Benke , Jean-Michel; Liu, Shuo; Ne ze , Nick
Wo king Pape
Time is knowledge: Wha esponse imes e eal
Discussion Pape s, No. 24-07
P o ided in Coope a ion wi h:
Depa men o Economics, Uni e si y o Be n
Sugges ed Ci a ion: Benke , Jean-Michel; Liu, Shuo; Ne ze , Nick (2024) : Time is knowledge: Wha
esponse imes e eal, Discussion Pape s, No. 24-07, Uni e si y o Be n, Depa men o Economics,
Be n
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Economics
Time is Knowledge:
Wha Response Times Re eal
Jean-Michel Benke , Shuo Liu, Nick Ne ze
24-07
Augus , 2024
Schanzenecks asse 1
CH-3012 Be n, Swi ze land
h p://www. wi.unibe.ch
DISCUSSION PAPERS
Time is Knowledge:
Wha Response Times Re eal
Jean-Michel Benke , Shuo Liu and Nick Ne ze ∗
Augus 2024
Abs ac
Response imes con ain in o ma ion abou economically ele an bu unobse ed
a iables like willingness o pay, p e e ence in ensi y, quali y, o happiness. He e, we
p o ide a gene al cha ac e iza ion o he p ope ies o la en a iables ha can be
de ec ed using esponse ime da a. Ou cha ac e iza ion gene alizes a ious esul s in
he li e a u e, helps o sol e iden i ica ion p oblems o bina y esponse models, and
pa es he way o many new applica ions. We apply he esul o es he hypo hesis
ha ma ginal happiness is dec easing in income, a p inciple ha is commonly accep ed
bu so a no es ablished empi ically.
Keywo ds: esponse imes, ch onome ic e ec , bina y esponse model, non-pa ame ic
iden i ica ion, dec easing ma ginal happiness
JEL Classi ica ion: C14, D60, D91, I31
∗Benke : Depa men o Economics, Uni e si y o Be n, jean-michel.benke @unibe.ch. Liu: Guanghua
School o Managemen , Peking Uni e si y, sh[email p o ec ed]. Ne ze : Depa men o Economics,
Uni e si y o Zu ich, nic[email p o ec ed]. We a e g a e ul o e y help ul commen s by Blaise Melly,
Cos anza Naguib, Ma in Vae h and semina pa icipan s a Uni e si y o Be n, Peking Uni e si y and
Xiamen Uni e si y. Shuo Liu acknowledges inancial suppo om he Na ional Na u al Science Founda ion
o China (G an No. 72322006).
1 In oduc ion
T adi ionally, economis s ha e igno ed choice p ocess da a like esponse imes. Only ecen ly
has he li e a u e ealized ha esponse imes can con ain aluable in o ma ion abou eco-
nomically ele an bu unobse ed a iables like, among o he s, willingness o pay o accep
(K ajbich e al., 2012; Co e and K ajbich, 2021), p e e ence in ensi y (Chab is e al., 2009;
Alós-Fe e , Feh and Ne ze , 2021; Alós-Fe e and Ga agnani, 2022), p oduc quali y (Ca d
e al., 2024), o happiness (Liu and Ne ze , 2023).
In his pape , we ake a sys ema ic app oach o s udy wha in o ma ion esponse imes
con ain. We do his in he con ex o a canonical bina y esponse model, which has been
used ex ensi ely in he economics li e a u e and is applicable o all he se ings desc ibed
abo e. The model posi s ha an unobse able la en a iable gene a es bina y choices. The
common obse a ion in he a o emen ioned s ands o li e a u e is ha decisions a e as e
when he alue o he la en a iable is la ge . The obse able esponse imes a e he e o e
in o ma i e abou he unobse able la en a iable.
We ph ase ou ques ion as one abou he iden i ica ion o he bina y esponse model. The
exis ing econome ics li e a u e has s udied ques ions along his line using assump ions on
he dis ibu ion o he la en a iable and exogenous a ia ion o obse ables (e.g., Manski,
1988; Ma zkin, 1992). We ake a complemen a y app oach and s udy he iden i ica ion o
dis ibu ional p ope ies using esponse ime da a. This allows us o ci cum en con o e sial
assump ions and o sol e iden i ica ion p oblems no ed in he li e a u e (e.g., Haile, Ho -
açsu and Kosenok, 2008; Bond and Lang, 2019). We p o ide a ull cha ac e iza ion o he
dis ibu ional p ope ies o la en a iables ha can be de ec ed wi h he help o esponse
ime da a, depending on he assump ions an analys is willing o make abou he ela ion
be ween he la en a iable and esponse ime.
The app oach ha we adop is mo e gene al han he exis ing li e a u e. In he con ex
o s ochas ic choice, Alós-Fe e , Feh and Ne ze (2021) ha e shown ha esponse ime
da a can be used o ob ain e ealed p e e ences wi hou making dis ibu ional assump ions
abou he andom u ili y componen and o imp o e ou -o -sample p edic ions. Se e al o
hei esul s ollow as immedia e co olla ies om ou cha ac e iza ion. We hen gene alize
hese esul s, o example allowing o addi ional indi idual he e ogenei y and dispensing
wi h unnecessa y symme y assump ions. In he con ex o happiness su eys, Liu and
Ne ze (2023) ha e shown ha esponse ime da a can help o sol e iden i ica ion p oblems
o o de ed esponse models. Once mo e, ou app oach yields se e al o hei esul s as
co olla ies and allows o u he gene aliza ion.
Ou app oach pa es he way o a ange o new applica ions. He e, we b ie ly highligh
1
h ee, each o which will be discussed in g ea e de ail in he heo e ical pa o he pape .
Fi s , we show how o de ec pola iza ion o poli ical a i udes (Lelkes, 2016) om simple
o dinal su ey ques ions. This ask would be di icul , i no impossible, wi hou esponse
imes (Vae h, 2023). Second, we demons a e how o in e p ope ies o demand unc ions
ha a e impo an o he op imal p icing o i ms (Johnson and Mya , 2006) om obse ed
pu chase decisions a a single p ice. Thi d, we show how o unco e co ela ional pa e ns
ha a e no di ec ly obse able o an analys because subjec s’ esponses may be dis o ed
when hey con lic wi h au ho i a ian go e nmen s o social no ms (Co man, Co man and
E icson, 2017; Gu ie and T eisman, 2020).
Ou heo e ical esul s make i possible o ackle long-s anding empi ical challenges and
deba es in he economics li e a u e. We showcase his po en ial by applying he esul s o
es he hypo hesis o dec easing ma ginal happiness o income, a p inciple ha is cen al o
edis ibu i e policies. Oswald (2008) and Kaise and Oswald (2022) ques ion he empi ical
ounda ion o his p inciple by a guing ha an obse ed conca e ela ionship be ween income
and sel - epo ed happiness may esul om a conca e epo ing unc ion a he han om
dec easing ma ginal happiness. Con en ional app oaches used in he happiness li e a u e
a e insu icien o es ablish he p inciple (Bond and Lang, 2019). In he empi ical pa
o he pape , we show ha he p inciple becomes es able wi h ou esponse ime-based
me hod. Ou analysis o exis ing su ey da a e eals ha he hypo hesis o dec easing
ma ginal happiness canno be ejec ed.
The cen al assump ion unde pinning ou analysis is he so-called ch onome ic unc ion
ha associa es each choice wi h a esponse ime. This unc ion is mono one, in he sense
ha a la ge absolu e alue o he la en a iable gene a e a as e decision, possibly a e
con olling o indi idual he e ogenei y. F om a heo e ical pe spec i e, such a mono one
ela ionship eme ges na u ally in e idence-accumula ion models (see, e.g., Chab is e al.,
2009; Fudenbe g, S ack and S zalecki, 2018; Ca d e al., 2024), whe e a s onge s imulus
gene a es as e decisions. The empi ical e idence o a mono one ch onome ic unc ion
in he labo a o y is as . Fo example, and among many o he s, Kellogg (1931), Moye
and Baye (1976), and Palme , Huk and Shadlen (2005) documen he e ec o choice
si ua ions wi h an objec i e s imulus, and Mo a (2005), Chab is e al. (2009), Kono alo
and K ajbich (2019), and Alós-Fe e and Ga agnani (2022) o alue-based en i onmen s.
Field e idence is also eme ging. Ca d e al. (2024) documen ha edi o ial decisions ake
longe when he submi ed pape ’s quali y implies a close decision. Using eBay da a on
ba gaining beha io , Co e and K ajbich (2021) show ha selle s’ esponse imes o an
o e sys ema ically depend on i s pe cei ed alue. In he con ex o an online su ey, Liu
and Ne ze (2023) demons a e ha as e esponses a e associa ed wi h a s onge sense o
2
app o al o he selec ed answe . The consis en suppo ac oss di e se se ings and s udies
unde sco es he alidi y and obus ness o he ch onome ic e ec as modeled in ou wo k.
To gain a mo e conc e e idea o ou main insigh , conside a s anda d decision-making
en i onmen whe e one o mul iple agen s choose be ween wo op ions. Choice is de e mined
by he ealiza ion xo an unobse able la en a iable, wi h x≤0gene a ing choice o op ion
aand x > 0gene a ing choice o op ion b. An analys obse ing he choices na u ally wonde s
wha can be lea ned abou he unde lying bina y esponse model, and in pa icula abou he
cumula i e dis ibu ion unc ion Go he la en a iable, om hose da a. Un o una ely,
he only p ope y o G ha is iden i ied wi hou addi ional assump ions o da a is i s alue
a ze o, G(0), which is gi en by he obse ed p obabili y o equency o choosing a. This
in o ma ion is ex emely limi ed and does, o example, no imply any hing abou he mean
o Gwi hou addi ional dis ibu ional assump ions. Now suppose ha he speed o he
decision is gi en by c(|x|) o a s ic ly dec easing ch onome ic unc ion c, assumed he e
o be iden ical o bo h choice op ions and all agen s jus o easy o exposi ion. Since a
choice o aa ises a ime o ea lie i xis su icien ly a below 0, whe e “su icien ly a ” is
de e mined by he ch onome ic unc ion, he obse ed p obabili y o equency o choosing
aa ime o ea lie pins down he alue o G(−c−1( )), and analogous o choices o b.
Obse ing he join dis ibu ion o esponses and esponse imes he e o e allows he analys
o iden i y a composi ion o he dis ibu ion Gand he ch onome ic unc ion c.
Consequen ly, i he analys had pe ec knowledge o he ch onome ic unc ion linking
alues o esponse imes, she could eco e he en i e la en dis ibu ion om he da a.
Howe e , such de ailed knowledge is no necessa y o in e ing only speci ic dis ibu ional
p ope ies. Ou main esul ully cha ac e izes which p ope ies (o hei iola ion) can be
de ec ed unde which assump ions on he ch onome ic unc ion. Fo example, de ec ing
p ope ies ha a e p ese ed unde mono one ans o ma ions equi es only knowledge o
mono onici y o he ch onome ic unc ion. This includes p ope ies such as ull suppo o ,
because ou app oach allows o se ings wi h mul iple la en a iables, i s -o de s ochas ic
dominance be ween dis ibu ions (as in Liu and Ne ze , 2023). Knowing mo e abou he
ch onome ic unc ion beyond i s mono onici y enables he de ec ion o a b oade class o
p ope ies. Fo example, i we es ic a en ion o ch onome ic unc ions ha a e iden ical
o bo h choice op ions, as in he abo e illus a ion, hen we can de ec p ope ies ha a e
p ese ed unde symme ic mono one ans o ma ions. This includes su icien condi ions
o he mean o a dis ibu ion o be posi i e (as in Alós-Fe e , Feh and Ne ze , 2021) and
o he anking o he means o mul iple dis ibu ions (as in Liu and Ne ze , 2020).
Ou main esul also p o ides a simple ecipe how o de ec o ejec any p ope y o
in e es . I in ol es cons uc ing a candida e dis ibu ion based on he empi ical da a using
3
a ep esen a i e ch onome ic unc ion ha he analys deems possible. Then, i he p ope y
o in e es holds o his candida e dis ibu ion, i mus hold o all ch onome ic unc ions
ha can be ob ained om he ep esen a i e one using any ans o ma ion unde which he
p ope y is p ese ed. An analogous s a emen applies when he p ope y is iola ed. We
also discuss an ex ension ha combines his app oach wi h di ec dis ibu ional assump ions
and we p o ide necessa y and su icien condi ions o a ionalizabili y o esponse ime da a
in his scena io.
The gene ali y o ou app oach lends i sel o a wide ange o applica ions. When applied
o a single dis ibu ion, i enables he de ec ion o p ope ies such as he sign o he mean,
inequali y, and unimodali y. These p ope ies play impo an oles in he con ex o e ealed
p e e ence heo y, op imal p icing, and poli ical analysis. Fo mul iple dis ibu ions, we can
de ec i s -o de s ochas ic dominance, he anking o means, likelihood- a io dominance,
and co ela ions wi h obse able a iables. These p ope ies ma e o he analysis o su ey
da a, ou -o -sample p edic ion o beha io , and mono one compa a i e s a ics.
The pape is o ganized as ollows. Sec ion 2 in oduces he o mal amewo k and p esen s
he main esul , along wi h wo ex ensions. Sec ion 3 applies he main esul and de i es
heo e ical condi ions o de ec ing he a ious dis ibu ional p ope ies o in e es discussed
abo e. Sec ion 4 uses se e al o hese esul s o empi ically es he hypo hesis ha ma ginal
happiness is dec easing in income. Sec ion 5 concludes. Addi ional ma e ial can be ound in
he Appendix.
2 Gene al Theo y
In his sec ion, we de elop ou gene al heo e ical amewo k and p esen ou main esul ,
which shows how and unde which condi ions dis ibu ional p ope ies can be de ec ed using
esponse ime da a.
2.1 Bina y Response Model
We i s in oduce he bina y esponse model (e.g. Manski, 1988). The e is a andom a iable
˜xwi h alues x∈R ha induce bina y esponses by compa ison wi h a decision h eshold.
We no malize he h eshold o ze o wi hou loss o gene ali y. Thus, he esponse is i= 0
i ˜x akes a alue x≤0and i= 1 i ˜x akes a alue x > 0. We desc ibe he dis ibu ion
o he la en a iable ˜xby a cumula i e dis ibu ion unc ion (cd ) G, which we assume
o be con inuous. I ollows ha he p obabili ies o he wo esponses a e p0=G(0) and
p1= 1 −G(0).
4
The model has di e en applica ions and in e p e a ions. Fo example, he la en a iable
could be a andom u ili y di e ence ˜x=u(1) −u(0) + ˜ϵ(1) −˜ϵ(0) be ween wo op ions,
inducing s ochas ic choices o a single agen (as in Alós-Fe e , Feh and Ne ze , 2021). In a
di e en applica ion, ˜xcould desc ibe he dis ibu ion o happiness in a popula ion o agen s,
inducing equencies o esponses o a bina y su ey ques ion abou li e happiness (as in Liu
and Ne ze , 2023). The same logic applies o o he su ey ques ions, whe e he esponses
could be d i en by a dis ibu ion o poli ical a i udes o o he p e e ence pa ame e s in he
popula ion. In ye ano he applica ion, ˜xcould cap u e he andom quali y o pape s ha
a e submi ed o a jou nal, inducing he edi o ’s decision o accep o ejec (as in Ca d e al.,
2024). The same logic applies o o he se ings whe e quali y de e mines a bina y decision,
such as whe he o in es in an inno a ion p ojec . Finally, he la en a iable ˜x= ˜ −p
could desc ibe he di e ence be ween willingness o pay o a p oduc among consume s and
he p oduc p ice, inducing he demand o he p oduc a p ice p(in he spi i o Co e and
K ajbich, 2021).
We now ollow Alós-Fe e , Feh and Ne ze (2021) and Liu and Ne ze (2023) and
assume ha he ealized alue xo ˜xno only de e mines he esponse bu also he esponse
ime, wi h la ge absolu e alues implying as e esponses, in line wi h he well-es ablished
ch onome ic e ec . Fo mally, we deno e by c:R→[ , ] he ch onome ic unc ion, whe e
0≤ < < ∞. The unc ion cmaps each ealized alue xin o a esponse ime c(x). I is
assumed o be con inuous, s ic ly inc easing on R−and s ic ly dec easing on R+whene e
c(x)> , and o sa is y c(0) = and limx→−∞ c(x) = limx→+∞c(x) = . Figu e 1 illus a es
wo examples o ch onome ic unc ions ha adhe e o all hese condi ions.
The es ic ion o c o x∈R−is deno ed c0. This unc ion c0has a well-de ined in e se
(c0)−1: ( , ]→R− ha is con inuous and s ic ly inc easing. We ex end i o by se ing
(c0)−1( ) = −∞ i c(x)> o all x∈R−and (c0)−1( ) = max{x∈R−|c(x) = }o he wise.
Analogously, he es ic ion o c o x∈R+is deno ed c1, wi h con inuous and s ic ly
dec easing in e se (c1)−1: ( , ]→R+, which we ex end o by se ing (c1)−1( )=+∞o
(c1)−1( ) = min{x∈R+|c(x) = }, as app op ia e.
In addi ion o he esponse p obabili ies, he model (G, c)also induces dis ibu ions o
esponse imes. We deno e by Fi he cd o esponse imes condi ional on a esponse o
i= 0,1. Since a esponse i= 0 a ime o ea lie a ises i x≤(c0)−1( ), we ob ain ha
p0F0( ) = G((c0)−1( )) (1)
o all ∈[ , ], whe e we use he con en ion G(−∞) = 0. Analogously, a esponse i= 1 a
5
•
•
¯
0x
esponse ime
c0(x)c1(x)
(a) c0(x)=0.1 + 1
1−2x, c1(x)=0.1 + 1
1+x.
•
•
¯
0x
esponse ime
c0(x)c1(x)
(b) c0(x) = c1(−x) = max{0.1x+ 1.1,0.1}.
Figu e 1: Examples o ch onome ic unc ions.
No es: The le panel in he igu e depic s an asymme ic ch onome ic unc ion c ha
asympo ically app oaches he as es esponse ime , while he igh panel shows a symme ic
one ha a ains a ini e absolu e alues o he la en a iable. In bo h panels, he ed and
blue cu es co espond o he es ic ions o c o R−and R+, espec i ely.
ime o ea lie a ises i x≥(c1)−1( ), so ha
p1F1( ) = 1 −G((c1)−1( )) (2)
o all ∈[ , ], whe e we use G(+∞)=1. The induced esponse- ime cd s Fia e con inuous
on [ , ]and sa is y Fi( )=1. In summa y, he bina y esponse model (G, c)induces he
da a (p, F) = (p0, p1, F0, F1)acco ding o (1) and (2).
2.2 De ec ing P ope ies
We now ask wha we can lea n om obse ed da a abou he unde lying bina y esponse
model and, in pa icula , abou he dis ibu ion Go he la en a iable. Taking obse ed
da a as gi en, di e en bina y esponse models could ha e gene a ed hose da a, so ha
in e ence abou he model is no s aigh o wa d. This is ue especially i he analys is no
willing o make po en ially s ong assump ions abou he o m o he ch onome ic unc ion
o he la en dis ibu ion. We ask whe he he e a e some p ope ies ha all models which
a e consis en wi h he da a sa is y. These espec i e p ope ies a e hen de ec ed om he
da a a he han assumed by he modelle .
Conside a p o ile o da a (pj, Fj)j= (p0
j, p1
j, F0
j, F1
j)jindexed by j∈J. The se Jcould
be a single on, e.g. when s udying he choices o a single agen be ween wo op ions, o he
6
(s ill assumed o be also bijec i e and hence con inuous). We deno e he se o all hese
ans o ma ions by Ψall . Suppose we wan o allow all ch onome ic unc ions ha app oach
asymp o ically in he limi bu ne e each . We impose no o he assump ions on hei
shape, such as symme y ac oss he wo di e en esponses. Deno e he se o all hese
unc ions by C∗
a.all (whe e as ands o asymp o ic) and obse e ha i is gene a ed by he
ep esen a i e membe (3) oge he wi h he ans o ma ions Ψall . We now cons uc an
empi ical unc ion Hacco ding o (5) using c∗ om (3), which yields
H(x) =
1−p1F1 +1
x+1/( − )i x > 0,
p0F0 +1
−x+1/( − )i x≤0.
(6)
Theo em 1 ells us ha ull suppo is de ec ed i His s ic ly inc easing in x, and a iola ion
o ull suppo is de ec ed i His no s ic ly inc easing in x. Exp essed di ec ly in e ms
o he obse ed da a, ull suppo is de ec ed i pi>0and Fihas ull suppo on [ , ], o
bo h i= 0,1. A iola ion o ull suppo is de ec ed o he wise.
Suppose ins ead ha we had easons o belie e ha all ch onome ic unc ions each
c(x) = o ini e absolu e alues o x, again wi hou making any o he assump ions on hei
shape. Deno e his se by C∗
.all (whe e s ands o ini e) and obse e ha i is gene a ed
by (4) oge he wi h Ψall . Theo em 1 now ells us ha we need o check whe he
H(x) =
1i ( − )< x,
1−p1F1( −x)i 0< x ≤( − ),
p0F0( +x)i −( − )≤x≤0,
0i x < −( − ),
(7)
is s ic ly inc easing in x. This is no he case, and we he e o e de ec ha he dis ibu ion
iola es ull suppo . In ui i ely, since he ch onome ic unc ions each bu he esponse
ime dis ibu ions ha e no a oms a , he la en dis ibu ions canno ha e ull suppo .
I we allow he union C∗
a.all ∪C∗
.all o ch onome ic unc ions, we can apply he ex ension
discussed in Subsec ion 2.5.1. I (6) is no s ic ly inc easing, we de ec ha all dis ibu ions
ha a e compa ible wi h he da a do no ha e ull suppo . I (6) is s ic ly inc easing, hen
we de ec nei he ull suppo no a iola ion o ull suppo , because he da a a e compa ible
wi h some dis ibu ions ha ha e ull suppo and o he s ha ha e no .
13
3.1.2 Sign o Mean
Ou i s economically ele an applica ion conce ns he sign o he mean. In a andom
u ili y applica ion whe e ˜x=u(1) −u(0) + ˜ϵ(1) −˜ϵ(0) and he e o s ha e mean ze o, he
mean equals u(1) −u(0). De ec ing he sign o he mean is he e o e he same as deducing
he agen ’s ue (non-dis o ed) o dinal p e e ence be ween he wo op ions.
The sign o he mean is in a ian o linea ans o ma ions o he o m ψ(x) = bx o b > 0.
Deno e he se o hese ans o ma ions by Ψlin. Un o una ely, he se s o ch onome ic
unc ions which can be gene a ed using Ψlin a e a he es ic i e. Fo example, when s a ing
om he symme ic linea unc ion (4), we can gene a e he se o all symme ic linea
unc ions. To de ec he sign o he mean u(1) −u(0) assuming his class, by Theo em 1 we
jus need o calcula e he sign o he mean o H, de ined in (7), which can easily be done
empi ically.6
We can achie e mo e obus de ec ion by wo king wi h a p ope y ha is su icien o
a posi i e mean ( he a gumen o a nega i e mean is analogous). Conside he asymme y
p ope y ha G(−x)≤1−G(x) o all x∈R+. This p ope y implies ha he mean o
Gis posi i e (see Alós-Fe e , Feh and Ne ze , 2021). I is in a ian o all ans o ma ions
ha a e symme ic a ound ze o bu no necessa ily linea . Deno e his se by Ψsym.
Suppose ha we once mo e allow ch onome ic unc ions ha app oach asymp o ically,
bu u he es ic a en ion o hose which a e symme ic ac oss esponses. The se o all
hese unc ions, deno ed C∗
a.sym, is gene a ed by (3) oge he wi h Ψsym. In ui i ely, his se
cap u es he assump ion ha he ch onome ic e ec is iden ical o he wo choice op ions.
We ob ain ha he desi ed asymme y o Gis de ec ed i H, de ined in (6), exhibi s he
desi ed asymme y. Taken oge he and exp essed di ec ly in e ms o he obse ed da a, i
ollows ha
p0F0( )≤p1F1( ) o all ∈[ , ](8)
is a su icien condi ion o a e ealed p e e ence u(0) ≤u(1). We ema k he e ha he same
condi ion ob ains when conside ing he se C∗
.sym o all symme ic ch onome ic unc ions
ha each and ha analogous s a emen s hold o e ealed s ic p e e ences.
Condi ion (8) is he same as in Theo em 1o Alós-Fe e , Feh and Ne ze (2021). They
discuss in de ail ha obse ing choice equencies p0≤p1is no su icien o a p e e ence
u(0) ≤u(1) o be e ealed wi hou he addi ional assump ions on e o dis ibu ions (be-
6Some dis ibu ions do no ha e a mean. This can be deal wi h ei he by using he p ocedu e desc ibed
in Subsec ion 2.5.2 o es ic he se o dis ibu ions o hose which ha e a mean, o by e ining he desi ed
p ope y, o example o “ he mean exis s and is posi i e.” Analogous a gumen s apply whene e a p ope y
is no well-de ined o all possible dis ibu ions.
14
yond mean ze o) ha a e made in con en ional logi o p obi models. Howe e , i he
inequali y holds o all esponse imes, as s a ed in (8), hen a p e e ence is obus ly e ealed
wi hou dis ibu ional assump ions. Alós-Fe e , Feh and Ne ze (2021) epo ha sligh ly
mo e han 60% o all s ochas ic choices in he da a o Cli he o (2018) do obus ly e eal a
p e e ence.7Alós-Fe e , Ga agnani and Feh (2023) use he same condi ion o show ha a
sizable ac ion o choices ha iola e s ochas ic ansi i i y in di e en expe imen s e eal
non- ansi i e p e e ences and can hus no be explained by ansi i e p e e ences oge he
wi h noise.
Ou app oach sugges s possible gene aliza ions o Alós-Fe e , Feh and Ne ze (2021).
Fo example, i we ha e easons o belie e ha he ch onome ic e ec is di e en o he wo
choice op ions, we can cons uc Hbased on a ep esen a i e ch onome ic unc ion c∗ ha
is asymme ic ac oss he op ions, and we ob ain a modi ied e sion o condi ion (8) which
e lec s ou p io knowledge o he asymme y. Assume, o example, ha he ep esen a i e
unc ion sa is ies c∗1(x) = m(c∗0(−x)) o some m: [ , ]→[ , ]and all x∈R+. Then,
p0F0( )≤p1F1(m( )) o all ∈[ , ]
is su icien o de ec ing u(0) ≤u(1). I esponses i= 1 a e a p io i known o be as e
han esponses i= 0, o malized by m( )≤ o all , hen he igh hand side is smalle
han in (8) and he inequali y is ha de o sa is y. The con e se is ue i esponses i= 0
a e as e . I we wan o allow o some deg ee o asymme y wi hou knowing de ails, we
can cons uc mul iple unc ions Hkusing di e en ep esen a i e unc ions ck∗wi h a ying
deg ees o asymme y. Fo a e ealed p e e ence, all modi ied e sions o condi ion (8) ha e
o hold simul aneously, ul ima ely esul ing in a equi emen ha he di e ence be ween he
le and he igh hand side o (8) mus be la ge enough. This would gi e ise o a mo e
demanding bu e en mo e obus es han in Alós-Fe e , Feh and Ne ze (2021).
As a side ema k—and o illus a e he logic o addi ional dis ibu ional assump ions
discussed in Subsec ion 2.5.2—le us inally assume ha any admissible Gis symme ic
a ound i s mean. In ha case, we can y o de ec he sign o he median ins ead o he
mean because he wo a e iden ical. Wha e e ep esen a i e ch onome ic unc ion c∗we
use in ou cons uc ion o H—and he e o e i espec i e o which C∗we wan o gene a e—
he sign o he median o Hequals he sign o p1−p0. This mi o s a well-known esul
(s a ed, o example, as P oposi ion 2in Alós-Fe e , Feh and Ne ze , 2021): unde he
assump ion o symme ic noise dis ibu ions, choice equencies e eal p e e ences wi hou
he need o ely on esponse ime da a.
7Alós-Fe e , Feh and Ne ze (2021) allow all ch onome ic unc ions ha a e symme ic. One mino
di e ence is ha hey assume he ch onome ic unc ion o be unbounded while we assume ha is ini e.
15
3.1.3 Inequali y
Ou me hod can also be used o de ec inequali y o dispe sion o a dis ibu ion, which has ap-
plica ions ac oss mul iple ields. Fo ins ance, wi hin he li e a u e on subjec i e well-being,
he e is a subs an ial in e es in unde s anding he inequali y o happiness (e.g. S e enson
and Wol e s, 2008). Simila ly, esea che s ha e s udied socie al pola iza ion by measu ing
he dispe sion o indi idual a i udes owa ds social and poli ical issues (DiMaggio, E ans
and B yson, 1996; E ans, 2003). In he con ex o ma ke compe i ion, he sp ead o con-
sume p e e ences has di ec implica ions o he op imal p icing and ad e ising s a egies
o i ms (Johnson and Mya , 2006; He i, Liu and Schmu zle , 2022). Un o una ely, s an-
da d measu es o inequali y like he Gini index equi e ca dinal in o ma ion, which makes
hei applica ion o o de ed esponse da a ques ionable (see he discussion in Du a and
Fos e , 2013). Response imes may se e as he sou ce o ca dinal in o ma ion, e en when
he decisions a e bina y such as consume s’ decisions o buy o no buy a p oduc (Co e
and K ajbich, 2021).
The Lo enz cu e is a con enien g aphical ep esen a ion o a dis ibu ion’s inequali y,
and in e es ing measu es o inequali y like he Gini index a e based on he Lo enz cu e
(A kinson, 1970; Cowell, 2011). Fo any dis ibu ion G, he associa ed Lo enz cu e is
de ined by
L(q, G) = Rq
0G−1(x)dx
R1
0G−1(x)dx o all q∈[0,1],
whe e G−1(x) := in {x|G(x)≥q}deno es he le in e se o G. In he con ex o subjec i e
well-being, L(q, G)could be unde s ood as he p opo ion o o al happiness alloca ed o he
leas happy 100qpe cen o he popula ion. How a he Lo enz cu e alls below he 45-
deg ee line is an indica ion o how unequal he dis ibu ion is. The cu e is in a ian o all
linea ans o ma ions Ψlin o he dis ibu ion G. The e o e, i we plo he Lo enz cu e o
an empi ical unc ion Hlike in (6) o (7), o based on any o he ep esen a i e unc ion c∗,
exac ly his Lo enz cu e (and any measu e based on i like he Gini index) is de ec ed o
he class o ch onome ic unc ions ha a e linea ly gene a ed om c∗. As be o e, we can
epea his p ocedu e o a ious di e en ep esen a i e unc ions ck∗ o ob ain bounds on
he ue Lo enz cu e o la ge se s o ch onome ic unc ions.
We jus ema k ha analogous a gumen s apply o o he dis ibu ional p ope ies ha
a e in a ian o linea ans o ma ions, like skewness o ku osis.
16
3.1.4 Unimodali y
We close he sec ion on single dis ibu ions wi h he p ope y ha he dis ibu ion is uni-
modal wi h mode a ze o, i.e., Gis con ex below ze o and conca e abo e ze o, and s ic ly
so excep when G(x)∈ {0,1}. Unimodali y is o in e es once mo e in poli ical applica ions
whe e Gdesc ibes he dis ibu ion o poli ical a i udes in a popula ion. Unimodali y e lec s
a cen e ed popula ion whe e mo e ex eme posi ions ecei e less suppo . An ongoing deba e
in poli ical science disag ees whe he poli ical a i udes ollow unimodal dis ibu ions o a e
pola ized and be e desc ibed by bimodal dis ibu ions (see Lelkes, 2016; Vae h, 2023). An-
alys s o en wan o lea n abou hese p ope ies om su ey esponses, bu as Vae h (2023)
poin s ou , o dinal esponses a e inadequa e o es o p ope ies like uni- o bimodali y o
he unde lying dis ibu ion.
The p ope y o unimodali y is in a ian o (sigmoid) ans o ma ions ha sa is y ψ(0) =
0and a e weakly con ex below ze o and weakly conca e abo e ze o, he se o which is
deno ed Ψsig. S a ing om a ep esen a i e ch onome ic unc ion c∗, he ans o ma ions
Ψsig allow us o gene a e all ch onome ic unc ions which a e weakly “mo e con ex” han
c∗on R−and on R+sepa a ely (because c∗is inc easing on R−bu dec easing on R+). We
will use his insigh in a sligh ly di e en way han be o e. Assume ha he obse ed cd s
Fia e s ic ly inc easing on [ , ]. Then conside he ep esen a i e ch onome ic unc ion
cons uc ed om he da a by
c∗(x) =
i 1< x,
(F1)−1(1 −x)i 0< x ≤1,
(F0)−1(1 + x)i −1≤x≤0,
i x < −1,
(9)
which is a membe o C∗
.all. Wi h his unc ion, he empi ical dis ibu ion Hbecomes
H(x) =
1i 1< x,
1−p1+p1xi 0< x ≤1,
p0+p0xi −1≤x≤0,
0i x < −1,
which is piece-wise linea . Applying any s ic ly sigmoid ans o ma ion o (9) gene a es a
ch onome ic unc ion ha is s ic ly mo e con ex ( o each esponse sepa a ely) and esul s
in a unimodal H. By he same logic, applying any s ic ly in e se-sigmoid ans o ma ion
17
gene a es a s ic ly mo e conca e ch onome ic unc ion and a dis ibu ion H ha is no
unimodal. Func ion (9) he e o e delimi s se s o ch onome ic unc ions o which we can
de ec o ejec unimodali y. I ollows om Theo em 1 ha unimodali y is de ec ed o all
hose unc ions ha a e mo e con ex han (9) and ejec ed o hose ha a e mo e conca e.
This app oach does no co e all possible ch onome ic unc ions bu may yield exp essi e
esul s. Fo example, i (9) plo ed om he da a is al eady s ongly con ex, hen i appea s
unlikely ha he ue ch onome ic unc ion is e en mo e con ex, and we may be able o
ejec he assump ion o unimodali y o he dis ibu ion.
3.2 Mul iple Dis ibu ions
3.2.1 Fi s -O de S ochas ic Dominance
As a i s applica ion in ol ing mo e han one dis ibu ion, conside he p ope y o (G1, G2)
ha G1 i s -o de s ochas ically domina es G2, i.e., G1(x)≤G2(x) o all x∈R. In he
con ex o happiness su eys, Bond and Lang (2019) ha e poin ed ou ha con en ional
p obi o logi models make ha assump ion when compa ing wo (o mo e) g oups o
su ey pa icipan s. The assump ion is c ucial o he esul s o hese models, as i yields a
anking o he g oups’ a e age happiness o any choice o he happiness scale. Wi hou he
assump ion, he sign o es ima ed pa ame e s can o en be lipped by using a di e en scale.
Fo example, ich su ey pa icipan s may be less happy on a e age han poo pa icipan s
despi e esponding o be happy mo e equen ly. I is di icul o es FOSD using only
esponse da a.
Fo ou app oach he e, obse e ha FOSD (and also i s iola ion) is in a ian o all
p o iles (ψ1, ψ2)o inc easing ans o ma ions ha sa is y ψ1=ψ2. Deno e he se o all
hese p o iles by Ψall.i (whe e is ands o iden ical ac oss he index j). Le C∗
a.all.ibe he
se o all p o iles o ch onome ic unc ions ha app oach asymp o ically and a e iden ical
ac oss indices. This se is gene a ed by he ep esen a i e ch onome ic unc ion (3) o all
j∈J oge he wi h Ψall.i. In ui i ely, i embodies he assump ion ha he ch onome ic
e ec is iden ical o all g oups bu o he wise un es ic ed. By Theo em 1, we now need o
check whe he H1 i s -o de s ochas ically domina es H2, whe e Hjis de ined as in (6) using
he obse ed da a (p0
j, p1
j, F0
j, F1
j)o g oup j= 1,2. Exp essed di ec ly in e ms o he da a,
we de ec FOSD i
p0
1F0
1( )−p0
2F0
2( )≤0≤p1
1F1
1( )−p1
2F1
2( )(10)
o all ∈[ , ], and a iola ion o FOSD o he wise. We ema k ha he same condi ion
18
ob ains when conside ing he se C∗
.all.io iden ical ch onome ic unc ions ha each and
ha an analogous s a emen holds when a s ic inequali y o some xis equi ed in he
de ini ion o FOSD.
Condi ion (10) equals condi ions (i)and (ii)o P oposi ion 2in Liu and Ne ze (2023).8
As hey poin ou , o = condi ion (10) educes o p0
1≤p0
2, which is he condi ion unde
which con en ional p obi o logi models conclude ha g oup j= 1 is happie han g oup
j= 2. To a i e a his conclusion wi hou making dis ibu ional assump ions, he inequali-
ies in (10) mus hold o all . Liu and Ne ze (2023) es hese inequali ies using da a om
an online su ey. They show ha he null hypo hesis o FOSD o en canno be ejec ed, in
pa icula in cases whe e he p obi model yields signi ican pa ame e es ima es, indica ing
ha he esul s o con en ional models o en seem o be obus a leas quali a i ely.
Ou app oach he e sugges s possible gene aliza ions. One wo y when compa ing e-
sponse imes ac oss indi iduals as in a su ey is ha hey may di e in hei decision speed.
Liu and Ne ze (2023) add ess his p oblem by no malizing indi idual esponse imes using
a baseline ques ion and by showing ha i.i.d. he e ogenei y does no a ec he necessa y
de ec ion condi ions used o es ing. A di e en app oach ha accoun s o g oup-speci ic
decision speed would be o cons uc he unc ions Hjbased on g oup-speci ic ep esen-
a i e ch onome ic unc ions c∗
j, u ilizing p io knowledge abou g oup di e ences. The
esul would be an asymme ic e sion o (10). Imp ecise knowledge o g oup di e ences
could once mo e be cap u ed by wo king wi h mul iple unc ions Hk
jand checking mul iple
co esponding condi ions, gi ing ise o a mo e demanding bu mo e obus es .
3.2.2 Ranking o Means
Suppose we wan o de ec whe he he mean o G1is la ge han he mean o G2. A i s
applica ion whe e he compa ison o means ma e s is once mo e he case o su eys, whe e
we wan o lea n whe he one g oup is happie han ano he on a e age. We will discuss a
second applica ion a he end o his subsec ion.
The anking o he means is in a ian o iden ical posi i e a ine ans o ma ions o he
o m ψj(x) = a+bx o b > 0. Since we can only use he subse o hose ans o ma ions
which sa is y a= 0 when gene a ing ch onome ic unc ions, we es ic a en ion o he
se Ψlin.i o iden ical linea ans o ma ions igh away. The se o ch onome ic unc ions
ha can be gene a ed by linea ans o ma ions is a he es ic i e, as we discussed be o e.
Howe e , i he assump ion holds, he analysis can be ema kably simple. As an example,
8P oposi ion 2in Liu and Ne ze (2023) cha ac e izes de ec ion o i s -o de s ochas ic dominance using
gene al o de ed esponse models o su eys wi h mo e han wo esponse ca ego ies. Thei condi ion (iii)
applies o he in e media e esponse ca ego ies and coincides wi h he demanding condi ion by Bond and
Lang (2019), as esponse imes a e no mono one and hence no in o ma i e in he in e media e ca ego ies.
19
i we ha e easons o belie e ha he ch onome ic unc ions a e symme ic, linea , and
iden ical o bo h j= 1,2, hen we can simply compa e he means o H1and H2, whe e Hj
is de ined as in (7) using he obse ed da a (p0
j, p1
j, F0
j, F1
j)o g oup j= 1,2.
I is again possible o achie e mo e obus de ec ion using condi ions ha a e su icien o
a anking o he means. One su icien condi ion is o cou se i s -o de s ochas ic dominance
o G1o e G2, which we discussed be o e. Hence, (10) is a su icien condi ion o de ec ing
ha he mean o G1is la ge han he mean o G2, using only he assump ion ha he
ch onome ic unc ion is he same in bo h g oups j= 1,2. Ano he su icien condi ion
would be he de ec ion ha he mean o G1is posi i e while he mean o G2is nega i e,
which we discussed in Subsec ion 3.1.2. I hus ollows immedia ely ha
p1
2F1
2( )−p0
2F0
2( )≤0≤p1
1F1
1( )−p0
1F0
1( )(11)
o all ∈[ , ]is also su icien o de ec ing a anking o he means.
A compa ison o condi ions (11) and (10) is ins uc i e. In inequali y (10), we compa e
esponse imes be ween he g oups j(by calcula ing a di e ence o he dis ibu ions) bu
no be ween he esponse ca ego ies i. The esul he e o e gene a es de ec ion unde he
assump ion ha he ch onome ic unc ion is iden ical o he wo g oups bu no necessa ily
symme ic ac oss he wo esponses. In inequali y (11), we compa e esponse imes be ween
esponse ca ego ies ibu no be ween g oups j. I he e o e equi es he assump ion ha he
ch onome ic unc ion is symme ic ac oss esponses bu no necessa ily iden ical be ween
he wo g oups. Fo mally, he condi ion ha we a e aiming o de ec wi h (11) is in a ian o
ans o ma ions ha a e symme ic a ound ze o bu possibly di e en be ween he g oups,
he se o which is deno ed Ψsym.d (whe e ds ands o di e en ac oss he index j). These
ans o ma ions can, o example, be used o gene a e he se s C∗
a.sym.d o C∗
.sym.d o p o iles
o ch onome ic unc ions wi h he jus -discussed p ope ies.9
We now discuss a hi d su icien condi ion o a anking o he means. Conside he
p ope y ha G1(x)+G1(−x)≤G2(x)+G2(−x) o all x∈R+. This condi ion implies ha
he mean o G1is la ge han he mean o G2(see Appendix B). Fu he mo e, he condi ion
is in a ian o ans o ma ions ha a e iden ical o bo h j= 1,2and symme ic a ound
ze o. Deno e his se by Ψsym.i.
S a ing om he ep esen a i e unc ions (3) o all j∈J, we can use Ψsym.i o gene a e
he se C∗
a.sym.i o all ch onome ic unc ions ha app oach asymp o ically and a e sym-
me ic ac oss esponses and iden ical ac oss indices. Using his se , Theo em 1 implies ha
9Condi ion (11) and he esul s in he nex wo pa ag aphs we e i s de i ed in ou ea lie wo king pape
Liu and Ne ze (2020) and a e unpublished as ye . O he esul s om Liu and Ne ze (2020) we e published
as Liu and Ne ze (2023).
20
ou desi ed inequali y condi ion is de ec ed i i holds o he unc ions H1and H2 ha a e
cons uc ed based on (6). Taken oge he and exp essed di ec ly in e ms o he obse ed
da a, i ollows ha
p0
1F0
1( )−p0
2F0
2( )≤p1
1F1
1( )−p1
2F1
2( )(12)
o all ∈[ , ]is ano he su icien condi ion o de ec ing ha he mean o G1is la ge han
he mean o G2. We ema k ha he same condi ion ob ains when allowing he espec i e
se C∗
.sym.i o ch onome ic unc ions ha each and ha an analogous s a emen holds
o de ec ing a s ic inequali y o means. Inequali y (12) is a weake equi emen han
he di ec ly compa able condi ions (10) o (11). Howe e , as i implemen s a compa ison
o esponse imes ac oss he g oups and ac oss he esponses, i gene a es de ec ion only
unde he s onge combina ion o assump ions equi ed in (10) and (11), namely ha he
ch onome ic unc ions a e symme ic ac oss esponses and iden ical be ween g oups.
We now discuss ano he applica ion ha in ol es he compa ison o wo means. Suppose
we obse e he choices o a single agen be ween he wo op ions xand zand be ween he
wo op ions yand z. Can we in e he agen ’s p e e ence be ween xand y om hese choices?
Some imes his is possible based on ansi i i y o p e e ences, o example i xis chosen o e
zand zis chosen o e y. I , by con as , bo h xand ya e chosen o e z, hen we canno
ank xand ydi ec ly. K ajbich, Oud and Feh (2014) no e, howe e , ha he p e e ence can
be deduced om esponse imes unde he assump ion o a mono one ch onome ic e ec .
When he choice o xo e zis as e han he choice o yo e z, hen u(x)−u(z)mus be
la ge han u(y)−u(z)and we can conclude ha u(y)≤u(x)(see also Echenique and Sai o,
2017). Alós-Fe e , Feh and Ne ze (2021) p o ide a gene aliza ion o his a gumen o
s ochas ic choice unde he assump ion o symme ic u ili y dis ibu ions. Following hei
se ing, suppose ha he andom u ili y di e ence be ween xand zis desc ibed by a cd
Gxz wi h mean u(x)−u(z), and he andom u ili y di e ence be ween yand zis desc ibed
by Gyz wi h mean u(y)−u(z). Deducing a e ealed p e e ence o xo e ycan now be
eph ased as de ec ing ha he mean o Gxz is la ge han he mean o Gyz.
We can pu se e al o ou abo e esul s o wo k. Fi s , one su icien condi ion is ha
Gxz i s -o de s ochas ically domina es Gyz, which we de ec (unde he abo e-desc ibed
assump ions on he ch onome ic unc ions) when
pz
xzFz
xz( )−pz
yzFz
yz( )≤0≤px
xzFx
xz( )−py
yzFy
yz( )
o all ∈[ , ], whe e lowe indices desc ibe he bina y choice p oblem and uppe indices
desc ibe he chosen op ion. Ano he su icien condi ion is ha he mean o Gxz is posi i e
21
while he mean o Gyz is nega i e, which we de ec (unde di e en assump ions on he
ch onome ic unc ions) when
py
yzFy
yz( )−pz
yzFz
yz( )≤0≤px
xzFx
xz( )−pz
xzFz
xz( )
o all ∈[ , ]. Finally, a weake su icien condi ion o de ec ing he ou -o -sample p e e -
ence (bu unde s ic e assump ions on he ch onome ic unc ions) is
pz
xzFz
xz( )−pz
yzFz
yz( )≤px
xzFx
xz( )−py
yzFy
yz( )
o all ∈[ , ]. To ou knowledge, none o hese condi ions has been s udied in he indi idual
choice con ex . We emphasize ha we ob ain he e ealed p e e ence be ween xand y
wi hou making any assump ions on he shape o he u ili y dis ibu ions, bu ema k ha
a e ealed p e e ence ansla es in o an ou -o -sample p edic ion o choice p obabili ies only
wi h addi ional dis ibu ional assump ions such as symme y.
We can also ollow Alós-Fe e , Feh and Ne ze (2021) and assume igh away ha Gxz
and Gyz a e symme ic a ound hei means. Since mean and median coincide in his case, we
can ins ead y o de ec whe he he median o Gxz is la ge han ha o Gyz. This p ope y
is in a ian o he se Ψall.i o all ans o ma ions ha a e iden ical o he wo dis ibu ions.
We can he e o e de ec he p ope y assuming ei he C∗
a.all.i o C∗
.all.i, which means ha
we only ha e o assume ha he ch onome ic e ec is he same in he wo bina y decision
p oblems. Conside hen he case whe e px
xz >1/2and py
yz >1/2, which unde symme y
implies ha bo h u(x)−u(z)a e u(y)−u(z)a e s ic ly posi i e, so ha a p e e ence be ween
xand ydoes no ollow om ansi i i y. De ine θxz and θyz as pe cen iles o he esponse
ime dis ibu ions when xo ywe e chosen o e z, espec i ely, as ollows:
Fx
xz(θxz) = 1
2px
xz
and Fy
yz(θyz) = 1
2py
yz
.
I is now an easy exe cise o show ha he median o Hxz is la ge han he median o Hyz,
whe e bo h unc ions a e cons uc ed ei he as in (6) o as in (7), i and only i
θxz ≤θyz.(13)
Analogous s a emen s hold o s ic inequali ies and o he case px
xz <1/2and py
yz <1/2.
Inequali y (13) is he condi ion s a ed in Theo em 2o Alós-Fe e , Feh and Ne ze (2021)
o a e ealed p e e ence u(y)≤u(x)unde he assump ion o symme ic dis ibu ions. As
hese au ho s discuss in de ail, (13) o malizes ha he choice o xo e zis as e han he
22
0 20000 40000 60000 80000 100000 120000 140000
0.70 0.75 0.80 0.85
income
a e age esponse
Figu e 2: The ela ionship be ween income and epo ed happiness.
No es: The eigh colo ed cu es co espond o he di e en me hods used o de e mine he
a e age income wi hin each bin.
whe e αis such ha αwL+(1−α)wH=wM. Following he logic o Bond and Lang (2019), we
can easily explain he da a in Figu e 2 wi h happiness dis ibu ions (GL, GM, GH) o which
(16) is iola ed, o example dis ibu ions ha become mo e igh -skewed as income g ows
and which he e o e go along wi h high a e age happiness among he ich. I is impossible
o ejec o e i y such dis ibu ional assump ions based on esponse da a alone.
Acco ding o (16), he p oblem o de ec ing o ejec ing dec easing ma ginal happiness is
a p oblem o anking he means o dis ibu ions, o which we ha e de eloped condi ions in
Subsec ion 3.2.2. The le -hand side o (16) is he a e age happiness in a mixed popula ion
composed o a ac ion αo subjec s wi h low income and a ac ion 1−αo subjec s wi h
high income. Unde he assump ion ha all income g oups ha e he same ch onome ic
unc ion, which mos o he c i e ia om Subsec ion 3.2.2 equi e anyway, we can he e o e
pool he wo ex eme income g oups wi h app op ia e weigh s and de ec o ejec (16) based
on esponse ime da a. We ollow Liu and Ne ze (2023) and no malize indi idual esponse
29
Me hod αMean Happiness p-Value
Pooled Middle (12) (10) (11)
10.500 98’787 129’335 0.9548 0.3380 0.0000
20.610 91’395 129’335 0.9501 0.4105 0.0000
30.660 88’036 129’335 0.9457 0.4397 0.0000
40.623 90’504 129’335 0.9481 0.4215 0.0000
50.660 88’003 129’335 0.9417 0.4405 0.0000
60.649 88’793 129’335 0.9427 0.4364 0.0000
70.646 88’985 129’335 0.9445 0.4350 0.0000
80.656 88’310 129’335 0.9482 0.4391 0.0000
Table 1: Summa y o he es s o conca i y.
imes by sub ac ing in logs a subjec ’s esponse ime o he ma i al s a us ques ion, which
accoun s o indi idual-speci ic speed and u he co obo a es he assump ion o iden ical
ch onome ic unc ions o he di e en income g oups.
I we a e willing o assume ha he iden ical ch onome ic unc ions a e also linea and
symme ic, we can use (7) o compu e one dis ibu ion HM o he middle income g oup and
one dis ibu ion HP o he pooled g oup o low and high incomes and simply compa e hei
means. Since HP=αHL+ (1 −α)HH, we can compu e he dis ibu ions HLand HHin he
low and high income g oups sepa a ely and hen o m a con ex combina ion, a he han
ac ually pooling he da a o compu e HP. Table 1 con ains he esul s o his app oach o
ou eigh di e en me hods o assigning income le els, which gi e weigh s αbe ween 0.50
and 0.66. The mean happiness o he middle income g oup clea ly exceeds ha o he pooled
g oup ac oss all me hods.11 This se es as a i s indica ion o conca i y o he ela ionship
be ween income and happiness bu is a om conclusi e. In pa icula , he assump ion o
a linea ch onome ic unc ion is p obably no less con o e sial han he assump ion o a
linea epo ing unc ion.
To ob ain esul s unde less s ingen assump ions, we can y o de ec i mean happiness
is la ge in he middle income g oup han in he pooled g oup based on inequali y (12).
This inequali y is su icien o de ec a anking o he means and equi es only ha he
ch onome ic unc ion is symme ic ac oss esponses and iden ical be ween g oups, no ha
11We use esponse imes ha a e no malized by he ma i al s a us ques ion bu no by aking he loga i hm
he e, as aking he loga i hm would co espond o a non-linea ans o ma ion o he ch onome ic unc ion
and his ma e s o he app oach. I is i ele an o he app oaches used la e in his subsec ion, so he e
we epo and depic all esul s using log no malized esponse imes.
30
i is linea . In ou con ex , he inequali y becomes
p0
MF0
M( )−p0
PF0
P( )≤p1
MF1
M( )−p1
PF1
P( )
o all ∈[ , ], whe e pi
PFi
P( ) = αpi
LFi
L( )+(1−α)pi
HFi
H( ). The condi ion in ui i ely ules
ou examples like he inc easingly igh -skewed dis ibu ions discussed abo e, as hese would
gene a e ela i ely as (slow) happy (unhappy) esponses in he pooled g oup.
Figu e 3 plo s he le -hand side and he igh -hand side o (12), again o all he eigh
me hods used o impu e incomes. Inequali y (12) is no sa is ied exac ly in he da a, because
he igh -hand side alls below he le -hand side o some small . This indica es ha he e
a e indeed some as happy esponses in he pooled ela i e o he middle g oup. Howe e , he
c ossing o he unc ions appea s o be mino , so ha he ques ion o s a is ical signi icance
a ises. To ob ain p- alues o he null hypo hesis ha (12) holds, we employ a boo s ap-
based me hod as in Liu and Ne ze (2023) ha es s on a es o con en ional i s -o de
s ochas ic dominance by Ba e and Donald (2003). Table 1 shows ha he p- alues a e
e y la ge. We clea ly canno ejec (12), which is a su icien condi ion o a anking o he
means in line wi h dec easing ma ginal happiness.
I we a e e en unwilling o accep he assump ion o symme y o he ch onome ic
unc ions, we can s ill es he s onge condi ion ha he happiness dis ibu ion o he
middle income g oup i s -o de s ochas ically domina es ha o he pooled g oup. This
condi ion is o malized by (10) and equi es ha he unc ions in Figu e 3 a e sepa a ed
by ze o. While no sa is ied exac ly in he da a, he p- alues o he null hypo hesis ha
(10) holds a e smalle han o (12) bu s ill la ge, as can be seen in Table 1.12 We canno
ejec he hypo hesis o i s -o de s ochas ic dominance and hence o a e y s ong su icien
condi ion o dec easing ma ginal happiness ha applies unde weak assump ions on he
ch onome ic unc ion.
Fo comple eness, Table 1 also epo s p- alues o he hypo hesis ha (11) holds, ano he
su icien condi ion o a anking o he means. This condi ion is o less in e es he e. Fi s ,
i is e y s ong and would de ec a anking only i one o he means happened o be smalle
and he o he la ge han ze o. Second, i s ad an age o allowing g oup-speci ic ch onome ic
unc ions has no bi e because ou app oach o pooling g oups equi es iden ical ch onome ic
unc ions (a e no maliza ion) anyway. The hypo hesis ha (11) holds is clea ly ejec ed.
To summa ize, ou esul s a e suppo i e o he idea ha ma ginal happiness is dec easing
in income, in a c oss-sec ional da a se . While ou es s a oid se e al o he pi alls no ed a
12The es again ollows Liu and Ne ze (2023) and implemen s he p ocedu e o Ba e and Donald
(2003) oge he wi h a join hypo hesis co ec ion by Romano and Wol (2016) o accoun o he ac ha
condi ion (10) con ains wo inequali ies. We e e he eade o Liu and Ne ze (2023) o mo e de ails.
31
−6 −4 −2 0 2 4
−0.15 −0.10 −0.05 0.00 0.05 0.10 0.15 0.20
log no malized esponse ime
cumula i e di e ence in esponse ac ions
Figu e 3: Empi ical condi ions o de ec ing he income-happiness ela ion.
No es: The cu es a he op ep esen he empi ical unc ions p1
MF1
M( )−p1
PF1
P( ), while
hose a he bo om ep esen p0
MF0
M( )−p0
PF0
P( ). Di e en colo s indica e he a ying
me hods used o de e mine he a e age income wi hin each bin.
he beginning o his sec ion, some limi a ions emain. Mos impo an ly, we we e no able
o es conca i y o µa any income le el bu only o he h ee income le els implied by he
bins used in he su ey, and he de ec ed ela ion is bi a ia e wi hou addi ional con ols.
5 Conclusion
The goal o his pape is o p o ide a sys ema ic accoun o he in o ma ion ha esponse ime
da a con ain. We app oach he p oblem by ph asing i as one o iden i ica ion in he con ex
o bina y esponse models. Ou main esul ela es he se o iden i iable dis ibu ional
p ope ies o he se o admissible ch onome ic unc ions. The undamen al idea is ha
he join dis ibu ion o esponses and esponse imes iden i ies a composi ion o he la en
dis ibu ion and he ch onome ic unc ion. P ope ies o he dis ibu ion ha a e p ese ed
unde a gi en se o ans o ma ions can he e o e be iden i ied i he ch onome ic unc ion
32
is known up o hese ans o ma ions. Se e al exis ing esul s in he li e a u e ollow as
co olla ies and can be gene alized. Many new esul s eme ge. To illus a e he applicabili y
o ou app oach, we empi ically es and canno ejec he hypo hesis o dec easing ma ginal
happiness o income.
Ou heo e ical applica ions in Sec ion 3 a e me ely examples o he scope o he me hod
and no an exhaus i e lis . Addi ional p ope ies ha one could s udy include gene al
linea ela ionships be ween obse able a iables and he la en a iable like in eg ession
models, as well as he ex en o which esponses a e po en ially dis o ed by he aming o
a decision p oblem. Simila ly, ou empi ical s udy in Sec ion 4 is only one s aigh o wa d
applica ion showing how he me hod can be used o con ibu e o long-s anding deba es.
O he applica ions ha we ha e in mind include he s udy o pola iza ion using su eys on
poli ical a i udes, as well as op imal p oduc p icing using da a on pu chase decisions om
online pla o ms. The e a e also se e al possible ex ensions o ou amewo k ha me i
in es iga ion. These include co ela ions be ween mul iple la en a iables, he case wi h
mo e han wo choice op ions, and he use o esponse imes in se ings whe e hese imes
a e a ec ed by addi ional ac o s like playe ypes o decision modes as in Rubins ein (2007,
2013, 2016).
33
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