Abstract
We introduce a semiparametric smooth coefficient estimator for recreation demand data that allows more flexible modeling of preference heterogeneity. We show that our sample of visitors each has an individual statistically significant price coefficient estimate leading to clearly nonparametric consumer surplus and willingness to pay (WTP) distributions. We also show mean WTP estimates that are different in economically meaningful ways for every demographic variable we have for our sample of beach visitors. This flexibility is valuable for future researchers who can include any variables of interest beyond the standard demographic variables we have included here. And the richer results, price elasticities, consumer surplus and WTP estimates, are valuable to planners and policymakers who can easily see how all these estimates vary with characteristics of the population of interest.
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Notes
Including the six observations has almost no impact on the estimation results in the current water quality scenario. For the improved water quality scenario, including them slightly changes the estimates of bandwidths, and the resulting mean WTP estimates are 18% higher. We return to this discussion in Sect. 5.
Silverman (1986) derives the optimal choice of bandwidth
$$\begin{aligned} h=\left( \frac{4{\hat{\sigma }}^5}{3n}\right) ^{\frac{1}{5}}\approx 1.06{\hat{\sigma }}n^{\frac{1}{5}},\end{aligned}$$where \({\hat{\sigma }}\) is the standard deviation of the samples. This approximation is called the normal distribution approximation or Gaussian approximation.
We first generate bootstrap error \(u_{i}^{*}\) based on the regression residual via Mammen’s two-point distribution. We then generate the bootstrap dependent variable \(y_{i}^{*}\) by adding the bootstrapped error to the fitted value from the regression, i.e. \(y_{i}^{*}=x_{i}{\hat{\beta }}(z_{i})+ u_{i}^{*}\). The bootstrap dependent variable \(y_{i}^{*}\) along with all other covariates are combined into a “bootstrap sample” which is used to estimate bootstrap coefficient \(\beta _{j}^{*}\). This process is repeated for a large number of times, 4000, and the standard deviation of all the bootstrap estimates of coefficient \(\beta _{j}\) is calculated as the standard error. The large number of draws was needed to insure convergence.
All the analysis in this paper is performed in R. The commands for estimation and testing are available in the ‘np’ package. Bootstrap and simulations are carried out using self-written codes.
To achieve stable results, a large number of simulations is needed; we used 4000 draws.
Following Bujosa et al. (2010), we also used the Wilcoxon non-parametric signed-rank test which produced results similar to the nonparametric equality of densities test. However, this test is usually used for observed outcomes, versus our estimated outcomes, so the results are not reported but available upon request.
For example, Ohio’s state gasoline tax is $0.28 per gallon while its eastern neighbor Pennsylvania has the nation’s highest state gasoline tax of $0.582, which is more than double.
The coefficient estimates are available upon request.
We use 500 Halton draws in the simulated maximum likelihood estimation.
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Appendix
Appendix
This appendix provides the summary of coefficient estimates from the semiparametric smooth coefficient mixed Poisson model when the two scenarios are estimated separately (Table 8) and jointly (Table 9).
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Liu, W., Egan, K.J. A Semiparametric Smooth Coefficient Estimator for Recreation Demand. Environ Resource Econ 74, 1163–1187 (2019). https://doi.org/10.1007/s10640-019-00362-7
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DOI: https://doi.org/10.1007/s10640-019-00362-7