Electronic Supplementary Material: Econometric Models
Preferences based on choices made in the choice experiment are modeled in terms of McFadden’s [1] Random Utility Model (RUM), allowing for a separation of utility () into a deterministic () and stochastic part (). CEs fall in the class of attribute-based methods in which the deterministic part of utility for individual i for good j in choice task t is described in (1) as a linear function of its attributes and other explanatory variables [2]:
(1)
In each choice task the respondent is presented with a limited set of product specifications, each proposing in this specific study an improvement in domestic water supply and quality. The stochastic term is assumed to follow an IID type 1 extreme value distribution.
To account for preference heterogeneity, the preference parameters for the choice attributes are allowed to vary across respondents, applying different mixing distributions. Equation (2) describes the mixed logit (ML) probability of individual i selecting alternative j in choice task t over other choice alternatives k. The utility coefficients vary across individuals (hence i) with density . This density can be a function of any set of parameters and represents in this case the mean and covariance of in the sample population.
(2)
ML-models assume heterogeneity to be continuous over the interval spanned by the assumed distribution for the preference parameters [3]. Treating preference parameters as random variables requires estimation through simulated maximum likelihood. Procedurally, the maximum likelihood algorithm searches for a solution by simulating draws from distributions with given means and standard deviations. Probabilities are calculated by integrating the joint simulated distribution.
Even if unobserved heterogeneity is accounted for in a ML-model, the model may fail to explain the sources of heterogeneity [4]. To this end, respondent specific household characteristics can be included in the utility function to improve the model fit [5]. Through the inclusion of a price attribute in the CE, WTP welfare estimates can be derived (e.g. [6]). The welfare measure represents the monetary value arising from a specified bundle of drinking water filter characteristics. This WTP welfare measure is compared to the directly stated WTP value in the CV[1].
The latter payment card WTP responses are modeled using Tobit regression in order to account for the limited dependent nature of the response variable. The WTP model can be written as follows (e.g. [7]):
if WTP > 0 (3)
otherwise (4)
where in equation (3) denotes the matrix of explanatory variables (in this case respondent characteristics) and the error term is assumed to be normal distributed with zero mean and variance 2. The estimates for the regression coefficients are obtained through maximum likelihood (ML) techniques. Since these Tobit estimates do not reflect marginal effects [8], they have to be adjusted using the cumulative standard normal distribution function.
References
[1] McFadden D, Train K. 2000. Mixed MNL models for discrete response. Journal of Applied Econometrics, 15: 447-470.
[2] Train K. 2003. Discrete Choice Methods with Simulation. Cambridge University Press: Cambridge, UK.
[3] McFadden D, Train KE. 2000. Mixed MNL models for discrete response. Journal of Applied Econometrics, 15(5): 447-470.
[4] Hynes S, Hanley N, Scarpa R. 2008. Effects on welfare measures of alternative means of accounting for preference heterogeneity in recreational demand models. American Journal of Agricultural Economics, 90(4): 1011-1027.
[5] Revelt D, Train K. 1998. Mixed logit with repeated choices: households’ choice of appliance efficiency level. Review of Economics and Statistics, 53: 647–657.
[6] Hensher DA, Rose JM, Greene WH. 2005. Applied choice analysis. A primer. Cambridge University Press: Cambridge, UK.
[7] Greene WH. 2003. Econometric Analysis. Prentice Hall International, Inc., Upper Saddle River, New Jersey.
[8] Halstead JM, Lindsay BE, Brown CM. 1991. Use of the Tobit model in contingent valuation: Experimental evidence from the Pemigewasset Wilderness Area. Journal of Environmental Management, 33: 7989.
[1] Because respondents are asked to choose between two alternative filter specifications in the CE compared to none of the two and the choice model relates to the utility of the two unlabeled hypothetical alternatives compared to the base level of having no improved filter, the estimated WTP welfare measure derived from the choice model is directly comparable to the CV-based estimated WTP.