of 48
Online Appendix
Meta-Analysis of Empirical Estimates of
Loss Aversion
Alexander L. Brown
Ferdinand M. Vieider
Taisuke Imai
Colin F. Camerer
Contents
A Data
1
A.1 Paper Search and Inclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
A.2 Coded Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
A.3 Approximation and Imputation of Missing Standard Errors . . . . . . . . . . .
6
A.4 Journals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
B Coefficient of Loss Aversion
11
C Bayesian Hierarchical Model
13
C.1 Modeling Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13
C.2 Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16
C.3 Robustness Checks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
D Additional Figures and Tables
22
E Frequentist Meta-Analysis
30
F Peer Prediction
32
G List of Studies Included in the Meta-Analysis
34
References
46
A Data
A.1 Paper Search and Inclusion
We searched for relevant papers on the scientific citation indexing database Web of Science.
We used, after several trial-and-error to fine-tune, the following combination of query terms.
©
­
­
­
«
(loss AND avers
*
)
OR "loss aversion coefficient"
OR "loss aversion index"
OR ("loss avers
*
" AND ("willingness to pay" OR "willingness to accept"))
ª
®
®
®
¬
AND
estimat
*
OR measur
*
OR experiment
*
OR survey

Figure A.1:
Keywords used in the search.
The initial search, made in the Summer of 2017, returned total hits of 1,547 papers. As
a first step of paper identification, we went through titles and abstracts and threw out 833
papers that were irrelevant to our study. We then read the remaining papers, applied our
inclusion criteria based on the content, and coded information. Finally, we posted a message
on the email list of the Economic Science Association to ask for relevant papers (in February
2018).
1st round: Web of Science
Articles searched on the basis of abstract
Excluded papers that do not collect empirical data
nor estimate loss aversion coefficient
Read through of article and application of
inclusion criteria
Excluded papers that do not collect empirical data
nor estimate loss aversion coefficient
2nd round: The ESA mailing list
Final set of papers (
=
150
)
=
1
,
547
=
714
Figure A.2:
Paper search and data construction.
1
A.2 Coded Variables
Table A.1:
List of coded variables.
Variable
Description
Atricle meta data
main
_
title
Title of the paper
main
_
lastnames
Last names of the authors
main
_
firstnames
First names of the authors
main
_
published
=
1
if published
main
_
yearpub
Year of publication
main
_
journal
Journal
main
_
affliations
Affiliations of the authors
Estimates
la
Reported loss aversion coefficient
la
_
type
Type of reported
la
_
aggmean
=
1
if reported
is aggregate-level
la
_
indmean
=
1
if reported
is individual-level mean
la
_
indmedian
=
1
if reported
is individual-level median
both
_
stats
=
1
if individual-level mean and median are reported
se
SE of
(reported or calculated)
se
_
imp
SE of
(reported, calculated, or imputed)
se
_
type
Type of SE (reported, calculated, or imputed)
se
_
calc
=
1
if SE is calculated from other information
se
_
calc
_
method
What information is used for SE calculation
Type of data
type
_
lab
_
exp
=
1
if laboratory experiment
type
_
field
_
exp
=
1
if field experiment
type
_
class
_
exp
=
1
if classroom experiment
type
_
online
_
exp
=
1
if online experiment
type
_
gameshow
=
1
if TV game show
type
_
field
_
other
=
1
if other field data
Type of the experiment/survey
loc
_
lab
=
1
if laboratory study
loc
_
field
=
1
if field study
loc
_
online
=
1
if online study
loc
_
class
=
1
if classroom study
Location of the experiment/survey
loc
_
country
Country
loc
_
state
State
loc
_
city
City
loc
_
<CONTINENT>
Continent dummy
2
Variable
Description
Subject pool
subject
_
children
=
1
if subjects are children
subject
_
uni
=
1
if subjects are university students/staffs
subject
_
elderly
=
1
if elderly population
subject
_
gen
=
1
if general population
subject
_
farmer
=
1
if subjects are farmers
subject
_
mixed
=
1
if mixed subject population
subject
_
unknown
=
1
if unknown population
subject
_
monkey
=
1
if subjects are Capuchin monkeys
Reward
reward
_
real
=
1
if real reward
reward
_
money
=
1
if monetary reward
reward
_
food
=
1
if food reward
reward
_
cons
_
good
=
1
if consumption goods
reward
_
env
_
good
=
1
if environmental goods
reward
_
health
=
1
if health
reward
_
mixed
=
1
if mixed type
reward
_
other
=
1
if other type of reward
Method
method
_
question
=
1
if questionnaire
method
_
seqbin
=
1
if sequential binary choice
method
_
mpl
=
1
if multiple price list
method
_
bdm
=
1
if BDM
method
_
matching
=
1
if matching
method
_
gp
=
1
if Gneezy-Potters
method
_
other
=
1
if other method
method
_
other
_
type
Description of the method (if
method
_
other
=
1
)
3
Variable
Description
Utility specifications
spec
_
u
_
est
=
1
if utility function is parametrically estimated
spec
_
u
_
crra
=
1
if CRRA is assumed
spec
_
u
_
crra
_
eq
=
1
if CRRA with common curvature is assumed
spec
_
u
_
crra
_
noneq
=
1
if CRRA with different curvatures is assumed
spec
_
u
_
cara
=
1
if CARA is assumed
spec
_
u
_
linear
=
1
if linear utility is assumed
spec
_
u
_
other
=
1
if other parametric form is assumed
spec
_
nonparametric
=
1
if
is nonparametrically recovered
Reference-point specifications
spec
_
rp
_
zero
=
1
if reference point is 0
spec
_
rp
_
statusquo
=
1
if reference point is status quo
spec
_
rp
_
expectation
=
1
if reference point is expectation
spec
_
rp
_
other
=
1
if other type of reference point is assumed
Loss aversion
loss
_
tversky
_
kahneman
=
1
if
is defined as in Tversky and Kahneman
loss
_
koebberling
_
wakker
=
1
if
is defined as in Köbberling and Wakker
loss
_
neilson
=
1
if
is defined as in Neilson
loss
_
wakker
_
tversky
=
1
if
is defined as in Wakker and Tversky
loss
_
bowman
=
1
if
is defined as in Bowman, Minehart and Rabin
loss
_
koszegi
_
rabin
=
1
if
is defined as in Kőszegi and Rabin
loss
_
other
=
1
if another definition of
is used
Notes
: See Online Appendix B for the definitions of loss aversion.
4
Variable
Description
Publication status
pub
_
regular
=
1
if published in a peer-reviewed journal
pub
_
econtopfive
=
1
if published in a “Top 5” journal in economics
pub
_
unpub
=
1
if not published in a peer-reviewed journal
journal
_
if
Journal impact factor (in 2018)
journal
_
if
_
std
Standardized journal impact factor (in 2018)
Journal topic/discipline
journal
_
category
Journal topic/discipline
cat
_
econ
=
1
if economics
cat
_
psych
=
1
if psychology
cat
_
neuro
=
1
if neuroscience
cat
_
agri
=
1
if agricultural sciences
cat
_
medical
=
1
if medical sciences
cat
_
mgt
=
1
if management
cat
_
transport
=
1
if transportation research
cat
_
multi
=
1
if multi-disciplinary
Notes
: Journal categories are based on the classification provided by The Master Journal List (
https://mjl.
clarivate.com/home
). Journal impact factors are downloaded from The Journal Citation Reports (
https:
//clarivate.com/webofsciencegroup/solutions/journal-citation-reports/
).
5
A.3 Approximation and Imputation of Missing Standard Errors
The dataset includes 192 estimates (out of 607) of loss aversion coefficient without corre-
sponding standard errors (SEs). In order to keep these observations in our meta-analysis, we
approximated and imputed missing SEs using other available information.
First, we calculated SEs of four observations from
-values of the two-sided test for the
null hypothesis
0
:
=
1
, from
se
=
|
1
|
Φ
1
(
1
)
,
where
Φ
1
is the quantile function of the standard normal distribution.
Second, we approximated 64 SEs from the inter-quartile range (IQR) and sample size, using
se
1
.
35
×
IQR
.
Note that the use of this approximation formula is legitimate if the parameters are normally
distributed in the population, which is a strong assumption in our dataset. Nevertheless,
obtaining even an “approximated” SE seemed preferable to dropping the observation entirely,
or to making other, even stronger, assumptions allowing us to keep the observation.
Finally, we imputed the remaining 124 missing SEs. The basic idea is to estimate the
parameters characterizing their distribution in the data,
log
(
se
) ∼ N(
se
,휎
2
se
)
. Using these
distributional parameters, we can then estimate the missing values in SE by letting
log
(
se
) ∼
N(
b
se
,
b
2
se
)
, where the subscripts
and
stand for
observed
and
missing
, respectively, and
(
b
se
,
b
se
)
are estimated quantities.
Implementing this estimation, we will thus obtain values for the missing observations in
SE that have the same mean and variance. We can, however, do much better than that if we
can find other variables in our dataset that are significantly associated with SEs (McElreath,
2016). As it turns out, the single best predictor of the SE is the loss aversion estimate itself.
Once it is controlled for, no other predictor—including the measurement type and the square
root of the number of observations—is significant. The loss aversion coefficient explains 51%
of the variance in SEs. By letting
se
=
se
+
se
, we can thus get much better imputation
results than by only using the distributional characteristics.
Figure A.4 shows the imputed standard errors juxtaposed with the observed standard er-
rors, and plotted against the loss aversion coefficient. The solid line indicates the regression
line of the SE on loss aversion in the subset of data for which we observe the SE. The estimates
of loss aversion with and without SEs exhibit systematic difference (
=
0
.
002
, Wilcoxon rank
sum test; Figure A.3 and Figure A.4B) but, as we would expect, the imputed SEs are no different
than the observed SEs on average (
=
0
.
458
, Wilcoxon rank sum test; Figure A.4C).
6
0.00
0.25
0.50
0.75
1.00
0
1
2
3
4
5
6
L
o
s
s
a
v
e
r
s
i
o
n
c
o
e
f
f
i
c
i
e
n
t
(
λ
)
CDF
SE reported (220)
SE imputed (57)
A
0.00
0.25
0.50
0.75
1.00
0
1
2
3
4
5
6
L
o
s
s
a
v
e
r
s
i
o
n
c
o
e
f
f
i
c
i
e
n
t
(
λ
)
CDF
SE reported (126)
SE imputed (33)
B
0.00
0.25
0.50
0.75
1.00
0
1
2
3
4
5
6
L
o
s
s
a
v
e
r
s
i
o
n
c
o
e
f
f
i
c
i
e
n
t
(
λ
)
CDF
SE reported (69)
SE imputed (34)
C
Figure A.3:
Empirical CDF of reported loss aversion coefficient
by the type of estimates and by the
type of SE.
Notes
: Solid lines correspond to observations with reported SEs and dashed lines correspond
to observations for which SEs are imputed.
−6
−4
−2
0
2
0
2
4
6
L
o
s
s
a
v
e
r
s
i
o
n
c
o
e
f
f
i
c
i
e
n
t
(
λ
)
log(SE)
SE Reported (352)
SE Imputed (124)
A
0.0
0.2
0.4
0.6
0
2
4
6
L
o
s
s
a
v
e
r
s
i
o
n
c
o
e
f
f
i
c
i
e
n
t
(
λ
)
SE reported
SE imputed
B
0.0
0.1
0.2
0.3
0.4
−3
0
3
log(SE)
SE reported
SE imputed
C
Figure A.4:
Imputation of standard errors.
Notes
: The solid black line in panel A is the regression line
of the standard errors on loss aversion in the data with observed standard errors. Panels B and C show
Kernel density estimates of the distributions of
and
log
(
se
)
. The Gaussian kernel with Silverman’s
rule of thumb for the bandwidth selection is applied. The
-axis in each panel is cut off at 6 for better
visual rendering, but the density estimation keeps all the relevant observations.
7
A.4 Journals
Table A.2:
List of journals and disciplines.
Journal
Category
1 Addiction
Substance Abuse
2 Addictive Behaviors
Psychology, Applied
3 American Economic Journal: Economic Policy
Economics
4 American Economic Journal: Microeconomics
Economics
5 American Economic Review
Economics
6 American Journal of Agricultural Economics
Agriculture/Agronomy
7 Behavioral Neuroscience
Neurosciences
8 Brain
Neurosciences
9 Cognition & Emotion
Psychology
10 Consciousness and Cognition
Psychology, Experimental
11 Current Biology
Cell Biology
12 Developmental Cognitive Neuroscience
Psychology, Development
13 Ecological Economics
Ecology
14 Economic Inquiry
Economics
15 Economics Letters
Economics & Business
16 Ekonomický časopis
Economics
17 Emotion
Psychology, Experimental
18 Environment and Development Economics
Economics
19 European Economic Review
Economics
20 European Journal of Operational Research
Operations Research & Management Science
21 European Journal of Transport and Infrastructure Research Social Sciences, General
22 European Review of Agricultural Economics
Economics & Business
23 Experimental Economics
Economics
24 Frontiers in Human Neuroscience
Psychology
25 Frontiers in Psychology
Psychology, Multidisciplinary
26 Games and Economic Behavior
Economics
27 International Economic Review
Economics
28 International Journal of Applied Behavioral Economics
Economics & Business
29 International Journal of Research in Marketing
Economics & Business
30 Journal of African Economies
Agricultural Sciences
31 Journal of Banking & Finance
Business, Finance
32 Journal of Behavioral and Experimental Economics
Economics
33 Journal of Behavioral Decision Making
Psychology, Applied
34 Journal of Behavioral Finance
Business, Finance
35 Journal of Business & Economic Statistics
Business & Economics
36 Journal of Consumer Research
Economics
37 Journal of Development Economics
Economics
38 Journal of Development Studies
Social Sciences, General
39 Journal of Economic Behavior & Organization
Economics
40 Journal of Economic Dynamics and Control
Economics
8
Journal
Category
41 Journal of Economic Psychology
Economics
42 Journal of Empirical Finance
Economics
43 Journal of Experimental Psychology: General
Psychology
44 Journal of Gambling Studies
Substance Abuse
45 Journal of Health Economics
Economics & Business
46 Journal of International Economics
Economics
47 Journal of Marketing Research
Economics
48 Journal of Mathematical Psychology
Psychology, Mathematical
49 Journal of Political Economy
Economics
50 Journal of Risk and Uncertainty
Business & Economics
51 Judgment and Decision Making
Psychiatry/Psychology
52 Management Science
Management
53 Marketing Science
Economics
54 Nature
Multidisciplinary Sciences
55 NeuroImage
Neurosciences
56 Neuron
Neurosciences
57 Neuropsychiatric Disease and Treatment
Psychiatry
58 Organizational Behavior and Human Decision Processes Management
59 PLOS Computational Biology
Biochemical Research Methods
60 PLOS ONE
Multidisciplinary Sciences
61 PNAS
Multidisciplinary Sciences
62 Proceedings of the Royal Society B: Biological Sciences
Evolutionary Biology
63 Psicológica
Psychology, Experimental
64 Psychiatry Research
Psychiatry/Psychology
65 Psychological Science
Psychology
66 Psychology and Aging
Gerontology
67 Quantitative Finance
Economics
68 Quarterly Journal of Economics
Economics
69 Rationality and Society
Social Sciences, General
70 Review of Economics and Statistics
Economics
71 Review of Managerial Science
Management
72 Revista Espanola de Financiacion y Contabilidad
Business, Finance
73 Science
Multidisciplinary Sciences
74 Theory and Decision
Economics
75 Tourism Management
Hospitality, Leisure, Sport & Tourism
76 Transportation Research Part B: Methodological
Transportation Science & Technology
77 Transportation Research Record
Transportation Science & Technology
78 World Development
Economics
Notes
: Journal categories are based on the classification provided by The Master Journal List (
https://mjl.
clarivate.com/home
).
9