Modern Economy, 2011, 2, 823-829
doi:10.4236/me.2011.25091 Published Online November 2011 (
Copyright © 2011 SciRes. ME
Accurately Forecasting Model for the Stochastic Volatility
Data in Tourism Demand
Ya-Ling Huang1, Yen-Hsien Lee2
1Department of Go lden-Ager Industry Management, Chaoyang University of Technology, Taiwan, China
2Department of Fin anc e, Chung Yuan Christian University, Taiwan, China
Received April 15, 2011; revised June 6, 2011; accepted July 7, 2011
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 short-term 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-
sion-makers 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 [1-3]. Short-term 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 mid-1990s
as more researchers adopted modern econometric tech-
niques, for example co-integration and error correction
models, for modeling and forecasting tourism demand
[5-7]. Additionally, previous studies have mainly fo-
cused on traditional econometric analysis, with the re-
gression model [8-10], Time-series model [11-13], 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
short-term 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 choice-based diffusion models suffer
limitations in forecasting demand for new technologies
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 long-term 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 short-term 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
short-term 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
n and GM involves the following steps:
Step 1. Creating a sequence of first-order accumu-
lated generating operation (AGO)
1 is defined as ’s first-order AGO sequence. That
111 1
00 0
11 1
(1),(2),,( )
kk k
xxx xn
kxk xk
 
 
Step 2. Defining parame ters a and b
The first-order differential equation of GM (1, 1) mo-
del is
d() ()
xt ax tb
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
 (3)
Furthermore, accumulated matrix B is
0.5 (1)(2)1
0.5 (2)(3)1
0.51( )1
xn xn
Meanwhile, the constant vector YN is
00 0
(2),(3),,() T
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.
ˆ1(1) ak
xk xe
 
Step 5. Obtaining the forecasting values based on
When 10
(1) (1)
, the sequence one-order inverse-
accumulated generating operation (IAGO) of reduction is
obtained as
kxkx k . Given 1,k
the sequence of reduction is obtained as fol-
2,, n
Copyright © 2011 SciRes. ME
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 one-order inverse-accumulated
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
 
()() ()kxkxk
, . (9)
2, ,kn
Step 2. Define FGM (1, 1)
The continuous residual series can be modeled using
Fourier series as:
ˆ()cos sin
ka akbk
 
 
 
 
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:
 nTa
ˆ()cos sin
for 2,
ka abk
, (11)
Equation (11) can be estimated as:
,,, ,, ,,
Caababab (14)
in Equation (15),
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
 
Finally, the original prediction series of FGM (a
) can
be corrected as:
()() ()
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-
Error =
kxkxk, (18)
Average error =
()() ()
cos2 sin2sin2
cos3 sin3sin3
cos sinsin
aa a
aa a
aa a
 
 
 
 
 
 
Copyright © 2011 SciRes. ME
Copyright © 2011 SciRes. ME
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
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
off-season while March is the on-season 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 short-term 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 short-term
and stochastic volatility predictions. FGM is estimated
for tourist arrivals to Taiwan and obtains an accurate
estimate when the sample data exhibit significant fluc-
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 short-term 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
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 short-term f tourism demand, thus draw-
ing up decision problems in the government and business
Year-round, 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) (Gil-Alana, 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, short-term trends and floating
demand can seriously influence decision-making 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 99-2410-H-
324-013 -MY2 and NSC 99-2410-H-033-023 -MY2).
Moreover, the authors would like to acknowledge and
thank Dr. In-Fun Lee for software support.
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