Discussion Paper No.295
An Empirical Investigation of Tourism Demand Variability:
The Gini Index and Entropy Measure Approach

Kazuhiko NAKAHIRA
Associate Professor
Faculty of Economics
Meikai University

Masahiro YABUTA
Professor
Faculty of Economics
Chuo University

December 2017

INSTITUTE OF ECONOMIC RESEARCH
Chuo University
Tokyo, Japan

An Empirical Investigation of Tourism Demand Variability:
The Gini Index and Entropy Measure Approach
Kazuhiko NAKAHIRA
Associate Professor, Faculty of Economics, Meikai University
Masahiro YABUTA
Professor, Faculty of Economics, Chuo University,
ABSTRACT
Focusing on the tourism in Japan, this paper examined the seasonality of tourism demand in Japan by utilizing
Gini coefficient and some kinds of entropy measures.
Regarding the estimated Gini coefficient, it is large in Hokkaido, Hiroshima, and Okinawa, and small in Tokyo
and Osaka. In addition, the east Japan great earthquake happened in 2011 might affect seasonal fluctuations of many
prefectures. After the earthquake, the seasonal fluctuations expressed by the Gini coefficient seem to vary from year
to year in spite of the general variation of the number of tourists.
Several points were worth considering through the analysis by estimated entropy measure for the Japanese
selected 10 prefectures in the sample period from the year 2008 to 2016. First, there is the one-time rapid increase inseasonality in tourism demand for major four prefectures in 2010 and rapid reversal in next year. In addition, theother two prefectures, have the same pattern in 2011 although the levels of fluctuations are relatively low.
The examination by utilizing the estimated Theil measure shows that the highest concentration of seasonality isrecorded for Tokyo and for the other prefectures located in the east and west areas in Japan just like the case ofentropy measure. In addition, Miyagi and Fukushima follow the same pattern in next year. Okinawa prefecture havethe highest seasonality in 2014 during the period under our study.
The evolution patterns of seasonality for the prefectures described by the estimated relative Theil measure arealmost the same as the ones measured by the two indicators described above.
The analysis by the entropy decompositions at the intra- and the inter-monthly levels are also conducted. The
estimated seasonality of Japanese selected 10 prefectures that could be explained by the intra-monthly part ofentropy measure revealed that the seasonality for the half of the total number of prefectures reﬂects a relative
increase in 2010, and the one for the other two prefectures follows the same pattern in next year. The graphs of theestimated inter-monthly part of entropy measure described the increase in tourism demand seasonality in 2010 likethe other measures. Tokyo experienced the relatively rapid reduction in seasonality in next year as a reactionary fall.
Okinawa had a certain degree of increase in seasonality again in 2013, and had a fluctuation of seasonality during thefollowing periods.
Finally, consideration of the tourism seasonality based on the relation between interest groups - origins anddestinations - by applying the entropy theory are implemented. The estimated entropy measures for annual amountof arrivals from origin (𝒊𝒊) to destination (𝒋𝒋) for the selected major six courses are examined. In particular, theseasonality for the routes Hokkaido→Tokyo, Kyoto→Tokyo, and Okinawa→Tokyo decreased rapidly in 2009, andincreased drastically in 2010 as a rebound. In short, the routes from the three major areas to Tokyo indicate temporal
growth of seasonality.
All the measures based on the entropy theory show a very similar pattern in that the tourism seasonality has the
temporal rapid growth in 2010 and the reactionary fall in 2011. Probably, the financial and economic crisis of global
economy occurred around 2008 and 2009 had a negative influence on Japanese tourism in 2010. By the effect of thecrisis, the number of visits or visiting frequency of tourism in Japan decreased, and its downturn might generate theconcentration of tourism in specific season or month as a result of selective behavior by tourists during the hardtimes.
Key words: seasonality in tourism demand, Gini index, Entropy measure
JEL Classification Code: C33, R58, Z32, Z38
1

1. Introduction
Some previous studies on seasonal variations of tourism mainly deal with technical aspects to grasp theseasonality of tourism demand accurately and policy aspects to consider the means for tourism development andmitigation of seasonal variations on tourism. For instance, Baum and Lundtorp (2001) give a comprehensive
explanation for seasonal variation of tourism, the fixed costs compensation, investment and employment issues,
maintenance of the supply chain, and the continuity of transportation. Butler (2001) overviews the cause and effectof the seasonal variations in tourism and points out the practical difficulties of the measures to mitigate the variations.
as a developed work of Butler (1994), known as a study of the tourism development cycle. Lundtorp (2001)
discusses some approaches including seasonal variation index and Gini coefficient to understand seasonal variationsof the tourism demand. To minimize the effects of the seasonal variation, many researchers including Kulendran andKing (1997), Lim and McAleer (2001), Goh and Law (2002), and Koc and Altinay (2007) tried to estimate thetourism demand accurately. In particular, Koc and Altinay (2007) show that different combination for a number ofvisitors and the tourism expenditure in Turkey could be linked and that the seasonality can be recognized through thetourism expenditure, by using X-12-ARIMA. Nadal et al. (2004) focuses on the UK and German tourists who visit
Baleares Islands, and insist that an increase in GDP or a rise in the relative price would decrease the Gini index. It isan example to show that the economic conditions such as the income and the relative price are decisive factors todetermine the number of visitors from the gravity theory perspective. Tourism seasonality also depends on thecharacteristics of the destination. Visitors for the nature-oriented destination such as the national park are influenced
by the climate condition compared with the ones for the culture-oriented destination. By analyzing the three tourism
destinations in Spain (Malaga, Granada, and Armenia), Morales (2003) insists that the culture-oriented policy suchas cultural exhibition and performance at the theater contributes to mitigate the tourism seasonality. Cuccia andRizzo (2011) show that the seasonality depends on the various cultural attractions by classifying the six destinations
in Italy into four clusters based on their cultural attractiveness.
The seasonality in tourism demand is an issue for the supply side of tourism rather than the demand side, and its
impact affects tourism management. The decision-making in investment and funds management by the manager of
the accommodation, caterings, attractions, and tour operators, such as flexible employment, tourism, anddifferentiation strategy, is essential to ensure stable management and earnings. In this respect, mitigation policy forthe seasonal variation is necessary for stable and profitable management as Lee et al. (2008) discuss. However,
because the causes of tourism seasonality in natural- and socio-economic factors do not correspond with one another,various strategies and individual measures need to be implemented.
By applying and expanding the contribution by the previous studies described above, this paper analyzes seasonalvariation in Japan by using the Gini coefficient and the entropy measure with the data of accommodation tourists inprefecture level. Because our data set includes information on origin and destination of tourists on prefectural levels,
we can investigate the factors of seasonal variation concretely.
The paper is composed of the following chapters. In chapter 2, recent monthly tourism seasonality in Japan is
explained. Chapter 3 examines seasonal fluctuation of tourism demand by the coefficient of variation and the Gini
coefficient. The analysis of seasonality by the entropy measure is described in chapter 4. Chapter 5 is for theconcluding remarks.
2. Tourism Seasonality and Measument
2.1 The Data
The paper uses the data obtained from the “Tourism Statistics Survey”1 of the Japan Tourism Agency, Ministry ofLand, Infrastructure, Transport and Tourism. This statistic is based on the number of guests in each accommodation
facility (hotels, ryokans (inns) and accommodations (companies, organizations, etc.), including the foreign residents.

1

The data on “Tourism Statistics Survey” was retrieved from the website of Japan Tourism Agency, Ministry of Land,
Infrastructure, Transport and Tourism: “http://www.mlit.go.jp/kankocho/siryou/toukei/shukuhakutoukei.
html”.

2

Figure 1: Monthly Tourism Development in Japan (2008-2016)
12000000

10000000

persons

8000000

6000000

4000000

2000000

201610201607

201604

201601

201510201507

201504

201501

201410201407

201404

201401

201310201307

201304

201301

201210201207

201204

201201

201110201107

201104

201101

201010201007

201004

201001

200910200907

200904

200901

200810200807

200801

2008040Year/month

Source: Accommodation travel statistics survey (http://www.mlit.go.jp/kankocho/siryou/toukei/
shukuhakutoukei.html)
This data set indicates the residential areas, nationality, and some other characteristics of tourists as the monthly
observation. The data used in this paper is based on the “Reference Table 2” in the “Tourism Statistics Survey”
which presents the number of tourists by classifying the origin and the destination on the prefectural base.
2.2 Monthly Tourism Seasonality in Japan
Figure 1 shows the tourism development in Japan from 2008 to 2016. The figure illustrates the monthly
fluctuations and the gradual increasing trend. Accommodation tourists tend to increase in August, and decrease inJanuary and February, making a regular pattern of the seasonality as we can see in this figure. Due to the impact of
the East Japan Great Earthquake in 2011, the number of tourists were declined significantly in April 2011.
3. Analysis of Seasonality by Coefficient of Variation and Gini Coefficient
3.1 Coefficient of Variation
The seasonal fluctuation of tourism demand is well known issue. In the case of Japan, tourism demand fluctuates
greatly in monsoon climate season. Seasonal fluctuation of tourism has a significant influence on regional economic
activities such as the operation of tourism-related service industry. In this sense, it is important to numericallyperceive tourism seasonality. Since the pioneering work of Lundtorp (2001), seasonal fluctuation has been explainedby grasping the regularity of the variations in the number of tourists. On the other hand, examination of seasonalvariation has been conducted in several ways, for example, Koenig-Lewis and Bischoff (2005) and Lambert andAronson (1993). Coefficient of variation (CV) and Gini coefficient (or Gini index) are regarded as the main methods
for this topic. CV means the magnitude of the variance around the mean value. It shows the extent of the variation by
concerning the average value of the variation. With the standard deviation (𝑠𝑠) and the mean value (𝜇𝜇), it is definedas
𝐶𝐶𝐶𝐶 =

𝑠𝑠𝜇𝜇(1)

3.2 Gini Coefficient
The Gini coefficient is applied mainly to represent the state of the income distribution in economic context to thepresent. However, the Gini coefficient is often used for analyzing tourism fluctuation. According to Lundtrop (2001),
the general expression of the Gini coefficient is:2
𝑛𝑛

Gini coefficient = ∑𝑛𝑛𝑖𝑖=1(𝑥𝑥𝑖𝑖 − 𝑦𝑦𝑖𝑖 )(2)3Figure 2: Development Pattern of Gini Coefficient
0.300

Hokkaido

0.250

Miyagi
Fukushima

GINI index

0.200

Tokyo
Aichi
Kyoto

0.150

Osaka
Hiroshima
Fukuoka

0.100

Okinawa
Abroad

0.050

0.000

Total

2008

2009

2010

2011

2012

2013

2014

2015

2016

Table1: Gini coefficient and Coefficient of Variation
Hokkaido Miyagi Fukushima Tokyo
2008
0.091
0.061
0.099
0.083
2009
0.105
0.081
0.049
0.092
2010
0.109
0.091
0.074
0.210
2011
0.138
0.179
0.139
0.101
2012
0.121
0.106
0.105
0.034
2013
0.114
0.065
0.141
0.051
2014
0.078
0.054
0.135
0.051
2015
0.099
0.060
0.065
0.034
2016
0.094
0.058
0.075
0.039
Average
0.106
0.084
0.098
0.077
STDdev
0.018
0.040
0.034
0.056
STD/Average
0.169
0.476
0.351
0.724

Aichi
0.077
0.081
0.247
0.089
0.043
0.058
0.053
0.049
0.062
0.084
0.063
0.743

Kyoto
0.083
0.104
0.106
0.088
0.085
0.096
0.081
0.073
0.051
0.085
0.017
0.196

Osaka Hiroshima Fukuoka Okinawa Abroad
0.059
0.104
0.073
0.110
0.081
0.102
0.105
0.093
0.088
0.082
0.231
0.254
0.211
0.106
0.134
0.081
0.072
0.055
0.093
0.131
0.060
0.076
0.061
0.091
0.057
0.057
0.104
0.052
0.136
0.082
0.067
0.086
0.058
0.154
0.080
0.053
0.094
0.061
0.083
0.060
0.042
0.104
0.054
0.073
0.050
0.083
0.111
0.080
0.104
0.084
0.058
0.055
0.051
0.026
0.030
0.695
0.495
0.636
0.253
0.357

Total
0.061
0.074
0.106
0.089
0.059
0.060
0.054
0.050
0.040
0.066
0.021
0.312

Source: Accommodation travel statistics survey(http://www.mlit.go.jp/kankocho/siryou/toukei/shukuhakutoukei.html,2017/08/25 access)𝑖𝑖
𝑛𝑛where n is the number of months (𝑛𝑛 = 12), 𝑥𝑥𝑖𝑖 is the rank of fractiles �𝑥𝑥𝑖𝑖 = �, and 𝑦𝑦𝑖𝑖 is the cumulated fractiles inthe Lorenz curve.
Figure 2 displays the estimation results of Gini coefficient on a monthly basis from the year 2008 to 2016 for theselected Japanese prefectures and for the accommodation tourists from abroad. Table 1 shows their numerical values.
Regarding the estimated Gini coefficient, the following points can be pointed out. First, pattern of seasonal
fluctuation varies by prefecture. It is large in Hokkaido, Hiroshima, and Okinawa, and small in Tokyo and Osaka.
Second, the east Japan great earthquake happened in 2011 might affect seasonal fluctuations of many prefectures
although it did not affect all areas in Japan. After the earthquake, the estimated Gini coefficients are generally in therange from 0.05 to 0.10. Thirdly, in spite of the general variation of the number of tourists, the seasonal fluctuations
expressed by the Gini coefficient seem to vary from year to year.
4. Analysis of Seasonality by Entropy Measure
4.1 Entropy Measure
The so-called Theil’s entropy measure, which is often used for the income distribution analysis, can capture thefeature of the objective in lower part of the distribution. It also can be applied to tourism demand analysis. Theil(1967) provides a tool for analyzing inequality by utilizing entropy theory. The general expression of Theil’s entropy
measure is the following form:
𝐻𝐻(𝛽𝛽) =1
𝑦𝑦 𝛽𝛽∑𝑖𝑖 𝑝𝑝𝑖𝑖 �� 𝑖𝑖� − 1� , 𝛽𝛽 ≠ 0,1
𝜇𝜇𝛽𝛽(𝛽𝛽−1)(3)

where the parameter 𝜷𝜷 is the weight which is given to distances between objectives at different parts of the
distribution taking any real value, 𝒑𝒑𝒊𝒊 is the relative weights of the observations (for example, months, seasons), 𝒚𝒚𝒊𝒊
4is the variable for observations (for instance, tourist demand, in the case of tourism analysis), and 𝝁𝝁 is the mean.2
With respect to the value of 𝜷𝜷, this measure is more sensitive to changes in the lower tail of the distribution, while
this measure is more sensitive to changes that affect the upper tail for higher values. The values of the general
expression class of measures take the value between 0 (perfect equality) and ∞ (or 1 in the case of normalization).
As 𝜷𝜷 tends to −∞ in the limit, Theil’s measure focuses on the lower end. In contrast, the higher 𝜷𝜷 is more sensitive
to changes at the top of the ranking. Actually, for the value which is greater than 2, it seems to be sensitive to thechanges among the highest value of the objective. In addition, careful attention should be paid to 𝑯𝑯(𝟎𝟎) and 𝑯𝑯(𝟏𝟏),and their algebraic expressions are:𝜇𝜇𝑦𝑦𝑖𝑖

𝐻𝐻(𝛽𝛽 = 0) = − ∑𝑖𝑖 𝑝𝑝𝑝𝑝 𝑙𝑙𝑙𝑙 � �
𝑦𝑦

(4)
𝑦𝑦

𝐻𝐻(𝛽𝛽 = 1) = − ∑𝑖𝑖 𝑝𝑝𝑝𝑝 � 𝜇𝜇𝑖𝑖 � 𝑙𝑙𝑙𝑙 � 𝜇𝜇𝑖𝑖 �(5)With respect to the property of decomposability which is applicable to the inequality literature, two different
approaches are given: the group decomposition and the source decomposition. First, the distributive observations are
divided into groups to distinguish which part of the total inequality should be attributed. In this context, theaggregate index, that is additively decomposable, could be described as the sum of the between-group component
which would measure the average dissimilarity among the groups and the within-group component which would
reﬂect the internal differences as a weighted average of the interior inequalities.
Generally, an index 𝐼𝐼 is additively decomposable if the decomposition format can be expressed as
𝐼𝐼 = ∑𝑔𝑔 𝑤𝑤𝑔𝑔 𝐼𝐼𝑔𝑔 + 𝐼𝐼𝑏𝑏
(6)
where 𝑤𝑤𝑔𝑔 is the weight for each group, 𝐼𝐼𝑔𝑔 describes the inner-inequality for each group, and 𝐼𝐼𝑏𝑏 is the inequality
among groups. The Theil Index is also additively decomposable so far as its format follows equation (6). According
to Shorrocks (1984), the group decomposition of 𝑇𝑇(0) generates less ambiguity if the structure of the model isexpressed as:
1
𝑔𝑔

1
𝑔𝑔

𝜇𝜇
𝑦𝑦𝑖𝑖

𝐻𝐻(0) = 𝐻𝐻(0)𝑤𝑤 + 𝐻𝐻(0)𝑏𝑏 = ∑𝑔𝑔 + ∑𝑔𝑔 𝑙𝑙𝑙𝑙 � �,

(7)

where 𝑇𝑇(0)𝑤𝑤 represents the aggregate between-groups inequality component, 𝑇𝑇(0)𝑏𝑏 is the internal inequality ingroup “𝑔𝑔”, 𝑦𝑦𝑖𝑖 is the tourism activity in month “𝑖𝑖”, and 𝜇𝜇 is the average annual average on tourism activity indicator.

4.2 Theil Measure
We can interpret the discussion in the previous section by a simple way. Theil’s entropy measure is simply
described by
1
𝑝𝑝𝑖𝑖

𝐻𝐻 = ∑𝑖𝑖 𝑝𝑝𝑖𝑖 𝑙𝑙𝑙𝑙 � �,

(8)where 𝑝𝑝𝑖𝑖 is the probability of an event. In this respect, Theil’s entropy measure shows that “entropy” is the weighted
summation of information value for each event. It takes the maximum value if all events occur with the sameprobability.
This measure has been applied to some kinds of social problem, for example, income distribution, tourism
seasonality. Let us consider a demand for tourism, 𝑥𝑥𝑖𝑖 instead of 𝑝𝑝𝑖𝑖 , and impose a normalization as ∑𝑖𝑖 𝑥𝑥𝑖𝑖 = 1. Then,the entropy for distribution of tourism demand takes the maximum value, 𝑙𝑙𝑙𝑙(𝑛𝑛), if tourism demands are distributed
equally (i.e. flat distribution) to each area of tourism where 𝑛𝑛 represents the number of area for the analysis. In thissense, “𝑙𝑙𝑙𝑙(𝑛𝑛) − 𝐻𝐻” describes the measure of inequality. Therefore, the so-called “Theil measure” for normalized
distribution 𝒙𝒙 = (𝑥𝑥1 ,∙∙∙, 𝑥𝑥𝑛𝑛 ) is expressed as
1
𝑥𝑥𝑖𝑖

𝑇𝑇(𝑥𝑥) = 𝑙𝑙𝑙𝑙(𝑛𝑛) − ∑𝑖𝑖 𝑥𝑥𝑖𝑖 𝑙𝑙𝑙𝑙 � � = ∑𝑖𝑖 𝑥𝑥𝑖𝑖 𝑙𝑙𝑙𝑙(𝑛𝑛𝑥𝑥𝑖𝑖 ).
2(9)

See Duro (2012) for a reference.

5

Since the total tourism demand is normalized to 1, 𝑥𝑥𝑖𝑖 virtually means the share of tourism demand for the 𝑖𝑖th area.
Therefore, Theil measure is a decreasing function of the weighted summation of reciprocal of tourism demand sharein logarithm. It takes a value of 0 (zero) when there is no seasonality in tourism (i.e. inequality of tourism demand)and a maximum value (𝑙𝑙𝑙𝑙(𝑛𝑛)) when all activities in tourism are concentrated in a single moment.
Theil measure satisfies the conditions of Pigou–Dalton principle in terms of equivalent income and constantrelative inequality-aversion since it depends only on the share of tourism demand. It used to express economic
inequality based on information theory. The entropy index is often applied to income inequality analysis based on
Pigou-Dalton's transfer principle. In this line of discussion, consider the case of the seasonality of tourism andassume that the number of tourists in the whole year be constant. Then, the PD condition is satisfied in that the levelof the seasonal variation changes if the number of visitors in a non-concentrated period decline while the visiting
frequency of the concentrated period increase. In other words, under the PD condition, the indicator of the seasonalvariation based on entropy theory is affected by the distribution of tourists in each month.
One of the problems that we have to take into account in the context of our empirical study is the fact that therange of Theil measure varies with the number of elements. For instance, in the case of annual data, there are 365
days for non-leap years and 366 days for leap years. To deal with this problem, relative Theil measure (relative
redundancy measure) is proposed.
𝑅𝑅 = 1 −𝐻𝐻𝑙𝑙𝑙𝑙(𝑛𝑛)(10)

It takes 0 (zero) in the case of no seasonality and 1 (one) if we have perfect seasonality, respectively.
4.3 Decomposition of Entropy Measure and Relation between Interest Groups
One of the properties of the entropy measure is its additive nature. In short, the total entropy can be decomposed
into the weighted summation of entropies within groups and among groups. The entropy within groups is𝑥𝑥

1

𝐻𝐻(𝑋𝑋𝜏𝜏 ) = ∑𝑥𝑥𝑡𝑡∈𝐴𝐴𝜏𝜏 𝑋𝑋𝑡𝑡 𝑙𝑙𝑙𝑙 � 𝑥𝑥𝑡𝑡 �.
𝜏𝜏

(11)

𝑋𝑋𝜏𝜏

On the other hand, the entropy among groups is defined as:
𝑋𝑋

1

𝜏𝜏
𝐻𝐻(𝑋𝑋1 ,∙∙∙, 𝑋𝑋𝑚𝑚 ) = ∑𝑚𝑚
𝜏𝜏=1 𝑋𝑋 𝑙𝑙𝑙𝑙 � 𝑋𝑋𝜏𝜏 �,

(12)

𝑋𝑋

where 𝑋𝑋𝜏𝜏 = ∑𝑥𝑥𝑡𝑡∈𝐴𝐴𝜏𝜏 𝑥𝑥𝑡𝑡 and 𝐴𝐴𝜏𝜏 is a subset for a 𝜏𝜏 = 1,∙∙∙∙ 𝑚𝑚 ≤ 𝑛𝑛.

Totally, the decomposition of the Theil’s entropy measure based on its additive nature is described as:
1
𝑝𝑝𝑡𝑡

𝑋𝑋

1

𝜏𝜏
𝐻𝐻 = ∑𝑛𝑛𝑡𝑡=1 𝑝𝑝𝑡𝑡 𝑙𝑙𝑙𝑙 � � = ∑𝑚𝑚
𝜏𝜏=1 ∑𝑥𝑥𝑡𝑡∈𝐴𝐴𝜏𝜏 𝑋𝑋 𝑙𝑙𝑙𝑙 � 𝑥𝑥𝑡𝑡 �𝑋𝑋= ∑𝑛𝑛𝜏𝜏=1 𝑋𝑋𝜏𝜏 𝐻𝐻(𝑋𝑋𝜏𝜏 ) + 𝐻𝐻(𝑋𝑋1 ,∙∙∙, 𝑋𝑋𝑚𝑚 ),

𝑋𝑋

(13)

where the first element is the weighted summation of entropies within groups and the second one represents theentropy among groups.
For the empirical analysis using a monthly dataset, 𝑋𝑋𝜏𝜏 is composed of 𝜏𝜏 = 1,∙∙∙ ,12 and the monthly temporal
aggregation is:
𝑋𝑋

𝑋𝑋

1

𝜏𝜏
𝜏𝜏
12
𝐻𝐻 = ∑12𝜏𝜏=1 𝑋𝑋 𝐻𝐻(𝑋𝑋𝜏𝜏 ) + ∑𝜏𝜏=1 𝑋𝑋 𝑙𝑙𝑙𝑙 � 𝑋𝑋𝜏𝜏 �,

(14)

𝑋𝑋

where the first addend represents the weighted intra-monthly entropy and the second one is the inter-monthly
entropy. In the empirical analysis, a part of the inter-monthly entropy is generated by the differences of the length ofmonth, since the days of each month are not all the same.3

3

See Rossello and Sanso (2017) for details.

6In addition to the above discussions, we can consider the relation between interest groups, for example, tourism
origin and destinations, by applying the entropy theory. The entropy measure for annual amount of arrivals fromorigin (𝑖𝑖) to destinations (𝑗𝑗) is described as:𝑥𝑥

𝑥𝑥

𝐻𝐻𝑖𝑖𝑖𝑖∙ �𝑥𝑥𝑖𝑖𝑖𝑖𝑖𝑖 � = ∑𝑛𝑛𝑡𝑡=1 𝑥𝑥𝑖𝑖𝑖𝑖𝑖𝑖 𝑙𝑙𝑙𝑙 �𝑥𝑥 𝑖𝑖𝑖𝑖∙ �,
𝑖𝑖𝑖𝑖∙

𝑖𝑖𝑖𝑖𝑖𝑖(15)

where 𝑥𝑥𝑖𝑖𝑖𝑖∙ = ∑𝑛𝑛𝑡𝑡=1 𝑥𝑥𝑖𝑖𝑖𝑖𝑖𝑖 . This measure is able to be utilized seasonality analysis for tourism.

4.4 Empirical Analysis by the Entropy Measures
In this section, empirical analysis by applying some kinds of the entropy measures described in above sections isconducted, The analysis of the seasonality in tourism demand is implemented with the monthly data set4 which is thesame as the one used in sections 2 and 3. The data on tourism arrivals is applied as the proxy variable for the tourism
demand. The entropy measure, the Theil measure, the relative Theil measure, the intra-monthly decomposition partof entropy measure, and the inter-monthly decomposition part of entropy measure are estimated. In addition, theentropy measure for annual amount of arrivals from origin to destination is considered.5
In our analysis, major 10 prefectures in terms of their annual tourism demands are focused on. In order of demand,the following prefectures are included: Hokkaido, Miyagi, Fukushima, Tokyo, Aichi, Kyoto, Osaka, Hiroshima,
Fukuoka, and Okinawa. These prefectures typically have strong tourism demands, and the reason for including
Fukushima is as follows. Fukushima prefecture suffered a certain damage by the Tohoku earthquake and tsunamithat hit Japan in March 2011. Whether it can recover the tourism demand is a continuous topic for discussion in
Japan. In this respect, Fukushima is turned into an object of our analysis.
Figure 4 displays the estimated entropy measures with respect to the tourism demand for the Japanese selected 10
prefectures in the sample period from the year 2008 to 2016. Several points are worth considering. First, there is the
one-time rapid increase in seasonality (decrease in the value of measure) in tourism demand for Tokyo, Osaka,
Hiroshima, and Aichi in 2010 and rapid reversal in next year. Second, the other two prefectures, Fukushima andMiyagi, have the same pattern in 2011 although the levels of fluctuations are relatively low. Third, the entropy
measure for Fukushima prefecture continually fluctuates during the sample period.
Figure 5 shows the seasonality which is identiﬁed by the estimated Theil measure during the period under ourstudy. Just like the case of entropy measure, the highest concentration of seasonality is recorded for Tokyo and forthe other prefectures located in the east and west areas in Japan. In addition, Miyagi and Fukushima follow the same
pattern in next year. Okinawa prefecture have the highest seasonality in 2014.
Figure 6 describes the seasonality estimated by the relative Theil measure for our sample period. The evolution
patterns of seasonality for the prefectures are almost the same as the ones measured by the two indicators describedabove.
The analysis can be taken further by applying the decomposition property of the Theil’s entropy measure based on
the additive nature described in the former section. Namely, the entropy decompositions at the intra- and the intermonthly levels are applied here. The estimated seasonality of Japanese selected 10 prefectures that could be
explained by the intra-monthly part of entropy measure is highlighted in Figure 7. As the graphs show, theseasonality for the half of the total number of prefectures reﬂects a relative increase in 2010, and the one for the othertwo prefectures follows the same pattern in next year. Except 2010 and 2011, the values of the intra-monthly
measure fluctuate in a relatively stable manner.
Figure 8 represents the estimated inter-monthly part of entropy measure with the lowest about 2.3 and the highest
about 3.12. The graphs of this measure describe the increase in tourism demand seasonality in 2010 like the othermeasures explained above. Tokyo, the capital city, experienced the relatively rapid reduction in seasonality in next

4

In any case, the adoption of the longer period would tend to increase the inequality inside the groups, and to reduce thebetween-group inequality.
5
With respect to the methodology in tourist applications, see Fernandez-Morales (2003) for example.

7

Figure 4: Estimated Entropy Measure

Figure 5: Estimated Theil Measure

Figure 6: Estimated Relative Theil Measure

8

Figure 7: Estimated intra-monthly decomposition part of entropy measure

Figure 8: Estimated inter-monthly decomposition part of entropy measure

Figure 9: Estimated entropy measure for annual amount of arrivals from origin to destinations

9

year as a reactionary fall. Okinawa had a certain degree of increase in seasonality again in 2013, and had afluctuation of seasonality during the following periods.
Since lengths of the months are not all the same, part of the inter-monthly entropy is generated by thesedifferences. It might be the cause of the fact that the summation of the intra- and the inter-monthly values is notequal to the one of the original entropy measure.
Results derived by the method of decomposition analysis may be the way of testing the usefulness of the temporal
aggregation. If we acknowledge that most of the annual seasonality inequalities are attributable to intra-groups
disparities, we might consider that informative relevance of the month-partitions would be limited. This is becauseinternal coherence within groups would be small, losing the signiﬁcance of these groupings. However, it does notseem to be the case of our analysis.
Finally, we consider the tourism seasonality based on the relation between interest groups, in short, origins anddestinations of tourism, by applying the entropy theory explained in section 4.3. The estimated entropy measures forannual amount of arrivals from origin (𝒊𝒊) to destination (𝒋𝒋) for the selected major six courses, Tokyo → Hokkaido,
Tokyo → Kyoto, Tokyo → Okinawa, Hokkaido → Tokyo, Kyoto → Tokyo, and Okinawa → Tokyo, aresummarized in Figure 9. In particular, the seasonality for the routes Hokkaido→Tokyo, Kyoto→Tokyo, andOkinawa → Tokyo decreased rapidly in 2009, and increased drastically in 2010 as a rebound. In short, the routes
from the three major areas to Tokyo indicate temporal growth of seasonality. The opposite directions did not always
set forth a trend like that.
All the measures described above show a very similar pattern in that the tourism seasonality has the temporal
rapid growth in 2010 and the reactionary fall in 2011. Probably, the financial and economic crisis of global economy
occurred around 2008 and 2009 had a negative influence on Japanese tourism in 2010. By the effect of the crisis, thenumber of visits or visiting frequency of tourism in Japan decreased, and its downturn might generate theconcentration of tourism in specific season or month as a result of selective behavior by tourists during the hardtimes.
5. Concluding Remarks
Focusing on the tourism in Japan, this paper examined the seasonality of tourism demand in Japan by utilizing
Gini coefficient and some kinds of entropy measures. Our analysis found the following facts.
Regarding the estimated Gini coefficient, pattern of seasonal fluctuation varies by prefecture. It is large in
Hokkaido, Hiroshima, and Okinawa, and small in Tokyo and Osaka. In addition, the east Japan great earthquake
happened in 2011 might affect seasonal fluctuations of many prefectures although it did not affect all areas in Japan.
After the earthquake, the estimated Gini coefficients are generally in the range from 0.05 to 0.10. Further, in spite of
the general variation of the number of tourists, the seasonal fluctuations expressed by the Gini coefficient seem tovary from year to year.
Several points were worth considering through the analysis by estimated entropy measure for the Japanese
selected 10 prefectures in the sample period from the year 2008 to 2016. First, there is the one-time rapid increase inseasonality (decrease in the value of measure) in tourism demand for major four prefectures in 2010 and rapid
reversal in next year. In addition, the other two prefectures, have the same pattern in 2011 although the levels offluctuations are relatively low.
The examination by utilizing the estimated Theil measure shows that the highest concentration of seasonality isrecorded for Tokyo and for the other prefectures located in the east and west areas in Japan just like the case ofentropy measure. In addition, Miyagi and Fukushima follow the same pattern in next year. Okinawa prefecture havethe highest seasonality in 2014 during the period under our study.
The evolution patterns of seasonality for the prefectures described by the estimated relative Theil measure arealmost the same as the ones measured by the two indicators described above.
The analysis can be taken further by applying the decomposition property of the Theil’s entropy measure based on
the additive nature - the entropy decompositions at the intra- and the inter-monthly levels. The estimated seasonalityof Japanese selected 10 prefectures that could be explained by the intra-monthly part of entropy measure revealed
10that the seasonality for the half of the total number of prefectures reﬂects a relative increase in 2010, and the one forthe other two prefectures follows the same pattern in next year. Except 2010 and 2011, the values of the intramonthly measure fluctuate in a relatively stable manner.
The graphs of the estimated inter-monthly part of entropy measure described the increase in tourism demandseasonality in 2010 like the other measures explained above. Tokyo, the capital city, experienced the relatively rapid
reduction in seasonality in next year as a reactionary fall. Okinawa had a certain degree of increase in seasonality
again in 2013, and had a fluctuation of seasonality during the following periods.
Finally, consideration of the tourism seasonality based on the relation between interest groups - origins anddestinations - by applying the entropy theory are implemented. The estimated entropy measures for annual amountof arrivals from origin (𝒊𝒊) to destination (𝒋𝒋) for the selected major six courses are examined. In particular, theseasonality for the routes Hokkaido → Tokyo, Kyoto → Tokyo, and Okinawa → Tokyo decreased rapidly in 2009,and increased drastically in 2010 as a rebound. In short, the routes from the three major areas to Tokyo indicate
temporal growth of seasonality.
All the measures based on the entropy theory described above show a very similar pattern in that the tourism
seasonality has the temporal rapid growth in 2010 and the reactionary fall in 2011. Probably, the financial andeconomic crisis of global economy occurred around 2008 and 2009 had a negative influence on Japanese tourism in2010. By the effect of the crisis, the number of visits or visiting frequency of tourism in Japan decreased, and its
downturn might generate the concentration of tourism in specific season or month as a result of selective behavior by
tourists during the hard times.
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