Belief updating: does the 'good-news, bad-news' asymmetry extend to purely financial domains?

Barron, Kai Article — Published Version Belief updating: does the 'good-news, bad-news' asymmetry extend to purely financial domains? Experimental Economics Provided in Cooperation with: WZB Berlin Social Science Center Suggested Citation: Barron, Kai (2021) : Belief updating: does the 'good-news, bad-news' asymmetry extend to purely financial domains?, Experimental Economics, ISSN 1573-6938, Springer, Dordrecht, Vol. 24, Iss. 1, pp. 31-58, https://doi.org/10.1007/s10683-020-09653-z This Version is available at: https://hdl.handle.net/10419/218847 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by/4.0/ Vol.:(0123456789) Experimental Economics (2021) 24:31–58 https://doi.org/10.1007/s10683-020-09653-z 1 3 ORIGINAL PAPER Belief updating: does the‘good‑news, bad‑news’ asymmetry extend topurely financial domains? KaiBarron1 Received: 20 February 2018 / Revised: 17 March 2020 / Accepted: 25 March 2020 / Published online: 13 April 2020 © The Author(s) 2020 Abstract Bayes’ statistical rule remains the status quo for modeling belief updating in both normative and descriptive models of behavior under uncertainty. Some recent research has questioned the use of Bayes’ rule in descriptive models of behavior, presenting evidence that people overweight ‘good news’ relative to ‘bad news’ when updating ego-relevant beliefs. In this paper, we present experimental evidence testing whether this ‘good-news, bad-news’ effect is present in a financial decision making context (i.e. a domain that is important for understanding much economic decision making). We find no evidence of asymmetric updating in this domain. In contrast, in our experiment, belief updating is close to the Bayesian benchmark on average. However, we show that this average behavior masks substantial heterogeneity in individual updating behavior. We find no evidence in support of a sizeable subgroup of asymmetric updators. Keywords Economic experiments· Bayes’ rule· Belief updating· Belief measurement· Proper scoring rules· Motivated beliefs JEL Classification C11· C91· D83 1 Introduction Throughout our lives, we are constantly receiving new information about ourselves and our environment. The way that we filter, summarize and store this new information is of critical importance for the quality of our decision making. Theories of human behavior under dynamic uncertainty are therefore enriched by an Electronic supplementary material The online version of this article (https ://doi.org/10.1007/s1068 3-020-09653 -z) contains supplementary material, which is available to authorized users. * Kai Barron kai.bar[email protected] 1 WZB, Reichpietschufer 50, 10785Berlin, Germany 32 K.Barron 1 3 understanding of how individuals process new information. Economists typically write down models where information is summarized in the form of probabilistic ‘beliefs’ over states of the world, and updated upon receipt of new information according to Bayes’ rule. However, the assumption that individuals process information in a statistically accurate way has increasingly been called into question, with many studies documenting systematic deviations.1 One important strand of this literature examines whether belief formation and updating is influenced by the affective content of the new information, i.e. whether individuals update their beliefs symmetrically in response to ‘good-news’ and ‘bad-news’ (see, for example, Eil and Rao 2011; Ertac 2011; Möbius etal. 2014; Coutts 2019). This literature tests an implicit assumption of Bayesian updating, namely that the only object that is relevant for predicting an individual’s belief is her information set—her beliefs are completely unaffected by the prizes and punishments she would receive in different states of the world. This fundamental assumption—that people update their beliefs symmetrically—is important because it underpins the entire orthodox approach to modelling uncertainty. In this paper, we test whether individuals update their beliefs symmetrically in response to ‘good-news’ and ‘bad-news’ when states differ only in the financial rewards they offer. To do this, we conduct a laboratory experiment in which subjects complete a series of belief updating tasks. In each task, there are two possible states of the world. Subjects are told the prior probabilities of each state, and then receive a sequence of partially informative binary signals. We elicit subjects’ beliefs after each signal. The financial rewards associated with the two states are either identical (symmetric), or different (asymmetric). This experimental design allows us to compare posterior beliefs insituations where the entire information set is held constant, but the rewards associated with the states of the world are varied. For example, we can compare how two groups of individuals revise their beliefs when both groups share the same prior belief and receive an equally informative signal, but for one group of individuals the signal is ‘good news’ and for the other the signal constitutes ‘bad news’. We can also conduct a similar exercise for a single individual, comparing contexts where she has identical priors and signals, but the signal constitutes ‘good news’ in one context, and ‘bad news’ in the other. These comparisons provide a clean test of the asymmetric updating hypothesis. The experiment considers belief updating in two contexts. In the symmetric treatment, subjects have an equal stake in each of the underlying states. In the two asym - metric treatments, a larger bonus payment is paid in one of the two states of the world. This design permits two separate tests of the asymmetric updating hypothesis. First, we can compare how the same individual responds to ‘good-news’ and ‘badnews’ within the asymmetric treatments. Second, we can conduct a between-subject 1 Some notable examples include the representativeness bias (Grether 1978, 1980, 1992), cognitive dissonance (Akerlof and Dickens 1982), anticipatory utility (Loewenstein 1987; Caplin and Leahy 2001; Brunnermeier and Parker 2005), base rate neglect (Kahneman and Tversky 1973; Holt and Smith 2009), confirmatory bias (Rabin and Schrag 1999), motivated belief formation (Benabou and Tirole 2002), correlation neglect (Enke and Zimmermann 2019), and selection neglect (Esponda and Vespa 2018; Barron etal. 2019; Enke 2019). 33 1 3 Belief updating: does the‘good-news, bad-news’ asymmetry… comparison of belief updating in the symmetric treatment and asymmetric treatments. Each individual in our experiment faces only one incentive environment. However, since we exogenously endow participants with a prior over the states of the world, we are able to repeat the exercise several times for each individual and study how they update from each of five different priors, p0 , chosen from the set {1 6 , 2 6 , 3 6 , 4 6 , 5 6} . The experimental design and analysis aim to address several challenges that are present when studying belief updating in the presence of state-dependent stakes. First, we use exogenous variation in the priors to ensure that the estimates are robust to the econometric issues that arise when a right-hand side variable (i.e. the prior) is a lagged version of the dependent variable (i.e. the posterior). Second, we avoid a second type of endogeneity issue, which arises when the underlying states are defined as a function of some personal characteristic of the individual (e.g. her relative IQ) that might also be related to how she updates (see Online Appendix C for further details). Third, we measure the influence that hedging has on belief elicitation when there are state-dependent stakes. Furthermore, we conduct several exercises to correct our estimates for this hedging influence—both experimentally, and econometrically. Fourth, our experimental design allows us to study belief updating from priors spanning much of the unit interval. Importantly, averaging across all subjects, the design generates a balanced distribution of ‘good’ and ‘bad’ signals. Our results show no evidence in favor of asymmetric updating in response to ‘good-news’ in comparison to ‘bad-news’ in the domain of financial outcomes. Several robustness exercises are carried out in support of this conclusion. Furthermore, we find that average updating behavior is well approximated by Bayes’ rule.2 This average behavior masks substantial heterogeneity in updating behavior at the individual level, but we find no evidence in support of a sizeable subgroup of asymmetric updators. The evidence reported here contributes to the recent literature studying the asymmetric updating hypothesis across different contexts (e.g. Eil and Rao 2011; Ertac 2011; Grossman and Owens 2012; Mayraz 2013; Möbius etal. 2014; Kuhnen 2015; Schwardmann and Vander Weele 2019; Gotthard-Real 2017; Charness and Dave 2017; Heger and Papageorge 2018; Buser etal. 2018; Coutts 2019). The results in this literature thus far are highly heterogeneous, with some papers finding a greater responsiveness to good-news, some a greater responsiveness to bad-news, and some no evidence of an asymmetry. The objective of the experiment discussed in this paper is not to isolate the contextual features that activate and deactivate asymmetric updating; rather the objective is to provide a clean test of the asymmetric updating hypothesis for contexts in which states differ only in their material rewards and have no direct ego-relevance. The reason for focusing on this context (where states yield different financial outcomes) is that it characterizes a large class of economically important decision problems under uncertainty—most economic models with uncertainty fit this description. This paper does not disentangle the reasons for why updating is asymmetric in some settings but 2 This is in line with, e.g. Holt and Smith (2009) and Coutts (2019), who find that average posteriors across all individuals are well approximated by posteriors obtained by applying Bayes’ rule. 34 K.Barron 1 3 not in others. Instead, by providing a clean test of the asymmetric updating hypothesis for one specific domain, the paper contributes to the growing collective body of evidence pertaining to asymmetric updating across a range of contexts. There are several candidate contextual and experimental design factors that could be generating the heterogeneous results observed in the literature as a whole. Section7 below offers a discussion of some of these candidate explanations, and asks whether the existing body of evidence can help us to detect a systematic pattern that organizes the results. The remainder of the paper proceeds as follows. Section2 outlines the theoretical framework, Sect.3 details the experimental design, Sect.4 provides some descriptive statistics, Sect.5 presents the empirical specification, Sect.6 discusses the related literature and Sect.7 concludes. 2 Theoretical framework andhypotheses The following section discusses a simple framework for belief updating that augments the standard normative Bayesian benchmark to allow for several commonly hypothesized deviations from Bayes’ rule. This framework is borrowed from Möbius etal. (2014) and is commonly used for analyzing belief updating descriptively. The framework provides a basis for the empirical approach that we will use to test whether agents update their beliefs asymmetrically in response to ‘good-news’ and ‘bad-news’. 2.1 A simple model ofbelief formation We consider a single agent who forms a belief over two states of the world, 𝜔∈{A,B} , at each point in time, t. One of these states of the world is selected by nature as the ‘correct’ (or ‘realized’) state, where state 𝜔=A is chosen with prior probability p0 (known to the agent). The agent’s belief at time t is denoted by 𝜋t∈[0, 1] , where 𝜋t is the agent’s belief regarding the likelihood that the state is 𝜔=A and 1−𝜋t is the agent’s belief that the state is 𝜔=B . In each period, the agent receives a signal, st∈{a,b} , regarding the state of the world, which is correct with probability q ∈( 1 2 ,1 ) . In other words, p(a | A)=p(b | B)=q> 1 2 . The history, Ht , is defined as the sequence of signals received by the agent in periods 1, …,t , with H0=� . Therefore, the history at time t is given by Ht=(s1, ..., st) . To study how individuals update their beliefs, we follow Möbius etal. (2014) in considering the following model of augmented Bayesian updating: The parameters 𝛿 , 𝛾a and 𝛾b can be interpretted as follows. If 𝛿=𝛾a=𝛾b=1 then the agent updates her beliefs according to Bayes’ rule. The 𝛿 parameter captures the degree to which the agent’s prior affects her updating. For example, if 𝛿>1 then (1) logit (𝜋t+1)=𝛿logit (𝜋t)+𝛾alog ( q 1 − q) ⋅1(st+1=a)−𝛾blog ( q 1 − q) ⋅1(st+1=b ) 35 1 3 Belief updating: does the‘good-news, bad-news’ asymmetry… the agent displays a confirmatory bias3, whereby she is more responsive to information that supports her prior. In contrast, if 𝛿<1 she is more responsive to information that contradicts her prior (i.e. base rate neglect4). The former would predict that beliefs will polarize over time, while the latter would predict that over time beliefs remain too close to 0.5. The parameters, 𝛾a and 𝛾b capture the agent’s responsiveness to information. If 𝛾a=𝛾b<1 then the agent is less responsive to information than a Bayesian. And if 𝛾a=𝛾b>1 , then she is more responsive than a Bayesian. For example, if 𝛾a =2 , then whenever the agent receives a signal st=a , she updates her belief exactly as much as a Bayesian would if he received two a signals, st={a,a} . The interpretation of the parameters is summarized in the first five rows of Table1. 2.1.1 Affective states In the preceding section, the affect or desirability of different states of the world played no role. However, in most situations in which individuals form beliefs, there are some states that yield an outcome that is preferred to the outcomes associated with other states—i.e. there are good and bad states of the world To allow for the possibility that individuals update their beliefs differently in response to good-news in comparison to bad-news, we relax the assumption that belief updating is orthogonal to the affect of the information.5 To do this, assume Table 1 Interpretation of parameters: a summary Belief updating distortion Parameter values Bayesian updating 𝛿=1 , 𝛾a =1 and 𝛾b=1 Confirmatory bias 𝛿>1 Base rate neglect 𝛿<1 Conservatism 𝛾j<1 for ∀j∈{a,b} Overresponsiveness 𝛾j>1 for ∀j∈{a,b} Optimistic updating (in ASYMMETRIC) 𝛾a>𝛾 b Pessimistic updating (in ASYMMETRIC) 𝛾a<𝛾 b 3 For a detailed discussion of the confirmatory bias, see Rabin and Schrag (1999). Essentially, it is the tendency to weight information that supports one’s priors more heavily than information that opposes one’s priors. In this case, when one’s prior regarding state 𝜔=A is greater than 0.5, i.e. 𝜋t > 0.5 , a participant who is prone to the confirmatory bias weights signals that support state 𝜔=A more heavily than signals that support state 𝜔=B ; and vice versa when her prior suggests state 𝜔=B is more likely, i.e. 𝜋t < 0.5 . 4 One can think of base rate neglect in this context as the agent forming her beliefs as if she attenuates the influence of her prior belief when calculating her posterior—i.e. acting as if her prior was closer to 0.5 than it actually was. 5 To avoid ambiguity, in the discussion below, the term ‘preference’ is usually used to refer to preferences over sure outcomes—never to a preference ordering over lotteries. We will also sometimes refer to ‘preferring’ one state of the world to another. This simply captures the idea that an individual prefers the realization of a state in which a good outcome is realized. 36 K.Barron 1 3 that each of the two states of the world is associated with a certain outcome—i.e. in state 𝜔=A , the agent receives outcome xA , and in state 𝜔=B , she receives xB . There are now two belief updating scenarios: • Scenario 1 ( symmetric ): the agent is indifferent between outcomes (i.e. xA∼xB ); and • Scenario 2 ( asymmetric ): the agent strictly prefers one of the two outcomes (i.e. xA≻xB ). The question of interest is whether the agent will update her beliefs differently in the symmetric and asymmetric scenarios. Under the assumption that the agent’s behavior is consistent with the model described above in Eq.1, this involves asking whether the parameters 𝛿 , 𝛾a and 𝛾b , differ between the two contexts. To guide our discussion, we consider the following two benchmarks. The first natural benchmark is Bayes’ rule, which prescribes that all three parameters equal 1 in both the symmetric and asymmetric contexts—statistically efficient updating of probabilities is unaffected by state-dependent rewards and punishments. According to Bayes’ rule, news is news, independent of its affective content. Hypothesis 1 (Bayesian updating) Individuals update their beliefs according to Bayes’ rule. Therefore, 𝛿=1 , 𝛾a =1 and 𝛾b=1 in both symmetric and asymmetric scenarios. The second benchmark that we consider is provided by the asymmetric updating hypothesis—that individuals respond more to ‘good-news’ than ‘badnews’. Here, in our simple framework there are two ways to identify asymmetric updating. First, if we only consider the behavior of individuals within the asymmetric scenario, we can ask whether there is an asymmetry in updating after signals that favor the more desirable state 𝜔=A (‘good-news’), relative to signals that favor the less desirable state 𝜔=B (‘bad-news’). For example, if 𝛾a>𝛾 b , this would indicate that the agent updates more in response to ‘good-news’. We refer to such an agent as an optimistic updater. Conversely, if we have 𝛾a<𝛾 b then the agent updates more in response to ‘bad-news’. We refer to such an agent as a pessimistic updater. Second, if we compare behavior between the symmetric and asymmetric scenarios, we can ask whether the parameters of Eq.1 differ according to the scenario. We use the postscript c∈{A,S} to distinguish the parameters in the two scenarios—i.e. 𝛿S , 𝛾S a and 𝛾S b in symmetric and 𝛿A , 𝛾A a and 𝛾A b in asymmetric . In the symmetric treatment, where the agent is completely indifferent between the two states, there is no reason to expect her updating to be asymmetric. Therefore, we assume that 𝛾S a =𝛾 S b =𝛾 S . Thus, the difference 𝛾A a −𝛾 S a reflects a measure of the increase in the agent’s responsiveness when information is desirable, relative to when information is neutral in terms of its affect. Similarly, 𝛾A b −𝛾 S b is a measure 37 1 3 Belief updating: does the‘good-news, bad-news’ asymmetry… of the increase in the agent’s responsiveness when information is undesirable, relative to the case in which information is neutral in affect. Hypothesis 2 (Asymmetric updating) Individuals update their beliefs asymmetrically, responding more to good than bad news. Therefore, within the asymmetric scenario, we will observe 𝛾A a >𝛾 A b . And in a comparison between the symmetric and asymmetric scenarios, we will observe 𝛾A a −𝛾 S a > 0 and 𝛾A b −𝛾 S b < 0 . Together, we can summarize the asymmetric updating hypothesis parameter predictions as follows: 𝛾A a >𝛾 S >𝛾 A b . 2.2 Belief elicitation andincentives To empirically test the hypotheses above using an experiment, we would like to be able to elicit our participants’ true beliefs. However, eliciting beliefs when studying the relationship between preferences and beliefs presents additional challenges. In particular, one needs to account for the inherent hedging motive faced by participants who have a stake in one state of the world (see Karni and Safra 1995 for a discussion). To obtain unbiased reported beliefs, we adopt the approach developed by Offerman etal. (2009), and extended to accommodate state-dependent stakes as in Kothiyal etal. (2011). The central idea behind this approach is to acknowledge that the incentive environment within which we elicit beliefs in the laboratory may exert a distortionary influence on reported beliefs. We therefore measure this distortionary influence of the incentive environment in a separate part of the experiment. Once we have constructed a mapping from true beliefs to reported beliefs within the relevant incentive environment, we can invert this function to recover the participant’s true beliefs from her reported beliefs. Our objective, therefore, is to recover the function that each individual uses to map her true beliefs to the beliefs that she reports within the given incentive environment. The incentive environment that we use in our experiment to elicit beliefs is the quadratic scoring rule (QSR).6 Online Appendix B.2 provides a detailed discussion of theway in which reported beliefs might be distorted under the quadratic scoring rule. In the absence of state-dependent stakes, it is well documented thatunder the 6 There are several reasons for adopting this approach: firstly, the QSR has the advantage that it ensures that the decision environment is clear and simple for the participants—essentially they are making a single choice from a list of binary prospects; secondly, the QSR has been widely used in the literature, implying that both the theoretical properties and empirical performance are well understood (see, e.g., Armantier and Treich 2013); thirdly, in a horse race between elicitation methods, Trautmann and van de Kuilen (2015) show that there is no improvement in the empirical performance of more complex elicitation methods over the Offerman etal. (2009) method, neither in terms of internal validity, nor in terms of behavior prediction. Out of the set of alternative elicitation techniques, the two that are most theoretically attractive are the binarized scoring rule, proposed by Hossain and Okui (2013), and the probability matching mechanism, described by Grether (1992) and Karni (2009). However, in the context of the current paper, we viewed neither of these approaches as being preferable to the Offerman etal. (2009) technique, since both of these approaches introduce an additional layer of probabilities and in the study of probability bias, this is an undesirable attribute of the elicitation strategy. 38 K.Barron 1 3 QSR a risk averse agent should distort their reported belief towards 50%. With statedependant stakes, a risk averse EU maximizer will face two distortionary motives in reporting her belief: (1) she will face the motive to distort her belief towards 50% as discussed above; and (2) in addition, there is a hedging motive, which will compel a risk averse individual to lower her reported belief, rt , towards 0% as the size of the exogenous stake increases.7 If the participants in our experiment are risk neutral expected utility maximizers, the reported beliefs, rt , that we elicit under the QSR will coincide with their true beliefs, 𝜋t , and no belief correction is necessary. However, to account for a hedging motive (e.g. due to risk aversion), we measure the size of the distortionary influence of the elicitation incentives at an individual level and correct the beliefs accordingly. The following section provides the intuition for how this correction works. 2.2.1 A Non‑EU ‘truth serum’ The Offerman et al. (2009) approach proposes correcting reported beliefs for the reporting bias generated by risk aversion or non-linear probability weighting. This approach involves eliciting subjects’ reported beliefs, r, corresponding to a set of risky events where both the participant and the analyst know the objective probabilities, p (known probability). This is done under precisely the same QSR incentive environment in which the subjects’ subjective beliefs, 𝜋 , regarding the events of interest are elicited (unknown probability). If a subject’s reported beliefs, r, differ from the known objective probabilities, p, this indicates that the subject is distorting her beliefs due to the incentive environment (e.g. due to risk aversion). The objective of the correction mechanism is therefore to construct a map, R, from the objective probabilities, p∈[0, 1] , to the reported beliefs, r, for each individual under the relevant incentive environment. Given this map, R, we can invert the function and recover the subject’s true beliefs from her reported beliefs about events with unknown probabilities, 𝜋 . In Online Appendix B.2, we offer a detailed discussion of how the Offerman etal. (2009) method operates, describes the underlying assumptions, and demonstrate how it can be augmented (as proposed in Kothiyal etal. 2011) to allow for the scenario where there are state-contingent stakes (i.e. x≠0 ). 3 Experimental design The experiment is designed to test the asymmetric updating hypothesis using both within-subject and between-subjects comparisons of updating behavior. The experiment consists of three treatment groups. The treatment T1:S ymmetric corresponds to Scenario 1 ( symmetric : no exogenous state-contingent stakes) and the other two 7 This assumes that the state-dependent payment is associated with the 100% state, not the 0% state. This assumption is made throughout the paper and the experiment. 45 1 3 Belief updating: does the‘good-news, bad-news’ asymmetry… 5.2 Core estimation specifications Our core estimation equations aim to test for systematic patterns in updating behavior, within the framework of Eq.1. First, we examine whether there are systematic deviations from Bayes’ rule in updating behavior, independently of having a stake in one of the two states of the world. Second, we assess the influence that having a stake in one of the two states of the world has by (1) testing for an asymmetry in updating within treatments where there is a state-contingent stake (i.e. T2: C ombined and T3: S eparate ); and (2) testing whether there are differences in updating behavior between the treatments with and without a state-contingent stake. The first estimation equation follows directly from Eq.1, allowing us to test the asymmetric updating hypothesis, and also to test for other common deviations from Bayes’ rule, such as a confirmatory bias or base rate neglect: where 𝜋 i,j,t=logit (𝜋i,j,t) and  q= log (q 1−q) ; j refers to a round of decisions; and the errors, 𝜖ijt , are clustered at the individual, i, level. To determine the belief updating pattern within each incentive environment, we estimate this equation separately for each treatment. Then to test for significant differences between the coefficients in different incentive environments, we pool our sample and interact the treatment variable with all three of the coefficients (2) 𝜋 i,j,t + 1=𝛿 𝜋 i,j,t+𝛾aq ⋅ 1(si,j,t + 1=a)−𝛾bq ⋅ 1(si,j,t + 1=b)+𝜖i,j,t + 1 Fig. 2 CDFs of corrected and uncorrected reported beliefs 46 K.Barron 1 3 in this equation (i.e. 𝛿 , 𝛾a , and 𝛾b ). This provides us with a test of whether the parameters differ between either of the two asymmetric treatments and the sym - metric treatment. 5.2.1 Endogeneity oftheLagged Belief One potential concern with the identification of the parameters of Eq.2 is that the right hand side contains lagged versions of the dependent variable. This implies that there is a possible endogeneity of the lagged beliefs, 𝜋i,j,t , if they are correlated with the error term (i.e. if E{𝜋 i,j,t𝜖i,j,t+1}≠0 ).14 If this is the case, it can result in biased and inconsistent estimators for the parameters of the regression. Our experiment was designed to avoid this issue by virtue of exogenously assigning the subjects’ entire information set. This allows us to use the exogenously assigned prior probability of state 𝜔=A being the true state, pi,j,t=0 , as well as the sequence of signals observed, st , to construct an instrument for the lagged belief, 𝜋i,j,t , in Specification 2. We do this by calculating the objective Bayesian posterior, given the agent’s information set at time t, and using this as an instrument for her belief, 𝜋i,j,t . The approach used here also avoids a second type of endogeneity issue that can arise when studying belief updating when the states of the world are a functions of personal characteristics (e.g. when examining beliefs regarding individual attributes, such as one’s own skills, IQ, or beauty) or personal choices. When this is the case, the conditional probability of observing a specific signal depends on the state of the world, and therefore can be correlated with personal characteristics (see Online Appendix C and earlier versions of this paper for further discussion of this issue). 6 Results Table3 reports the results from estimating Eq.2 for each of the treatment groups separately. These estimates describe the updating behavior of the average individual in each of the three treatment groups. Within each treatment group, we report the results for both the OLS (top panel) and the IV (bottom panel) estimations discussed above. Columns (1a), (2a) and (3a) use the uncorrected reported beliefs, while columns (1b), (2b) and (3b) use the corrected beliefs. Every coefficient in the table is statistically different from 0 at the 1% level. Since our primary interest is in testing whether the coefficients are different from 1, in this table we use asterisks to reflect the significance of a ttest of whether a coefficient is statistically different from 1. Perhaps the most striking features of this table are: (1) the similarity in the updating patterns across the three treatment groups; and (2) that for the average individual, the observed updating behavior is close to Bayesian in all three treatment groups. The pvalues from the test of the null hypothesis, H0∶𝛾a=𝛾b , show that 14 For example, this would be the case if there is individual heterogeneity in the way individuals respond to information. We provide evidence below that this individual heterogeneity is present. 47 1 3 Belief updating: does the‘good-news, bad-news’ asymmetry… in none of the three treatment groups do we observe a statistically significant difference (at the 5% level) between the responsiveness to the signals in favor of 𝜔=A and 𝜔=B (i.e. we don’t observe asymmetric updating). Both the OLS results in the top panel and the IV results in the bottom panel indicate that the responsiveness to new information was, on average, not statistically different to that of a Bayesian, since both 𝛾a and 𝛾b are not significantly different to 1 at the 5% level. The primary difference between the OLS results and the IV results is that, while the OLS estimates suggest a small degree of base rate neglect across all three treatments ( 𝛿<1 ), once we control for the possible sources of endogeneity discussed above using our instrumental variable strategy, the estimates are no longer indicative of base rate neglect. Since the OLS estimates may be biased, the IV estimates represent our preferred results. The first stage regression results for the IV estimation are reported in the Appendices in Table8, indicating that we Table 3 Average updating behavior across treatments (1) Standard errors in parentheses (clustered at the individual level) (2) All coefficients are significantly different from 0 at the 1% level. Therefore, ttests of the null hypothesis ( H0 : coefficient = 1) are reported: * = 10%, ** = 5%, *** = 1% (3) The rows corresponding to p ( H0 ∶ 𝛾a = 𝛾b ) report the pstatistic from a ttest of the equality of the coefficients 𝛾a and 𝛾b (i.e. a test of the asymmetric updating hypothesis) T1: SYMMETRIC T2: COMBINED T3: SEPARATE Reported Corrected Reported Corrected Reported Corrected (1a) (1b) (2a) (2b) (3a) (3b) OLS 𝛿 0.90 0.90 0.86 0.86 0.91 0.93 (0.03)*** (0.03)*** (0.04)*** (0.04)*** (0.03)*** (0.02)*** 𝛾a 1.12 1.09 1.05 1.06 1.25 1.16 (0.12) (0.11) (0.12) (0.12) (0.14)* (0.11) 𝛾b 1.21 1.17 1.19 1.12 1.20 1.13 (0.13)* (0.11) (0.15) (0.13) (0.12) (0.10) p( H0∶𝛾a=𝛾b ) 0.30 0.32 0.20 0.48 0.58 0.73 N1075 1075 1285 1285 1140 1140 R2 0.71 0.73 0.73 0.74 0.81 0.84 IV 𝛿 0.99 0.99 1.00 0.99 0.99 0.99 (0.03) (0.03) (0.02) (0.02) (0.03) (0.02) 𝛾a 1.12 1.09 1.08 1.02 1.22 1.14 (0.12) (0.11) (0.12) (0.11) (0.13)* (0.11) 𝛾b 1.21 1.16 1.11 1.12 1.18 1.11 (0.12)* (0.11) (0.13) (0.12) (0.12) (0.10) p( H0∶𝛾a=𝛾b ) 0.30 0.31 0.76 0.25 0.63 0.74 N1075 1075 1285 1285 1140 1140 1st Stage F 61.38 84.04 107.98 107.01 76.90 95.45 48 K.Barron 1 3 don’t have a weak instrument issue. Importantly, however, the form of endogeneity addressed by IV estimation only pertains to potential biases in the 𝛿 parameter, since it addresses endogeneity in the prior belief variable. It is therefore reassuring that for the estimates of our primary parameters of interest, namely 𝛾a and 𝛾b , the estimates are largely consistent across all the estimation specifications reported in Table3. A reason for this is that the signals, si,j,t+1 , that subjects receive are always completely exogenous, both across rounds, and with respect to the subject’s personal characteristics. Furthermore, the distribution of signals observed is also exogenous, and balanced in expectation. This helps to avoid other sources of endogeneity (see, e.g., Online Appendix C) and to alleviate the influence of other potential confounding belief updating biases (see, e.g., Coutts (2019) for a discussion of the influence of signal distributions on belief updating). It is worth noticing that although we observe a substantial difference in the levels of the corrected and uncorrected beliefs in Fig.2 above, the estimates for updating in Table3 are quite similar for the corrected and uncorrected beliefs. One explanation for this apparent inconsistency is the following. If the degree of hedging by an individual is similar for both the prior and posterior belief, then the correction could have a sizeable effect on the levels of both beliefs, but not result in the large difference in the estimated updating parameters, since updating pertains to the change in the belief rather than the level. 6.1 A model free test oftheasymmetric updating hypothesis In order to alleviate the potential concern that these results are dependent on the functional form of our empirical specification, we conduct a model-free test of the the asymmetric updating hypothesis. Perhaps the simplest and most direct test of this hypothesis is obtained by directly comparing the posterior beliefs formed in two scenarios where the information set is identical, but the rewards associated with one of the states of the world is varied. Our data is well suited for conducting this exercise. We do this by considering a comparison of information-set-equivalent posterior beliefs after individuals have received only (1) the exogenous prior and (2) a single ball draw. This allows us to test the asymmetric updating hypothesis while remaining agnostic regarding the process that guides belief updating, testing only whether it is symmetric. Our data allows us to conduct two comparisons of information-setequivalent posterior beliefs—a within-subject and a between-subject comparison. First, we can compare posterior beliefs, 𝜋1 , formed with identical information sets {p0,s1} between treatment groups, where the payments associated with states of the world differ. For example, we can compare the average posterior formed after an identical prior, e.g. p 0= 1 6 , and an identical signal, e.g. s1=a (i.e. a blue ball), across treatments. Second, we can compare information-set-equivalent posterior beliefs within treatment groups. This comparison involves comparing 𝜋1 after {p0=p,s1=s} 49 1 3 Belief updating: does the‘good-news, bad-news’ asymmetry… with 1−𝜋1 after {p0=1−p,s1=sc} where sc is the complementary signal to s.15 For example, we can compare the posterior, 𝜋1 , formed after a prior of p0= 1 6 and the signal s1=a (i.e. a blue ball), with 1−𝜋1 after a prior of p0= 5 6 and the signal s1=b (i.e. a red ball). To see why this comparison involves a comparison of information-set-equivalent posterior beliefs, recall that the experiment is designed to be completely symmetric in terms of information, with the informativeness of a red ball exactly the same as a blue ball. Therefore, if an individual updates symmetrically, then 𝜋 1 | {p 0 =p,s 1 =s} = 1 − 𝜋1 | {p 0 =1−p,s 1 =sc } . This prediction does not rely on Bayes’ rule (although it is an implication of Bayes’ rule), but rather only requires symmetric updating, and therefore it provides us with a non-parametric test of the asymmetric updating hypothesis. Figure3 depicts both of these comparisons, with each group of six bars collecting together the relevant information-set-equivalent groups. Each bar presents the mean posterior belief for that group, as well as a 95% confidence interval around the mean. Each group is labeled on the xaxis by the prior belief associated with the ‘red’ bars, which correspond to the information sets that include a red ball as a signal (i.e. s1=b ). The ‘blue’ bars report the mean of 1−𝜋1 for information sets containing a blue ball (i.e. s1=a ) and for these bars the xaxis label corresponds to 1−p0 . Within each group, the first two bars represent the average posterior beliefs in T1: S ymmetric ; the second pair of bars depict the same for T2: C ombined ; and the third pair of bars for T3: S eparate . Fig. 3 Comparison of beliefs after the receipt of a single signal and an exogenous prior 15 Therefore, if s=a then sc=b and vice versa. 50 K.Barron 1 3 The results displayed in Fig. 3 show that there are no systematic differences between posterior beliefs within information-set-equivalent groups, neither within nor between treatment groups. Furthermore, when testing non-parametrically whether there are differences within or between treatment groups for information-set-equivalent groups, none of the 45 relevant binary comparisons16 are significant at the 5% significance level under a Mann–Whitney test, suggesting that we cannot reject the hypothesis that the posterior beliefs within information-set-equivalent groups are drawn from the same distribution. This lends support to the results described above which indicate that we fail to find evidence in support of the asymmetric updating hypothesis. 6.2 Robustness exercises In addition to this model-free test, to check for the robustness of the belief updating results for the average individual presented in Table 3, we conducted several robustness exercises. These exercises, and their corresponding results, are discussed in detail in Online Appendix A. The first exercise examines whether the results from the main specification described in Eq.2 are robust to first differencing the dependent variable (i.e. this imposes the assumption that 𝛿=1 ). The second subsection extends the main empirical specification to allow for individual-specific updating parameters. For both of these specifications, an ex post power analysis is conducted, reporting the MDE for a significance level of 𝛼=0.05 and a power of 𝜅=0.8 . The third subsection pools all the observations across the three treatments together, and then tests whether the average updating parameters differ across treatments by interacting treatment group dummies with the regressors of the main specification described in Eq.2. The results from all of these exercises are highly consistent with those in Table3 and fail to provide any evidence in favor of an asymmetry in updating. 6.3 Heterogeneity inupdating behavior In order to investigate whether the aggregate results are masking heterogeneity in updating behavior, we estimate Specification2 at the individual level and collect the parameters. The distributions of these individual level parameters are reported in Fig.4. Perhaps the most conspicuous feature of this figure is the fact that all three treatment groups display such similar parameter distributions in each of the panels—i.e. for each of the parameters. Testing for differences between the underlying distributions from which the parameters are drawn in the different treatment groups fails to detect any statistically significant differences 16 For these comparisons, for each exogenous prior, we test the following binary comparisons: (1) within treatment group, we test between those that received the s1=a and s1 = b signals ( 3×5 binary comparisons); (2) for those that received the same signal, s1 , we test between treatment groups ( 6×5 binary comparisons). 51 1 3 Belief updating: does the‘good-news, bad-news’ asymmetry… in any of the four panels.17 The upper-right panel shows that the majority of individuals have an estimated 𝛿 parameter in the interval [0.6, 1.1] , with a large proportion of these concentrated around 1 in all three treatment groups. The two left-hand panels show that there is substantially more individual heterogeneity in the estimated 𝛾a and 𝛾b parameters, which are dispersed over the interval [0, 3.5] in all three treatments. With such a large degree of variation in the individual level parameters, a natural conjecture to make is that, while we do not observe asymmetric updating at the aggregate level, it is entirely plausible that there may be a subsample of individuals who are optimistic updaters and another subsample of individuals who are pessimistic updaters. If these two subsamples are of a similar size and their bias is of a similar magnitude, we would observe no asymmetry at the aggregate level. The lower-right panel of Fig.4 suggests that this is not the case by plotting the distribution of the individual level difference between the 𝛾a and Fig. 4 Distributions of individual level updating parameters 17 More specifically, for each of the four panels in Fig.4, we conducted three binary Mann–Whitney tests comparing each possible pair of treatments. In total, these twelve statistical tests failed to detect any significant difference in the underlying parameter distributions between the treatment groups at the 10% level. In addition, another twenty-four similar tests for differences in the mean (ttest) and median (chi-squared) of the underlying distributions between treatments fail to detect any statistically significant differences at the 10% level. 52 K.Barron 1 3 𝛾b parameters. The majority of the distribution is concentrated in a narrow interval around 0 for all three treatment groups, suggesting that there is no asymmetry for any sizable subsample. Furthermore, this conclusion is supported by the fact there are no significant differences between the distributions of updating parameters observed across the three treatments in any of the four panels, since the motive for a ‘good-news, bad-news’ effect is switched off in the T1: S ymmet - ric treatment. 7 Discussion 7.1 Heterogeneous results observed intheasymmetric updating literature A central question that emerges from the discussion above is why we observe no evidence of a ‘good-news, bad-news’ effect here, while some other influential contributions to this literature have found evidence for such an effect. More generally, Benjamin (2019) points out that the evidence in this nascent literature is so far very mixed. In the economics literature, three papers find evidence in favor of stronger inference from good news: Eil and Rao (2011), Möbius etal. (2014) and Charness and Dave (2017).18 In contrast, there are three papers that find evidence of stronger inference from bad news: Ertac (2011), Kuhnen (2015) and Coutts (2019). Furthermore, in addition to the current paper, there are four other papers that find no evidence in favor of a preference-biased asymmetry in belief updating: Grossman and Owens (2012), Schwardmann and Vander Weele (2019), Gotthard-Real (2017) and Buser etal. (2018).19 In the psychology literature, however, there appears to be a near-consensus arguing that there is an asymmetry in belief updating in favor of good news (see, e.g., Sharot etal. 2011, 2012; Kuzmanovic etal. 2015; Marks and Baines 2017, amongst others). A notable exception is Shah etal. (2016) who argue that many of the contributions to this literature suffer from methodological concerns; Garrett and Sharot (2017) offer a rebuttal, claiming that optimistically biased updating is robust to these concerns. The aim of the experiment discussed in this paper is not to isolate the contextual features that generate these heterogeneous results. Rather, it is to contribute robust evidence on updating for one particular domain—belief updating when states differ in terms of the financial rewards they bring. Nevertheless, it is important to consider the body of evidence as a whole to assess whether this reveals a systematic pattern that might organize these heterogeneous results. Below, I offer a discussion of some of the candidate explanations for the heterogeneity. In general, these explanations fall into two categories: (1) the hypothesis that contextual factors mediate 18 Mayraz (2013) also presents evidence in favor of individuals forming motivated beliefs that are distorted towards more desirable states, however in his experiment one cannot calculate the Bayesian posterior, so it is less comparable to the other studies in this literature. 19 Additionally, while Eil and Rao (2011) found evidence of an asymmetry in favor of good-news in the domain of beliefs about one’s own Beauty, they did not find evidence of an asymmetry in the domain of beliefs about one’s own IQ. 53 1 3 Belief updating: does the‘good-news, bad-news’ asymmetry… asymmetric updating: belief updating is influenced by preferences, but this preference-biased updating is switched on or off by contextual factors. (2) the hypothesis that asymmetric updating is sometimes misidentified: belief updating is not actually influenced by preferences, but rather, what appears to be asymmetric updating is driven by a different cognitive bias (e.g. prior-biased inference). The majority of the explanations discussed below fall into the first category. 7.1.1 Information structure One avenue of enquiry for attempting to reconcile the results is to consider the differences in the information structures across experiments. For example, while several of the studies adopt a two-state bookbag-and-poker-chip experimental paradigm, Ertac (2011) uses a three-state structure with signals that are perfectly informative about one state, and Eil and Rao (2011) consider a ten-state updating task with binary signals. However, this does not seem to be driving the differences in results, since we observe heterogeneous results amongst papers with similar information structures—e.g. restricting attention only to the papers with two-state structures with binary signals (e.g. the current paper, Möbius etal. 2014; Gotthard-Real 2017; Coutts 2019) yields mixed results. 7.1.2 Priors Focusing only on two-state experiments, there is substantial variation inthe average prior belief across experiments, with Coutts (2019) (by design) observing relatively low average priors in comparison to Möbius etal. (2014), for example. If belief updating is influenced by prior beliefs (e.g. a confirmatory bias), then what looks like preference-biased belief updating may be driven by a completely different cognitive deviation from Bayesian updating, namely prior-biased updating (see, e.g., Benjamin 2019, Section8). However, if we look at the papers that find evidence for preference-biased updating, Charness and Dave (2017) do find evidence in favor of prior-biased updating, while Eil and Rao (2011) and Möbius etal. (2014) do not find evidence of prior-biased updating. This speaks against the explanation that what appears to be asymmetric belief updating due to preferences is actually driven by a confirmatory bias. 7.1.3 Ambiguity One important dimension in the belief updating literature is whether subjects are provided with exogenous point estimate priors or update from subjectively formed prior beliefs. There are advantages and disadvantages to each approach. The former 54 K.Barron 1 3 brings increased experimental control and improved causal identification, but this comes at the expense of reduced realism and perhaps a slightly less natural setting.20 This discussion highlights a key assumption that is typically made in this literature, namely that subjects are probabilistically sophisticated and therefore update as if they hold a point estimate prior. In cases where subjects must form their own subjective prior belief, this assumption is not innocuous. If, instead, the beliefs subjects hold are not precise point estimates (e.g. if subjects hold ambiguous prior beliefs over an interval), then simple Bayesian updating may no longer be the most appropriate benchmark model. First, there are several competing theoretical models of belief updating in the presence of ambiguity with differing predictions, e.g. full Bayesian updating (Jaffray 1989; Pires 2002) and maximum likelihood updating (Gilboa and Schmeidler 1993), and some recent experimental evidence testing between them (Ngangoue 2018; Liang 2019). Second, one might postulate that there is greater scope for motivated reasoning when one is updating beliefs from ambiguous priors (or ambiguous signals) in comparison to belief updating from exogenously endowed point estimate priors (and signals with clearly defined informativeness). However, the existing evidence suggests that the presence or absence of ambiguity is not a unifying explanation for the differing results. Even within the set of papers with home-grown subjective priors (e.g. Eil and Rao 2011; Ertac 2011; Möbius etal. 2014, and Coutts 2019), the results are very mixed. 7.1.4 Domain ofbelief updating Typically, we study belief formation as if it is domain-independent. However, it seems natural to consider the possibility that humans evolved to process information about their physical environment differently from information about their self and their social environment. For example, the mental processes involved in forming a belief about the likelihood of future rainfall may be fundamentally different to those involved in forming a strategic belief about the probability that another individual will be trustworthy in a specific scenario. Some papers in this literature have explored this question by asking whether we update differently about a given fundamental characteristic of one’s own self in comparison to the same fundamental characteristic of another individual (e.g., Möbius etal. 2014; Coutts 2019). Furthermore, recent theoretical and experimental work has studied how individuals attribute outcomes to their self versus an external fundamental from their physical or social environment (see, e.g., Heidhues etal. 2018; Hestermann and LeYaouanq 2020; Coutts etal. 2019). The influence of the domain on belief updating may matter for the asymmetric updating literature because this literature considers updating scenarios pertaining to both the environment (with ‘good’ states typically represented by high monetary payments) and to the self (where ‘good’ states pertain to a desirable individual 20 An interesting recent contribution by LeYaouanq and Schwardmann (2019) proposes a methodology for studying belief updating in more natural settings, while still maintaining experimental control and permitting a comparison with Bayesian updating.