Estimating Subjective Probabilities

This is one major paper in a portfolio of 3 papers on using scoring rule methods for belief elicitation. The focus of this paper is on the econometric techniques for joint estimation of beliefs and risk attitudes . The other two focus on (1) theory connecting optimal decisions, scoring rules and betting markets and on (2) experimental methods of implementing scoring rules. This paper was submitted to Econometrica. It was not accepted there and after taking into account referee commnets we have redrafted the paper and sent it to IER.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