When dichotomous choice CVM data is of low quality, the measure of central tendancy is sensitive to assumptions. As I showed in a paper presented earlier this year (Landry and Whitehed 2020), with the highest quality data it makes no difference the WTP estimator that is used. The Turnbull, Kristrom, linear logit (under both zero WTP assumptions), and linear probability models all produce the same estimate.
As data quality falls, however, the choice of WTP estimate can matter a great deal. In this situation, so as to avoid sponsor and other biases, it is important for the CVM researcher to present the full range of WTP estimates and avoid the impression that results have been cherry picked. This range of WTP provides a more complete depiction of analyst uncertainty and allows for sensitivity and other analyses.
I have grown accustomed to intense suspicion whenever I see hypothesis tests conducted with only the Turnbull WTP estimate. First, it is a lower bound WTP estimate and potential differences are minimized. Second, its standard errrors are smaller (relative to the mean) than parametric WTP estimates. This second observation is due to the way that the standard errors are calculated and to the fact that the data are smoothed when there are non-monotonicities. As Haab and McConnell (1997, p 253) explained (emphasis added): "We demonstrate that the Turnbull model ... provides a straightforward alternative to parametric models, so long as one simply wants to estimate mean willingness to pay." When hypothesis tests are being conducted, a range of WTP estimates should be used to determine if the results are robust to estimation method.
So, is it reasonable to include the linear-in-bid parametric model in this collection of WTP estimates? Hanemann (1984, 1989) showed that in a linear utility model, U = a(Q) + bY where Q is a good and Y is income, the mean (and median) willingness to pay is WTP = -a*/b, where a* is the change in utility from changes in Q and b is the marginal utility of income. One benefit of this estimate is that it is insensitive to fat tails. However, this estimate allows for negative WTP values unless the probability of a yes response to a dichotomous choice question is 100% when the bid amount is zero. Negative WTP values can enter into the analysis in two ways. First, the WTP estimate itself can be negative. This will occur when the probability of a yes response at the lowest bid amount is less than 50% (it is this possibility that, I think, motivated Haab and McConnell). The second possibility is that the empirical distribution of WTP can include negative values. This is of little consequence to the analysis unless the confidence interval includes zero. Both circumstances arise with the DMT (2015) data.
DMT (2020) dismiss outright the possibility of negative WTP. Their dismissal is consistent with Haab and McConnell's argument that since public goods are freely disposable, negative WTP is only an empirical artifact of a distributional assumption. But, with government policy free disposal is not always possible. In the case of a clean up of natural resource damages, the clean up could be considered a wasteful intrusion into a private business decision. Bohara, Kerkvliet and Berrens (2001) discuss how and why negative WTP values might arise, along with empirical examples. Considering this, I would not be surprised if some of the respondents to CVM scenarios demanded compensation for environmental cleanup.
There have been a number of suggestions about how to handle negative willingness to pay. Many of these involve obtaining more data with follow-up questions (Landry and Whitehead). Unfortunately, the DMT (2015) survey data does not have any of this supplemental information. In that case, in my opinion, an assumption that negative WTP is a possibility can not be ruled out. Inclusion of the linear model allowing for negative WTP, as long as it is presented along with other estimates, should not be dismissed outright.
DMT (2020) state: "This means that adding-up passed in his calculations on linear models not because of the data but because of his implausible additional assumption that many people have a negative WTP for the environmental programs." It is not true that the linear model finds that "many" people have negative willingness to pay in each of the scenarios. According to the Krinsky-Robb WTP simulation, the percentage of negative WTP values for the whole, first, second, third, and fourth scenarios in the DMT (2020) data are 2%, 0.01%, 77%, 25% and 0.83%. The WTP from the second scenario is negative (situation 1 above). The WTP from the third scenario has a Delta method confidence interval that includes zero (situation 2 above).
If the negative mean WTP from the second scenario is set equal to zero then the difference in WTP for the whole and the sum of its parts is statistically significant at the p=0.088 level with the Delta Method confidence intervals. The Krinsky-Robb confidence interval is [68, 788] which includes the sum of the WTP for the parts with WTP from the second scenario set equal to zero ($467) indicating that the adding-up test is supported. It is still my contention that the adding-up passed in the (untruncated) linear model not because of the data.
My conclusion is that the negative WTP values do not have an important effect on the adding-up tests. Dismissing these tests because negative WTP values are implausible ignores the literature and the empirical evidence.
References
Bohara, Alok K., Joe Kerkvliet, and Robert P. Berrens. "Addressing negative willingness to pay in dichotomous choice contingent valuation." Environmental and Resource Economics 20, no. 3 (2001): 173-195.
Landry, Craig, and John Whitehead, "Estimating Willingness to Pay with Referendum Follow-up Multiple-Bounded Payment Cards," paper presented at the 2020 W-4133, Athens, GA, February.
Haab, Timothy C., and Kenneth E. McConnell. "Referendum models and negative willingness to pay: alternative solutions." Journal of Environmental Economics and Management 32, no. 2 (1997): 251-270.
Hanemann, W. Michael. "Welfare evaluations in contingent valuation experiments with discrete responses." American journal of agricultural economics 66, no. 3 (1984): 332-341.
Hanemann, W. Michael. "Welfare evaluations in contingent valuation experiments with discrete response data: reply." American journal of agricultural economics 71, no. 4 (1989): 1057-1061.
