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Centre for Economic and Business Research ÿkonomi- og Erhvervsministeriets enhed for erhvervs- konomisk forskning og analyse Discussion Paper Behavioral Econometrics for Psychologists Steffen Andersen

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Centre for Economic and Business Research ÿkonomi- og Erhvervsministeriets enhed for erhvervs- konomisk forskning og analyse Discussion Paper Behavioral Econometrics for Psychologists Steffen Andersen Glenn W. Harrison Morten Igel Lau Elisabeth E. Rutström Behavioral Econometrics for Psychologists by Steffen Andersen, Glenn W. Harrison, Morten Igel Lau and Elisabet E. Rutström August 2007 Abstract. We make the case that psychologists should make wider use of structural econometric methods. These methods involve the development of maximum likelihood estimates of models, where the likelihood function is tailored to the structural model. In recent years these models have been developed for a wide range of behavioral models of choice under uncertainty. We explain the components of this methodology, and illustrate with applications to major models from psychology. The goal is to build, and traverse, a constructive bridge between the modeling insights of psychology and the statistical tools of economists. Centre for Economic and Business Research, Copenhagen Business School, Copenhagen, Denmark (Andersen); Department of Economics, College of Business Administration, University of Central Florida, USA (Harrison and Rutström) and Department of Economics and Finance, Durham Business School, Durham University, United Kingdom (Lau). and Harrison and Rutström thank the U.S. National Science Foundation for research support under grants NSF/IIS , NSF/HSD and NSF/SES , we all thank the Danish Social Science Research Council for research support under project # An earlier version was presented by Harrison as an invited lecture at the Annual Conference of the French Economic Association: Behavioral Economics and Experimental Economics, Lyon, May 23-25, 2007, and we are grateful for the invitation, hospitality and comments received there. Table of Contents 1. Elements of Structural Estimation Structural Estimation Assuming EUT Stochastic Errors Probability Weighting and Rank-Dependent Utility Loss Aversion and Sign-Dependent Utility Original Prospect Theory Cumulative Prospect Theory Will the True Reference Point Please Stand Up? Mixture Models and Multiple Decision Processes Recognizing Multiple Decision Processes Implications for the Interpretation of Process Data Comparing Latent Process Models Dual Criteria Models from Psychology The SP Criteria The A Criteria Combining the Two Criteria Applications The Priority Heuristic Conclusion References Appendix A: Experimental Procedures for Binary Lottery Choice Task...-A1- Appendix B: Estimation Using Maximum Likelihood...-A3- B1. Estimating a CRRA Utility Function...-A4- B2. Loss Aversion and Probability Weighting...-A9- B3. Adding Stochastic Errors...-A11- B4. Extensions...-A13- Appendix C: Instructions for Laboratory Experiments...-A14- C1. Baseline Instructions...-A14- C2. Additional Instructions for UK Version...-A21- Economists tend not to take the full range of theories from psychology as seriously as they should. Psychologists have much more to offer to economists than the limited array of models or ad hoc insights that have been adopted by behavioral economists. One simple reason for this lack of communication is that psychologists tend to estimate their models, and test them, in a naive fashion that makes it hard for economists to evaluate the broader explanatory power of those models. Another reason is that psychologists tend to think in terms of the process of decisionmaking, rather than the characterization of the choice itself, and it has been hard to see how such models could be estimated in the same way as standard models from economics. We propose that psychologists use structural maximum likelihood estimation to address these barriers to trade between the two disciplines (or that economists use these methods to evaluate models from psychology). Recent developments in behavioral econometrics allow much richer specifications of traditional and non-traditional models of behavior. It is possible to jointly estimate parameters of complete structural models, rather than using one experiment to pin down one parameter, another experiment to pin down another, and losing track of the explanatory power and sampling errors of the whole system. It is also possible to see the maximum likelihood evaluator as an intellectual device to write out the process in as detailed a fashion as desired, rather than relying on pre-existing estimation routines to shoe-horn the model into. The mainstream models can also be seen as process models in this light, even if they do not need to be interpreted that way in economics. In section 1 we review the basic elements of structural modeling of choice under uncertainty, using expected utility theory from mainstream economics to illustrate and provide a baseline model. In section 2 we illustrate how one of the most important insights from psychology (Edwards [1962]), the possibility of probability weighting, can be incorporated. In section 3 we demonstrate the effects of including one of the other major insights from psychology (Kahneman and Tversky [1979]), the possibility of sign dependence in utility evaluation. In particular, we demonstrate how circularity in the use of priors about the true reference point can dramatically affect the empirical inferences one might make about the prevalence of loss aversion. The introduction of alternative structural models -1- leads to a discussion of how one should view the implied hypothesis testing problem. We advocate a mixture specification in section 4, in which one allows for multiple latent data-generating processes, and then uses the data to identify which process applies in which task domain and for which subjects. We then examine in section 5 how the tools of behavioral econometrics can be applied to the neglected model from psychology proposed by Lopes [1995]. Her model represents a novel way to think about rank-dependent and sign-dependent choices jointly, and directly complements literature in economics. Finally, in section 6 we examine the statistical basis of the claims of Brandstatter, Gigerenzer and Hertwig [2006] that their Priority Heuristic dramatically outperforms other models of choice under uncertainty from economics and psychology. We are not saying that psychologists are ignorant about the value or methods of structural maximum likelihood modeling, that every behavioral economist uses these methods, or indeed that they are needed for every empirical issue that arises between economists and psychologists. Instead, we are arguing that many needless debates can be efficiently avoided if we share a common statistical language for communication. The use of experiments themselves provides a critical building block in developing that common language: if we can resolve differences in procedures then the experimental data itself provides an objective basis for debates over interpretation to be meaningfully joined Elements of Structural Estimation 1.1 Structural Estimation Assuming EUT Assume for the moment that utility of income is defined by U(x) = x r (1) where x is the lottery prize and r is a parameter to be estimated. For r=0 assume U(x)=ln(x) if needed. Thus 1-r is the coefficient of Constant Relative Risk Aversion (CRRA): r=1 corresponds to risk neutrality, r 1 to risk loving, and r 1 to risk aversion. 2 Let there be K possible outcomes in a 1 Hertwig and Ortmann [2001][2005] evaluate the differences systematically, and in a balanced manner. 2 Funny things happen to this power function as r tends to 0 and becomes negative. Gollier [2001; p.27] notes the different asymptotic properties of CRRA functions when r is positive or r is negative. When r 0, utility goes from 0 to 4 as income goes from 0 to 4. However, when r 0, utility goes from minus 4 to 0 as income goes from 0 to 4. Wakker [2006] extensively studies the properties of the power utility function. An alternative form, U(x) = x (1-F) /(1-F), is -2- lottery. Under EUT the probabilities for each outcome k, p(k), are those that are induced by the experimenter, so expected utility is simply the probability weighted utility of each outcome in each lottery i: EU i = 3 k=1,k [ p(k) U(k) ]. (2) The EU for each lottery pair is calculated for a candidate estimate of r, and the index LEU = EU R - EU L (3) calculated, where EU L is the left lottery and EU R is the right lottery. This latent index, based on latent preferences, is then linked to the observed choices using a standard cumulative normal distribution function M(LEU). This probit function takes any argument between ±4 and transforms it into a number between 0 and 1 using the function shown in Figure 1. Thus we have the probit link function, prob(choose lottery R) = M(LEU) (4) The logistic function is very similar, as illustrated in Figure 1, and leads instead to the logit specification. 3 Even though Figure 1 is common in econometrics texts, it is worth noting explicitly and understanding. It forms the critical statistical link between observed binary choices, the latent structure generating the index y*, and the probability of that index y* being observed. In our applications y* refers to some function, such as (3), of the EU of two lotteries; or, later, the prospective utility of two lotteries. The index defined by (3) is linked to the observed choices by specifying that the R lottery is chosen when LEU ½, which is implied by (4). Thus the likelihood of the observed responses, conditional on the EUT and CRRA specifications being true, depends on the estimates of r given the above statistical specification and the observed choices. The statistical specification here includes assuming some functional form for the cumulative density function (CDF), such as one of the two shown in Figure 1. If we ignore often estimated, and in this form F is the CRRA. 3 In some cases a parameter is used to adjust the latent index LEU defined by (3). For example, Birnbaum and Chavez [1997; p.187, eq. (14)] specify that prob(choose lottery R) = M(V LEU) and estimate V. This is formally identical to a so-called Fechner error specification, discussed below in 2.1 (see equation (3 ), and set : there to 1/V). -3- responses that reflect indifference 4 the conditional log-likelihood would be ln L(r; y, X) = 3 i [ (ln M(LEU) * y i = 1) + (ln M(1!LEU) * y i =!1) ] (5) where y i =1(!1) denotes the choice of the Option R (L) lottery in risk aversion task i, and X is a vector of individual characteristics reflecting age, sex, race, and so on. The latent index (3) could have been written in a ratio form: LEU = EU R / (EU R + EU L ) (3') and then the latent index would already be in the form of a probability between 0 and 1, so we would not need to take the probit or logit transformation. This specification has been used, with some modifications to include stochastic errors, in Holt and Laury [2002]. Appendix A reviews experimental procedures for some canonical binary lottery choice tasks we will use to illustrate many of the models considered here. These data amount to a replication of the classic experiments of Hey and Orme [1994], with extensions to collect individual demographic characteristics and to present subjects with some prizes framed as losses. Details of the experiments are reported in Harrison and Rutström [2005][2007]. Subjects were recruited from the undergraduate and graduate student population of the University of Central Florida in late 2003 and throughout Each subject made 60 lottery choices and was paid for 3 of these, drawn at random. A total of 158 subjects made choices. Some of these had prizes of $0, $5, $10 and $15 in what we refer to as the gain frame (N=63). Some had prizes framed as losses of $15, $10, $5 and $0 relative to an endowment of $15, ending up with the same final prize outcomes as the gain frame (N=58). Finally, some subjects had an endowment of $8, and the prizes were transformed to be -$8, -$3, $3 and $8, generating final outcomes inclusive of the endowment of $0, $5, $11 and $16. Appendix B reviews procedures and syntax from the popular statistical package Stata that can be used to estimate structural models of this kind, as well as more complex models discussed later. 5 The goal is to illustrate how experimental economists can write explicit maximum likelihood 4 Relatively few subjects use this option. The extension to handling it in models of this kind is discussed in Harrison and Rutström [2007; 2.2]. 5 Appendix B is available in the working paper version, Andersen, Harrison, Lau and Rutström [2007b], available online at -4- (ML) routines that are specific to different structural choice models. It is a simple matter to correct for stratified survey responses, multiple responses from the same subject ( clustering ), 6 or heteroskedasticity, as needed, and those procedures are discussed in Appendix B. Panel A of Table 1 shows maximum likelihood estimates obtained with this simple specification. The coefficient r is estimated to be 0.776, with a 95% confidence interval between and This indicates modest degrees of risk aversion, consistent with vast amounts of experimental evidence for samples of this kind. Extensions of the basic model are easy to implement, and this is the major attraction of the structural estimation approach. For example, one can easily extend the functional forms of utility to allow for varying degrees of relative risk aversion (RRA). Consider, as one important example, the Expo-Power (EP) utility function proposed by Saha [1993]. Following Holt and Laury [2002], the EP function is defined as U(x) = [1-exp(- x 1-r )]/ , (1') where and r are parameters to be estimated. RRA is then r + (1-r)y 1-r, so RRA varies with income if 0. This function nests CRRA (as 6 0) and CARA (as r 6 0). It is also simple matter to generalize this ML analysis to allow the core parameter r to be a linear function of observable characteristics of the individual or task. For example, assume that we collected information on the sex of the subject, and coded this as a binary dummy variable called Female. In this case we extend the model to be r = r 0 + r 1 Female, where r 0 and r 1 are now the parameters to be estimated. In effect the prior model was to assume r = r 0 and just estimate r 0. This extension significantly enhances the attraction of structural ML estimation, particularly for responses pooled over different subjects, since one can condition estimates on observable characteristics of the 6 Clustering commonly arises in national field surveys from the fact that physically proximate households are often sampled to save time and money, but it can also arise from more homely sampling procedures. For example, Williams [2000; p.645] notes that it could arise from dental studies that collect data on each tooth surface for each of several teeth from a set of patients or repeated measurements or recurrent events observed on the same person. The procedures for allowing for clustering allow heteroskedasticity between and within clusters, as well as autocorrelation within clusters. They are closely related to the generalized estimating equations approach to panel estimation in epidemiology (see Liang and Zeger [1986]), and generalize the robust standard errors approach popular in econometrics (see Rogers [1993]). Wooldridge [2003] reviews some issues in the use of clustering for panel effects, noting that significant inferential problems may arise with small numbers of panels. -5- task or subject. We illustrate the richness of this extension later. For now, we estimate r 0 =0.83 and r 1 = -0.11, with standard errors of and respectively. So there is some evidence of a sex effect, with women exhibiting slightly greater risk aversion. Of course, this specification does not control for other variables that might be confounding the effect of sex. 1.2 Stochastic Errors An important extension of the core model is to allow for subjects to make some errors. The notion of error is one that has already been encountered in the form of the statistical assumption that the probability of choosing a lottery is not 1 when the EU of that lottery exceeds the EU of the other lottery. This assumption is clear in the use of a link function between the latent index LEU and the probability of picking one or other lottery; in the case of the normal CDF, this link function is M(LEU) and is displayed in Figure 1. If the subject exhibited no errors from the perspective of EUT, this function would be a step function in Figure 1: zero for all values of y* 0, anywhere between 0 and 1 for y*=0, and 1 for all values of y* 0. By varying the shape of the link function in Figure 1, one can informally imagine subjects that are more sensitive to a given difference in the index LEU and subjects that are not so sensitive. Of course, such informal intuition is not strictly valid, since we can choose any scaling of utility for a given subject, but it is suggestive of the motivation for allowing for errors, and why we might want them to vary across subjects or task domains. Consider the error specification used by Holt and Laury [2002], originally due to Luce [1959], and popularized by Becker, DeGroot and Marschak [1963]. The EU for each lottery pair is calculated for candidate estimates of r, as explained above, and the ratio LEU = EU R 1/: / (EU L 1/: + EU R 1/: ) (3O) calculated, where : is a structural noise parameter used to allow some errors from the perspective of the deterministic EUT model. The index LEU is in the form of a cumulative probability distribution function defined over differences in the EU of the two lotteries and the noise parameter :. Thus, as : 6 0 this specification collapses to the deterministic choice EUT model, where the -6- choice is strictly determined by the EU of the two lotteries; but as : gets larger and larger the choice essentially becomes random. When :=1 this specification collapses to (3'), where the probability of picking one lottery is given by the ratio of the EU of one lottery to the sum of the EU of both lotteries. Thus : can be viewed as a parameter that flattens out the link functions in Figure 1 as it gets larger. This is just one of several different types of error story that could be used, and Wilcox [2007] provides a masterful review of the implications of the alternatives. 7 There is one other important error specification, due originally to Fechner [1860] and popularized by Becker, DeGroot and Marschak [1963] and Hey and Orme [1994]. This error specification posits the latent index LEU = (EU R - EU L )/: (3 ) instead of (3), (3') or (3O). In our analyses we default to the use of the Fechner specification, but recognize that we need to learn a great deal more about how these stochastic error specifications interact with substantive inferences (e.g., Loomes [2005], Wilcox [2007], Harrison and Rutström [2007; 2.3]). Panel B of Table 1 illustrates the effect of incorporating a Fechner error story into the basic EUT specification of Panel A. There is virtually no change in the point estimate of risk attitudes, but a slight widening of the confidence interval. Panel C illustrates the effects of allowing for observable individual characteristics in this structural model. The core coefficients r and : are each specified as a linear function of several characteristics. The heterogeneity of the error specification : is akin to allowing for heteroskedasticity, but it is important not to confuse the structural error parameter : from the sampling errors associated with parameter estimates. We observe that the effect of sex remains 7 See Harless and Camerer [1994], Hey and Orme [1994] and Loomes and Sugden [1995] for the first wave of empirical studies including some formal stochastic specification in the version of EUT tested. There are several species of errors in use, reviewed by Hey [1995][2002], Loomes and Sugden [1995], Ballinger and Wilcox [1997], and Loom

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