A HYBRID EMD-MA FOR FORECASTING STOCK MARKET INDEX

Transcription

1 ITALIAN JOURNAL OF PURE AND APPLIED MATHEMATICS N ( ) 313 A HYBRID EMD-MA FOR FORECASTING STOCK MARKET INDEX Ahmad M. Awajan Mohd Tahir Ismail School of Mathematical Sciences University Science Malaysia awajanmath@yahoo.com mtahir@cs.usm.my S. Al Wadi Department of Risk Management and Insurance University of Jordan Jordan sadam alwadi@yahoo.co.uk Abstract. Nowadays, stock market data forecasting has drawn a high attention in the field of nonstationary and nonlinear time series data with a high heteroscedasticity, since improving the forecasting accuracy is a hot topic for the researchers. Therefore, in this article the authors are proposed a new methodology via combining Empirical Mode decomposition and Moving Average model as a modified method to improve forecasting accuracy in content of stock market data. The strength of this proposed methodology lies in its ability to forecast nonlinear and non-stationary financial data without a need to use any transformation method. Moreover, this method provides a better model with sufficient forecasting accuracy. The daily stock market data of fourteen countries is applied to show the forecasting performance of the proposed method. Based on the five forecast accuracy measures, the results indicate that proposed forecasting method performance is superior to four selected forecasting techniques. Keywords: Stock market index forecasting, Nonlinear and non-stationary time series, Empirical mode decomposition, Combined forecasting Model, Heteroscedasticity time series. In financial time series analysis, one of the primary issues is modeling and forecasting financial time series data specifically stock market index. Usually, the transformation of a financial time series, rather than its original scale, is taken for describing its dynamics. Proper transformation is necessary to convert original non-stationary processes to stationary processes and subsequently to utilize mathematical and statistical properties for stationary processes. The hybrid models combine strengths of few traditional models to get a better forecasting accuracy. Recently, several hybrid models were applied EMD in the literature for time series forecasting. That by using EMD to decompose the non-stationary and non-linear time series data into Intrinsic Mode Functions (IMFs) and residual components. And then use forecasting model to forecast. Corresponding author

2 AHMAD M. AWAJAN, MOHD TAHIR ISMAIL, S. AL WADI 314 each component. Then all these forecasted values were aggregated to produce the final forecasted value of the original time series. Such as in [1] used a hybrid EMD-ARIMA (Autoregressive integrated moving average) to forecasting the monthly prices of rice data. Also, [14] also used the same methodology, but with wind speed data. A hybrid EMD-AR (Autoregressive) model was developed by coupling an AR model with the EMD technique in [8]. A hybrid EMD-LSSVR (least squares support vector regression) forecasting models has been applied on foreign exchange rate in [15]. While in [22] used a hybrid of EMD, LS-SVM (Least Squares-Support Vector Machines) and AR model with Kalman filter to predict wind speed data. Therefore, the significant of this research article can be summarize as after intensive research in the financial forecasting literature, there are plenty research papers have been conducted in forecasting in content of stock market data such as [3], [2] and [12], also most of the articles have used the mentioned models directly without any combination such as [25]. With regard to all those literature reviews, this study attempts to employ the proposed method to forecast the daily stock market data of fourteen countries. Four selected forecasting models are used in the proposed method comparison to assess its performance of forecasting. Experimental results show that the proposed method is superior to existing method in terms of five accuracy forecasting measure. Section 2 introduces methods are used in methodology in this paper which are EMD, IMF and Moving Average Model. In this section introduces statistical techniques for consideration method. Section 3 presented the proposed methodology. Section 4 analyzes the daily stock market time series data of four countries with a discussion the result showing the capability of proposed forecasting method. Finally, in Section 5 some concluding remarks are addressed. 1. Methodology In this section, the various steps for the implementation of the proposed forecasting method are described in detail. Which are Empirical Mode Decomposition, Moving Average Model and statistical techniques for consideration method. 1.1 Empirical mode decomposition (EMD) EMD was described by [10], and this method has been modified by [16] and [13].The main idea of EM D is the decomposing of nonlinear and non-stationary time series data into several of simple time series. And it analyzing time series with keeping the time domain of the signal. It supplies an strong and adaptive process to decompose a time series into a combination of time series that known as intrinsic mode functions (IM F ) and residual. Later, the original signal can

3 A HYBRID EMD-MA FOR FORECASTING STOCK MARKET INDEX 315 be constructed back as the following: n (1.1) x(t) = IMF i (t) + r(t) i=1 where x(t) represents the original time series, r(t) represents the residue of the original time series data decomposition and IMF i represent the i th intrinsic mode function (IMF ) series. In order to estimate these IMF s, the following steps should be initiated and the process is called the sifting process of time series x(t) [20] are shown below: 1. Start the first step by taking the original time series x(t) for sifting process and assuming the iteration index value is i = Then, evaluate all of local extrema values of the time series x(t). 3. After that, form the local maxima (local upper) envelope function u(t) by connecting all local maxima values using a cubic spline line. In a similar way, form the local minimum (local lower) envelope function l(t), and then form the mean function m(t) by using this following u(t) + l(t) (1.2) m(t) = 2 4. Next, define a new function h(t) using the mean envelope m(t) and the signal x(t) on this formula (1.3) h(t) = x(t) m(t) Check the function h(t) is an IMF, according to IMF conditions (shown in the second part of this section). If the function h(t) has satisfied IM F conditions, then go to step 5. If not, go back to step 2 and renew the value of x(t) such that became h(t), repeat steps 2 again until In step 5, firstly save the result of the IMF obtain from the last step. Secondly, renew the iteration index value such that became i = i + 1. Thirdly attain the residue function r(t) using the IM F and the signal x(t) on the formula (1.4) IMF i (t) = h(t) r i+1 (t) = x(t) IMF i (t). 6. Finally, make a decision whether the residue function r(t) that acquire from step 5 is a monotonic or constant function. Then, save the residue and all the IMF s obtained. If the residue is not monotonic or constant function, return to step 2. The steps 1 to 6 which were discussed above allow the sifting process (EMD algorithm) to separate time-altering signal features.

4 AHMAD M. AWAJAN, MOHD TAHIR ISMAIL, S. AL WADI Intrinsic Mode Function (IMF) Based on the EMD algorithm presented in the previous section, the IMF produces by the sifting process need to satisfy two conditions [20] which are (1.5) N um[extreme] N um[cross zero] <1. where Num.extreme represents the number of local extreme points (all local maxima and all local minima), also Num[cross-zero] represent the number of cross-zero points (1.6) m(t) = u(t) + l(t) <ε, 2 where u(t) represents the envelope function generated by using cubic spline line on all local maxima, l(t) represents the envelope function generated by using a cube spline line on all local minima, m(t) represents the mean function that it was obtained by evaluating the mean of u(t) and l(t), and ε is a very small positive number that close to zero, sometime equal zero. 1.3 Moving Average Model (MA) A moving-average model is a model of linear regression uses past errors of the time series to describe the present observation. The moving average model of order q is denoted by MA(q). Mathematically; MA(q) of time series X t is formally given by [7]: (1.7) X t = µ + ε t + q θ j ε t j = µ + ε t + θ 1 ε t θ q ε t q j=1 where µ is the mean of the series, and θ 1,..., θ q are the parameters may be positive or negative. And the ε t,ε t 1,..., ε t q, are white noise error terms. The value of q is called the order of the MA model. This can be written using the backshift operator B as (1.8) X t = µ + (1 + θ 1 B + + θ q B q )ε t. The maximum of order q is selected by using the autocorrelation function(acf ) (see [23]). And the parameters θ 1,..., θ q are selected by using the Corrected Akaike s information criterion (AICc) by [11]. 1.4 Statistical techniques for consideration method In this study the proposed forecasting method is compered with four methods. Moving Average model, Holt-winter model was presented in [9] and [24], ARIMA models were presented in [18] and Random Walk method are used in order to validate the forecasting performance of proposed forecasting model.

5 A HYBRID EMD-MA FOR FORECASTING STOCK MARKET INDEX 317 These statistical methods were selected based on their performance in forecasting competitions and other empirical applications, as well as on their ability to capture salient features of the data. ARIMA (autoregressive integrated moving average) or Box-Jenkins models are generally denoted ARIMA(p, d, q). Where parameters p, d, and q are non-negative integers, p is the order of the Autoregressive model, d is the degree of differencing, and q is the order of the Moving-average model. Recently ARIMA is employed in several of studies in forecasting financial time series such as [4]. He applied ARIMA to forecast the cultivated area and production of maize in Nigeria. A random walk (RW) is a process where the current value of a variable is composed of the past value with adding an error term defined as a white noise. It was first studied several hundred years ago as models for games of chance. Recently, In [17] have been shown that the random walk model turns out to be a hard to beat benchmark in forecasting the CEE exchange rates. And in [21] was applied a variable drift term with the random walk process. This was estimated using a Kalman filter. This simple statistical process was shown to perform better than all the three models that he was selected in out of-sample forecasts. 2. Propose methodology and data In this section, the various steps for the implementation of the proposed forecasting method are described in detail. Which are Empirical Mode Decomposition and Moving Average Model. 2.1 Data In this study, nonlinear and non-stationary time series data from the daily stock market of fourteen countries are used. These countries are Australia, Denmark, Estonia, Finland, France, India, Lithuania, Malaysia, Netherlands, Norwegian, Slovenia, Switzerland, Thailand and UK. While Figure 4 shows the time series plot of these countries. Table 1 presents these countries with the Basic statistics for each country, where S.Deviation is Standard Deviation, Nimf the number of IMF s and N is Number of observations. Moreover, Table 1 presents the p-value of KPSS (Kwiatkowski-Phillips-Schmidt-Shin by [5]), RESET (Ramsey Regression Equation Specification Error Test by [19]), BP (Breusch-Pagan test by [6] for nonstationary, nonlinear and heteroscedasticity, respectively. According to this value all stock market are significantly nonlinear, nonstationary with high heteroscedasticity. The data are extracted from the Yahoo finance website. The daily closing prices are used as a general measure of the stock market over the past six years. The whole data set - for each country - covers the period from 9 February 2010 to 7 January The data set is divided into two parts. The first part (m observations) is used to determine the specifications of the models and parameters. The second part, on the other hand, (h observations)

6 AHMAD M. AWAJAN, MOHD TAHIR ISMAIL, S. AL WADI 318 is reserved for out-of-sample evaluation. This part is used comparison of performances among various forecasting models. Malaysia stock market data are taken as example. Where the number observation is N= 1459, the first part is m = 1458, 1457, 1456, 1455, 1494 and 1493 and the second part is h=1, 2, 3, 4, 5 and 6 respectively, are used. 2.2 Propose Methodology The proposed methodology consists of four stages. Before began the proposed method, the time series data were divided into two parts. The first part is the training part to train the proposed method. This part is used to modeling to get the forecasting value of time series. The second part is the testing part. This part is used to compering with the forecasting value. After that, the time series are ready to follow this stage:- 1. Firstly, the use of empirical mode decomposition (EM D) on the time series. In this stage, the several Intrinsic Mode Functions (IM F s) and residue are obtained. 2. Secondly, the KPSS test [5] is applied on each of IMF s and residue to select the order of difference (d). While each component has own d. ACF is applied on all component to select the order of moving average model (q). If q of time series goes to infinity, the d is exceeded until q be finite. After that, the θ 1,..., θ q are selected by using the Corrected Akaike s information criterion (AICc) for each component. 3. Thirdly, by using the result p, q and θ 1,..., θ q - were found in the last stage - each component is modeling by MA(q) to forecasting h days ahead. 4. Finally, in the this stage all the forecasting results for IMF s and residue are added up to get the forecasting for the time series. After that the forecasting results are compared with the forecasting result of random-walk with draft, Holt-winter, ARIMA and moving average model without EM D. Figure 2 summarizes all the proposed methodology steps. 3. Result and discussion In this study, stock market data of fourteen countries are used to present the forecasting accuracy of the proposed forecasting method. Four Models are used in order to validate the forecasting performance of proposed forecasting method. Root Mean Squared Error (RMSE), Mean Absolute Error (MAE), Mean Absolute Percentage Error (MAPE), Mean Absolute Scaled Error(MASE) and Theil s U-statistic (TheilU) will be utilized to evaluate the forecasting accuracy for each method. Equations 3.1, 3.2, 3.3, 3.4 and 3.5 are showed the formula of RMSE,

7 A HYBRID EMD-MA FOR FORECASTING STOCK MARKET INDEX 319 MAE, MAPE, MASE and TheilU respectively. Where ŷ i is the forecast value of the variable y at time period i from knowledge of the actual series values. (3.1) (3.2) (3.3) (3.4) (3.5) RMSE = 1 n MAE = 1 n MAP E = 1 n MASE = 1 n T heilu = n (y i ŷ i ) 2, i=1 n y i ŷ i, i=1 n y i ŷ i.100%, y i ( n y i ŷ i i=1 i=1 n 1 i=1 n 1 i=1 1 n n 1 i=2 y i y i 1 ( ) ŷi+1 y 2 i+1 y i ( yi+1 y i ) 2. y i ), Table 2, 3, 4, 5, 6, 7 and Table 8 presented the comparison results of applied proposed forecasting method with four forecasting methods on Australia with Denmark, Estonia with Finland, France with India, Lithuania with Malaysia, Netherlands with Norwegian, Slovenia with Switzerland and Thailand with UK, respectively. When each table presents the result of five measure error for forecasting at h =1, 2, 3, 4, 5 and 6 days ahead. When the four forecasting methods are used to compare with proposed forecasting method. These methods are Holt-winter,moving average, ARIMA and random-walk with draft. The five error measures of RMSE, MAE, MAPE, MASE and TheilU are used to find the forecasting accuracy. The results obtained indicates that the forecast accuracy for proposed forecasting method (denoted by EMD-MA) is generally better than the four selected forecasting models. 4. Conclusion In this article, the forecasting accuracy has improved in financial time series area since the forecasting accuracy still remains as one of the most difficult area due to the non-stationary and non-linear data. In this study, a new hybrid method has composited which are EMD and MA for modeling and improving forecasting accuracy in content of stock market data for fourteen countries based on several comparison forecast accuracy measurements. Then the findings indicate that proposed is able to outperform the four forecasting models. Thus, this paper has strengthened the idea that proposed forecasting method is suitable for stock market data. These results which will be useful to predict the future event and some structure breaks event.

8 AHMAD M. AWAJAN, MOHD TAHIR ISMAIL, S. AL WADI 320 Table 1: Basic statistics of time series Mean Median SD Skewness Kurtosis N. N KPSS RESET BP Country IMF p.value p.value p.value Australia <.01 <.01 <.01 Denmark <.01 <.01 <.01 Estonia <.01 <.01 <.01 Finland <.01 <.01 <.01 France <.01 < India <.01 <.01 <.01 Lithuania <.01 <.01 <.01 Malaysia <.01 <.01 <.01 Netherlands <.01 <.01 <.01 Norwegian <.01 <.01 <.01 Slovenia <.01 <.01 <.01 Switzerland <.01 <.01 <.01 Thailand <.01 <.01 <.01 UK <.01 <.01 <.01

9 A HYBRID EMD-MA FOR FORECASTING STOCK MARKET INDEX 321 Figure 1: Time series plot

10 AHMAD M. AWAJAN, MOHD TAHIR ISMAIL, S. AL WADI 322 Figure 2: Flowchart of empirical mode decomposition with moving average estimation process

11 A HYBRID EMD-MA FOR FORECASTING STOCK MARKET INDEX 323 Table 2: Presented the RMSE, MAE, MASE, MAPE and TheilU of proposed method and four forecasting methods for forecasting at h = 1, 2, 3, 4, 5 and 6 for stock market data of Australia and Denmark. Countries Method h=1 h=2 h=3 h=4 h=5 h=6 Australia Holt-Winter MA MAE ARIMA R.Walk EMD-MA Holt-Winter MA RMSE ARIMA R.Walk EMD-MA Holt-Winter MA MAPE ARIMA R.Walk EMD-MA Holt-Winter MA TheilU ARIMA R.Walk EMD-MA Holt-Winter MA MASE ARIMA R.Walk EMD-MA Denmark Holt-Winter MA MAE ARIMA R.Walk EMD-MA Holt-Winter MA RMSE ARIMA R.Walk EMD-MA Holt-Winter MA MAPE ARIMA R.Walk EMD-MA Holt-Winter MA TheilU ARIMA R.Walk EMD-MA Holt-Winter MA MASE ARIMA R.Walk EMD-MA

12 AHMAD M. AWAJAN, MOHD TAHIR ISMAIL, S. AL WADI 324 Table 3: Presented the RMSE, MAE, MASE, MAPE and TheilU of proposed method and four forecasting methods for forecasting at h = 1, 2, 3, 4, 5 and 6 for stock market data of Estonia and Finland. Countries Method h=1 h=2 h=3 h=4 h=5 h=6 Estonia Holt-Winter MA MAE ARIMA R.Walk EMD-MA Holt-Winter MA RMSE ARIMA R.Walk EMD-MA Holt-Winter MA MAPE ARIMA R.Walk EMD-MA Holt-Winter MA TheilU ARIMA R.Walk EMD-MA Holt-Winter MA MASE ARIMA R.Walk EMD-MA Finland Holt-Winter MA MAE ARIMA R.Walk EMD-MA Holt-Winter MA RMSE ARIMA R.Walk EMD-MA Holt-Winter MA MAPE ARIMA R.Walk EMD-MA Holt-Winter MA TheilU ARIMA R.Walk EMD-MA Holt-Winter MA MASE ARIMA R.Walk EMD-MA

13 A HYBRID EMD-MA FOR FORECASTING STOCK MARKET INDEX 325 Table 4: Presented the RMSE, MAE, MASE, MAPE and TheilU of proposed method and four forecasting methods for forecasting at h = 1, 2, 3, 4, 5 and 6 for stock market data of France and India. Countries Method h=1 h=2 h=3 h=4 h=5 h=6 France Holt-Winter MA MAE ARIMA R.Walk EMD-MA Holt-Winter MA RMSE ARIMA R.Walk EMD-MA Holt-Winter MA MAPE ARIMA R.Walk EMD-MA Holt-Winter MA TheilU ARIMA R.Walk EMD-MA Holt-Winter MA MASE ARIMA R.Walk EMD-MA India Holt-Winter MA MAE ARIMA R.Walk EMD-MA Holt-Winter MA RMSE ARIMA R.Walk EMD-MA Holt-Winter MA MAPE ARIMA R.Walk EMD-MA HW MA TheilU ARIMA R.Walk EMD-MA Holt-Winter MA MASE ARIMA R.Walk EMD-MA

14 AHMAD M. AWAJAN, MOHD TAHIR ISMAIL, S. AL WADI 326 Table 5: Presented the RMSE, MAE, MASE, MAPE and TheilU of proposed method and four forecasting methods for forecasting at h = 1, 2, 3, 4, 5 and 6 for stock market data of Lithuania and Malaysia. Countries Method h=1 h=2 h=3 h=4 h=5 h=6 Lithuania Holt-Winter MA MAE ARIMA R.Walk EMD-MA Holt-Winter MA RMSE ARIMA R.Walk EMD-MA Holt-Winter MA MAPE ARIMA R.Walk EMD-MA Holt-Winter MA TheilU ARIMA R.Walk EMD-MA Holt-Winter MA MASE ARIMA R.Walk EMD-MA Malaysia Holt-Winter MA MAE ARIMA R.Walk EMD-MA Holt-Winter MA RMSE ARIMA R.Walk EMD-MA Holt-Winter MA MAPE ARIMA R.Walk EMD-MA Holt-Winter MA TheilU ARIMA R.Walk EMD-MA Holt-Winter MA MASE ARIMA R.Walk EMD-MA

15 A HYBRID EMD-MA FOR FORECASTING STOCK MARKET INDEX 327 Table 6: Presented the RMSE, MAE, MASE, MAPE and TheilU of proposed method and four forecasting methods for forecasting at h = 1, 2, 3, 4, 5 and 6 for stock market data of Netherlands and Norwegian. Countries Method h=1 h=2 h=3 h=4 h=5 h=6 Netherlands Holt-Winter MA MAE ARIMA RW EMD-MA Holt-Winter MA RMSE ARIMA RW EMD-MA Holt-Winter MA MAPE ARIMA RW EMD-MA Holt-Winter MA TheilU ARIMA RW EMD-MA Holt-Winter MA MASE ARIMA RW EMD-MA Norwegian Holt-Winter MA MAE ARIMA R.Walk EMD-MA Holt-Winter MA RMSE ARIMA R.Walk EMD-MA Holt-Winter MA MAPE ARIMA R.Walk EMD-MA Holt-Winter MA TheilU ARIMA R.Walk EMD-MA Holt-Winter MA MASE ARIMA R.Walk EMD-MA

16 AHMAD M. AWAJAN, MOHD TAHIR ISMAIL, S. AL WADI 328 Table 7: Presented the RMSE, MAE, MASE, MAPE and TheilU of proposed method and four forecasting methods for forecasting at h = 1, 2, 3, 4, 5 and 6 for stock market data of Slovenia and Switzerland. Countries Method h=1 h=2 h=3 h=4 h=5 h=6 Slovenia Holt-Winter MA MAE ARIMA R.Walk EMD-MA Holt-Winter MA RMSE ARIMA R.Walk EMD-MA Holt-Winter MA MAPE ARIMA R.Walk EMD-MA Holt-Winter MA TheilU ARIMA R.Walk EMD-MA Holt-Winter MA MASE ARIMA R.Walk EMD-MA Switzerland Holt-Winter MA MAE ARIMA R.Walk EMD-MA Holt-Winter MA RMSE ARIMA R.Walk EMD-MA Holt-Winter MA MAPE ARIMA R.Walk EMD-MA Holt-Winter MA TheilU ARIMA R.Walk EMD-MA Holt-Winter MA MASE ARIMA R.Walk EMD-MA

17 A HYBRID EMD-MA FOR FORECASTING STOCK MARKET INDEX 329 Table 8: Presented the RMSE, MAE, MASE, MAPE and TheilU of proposed method and four forecasting methods for forecasting at h = 1, 2, 3, 4, 5 and 6 for stock market data of Thailand and UK. Countries Method h=1 h=2 h=3 h=4 h=5 h=6 Thailand Holt-Winter MA MAE ARIMA R.Walk EMD-MA Holt-Winter MA RMSE ARIMA R.Walk EMD-MA Holt-Winter MA MAPE ARIMA R.Walk EMD-MA Holt-Winter MA TheilU ARIMA R.Walk EMD-MA Holt-Winter MA MASE ARIMA R.Walk EMD-MA UK Holt-Winter MA MAE ARIMA R.Walk EMD-MA Holt-Winter MA RMSE ARIMA R.Walk EMD-MA Holt-Winter MA MAPE ARIMA R.Walk EMD-MA Holt-Winter MA TheilU ARIMA R.Walk EMD-MA Holt-Winter MA MASE ARIMA R.Walk EMD-MA

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