Forecasting Tourism Demand in Selected Caribbean Countries Using Optimised Grey Forecasting Models

Abstract

For many Caribbean countries, tourism represents a major source of foreign exchange and is the main contributor to gross domestic product. Therefore, the timely production of reliable estimates for tourism demand is paramount, especially in light of its seasonal nature. This article utilises a quasi-optimisation algorithm to produce optimised values for the generation coefficient (weights) of the grey GM(1,1) forecasting model, together with two modified grey GM(1,1) models. With data for 2008–2015, the parameter-optimised models were then used to produce in-sample and out-of-sample forecasts of tourism demand (stay-over tourist arrivals) for selected Caribbean countries. These estimates were also compared to those produced by the standard versions of the models. The results indicate that while the standard models can generate reasonably accurate results, in some instances, the performance of the models improved noticeably with the use of the optimised parameters. Overall, most of the in-sample and out-of-sample mean absolute percentage errors are well below 5%, indicating a high level of forecasting accuracy.

Full text

Social and Economic Studies 70: 1&2 (2021): 35–53 ISSN 0037-7651
Forecasting Tourism Demand in Selected
Caribbean Countries Using Optimised Grey
Forecasting Models
altHEa diannE la foucadE, SamuEl
GaBRiEl, EWan Scott, cHaRmainE mEtiViER,
and cHRiStinE laPtiStE
Abstract: For many Caribbean countries, tourism represents a major source of
foreign exchange and is the main contributor to gross domestic product. Therefore, the
timely production of reliable estimates for tourism demand is paramount, especially
in light of its seasonal nature. This article utilises a quasi-optimisation algorithm to
produce optimised values for the generation coefcient (weights) of the grey GM(1,1)
forecasting model, together with two modied grey GM(1,1) models. With data
for 2008–2015, the parameter-optimised models were then used to produce in-
sample and out-of-sample forecasts of tourism demand (stay-over tourist arrivals) for
selected Caribbean countries. These estimates were also compared to those produced
by the standard versions of the models. The results indicate that while the standard
models can generate reasonably accurate results, in some instances the performance
of the models improved noticeably with the use of the optimised parameters. Overall,
most of the in-sample and out-of-sample mean absolute percentage errors are well
below 5%, indicating a high level of forecasting accuracy.
Keywords: grey forecasting, parameter optimisation, tourism, tourism
demand.
INTRODUCTION
The tourism industry occupies a prominent position in the economic
sector of several countries and is often a chief driver of economic
growth. Indeed, tourism can yield dividends in the form of revenues
for the destination country and provides a viable source of quality
employment for its residents. In the global arena, the tourism industry
offers an avenue through which external scal imbalances may be
improved and the diversication of the destination country’s local
economy advanced. This is especially so in the context of the Caribbean
region where, for many countries, tourism contributes more to gross
domestic product than any other sector of the economy.

36 Social and Economic StudiES
Preservation and advancement of tourism-related benets demand
informed and prudent decision-making on the part of all stakeholders.
Investment in infrastructure, accommodations, transportation
services, local attractions, and other services depends critically on
accurate forecasts of the demand for the tourism product. Further, the
government and other stakeholders must be able to anticipate potential
shortfalls in tourism-related revenues stemming from a cooling of
demand and to act accordingly (Uysal and Crompton 1985). In the
Caribbean where large amounts of quality data are often unavailable,
the identication of region-appropriate methods of tourism demand
estimation is paramount.
LITERATURE REVIEW
The approaches for estimating the future demand for the tourism
product span a wide spectrum. Broadly speaking, these can be split into
qualitative or quantitative approaches. Qualitative approaches include
the traditional methods, which analyse national and regional “holiday
surveys or execute survey inquiries of potential visitors in tourism-
generating areas” (Uysal and Crompton 1985, 8). Other qualitative
approaches include the Delphi model, which “seeks to arrive at an
agreement among industry professionals through administering of a
series of questionnaires, collecting judgments and providing feedback
from each series of questionnaires…” (Uysal and Crompton 1985, 8).
Additionally, the judgement-aided model seeks to establish a consensus
among industry experts, which is done through the assembly of a panel
of experts in the form of seminars or meetings (Uysal and Crompton
1985). These approaches have one specic advantage in that they do
not place signicant data demands on the process. However, they do
not produce statistically testable estimates and are therefore seen as
less reliable.
There are several published works by international researchers on
forecasting and modelling of the demand for tourism using quantitative
methods. These include Error Correction Models (Daniel and Ramos
2002; Dritsakis 2004; Kulendran and Wilson 2000), Almost Ideal
Demand Systems and Linearised Almost Ideal Demand Systems (De
Mello and Fortuna 2005; De Mello and Nell 2005; Divisekera 2003).
Other works (Gil-Alana 2005; Lim and Mcaleer 2001; Papatheodorou
and Song 2010) have found varying degrees of success in forecasting
and modelling the demand using the time-series methods of
autoregressive integrated moving average (ARIMA) and multivariate
ARIMA. Claveria, Monte, and Torra (2013, 2014, 2015) explored more

GREy foREcaStinG of touRiSm dEmand 37
contemporary approaches, such as neural networks that harness the
power of articial intelligence to improve prediction accuracy.
Likewise, Caribbean authors have made some attempts to quantify
the demand for Caribbean tourism. Carey (1991) made use of
generalised least squares to estimate a log function of variables that
are believed to inuence the demand for Caribbean tourism arrivals (a
measure of Caribbean tourism demand). In his study on the impact of
climate change on Caribbean tourism, Moore (2010) saw merit in the
panel autoregressive distributed lag approach, and more recently, the
augmented gravity approach was put to work by Lorde, Li, and Airey
(2015) as they sought to model tourism in the region.
From the above-mentioned approaches, two categories emerge.
The rst is the time-series category, where models are used to explain
changes in the variable of interest based on past observations and
trends of the variable itself while accounting for errors by the inclusion
of a disturbance term. The other category, econometric models,
investigates the causal relationship between the variable of interest and
its determining factors, among other things (Song and Li 2008). The
ability of econometric models to examine causal relationships between
the predicted variable and its determinants (predictors) stands out as
a major advantage over time-series approaches. Econometric models
can also be one of the instruments through which policies may be
created and evaluated.
Notwithstanding their obvious virtues, econometric models and
most time-series models generally require large amounts of detailed
data—a requirement that is often difcult to meet, especially in the
context of developing countries in general and small island developing
states in particular. Furthermore, these models lack the ability to
incorporate in their analysis the inherent inaccuracies and uncertainties
that may exist in the tourism industry. Grey system theory and
specically grey forecasting models boast of the ability to overcome
these shortcomings (Ho 2012).
Grey forecasting
Established by Julong Deng in 1982, the grey system theory is a
relatively new methodology that focusses on the study of problems
involving small samples and poor information. It deals with uncertain
systems with partially known information by generating, expanding,
and extracting useful information from what is available (Deng 1982);
“it looks for realistic patterns based on the modelling of small amounts
of available data” (Ho 2012, 371). Thus, it allows for the effective

38 Social and Economic StudiES
monitoring and description of systems’ behaviour and the laws that
govern changes within them.
Unlike the econometric and time-series models, which address
uncertainty by the use of large samples that adhere to some specic
distributional assumption, the grey system theory centres on problems
with small samples and poor information—characteristics that will
prove to be difcult for approaches of probability and statistics to
manage (Liu et al. 2012). According to Chin-tsai, In-fun, and Ya-ling
(2009, 372), “…production forecasting depends more on limited
current data rather than large amounts of historical data.” In fact,
they argue that with the use of these models, unknown systems can
be characterised with only four data points. Indeed, the grey system
family of forecasting models has been applied widely with success—
for example, in the electronics industry (Hsu 2003, 2009, 2011), the
energy industry (Guo, Wu, and Wang 2011; Lee and Shih 2011; Hsu
and Chen 2003), the nancial sector (Huang and Jane 2009; Wang
2002; Wu and Chen 2011), the medical eld (Hsu and Chen 2007;
Shen et al. 2013), medical tourism forecasting (La Foucade et al.
2019), and tourism forecasting (Huang 2012; Chin-tsai, In-fun, and
Ya-ling 2009).
According to Wang (2004), the grey GM(1,1) forecasting model
is one among the basic time-series forecasting models that came
out of the grey system theory. GM stands for grey model, and (1,1)
comes from the fact that the model uses a rst (1) order differential
equation and a single (1) variable. The grey forecasting model builds
differential equations with coefcients that are time varying so that
“the model is renewed as new data become available to the prediction
model” (Kayacan, Ulutas, and Kaynak 2010, 1785). Essentially, this
approach is characterised by the need for less data. The smoothing of
the randomness in the original series is achieved by applying the rst-
order accumulating generation operator (1-AGO). A solution for the
differential equation GM(1,1) model is found to enable predictions of
the future values of the system. Once these values are found, the inverse
accumulating generation operator is employed to forecast values of the
original observations (Kayacan, Ulutas, and Kaynak 2010).
METHODOLOGY
The grey forecasting models
The general procedure for the implementation of the grey forecasting
models is as follows (Kayacan et al. 2010; Liu and Forrest 2010).
Assume a sequence of raw data dened as:

GREy foREcaStinG of touRiSm dEmand 39
Xxxxnn
() () () ()
((), (),..., ( )),
0000
12 4

, (1)
where X(0) is a non-negative data series and n is the sample size. The
application of the 1-AGO to the X(0) series yields the sequence X(1),
dened as follows:
Xxxxnn
() () () ()
((), (),..., ( )),
1111
12 4

, (2)
where
xk xi
kn
i
k
()
(),(), ,,,...,
10
1
123


. (3)
The next step is to obtain the sequence Z(1) of the generated series X(1),
where Z(1)(k) is the average value of adjacent (neighbour) data points,
specically:
Zk xk xk
kn
() () ()
() () (),,,...,
111123

. (4)
In this case, α is the generation/production coefcient or weight where
0 < α < 1. When α > 0.5, the generation process places more emphasis
on the more recent data and less on the older data. When α < 0.5, more
importance is assigned to the older data and less to the more recent
data. An α equal to 0.5 indicates equal weighting, the assumption of
the basic GM(1,1) (Mohammadi and Zade 2011).
Next, the grey difference equation of GM(1,1) is dened by Deng
(1982) and Liu, Lin, and Forrest (2010) as follows:
zkaz
kb
() ()
() ()
01

. (5)
If â = (a,b)T is a sequence of parameters and
Y
x
x
xn
B
z
z


















()
()
()
()
()
()
()
()
,
()
()
0
0
0
1
1
2
3
2
3
















zn
()
()
1
1
1
1
, (6)
then the least squares estimate of the GM(1,1) model—that is, equation
(5)—satises
â = (BT)–1BTY.
If
(,)()ab
BB
Y
TTT
1, (7)
then the whitenisation1 (or image) equation of the GM(1,1) model (Liu, Lin,
and Forrest 2010, 108) is given by
1 Broadly speaking, whitenisation is a process that moves a grey (uncertain,
inaccurate) number to a white (certain, accurate) number.

40 Social and Economic StudiES
dx
dt
ax b
()
()
1
1

. (8)
In other words, the model’s parameters b and a can be estimated via
ordinary least squares (OLS) as dened by equation (7). In equation (8),
t represents the independent variable, a represents the development
coefcient, and b denotes the grey action quantity. The time response
function or the solution to the whitenisation equation (8) is given by
xt x
b
ae
b
a
at() ()
() ()
11
1





, (9)
The time response sequence of the GM(1,1) model in equation (5) is
given by
xk
ˆxb
aeb
a
kn
ak() ()
() () ,,,...,
10
11 12 . (10)
To obtain forecast values of the original data series at time (k + 1), the
inverse accumulating generation operator is used to form the following
model:
x
kxkxkx
kexb
a
a
() () () () ()
()
() () () ()
()()
0111
10
11
111ek n
ak ,,,...12
ˆˆˆˆ
x
kxkxkxke
x
b
a
a
() () () () ()
()
() () () () ()
()
011110
1111
1ek n
ak ,,,...12
ˆˆˆˆ
. (11)
Forecast values of the original series at time K + H are given by the
following formula:
x
kH ex
b
a
e
aa
KH() ()
()
()()()
00
1
11
ˆ
. (12)
The main means by which the accuracy of the model can be examined
are given by the following formulae (Kayacan et al. 2010; Liu and
Forrest 2010):
() () ()*
() ()
kxkxk
00
100
ˆϵ
. (13)
MAPE
n
k
xk
k
n
1
1100
0
2
()
()
*
()
ϵ
, (14)
where APE denotes the absolute percentage error and MAPE
represents the mean absolute percentage error. Tsai (2012) outlined,

GREy foREcaStinG of touRiSm dEmand 41
as seen in Table 1, the evaluation criteria for the model on the basis of
the MAPE.
Table 1: MAPE criteria for model evaluation
Residual test:
MAPE n
k
xk
k
n
1
1100
0
2
()
()
*
()
ϵ
Criteria (MAPE, %) Interpretation
<10 High forecasting accuracy
10–20 Good forecasting accuracy
21–50 Reasonable forecasting accuracy
>50 Weak and inaccurate predictability
Furthermore, since the introduction of the original model, many
proposed improvements have emerged. Peirong, Xianju, and Hongbo
(2007) showed that the basic grey GM(1,1) model is biased and
proceeded to suggest what they called an unbiased GM(1,1) model
dened as follows:
Equation (8) can also be written as
21
1
2
1
()e
e
A
e
a
a
a
, from which the
following can be obtained:
aa
a
A
b
a
ln ,
2
2
2
2
ˆˆ
ˆˆ
where â and b
̂ are the estimated parameters from the GM(1,1) model,
and a and A are the parameters of the unbiased GM(1,1) model. Using
a and A, the unbiased GM(1,1) model is established as follows:
xkAek n
ak
()
() ,,,...,
()
0
123
ˆ''
x x
() ()
() ()
00
11
ˆ
Ji et al. (2011) later proposed a modied version of the unbiased
GM(1,1) model in which each value of the original data series
is transformed into its 2-th root. According to the authors, this
transformation reduces the growth rate of the original data sequence,
making for improved conditions for model construction.
Given the original data series, X(0)=(x(0)(1), x(0)(2),…, x(0)(n)) and
x(0)(k) ≥ 0, the modied unbiased GM(1,1) model is built as follows:
(1) Calculate the square root of each value of the original series:

42 Social and Economic StudiES
xkxkk n
00
2
() ()
() (),,...,
'.
(2) Using the transformed data series, x'(0), generate x(1)(k) by
applying the 1-AGO:
x
kxmk n
m
k
() ()
() (),,,...,
10
1
12
'.
(3)–(5) Remain the same as steps (4)–(7) of the original GM(1,1)
model.
(6) Build the model using the transformed data series:
x x
() ()
() ()
00
11
ˆ'
.
xkAek n
a
k
()
() ,,,...,
()
0
123
ˆ''
'.
(7) Get predicted values of the original data series by
retransforming the data:
x x
() ()
() (())
002
11
''
ˆ
.
xxk
kn
() ()
() ,,,...,
10
223
''ˆ .
Despite modications to the original GM(1,1) model, Zhang et
al. (2014), Liu et al. (2014), Ma et al. (2011), and Wang and Hsu
(2008) indicated that under certain circumstances, further and more
signicant improvements may be achieved if the optimal value for the
generation coefcient alpha (α) can be identied. Specically, when
the OLS process produces very large development coefcients α in
equation (8), the decision to set alpha equal to 0.5 may be a suboptimal
choice (Ma et al. 2011), and the optimisation of this parameter tends
to improve forecasting accuracy.
Quasi-optimisation method
While the genetic and particle swarm optimisation algorithms are among
the more popular methods of identifying the optimum parameters,
these methods require, in most cases, signicant programming
capabilities that are not always accessible. Given this, the following
quasi-optimisation method is suggested. The optimised parameter, α, is
the one that produces the minimum value of the in-sample MAPE, i.e.,
the objective function:
min MAPE n
PE k
n
xk xk
xk
*
k
n
k
n
1
1
1
1
100
2
00
0
2
() () ()
()
() ()
()
ˆ
.
As stated above, 0 < α < 1 and as such, this approach initialises with
the generation of alphas within the range 0.01 < α < 0.99999. It begins
with an initial value of 0.01, increments by 0.0001, and ends at 0.99999.

GREy foREcaStinG of touRiSm dEmand 43
Next, the algorithm iterates n times2 where each of the generated alphas
is entered into the grey forecasting model to produce n values for the
MAPE. The algorithm then plots the graph of the MAPEs and the
generated alphas and yields the value of alpha that is associated with
the minimum MAPE. This process produces an initial quasi-optimal
value that will give an indication of the general region where a more
precise quasi-optimal value lies. Once this is done, a second round of
alpha generation commences. This time the incremental value and the
upper and lower bounds are reduced. This has a zooming effect that
identies a more accurate measure for the optimal value of alpha and
reduces the need for a lengthy iterative process. It is at this stage that
the quasi-optimal alpha is identied. A graphical representation of this
method of parameter optimisation is illustrated in Figure 1.
Figure 1: Quasi-optimisation process
Generate αi, 0 < α < 1,
i = 1 ... n
Use αi to estimate
the grey model
Compare grey forecasts
with original data
Plot graph/identify
quasi-optimal alpha
Generate i grey forecasts
Calculate MAPEi
Source: Adapted from Ma et al. (2011).
RESULTS
Data was collected from various national and regional tourism sites for
eight Caribbean countries, of which six were from the Organisation of
Eastern Caribbean States (OECS), namely, Grenada (GND), Dominica
(DOM), Antigua and Barbuda (ANT), St. Lucia (SLU), St. Vincent and
the Grenadines (SVG), and St. Kitts and Nevis (SKN). Tourism data
from Jamaica (JAM) and The Bahamas (BAH) was also analysed.
Figure 2 depicts the number of stay-over visitors entering selected
Caribbean countries between 2008 and 2015. For the most part,
2 In this case, n is the number of generated alphas between 0.01 and 0.99999.

44 Social and Economic StudiES
the data shows a general upward trend in the number of tourists
entering the countries, except for DOM and SVG, which appear to
have experienced a slump in demand for that segment of the tourism
market. Moreover, SLU welcomed an average of 312,636 stay-over
visitors over that period, the largest number among the OECS
countries, whereas DOM recorded the lowest average number of stay-
over visitors (73,566) during the same period. Overall, JAM saw the
strongest average demand for the product as revealed by the 1,958,689
tourists who visited the island on a stay-over basis during that period.
Figure 2: Series plot of stay-over visitors in selected Caribbean countries (2008–
2015)
Stay-over visitors in selected OECS countries
2008
0.5
1
1.5
2.5
3.5 ×105
2
3↑ St. Lucia (SLU)
↓ Antigua and Barbuda (ANT)
↓ Grenada (GND)
↓ St. Vincent and the Grenadines (SVG)
↑ Dominica (DOM)
↑ St. Kitts and Nevis (SKN)
20102009 2011 2012
Yea r 2013 2014 2015
Stay-over visitors
Stay-over visitors in Jamaica (JAM) and The Bahamas (BAH)
2009
1
1.2
1.4
1.6
1.8
2
×106
↑ JAM
↓ BAH
2010 2011 2012
Yea r 2013 2014 2015
Stay-over visitors

GREy foREcaStinG of touRiSm dEmand 45
As previously indicated, the analysis was conducted with the
generation of 9,900 alpha values, beginning with an initial alpha of
0.01 and incrementing by 0.0001 to create the remaining values, up
to the nal value of 0.99999. Each alpha was entered in the grey
forecasting models, where in-sample forecasts and MAPEs were
produced, along with the identication of the alpha that is associated
with the minimum MAPE. Once this was done, the upper and lower
limits were reduced to closely border the initial quasi-optimal alpha
and the increments reduced to 0.000001. This adjustment resulted in
the generation of 300,001 alphas (potential solutions), each of which
was utilised in a second round of analyses, where in-sample MAPEs,
four-periods-ahead out-of-sample forecasts, plots and out-of-sample
MAPEs were produced. With the aid of the plots of MAPEs and the
generated alphas and the min Matlab function, the quasi-optimal alpha
was identied. Example plots are shown in Figure 3.
Figure 3: Plots of MAPEs and generated alphas (weights) showing location of
quasi-optimal alpha
0
0
100
200
300
400
500
600
0
5
10
15
20
25
30
35
40
45
50
0.1 0.2 0.3 0.4
ANT
Optimal alpha (weights): 0.41696
Minimum point magnified
Alphas (weights) Alphas (weights)
MAPEs
MAPEs
Minimum point magnified
BAH
Optimal alpha (weights): 0.30581
0.5
0.3
2
3
4
5
6
0.4 0.5 0.6 0.7 0.8 0.3
1.2
1.3
1.4
1.5
1.6
1.7
0.4 0.5 0.6 0.7 0.8
0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Tables 2–4 exhibit the in-sample and out-of-sample APE and the in-
sample and out-of-sample MAPE for the eight countries, using the
standard alpha (0.5) and the optimised alpha. Table 2 presents results
produced by the original grey GM(1,1) forecasting model, while Tables
3 and 4 outline the results obtained by the modied models proposed
by Peirong, Xianju, and Hongbo (2007) and Ji et al. (2011), respectively.
All models performed well across every country with the exception
of SLU.3 Both the in-sample and out-of-sample MAPEs were below
5% in most cases, with the out-of-sample MAPE dipping to a low of
3 The SLU data recorded the poorest performance across all three models, with
out-of-sample MAPEs ranging between 9.15% and 11.19%.

46 Social and Economic StudiES
Table 2: MAPE using standard and optimal alphas: original GM(1,1) model
Year ANT BAH DOM GND JAM SLU SVG SKN
α values α values α values α values α values α values α values α values
0.5 0.416956 0.5 0.305807 0.5 0.550091 0.5 0.403764 0.5 0.524171 0.5 0.528144 0.5 0.671638 0.5 0.509063
2008
2009 1.14 0.66 1.04 0.57 0.39 0.22 1.95 1.14 0.58 0.36 1.36 0.88 0.95 0.51 0.37 0.22
2010 2.29 3.10 2.04 2.77 0.75 1.00 3.96 5.34 1.09 1.45 2.41 3.16 2.00 2.74 0.72 0.94
2011 1.08 0.00 1.06 0.00 0.37 0.00 1.80 0.00 0.51 0.00 1.08 0.00 0.99 0.00 0.31 0.00
I.S.
MAPE
1.50 1.25 1.38 1.12 0.50 0.41 2.57 2.16 0.73 0.60 1.62 1.35 1.31 1.08 0.46 0.39
2012 1.87 0.51 5.05 6.32 3.04 3.49 1.70 4.08 1.94 1.27 8.86 7.38 2.71 1.46 2.26 1.87
2013 1.12 2.83 0.68 2.30 1.90 2.46 3.40 6.35 4.04 3.21 10.86 9.02 0.21 1.34 3.97 3.49
2014 0.13 2.12 4.71 6.54 4.65 5.29 10.82 7.82 3.67 2.69 10.48 8.33 0.13 1.97 3.06 2.51
2015 1.17 3.49 8.37 10.39 5.50 4.69 13.59 10.25 4.83 3.69 14.56 11.99 7.08 5.11 4.23 3.58
O.S.
MAPE
1.07 2.24 4.71 6.38 3.77 3.98 7.38 7.13 3.62 2.72 11.19 9.18 2.53 2.47 3.38 2.86
Note: I.S. = in-sample; O.S. = out-of-sample.

GREy foREcaStinG of touRiSm dEmand 47
Table 3: MAPE using standard and optimal alphas: modied GM(1,1) model proposed by Peirong, Xianju, and Hongbo (2007)
Year ANT BAH DOM GND JAM SLU SVG SKN
α values α values α values α values α values α values α values α values
0.5 0.398507 0.5 0.298358 0.5 0.538126 0.5 0.384703 0.5 0.525948 0.5 0.512382 0.5 0.650872 0.5 0.405143
2008
2009 1.38 0.82 0.93 0.61 0.30 0.18 2.31 1.39 0.62 0.38 0.76 0.55 0.83 0.46 1.55 0.00
2010 2.04 3.01 2.93 2.76 0.83 1.03 3.58 5.21 1.05 1.44 2.99 3.32 2.12 2.76 2.61 0.06
2011 1.32 0.00 0.74 0.00 0.28 0.00 2.16 0.00 0.55 0.00 0.48 0.00 0.87 0.00 1.61 2.02
I.S. MAPE 1.58 1.28 1.39 1.12 0.47 0.40 2.68 2.20 0.74 0.61 1.41 1.29 1.28 1.07 1.92 0.69
2012 2.11 0.42 5.01 6.34 3.12 3.47 1.33 4.21 1.98 1.26 8.21 7.55 2.59 1.49 0.31 5.08
2013 0.87 3.00 0.65 2.34 1.8 2.41 3.02 6.62 4.08 3.19 10.20 9.38 0.09 1.29 1.99 7.94
2014 0.12 2.37 4.67 6.59 4.73 5.22 11.15 7.47 3.71 2.66 9.83 8.86 0.25 1.89 1.10 8.11
2015 0.92 3.83 8.34 10.46 5.41 4.78 13.91 9.79 4.87 3.65 13.89 12.73 6.97 5.21 2.24 10.47
O.S. MAPE 1.00 2.41 4.67 6.43 3.81 3.97 7.35 7.02 3.66 2.69 10.53 9.63 2.48 2.47 1.41 7.90
Note: I.S. = in-sample; O.S. = out-of-sample.

48 Social and Economic StudiES
Table 4: MAPE using standard and optimal alphas: modied GM(1,1) model proposed by Ji et al. (2011)
Year ANT BAH DOM GND JAM SLU SVG SKN
α values α values α values α values α values α values α values α values
0.5 0.404264 0.5 0.302979 0.5 0.545136 0.5 0.388897 0.5 0.525782 0.5 0.522297 0.5 0.665959 0.5 0.464728
2008
2009 1.25 0.73 1.06 0.60 0.34 0.20 2.10 1.24 0.58 0.35 0.99 0.63 0.91 0.51 0.50 0.00
2010 2.17 3.06 2.02 2.76 0.79 1.02 3.78 5.29 1.07 1.45 2.70 3.29 2.06 2.74 1.56 0.73
2011 1.21 0.00 1.07 0.00 0.33 0.00 2.02 0.00 0.55 0.00 0.85 0.00 0.93 0.00 0.53 0.64
I.S. MAPE 1.54 1.26 1.38 1.12 0.48 0.41 2.63 2.18 0.73 0.60 1.52 1.30 1.30 1.08 0.87 0.46
2012 2.02 0.47 5.05 6.34 3.07 3.48 1.43 4.13 1.99 1.28 8.69 7.51 2.64 1.46 1.42 2.96
2013 0.95 2.91 0.68 2.33 1.93 2.43 3.09 6.45 4.11 3.22 10.77 9.30 0.13 1.35 3.13 5.05
2014 0.06 2.24 4.72 6.58 4.67 5.25 11.12 7.69 3.74 2.72 10.47 8.74 0.22 1.97 2.25 4.50
2015 0.96 3.64 8.38 10.45 5.48 4.74 13.92 10.07 4.94 3.73 14.63 12.56 6.98 5.10 3.42 6.05
O.S. MAPE 1.00 2.31 4.71 6.42 3.79 3.98 7.39 7.09 3.70 2.74 11.14 9.53 2.49 2.47 2.56 4.64
Note: I.S. = in-sample; O.S. = out-of-sample.

GREy foREcaStinG of touRiSm dEmand 49
1% in the case of ANT across all three models. In 11 of the 16 analyses
(69%), out-of-sample MAPEs of less than 5% were recorded, while in
15 of the 16 analyses (94%), a MAPE of less than 10% was produced
across all models.
According to the evaluation criteria outlined in Table 1, these
results indicate that all three models demonstrated a high degree of
forecast accuracy. Furthermore, the use of optimised alpha did not
always translate into improved performance. In fact, in roughly 50%
of the times, the optimised alpha resulted in higher out-of-sample
MAPE, even as the in-sample MAPE fell. This highlights the fact
that the minimisation of the in-sample MAPE does not guarantee
improved out-of-sample performance. As Ma et al. (2011) outlined,
there are instances where an alpha of 0.5 is indeed optimal for the
overall performance of the model.
Moreover, the modied model proposed by Peirong, Xianju, and
Hongbo (2007) registered a superior performance than the other
models. This model recorded 9 instances with the lowest out-of-
sample MAPE across all alphas, compared to 6 for the original grey
GM(1,1) model and 3 in the case of the model put forward by Ji et al.
(2011). The varied performance of the country data emphasises the
importance of the nature of the series and its usefulness in the models.
Series that are characterised by a strong trend and little signicant
variations tend to perform best with the grey family of models.
CONCLUSION
In the Caribbean where large amounts of quality data are often
unavailable, critical analyses to facilitate informed decision-making in
the tourism sector and, indeed, other sectors are often left undone.
This article demonstrates that the grey family of forecast models may
offer a viable investigative option to produce substantially accurate
forecasts, with the use of only four data points. Largely, the models
have shown substantial potential for use as an alternative to traditional
econometric methods in instances where data is scarce and forecasting
is necessary.
REFERENCES
Carey, Kathleen. 1991. “Estimation of Caribbean Tourism Demand: Issues in
Measurement and Methodology.” Atlantic Economic Journal 19 (3): 32–44.
Chin-tsai, Lin, Lee In-fun, and Huang Ya-ling. 2009. “Forecasting Thailand’s
Medical Tourism Demand and Revenue from Foreign Patients.” The Journal
of Grey System 4: 369–76.

50 Social and Economic StudiES
Claveria, Oscar, Enric Monte, and Salvador Torra. 2013. “Tourism Demand
Forecasting with Different Neural Networks Models.” 21. Research Institute of
Applied Economics and Regional Quantitative Analysis Research Group. Barcelona.
________. 2014. “A Multivariate Neural Network Approach to Tourism Demand
Forecasting.”
________. 2015. “Tourism Demand Forecasting with Neural Network Models:
Different Ways of Treating Information.” International Journal of Tourism
Research 17: 492–500. doi:10.1002/jtr.
Daniel, Ana Cristina M., and Francisco F. R. Ramos. 2002. “Modelling Inbound
International Tourism Demand to Portugal.” International Journal of Tourism
Research 4 (3): 193–209. doi:10.1002/jtr.376.
De Mello, Maria M., and Natércia Fortuna. 2005. “Testing Alternative Dynamic
Systems for Modelling Tourism Demand.” Tourism Economics 11 (4): 517–37.
doi:10.5367/000000005775108719.
De Mello, Maria M., and Kevin S. Nell. 2005. “The Forecasting Ability of a
Cointegrated VAR System of the UK Tourism Demand for France, Spain
and Portugal.” Empirical Economics 30 (2): 277–308. doi:10.1007/s00181-005-
0241-0.
Deng, J. 1982. Grey System Fundamental Method. Wuhan, China: Huazhong University
of Science and Technology.
Divisekera, Sarath. 2003. “A Model of Demand for International Tourism.”
Annals of Tourism Research 30 (1): 31–49. doi:10.1016/S0160-7383(02)00029-4.
Dritsakis, Nikolaos. 2004. “Cointegration Analysis of German and British Tourism
Demand for Greece.” Tourism Management 25 (1): 111–19. doi:10.1016/S0261-
5177(03)00061-X.
Gil-Alana, L. A. 2005. “Modelling International Monthly Arrivals Using Seasonal
Univariate Long-Memory Processes.” Tourism Management 26 (6): 867–78.
doi:10.1016/j.tourman.2004.05.003.
Guo, J. J., J. Y. Wu, and R. Z. Wang. 2011. “A New Approach to Energy
Consumption Prediction of Domestic Heat Pump Water Heater Based on
Grey System Theory.” Energy and Buildings 43 (6): 1273–79. doi:10.1016/j.
enbuild.2011.01.001.
Ho, PHK. 2012. “Comparison of the Grey Model and the Box Jenkins Model
in Forecasting Manpower in the UK Construction Industry.” Arcom.ac.uk,
September: 3–5. http://www.arcom.ac.uk/-docs/proceedings/ar2012-0369-
0379_Ho.pdf.
Hsu, Che-Chiang, and Chia-Yon Chen. 2003. “Applications of Improved Grey
Prediction Model for Power Demand Forecasting.” Energy Conversion and
Management 44 (14): 2241–49. doi:10.1016/S0196-8904(02)00248-0.
Hsu, Chin-tsai Lin Pi-fang, and Bi-yu Chen. 2007. “Comparing Accuracy of GM
(1, 1) and Grey Verhulst Model in Taiwan Dental Clinics Forecasting.” Journal
of Grey System 1: 31–38.

GREy foREcaStinG of touRiSm dEmand 51
Hsu, Li-Chang. 2003. “Applying the Grey Prediction Model to the Global
Integrated Circuit Industry.” Technological Forecasting and Social Change 70 (6):
563–74. doi:10.1016/S0040-1625(02)00195-6.
________. 2009. “Forecasting the Output of Integrated Circuit Industry Using
Genetic Algorithm Based Multivariable Grey Optimization Models.” Expert
Systems with Applications 36 (4): 7898–903. doi:10.1016/j.eswa.2008.11.004.
________. 2011. “Using Improved Grey Forecasting Models to Forecast the
Output of Opto-Electronics Industry.” Expert Systems with Applications 38 (11):
13879–85. doi:10.1016/j.eswa.2011.04.192.
Huang, Kuang Yu, and Chuen-Jiuan Jane. 2009. “A Hybrid Model for Stock
Market Forecasting and Portfolio Selection Based on ARX, Grey System and
RS Theories.” Expert Systems with Applications 36 (3): 5387–92. doi:10.1016/j.
eswa.2008.06.103.
Huang, Ya-ling. 2012. “Forecasting the Demand for Health Tourism in Asian
Countries Using a GM(1,1)-Alpha Model.” Tourism and Hospitality Management
18 (2): 171–81.
Ji, Peirong, Jian Zhang, Hongbo Zou, and Wenchen Zheng. 2011. “A Modied
Unbiased GM(1,1) Model.” Grey Systems: Theory and Application 1 (3): 192–201.
doi:10.1108/20439371111181206.
Kayacan, Erdal, Baris Ulutas, and Okyay Kaynak. 2010. “Grey System Theory-
Based Models in Time Series Prediction.” Expert Systems with Applications 37
(2): 1784–89. doi:10.1016/j.eswa.2009.07.064.
Kulendran, Nada, and Kenneth Wilson. 2000. “Modelling Business Travel.”
Tourism Economics 6 (1): 47–59. doi:10.5367/000000000101297460.
LaFoucade, Althea, Samuel Gabriel, Ewan Scott, Karl Theodore, and Charmaine
Metivier. 2019. “A Survey of Selected Grey Forecasting Models with
Application to Medical Tourism Forecasting.” Theoretical Economics Letters 09:
1079–92. https://doi.org/10.4236/tel.2019.94070.
Lee, Shun-Chung, and Li-Hsing Shih. 2011. “Forecasting of Electricity Costs Based
on an Enhanced Gray-Based Learning Model: A Case Study of Renewable
Energy in Taiwan.” Technological Forecasting and Social Change 78 (7): 1242–53.
doi:10.1016/j.techfore.2011.02.009.
Lim, Christine, and Michael Mcaleer. 2001. “Monthly Seasonal Variations: Asian
Tourism to Australia.” Animals of Tourism Research 28 (1): 68–82.
Liu, Li, Qianru Wang, Ming Liu, and Lian Li. 2014. “An Intelligence Optimized
Rolling Grey Forecasting Model Fitting to Small Economic Dataset.” Abstract
and Applied Analysis 2014: 1–10. doi:10.1155/2014/641514.
Liu, Sifeng, Jeffrey Forrest, and Y. Yang. 2012. “A Brief Introduction to
Grey Systems Theory.” Grey Systems: Theory and Applications 2 (2): 89–104.
doi:10.1108/20439371211260081.
Liu, S, Y Lin, and JYL Forrest. 2010. Grey Systems: Theory and Applications. Springer-
Verlag Berlin Heidelberg.

52 Social and Economic StudiES
Lorde, T., G. Li, and D. Airey. 2015. “Modeling Caribbean Tourism Demand:
An Augmented Gravity Approach.” Journal of Travel Research 55 (7): 946–56.
doi:10.1177/0047287515592852.
Ma, Dan, Qiang Zhang, Yan Peng, and Shaojie Liu. 2011. “A Particle Swarm
Optimization Based Grey Forecast Model of Underground Pressure for
Working Surface.” Electronic Journal of Geotechnical Engineering 16: 811–30.
Mohammadi, Ali, and Sara Zeinodin Zade. 2011. “Applying Grey Forecasting
Method to Forecast the Portfolio’s Rate of Return in Stock Market of Iran.”
Australian Journal of Business and Management Research 1 (7): 1–16.
Moore, Winston Ricardo. 2010. “The Impact of Climate Change on
Caribbean Tourism Demand.” Current Issues in Tourism 13 (5): 495–505.
doi:10.1080/13683500903576045.
Papatheodorou, Andreas, and Haiyan Song. 2010. “International Tourism
Forecasts: Time-Series Analysis of World and Regional Data.” Tourism
Economics 11 (1): 11–23.
Peirong, Ji, Luo Xianju, and Zou Hongbo. 2007. “A Study on Properties of
GM(1,1) Model and Direct GM(1,1) Model.” In IEEE International Conference
on Grey Systems and Intelligent Services, 399–403. IEEE.
Shen, Xuejun, Limin Ou, Xiaojun Chen, Xin Zhang, and Xuerui Tan. 2013.
“The Application of the Grey Disaster Model to Forecast Epidemic Peaks of
Typhoid and Paratyphoid Fever in China.” PloS One 8 (4): e60601. doi:10.1371/
journal.pone.0060601.
Song, Haiyan, and Gang Li. 2008. “Tourism Demand Modelling and Forecasting—A
Review of Recent Research.” Tourism Management 9 (2): 203–20.
Tsai, Chen-fang. 2012. “The Application of Grey Theory to Taiwan Pollution
Prediction.” In Proceedings of the World Congress on Engineering, II: 2–7.
Uysal, Muzaffer, and John L Crompton. 1985. “An Overview of Approaches
Used to Forecast Tourism Demand.” Journal of Travel Research 23: 7–15.
doi:10.1177/004728758502300402.
Wang, Chao-Hung. 2004. “Predicting Tourism Demand Using Fuzzy Time Series
and Hybrid Grey Theory.” Tourism Management 25 (3): 367–74. doi:10.1016/
S0261-5177(03)00132-8.
Wang, Chao-Hung, and Li-Chang Hsu. 2008. “Using Genetic Algorithms Grey
Theory to Forecast High Technology Industrial Output.” Applied Mathematics
and Computation 195 (1): 256–63. doi:10.1016/j.amc.2007.04.080.
Wang, Y. 2002. “Predicting Stock Price Using Fuzzy Grey Prediction
System.” Expert Systems with Applications 22 (1): 33–38. doi:10.1016/S0957-
4174(01)00047-1.
Wu, Hong, and Fuzhong Chen. 2011. “The Application of Grey System Theory
to Exchange Rate Prediction in the Post-Crisis Era.” International Journal of
Innovative Management, Information & Production 2 (2): 83–89.

GREy foREcaStinG of touRiSm dEmand 53
Zhang, Liping, Yanling Zheng, Kai Wang, Xueliang Zhang, and Yujian Zheng.
2014. “An Optimized Nash Nonlinear Grey Bernoulli Model Based on Particle
Swarm Optimization and Its Application in Prediction for the Incidence of
Hepatitis B in Xinjiang, China.” Computers in Biology and Medicine 49: 67–73.
doi:10.1016/j.compbiomed.2014.02.008.