables that may correlate to the dependent variables e.g., the gold price varies with the index of global inflation, stock price, and so on. Hence, many studies replace the autoregression algorithm in the MS-GARCH with a more suitable regression model. Hence, the new model is called the Markov switching regression (MSR) (Hass et al., 2004).

In this paper, we propose a new model for filtering, smoothing and predicting nonlinear and nonstationary time series signals. It comprises Differential EMD algorithm, and SVR model. The Differential EMD is able to smooth and reduce noise, whereas the SVR model can predict exchange rates such as the euro-USD rate. Our simulations show that our proposed model outperforms both the state-of-the-art Markov switching GARCH (MS-GARCH) model and the MSR model.

Section 2 is a collection of exchange rates and other relevant datasets, including tests. This serves to confirm that the data are nonlinear and nonstationary. In Section 3, we introduce mathematical considerations of the Differential EMD and SVM model compared to the MS-GARCH and MSR models. Section 4 consists of simulations and results of the proposed model and other relevant models, including their measurements. Finally, Section 5 concludes this paper and summarises the discussion.

Section snippets

Data collection and testing

Foreign exchange (FX) data are numbers with two to six decimal points coded in ASCII, and are nonlinear and nonstationary with exogenous influence. Normally, official FX data cannot be obtained publicly, but it is available from financial institutions and media companies, such as Bloomberg and Olsen Associates. The datasets used in this paper are the euro-USD exchange rates, trade exchange rates, commodities indices, major foreign exchange rates, and other macroeconomic data such as interest

Mathematical considerations of the Differential EMD and SVR model

As proposed, the EMD algorithm smoothes and filters nonlinear and nonstationary time series signals, whereas the SVR model in general works as a nonlinear regression estimator. The following sections are mathematical considerations of the combination of the Differential EMD and SVR models.

Simulations and results

This section discusses the simulation works and results of the Differential EMD and SVR model in predicting the euro-USD exchange rates. Simulation works begin with using the EMD algorithm that requires only the Dependent variables, detailed in Section 4.1. Following to this, it is the simulation work of the proposed Differential EMD algorithm in Section 4.2. Both are displayed in Fig. 1, Fig. 2. To demonstrate the performance of ‘the proposed Differential EMD algorithm combined with the SVR

Conclusions and discussion

Predicting nonlinear and nonstationary euro-USD exchange rates is very challenging, since there are many correlated independent variables involved. This paper introduced a new model comprised the Differential EMD algorithm and the SVR model. While the Differential EMD algorithm functions as a filter for nonlinear and nonstationary time series systems, the SVR model predicts the future value of the exchange rates. Independent and dependent variables were retrieved from the Bloomberg terminal

Acknowledgments

This work is fully inspired by the collaborations of The Department of Electrical and Electronic Engineering and Centre of Bio-inspired Technology, Imperial College London. The authors are grateful to Dr. Peerapol Yuvapoositanon, Wong Fan Voon, and Janpen Jantorntrakul who read, edit and provide helpful suggestions.

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