Desvousges, Mathews and Train (2015) [DMT] find that their contingent valuation method (CVM) survey does not pass the adding up test. In a previous post (Whitehead, Sept. 20, 2016) I examined the data used by Desvousges, Mathews and Train (2015) and find that "it is ... not clear if [the data] passes the most basic validity test in contingent valuation over 89% of the range of bids."
There are other things wrong with the paper. DMT find that “WTP for the whole is statistically different from the WTP for the second and third increments and is not statistically different from the WTP for the first and fourth increments” and “The sum of the four increments … is about three times as large as the value of the whole” (p. 566).
A nonparametric estimator is needed since the DMT data exhibits relatively low internal consistency over the bid range and fat tails (Desvousges, Mathews and Train 2015, Parsons and Myers 2016, Appendix A). Given these limitations, following Chapman (2009), DMT choose the ABERS nonparametric estimator for willingness to pay (Ayer et al. 1955). The ABERS estimates in DMT are similar to the more familiar Turnbull nonparametric WTP estimates, which is a lower bound (Haab and McConnell 2003).  The Turnbull nonparametric WTP estimate is the area under the step function formed by the unconditional probability of a yes response at each bid amount.
In contrast, the Kriström (1990) nonparametric WTP estimate is the area under the demand function formed by assuming a probability of a yes response of one at a zero bid, using linear interpolation to determine an estimate of the bid amount that leads to a probability of zero (i.e., the choke price) and using linear interpolation for the slope of the survival function between bid amounts.
In both the Turnbull and Kriström estimators when the unconditional probability of a yes response is not monotonically decreasing, as in the DMT data, the probability of a yes response is pooled back to the lowest bid amount over the range of non-monotonically increasing bid amounts. These probabilities are presented in Table 1. The upper bid amounts that drive the probability of a yes response to zero (i.e., the choke prices) are 2096, 560, 677, 608 and 561 for the whole and first, second, third and fourth increments. These are estimated by linear interpolation in R following Kriström (1990) (Aizaki, Nakatani and Sato, 2014).
Using these survival functions I estimate the Kriström WTP estimates (Table 2). The 95% confidence intervals are computed using the formulas for the variance developed by Bowman, Bostedt and Kriström (1999). The variance for the sum of the four parts is the square root of the sum of the variances of the four parts. In contrast to DMT I find that WTP for the whole is statistically different from WTP for each of the incremental parts. The sum of the WTP for the four increments, $760 (standard error = 103), is not statistically different than the value of the whole, $576 (standard error = 84) (Figure 1).
The DMT data passes the scope and adding up tests with an alternative estimate of WTP than that used by DMT. This result is driven by the flatness of the survival function for the whole scenario over the higher end of the bid range ($125-$405). Given the fat tails problem this estimate of the choke price is arbitrary but no less arbitrary than truncation at the highest bid amount as in DMT. The ABERS/Turnbull WTP estimate used by DMT is appropriate for natural resource damage assessment where a lower bound estimate is desired (Chapman 2009). The ABERS/Turnbull WTP is less appropriate for validity testing of the CVM.
Figure 1. The sum of the parts is equal to the whole
 I find differences between 0% and 13% between the WTP estimates presented by DMT and the Turnbull estimates computed in R (Aizaki, Nakatani, and Sato, 2014).
Aizaki, Hideo, Tomoaki Nakatani, and Kazuo Sato. Stated Preference Methods Using R. CRC Press, 2014.
Ayer, Miriam, H. D. Brunk, G. M. Ewing, W. T. Reid, and Edward Silverman. 1955. “An Empirical Distribution Function for Sampling with Incomplete Information.” Annals of Mathematical Statistics 26 (4): 641–47.
Boman, Mattias, Göran Bostedt, and Bengt Kriström. "Obtaining welfare bounds in discrete-response valuation studies: A non-parametric approach." Land Economics (1999): 284-294.
Chapman, David, Richard Bishop, Michael Hanemann, Barbara Kanninen, Jon Krosnick, Edward Morey and Roger Tourangeau. 2009. Natural Resource Damages Associated with Aesthetic and Ecosystem Injuries to Oklahoma’s Illinois River System and Tenkiller Lake.
Desvousges, William, Kristy Mathews, and Kenneth Train. “An Adding Up Test on Contingent Valuations of River and Lake Quality.” Land Economics 91(2015): 556-571.
Haab, Timothy C., and Kenneth E. McConnell. Valuing Environmental and Natural Resources: The Econometrics of Non-market Valuation. Edward Elgar Publishing, 2002.
Kriström, Bengt. "A non-parametric approach to the estimation of welfare measures in discrete response valuation studies." Land economics 66, no. 2 (1990): 135-139.
Parsons, George R., and Kelley Myers. "Fat tails and truncated bids in contingent valuation: An application to an endangered shorebird species." Ecological Economics 129 (2016): 210-219.