Flexible Functional Form Estimation of Willingness to Pay Using Dichotomous Choice Data

Journal of Environmental Economics and Management 43, 267᎐279 Ž2002. doi:10.1006rjeem.2000.1177, available online at http:rrwww.idealibrary.com on

Flexible Functional Form Estimation of Willingness to Pay Using Dichotomous Choice Data Joseph C. Cooper 1 Resource Economics Di¨ ision, Economic Research Ser¨ ice, 1800 M Street, Northwest, Washington, DC 20036-5831 E-mail: [email protected] Received November 10, 1999; revised May 24, 2000; published online July 11, 2001 This paper introduces fully flexible Žsemi-nonparametric. methods for calculating willingness to pay ŽWTP. bounded between zero and the respondent’s income for dichotomous choice contingent valuation. A method to estimate median WTP using fully flexible functional forms is also presented. Finally, the numerical illustration demonstrates the importance of considering Jensen’s Inequality in calculating WTP with nonlinear models. 䊚 2001 Elsevier Science ŽUSA .

Key Words: contingent valuation; discrete choice; nonparametric; semi-nonparametric.

INTRODUCTION Nonparametric and semi-nonparametric ŽSNP. estimation techniques for the discrete choice contingent valuation format are receiving increasing interest in the published nonmarket valuation literature. This interest is not surprising given the concern for potential biases associated with incorrect specifications of functional forms and distributions in parametric approaches. Nonparametric smoothing techniques represent a set of flexible tools for analyzing unknown regression relationships. Alternatives to nonparametric methods are expanding parameter space, or semi-nonparametric ŽSNP., methods, which are halfway between parametric and nonparametric inference procedures Že.g., Fenton and Gallant w6x.. An advantage of SNP over nonparametric methods is that they allow the researcher to reduce the potential for misspecification bias associated with parametric techniques, while at the same time accounting for explanatory variables more easily than nonparametric methods. The cost of SNP methods is that they are more complex than the parametric approaches and they require careful fitting to the data. Creel and Loomis w4x and Chen and Randall w2x provide the foundation for SNP applications to DC Ždichotomous choice. CVM, with a focus on coefficient estimation.2 The main contribution of this paper is to suggest important refinements to calculating willingness to pay ŽWTP. given the semi-nonparametric regression results. Most importantly, this paper addresses how to use the SNP approach to 1

The views expressed are the author’s and do not necessarily represent the policies or views of his institution. 2 An anonymous reviewer points out that the term ‘‘semi-parametric’’ is perhaps more apt than ‘‘semi-nonparametric.’’ However, I use the latter term for consistency with papers cited in the references. 267 0095-0696r01 $35.00 䊚 2001 Elsevier Science ŽUSA . All rights reserved.

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produce a WTP measure for a normal good that is compatible with economic theory, i.e., that WTP is nonnegative and bounded from above by income. Second, this paper discusses how to calculate median WTP Žy⬁ F WTP s ⬁. with the SNP approach, which can be useful for specification testing, and is consistent with standard discrete choice estimation procedures, which do not place any restrictions on the estimated coefficients. It may also be useful in situations where bounding WTP is not practical or desired. The rest of the paper is organized as follows: Section I gives a brief review of the utility theoretic background for estimating WTP using dichotomous choice CVM; Section II is an overview of SNP DC models; Section III discusses how to impose bounds on WTP calculation given SNP coefficient estimates; Section IV addresses calculation of median WTP given SNP coefficient estimates; and the last section provides concluding comments. I. UTILITY THEORETIC BACKGROUND FOR ESTIMATING WTP USING DICHOTOMOUS CHOICE CVM For brevity, this paper focuses on application of SNP to the single-bound approach, given that the concepts discussed here apply to multiple-bound models as well. Hanemann w12, 13x provides the utility theoretical background for the DC CVM approach. The theoretical background is well known, but a brief summary is useful as a basis for introducing the notation to be used in subsequent sections. Say that an individual is confronted with the possibility of obtaining a change in a good q from q0 to q1 ) q0 , and the indirect utility in the base case is V Ž q0 , y, z, ␧ ., where y is income, z is a vector of market commodities, prices, and characteristics of the individual, and ␧ is some stochastic component that is unobservable to the researcher. If the individual views this change as an improvement, then V Ž q1 , y, z, ␧ . G V Ž q0 , y, z, ␧ .. If the individual is told this change will cost $ A, and the individual is a utility maximizer, then the individual will pay $ A Ž‘‘yes’’. only if V Ž q1 , y y A, z, ␧ . G V Ž q0 , y, z, ␧ ., and ‘‘no’’ otherwise. The compensating variation measure C is the value that solves ⌬V Ž C, q1 , q0 , y, z, ␧ . s V Ž q1 , y y C, z, ␧ . y V Ž q0 , y, z, ␧ . s 0. Given this solution, C s C Ž q1 , q0 , y, x, ␧ . is the maximum WTP for the change from q0 to q1. The relationships above can be equivalently expressed in a probabilistic framework as P ‘‘yes’’ to $ A4 s P ¨ Ž q1 , y y A, z, ␧ . G ¨ Ž q0 , y, z, ␧ .4 s P C Ž q1 , q0 , y, z, ␧ . G A4 s P ⌬V G 04 . If for estimation purposes ⌬V s ⌬V Žx, ␪ ., where x s  A, z, y, ␧ 4 , then the standard discrete choice likelihood function can be used to estimate the parameter vector ␪, N

ln L Ž d ¬ x, ␪ . s

Ý Ž d i ln P Ž ⌬V Ž x i , ␪ . .

q Ž 1 y d i . ln 1 y P Ž ⌬V Ž x i , ␪ . .

.,

is1

Ž 1. where d i s 1 for a yes, and 0 otherwise, and d is the Ž N = 1. vector of yesrno indicators. Common cumulative distribution functions used in estimation are the logistic, Weibull, and the normal, where for the latter, Prw C Ž q1 , q0 , y, z, ␧ . F A x s ⌽␧ wy⌬¨ Žz, A, y, ␪ .x, and where ⌽␧ s 1 y P yes4 . Hanemann w12x notes that unrestricted WTP Ž Cu ., for example, can be found using the following formula for

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the mean of a random variable, Cu s

⬁

⬁

Hy⬁ Af Ž A . dA s H0

1 y F Ž A . dA y

0

Hy⬁ F Ž A . dA,

Ž 2.

where F Ž A. s F wy⌬¨ Žz, A, y, ␪ .x is the cumulative density function for WTP. Economic theory suggests that a person’s maximum willingness to pay for an item is bounded from above by income. If the individual is indifferent to the good or sees it as an improvement, then WTP is bounded from below by 0, or taking the two bounds together, 0 F C Ž q1 , q0 , y, z, ␧ . F y. Hence, desirable properties from an economic standpoint for the probability function are that P ‘‘yes’’¬ $ A s 04 s 1 and P ‘‘yes’’¬ $ A s y4 s 0.3 II. OVERVIEW OF SEMI-NONPARAMETRIC APPROACHES TO COEFFICIENT ESTIMATION Several semi-nonparametric approaches to statistical analysis of dichotomous choice data can be found in the econometric literature. This paper focuses on two approaches: Ž1. the likelihood function uses a conventional CDF, e.g., the normal or the logistic, but with terms appended to the random utility model ⌬V to make the function flexible; and Ž2. the likelihood function uses a simple, conventional random utility model, but with a flexible CDF. The former has been applied to CVM by Creel and Loomis w4x with their semi-nonparametric distribution-free approach. An example of the latter, although not in a CVM application, is the Hermite model of Gabler et al. w10x. In general, the first approach is the most feasible for general use, given that the objective function is concave, which simplifies solving it for the optimal coefficients. 4 In the standard parametric DC CVM models, F is restricted to known CDFs, such as the normal, and ⌬V is restricted to known functional forms, such as the linear y⌬V s x X ␪, where row vector x contains all arguments of the utility difference model. The semi-nonparametric distribution-free model, on the other hand, avoids these restrictions by not attempting to individually model F and ⌬V, but only the compound function F Ž ⌬VF ., where ⌬VF is a version of the Fourier functional form for nonperiodic functions Že.g., Gallant w7x.. The Fourier functional form is one of the few functional forms known to have Sobolev flexibility, which means that the difference between a model hŽx, ␪ . and the true function f Žx. can be made arbitrarily small for any value of x as the sample size becomes large ŽGallant w8x.. 3 The probability condition P‘‘yes’’¬ $ A s 04 s 1 is desirable if we assume that the respondent is not indifferent to the good at $0 bid. An anonymous reviewer points out that if some of the respondents are indifferent to provision of the good at this bid point, these respondents may respond ‘‘yes’’ or ‘‘no’’ to this bid, and hence, P‘‘yes’’¬ $ A s 04 can be less than 1. 4 The alternative presented by Chen and Randall w2x, in which the SNP flexible functional form is estimated simultaneously with an SNP density function, is only briefly touched upon here. Convergence of the coefficient values may be difficult to achieve with this model given its nonconcavity, large number of parameters, and heavier demands for a large number of observations. The focus here is on models easily implemented for applied analysis, although convergence is not so easy to achieve with the Hermite model either. However, our discussion of bounding the CV measure applies to the Chen and Randall model as well.

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Given that a detailed discussion of the Fourier transformation is outside the scope of this paper, it should suffice to say that there exists a vector ␪ k such that ⌬V Žx, ␪ k . can be made arbitrarily close to a continuous unknown function ⌬UŽx. for any value of x as the sample size becomes large. Given that F Ž ⌬VF . is a monotone transformation of ⌬VF , any convenient CDF will do for the purposes of estimation. Creel and Loomis w4x present a specification of ⌬VF that omits the quadratic term, ⌬VF Ž x i , ␪ k . s x Xi ␤ q

M

J

Ý Ý Ž ¨ jm cos

X X jrm s Ž x i . y wjm sin jrm i sŽx i .

.,

Ž 3.

ms1 js1

where the p = 1 vector x contains all arguments of the utility difference model, k is the number of coefficients in ␪, which consists of the ␤ , ¨ jm , and wjm coefficients to be estimated, M and J are positive integers, and rm is a p = 1 vector of positive and negative integers that forms indices in the conditioning variables and that determine which combinations of variables in x form each of the transformed variables. The integer m is the sum of absolute value of the elements in the multi-indexes in vector rm and J is order of the transformation and is basically the number of inner-loop transformations of x. For example, if x contains 3 variables and M s J s 1, then the rm vectors are Ž1, 0, 0., Ž0, 1, 0., and Ž0, 0, 1., resulting in k s 9 Žnot counting the constant.. The p = 1 function sŽx. prevents periodicity in the model by rescaling x so that it falls in the range w0, 2␲ y 0.000001x ŽGallant w7x.. This rescaling of x is achieved by subtracting from x its minimum value, then dividing this difference by the maximum value, and then multiplying the resulting value by w2␲ y 0.000001x.5 For example, if bid is the only explanatory variable, then rm is a Ž1 = 1. unit vector and maxŽ M . equals 1. If furthermore, J s M and bid has more than three unique values, then ⌬V Ž A, ␪ k . s ␣ q ␦ A q ␦¨ cos s Ž A . q ␦w sin s Ž A . .

Ž 4.

If a variable has only three unique values, then only the ¨ or w transformation may be performed. With two values, none of the transformations can be performed. The semi-nonparametric distribution-free specification above takes an SNP functional form and estimates it by specifying an arbitrary known distribution. An alternative is to do the reverse and specify an arbitrary functional form, such as the basic linear model, and estimate it with an SNP distribution. Doing so considerably simplifies the process of calculating certain WTP values, although with some loss in flexibility. Gallant and Nychka w9x demonstrate an SNP distribution based on a Hermite polynomial expansion that can approximate any smooth density that has a moment generating function with tails no fatter than that of a t-distribution. Gabler et al. w10x apply this concept to binary choice models. In our notation, letting ⌬VH s g Žx, ␤ . q ␧ , the CDF ⍀ ␧ s ⍀ w g Žx, ␤ .x is ⍀ g Ž x, ␤ . s 5

Hg⬁Ž x , ␤ . h Ž ␧ . d ␧ ⬁ Hy⬁ hŽ ␧ . d␧

,

Ž 5.

In addition to appending x ␤ to the Fourier series in Eq. Ž4., Gallant suggests appending quadratic terms when modeling nonperiodic functions. Our experiments suggest that inclusion of the quadratic terms as well in the regressions had little impact on the WTP estimates. Hence, we leave them out for the sake of efficiency.

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where hŽ ␧ . s ŽÝ i,K js0 ␣ i2␧ iqj . exp yŽ ␧r␦ . 2 4 and where K is the order of the Hermite polynomial expansion. Nonnegativity of the function is assured by the squaring of the first term and by taking the exponential of the second term. If K is allowed to increase with the sample size, then the binary choice objective function will be a likelihood function, and ␤ will be consistent. If K is arbitrarily set prior to estimation, then pseudo-maximum likelihood is appealed to for obtaining asymptotic properties. In estimation, restrictions need to be placed on ␣ and ␦ to impose E Ž ␧ . s 0 and to guarantee that the integral of the density over the range y⬁ to ⬁ is 1, the latter of which is achieved by inclusion of the denominator in the equation above. When parameter ␣ 0 is set to Ž2␲ .y1 r4 and ␦ to '2 , the CDF becomes the standard normal for K s 0. Detailed discussion of the choice of K and the restrictions on ␣ and ␦ are beyond the scope of this paper, but are covered in Gabler et al. w10x. In practice, K s 3 appears to provide sufficient flexibility ŽGabler et al. w10x.. If g Žx, ␤ . s x ␤ , it is a simple matter to calculate the median WTP estimator. The solution ⌬¨ hŽ C, q1 , q0 , y, x, ␧ . s 0 is the same as the linear case. Letting x i ␤ s zXi ␶ q ␸ A i , then the individual i’s semi-nonparametric hermite WTP s yzXi ␶r␸ . The CVM application of the semi-nonparametric hermite to the existing SNP CVM methods is more parsimonious in coefficients than the approaches of Creel and Loomis w4x and Chen and Randall w2x, and unlike the former, allows closed form solutions to WTP in some cases. On the downside, the semi-nonparametric hermite imposes a functional form for g Žx, ␤ . that may not be desirable from an econometric standpoint, i.e., ␧ does not allow for possible heteroskedasticity across the respondents. The homoskedasticity of ␧ can be relaxed by making the ␣ parameters functions of some set of exogenous variables, but this approach would be too general in that it would cover heteroskewness and heterokurtosis as well as heteroskedasticity ŽGabler et al. w10x.. In addition, unlike the semi-nonparametric distribution-free approach, but like the Chen and Randall w2x approach, the semi-nonparametric Hermite likelihood function is not globally concave. Another possible SNP CVM approach is to combine the semi-nonparametric Hermite and semi-nonparametric distribution-free models, i.e., substitute the Fourier series expansion in Eq. Ž3. in place of g Žx, ␤ . s x ␤ in ⍀ ␧ w g Žx, ␤ .x, which differs from Chen and Randall w2x in that they use a known CDF but take the polynomial transformation of ⌬VF .6

Like SNPDF, but unlike Chen and Randall w2x or SNPH, the SNPHDF Žsemi-nonparametric Hermite distribution free. model will not guarantee the characteristic of monotonicity of P yes’’ to $ A4 with respect to A. Chen and Randall assume the marginal utility of a dollar is constant, allowing them to separate A from the functional form as P ␧ s Arexpw hŽz, ␪ .x4, which by construction, is monotonic in A, where hŽz, ␪ . is a Fourier transformation and CDF P.4 is a polynomial expansion. Less parameters may be needed to model the distribution using SNPHDF than Chen and Randall, and hence, it may reach convergence easier than the latter model. In small samples, where degrees of freedom or multicolinearity problems may only allow small J and M transformation terms, the SNPHDF may exhibit increased flexibility over the SNPDF approach. At any rate, since the SNPH model nests the probit, the SNPHDF model nests the SNPDF. For the data used in the empirical example, the SNPHDF mean WTP estimate with the length and order of the Fourier terms equal to one is 2% lower than the SNPH estimate. 6

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III. IMPOSING BOUNDS ON WTP CALCULATION WITH SEMI-NONPARAMETRIC MODELS Bounded estimation of WTP in parametric models is addressed in Ready and Hu w19x and Haab and McConnell w11x. In the context of WTP calculation for the SNP model, this paper focuses on three cases of provision of some good: Ž1. the respondent considers the change to be an improvement; Ž2. the respondent considers the change as either an improvement or is indifferent to it; and Ž3. no restrictions are assumed on the respondent’s views regarding the change Žcovered in the next section.. Hanemann and Kanninen w14x, in their review of many potential bounded parametric models, refer to Ž1. as the canonical case in reference to other cases being modifications of this canonical case. Case 1. As noted in Hanemann and Kanninen w14x, the cannonical case can be estimated using a truncated distribution. Here the truncated approach is generalized so that it can be made SNP. The SNP double truncated probability of a ‘‘yes’’ to offer A is Prob  ‘‘yes’’4 s

⌽ ⌬VF Ž A . y ⌽ ⌬VF Ž y . ⌽ ⌬VF Ž 0 . y ⌽ ⌬VF Ž y .

if 0 F A F y,

Ž 6.

where ⌬VF Ž y . is the left truncation and ⌬VF Ž0.. The term ⌬VF Ž y . is the specification of ⌬VF with variable A replaced by y and in ⌬VF Ž0. by $0. Since ⌬VF Ž y . s ⌬VF Ž A. s ⌬VF Ž0., in the context of a truncated normal distribution, ⌬VF Ž y . is the left truncation and ⌬VF Ž0. the right. For estimation, the probability specification above is substituted into Eq. Ž1.. Given the coefficients estimates, mean WTP is calculated as EŽ C . s s

y

H0

⌽ ⌬VF Ž b . y ⌽ ⌬VF Ž y . ⌽ ⌬VF Ž 0 . y ⌽ ⌬ F V Ž y . 1

y

⌽ ⌬VF Ž 0 . y ⌽ ⌬VF Ž y .

H0

db ⌽ ⌬VF Ž b . y ⌽ ⌬VF Ž y .

db, Ž 7 .

where the integral must be calculated numerically. Case 2. Now suppose one wants to allow for indifferenceᎏwith some positive probability, the individual has a zero WTP for the change in q. Indifference is equivalent to a probability mass at C s 0. The other way to introduce indifference is to use a response probability model with censoring at C s 0. A model satisfying C g w0, y x with spikes at A s 0 and A G y is

¡1 ¢0

Prob  ‘‘yes’’4 s~ P

⌬V Ž A .

if A s 0 if 0 - A - y if A G y.

Ž 8.

Since this probability function can be applied after the coefficient estimation stage, where P w ⌬V Ž A.x can be either that for the semi-nonparametric distribution-free or the semi-nonparametric hermite, the standard logit or probit MLE can be used for estimation.

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Median WTP Calculation with Semi-nonparametric Models In some cases, such as for specification testing, when valuing goods that have a potential for nuisance value Ži.e., unbounded from below., and when the respondent income is unknown, it may be desirable to estimate unbounded WTP. However, if the lower and upper bounds of the WTP estimators discussed above are outside of the bid range, evaluating the integral Že.g., Eq. Ž7.. requires forecasting outside the estimated region of the distribution. Because the SNP functional form scales the variables, forecasting outside the estimated range SNP models can produce especially undesireable results ŽWeaver w22x., such as the density not being monotonic in A or otherwise not meaningful. For this reason, Creel and Loomis w4x use Cr s H0max A w1 y F Ž A.x dA.7 Note that this criticism of the SNP response function is not meant to suggest that the shape of the parametric response function outside the range of the data is less biased than the SNP one, but that at least the parametric one is guaranteed to be monotonic and have predictable characteristics. Whether or not the bounded estimators discussed above can be used with SNP coefficients is an empirical question. For example, for the data used for the numerical illustration, the SNP-based Prob ‘‘yes’’ to A4 falls towards zero outside the observed bid range faster than does the parametric-based density function Žwhich is a reason why the SNP WTP estimates are lower than the parametric WTP estimates. and is monotonic. Hence, in this case, the SNP model is more conservative than the parametric model, and concerns associated with extrapolation of the SNP model are minimal. In general, one approach is to minimize extrapolation effects is to include in the survey at least one bid offer for which the probability of a yes is very low, say less than 1%. Such a bid can be found through pre-testing. For calculation of WTP, the probability of a yes can be set to zero above this bid. Inclusion of such a bid is not optimal from the standoint of efficiency of the coefficient estimates, but the potential tradeoff is lowered bias in the WTP estimate.8 Further research into this approach or others that address bid design issues with bounded WTP measures would be fruitful. If the density function is not monotone outside of the bid range, then the Creel and Loomis w4x WTP estimator may be preferable to the income-bounded estimators. An alternative in this situation is to calculate median WTP, a measure that is consistent with estimation of the coefficients in that both assume the random variable is unrestricted. For distributions for which F Ž0. s 0.5, the median can be found by solving ⌬V Ž q1 , q0 , y, z, A, ␧ . s 0 with respect to A. With linear RUMs and distributions that are symmetric around zero, median WTP is the same as the unrestricted mean Ž Cu . measure. The median WTP can be found analytically for the simple linear specification of ⌬V that is common in the literature and for semi-nonparametric Hermite as discussed earlier, but not for ⌬VF . In the simplest X X An alternative restricted WTP estimate, or Cˆr Ži.e., 0 F Cˆr - ⬁., would integrate out to A s ⬁ X instead of max A. The relationship of Cˆu to this Cˆr integrated between 0 and ⬁ is ambiguous. If ⬁ 0 ˆ ˆX w Ž .x Ž . Hma x B I d 1 y F b db ) Hy⬁ F b db, then C u is greater than C r . 8 A less desirable ex post alternative is to assume that each respondent will respond with a ‘‘no’’ to a bid equal to the respondent’s income Žor some large fraction thereof., and to append the data set with the responses to the BID s INCOME data. However, as this approach doubles the size of the data set without actually adding new information, the covariance matrix will require some form of rescaling. 7

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Fourier case, the solution is to find A satisfying ⌬VF Ž A, ␪ˆ. s ␣ ˆ q ␦ˆA q ␦ˆ¨ cos sŽ A. q ␦ˆw sin sŽ A. s 0. Unlike a numerical integration of the CDF between y⬁ and ⬁, numerically solving ⌬VF Ž A, ␪ˆ. s 0 keeps the analysis inside the estimated region of the distribution when AU falls within the bid range  minimum ˆ U F maximum bid4. In general, most well-designed surveys had a wide bid F WTP enough bid range that the mean WTP is somewhere within the range of the available bids. Given that the solution to LŽ AU ¬ z, y, ␪ˆ. s ⌬VF Ž A, z, y, ␪ˆ. s 0 is not a simple task to find analytically, it can be solved numerically. Using a grid search over WTP is the simplest approach, while an optimization approach Že.g., the method of annealing. may be more efficient at higher levels of resolution. If AU is within the data bounds, then the potential ill effects of extrapolation are not an issue. If it is not within bounds, either less flexible model specifications or more and better data may be needed. As mentioned earlier, the unbounded SNP WTP estimator serves as a useful specification check on the bounded model. Since the SNP models consistently estimate P x, ␪ 4 under certain conditions ŽCreel and Loomis w4x; Gabler et al. w10x., if the tails of the bounded density differ substantially from the unbounded one, this implies that: Ž1. the sample was not large enough to establish the desirable properties P yes to bid s $04 s 1 and P yes to bid s $income4 s 0; Ž2. the sample responses are biased, whether due to a suboptimal survey question or sample design; or Ž3. regarding the lower bound, respondents truly can hold a nuisance value for the good in question. On the other hand, in comparing the parametric median and bounded mean WTP models, statistical differences may be due to incorrect specification of the functional form or the density function, and these cannot be separably identified from the three points above. IV. NUMERICAL ILLUSTRATION Since the magnitude of the empirical differences between the benefit estimators is dependent on the specific data set analyzed, little is added by an in-depth empirical comparison of the various benefit estimators discussed here. This numerical illustration is only intended to show the reader how an application might look in practice with a real world data set. The data set is discussed in detail in Loomis and duVair w17x. Table A-II in the Appendix presents the bid values and the associated response probabilities. While the data demonstrate a monotonic trend in the Žnonparametric. responses probabilities to each bid, the change in probability from one bid point to another is not strictly monotonic, suggesting that a flexible estimation approach may be useful. Table I presents the WTP results, both for the representative consumer and the sample Žthe regression results are presented in the Appendix along with a discussion of some issues regarding presentation of SNP results ..9 For comparison, 9

A distinction not usually noted in the CVM literature is if the WTP specification is not linear in the variables. Then WTP for a representative individual can diverge from average WTP across individuals. For the representative individual WTP s WTPŽ x, ␪ˆ. while sample WTP s Ž1rn.Ý nis 1WTPŽ x i , ␪ˆ.. For example, by Jensen’s Inequality, if the WTP function is convex, then the sample WTP is greater than or equal to the WTP for the mean individual. The differences are minor for many parametric models, but may not be for models using the Fourier transformation, which can be highly nonlinear in the variables.

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TABLE I WTP Estimates ŽMean WTP Results Except Where Noted. Benefit estimates Ž$. Case Double spike

Double truncation Median

Nonparametric b

Approach

Sample

Representative consumer Ž90% BCa confidence intervals.

Parametric SNPDF-I SNPDF-II SNPH Parametric SNPDF-I Parametric SNPDF-I SNPH Turnbull Kernel

481.30 416.28 393.62 464.22 483.99 416.28 164.73 261.31 ᎏa ᎏd ᎏd

436.94 Ž414.77, 459.77. 403.72 Ž379.53, 428.65. 355.39 Ž327.95, 383.67. 417.50 456.34 Ž427.38, 486.50. 404.40 Ž341.08, 470.37. c 149.95 Ž117.37, 197.43. 159.80 118.62 391.97 Ž347.80, 437.51. 389.58 Ž346.46, 434.03.

a

WTP estimate across sample is the same as WTP for representative consumer. Computer programs for Turnbull and kernel estimation are also available from the author. c Skewness is evident, e.g., the empirical percentile 90% CI is Ž$370.08, $423.50.. d Not applicable. b

nonparametric results using both the Turnbull ŽTurnbull w21x; Kristrom ¨ w16x. and Ž w x. kernel Kappenman 15 approaches are presented at the bottom of Table I. Confidence intervals were not estimated for the several cases where the WTP measure is particularly intensive in terms of computational time. The truncated Žcase 1. and spike Žcase 2. results tend to be in the same general ballpark. The median WTP results are substantially lowerᎏmean WTP tends to be lower than mean WTP in CVM studies that estimate bothᎏand tend to vary substantially among themselves, although the parametric and semi-nonparametric distributionfree Ž‘‘SNPDF-I’’ in Table I. estimates for the representative consumer are quite close. The variation is probably due to the fact that, for this data set, the empirical Žnonparametric. response probabilities are far from being monotonic with respect to bid in the region around the fiftieth percentile, which is in the $100 to $300 bid range, and hence, the median WTP value is not easy to pin down.10 The relatively large differences between sample and representative consumer WTP in some cases suggest that the implications of Jensen’s inequality should not be ignored.

10

An anonymous reviewer points out that yet another means of estimation, that of Manski’s semi-parametric MSCORE estimator, is available in Limdep 7. Using this Limdep estimator, the median WTP value is $122.48 Žwhere WTP s expŽy␣ ˆr␤ˆ. given that lnŽBID. is used in the regression., which is not out of line with the other median estimates. I do not calculate the mean WTP values as the CDF for this estimator is unknownᎏand is not asymptotically normalᎏand hence, I cannot use the bounded WTP measures.

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CONCLUSION Bounded estimation of WTP requires integration of the function across the price range from 0 to income. However, CVM surveys rarely provide bid offers in the far reaches of the tails of the distribution, and in fact, excluding these bid offers may be appropriate with parametric regression approaches given their potential sensitivity to outliers. Parametric models may no more realistically model the tails outside the range of the data than do semi-nonparametric ŽSNP. models, but at least their arbitrary functional forms allow well-behaved tails. Because an SNP response function may behave erratically outside the range of the data, it is a good idea to use a wide range of bids if one desires a bounded SNP measure. As long as estimated WTP is with the range of the data, the median SNP WTP measure presented here can be used in cases when the SNP-based density function is not monotonic outside the range of the data. The median SNP WTP estimator converges on the bounded estimator as the quality of the data increases and Prob ‘‘yes’’ to BID s $04 and Prob ‘‘yes’’ to BID s $income4 converge on 1 and 0, respectively.11 Finally, just because bounded SNP benefit estimation routines are readily available does not mean parametric models are no longer of interest.12 For instance, benefit transfer exercises cannot be easily conducted with published SNP results. Perhaps the main value of the SNP models discussed here is as specification checks on parametric bounded estimators Že.g., Boyle et al. w1x; Haab and McConnell w11x; Ready and Hu w19x; Hanemann and Kanninen w14x.. Parametric results are more efficient and more convenient than are SNP results, and we can feel more comfortable using the former given their validation by the latter. APPENDIX Other Semi-nonparametric CVM Issues Taken individually, Fourier coefficients do not have an economic interpretation, and there is little point filling a table with them, especially if the number of parameters is large. To give those regression coefficients an economic interpretation, they must be re-expressed in terms of the base variables. One possible way to add economic content is to numerically generate graphs of the relationship between the benefit measure and the explanatory values ŽCreel and Loomis w4x.. Another way to do this is to evaluate ⭸ ⌽w h k Ž x, ␪ .xr⭸ x, or if sensitivity of the likelihood function is of interest, ⭸ ln ⌽w h k Ž x, ␪ .xr⭸ x s  ␾ w h k Ž x, ␪ .xr⌽w h k Ž x, ␪ .x4 ⭈ ⭸ h k Ž x, ␪ .xr⭸ x, noting that

⭸ h k Ž x, ␪ k . ⭸x 11

A

sbq2

½

J

Ý Ý j Ž ¨ j␣ cos

␣s1

js1

jr␣X s Ž x . q wj ␣ sin jr␣X s Ž x .

. r␣

5

. Ž 9.

The Probyes to BID s $04 s 1 assumes that respondents will ‘‘buy’’ the good at the zero bid

level. 12

Computer programs for implementing the routines discussed here are freely available for downloading from www.ers.usda.govrmodelsrjcooper.

277

WTP WITH FLEXIBLE FUNCTIONAL FORMS

Monte Carlo Analysis of the Parametric and Semi-nonparametric Approaches Given the relative complexity of fitting the semi-nonparametric models, one may question whether or not there is an advantage to using them. This question is addressed by a Monte Carlo comparison of the basic linear model with the semi-nonparametric distribution free model ŽSNPDF-I in Table A-I.. To impart realism into the Monte Carlo, the data are generated using the actual coefficient estimates. Two cases are conducted: Ž1. the underlying true RUM is the basic linear model, normally distributed, and with coefficients from Table A-I; and Ž2. the true underlying distribution and RUM are those generated by the SNPDF-I coefficients in Table A-I. For the Monte Carlo analysis, 1000 simulated data sets were created for each run. To assign a bid offer to each observation in a data set of size N, the bid sets in Table A-II were randomly drawn with equal probability of assignment to each simulated data point. The dependent variable for each data point was generated as follows: for each of the i s 1, . . . , N observations, Pi Žthe probability of answering yes to the first bid. is generated using Pi s ProbŽ yi s 1. s ⌽ Ž ⌬Vi ., where ⌽ Ž.. is the normal CDF Že.g., Cooper w3x.. Next, for each observation i s 1, . . . , N, using a Bernoulli distribution with parameter Pi , a 0 or 1 response is generated. The resulting response vector and the bid variable form the simulated data set. Table A-III demonstrates that the SNP model does better in terms of bias than the parametric model when the true RUM is drawn from the Fourier coefficients. When the true RUM is the simple parametric case, the SNP does almost as well as the parametric model. Hence, the SNP model appears to be a good choice for estimation. Of course, the simulation results are only applicable to the data set and coefficients used in the analysis, and the results cannot be generalized.

TABLE A-I Parametric and Semi-nonparametric Regression Results

⭸ ln Pr⭸ x Ž90% BCa confidence intervals.

Coefficient Ž t-stat.

Variable

Parametric

SNPDF-I a

SNPDF-II b

Parametric

SNPH c

Constant

ᎏ

ᎏ

ᎏ

y0.2208 Žy0.2628, y0.179. 3.131e-06 Ž1.26e-06, 5.07e-06.

y0.0792 Žy0.1896, 0.0263. 1.090e-06 Žy2.81e-06, 5.12e-06. y488.57 0.126 108.45

y0.2126 Žy0.4413, 0.00716. y1.063e-05 Žy2.005e-05, y1.55e-06. y478.63 0.150 128.34

1.3697 Ž7.67. y0.3173 Žy9.29. 4.501e-006 Ž2.75.

4.00558 Ž7.66. y1.18803 Žy7.84. ᎏ ᎏ

y493.34 0.117 98.92

y494.139 0.115 97.32

LnŽBID. INCOME

Log-L Efron’s R 2 Chi-sq. a

y493.34 0.117 98.92

SNPDF-I: order s length s 1. SNPDF-II, order s 2; length s 1. c Hermite density coefficients are ␦ s 1.41421,  ␣ 0 , ␣ 1 , ␣ 2 , ␣ 3 4 s 0.63162, 0.39092, y0.58734, y0.087414. Income was excluded from the Hermite ŽSNPH. regression as doing so provided a better match between the actual and predicted response probabilities. b

278

JOSEPH C. COOPER TABLE A-II Data Set Used for the Example Ž791 Observations.

Bid Ž$.

Sample size

No. of ‘‘Yes’’

Percent ‘‘Yes’’

5 7 10 12 15 20 22 28 30 35 40 45 50 55 60 65 70 80 90 100 110 120 125 140 160 175 195 225 250 280 320 450 500 600 650 750 850 900 950 1100

19 18 39 18 20 19 14 14 20 19 38 18 40 20 38 17 16 21 18 35 15 16 15 16 16 15 11 13 15 17 18 18 18 19 18 18 18 19 17 18

17 18 31 13 13 8 11 9 15 14 26 12 26 10 24 11 8 8 10 24 10 12 7 7 8 6 7 9 5 5 5 11 10 10 3 6 2 7 2 2

89.5 100.0 79.5 72.2 65.0 42.1 78.6 64.3 75.0 73.7 68.4 66.7 65.0 50.0 63.2 64.7 50.0 38.1 55.6 68.6 66.7 75.0 46.7 43.8 50.0 40.0 63.6 69.2 33.3 29.4 27.8 61.1 55.6 52.6 16.7 33.3 11.1 36.8 11.8 11.1

TABLE A-III Monte Carlo Analysis of the Parametric and Semi-nonparametric Approaches Model

10th perc.

90th perc.

Mean

Median

Ži. True random utility model is the Fourier Žtrue WTP s $403.72. Parametric 417.29 458.29 437.00 436.65 Semi-nonparametric 378.32 428.17 403.74 404.01 Žii. True random utility model is ⌬V s ␣ q ␤ lnŽBID. Žtrue WTP s $436.94. Parametric 417.32 460.61 437.54 437.10 Semi-nonparametric 402.87 491.19 440.94 437.09

WTP WITH FLEXIBLE FUNCTIONAL FORMS

279

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