J Syst Sci Complex (2014) 27: 1–8
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 forecasting 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 market in the globe. In a world where majority of the nations operate open economies, exchange rates continue to have a significant impact on the economic stability of a given country. As such, obtaining accurate exchange rate forecasts are imperative for currency traders, importers and 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 future exchange rate movements. Importers and exporters can benefit via better management decisions on the timing of imports and exports to coincide with accurate forecasts which can enhance the firms profitability. For governments, accurate foreign exchange forecasts can enable the sound BENEKI Christina Department of Business Administration, Technological Educational Institute of Kefalonia. 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.
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BENEKI CHRISTINA · YARMOHAMMADI MASOUD
management of a nation’s foreign exchange reserves which has a direct impact on monetary policy in a given country. Over the years both researchers and academics have endeavoured to develop the best forecasting model to predict the highly unpredictable foreign exchange market. In the process, both parametric and nonparametric forecasting techniques have been evaluated. It is not the intention 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 as artificial neural networks (NN) and singular spectrum analysis (SSA) for forecasting exchange rates in the United Kingdom, European Union, and China. As nonparametric techniques are not bound by any assumptions, it is more likely to provide an accurate representation of the true 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 Recurrent SSA (RSSA). In this paper, we evaluate both Vector and (for the first time) Recurrent SSA for forecasting exchange rates. The SSA technique itself is experiencing a rapid growth in popularity with diverse applications in a variety of fields (see, for example, [2–10]). On the other hand, neural network models have been evaluated for exchange rate forecasting both historically and more recently (see, for example, [11–19]). Figure 1 illustrates the exchange rates for UK, EU, and China. It is clear from the figure that the distributions of the exchange rates would not meet the parametric assumptions, and thereby further support the case for employing nonparametric models to forecast exchange rates in the future. The remainder of this paper is organized as follows. Section 2 explains briefly the forecasting methods while Section 3 reports the empirical results. The paper concludes with some conclusions in Section 4.
2 2.1
Forecasting Models Neural Networks (NN)
Here, we use the nnetar forecasting function which a system of feed-forward neural networks with lagged inputs and one hidden layer. The function trains 25 neural networks by adopting random starting values and then obtains the mean of the resulting predictions to compute the forecasts. 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 we provide a brief introduction to the process involved, and in doing so we mainly follow [23].
1.54
1.58
1.62
Please provide the 3 running title
1.50
UK Daily Exchange Rate
FORECASTING EXCHANGE RATES
2012.0
2012.2
2012.4
2012.6
2012.8
2013.0
2012.6
2012.8
2013.0
2012.6
2012.8
2013.0
1.30 1.20
EU Daily Exchange Rate
Year
2012.0
2012.2
2012.4
6.35 6.25
China Daily Exchange Rate
Year
2012.0
2012.2
2012.4 Year
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 . Let K = T − L + 1, where L (L ≤ T /2) is some integer called the window length. The first stage is known as decomposition. Here, we define the matrix X = (xij )L,K i,j=1 = [X1 , X2 , · · · , XK ], where Xj = (yj , yj+1 , · · · , yL+j−1 )T . In doing so we are transforming a one dimensional time series into a multidimensional time series which is referred to as the embedding step. The next step, singular value decomposition (SVD) of XX T provides us with the collections of L eigenvalues (λ1 ≥ λ2 ≥ · · · λL ≥ 0) and the corresponding eigenvectors U1 , U2 , · · · , UL , where Ui is the normalized eigenvector corresponding to the eigenvalue λi (i = 1, 2, · · · , L). In the second stage which is known as reconstruction, we first group the eigenvalues in order to reduce the 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 perform diagonal averaging. This provides an approximation of the original series with less noise which can be used to forecast new data points. The forecasting algorithm of SSA can be applied to any time series that approximately satisfies the linear recurrent formulae (LRF)[21] . The series YT satisfies an LRF of order d if there are numbers a1 , a2 , · · · , ad such that yi+d =
d k=1
ak yi+d−k ,
1 ≤ i ≤ T − d.
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BENEKI CHRISTINA · YARMOHAMMADI MASOUD
To obtain the coefficients a1 , a2 , · · · , ad we use the eigenvectors obtained from the SVD step or characteristic 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, and China between the dates 03/01/2012 and 01/03/2013. In order to train the forecasting models, rd approximately 23 of the observations (i.e., 192 observations) were used while approximately 1 rd of the observations (i.e., 100) were left aside for testing the forecasting accuracy of the 3 models. The following measures were adopted to compare and contrast between the forecasting models. 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 forecasting literature (see, for example, [2–6] and [24]). In order to save space, here we only provide the RMSE ratio of SSA to that of NN: 1/2 N 2 ( y − y ) T +i,i T +h,i i=1 SSA = RRMSE = 1/2 , N NN 2 ( y − y ) T +h,i T +h,i i=1 where, yT +h is the h-step ahead forecast obtained by SSA, yT +h is the h-step ahead forecast from the NN model, and N is the number of the forecasts. If RRMSE < 1, then the SSA ETS percent. outperforms NN by 1 − ARIMA Direction of Change (DC) DC is a measure of the percentage of forecasts that accurately predict the direction of change[4] . Here, the concept of DC is explained in brief. A detailed account can be found in [4]. In the univariate case, for forecasts obtained using XT , let DXi be equal to 1 if the forecast is able to correctly predict the actual direction of change and 0 otherwise. Then, X = n DXi /n shows the proportion of forecasts that correctly identify the direction of D i=1 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 been multiplied 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 three countries at all horizons. Instead, we see that for forecasting exchange rates in UK, the Basic VSSA model is able to outperform both NN and Basic RSSA at horizons of h = 1 and 3 steps ahead. For EU, we see that the NN model outperforms both Basic VSSA and Basic RSSA at forecasting the exchange rate at all horizons. The results for China show a completely different outlook as the NN model’s forecasting accuracy appears to have deteriorated to a great extent and both Basic VSSA and Basic RSSA models outperform the NN model. In China, for 1 day
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FORECASTING EXCHANGE RATES
ahead forecasts of exchange rates the Basic VSSA model outperforms Basic RSSA marginally whilst for 3 day ahead forecasts, the Basic RSSA model is seen to outperform the Basic VSSA model marginally. However, in order to evaluate the validity of these conclusions it is important to test the results for statistical significance. Here, we use the modified Diebold-Mariano test in [25] to test the RRMSE results. Table 1 Out-of-sample forecasting results for exchange rates NN Series
Basic VSSA
Basic RSSA
BasicVSSA NN
BasicVSSA BasicRSSA
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% better than 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 exchange rates, the Basic VSSA model is 68% better than NN. However, the results also indicate that for forecasting exchange rates in UK and EU there is no real difference between the accuracy of any of the models and that the conclusions derived earlier based on the RMSE alone are likely to be chance occurrences and not statistically valid. This further highlights the importance of testing statistical results for significance as otherwise we would be resorting to conclusions which are unlikely to be valid in the real world. Finally, we test the ability of the forecasting models at predicting the actual direction of change 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 Basic VSSA and Basic RSSA models. For UK, the Basic VSSA model provides the best DC prediction at h = 1 and 3 days ahead. For EU, Basic VSSA has the most favourable DC prediction at one 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 step ahead whilst Basic VSSA provides the best DC prediction at three steps ahead. Once again we test our results for statistical significance. Accordingly, Table 2 shows that all conclusions on DC which were made previously can be attributed to chance occurrences. However, we are able to conclude with 95% confidence that for the Chinese exchange rate forecast, the NN model provides the worst DC prediction at 3 steps ahead with only 48% accuracy.
BENEKI CHRISTINA · YARMOHAMMADI MASOUD
6
Table 2 Direction of change results for exchange rates NN Series
Basic VSSA
Basic RSSA
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.
Actual
NN forecast
NN forecast 6.30
Actual
VSSA forecast RSSA forecast
6.28
China: h=3 days ahead forecast
6.28 6.26
6.22
6.24 6.22
China: h=1 day ahead forecast
RSSA forecast
6.26
VSSA forecast
6.24
6.30
Figure 2 illustrates the forecasting results for the statistically significant outcomes which were visible in the results for China at h = 1 and 3 days ahead. The figure shows clearly that the two SSA models do indeed provide a better fit for the Chinese exchange rates forecasts and it also shows the accuracy of all three models deteriorating as the horizon increases from 1 step ahead to 3 steps ahead.
2012.2
2012.6 Year
2013.0
2012.2
2012.6
2013.0
Year
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 and VSSA models at forecasting exchange rates for UK, EU and China. Furthermore, we find no statistically significant evidence to outline either one of the forecasting models employed in this
FORECASTING EXCHANGE RATES
7
study to be the best for forecasting exchange rates in UK and EU. However, we are able to conclude with 95% confidence that for forecasting the Chinese exchange rate, the Basic VSSA and Basic RSSA models outperform the NN model at horizons of h = 1 and 3 days ahead. The study also finds with statistically significant evidence that the Basic RSSA model can achieve 60% accuracy in terms of DC prediction when forecasting the Chinese exchange rate at h = 1 step ahead, and also that the NN model provides the least favourable DC prediction for China at h = 3 steps ahead. The rest of the DC results were found to be attributable to chance occurrences. Future research should consider incorporating more forecasting techniques in order to evaluate whether statistically significant outcomes could then be achieved. It would also be interesting to see how a hybrid model combining Neural Networks and Singular Spectrum Analysis would fare at forecasting exchange rates for UK, EU and China in the future.
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