Estimating Subjective Probabilities

Subjective probabilities play a role in many economic decisions. There is a large theoretical literature on the elicitation of subjective probabilities, and an equally large empirical literature. However, there is a gulf between the two. The theoretical literature proposes a range of procedures that can be used to recover subjective probabilities, but stresses the need to make strong auxiliary assumptions or "calibrating adjustments" to elicited reports in order to recover the latent probability. With some notable exceptions, the empirical literature seems intent on either making those strong assumptions or ignoring the need for calibration. We illustrate how the joint estimation of risk attitudes and subjective probabilities using structural maximum likelihood methods can provide the calibration adjustments that theory calls for. This allows the observer to make inferences about the latent subjective probability, calibrating for virtually any well-specified model of choice under uncertainty. We demonstrate our procedures with experiments in which we elicit subjective probabilities. We calibrate the estimates of subjective beliefs assuming that choices are made consistently with expected utility theory or rank-dependent utility theory. Inferred subjective probabilities are significantly different when calibrated according to either theory, thus showing the importance of undertaking such exercises. Our findings also have implications for the interpretation of probabilities inferred from prediction markets.

Inferring Beliefs as Subjectively Uncertain Probabilities

We propose a method for estimating subjective beliefs, viewed as a subjective probability distribution. The key insight is to characterize beliefs as a parameter to be estimated from observed choices in a well-defined experimental task, and to estimate that parameter as a random coefficient. The experimental task consists of a series of standard lottery choices in which the subject is assumed to use conventional risk attitudes to select one lottery or the other, and then a series of betting choices in which the subject is presented with a range of bookies offering odds on the outcome of some event that the subject has a belief over. Knowledge of the risk attitudes of subjects conditions the inferences about subjective beliefs. Maximum simulated likelihood methods are used to estimate a structural model in which subjects employ subjective beliefs to make bets. We present evidence that some subjective probabilities are indeed best characterized as probability distributions with non-zero variance.

Behavioral econometrics for psychologists ?

We make the case that psychologists should make wider use of structuraleconometric methods. These methods involve the development of maximumlikelihood estimates of models, where the likelihood function is tailored to thestructural model. In recent years these models have been developed for a wide rangeof behavioral models of choice under uncertainty. We explain the components ofthis methodology, and illustrate with applications to major models from psychology.The goal is to build, and traverse, a constructive bridge between the modelinginsights of psychology and the statistical tools of economists

Discounting behavior : a reconsideration

We re-evaluate the theory, experimental design and econometrics behind claims that individualsexhibit non-constant discounting behavior. Theory points to the importance of controlling for thenon-linearity of the utility function of individuals, since the discount rate is defined over time-datedutility flows and not flows of money. It also points to a menagerie of functional forms tocharacterize different types of non-constant discounting behavior. The implied experimental designcalls for individuals to undertake several tasks to allow us to identify these models, and to severaltreatments such as multiple horizons and the effect of allowing for a front end delay on earlierpayments. The implied econometrics calls for structural estimation of the theoretical models,allowing for joint estimation of utility functions and discounting functions. Using data collectedfrom a representative sample of 413 adult Danes in 2009, we draw striking conclusions. Assumingan exponential discounting model we estimate discount rates to be 5.6% on average: this issignificantly lower than all previous estimates using controlled experiments. We also find noevidence to support quasi-hyperbolic discounting or “fixed cost” discounting, and only modestevidence to support other specifications of non-constant discounting. Furthermore, the evidence fornon-constant discounting, while statistically significant, is not economically significant in terms ofthe size of the estimated discount rates. We undertake extensive robustness checks on thesefindings, including a detailed review of the previous, comparable literature

Intertemporal utility and correlation aversion

Convenient assumptions about qualitative properties of the intertemporal utility function have generated counter-intuitive implications for the relationship between atemporal risk aversionand the intertemporal elasticity of substitution. If the intertemporal utility function is additively separable then the latter two concepts are the inverse of each other. We review a simple theoretical specification with a long lineage in the literature on multi-attribute utility, and demonstrate the critical role of a concept known as intertemporal risk aversion or intertemporal correlation aversion.This concept is the intertemporal analogue of a more general concept applied to two attributes of utility, but where the attributes just happen to be the time-dating of the good. In the context of intertemporal utility functions, the concept provides an intuitive explanation of possible differences between (the inverse of) atemporal risk aversion and the intertemporal elasticity of substitution. We use this theoretical structure to guide the design of a series of experiments that allow us to identify and estimate intertemporal correlation aversion. Our results show that subjects are correlation averse over lotteries with intertemporal income profiles, and that the convenient additive specification of the intertemporal utility function is not an appropriate representation of preferences over time

Non-linear mixed logit

We develop an extension of the familiar linear mixed logit model to allowfor the direct estimation of parametric non-linear functions defined over structuralparameters. Classic applications include the estimation of coefficients of utilityfunctions to characterize risk attitudes and discounting functions to characterizeimpatience. There are several unexpected benefits of this extension, apart from theability to directly estimate structural parameters of theoretical interest

Discounting behavior and the magnitude effect

We evaluate the claim that individuals exhibit a magnitude effect in their discounting behavior,which is said to occur when higher discount rates are inferred from choices made with lowerprincipals, all else being equal. If the effect is robust, as claimed, we should be able to see it usingprocedures that are more familiar to economists. Using data collected from a representative sampleof adult Danes, we find statistically significant evidence of a small magnitude effect, at levels that are much smaller than is typically claimed. This evidence only surfaces if one carefully controls forunobserved individual heterogeneity in the population. And it disappears completely if we includediscounting choices in which both options have some time delay