Loss aversion and learning to bid∗
Dennis A.V. Dittricha, Werner Güthb,
Martin G. Kocherc, Paul Pezanis-Christoud
February, 2005
Abstract
Bidding challenges learning theories since experiences with the same bid vary stochastically: the same choice can result in either a gain or a loss. In such an environment the
question arises how the nearly universally documented phenomenon of loss aversion affects the adaptive dynamics. We analyze the impact of loss aversion in a simple auction
for different learning theories. Our experimental results suggest that a version of reinforcement learning which accounts for loss aversion fares as well as more sophisticated
alternatives.
Keywords: loss aversion, bidding, auction, experiment, learning
JEL-Classification: C91, D44, D83
∗ We are grateful for the assistance of Torsten Weiland and Ben Greiner who helped in programming the experiment.
a Max Planck Institute for Research into Economic Systems, Jena; email: dittrich@mpiew-jena.mpg.de
b Max Planck Institute for Research into Economic Systems, Jena; email: gueth@mpiew-jena.mpg.de
c University of Innsbruck; email: Martin.Kocher@uibk.ac.at
d BETA-Theme, Université Louis Pasteur, Strasbourg; email: ppc@cournot.u-strasbg.fr
1. Introduction
There are very rare situations where bidding games, usually referred to as auctions, are dominance solvable, like in the random price mechanism (see Becker, de Groot, and Marshak, 1964)
or in second-price auctions (see Gandenberger, 1961, and Vickrey, 1961). Apart from such special setups bidding poses quite a challenge for learning to bid. According to the usual first-price
rule (the winner is the bidder with the highest bid, and she pays her bid to buy the object), one
has to underbid one’s own value (bid shading) to guarantee a positive profit in case of buying.
Such underbidding may cause feelings of loss when the object has not been bought, although
its price was lower than one’s own value. In a similar vein, even when winning the auction,
feelings of loss may be evoked to think one could also have won by a lower bid and could thus
have achieved a larger profit. Given that the values of the interacting bidders are stochastic,
whether one experiences a loss or a gain is highly stochastic as well.
In this paper we investigate two important issues. First, we are going to reassess the issue
of learning in auctions. Although bidding experiments are numerous in economics (see Kagel,
1995, for an early survey), and quite a few of them provide ample opportunities for learning
(e. g., Garvin and Kagel, 1994; Selten and Buchta, 1998; Armantier, 2002, and Güth et al.,
2003), we believe that, so far, the number of repetitions in these studies has been too low to
fully account for possible learning dynamics. In stochastic environments, however, learning
is typically slow, and the number of repetitions or past bidding experiences are important
dimensions.
Thus, we experimentally implement a rather simple one-bidder contest (Bazerman and
Samuelson, 1983, and Ball, Bazerman, and Carroll, 1991) that allows for a considerable number of repetitions. To directly identify the path dependence of bidding behavior (i. e., why and
how bidding behavior adjusts in the light of past results) without any confounding factors, we
deliberately choose a bidding task that lacks strategic interaction.
Second, we experimentally test for the impact of loss aversion (Kahneman and Tversky,
2
1979; Tversky and Kahneman, 1992) on the learning dynamics in this bidding task.1 Loss aversion is referred to as the behavioral tendency of individuals to weigh losses more heavily than
gains. Obviously, in a first-price auction all possible choices can prove to be a success (yield a
gain) or a failure (imply a loss), at least in retrospect. Therefore, the usual and robust finding
of loss aversion should be reflected in the adaptation dynamics. More specifically, our main
hypothesis on the impact of loss aversion claims that, in absolute terms, an actual loss will
change bidding dispositions more than an equally large gain. To the best of our knowledge,
loss aversion has so far not been considered when specifying learning dynamics in situations
comparable to ours.
In the basic bidding task (originally studied experimentally by Samuelson and Bazerman,
1985, and more recently by Selten, Abbink, and Cox, 2002), to which subjects are exposed
in our experiment, the only bidder and the seller have perfectly correlated values. Put differently, the seller’s valuation is always the same constant and proper share of the bidder’s value.
Whereas the seller knows her value, the bidder only knows how it is randomly generated.
The bid of the potential buyer determines the price if the asset is sold. In our experiment
(like in earlier studies) the seller is captured by a robot strategy accepting only bids exceeding
her evaluation. Thus, the bidder has to anticipate that, whenever her bid is accepted, it must
have exceeded the seller’s valuation. Neglecting this fact may cause a winner’s curse as in standard common-value auctions involving more than one bidder. If the seller’s share or quota in
the bidder’s value is high, such a situation turns out to be a social dilemma: According to the
solution under risk neutrality, the bidder abstains from bidding in spite of the positive but –
due to the high quota – relatively low surplus for all possible values. However, if the quota is
low, the surplus is fully exploited: the optimal bid exceeds the highest valuation of the seller
and thereby guarantees trade. Earlier studies have only focused on the former possibility. In
our experiment each participant, as bidder, repeatedly experiences both low and high quotas.
1 There is ample evidence for loss aversion in both risky and risk-free environments (see Starmer, 2000, for a recent
overview).
3
In a setting where bids can vary continuously it is rather tricky to explore how bidding
behavior is adjusted in the light of past results. Traditional learning models like reinforcement learning, also referred to as stimulus-response dynamics or law of effect (see Bush and
Mosteller, 1955), cannot readily be applied. Therefore, we deviated from earlier studies by offering participants only a binary choice, namely to abstain from or engage in bidding. In order
to observe directly the inclination to abstain or engage, respectively, we allowed participants
to explicitly randomize their choice among the two possible strategies (for earlier experiments
offering explicitly mixed strategies, see Ochs, 1995; Albers and Vogt, 1997; Anderhub, Engelmann, and Güth, 2002).
More specifically, each participant played 500 rounds of the same high- and low-quota
game where a low quota suggests an efficient benchmark solution and a quota an inefficient
one. As mentioned earlier, we consider such extensive experience necessary since learning in
stochastic settings is typically slow. We also wanted sufficiently many loss and gain experiences
for each participant when trying to explore the path dependence of bidding behavior and how
adaptation is influenced by loss aversion.2
The paper proceeds as follows: The next section introduces the basic model in its continuous and its binary forms. Section three derives our hypothesis on the impact of loss aversion on reinforcement learning and on experience-weighted attraction learning, respectively.
In Section four we discuss the details of the experimental design and the laboratory protocol. Section five presents the results for the estimation of the learning models, and Section six
concludes.
2. The model
Before introducing the binary (’mini’-)bidding task, let us briefly discuss its continuous version. Let v ∈ [0, 1] denote the value of the object for the sole bidder who will be the only de2 It is straightforward that the bidding tasks participants are exposed to is an ideal environment to study learning
since other, possibly confounding behavioral determinants like social preferences cannot occur.
4
cision maker in this game. The seller’s evaluation is linearly and perfectly correlated to v, i. e.,
qv with q ∈ (0, 1) and q 6= 1/2. We assume that quota q is commonly known. Let b ∈ [0, 1]
denote the bid. It is assumed that v ∈ [0, 1] is realized according to the uniform density on
[0, 1], that the seller knows v, whereas the bidder only knows how v is randomly generated and
that the object is sold whenever b > qv.
Since the risk neutral buyer earns v − b in case of b > qv and 0 otherwise, her payoff
expectation is
E(v − b) =

2 R


b
1

(v − b) dv = q 2q − 1 for 0 6 b 6 q

b>qv

R1


 (v − b) dv = 1/2 − b for q < b < 1.
(1)
0
Thus, any positive bid b for b 6 q < 1/2 implies a positive expected profit, and the optimal
bid b∗ depends on q via
b∗ = b∗ (q) =



q for 0 < q < 1/2
(2)


0 for 1/2 < q < 1
so that E(v − b∗ ) = 0 for 1/2 < q < 1 and E(v − b∗ ) = 1/2 − q for 0 < q < 1/2, since q is the
highest possible evaluation by the seller. In case of q > 1/2, the positive expected surplus of
Z1
(1 − q)v dv =
1−q
2
(3)
0
is lost, whereas in case of q < 1/2 it is fully exploited.
In the mini-version, used for the experimental implementation, the assumption b ∈ [0, 1]
is substituted for b ∈ {0, B} with 0 < B < q. More specifically, in the experiment all participants will repeatedly encounter two bidding tasks, namely the case of q = q with 0 < q < 1/2
for which b∗ = B and the case of q = q with 1/2 < q < 1 for which b∗ = 0. Actually, it will be
5
assumed that the same B < q(1 + q)−1 < q applies regardless whether q = q or q = q. Thus,
the two bidding alternatives do not vary with q, and even in case of q = q, bidding b = B
does not guarantee trade and thereby efficiency.
Let us finally investigate the stochastic payoff effects for bidding b = 0 or b = B. In case
of b = 0, the bidder never buys. However, since she will be informed about v, she should in
retrospect understand that, compared to b = B, her past bid b = 0 is justified retrospectively
if B > v, which happens with probability P0+ = B. If, on the other hand, v > B > qv, which
happens with probability P0− = (B/q) − B, having bid b = 0 counts as a loss of v − B in
retrospect. Finally, none of the two applies if there is no trade even if b = B, i. e. if qv > B
which occurs with probability P0n = 1 − (B/q).
Similarly, bidding b = B implies a success of v − B if v > B > qv, what happens with
+
−
probability PB
= B/q − B, and a loss of v − B if B > v, which occurs with probability PB
= B.
n
In case qv > B, which happens with probability PB
= 1 − (B/q), trade is excluded. The
−
+
, P0− = PB
and
assumption 0 < B < q(1 + q)−1 guarantees that all probabilities P0+ = PB
n
P0n = PB
= Pn are positive. Thus, regardless whether q = q or q = q applies, bidding b = 0
or b = B implies positive, negative as well as neutral experiences (at least in retrospect).
3. Prospect theory, learning, and loss aversion
The above-described environment, simple as it may seem at first sight, poses quite a challenge
for learning theories. Obviously, whatever the bidder chooses, she can in retrospect experience
both an encouragement and regret with positive probability. As already mentioned, we are
interested in two specific research questions: (i) Will regret be more decisive than positive
encouragement as suggested by (myopic) loss aversion (see Benartzi and Thaler, 1995, as well
as Gneezy and Potters, 1997)? (ii) Will learning bring about an adjustment to bidding b = 0
when q = q and b = B when q = q?
6
3.1. Prospect theory
For a rigorous investigation of those questions, it is necessary to specify in detail how (myopic) loss aversion should be accounted for in learning theories where, of course, the fact of
repeatedly confronting the same or a similar task might weaken or even question the idea of
loss aversion. Let us, however, first derive the benchmark solution based on prospect theory
claiming risk aversion in the gain region and risk seeking in the loss region (Kahneman and
Tversky, 1979, and Tversky and Kahneman, 1992). To limit the number of parameters, assume
V(m) =



mα for a gain m > 0
(4)


−K(−m)α for a loss m < 0
where K(> 1) captures the stronger concern for losses than for gains and α ∈ [0, 1] the bidder’s
constant relative risk attitude.
For the continuous version of our bidding task this yields
Z
Z
α
(b − v)α dv.
(v − b) dv − K
V=
v>b>qv
(5)
b>qv
b>v
Solving the integrals, one obtains
V=
(1 − q)α+1 − qα+1 K α+1
b .
(α + 1)qα+1
(6)
Thus, the optimal bid b+ is
b+ =



0 if K > (1 − q)α+1 q−(α+1)
(7)


q otherwise.
In experimental economics one usually observes a K-parameter close to K = 2 (see Langer
7
and Weber, forthcoming, for a discussion).3 For such a value and the experimental specifications of q that we will use, the condition K > (1 − q)α+1 q−(α+1) always applies. Prospect
theory can therefore account for abstaining from bidding independently of the experimentally
used q-values, but not for general bidding activity.
3.2. Reinforcement learning
In general, reinforcement learning relies on the cognitive assumption that what has been good
in the past, will also be good in the future. In such a context, one does not have to be aware of
the decision environment except for an implicit stationarity assumption justifying that earlier
experiences are a reasonable indicator of future success.
Our decision environment implies straightforwardly that subject’s behavior is not only
influenced by actual gains and losses but also by retrospective gains and losses. These can be
derived unambiguously when, after each round, the bidder learns about the value v ∈ [0, 1],
regardless whether the object has been bought or not.
For t = 1, 2, . . . let Ct (0) denote the accumulated retrospective gains from bidding b = 0
by
Ct (0) = δ0t (B − vt ) − ρ0t k(vt − B) + dCt−1 (0)
(8)
where d with 0 < d < 1 discounts earlier experiences and k > 1 captures the idea that
retrospective losses (if vt > B) of bidding b = 0 matter more than retrospective gains (if
B > vt ). Here, of course, δ0t = 1 if B > vt and δ0t = 0 otherwise, and ρ0t = 1 if vt > B > qvt
and ρ0t = 0 otherwise. Similarly,
B
Ct (B) = δB
t (vt − B) − ρt k(B − vt ) + dCt−1 (B)
(9)
3 Benartzi and Thaler (1995) show that for a K-parameter close to K = 2, the empirically observed equity premium
would be consistent with an annual evaluation of investments.
8
is the accumulated realized gains from bidding b = B, where δB
t = 1 if vt > B > qvt and
B
B
δB
t = 0 otherwise, as well as ρt = 1 if B > vt and ρt = 0 otherwise.
In the tradition of reinforcement learning, these accumulated gains are assumed to determine the probabilistic choice behavior (see Bush and Mosteller, 1955; Roth and Erev, 1996;
Camerer and Ho, 1999, and Ho, Camerer, and Chong, 2002). If Pt denotes the probability of
selecting bt = B in round t and bt = 0 is chosen with probability 1 − Pt , we postulate
Pt =
exp(Γt )
1 + exp(Γt )
(10)
with
and
Γt = β0 + β1 I(q, q)Ct + β2 I(q, q)Ct
Ct = Ct (B) + αCt (0),
where I(x, y) is an indicator function that equals zero if x 6= y and 1 if x = y. For all βi as
well as for α and the actual choice sequence before round t, i. e. for {(bτ , vτ )}τ=1,...,t−1 for all
rounds t = 1, 2, . . ., the condition Pt ∈ [0, 1] is always satisfied. Thus the parameters to be
estimated are βi , α, d(∈ [0, 1]), and k(> 1).
3.3. Functional experience-weighted attraction learning and loss aversion
Ho, Camerer, and Chong (2002) proposed a one-parameter theory of learning in games that
appears to be especially suitable for our environment: Functional experience-weighted attraction (fEWA) is supposed to predict the path dependence of individual behavior in any normalform game and takes into account both past experience and expectations. Ho, Camerer, and
Chong (2002) claim themselves that it can be easily extended to extensive form games and
games with incomplete information. In the following, we describe how we adapted their model
to fit it to our experimental setting and to incorporate loss aversion.
9
Denote a bidder’s jth choice by sj , her round t choice st , and her payoff by π(sj ). Let
I(x, y) be an indicator function that equals zero if x 6= y and 1 if x = y. Using the notation
of Ho, Camerer, and Chong (2002), a bidder’s attraction Ajt for choice j at time t can then be
defined as
Ajt
φNt−1 Ajt−1 + [δ + (1 − δ)I(sj , st )]πjt (sj )
=
Nt−1 φ(1 − κt ) + 1
(11)
where Nt = Nt−1 φ(1 − κt ) + 1. Here κ represents an exploitation parameter that is defined as a normalized Gini coefficient on the choice frequencies, which is introduced to control
the growth rate of attractions. For the case at hand we will either set it equal to 1 or compute it
for each q-regime separately. The parameter φ reflects profit discounting or, equivalently, the
decay of previous attractions due to forgetting. It is analogous to the d parameter of equation
(9) of our reinforcement learning model and will be freely estimated, as in the original EWA
model of Camerer and Ho (1999). The δ parameter is the weight placed on retrospective gains
and losses (foregone payoffs) and we assume that it is either equal to 1 or to φ. Being fully responsive to foregone payoffs, i.e., δ = 1, implies that the decision maker’s behavior is not only
reinforced but rather driven by her payoff expectations, which suggests a thorough cognitive
retrospective analysis of bidding behavior. By tying it to the decay of previous attractions, we
are going to relax this assumption in the heuristic way proposed by Ho, Camerer, and Chong
(2002).
To augment the model in equation (11) by loss aversion, we introduce a loss weight l,
and we define by vt the realized losses or non-realized gains at time t, depending on whether
bidding B generated a loss. The bidder’s attraction AB
t for the choice b = B at time t then
becomes
10
AB
t =
B
B
B
φNt−1 AB
t−1 + [δ + (1 − δ)I(s , st )]πt + l[δ + (1 − δ)I(s , st )]vt
.
Nt−1 φ(1 − κt ) + 1
(12)
With such a specification, a loss weight of l = 0 indicates that losses influence the probability of choosing b = B to the same extent as gains. The probabilistic choice behavior is then
assumed to follow
Pt =
exp(Γt )
1 + exp(Γt )
(13)
with
B
Γt = β0 + β1 I(q, q)AB
t + β2 I(q, q)At .
Thus, the parameters to be estimated are βi , φ(∈ [0, 1]), and l(> 0).
Notice that, in contrast to the reinforcement learning model presented in the previous
section, the fEWA model accounts for retrospective gains and losses from having bid B but
cannot account for those from having bid 0, i. e., abstaining.
4. Laboratory protocol
Participants in the experiments were instructed that there were two roles to be assigned in
the experiment. They themselves would act as buyers, and the computer would follow a fixed
robot strategy as a seller. The instructions informed participants about parameter values, the
distribution of the values, and the rule followed by the robot seller.4 We chose B = 25 and a
4 A transcript of the German instructions can be found in the Appendix.
11
uniformly distributed value v ∈ [0, 100] as well as q = 0.6 and q = 0.4. For such parameters,
it is optimal to always bid b = B if q = 0.4 and to bid b = 0 if q = 0.6.
To observe propensities Pt for bidding b = B, we directly elicited mixed strategies by
asking participants to allocate x0 of 10 balls to the b = 0-box and the remaining 10 − x0
balls in the b = B-box, where, of course, 0 6 x0 6 10 must hold. The advantage is that
one avoids the randomness of the actual choice between bt = 0 and bt = B when testing
for each individual participant how well the adaption dynamics postulated in (10) capture her
idiosyncratic learning behavior.
In spite of the 500 rounds of play, the whole experiment took less than two hours to complete, which included the time needed to read the instructions. As already mentioned, we allowed for a sufficiently large number of repetitions to be able to test for learning effects since
the assessment of loss aversion in a stochastic environment obviously requires many repetitions. At the end of each round, participants were informed about the value v, whether trade
occurred, and about earnings for both possible choices, i. e., for b = B and b = 0. The seller’s
quota, q = 0.4 or q = 0.6, was displayed on each decision screen and changed every ten
rounds.
The conversion rate from experimental points to euros (1:200) was announced in the instructions. Participants received an initial endowment of 1500 experimental points (¤ 7.50)
to avoid bankruptcy during the experiment. As a matter of fact, such a case never occured.
Overall, 42 subjects participated in the two sessions that we conducted. The experiment was
computerized using z-tree (Fischbacher, 1999) and was conducted with students from various
faculties of the University of Jena.
5. Experimental results
We start by presenting some descriptive results on subjects’ behavior. We then discuss the
econometric estimations of the loss aversion parameter within our reinforcement learning
12
framework and within the more elaborate experience-weighted attraction model à la Ho, Camerer,
and Chong (2002).
5.1. Descriptive results
O 1
Only 4 (out of 42) subjects always played the optimal pure strategy when q =
0.4, and only 2 subjects always played the optimal pure strategy of bidding 0 when q = 0.6. No
single participant chose the optimal strategy in any round. Subjects performed significantly better
when q = 0.4 than when q = 0.6.
Table 1 plots the 500 decisions of each participant together with the individual average
deviations from optimal choices. Black dots represent actual choices, grey bars indicate deviations from the normative solution. Mean deviations from the normative solution in columns
three and four of Table 1 are computed for each block of ten rounds. The table allows discerning straightforwardly how subjects fare in the experiment in comparison to the theoretical
prediction and to what extent deviations from the normative solution decline over time, i. e.,
to what extent learning occurres. As can be seen rather easily, a few participants play optimally
almost always when q = 0.6 (i. e., subjects 2, 19, 26, and 39), and a few more do so when
q = 0.4 (i. e., subjects 2, 9, 19, 26, 27, 28, and 39).
5
In general, mean deviations from the
normative solution are significantly larger for the q-regime than for the q-regime for most of
the subjects, which indicates a behavioral tendency to bid even if it is not optimal. This bias toward bidding resembles the well-documented phenomenon of the winner’s curse in standard
first-price auctions. Another obvious regularity of the bidding data – that will have to be taken
into account in the estimation of our learning models – is a considerable heterogeneity across
subjects.
In order to give an impression of the strategies played by our subjects, Figure 1 shows the
relative frequency of probability choices (0,0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9,1) contingent on
5 Obviously, those observations will have to be dropped when estimating learning dynamics. Note that subject 19
did not manage to play the last 100 rounds in the given time for the experiment.
13
0.6
0.4
0.2
0.0
Relative Frequency
q = 0.4
q = 0.6
0
10
20
30
40
50
60
70
80
90 100
Probability of Bidding
Figure 1: Relative frequency of probability choices
Table 1: Choices and Deviations from the Normative Prediction
Subject
ID
Choices
1
Round
Mean deviation
500
1
2
3
4
5
6
7
8
9
10
11
12
14
q
q
Table 1: Choices and Deviations from the Normative Prediction
Subject
ID
Choices
1
Round
Mean deviation
500
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
15
q
q
Table 1: Choices and Deviations from the Normative Prediction
Subject
ID
Choices
1
Round
Mean deviation
500
q
q
35
36
37
38
39
40
41
42
Note:
Black dots represent actual choices, grey bars are deviations from the normative solution. Mean deviations from the normative solution in columns
three and four are computed for each block of ten rounds.
q. When q = 0.4, the optimal pure strategy of bidding B represents 58 % of all probability
choices, whereas the zero bid option represents only 3 % of them. When q = 0.6, the optimal
zero bid strategy represents 18 % of all probability choices. The fact that in this treatment 34 %
of all probability choices are “Bidding B for sure” either indicates that participants did not
perceive the expected revenue implications of such a choice, or that they were extremely risk
seeking.
To evaluate the degree of normative optimality of a bidder’s choices from a behavioral
point of view, we denote by fqa the frequency of probability choices a = 0, 0.1, 0.2, . . . , 1 in
regime q. We then measure her performance by the percentage sq of optimal choices in 250
bidding rounds for each q-regime, that is:
16
1 X 0.4
=
f
·a
25000 a=0 100a
1
s0.4
1 X 0.6
=1−
f
·a
25000 a=0 100a
1
and s0.6
(14)
0.4
s0.4 takes its minimum value (0) when f0.4
0 = 250 and its maximum value (1) when f1 =
250. Likewise, s0.6 takes its minimum value (0) when f0.6
1 = 0 and its maximum value (1) when
f0.6
0 = 250. Finally, we define a bidder’s overall performance index S ∈ [0; 1] as
1
1
S= +
2 50000
1
X
f0.4
100a · a −
a=0
1
X
!
f0.6
100a · a .
(15)
a=0
Figure 2 reports the distributions of s0.4 , s0.6 , and S. The average of s0.4 is 0.79 (s.d.: 0.19)
whereas s0.6 averages at 0.41 (s.d.: 0.28). The difference is significant at α < .001 according to a
one-tailed Wilcoxon Rank Sum test, confirming the visual expression from Table 1 that choices
are closer to the theoretical prediction in case of q = 0.4. This difference in performance is
also reflected in the average earnings under the two regimes: they are equal to 825.76 points
when q = 0.4 (s.d.: 223.96; Min: 269; Max: 1274) and to -213.66 points when q = 0.6 (s.d.:
127.985; Min: -547; Max: 20).
O 2
On average, subjects performed significantly better in the last rather than in
the first 100 rounds of the experiment. This holds for both q-regimes. In the last 100 rounds, four
bidders always used the optimal pure strategy in each q-regime.
Figures 3 and 4 report the same data as 1 but only for the first and last 100 rounds of
play (each of these 100 rounds are made up of 50 rounds with q = 0.4 and 50 rounds with
q = 0.6). In the last 100 rounds, 24 % of participants (ten bidders) always played the optimal
strategy when q = 0.4, and 10 % of participants (four bidders) did so when q = 0.6. The
latter outcome is in line with the 7 % reported by Ball, Bazerman, and Carroll (1991) for a
game that corresponds to our q = 0.6-regime which spanned only 20 rounds and for which
17
1.0
0.8
0.6
0.4
0.2
0.0
Cumulative Density
Performance for q = 0.4
Performance for q = 0.6
Performance for all q
0.2
0.4
0.6
0.8
1.0
Performance Score
Figure 2: Cumulative density of performance indices
no significant change in behavior was found. The performance indices sq are also significantly
greater in the last 100 rounds in both q-regimes (p < 0.001, according to Wilcoxon Rank
Sum tests). In fact, the relative frequency of "bidding B for sure" increases by 16.25 percentage
points between the first and last 100 rounds when q = 0.4, whereas those of "bidding 0 for
sure" when q = 0.6 increase by a smaller degree, i. e., 1.6 percentage points.
The average overall performance index S is equal to 0.597 for the first 100 rounds and
increases significantly (p = 0.009) to 0.648 for the last 100 rounds so that in the aggregate,
behavior gets closer to the optimum as the experiment proceeds. Nevertheless, as only four
bidders always use the optimal strategy in both q-regimes, the well-established pattern of the
winner’s curse still remains predominant.
18
0.6
0.4
0.2
0.0
Relative Frequency
First 100 Periods
Last 100 Periods
0
10
20
30
40
50
60
70
80
90 100
Probability of Bidding
0.6
0.2
0.4
First 100 Periods
Last 100 Periods
0.0
Relative Frequency
Figure 3: Relative frequency of probability choices during first and last 100 periods for q = 0.4
0
10
20
30
40
50
60
70
80
90 100
Probability of Bidding
Figure 4: Relative frequency of probability choices during first and last 100 periods for q = 0.6
5.2. Estimation of the learning functions and loss aversion parameter
As learning patterns are obviously rather heterogeneous in our population of 42 bidders, and
as we have data on individual behavior for a sufficiently large number of rounds, we estimate learning models for each participant separately. Clearly, estimating a random coefficients
model would also be possible, but this would (i) require specific assumptions on the distribution of each random coefficient and (ii) would not enable us to identify and assess individual
learning behavior and loss aversion.
19
Estimation is done by minimizing residual deviance.
2
Xh
i
Pt (log Pt − log P̂t ) + (1 − Pt )(log(1 − Pt ) − log(1 − P̂t ))
(16)
t
where Pt denotes the observed probability of bidding b = B and P̂t the expected probability
of bidding b = B from either (10) or (13).6
5.2.1. Reinforcement learning with retrospective loss aversion
In a first model, there is no loss aversion and no retrospective evaluation of bidding 0 so that
in (10), k = 1 and α = 0. We then reestimate (10) with no restriction on α. Finally, we allow
for loss aversion by estimating (10) with no restriction on both k and α. The estimated three
models (Model 1, 2, and 3, respectively) are reported in Tables 7, 8, and 9 in the Appendix.
Table 2 summarizes the explanatory power of retrospective gains and losses from "bid 0" by
comparing the outcomes of Model 2 to those of Model 1. For this comparison we discarded the
data of ten individuals for whom the estimation procedure could not converge due to a lack of
variation (i.e., individuals who did not learn anything during the experiment). A joint test on
the comparisons of the 32 remaining individuals indicates that including retrospective gains
and losses, as defined by equation 8, improves the model’s explanatory power significantly.
Including a loss aversion parameter in the reinforcement learning model yields a significant
improvement in terms of the model’s goodness-of-fit, which is confirmed by the joint test
comparing Models 2 and 3.
Table 3 reports some summary statistics for the main parameters. They were generated
by taking into account only those individuals for whom the model likelihood ratio test was
significant at the 1 % level, i. e. for 21 subjects (see Table 9). As can be easily discerned in
Table 3, the estimate of the profit discounting factor d is slightly smaller than 1, meaning that
6 The R procedures (see R Development Core Team, 2004) that were used for the estimation can be obtained from
the authors upon request.
20
Table 2: Reinforcement Learning Model Comparison
Model 1 vs. 2
α = 0 vs. α 6= 0
ID ∆ Residual deviance p-value
1
3
4
5
6
7
10
11
12
13
14
15
16
17
18
20
21
22
24
25
29
30
31
32
33
34
35
36
38
40
41
42
Joint test
Model 2 vs. 3
k = 1 vs. k 6= 1
∆ Residual deviance p-value
0.1544
0.2946
0.2704
0.0726
0.1058
4.9449
0.5342
0.2902
18.4784
0.0579
0.8995
1.6431
8.8800
6.0536
0.7973
0.1099
6.1838
0.0972
8.1094
21.7328
0.4391
1.1104
16.0136
7.0901
2.3987
6.9365
3.9780
3.2591
15.9563
0.8099
0.7655
2.5330
0.6944
0.5873
0.6030
0.7876
0.7449
0.0262
0.4649
0.5901
<0.0001
0.8098
0.3429
0.1999
0.0029
0.0139
0.3719
0.7403
0.0129
0.7553
0.0044
<0.0001
0.5075
0.2920
0.0001
0.0078
0.1214
0.0084
0.0461
0.0710
0.0001
0.3682
0.3816
0.1115
0.0002
0.2025
0.4491
0.6274
1.8389
15.9013
3.7280
0.2431
6.1161
0.3114
0.0019
0.0111
0.1401
5.7064
7.4571
0.0506
1.4109
0.0057
4.3111
0.1576
5.0221
0.0331
23.5405
7.2050
15.959
0.1089
1.8963
1.2497
2.6395
15.9504
10.9593
0.7135
0.9898
0.6527
0.5028
0.4283
0.1751
0.0001
0.0535
0.6220
0.0134
0.5768
0.9649
0.9159
0.7082
0.0169
0.0063
0.8220
0.2349
0.9398
0.0379
0.6914
0.0250
0.8555
<0.0001
0.0073
0.0001
0.7414
0.1685
0.2636
0.1042
0.0001
0.0009
0.3983
140.9999
<0.0001
133.9478
<0.0001
21
5
Table 3: Summary Statistics for the Final Reinforcement Learning Model (Model 3)
3
2
1
0.995
0.823
0.894
0.954
0
Median
Mean
Trimmed mean (90%)
Trimmed mean (80%)
Density
4
d
−0.2
1.073
61.348
0.973
0.868
0.2
0.1
Density
k
0
2
4
6
0.0 0.1 0.2 0.3 0.4
−2
Median
Mean
Trimmed mean (90%)
Trimmed mean (80%)
1.0
0.0
0.071
0.897
0.082
0.124
Density
Median
Mean
Trimmed mean (90%)
Trimmed mean (80%)
0.6
0.3
α
0.2
−2
0
2
4
6
8
participants tend to discount only moderately, i. e., they account for the entire history when
making a decision. This is quite reasonable since the experimental environment has been rather
stable (except for the q-switches).
The parameter α is usually positive suggesting that retrospective gains and losses from “bid
0” in the previous round actually affect learning. However, given the spread of the estimates,
there is a lot of heterogeneity among subjects. Finally, we find no strong evidence of a loss
aversion parameter k > 1 as its median value is greater but almost equal to 1 and its trimmed
mean is smaller than 1. The considerable discrepancy between the median and the mean values
of k suggests again an important heterogeneity in behavior.
5.2.2. Experience-weighted attraction models with(out) loss aversion
We now analyze loss aversion within the experience-weighted attraction framework. To this
end, we estimate intermediate models with various restrictions on the parameters to assess
22
their separate impact on learning. In the first model, the loss aversion parameter l is restricted
to l = 0, δ to δ = 1 and κ to κ = 1. This reduces the EWA model to a reinforcement learning
model with profit discounting and equal weights of actual and foregone payoffs. We will refer
to this model as Model 1’. We then relax the restriction on the loss aversion parameter (Model
2’). In Models 3’ and 4’, we estimate a version of the fEWA model proposed by Ho, Camerer,
and Chong (2002) that uses the functional form of κ (a Gini coefficient on choice frequencies)
and, given the stable strategic environment of our individual decision-making experiment,
with the additional restriction that δ = φ. The loss aversion parameter l is restricted to l = 0
in Model 3’ and is estimated freely in Model 4’. Estimation results are reported in Tables 10 to
13 in the Appendix. A summary of the explanatory power of loss aversion for learning to bid
is reported in Table 4.
After discarding the data of ten individuals for whom the EWA estimation procedures did
not converge, we find that including loss aversion in an EWA-type of reinforcement learning
model significantly increases the model’s goodness of fit (Model 1’ vs Model 2’). A similar conlusion can be reached when the growth rate of the attraction of bidding B is controlled by κ
(Model 3’ vs Model 4’). The statistics on the estimated parameters are reported in Tables 5 and
6 and confirm our previous finding that participants tend to discount past profits only moderately. Interestingly, we now find that the loss aversion parameter l is mostly positive, thereby
indicating that individuals tend to react more strongly to losses than to gains. Compared to the
more mitigated outcome of our previous model (where the loss aversion parameter accounted
for actual and retrospective losses from both bidding and not bidding), we conclude that although the inclusion of a loss aversion parameter significantly improves the goodness of fit in
both types of models, it is aversion to actual losses from bidding B that matters most in this
bidding game.
Note that Model 4’ uses more data than Model 2’ (i. e., it includes retrospective gains and
losses), although it has the same number of freely estimated parameters. Surprisingly, a pair-
23
Table 4: fEWA Model Comparison
Model 1’ vs. 2’
l = 0 vs. l 6= 0
ID ∆ Residual deviance p-value
1
3
4
5
6
7
10
11
12
13
14
15
16
17
18
20
21
22
24
25
29
30
31
32
33
34
35
36
38
40
41
42
Joint test
Model 3’ vs. 4’
l = 0 vs. l 6= 0
∆ Residual deviance p-value
1.2283
1.4229
4.6326
3.5452
4.3095
0.0096
15.8465
0.3823
9.4439
5.6123
0.0263
0.0096
0.0686
28.3936
3.7105
0.1298
24.5608
0.0000
11.8696
11.1497
17.2196
1.9858
1.0861
10.3973
22.8017
6.2305
1.4409
5.4314
33.6736
2.9117
12.4242
1.1212
0.2677
0.2329
0.0314
0.0597
0.0379
0.9219
0.0001
0.5364
0.0021
0.0178
0.8711
0.9221
0.7934
<0.0001
0.0541
0.7187
<0.0001
0.9999
0.0006
0.0008
<0.0001
0.1588
0.2973
0.0013
<0.0001
0.0126
0.2300
0.0198
<0.0001
0.0879
0.0004
0.2897
0.0986
0.4269
2.8800
3.0013
4.8293
0.1759
11.4128
0.3521
9.2802
0.6737
0.0147
11.1774
2.0021
28.5713
3.3562
0.0817
24.2655
0.0635
10.7218
18.1861
18.3205
2.2000
1.3126
10.9121
26.2552
4.3010
1.3047
5.475
33.3827
4.945
13.2883
1.0561
0.7535
0.5135
0.0897
0.0832
0.0280
0.6749
0.0007
0.5529
0.0023
0.4118
0.9034
0.0008
0.1571
<0.0001
0.0670
0.7751
<0.0001
0.8010
0.0011
<0.0001
<0.0001
0.1380
0.2519
0.0010
<0.0001
0.0381
0.2534
0.0193
<0.0001
0.0262
0.0003
0.3041
243.0757
<0.0001
254.3245
<0.0001
24
4
Table 5: Summary Statistics for the Final fEWA Model 2’
2
1
0.993
0.795
0.858
0.930
0
Median
Mean
Trimmed mean (90%)
Trimmed mean (80%)
Density
3
φ
0.0
1.2
0.2
0.0
0.289
-0.381
0.288
0.316
Density
Median
Mean
Trimmed mean (90%)
Trimmed mean (80%)
0.8
0.4
l
0.4
−2
0
1
2
3
Median
Mean
Trimmed mean (90%)
Trimmed mean (80%)
0.992
0.834
0.893
0.953
Density
φ
0 1 2 3 4 5 6 7
Table 6: Summary Statistics for the Final fEWA Model 4’
Median
Mean
Trimmed mean (90%)
Trimmed mean (80%)
0.285
0.115
0.321
0.312
Density
l
0.4
0.8
1.2
0.00 0.10 0.20 0.30
0.0
−2
0
1
2
3
wise comparison of the residual deviances reveals no improvement over Model 2’ (t-test, onesided, p = 0.211). Thus, the simpler reinforcement-type learning models capture the learning
dynamics as effectively as the more sophisticated fEWA versions do.
6. Conclusion
One of the most robust findings of auction experiments is the winner’s curse in common
value auctions. However, all reasonable adaptation or learning dynamics suggest that ’winners’
25
should learn to update their profit expectations more properly. In the simple bilateral trade
situation at hand, the winner’s curse would mean that the buyer neglects that the price of B
will be accepted only if it exceeds the seller’s evaluation of the asset (qv). We designed our
experiment so that the buyer’s optimal strategy, in terms of expected revenues, is to abstain
from bidding B when the seller’s quota (q) is high, and to bid B with probability one when it
is low.
Although we have provided ample opportunity for learning (250 rounds for each q-regime),
bidding B with a positive probability when the seller’s quota is high remains a strong pattern
throughout the experiment and weakens only slightly over time. In contrast, learning to bid B
with probability one when the seller’s quota is low was much faster. This could be partly due to
an efficiency motive. Efficiency always requires trade if the seller were real. For the real bidder,
the bid B generates for any possible positive v a larger surplus ((1−q)v) when the seller’s quota
is low than when it is high. As a consequence, bidding B is more profitable for lower quotas
than for higher ones, and is therefore more strongly reinforced.
In general, the bidding task seems a far greater challenge for learning models. There is
considerable individual heterogeneity of learning dynamics. Moreover, there is a small fraction
of people who almost instantaneously converge to optimal bidding. Lastly, learning to avoid
the winner’s curse is, if it exists at all, very weak. Either the participants understand sooner or
later that bidding is dangerous when q is high, or they never recognize it clearly and are only
slightly discouraged by (the more frequent) loss experiences.
Altogether, the important conclusions concerning learning are that loss aversion significantly influences the learning dynamics, both in a simple reinforcement learning model and in
a more sophisticated fEWA model, and that individual parameter estimates are mostly in line
with the loss aversion hypothesis when the model tested accounts for actual and retrospective
losses, as is the case of fEWA. However, in terms of goodness of fit, this model does not explain the behavioral adjustments in the bidding task better than reinforcement learning does.
26
Finally, retrospective gains and losses have a limited influence on learning to bid.
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A. Appendix
A.1. Instructions (originally in German; not necessarily for publication)
Welcome to the experiment!
Please do not speak with other participants from now on.
Instructions
This experiment is designed to study decision making. You can earn ‘real’ money, which
will be paid to you in cash at the end of the experiment. The ‘experimental currency unit’ will
be ‘experimental points (EP)’. EP will be converted to euros according to the exchange rate
in the instructions. During the experiment you and other participants will be asked to take
decisions. Your decisions will determine your earnings according to the instructions. All participants receive identical instructions. If you have any questions after reading the instructions,
raise your hands. One of the experimenters will come to you and answer them privately. The
experiment will last for 2 hours maximum.
Roles
29
You are in the role of the buyer; the role of the seller is captured by a robot strategy of the
computer. The computer follows an easy strategy that will be fully described below.
Stages
At the beginning of each round you can decide whether you want to ‘bid’ for a good or
abstain. For this you can apply a special procedure that we will introduce in a moment. Your
bid B is always 25 EP. The value of the good v will be determined independently and by chance
move in every round. v is always between (and including) 0 EP and 100 EP. The chosen values
are equally distributed, which means that every integer number between (and including) 0 EP
and 100 EP is the value of the good with equal probability. The computer as the seller only sells
the good when the bid is higher than the value multiplied by a factor q.
Mathematically, the computer only sells if B > qv.
The factor q takes on the values 0.4 or 0.6. At the beginning of each round you will be
informed about the value of q in that round. Every 10 rounds q changes. Keep in mind that
you are the only potential buyer of the good. Therefore, this is a very special case of an auction.
Special procedure to choose to bid or to abstain
The choice between bidding or abstaining is made by assigning balls to the two possibilities. You have 10 balls that have to be assigned to the two possibilities. The number of assigned
balls corresponds with the probability that a possibility is chosen. Thus, each ball corresponds
with a probability of 10 percent. All possible distributions of the 10 balls to the two possibilities
are allowed.
Information at the end of each round
At the end of each round you receive information on the value v, whether the good was
sold by the computer (or whether the good would have been sold in case you had bid). Additionally, you see your round profit.
Profit
• If you did not bid, your round profit is 0.
30
• If the bid was smaller than the value of the good multiplied by the factor (B < qv), the
computer does not sell. No transaction takes place and your round profit is 0.
• If you bid and B > qv, then your round profit depends on the value of the good. In case
of a value of the good that is higher than the bid v > B, you make a profit. In case of
a value of the good that is lower than the bid, you make a loss in that round. Losses in
single rounds can, of course, be balanced by profits from other rounds.
At the beginning of the experiment you receive an endowment of 1500 EP. At the end of the
experiment, experimental points will be exchanged at a rate of 1:200. That means 200 EP = 1
¤.
Rounds
There are 500 rounds in the experiment. In each round you have to take the same decision
(assign the balls to the two possibilities of bidding or abstaining) as explained above. Only
factor q will be changed every 10 rounds.
Means of help
You find a calculator on your screen. You are, of course, allowed to write notes on the
instruction sheets.
Thank you for participating!
31
32
-0.1817
(-0.850)
0.2736
(1.008)
1.0019
(791.170)
0.8081
(6.930)
1.0073
(371.731)
0.9316
(46.595)
0.4184
(2.715)
0.9915
(56.611)
0.6792
(4.820)
-0.2689
(-0.266)
0.9833
(126.366)
0.7943
(6.698)
1
15
14
13
12
11
10
7
6
5
4
3
d
ID
1.7662
(13.536)
0.7217
(7.307)
1.3759
(5.597)
1.7516
(10.157)
2.0418
(9.510)
0.3437
(2.314)
0.6313
(6.146)
0.1007
(0.631)
0.8513
(7.132)
-0.3000
(-3.278)
0.4041
(2.020)
1.2297
(10.002)
Constant
A.2. Estimation Results (not for publication)
0.0508
(2.099)
0.0600
(2.791)
0.0173
(4.581)
0.0144
(1.077)
0.0123
(5.302)
-0.0291
(-3.206)
-0.0417
(-1.825)
0.0047
(1.067)
0.0156
(0.976)
0.0150
(0.667)
-0.0036
(-0.889)
-0.0101
(-0.693)
β1
0.0348
(2.299)
-0.0001
(-0.007)
0.0035
(3.277)
-0.0243
(-2.445)
0.0002
(1.109)
0.0085
(2.564)
-0.0411
(-3.546)
0.0014
(0.744)
-0.0203
(-2.211)
-0.0026
(-0.182)
0.0083
(2.603)
-0.0154
(-1.707)
β2
864.35
954.69
1276.36
657.81
1362.63
671.58
1079.36
733.68
790.48
1005.43
1025.99
643.83
Null
Deviance
858.96
924.04
1275.86
651.39
1351.28
649.67
1058.17
662.15
777.22
764.37
1017.38
633.60
Residual
Deviance
Table 7: Estimation results for Model 1
0.4412
0.0016
0.9686
0.3608
0.1284
0.0120
0.0141
0.0000
0.0848
0.0000
0.2306
0.1636
0.0054
0.0305
0.0005
0.0064
0.0113
0.0220
0.0211
0.0715
0.0132
0.2333
0.0086
0.0102
Estrella
P(> χ )
R2
2
33
-0.1598
(-1.063)
0.9702
(88.896)
-0.0229
(-0.095)
0.9951
(246.349)
0.8207
(10.513)
0.9995
(246.236)
0.9402
(24.589)
-0.3207
(-0.951)
0.9971
(232.382)
0.4263
(1.803)
0.9836
(323.234)
0.9816
(114.093)
0.6136
(2.983)
16
33
32
31
30
29
25
24
22
21
20
18
17
d
ID
1.4953
(12.371)
0.7869
(5.740)
1.4109
(12.066)
0.6018
(2.536)
-0.2688
(-2.691)
-0.0846
(-0.422)
0.8997
(6.768)
0.9011
(8.935)
0.5825
(4.308)
2.4052
(13.609)
-1.4311
(-6.417)
0.6442
(1.928)
1.3517
(11.785)
Constant
β2
0.0410
0.0513
(1.784) (3.569)
0.0258 -0.0006
(2.634) (-0.226)
0.0391
0.0356
(1.848) (2.516)
0.0069
0.0076
(2.436) (3.144)
-0.0049
0.0180
(-0.329) (2.485)
-0.0015
0.0049
(-0.641) (2.591)
0.0172
0.0002
(1.725) (0.080)
0.0566
0.0052
(2.788) (0.508)
-0.0046
0.0056
(-1.873) (2.632)
0.0649
0.0525
(2.364) (2.645)
-0.0138
0.0336
(-1.947) (6.632)
0.0096
0.0089
(1.099) (1.556)
-0.0334
0.0013
(-1.591) (0.130)
β1
595.38
643.98
783.69
392.98
913.47
612.69
1156.32
1109.81
1214.59
994.84
766.89
1202.12
512.72
Null
Deviance
591.82
575.48
486.45
377.89
889.16
603.66
1151.95
1033.66
1203.06
844.88
756.23
1192.46
493.62
Residual
Deviance
Table 7: Estimation results for Model 1
0.6194
0.0000
0.0000
0.0563
0.0069
0.2110
0.5345
0.0000
0.1237
0.0000
0.1492
0.1846
0.0228
P(> χ2 )
0.0036
0.0695
0.3038
0.0152
0.0243
0.0091
0.0044
0.0751
0.0115
0.1468
0.0107
0.0096
0.0193
Estrella
R2
34
0.9924
(199.553)
0.9968
(107.95)
0.9992
(386.720)
1.0022
(146.288)
0.5158
(2.337)
0.987
(290.577)
0.5802
(0.976)
34
42
41
40
38
36
35
d
ID
β1
0.2481
0.0001
(0.820) (0.023)
2.1328
0.0086
(7.187) (1.347)
1.2656
0.0089
(5.109) (3.506)
2.2389 -0.0074
(10.202) (-1.689)
1.0945 -0.0415
(9.918) (-1.852)
-0.3127
0.0133
(-1.604) (1.630)
1.3141
0.0031
(10.920) (0.192)
Constant
0.0052
(2.667)
0.0023
(1.368)
0.0011
(1.643)
0.0018
(1.616)
0.0063
(0.725)
0.0194
(4.520)
0.0118
(0.840)
β2
895.63
942.39
778.32
352.90
990.60
498.87
894.65
Null
Deviance
894.39
724.17
773.22
339.72
867.97
423.61
824.34
Residual
Deviance
Table 7: Estimation results for Model 1
0.8919
0.0000
0.4662
0.0862
0.0000
0.0000
0.0000
P(> χ2 )
0.0012
0.2134
0.0051
0.0133
0.1207
0.0775
0.0699
Estrella
R2
35
-0.2002
(-0.953)
0.2121
(0.773)
1.0054
(243.536)
0.8157
(7.507)
1.0026
(118.500)
0.9399
(34.705)
0.4574
(3.123)
0.9914
(45.434)
0.1093
(0.780)
-0.0900
(-0.088)
0.9804
(133.639)
0.924
(13.366)
-0.0601
(-0.557)
1
16
15
14
13
12
11
10
7
6
5
4
3
d
ID
1.7692
(13.522)
0.7322
(7.277)
1.4413
(5.984)
1.7407
(10.146)
1.9838
(8.436)
0.6518
(2.411)
0.6352
(6.117)
0.0946
(0.512)
0.7687
(7.622)
-0.3065
(-3.199)
0.0744
(0.186)
1.4256
(7.401)
1.5139
(12.140)
Constant
-0.4527
(-0.551)
-0.3594
(-0.750)
0.3320
(0.802)
0.2154
(0.381)
-0.8311
(-0.619)
1.7568
(1.046)
-0.3399
(-0.982)
-0.8252
(-0.278)
17.7032
(0.666)
0.5845
(0.289)
-1.1882
(-1.313)
-5.2126
(-0.689)
-3.0953
(-2.556)
α
0.0502
(2.077)
0.0607
(2.822)
0.0163
(4.799)
0.0138
(1.058)
0.0135
(4.665)
-0.0219
(-1.351)
-0.0451
(-2.112)
0.0028
(0.394)
-0.008
(-0.678)
0.0123
(0.450)
-0.0051
(-1.270)
0.0032
(0.635)
0.0369
(2.207)
β1
0.0349
(2.346)
-0.0008
(-0.076)
0.0031
(3.013)
-0.0243
(-2.578)
0.0004
(0.889)
0.0027
(0.426)
-0.0375
(-3.036)
0.0008
(0.325)
-0.0062
(-0.656)
-0.0051
(-0.431)
0.0091
(2.753)
-0.0036
(-0.601)
0.0536
(3.850)
β2
512.72
864.35
954.69
1276.36
657.81
1362.63
671.58
1079.36
733.68
790.48
1005.43
1025.99
643.83
Null
Deviance
Table 8: Estimation results for Model 2
475.86
855.68
922.25
1275.74
614.44
1350.70
648.60
1048.28
661.94
777.08
763.83
1016.79
633.30
Residual
Deviance
0.0010
0.3623
0.0027
0.9891
0.0002
0.2016
0.0216
0.0037
0.0000
0.1526
0.0000
0.3312
0.2609
P(> χ2 )
0.0374
0.0087
0.0323
0.0006
0.0438
0.0119
0.0231
0.0309
0.0717
0.0134
0.2338
0.0092
0.0105
Estrella
R2
36
1.0065
(169.029)
0.1534
(0.584)
0.9938
(206.353)
0.9170
(29.875)
1.0008
(233.816)
0.9495
(12.783)
0.2182
(1.998)
0.9979
(259.432)
0.3247
(1.350)
1.0030
(453.062)
0.9812
(154.361)
0.5449
(2.317)
0.9867
(242.560)
17
34
33
32
31
30
29
25
24
22
21
20
18
d
ID
0.3715
(3.247)
1.3936
(11.536)
0.5413
(2.022)
-0.0383
(-0.282)
-0.1570
(-0.631)
0.9166
(7.954)
0.9133
(8.645)
0.4892
(2.919)
2.4432
(13.761)
0.5807
(2.369)
1.1760
(4.216)
1.4158
(12.217)
-0.1709
(-0.808)
Constant
1.2755
(11.534)
-0.8383
(-1.071)
-0.2151
(-0.426)
1.3142
(2.960)
-0.7012
(-0.428)
-642.2003
(-34.980)
-10.4398
(-0.774)
-0.533
(-0.831)
-1.0994
(-1.478)
3.3416
(3.449)
-1.4824
(-2.067)
3.5832
(0.855)
1.4371
(4.844)
α
β2
0.0136
0.0029
(2.052) (2.452)
0.0319
0.0362
(1.354) (2.647)
0.0063
0.0081
(1.901) (3.076)
-0.0081
0.0139
(-1.160) (2.922)
-0.0011
0.0047
(-0.706) (2.592)
0.0000
0.0000
(0.814) (0.561)
0.0151
0.0125
(0.710) (0.905)
-0.0040
0.0055
(-1.775) (2.797)
0.0722
0.0507
(2.911) (2.555)
-0.0038
0.0059
(-2.702) (3.316)
0.0138
0.0046
(2.658) (1.870)
-0.0180 -0.0098
(-0.837) (-1.283)
0.0117
0.0138
(2.031) (4.193)
β1
894.65
595.38
643.98
783.69
392.98
913.47
612.69
1156.32
1109.81
1214.59
994.84
766.89
1202.12
Null
Deviance
Table 8: Estimation results for Model 2
810.46
587.02
561.30
454.42
375.66
888.28
560.19
1135.73
1033.47
1190.70
844.66
754.63
1180.36
Residual
Deviance
0.0000
0.3824
0.0000
0.0000
0.0702
0.0134
0.0000
0.0357
0.0000
0.0177
0.0000
0.1899
0.0279
P(> χ2 )
0.0836
0.0084
0.0841
0.3375
0.0175
0.0251
0.0534
0.0205
0.0753
0.0238
0.1470
0.0122
0.0217
Estrella
R2
37
0.9909
(177.051)
1.0001
(421.061)
0.9830
(103.276)
0.5280
(2.961)
0.9862
(262.811)
0.9423
(24.538)
35
42
41
40
38
36
d
ID
α
β1
1.3930
2.1080
0.0237
(3.559)
(4.629) (4.042)
1.3305
2.4618
0.0075
(5.629)
(2.187) (3.400)
2.793
-7.6377 -0.0146
(9.610)
(-2.927) (-1.687)
1.0695
-1.0336 -0.0411
(9.352)
(-1.049) (-1.947)
-0.1858
-0.4673
0.0115
(-0.892)
(-0.923) (1.671)
1.4983 3162.4744
0.0000
(10.854)
(3.043) (-2.273)
Constant
0.0064
(2.176)
0.0017
(1.406)
0.0085
(2.641)
0.0050
(0.636)
0.0186
(4.468)
0.0000
(2.840)
β2
895.63
942.39
778.32
352.90
990.60
498.87
Null
Deviance
Table 8: Estimation results for Model 2
889.32
722.63
771.60
307.81
861.45
415.65
Residual
Deviance
0.5326
0.0000
0.4994
0.0002
0.0000
0.0000
P(> χ2 )
0.0063
0.2149
0.0067
0.0463
0.1270
0.0860
Estrella
R2
38
16
15
14
13
12
11
10
7
6
5
4
3
1
ID
k
-0.2008
0.9848
(-0.948)
(1.182)
0.2423
2.0438
(0.891)
(0.629)
1.0016
1.3453
(235.795)
(3.433)
0.8015
1.5633
(7.100)
(2.474)
1.0047
6.1333
(358.70)
(1.265)
0.9760
-0.4556
(120.647)
(-1.371)
0.5768
4.7332
(4.580)
(2.438)
0.9962
0.1575
(113.585)
(0.507)
0.1167
0.1868
(1.092)
(1.391)
-0.2464 1123.6293
(-0.362)
(2.296)
0.9809
1.0369
(95.173)
(1.763)
0.9309
0.9029
(11.148)
(1.530)
-0.0409
1.2548
(-0.371)
(2.358)
d
1.7681
(12.347)
0.7482
(7.189)
1.6335
(4.664)
1.6714
(8.656)
2.4142
(8.622)
2.8197
(5.277)
0.4159
(2.888)
0.1124
(0.574)
0.9177
(7.678)
-0.2946
(-3.007)
0.0750
(0.188)
1.4302
(7.663)
1.5343
(11.692)
Constant
β1
-0.4573
0.0508
(-0.535) (1.289)
-0.4439
0.0337
(-0.609) (0.665)
0.0706
0.013
(0.134) (4.075)
0.0332
0.0085
(0.083) (0.850)
-1.3038
0.001
(-1.168) (1.387)
2.7452 -0.0121
(1.742) (-1.758)
0.0517 -0.0080
(0.449) (-1.569)
-5.1509
0.0005
(-0.890) (0.637)
26.9258 -0.0086
(0.602) (-0.602)
-0.2802
0.0000
(-0.208) (6.296)
-1.2201 -0.0048
(-1.199) (-0.810)
-6.8641
0.0026
(-0.325) (0.311)
-3.3013
0.0331
(-2.431) (2.019)
α
0.0352
(1.774)
0.0023
(0.251)
0.0051
(2.304)
-0.0246
(-2.827)
0.0003
(0.435)
-0.0052
(-2.299)
-0.0275
(-2.474)
0.0011
(0.522)
-0.0071
(-0.599)
0.0000
(0.435)
0.0089
(2.103)
-0.0030
(-0.333)
0.0478
(2.890)
β2
512.72
864.35
954.69
1276.36
657.81
1362.63
671.58
1079.36
733.68
790.48
1005.43
1025.99
643.83
Null
Deviance
Table 9: Estimation results for Model 3
475.58
855.66
922.24
1275.12
602.20
1350.21
641.14
1016.48
658.26
775.82
762.93
1016.39
633.30
Residual
Deviance
0.0023
0.5004
0.0062
0.9870
0.0000
0.2863
0.0095
0.0000
0.0000
0.1974
0.0000
0.4408
0.3840
P(> χ2 )
0.0377
0.0087
0.0323
0.0012
0.0564
0.0124
0.0307
0.0622
0.0754
0.0146
0.2347
0.0096
0.0105
Estrella
R2
39
34
33
32
31
30
29
25
24
22
21
20
18
17
ID
k
1.0005
-0.3975
(246.846)
(-0.298)
0.9928 796.6397
(464.660)
(23.563)
0.9945
1.2129
(203.067)
(1.808)
0.9068
1.4012
(26.515)
(4.219)
1.0012
1.0727
(180.817)
(1.613)
1.0035
8.2077
(422.765)
(3.149)
0.2021
1.3001
(1.956)
(2.237)
0.9952
0.3826
(151.811)
(2.555)
0.3545
1.1699
(1.302)
(1.633)
0.9968
-23.2483
(71.116)
(-0.377)
0.8546
-1.0078
(23.379)
(-2.794)
0.7353 1290.2667
(12.731)
(2.718)
0.9908
1.1653
(141.01)
(5.045)
d
-0.2491
(-1.033)
1.7032
(12.387)
0.5609
(1.998)
-0.0373
(-0.310)
-0.1605
(-0.659)
0.4854
(2.591)
0.9102
(8.462)
0.3559
(1.324)
2.4713
(11.747)
1.3334
(0.711)
-0.0669
(-0.241)
0.6749
(3.865)
0.0679
(0.152)
Constant
1.3106
(0.767)
-1.0300
(-8.770)
-0.2210
(-0.376)
0.8399
(2.201)
-0.6839
(-0.402)
-7.1516
(-4.816)
-8.6853
(-1.316)
-0.5476
(-2.591)
-1.1412
(-1.343)
1.6332
(0.279)
-0.4118
(-1.674)
0.466
(1.902)
1.3499
(6.712)
α
0.0024
(1.268)
-0.0000
(-4.575)
0.0049
(1.018)
-0.0077
(-0.957)
-0.0011
(-0.687)
0.0010
(2.332)
0.0180
(1.210)
-0.0202
(-1.747)
0.0636
(1.624)
-0.0006
(-1.514)
0.0584
(3.268)
-0.0000
(-2.840)
0.0136
(2.289)
β1
0.0008
(1.403)
0.0000
(7.645)
0.0091
(2.124)
0.0151
(3.180)
0.0048
(2.406)
0.0000
(2.684)
0.0128
(1.546)
0.0037
(1.323)
0.0461
(1.815)
0.0002
(0.223)
0.0379
(4.112)
-0.0001
(-3.521)
0.012
(2.958)
β2
894.65
595.38
643.98
783.69
392.98
913.47
612.69
1156.32
1109.81
1214.59
994.84
766.89
1202.12
Null
Deviance
Table 9: Estimation results for Model 3
810.24
555.10
546.89
407.34
375.60
878.24
559.88
1127.11
1033.45
1187.87
844.56
739.72
1168.94
Residual
Deviance
0.0000
0.0012
0.0000
0.0000
0.1220
0.0035
0.0001
0.0122
0.0000
0.0202
0.0000
0.0185
0.0053
P(> χ2 )
0.0838
0.0406
0.0991
0.3874
0.0175
0.0351
0.0537
0.0290
0.0753
0.0266
0.1471
0.0272
0.0329
Estrella
R2
40
1.0046
(128.696)
1.0047
(124.280)
0.9964
(142.649)
1.0108
(91.242)
0.8785
(21.224)
0.9423
(26.620)
35
42
41
40
38
36
d
ID
1.6008
(4.153)
0.4024
(4.443)
1.9930
(3.081)
0.7179
(3.756)
-1.3977
(-1.668)
0.4975
(1.224)
k
α
β1
1.3851
4.7046
0.0123
(4.149)
(2.806) (2.301)
1.2825
0.2218
0.0477
(5.266)
(0.411) (1.572)
1.9640
-5.4936 -0.0095
(4.372)
(-1.830) (-1.671)
1.0065
0.7123
0.0022
(4.645)
(1.824) (0.551)
-0.3146
0.5150 -0.0106
(-0.655)
(0.637) (-0.865)
1.4983 3162.4744 -0.0000
(11.239)
(1.419) (-4.126)
Constant
0.0041
(2.106)
0.0004
(0.701)
0.0064
(2.210)
0.0025
(1.061)
0.0499
(2.237)
0.0000
(0.304)
β2
895.63
942.39
778.32
352.90
990.60
498.87
Null
Deviance
Table 9: Estimation results for Model 3
887.89
700.72
739.70
302.53
858.95
411.86
Residual
Deviance
0.5688
0.0000
0.0017
0.0001
0.0000
0.0000
P(> χ2 )
0.0077
0.2358
0.0386
0.0519
0.1294
0.0900
Estrella
R2
41
-0.2204
(-1.049)
0.1461
(0.520)
0.9944
(275.804)
0.8545
(4.725)
1.0018
(291.874)
0.9357
(31.562)
0.5356
(4.043)
0.9919
(64.471)
0.1221
(0.413)
0.6189
(0.684)
0.9816
(144.29)
0.8099
(8.883)
-0.1085
(-0.899)
1
16
15
14
13
12
11
10
7
6
5
4
3
φ
ID
β2
0.0326
0.0450
(2.342) (2.030)
-0.0014
0.0467
(-0.156) (2.769)
0.0054
0.0113
(3.629) (3.172)
-0.0142
0.0150
(-1.119) (1.383)
0.0004
0.0134
(1.200) (4.544)
0.0059 -0.0118
(1.963) (-2.028)
-0.0252 -0.0371
(-2.855) (-2.416)
0.0007
0.0024
(0.702) (1.054)
0.0091
0.0605
(0.614) (3.563)
0.0020
0.0031
(0.286) (0.249)
0.0088 -0.0054
(2.881) (-1.321)
-0.0168
0.0024
(-2.328) (0.211)
0.0616
0.0518
(4.345) (2.513)
β1
1.7686
(13.532)
0.7360
(7.286)
0.8665
(3.650)
1.7838
(7.652)
1.9897
(9.063)
0.4734
(2.668)
0.6261
(6.003)
0.0895
(0.470)
0.7597
(7.027)
-0.3112
(-2.977)
0.0742
(0.204)
1.3317
(9.253)
1.5110
(12.182)
Constant
512.72
864.35
954.69
1276.36
657.81
1362.63
671.58
1079.36
733.68
790.48
1005.43
1025.99
643.83
Null
Deviance
482.63
855.64
922.35
1276.17
643.11
1350.65
651.18
1072.78
661.75
781.12
771.05
1018.02
633.76
Residual
Deviance
Table 10: Estimation results for fEWA Model 1’
0.0000
0.0334
0.0000
0.9788
0.0021
0.0074
0.0001
0.0865
0.0000
0.0249
0.0000
0.0468
0.018
P(> χ2 )
0.0564
0.0128
0.0460
0.0003
0.0283
0.0160
0.0398
0.0092
0.1035
0.0134
0.3202
0.0115
0.0149
Estrella
R2
42
0.9609
(54.371)
0.1832
(0.851)
0.9941
(226.358)
0.7420
(4.070)
1.0015
(336.966)
0.9420
(42.303)
0.0188
(0.124)
0.9979
(293.765)
0.9591
(39.078)
1.0028
(581.207)
0.9836
(165.022)
0.9805
(72.548)
0.9946
(265.037)
17
34
33
32
31
30
29
25
24
22
21
20
18
φ
ID
β2
-0.0018
0.0135
(-0.681) (2.056)
0.0350
0.0295
(2.915) (1.468)
0.0074
0.0031
(3.343) (2.134)
0.0055
0.0053
(1.077) (0.430)
0.0043 -0.0010
(2.754) (-0.994)
0.0001
0.0200
(0.049) (2.754)
0.0225
0.0855
(2.168) (4.846)
0.0056 -0.0034
(2.972) (-2.233)
0.0147 -0.0114
(1.963) (-1.152)
0.0033
0.0081
(3.377) (4.570)
0.0041
0.0163
(1.532) (3.897)
0.0057
0.0020
(1.236) (0.605)
0.0039 -0.0013
(2.594) (-0.634)
β1
0.8211
(5.232)
1.3894
(11.555)
0.3418
(1.782)
-0.1971
(-1.868)
-0.1934
(-0.831)
1.0217
(6.182)
0.9462
(8.971)
0.4087
(2.454)
1.7903
(5.186)
1.1016
(4.048)
1.2371
(3.923)
1.1718
(4.394)
0.0470
(0.140)
Constant
894.65
595.38
643.98
783.69
392.98
913.47
612.69
1156.32
1109.81
1214.59
994.84
766.89
1202.12
Null
Deviance
826.59
580.31
561.45
426.50
377.07
886.52
580.47
1141.06
1033.20
1212.89
845.67
754.57
1197.07
Residual
Deviance
Table 10: Estimation results for fEWA Model 1’
0.0000
0.0018
0.0000
0.0000
0.0012
0.0000
0.0000
0.0016
0.0000
0.6357
0.0000
0.0064
0.1675
P(> χ2 )
0.1001
0.0260
0.1467
0.5551
0.0258
0.0390
0.0635
0.0207
0.1045
0.0024
0.2036
0.0181
0.0069
Estrella
R2
43
1.0028
(126.728)
1.0011
(376.792)
0.9994
(120.157)
0.5281
(2.955)
0.9856
(266.359)
0.9722
(23.881)
35
42
41
40
38
36
φ
ID
β2
0.0013
0.0036
(1.296) (1.121)
0.0009
0.0071
(1.819) (3.328)
0.0026 -0.0091
(1.576) (-1.670)
0.0053 -0.0413
(0.687) (-2.216)
0.0182
0.0073
(4.588) (2.085)
-0.0005
0.0078
(-0.202) (1.258)
β1
2.1174
(7.393)
1.1399
(5.045)
2.186
(9.968)
1.0684
(9.342)
-0.2155
(-1.054)
1.5618
(4.451)
Constant
895.63
942.39
778.32
352.90
990.60
498.87
Null
Deviance
891.57
719.81
771.64
337.54
874.41
425.34
Residual
Deviance
Table 10: Estimation results for fEWA Model 1’
0.2559
0.0000
0.0826
0.0015
0.0000
0.0000
P(> χ2 )
0.0057
0.3120
0.0103
0.0229
0.1646
0.1273
Estrella
R2
44
-0.1358
(-0.550)
0.2411
(0.787)
0.9938
(327.949)
0.7803
(6.318)
1.0036
(271.693)
0.9335
(24.300)
0.5632
(5.817)
0.9903
(59.789)
0.0816
(0.337)
0.9864
(83.589)
0.9805
(105.661)
0.8073
(8.397)
-0.1065
(-0.900)
1
16
15
14
13
12
11
10
7
6
5
4
3
φ
ID
β1
β2
1.0992
0.0239
0.0278
(0.735) (1.584) (1.378)
1.3454
0.0028
0.0284
(0.611) (0.331) (1.091)
0.4576
0.0068
0.0103
(2.280) (3.725) (3.775)
0.7579 -0.0233
0.0064
(1.795) (-2.656) (0.745)
1.9132 -0.0025
0.0018
(6.721) (-0.933) (1.712)
0.0396
0.0060 -0.0114
(0.097) (1.733) (-1.605)
1.6004 -0.0416 -0.0152
(4.228) (-4.340) (-1.950)
0.4990
0.0012
0.0021
(0.736) (0.674) (0.866)
-0.9611
0.0243
0.0876
(-3.285) (1.858) (3.265)
70.9783
0.0000
0.0001
(1.661) (15.542) (7.371)
-0.0489
0.0089 -0.0060
(-0.166) (2.897) (-1.035)
-0.0588 -0.0166
0.0018
(-0.093) (-2.137) (0.131)
-0.0988
0.0645
0.0543
(-0.272) (3.485) (2.287)
l
1.8372
(12.291)
0.7751
(6.558)
1.4308
(3.575)
1.6319
(8.679)
2.5455
(7.481)
0.4812
(2.476)
0.4206
(3.250)
0.1632
(0.689)
0.6172
(5.438)
0.1676
(0.920)
0.0675
(0.185)
1.3332
(9.305)
1.4949
(10.812)
Constant
512.72
864.35
954.69
1276.36
657.81
1362.63
671.58
1079.36
733.68
790.48
1005.43
1025.99
643.83
Null
Deviance
482.56
855.63
922.32
1270.56
633.67
1350.26
635.33
1072.77
657.45
777.57
766.42
1016.60
632.54
Residual
Deviance
Table 11: Estimation results for fEWA Model 2’
0.0000
0.0685
0.0000
0.2142
0.0001
0.0148
0.0000
0.1592
0.0000
0.0118
0.0000
0.0522
0.0234
P(> χ2 )
0.0566
0.0128
0.0461
0.0079
0.0464
0.0165
0.0704
0.0092
0.1096
0.0184
0.3259
0.0136
0.0167
Estrella
R2
45
34
33
32
31
30
29
25
24
22
21
20
18
17
ID
l
0.9996 -1.5034
(50.801) (-0.256)
0.2013
1.0257
(0.884)
(1.633)
0.9944
0.1672
(227.199)
(0.367)
0.8691
1.7850
(16.191)
(3.644)
1.0015
0.0000
(284.122)
(0.000)
0.9910 -1.2416
(133.168) (-3.398)
0.1363 -0.7644
(1.072) (-4.670)
0.9946 -0.6824
(181.524) (-9.515)
0.9445
0.4827
(16.551)
(1.066)
1.0023
0.2889
(536.462)
(1.121)
0.9930
1.2999
(293.928)
(4.349)
0.7099 -15.7199
(7.495) (-0.496)
1.0084
1.6891
(224.68)
(5.891)
φ
β2
0.0007
0.0042
(0.232) (0.137)
0.0327
0.0170
(2.964) (1.376)
0.0084
0.0028
(2.178) (1.778)
0.0137
0.0039
(2.880) (0.906)
0.0043 -0.0010
(2.423) (-0.948)
0.0016
0.0086
(1.689) (1.618)
0.0277
0.1431
(2.582) (4.425)
0.0021 -0.0228
(1.419) (-3.781)
0.0175
0.0021
(1.497) (0.167)
0.0043
0.0074
(2.562) (4.132)
0.0049
0.0054
(2.379) (2.943)
0.0066
0.0037
(0.461) (0.417)
0.0073 -0.0001
(4.088) (-0.583)
β1
-0.1818
(-0.552)
1.4988
(11.096)
0.3948
(1.645)
0.1445
(0.811)
-0.1934
(-0.815)
-0.0244
(-0.087)
0.8111
(6.590)
0.7462
(2.602)
2.2892
(4.185)
1.2253
(3.888)
1.9164
(4.970)
0.6804
(3.914)
0.1563
(0.831)
Constant
894.65
595.38
643.98
783.69
392.98
913.47
612.69
1156.32
1109.81
1214.59
994.84
766.89
1202.12
Null
Deviance
820.36
557.51
551.05
425.42
375.09
869.30
569.32
1129.20
1033.20
1188.32
845.54
750.86
1168.67
Residual
Deviance
Table 11: Estimation results for fEWA Model 2’
0.0000
0.0000
0.0000
0.0000
0.0013
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0030
0.0000
P(> χ2 )
0.1090
0.0651
0.1648
0.5565
0.0290
0.0636
0.0853
0.0366
0.1045
0.0363
0.2037
0.0236
0.0451
Estrella
R2
46
1.0076
(318.644)
1.0001
(247.362)
1.0098
(233.020)
0.7149
(2.649)
0.9153
(39.651)
0.9961
(62.135)
35
42
41
40
38
36
φ
ID
1.858
(5.626)
0.9233
(3.358)
1.0450
(21.385)
-0.7257
(-2.133)
-2.2752
(-2.867)
-0.4573
(-2.649)
l
β2
0.0028
0.0005
(1.550) (1.438)
0.0040
0.0025
(2.017) (1.580)
0.0212 -0.0024
(4.326) (-1.300)
0.0070 -0.0473
(1.155) (-1.583)
0.0341 -0.0010
(3.219) (-0.213)
-0.0003
0.0155
(-0.416) (1.674)
β1
1.9851
(9.973)
1.0202
(3.740)
2.5618
(6.489)
1.0715
(6.717)
-0.5744
(-2.529)
1.6132
(5.469)
Constant
895.63
942.39
778.32
352.90
990.60
498.87
Null
Deviance
890.45
707.39
768.73
303.87
868.98
423.90
Residual
Deviance
Table 11: Estimation results for fEWA Model 2’
0.2700
0.0000
0.0478
0.0000
0.0000
0.0000
P(> χ2 )
0.0072
0.3278
0.0148
0.0740
0.1719
0.1298
Estrella
R2
47
-0.1401
(-0.692)
0.2345
(1.144)
0.9941
(277.973)
0.8744
(6.031)
1.0052
(369.252)
0.9497
(58.876)
0.4564
(2.988)
0.9916
(61.710)
0.6078
(4.407)
-0.2701
(-0.415)
0.9819
(134.676)
0.8014
(8.013)
-0.0073
(-0.050)
1
16
15
14
13
12
11
10
7
6
5
4
3
φ
ID
0.0327
(2.241)
-0.0010
(-0.074)
0.0080
(3.729)
-0.0196
(-1.296)
0.0004
(1.415)
0.0134
(2.949)
-0.0374
(-3.030)
0.0008
(0.672)
-0.0138
(-1.555)
-0.0036
(-0.305)
0.0121
(2.642)
-0.0220
(-2.178)
0.0517
(3.493)
β1
0.0492
(2.048)
0.0701
(2.865)
0.0221
(3.180)
0.0213
(1.515)
0.0197
(6.186)
-0.0248
(-3.204)
-0.0485
(-2.221)
0.0029
(1.051)
0.0419
(2.374)
0.0128
(0.600)
-0.0125
(-1.028)
-0.0017
(-0.110)
0.0359
(1.573)
β2
1.7637
(13.530)
0.7299
(7.269)
0.7634
(3.349)
1.8095
(7.547)
1.9257
(9.006)
0.2404
(1.278)
0.6332
(6.092)
0.0910
(0.485)
0.8592
(7.217)
-0.3024
(-3.304)
0.1813
(0.509)
1.2969
(9.370)
1.4764
(12.228)
Constant
512.72
864.35
954.69
1276.36
657.81
1362.63
671.58
1079.36
733.68
790.48
1005.43
1025.99
643.83
Null
Deviance
495.06
856.92
923.66
1275.79
649.48
1350.74
648.99
1062.86
661.61
780.83
776.11
1017.07
633.89
Residual
Deviance
Table 12: Estimation results for fEWA Model 3’
0.0005
0.0593
0.0000
0.9036
0.0396
0.0077
0.0000
0.0009
0.0000
0.0218
0.0000
0.0305
0.0191
P(> χ2 )
0.0331
0.0109
0.0442
0.0008
0.0161
0.0159
0.0440
0.0230
0.1037
0.0138
0.3140
0.0129
0.0147
Estrella
R2
48
0.9612
(53.778)
0.1264
(0.503)
0.9908
(199.425)
0.1561
(0.309)
1.0035
(376.891)
0.9433
(44.761)
0.3645
(2.470)
0.9980
(280.264)
0.9524
(27.908)
0.9997
(541.895)
0.9853
(151.296)
0.9803
(74.124)
0.9884
(193.511)
17
34
33
32
31
30
29
25
24
22
21
20
18
φ
ID
β2
-0.0019
0.0247
(-0.408) (1.999)
0.0382
0.0392
(2.496) (1.792)
0.0121
0.0064
(3.085) (2.348)
0.0206
0.0057
(1.658) (0.216)
0.0054 -0.0016
(2.890) (-0.877)
-0.0006
0.0315
(-0.138) (2.879)
0.0063
0.0638
(0.599) (3.234)
0.0072 -0.0071
(2.905) (-2.041)
0.0190 -0.0104
(1.716) (-0.787)
0.0052
0.0165
(3.799) (4.177)
0.0036
0.0182
(1.236) (4.089)
0.0061
0.0022
(1.273) (0.579)
0.0086 -0.0034
(2.745) (-0.374)
β1
0.7963
(5.131)
1.401
(11.800)
0.4015
(2.079)
-0.1856
(-1.968)
-0.1542
(-0.690)
1.0581
(6.285)
0.9390
(8.717)
0.4565
(2.808)
1.8649
(5.192)
0.9400
(3.412)
1.3029
(3.631)
1.1637
(4.452)
0.1383
(0.458)
Constant
894.65
595.38
643.98
783.69
392.98
913.47
612.69
1156.32
1109.81
1214.59
994.84
766.89
1202.12
Null
Deviance
825.35
580.27
560.33
424.89
376.96
888.09
600.26
1139.64
1034.05
1211.66
849.35
755.89
1196.93
Residual
Deviance
Table 12: Estimation results for fEWA Model 3’
0.0000
0.0017
0.0000
0.0000
0.0011
0.0000
0.0061
0.0008
0.0000
0.4031
0.0000
0.0118
0.1579
P(> χ2 )
0.1019
0.0260
0.1487
0.5572
0.0260
0.0368
0.0246
0.0226
0.1034
0.0041
0.1988
0.0162
0.0071
Estrella
R2
49
1.0022
(129.992)
1.0020
(393.099)
0.9970
(109.108)
0.5615
(3.408)
0.9856
(267.117)
0.9766
(28.120)
35
42
41
40
38
36
φ
ID
β2
0.0014
0.0049
(1.364) (1.162)
0.0010
0.0102
(1.804) (3.403)
0.0047 -0.0105
(1.531) (-1.672)
0.0076 -0.0518
(0.834) (-2.210)
0.0188
0.0082
(4.606) (2.067)
-0.0010
0.0099
(-0.374) (1.286)
β1
2.1490
(7.255)
1.1578
(5.123)
2.1670
(9.919)
1.0644
(9.237)
-0.2212
(-1.089)
1.6046
(4.196)
Constant
895.63
942.39
778.32
352.90
990.60
498.87
Null
Deviance
891.65
720.09
771.43
336.45
874.31
425.14
Residual
Deviance
Table 12: Estimation results for fEWA Model 3’
0.2641
0.0000
0.0755
0.0009
0.0000
0.0000
P(> χ2 )
0.0055
0.3116
0.0107
0.0246
0.1647
0.1276
Estrella
R2
50
-0.1318
(-0.606)
0.2969
(1.380)
0.9936
(320.92)
0.7901
(5.648)
1.0089
(289.350)
0.9544
(53.241)
0.5863
(6.062)
0.9905
(60.673)
0.4560
(4.437)
-0.2696
(-0.472)
0.9809
(90.893)
0.9981
(106.373)
-0.1071
(-0.590)
1
16
15
14
13
12
11
10
7
6
5
4
3
φ
ID
β1
β2
0.3256
0.0287
0.0393
(0.268) (1.521) (1.119)
7.3070
0.0031
0.0102
(0.040) (0.058) (0.045)
0.3699
0.0096
0.0209
(1.792) (3.767) (3.582)
0.7159 -0.0307
0.0088
(1.509) (-2.769) (0.725)
2.0133 -0.0027
0.0016
(12.162) (-0.858) (1.838)
-0.1100
0.0125 -0.0277
(-0.443) (2.555) (-2.588)
2.1123 -0.0390 -0.0137
(3.157) (-4.011) (-1.915)
0.4799
0.0013
0.0024
(0.690) (0.657) (0.876)
-0.9761
0.0089
0.1403
(-5.478) (0.745) (2.837)
70.9783
0.0001
0.0003
(0.293) (0.388) (0.294)
-0.0440
0.0122 -0.0140
(-0.124) (2.625) (-0.782)
-0.9539 -0.0018 -0.0150
(-1.887) (-1.116) (-1.879)
1.6585
0.0266
0.0203
(0.761) (1.326) (1.272)
l
1.7859
(11.955)
0.8103
(1.684)
1.2148
(3.089)
1.6376
(8.097)
2.5151
(7.835)
0.2056
(0.990)
0.4074
(3.014)
0.1580
(0.719)
0.6038
(4.449)
-0.2948
(-3.119)
0.1728
(0.478)
2.1350
(5.088)
1.5908
(11.242)
Constant
512.72
864.35
954.69
1276.36
657.81
1362.63
671.58
1079.36
733.68
790.48
1005.43
1025.99
643.83
Null
Deviance
493.06
845.74
923.65
1275.12
640.20
1350.38
637.58
1062.69
656.78
777.83
773.23
1016.64
633.79
Residual
Deviance
Table 13: Estimation results for fEWA Model 4’
0.0006
0.0009
0.0000
0.8711
0.0015
0.0156
0.0000
0.0022
0.0000
0.0131
0.0000
0.0531
0.0398
P(> χ2 )
0.0369
0.0273
0.0442
0.0017
0.0339
0.0163
0.0661
0.0233
0.1106
0.0180
0.3175
0.0135
0.0149
Estrella
R2
51
34
33
32
31
30
29
25
24
22
21
20
18
17
ID
l
0.9986
-1.4744
(170.452) (-1.171)
0.2056
3.0046
(0.853)
(0.780)
0.9913
0.1138
(194.756)
(0.289)
0.8960
1.9904
(21.937)
(3.472)
1.0030
-0.0973
(310.801) (-0.242)
0.9918
-1.1544
(153.618) (-4.694)
0.4058
-0.8996
(4.103) (-8.349)
0.9950
-0.7010
(179.894) (-10.500)
0.9447
0.4787
(19.908)
(1.141)
0.9986
0.2847
(438.563)
(1.216)
0.9948
1.4478
(328.871)
(4.270)
0.6428
-6.6366
(6.081) (-1.379)
1.0014
1.2711
(138.743)
(2.541)
φ
β2
0.0014
0.0088
(1.103) (0.691)
0.0199
0.0101
(1.093) (0.926)
0.0128
0.0059
(2.589) (2.011)
0.0240
0.0095
(2.689) (1.490)
0.0051 -0.0017
(2.261) (-0.788)
0.0025
0.0170
(1.836) (2.090)
0.0187
0.2098
(1.567) (3.069)
0.0025 -0.0530
(1.374) (-3.916)
0.0202
0.0030
(1.696) (0.245)
0.0066
0.0164
(3.021) (3.817)
0.0043
0.0053
(2.211) (3.159)
0.0220
0.0142
(1.126) (0.941)
0.0118 -0.0008
(3.828) (-0.393)
β1
-0.1790
(-0.725)
1.5448
(10.487)
0.4468
(1.806)
0.2319
(1.210)
-0.1413
(-0.587)
-0.0189
(-0.066)
0.7346
(5.289)
0.8309
(2.942)
2.3079
(4.739)
1.1479
(3.049)
2.0289
(5.157)
0.6504
(3.789)
0.1161
(0.388)
Constant
894.65
595.38
643.98
783.69
392.98
913.47
612.69
1156.32
1109.81
1214.59
994.84
766.89
1202.12
Null
Deviance
821.05
554.02
549.41
423.57
374.76
869.76
582.07
1128.92
1033.99
1187.40
849.26
752.53
1168.36
Residual
Deviance
Table 13: Estimation results for fEWA Model 4’
0.0000
0.0000
0.0000
0.0000
0.0011
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0063
0.0000
P(> χ2 )
0.1080
0.0710
0.1677
0.5589
0.0296
0.0629
0.0603
0.0369
0.1035
0.0376
0.1989
0.0211
0.0456
Estrella
R2
52
1.0052
(224.263)
1.0012
(263.307)
1.0098
(247.219)
0.7205
(3.166)
0.9095
(36.255)
0.9956
(59.741)
35
42
41
40
38
36
φ
ID
β1
β2
1.5000
0.0026
0.0011
(2.614) (1.439) (0.980)
0.9748
0.0048
0.0034
(3.780) (2.007) (1.657)
1.0479
0.0373 -0.0023
(23.184) (4.211) (-1.299)
-0.7550
0.0108 -0.0737
(-3.172) (1.390) (-1.674)
-2.3435
0.0372 -0.0015
(-2.944) (3.173) (-0.254)
-0.4521 -0.0004
0.0204
(-2.395) (-0.397) (1.620)
l
2.089
(12.854)
1.0189
(3.712)
2.6098
(6.422)
1.0687
(6.420)
-0.5673
(-2.463)
1.6104
(5.440)
Constant
895.63
942.39
778.32
352.90
990.60
498.87
Null
Deviance
890.59
706.81
766.49
303.07
868.83
423.83
Residual
Deviance
Table 13: Estimation results for fEWA Model 4’
0.2840
0.0000
0.0186
0.0000
0.0000
0.0000
P(> χ2 )
0.0070
0.3285
0.0183
0.0752
0.1721
0.1299
Estrella
R2