forecasting exchange rates: an optimal approach

J Syst Sci Complex (2014) 27: 21–28

FORECASTING EXCHANGE RATES: AN OPTIMAL

APPROACH∗

BENEKI Christina · YARMOHAMMADI Masoud

DOI: 10.1007/s11424-014-3304-5

Received: 25 April 2013 / Revised: 30 September 2013

c©The Editorial Office of JSSC & Springer-Verlag Berlin Heidelberg 2014

Abstract This paper looks at forecasting daily exchange rates for the United Kingdom, European

Union, and China. Here, the authors evaluate the forecasting performance of neural networks (NN),

vector singular spectrum analysis (VSSA), and recurrent singular spectrum analysis (RSSA) for fore-

casting exchange rates in these countries. The authors find statistically significant evidence based on

the RMSE, that both VSSA and RSSA models outperform NN at forecasting the highly unpredictable

exchange rates for China. However, the authors find no evidence to suggest any difference between the

forecasting accuracy of the three models for UK and EU exchange rates.

Keywords China, European union, exchange rates, forecasting, neural networks, recurrent singular

spectrum analysis, United Kingdom, vector singular spectrum analysis.

1 Introduction

The foreign exchange market is known to be the most liquid, and largest, financial marketin the globe. In a world where majority of the nations operate open economies, exchangerates continue to have a significant impact on the economic stability of a given country. Assuch, obtaining accurate exchange rate forecasts are imperative for currency traders, importersand exporters of goods and services, and the governments which are managing the economies.For currency traders, accurate forecasts can reduce their risk as they seek to profit on futureexchange rate movements. Importers and exporters can benefit via better management decisionson the timing of imports and exports to coincide with accurate forecasts which can enhance thefirms profitability. For governments, accurate foreign exchange forecasts can enable the sound

BENEKI Christina

School of Business and Economics, Department of Business Administration, Technological Educational Institute

of Ionian Islands, 31100 Lefkada, Greece. Email: [email protected].

YARMOHAMMADI Masoud

Department of Statistics, Payame Noor University, 19395-4697 Tehran, Islamic Republic of Iran.

Email : [email protected].∗This research was supported by a grant from Payame Noor University, Tehran-Iran.�This paper was recommended for publication by Editor WANG Shouyang.

22 BENEKI CHRISTINA · YARMOHAMMADI MASOUD

management of a nation’s foreign exchange reserves which has a direct impact on monetarypolicy in a given country.

Over the years both researchers and academics have endeavoured to develop the best fore-casting model to predict the highly unpredictable foreign exchange market. In the process,both parametric and nonparametric forecasting techniques have been evaluated. It is not theintention of this paper to evaluate the forecasting performance of all such models. Instead,here we consider two nonparametric time series analysis and forecasting techniques known asartificial neural networks (NN) and singular spectrum analysis (SSA) for forecasting exchangerates in the United Kingdom, European Union, and China. As nonparametric techniques arenot bound by any assumptions, it is more likely to provide an accurate representation of thetrue scenario in comparison to parametric techniques. The singular spectrum analysis (SSA)technique was introduced for exchange rate forecasting through the work of Hassani, et al.[1].There exist two variations of the basic univariate SSA known as Vector SSA (VSSA) and Re-current SSA (RSSA). In this paper, we evaluate both Vector and (for the first time) RecurrentSSA for forecasting exchange rates. The SSA technique itself is experiencing a rapid growthin popularity with diverse applications in a variety of fields (see, for example, [2–10]). Onthe other hand, neural network models have been evaluated for exchange rate forecasting bothhistorically and more recently (see, for example, [11–19]). Figure 1 illustrates the exchangerates for UK, EU, and China. It is clear from the figure that the distributions of the exchangerates would not meet the parametric assumptions, and thereby further support the case foremploying nonparametric models to forecast exchange rates in the future.

The remainder of this paper is organized as follows. Section 2 explains briefly the fore-casting methods while Section 3 reports the empirical results. The paper concludes with someconclusions in Section 4.

2 Forecasting Models

2.1 Neural Networks (NN)

Here, we use the nnetar forecasting function which a system of feed-forward neural networkswith lagged inputs and one hidden layer. The function trains 25 neural networks by adoptingrandom starting values and then obtains the mean of the resulting predictions to compute theforecasts. A detailed explanation on the underlying dynamics of this neural network model, see[20].

2.2 Singular Spectrum Analysis (SSA)

A detailed description on the theory underlying SSA is found in [21] and [22]. Here weprovide a brief introduction to the process involved, and in doing so we mainly follow [23].

FORECASTING EXCHANGE RATES 23

Year

UK

Dai

ly E

xcha

nge

Rat

e

2012.0 2012.2 2012.4 2012.6 2012.8 2013.0

1.50

1.54

1.58

1.62

Year

EU D

aily

Exc

hang

e R

ate

2012.0 2012.2 2012.4 2012.6 2012.8 2013.0

1.20

1.30

Year

Chi

na D

aily

Exc

hang

e R

ate

2012.0 2012.2 2012.4 2012.6 2012.8 2013.0

6.25

6.35

Figure 1 Daily exchange rates (03/01/2012–01/03/2013)

Consider the real-valued nonzero time series YT = (y1, y2, · · · , yT ) of sufficient length T . LetK = T − L + 1, where L (L ≤ T/2) is some integer called the window length. The first stageis known as decomposition. Here, we define the matrix X = (xij)

L,Ki,j=1 = [X1, X2, · · · , XK ],

where Xj = (yj , yj+1, · · · , yL+j−1)T. In doing so, we are transforming a one dimensional timeseries into a multidimensional time series which is referred to as the embedding step. Thenext step, singular value decomposition (SVD) of XXT provides us with the collections of L

eigenvalues (λ1 ≥ λ2 ≥ · · ·λL ≥ 0) and the corresponding eigenvectors U1, U2, · · · , UL, whereUi is the normalized eigenvector corresponding to the eigenvalue λi (i = 1, 2, · · · , L). In thesecond stage which is known as reconstruction, we first group the eigenvalues in order to reducethe noise level in the original noisy series. To do this, we select r singular values from L.Finally, in order to convert the matrix of selected components into a time series we performdiagonal averaging. This provides an approximation of the original series with less noise whichcan be used to forecast new data points. The forecasting algorithm of SSA can be applied toany time series that approximately satisfies the linear recurrent formulae (LRF)[21]. The seriesYT satisfies an LRF of order d if there are numbers a1, a2, · · · , ad such that

yi+d =d∑

k=1

akyi+d−k, 1 ≤ i ≤ T − d.


To obtain the coefficients a1, a2, · · · , ad we use the eigenvectors obtained from the SVD step orcharacteristic polynomial (for more information relating to this procedure see, [21]).

3 Empirical Results

The data relates to daily exchange rates in the United Kingdom, European Union, andChina between the dates 03/01/2012 and 01/03/2013. In order to train the forecasting models,approximately 2

3

rd of the observations (i.e., 192 observations) were used while approximately13

rd of the observations (i.e., 100) were left aside for testing the forecasting accuracy of themodels. The following measures were adopted to compare and contrast between the forecastingmodels.

3.1 Measures for Evaluating the Forecasting Accuracy

Root Mean Squared Error (RMSE)

The RMSE is now a popular measure of forecasting accuracy and frequently cited in fore-casting literature (see, for example, [2–6] and [24]). In order to save space, here we only providethe RMSE ratio of SSA to that of NN:

RRMSE =SSANN

=

(∑Ni=1(yT+i,i − yT+h,i)2

)1/2

(∑Ni=1(yT+h,i − yT+h,i)2

)1/2,

where, yT+h is the h-step ahead forecast obtained by SSA, yT+h is the h-step ahead forecastfrom the NN model, and N is the number of the forecasts. If RRMSE < 1, then the SSAoutperforms NN by 1 − ETS

ARIMA percent.

Direction of Change (DC)

DC is a measure of the percentage of forecasts that accurately predict the direction ofchange[4]. Here, the concept of DC is explained in brief. A detailed account can be foundin [4]. In the univariate case, for forecasts obtained using XT , let DXi be equal to 1 if theforecast is able to correctly predict the actual direction of change and 0 otherwise. Then,DX =

∑ni=1 DXi/n shows the proportion of forecasts that correctly identify the direction of

change in the actual series.

3.2 Forecasting Results

Table 1 reports the RMSE’s for the out-of-sample forecasting results. All RMSE’s have beenmultiplied by 103 in order to enable easy comparison between the forecasting models. Firstly,based on the RMSE, the results show that we cannot identify one model to be best for all threecountries at all horizons. Instead, we see that for forecasting exchange rates in UK, the BasicVSSA model is able to outperform both NN and Basic RSSA at horizons of h = 1 and 3 stepsahead. For EU, we see that the NN model outperforms both Basic VSSA and Basic RSSA atforecasting the exchange rate at all horizons. The results for China show a completely differentoutlook as the NN model’s forecasting accuracy appears to have deteriorated to a great extentand both Basic VSSA and Basic RSSA models outperform the NN model. In China, for 1 day


ahead forecasts of exchange rates the Basic VSSA model outperforms Basic RSSA marginallywhilst for 3 day ahead forecasts, the Basic RSSA model is seen to outperform the Basic VSSAmodel marginally. However, in order to evaluate the validity of these conclusions it is importantto test the results for statistical significance. Here, we use the modified Diebold-Mariano testin [25] to test the RRMSE results.

Table 1 Out-of-sample forecasting results for exchange rates

NN Basic VSSA Basic RSSA BasicVSSANN

BasicVSSABasicRSSA

Series 1 3 1 3 1 3 1 3 1 3

UK 7.62 12.8 6.80 11.5 7.10 11.7 0.89 0.89 0.96 0.98

EU 6.77 11.0 7.00 11.8 7.10 11.8 1.03 1.07 0.99 1.00

China 15.7 25.0 5.19 7.95 5.21 7.92 0.33** 0.32* 0.996 1.004

Note: * indicates statistical significance based on the DM test at p=0.01.

** indicates statistical significance based on the DM test at p=0.05.

Based on the RRMSE, we can see that only two outcomes are statistically significant.Accordingly, we can conclude with 95% confidence that the Basic VSSA model is 67% betterthan NN at forecasting the Chinese exchange rate at a horizon of one day ahead. Furthermore,we are able to conclude with 99% confidence that for 3 days ahead forecasts of Chinese exchangerates, the Basic VSSA model is 68% better than NN. However, the results also indicate thatfor forecasting exchange rates in UK and EU there is no real difference between the accuracy ofany of the models and that the conclusions derived earlier based on the RMSE alone are likelyto be chance occurrences and not statistically valid. This further highlights the importanceof testing statistical results for significance as otherwise we would be resorting to conclusionswhich are unlikely to be valid in the real world.

Finally, we test the ability of the forecasting models at predicting the actual direction ofchange in the exchange rate time series. The results are reported in Table 2. Based on the DC,for all three countries, the NN model provides the worst DC prediction in comparison to BasicVSSA and Basic RSSA models. For UK, the Basic VSSA model provides the best DC predictionat h = 1 and 3 days ahead. For EU, Basic VSSA has the most favourable DC prediction atone step ahead, but NN appears to have the best DC prediction ability at three steps ahead.Finally, for China we can see that Basic RSSA records the best DC prediction at h = 1 stepahead whilst Basic VSSA provides the best DC prediction at three steps ahead. Once again wetest our results for statistical significance. Accordingly, Table 2 shows that all conclusions onDC which were made previously can be attributed to chance occurrences. However, we are ableto conclude with 95% confidence that for the Chinese exchange rate forecast, the NN modelprovides the worst DC prediction at 3 steps ahead with only 48% accuracy.


Table 2 Direction of change results for exchange rates

NN Basic VSSA Basic RSSA

Series 1 3 1 3 1 3

UK 0.43 0.57 0.51 0.62 0.49 0.51

EU 0.48 0.69 0.52 0.51 0.46 0.51

China 0.52 0.48* 0.54 0.57 0.62* 0.54

Note: * indicates statistical significance based on a t-test at p=0.05.

Figure 2 illustrates the forecasting results for the statistically significant outcomes whichwere visible in the results for China at h = 1 and 3 days ahead. The figure shows clearly thatthe two SSA models do indeed provide a better fit for the Chinese exchange rates forecasts andit also shows the accuracy of all three models deteriorating as the horizon increases from 1 stepahead to 3 steps ahead.

Year

Chi

na: h

=1 d

ay a

head

for

ecas

t

2012.2 2012.6 2013.0

6.22

6.24

6.26

6.28

6.30

ActualNN forecastVSSA forecastRSSA forecast

Year

Chi

na: h

=3 d

ays a

head

fore

cast

2012.2 2012.6 2013.0

6.22

6.24

6.26

6.28

6.30

ActualNN forecastVSSA forecastRSSA forecast

Figure 2 Daily exchange rates forecasts for China

4 Conclusion

This paper marks the introduction of Recurrent SSA for exchange rate forecasting. However,the result indicate that there is no statistically significant difference between the RSSA andVSSA models at forecasting exchange rates for UK, EU and China. Furthermore, we find nostatistically significant evidence to outline either one of the forecasting models employed in this


study to be the best for forecasting exchange rates in UK and EU. However, we are able toconclude with 95% confidence that for forecasting the Chinese exchange rate, the Basic VSSAand Basic RSSA models outperform the NN model at horizons of h = 1 and 3 days ahead. Thestudy also finds with statistically significant evidence that the Basic RSSA model can achieve60% accuracy in terms of DC prediction when forecasting the Chinese exchange rate at h = 1step ahead, and also that the NN model provides the least favourable DC prediction for Chinaat h = 3 steps ahead. The rest of the DC results were found to be attributable to chanceoccurrences.

Future research should consider incorporating more forecasting techniques in order to eval-uate whether statistically significant outcomes could then be achieved. It would also be inter-esting to see how a hybrid model combining Neural Networks and Singular Spectrum Analysiswould fare at forecasting exchange rates for UK, EU and China in the future.

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forecasting exchange rates: an optimal approach

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