When dichotomous choice CVM data has a negative WTP problem, one of the standard corrections is to estimate a log-linear model and present the median WTP. With many estimated log-linear models the mean WTP is undefined. This is because the log-linear model flattens the estimated survival curve and, in contrast to a linear model, the probability of a no response does not approach zero at any reasonable bid amount. The median WTP estimate from the log-linear model tends to be a useful supplement to the welfare measures available from the linear model.
Desvousges, Mathews and Train (2020) rightly argue that (1) the sum of medians is not equal to the median of the sums and (2) the mean WTP estimates for each of their five scenarios are basically infinity (or in the millions of dollars when the data are estimated with a log-linear probit). Given this empirical fact, there seems to be no median estimate available for the sum of the four individual WTP amounts (that could be compared to the median of the whole). However, DMT (2020) are able to estimate the “median of the sum of WTPs through simulation” to be $4904. They then explain that since $4904 is 24 times that of the median of the whole scenario, $201, is “clearly a violation of adding up”. Given that the confidence interval from the Delta Method, [-47, 449], does not include $4904 we could reject the notion that the willingness to pay estimates pass this version of the adding-up test.
Previously however, I’ve argued that these standard errors are likely the wrong ones to use since the cost parameter is measured without much precision. In this case the Krinsky-Robb confidence intervals are more appropriate. The Krinsky-Robb confidence interval for the median WTP estimate for the whole scenario is [54, 9558]. Since the median of the sum of the WTP estimates from the four adding-up scenarios, estimated by DMT (2020) to be $4904, lies within the 95% Krinsky-Robb confidence interval then we fail to reject the adding-up hypothesis at the 95% confidence level. In contrast, the 95% Krinsky-Robb confidence interval estimated with the whole scenario data from Chapman et al. (2009), which I’ve argued is higher quality data, is relatively tight around the median of $167: [134, 217].
The only other adding-up test that can be conducted with median WTP estimates is to compare the median for the whole with the sum of the medians for the four parts. In this test I found that the median WTP estimates passed the adding up test using the confidence intervals from the Delta Method (Whitehead 2020). It still seems to me that this test is a useful supplement when one is inclined to consider the robustness of the adding-up test conducted with only the Turnbull estimator (as in DMT 2015). Otherwise, we’re treating mean WTP estimates in the millions as if they are meaningful.
To answer the question in the title of this post, the log-linear model is not meaningless. In fact, it is a better model statistically than the linear model for the first and third WTP scenarios in DMT (2015). Information gleaned from the log-linear model provides insights into the quality of the DMT (2015) data.
While I argue with DMT over the minutiae of these tests and different estimators, the reader shouldn’t lose site of how silly the debate over DMT (2015) has become. The bottom line is that the DMT (2015) data is of low quality data and do not rise to the threshold that is needed to support an adding-up test, which requires estimates of willingness to pay as ratios of coefficients.