Abstract
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.
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Ahn, D., Choi, S., Gale, D., & Kariv, S. (2009). Estimating ambiguity aversion in a portfolio choice experiment. Working paper, Department of Economics, University of California at Berkeley, Feb 2009.
Aitchison J., Begg C.B. (1976) Statistical diagnosis when basic cases are not classified with certainty. Biometrika 63(1): 1–12
Andersen, S., Fountain, J., Harrison, G. W., & Rutström, E. E. (2010). Estimating subjective probabilities. Working Paper 2010–2006, Center for the Economic Analysis of Risk, Robinson College of Business, Georgia State University.
Andersen, S., Harrison, G. W., Hole, A. R., Lau, & Ruström, E. E. (2011). Non-linear mixed logit. Theory and Decision. doi:10.10071s112.38-011-9277-00.
Andersen S., Harrison G. W., Lau Morten I., Rutström E. E. (2008) Eliciting risk and time preferences. Econometrica 76(3): 583–618
Binswanger H. P. (1981) Attitudes toward risk: Theoretical implications of an experiment in rural India. Economic Journal 91: 867–890
de Finetti, B. (1937). La Prévision: Ses Lois Logiques, ses sources subjectives. Annales de l’Institut Henri Poincairé, 7, 1–68; English translation as “Foresight: Its logical laws, its subjective sources,” in H. E. Kyburg & H. E. Smokler (Eds.), Studies in subjective probability. Huntington, NY: Robert E. Krieger, 1980, Second Edition.
de Finetti B. (1970) Logical foundations and measurement of subjective probability. Acta Psychologica 34: 129–145
Drukker D.M., Gates R. (2006) Generating Halton sequences using mata. Stata Journal 6(2): 214–228
Eckel C. C., Grossman P. J. (2002) Sex differences and statistical stereotyping in attitudes toward financial risk. Evolution and Human Behavior 23(4): 281–295
Ellsberg D. (1961) Risk, ambiguity, and the savage axioms. Quarterly Journal of Economics 75: 643–669
Epstein R. A. (1977) The theory of gambling and statistical logic. Academic Press, San Diego
Ergin H., Gul F. (2009) A theory of subjective compound lotteries. Journal of Economic Theory 144(3): 899–929
Ghirardoto P., Maccheroni F., Marinacci M. (2004) Differentiating ambiguity and ambiguity attitude. Journal of Economic Theory 118: 133–173
Gilboa I., Postlewaite A. P., Schmeidler D. (2008) Probability and uncertainty in economic modeling. Journal of Economic Perspectives 22(3): 173–188
Gilboa I., Schmeidler D. (1989) Maxmin expected utility with a non-unique prior. Journal of Mathematical Economics 18: 141–153
Grant S., Kajii A., Polak B. (1998) Intrinsic preference for information. Journal of Economic Theory 83: 233–259
Halevy Y. (2007) Ellsberg revisited: An experimental study. Econometrica 75: 503–536
Harrison G. W., Rutström E. E. (2008) Risk aversion in the laboratory. In: Cox J. C., Harrison G. W. (eds) Risk aversion in experiments. Emerald, Bingley, UK
Hey, J. D., Lotito, G., Maffioletti, A. (2007). Choquet OK? Discussion Paper No. 2007/12, Department of Economics and Related Studies, University of York.
Hey, J. D., Lotito, G., Maffioletti, A. (2008). The descriptive and predictive adequacy of theories of decision making under uncertainty/ambiguity. Discussion Paper No. 2008/04, Department of Economics and Related Studies, University of York.
Hey J. D., Orme C. (1994) Investigating generalizations of expected utility theory using experimental data. Econometrica 62(6): 1291–1326
Hole A. R. (2007) Fitting mixed logit models by using maximum simulated likelihood. Stata Journal 7(3): 388–401
Holt C. A., Laury S. K. (2002) Risk aversion and incentive effects. American Economic Review 92(5): 1644–1655
Huber J., Train K. (2001) On the similarity of classical and Bayesian estimates of individual mean partworths. Marketing Letters 12(3): 259–269
Klibanoff P., Marinacci M., Mukerji S. (2005) A smooth model of decision making under ambiguity. Econometrica 73(6): 1849–1892
Lesaffre E., Rizopoulos D., Tsonaka R. (2007) The logistic transform for bounded outcome scores. Biostatistics 8(1): 72–85
Machina M. J. (2004) Almost-objective uncertainty. Economic Theory 24: 1–54
Mathieson J. E., Winkler R. L. (1976) Scoring rules for continuous probability distributions. Management Science 22(10): 1087–1096
Nau R. F. (2006) Uncertainty aversion with second-order utilities and probabilities. Management Science 52: 136–156
Nau R. F. (2007) Extensions of the subjective expected utility model. In: Ward E., Ralph M., von Detlof W. (eds) Advances in decision analysis: From foundations to applications. Cambridge University Press, New York
Neilson W. S. (2010) A simplified axiomatic approach to ambiguity aversion. Journal of Risk and Uncertainty 41: 113–124
Oehlert G. W. (1992) A note on the delta method. The American Statistician 46(1): 27–29
Quiggin J. (1993) Generalized expected utility theory: The rank-dependent model. Kluwer Academic, Norwell, MA
Revelt D., Train K. (1998) Mixed logit with repeated choices: Households’ choices of appliance efficiency levels. Review of Economics and Statistics 80: 647–657
Saha A. (1993) Expo-power utility: A flexible form for absolute and relative risk aversion. American Journal of Agricultural Economics 75(4): 905–913
Savage L. J. (1971) Elicitation of personal probabilities and expectations. Journal of American Statistical Association 66: 783–801
Segal U. (1987) The Ellsberg paradox and risk aversion: An anticipated utility approach. International Economic Review 28(1): 175–202
Segal U. (1990) Two-stage lotteries without the independence axiom. Econometrica 58(2): 349–377
Smith V. L. (1969) Measuring nonmonetary utilities in uncertain choices: The Ellsberg Urn. Quarterly Journal of Economics 83(2): 324–329
Train K. E. (2003) Discrete choice methods with simulation. Cambridge University Press, New York
Wilcox N. T. (2008) Stochastic models for binary discrete choice under risk: A critical primer and econometric comparison. In: Cox J., Harrison G. W. (eds) Risk aversion in experiments. Emerald, Bingley, UK
Wilcox N.T. (2011) ‘Stochastically More Risk Averse:’ A Contextual Theory of Stochastic Discrete Choice Under Risk. Journal of Econometrics 162(1): 87–104
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Andersen, S., Fountain, J., Harrison, G.W. et al. Inferring beliefs as subjectively imprecise probabilities. Theory Decis 73, 161–184 (2012). https://doi.org/10.1007/s11238-011-9276-1
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DOI: https://doi.org/10.1007/s11238-011-9276-1