# RECORD ba26780937630bfce76ea1bcc89a7777faa87ed0
# TITLE: Earthquake Risk Embedded in Property Prices: Evidence From Five Japanese Cities
# YEAR: 2018.0  VENUE: Journal of the American Statistical Association  DOI: 10.1080/01621459.2021.1928512
# AUTHORS: Masako Ikefuji; R. Laeven; J. Magnus; Y. Yue
# PAGES: 13  CANDIDATE TABLE PAGES (1-based): [7, 9, 11]
# ABSTRACT: Abstract We analyze the impact of short-run (90 days) and long-run (30 years) earthquake risk on real estate transaction prices in five Japanese cities (Tokyo, Osaka, Nagoya, Fukuoka, and Sapporo), using quarterly data over the period 2006–2015. We exploit a rich panel dataset (331,343 observations) with property characteristics, ward attractiveness information, macroeconomic variables, and long-run seismic hazard data, supplemented with short-run earthquake probabilities generated from a seismic excitation model using historical earthquake occurrences. We design a hedonic property price model that allows for subjective probability weighting, employ a multivariate error components structure, and develop associated maximum likelihood estimation and variance computation procedures. Our approach enables us to identify the total compensation for earthquake risk embedded in property prices, to decompose this into pieces stemming from short-run and long-run risk, and to distinguish between objective and subjectively weighted (“distorted”) earthquake probabilities. We find that objective long-run earthquake probabilities have a statistically significant negative impact on property prices, whereas short-run earthquake probabilities become statistically significant only when we allow them to be distorted. The total compensation for earthquake risk amounts to an average –2.0% of log property prices, slightly more than the annual income of a middle-income Japanese household. Supplementary materials for this article, including a standardized description of the materials available for reproducing the work, are available as an online supplement.


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Earthquake Risk Embedded in Property Prices:
Evidence From Five Japanese Cities
Masako Ikefuji, Roger J. A. Laeven, Jan R. Magnus & Yuan Yue
To cite this article: Masako Ikefuji, Roger J. A. Laeven, Jan R. Magnus & Yuan Yue (2021):
Earthquake Risk Embedded in Property Prices: Evidence From Five Japanese Cities, Journal of the
American Statistical Association, DOI: 10.1080/01621459.2021.1928512
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6
M. IKEFUJI ET AL.
the Koji-Chika dataset published by the Ministry of Land,
Infrastructure, Transport, and Tourism (MLIT). The Koji-
Chika dataset provides fictional sales prices (as produced by
“experts”) and they are only available at annual intervals,
which we consider to be too long for our purpose. We use
a different dataset, which provides self-reported transaction
prices at three-months intervals. This dataset, also provided
by the MLIT, is known as the “real estate transaction-price
information;” see https://www.land.mlit.go.jp/webland_english/
servlet/MainServlet. The information in this dataset is based
on the results of a questionnaire survey of persons involved
in real estate transactions conducted by MLIT, compiled and
published quarterly. We thus know the transaction price and
the transaction date (quarter), and also in which district the
property lies and the name of the nearest station. In addition,
many property characteristics are provided, of which we shall
only consider: total area in square meters, total floor area in
square meters, distance to nearest station measured in walking
minutes, age of the building (if applicable), building structure
with varying degrees of earthquake resistance (reinforced
concrete, steel, or wood), purpose of city planning in the urban
control area, maximum building coverage ratio (BCR), and
maximum floor area ratio (FAR). Furthermore, as there were
major changes in the regulations on earthquake-resistance
building standards in 1981 and 2000, our regression design
also features built-1981–2000 and built-after-2000 dummies.
Different types may have different regressors. For example, the
equation for land only does not have “building structure” or
“building age” as a regressor; and the equation for condos does
not use “building structure” as a regressor.
Economic indicators. Property prices are affected by general
economic conditions. To incorporate possible effects of these
economic conditions, we have selected two national macroeco-
nomic indicators: GDP and CPI.
Long-run earthquake risk. We consider two measures of
earthquake risk: short-run risk (i.e., seismic excitation; see
Section 2.1) and long-run risk. Long-run earthquake risk is
defined as the probability of an earthquake exceeding certain
intensity thresholds in the next 30 years in a given area, provided
by the Japan Seismic Hazard Information Station (JSHIS). We
select the threshold intensities “5-lower” (medium risk) and
“6-lower” (high risk) in our analysis. The JSHIS probabilities
are provided in various mesh sizes, varying from one square
km to 250 square meters. For each district, we identify its
center and then define the risk of that district as the JSHIS risk
associated with the smallest available mesh in which this center
lies. Although the JSHIS exceedance probabilities are updated
every one or two years, we take the average of the JSHIS risk data
over all available years, thus obtaining a time-invariant measure
of long-run risk for each district, not influenced by short-
term deviations. Clearly, new events happening or new insights
within this time window may have impacted the modeling of
JSHIS, and lead to different predictions. However, the variation
in the long-term earthquake probabilities provided by JSHIS
is mostly modest. These probabilities are included as objective
measures of long-run earthquake risk in our regression design,
at the district level. Choosing a district of relative safety may
be viewed as a form of self-insurance. Therefore, provided
this information, which is publicly available, is known among
Table 2. Seismic hazard probabilities per city, averaged over districts and time
(2005–2014).
City
Mean
Min
25%
50%
75%
max
sd
Exceeding intensity level “5 lower”
Tokyo
1.00
0.99
0.99
1.00
1.00
1.00
0.00
Osaka
0.93
0.90
0.92
0.94
0.95
0.97
0.02
Nagoya
0.96
0.91
0.94
0.97
0.98
0.98
0.02
Fukuoka
0.39
0.06
0.30
0.42
0.48
0.56
0.12
Sapporo
0.33
0.05
0.21
0.33
0.44
0.51
0.12
Exceeding intensity level “6 lower”
Tokyo
0.35
0.16
0.22
0.28
0.49
0.59
0.13
Osaka
0.37
0.22
0.30
0.39
0.44
0.52
0.09
Nagoya
0.56
0.21
0.41
0.61
0.67
0.77
0.14
Fukuoka
0.03
0.00
0.02
0.03
0.03
0.05
0.01
Sapporo
0.01
0.00
0.01
0.01
0.02
0.03
0.01
consumers, we would expect higher property prices in relatively
safe areas all else being equal.
If the intensity is “5 lower,” then according to the Japan
Meteorological Agency, many people will be frightened and feel
the need to hold on to something. Hanging objects (such as
lamps) will swing violently, books may fall from bookshelves,
and unstable furniture may topple over. Windows may break,
electricity poles may move, and roads may sustain damage.
There may be cracks in the walls of wooden properties. If the
intensity is “6 lower,” then the effects will be more severe. It will
be difficult to remain standing, unsecured furniture will move
and topple over, and cracks in walls, crossbeams, and pillars will
appear not only in wooden properties but also in properties built
from reinforced concrete.
Summary statistics are shown in Table 2. It is clear from
Table 2 that Tokyo, Nagoya, and Osaka are high-risk cities
with respect to “small” earthquakes. In fact, it is almost certain
that an earthquake will occur in Tokyo with an intensity more
severe than “5 lower” within the next 30 years. Regarding the
occurrence of “severe” earthquakes (“6 lower”), Nagoya is more
exposed than Tokyo and Osaka, and much more exposed than
Fukuoka and Sapporo. The variation in probabilities of severe
earthquakes in Tokyo, Osaka, and Nagoya is also much larger
than in the other two cities. Fukuoka and Sapporo are not
likely to have severe earthquakes, but there is still considerable
probability (and variation) of smaller earthquakes. This suggests
that it is important to use both thresholds, 5-lower and 6-lower,
in characterizing the distribution of long-run earthquake risk
for our purpose. This also guarantees sufficient variation of
long-run probabilities in the hedonic price model discussed in
Section 4.
4. The Model
The dependent variable is log-property price, and we denote the
hth observation of type k in district i during quarter t as y(h,k)
it
.
The most common method of modeling the property market
is hedonic pricing, pioneered by Rosen (1974) who argued that
an item’s total price can be thought of as the sum of the price
of each of its homogeneous characteristics, so that the effect of
each characteristic on the price can be determined by regressing
(log)price on these characteristics. We shall follow the hedonic
approach. In our case, the (log)price is determined by charac-
teristics of the property itself (size, age, etc.), the surrounding


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JOURNAL OF THE AMERICAN STATISTICAL ASSOCIATION
7
environment (location, crime rate, schools, etc.), earthquake
risk factors, and macroeconomic influences.
The district i determines the city c(i), which takes values
1, . . . , 5 depending on the city in which district i is situated.
Also, the time variable t determines in which quarter q(t) the
transaction took place, taking values 1, . . . , 4 depending on
whether t refers to the first, second, third, or fourth quarter.
The number of observations varies per district, type and quarter,
and this affects the precision. We let H(k)
it
denote the number
of observations on each type k = 1, 2, 3 in district i during
quarter t. We model the difference between cities by a shift αc(i)
in the intercept term, but we assume that all other parameters
are the same between cities. The difference between cities is thus
completely captured by αc(i).
Our model can now be written as
y(h,k)
it
= α(k)
0
+ αc(i) + γq(t) + x(k)
i·
′β1 + x(k)
·t
′β2
+x(h,k)
it
′β3 + rit(ψ)′β4 + u(h,k)
it
,
(3)
where xi· denotes a variable that is constant over time, but varies
over districts (attractiveness variables), x·t denotes a variable
that is constant over districts, but varies over time (economic
indicators), xit denotes a variable that varies over districts and
over time (property characteristics), and rit denotes the risk data
(same for each type k) given by the (distorted) short- and long-
run earthquake probabilities. The reference dummies are the
city dummy for Tokyo and the quarter dummy for Q4; these are
set to zero.
The property price is log-linear in the (distorted) short-
run and long-run earthquake probabilities, instead of in
the logarithm of these earthquake probabilities. This is due
to two main reasons: statistical performance and economic
motivation. Indeed, in our analysis, the model specification
without a log-transformation of earthquake probabilities turns
out to statistically significantly outperform a specification with
log-transformed probabilities. Economically, a specification
with log-prices and probabilities (the latter without log-
transformation) occurs naturally in a simple (rank-dependent)
expected utility model (Von Neumann and Morgenstern 1944;
Quiggin 1982) with the canonical log-utility function. This
utility function is a special case of interest in the constant relative
risk aversion family of utility functions, by far the most widely
applied family of utility functions in economics and decision
sciences (Wakker 2008). Indeed, the certainty equivalent loss,
that is, the compensation for risk, in this canonical risk model
is log-linear in (distorted) probabilities.
Although the model appears to be linear in the parameters
this is not completely the case, because the risk variable rit is
a nonlinear function of one or more ψ’s which appear in the
probability weighting function w(p) discussed in Section 2. This
complicates the estimation, and we shall discuss this issue in the
next section. To obtain a (balanced) panel we average over h, and
obtain
¯y(k)
it
= α(k)
0
+ αc(i) + γq(t) + x(k)
i·
′β1 + x(k)
·t
′β2
+¯x(k)
it
′β3 + rit(ψ)′β4 + ¯u(k)
it ,
(4)
where we average over H(k)
it
items, which thus depends on how
many properties of type k there are in a given district. Next, we
combine the three types of property into one 3 × 1 vector:
¯yit = α0+(αc(i)+γq(t)) ı+X∗
i·β1+X∗
·tβ2+X∗
itβ3+ırit(ψ)′β4+¯uit,
(5)
where ı = (1, 1, 1)′, which we write more succinctly as
¯yit = ¯Xitβ + ¯uit
(i = 1, . . . , N; t = 1, . . . , T),
(6)
where ¯yit is a p × 1 vector of random observations, explained by
regressors ¯Xit = ¯Xit(ψ), an unknown parameter vector β, and
random errors ¯uit (p × 1). In our case p = 3.
5. Estimation Method
We wish to estimate the parameters and their precisions of
the model thus described. This is essentially, apart from the
nonlinearity caused by ψ, a linear mixed model with a large
variance matrix. Such models arise in many contexts (genomics:
Lippert et al. 2011; Heckerman et al. 2016; spatial statistics:
Dutta and Mondal 2015, 2016), and various estimation pro-
cedures have been developed; see also Demidenko (2013). In
our setup, we will employ a (multivariate) error components
structure, since this has a natural interpretation in our context.
For our purpose, we need not only the estimate of ψ, which
could conceivably be obtained via profile-likelihood, but also
its (asymptotic) estimated variance. This means that we need
a comprehensive approach to our estimation problem, which
provides all estimated variances jointly, as follows.3
The errors are assumed to follow a p-variate three-error
components structure,
¯uit = ζi + ηt + ϵit,
(7)
a sum of three independent components each of which is iid
with zero means and variances
var(ζi) = 
ζ,
var(ηt) = 
η,
and
var(ϵit) = 
ϵ,
(8)
where 
ζ and 
η are positive semidefinite, and 
ϵ is positive
definite, all of order p × p. Multivariate two-error components
were first employed by Chamberlain and Griliches (1975) using
maximum likelihood techniques. Multivariate three-error com-
ponents were first considered by Avery (1977) who derived
a feasible Aitken estimator, which is however not maximum
likelihood and turns out to be asymptotically inefficient. Balt-
agi (1980) derived an alternative estimator, also not maximum
likelihood, which is asymptotically efficient. Magnus (1982)
discussed the estimation and testing of the multivariate two-
and three-error components models in a maximum likelihood
context.
Our error structure implies that
E(¯uit ¯u′
js) =
⎧
⎪⎪⎪⎨
⎪⎪⎪⎩

ζ + 
η + 
ϵ
if i = j and t = s,

ζ
if i = j and t ̸= s,

η
if i ̸= j and t = s,
0
if i ̸= j and t ̸= s.
(9)
3The multivariate error components regression R package mvecr that we
have developed for this article is available from the article’s github page.


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8
M. IKEFUJI ET AL.
Let
Y =
⎛
⎜⎜⎜⎝
¯y11 ¯y12 . . . ¯y1T
¯y21 ¯y22 . . . ¯y2T
...
...
...
¯yN1 ¯yN2 . . . ¯yNT
⎞
⎟⎟⎟⎠,
U =
⎛
⎜⎜⎜⎝
¯u11 ¯u12 . . . ¯u1T
¯u21 ¯u22 . . . ¯u2T
...
...
...
¯uN1 ¯uN2 . . . ¯uNT
⎞
⎟⎟⎟⎠,
(10)
and
¯X(t) =
⎛
⎜⎜⎜⎝
¯X1t
¯X2t
...
¯XNt
⎞
⎟⎟⎟⎠,
X =
⎛
⎜⎜⎜⎝
¯X(1)
¯X(2)
...
¯X(T)
⎞
⎟⎟⎟⎠.
(11)
Then, we can write Equation (6) in stacked form as
y = Xβ + u,
(12)
where y = vecY and u = vecU. We shall assume that y is
normally distributed with mean μ = Xβ and variance (θ) =
var(u), so that β refers to the mean parameters and θ to the
variance parameters.
Maximum likelihood estimation for the model in Equation
(12), via optimization of the corresponding loglikelihood under
normality, and associated variance computation pose several
nontrivial challenges. One complication lies in the fact that
the nonrandom matrix X depends on a parameter (vector) as
well, so that X = X(ψ). Another complication is the high
dimensionality of the variance matrix  = (θ). Appendix A
(supplementary material) provides our detailed treatment of
how to overcome these and related challenges.
Although the variance matrix  = var(u) is of a very large
dimension, the error components structure allows us to write it
in a convenient form, allowing simple expressions for its inverse
and determinant, and also for quadratic forms like v′−1v and
X′−1X. The relevant formulas are provided in Appendix A
(supplementary material).
In a nutshell, estimation of the parameters then proceeds as
follows. For given ψ, we maximize the concentrated likelihood
with respect to the variance parameters θ, where using the
explicit expression for the gradient will speed up the optimiza-
tion. Performing a grid search on ψ we obtain the maximum
likelihood estimates ˆθ and ˆψ. Next, we find ˆβ and finally the esti-
mated variances of ˆβ and ˆψ; see the development in Appendix A
(supplementary material), in particular (A.7)–(A.12).
6. Estimation Results
Our primary interest is in earthquake risk and its impact on
property prices. More specifically, we wish to answer three ques-
tions: (i) do objective long-run earthquake probabilities have an
effect on property prices, (ii) do objective short-run earthquake
probabilities have an effect on property prices, in addition to the
effect of long-run probabilities, and (iii) do potentially distorted
short-run earthquake probabilities have an effect on property
prices, in addition to the effect of long-run probabilities?
Table 3. Estimation results under various risk assumptions.
Variable
LR
LR and
Base
only
objective SR
model
Intercepts
Land & building
3.7592
4.5593
4.3812
Land only
3.5949
4.3940
4.2155
Condo
3.1025
3.9024
3.7244
City Dummies
Osaka
−0.2273
−0.2625
−0.2615
Nagoya
−0.3801
−0.4100
−0.4139
Fukuoka
−0.8770
−0.9133
−0.9108
Sapporo
−1.2050
−1.2458
−1.2388
Ward
Immigrants
6.7245
6.7224
6.7218
Attractiveness
Crime
−0.0437
−0.0436
−0.0436
Unemployment
−4.3360
−4.3395
−4.3399
Executives
3.3426
3.3447
3.3464
Economic
log(GDP)
0.5606
0.5220
0.5229
indicators
log(CPI)
1.5347
1.4687
1.5030
Property
Area (m2)
0.0025
0.0025
0.0025
Characteristics
Floor area (m2)
0.0006
0.0006
0.0006
Distance to nearest station
−0.0145
−0.0145
−0.0145
Age
−0.0121
−0.0121
−0.0121
Built 1981–2000
0.1674
0.1658
0.1652
Built after 2000
0.4136
0.4126
0.4123
Structure: reinf. concrete
0.4348
0.4344
0.4343
Structure: steel
0.1867
0.1867
0.1867
Structure: wood
−0.1264
−0.1266
−0.1266
Urban control
−0.8972
−0.8967
−0.8967
max BCR
−0.0019
−0.0019
−0.0019
max FAR
0.0004
0.0004
0.0004
Risk
Long run 45–55
−0.1433
−0.1427
−0.1427
Long run 55+
−0.5037
−0.5039
−0.5041
Short run
–
−0.0915‡
−0.0514
ˆψ
–
–
3.74†

 log L
−68.5
−15.8
–
Before we answer these questions and comment on our esti-
mates in Table 3, we explain our econometric modeling strategy.
This strategy is based on two ingredients. First, we aim for par-
simony. We want the smallest model that captures the essence
of our story. This means that sometimes regressors have been
deleted from our model even when the associated parameters
are “significant.” Significance does not imply importance, and
importance is what interests us. Second, we make a distinction
between focus and auxiliary regressors. The focus regressors are
the effects that we are interested in or are part of the minimum
set that would make up a credible model, while the auxiliary
regressors are only in the model because they improve the
estimation of the focus parameters.
Since we have many observations, most estimates are likely to
be significant at the usual 1.96 level. We provide more informa-
tion about the results by strengthening the significance require-
ment on the t-values. Thus, a ‡ will indicate that |t| ≤1.96,
which we interpret as not significant, while † indicates signifi-
cance with 1.96 < |t| ≤4.00. Estimates without superscript are
therefore significant with |t| > 4.00. The choice of 4.00 is some-
what arbitrary and chosen a posteriori in order to provide more
information about the precision of our estimates, in particular
our parameter estimates pertaining to the risk variables. (All
t-values test the null hypothesis that the parameter of interest
equals zero, except the t-value of ˆψ which tests the null that
ψ = 1.)
Ideally, one would like to combine the concepts of model
selection and estimation into one procedure. This is called


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JOURNAL OF THE AMERICAN STATISTICAL ASSOCIATION
9
“model averaging.” Also, one would like to combine the concepts
of significance and importance into one procedure, and the
so-called “fence method” provides the required flexibility by
allowing the criterion of optimality for the selection within
the fence to incorporate scientific, economical, political, and
even legal concerns, in addition to statistical ones (Jiang et al.
2008; Jiang and Nguyen 2015). Both issues are, however, not
fully developed yet and their application is beyond us in this
nonstandard regression model.
Now consider the first question. The results are presented in
Table 3 under the heading “LR only” and we see that all estimates
are significant (i.e., appear without superscript), that is, their t-
value (in absolute terms) exceeds 4.00. Regarding the long-run
risk effects, we remark that long run 45–55 (medium risk) indi-
cates the JSHIS probability that an earthquake occurs in the next
30 years of higher intensity than 5-lower and lower intensity
than 6-lower; and that long run 55+ (high risk) indicates the
JSHIS probability that in the next 30 years an earthquake occurs
of intensity 6-lower or higher. Both medium risk and high risk
appear to have a significant negative impact on property prices.
The higher risk level has a more severe impact, which is intu-
itively reasonable. Hence, long-run risk matters. This answers
the first question.
Next, we consider the second question: given that long-run
risk plays a role, do objective short-run probabilities also have
an effect on property prices? The results are presented in the
next column of Table 3 under the heading “LR and objective SR”.
Apparently they do not: the effect of the risk variable short run,
while negative as we would expect, is not significantly different
from zero.
Finally, we consider the third question: given that long-run
risk plays a role, do potentially distorted short-run probabil-
ities also have an effect on property prices? The results are
displayed in the final column of Table 3 under the heading “Base
model”. Apparently they do: after distortion, short-run probabil-
ities have a significant negative effect on property prices.
The difference between objective and distorted short-run
risk is that short-run probabilities are now allowed to be
distorted using a probability weighting function, in this case
the one-parameter weighting function (2) proposed by Prelec
(1998), which yields the highest likelihood. The parameter ψ in
the Prelec function is estimated to be 3.74 and is significantly
different from unity, since the absolute value of its t-value lies
between 1.96 and 4.00; in fact |t| = 2.91.
As shown in Figure 3, the estimated probability weighting
function has an S-shaped pattern where small probabilities are
underweighted and large probabilities are overweighted, which
is in contrast to the inverse S-shaped probability weighting
function often found in experiments. This contrast may be
explained by the fact that with a positive background intensity
of earthquakes, temporary deviations of short-run earthquake
probabilities caused by seismic excitation are not evaluated (and
overweighted) with respect to a reference probability of zero
but with respect to a positive reference probability level. This
applies in particular to Tokyo where the 90-day probability of
an earthquake exceeding the magnitude threshold of 5.5 never
drops below 35% in the period that we analyze.
In summary: long-run risk matters, objective short-run risk
does not, but distorted short-run risk does. In addition, all
Figure 3. Estimated probability weighting of short-run probabilities, Prelec proba-
bility weighting function, ˆψ = 3.74.
nonrisk parameter estimates in the second and third columns
are similar to the ones in the first column and all are significant
(with a t-value larger than 4.00 in absolute value).
We briefly comment on these other (nonrisk) parameters in
the base model:
Intercept and city dummies. Tokyo, of course, is the most
expensive city to buy property. If we set the property price level
of Tokyo at 1.00, then the average property price levels of the
other cities are 0.77 in Osaka, 0.66 in Nagoya, 0.40 in Fukuoka,
and 0.29 in Sapporo. (Recall that we do not regress price but
log-price on these dummies.) Also, if we set the price of land
and building at 1.00, then the average price of the other types of
property are 0.85 for land only and 0.52 for condos.
Ward attractiveness. As discussed in Section 3, we selected 11
characteristics for each ward, divided into 6 categories. Only
four of these 11 characteristics appear in our base model:
percentage of immigrants (representing population); number
of criminal offenses (representing safety); and unemployment
ratio and percentage of executives (representing employment).
Executives make a ward more attractive, while crime and
unemployment make it less attractive. Immigrants too make
a ward more attractive, which makes sense if we realize that
the word “immigrant” refers to somebody moving into the
ward from another municipality, usually within Japan. Hence,
the more people move in from other areas in Japan, the more
attractive the ward apparently is.
Economic indicators. Property prices are affected by general
economic conditions, and two indicators appear in Table 3 and
in our base model: log(GDP) and log(CPI), both of which have a
positive effect on property prices. The inclusion of log(CPI) has
the additional advantage that if we wish to explain real (rather
than nominal) property prices, then all results remain the same
except that the effect of log(CPI) is now 0.503 rather than 1.503.
Hence, CPI has a positive effect not only on nominal but also
on real property prices.
Property characteristics. A large (floor) area and proximity
to the nearest station contribute positively to the price. New


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10
M. IKEFUJI ET AL.
buildings are preferred to old ones, where we have included two
dummies because major changes occurred in the regulations on
earthquake-resistance standards in both 1981 and 2000. As a
result, buyers prefer a house built between 1981 and 2000 over
a house built before 1981, and they like a house built after 2000
even better. Regarding the structure, wood is not desirable, steel
is desirable, but reinforced concrete is preferred. Urban control
signifies restrictions on development possibilities, and this has
a negative effect on prices.
For all three property types, the designated maximum BCR
and the maximum FAR are provided. These ratios are legally
allowed maxima, different for each piece of land. The BCR is
the percentage of the building area to the site area; the FAR is
the percentage of the total floor area to the site area. We use
both ratios in our regression and find a negative effect of BCR
and a positive effect of FAR. Shimizu and Nishimura (2006)
and Nakagawa, Saito, and Yamaga (2009) used only FAR and
found mixed effects and a positive effect, respectively. Hidano,
Hoshino, and Sugiura (2015) used both ratios (as we do) and
found a negative effect of BCR and a mixed effect of FAR.
Quarterly effects. Estate agents sometimes tell customers that
some months are better to buy or sell than others. Our results
(in quarters, not months) are ambiguous, which is why we
have omitted the quarter dummies from our regression. We
return to this issue in our sensitivity analysis, see Appendix B
(supplementary material).
Error components. We estimated the coefficients using the mul-
tivariate three-error components structure, as described in Sec-
tion 5. It turns out that
tr(
ϵ) = 0.406 > tr(
ζ) = 0.129 ≫tr(
η) = 0.002,
and as a result we set 
η = 0, so that we end up with a two-error
components structure. The effect of this is negligible and will
be discussed further in our sensitivity analysis in Appendix B
(supplementary material).
Appendix C (supplementary material) provides a detailed
analysis of the corresponding impact of each of the explanatory
variables component on the (log) property prices, and the
premia for earthquake risk embedded in property prices. We
find in particular that the joint impact of long-run and distorted
short-run earthquake risk amounts to an average −2.0% of
log property prices, which translates in monetary terms into
a marginal effect of around −7 million Japanese yen per
property—slightly more than the average annual income of a
middle-income Japanese household in the period 2006/Q2 to
2015/Q3 we analyze. The impacts of long-run and distorted
short-run earthquake risk are nearly the same for all property
types but differ substantially among cities. Furthermore, the
earthquake risk variables stand almost on equal footing with
ward characteristics in explaining dispersion in property prices.
7. An Extension
In our base model, we use objective long-run probabilities and
distorted short-run probabilities based on the Prelec probability
weighting function. This raises various questions. First, one
could argue that we should allow long-run probabilities to be
distorted too; and second, we could experiment with different
probability weighting functions.
In Table 4, we experiment with an alternative functional
form for the short-run risk variable, namely the weighting func-
tion (1) introduced by Tversky and Kahneman (1992). In par-
ticular, in column 2 (Dist. SR, TK) we replace the Prelec func-
tion applied to the short-run earthquake probabilities with the
Tversky-Kahneman probability weighting function. The estima-
tion results are similar to the base model, but somewhat less pre-
cise, and the loglikelihood decreases. The Tversky-Kahneman
probability weighting function is found to be S-shaped, just like
the Prelec function, confirming the robustness of this finding.
Next, we also allow long-run risk to be distorted using both
the Prelec and the Tversky-Kahneman weighting functions. The
model contains two related time-invariant long-run probabili-
ties and we quite naturally assume that these two probabilities
share the same weighting function with the same parameter γ .
(In particular, distorted long run 45–55 is computed as distorted
long run 45+ minus distorted long run 55+, consistent with
Choquet integration.) In columns 3 and 4 of Table 4 we allow
both long-run risk and short-run risk to be distorted.
The model with the higher likelihood is the one with an
inverse S-shaped Tversky-Kahneman weighting function for
Table 4. Sensitivity and extension—probability weighting functions.
Base
Dist. SR
Dist. LR
Dist. LR
TK
Prelec
TK
Area (m2)
0.0025
0.0025
0.0025
0.0025
Floor area (m2)
0.0006
0.0006
0.0006
0.0006
Distance to nearest station
−0.0145
−0.0145
−0.0143
−0.0143
Age
−0.0121
−0.0121
−0.0121
−0.0121
Long run 45–55
−0.1427
−0.1427
−0.8644
−0.4856
Long run 55+
−0.5041
−0.5039
−1.3838
−1.5028
Short run
−0.0514
−0.0733‡
−0.0517
−0.0518
ˆψ
3.74†
1.40‡
3.78†
3.77†
ˆγ
–
–
0.17
0.32

 log L
–
−14.6
152.8
167.6
Figure 4. Implied probability weighting functions of long-run and short-run earth-
quake risk.


===== PAGE 12 (score 9, numeric-paren 0, table-words 0, var-words 9) =====

JOURNAL OF THE AMERICAN STATISTICAL ASSOCIATION
11
long-run risk and an S-shaped Prelec weighting function for
short-run risk, as shown in Figure 4. We note that the Prelec
function for long-run risk, although yielding a lower loglikeli-
hood than the Tversky-Kahneman weighting function, is also
found to be inverse S-shaped, which is again reassuring for
the robustness of our results. Thus, in an extension of our
base model that allows for distortion of time-invariant long-
run earthquake probabilities we find evidence of a conven-
tional inverse S-shaped probability weighting function for long-
run earthquake probabilities. This means that when purchasing
property in Japan, people tend to overweight small long-run
probabilities and underweight large long-run probabilities.
8. Conclusion
We have studied the impact of earthquake risk on Japanese
property prices using a rich panel dataset. We have not only
allowed for time-invariant long-run earthquake probabilities to
impact property prices, but we have also analyzed the impact
of short-run earthquake probabilities generated from a seismic
excitation model. We have designed a hedonic property prices
model that accommodates probability weighting, employing a
multivariate error components structure, and have developed
the associated maximum likelihood and variance computation
procedures. We have shown that long-run earthquake proba-
bilities negatively impact property prices and increasingly so at
higher risk levels. We have also shown that short-run earthquake
probabilities have a negative impact on property prices, and that
this effect becomes statistically significant only after we allow for
probability weighting.
The probability weighting function associated with short-
run earthquake probabilities is found to be S-shaped. That
stands in contrast to the familiar inverse S-shaped probability
weighting functions predominantly found in experiments. The
shape we find may be explained by the fact that in our setting
there is a nonnegligible positive background arrival rate of
earthquakes. People may therefore tend to evaluate earthquake
probabilities, and overweight their temporary deviations under
seismic excitation, not with respect to zero but with respect
to a positive reference probability level. This remarkable find-
ing calls for the development of reference-dependent models
for probabilities to augment the large literature on reference-
dependent models for changes in wealth levels.
Our objective short-run earthquake probabilities are based
on a purely temporal ETAS model. Within the spatial windows
used for estimation of the ETAS model, it is assumed that
the earthquake intensity functions, and people’s perception of
seismic risk, are homogeneous (see also our Data Documenta-
tion). A refinement that accounts for spatial variation would be
desirable, but is subject to statistical and computational compli-
cations. Addressing the potential limitations inherent in using
a purely temporal ETAS model in this context is an interesting
and challenging problem for future research.
Acknowledgments
We are very grateful to the editor, associate editor and three referees for
thoughtful comments and suggestions that have significantly improved the
article. Earlier versions of this article were presented at Keio University
in Tokyo, the Technische Universität Wien, the Universities of Oxford,
Madrid and Fudan, and at the 2018 Econometric Society Australasian
Meeting in Auckland and the 2018 China Meeting of the Econometric
Society in Shanghai. We thank the participants and in particular Yoshitsugu
Kanemoto for helpful comments.
Funding
This research was supported in part by the Japan Society for the Promotion
of Science (JSPS) under grant 16K03565 (Ikefuji) and the Netherlands
Organization for Scientific Research (NWO) under grant VIDI (Laeven).
Supplementary Material
This article comes with three supplementary files. First, the Appendix
which contains various technical results, a discussion of the robustness of
our estimation results, an analysis of the influence of each component to
the total property prices and the implied premia for earthquake risk, a brief
literature review, and an auxiliary figure. Second, the Data Documentation,
which contains a detailed description of the data, and is available from
https://bit.ly/3qHcTQ3. And third, the cleaned and compiled datasets and
all codes to implement the procedures developed in this article, including
our multivariate error components regression R package mvecr, available
from https://github.com/yy112/earthquake-risk.
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