Modern Economy, 2011, 2, 823829 doi:10.4236/me.2011.25091 Published Online November 2011 (http://www.SciRP.org/journal/me) Copyright © 2011 SciRes. ME 823 Accurately Forecasting Model for the Stochastic Volatility Data in Tourism Demand YaLing Huang1, YenHsien Lee2 1Department of Go ldenAger Industry Management, Chaoyang University of Technology, Taiwan, China 2Department of Fin anc e, Chung Yuan Christian University, Taiwan, China Email: ylhuang@cyut.edu.tw, yh@cycu.edu.tw Received April 15, 2011; revised June 6, 2011; accepted July 7, 2011 Abstract This study attempts to enhance the effectiveness of stochastic volatility data. This work presents an empirical case involving the forecasting of tourism demand to demonstrate the efficacy of the accuracy forecasting model. Work combining the grey forecasting model (GM) and Fourier residual modification model to refine the forecasting effectiveness for the stochastic volatility data, which can estimate fluctuations in historical time series. This study makes the following contributions: 1) combining the grey forecasting and Fourier re sidual modification models to refine the forecasting effectiveness for the stochastic volatility data, 2) pro viding an effective method for forecasting the number of international visitors to Taiwan, 3) improving the accuracy of shortterm forecasting in cases involving sample data with significant fluctuations. Keywords: Fourier Residual Modification Grey Forecasting Model (FGM), Grey Forecasting Model (GM), Stochastic Volatility, Tourism Demand 1. Introduction International tourism has increased rapidly during the past two decades, and has strong implications for deci sionmakers in government and business. Tourism de mand forecasting has been a key issue in national tour ism industry development and has attracted increased attention during the past decade. In relation to govern ment agencies, accurate forecasts of tourism flows can assist in forecasting in various areas relevant to policy making, including price regulation, environmental qual ity control, and infrastructure provision. For tourism businesses attempting maximizing their profits, accurate forecasts can avoid the financial costs associated with excess capacity and the opportunity costs associated with unfilled demand. Accurate forecasting of tourism de mand is important in tourism planning by both the public and business sectors owing to the perishable nature of tourism products. This shows that the forecasting meth odology is most relevant and most easily applicable in management, and that demand forecasting is the key ob jective of management [13]. Shortterm trends and fluc tuating demand can significantly influence decision making regarding purchasing, inventory and logistics [4]. Accurate forecasting enables companies to control chan geable markets, reduce inventory costs, improve cus tomer service and enhance their competitiveness. Along with the development of forecasting techniques, numerous quantitative methods have been applied to forecast tourism demand. Before the 1990s, traditional regression approaches dominated the tourism forecasting literature, but this situation changed after the mid1990s as more researchers adopted modern econometric tech niques, for example cointegration and error correction models, for modeling and forecasting tourism demand [57]. Additionally, previous studies have mainly fo cused on traditional econometric analysis, with the re gression model [810], Timeseries model [1113], and an ARIMA model providing one example [11,14,15]. These models are highly effective for forecasting long term tourism demand, but perform poorly in forecasting shortterm and fluctuating data sets [16]. The original Fourier model was developed by Brigham [17] and this work extends the Fourier model developed by Hsu [18] and Lin & Lee [16] using the Fourier series model to enhance the forecasting accuracy. Fourier series can yield accurate estimates when the sample data are significantly fluctuating [19]. Moreover, Lee et al., [20] noted that some choicebased diffusion models suffer limitations in forecasting demand for new technologies
Y.L. HUANG ET AL. 824 because such models generally depend on historical sales data or adopter data, and thus it is difficult to explain the diffusion of newly introduced technology based on a limited number of data observations is difficult. Regard ing the assessment of forecasts of tourism demand, most studies have focused on forecasting longterm tourism demand, and the problem of accurately estimating de mand when the sample data is highly variable has been largely ignored. In practice, changes in tourism demand are especially concerning to tourism businesses, particu larly in relation to shortterm predictions, because in an environment as competitive as the tourism industry, business strategies must be require frequent adjustment in response to dynamic changes in demand. The need to accurately forecast international tourist arrivals is an essential strategic requirement in the tour ism industry, especially for host countries that invest heavily in promotional strategies and are motivated by the economic benefits brought by inbound visitors. In this work, the forecasting method mainly considers shortterm tourism demand. This study applied the Fou rier residual modification Grey forecasting model (FGM (1, 1)) to forecast overseas passenger numbers. The grey forecasting theory focuses on the system model under uncertainty and information integrality, and can make forecasts based on small quantities of data. Tourism de mand is further estimated using Fourier series. The FGM (1, 1) also provides an effective method of dealing pre dictably with fluctuating data sets. FGM (1, 1) thus pro vides a more effective forecasting method for solving the problem, namely the difficulty of estimating seasonal variation in international traveler numbers. To summarize, this work develops a model that can improve the accuracy of forecasts of fluctuating tourism demand. This study applied the FGM (1, 1) approach to enhance the ability to forecast international tourist num bers to Taiwan. For this study, the rest of the paper is organized as follows. Section 2 introduces methodology. Section 3 describes data source and analyzes the empiri cal findings. Finally, the conclusions are presented in Section 4. 2. Methodology 2.1. Grey Forecasting Model (GM) Grey system theory was developed by Deng [21]. The Grey forecasting model (GM) represents the core of Grey system theory which treats all variables as Grey quantities within a certain range. GM then gathers avail able data to determine the internal regularity. In manag ing the disorganized primitive data the model then ex amines the nature of the internal regularity. The model was devised by transforming the arranged sequence into a differential equation. The algorithm of GM (1, 1) is described as follows [22,23]. Assuming the raw data series to be 00 0 (1), (2),xx x 0 ,() n and GM involves the following steps: Step 1. Creating a sequence of firstorder accumu lated generating operation (AGO) 1 is defined as ’s firstorder AGO sequence. That is, x0 x 111 1 12 00 0 11 1 (1),(2),,( ) (),(),,() n kk k xxx xn kxk xk (1) Step 2. Defining parame ters a and b The firstorder differential equation of GM (1, 1) mo del is 1 1 d() () d xt ax tb t (2) where t denotes the independent variables in the system, a represents the developed coefficient, b is the Grey con trolled variable, and a and b denote the model parameters requiring determination. Step 3. Calculating the values of a and b The values of a and b becomes by the ordinal least square method (OLS) as 1 TT aBB BY b (3) Furthermore, accumulated matrix B is 11 11 11 0.5 (1)(2)1 0.5 (2)(3)1 0.51( )1 xx xx B xn xn (4) Meanwhile, the constant vector YN is 00 0 (2),(3),,() T N Yxx xn (5) Step 4. Defi n ing the predi c ti o n model The approximate relationship can be obtained as fol lows by substituting a obtained in the differential equa tion, and solving raw data series. 10 ˆ1(1) ak a xk xe ba b (6) Step 5. Obtaining the forecasting values based on IAGO When 10 ˆˆ (1) (1) xx , the sequence oneorder inverse accumulated generating operation (IAGO) of reduction is obtained as 01 ˆˆˆ 11 1 () kxkx k . Given 1,k the sequence of reduction is obtained as fol 2,, n Copyright © 2011 SciRes. ME
825 Y.L. HUANG ET AL. lows: 00 00 ˆˆ ˆˆ (2), (3),, () xx xn (7) 2.2. Fourier Residual Modification Grey Forecasting Model (FGM) The application of physical system is subject to influence by causes its stability in the real world is not examined [24]. The grey model exhibited variable speed. The grey model develops with corresponding regular exponents during different periods and the forecasting accuracy is low. Therefore, GM may be unable to forecasting the instable variance. The Fourier residual modification Grey forecasting model (FGM (1, 1)) was proposed by Tan & Chang [24]. The method of GM residual correction adopted by AGO and the regularity of the origin series reduces the forecasting accuracy. The accuracy of predictions made using GM (1, 1), this study applied the FGM (1, 1) approach to increase its prediction capabilities. Figure 1 shows the process of FGM (1, 1), while the following illustration details the method used to establish the steps of FGM for the resid ual correction, which are detailed as follows [16,19,24]. Step 1. Obtain the residual series from operating GM (1, 1) When, the sequence oneorder inverseaccumulated generating operation (IAGO) of reduction is obtained in the form of Equation (7) and the residual series is de fined as: 00 00 (2), (3),, ()T n (8) where 000 ˆ ()() ()kxkxk , . (9) 2, ,kn Step 2. Define FGM (1, 1) The continuous residual series can be modeled using Fourier series as: 00 1 12π2π ˆ()cos sin 2 a k ii iaa ii ka akbk TT (10) In Equation (10), a and indicates the pe riod of the residual series, . And the continu ous residual series can be modeled by Fourier series as: 0ka T 1 nTa 00 1 2π ˆ()cos sin 2 for 2, a k i iaa ii ka abk TT 12π 3,,. i k kn , (11) Equation (11) can be estimated as: 0 PC (12) where 1 0 TT CPPP (13) 01122 ,,, ,, ,, aa T kk Caababab (14) in Equation (15), (1)2 a kn 1 and means the minimoum deployment frequency of Fourier series. Step 3. Correct original prediction series The modeled residual series can be modeled as: 00 00 (2), (3), , ()T n (16) Finally, the original prediction series of FGM (a ) can be corrected as: 000 ˆ ˆˆ ()() () a kxk k , . (17) 2,3, ,kn 2.3. Evaluate Accuracy of Prediction Following generating and developing the above model, further tests are necessary to clarify the forecast and ac tual values. To demonstrate the efficiency of the pro posed forecasting model, this study adopts the residual error test method to compare the actual and forecast val ues. Herein, Equations (18) and (19) are used to calculate the residual and average residual error of Grey forecast ing. Error = 000 ˆ ()()() kxkxk, (18) 2k Average error = 000 1 ˆ ()() () n k kxkxkn (19) 2π 12π12π1 cos2 sin2sin2 2 2π 12π12π1 cos3 sin3sin3 2 2π 12π12π1 cos sinsin 2 a aa a a aa a a aa a k TT T k pTT T k nn TT T n (15) Copyright © 2011 SciRes. ME
Y.L. HUANG ET AL. Copyright © 2011 SciRes. ME 826 Figure 1. FGM (1, 1) procedure for forecasting tourist demand. Table 1. The number of tourist visit to Taiwan from 2006 to 2007. Table 2. Residual value of tourist visit to Taiwan from 2006 to 2007 by GM. Year Month Asia AmericaEurope OceaniaAfrica 2006 Sep. 164,980 33,479 16,911 4915 700 Oct. 181,050 43,770 21,610 5500 880 Nov. 193,510 44,260 22,450 5890 710 Dec. 190,390 47,950 17,170 6670 670 2007 Jan. 173,730 36,280 17,370 5970 590 Feb. 147,280 37,920 16,090 4750 760 Mar. 210,610 45,880 23,910 6130 880 Apr. 175,030 42,610 21,200 6070 940 May. 171,540 39,770 17,820 5200 570 Jun. 175,540 48,200 21,290 6370 830 Jul. 148,140 46,350 20,650 6380 670 Aug. 171,200 40,850 19,870 5900 820 Sep. 177,530 35,334 18,166 6353 923 Oct. 181,170 46,233 23,920 6751 706 YearMonthAsia America Europe OceaniaAfrica 2006Oct.–2917 594 2080 –154 138 Nov.10,790 1160 2824 181 –36 Dec.8908 4926 –2552 906 –80 2007Jan.–6522 –6669 –2449 150 –164 Feb.–31,750–4953 –3827 –11272 Mar.32,794 3082 3895 197 119 Apr.–1581 –113 1087 79 175 May.–3874 –2878 –2392 –849 –199 Jun.1315 5627 979 263 57 Jul. –24,9043852 239 214 –107 Aug.–671 –1573 –642 –326 39 Sep.6823 –7015 –2446 67 138 Oct.11,620 3959 3206 404 –83 3. Applying FGM (1, 1) to Undulated Data 3.1. Estimate Residual Series of Tourists from Five Continents This research adopts data provided by the Taiwan Tour ism Bureau on the number of tourist visits to Taiwan. The data include number of tourist visits to Taiwan from Asia, America (include North and South America), Euro pe, Oceania and Africa. The data period lasts from Au gust 1, 2006 to July 31, 2007. Table 1 lists the number of tourist visits to Taiwan from Asia, America, Europe, Oceania and Africa from 2006 to 2007. This study applies a novel method, namely the Fourier residual modification Grey forecasting model (FGM (1, 1)), to forecast tourism demand in Taiwan. Table 2 lists residual value for international tourist arrivals from five ontinents to Taiwan from 2006 to 2007 by GM. c
827 Y.L. HUANG ET AL. Table 3. Forecasting Asian tourism demand to Taiwan using GM and FGM. GM (1, 1) FGM (1, 1) Year Month t Actual numbers Forecasted numbers Residual valueResidual Error (%)Forecasted numbersForecasted numbers Residual Error (%) 2006 Nov. 1 193,510 182,720 10,790 5.58 8,518 191,238 1.17 Dec. 2 190,390 181,482 8908 4.68 11,180 192,662 1.19 2007 Jan. 3 173,730 180,252 –6522 3.75 –8794 171,458 1.31 Feb. 4 147,280 179,030 –31,750 *21.56 –29,478 149,552 1.54 Mar. 5 210,610 177,816 32,794 *15.57 30,522 208,338 1.08 Apr. 6 175,030 176,611 –1581 0.90 691 177,302 1.30 May. 7 171,540 175,414 –3874 2.26 –6146 169,268 1.32 Jun. 8 175,540 174,225 1315 0.75 3587 177,812 1.29 Jul. 9 148,140 173,044 –24,904 *16.81 –27,176 145,868 1.53 Aug. 10 171,200 171,871 –671 0.39 1601 173,472 1.33 Sep. 11 177,530 170,707 6823 3.84 4551 175,258 1.28 Oct. 12 181,170 169,550 11,620 6.41 13,892 183,442 1.25 Average residual error (%) 6.47 1.30 Accuracy (%) 93.53 98.70 Note: 1. the numbers means tourist numbers from Asia. 2. * maximal residual error. 3.2. Empirical Analysis—FGM of Tourists from Asia for Forecasting Tourism Demand in Taiwan This study uses the example of tourist visits to Taiwan from Asia. Monthly visitor numbers to Taiwan during the on and off seasons are displayed or each continent. Past information shows that February and July fall in the offseason while March is the onseason for international tourist visitors to Taiwan from Asia. Table 3 shows the actual number, forecast number, residual error and aver age accuracy rates for tourists from Asia by GM (1, 1) and FGM (1, 1). Numbers of tourist visits to Taiwan from Asia pre dicted with GM. The average residual error is 6.47%, there are several maximal residual errors are 21.56% (Feb.), 15.57% (Mar.) and 16.81% (Jul.) and the minimal residual error is 0.39%. Though the accuracy rate is 93.53%, GM (1, 1) is unsuitable for predicting stochastic volatility data. The number of tourist visit to Taiwan from Asia pre dicted using FGM. The average residual error is 1.3%, the maximal residual error is 1.54% and the minimal residual error is 1.08%. Although the minimal residual error of FGM exceeds the minimal residual of GM, the accuracy rate is 98.7%. The above statistics indicate the shortterm efficiency of FGM. Figure 2 compares actual value, GM forecast ing value and FGM forecasting value. The results show that the curve of GM appears flat and curve of FGM fol lows undulated curve of actual data and has an undulat ing appearance. FGM is more effective in predicting sto chastic volatility data. From the results, this study de Figure 2. Tourist demand forecasting using GM and FGM for comparison. velops an accuracy forecasting model for improving the effectiveness of tourism demand in making shortterm and stochastic volatility predictions. FGM is estimated for tourist arrivals to Taiwan and obtains an accurate estimate when the sample data exhibit significant fluc tuation. 4. Conclusions and Discussions This study makes the following contributions: 1) com bining the grey forecasting and Fourier series models to refine the forecasting effectiveness for the stochastic volatility data, 2) providing an effective method for fore casting the number of international visitors to Taiwan, 3) improving the accuracy of shortterm forecasting in cases involving sample data with significant fluctuations. De mand forecasting using the proposed model, thus is ex pected to reduce uncertainty regarding future markets and justify investment in new technology development. Copyright © 2011 SciRes. ME
Y.L. HUANG ET AL. 828 This study applied FGM (1, 1) to tourism demand esti mation and empirically tested and using raw data from Taiwan. FGM (1, 1) applies the Fourier series model to revise the residual values of GM (1, 1) and then to accu rately forecast shortterm f tourism demand, thus draw ing up decision problems in the government and business sectors. Yearround, international tourist arrivals to Taiwan have fluctuated between the on and off seasons. FGM (1, 1) can deal with the rising and falling tendency of data. Generally, consistent with recent research identify which model generates the best forecasts based on the value of calculated mean absolute percentage errors (MAPE) or root mean square errors (RMSE) (GilAlana, 2005; Chu, 2004). However, ARIMA is superior to other models for forecasting tourism demand [5,12,25,] and infers that ARIMA receives a coefficient of MAPE less than 10% [26,27]. Nevertheless, this work devises a FGM (1, 1) procedure for forecasting tourist arrivals in Taiwan. The analytical results indicated that using this FGM (1, 1) for forecasting achieves accuracy exceeding 98% and dis plays good fit performance. The test results reveal that the proposed method can accurately evaluate interna tional traveler numbers when the sample data exhibit significant fluctuations using FGM (1, 1). The FGM (1, 1) is demonstrated to be more reliable via posterior checks and to yield more accurate prediction results than the ARIMA and multiple regression models. Results of this study are important for tourism industry operators and government agencies interested in fore casting tourism demand. Where possible and appropriate, the use of indirect methods is recommended. Forecasting based on indirect methods with a manageable number of components and FGM (1, 1) can provide more accurate forecasts of numbers of international tourists. Addition ally, FGM (1, 1) can provide much more diverse and detailed information than the direct method. This work reaffirms that accurate forecasting of tourism demand is important in tourism planning by both public and busi ness sectors due to the highly perishable nature of tour ism products. Especially, shortterm trends and floating demand can seriously influence decisionmaking regard ing purchasing, inventory and logistics. The study results suggest that applying FGM (1, 1) to forecast short term tourism demands can achieve excel lent predictions. GM (1, 1) can to estimate the residual values from international tourists numbers base on 12 term, and then FGM (1, 1) to revise the predict values of international tourists arrive in Taiwan in next 12 term. Finally, FGM (1, 1) is applied to forecast international tourist numbers, and then accuracy values for predicting tourism demand are obtained, with residual values first being obtained and then these values being revised as necessary. However, further research on this issue is warranted for further generalizing the study results. 5. Acknowledgements The authors are thanks the National Science Council, Taiwan, R.O.C. for partial support (NSC 992410H 324013 MY2 and NSC 992410H033023 MY2). Moreover, the authors would like to acknowledge and thank Dr. InFun Lee for software support. 6. References [1] L. Škuflić and I. Štoković, “Demand Function for Croa tian Tourist Product: A Panel Data Approach,” Modern Economy, Vol. 2, 2011, pp. 4953. [2] Z. S. Hua, B. Zhang, J. Yang and D. S. 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