1. Designing experiments for efficient WTP estimates Ric Scarpa Prepared for the Choice Modelling Workshop 1st and 2nd of May Brisbane Powerhouse, New Farm Brisbane

2. The background • Profession is exploring designs that incorporate a-priori beliefs on taste intensities • As soon as one moves away from zero taste intensities (indeed an unlikely prior) large efficiency gains are achievable • In non-market valuation the focus is not on accurate taste intensities estimation, but on WTP estimation

3. Structure of the talk • Background on estimation of WTP variance • C-efficiency extended to choice modeling • Some challenges • Some criteria for efficient design • Results from an exploration in the MNL context with utility in the preference space • Future research

4. Variance of WTP from preference-space RUMs • In a typical RUM setting specification of utility is carried out in the preference space • U=kkxk+ $x$ • WTPk =-k / $, with all  estimated by ML • Under ML assumptions  ~ N(b, ACV) • Var(WTPk ) = Var (-k / $) • Received wisdom indicates that approximations to this can be achieved by the so called Krinsky and Robb procedure

5. Krinsky and Robb… an accident? • It is unclear as to why this has become the state of practice and substituted the delta method • Examining the literature one finds: • Krinsky, I. & Robb, A.L. On Approximating the Statistical Properties of Elasticities Review of Economics and Statistics, 1986, 68, 715-719 detection of a problem with delta method • Krinsky, I. & Robb, A.L. On Approximating the Statistical Properties of Elasticities: A Correction Review of Economics and Statistics, 1990, 72, 189-190  acknowledgement that they made a mistake in the first paper • Env. economics literature never took in or referred to the 1990 “Correction” and went on practicing K&R 86, which is basically a parametric bootstrap • However, the delta method works, as indicated by the correction and illustrated in other fields

6. Delta method and C-efficiency • DM offers a close form approximation via Taylor series that can be used to measure C-efficiency using priors on beta • C-efficiency is a form of M-efficiency (M=managerial) in that it is the variance of a managerial quantity that is of importance • It was proposed in the early 90’ by B. Kanninen in CVM design

7. Delta method • Based on Slutsky’s theorem and the properties of the ML estimator • Take any continuous function at least twice differentiable g(). Using the first two terms of a Taylor series approximation to expand it around the estimates one obtains: Where is the vector of K first derivatives, the gradient of g(.), and ' indicates the transpose.

8. Differences with CVM • Of course in SP CM we have k attributes and this complicates things as the dimension of WTP is k-1 • One can use an algorithm that finds the design allocation of attribute in alternatives and their levels that minimize the sum of variances of WTP • This search is unlikely to produce a balanced outcome because attributes with large WTP will have larger variances • For example, the minimum may be reached by achieving a very small variance for one attribute while leaving the variance for another attribute higher than what is desirable. • A potential solution is to use weighting in the construction of the C-error, assigning higher weights to attributes whose variance one wants to reduce most

9. Additional criteria in design • Min-max and max-min criteria can be useful in this context • maximizing the minimum t-value for the WTP: • or equivalently, that of minimizing the number of design replicates necessary to achieve the desired significance level for WTP:

10. Differences with D-efficiency • WTP-efficiency typically does not require maximization outcomes to be linked with continuous variables in the design (e.g. cost) to be placed at the extreme levels of the range • It is dependent on the covariances of the taste intensities, and they typically involve more trade-offs than D-efficient designs

11. Specificity of method in env. economics practice • Most env. econ. Studies include a SQ alternative • Efficiency measures differ in the presence or absence of a SQ constant • Recent studies have also started to pay attention to the effect of scale (Gumbel heteroskedasticity) • Finally, in WTP-based benefit transfer exercises from many different locations the pivot (reference points of respondent may vary) and this also has an impact on optimal WTP-design depending on whether the objective is WTP prediction or choice prediction

13. How does C-efficiency trade off other criteria in practice? • Results of a desk study done with J. Rose (ITLS Sydney)

15. Future research • Adaptive learning design (sequential updating, best with CAPI) • Individual-specific design (pivoting and individual scale factors matter) • Fast implementation designs (one balancing attribute on fixed frame e.g. cost as for Kanninen 2002) • Behaviourally valid designs (maximizing respondent efficiency)