Discussion Paper No. 103
Estimating the value of coral reefs in Kume Island,
Japan, using a contingent behavior method:A Poisson-Inverse Gaussian approachwith on-site correction
Katsuhito Nohara, Masaki Narukawa
and Akira Hibiki
October, 2019

Data Science and Service Research
Discussion Paper

Center for Data Science and Service Research

Graduate School of Economic and Management
Tohoku University27-1 Kawauchi, Aobaku

Sendai 980-8576, JAPANEstimating the value of coral reefs in Kume Island, Japan, using acontingent behavior method: A Poisson-Inverse Gaussian approachwith on-site correction
Katsuhito Noharaa,*, Masaki Narukawab and Akira Hibikic
aHokusei Gakuen University, bOkayama University, and cTohoku University

Abstract
Coral reefs face a critical crisis because of rising ocean temperature, human resource use, run-off of redsoil, among others. Since the recreational and tourism value of reef, in particular, have great potential, thedegradation of reef quality may have a great effect on the tourism industry of Okinawa Prefecture, whichlargely depends on it. This study employs a contingent behavior approach to estimate the effect of reef
extinction on recreational demand for Kume Island, Okinawa, Japan. Moreover, the Poisson-Inverse
Gaussian model with correction for on-site sampling issues proposed in this study can potentially derivemore accurate estimates of consumer surplus. The results show that the CS losses in the case of coral reef
extinction become about 627.78 million yen per year.

Keywords: Contingent behavior, Coral reef; Count data model; On-site sampling; Poisson-Inverse
Gaussian model; Random-effects model

1. Introduction
Currently, coral reefs worldwide are facing serious threats. The Paris Agreement proposed topursue efforts to limit the temperature increase to 1.5 °C above pre-industrial levels. However, even if
we can achieve that target, 70 to 90 percent of coral reefs are projected to decline with high statisticalconfidence by the middle of this century (IPCC, 2018). The Okinawa Prefecture has 80 percent of thecoral reefs of Japan. However, they now tend to decrease because of factors including coral reefs
bleaching, primarily due to climate-induced ocean warming, feeding damage by acanthaster, and runoff of red soil. (Hongo and Yamano, 2013). The Ministry of the Environment started an investigationof the coral reef communities in 2017 to capture their condition by using images from an artificial
satellite and results from a field study. In 1991, the area covered by more than 50 percent of coral reefs
filled 5.5 percent of all the area in the surrounding waters of Ishigaki Island and Iriomote Island.
However, from the recent investigation, its coverage reduced by approximately 0.1 percent. A* Corresponding author at: School of Economics, Hokusei Gakuen University, 2-3-1 Ohyachi-Nishi, Atsubetsu-ku, Sapporo, 004-

8631, Japan. E-mail address: noahra@hokusei.ac.jp

1

supplementary investigation in 2018 concluded that coral reefs bleaching occurred at all observation
spots (https://www.env.go.jp/press/105494-print.html). However, in 2010, the World Wildlife Fund(WWF) reported that a diver and fisherman had discovered a new coral reef community in Kume
Island (https://www.wwf.or.jp/activities/achievement/3434.html). Although its scholarly value isremarkably high, it might disappear due to ocean warming or other external factors. As coral reef
bleaching occurs in many other remote islands of the Okinawa Prefecture, it seems to be an urgent
issue to conserve coral reefs in Kume Island even if those conditions are not terrible at present.
In general, coral reefs provide many ecological goods and services, such as food provision,
shoreline protection, erosion regulation, biogeochemical cycling, and tourism and recreational
opportunities (Elliff and Kikuchi, 2017; Robles-Zavala and Reynoso, 2018). Additionally, manystudies point out that coral reefs have multiple ecosystem functions which support tourism benefits
such as the generation of fine sand beaches, the maintenance of islands, protection from storm, andthe production of seafood (Perry et al., 2015; Kench, 2014; Perry et al., 2011; Ferrario et al., 2014;
Cabral and Geronimo, 2018). The tourism or recreation value of coral reefs in the world have thehighest potential net benefits, and the net present value includes the value of fisheries, coastal
protection, and biodiversity (Cesar et al., 2003; van Beukering et al., 2011). Therefore, the degradationof coral reefs would seriously affect the tourism industry. According to the Okinawa Prefectural
Government1 in 2016, a ripple effect on other industries of the tourism industry will be approximately
1.14 trillion yen. As the Okinawa Prefecture largely depends on the tertiary industry, the detriment to
the tourism industry would lead to a sluggish regional economy.
A large number of studies have attempted to estimate the monetary value of coral reefs in several
parts of the world by using non-market valuation methods. The travel cost method (TCM) using
revealed preference (RP) data is one of the widely accepted techniques to assess the value of outdoor
recreational activities. However, it is difficult to estimate the difference in consumer surplus (CS)when the quality of the environment has changed. Therefore, combining contingent behavior (CB)classified as stated preference (SP) data with TCM (TCM + CB) has recently been attempted. The CB
model asks individuals to state their intended visit frequency if environmental quality changes undera hypothetical situation (Lienhoop and Ansmann, 2011; Pueyo-Ros et al., 2018). TCM+CB is often
applied to quality changes to estimate benefits which includes sports fishing, recreational fishing,coastal wetland, swimming, cave diving, and winter outdoor recreation (Alberini et al., 2007; Prayaga
et al., 2010; Pueyo-Ros et al., 2018; Deely et al., 2019; Lankia et al., 2019; Morgan and Huth, 2011;
Filippini et al., 2018). Apart from these, some studies are adopting TCM + CB to valuate coral reefs.
Bhat (2003) estimates the recreational benefits if the quality of coral reefs is improved by the so-called
gamma distributed Poisson random-effects model in TCM + CB, which indicates that the number of

Okinawa Prefectural Government, http://www.pref.okinawa.jp/site/chijiko/kohokoryu/foreign/english/index.html, Accessed date:
16 April 2018.
1

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trips will increase by about 43 percent and the change in CS per person will be US$3,080 under the
scenario of 100 percent improvement of coral quality. Folkersen et al. (2018) employ TCM + CB toestimate the effect of deep sea mining on future trip demand in Fiji, although they focus on the numberof planned future trips with and without deep sea mining. This approach means that irrespective ofwhether degradation of coral reefs would occur, the recreational use-value of coral reefs is limited todiving and snorkeling. However, Kragt et al. (2009) estimate the effects of Great Barrier Reef
degradations on trip demand by using only CB data in a negative binomial random-effects model.
Although almost all of the previous studies estimate the effects of environmental improvements on
trip demand, they assess the effects of environmental degradation on recreational demand. That is,they use the number of future trips as SP data under the hypothetical scenario of decline in reef quality.
Following their approach, we use only CB data by asking respondents about their future trips under
the scenario of both its current state and extinction. Note that this study focuses on not the
improvement of reef quality but the extinction of coral reefs because of two reasons. First, for example,
they argue that using the number of planned trips at current and degraded reef quality is more suitablein the case of Great Barrier Reef quality decline, from which they consider an 80 percent reduction ofcoral reefs as a hypothetical scenario. However, it seems difficult for respondents to imagine the effects
of reef degradation, such as an 80 percent loss on their future trips even if they are shown pictures.
Second, we pay much attention to the fact that coral reefs were imminently threatened with extinctionin the past, and this situation is worsening year by year. For instance, multiple events of coral bleaching
have been recorded in most regions since the mass bleaching event of 1998, which cause a 100 percent
coral depletion in some regions.2 Given the state of recent coral reefs, such a scenario is not unrealistic.
Furthermore, the extinction of coral reefs does not necessarily induce zero recreation demand,
although it may affect water quality and landscape. According to a public opinion poll carried out by
the Cabinet Office (2014), for example, most people do not recognize the ecosystem services of coral
reef such as recreation or tourism included in cultural services; only 19 percent do. Meanwhile, inJapan, the recreational value of coral reefs has been little investigated. Oh (2004) and Tamura (2006)
estimate the non-use values of coral reefs in the Kerama Islands and around the Akajima sea area byusing CVM, respectively. The problem seems to lie in the fact that there are no studies on estimating
the effects of the decline in reef quality on future recreational demand in Japan.
It should also be emphasized that the previous studies estimating the value of coral reefs have notconsidered the possibility of employing a more suitable statistical approach. For example, although
Prayaga et al. (2010) and Pueyo-Ros et al. (2018) have not estimated the value of coral reefs directly,
they adopt a pooled TCM + CB model which cannot capture individual-specific effects in count data.
Despite that, Bhat (2003) collects the data through an on-site survey; however, an estimation problem
related to the sampling is not dealt with. Statistical analysis of such on-site count data should be
2

United Nations Environment Programme, Coral Reef Unit, http://coral.unep.ch/Coral_Reefs.html, Accessed date: 16 April 2018.

3

controlled for truncation and endogenous stratification, as advocated by Shaw (1988), who addresses
these issues in the Poisson regression model. As stated above, Kragt et al. (2009) analyze CB data bya negative binomial random-effects model; however, their model is not adjusted for on-site sampling.
As far as collecting data through an on-site survey, even if only CB data is used in the estimation, it
must be corrected for its issues. In addition, as argued by Guo and Trivedi (2002), Sarker and Surry(2004), and Cameron and Trivedi (2013), the capability of negative binomial to capture overdispersion
would be limited and inadequate if the data has a distribution with a so-called long (heavy) tail.
Therefore, a reliable statistical inference cannot be made. However, Willmot (1987) and Dean et al.
(1989) consider the Poisson-Inverse Gaussian (PIG) model to be an easier and more usable parametricmodel because, in an analysis of insurance data, it reflects more long-tailed count data than thenegative binomial model, even with the same number of parameters. Guo and Trivedi (2002) apply
the PIG model to an analysis of patent data. Since this paper uses trip number data from an on-site
survey, our estimation approach is based on the PIG model and add Shaw’s (1988) correction for onsite sampling issues. Moreover, to analyze the CB data, we expand it into a random-effects model thatcan use pseudo panel data, as in Beaumais and Appéré (2010). Although Narukawa and Nohara (2018)
have recently proposed an estimation approach to panel count data (truncated at zero), to utilize TCM+ CB, they presume that the data are collected via a web-based off-site survey. Our approach is clearly
different from theirs since we consider the PIG model adjusted for an on-site survey. To our knowledge,
this is the first attempt to construct the PIG approach using on-site sampling data, which is anothercontribution of this study.
This study estimates the changes of consumer surplus in the Kume Island trip resulting from adecline in reef quality using the PIG approach while controlling for on-site sampling in CB. The
remainder of this study is composed as follows. In the next section, we explain the data collection
procedure. Section 3 proposes the estimation approach based on the PIG model for an on-site survey.
Section 4 provides the estimation results and the welfare estimates related to loss of reef quality.
Section 5 concludes and presents the topics for future research.

2. Survey design and data
2.1 Data collection
Kume Island is located about 90 km west from the main island of Okinawa (Fig.1) and blessedwith a richness of natural resources that yields much potential ecosystem services. However, not somany tourists visit as compared to other isolated islands of the Okinawa Prefecture, such as Ishigaki
Island or Miyako Island. However, the rich natural resources have been largely preserved and4

Main island of Okinawa

Japan

Kume Island

Miyako Island

Iriomote Island

Ishigaki Island

Source: http://www.craftmap.box-i.net/ken.php

Fig. 1 Location of Kume island

untouched due to the absence of large-scale development in Kume Island. Table 1 shows the top five
most visited isolated islands in 1985 and the corresponding number of tourists in 1985 and 2015, which
is obtained from the Okinawa Prefectural Government (2018). Although the increasing rate of tourists
in the past thirty years in Kume Island is of least value among the isolated islands, many tourists will
likely be attracted to Kume Island in the near future because there remain precious reefs as stated in
Section 1.
Our survey was conducted for a week in September 2015 at the Kumejima airport; it included
weekdays and weekend.3 The sample was randomly selected at the airport, as well as from limited
visitors who had come from other prefectures to end their trips in Kume Island. We presented
photographic material to respondents, which was provided by a scientist, Dr. Hiroya Yamano, to help
respondents comprehend the condition of the reef going extinct. We obtained 302 respondents, but not
all were used for analysis due to missing information. Thus, 168 respondents were included in theempirical analysis of this study. The information was gathered by distributing a survey questionnaire
to visitors who came on a trip to Kume Island. The questionnaire included accompanying persons and3

The survey questions used in this study are available upon request.

5

Table 1 The number of tourists for 30 years in isolated islands of Okinawa Prefecture
Island name

1985

2015

The rate of increase (%)

Ishigaki

250,072

11,477,964

4489.86

Miyako122,715

511,665316.95

Kume81,268102,797

26.49

Iriomote

71,405

380,573432.98Ie

58,000135,739134.03

their age, activities enjoyed during the trip, the interest for natural resources in Kume Island, the visit
duration in Kume Island, travel mode, several demographic characteristics, and the number of trips
planned for the next ten years at the current reef quality given the extinction of coral reef. It was
expected that these variables would affect visitors’ trip frequency significantly. Moreover, we gathereddata on whether visitors had stayed at Naha city to locate the main destination of the trip. Since there
were no direct flights between Kume Island and other domestic regions, and all visitors had to transit
at Naha airport, we can exclude the possibility of a substitute recreation site affecting the demand for
Kume Island and apply the framework of a single-site demand function. Although 19 percent ofrespondents made a stop in Naha city and all of them had an overnight stay there, their length of stayat Kume Island was greater than that of Naha city. Concerning CB questions, Kragt et al. (2009) and
Folkersen et al. (2018) ask respondents about the planned trips for the next five years under ahypothetical scenario. However, we set its period to the next ten years because it would be unrealistic
for the extinction of coral reefs to occur in such a short term.
From the above survey design and the collected data, the variables used in our analysis are
summarized in Table 2. In general, the recreation benefits of the quality changes are measured as CS,
which is the area between the RP trip demand curves and the SP trip demand curves (Whitehead et al.,2000). In other words, respondents provide the actual number of trips (the observed behavior data)
under the current reef quality and the planned reef visits (the contingent behavior data) based on ahypothetical reef quality. However, as discussed by Bockstael et al. (1989) and Kragt et al. (2009),
since incorporating the actual number of trips in the recreational demand function would result inbiased estimates of CS, it seems more appropriate for its estimation to employ the difference between
the planned recreational demand at the current reef quality, as well as the degraded reef quality. Thus,
we estimate CS by using the number of planned trips at the current reef quality and at the extinctionof coral reefs as dependent variables in the subsequent empirical model.

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Table 2 Definition of variables used in the model
Variable

Definition

Mean

SD

Visit_SP0

Number of planned recreational trips to Kume Island in the next ten years

3.417

3.287

0.452

1.866

7.017

1.525

at the current reef quality;
Visit_SP100

Number of planned recreational trips to Kume Island in the next ten years
at the degraded reef quality (100 percent loss)TC

Per-person travel costs to access Kume Island (¥10,000)

SP100

Dummy variable denoting trip counts elicited through a contingent
behavior question

IncomeHousehold income (¥1,000,000)

5.500

3.864

Education

Dummy variable = 1 if the respondent has graduated from the university

0.601

0.491

Accompany

Number of accompanying persons

2.571

1.610

Alone

Dummy variable = 1 if the respondent takes a trip alone

0.083

0.277

Kume 1

Dummy variable = 1 if the respondent experiences marine leisure during

0.845

0.363

Kume 2the trip

0.143

0.351

Kume 3

Dummy variable = 1 if the respondent experiences diving during the trip

0.006

0.077

Dummy variable = 1 if the respondent participates in activities in a natural
Interesting 1

environment during the trip

0.637

0.482

Interesting 2

Dummy variable = 1 if the respondent is interested in coral reef

0.673

0.471

Interesting 3

Dummy variable = 1 if the respondent is interested in marine species

0.054

0.226

Dummy variable = 1 if the respondent is interested in endemic insects inInteresting 4

Kume Island

0.821

0.384

Naha stay

Dummy variable = 1 if the respondent is interested in landscape

0.185

0.389

Days

Dummy variable denoting stay at Naha city (yes = 1)

3.690

1.137

Experience 1

Trip length in Kume Island

0.976

0.153

Dummy variable = 1 if the respondent has prior experience of marine
Experience 2

activities except for diving

0.446

0.499

Experience 3

Dummy variable = 1 if the respondent has prior experience of diving

0.054

0.226

Dummy variable = 1 if the respondent has prior experience of activities
in a natural environment

2.2 Travel costs
The travel costs are computed as the round trip costs from origin to destination. Specifically, we
calculated them by summing up 1) the costs from the nearest public office to the nearest airport and2) airfares from that airport to the Kumejima airport. First, when respondents used their own car
between their house and the nearest airport, the costs were defined as the petrol cost at that time (1357yen) which corresponds to the Price Survey of Oil Products published by Agency for Natural
Resources and Energy (2015).4 The average runnable distance per liter (26.1 km/l) of passenger cars
using petrol, based on the List of Vehicle Fuel Consumption published by Ministry of Land,
Infrastructure, Transport and Tourism (2015a),5 was adopted for the calculation of fuel consumption.
If respondents used the highway for time savings, we assumed that they referred to Drive Plaza6 toinfer their cost. The distance from a respondent’s house to the nearest airport was calculated using
Google Maps.7 When the respondents used rental cars between their house and the nearest airport, weregarded that costs as the sum of the price of a rental car and the petrol cost. The rental car fee wascalculated by using the price list of the nearest rental car shop from a respondent’s house. We assumed
that the price of a rental car is the one-way car rental fee.8,9 When respondents used taxi services or
public transportation between their house and the nearest airport, the cost was calculated by summingup each fee from the appropriate internet site.10,11 Second, airfares from the nearest airport to the
Kumejima airport were calculated by using the Airplane Passenger Survey published by the Ministryof Land, Infrastructure, Transport and Tourism (2015b). 12 We adopted the discount which mostpassengers utilized at each air route. The opportunity cost of time between respondents’ house andKumejima airport was considered to be one-third of the wage rate following many previous studies.

3. Model estimation
3.1 A PIG model with on-site correction
Let 𝑦𝑖 and 𝐱 𝑖 = (𝑥1𝑖 , ⋯ , 𝑥𝑘𝑖 )′ denote the number of trips by individual 𝑖 and the 𝑘 dimensional explanatory variable vector, which includes a constant, respectively. It, then, follows fromthe exponential mean specification (Cameron and Trivedi, 2013, p. 71) that the conditional mean of
𝑦𝑖 is defined as𝜆𝑖 = E(𝑦𝑖 |𝐱 𝑖 ) = exp(𝐱 ′𝑖 𝜷) ,𝑖 = 1, ⋯ , 𝑁,(1)where 𝜷 is the parameter vector. If 𝑦𝑖 is independently Poisson distributed with the above mean
4

Agency for Natural Resources and Energy., 2015. The Price Survey of Oil Products.
http://www.enecho.meti.go.jp/statistics/petroleum_and_lpgas/pl007/results.html#headline4, Accessed date: 22 February 2018.
5
Ministry of Land, Infrastructure, Transport and Tourism., 2015a. The List of Vehicle Fuel Consumption.
http://www.mlit.go.jp/jidosha/jidosha_fr10_000031.html, Accessed date: 22 February 2018.
6
Drive Plaza. http://www.driveplaza.com/dp/SearchTop, Accessed date: 22 February 2018.
7

Google Maps. https://www.google.co.jp/maps, Accessed date: 21 February 2018.

8

Nippon Rent-a-car. https://www.nrgroup-global.com/en/, Accessed date: 21 February 2018.

9

Niconico Rent a car. https://niconicorentacar.jp/, Accessed date: 21 February 2018.

10

TaxiSite. http://www.taxisite.com/ (in Japanese), Accessed date: 22 February 2018.

11

Google Maps Transit. http://maps.google.com/landing/transit/index.html, Accessed date: 22 February 2018.

12

Ministry of Land, Infrastructure, Transport and Tourism., 2015. Airplane Passenger Survey,
http://www.mlit.go.jp/koku/koku_tk6_000001.html (in Japanese). Accessed date: 22 February 2018.

8

parameter 𝜆𝑖 , Equation (1) is the well-known standard Poisson regression model. However, thisspecification has the so-called equidispersion property, which means that the conditional varianceequals its mean. Thus, to relax this restrictive property, we introduce 𝜈𝑖 , which expresses theunobserved heterogeneity of individuals to Equation (1) as follows: 𝜇𝑖 = 𝜆𝑖 𝜈𝑖 , where 𝜈𝑖 isindependent of 𝑦𝑖 , and thus E(𝜇𝑖 |𝜆𝑖 ) = 𝜆𝑖 because we can assume that E(𝜈𝑖 ) = 1 without loss ofgenerality. Thus, unobserved heterogeneity is multiplicatively incorporated into the exponentialconditional mean. Now, assuming that 𝑦𝑖 follows the Poisson distribution of the mean parameter 𝜇𝑖 ,and letting 𝑔(𝜈𝑖 ) denote the probability density function of 𝜈𝑖 , the (marginal) probability density
function of 𝑦𝑖 , which is called a mixed Poisson distribution, is shown as∞𝑓(𝑦|𝐱) = ∫0exp(−𝜆𝜈) (𝜆𝜈)𝑦𝑔(𝜈)𝑑𝜈 ,𝑦!

(2)where the subscript 𝑖 for an individual is omitted for notational simplicity. This expression is a
generalization of the standard Poisson regression model, although specifying 𝑔(𝜈) is necessary toobtain the explicit form of the density. The most popular example is to assume that 𝜈 follows a
gamma distribution; that is, the mixed Poisson distribution (2) is the Poisson-gamma mixture, whichleads to the well-known negative binomial model.
This study considers the PIG model of Dean et al. (1989), in which 𝜈 follows an inverse
Gaussian (IG) distribution. Since E(𝜈) = 1, the probability density function of an IG distribution is
given by(𝜈 − 1)2
1𝑔(𝜈) = √exp(−),
2𝜋𝜏𝜈 32𝜏𝜈
where Var(𝜈) = 𝜏 > 0 is a shape parameter and unknown. Thus, we have a Poisson inverse Gaussian
mixture as the mixed Poisson distribution (2). From the explicit expression of a PIG distribution shownby Willmot (1987), the conditional probability mass function for the PIG model can be obtained from
Equations (3) and (4) below. If y > 0,𝑦−1𝑦+𝑘𝑝(0)𝜆𝑦Γ(𝑦 + 𝑘)𝜏 𝑘ℎ(𝑦|x) =∑( ) (1 + 2𝜏𝜆)− 2 ,Γ(𝑦 + 1)
Γ(𝑦 − 𝑘)Γ(𝑘 + 1) 2

(3)

𝑘=0

whereas if 𝑦 = 0,𝑝(0) = exp (𝜏 −1 (1 − √1 + 2𝜏𝜆)) .

(4)Note that as the shape parameter 𝜏 → 0, the PIG model approaches the standard Poisson regression
model, and, thus, 𝜏 is the parameter describing overdispersion.
Since the count data are collected via an on-site survey, there are two problems: truncation andendogenous stratification. This problem exists because non-visitors are excluded, which means that9the sample is zero-truncated, and visitors who make frequent trips to the site are covered byoversampling. The endogenous stratification problem is one of the particular forms of the so-called
choice-based sampling and causes biased and inconsistent estimators of parameters, which may leadto serious mistakes in the statistical inference. Following Shaw (1988), we derive a probability massfunction of the PIG model that allows for on-site sampling. Shaw’s correction for the conditional
probability density function to control for the effects involved in on-site sampling is given byℎ 𝑆 (𝑦|𝐱) = ℎ(𝑦|𝐱)𝑤(𝑦, 𝜆),𝑤(𝑦, 𝜆) =𝑦.
E(𝑦|𝐱)(5)

Thus, by applying Equation (5), we can construct a log-likelihood function suitable for the on-site
sampling data, as shown in Equation (6):𝑁

𝑁
𝑆

∑ log ℎ (𝑦𝑖 |𝐱𝑖 ; 𝜽) = ∑ log (
𝑖=1

𝑖=1

𝑦𝑖
ℎ(𝑦𝑖 |𝐱 𝑖 ; 𝜽))
𝜆𝑖

𝑁

𝑦−1

𝑖=1

𝑘=0

𝑦+𝑘
𝜆𝑖
Γ(𝑦 + 𝑘)𝜏 𝑘
= ∑ {log+ 𝜏 −1 (1 − √1 + 2𝜏𝜆) log (∑( ) (1 + 2𝜏𝜆)− 2 )} . (6)Γ(𝑦𝑖 )
Γ(𝑦 − 𝑘)Γ(𝑘 + 1) 2

Here, 𝜽 = (𝜷′ , 𝜏)′ is the unknown parameter. Thus, we obtain the maximum likelihood estimators
based on the PIG model under on-site sampling.
3.2 Expansion to the random effects model
Given that this study aims to measure the recreational benefits, it is necessary to analyze the TCM+ CB data. Thus, it is not desirable to analyze each response from a given individual as a univariate
count data because ignoring the multivariate dependence will cause an efficiency loss of the estimatorsand may also affect their consistency. The most natural expansion is to handle it as a multivariate countdata, as in Egan and Herriges (2006). However, it is not easy to obtain the estimates because thelikelihood function is usually complicated, and its computational burden may be heavy. As analternative estimation method, their study proposes the use of the seemingly unrelated negative
binomial (SUNB) model of Winkelmann (2000) because it avoids computational complexity eventhough the correlation structure is restrictive. However, Beaumais and Appéré (2010) view
multivariate data as a pseudo-panel data. This view implies that the time index of the standard panel
data model is regarded as the number of scenarios that accompanies the CB data. Thus, they propose
an estimation method invoking the gamma-distributed Poisson random-effects (RE-PGM) model ofHausman et al. (1984), in which each of the random effects is independently and identically distributed
as gamma. Following their pseudo-panel approach, we first introduce the inverse Gaussian distributed
Poisson random effects (RE-PIG) model, which is the expansion of the univariate PIG model. Then,
to analyze on-site sampling data, we correct for its sampling effects in a way similar to that given in
Section 3.1.
10′

Let 𝑦𝑖𝑗 be the number of trips in scenario 𝑗 for individual 𝑖 , and let 𝐱 𝑖𝑗 = (𝑥1𝑖𝑗 , ⋯ , 𝑥𝑘𝑖𝑗 )denote the 𝑘-dimensional explanatory variable vector, including a constant in scenario 𝑗. Similar to
Section 3.1, we assume that the conditional mean, which is denoted by 𝜆𝑖𝑗 and satisfies E(𝜇𝑖𝑗 |𝜆𝑖𝑗 ) =𝜆𝑖𝑗 , can be described as follows:
𝜇𝑖𝑗 = exp(𝐱 ′𝑖𝑗 𝜷) 𝜈𝑖 ,𝑖 = 1, ⋯ , 𝑁,

𝑗 = 1, ⋯ , 𝐽

The characteristic feature of this specification is that 𝜈𝑖𝑗 , which denotes the heterogeneity of
individuals in a scenario, is considered a random effect that is not dependent on scenario 𝑗; thus,𝜈𝑖𝑗 = 𝜈𝑖 . Hence, although the random effect is denoted by a random variable that follows a common
IG distribution, note that it restricts the correlation structure. The number of trips for each individual′

is now a multivariate count data; thus, we introduce some new notations: 𝐲𝑖 = (𝑦𝑖1 , ⋯ , 𝑦𝑖𝐽 ) and′𝐱̃ 𝑖 = (𝐱 𝑖1 , ⋯ , 𝐱 𝑖𝐽 ) . Then, by expanding Equation (2) in Section 3.1 to the present context, theconditional probability density function of the RE-PIG model is given by𝐽∞ℎ(𝒚|𝐱̃) = ∫ ∏
0

𝑗=1

𝐽

𝑦

exp(−𝜇𝑗 ) 𝜇𝑗 𝑗𝑦𝑗 !

𝑦𝑔(𝜈)𝑑𝜈 = ∏
𝑗=1

𝜆𝑗 𝑗

∞

∗

∫ exp(−𝜈 𝜆𝐽∗ )𝜈 𝑦𝐽 𝑔(𝜈)𝑑𝜈,

𝑦𝑗 ! 0where 𝑦𝐽∗ = ∑𝐽𝑗=1 𝑦𝑗 , 𝜆𝐽∗ = ∑𝐽𝑗=1 𝜆𝑗 , and the subscript 𝑖 denoting an individual is omitted fornotational simplification. Since 𝑔(𝜈) is the density function of the IG distribution, it follows fromthe same argument in Section 3.1 that, after some calculation, we obtain the conditional probability
mass function for the RE-PIG model as follows:𝑦𝐽∗ −1

∗

𝐽

𝑦

𝐽

𝑦

𝑦𝐽 +𝑘
𝜆𝑗 𝑗
𝜆𝑗 𝑗
𝜏 𝑘
−
∗
∗
2
ℎ(𝒚|𝐱̃) = 𝑞(0) ∑∏= 𝑞(𝑦𝐽 ) ∏ ,( ) (1 + 2𝜏𝜆𝐽 )
𝑦𝑗 !
𝑦𝑗 !
Γ(𝑦𝐽∗ − 𝑘)Γ(𝑘 + 1) 2

Γ(𝑦𝐽∗ + 𝑘)

𝑘=0

𝑗=1

𝑗=1

where 𝑞(0) = exp(𝜏 −1 (1 − √1 + 2𝜏𝜆𝐽∗ )).
Next, it is necessary to allow for the fact that 𝐲𝑖 is assumed to be collected via an on-site survey.
Since there is typically one variable with on-site sampling in 𝐲𝑖 , which we set at 𝑦𝑖1 , it is sufficient
to control for the sampling effects only for variable 𝑦1 . Thus, considering this point, the conditional
probability mass function with on-site correction is written as
𝐽ℎ

𝑆 (𝒚|𝐱
̃)

𝑦

𝑦 −1
𝜆𝑗 𝑗
𝑞(𝑦𝐽∗ )𝜆1 1=∏.
(𝑦1 − 1)!
𝑦𝑗 !

(7)

𝑗=2

Hence, we can construct a log-likelihood function from Equation (7) in the same way as in Equation(6) in Section 3.1 and obtain the maximum likelihood estimators of parameters, which are given by𝑆̃ 𝑖 ; 𝜽) with respect to the unknown parameters 𝜽 = (𝜷′ , 𝜏)′. Note thatmaximizing ∑𝑁
𝑖=1 log ℎ (𝒚𝑖 |𝐱

the proposed estimation approach has a similar correlation structure to the SUNB model and the REPGM model; thus, the correlation structure among the multivariate count data (that is, the overscenarios) is restricted to be positive and is mainly determined by only one parameter.
113.3 Empirical modelThis section introduces our model for empirical analysis, in which dependent variables areconstructed from the CB data only; thus, the proposed estimation approach is also capable of dealing
with such a case. Following the variable definition from the on-site survey as described in Table 2, therecreational demand function for Kume Island can be specified as:
𝜆𝑖𝑗 = exp(𝛽0 + 𝛽1 𝑇𝐶𝑖𝑗 + 𝛽2 𝐼𝑛𝑐𝑜𝑚𝑒𝑖𝑗 + 𝛽4 𝐸𝑑𝑢𝑐𝑎𝑡𝑖𝑜𝑛𝑖𝑗 + 𝛽5 𝐴𝑐𝑐𝑜𝑚𝑝𝑎𝑛𝑦𝑖𝑗 + 𝛽6 𝐴𝑙𝑜𝑛𝑒𝑖𝑗+𝛽7 𝐾𝑢𝑚𝑒1𝑖𝑗 + 𝛽8 𝐾𝑢𝑚𝑒2𝑖𝑗 + 𝛽9 𝐾𝑢𝑚𝑒3𝑖𝑗 + 𝛽10 𝐼𝑛𝑡𝑒𝑟𝑒𝑠𝑡𝑖𝑛𝑔1𝑖𝑗 + 𝛽11 𝐼𝑛𝑡𝑒𝑟𝑒𝑠𝑡𝑖𝑛𝑔2𝑖𝑗+𝛽12 𝐼𝑛𝑡𝑒𝑟𝑒𝑠𝑡𝑖𝑛𝑔3𝑖𝑗 + 𝛽13 𝐼𝑛𝑡𝑒𝑟𝑒𝑠𝑡𝑖𝑛𝑔4𝑖𝑗 + 𝛽14 𝑁𝑎ℎ𝑎𝑖𝑗 + 𝛽15 𝐷𝑎𝑦𝑠𝑖𝑗+𝛽16 𝐸𝑥𝑝𝑒𝑟𝑖𝑎𝑛𝑐𝑒1𝑖𝑗 + 𝛽17 𝐸𝑥𝑝𝑒𝑟𝑖𝑎𝑛𝑐𝑒2𝑖𝑗 + 𝛽18 𝐸𝑥𝑝𝑒𝑟𝑖𝑎𝑛𝑐𝑒3𝑖𝑗 ),whichimplies

𝑗 = 1, 2,that

𝐱 𝑖𝑗 =(1, 𝑇𝐶𝑖𝑗 , 𝐼𝑛𝑐𝑜𝑚𝑒𝑖𝑗 , 𝐸𝑑𝑢𝑐𝑎𝑡𝑖𝑜𝑛𝑖𝑗 , 𝐴𝑐𝑐𝑜𝑚𝑝𝑎𝑛𝑦𝑖𝑗 , 𝐴𝑙𝑜𝑛𝑒𝑖𝑗 , 𝐾𝑢𝑚𝑒1𝑖𝑗 , 𝐾𝑢𝑚𝑒2𝑖𝑗 ,′𝐾𝑢𝑚𝑒3𝑖𝑗 , 𝐼𝑛𝑡𝑒𝑟𝑒𝑠𝑡𝑖𝑛𝑔1𝑖𝑗 , ⋯ , 𝐼𝑛𝑡𝑒𝑟𝑒𝑠𝑡𝑖𝑛𝑔4𝑖𝑗 , 𝑁𝑎ℎ𝑎𝑖𝑗 , 𝐷𝑎𝑦𝑠𝑖𝑗 , 𝐸𝑥𝑝𝑒𝑟𝑖𝑎𝑛𝑐𝑒1𝑖𝑗 , ⋯ , 𝐸𝑥𝑝𝑒𝑟𝑖𝑎𝑛𝑐𝑒3𝑖𝑗 )and

𝜷 = (𝛽0 , 𝛽1 , ⋯ , 𝛽18 )′

in

the

frameworkofSection

3.2.

Note

that

𝐲𝑖 =

(𝑉𝑖𝑠𝑖𝑡_𝑆𝑃0𝑖 , 𝑉𝑖𝑠𝑡_𝑆𝑃100𝑖 )′ represents the CB data under the hypothetical scenarios, the current reef
condition, and reef extinction (cf. Kragt. et al., 2009). That is, 𝑦1 is subject to the on-site correction
because, to collect the data, an on-site survey is employed as mentioned in Section 2.1, and it seems
natural that the number of visits will not decrease under the current reef quality. The minimum numberof planned trips under the current reef quality is 1 from the on-site survey data. However, 𝑦2 indicates
CB data in which the hypothetical scenario of coral reef extinction may lead to a decrease in thenumber of planned trips.
From the empirical model as specified above, per-person recreational value of a site qualitychange is measured as
ΔCS =𝜆2 − 𝜆1,
𝛽1(8)where 𝜆2 is the number of planned trips associated with a change in reef quality (extinction), 𝜆1 isthe number of planned trips under current reef quality, and the coefficient of travel cost is assumed toremain the same after a quality change. In the subsequent section, we compute the estimated ΔCS byreplacing 𝜆1, 𝜆2, and 𝛽1 with their predicted or estimated values 𝜆̂1, 𝜆̂2, and 𝛽̂1 in Equation (8).
Note that for the predicted number of the trips, 𝜆̂𝑗 , the evaluation at the mean of the independent
variables is adopted in the same manner as the previous studies (Whitehead et al., 2000).

4. Estimation results
We estimate the parameters in the recreational demand function constructed in the previous
section by the two types of econometric approaches–the RE-PGM and RE-PIG models with on-site12

correction. All computations in this section are conducted using Ox (Doornik, 2009). Table 3 reports
the estimation results of the empirical model by the two approaches. First, the travel cost coefficients(𝑇𝐶), which is our primary interest, are negative as expected and significant at the 5% level in both
approaches. Moreover, both of the likelihood ratio (LR) statistics reject the null hypothesis that all of
the coefficients except for the constant are zero at the 1% significance level. Although there are only

Table 3 Results of RE-PGM and RE-PIG models with on-site correction
RE-PGM

RE-PIG

Variable

Coeff.

SE

Coeff.

SETC-0.138**0.061-0.125**

0.063

SP100

-1.676***0.125-1.676***

0.125

Income0.055**

0.025

0.053**

0.025

Education-0.468**0.184-0.452**

0.187

Alone

1.378***

0.321

1.361***

0.331

Accompany

0.222***

0.059

0.221***

0.054

Kume1

0.591**

0.285

0.545*

0.294

Kume2

0.987***

0.296

0.972***

0.299

Kume3

1.630*

0.963

1.887*

1.071

Interesting1-0.1480.190-0.1380.194

Interesting2-0.1820.205-0.1780.208

Interesting3

0.986***

0.357

0.959***

0.362

Interesting4

0.131

0.244

0.139

0.247

Naha stay

0.164

0.234

0.213

0.242

Days

0.086

0.074

0.096

0.078

Experience1-0.3490.562-0.3490.569

Experience2-0.2250.196-0.232

0.200

Experience3-0.6810.431-0.6390.443

Constant-0.6470.854-0.3670.799

𝛼 or 𝜏

1.905*

1.047

0.977***

0.281

Log-likelihood-466.6-465.2LR

318.0***

310.5***

AIC

973.3

970.5

Note: ***, **, and * indicate significance at the 1%, 5%, and 10% levels, respectively.

13

slight differences in the significance level between the RE-PGM and RE-PIG models, each of thecoefficients for SP100, Income, Education, Alone, Accompany, Kume, and Interesting 3 is statisticallysignificant at the 10% or lower levels. In particular, the estimates of SP100 support the anticipationthat the number of planned trips at the degraded quality will be less than that at the current quality.
Further, the coefficients associated with Kume show statistically positive signs, indicating that theexperience of activities during the trip has increasing effects on future recreational demand. Since
Days and the other dummy variables except for Interesting 3 are not statistically significant in both
approaches, visitors’ interest regarding natural resources in Kume Island and past experiences aboutmarine activities do not seem to affect their trip decision making. Focusing on the overdispersion
parameters, 𝛼 and 𝜏, we find that they are statistically different from zero at the 10% and 1% levels,
respectively, which implies that ignoring unobserved heterogeneity will incur efficiency loss of theestimators and may also make them inconsistent. Thus, it seems that the random-effects model
approaches with on-site correction in the framework of pseudo panel data offer more reliable
parameter estimates. Next, to compare the performance between the RE-PGM and RE-PIG models,the Akaike information criterion (AIC; Akaike, 1973) for each approach are reported in Table 3. Since
the AIC of the RE-PIG model is slightly smaller than that of the RE-PGM model, in addition to thefact that the significance levels of 𝛼 and 𝜏 are largely different from each other, it is conjectured
that the former approach is more appropriate to analyzing our on-site sampling data than the latterapproach. This conjecture implies that the IG distribution would be able to capture overdispersion orunobserved heterogeneity than the gamma distribution more adequately.
Following Equation (8) and the related discussion in Section 3.3, we can calculate the per-person
CS (ΔCS) for ten years in Table 4, where the 90% confidence intervals of the estimates using theKrinsky-Robb procedure (Haab and MacCnonell, 2002; González-Sepúlveda and Loomis. 2011) arealso reported. Note that Table 4 includes the estimates obtained by using the RE-PGM and RE-PIG
models while ignoring the on-site sampling issues to examine the effects of on-site correction. The
estimation results of the empirical model corresponding to Table 3 by these approaches are provided
in the Appendix. It is easily seen from the results that the annual CS per-person trip by the RE-PGM

Table 4 Estimation results for CS loss
RE-PGM

RE-PIG

RE-PGM without
on-site correction

RE-PIG without
on-site correction

ΔCS (ten years)

3.769

6.107

22.61825.476

90% CI-LB

1.598

2.864

13.34213.749

90% CI-UB

13.19323.04259.00568.032
Unit: ¥10,000

14model (3,796 yen) is smaller than that by the RE-PIG model (6,107 yen), and, although both of theconfidence intervals are asymmetric, the latter has a wider range than the former. We find that a similartendency also exists in both models without on-site correction. These features seem to reflect theunderestimation caused by the inadequacy of the RE-PGM model specification on unobserved
heterogeneity as discussed above. Thus, in terms of the model evaluation, it is preferred to adopt theresult of the RE-PIG model in the following discussion. For comparison, Kragt et al. (2009)
demonstrated that the annual CS per-person trip was 83.5 Australian dollars in their study, though the
per-person recreational value of a site quality change using Equation (8) was not explicitly provided.
Thus, converting Australian dollars into yen by using the exchange rate at that time, we find that thisamount is approximately 7,097 yen, noting that the hypothetical scenarios (the degraded reef quality)
are not the same. It is clear from Table 4 that the CS estimates based on the models without on-sitecorrection are considerably larger than those of the corrected models. Given this fact, there is apossibility that Kragt et al. (2009) will overestimate the CS loss because they did not deal with the onsite sampling issues. It is, thus, crucial to measure recreational values via an on-site survey to control
for on-site sampling and adequately specify unobserved heterogeneity or overdispersion.
As the total number of visitors to Kume Island in 2015 is 102,797, the CS loss in the case of coral
reef extinction become about 627.78 million yen per year based on our RE-PIG estimate. According
to the original budget from the Okinawa Prefectural Government of 2015 and 2019,13 reproduction
project for conservation of coral reefs in 2015 includes 233.52 million yen, whereas that amountdecreases by 69.3 million yen in 2019. As the original budget in 2015 to protect coral reefs is notenough given our results, it might be necessary to reconsider the optimal allocation of budget and theeffective use of it.

5. Concluding remarks
In Japan, coral reefs in the Okinawa Prefecture are seriously damaged, and their distributional
area decreases year by year. However, there remains a coral reef community which has remarkably
high scholarly value in Kume Island. Thus, this paper focuses on Kume Island and estimates therecreational demand function by using only CB data. Moreover, we propose the PIG model adjusted
for an on-site survey and expand it to the random-effects model as an estimation approach. From theempirical analysis, we estimate the CS losses under the hypothetical scenario of current coral reef
quality and extinction and show that the annual CS per-person trip is 6,107 yen by the RE-PIG model.
To avoid the overestimation of CS, the comparative study suggests that, choosing the appropriate
estimation approach and the correct for on-site sampling issues is a requirement. However, note that

13

Okinawa Prefectural Government. https://www.pref.okinawa.jp/site/somu/zaisei/17312.html, Accessed date: 30 May 2019.

15

there is still some room for improving the estimation accuracy because the sample size may be small.
Furthermore, from the estimated CS loss, the coral reef extinction in Kume Island might cause
serious economic damage to recreational demand in the Okinawa Prefecture. According to the reportabout the action plan to conserve coral reef ecosystems in Japan 2016-2020, published by the Ministryof the Environment (2015) in Japan, three priority issues are selected, and one of them is the promotionof sustainable tourism in coral reef ecosystems. This report also mentions that coral reef tourism isvery popular and is an industry which produces the highest economic value in the coral reef areas.
Finally, it concludes that the coral reef will become more and more important in terms of thedevelopment of the tourism industry in the region because conservation of the coral reef ecosystemenhances its value as a tourism resource. Thus, it is agreeable that the protection of the coral reef basedon economic valuation leads to the desirable promotion of sustainable tourism. Additionally, in Kume
Island, the reproduction project for the protection of coral reefs was organized in 2019 to promotesustainable activities which aimed at recuperation from coral reef bleaching or death. The contents ofthis project include cultivation, monitoring, and enlightening people on coral reefs. It seems valid topresume that our results present the necessity of cost-effective policy measures as soon as possible tosupport, such as a local project. It is hoped that our study would be of some benefit to the conservationof coral reefs in Kume Island.
From the methodological viewpoint, there is a potential concern that the PIG model with
controlling for on-site sampling is extended to latent class or random parameter approaches, which
was considered and proposed by Hynes and Greene (2013, 2016) based on the negative binomial
model. They applied these approaches to the panel dataset of beach users and showed that theunobserved heterogeneity in the framework of their contingent behavior travel cost model can beadequately accounted for even if the data are collected through an on‐site survey. These directions
may, thus, make it possible to more flexibly cover a wide range of specification on unobserved
heterogeneity in a pseudo-panel data and would be of value to the field regarding welfare estimationof recreation, which is scope for future research.

Acknowledgments
This study was funded by the National Institute for Environmental Studies (NIES) and, also,

supported by Japan Society for the Promotion of Science (JSPS), Grant-in-Aid for Young Scientists(B) (Grant Number 17K18038).

16

Appendix
Table A1 Results of RE-PGM and RE-PIG models without on-site correction
RE-PGM

RE-PIG

Variable

Coeff.

SE

Coeff.

SETC-0.113***0.043-0.100**

0.045

SP100

-2.002***0.122-2.022***

0.122

Income0.039**

0.017

0.036**

0.017

Education-0.285**0.125-0.271**

0.129

Alone

1.094***

0.215

1.055***

0.228

Accompany

0.154***

0.035

0.152***

0.034

Kume1

0.368*

0.191

0.330*

0.198

Kume2

0.657***

0.199

0.641***

0.205

Kume3

1.379**

0.616

1.573**

0.656

Interesting1-0.1360.130-0.1200.135

Interesting2-0.1260.139-0.1270.144

Interesting3

0.759***

0.237

0.701***

0.247

Interesting4

0.126

0.167

0.125

0.171

Naha stay

0.164

0.159

0.188

0.166

Days

0.077

0.049

0.082

0.051

Experience1-0.2580.381-0.257

0.393

Experience2-0.1300.134-0.134

0.139

Experience3-0.617**0.299-0.571*

0.310

Constant

0.984*

0.520

0.917*

0.543

𝛼 or 𝜏

0.232*

0.051

0.279***

0.068

Log-likelihood-509.0-506.4LR

512.8***

492.7***

AIC

1058.01052.8

Note: ***, **, and * indicate significance at the 1%, 5%, and 10% levels, respectively.

17

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