Individual Decision-Making under Uncertainty
--An Experimental Panel Data Study
Chinn-Ping Fan1, Mei-Hua Tsai2 and Bih-Shiow Chen3
Preliminary Version
Abstract
Decision-making under uncertainty attracts the research interests of many experimental
economists. With a good experiment laboratory, it is now quite easy to gather enough data to run
estimations for each single individual subject. However, for individual estimations to be
meaningful, we need to be assured that, at individual level, the behavior patterns have certain
degree of stability. This research tries to tackle this question with panel experimental data.
We designed lottery experiment with monetary rewards and recruited 27 undergraduate
students in the Department of Economics and Law School of Soochow University, Taipei,
Taiwan. The z-Tree program was employed to operate the experiment in the computer lab. In
2005, each subject made 120 pricing decisions, 96 positive reward lotteries and 24 negative ones.
In 2006, each subject made 108 pricing decisions, 54 positive and 54 negative reward lotteries.
We estimate, for each individual, utility functions and probability weighting functions with this
panel data design. We tested for various functional forms of expected utility theory and prospect
theory. With this design, we try to answer the following question: in the individual level, do there
really exist stable behavior patterns.
1
Corresponding author. Professor, Department of Economics, Soochow University. E-mail: [email protected].
Financial support from The National Science Foundation, Taiwan, ROC, under Grand No. 93-2415-H-031-001 is
gratefully acknowledged..
2
Lecturer, Department of International Trade, Chin Min Institute of Technology, Miaoli, Taiwan,
E-mail:[email protected].
3
Associate Professor, Department of Economics, Soochow University. E-mail: [email protected].
1. Introduction
Decision making under uncertainty is one of the research topics that well demonstrate the
possible contributions of experimental economics. On the one hand, there are expected utility
theory, prospect theory, and many others. And on the other hand, there are various experimental
designs and strategies to analyze human behavior and test the competing theories. When
studying individual decision making behavior, the experiment design may involve different
degrees of “
individuality”
. Some researches take the whole group of subjects as the unit of
analysis and study the decision-making behavior of a representative individual, for example, see
Wakker and Deneffe (1996), Bleichrodt (2001) and Wu, Zhang and Abdellaous (2005). And with
the progress of computerized experiment, it becomes easy to obtain more data of a single subject.
With enough observations, scholars can now take a single individual as the unit of analysis and
run estimation for each individual subject. Tversky and Kahneman (1992), Hey and Orme (1994),
Tversky and Fox (1995), Gonzalez and Wu (1999), Killa and Webber (2001), and Blondel (2002),
to list just a few, analyze experimental data at individual level.
Since theories concern “
individual”behavior, so analyzing the behavior of a single subject
seems closer to the theoretical structure. But a prerequisite for this approach is that the behavior
pattern at the individual level is somewhat stable, or at least, with not too many changes over
time. The current paper examines if and how individual behavior patterns change over time. We
recruited subjects to participate in a lottery experiment in 2005. Our estimation of the
experimental data could be considered as a behavior snap shot for an individual subject. With
this snap shot, we estimated a subjects’probability weighting function and monetary utility
function. Then, one year later, we recruited this same subject again to participate in a similar
lottery experiment. With exactly the same estimation procedure, we took the snap shot for the
same subject a second time. How are these two snap shots different? With this panel data design,
we want to know if the estimation are stable over time or does there exist structural changes?
Among the many competing theories, we focus on the estimation and comparison of
expected utility theory (abbreviated EU) and prospect theory (abbreviated PT). In this paper, an
uncertainty
is
represented
as
a

L x, p  where x  x1 , x2 ,..., xn
lottery
1
 and
p p1,p2 ,..., pn
are,
respectively, the outcome vector and probability vector4. The utility
I
function over lottery could be written in the general format of: U 
, where
L 
pi 
u xi 
i 1

pi is the decision weight function and u 
xi is the monetary utility function. According to
EU, 
pi is the identity function, i.e., probabilities enter into U 
L as linear weights. But
Kaheman and Tversky (1979) observed that people often overestimate small probabilities and
underestimate large probabilities. They first suggested the nonlinear probability weighting
function w
pi 
. And later, to accommodate multi-outcome lotteries, Tversky and Kahneman
i
 i 1 
(1992) constructed decision weights 
pi 
w p j w p j  on the basis of probability
j 1  j 1 

 

weight w
pi 
.on ranked outcomes.
Previous researches suggested many functional forms for the components of U 
L . Among
them, we estimate, for the probability weighting function wpi
1. w( pi ) 

pi

1


pi (1 pi )
of PT:
(Tversky & Kahneman(1992), denoted as the 92 form);
 pi
2. w( pi ) 
(Tversky & Fox(1995) , denoted as the 95 form); and

 pi
1 pi 
r 

3. w( pi ) exp
- - ln pi  (Prelec (1998) , denoted as the 98 form).


Since EU assumes identity probability weights, it could be written as:
4. wpi p i (denoted as the EU form).
For the monetary utility function u xi 
, we estimate:
1. the quadratic function u xi 
axi2 bxi (the Q form), and
2. the power function u xi 
xi (the P form).
4
The outcomes are ranked, so that x1 x 2 ... x n .
2
A total of eight U ( L ) models could be constructed from the above-listed candidates of
u xi
and wpi . These eight
U (L ) models will be abbreviated as: EUQ, EUP (for EU),
92Q, 92P, 95Q, 95P, 98Q, and 98P (for PT). The first two characters denote the functional form
of wpi
and the third u xi . For each individual subject, all eight
U (L ) models will be
estimated.
The main purpose of this research is to examine the possible changes of individual subject’
s
behavior patterns. By comparing the estimation results of 2005 and 2006, we ask the following
questions:
Q1. Change of Model: Among the eight U (L ) models, does there exist one model that
could explain subject’
s behavior in both years? Are there structural changes?
Q2. Change of Theory: Could one theory explain subjects’behavior of both years? Between
EU and PT, is there a better theory that could explain subject’
s behavior in both years?
Q3. Change of u xi
functional form: Between the quadratic and power forms, is there a
functional form that is appropriate for explaining subject behavior in both years?
Q4. Change of risk-attitude: What were subjects
’risk attitudes? Did they change between
2005 and 2006?
Section 2 reports the experimental design and procedures, Section 3 analyzes the
experimental results, and Section 4 concludes this paper.
2. Experimental Design and Procedure
The experiment was conducted with the z-Tree program in the experiment lab in the
Department of Economics, Soochow University, Taipei, Taiwan. In June 2005, we recruited 33
3
first-year undergraduate students from Business School and Law School of Soochow University.
In June 2006, we successfully recruited 27 of the original 33 subjects to participate in the second
year experiment.
In both years, subjects always received three-outcome lotteries, so x ( x1 , x2 , x3 ) and
p ( p1 , p 2 , p3 ) . In 2005, lotteries consisted of 5 outcome vectors with monetary values
ranging from NT$-150 to NT$200. Together with 24 probability vectors, the 2005 experiment
involved 120 lotteries, 96 gains and 24 losses. In 2006, we designed 18 outcome vectors with
values ranging from NT$-420 to NT$420 and 6 probability vectors. There were a total of 108
lotteries, 54 gains and 54 losses5.
Each year subjects received a show-up fee of NT$200. In 2005, 3 gain and 1 loss lotteries
were randomly selected to determined subject monetary payoff. And in 2006, 2 gain and 1 loss
lotteries were selected. These rules for calculating monetary rewards were explained in the
instruction to the subjects. Subjects were also told that the random selection of lotteries would be
conducted only after they completed all decision making. Subjects were required to make many
decisions. To avoid boredom, there were half-time breaks in both years. An average session took
about 90~120 minutes. Subjects’monetary rewards varied form sessions to sessions, the range is
between NT$120 to NT$800.
We use the BDM proceduret
oe
l
i
c
i
ts
ubj
e
c
t
s
’va
l
ua
t
i
onsoft
hel
ot
t
e
r
i
e
s
. A gain lottery like
Figure 1 appeared in the computer screen, and subjects were required to determine the minimum
acceptable price (WTA, willingness to accept) for selling this lottery. A market price for this
lottery would later be determined by some random mechanism. If the randomly chosen market
price was greater than or equal to WTA, the lottery was successfully sold and the subject received
the market price as reward. And the lottery was not sold if the market price was smaller than
5
Appendix I contains the details of lottery design of both years. These lotteries will be denoted as 05-gain, 05-loss,
06-gain, and 06-loss.
4
WTA. It is still possible for the subjects to receive reward according to the probability structure of
this lottery.
I would like to sell this lottery
for
NT dollars.
Reward
$200
$50
$0
Probability
0.08
0.02
0.90
Figure 1 A sample gain lottery
The loss lotteries were described as possible financial damages from natural disasters. And
subjects determine how much they are willing to pay (WTP) to purchase insurance policy for this
natural disaster. The market price of the insurance premium was also determined randomly. If
WTP was smaller than the market premium, the subject failed to purchase the insurance policy,
so he might later incur various losses. And if WTP was greater than or equal to the market
premium, a subject would successfully purchase an insurance policy. This subject should paid the
market insurance premium and any future losses would be fully covered by the insurance.
I would like to pay
NT
dollars to purchase insurance policy.
Losses
Probability
Figure 2. A sample loss lottery
5
$0
$－50
$－150
0.55
0.25
0.20
3. Experimental Results
3.1 Change of model
Not knowing which U (L ) model is the best fit, so we conducted the likelihood ratio test
of structural change for all eight models, for all subjects. Also, considering the possibility that
subjects may behave differently for gains and losses, so the testing were conducted separately for
positive- and negative-outcome lotteries. The complete test results are listed in Appendix 2.
Within the 4326 test statistics, only 10 have p-values greater than 5%. Therefore, the
general conclusion seems to be that our subjects do not have stable U (L ) models between
2005 and 2006, we mostly observe structural changes.
This overwhelming degree of structural change is not a surprising result. It may be highly
unrealistic to expect a subject to have exactly the same U (L ) functional form between the two
years. And the difference in the lottery designs and the experience factor may also be the causes.
These factors, of course, could be tested by future study. For our current research, the important
implication of the structural change results is that it is inappropriate to pool together the data of
2005 and 2006. Therefore, each subject’
s U (L ) functions for 2005 and 2006 will be estimated
separately and then examined for possible differences.
3.2 Change of Theory
Subjects’single year behavior pattern is now investigated. We first run all eight U (L )
estimations, and according to the following rules, we pick the most appropriate models for each
subject. Models that does not converge or have non-significant parameters will be rejected first,
and the rest are arranged according to their AIC7 values. The complete results of estimated
6
7
There were 27 subjects, 8 U 
L models, and gain and loss lotteries.
AIC refers to Akaike information criterion, see Amemiya (1980).
6
parameters and AIC values are available from the authors upon request, and Appendix 3 contains
the AIC ranking of the eight U (L ) models for all subjects.
Between EU and PT, we want to determine a theory that provides a better fit for each subject.
We could, of course, simply choose the model with the smallest AIC value. But in order to build
a more solid foundation for our conclusion, we formally test for the nonlinearity of wpi 
. And
to do this, we need to fix a u xi
function first. Let’
s use S7-05-gain as an example to explain
our procedure. Table A3.1 in Appendix 3 shows that EUQ is better between the two EU models;
and 98Q is the best among the PT models. So we are certain that quadratic form would be a
better choice of u xi
function.
Then, the only difference between EUQ and 98Q is the
nonlinear probability weighting function wpi
.
So we conduct a Wald test, the null
hypothesis is: 1 , and the alternate hypothesis is: 1 .8 The Wald test comes up with a
Χ2 statistics of 1.33 and a p-value of 0.2483. Therefore, we conclude that that EU (with
quadratic u xi
function) is the better theory for S7-05-gain.
For some subjects, we need to test for the nonlinearity of wpi
for both the quadratic and
power u xi 
. For example, S1-05-gain had EUQ and 95P as the best models within the two
theoretical categories. Therefore, we conducted two Wald tests; EUQ vs. 95Q and EUP vs. 95P.
The nonlinearity of wpi
was confirmed in both tests, so we are assured that PT is the better
theory. And finally we choose 95P because of the AIC ranking.
For each subject, the above procedure is repeated four times: 05-gain, 05-loss, 06-gain, and
06-loss. Table 1 reports the model selection results.
The  refers to the parameter in the probability weighting function of model 98Q. If 1 , the model is
reduced to a EU model.
8
7
Table 1 Single Year Model Selection
Subject
S1
S2
S3
S4
S5
S6
S7
S8
S9
S10
S11
S12
S13
S14
S15
S16
S17
S18
S19
S20
S21
S22
S23
S24
S25
S26
S27
Gain
2005
95P
92Q
EUQ
95Q
EUQ
98Q
EUQ
EUP
EUP
EUQ
95Q
95P
92Q
EUP
95P
EUP
EUP
98P
EUP
92Q
EUQ
EUP
95Q
95P
92P
EUP
95Q
Loss
2006
95Q
EUP
95P
92P
95Q
92Q
92P
92Q
95P
98P
95P
EUP
95Q
95P
95P
EUP
95P
95P
EUQ
EUP
EUQ
92P
EUQ
98P
EUQ
EUP
95P
2005
98P
92P
98Q
98P
EUP
EUQ
98P
EUP
95P
EUQ
98P
92P
98P
98P
98P
98P
EUP
98P
92P
92Q
EUP
EUP
98P
92P
EUP
98P
92Q
2006
92P
EUP
EUP
EUP
98Q
EUQ
EUP
92P
92P
95Q
95P
EUP
EUP
92P
98P
92P
95P
98P
92P
EUP
EUQ
92Q
98P
92P
95Q
EUP
92Q
The result in Table 1 could be analyzed as follows. First, for single year estimation, we want
to know which theory provides a better fit. The answer is listed in Table 2. It appears that PT has
a slight advantage over EU, and especially so for the loss region.
8
Table 2. Single-year theory selection—subject percentage
Gain
2005
2006
EU
48.15%
33.33%
PT
51.85%
66.67%
Loss
2005
29.63%
70.37%
2006
37.04%
62.96%
The focus of this research is on the consistency of subject’
s behavior pattern, so we are more
interested in observing how things change between the two years. For the gain (loss) region, 4 (7)
subjects9 have exactly the same model for both year. These proportions numbers are quite small.
But for examining theory consistency, we want to be less restrictive; we only distinguish
between EU and PT, and allowing functional forms of wpi
and u xi to be different. For
example, for gain lotteries, subject S18 had 98P in 2005 and 95P in 2006. Although with
different wpi
function; this subject is still classified as “Both years PT”in Table 3.
Table 3. Theory consistency—subject percentage
Gain
Both-year EU
14.81%
Both-year PT
33.33%
Change
51.85%
Loss
7.41%
40.74%
51.85%
Table 3 tells us that the degree of theory consistency is not strong, over half of the subjects
belong to the “
Change”group, who had different best theory in the two years. But comparing the
two theories considered in this research, the advantage of PT is more prominent than Table 3. PT
has a much stronger degree of consistency than EU and again, especially for the loss lotteries.
Over forty percents of the subjects had PT as the better theory for both years.
9
These are, respectively, for gain region: S15 with 95P; S16 and S26 with EUP, and S21 with EUQ; and for the loss
region, S6 with EUQ, S15, S18, and S23 with 98P, S19 and S24 with 92P, and S27 with 92Q.
9
3.3 u xi
function consistency
The above two tables distinguish between EU and PT by examining the nonlinearity of
wpi 
. We now want to compare the other component of U 
L
; the u xi
we rearrange the results of Table 1 to analyze the single-year u xi
function. Again
functional form selection
and its consistency. The first column of Table 1 shows that for 05-gain, 14 subjects (51.85%) had
quadratic u xi
function in their selected models, and Table 4 reports the complete results of
single-year u xi
function selection. It appears that power function is the better choice for the
loss region, but this superiority is not that obvious for gains.
Table 4. Single-year u xi
function selection—subject percentage
Gain
2005
51.85%
48.15%
Quadratic
Power
Loss
2006
37.04%
62.96%
Table 5 reports the consistency of the u xi
2005
18.52%
81.48%
2006
25.93%
74.07%
function form. For gain region, the degree of
consistency is not high, about half of the subjects had different u xi
functional forms in the
two years. But for the loss region, the superiority of the power function is clearly demonstrated
in Table 5. Two thirds of the subjects consistently had power functions for both years.
Table 5. u xi
function consistency
Gain
22.22%
37.04%
40.74%
Both-year Quadratic
Both-year Power
Change
Loss
11.11%
66.67%
22.22%
Comparing Tables 3 and 5, we find that Table 3 has higher proportions in the “
Change”
cells. So it appears that the u xi
nonlinearity of the wpi
 function
had higher degree of consistency and the
function is less time-consistent.
10
3.4 Risk attitude consistency
Finally, we want to examine the risk attitudes of our subjects based on the model selection
results of Table 1. The Arrow-Pratt absolute risk aversion coefficients10 are first calculated for
the median outcome. We than formally test the null hypothesis of risk neutrality. Appendix 5
contains the complete list of risk aversion coefficients and tests results. With the data in
Appendix 5, we again examine subjects’single-year risk attitudes and its time consistency.
Table 6. Single-year risk attitude—subject percentage
Gain
2005
2006
Risk Averse
77.78%
14.81%
Risk Neutral
18.52%
62.96%
Risk Loving
3.70%
22.22%
Loss
2005
66.67%
22.22%
11.11%
2006
40.74%
51.85%
7.41%
Table 6 seems to suggest that subjects’risk attitudes changed over time; they were less risk
averse and more risk neutral in 2006. For both gains and losses, over half of the subjects were
risk neutral in 2006, but these proportions were much smaller in 2005. The experience factor
may have caused this difference, but of course, future study is necessary to settle this issue.
Table 7 examines the time consistency of subjects’risk attitudes. For gains, over seventy
percents of the subjects changed their risk attitudes, but this proportion drops to below fifty for
losses. Similar to the consistency results of wpi
consistency is much stronger for loss lotteries.
10
''
r ( x )  u ( x )
u '(x)
.
11
and u xi functions, the degree of time
Table 7. Risk attitude consistency
Both-year Risk Averse
Both-year Risk Neutral
Both-year Risk Loving
Change
Gain
14.81%
11.11%
0.00%
74.07%
Loss
33.33%
14.81%
3.70%
48.15%
Comparing the proportions of changes in Tables 3, 5, and 7, we find that the risk attitudes
have more changes, rough speaking, than wpi
and u xi . A little surprised by this result,
we look more closely into the data and come up with this final piece of interesting finding. Our
27 subjects consisted of 18 female and 9 male students. For gain lotteries, within the 18 female
subjects, 14 (77.78% of the female subjects) changed their risk attitudes. We follow this
procedure and calculate the relationship between risk attitude changes and sex in Table 8. It
appears that females had a higher proportion of risk attitude changes.
Table 8. Risk attitude change and sex
Positive
Female
Male
No Change
22.22%
33.33%
Change
77.78%
66.67%
Negative
Female
38.89%
61.11%
Male
77.78%
22.22%
4. Conclusion
This paper studies the possible changes of individual behavior patterns with panel
experimental data. We test for the nonlinearity of wpi 
, compare the functional forms of
u xi
and calculate subjects’risk aversion coefficients. Besides single-year estimations, we are
more interested in examining the consistency of subjects’behavior pattern. In general, we
12
observe a strong tendency of changes in all three respects. These results will be further tested by
our ongoing research. At the current stage of our study, the conclusion is that the degree of time
consistency in subjects’behavior pattern is quite weak. The two snap shots we took in 2005 and
2006 looks quite different.
This weak consistency may have resulted from the following reasons. First, the lottery
designs in the two years are quite different, second, there is the experience factor, and finally, it
is also possible that, for many people, there simply does not exist stable behavior patterns.
The firs factor seems an unlikely candidate for the following reason. The difference in the
lottery structures is much more prominent for losses. The 2005 experiment consisted of only 24
loss lotteries, with one outcome vector x 0,50,. 150 and twenty four probability vectors;
while the 2006 experiment had 54 loss lotteries, with seven outcomes vectors (with thirteen
non-positive values ranging from 0 to -420) and six probability vectors. However, our analysis
shows that the time consistency is stronger for the losses. So the greater difference in the lottery
structure did not prevent the stronger consistency of the losses.
The experience effect is a tricky and interesting factor that needs further study. We observed
that the subjects were more risk averse in the first year, but in the second year, as experienced
participants, they move toward risk neutrality. Is this a general tendency? We need more study to
answer this question.
And the final possibility serves as a reminder that we should be very careful in the
interpretation of experimental results. What we observe in one year may not look the same one
year later.
13
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14
Appendix 1 Lottery design
2005
Outcome vector
x1
x2
200
100
200
50
150
50
150
100
0
-50
Probability vector
p1
p2
95
90
90
85
85
80
0
9
2
13
4
17
5
1
8
2
11
3
33
30
40
20
35
50
0
9
2
13
4
17
95
90
90
85
85
80
0
9
2
13
4
17
33
35
35
30
25
20
x3
x1
2006
Outcome Vector
x2
50
0
0
0
-150
420
420
385
385
350
350
70
210
105
140
175
210
35
0
0
0
35
140
p3
315
315
280
-35
0
0
0
-35
-140
-175
-105
-70
280
245
245
-70
-210
-105
-140
-175
-210
-280
-245
-245
Probability vector
p2
5
5
70
20
30
35
175
105
70
-420
-420
-385
-385
-350
-350
-315
-315
-280
5
1
8
2
11
3
5
1
8
2
11
3
95
90
90
85
85
80
34
35
25
50
40
30
p1
5
80
10
30
60
40
15
x3
p3
90
15
20
50
10
25
Appendix 2 Likelihood ratio test for structural change, eight U (L ) models
Table A2.1 Likelihood ratio statistics for structural change, gain lotteries
Subject
S1
S2
S3
S4
S5
S6
S7
S8
S9
S10
S11
S12
S13
S14
S15
S16
S17
S18
S19
S20
S21
S22
S23
S24
S25
S26
S27
EUQ
EUP
92Q
92P
95Q
95P
98Q
98P
58.28
18.73
364.32
120.34
99.95
39.49
94.12
115.84
17.92
165.41
342.78
54.83
180.64
87.83
29.79
88.88
40.16
17.04
28.84
203.20
80.05
58.24
19.04
413.96
149.93
159.33
33.51
59.70
127.69
27.74
197.02
364.59
55.14
176.68
135.85
28.36
129.23
55.91
70.55
14.11
319.27
124.86
53.07
21.12
55.83
27.28
28.26
112.14
244.80
65.36
137.28
53.40
27.56
77.33
18.73
16.63
28.31
177.49
70.34
4.40
90.55
18.43
57.99
77.59
46.14
72.15
20.86
415.43
159.34
162.33
20.05
44.91
132.20
35.67
202.57
393.20
72.44
192.78
136.91
26.58
129.89
57.20
5.37
52.71
299.43
76.74
4.83
199.61
26.66
75.97
187.33
47.87
79.59
15.42
324.76
130.57
56.44
NC
54.61
113.89
37.76
112.42
291.25
87.17
139.78
60.08
30.10
73.42
26.52
23.66
31.94
165.21
74.35
14.30
107.50
28.63
53.65
76.74
47.74
77.95
20.87
424.51
167.46
167.02
26.28
42.84
142.05
36.55
205.66
454.37
92.05
191.79
139.68
29.60
125.72
62.01
62.32
14.00
326.78
118.69
55.97
12.54
64.12
123.85
22.51
120.34
263.45
66.86
142.50
52.85
29.17
75.79
20.20
18.71
27.59
180.77
NC
NC
94.55
16.56
63.36
74.34
39.90
64.24
20.28
416.20
157.41
164.30
14.29
48.05
135.92
23.68
204.39
NC
74.80
193.91
137.76
26.10
130.09
59.34
13.63
NC
299.46
86.42
NC
NC
26.59
78.49
NC
43.01
5.66
98.27
32.37
78.08
94.52
74.70
1.04
52.74
283.47
69.42
3.95
186.78
27.75
78.40
187.13
56.11
2.43
49.68
297.88
80.86
23.38
207.23
34.41
73.31
NC
42.22
* NC means non-convergent, larger bold italic numbers indicate that the null hypothesis can not
be rejected at 5% significant level, and all the other cells are significant.
NC means non-convergent, larger bold italic numbers indicate that the null hypothesis (structure
does not change) can not be rejected at 5% significant level, and all the other cells accept
alternative hypothesis that structure does change.
16
Table A2.2 Likelihood ratio statistics for structural change, loss lotteries
Subject
EUQ
EUP
2.21
3.40
S23
8.18
11.45
22.80
11.45
18.56
71.47
16.32
32.98
15.08
38.85
27.33
15.42
16.69
28.67
16.78
15.07
16.23
44.98
15.30
28.02
15.08
15.85
19.69
10.42
13.89
18.53
44.28
15.80
36.42
14.54
21.56
21.50
12.46
14.36
27.69
17.85
15.49
15.23
55.61
16.01
27.55
12.95
14.74
20.90
S24
4.34
1.33
S25
66.45
20.12
33.94
52.10
18.16
13.03
S1
S2
S3
S4
S5
S6
S7
S8
S9
S10
S11
S12
S13
S14
S15
S16
S17
S18
S19
S20
S21
S22
S26
S27
92Q
51.00
NC
34.82
27.70
17.61
73.81
18.59
28.01
16.21
30.56
21.59
21.99
24.32
26.82
23.70
20.39
10.24
43.65
11.86
29.33
6.92
12.74
42.15
6.92
67.37
27.75
35.49
92P
95Q
95P
98Q
98P
70.71
44.13
25.66
27.53
NC
50.26
17.79
39.60
15.20
19.35
25.11
21.90
25.03
29.21
26.07
20.18
11.54
67.14
15.71
34.80
13.99
11.52
48.20
7.02
51.86
28.78
24.98
60.71
NC
36.17
NC
NC
71.45
19.92
36.17
19.28
35.97
24.40
22.50
24.21
27.68
25.19
19.94
10.72
56.60
11.39
27.88
19.40
NC
49.43
57.99
42.95
29.08
28.07
12.97
NC
17.91
41.31
19.44
18.77
28.78
22.57
25.23
29.16
28.16
21.27
11.73
70.55
16.20
36.56
15.26
NC
51.88
54.02
43.04
35.85
29.59
NC
79.48
18.90
27.47
16.70
29.63
21.53
22.02
24.40
25.87
24.63
16.76
9.90
56.99
11.23
29.43
NC
14.27
46.74
7.58
7.66
6.47
45.60
30.66
36.05
39.86
31.21
30.28
67.78
28.80
34.91
53.94
42.77
29.53
28.16
NC
49.31
18.25
36.52
15.71
18.20
25.40
21.94
25.13
28.59
26.63
20.07
11.12
69.46
14.86
35.24
14.45
13.87
50.88
6.52
52.27
29.20
26.59
* NC means non-convergent, larger bold italic numbers indicate that the null hypothesis can not
be rejected at 5% significant level, and all the other cells are significant.
17
Appendix 3 AIC ranking of eight U (L ) models
Table A3.1. AIC ranking, gain lottery
2006
2005
Subject EU EUP 92Q 92P 95Q 95P 98Q 98P Subject EUQ EUP 92Q 92P 95Q 95P 98Q 98P
S1 Q
7
8
5
3
2
1
6
4 S1 6
4
3
2
1 NC 7
5
S2 3
7
1
1
5 NS 4
6 S2 2
1
6
2
8
5
7
4
S3 3
6
5
7
1
2
4
7 S3 7
5
4
2 NS 1
5
3
S4 5
8
4
7
1
2
3
6 S4 7
6
2
3
1
4
5
8
S5 1
2
3
5
7
8
3
5 S5 8
7
3
6
1
4
2
5
S6 6
5
3
4 NS NS 1
2 S6 3
4
1
2 NS NS NS NS
S7 1
8
3
6
4
7
2
5 S7 7
8
3
1
5
4
2
6
S8 5
4
5
3 NS NS 2
1 S8 4
5
1
2
3 NC 6
7
S9 6
4
8
7
2
1
3
4 S9 8
7
4
3
2
1
6
5
S10 3
8
4
5
1
2
6
6 S10 7
8
5
2
6
3
4
1
S11 7
8
5
6
1
2
3
4 S11 6
7
4
5 NS 1
2
3
S12 7
8
6
5
2
1
4
3 S12 2
1
6
3
8
6
5
4
S13 4
8
1
5
3
7
2
6 S13 7
8
4
3
1
2
6
5
S14 5
2
7
3
4
1
8
6 S14 8
7
3
5
2
1
4
6
S15 8
7
6
5
2
1
3
4 S15 6
7
5
4 NS 1
3
2
S16 2
1
6
4
8
7
5
3 S16 3
1
6
2
7 NS 5
4
S17 2
1
5
4
8
7
5
3 S17 8
7
4
5
2
1
3
6
S18 4 NC 3
5 NS NS 2
1 S18 8
7
3
5
2
1
4
6
S19 3
1
5
2
8
6
7
4 S19 2
3
4
6 NS NS 5
1
S20 7
8
1
5
3
6
2
4 S20 5
1
7
2
6
3
8
4
S21 2
5
1
4 NS NS NS 3 S21 3
4
6
7 NS 2
5
1
S22 2
1
4
3 NS NS NC NC S22 8
6
4
1
3
2
7
5
S23 7
8
5
6
1
2
4
3 S23 3
7
5
8
1
2
6
4
S24 2
7
3
6 NS 1
4
5 S24 8
6
4
3
7
5
2
1
S25 6
3
4
1 NS NS 5
2 S25 1
8
3
6
4
7
2
5
S26 NS 1 NS NS NC NC NS NS S26 2
1
5
3 NS NS 6
4
S27 7
8
6
5
1
2
4
3 S27 6
7
5
4 NS 1
2
3
* NS refers to models with non-significant parameters, NC means non-convergent, and the other
cells are ranked according to their AIC values in ascending order. So “
1”means the best model,
i.e., with smallest AIC value.
18
Table A3.2. AIC ranking, loss lottery
2005
Subject EU EUP 92Q 92P 95Q 95P
S1 Q
7
8
6
5
3
2
S2 2
3 NC 1 NC NS
S3 5
6
2
4 NS NS
S4 4
3 NS 2 NC NS
S5 2
1 NS 3
6 NS
S6 1
6
3
4 NS NS
S7 7
3
5
2
8
5
S8 5
2
7
3
8
4
S9 7
6 NS 4
2
1
S10 1
7
2
3
6 NS
S11 5
6
4
2 NS NS
S12 7
8
4
1
6
3
S13 7
8
4
2
6
5
S14 6
5 NS 2
4
3
S15 4
3 NS 2 NS NS
S16 5
6
4
2 NS NS
S17 4
1
5
2 NS NS
S18 5
6
4
2 NS NS
S19 4
1
5
2 NS NS
S20 4
6
1
5 NS NS
S21 4
1 NS 2 NS NS
S22 5
2
3
1 NS NS
S23 6
7
5
2 NS 3
S24 3
4 NS 1 NS NS
S25 2
1
3
5 NS NS
S26 3
4 NS 2 NS NS
S27 6
7
1
5
3 NS
98Q 98P
4
1
NS NS
1
3
NS 1
5
3
2
5
4
1
6
1
5
3
4
5
3
1
5
2
3
1
NS 1
NS 1
3
1
6
3
3
1
6
3
2
3
5
3
6
4
4
1
NS 2
6
4
NS 1
2
4
2006
Subject EUQ EUP 92Q 92P 95Q 95P 98Q 98P
S1 8
7
6
1
2
3
5
4
S2 4
2
5
1
8
6
7
3
S3 4
1
3
8
6
7
2
5
S4 4
2
7
3
8
6
5
1
S5 7
6
4
5 NC 3
1
2
S6 2
5 NS 4 NS NS 1
3
S7 4
1
5
3
2
8
7
6
S8 8
6
7
1
3
2
5
4
S9 8
7
6
1
3
2
4
5
S10 4
8
3
7
1
6
2
5
S11 8
7
4
5
2
1
3
6
S12 6
2
4
8
7
3
5
1
S13 4
1
5
2
8
7
5
2
S14 8
7
6
1
3
2
5
4
S15 8
7
3
5
5
4
2
1
S16 6
2
3
1
7
4
8
5
S17 8
7
6
2
4
1
5
3
S18 5
7
6
8
4
3
2
1
S19 8
7
5
1
3
2
4
6
S20 2
1
5
3
8
7
6
3
S21 4
5
7
6
1
2 NC 3
S22 4
6
1
2 NC NC 5
3
S23 5
8
5
7
4
3
2
1
S24 8
4
7
1
6
1
5
3
S25 3
5
8
6
1
2
4
7
S26 4
1
5
2
8
5
5
2
S27 8
7
1
6
4
3
2
5
19
Appendix 4. Wald Test for Nonlinearity of w( pi
)
TableA.4.1 Wald test for the nonlinearity of w( pi ) , gain lottery
2005
Sub
H1
S1
S1
S2
S3
S4
S5
S5
S6
S6
S7
S8
S9
S10
S11
S12
S12
S13
S14
S15
S16
S17
S18
S19
S20
S21
S22
S23
S24
S24
S25
S27
95Q
95P
92Q
95Q
95Q
98Q
92Q
98Q
98P
98Q
98P
95P
95Q
95Q
95Q
95P
92Q
95P
95P
98P
98P
98Q
92P
92Q
92Q
92P
95Q
92Q
95P
92P
95Q
2006X
2
9.83
12.96
8.10
5.83
12.29
0.09
0.07
148.32
95.35
1.33
1.80
3.98
4.09
121.59
53.57
58.95
24.50
3.31
41.88
1.06
0.04
53.03
0.29
35.23
0.07
0.02
32.66
1.72
6.43
6.78
120.85
p value
0.0073
0.0015
0.0044
0.0542
0.0021
0.7662
0.7948
0.0000
0.0000
0.2483
0.1793
0.1366
0.1293
0.0000
0.0000
0.0000
0.0000
0.1907
0.0000
0.3021
0.8419
0.0000
0.5871
0.0000
0.7911
0.8974
0.0000
0.1893
0.0401
0.0092
0.0000
20
Sub
H1
S1
S1
S2
S3
S4
S4
S5
S5
S6
S7
S7
S8
S9
S10
S10
S11
S11
S12
S13
S14
S15
S15
S16
S17
S18
S19
S19
S20
S21
S21
S22
S23
S24
S25
S26
S27
S27
95Q
92P
92P
95P
95Q
92P
95Q
95P
92Q
98Q
92P
92Q
95P
98Q
98P
98Q
95P
92P
95Q
95P
98Q
95P
92P
95P
95P
92Q
98P
92P
98Q
98P
92P
95Q
98P
98Q
92P
98Q
95P
X2
89.35
37.68
0.59
7.28
4.93
5.31
67.21
31.83
218.36
47.59
254.57
9.97
27.97
21.52
30.38
391.13
140.39
0.08
8.25
10.48
44.29
35.42
1.28
32.50
6.64
0.08
3.73
0.94
1.13
3.18
9.56
2.63
11.44
0.85
0.32
244.40
186.61
p value
0.0000
0.0000
0.4416
0.0262
0.0852
0.0212
0.0000
0.0000
0.0000
0.0000
0.0000
0.0016
0.0000
0.0000
0.0000
0.0000
0.0000
0.7786
0.0162
0.0053
0.0000
0.0000
0.2584
0.0000
0.0361
0.7829
0.0536
0.3316
0.2872
0.0745
0.0020
0.2686
0.0007
0.3564
0.5709
0.0000
0.0000
TableA4.2 Wald test for the nonlinearity of w( pi ) , loss lottery
2005
Sub
H1
S1
S1
S2
S3
S4
S5
S5
S6
S7
S8
S9
S10
S11
S11
S12
S12
S13
S13
S14
S15
S16
S16
S17
S18
S18
S19
S20
S21
S22
S23
S23
S24
S25
S25
S26
S27
95Q
98P
92P
98Q
98P
92P
98P
98Q
98P
98P
95P
92Q
98Q
98P
92Q
92P
98Q
98P
98P
98P
98Q
98P
92P
98Q
98P
92P
92Q
92P
92P
98Q
98P
92P
92Q
98P
98P
92Q
2006
X
2
1041.35
378.66
147.92
57.74
53.36
0.00
0.00
2.62
5.07
3.42
104.46
1.45
86.52
96.89
33.84
49.71
16.82
29.34
20.76
55.94
6.78
12.80
1.76
22.06
27.14
3.90
16.08
2.59
3.66
136.62
155.28
29.25
2.50
1.84
29.22
85.71
p value
0.0000
0.0000
0.0000
0.0000
0.0000
0.9750
0.9748
0.1054
0.0244
0.0644
0.0000
0.2283
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0092
0.0004
0.1842
0.0000
0.0000
0.0483
0.0001
0.1074
0.0559
0.0000
0.0000
0.0000
0.1138
0.1751
0.0000
0.0000
21
Sub
H1
S1
S2
S3
S3
S4
S5
S5
S6
S7
S7
S8
S9
S10
S11
S12
S13
S13
S14
S15
S16
S17
S18
S18
S19
S20
S21
S22
S23
S23
S24
S24
S25
S26
S26
S27
S27
92P
92P
98Q
98P
98P
98Q
98P
98Q
95Q
92P
92P
92P
95Q
95P
98P
92P
98P
92P
98P
92P
95P
98Q
98P
92P
92P
95Q
92Q
98Q
98P
92P
95P
95Q
92P
98P
92Q
95P
X2
78.88
1.07
3.15
0.94
1.73
5.08
6.24
2.77
7.36
1.08
59.34
59.17
27.90
292.22
3.82
0.29
0.37
82.07
12.55
10.83
44.56
4.17
6.03
39.89
0.06
1.63
12.38
5.31
8.08
24.18
8.41
7.81
0.03
0.06
85.18
47.41
p value
0.0000
0.3000
0.0760
0.3330
0.1879
0.0242
0.0125
0.0958
0.0252
0.2993
0.0000
0.0000
0.0000
0.0000
0.0505
0.5924
0.5442
0.0000
0.0004
0.0010
0.0000
0.0411
0.0140
0.0000
0.8000
0.4425
0.0004
0.0212
0.0045
0.0000
0.0149
0.0202
0.8517
0.7991
0.0000
0.0000
Appendix 5 Risk Attitudes
Table A5.1 Risk attitudes, gain lottery
2005
Subject
2006
Sex
S1
F
95P
0.000267
0.0005
Risk
Attitudes*
*
RA
95Q
0.001536
0.0755
Risk
Attitudes
**
RN
S2
F
92Q
0.009629
0.2394
RN
EUP
-0.000012
0.6306
RN
S3
F
EUQ
0.013851
0.0001
RA
95P
0.000076
0.4362
RN
S4
M
95Q
0.007642
0.0071
RA
92P
0.000037
0.2318
RN
S5
M
EUQ
0.003596
0.0648
RN
95Q
-0.001879
0.0254
RL
S6
F
98Q
0.001206
0.7962
RN
92Q
-0.004552
0.0000
RL
S7
F
EUQ
0.007526
0.0000
RA
92P
-0.000104
0.0104
RL
S8
F
EUP
0.001229
0.0000
RA
92Q
0.002348
0.1145
RN
S9
F
EUP
0.000231
0.0000
RA
95P
0.000021
0.4549
RN
S10
M
EUQ
0.006361
0.0035
RA
98P
-0.000019
0.4136
RN
S11
M
95Q
0.011913
0.0000
RA
95P
0.000070
0.1852
RN
S12
M
95P
0.000296
0.0003
RA
EUP
-0.000046
0.0653
RN
S13
M
92Q
0.008644
0.0001
RA
95Q
0.003223
0.0037
RA
S14
F
EUP
0.000695
0.0000
RA
95P
0.000048
0.1274
RN
S15
F
95P
0.000335
0.0046
RA
95P
0.000118
0.1325
RN
S16
F
EUP
0.001404
0.0000
RA
EUP
0.000038
0.2706
RN
S17
F
EUP
0.000631
0.0000
RA
95P
0.000120
0.0080
RA
S18
F
98P
-0.000392
0.0000
RL
95P
0.000001
0.9797
RN
S19
F
EUP
0.000251
0.0000
RA
EUQ
-0.001346
0.0676
RN
S20
F
92Q
0.012757
0.0020
RA
EUP
0.000007
0.5825
RN
S21
F
EUQ
0.037359
0.0000
RA
EUQ
-0.001621
0.0345
RL
S22
M
EUP
-0.000099
0.0755
RN
92P
-0.000021
0.3584
RN
S23
M
95Q
0.004051
0.0106
RA
EUQ
-0.002055
0.0000
RL
S24
M
95P
0.001162
0.0000
RA
98P
0.000163
0.0000
RA
S25
F
92P
0.001322
0.0000
RA
EUQ
0.005603
0.0110
RA
S26
F
EUP
0.003833
0.0000
RA
EUP
-0.000059
0.0260
RL
S27
F
95Q
0.003916
0.0981
RN
95P
0.000055
0.4602
RN
Model
r (x) *
p-value
u '' ( x)
*:Arrow-Pratt absolute risk aversion coefficient r ( x) 
r (x) *
Model
.
u ' ( x)
**: RA, RN and RL denote, respectively, risk averse, risk neutral, and risk loving.
22
p-value
Table A5.2 Risk attitudes, loss lottery
2005
Subject
2006
Sex
S1
F
98P
-0.000808
0.0000
Risk
Attitudes*
*
RL
92P
-0.000002
0.9247
Risk
Attitudes
**
RN
S2
F
92P
-0.000061
0.8941
RN
EUP
0.000018
0.5969
RN
S3
F
98Q
0.016389
0.0009
RA
EUP
-0.000020
0.3374
RN
S4
M
98P
0.001362
0.0000
RA
EUP
0.000102
0.0000
RA
S5
M
EUP
0.000897
0.0004
RA
98Q
0.003006
0.0003
RA
S6
F
EUQ
0.022488
0.0000
RA
EUQ
0.052338
0.1051
RN
S7
F
98P
0.000185
0.3801
RN
EUP
0.000084
0.0001
RA
S8
F
EUP
0.001363
0.0000
RA
92P
0.000058
0.0032
RA
S9
F
95P
0.000415
0.0356
RA
92P
0.000036
0.1226
RN
S10
M
EUQ
0.010190
0.0015
RA
95Q
0.004494
0.0000
RA
S11
M
98P
0.000549
0.0192
RA
95P
0.000092
0.0109
RA
S12
M
92P
-0.000104
0.4798
RN
EUP
-0.000020
0.2796
RN
S13
M
98P
0.000566
0.0045
RA
EUP
0.000000
0.9748
RN
S14
F
98P
0.000349
0.0632
RN
92P
0.000045
0.0304
RA
S15
F
98P
0.001180
0.0000
RA
98P
0.000059
0.0065
RA
S16
F
98P
0.001063
0.0041
RA
92P
-0.000006
0.8408
RN
S17
F
EUP
0.000388
0.1130
RN
95P
0.000072
0.0520
RN
S18
F
98P
0.002876
0.0000
RA
98P
0.000019
0.4790
RN
S19
F
92P
0.000448
0.1317
RN
92P
0.000028
0.3161
RN
S20
F
92Q
0.017040
0.0001
RA
EUP
0.000056
0.0000
RA
S21
F
EUP
0.001756
0.0006
RA
EUQ
0.002823
0.1498
RN
S22
M
EUP
-0.000576
0.0370
RL
92Q
-0.001938
0.0157
RL
S23
M
98P
0.000618
0.0000
RA
98P
0.000007
0.7139
RN
S24
M
92P
0.001287
0.0001
RA
92P
0.000324
0.0000
RA
S25
F
EUP
0.004507
0.0000
RA
95Q
0.005233
0.0333
RA
S26
F
98P
-0.001120
0.0000
RL
EUP
-0.000024
0.5023
RN
S27
F
92Q
0.017056
0.0000
RA
92Q
-0.001094
0.0411
RL
Model
r (x) *
p-value
u '' ( x)
*:Arrow-Pratt absolute risk aversion coefficient r ( x) 
r (x) *
Model
.
u ' ( x)
**: RA, RN and RL denote, respectively, risk averse, risk neutral, and risk loving.
23
p-value