Selection of the Appropriate Lag Structure of Foreign Exchange Rates Forecasting Based on Autocorrelation Coefficient

Transcription

1 Selection of the Appropriate Lag Structure of Foreign Exchange Rates Forecasting Based on Autocorrelation Coefficient Wei Huang 1,2, Shouyang Wang 2, Hui Zhang 3,4, and Renbin Xiao 1 1 School of Management, Huazhong University of Science and Technology, Wuhan, , China rbxiao@163.com 2 Institute of Systems Science, Academy of Mathematics and Systems Sciences, Chinese Academy of Sciences, Beijing, , China {whuang, sywang}@amss.ac.cn 3 School of Knowledge Science, Japan Advanced Institute of Science and Technology, Asahidai 1-1, Tatsunouchi, Ishiawa, , Japan zhang-h@jaist.ac.jp 4 School of Computer Science, Southwest University of Science and Technology, Mianyang, Sichuan, , China Abstract. We propose a new criterion, called autocorrelation coefficient criterion (ACC) to select the appropriate lag structure of foreign exchange rates forecasting with neural networs, and design the corresponding algorithm. The criterion and algorithm are data-driven in that there is no prior assumption about the models for time series under study. We conduct the experiments to compare the prediction performance of the neural networs based on the different lag structures by using the different criterions. The experiment results show that ACC performs best in selecting the appropriate lag structure for foreign exchange rates forecasting with neural networs. 1 Introduction Financial time series forecasting is one of the most challenging problems among the many applications of neural networs. Several modeling issues have been discussed [1-7]. In many situations, we have to predict the future value of foreign exchange rates through past measurements of it in the following way: ˆ = F ( yt s, 1 y t + n yt s 2,, y, ) (1) t s i where y ˆ t + n is the predicted value when we mae a prediction of n periods ahead from the present period t ; y j is the actual value at period j ; s i is the lag period from the present period t ; F ( ) is a nonlinear function determined by the neural networs. The problem is to select the appropriate lag structure{ s 1, s2,..., si,... }. In previous wor, the criteria of selecting the lag structure include Aaie Information Criterion (AIC), Hannan-Quinn Criterion (HQC), Bayesian Information Criterion J. Wang et al. (Eds.): ISNN 2006, LNCS 3973, pp , Springer-Verlag Berlin Heidelberg 2006

2 Selection of the Appropriate Lag Structure of Foreign Exchange Rates Forecasting 513 (BIC), Schwarz Information Criterion (SIC), the general-to-specific sequential Lielihood Ratio test (LR) and the specific-to-general sequential Portmanteau test. For monthly VAR models, AIC produces the most accurate structural and semi-structural impulse response estimates for realistic sample sizes. For quarterly VAR models, HQC appears to be the most accurate criterion with the exception of sample sizes smaller than 120, for which SIC is more accurate. For persistence profiles based on quarterly vector error correction (VEC) models, the SIC is the most accurate criterion for all realistic sample sizes. Sequential Lagrange-multiplier and lielihood ratio tests cannot be recommended [8]. AIC, SIC, HQC have an important role to play in model selection for nonlinear threshold models, while Generalized Information criterion (GIC) and especially Informational Complexity Criterion (ICOMP) prove less reliable [9]. A class of Modified Information Criteria (MIC) was considered which account for the fact that the bias in the sum of the autoregressive coefficients is highly dependent on the lag order [10]. For symmetric lag and asymmetric lag models, the results are more ambiguous in terms of forecasting performance in comparisons of AIC, SIC, Posterior Information Criterion (PIC), and Keating s modification of the AIC and SIC (KAIC and KSIC, respectively) [11]. It is thus difficult to generalize about the preferred lag selection technique when there is uncertainty about the true lag length and whether the lags are symmetric or asymmetric. AIC and BIC as well as several extensions have been used as information-based in-sample model selection criteria in selecting neural networs for foreign exchange rates forecasting [12]. However, the in-sample model selection criteria are not able to provide a reliable guide to out-of-sample performance and there is no apparent connection between in-sample model fit and out-of-sample forecasting performance. A nonparametric version of the Final Prediction Error (FPE) was analyzed for lag selection in nonlinear autoregressive time series under relative general conditions including heterosedasticity [13]. There are assumptions that can t necessarily be satisfied in most cases. It requires suitable ernel and bandwidth choices to compute the lag selection criteria. Our contribution is to propose a new criterion of selecting the appropriate lag structure of foreign exchange rates forecasting with neural networs and design the corresponding algorithm. The remainder of this paper is organized as follows. Section 2 describes the new criterion and algorithm. In Section 3, we conduct the experiments to compare the prediction performance of the neural networs based on the different lag structures by using the different criterions. Finally, conclusions are given in Section 4. 2 Autocorrelation Coefficient Criterion and Algorithm In foreign exchange rates forecasting with neural networs, the contribution of one input variable to the output variable is affected by the other input variable. The input variables should be as predictive as possible. On the other hand, the input variables should not be much correlated, because the correlated input variables may degrade the prediction performance by interacting with each other and producing a biased effect [14]. Actually, the correlated input variables contribute the similar information for the output variable of neural networs. Therefore, the neural networs get confused and do not now to use which one. Considering the above features of foreign exchange rates forecasting with neural networs, we propose a new criterion of selecting the lag

3 514 W. Huang et al. structure of foreign exchange rates forecasting, called autocorrelation coefficient criterion (ACC) as follows: (1) The absolute value of autocorrelation coefficient between the lag period and the forecasting period ahead should be as large as possible. In this case, the input variable is more correlated to the output variable. Therefore, the input variable contributes more predictive information for the output variable. (2) The sum of absolute value of autocorrelation coefficients between the lag period and the other selected lag periods should be as small as possible. In this case, the input variable will not be much correlated to the other selected input variables. One of the main features of foreign exchange rates is that as the lag period becomes long, the absolute value of autocorrelation coefficient will becomes small. Accordingly, we design an algorithm to select the lag structure of foreign exchange rates forecasting with neural networs as follows: Step 1. Let i =1, s 1=0; and set the maximum lag period N, the forecasting period ahead n. Step 2. Let s i+ 1 = r + n arg s < N Max, and i = i + 1. i i r s j j = 1 Step 3. If s i less than N 1, go to Step 2; otherwise, exit and the appropriate lag structure is { s 1, s2,..., s i }. In order to further compare the effect of the different lag structures on the prediction performance, we add another two criterions of selecting the lag periods of foreign exchange rates forecasting, namely short criterion (SC) and long criterion (LC) as follows: (1) Short criterion (SC) If the lag structure based on ACC is { s 1, s2,..., s m }, the lag structure based on SC is { 0,1,..., m 1}. (2) Long criterion (LC) If the initial parameter maximum lag period of ACC is N, the lag structure 0,1,..., N. based on LC is { } 3 Experiments Analysis We conduct the experiments to compare the prediction performance of the neural networs based on the different lag structures by using the different criterions. 3.1 Neural Networ Model We employ the popular three layers bac-propagation networ with adaptive learning rate and momentum. The logistic function is used for all hidden nodes as the activation function. The linear activation function is employed for the output node. For time

4 Selection of the Appropriate Lag Structure of Foreign Exchange Rates Forecasting 515 series forecasting problem, the networ inputs are the past, lagged observations of the data and the output is the future value. Therefore, the number of input nodes corresponds to the number of past lagged data. Generally speaing, too many nodes in the hidden layer produce a networ that memorizes the input data and lacs the ability to generalize. Parsimony is a principle for designing neural networs. Hence, the number of hidden nodes is equal to the number of input nodes. 3.2 Performance Measure Normalized mean squared error (NMSE) is used to evaluate the prediction performance of neural networs. Given a set P comprising pairs of the actual value x and predicted value xˆ, the NMSE can be defined as follows: NMSE = P P ( x ( x xˆ ) x ) 2 2 (2) where x is the mean of actual values. 3.3 Data Preparation From Pacific Exchange Rate Service provided by Professor Werner Antweiler, University of British Columbia, Canada, we obtain 1005 daily observations covering the period from 2001 to 2004 for the three exchange rates respectively. For each exchange rate, we select the appropriate size of training set by using the method in [15]. The test set contains 209 daily observations covering the period from Jan 2005 to Oct, 2005 for the three exchange rates respectively. 3.4 Results We set the initial parameters of ACC: (1) the maximum lag period N = 10; (2) the forecasting period ahead n = 1. Table 1 shows the lag structures of the three exchange rates forecasting by using the different criterions. Table 2 shows the three exchange rates prediction performance of the neural networs based on the different lag structures by using the different criterions. The value of NMSE under ACC is the Table 1. The lag structures of the three exchange rates forecasting by using the different criterions (EUR, GBP and JPY against USD) Criterion of selecting lag structure Lag structure of EUR Lag structure of GBP Lag structure of JPY ACC {0, 4, 6, 9} {0, 4, 6, 9} {0, 4, 6, 9} AIC {0, 1, 6, 8} {0, 1, 6, 8} {0, 1, 6, 8} BIC {0} {0} {0} HQC {0, 1, 6, 8} {0, 1, 6, 8} {0, 1, 6, 8} SC {0, 1, 2, 3} {0, 1, 2, 3} {0, 1, 2, 3} LC {0, 1,, 10} {0, 1,, 10} {0, 1,, 10}

5 516 W. Huang et al. Table 2. The three exchange rates prediction performance of the neural networs based on the different lag structures by using the different criterions (EUR, GBP and JPY against USD) Criterion of selecting lag structure NMSE of EUR NMSE of GBP NMSE of JPY ACC AIC BIC HQC SC LC smallest among the different criterions in each exchange rates forecasting. It shows that ACC performs best in selecting the appropriate lag structure. ACC doesn t require any assumptions, completely independent of particular class of model. The criterion maes full uses of information among sample observations even if the underlying relationships are unnown or hard to describe. It integrated the features of autocorrelation coefficient of foreign exchange rates and the requirements of inputs for neural networs. 4 Conclusions In this paper, we propose a new criterion, called autocorrelation coefficient criterion (ACC) to select the lag structure of foreign exchange rates forecasting with neural networs, and design the corresponding algorithm. The criterion and algorithm are data-driven in that there is no prior assumption about the models for time series under study. We conduct the experiments to compare the prediction performance of the neural networs based on the different lag structures by using the different criterions. The experiment results show that ACC outperforms the other criterions in selecting the lag structure for foreign exchange rates forecasting with neural networs. Acnowledgements This wor is partially supported by National Natural Science Foundation of China (NSFC No ) and the Key Research Institute of Humanities and Social Sciences in Hubei Province-Research Center of Modern Information Management. References 1. Lai, K.K., Yu, L.A., Wang, S.Y.: A Neural Networ and Web-Based Decision Support System for Forex Forecasting and Trading. Lecture Notes in Artificial Intelligence, Vol Springer-Verlag Berlin Heidelberg (2004) Huang, W., Lai, K.K., Naamori, Y., Wang, S.Y.: Forecasting Foreign Exchange Rates with Artificial Neural Networs: a Review. International Journal of Information Technology & Decision Maing, 3(1) (2004)

6 Selection of the Appropriate Lag Structure of Foreign Exchange Rates Forecasting Zhang, H, Ho, T. B., Huang, W.: Blind Feature Extraction for Time-Series Classification Using Haar Wavelet Transform. Lecture Notes in Computer Science, Vol Springer- Verlag Berlin Heidelberg (2005) Yu, L.A., Wang, S.Y., Lai, K.K.: Adaptive Smoothing Neural Networs in Foreign Exchange Rate Forecasting. Lecture Notes in Computer Science, Vol Springer-Verlag Berlin Heidelberg (2005) Yu, L.A., Wang, S.Y., Lai, K.K.: A Novel Nonlinear Ensemble Forecasting Model Incorporating GLAR and ANN for Foreign Exchange Rates. Computers & Operations Research, 32 (2005) Yu, L.A., Lai, K.K., Wang, S.Y.: Double Robustness Analysis for Determining Optimal Feedforward Neural Networ Architecture. Lecture Notes in Computer Science, Vol Springer-Verlag Berlin Heidelberg (2005) Huang, W., Naamori, Y., Wang, S.Y.: Forecasting Stoc Maret Movement Direction with Support Vector Machine. Computers & Operations Research, 32 (2005) Ivanov, V., Kilian, L.: A Practitioner s Guide to Lag-Order Selection for Vector Autoregressions. Woring paper, Centre for Economic Policy Research, (2000) Kapetanios, G.: Model Selection in Threshold Models. Journal of Time Series Analysis, 22 (2001) Ng, S., Perron, P.: Lag Length Selection and the Construction of Unit Root Tests with Good Size and Power. Boston College Woring Papers in Economics 369, Boston College Department of Economics (2000) 11. Ozcice, O., Mcmillian, W.D.: Lag Length Selection in Vector Autoregressive Models: Symmetric and Asymmetric Lags. Applied Economics, 31 (1999) Qi, M., Zhang, G.P.: An Investigation of Model Selection Criteria for Neural Networ Time Series Forecasting. European Journal of Operational Research, 132 (2001) Tschernig, R., Yang, L.: Nonparametric Lag Selection for Time Series. Journal of Time Series Analysism, 21 (2000) Zhang, G.P.: Neural Networs in Business Forecasting. Idea Group Inc., (2003) 15. Huang, W., Naamori, Y., Wang, S.Y., Zhang, H.: Select the Size of Training Set for Financial Forecasting with Neural Networs. Lecture Notes in Computer Science, Vol Springer-Verlag Berlin Heidelberg (2005)