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Measuring “Correlation Neglect” Experimental Procedures and Applications Mémoire Blanchard Conombo Maîtrise en Économique Maître ès sciences (M.Sc.) Québec, Canada © Blanchard Conombo, 2017

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Page 1: Measuring “Correlation Neglect” Experimental Procedures ...€¦ · Measuring “Correlation Neglect” Experimental Procedures and Applications Mémoire Blanchard Conombo Maîtrise

Measuring “Correlation Neglect”Experimental Procedures and Applications

Mémoire

Blanchard Conombo

Maîtrise en ÉconomiqueMaître ès sciences (M.Sc.)

Québec, Canada

© Blanchard Conombo, 2017

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Résumé

Des études économiques récentes ont identifié la difficulté des personnes à prendre des déci-sions optimales lorsqu’il existe une corrélation entre différentes variables d’état (aléatoires),que l’on appelle maintenant dans la littérature «inattention envers la corrélation». Dans cetarticle, nous supposons que l’inattention envers la corrélation est un trait individuel d’unepersonne et nous proposons différentes mesures de cette caractéristique. Nous comparons dif-férentes mesures en termes de corrélation à partir des résultats d’expériences de laboratoire.Nous présentons les applications de ces mesures dans des domaines précis.

Mots clés : heuristiques et biais, inattention envers la corrélation, mesure.

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Abstract

Recent economic studies identified the difficulty of persons to make optimal decisions whenthere is correlation between (random) state variables, now referred to in the literatures as“correlation neglect.” In this article, we presume correlation neglect to be an individual traitof a person and propose different measures of this characteristic. We compare different mea-sures in terms of their correlation based on results from laboratory experiments. We presentapplications of the measures in the field.

Keywords : heuristics and biases, correlation neglect, measurement.

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Table des matières

Résumé iii

Abstract v

Table des matières vii

Liste des tableaux ix

Liste des figures xi

Remerciements xvii

Introduction 1

1 Literature 3

2 Experiment 72.1 Experimental Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 Treatments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3 Empirical Measure 113.1 Simple Set Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

4 Results 134.1 Descriptive Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134.2 Heterogeneity in correlation neglect . . . . . . . . . . . . . . . . . . . . . . . 134.3 Cognitive effort and learning over time . . . . . . . . . . . . . . . . . . . . . 174.4 Mechanisms underlying correlation neglect . . . . . . . . . . . . . . . . . . . 18

5 Applications 215.1 Mother’s age at first childbearing and Child mortality . . . . . . . . . . . . 215.2 Financial Allocation Decisions . . . . . . . . . . . . . . . . . . . . . . . . . . 225.3 Pre-election Polls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

Conclusion 25

Bibliographie 27

A Annexe 29A.1 Enke and Zimmerman : correlation between signals . . . . . . . . . . . . . . 29

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A.2 Convergence of Correlation in the Simple Set-Up . . . . . . . . . . . . . . . 29A.3 Others Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31A.4 Instructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

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Liste des tableaux

2.1 Experimental Design : Variation of experimental parameters and correlationcoefficients. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2 beliefs prediction for joint distribution . . . . . . . . . . . . . . . . . . . . . . . 92.3 Structure of portfolio choice problem. . . . . . . . . . . . . . . . . . . . . . . . 9

4.1 Descriptive statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134.2 Median correlation by treatment and informational presentation . . . . . . . . . 144.3 Structural . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164.4 Empirical without order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164.5 Empirical with order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164.6 Cognitive effort and learning over time . . . . . . . . . . . . . . . . . . . . . . . 174.7 Correlation neglect and informational presentation . . . . . . . . . . . . . . . . 184.8 correlation=0.33 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.9 correlation=0.20 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.10 Correlation=0.11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.11 Correlation=0.005 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.12 Number of historical information over time . . . . . . . . . . . . . . . . . . . . 204.13 Correlation neglect and number of past observations . . . . . . . . . . . . . . . 20

5.1 Joint distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225.2 subjects choices for each type of individual . . . . . . . . . . . . . . . . . . . . 23

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Liste des figures

4.1 Heterogeneity in Correlation Neglect . . . . . . . . . . . . . . . . . . . . . . . . 15

A.1 Median Correlation Neglect- Structural vs Empirical Without Order . . . . . . 31A.2 Median Correlation Neglect- Structural vs Empirical With Order . . . . . . . . 31A.3 Median Correlation Neglect- Empirical With Order vs Empirical Without Order 32A.4 Mean correlation neglect per value of true correlation . . . . . . . . . . . . . . . 32A.5 Histograms of the difference between original median correlation neglect para-

meters and modified median correlation neglect parameters when excluding oneparameter that is closest to the original median correlation neglect parameter. . 33

A.6 Presentation - Structural . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34A.7 Presentation - Empirical Without Order . . . . . . . . . . . . . . . . . . . . . . 35A.8 Presentation - Empirical With Order . . . . . . . . . . . . . . . . . . . . . . . . 36A.9 Presentation - Empirical With Order . . . . . . . . . . . . . . . . . . . . . . . . 37

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To my father Daniel Conombo,Rest in peace

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Le travail est la vie elle-même, etla vie est un continuel travail

Émile Zola

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Remerciements

Je tiens à exprimer ma profonde gratitude à mes directeurs, les Professeurs Sabine Kröger etCharles Bellemare pour avoir accepté de superviser et de financer cette recherche, pour leurgrande disponibilité et pour leurs avis éclairés. Je remercie également le Professeur SylvainDessy pour ses multiples conseils.

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Introduction

A decision maker’s ignorance of the correlation between two variables, the so called “correla-tion neglect,” has been demonstrated in many different choice contexts, i.e., portfolio choice(Kallir and Sonsino, 2009), investment decisions (Eyster and Weizsäcker, 2011), predictions ofoutcomes of random draws (Enke and Zimmermann, 2015). All those studies use the experi-mental comparison between treatments with and treatments without correlation between twovariables to show correlation neglect to exist in a particular context.

In real life as well as in the experimental studies above, the way varies in which persons canidentify and are presented with the correlation between two variables. Enke and Zimmermann(2015) present the correlation by the structure of the situation. In Eyster and Weizsäcker(2011), a decision maker can observe realizations of the different variables but has no informa-tion on the structure. Finally, Kallir and Sonsino (2009) give both information on the structureand realizations.Whether either presentations are equivalent or which presentation allows better to identifythe correlation is not a straightforward question and needs an empirical investigation.

This study proposes a general measure, in the same spirit as measures of individual riskaversion (e.g.,Cohen et al. (1987) ; Holt and Laury. (2002)), that to the best of our knowledgedo not exist so far for individual correlation neglect.

The paper is structured as follows. Section 1 summarizes the existing literature. In section2 we present an experimental design allowing us to assess the severity of correlation neglect.Section 3 presents our empirical approach how to model and estimate correlation neglect. Insection 4 we present results from a series of laboratory experiments allowing us to compareboth presentations. In section 5 we show how these abstract measures can be easily adaptedto applications in the field, when the aim is to assess the severity of correlation neglect ofdecision makers for particular situations. Section 6 concludes.

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Chapitre 1

Literature

Recent literature analysed the psychological phenomenon of correlation neglect (Eyster andWeizsäcker, 2011; Enke and Zimmermann, 2015; Levy and Razin, 2015; Ortoleva and Snow-berg, 2015). Correlation neglect implies that individuals underappreciate the correlation bet-ween state variables or different events they observe. This cognitive bias has negative spilloversfor individuals decisions making and lead to overconfidence in market settings which predictsbubbles and crashes. A small set of previous experiments have examined individuals’ responsesto correlations in informational sources and have found that subjects have limited attentionand find it cognitively challenging to work with joint distributions of random variables (Eysterand Weizsäcker, 2011). However, in this literature, there is a conceptual difference in the mea-ning of correlation neglect (structure of the environment and empirical analysis of historicaldata). The literature on boundedly rational and beliefs formation in networks uses the struc-ture of the information and analyses a double-counting problem in the informations sourceswhen people update their beliefs about a state variable (Enke and Zimmermann, 2015). Asresult, subjects in experiments on group communication (DeMarzo et al., 2003) or politicalcompetition and voting behavior (Ortoleva and Snowberg, 2015) overweight the impact ofinformational redundancies in their beliefs.

Enke and Zimmermann (2015) used the structure of decision problem and analysed doublecounting problem in beliefs formation. They find that experimental subjects in a relative simplesetting neglect correlations in information sources when forming beliefs with heterogeneity atthe individual level. They suggest a measure of individual correlation neglect in the betweensubjects design under the assumption that signals are drawn from a truncated discretizednormal distribution with mean µ - the true value of the state - and standard deviation σ = µ

2 .

Truncation implies that signals belong to the interval [0, 2µ] in order to avoid negative signalsand then negative correlations. But in real life, people face many situations involving negativeand positive informations (signals) about events. Informations might be positively or negativelycorrelated. In their experiment, two computers, A and B, generate two “iid” unbiased signalssA and sB with (sh ∼ N(µ, (µ2 )2), h ∈ {A,B}). Subjects observe these signals as numbers that

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they must use to estimate the number of items in an imaginary container. In the correlatedtreatment, subjects observe the realizations of a computer A (sA) and the mean realization of

two computers sB =sA + sB

2so that the two signals are correlated with a correlation of 71% 1.

Subjects in the control condition observe two independent signals (sA and sB). For rationalestimation of the number of item, subjects in the correlated treatment must take into accountthe information about the correlation of two signals given the structure of the environmentwhen the unbiased estimate of the number of items is the empirical mean of two signals incontrol condition. Therefore, a rational subject must extract sB from sB and compute themean of sA and sB as an estimate of the number of items in the container. The following ruleis used for each subject in the correlated treatment when he tries to extract the right signalsB :

sB = χsB + (1− χ)sB

χ is an individual measure of correlation neglect that captures subjects ability to extract theright signal into sB when authors consider only the structure of the environment.

{χ = 0 for rational subjectχ = 1 for full correlation neglecter

As result, people’s beliefs in the correlated treatment deviate from rationality because sub-jects neglect informational redundancies. Their individual measure of correlation neglect canbe apply in differents settings on informational structure (e.g Group communication, VotingBehavior, Political competitions, etc.). While a part of literature focuses on the structure ofthe environment, the other analyses people’s limited attention on joint distribution of ran-dom variables when engaging in empirical analysis of historical and empirical data (Kallir andSonsino, 2009; Eyster and Weizsäcker, 2011).

Kallir and Sonsino (2009) find that changing the correlation of a portfolio-choice problemleads to little or no change in participants decision making. In their experiment, subjectsobserved historical data on the joint distribution of the realized returns of two virtual assetswith different levels of correlation for 12 preceding periods ; and they have to predict therealized returns of the first asset in four additional observations when observing the returnsof the second asset. In this predictions-allocations problems, the results show that subjectsrecognize shifts in correlation in their prediction tasks but fail to account for this correlationin their allocations decisions. Therefore, correlation neglect predicts no change in participants’behavior. They shed additional light on the cognitive nature of the bias that is consistent with

1. See appendix A

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the interpretation of correlation neglect as deriving from limited attention. No formal measureof individual correlation neglect has been suggested.

Eyster and Weizsäcker (2011) focus on the impact of correlation neglect on financial decisionmaking because an investor who fails to account for the correlation when allocating his financialportfolio can hold a portfolio that contains undesirable risks. They suggest a measure ofindividual correlation neglect in a series of controlled experiments using a framing variationin which each participant faces two versions of the same portfolio-choice problem. The assetsin the correlated frame are linear combinations of those in the uncorrelated frame and spanexactly the same set of earnings distributions. Under the hypothesis that people correctlyperceive the covariance structure, the framing variations does not affect behavior. By ensuringthat participants understand the payoff structure and the co-movements of the assets returns,they find that behaviors change strongly. People ignore the correlation and treat correlatedassets as independent following sometimes a simple "1/N heuristic" which is investing equalshares of financial portfolio into all available assets. They measure people "ignorance" usingthe following transformation of the matrix of variance-covariance with penalties on varianceand covariance terms.

V =

((σ21)l k.sgn(σ12)|σ12|l

k.sgn(σ12)|σ12|l (σ22)l

)

k and l represent the parameters of correlation neglect and variance neglect (respectively) andare estimated for each subject in the experiment. Then, they classify subjects in 3 differentsgroups according the severity of correlation neglect.

This conceptual difference according the context and the structure of the decision problem andvarious experimental approaches (within and between-subjects designs) raises the difficulty toapply a measure of one paper to evaluate correlation neglect in another. To the best of ourknowledge, there exists no a single measure of correlation neglect that can be applied indifferent contexts.

Our paper is related to this literature on correlation neglect and we propose a measure thatcan be used as general measure of correlation neglect regardless of contexts. We need not toassume any hypothesis about individuals’ preferences, but only assessing their beliefs aboutthe distribution of state variables in our experiment. Then, we compute a measure of individualcorrelation neglect.

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Chapitre 2

Experiment

A decision maker can identify correlation between state variables either by engaging in empi-rical analysis of historical data or by analyzing the structure of the environment. Correlationneglect is present, when information, that can help to identify the correlation, is available butis ignored in the decision process. It is important to distinguish between the two dimensionsbecause it allows to localise the cognitive shortcoming of judgement and because it mightaffect the choice of policy if one wants to mitigate the ignorance.

Following this reasoning, we propose to measure correlation neglect in two ways : first, thecorrelation of state variables is presented by the structure of the decision problem (e.g Eysterand Weizsäcker (2011), Enke and Zimmermann (2015)), or second, by observing realizationsof both variables (e.g Kallir and Sonsino (2009)).

As in the literature on correlation neglect, we presume that ignoring the correlation betweentwo random variables affects beliefs about the joint distribution of those variables. Howe-ver, different to the existing literature, we propose to measure correlation neglect directly byeliciting (subjective) beliefs and not indirectly via observed choices.

2.1 Experimental Design

The basic set-up of our experiment consists of two urns, Urn 1 and Urn 2, containing N1 andN2 balls, respectively. Balls are either blue B or green G with B1+G1 = N1 and B2+G2 = N2

and the distribution is represented by the ratio of blue balls, b1 = B1/N1 and b2 = B2/N2. B1

and G1 are respectively the number of blue and green balls inUrn 1 while B2 and G2 representthe number of blue and green balls in Urn 2. Then a number of balls D1 are drawn fromUrn 1 without replacement and placed in Urn 2, from which then D2 balls are drawn againwithout replacement. This procedure is repeated S times, each time resetting both urns to theoriginal set-up. The task of the participant is to give a personal evaluation of the followingthree distributions : distribution of variable X : representing the distribution of blue balls

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(XB) or green (XG) in D1 over S, the distribution of variable Y , representing the distributionof blue (YB) or green (YG) balls in D2 over S, and the distribution of variable Z, representingtheir joint distribution over S.

Simple Set Upno Urn 1 Urn 2 E[X] E[Y ] E[XY ] Cov Corr

N1 b1 D1D1N1

N2 b2 D2D2

(N2+D1)ρ

X = XB, Y = YB1 2 0.5 1 0.5 2 0.5 1 0.33 0.5 0.5 0.33 0.08 0.332 2 0.5 1 0.5 4 0.5 1 0.20 0.5 0.5 0.30 0.05 0.203 2 0.5 1 0.5 8 0.5 1 0.11 0.5 0.5 0.28 0.03 0.114 2 0.5 1 0.5 200 0.5 1 0.005 0.5 0.5 0.25 0.005 0.005

X = XB, Y = YG5 2 0.5 1 0.5 2 0.5 1 0.33 0.5 0.5 0.16 -0.08 -0.336 2 0.5 1 0.5 4 0.5 1 0.20 0.5 0.5 0.20 -0.05 -0.207 2 0.5 1 0.5 8 0.5 1 0.11 0.5 0.5 0.22 -0.03 -0.118 2 0.5 1 0.5 200 0.5 1 0.005 0.5 0.5 0.245 -0.005 -0.005

Table 2.1 – Experimental Design : Variation of experimental parameters and correlationcoefficients.

A simple case of our experiment is shown by the following example that is summarized inTable 2.1. The urns contain 2 balls, one blue and one green, each. One ball is drawn fromeach urn, with a total of S=100 repetitions. xB is the number of times out of 100 repetitionsa blue ball would be drawn first, yB is the number of times out of 100 a blue ball would bedrawn second and zB is the number of times out of 100 where both draws would be blue. Thecorresponding questions eliciting the distribution of those variables are :

1. “What are the chances out of 100 that a blue ball is drawn from the first urn ?”

2. “What are the chances out of 100 that a blue ball is drawn from the second urn ?”

3. “What are the chances out of 100 that a blue ball is drawn from both urns ?”

With the response “xB out of 100,” question 1 elicits E[X] = Pr[X = B] = Pr[D1 = B] =

xB/100 = xB. Response to question 2 reveals E[Y ] = Pr[Y = B] = Pr[D2 = B] = yB/100 =

yB and to question 3, E[Z] = E[XY ] = Pr[X = B, Y = B] = Pr[D1 = B,D2 = B] =

zB/100 = zB. The correlation is obtained simply by ρ = (zB−xB yB)/√xB yB(1− xB)(1− yB).

Given the experimental design of the simple example, the theoretical correlation between thetwo random variables is 0.33.

With this structure, we introduced a correlation between X and Y when taking a ball from thefirst urn and put it in the second urn. Because ignoring this correlation affects beliefs aboutthe joint distribution of X and Y , Table 2.2 presents beliefs’ prediction for rational subjectsand full correlation neglecters for 4 parameterizations presented in Table 2.1. Someone who

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fully neglects the structure of correlation thinks that there is no link between X and Y andthen :

P [X = xi, Y = yj ] = P [X = xi]× P [Y = yj ] for i, j ∈ {Blue,Green}

P[X=1, Y=1]Treatment correlation Rational beliefs Full Corr. neglect beliefs

0,33 0,33 0,250,2 0,3 0,250,11 0,27 0,250,005 0,25 0,25

Table 2.2 – beliefs prediction for joint distribution

Our design is structured so that for all beliefs elicitations problem E[X] = E[Y ] = 12 . The

Simple Set Up allows the correlations to lay between −0.33 and 0.33 1. In the Simple Set Upwe restrain N1 = 2, b1 = b2 = 0.5 and D1 = D2 = 1. By varying N2, the size of the second urn,we can manipulate the level of correlation to be between 0 and 0.33 2. And by varying whetherthe subjective expectation for the second draw concern the same color as the one in the firstdraw or the other, we manipulate the direction of the correlation to be positive or negative.Rows (1) - (8) of Table 2.1 shows 8 possible parameterizations resulting in correlations of{−0.33,−0.20,−0.10,−0.005, 0.005, 0.10, 0.20, 0.33} covering uniformly the range of possiblevalues.

This first design allows to measure people understanting of the correlation when they facesituations which introduce the correlation by taking one variable as its combination with theother one. For instance, Eyster and Weizsäcker (2011) construct portfolio choice problems withstate-dependent returns using framing variation in which each participant faces two versionsof the same portfolio-choice problem. Across the two framing variations, they switch assetcorrelation on and off as presented in table 2.3

State-dependent returns{X(1), X(2), X(3), X(4)} {Y(1), Y(2), Y(3), Y(4)}

portfolio 1 A = {12, 24, 12, 24} B = {12, 12, 24, 24}portfolio 2 C = {12, 24, 12, 24} D = {12, 18, 18, 24}

Table 2.3 – Structure of portfolio choice problem.

In portfolio 1 there is no correlation between asset A and B. Portfolio 2 is constructed such

that the returns of C = A and D =A+B

2, thus introducing the correlation between C

1. By switching the color of the ball drawn in the second urn we allow correlation to be positive or negative2.

limN2→∞

ρX,Y = 0

(proof in appendix A.2)

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and D. Under the hypothesis that people correctly perceive the correlation structure, thisframing variation does not affect their behaviour. Our simple design allows to measure peopleunderstanding of the correlation in this kind of situations before making their decisions.

Instead of observing the structure of the environment (decision problem), subjects may facesituations in which they observe historical data of state variables. This situation is illustratedin Kallir and Sonsino (2009) where subjects observe the joint distribution of realized returnsof two virtual assets with two levels of returns (high and low) for 12 preceding periods. Theyconsider five predictions problems involving five different levels of correlation between assetsreturns and subjects are requested to predict returns for 4 additional periods under the as-sumption that future returns are sampled from the empirical distribution.

We integrate this situation in our experiment and subjects observe S draws from the first andthe second urn simultaneously. We allow the number of draws S to be endogenous to eachparticipant in the experiment. Then, each subject can decide on the number of draws that hewants to observe. By endogenizing S, the number of draws, participants control the quantityof information that they have, a possible source of debiasing. The task of the subject is to givehis personal evaluation of following distributions : distribution of variable X : representingthe distribution of blue (XB) or green (XG) balls in D1 over 100, the distribution of variableY , representing the distribution of blue (YB) or green (YG) balls in D2 over 100, and thedistribution of variable Z, representing their joint distribution over 100. The correspondingquestions eliciting the distribution of those variables are :

1. “In how many out of 100 draws do you think that a blue ball or a green ball is drawnfrom the first urn ?”

2. “In how many out of 100 draws do you think that a blue ball or a green ball is drawnfrom the second urn ?”

3. “In how many out of 100 draws do you think that a blue ball is drawn from the firsturn and a green ball from the second urn ?”

4. “In how many out of 100 draws do you think that a green ball is drawn from the firsturn and a blue ball from the second urn ?”

2.2 Treatments

We consider three different presentations. First, a presentation of the structure of thesituation, but no demonstration of realizations, i.e., S = 0 and questions as in the structuralpresentation above. Second, no information on the structure, but a time series showing actualrealizations of a certain amount S of draws and questions as in the empirical presentation.Third, no information on the structure, but a time series showing joint distributions ofvariables as relative frequencies in matrix form and questions as in the empirical presentation.

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Chapitre 3

Empirical Measure

In this section, we present our measure of individual correlation neglect and the others mea-sures in the literature.

3.1 Simple Set Up

In this paper, we presume correlation neglect to be an individual trait of a person and wepropose a measure of this caracteristic. To compute a subjective measure of “individualcorrelation” when asking for their beliefs about the joint distribution of state variables inthe Empirical treatment, we use a measure of correlation for bivariate data, the so called“Phi Coefficient.” This is one of the straightforward and usefull methods to assess thecorrelation between two bivariate variables and it has the same interpretations as pearson’scorrelation. The “Individual Phi Coefficient” for each subject is compared to the true value ofthe correlation allowed by our experiment and in the same treatment.During the experiment and in empirical treatments, subjects answer differents questions aboutbivariate variables and for each subject, we construct a 2×2 matrix corresponding to hisanswers. Let subject “i” when answer to questions in treatment “j” forms the following 2 × 2Matrix.

Urn 2 : Variable YBlue Green Total

Urn 1 : Variable X Blue aji bji ejiGreen cji dji f jiTotal gji hji n = 100

In the experiment, eji and fji represent the subjectives distributions of variableX for individual

i. gji and hji represent the distribution of variable Y (representing the color of the ball drawnin the second urn) in the same treatment. With this presentation, the value of correlation forindividual i in treatment j, is computed as follow :

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φji =aji × d

ji − c

ji × b

ji√

(eji × fji × g

ji × h

ji )

Since φji is a subjective value of the correlation for individual i in the treatment j, this iscompared to the right value of correlation in the same treatment ( called φj). Our measure isdefined as follow :

χji = φj − φji

χji quantifies individual i correlation neglect in treatment j. This framework allows χji to be-long in the interval [−2, 2] where near to 0 represents rational subjects.

In the structurals treatments we don’t need to use the definition of “Phi Coefficient ;” subjectsresponses are used to compute their subjective correlation in the corresponding treatmentusing the formula :

ρi =covi(X,Y )

σi,Xσi,Y=

P [X = x, Y = y]− E[X]E[Y ]√P [X = x](1− P [X = x])× P [Y = y](1− P [Y = y])

This value is then compared to the theoretical value of correlation as describe above to quantifytheir neglect.

For the purpose of some statistical analysis, because our beliefs formation tasks allow for 3differents presentations of the information, we compute a single measure of individual correla-tion neglect (median correlation neglect) for each subject and each informational presentation.We compute this measure by taking a median of j correlation neglect parameters at the sameinformational presentation (k) :

χki = med(φj,k − φj,ki )

Then, each subject has one value of correlation neglect parameter by type of informationalpresentation.

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Chapitre 4

Results

4.1 Descriptive Statistics

In this experiment 17 participants were enrolled to participate in the 12 belief formation tasksfor a total of 204 observations. The experiment was run using the experimental software z-Treeand a session lasted about 90 minutes. Subjects were paid the same amount of 25 CAN. Table4.1 presents descriptive statistics of participants. The mean age is 30 years and about 41% of

Table 4.1 – Descriptive statistics

Variables Mean Std.Dev Min MaxAge 30 6 22 47Gender

Male 0.59 0.49 0 1Female 0.41 0.49 0 1

subjects are women. The subjects are either undergraduate or graduate. By pooling the dataacross treatments, table 4.2 presents summary statistics for all treatments and reveals thatsubjects evaluation of the correlation by presentation differ to the true value of correlation insome treatments. Thus, table 4.2 presents a sufficient amount of correlation neglect betweenindividuals and treatments. We focus our analysis on correlation neglect at individual-level inthe next section.

4.2 Heterogeneity in correlation neglect

Because table 4.2 reveals that some subjects neglect correlation, we develop a measure ofindividual correlation neglect in order to investigate this heterogeneity. Our design allows usto estimate individual’s correlation neglect parameter by informational presentation k andtreatment j, χki = med(φj,k − φj,ki ) for each level of correlation. At the individual level,

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Table 4.2 – Median correlation by treatment andinformational presentation

True correlation Median Subjective Correlationvalue structural Empirical Empirical

without order with order0.33 0.00 0.15 0.200.20 0.20 0.18 0.200.11 0.12 0.00 0.000.005 0.00 0.00 0.00

The table presents a summary statistics of subjective correlationsacross treatments for each presentation.

graphics 6-8 reveal that neglecting the correlation somewhat depends on the presentation ofinformation 1.

Figure 1 provides kernel density estimates of the distribution of these median correlationneglect parameters by informational presentation. The plots reveal 2 spikes for structuralpresentation ; one around zero for the vast majority of subjects (they behaves approximatelyrational) and one around 0.2 (correlation neglecters). For the others types of informationalpresentation, we observe 3 spikes when the majority of subjects also behave rational with thespike around 0. The others spike suggest the presence of different types of individuals.

This procedure however ignores the variability in subjects tendency to neglect correlation.Figure 1 suggests the existence of different types of individuals who neglect the correlationat different level of informational presentation. For each type of presentation, some partici-pants behave like if they completely ignore the correlation between variables. For the purposeof finite mixture model, we suppose that each participant is characterized by a set of two-dimensional types (χkt , σ

kt ) with t ∈ {1, ..., T}, and k ∈ {1, 2, 3}, where the population weights

πt are estimated along with (χt, σt). σkt is the variance of individual type t in informational

presentation k. The correlation neglect parameter of subject i in round j for presentation kcan be expressed as χj,ki = χkt +µj,ki , µj,ki ∼ N(0, σkt ). The likelihood contribution of individuali in presentation k is given by :

Li(χk, σk, πk) =

T∑t=1

πt

17∏j=1

P [χj,ki = χkt + µj,ki |χkt , σ

kt ]

The grand likelihood is obtained by summing the logs of the individual likelihood contributions.The model generates different results depending on the number of types T included. Theestimations are ran for up to T = 5 and we report results for up to T = 3 for the two differenthistorical data presentations (with and without order) and for up to T = 4 for the structuralpresentation of the information. For the others types, the results are significantly similar forthe case T = 3 and T = 4 but the likelihoods are scarcely improved relatively.

1. We investigate this issue in section 4.4.1

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01

23

4D

en

sity

-.2 0 .2 .4 .6 .8

structural Empirical without order

Empirical with order

Figure 4.1 – Heterogeneity in Correlation Neglect

The estimates show that when we allow for one type of individuals in the experiment and foreach informational presentation, the variance estimated is high. This mean that, the modelwith one class of individuals masks a considerable degree of heterogeneity. By allowing theexistence of two types of individuals, the model fit increases but the variance is still higher insome groups. The model suggests that the data can be explained as a mixture of two differentsgroups of subjects. For one group, the estimates generate a correlation neglect parameter closeto 0 (rational subjects). The second group is characterized by a large amount of correlationneglect. The high variances in some groups predict the presence of further sub-populations inthe data. We allow for three types of individuals in the experiment. If we allow for more thanthree types in our historical presentations and more than four types in structural presentationthe models fits increase but not dramatically and the parameters estimated for some groupsremain unchanged. This individual-level analysis shows that subjects tendency to ignore cor-relation masks a considerable heterogeneity and the results are similar to the inference fromFigure 4.1.

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Table 4.3 – Structural

Parameter Goodness of fitModel Type χ σ π (%) LL AIC BICT=1 t=1 0.04 0.14 100 40.12 -78.25 -76

(0.02) (0.02)t=1 -0.28 0 6

T=2 (0) (0) (0.03) 724.2391 -1442.478 -1435.82t=2 0.06 0.11 94

(0.014) (0.01) (0.03)t=1 -0.28 0 6

(0) (0) (0.03)T=3 t=2 -0.02 0.05 59 735.9765 -1459.953 -1446.636

(0.008) (0.006) (0.06)t=3 0.18 0.044 35

(0.01) (0.0065) (0.06)t=1 -0.28 0 6

(0) (0) (0.03)t=2 0.004 0.002 47

T=4 (0.0004) (0.0003) (0.06) 824.5472 -1631.094 -1611.119t=3 0.18 0.04 35

(0.009) (0.006) (0.06)t=4 -0.12 0.013 12

(0.005) (0.034) (0.04)

17 subjects, standard errors in parentheses.

Table 4.4 – Empirical without order

Parameter Goodness of fitModel Type χ σ π (%) LL AIC BICT=1 t=1 0.13 0.16 100 26.600 -51.198 -48.978

(0.02) (0.03)t=1 0.05 0.09 77

T=2 (0.013) (0.01) (0.05) 39.523 -69.046 -57.948t=2 0.38 0.05 23

(0.014) (0.01) (0.05)t=1 -0.014 0.044 47

(0.008) (0.005) (0.06)T=3 t=2 0.38 0.055 23 91.186 -166.372 -148.616

(0.014) (0.05) (0.05)t=3 0.15 0.003 30

(0.0008) (0.0005) (0.055)

17 subjects, standard errors in parentheses.

Table 4.5 – Empirical with order

Parameter Goodness of fitModel Type χ σ π (%) LL AIC BICT=1 t=1 0.13 0.21 100 8.073 -14.14576 -11.926

(0.03) (0.05)t=1 0.05 0.07 75.7

T=2 (0.012) (0.009) (0.075) 36.106 -62.213 -51.116t=2 0.4 0.27 24.3

(0.09) (0.05) (0.075)t=1 0.055 0.07 82.5

(0.01) (0.0075) (0.05)T=3 t=2 0.86 0 6 185.695 -359.390 -346.073

(0.00) (0.00) (0.03)t=3 0.34 0.033 11.5

(0.011) (0.008) (0.04)

17 subjects, standard errors in parentheses.

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4.3 Cognitive effort and learning over time

This section investigates the relationship between correlation neglect and subjects’ responsetimes commonly used as proxy of cognitive effort (Rubinstein (2007)). Because of cognitivecosts, subjects might develop a solution strategy and opt for simplifying heuristic. Table 4.6provides the results of panel regression with random effects and heteroskedasticity-robuststandard errors of correlation neglect parameter for each treatment on subjects’ responsetime. Then, we check if subjects learn about correlation over time. The results show thatcorrelation neglect is not associated with response time. A longer time spent on a task doesn’taffect subject’s tendency to neglect correlation.

Table 4.6 – Cognitive effort and learningover time

Dependent variable : Correlation Neglect Parameter

Time Treatment -0.00002(0.00003)

Time Trend -0.002(0.009)

Time Treatment# Time Trend 0.00001(0.00002)

1 if Female 0.18(0.13)

Age 0.093**(0.038)

Age2 -0.0015***(0.0006)

Education -0.014(0.045)

Constant -1.36**(0.63)

Controls variables include age, gender, level of educa-tion. *p < 0.10, **p < 0.05, ***p < 0.01.

Do subjects learn about correlation over time ? the results suggest that correlation neglectdoesn’t become smaller over time.

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4.4 Mechanisms underlying correlation neglect

4.4.1 Correlation neglect and informational presentation

As shown in table 4.7 and graphics A.1-A.4 (appendix A), correlation neglect is associated withinformational presentation. Some presentations allow to better recognize correlation betweenstates variables than others. To investigate this issue, we regress individual correlation neglectparameter on informational presentation.

Table 4.7 – Correlation neglect and informationalpresentation

Median correlation neglect(1) (2)

2 if empirical without order 0.14*** 0.065*(0.04) (0.04)

3 if matrix form 0.04 0.001(0.04) (0.04)

Constant 0.0025 0.05(0.03) (0.17)

controls No Yes

Obs. 204 204Pseudo R2 0.05 0.08

Median regression, standard errors in parentheses. Controls va-riables include age, gender, level of education. *p < 0.10, **p <0.05, ***p < 0.01.

The results show that relative to the baseline presentation (structural presentation), obser-ving historical data without ordering significantly increases subjects tendency to neglect thecorrelation. There is no significant difference between structural presentation and historicaldata presentation with ordering possibility. This results reveal that structural and empiricalwith order presentations are better to reduce individuals’ correlation neglect.

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4.4.2 Number of historical informations

Our endogenous treatment allows subjects to decide the number of past observations theyobserve. A good strategy is to observe all the data up to 100 draws and report the samedistribution in his beliefs formation tasks. Tables 4.8-4.11 give the number of subjects whosimultaneously choose a certain number of draws in the two empirical treatments (with andwithout ordering) for each level of true correlation. The tables reveal that the majority ofindividuals (in bold) observe up to 100 draws from the 2 informational presentations. However,we are not able to tell if the same subject choose the same number in all empirical treatmentduring the experiment. We investigate this issue by regressing the number of observationsobserved by each subject on time trend. The results in table 4.12 suggest that the number ofobserved informations doesn’t increase significantly over time. Some of subjects keep the samenumber of observations during experiment. The results also suggest that level of educationand gender affect the number of informations. Most educated individuals tend to ask for moreinformations than less educated one and women ask for less informations than men.

Table 4.8 – correlation=0.33Emp. With order10 20 100 Total

10 2 1 0 320 3 0 1 4

Emp. Without order 30 1 0 2 3100 0 0 7 7Total 6 1 10 17

Table 4.9 – correlation=0.20Emp. With order

10 20 40 100 Total10 1 1 0 0 2

Emp. Without order 20 2 1 1 2 630 0 0 0 1 160 1 0 0 1 2100 1 0 0 5 6Total 5 2 1 9 17

Table 4.10 – Correlation=0.11Emp. With order

10 20 30 100 Total10 3 1 1 1 6

Emp. Without order 20 1 2 0 1 430 0 0 0 1 1100 0 0 0 6 6Total 4 3 1 9 17

We are also interest to investigate the effect of the number of historical observations allow byour two empirical presentations on subjects tendency to neglect the correlation. In table 4.13,we run a median regression of correlation neglect parameter on the number of informations.The table reveals that more information tend to reduce correlation neglect. So, it is importantto observe the maximum of information available before forming his beliefs.

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Table 4.11 – Correlation=0.005Emp. With order10 20 100 Total

10 1 0 0 1Emp. With order 20 1 2 2 5

30 1 0 0 140 0 0 1 190 0 0 1 1100 3 1 4 8Total 6 3 8 17

Table 4.12 – Number of historical information over time

Dependent variable : Number of histirical informationsEmpirical without order Empirical with order Emp. without order + Emp. with order

(1) (2) (3) (4) (5) (6)Time 0.66 0.57 0.07 -0.02 0.00 0.31

(1.23) (1.2) (0.97) (0.0001) (0.45) (0.74)Age -0.88 -2.4*** -1.65***

(0.83) (0.8) (0.57)1 if female -18.2 -17.75* -17.8**

(11.13) (10.6) (7.65)Educational level 2.23 9.22* 5.8

(5.62) (5.35) (3.86)constant 48.70*** 80*** 60*** 127*** 52.9*** 103.4***

(10) (28) (11.5) (26.7) (0.04) (19.22)

Obs. 68 68 68 68 136 136R2 0.004 0.11 0.0001 0.30 0.0023 0.19

Median regression, standard errors in parentheses. Controls variables include age, gender, level of education. *p < 0.10,**p < 0.05, ***p < 0.01.

Table 4.13 – Correlation neglect and number of past observations

Dependent variable : Correlation Neglect ParameterEmpirical without order Empirical with order

(1) (2) (3) (4)Numb. Past observations -0.002 -0.002* -0.002** -0.0008

(0.0008) (0.0007) (0.0006) (0.0005)Constant 0.14** 0.56*** 0.17*** 0.06

(0.06) (0.16) (0.05) (0.13)controls No Yes No Yes

Obs. 68 68 68 68Pseudo R2 0.03 0.11 0.07 0.08

Controls variables include age, gender, level of education. *p < 0.10, **p < 0.05,***p < 0.01.

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Chapitre 5

Applications

5.1 Mother’s age at first childbearing and Child mortality

Young maternal age is associated with adverse birth outcomes for child and mother (Omaret al. (2010) ; Wang et al. (2012) ; Shrim et al. (2011) ; Kang et al. (2015) ; Fraser et al. (1995)).Measuring people “ignorance" of this correlation may help for news stragies to reduce adversebirth outcomes for children and mothers. Recent literature (Diarra and Dessy (2017)) on thedeterminants of the demand of child bride suggests that people ignores the correlation betweenwomen’s age at first birth and the level of mother mortality, or fails to factor this informationin their marriage decision. Our design allows us to measure individual correlation neglect byeliciting their expectation about the age of bride at first childbearing and the level of infantmortality 1.We elicit these distributions using natural frequency questions and asking for probability foroutcome intervals. People are asked to share their evaluation out of 100 randomly selectedwomen at age 12 or less, how many will have their first child at different age groups. Thisquestion elicits their subjective distribution of age at first birth.Then, people are told to share their expectation out of 100 randomly selected newborn-babies,how many will died before reaching their first birthday. This question assesses subjective dis-tribution about infant mortality.finally, we elicit beliefs about the joint distribution of two variables. Subjects have to answerthese questions in the following way :

A.) I think that out of the 100 randomly selected women at age 12 or less,

(1) “...d1... will have their first birth before 18 years old.”

(2) “...d0... will have their first birth from 18 years old. ”

B.) I think that out of the 100 randomly selected newborn-babies,

1. Because mother mortality is not observed in the data in order to compute an empirical correlation withmother mortality

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(1) “...b1... will died before their first birthday,”

(2) “...b0... will be alive until their first birthday,”

C.) I think that out of the ...d1... women who will have their first birth before 18 years old,

(1) “...d1b1... will have their baby died before reaching age 1,”

(2) “...d1b0... will have their baby alive until their first birthday,”

D.) I think that out of the ...d0... women who will have their first birth from 18 years old,

(1) “...d0b1... will have their baby died before reaching age 1,”

(2) “...d0b0... will have their baby alive until their first birthday,”

5.2 Financial Allocation Decisions

In order to measure people perception of the correlation in assets returns when making theirfinancial allocation decisions, our measure using beliefs elicitation can be apply. For simplicity,like Kallir and Sonsino (2009), portfolio contains two assets A and B with two levels of returns :"high" and "low". Information on the joint distribution of returns is present in the form ofempirical frequencies for 12 preceding periods as presented in table 5.1. Subjects are toldto predict returns for 100 additional periods, under the assumption that future returns aresampled from the empirical distribution.

Asset Bhigh low

Asset A high 5/12 1/12low 1/12 5/12

Table 5.1 – Joint distribution

The corresponding questions eliciting the joint distribution of 100 future returns for assets Aand B are :

1. “How many out of the 100 additional periods do you think that assets A and B will havesimultaneously high returns levels ?”

2. “How many out of the 100 additional periods do you think that assets A and B will havesimultaneousl low returns levels ?”

3. “How many out of the 100 additional periods do you think that asset A will have highreturn and low return for asset B ?”

4. “How many out of the 100 additional periods do you think that asset A will have lowreturn and high return for asset B ?”

By varying the correlation in assets returns and asking for the same questions as above, wecan analyse subjects responses to change in correlation.

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5.3 Pre-election Polls

Subjects are told to predict the results of an election. They have a choice between two optionsA and B. Each option has two levels of cost : high cost or low cost. The two options representthe two candidates for presidential race. There is 4 types of individuals in this populationcharacterized by 4 different colors : blue, dark green, pale green and yellow. Each individualbelongs to only one type of color. Subjects in the experiment observe the decisions of 100individuals, randomly selected in the whole population (25 individuals per color). The resultsare presented in table 5.2. Subjects are told that this table represents the joint distributionbetween the choice of an option and his cost. Subjects have to predict to predict the choices of100 additional people sampled in the same population such that 25 persons were drawn in eachgroup. The following questions are used to assess their beliefs for 100 additional observations :

Blue Pale Greenhigh cost low cost Total high cost low cost Total

Option A 2 20 22 Option A 12 3 15Option B 1 2 3 Option B 5 5 10Total 3 22 25 Total 17 8 25

Dark Green Yellowhigh cost low cost Total high cost low cost Total

Option A 1 11 12 Option A 4 1 5Option B 1 12 13 Option B 18 2 20Total 2 23 25 Total 22 3 25

Table 5.2 – subjects choices for each type of individual

1. “How many out of the 100 additional people do you think, will choose option A and willpay high costs ?”

2. “How many out of the 100 additional people do you think, will choose option A and willpay low costs ?”

3. “How many out of the 100 additional people do you think, will choose option B and willpay high costs ?”

4. “How many out of the 100 additional people do you think, will choose option B and willpay low costs ?”

Subjects who perceive the correlation between variables should fill in exactly by summing thenumber of individuals across groups who are in the same situation in the historical distributionthat they observed. This design allow to measure, for each subject, his level of correlationneglect.

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Conclusion

This paper has provided a sequence of belief formation tasks by varying the level of correlationbetween tasks and has demonstrated that people neglect correlation when they face some typesof informational presentation. In this paper, we suppose correlation neglect to be an individualcharacteristic of a person and we suggest an empirical measure. Our measure is based on asequence of laboratory experiments and tests for 3 types of informational presentation. First,subjects learn about the structure of decision problem and make their predictions ; second,they observe historical data of state variables without ordering possibility and third, they havea possibility to sort historical data.

Our results suggest a good amount of heterogeneity in correlation neglect at individual-levelanalysis. Two types of presentation allow to reduce subjects’ tendency to neglect correlation :structural presentation and historical presentation with ordering. Women are likely to neglectcorrelation more than men and the level of education does not significantly affect their neglect.

Empirical data analysis suggests that individuals may observe all the information available inorder to form their beliefs about state variables. Subjects don’t learn about correlation overtime and cognitive effort doesn’t affect the correlation neglect parameter.

Our strategy, because we are able to measure correlation neglect at individual-level withoutmaking any ancillary hypothesis about individual’s preferences and without observing theirdecisions making process can be use as a general measure of correlation neglect when askingpeople’s expectations about the distribution of state variables.

Although we propose a simple measure of individual correlation neglect that can be appliedin various domains, some authors suggest a measure that is specific to a domain (investmentdecision and portfolio choice problem, auctions market setting, etc.). These measures presentthe correlation either by the structure of information (correlation neglect in informationalsource or common source of information) or by observing historical data (e.g returns) andcompute correlation neglect based on individual’s investment decision.

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Bibliographie

Cohen, M., J.-Y. Jaffray, and T. Said (1987). Experimental comparisons of individual behaviorunder risk and under uncertainty for gains and for losses. Organizational Behavior andHuman Decision Processes 39, 1–22.

DeMarzo, P., D. Vayanos, and J. Zwiebel (2003). Persuasion bias, social influence, and unidi-mensional opinions. Quaternaly Journal of Economics 118 (3), 909–968.

Diarra, S. and S. Dessy (2017). The determinants of the demand for child brides in sub-saharanafrica. Working paper .

Enke, B. and F. Zimmermann (2015). Correlation neglect in belief formation. University ofZurich. Mimeo.

Eyster, E. and G. Weizsäcker (2011). Correlation neglect in financial decision making. Workingpaper .

Fraser, A., J. Brockert, and R. Ward (1995). Association of young maternal age and adversereproductive outcomes. The New England Journal of Medecine 332 (17).

Holt, C. A. and S. K. Laury. (2002). Risk aversion and incentive effects. American EconomicReview 92 (5), 1644–1655.

Kallir, I. and D. Sonsino (2009). The neglect of correlation in allocation decisions. SouthernEconomic Journal 75 (4), 1045–1066.

Kang, G., J. Lim, A. Sugam, and L. Y. Lee (2015). Adverse effects of young maternal age onneonatal outcomes. Singapore Medical Journal 56 (3), 157–163.

Levy, G. and R. Razin (2015). Correlation neglect, voting behavior, and information aggrega-tion. American Economic Review 105 (4), 1634–1645.

Omar, K., S. Hasim, N. A. Muhammad, and A. Jaffar (2010). Adolescent pregnancy outcomesand risk factors in malaysia. International Journal of Gynecology and Obstetrics 111, 220–223.

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Ortoleva, P. and E. Snowberg (2015). Overconfidence in political behavior. American Econo-mic Review 105 (2), 504–535.

Rubinstein, A. (2007). Instinctive and cognitive reasoning : A study of response times. Eco-nomic Journal 117 (523), 1243–1259.

Shrim, A., S. Ates, A. Mallozzi, and R. Brown (2011). Is young maternal age really a risk factorfor adverse pregnancy outcome in a canadian tertiary referral hospital ? North AmericanSociety for Pediatric and Adolescent Gynecology 24, 218–222.

Wang, S., L. Wang, and M. Lee (2012). Adolescent mothers and older mothers :who is at highrisk for adverse birth outcomes ? Public Health 126, 1038–1043.

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Annexe A

Annexe

A.1 Enke and Zimmerman : correlation between signals

Computers generate two signals sA and sB =sA + sB

2, where sh ∼ N(µ, (µ2 )2) for h ∈ {A,B}.

corr(sA, sB) =cov(sA,

sA + sB2

)

σsAσsB= 1

2

(σsA)2

σsAσsB= 1

2

σsAσsB

σsB =√

(σsB )2 =

√V ar(

sA + sB2

) =√

14(µ2 )2 + 1

4(µ2 )2 = µ2

√12 =

√12σsA .

Finally, the spearman’s correlation coefficient between sA and sB is define as follow :

corr(sA, sB) =√22 ≈ 71%

A.2 Convergence of Correlation in the Simple Set-Up

The linear correlation between X and Y is defined as follow :

ρX,Y =cov(X,Y )

σXσY=E[XY ]− E[X]E[Y ]

σXσY

E[X] = P [X = B] =B1

N1=

N12

N1=

1

2

E[Y ] = P [Y = B] =B1

N1∗ B2 + 1

N2 + 1+G1

N1∗ B2

N2 + 1=

1

2

with G1 = N12

29

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σX =√P [X = B](1− P [X = B]) =

1

2

σY =√P [Y = B](1− P [Y = B]) =

1

2

E[XY ] = P [X = B;Y = B] =B1

N1∗ B2 + 1

N2 + 1=

1

4∗ N2 + 2

N2 + 1

ρX,Y =1

N2 + 1

limN2→∞

ρX,Y = 0

limN1→∞

ρX,Y =1

N2 + 1

30

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A.3 Others Results

-1-.5

0.5

11.

5M

edia

n C

orr.

Neg

lect

- St

ruct

ural

-1 -.5 0 .5 1 1.5Median Corr. Neglect - Empirical without order

Figure A.1 – Median Correlation Neglect- Structural vs Empirical Without Order

-1-.5

0.5

11.

5M

edia

n C

orr.

Neg

lect

- St

ruct

ural

-1 -.5 0 .5 1 1.5Median Corr. Neglect - Empirical with order

Figure A.2 – Median Correlation Neglect- Structural vs Empirical With Order

31

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-1-.5

0.5

11.

5M

edia

n C

orr.

Neg

lect

- Em

piric

al w

ithou

t ord

er

-1 -.5 0 .5 1 1.5Median Corr. Neglect - Empirical with order

Figure A.3 – Median Correlation Neglect- Empirical With Order vs Empirical Without Order

-.2-.1

0.1

.2M

ean

Bias

0 .1 .2 .3 .4Value of Correlation

Structural Empirical without order

Empirical with order

Figure A.4 – Mean correlation neglect per value of true correlation

32

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0.2

.4.6

.80

.2.4

.6.8

-.5 0 .5

-.5 0 .5

structural Empirical without order

Empirical with orderFra

ctio

n

Figure A.5 – Histograms of the difference between original median correlation neglect pa-rameters and modified median correlation neglect parameters when excluding one parameterthat is closest to the original median correlation neglect parameter.

33

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A.4 Instructions

Figure A.6 – Presentation - Structural

34

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Figure A.7 – Presentation - Empirical Without Order

35

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Figure A.8 – Presentation - Empirical With Order

36

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Figure A.9 – Presentation - Empirical With Order

37