Estimating parametric loss aversion with prospect theory: Recognising and dealing with size dependence

Highlights

• We introduce an approach to generalise loss aversion estimation.

• We remove the need to impose unnecessary restriction on model estimation.

• We find support for Prospect Theory and evidence of mild loss aversion.

• Our results support the need for allowing respondent heterogeneity.

• Researchers should avoid a priori imposing average conditions on participants.

Abstract

Parameteric identification of loss aversion requires either the imposition of rotational symmetry on the utility function or a point dependent normalization condition. In this paper, we propose a new approach in which point dependence is reduced by integration over normalization points. To illustrate our approach, we consider a sample of Ghanaian farmers’ risk preferences over the gain, loss and mixed domains. Using Bayesian econometric methods, we find support for Prospect Theory albeit with substantial behavioral variation across individuals plus mild overweighting of losses compared to gains. We also show that the majority of respondents are mildly loss averse especially as the size of the payoffs increase.

Introduction

In this paper, we propose an approach to estimating loss aversion that recognises that loss aversion cannot be divorced by the scale/size of the gains and losses being compared. We make explicit that, in general, loss aversion cannot be decoupled from the scale of losses versus gains without imposing restrictions that may not be supported by data. A potential solution to this problem is to aggregate over scales to arrive at a global measure of loss aversion. In our empirical study, we demonstrate and provide practical guidance on how to undertake calculations to that purpose. Our approach requires no consideration of scale prior to estimation, but it makes clear that scale is, in general, necessary for ex-post interpretation of loss aversion measures.

The framework we adopt in this paper is a familiar one. Following Tversky and Kahneman (1992), we define utility (or ‘value’) as a piecewise function governed by three parameters α, β and λ, where α and β govern utility curvature in the gain and loss domains and λ pivots utility at zero. It is common within this framework to impose the restriction that $\alpha =\beta$. The appeal of this restriction is that it simplifies the interpretation of λ, typically interpreted as an index of loss aversion. The importance of this restriction is apparent from the way in which Prospect Theory (PT) has been implemented and interpreted since Tversky and Kahneman (1992). For example, loss aversion as formulated by Köbberling and Wakker (2005) requires symmetry for the constant relative risk aversion (CRRA) utility function and if λ is identified from a normalization restriction, then its value will be dependent on a point of normalization unless symmetry is imposed. From an econometric perspective, we argue that researchers should not automatically impose utility ‘symmetry’ within PT models just because of the apparent simplicity it affords in the interpretation of λ.

From a statistical perspective, there is no reason to impose symmetry without testing the restriction first. Indeed, there are studies that provide statistical evidence against symmetry. But, because of the issues surrounding the definition and interpretation of loss aversion, these results are often reported with strong caveats. For example Andersen et al. (2010) state:

“...there is a significant theoretical trade-off if one maintains this difference between α and β, stressed by Köbberling and Wakker (2005, Section 7), so this is not the sort of constraint that one should decide on purely empirical grounds.” (p. 561).

Furthermore, rather than rejecting symmetry when it is inconsistent with the data, some researchers take a different stance. For example, Scheibehenne and Pachur (2015) support the imposition of the equality restriction;

“In our implementation of CPT, we set $\alpha =\beta$ because (Nilsson et al., 2011) found that estimating α and β separately can lead to a serious mis-estimation of λ .” (p. 394).

Here, we take the view that symmetry should be tested empirically, not simply imposed a priori. Furthermore, if symmetry is not imposed, this does not preclude examining the extent of loss aversion, providing loss aversion is seen as being dependent on the size of prospect payoffs.

We propose an empirical approach that recognizes that, without symmetry, the identification of λ requires a normalization method along with a point of normalization (a point made by Köbberling and Wakker (2005)). Specifically, as an alternative to imposing symmetry or the selection of a single normalization point (which is standard practice in the literature, as discussed by Abdellaoui et al. (2007)), we propose and implement a method that averages over a weighted set of payoffs based on the use of a Gamma ‘importance weighting’ function. Our approach yields a “global” measure of loss aversion conditional on weightings selected by the researcher. As recognised in the literature, existing global measures of loss aversion offer attractive properties however, as observed by Abdellaoui et al. (2007) they have significant limitations when used empirically. As we demonstrate, size dependence is mitigated by our approach, but does not disappear altogether. In addition, our method also addresses the importance of ‘scale’ dependence in the evaluation of losses versus gains when symmetry of the utility function is not satisfied. This matters as numerous works that report degrees of loss aversion are divorced from the magnitude of the gains and lossses being compared. While some literature recognises this not to be a universal feature (e.g. Abdellaoui et al., 2007; p.1662), we believe that a clear articulation of this point is absent. For example, we do not find that (Abdellaoui et al., 2007) clearly articulate that if a comparison of loss aversion parameters between studies is being made that researchers must use the same “real” point of comparison, not “nominal” values. However, they do note that numerous authors choose to evaluate risk aversion at different nominal values. Therefore, if the size conditionality of loss aversion was widely recognised, comparisons of loss aversion that ignore this feature would not be common within the literature. The issues we discuss are not confined to a particular utility function, but we specifically deal with two utility specifications: constant relative and constant absolute risk aversion (CRRA and CARA).¹

This article therefore makes three points explicit that pertain to parametric estimates of loss aversion coefficients:

1) In general loss aversion coefficients are size dependent (i.e. the real size of the comparative gains and losses matters);

2) In a parametric context, size dependence can be avoided by assuming a symmetry condition but there is no other compelling reason to do so; and

3) Without a symmetry condition researchers must either pick a point (size) at which loss aversion is evaluated, or alternatively a range of points or some sort of “average” over these points.

Our recommendation is that researchers “average” over the size of payoffs, and do so using a Gamma weighting scheme, admitting that some size comparisons matter more than others, and making this judgement is unavoidable in all empirical contexts. In developing our approach, we make explicit the need for researchers to be aware about their normalization restrictions, and also be cognizant of the requirement that normalizations require real and not just nominal equivalence (of λ estimates) in order to be comparable from one study to another. The applied literature frequently overlooks this requirement, while reporting parameter values that are taken to support ‘loss aversion’.

Apart from the methodological contribution, our research adds to the literature that considers risk preferences in the field using PT (e.g. Liu, Huang, 2013, Tanaka, Camerer, Nguyen, 2010 and Ward and Singh, 2015). We employ data from a survey of Ghanaian farmers. We estimate the data employing alternative utility specifications (CRRA and CARA) whilst allowing for heterogeneity amongst respondents by using a Hierachical Bayesian Logit (HBL). We also adopt a Bayesian log marginal likelihood (LML) approach to model discrimination using the HBL.

Our results unambiguously show that the CRRA is preferred to CARA, and ‘symmetry’ is rejected for the CRRA form (though not by the CARA). The values of λ differ by utility specification, although our approach to integration of normalizations narrows the differences. Moreover, when we consider the implications for certainty equivalents, we find that the differences are actually quite small. The fact that we should not necessarily expect similar estimates of λ when using alternative functional forms is also something we discuss.