Estimating subjective probabilities

Abstract

ABSTRACT. 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. † Department of Economics, Copenhagen Business School, Copenhagen, Denmark (Andersen); Department of Economics, University of Canterbury, Christchurch, New Zealand (Fountain); Department of Risk Management & Insurance and CEAR, Robinson College of Business, Georgia State University, USA (Harrison); and Robinson College of Business, Georgia State University, USA (Rutström). E-mail contacts: sa.eco@cbs.dk, john.fountain@canterbury.ac.nz, gharrison@gsu.edu and erutstrom@gmail.com. We thank the U.S. National Science Foundation for research support under grants NSF/HSD 0527675 and NSF/SES 0616746. We are grateful to John Duffy, Peter Wakker, Randall Walsh and Nathaniel Wilcox for discussions.-1-Subjective probabilities about some event are operationally defined as those probabilities that lead an agent to make certain choices over outcomes that depend on that event. These choices could be as natural as placing a bet on a horse race, or as structured as responding to the payoffs provided by some scoring rule. In order to infer subjective probabilities from observed choices of this kind, however, one either has to make some strong assumptions about risk attitudes or jointly estimate risk attitudes and subjective probabilities. We show how the latter can be implemented by pairing several tasks together, some of which identify risk attitudes and some of which identify the interplay between risk attitudes and subjective probabilities. Joint estimation of a structural model of choice across these two types of ta...

How do risk attitudes affect measured confidence?

“…Offerman et al (2009);Trautmann and van de Kuilen (2015) and Andersen et al (2014) elicit subjects' beliefs about uncertain events in a twostep process, using estimates of individual risk attitudes to filter out the effect of risk attitudes from measured beliefs. Our experiment also uses estimated risk attitudes to filter out the effect of risk attitudes from beliefs but a key difference is that we are concerned with biases in subjective estimates of confidence in own performance (not biases in assessments of naturally determined chance events).…”

Scoring rules for subjective probability distributions

“…785] and Kadane and Winkler [1988]. 2 See Offerman, Sonnemans, van de Kuilen and Wakker [2009] or Andersen, Fountain, Harrison and Rutström [2010].…”

The pricing kernel puzzle: survey and outlook

“…Closely related is the work by Bartunek and Chowdhury (1997) who use power utility and Benth, Groth, and Lindberg (2010) who use exponential utility instead; both papers calibrate to options data. While the equity premium literature equates the forward looking physical probability distribution with the historical distribution and determines the intertemporal rate of substitution solely through the risk aversion coefficient, Andersen, Fountain, Harrison, and Rutstroem (2014) suggest to elicit physical probabilities, too, and Sprenger (2012a, 2012b) and Laury, McInnes, Swarthout, and Von Nessen (2012) additionally estimate time preferences. 2 2 An interesting observation reconciling the two estimation methods (surveys/experiments vs. market based) can be found in Haug, Hens, and Woehrmann (2013) who argue that the typical inclusion of background risk in market studies leads to larger risk aversion coefficients than in surveys or experiments which tend to ignore background risk.…”