Parametric preference functionals under risk in the gain domain: A Bayesian analysis

Abstract

The performance of rank dependent preference functionals under risk is comprehensively evaluated using Bayesian model averaging. Model comparisons are made at three levels of heterogeneity plus three ways of linking deterministic and stochastic models: differences in utilities, differences in certainty equivalents and contextual utility. Overall, the “best model”, which is conditional on the form of heterogeneity, is a form of Rank Dependent Utility or Prospect Theory that captures most behaviour at the representative agent and individual level. However, the curvature of the probability weighting function for many individuals is S-shaped, or ostensibly concave or convex rather than the inverse S-shape commonly employed. Also contextual utility is broadly supported across all levels of heterogeneity. Finally, the Priority Heuristic model is estimated within a stochastic framework, and allowing for endogenous thresholds does improve model performance although it does not compete well with the other specifications considered.

Appendices

Appendix A1: Transformations

The parameters of interest in the models take 𝜃 in only one of two forms. That is, we parameterise our model by using $\theta =t_{1}( \vartheta ;\delta _{l},\delta _{u}) =\delta _{l}+ ( \delta _{u}-\delta _{l}) \frac {e^{\vartheta }}{1+e^{\vartheta }}$ or 𝜃=t ₂(𝜗)= exp(𝜗) where 𝜗∈R. In the case of t ₁(𝜗;δ _l ,δ _u ) the transformed parameter lies within the interval (δ _l ,δ _u ). We set the values for {δ _i }a priori in accordance with the inequality constraints. The priors for parameters of the form t ₁(𝜗;δ _l ,δ _u ) are ( 𝜗∼N(0,ζ)) where they are assigned a variance ζ equal to $\frac {9}{4}$ , yielding an approximately uniform prior within the specified interval, although there is less mass at the very extremes. Thus, in a sense we are being ‘non-informative’ about the values except that we have specified the interval over which the parameters lie. For parameters of the form t ₂(𝜗) we assume that 𝜗 is normally distributed so that the implied prior distribution for the transformed parameter is log-normal.

Appendix A2: Pratt coefficients

The v-forms in the text are as follows:

$$\begin{array}{@{}rcl@{}} \text{POWER-I} &:& pc=\frac{\left( 1-\alpha_{1}\right) }{x}:\alpha_{1}>0 \\ \text{EXPO-I} &:& pc=\alpha_{2}:\alpha_{2}>0 \\ \text{LOG} &:& pc=\frac{\alpha_{3}}{(1+\alpha_{3}x)}:\alpha_{3}>0 \\ \text{QUAD} &:& pc=\frac{\alpha_{4}}{1-\alpha_{4}x}:\ \alpha_{4}>0,\ \alpha_{4}<\frac{2}{x_{\max }} \\ \text{POWER-II} &:& pc=\frac{\left( 1-\alpha_{5}\right) }{\alpha_{6}+x}:\ \alpha_{5}>0,\ \alpha_{6}>0 \\ \text{EXPO-II} &:&pc=\left( \left( \alpha_{7}-1\right) x^{-1}+\alpha_{7}\alpha_{8}x^{\alpha_{8}-1}\right) :\alpha_{7}>0,\text{ }0.5<\alpha_{8}<1.5 \end{array} $$