1. Loss Aversion Index Recovery in The Cross
Section of Subjective Well Being and Happiness
On Economic Growth
G. Charles-Cadogan
University of Cape Town
Presentation:
Network for Integrated Behavioural Science (NIBS)
International Interdisciplinary Conference On
“Assessing well-being when preferences are incoherent”
University of East Anglia, Norwich
April 4-6, 2016

2. Outline
Overview of subjective well being (SWB) as alternative to GDP
Review of emerging research on asymmetric effects of loss aversion to
decline in SWB
Microfoundations for econometric theory and recovery of loss
aversion to decline in SWB
Econometric estimation and testing for impact of loss aversion on
SWB cross sectional regressions
Conclusion and summary: A loss aversion index can be recovered and
tested from published papers using simple recovered ratio procedures.
Extant cross-sectional models are misspecifed due to simultaneity
bias, and they overestimate the impact of economic growth on SWB.
Loss aversion identifies clientele effects in SWB. We propose an
econometric theory to solve the simultaneity bias problem.

3. Motivation for study
☛ Subjective well being (SWB)
Many applied papers report empirical associations between
happiness and other variables. Few papers treat happiness economics
in relation to its origins, definitions, theory, and methods
[MacKerron, 2012].
Happiness–A self reported measure of SWB from surveys.
SWB measures features of individuals perceptions of their experiences,
not their utility as economists typically conceive of it
[Kahneman and Krueger, 2005].
Life satisfaction depends on relative income compared to social
reference group and weather! [Feddersen et al., 2015].
Those perceptions are a more accurate gauge of actual feelings if they
are reported closer to the time of, and in direct reference to, the actual
experience.
☛ Happiness-income paradox [Easterlin, 1974, p. 113]:
An increase in national income per capita does not increase the overall
level of happiness in the economy.
People compare themselves to their neighbours or some aspiration or
reference level other than per capita income alone.

4. Explosive growth in research on economics of happiness
Source: [MacKerron, 2012, Fig. 1, p. 706]
Number of EconLit Journal Articles with Titles Including the Terms
’Happiness’, ’Wellbeing’, ’Well-being’ or ’Life Satisfaction’, by Year
The series plotted with filled circles excludes articles from the Journal of
Happiness Studies.

5. Some policy implications of misspecified SWB models
Longitudinal studies that fail to account for loss aversion may
overestimate the positive effects of income on SWB.
Such failure could carry important policy implications regarding
raising individual and national income and well-being. [Boyce, 2010].
Income inequality implies that as reference income increases, loss
aversion within relatively low income groups may cause them to seek
greater access to credit and subprime loans in order to maintain
well-being–thereby reducing public welfare. [Treeck, 2014].
Loss aversion implies that instead of overall economic growth
national agenda should focus on closing income inequality gaps in
order to increase SWB.

6. SWB as predictor of social unrest I–Arab Spring
30
25
20
15
10
5
0
7 000
6 500
6 000
5 500
5 000
4 500
2005 2006 2007 2008 2009 2010
%
5 158
4 762
5 508
5 904
6 114
6 367
Thriving (left scale) GDP per capita (PPP) (right scale)
US dollars
25%
29%
13% 13%
12%
Source: OECD (2013), OECD Guidelines on Measuring Subjective
Well-being. SWB data for Egypt are from Gallup. GDP per capita (PPP)
estimates are from the International Monetary Fund’s World Economic
Outlook Database. The “Arab Spring” occurred in February 2011 over
perceptions of worsening living conditions despite rising GDP per capita.

7. Myopic loss aversion (MLA) to decline in SWB
Loss aversion redux: Anticipated losses have a greater influence
on choice and predicted feelings about an outcome than anticipated
gains of the same magnitude [Boyce et al., 2013]. It is the tendency
for individuals to be more sensitive to reductions in their levels of
well-being than to increases [Benartzi and Thaler, 1995].
Myopia is characterized by short term evaluation. For example,
deviation from optimal consumption plans [Stroz, 1956];
consumption tracking income [Shea, 1995].
Myopic loss aversion (MLA) is characterized by loss aversion over
short evaluation periods [Benartzi and Thaler, 1995].
Myopic loss aversion to decline in SWB was identified in
[Vendrik and Woltje, 2007] but the MLA index not estimated.

8. Literature on loss aversion and subjective well being
There several emergent papers on the subject of loss aversion to decline in
SWB. None of them estimate a MLA index directly.
The most comprehensive study so far is by [De Neve et al., 2015]
who used millions of observations for data at the international level.
➵ Implied MLA index in range [1.5, 6.31] wherein measures of life
satisfaction are more sensitive to negative economic growth compared
to positive economic growth.
➵ MLA implicates growth policy and our understanding of the long-run
relationship between GDP and subjective well-being.
[Boyce et al., 2013] implied MLA index of 2.2 and 2.6
[Di Tella et al., 2010] implied MLA index 4.0, 5.0
[Vendrik and Woltje, 2007] implied MLA index 1.31
[Charles-Cadogan, 2015a] directly estimated a time series of MLA
indexes for the relative income hypothesis concentrated in the range
[1.27, 7.79].

9. Research questions posed
☛ Assuming that self reported (as opposed to elicited) ratings on one’s
life satisfaction in a single item in a survey are admissible measures of
utility, we try to answer the following questions under rubric of
relative income and subjective well being:
Is loss aversion to decline in happiness estimable?
If so, are extant cross sectional models of SWB misspecified by virtue
of analysts failure to explicitly include a loss aversion index parameter
in their specifications?
What would the econometric theory for embedding and recovering
the MLA index in SWB regressions look like?
Are wealthier people more loss averse to a decline in their happiness
compared to poorer people?
Is the myopic loss aversion index for decline in happiness identifiable in
extant cross-sectional regressions on SWB?

10. The robust ratio of slopes formula for MLA index
[Tversky and Kahneman, 1992] introduced a robust ratio of slopes
formula for MLA via stochastic choice.
Subjects were presented with two simple lotteries L1 = a, 1
2; b, 1
2
and L2 = c, 1
2; x, 1
2 . Vizly, for L1, probability of loss of amount a is
1
2 and probability of gain of amount b is 1
2 and similarly for L2.
Value of x which subjects chose to establish equivalence between
lotteries, i.e., L1 ∼ L2, was noted for a, b, c combination.
The ratio λθ = x−b
c−a was used as a robust estimator of MLA
[Tversky and Kahneman, 1992, p. 310] noted that “when the
possible loss is increased by k the compensating gain must be
increased by about 2k. So λθ is a ratio of the slope of gains over
the slope of losses.

11. Relative PCE growth for South Africa 1960:Q1–2014:Q1
-0.15
-0.1
-0.05
0
0.05
0.1
1965/01
1966/04
1968/03
1970/02
1972/01
1973/04
1975/03
1977/02
1979/01
1980/04
1982/03
1984/02
1986/01
1987/04
1989/03
1991/02
1993/01
1994/04
1996/03
1998/02
2000/01
2001/04
2003/03
2005/02
2007/01
2008/04
2010/03
2012/02
2014/01
ConsumptionGrowthRate
ZA Rolling 5yrs Impact of Loss Aversion to Decline in Growth of Standard of Living
ZA (Semidurable) Personal Consumption Expenditure Growth Rate in 2010 Prices
Source: [Charles-Cadogan, 2015a]
Relative PCE growth over 20Q (5-years) standard of living sliding window.
Psychological pain associated with asymmetric response to negative growth magnified
by loss aversion.

12. Local regression estimator for MLA index
GainsLosses O
Microfoundations of cross-sectional regression model with asymmetric
growth in an ǫ-neighbourhood Bǫ(xr ) of the reference point xr .

13. Canonical model of subjective well being (SWB)
Ui = f Ci
∑
j
αij Cj
where Ci is the consumption expenditure of the i -th consumer, and
Ui is her utility index [Duesenberry, 1949, p. 32]; [Easterlin, 1974,
p. 113].
αij is the weight assigned by i to j’s consumption. According to
Easterlin, if i chooses the weights αij = 1
n then she is comparing
herself to the average consumption expenditure of her neighbours.
Uik = α ln(xik) + β ln xik
xi ·
+ zTγγγ + ǫik
is i-th cross sectional regression (usually ordered probit) over
observations k = 1, . . . , K; Uik is ordinal scaled item response from
surveys; zzz is a vector of demographic variables; xi · is an assigned
reference point; β is impact of growth on SWB.
Model typically pays homage to
[Mundlak, 1978, Hausman and Taylor, 1981].

14. Clientele Effects Of Economic Growth On SWB
The relative impact of a unit change in average economic growth ( ¯x) on average SWB
( ¯U):
̺Neo
i =
(∂ ¯Ui./∂ ¯xG
i. )
(∂ ¯Ui./∂ ¯xL
i.)
=
β/ ¯xG
i.
β/ ¯xL
i.
=
¯xL
i.
¯xG
i.
̺BE
i =
(∂ ¯Ui./∂ ¯xG
i. )
(∂ ¯Ui./∂ ¯xL
i.)
=
βG / ¯xG
i.
βL/ ¯xL
i.
=
βG
βL
¯xL
i.
¯xG
i.
=
1
λi
¯xL
i.
¯xG
i.
2
If ¯xG
i. = ¯xL
i., then ̺Neo
i = 1 and ̺BE
i = 1/λi .
Under ̺Neo
i the impact of a unit increase in economic growth is offset by a unit
decrease in economic growth.
Under ̺BE
i the impact of a unit change in economic growth is asymmetric. The
gain in SWB for a unit increase in economic growth is a fraction of the loss in
SWB for a unit decrease in economic growth.
Gain seeking in low income countries, i.e., 0 < λi < 1 magnifies SWB by a
factor much greater than that for higher income countries characterized by loss
aversion, i.e., λi > 1. Thus, risk attitudes provide a theoretical explanation for
why low income countries are more optimistic than high income countries.
Cf. [Humphrey and Verschoor, 2004, Graham, 2008].

15. Recovery of MLA index in 2SLS procedure for β
Rewrite canonical regression as
Uik = α ln(xik ) + βG gG
ik I{xik> ˜xi ·} − βLgL
ikI{xik< ˜xi ·} + zT γγγ + ǫik , ǫ ∼
(0, σ2
ǫ ), H0 : βG = βL = β vs Ha : H0 not true
Incorporate the identifying restriction for MLA
gL
ik = λgG
ik + ηik , ηik ∼ (0, σ2
η ), k = 1, . . . , K
Theoretical first stage regression yields ¯g
L
ik = λ ¯gG
ik
Estimation of second stage regression yields
Uik = α ln(xik ) + βG gG
ik I{xik>xi ·} − θgL
ikI{xik<xi ·} + zTγγγ
where
λ =
θ
βG
Run auxiliary regression ǫi · = βLηi ·I{xik<xi ·} + νi ·, νi · ∼ (0, σ2
ν )
from stages 1 and 2
βL =
∑i ǫi ·ηi ·I{xik<xi ·}
∑i η2
i ·I{xik <xi ·}
βL is asymptotically consistent and efficient [White, 2001, p. 144] so
λ is a conservative [under] estimate of λ.

16. Canonical model overestimates impact of growth on SWB
yyy = Xααα + Zβββ + ǫǫǫ, Z = gggG gggL , βββ = βG βL
T
ηηη = Zφφφ, φφφ = [−λ 1]T , ǫǫǫ = βLηηη + ννν
MX = I − X(XT X)−1XT , MZ = I − Z(ZT Z)−1ZT
ααα = XT MZ X
−1
XT MZyyy
βG = gggGT
MXgggG
−1
gggGT
MXyyyI{gggG >000}
βL = φφφ
T
ZT Zφφφ
−1
φφφ
T
Zǫǫǫ embeds MLA theory for λ,
βL = gggLT
MXgggL
−1
gggLT
MXyyyI{gggL<000} no MLA theory,
V βL = E βL − βL
2
= φφφT ZT Zφφφ
−1
σ2
ν
V βL = E βL − βL
2
= gggLT
MXgggL
−1
σ2
ǫ I{gggL<000}
V βL = gggLT
gggL + λ2gggGT
gggG
−1
σ2
ν < V (βL) =
gggLT
gggL−gggLT
X(XT X)−1XTgggL
−1
σ2
ǫ I{gggL<000}
=⇒ βL < βL a.s. in Hilbert space L2(R) or weakly in R

17. Recovery Theorem for MLA index for SWB
THEOREM. Under regularity conditions for least squares write the canonical SWB regression
under H0 = βG = βL = β and Ha : βG = β and βL = θ, β = θ
yyy = Xδδδ + ǫ, so that δδδ = XT
X
−1
XT
yyy,
where X = [x zzzT gggGT
gggLT
] = X : Z is the design matrix with transposed (T) row
vectors in bold, yyy is a vector of utility scores (ordinal or otherwise),
δδδ = [α γγγ β θ]T
is the vector of parameters to be estimated; ǫ ∼ (0, σ2
ǫ IIIn) where IIIn is the n × n identity matrix;
√
n ˆδδδ − δδδ −→
d
(000, ΣΣΣδ) where ΣΣΣδ is the variance-covariance matrix of δδδ given by
ΣΣΣδ = σ2
ǫ XT
X
−1
=
XT X XT Z
ZT X ZT Z
−1
=
⋆ ⋆
⋆ ΣΣΣ(θ, β)
where ⋆ denotes sub-matrices for parameters that are not of interest. And σ2
θ , σ2
β, σ2
θβ are the
variance and covariance components of the sub-matrix of interest ΣΣΣ(θ, β) =
σ2
θ σθβ
σθβ σ2
β
and
ˆθ
ˆβ
∼ N
θ
β , ΣΣΣ(θ, β) . We claim that limn P λn = limn P
θn
βn
= λ and the asymptotic
distribution √
n(ˆˆλn − λ) −→
d
N 0,
σ2
θ − 2λσθβ + λ2σ2
β
β2
exist provided |β| > 0. By definition σθβ = 0.

18. Geometry of identifying restriction for MLA
O
45o
[Tversky and Kahneman, 1992] compensating gain hypothesis posits:
“when the possible loss is increased by k the compensating gain must be
increased by about [λ]k”. This is equivalent to
tan(ψ(λ)) = λ gG ⋆ gL −1 = 1. The vector of losses dominates the
space and permeates the corresponding regressions.

19. Variance stabilizing transform for λn
√
n(ˆˆλn − λ) −→
d
N 0,
σ2
θ −2λσθβ+λ2σ2
β
β2 , σθβ = 0
Confound: Asymptotic variance λn depends on λ
A variance stabilizing transformation is needed to eliminate the
variance confound [Bar-Lev and Enis, 1990].
The computed variance stabilizing transformation (VST) is
g(λ) =
σ
σ∗
β
ln tan
π
4
+
ψ
2
tan(ψ) =
λσ∗
β
σ∗
θ
, tan
ψ
2
=
λσ∗
β
σ∗
θ + τ(λ)
√
n g λn − g(λ) −→
d
N(0, σ2) where σ2 = τ2(λ)g′2
(λ),
τ2(λ) = σ∗2
θ + λ2σ∗2
β , and σ∗
θ = σθ/β, σ∗
β = σβ/β
g(λ) is related to the distribution function F(u) = 1
2 + 1
π arctan(u)
for a Cauchy density function of type f (u) = [π(1 + u2)]−1. So the
VST is in the α-stable class, i.e.,
f (λ; σ∗2
, σ∗2
) = 2 1 + λ2σ∗2
σ∗−2 −1

20. Sample distribution of MLA index
Figure : Empirical distribution
of US MLA index
0 5 10 15 20 25 30 35
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
Reference dependent MLA USA
Cauchytransformwithmedian=2.25
Figure : Theoretical Levy
distribution
The empirical US MLA index was fitted by interpolation from a method of moments estimator
applied to US real income and nondurable consumption data in [?]. The theoretical Levy
distribution is
f (x; µ, c) =
c
2π
(x − µ)−3/2
exp −c(2(x − µ)−1
)
with measure of location µ and scale parameter c for tail efffects.

21. Levy scale parameter and asymmetric growth exposure
Assign the standard deviations of growth exposure to the abstract
norms, i.e., gL = σ∗
θ , gG = σ∗
β
Let c = σ∗
θ /σ∗
β be a scale parameter induced by the norms for
growth exposure.
Levy density now has the form f (x) = (c
√
2π)
−1
x− 3
2 exp(−2x)
when measure of location µ = 0.
If tan(ψ(λ)) = λ gG ⋆ gL −1 = 1, then λ = c and λ = 2 when
c = 2.
The scale parameter c for the Levy density is informative about
the MLA index asymmetric growth exposure under Ha.
f (λ; σ∗2
β , σ∗2
θ ) = 2 1 +
λ2
c2
−1

22. Exact test for identifying MLA index restriction
Theorem
The exact test for the myopic loss aversion (MLA) index identifying
restriction is given by
Z =
4 λ2
λ2
H0
− ln σν
σǫ
− 1
√
2
∼ N (0, 1)
where Z is a standard normal random variable, λ is the observed or
estimated MLA index, λH0
= gggL / gGgG
gG is the given MLA index under
the null, and gG and gL are suitable normed vectors of gains and
losses in growth, respectively.
Remarks. The test statistic is derived from the LR statistic
−2 ln ϑ = -2 ln(V ( ˆˆβL)/V ( ˆβL)) ∼ χ2
1 where χ2
1 ∼ (1, 2), and
Z = χ2
1 − 1 /
√
2. It is based on H0 : βG = βL = β and
Ha : βG = β and βL = θ, β = θ. A correction factor 3
√
2 was used for
consistency under the null, and σν < σǫ by definition.

23. Comparison of recovered MLA index for SWB
Authors Specification Data source MLA index λ
[Abdellaoui et al., 2015] − U(−x)
U(x)
Experiment
hypothetical
choices 2.21a, [1.06, 5.52]b ∗∗
[Hardie et al., 1993] Multinomial logit
Supermarket
scanner panel data 2.695c , 1.660d
[Rieger et al., 2011]
Robust Ratio method
λθ
Experiment
hypothetical
choices 2.0e , 1.65i
[Charles-Cadogan, 2015c]
Method of moments and
robust ratio
˜λ = ¯gL/ ¯gG
Time series of
consumption and
income
0.67h∗∗[0.34, 1.23]b,
1.92i [1.27, 7.79]b
[Tversky and Kahneman, 1992] Nonlinear least squares
Experiment
hypothetical
choices 2.25a
[Ferrer-i-Carbonell, 2005] RRMe,j ˆˆλ = ˆθ/ ˆβ Survey panel data 2.39e , 5.72f ∗∗, 1.05g ∗∗
[Shea, 1995] RRMi,j ˆˆλ = ˆθ/ ˆβ
Time series of
consumption and
income 3.07k , 3.16k , 11.11k , 3.55k ∗∗
[Vendrik and Woltje, 2007] RRMi,j ˆˆλ = ˆθ/ ˆβ Survey panel data 1.31e
[Boyce et al., 2013] RRMi,j ˆˆλ = ˆθ/ ˆβ Survey panel data 2.2e , 2.6e
[De Neve et al., 2015] RRMi,j ˆˆλ = ˆθ/ ˆβ Survey panel data 1.51i , 5.81m∗∗, 6.34n∗∗
[Di Tella et al., 2010] RRMi,j ˆˆλ = ˆθ/ ˆβ Survey panel data 4.0f , 5.0f ∗∗.
[Fishburn and Kochenberger, 1979]
Robust Ratio method
λθ Metastudy 4.85i ∗∗[2.82, 7.72]b
** Exact test significant at p=0.05 for H0 : λ = 2.25, assuming ln(σν/σǫ) = 0, RRM=Recovered Ratio Method
a=median value, b=interquartile range, c=loss of quality, d=loss of price
e=Germany, f=West Germany, g=East Germany, h=South Africa, i=Unites States
j=Authors own estimates obtained from recovered ratio method
k=Estimates obtained from model with four different lists of instrumental variables
l=Robust loss aversion index estimator, m=Global, n=Europe

24. Distribution of MLA index by data source
0 2 4 6 8 10 12
1
2
3
4
5
6
7
8
9
10
11
Myopic loss aversion index
Metastudy
Time series of consumption and
income
Survey panel data
Experiment-hypothetical choices
Supermarket scanner panel data
2.25

25. Conclusion
☛ Theory shows that the canonical SWB regression model overestimates
the impact of growth on SWB by failing to control for MLA.
☛ The recovery theorem for MLA index for decline in SWB introduced
here is new to the literature.
☛ Application of the theorem to published results produced admissible
MLA index estimates.
☛ The α-stable feature of the recovered MLA index estimator
complicates statistical inference but establishes a relation between
tail thickness and asymmetric exposure of SWB to income growth.
☛ Further research on econometric theory of myopic loss aversion to
decline in SWB is needed to:
derive sharp bounds for recovered MLA index estimator
characterize the geometry of MLA index in gain loss space

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