Essays on uncertainty, beliefs updating and portfolio choice

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Essays on Uncertainty, Beliefs Updating and Portfolio

Choice

Thèse

Kouamé Marius Sossou

Doctorat en économique

Philosophiæ doctor (Ph. D.)

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Essays on Uncertainty, Beliefs Updating and Portfolio

Choice

Thèse

Kouamé Marius Sossou

Sous la direction de:

Charles Bellemare, directeur de recherche Sabine Erika Kröger, codirectrice de recherche

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Résumé

Cette Thèse, structurée en trois essais, étudie de façon expérimentale l’effet de l’incertitude sur la prise de décision avec des applications aux choix de portefeuille.

Le premier essai étudie les règles que les agents économiques utilisent pour, d’une part, rappor-ter des distributions de probabilités sur des événements futurs en présence de l’incertitude et d’autre part, mettre à jour cette incertitude lorsque de nouvelles informations sont disponibles. Plusieurs règles de rapport de distribution de probabilité et de mise à jour de l’incertitude sont présentées et notre analyse empirique se focalise sur la caractérisation de l’hétérogénéité de ces règles dans la population considérée. Les résultats révèlent que deux règles sont principale-ment utilisées pour rapporter des distributions de probabilités: 65% des individus rapportent des distributions en pondérant correctement les distributions possibles par leur incertitude exprimée, alors que 22% rapportent des distributions proches de la distribution qu’ils per-çoivent comme la plus probable. Par contre, nous observons une hétérogénéité considérable dans la façon dont les individus mettent à jour leur incertitude. En général, les individus ont tendance à attribuer un poids relativement faible aux nouvelles informations, entrainant une persistance de l’incertitude pour un nombre important d’individus. Des analyses contrefac-tuelles suggèrent que cette persistance pourrait être présente dans des contextes non couverts par notre étude.

Le deuxième essai étudie la fréquence optimale d’évaluation de portefeuille en présence d’incertitude. Cet essai met en évidence que l’ambiguïté et l’aversion à la perte ont des effets opposés sur les marchés financiers et peuvent coexister en présence d’incertitude. Nous présentons un de-sign expérimental dans lequel les investisseurs font des choix répétés de portefeuille en faisant initialement face à l’incertitude concernant la distribution des rendements de l’un des actifs disponibles. Nous exploitons des variations exogènes de la fréquence d’évaluation ainsi que les

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variations temporelles des anticipations probabilistes par rapport aux distributions possibles de rendements pour identifier conjointement les préférences des investisseurs envers l’ambiguïté, la perte et le risque, ainsi que les règles qu’ils utilisent pour mettre à jour leur incertitude. Les résultats obtenus de l’estimation d’un modèle économétrique structurel suggèrent sept diffé-rentes catégories d’investisseurs. En général, nous observons que les investisseurs sont averses à l’ambiguïté, averses à la perte, averses au risque dans le domaine des gains, mais ils aiment prendre du risque dans le domaine des pertes. Nous concluons notre analyse en utilisant nos estimations pour prédire la distribution des périodes d’évaluation optimales dans notre échantillon. Nos résultats suggèrent qu’environ 70% des investisseurs préfèrent une fréquence d’évaluation plus élevée, reflétant l’effet dominant de l’ambiguïté sur l’aversion à la perte. Le troisième et dernier essai examine si les préférences temporelles expliquent les décisions financières prises par les personnes âgées. Nous développons dans un premier temps un modèle structurel de prise de décision inter-temporelle en tenant compte de l’incertitude de fin de vie à laquelle font face ces personnes. Ensuite, nous estimons les préférences temporelles en utilisant les données expérimentales de choix inter-temporelles auprès d’un échantillon représentatif de séniors américains. Enfin, nous examinons comment les paramètres estimés sont liés à la composition réelle de portefeuille de ces aînés. Nos résultats indiquent que les préférences temporelles des personnes âgées sont très hétérogènes et que seule une petite partie de cette hétérogénéité peut être expliquée par les caractéristiques sociodémographiques standard telles que l’âge, le sexe, l’éducation, l’état matrimonial, le revenu, la richesse, etc. Nous observons que les personnes âgées qui ont un facteur d’escompte plus élevé sont plus susceptibles de posséder des comptes de retraite et des actifs risqués. De même, ces personnes réduisent la part de leur richesse allouée aux actifs sûrs et augmentent celle allouée aux actifs risqués. Ces résultats suggèrent que les préférences temporelles affectent les choix d’investissement des actifs sûrs vers d’autres actifs financiers.

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Abstract

This Thesis, consisting of three chapters, studies the effects of uncertainty on decision-making with portfolio choice applications.

Chapter 1 studies how experimental subjects report subjective probability distributions in the presence of ambiguity characterized by uncertainty over a fixed set of possible probability distributions generating future outcomes. The level of distribution uncertainty varies accord-ing to the observed outcomes and the rules used by the subjects to update the distribution uncertainty. This chapter introduces several reporting and updating rules and our empirical analysis focuses on estimating the sample distribution of these rules. Two dominant reporting rules emerge from our analysis: we find that 65% of subjects report distributions by properly weighting the possible distributions using their expressed uncertainty, while 22% of subjects report distributions close to the distribution they perceive as most likely. Further, we find significant heterogeneity in how subjects update their expressed uncertainty. On average, subjects tend to overweight the importance of their prior uncertainty relative to new infor-mation, leading to ambiguity that is substantially more persistent than would be predicted using Bayes’ rule. Counterfactual simulations suggest that this persistence will likely hold in settings not covered by our experiment.

Uncertainty in financial markets is a natural consequence of investors being unaware of ob-jective probabilities of asset returns. Chapter 2 highlights that ambiguity and loss aversion have opposite effects on financial markets and can coexist in the presence of uncertainty. This chapter addresses the normative question of the optimal portfolio evaluation frequency for an investor in order to minimize the effect of myopia, but to learn about the investment op-portunities in the market. Towards this end, we present a new experimental design in which investors are asked to make repeated portfolio choices facing initial ambiguity concerning the

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distribution of returns of one of the available assets. We exploit exogenous variations in eval-uation frequency along with time variation of probabilistic beliefs over the possible return distributions to jointly identify ambiguity, loss, and risk aversion along with rules investors use to update their ambiguity. Estimates from a structural model suggest seven different classes of investors. Investor class membership depends on loss aversion, ambiguity aversion as well as risk aversion preferences. Further, we find that at the aggregated level, investors are loss averse, ambiguity averse and they display risk aversion over gains and risk seeking over losses. We conclude our analysis by using our model estimates to predict the distribution of optimal evaluation periods for our sample. Our predictions suggest that approximatively 70% of investors prefer the highest possible evaluation period frequency.

Finally, Chapter 3 investigates whether or not the discount factor of the elderly affects their portfolio choices. We estimate time preferences using inter-temporal choice data from a hypo-thetical experiment in a representative sample of American elders and a structural model of decision-making accounting for lifetime uncertainty. Our results indicate considerable hetero-geneity in the elderly population. Moreover, we find that older people who display a higher discount factor are more likely to own retirement accounts and risky assets. These older people also tend to decrease the share of financial wealth held in safe assets and increase the share of financial wealth held in risky assets. These findings suggest that time preferences affect investment choices from safe assets toward other financial assets, all else being equal.

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List of Tables

1.1 Predicted convergence rates in the last experimental period for the baseline and

follow-up experiment . . . 29

1.2 OLS regressions of model parameters on other model parameters. . . 31

1.3 Summary of parameters and their relation. Benchmark cases and estimation results.. . . 33

1.4 OLS regressions of model parameters on other model parameters. Incentives is a binary variable taking the value of 1 when incentives were used to elicit beliefs about distribution uncertainty, 0 otherwise. . . 44

2.1 Estimated preference parameters of the latent class model with seven classes (z ratios in parentheses). . . 69

2.2 OLS regressions of determinants of individual level parameters . . . 70

2.3 Frequency of investors dropped as a function of the treatment sequence and the length of the investment interval in treatments L and M. . . 75

2.4 Latent Class Models with different numbers of classes . . . 75

2.5 Distribution of posterior probabilities. . . 76

2.6 OLS regressions of determinants of individual level parameters . . . 78

2.7 Latent Class Models with different numbers of classes . . . 79

2.8 Estimate latent class model with seven classes (z ratios in parentheses) . . . 79

2.9 Spearman pairwise correlation matrix of preference parameters . . . 79

3.1 Summary statistics for the analysis sample. . . 87

3.2 Estimated determinants of the discount factor . . . 94

3.3 Probit models for ownership probabilities. The dependent variable is the prob-ability of owning particular types of assets. Marginal effects significant at the 10% level or better are in bold. . . 96

3.4 Tobit regressions of portfolio shares. The dependent variable is the share of financial wealth held in a particular asset. Marginal effects significant at the 10% level or better are in bold. . . 99

3.5 Probit models for ownership probabilities. The dependent variable is the prob-ability of owning particular types of assets. Marginal effects significant at the 10% level or better are in bold. . . 105

3.6 Tobit regressions of portfolio shares. The dependent variable is the share of financial wealth held in a particular asset. Marginal effects significant at the 10% level or better are in bold. . . 106

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List of Figures

1.1 Screenshot of main decision screen : treatment D . . . 12

1.2 Screenshot of main decision screen : treatment S . . . 12

1.3 Distribution of |∆| (difference between reported distribution and benchmark) for values of max{µj_t} < 0.6 (left panel), for values of 0.6 ≤ max{µj_t} < 0.9

(middle panel), and for values of max{µj_t} ≥ 0.9 (right panel).. . . 16

1.4 Quantile curves (10%, 25%, 50%, 75%, 90%) of likelihood µj_t assigned to the true distribution over all 100 periods for treatment S (left graphs) and treat-ment D (right graphs) using the observed data (top graphs) and the Bayesian simulations with initial likelihoods set to 1/3 for all three possible distributions

(bottom graphs). . . 18

1.5 Left panel plots the probability measure ω(µj_t) against µj_t for α = 1, β = 1 (full line), α = 10, β = 4 (dashed line), and α = 0.01, β = 1.1 (dotted line). Right panel plots c_tagainst max{µj_t} for γ = −10 (dotted line), γ = 1 (full line), and

γ = 10 (dashed line). . . 21

1.6 Distributions of estimated reporting rule parameters (α, β, γ). . . 25

1.7 Distributions of estimated updating rule parameters δ₁ and δ₂. . . 27

(bottom graphs). Curves plotted using data without incentives. . . 36

(middle panel), and for values of max{µj_t} ≥ 0.9 (right panel) without incentives. 37

1.10 Distributions of estimated reporting rule parameters (α, β, γ) without incentives 38

1.11 Distributions of estimated updating rule parameters δ₁ and δ₂ without incentives 39

(bottom graphs). Curves plotted using data with incentives. . . 40

(middle panel), and for values of max{µj_t} ≥ 0.9 (right panel) using incentives . 41

1.14 Distributions of estimated reporting rule parameters (α, β, γ) using incentives. . 42

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2.1 Screenshot of the decision screen. . . 51

2.2 Left panel plots the distribution of experimental periods starting from which investors correctly assign a subjective probability greater than 90% on the true return distribution of the stock in treatment H. Right panel plots the

corre-sponding distribution assuming investors are Bayesian. . . 68

2.3 Distribution of experimental periods starting from which investors correctly assign a subjective probability greater than 90% on the true return distribution of the stock in treatments M and L as a function of the number of periods of

the investment interval. . . 68

2.4 Distribution of experimental periods predicted using Bayes rule starting from which investors correctly assign a subjective probability greater than 90% on the true return distribution of the stock in treatments M and L as a function

of the number of periods of the investment interval. . . 69

2.5 Distributions of estimated updating rule parameters δ1 and δ2 . . . 70

2.6 Sampling distributions of ¯h∗ for the seven classes and the investor population

as a whole . . . 71

2.7 Screenshots of the feedback screen for treatments H and M (left) and for

treat-ment L (right). . . 74

2.8 Record sheet with examples of recording tables for all three treatments. Ex-ample for low flexibility/frequency with five periods (left hand) and with three

periods (right hand). . . 74

2.9 Distribution of optimal portfolio evaluation frequency in the population of

in-vestors assuming updating using Bayes rule. . . 80

3.1 Tree for the staircase method of the time preferences task (numbers = payments

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Dedicated to Pélagie, Marc and Fred

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"Information is the resolution of uncertainty"

Claude Elwood Shannon (April 30, 1916 – February 24, 2001), American mathematician, electrical engineer, and cryptographer.

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Acknowledgements

First of all, I would like to express my immense gratitude to my advisors Prof. Charles Bellemare and Prof. Sabine Erika Kröger for their guidance and support throughout the course of this research. Thank you for always having your doors open for impromptu meetings, for your caring and parental attitudes, and many more things. Collaborating with you has been an invaluable and very rewarding experience in my development as a researcher. I was extremely lucky to have you both as advisors.

Special thanks to committee members Stephen Gordon, Guy Lacroix and Luc Bissonnette for their keen interest in my research. I am grateful to Alexander Christopher Sebald for acting as external reviewer. I would also like to thank Marion Goussé and Bertrand Achou for their much-appreciated advice on my research on various occasions. My deep gratitude to Carleen Gruntman for taking time out from her busy schedule to proofread important parts of this dissertation.

Thank you to my classmates and friends Daouda Belem and Thomas Golo for their support throughout these years. I thank other fellow colleagues in the PhD program in economics, es-pecially Koffi Akakpo, Guy Morel Kossivi Amouzou Agbe, Jean-Louis Bago, Finagnon Antoine Dedewanou, Morvan Nongni Donfack, Elfried Faton, Elysee Aristide Houndetoungan, Mélissa Huguet, Rolande Kpekou Tossou and Ibrahima Sarr for creating a positive atmosphere in the department. I would also like to thank Mélanie Brière, Blanchard Conombo, Rudy Hamel, Nathan Roger Lea Jombi, and Ibrahima Sarr for their support in carrying out the experiments presented in this thesis.

During my PhD, I spent 3 months at Munich Graduate School of Economics in Germany as an academic visitor. It was a valuable research experience that enriched my research perspective. I would like to thank Prof. Joachim Winter and his secretary Julia Zimmermann for hosting

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me in Munich. I had the pleasure to meet some amazing people in Munich who made my stay not only scientifically stimulating, but also very enjoyable. Thanks to Audrey Au Yong Lyn, Sebastian Bordt, Manfei Li, Anne Niedermaier, Pavel Obraztcov, Vera-Maria Sommer, Fabian Stürmer-Heiber, Leonhard Vollmer and Johannes Wimmer.

More fundamentally, I would like to thank my family in Benin, especially my parents, my brothers and sisters, my uncle Maximen Sossou, and Mr Séverin Nsia for their encouragement and unconditional support. I am indebted to my friends who supported my family in Quebec during these years of doctoral studies: Eusebe Ahossi, Sègbédji Parfait Aïhounhin and his wife, Finangnon Antoine Dedewanou, Moïse Dovonou and his wife, Fernand Kasongo Kalenga and his wife, Rolande Kpekou Tossou, Bignon Aurelas Tohon and his wife, Yolande Tohoundjo and her husband, and Bienvenu Tossou.

My thesis is dedicated to my wife Pélagie Gnimavo and our children Marc and Fred for their patience, sacrifices, encouragement and unwavering support.

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Foreword

The three chapters of this thesis are separate articles submitted or in preparation for sub-mission to peer-reviewed scientific journals. The first two chapters were written jointly with my thesis director, Charles Bellemare, and my co-supervisor, Sabine Erika Kröger, all full professors in the Department of Economics at Laval University. I am the principal author of the three chapters presented in this thesis.

The first chapter entitled "Reporting probabilistic expectations with dynamic uncertainty about possible distributions" has been published in the Journal of Risk and Uncertainty, vol-ume 57, pages 153-176. The second chapter titled "Optimal Evaluation Period in a Portfolio Choice Experiment with Ambiguity and Loss Aversion" is submitted to the Journal of Econo-metrics, and the third chapter entitled "Time Preferences and Financial Decisions among the Elderly" is in preparation to be submitted to the Economic Inquiry.

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Introduction

Uncertainty is an ever-present factor in our daily lives. In many circumstances, decisions have to be made in uncertain environments –that is, choosing actions often based on imperfect observations, with unknown outcomes. For instance, people can face uncertainty about their future income, the return on different investments and their health when making consumption and savings decisions. Firms face demand uncertainty when making production decisions. There is also uncertainty about future productivity, growth, inflation and unemployment when governments set up new policies. Since the seminal works ofKnight(1921) followed byEllsberg

(1961), there is a well-known distinction between risk and uncertainty. Risk refers to events for which the probabilities of the future outcomes are known, while uncertainty or ambiguity refers to events for which the probabilities of the future outcomes are unknown.

Decision making under uncertainty requires forming beliefs that integrate prior and new infor-mation. Theoretical models of decision making under uncertainty assume that decision makers assign subjective probabilities to events and update their subjective probabilities using Bayes’ rule when new information becomes available1. However, empirical evidence suggests that human decision makers typically deviate from Bayesian updating (Kahneman and Tversky

(1972); Grether (1980); Ouwersloot et al. (1998); Hoffman et al. (2011) among others). In addition, much of the existing empirical research on decision making under uncertainty was done in simple settings where individuals are not given opportunity to learn and update their beliefs to new information (see Etner et al.(2012) andTrautmann and Van De Kuilen(2015) for recent surveys). The few empirical studies which have looked at behavior under uncer-tainty in a dynamic setting have not been able to clearly separate the effects of unceruncer-tainty, updating beliefs, uncertainty preferences on choices (See for instance Cohen et al. (2000);

1

SeeGilboa and Schmeidler(1993);Epstein and Schneider(2007);Miao(2009);Ju and Miao(2012);Choi

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Dominiak et al.(2012); Qiu and Weitzel(2016);Baillon et al.(2017)).

These limitations have motivated this dissertation, in which I study how decision makers form and update their beliefs and change their attitudes towards ambiguity when more information becomes available. The fundamental issues are: (i) How do agents facing uncertainty update their beliefs as new information arrives over time? (ii) What are the effects of uncertainty on behavior? I provide answers to these questions by proposing updating rules under uncertainty and investigate their implications for portfolio choice.

In what follows, Chapter 1 analyzes how experimental subjects report distributions of future events in the presence of uncertainty and how they update this uncertainty as new information is revealed about the stochastic process generating outcomes. In the experiment designed in this chapter, subjects face initial uncertainty concerning a fixed set of possible distributions of market returns and told that one of these distributions has been selected to generate out-comes. Subjects observe new draws from the distribution selected to generate outcomes and are required to express their distribution uncertainty by assigning likelihoods to each possible distribution at selected periods of the experiment. They are next required to report a unique distribution, given the information available. This chapter introduces several reporting and updating rules and our empirical analysis focuses on estimating the sample distribution of these rules. Our results show that reporting and updating rules used by subjects are hetero-geneous in our sample, but relatively robust across our treatment variations. Two dominant reporting rules emerge from our analysis: we find that 65% of subjects report distributions by properly weighting the possible distributions using their expressed uncertainty over them, while 22% of subjects report distributions close to the distribution they perceive as most likely. These results indicate that a sizeable fraction of subjects are able to aggregate their uncertainty in order to report informative distributions of future outcomes. Moreover, our analysis suggests that distribution uncertainty can be persistent for a significant number of subjects. Our econometric model attributes this persistence to the relatively low weight that subjects place on new information about the stochastic process generating outcomes. Coun-terfactual simulations suggest that this persistence will likely hold in settings not covered by our experiment.

Chapter 2 analyzes data from a new experimental design where investors are asked to make repeated portfolio choices facing initial ambiguity concerning the distribution of returns of one

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of the available assets. Throughout the experiment, investors have the opportunity to learn about the return prospects by evaluating their portfolio performance. Frequent portfolio eval-uation enhances learning, resolves ambiguity and results in a risky environment. In contrast to this advantage of frequent portfolio evaluation stands the empirical and laboratory evidence that investors in risky and uncertain investment environments invest more cautiously when they receive feedback more frequently 2. Benartzi and Thaler (1995) explain this conservative behavior as a joint phenomenon of loss aversion and narrow bracketing (myopia). Therefore, in contrast to ambiguity averse investors, myopic loss averse investors benefit from lower port-folio evaluation frequency. Ambiguity aversion and myopic loss aversion can co-exist in the presence of uncertainty and this chapter addresses the normative question of the optimal port-folio evaluation frequency. Our analysis exploits exogenous variations in investment periods along with time variation of expressed uncertainty, and portfolio choices to identify the joint distribution of key behavioral components including risk aversion, ambiguity aversion, loss aversion as well as probabilistic belief updating rule. This joint distribution is modelled using a finite mixture approach placing limited shape restrictions. Overall, we find that investors are slow to update their probabilistic beliefs regarding the return prospects and thus slow to eliminate ambiguity. Estimates from a structural model suggest seven (7) different classes of investors. Investor class membership depends on loss aversion, ambiguity aversion as well as risk aversion preferences. Moreover, in general, we find that investors are loss averse, ambigu-ity averse and they display risk aversion over gains and risk seeking over losses. We conclude our analysis by using our model estimates to predict the distribution of optimal evaluation periods for our sample. Our results suggest that approximatively 70% of investors prefer higher evaluation period frequency, reflecting the dominating effect of ambiguity aversion over myopic loss aversion.

The general representation of uncertainty calls for modelling decision makers behaviors in terms of preferences. In addition to probability beliefs, risk and ambiguity attitudes, these preferences are also captured by times preferences (Dreze,1990). Following up on Chapter 1 and 2, the Chapter 3 presents an independent study by investigating whether time preferences are helpful to explain the financial decisions made by the elderly. Specifically, it develops a structural model of decision-making accounting for lifetime uncertainty. The model allow us

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to estimate time preference parameters using inter-temporal choice data from a hypothetical experiment in a representative sample of American elders. The estimated parameters are then related to the composition of actual portfolio (in term of the types of financial assets held and the share of financial wealth held in each asset type) of this cohort. Our results indicate considerable heterogeneity in the elderly population. We find that the discount factor of the elderly increases with wealth and income. Women and better-educated have higher discount factors, as do individuals with higher cognitive skills. An indicator of good health condition is associated with a high discount factor. Moreover, we find that older people who display a higher discount factor are more likely to own retirement accounts and risky assets. We also find that higher discount factor reduces the share of financial wealth in safe assets and increases the share of financial wealth in risky assets. These findings suggest that time preferences affect investment choices from safe assets toward other financial assets, all else equal.

This thesis mainly contributes to the field of structural econometrics using experimental data by analyzing the effects of uncertainty on decision-making and its implications for portfolio choice. First, it contributes to the literature which analyzes how uncertainty affects mea-surement of subjective probabilities (Chapter 1). Second, it contributes to a small amount of literature which analyzes how subjects facing ambiguity update their beliefs using new information (Chapter 1 and 2). Third, it significantly extends previous work on the joint determinants of portfolio choices (Chapter 2). Finally, it sheds light on the financial decision making of elderly people, particularly the impact of time preferences on portfolio composition (Chapter 3).

Overall, this dissertation indicates that individuals’ attitudes towards uncertainty captured by probability beliefs, loss, risk and ambiguity attitudes as well as time preferences are very heterogeneous, and only a small part of this heterogeneity can be explained by observable characteristics. The results from this dissertation suggest that uncertainty can have enduring economic effects for a sizeable share of people. As a result, findings from this dissertation could help both governments, firms and financial institutions better predict the impact of informa-tion disseminainforma-tion on behavior in various areas, including portfolio choice, voting behavior, schooling choice, and retirement decisions for which uncertainty is likely to be important.

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Bibliography

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Chapter 1

Reporting probabilistic expectations

with dynamic uncertainty about

possible distributions

Résumé

Dans cet article, nous étudions la manière dont les individus rapportent des distributions de probabilités d’événements futurs en présence d’incertitude et la manière dont ils mettent à jour cette incertitude à mesure que de nouvelles informations sont révélées sur le processus stochastique régissant ces événements. Plusieurs règles de rapport de distribution de proba-bilité et de mise à jour de l’incertitude sont présentées et notre analyse empirique se focalise sur la caractérisation de l’hétérogénéité de ces règles dans la population considérée. Les ré-sultats révèlent que deux règles sont principalement utilisées pour rapporter des distributions de probabilités: 65% des individus rapportent des distributions en pondérant correctement les distributions possibles par leur incertitude exprimée, alors que 22% rapportent des dis-tributions proches de la distribution qu’ils perçoivent comme la plus probable. Par contre, nous observons une hétérogénéité considérable dans la façon dont les individus mettent à jour leur incertitude. En général, les individus ont tendance à attribuer un poids relativement faible aux nouvelles informations, entrainant une persistance de l’incertitude pour un nombre important d’individus. Des analyses contrefactuelles suggèrent que cette persistance pourrait être présente dans des contextes non couverts par notre étude.

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Abstract

We study how experimental subjects report subjective probability distributions in the presence of ambiguity characterized by uncertainty over a fixed set of possible probability distributions generating future outcomes. Subjects observe draws from the true but unknown probability distribution generating outcomes at the beginning of each period of the experiment and state at selected periods a) the likelihoods that each probability distribution in the set is the true distribution, and b) the likelihoods of future outcomes. We estimate heterogeneity of rules used to update uncertainty about the true distribution and rules used to report distributions of future outcomes. We find that approximately 65% of subjects report distributions by properly weighting the possible distributions using their expressed uncertainty, while 22% of subjects report distributions close to the distribution they perceive as most likely. We find significant heterogeneity in how subjects update their expressed uncertainty. On average, subjects tend to overweight the importance of their prior uncertainty relative to new information, leading to ambiguity that is substantially more persistent than what would be predicted using Bayes’ rule. Counterfactual simulations suggest that this persistence will likely hold in settings not covered by our experiment.

JEL codes: D03, D84, C50

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1.1 Introduction

Measurement of subjective probabilities is increasingly used to provide quantitative evidence on how individuals form expectations, and to better predict various economic decisions. This approach has recently been applied in several fields, such as portfolio choices analysis ( Do-minitz and Manski,2011, 2007), schooling decisions (Wiswall and Zafar, 2015), social inter-actions (Bellemare et al., 2008), health decisions (Rappange et al., 2015), voting decisions (Delavande and Manski, 2015), auctions (Neri, 2015), and higher order beliefs (Manski and Neri, 2013).1 The general protocol used to measure subjective probabilities requires that respondents report (or approximate in the case of a continuous random variable) a unique well-defined probability distribution over events. The interpretation of these reported dis-tributions is however unclear when respondents face uncertainty about the distribution to report. For example, an investor may not have a clear distribution of market returns in mind for newly-introduced financial assets, while a student may think that the probability of finding a job for a given college major is between 10% and 30%. A consequence of distribution un-certainty is that the likelihoods of events may not be unique. Even so, respondents who take part in a protocol on measuring subjective distributions typically report a unique distribution even when events with ambiguous probabilities are plausible.

In this paper we design an experiment in which subjects are initially presented with three possible distributions of market returns and are told that one of these distributions has been selected to generate outcomes during the treatment. Our design contains two treatments and each treatment consists of 100 periods. Both treatments differ only with respect to the three possible distributions presented to each subject. We consider sets of distributions which are similar (treatment S) or dissimilar (treatment D), allowing for slow and fast resolution of un-certainty. In each treatment, subjects observe a new draw taken from the distribution selected to generate outcomes at the start of each period. They can also visualize the distribution of draws generated since the beginning of the treatment. At selected periods, subjects are re-quired to first express their distribution uncertainty by assigning likelihoods to each possible distribution. Subjects are next required to report a unique distribution, given the informa-tion available. The level of distribuinforma-tion uncertainty varies during a treatment according to

1

Early surveys of subjective probability measurement includevan Lenthe(1993);McClelland and Bolder (1994), andManski(2004).

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the draws observed and the rules used to update distribution uncertainty. Note that subjects face distribution uncertainty until they express knowing the true distribution (the distribution selected to generate outcomes) with probability 1.

Our empirical analysis focuses on estimating the sample distribution of reporting rules and updating rules. All rules are estimated at the individual level, allowing us to determine whether or not reporting rules and uncertainty updating rules are correlated in our sample. The reporting rules we consider differ with respect to the way subjects overweight or down-weight specific elements of their distribution uncertainty. Several notable reporting rules emerge as special cases of our specification. Subjects may, for example, report the distribution they perceive is most likely to be true. They can also report a properly-weighted average of all three possible distributions, where the weights reflect their distribution uncertainty. They can alternatively report a uniform distribution, assigning the same probability to each event. This would extend the 50% reporting rule believed to characterize belief elicitation over binary events in the presence of ambiguity (Bruine de Bruin et al., 2002) to other settings. Our econometric model is flexible enough to capture these and other related reporting rules. The updating rules we consider generalize Bayes’ updating rule, allowing subjects to differ with respect to the importance they attribute to their prior distribution uncertainty and to the new information made available about the stochastic process. More precisely, our model allows for “base rate neglect” (Kahneman and Tversky, 1973) and “conservatism” (Edwards, 1982) as possible deviations from Bayes’ rules which may characterize updating behavior. Related work includes Grether(1980) who estimated similar updating rules in the context of risk. We extend this approach to model updating of second-order probabilities in our experiment. Our work contributes to two different strands of literature. First, we contribute to the liter-ature analyzing how ambiguity affects the measurement of subjective probabilities. Existing research suggests ambiguity may generate specific reporting patterns, including additivity vi-olations (Offerman et al.,2009) and excessive use of uniform reporting rules (50% probability statements) for binary events (seeBruine de Bruin et al.(2002) ). These results suggest that ambiguity can reduce the quality and informativeness of reported distributions. However, none of these studies relate direct measurements of the level of ambiguity to the reported probability distributions. To our knowledge, our study is the first to present direct evidence on the reporting rules used by subjects when facing specific levels of ambiguity. What is more,

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the effects of ambiguity are possibly dynamic as uncertainty is reduced when outcomes can be observed over time. Observing outcomes provides the opportunity to learn about the under-lying outcome generating process and allows individuals to update their level of distribution uncertainty.

Second, our paper contributes to a small literature analyzing how subjects facing ambiguity update their (non-standard) beliefs using new information. Cohen et al. (2000) and Dominiak et al. (2012) conduct dynamic two-period Ellsberg urn experiments and compare Bayesian and Maximum Likelihood updating rules where ambiguity is modelled using a multiple prior framework. Qiu and Weitzel (2016) measure beliefs with multiple priors and preferences to discriminate experimentally between different models of decision making under ambiguity with multiple priors. In contrast, we focus on how subjects update their uncertainty over the possible (prior) distributions in a set, rather than how they update possible distributions in the set itself. As such, our approach is closest in spirit to updating of “second-order” probabilities defined in the smooth ambiguity model of Klibanoff et al. (2005).2 Finally, Baillon et al.

(2017), Moreno and Rosokha (2016) and Engle-Warnick and Laszlo (2017) study the effect of learning new information on ambiguity. Baillon et al. find that subjects facing ambiguity tend towards (without reaching) expected utility maximization as they learn new information about the underlying stochastic processes, the result of diminishing likelihood insensitivity. Moreno and Rosokha find evidence that subjects overweight new signals.

Our first main set of results concern the reporting rules used by subjects. We find that re-ported distributions are generally informative of the distribution uncertainty facing subjects. Two dominant reporting rules emerge from our analysis. First, we find that 65% of subjects report a distribution of future draws that properly weights their expressed distribution uncer-tainty. On the other hand, 22% of subjects tend to report distributions that overweight the distribution they perceive as most likely. We find that the tendency of overweighting selected possible distributions is relatively insensitive to the level of distribution uncertainty. Our sec-ond main set of results concerns how subjects update their distribution uncertainty over time. Model estimates suggest that on average, subjects update their distribution uncertainty by placing a relatively lower weight on new information relative to their most recently updated

2

Interpreting second-order probabilities as ambiguity is common but also criticized, see Trautmann and

van de Kuilen(2015a) for a discussion. Moreover,Qiu and Weitzel(2016) find the smooth model of ambiguity

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prior. It follows that distribution uncertainty is predicted to be persistent over time for a significant share of subjects. More precisely, simulations based on the estimated parameters show that only about 33% of the subjects converge before period 100. These results suggest that ambiguity can have enduring economic effects for a sizeable share of individuals. We also conduct counterfactual simulations to assess how likely the observed persistence in our data will be in settings not covered by our experiment (fewer and less precise possible distribu-tions). Our simulations suggest the persistence of ambiguity is likely to hold more generally. Finally, we find that reporting rules and updating rules are correlated at the individual level. In particular, subjects who update by placing a relatively higher weight on recent information are relatively more prone to report the most likely possible distribution.

The paper is organized as follows. Section1.2presents the experimental design and procedures. Section 1.3 presents the data. Section 1.4 presents our models of uncertainty updating and reporting rules. Section 1.5presents our estimation results. Section1.6 concludes.

1.2 Experiment

We use a within-subjects design with two treatments, denoted S (for similar distributions) and D (for dissimilar distributions). The order of treatments was randomly determined and varied across subjects. Figure 1.1 presents the main decision screen in treatment D. The left hand part of the screen presents three ”possible” continuous distributions, all summarized over 6 equal-length intervals {] − 30, −20], ] − 20, −10], ] − 10, 0], ]0, 10], ]10, 20], ]20, 30]}. Subjects were instructed to treat the intervals as representing negative and positive profits of a company. Subjects were presented with the same three possible distributions, but the order of presentation on screen (top, middle, bottom) was randomly varied across subjects. Subjects were informed that one of the three possible distributions had been randomly selected to be the true distribution – the distribution used to generate draws for all periods of the treatment. Randomisation of the true distribution was at the subject level. As a result, subjects were assigned different true distributions from the three possible distributions.3

Each treatment consisted of 100 periods. In the initial period (period 0), subjects observe

3

In practice each possible distribution had an equal chance of being designated the true distribution. Participants were not informed about the probability with which one of them would be drawn. This was to ensure genuine distribution uncertainty.

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Figure 1.1: Screenshot of main decision screen : treatment D

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the three possible distributions and are then asked to complete two tasks. The first task (the uncertainty task) required subjects to express their distribution uncertainty by stating the likelihood that each possible distribution is the true distribution. Subjects entered this information in the appropriate boxes appearing on the middle part of the screen. The computer program made sure the three reported likelihoods summed up to 1. The second task (the reporting task) required that subjects report a unique probability distribution concerning the likelihood that future draws fall within each of six intervals. The six intervals used in the reporting task coincided with the intervals over which all three possible distributions were defined. Subjects had to state for every interval the likelihood with which they expected the next period’s profit to fall within the boundaries. Again, the computer program made sure the six probabilities summed up to 1.4

Once the initial period was completed, subjects could visualize a sequence of 9 draws (1 draw per period for 9 periods) taken from the true distribution. Draws appeared sequentially on the screen. A separate histogram appeared on screen plotting the distribution of all draws taken from the true distribution since period 1. This histogram was automatically updated after each draw and served as a visual tool to help subjects to represent the empirical distribution of all realized draws since the treatment began. Relative to period 0, the information set of each subject contains 9 new draws from the true distribution at the beginning of period 10. Subjects were asked to repeat the uncertainty and reporting tasks. Subjects were then asked to visualize ten additional new draws (in addition to all previous draws) from the true distribution before being asked to repeat the uncertainty and the reporting tasks at the beginning of period 20. The treatment continued until the last period (period 100), before which subjects were asked for a last time to complete the uncertainty and reporting task. Subjects could at any time visualize the history of past draws when clicking a button “Historique” on the screen.

Our design differs in several points from that of other laboratory experiments investigating ambiguity, e.g., Qiu and Weitzel(2016) or Moreno and Rosokha (2016). First, instead of the traditional design using urns filled with differently colored balls of unknown proportions, ambi-guity is defined over a continuous random variable. Moreover, we limit the number of possible priors and allow for learning over 100 periods. Second, we elicit beliefs directly, whereas the literature almost exclusively uses observed choice data to recover underlying beliefs. Finally,

4

Graphical interfaces have been found to be effective in helping subjects report accurate probability dis-tributions. SeeGoldstein and Rothschild(2014) for recent experimental evidence.

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our design creates a natural transition from an ambiguous to a risky environment as subjects can learn about the true distribution used to generate draws.

The data contain the expressed uncertainty and reported distributions of each subject at all 11 periods where this information was elicited (periods 0, 10, 20, . . . , 100). We also observe for each subject the true distribution and the corresponding draws generated during the treat-ment. Treatment S differed from treatment D only with respect to the possible distributions, presented in Figure1.1and1.2. It is expected that the greater similarity between the possible distributions in treatment S will require a higher number of draws before subjects identify the true distribution with probability 1.

The experiment was computerized using z-Tree (Fischbacher, 2007) and was conducted in January 2013 at the Laboratory of Experimental Economics at Université Laval. We recruited subjects from a database that contains approximately 400 persons interested in participating in economic experiments. Subjects were mainly students enrolled in social science programs. We conducted 5 sessions with an average of 18 subjects per session and a total number of 89 subjects. Additional sessions were conducted in December 2017 and January 2018 with 34 subjects replicating the experiment described above but using a quadratic scoring rule to incentivize the elicitation of the expressed uncertainty about the true distribution. The Appendix (page 36) presents all tables and figures of this paper separately when incentives are used or not. We find that results are largely robust to the use of incentives. The Appendix also presents formal regression analysis confirming the lack of significant statistical differences between the behavioral parameters analyzed in this paper when incentives are used or not. Data from all these sessions are consequently pooled and analyzed below.5

At the beginning of the experiment, subjects watched video instructions presenting the deci-sion screens and the tasks they would be required to complete. They could also access online instructions and a calculator by clicking on corresponding buttons on the screen.6 At the end of the experiment, subjects were asked to complete a questionnaire with standard demo-graphic questions. They were also asked to explain how they made their decisions during the

5

These results add to recent work suggesting that non-incentivized belief elicitation provides reliable data

(seeArmantier and Treich(2013) andTrautmann and van de Kuilen(2015b)).

6_{We added a treatment in which participants could hit a button “I don’t know” when they could not or}

did not want to express their judgement concerning the true distribution or the prediction task. This option was used only ten times (0.56%) for all elicitation periods and all subjects, and we did not find differences in responses between both treatment groups, and thus we will not discuss this treatment in great detail.

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experiment. Subjects completed a numeracy questionnaire aimed at measuring their under-standing of probabilities. The experiment lasted on average 90 minutes. For the non-incentive treatment, subjects received $35 for their participation in the experiment, while the average reward for the incentive treatment was $32.95. All subjects received a $5 show-up fee, and an additional $5 if they answered the numeracy questionnaire.

1.3 Data

In this section we present a descriptive analysis of the data. Let T = {0, 9, 19, . . . , 99} denote the set of periods where subjects completed the uncertainty and reporting tasks. Let µj_t denote the probability placed on distribution j being the true distribution at period t ∈ T . Distribution uncertainty persists until a subject reports µj_t = 1 for a given j. Let πj_k denote the probability that a draw falls in interval k given possible distribution j, withP6

k=1π

j

k= 1.

Finally, let πr_k,t denote a subject’s reported probability that a draw falls in interval k at time t. Note that reported distributions are indexed by t to reflect the fact that subjects could modify them at each t ∈ T .

We start by presenting a brief descriptive analysis of reported distributions. We find very little evidence that subjects use a uniform reporting rule in our experiment. In particular, 5.7% of reported distributions in period 0 and 0.9% of all reported distributions in the experiment are approximately uniform – with all six reported probabilities jointly falling within 1/6 ± 0.01. Uniform reporting rules with binary events (assigning 0.5 to each event) are widely believed to reflect expressions of ambiguity (seeBruine de Bruin et al.(2002)). Our data suggests that this rule is not prevalent in our context. One possible reason may be that 1/6 (contrary to 0.5) is complicated to report. Alternatively, our design involves a limited form of distribution uncertainty with at most three probabilities that a given draw falls in each of the 6 intervals. Our data does not rule out uniform reporting in situations with higher levels of ambiguity. One natural benchmark is to assume that subjects report distributions by weighting each possible distribution using their expressed uncertainty µj_t. Let |∆| =P6

k=1|πrk,t− P3 j=1µ j tπ j k|

denote a simple measure of the difference between a reported distribution and this benchmark. It is insightful to relate |∆| to the level of distribution uncertainty facing subjects. Subjects who express knowing the true distribution set µj_t = 1 for the possible distribution they believe

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0 .1 .2 .3 .4 .5 .6 .7 .8 Fraction 0 .5 1 1.5 2 |∆| max {µjt} < 0.6 0 .1 .2 .3 .4 .5 .6 .7 .8 Fraction 0 .5 1 1.5 2 |∆| 0.6 ≤ max {µjt} < 0.9 0 .1 .2 .3 .4 .5 .6 .7 .8 Fraction 0 .5 1 1.5 2 |∆| max {µjt} ≥ 0.9

Figure 1.3: Distribution of |∆| (difference between reported distribution and benchmark) for values of max{µj_t} < 0.6 (left panel), for values of 0.6 ≤ max{µj_t} < 0.9 (middle panel), and for values of max{µj_t} ≥ 0.9 (right panel).

to be the true distribution, and µj_t = 0 for the other two distributions. In other cases, µj_t 6= 1 for j = 1, 2, 3 and subjects face distribution uncertainty. A simple measure of distribution un-certainty is given by max{µj_t}. Note that max{µj_t} = 1 in absence of distribution uncertainty. Low values of max{µj_t} (e.g., max{µj_t} = 0.34) on the other hand imply that all possible distributions are perceived as likely.

Figure 1.3 presents the distribution of |∆| for three ranges of values of max{µj_t}. The first panel focuses on observations where max{µj_t} < 0.6. This corresponds to cases where subjects face relatively high distribution uncertainty and do not assign a value of µj_t greater than 0.6 to any of the three possible distributions. The second panel conditions on values of 0.6 ≤ max{µj_t} < 0.9. The last panel on the right corresponds to observations where subjects face low distribution uncertainty, with one possible distribution having µj_t ≥ 0.9. We find

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that deviations from the benchmark rule (i.e., |∆| > 0) are important in the first two panels, suggesting that reporting rules other than the benchmark rule are used. However, deviations from our benchmark are less important when distribution uncertainty is low (right panel). There, we find that close to 50% of all reported distributions satisfy |∆| = 0. The concentration of |∆| (close to 0) in the last panel reflects the fact that exact weighting using µj_t becomes natural in cases where max{µj_t} is close to 1. Then, subjects simply report the corresponding distribution (e.g., π_k,tr = π_kj for all k when µj_t = 1).

We next present descriptive evidence on uncertainty updating. The top two graphs of Figure

1.4 present various quantiles of the distribution of the weight µj_t assigned to the true distri-bution over all periods of the experiment in each treatment. As expected, the distridistri-bution of µj_t attached to the true distribution is centered around 1/3 in the early periods of both treatments, when subjects have little information about the true distribution. The quantile functions generally increase with the number of periods, an indication that subjects eventu-ally assign a higher likelihood to the true distribution as information is accumulated. Yet, there remains a sizeable fraction of subjects who continue to report distribution uncertainty at the last period – at least 25 percent of subjects place less than a 50% likelihood on the true distribution at the last period of each treatment. One way to get further insights in the evolution of distribution uncertainty over time is to contrast the likelihoods assigned to the true distribution with some standard benchmarks such as Bayes’ rule. The bottom two graphs of Figure 1.4 plot the quantile functions of the predicted likelihoods assigned to true distributions computed for each subject using Bayes’ rule and assigning a likelihood of 1/3 to all possible distributions in period 0. We find very little heterogeneity across subjects, with more than 75% of subjects assigning a value µj_t close to 1 to the true distribution by period 40. Again, we find little difference across treatments. The differences with the observed data reported in the top graphs of Figure 1.4suggest that subjects use other updating rules than Bayes’ rule, and assign prior likelihoods to each possible distributions other than 1/3.

The previous descriptive analysis provides some interesting insights. First, subjects appear to use different reporting rules. Second, heterogeneity of µj_t and its evolution over time can potentially be explained by the heterogeneity of the priors at the start of each part (seen from the left panel of Figure1.4), by the heterogeneous updating rules used by subjects during the experiment, and by the different sequences of draws observed by each subject. The econometric

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0 .2 .4 .6 .8 1 0 20 40 60 80 100 0 20 40 60 80 100 Treatment S Treatment D 10 % 25 % 50 % 75 % 90 %

Weight on true distribution

Periods 0 .2 .4 .6 .8 1 0 20 40 60 80 100 0 20 40 60 80 100 Treatment S Treatment D 10 % 25 % 50 % 75 % 90 %

Weight on true distribution

Periods

Figure 1.4: Quantile curves (10%, 25%, 50%, 75%, 90%) of likelihood µj_t assigned to the true distribution over all 100 periods for treatment S (left graphs) and treatment D (right graphs) using the observed data (top graphs) and the Bayesian simulations with initial likelihoods set to 1/3 for all three possible distributions (bottom graphs).

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model presented in the next section will allow us to disentangle the role played by each of these elements.

1.4 Model

In this section, we model how subjects report distributions of future draws given the available information and how they update their distribution uncertainty. We focus on parsimonious reporting and updating rules which can be estimated at the individual level.

1.4.1 Modelling reporting behavior

Subjects were asked to report at each t ∈ T a distribution of future draws. We model the relationship between the reported distributions {πr_k,t : k = 1, 2, . . . , 6}, the possible distribu-tions {π_kj : j ∈ {1, 2, 3}, k = 1, 2, . . . , 6}, and the subjective uncertainty expressed about the true distribution, captured by {µj_t : j ∈ {1, 2, 3}}. There are different intuitive rules subjects can use to report a distribution of future draws given the available information. We describe the most intuitive rules, summarized in Table 1.3(page 33). First, subjects can weight pos-sible distributions using the uncertainty they expressed µj_t. This requires computing for each interval k π_k,tr = 3 X j=1 µj_tπ_kj for all k (1.1)

We refer to this benchmark reporting rule as “Empirical mean,” as it uses unbiased subjective uncertainty weights. Second, subjects may report a distribution amplifying their expressed uncertainty. For example, a subject may report the possible distribution perceived as most likely (i.e., the distribution associated with the mode of µj_t), leading to the following reporting rule

π_k,tr = πj∗_k for all k when µj∗_t = max n

µj_t o

(1.2) This is the “Modal” reporting rule. In cases where µj_t = max

n µj_t

o

holds for two potential distributions (say µj_t = 0.45 for j = 1, 2 and µ3_t = 0.1), subjects could report one of the two distributions, or average both distributions. A third reporting rule is given by

π_k,tr =

3

X

j=1

1/3 · π_kj for all k (1.3)

with subjects computing for each interval the arithmetic mean over the corresponding proba-bilities of each possible distribution, ignoring their expressed uncertainty. Hence, we refer to

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this rule as “Arithmetic mean.” Finally, a fourth reporting rule is given by the “Uniform" rule

π_k,tr = 1/6 (1.4)

where the same probability is assigned to each of the 6 intervals. As discussed in Section 1.3, there is very little support in the data for this rule and will not be considered in our analysis. Our analysis is based on the following specification

πr_k,t= 3 X j=1 wµj_tπj_k (1.5) where w µj_t

represents a function of the expressed uncertainty µj_t. Reporting rules (1.1) to (1.3) have the structure in (1.5) for given wµj_t that we model as a linear combination of the prior µj_t and the weighted prior ω(µj_t).

Deviations from the “Empirical mean” benchmark rule are captured by the weighting function ωµj_t,7 ω µj_t = e−β(− ln(µjt)) α (1.6)

This simple function is flexible enough to capture the reporting rules we consider in this paper. Parameter α controls the type of weighting (overweighting or down-weighting) used by a subject. When 0 < α < 1, ω(µ) has an inverse S shape, implying low expressed uncertainty is overweighted and down-weighted if it is high. The converse holds if α > 1 with S-shaped ω(µ). Parameter β controls the location of the point, where the change from overweighting to down-weighting or the change from down-weighting to overweighting is made. This point, named the “Switching point” is given by µ∗= e−(β1)

1 α−1

, α 6= 1 (seeAl-Nowaihi and Dhami (2010)). For example, if 0 < α < 1, subjects overweight expressed uncertainty when µj_t < µ∗. Figure1.5 (left graph) plots ω

µj_t

for three different combinations of (α, β). The function is linear when (α = 1, β = 1), which implies that w

µj_t = µj_t for all j – subjects use the benchmark rule for any level of expressed uncertainty (independent of c_t). Values (α = 10, β = 4) approximate reporting rule (1.2) – the reported distribution overweights the most likely possible distributions. Reporting rule (1.3) on the other hand is approximated by values of (α = 0.01, β = 1.1). These values generate a flat weighting function with subjects

7

This function was initially proposed byPrelec(1998) to model how individuals overweight or down-weight objective probabilities, a central element of prospect theory. We use it differently than originally intended to weight subjective probabilities that a person associates with a particular prior distribution.

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(α=1,β=1) (α=0.01,β=1.1) (α=10,β=4) 0.0 0.2 0.4 0.6 0.8 1.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 µ_tj w ( µt j) (α=1,β=1) (α=0.01,β=1.1) (α=10,β=4) γ= −10 γ= 1 γ= 10 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 max {µ t j_} ct γ= −10 γ= 1 γ= 10

Figure 1.5: Left panel plots the probability measure ω(µj_t) against µj_t for α = 1, β = 1 (full line), α = 10, β = 4 (dashed line), and α = 0.01, β = 1.1 (dotted line). Right panel plots c_t against max{µj_t} for γ = −10 (dotted line), γ = 1 (full line), and γ = 10 (dashed line).

reporting for each interval the arithmetic mean across the possible distributions. Other values of (α, β) generate different reporting rules, some of which will be discussed in Section 1.5. Descriptive evidence provided in Section 1.3 suggests that deviations from the “Empirical mean” benchmark reporting rule (1.1) tend to vary across periods. Notably, we observe fewer deviations from this when subjects face lower distribution uncertainty (i.e., at higher values of maxnµj_to). We capture these features in the data by specifying

wµj_t= ctµjt+ (1 − ct) ω

µj_t (1.7)

where the time-varying function c_t∈ [0, 1] balances the expressed uncertainty µj_t and possible deviations from µj_tcaptured by ω

µj_t

. Values of c_tclose to 1 correspond to reporting behavior using the “Empirical mean” benchmark rule, with little weight placed on possible distortions captured by ω

µj_t

. Values of c_t close to 0 correspond to reporting behavior using possible distortions captured by ωµj_t. We specify

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ct=

maxnµj_toγ− (1/3)γ

1 − (1/3)γ (1.8)

The parameter γ controls how quickly ct increases from 0 to 1 as distribution uncertainty

decreases (maxnµj_to → 1). This specification allows reporting behavior in periods with relatively high distribution uncertainty (characterized by low values of max

n µj_t

o

) to differ from reporting behavior in periods with low levels of distribution uncertainty (high values of max

n µj_t

o

). Subjects with very negative values of γ rely on deviations given by ω

µj_t

only when maxnµj_to is small and subjects are still unsure about the true distribution. Subjects with very positive values of γ on the other hand rely on ω

µj_t

until uncertainty about the true distribution is almost resolved – c_tfor example remains low even when maxnµj_tois equal to 90%.

The model above is parsimonious while allowing reporting behavior to vary across periods and level of distribution uncertainty through c_t. This parameter also prevents estimates of the time-invariant parameters α and β to be biased towards 1. This bias occurs when subjects get rid of their ambiguity (µj_t = 1) and simply start reporting (as they should) the distribution they believe is the correct one. These reported distributions are consistent with (α = 1, β = 1) values, but these do not capture weighting under ambiguity as ambiguity is no longer present when µj_t = 1. Pooled estimation of (α, β) using all periods would thus err towards (α = 1, β = 1) for subjects who quickly get rid of their ambiguity, distorting how they report distributions in the presence of ambiguity. Our specification adds the parameter c_tto control for this issue. Alternatively, it would be possible to allow (α, β) to vary across periods and values of µj_t, perhaps with period specific parameters. This would require a different specification, one that likely involves possibly many more parameters to estimate at the individual level.

Reporting rules estimation approach

We estimate (α, β, γ) for each subject using a two-step approach. In the first step, we estimate the weights w_bi,t(i = 1 , 2 , 3) that a subject used to construct their reported distribution by

solving the following problem for each period t ∈ T ,

min b w1,t,bw2,t,wb3,t 6 X k=1 πr_k,t−w_b1,t· πk1−wb2,t· π 2 k−wb3,t· π 3 k 2

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This yields estimated weights assigned to each possible distribution consistent with the re-ported distribution at time t. Note that these weights may differ from the expressed uncer-tainty µj_t. We estimate (α, β, γ) in the second step by solving8

min (α,β,γ) X t∈T 3 X j=1 b wj,t− w µj_t 2

We restrict estimates (α, β, γ) to be the same for treatments S and D, allowing us to pool the data from both treatments and increase statistical precision. Relaxing this restriction and estimating separate parameters for both treatments yields similar results.

Identification of the model parameters relies on differences in reporting behavior across the different periods. In particular, reported distributions in early periods (characterised by low values of max{µj_t} and ct) identify the parameters (α, β). Reported distributions and expressed

uncertainty in later periods identify γ (and thus c_t).

1.4.2 Modelling distribution uncertainty updating

We model how subjects update their expressed uncertainty µj_t given the draws observed during the experiment in the following way. Let d_k,t−1 denote 6 mutually-exclusive binary variables (with P6

k=1dk,t−1 = 1), each taking a value of 1 if the draw of the previous period falls

in interval k, and 0 otherwise. We assume that subjects update their uncertainty using the following modified Bayesian rule

µj_t = µj_t−1 δ1 P6 k=1dk,t−1π_k,t−1j δ2 P3 j=1 µj_t−1δ1P6 k=1dk,t−1πjk,t−1 δ2 (1.9)

for j = 1, 2, 3. The parameters (δ₁, δ2) control the weight placed by subjects on their prior

uncertainty µj_t−1 and on the most recent information (draw). Subjects relying predominantly on the most recent draw would place a relatively lower weight on their prior (δ2> δ1), leading

to behavior originally referred to as base rate neglect (see Kahneman and Tversky (1973)). Conversely, (δ2 < δ1) would imply insensitivity to recent information and a reluctance to

8

We also considered a one-step approach, directly estimating (α, β, γ) by solving the following minimization problem min (α,β,γ) X t∈T 6 X k=1 πrk,t− w µj_tπ1k− w µj_tπ2k− w µj_tπk3 2