Distributional Assumptions and the Estimation of Contingent Valuation Models

Abstract

Contingent valuation methods are well-established techniques for measuring the value of goods and services not transacted in markets and have been applied in many different settings. Some of these applications include estimating the value of outdoor recreation, reducing risk, decreasing pollution, or reducing transportation time. The parameter estimates depend upon the survey design, the model specification, and the method of estimation. Distributional misspecification or heteroskedasticity can lead to inconsistent estimators. This paper introduces a partially adaptive estimation procedure, based on two families of flexible probability density functions [the generalized beta of the second kind (GB2) and the skewed generalized t (SGT)], to adjust for distributional misspecification and accommodate possible heteroskedasticity. Using a linear link function, these methods are applied to the problem of estimating the willingness to pay to protect Australia’s Kakadu Conservation Zone from mining. In this application, the assumption of homoskedasticity is not rejected for the GB2 family, but is rejected for the SGT. A Monte Carlo simulation confirms the importance of the homoskedasticity assumption as well as the impact of the bid design. For this example, many of the more flexible distributions are in fairly close agreement with some of their special cases. However, this application illustrates how flexible nested distributions can be used to accommodate diverse distributional characteristics, including possible heteroskedasticity.

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Fig. 1

Adapted from McDonald and Xu (1995)

Fig. 2

Adapted from Hansen et al. (2010)

Fig. 3
Fig. 4

Notes

  1. 1.

    Carson (2011) has compiled an extensive review of the CV literature, including more than 7000 papers covering 130 countries over a 50-year time period.

  2. 2.

    Three common formats in the CV literature which result in the WTP being reported in intervals are the single-bound, double-bounded, and one-and-one half bound formats. Cooper et al. (2002) explore the relative efficiency of each, and corresponding likelihood functions.

  3. 3.

    They use the inverse hyperbolic sine distribution and for the homoskedastic case find that estimator precision depends on the flexibility of the assumed distribution and increases with a reduction in the size of the intervals, with the partially adaptive procedures being able to compensate for asymmetry and thick-tailed error distributions. The impact of heteroskedasticity is not explored.

  4. 4.

    g(y) will be y or ln(y), depending upon the assumed distribution for the data.

  5. 5.

    Spike models provide an important approach to explicitly accommodate zero values which are common in CV applications. The corresponding log-likelihood function, with only interval censoring, can be expressed as \( \sum_{{i:\lambda_{i} = 1}} \ln \left[ {F(g(U_{i} ) - X_{i} \beta ;\theta ) - F(g(L_{i} ) - X_{i} \beta ;\theta )} \right] + \sum\nolimits_{i} {\ln \left[ {p_{i}^{{\lambda_{i} }} (1 - p_{i} )^{{1 - \lambda_{i} }} } \right]} \) where \( \lambda_{i} = 0 \) for WTP = 0 and 1 otherwise and pi denotes the probability that the ithobservation has a positive WTP. Thus, the estimation of the β vector is similar to that of maximizing (2.2). Different assumptions can be made about the behavior of pi, for example, a constant or a probit or logit specification (Kristrӧm 1997; McFadden 1994; Reiser and Shechter 1999).

  6. 6.

    See Theodossiou (1998), Hansen (1994), and McDonald and Newey (1988) for additional information about these and related distributions.

  7. 7.

    The use of mixture distributions (Caudill 2012) provides an alternative estimation approach that is not explored in this paper. As noted earlier, Cook and McDonald (2013) use the inverse hyperbolic sine distribution and do not allow heteroskedasticity, as is considered in this paper.

  8. 8.

    The notation for the parameters has been modified from that traditionally used in the GB2 specification to accommodate the CV application. Replacing δ with ln(β) and σ with 1/a in Eq. (2.3) is the more common form for the GB2 (McDonald 1984).

  9. 9.

    This specification models the variance of log(y) because the variance of log(y) is given by \( \sigma^{2} [\varPsi^{{\prime }} (p) + \varPsi^{{\prime }} (q)] \) where \( \varPsi ( \cdot ) \) is the digamma function.

  10. 10.

    To account for zero values in a spike model specification, these results are multiplied by pi.

  11. 11.

    Closed-form expressions for the medians for other distributions discussed in this paper are not available and require numerical methods to evaluate. Different quantiles might also be useful for some practical applications.

  12. 12.

    Allowing σ to be a function of the explanatory variables can model heteroskedasticity.

  13. 13.

    The AIC and BIC are defined by \( AIC = 2(k - \ell ) \) and \( BIC = k\log (n) - 2\ell \).

  14. 14.

    The lower bound is assumed to be zero with a strict inequality.

  15. 15.

    The sample mean for adjusted income reported in Carson et al. (1994) was 21.514, compared to 21.657 for the unadjusted income data set used in this paper.

  16. 16.

    Ready and Hu (1995) consider some approaches to what is referred to as the fat tail problem, which also tends to yield highly variable results.

  17. 17.

    The robust standard errors are calculated using the sandwich formulation with the formulas given in “Appendix 2”.

  18. 18.

    The mean WTP for the GG is defined (Eq. 2.4b) but is equal to 58,764.

  19. 19.

    Harrison and Kristrӧm (1995) discuss this issue.

  20. 20.

    e is evaluated for (envcon, age, income, major) = (1, 30 years, $20, 1) and (0,45 years, $30, 1) with the value of other explanatory variables being selected to be the means shown in Table 1.

  21. 21.

    The scale parameter is selected to be \( \sigma = e^{X\gamma } \).

  22. 22.

    δ0 = 3.9, δ1 = 0.015, σ = 2.42, p = 1.3, and q = 2.09. The mean is not defined for these values of p and q. Different values of p and q were selected so that the mean of Y is defined and equal to 72 for an income of 25.

  23. 23.

    A likelihood ratio test of \( H_{0} :GB2 = Burr3 \) for income is statistically insignificant and the corresponding expected value is 24.5.

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Acknowledgement

The authors express appreciation to Jason Cook, Will Cockriel, Carla Johnston, Sean Kerman, and Sean Musso, for research assistance. The author is also grateful to Richard Carson for providing the data used in the example in this paper. Helpful comments from Nicolai Kuminoff, Rulon Pope, and Mark Showalter are also appreciated.

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Appendices

Appendix 1: Selected cdfs

$$ \begin{aligned} F_{GB2} (y;\delta ,\sigma ,p,q) = &B_{{z_{GB2} }} (p,q) \\ F_{GG} (y;\delta ,\sigma ,p) = &\varGamma_{{z_{GG} }} (p) \\ F_{LN} (y;\mu ,\sigma ) = & \frac{1}{2} + \frac{{sign(z_{LN} )}}{2}\varGamma_{{z_{LN}^{2} /2}} \left( {\frac{1}{2}} \right) \\ F_{BR3} (y;\delta ,\sigma ,p) = & \frac{{e^{(\ln (y) - \delta )p/\sigma } }}{{\left( {1 + e^{(\ln (y) - \delta )/\sigma } } \right)^{p} }} \\ F_{BR12} (y;\delta ,\sigma ,q) = & 1 - \frac{1}{{\left( {1 + e^{(\ln (y) - \delta )/\sigma } } \right)^{q} }} \\ F_{W} (y;\delta ,\sigma ) = & 1 - e^{{ - e^{(\ln (y) - \delta )/\sigma } }} \\ F_{SGT} (y;\mu ,\lambda ,\sigma ,p,q) = &\frac{1 - \lambda }{2} + \frac{sign(y - \mu )(1 + \lambda sign(y - \mu ))}{2}B_{{z_{SGT} }} \left( {\frac{1}{p},q} \right) \\ F_{SGED} (y;\mu ,\lambda ,\sigma ,p) = & \frac{1 - \lambda }{2} + \frac{sign(y - \mu )(1 + \lambda sign(y - \mu ))}{2}\varGamma_{{z_{SGED} }} (p), \\ \end{aligned} $$

where \( B_{z} ( \cdot , \cdot ) \) and \( \varGamma_{z} ( \cdot ) \) denote the incomplete beta and gamma functions and

$$ \begin{aligned} z_{GB2} = & \frac{{e^{(\ln (y) - \delta )/\sigma } }}{{1 + e^{(\ln (y) - \delta )/\sigma } }} \\ z_{GG} = & e^{(\ln (y) - \delta )/\sigma } \\ z_{LN} = & \frac{\ln (y) - \mu }{\sigma } \\ z_{SGT} = & \frac{{|y - \mu |^{p} /(1 + \lambda sign(y - \mu ))^{p} q\sigma^{p} }}{{1 + |y - \mu |^{p} /(1 + \lambda sign(y - \mu ))^{p} q\sigma^{p} }} \\ z_{SGED} = & |y - \mu |^{p} /(1 + \lambda sign(y - \mu ))^{p} q\sigma^{p} . \\ \end{aligned} $$

As noted above the Burr3, Burr12, and Weibull cdfs can be simplified and expressed in a closed-form whereas, the general forms for the GB2, GG, LN and SGT involve infinite series.

Appendix 2

See Tables 4, 5, 6, 7, and 8.

Table 4 Kakadu: variable definitions and means (range)
Table 5 Kakadu: GB2 estimation results: homoskedasticity
Table 6 Kakadu: SGT estimation results: homoskedasticity
Table 7 Kakadu: GB2 estimation results: heteroskedasticity
Table 8 Kakadu: SGT estimation results: heteroskedasticity

Appendix 3: Calculation of the Standard Errors

The standard errors can be evaluated from the gradient and Hessian matrices corresponding to the log-likelihood function

$$ \begin{aligned} \ell = \sum\limits_{{i:y_{i}^{*} < U_{i} }} {\ln } [F(\varepsilon_{{u_{i} }} = &g(U_{i} ) - X_{i} \beta ;\theta )] \\ \quad + \sum\limits_{{i:L_{i} \le y_{i}^{*} \le U_{i} }} {\ln } [F(\varepsilon_{{u_{i} }} = &g(U_{i} ) - X_{i} \beta ;\theta ) - F(\varepsilon_{{l_{i} }} = g(L_{i} ) - X_{i} \beta ;\theta )] \\ \quad \sum\limits_{{i:L_{i} < y_{i}^{*} }} {\ln } [1 - F(\varepsilon_{{l_{i} }} = & g(L_{i} ) - X_{i} \beta ;\theta )]. \\ \end{aligned} $$

The gradient is given by

$$ \begin{aligned}\frac{d\ell }{d\beta } =& \sum_{{i:y_{i}^{*} < U_{i} }} \frac{{ - f(\varepsilon_{{u_{i} }} ;\theta )}}{{F(\varepsilon_{{u_{i} }} ;\theta )}}X_{i}^{{\prime }} + \sum_{{i:L_{i} \le y_{i}^{*} \le U_{i} }} \frac{{ - (f(\varepsilon_{{u_{i} }} ;\theta ) - f(\varepsilon_{{l_{i} }} ;\theta ))}}{{(F(\varepsilon_{{u_{i} }} ;\theta ) - F(\varepsilon_{{l_{i} }} ;\theta ))}}X_{i}^{{\prime }}\\ &+ \sum_{{i:L_{i} < y_{i}^{*} }} \frac{{f(\varepsilon_{{l_{i} }} ;\theta )}}{{(1 - F(\varepsilon_{{l_{i} }} ;\theta ))}}X_{i}^{{\prime }}\end{aligned} $$

and the Hessian is given by

$$ \begin{aligned} \frac{{d^{2} \ell }}{{d\beta d\beta^{{\prime }} }} & = \sum\limits_{{i:y_{i}^{*} < U_{i} }} {\left[ {\frac{{f'(\varepsilon_{{u_{i} }} ;\theta )}}{{F(\varepsilon_{{u_{i} }} ;\theta )}} - \left( {\frac{{f(\varepsilon_{{u_{i} }} ;\theta )}}{{F(\varepsilon_{{u_{i} }} ;\theta )}}} \right)^{2} } \right]X_{i}^{'} X_{i} } \\ & \quad + \sum\limits_{{i:L_{i} \le y_{i}^{*} \le U_{i} }} {\left[ {\frac{{f'(\varepsilon_{{u_{i} }} ;\theta ) - f'(\varepsilon_{{l_{i} }} ;\theta )}}{{F(\varepsilon_{{u_{i} }} ;\theta ) - F(\varepsilon_{{l_{i} }} ;\theta )}} - \left( {\frac{{f(\varepsilon_{{u_{i} }} ;\theta ) - f(\varepsilon_{{l_{i} }} ;\theta )}}{{F(\varepsilon_{{u_{i} }} ;\theta ) - F(\varepsilon_{{l_{i} }} ;\theta )}}} \right)^{2} } \right]X_{i}^{'} X_{i} } \\ & \quad + \sum\limits_{{i:L_{i} < y_{i}^{*} }} {\left[ {\frac{{f'(\varepsilon_{{l_{i} }} ;\theta )}}{{1 - F(\varepsilon_{{l_{i} }} ;\theta )}} - \left( {\frac{{f(\varepsilon_{{l_{i} }} ;\theta )}}{{1 - F(\varepsilon_{{l_{i} }} ;\theta )}}} \right)^{2} } \right]X_{i}^{'} X_{i} .} \\ \end{aligned} $$

The standard errors used in this paper for the main estimation results were calculated using analytic expressions for the gradient and Hessian to evaluate the corresponding sandwich standard errors.

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McDonald, J.B., Walton, D.B. & Chia, B. Distributional Assumptions and the Estimation of Contingent Valuation Models. Comput Econ 56, 431–460 (2020). https://doi.org/10.1007/s10614-019-09930-x

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Keywords

  • Willingness to pay
  • Partially adaptive estimation
  • Skewed generalized t
  • Generalized beta of the second kind
  • Heteroskedasticity