A Comparison of Willingness to Pay Estimation ... - Springer Link

Environmental and Resource Economics 20: 331–346, 2001. © 2001 Kluwer Academic Publishers. Printed in the Netherlands.

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A Comparison of Willingness to Pay Estimation Techniques From Referendum Questions Application to Endangered Fish KELLY L. GIRAUD1 , JOHN B. LOOMIS2 and JOSEPH C. COOPER3 1 University of Alaska Fairbanks, Department of Economics, Fairbanks, AK 99775, USA; 2 Colorando State University, Department of Agricultural and Resource Economics, Fort Collins, CO 80523, USA; 3 Economic Research Service (USDA), 1800 M Street, NW, Washington, DC

20036, USA Accepted 18 April 2001 Abstract. Referendum style willingness to pay questions have been used to estimate passive use values. This referendum question format method may be problematic for many reasons, including the statistical techniques used to estimate willingness to pay from discrete responses. This paper compares a number of parametric, semi-nonparametric and nonparametric estimation techniques using data collected from US households regarding Federal protection of endangered fish species.The advantages and disadvantages of the various statistical models used are explored. A hypothesis test for statistical equality among estimation techniques is performed using a jackknife bootstrapping method. When the equality test is applied, the modeling techniques do show significant differences in some possible comparisons, but only those that are nonparamentric. This can lead to conflicting interpretations of what the data show. Resource managers and policy analysts need to use caution when interpreting results until an industry standard can be developed for estimating willingness to pay from closed ended questions. Key words: contingent valuation, endangered species, nonmarket economics, willingness to pay JEL classification: Q2, C2 Abbreviations: Willingness to Pay (WTP), Contingent Valuation Method (CVM), Semi-nonparametric Distribution-free (SNPDF), Critical Habitat Units (CHUs)

1. Introduction If is often difficult to estimate the willingness to pay for many nonmarket resources such as public goods. Indirect means, like revealed preference and hypothetical market techniques are often employed. However, some qualities of the public goods elude even the best nonmarket estimations because they have non-use or passiveuse characteristics (i.e. existence, option, and bequest values). Until recently, the only method to estimate willingness to pay for passive use values is the contingent valuation method (Flores 1996). Today there are other stated preference techniques, like conjoint analysis, contingent ranking and choice experiments. Nonetheless, contingent valuation remains the most popular way to estimate passive use values.

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However, the contingent valuation method (CVM) is fraught with controversy over the inconsistencies of it’s estimates (Cummings, Harrison and Rutstrom 1995; Diamond and Hausman 1994; Neill et al. 1994). Great efforts have been made to reduce the extent of estimation inconsistencies through survey design and implementation. The Blue Ribbon panel on contingent valuation set up by the National Oceanic and Atmospheric Association (DOC 1993) along with Mitchell and Carson (1989) have been interpreted as industry standards for designing an implementing CVM studies. The Blue Ribbon Panel recommended that WTP questions be worded in a referendum format, i.e. ask the respondents if they would vote in favor of paying a given dollar amount for given good(s) or service(s). The referendum CVM was introduced by Hoehn and Randall in 1987, based from the single bounded dichotomous choice format introduced by Bishop and Heberlein in 1979. The discrete choice responses have traditionally been modeled using logistical or cumulative normal equations, known as logit and probit, respectively. Mean WTP estimates are constructed from the distribution described by the logit or probit equation. Innovations have also been developed to calculate mean WTP without the constraint of a given distributionn (Haab and McConnel 1997; Creel and Loomis 1997; Creel 1995; Kristrom 1990; Turnbull 1976). To date, there is no single clearly superior method to compute WTP estimates from discrete responses. This paper compares alternative methods to calculate willingness to pay (WTP) from dichotomous choice data, with an emphasis on the advantages and disadvantages of each, as well as robustness of results for one data set. This is done by use of a jackknife bootstrapping technique, which pulls 1,000 draws from a data set. All of the techniques, mean WTP, as well as the 95% confidence intervals were estimated using GAUSS (1997). For a more rigorous explanation of these estimation techniques, with the exception of the Semi-nonparametric Distribution-Free Modeling, please see Hanemann and Kanninen (1999).

2. Theory The CV response probabilities are related to the underlying WTP distribution by (Hanemann 1984): πiN ≡ Pr{NotoBi∗ } ≡ Pr{Bi∗ > Ci } = G(Bi∗ ; θ)

(1)

While the individual is assumed to know his or her own WTP, the bid offer Cj , to the analyst WTP is a random variable with a given cumulative distribution function (cdf) denoted G(Cj ; θ) where θ represents the parameters of this distribution, which are to be estimates on the basis of the responses to the CVM survey. The parameters will be functions of the variables in Xi , but this is left implicit in G(Ci ; θ). For example, there can be a mean of the WTP distribution which depends on covariates, µ = Xβ, and a variance, σ 2 . In this case, θ = (β, σ 2 ).

ESTIMATION TECHNIQUES FROM REFERENDUM QUESTION

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2.1. PARAMETRIC MODELING Traditionally, dichotomous choice WTP responses (Y) are regressed against a constant, the bid amount (BID), and a vector of socioeconomic variables (X) using a logistic function: Y =

1 1 + exp[−(β0 + β1 BID + X β2 )

(2)

The logistic function estimates the probability that an individual is willing to pay for a good or service, given a bid amount and a set of socioeconomic characteristics. The variable Y is binomial, taking on a value of 1 for a ‘yes’ response, and 0 for a ‘no’ response. In estimating this function, the probability of a ‘yes’ response can be modeled for varying bid amounts (dollar values) and socioeconomic characteristics (such as income, education, age, etc.) While the linear-in-bid model is consistent with a utility difference formulation, a variation on this procedure is to use the log of the BID amount, as in Equation (3). The log transformation has the advantage of restricting mean WTP to fall between zero and infinity, whereas this is not the true with the linear-in-bid model. 1 (3) Y = 1 + exp[−(β0 + β1 ln BID + X β2 ) Mean WTPs are calculated using regression coefficients, β2 , β0 and β1 . Mean WTP is calculated using a formula from Hanemann (1989): Median WTP =

(β0 + X β2 ) |β1 |

(4)

WTP from the log-logistic function shown in Equation (3) has corresponding expressions (Hanemann 1984): Mean WTP = −e−β0 /β1 Median WTP = e−β0 /β1

π/β1 sin(−π/β1)

(5) (6)

There is an important difference between the mean and median WTP estimates for the log logit model. The median value is less affected by extreme outliers, but it places less importance on the votes of those with strong preferences. In the case of the log-logit estimation, the mean WTP is sensitive to changes in the distribution resulting from outliers (Hanemann 1984). If the coefficient on the bid amount is great than −1, this functional form will lead to an undefined estimate of mean WTP. The probit model leads to a similar functional form as Equation (2), but it uses a cumulative normal distribution, shown in Equation (7): β0 +β1 BID+X β2 1 2 exp(−s /2 )ds (7) Pi = √ 2π −∞

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where Pi is the probability of respondent i saying yes to a given bid amount, and s is a standard normal variable. The mean and median WTP for the probit are calculated using the same formula as the logit model, Equation (4). Again, they are equal because of the symmetrical probability distribution function inherent in probit modeling. Advantages of the probit over the logit model include less need for a large sample. Studenmund (1992) suggests that logit models should have a sample size of at least 500 observations. The probit model specification is more cumbersome, and probit’s historic disadvantage has been the computer time required to run a model. However, with software and hardware advances, this is no longer the case. The next estimation method is similar to Logit, but uses a slightly asymmetric distribution known as the Weibull distribution: Pi = exp(−exp(β0 + β1 BID + X β2 ))

(8)

Coefficients from the logit, probit and Weibull functions can be used to generate confidence intervals by a method developed by Park, Loomis and Creel (1991) from the Krinsky and Robb method (1986). These approaches make thousands of draws of the coefficients using the variance-covariance matrix, and WTP is calculated for each draw of the coefficients. These estimators are appealing because of the explanatory power of including socioeconomic variables, while non-parametric estimates assume a homogenous sample. If a given sample is not a perfect representation of the population, a parametric WTP function can be projected onto the entire population using demographic information obtained from sources like the Census Bureau. The disadvantage of parametric approaches is that the data is forced into fitting a given distribution. Non-parmetric estimation techniques are distribution-free, so the researcher gets a more realistic sense of the cumulative density functions and mean WTP of the sample. Duffield and Patterson (1991) suggest that the non-parametric models should be used in survey bid design, not necessarily as a replacement for the parameter approaches to estimating the final WTP. 2.2. SEMI - NONPARAMETRIC DISTRIBUTION - FREE MODELING The semi-nonparametric distribution-free (SNPDF) approach, first applied to single bounded data by Creel and Loomis (1997), is applied to our data set. Chen and Randall (1997) present an alternative model for single bound data similar to that of Creel and Loomis. The goal of the SNPDF approach is to reduce the sensitivity of our econometric analysis to specific parametric assumptions regarding the form of the WTP distribution. A simple way to motivate the SNPDF approach is to observe that, with the logistic WTP distribution, the CVM response probabilities can take the form: Pi = G(Bi∗ ; θ) ≡ F (Bi∗ )

(9)


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∗

where F(B∗i ) = [1 + e−Bi ]−1 is the standard logistic cumulative density function (cdf) and (β) ≡ −α + βB

(10)

is what Hanemann (1984) calls a utility difference function, which is increasing in the bid price, B. The SNPDF approach retains the logistic cdf in the response probabilities such as (9), but replaces the linear utility difference with a Fourier flexible form (e.g. Gallant 1982), where (omitting quadratic term as in Loomis and Creel): V (x, θk ) = xβ +

J A

(vj α cos[j k α s(x)] − wj α sin[j k α s(x)])

(11)

α=1 j =1

the vector x contains all arguments of the utility difference model, A and J are positive integers, and kα are vectors of positive and negative integers that form indices in the conditioning variables, after shifting and scaling of x by s(x).1 There exists a coefficient vector such that, as the sample size becomes large, (x) in (11) can be made arbitrarily close to a continuous unknown utility difference function for any value of x. In our particular specification, the bid price is the only explanatory variable, so that kα is a (1 × 1) unit vector and max (A) equals 1. We choose the same value for integer J as do Creel and Loomis, leading to V (B) = γ + δB + δv coss(B) + δw sins(B)

(12)

where s(B) prevents periodicity in the model and is a function that shifts and scales the variable to lie in an interval less than 2π (Gallant).2 Specifically, the variable is scaled by subtracting its minimum value, then dividing by the maximum value, and then multiply the resulting value by 2π − 0.00001, which produces a final scaled variable in the interval [0, 2π − 0.0001]. When δv = δw = 0, (12) reduces with δ = β and γ = −α: the logistic WTP model is nested within the SPNDF model. 2.3. NON - PARAMETRIC MODELING In 1976, Turnbull pioneered a distribution-free strategy, which when applied to CVM eliminates the possibility of negative WTP estimates. Negative WTP estimates can be problematic in linear parametric models. With the Trunbull technique, the data is separated into intervals based on the bid amounts. The probability density function for each bid amount, pi , is defined as the percentage of respondents voting “no” to a given bid amount (cj −1 ), this percentage must be less than or equal to the percent of “no” votes to the next higher bid amount (cj ). It is therefore the estimates probability that a respondent’s maximum WTP falls within the interval cj −1 to cj (Haab and McConnell 1997): Pi = P (cj −1 < WTP ≤ cj )

(13)

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The likelihood function can be written as: j M n [Nj ln( pi ) + Kj ln(1 − pi )] L(p; N, K) = j =1

i=1

(14)

i=1

where N is the number of respondents who respond ‘no’ to cj , and K is the number of respondents who say ‘yes’, M is the number of bid amount used. This function is not as difficult to calculate as it first appears and Kristrom (1990) describes it as a “computation that can be done on the back of an envelope.” The method requires a monotonically decreasing probability sequence that can be calculated using a simple algorithm: pi =

ki ni

(15)

In this case, i denotes individual i = 1, 2, . . ., N + K. If the data set is not monotonic, then adjacent values are pooled as the Equation (16) until it is. pi =(ki −ki−1 )/(ni −ni−1 )

(16)

This will lead to the probability density function, and subsequently to the cumulative density function. The mean WTP is calculated by multiplying the monotonic probabilities (pi ) by the bid amounts (cj ). This simple technique has been estimated by a number of researchers (Haab and McConnell 1997; Duffield and Patterson 1991; Kristrom 1990). Slight variations of the Turnbull estimation include the assumption of a piecewise linear functional form, with an arbitrarily chosen upper bound. Haab and McConnell point out that this can lead to different estimates of mean WTP, depending on the range of the bid amounts. They assume that the pi ’s are distributed asymptotically normal, so that more efficient confidence intervals can be constructed if the mean and variance of the sample are calculated. The variance of the lower bound estimate is given by:   M+1 M+1 M pj cj −1  = cj2−1 (V (Fj ) + V (Fj −1 )) − 2 cj cj −1 V (Fj ) (17) V j =1

j =1

Where Fj is the cdf for bid cj . Further, V(Fj ) is given by: −1 ∂ 2L Fj (1 − Fj ) = V (Fj ) = − 2 Nj + Kj ∂Fj

j =1

(18)

While the non-parametric is distribution free, it has at least two drawbacks when compared to parametric approaches. First, large sample sizes are needed at each bid amount to reduce the chances of non-monotonically and resulting loss in efficiency of the estimator from pooling adjacent bid responses. Second, coefficients, such as demographics cannot be included when estimating WTP.


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3. Comparisons of Benefits Estimates While each approach has conceptual advantages and disadvantages, a key question remains: does the selection of a particular estimator result in significant differences in WTP? After all, benefit-cost analysis is a policy driven empirical undertaking. At a scientific level, we can test the hypothesis that WTP of the alternative estimators are significantly different. Specifically: HO : WTPi − WTPj = 0 HA : WTPi − WTPj = 0

(19) (20)

For each of the models j and i, where j = i. Poe et al. (1994) have demonstrated that the overlapping of confidence intervals is not appropriate for determining if two estimated distributions are significantly different. Poe et al. suggest a thorough convolutions approach. This paper implements a pairwise comparis on of benefit estimates that result from a jackknife bootstrapping procedure. For actual policy and benefit-cost analysis, decision makers are perhaps more interested in whether any differences in estimators would change the economic efficiency recommendations regarding a particular policy action. Thus even if Equation (19) is rejected, and if the significant differences in WTP are small enough not to change the outcome of the benefit-cost analysis, then selection of the estimator is not crucial. Alternatively, if the variance of WTP is large, policy relevant differences in WTP estimates may not be statistically different, but could result in reversals of efficiency recommendations from one of benefits exceeding costs to one of costs exceeding benefits. Because it is the welfare measures, and not the coefficient estimates, that are generally of primary interest, it is useful to compare the estimated welfare that are derived from each regression. Furthermore, because the welfare measures are nonlinear functions of the coefficients, the observations from the coefficient analysis above may not hold for the welfare estimates. Corresponding point estimates of mean WTP calculated may be estimated using: max BID R {1 − F [V (B)]}dB (21) C = 0

where δV(B) as in Creel and Loomis (1997). Standard errors associated with these point estimates can be derived via the jackknife method (described below) with 1,000 repetitions in each case.3 The empirical 95% confidence intervals for median WTP are based on the jackknife output. Efron’s (1987) Bias Corrected Accelerated (BCa) 95% confidence intervals adjust the jackknife output for potential non-normalities. The Fourier form is SNPDF produces estimators are inherently highly collinear, so hypothesis tests derived from the covariance matrix are not trustworthy. Instead, confidence intervals for the regression results are produced with a jackknife approach (Cooper 2000). Specifically, 1,000 simulated data sets are generated by

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randomly drawing (with replacement) observations from the real data set to create simulated data sets with the same sample size as the real data set. The bootstrap is the most general method for estimating confidence intervals since it uses the actual sample as the population for the choice sets. The confidence intervals presented in Table IV are constructed from the regression results on each simulated data set and are of the bias corrected accelerated (BCa) type (Efron 1987), which gives the bootstrap results an interpretation analogous to t-statistics by making the estimated confidence interval symmetric around the mean. Gains from increasing the number of transformations with the SNPDF approach are addressed in Cooper (2000). To investigate the hypothesis using this approach, we rely on the following case study. 4. The Data During the summer of 1996, a sample of 1600 respondents received a mail survey concerning a program to protect Critical Habitat Units (CHUs) of nine endangered fish species which live in several rivers in the southwestern United States. The respondents are asked if their household would pay a given bid amount to maintain these CHUs. The payment vehicle was a federal tax that would go into a trust fund could be used only for the endangered fish recovery: If the Four Corners Region Threatened and Endangered Fish Trust Fund was the only issue on the next ballot and it would cost your household $ every year, would you vote in favor of it? YES NO

The bid amounts range from $1 to $350, as can be seen in Table I. The survey followed most recommendations made by Mitchell and Carson (1989) and the NOAA panel (D.O.C. 1993). One notable exception to the recommendations of the NOAA panel is that a mail survey was employed, rather than in-person interviews. The mail survey followed the Dillman Total Design Method (Dillman 1978). The response rate was 53.7%, which is 715 surveys (accounting for undeliverable surveys). The respondents in this survey differ from the characteristics of the general population. This is not unusual situation. Table II compares the demographics of the two samples to 1990 estimates of the population levels of these variables (with income adjusted to 1995 levels). As is typical in mail surveys, the sample education level, income and age is higher than the state level. Due to Survey Sampling Inc. drawing the majority of names from the phone books, which are traditionally listed under the male’s name, the sample over-represented males relative to females, about 70 percent to 30 percent compared to about 49 percent males in the population. Respondents age and education are somewhat higher than that of the general population. This can be seen as advantage. The respondents who choose to complete a survey may likely be part of the population segment that chooses to vote. Either way, it is important to take the socioeconomic characteristics into consideration when making general assumptions of willingness to pay.

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Table I. Responses from the survey by bid amount. Bid

# sent

# returned

% returned

# YES

% YES

$1 3 5 10 15 20 30 40 50 75 100 150 200 350

116 116 114 114 114 114 114 114 114 114 114 114 114 114

61 52 54 54 46 57 55 50 41 48 58 45 50 46

52.59 44.83 47.37 47.37 40.35 50.00 48.25 43.86 35.96 42.11 50.88 39.47 43.86 40.35

40 41 39 34 26 33 31 30 16 19 24 14 18 18

65.57 78.85 72.77 62.96 56.52 57.89 56.36 60.00 39.02 39.58 41.38 31.11 36.00 39.13

Table II. Socioeconomic characteristics of survey sample compared to the general population.

Average age of households Education, 25 years and older2 Mean income, for all households Household size Percentage of males Percentage of females

United States1

Sample

43 12.1 $44,7003 2.64 48.7 51.3

51.9–55.4 14.5–15.1 $49081–$57644 2.35–2.64 60.8–71.0 29.0–39.2

1 Source: U.S. Bureau of the Census 1994. 2 Education in years, i.e. level 12 is high school, 14 is associates degree,

16 is college graduate. 3 Adjusted to 1995 level using Producer Price Index, Employment Price Index, Private Industries.

5. Results 5.1. STATISTICAL RESULTS The coefficients from the models that were estimated are presented in Table III. The latent variable (yes = 1 and no = 0) is regressed against a Constant term, the BID amount, the respondents highest level of completed Education (in years), the respondent’s household Income, and responses to a group of opinion and knowledge holding questions. Projob and Protect are the result of a series of

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likert scale statements which were rated from strongly agree to strongly disagree.4 The Projob statements included things such as, “Businesses should be allowed to extract natural resources from Federal Lands”. The Protect statements included things such as, “Plants and animals have as much right as humans to exist”. Knowfish is a variable that is equal to 1 if the respondent has read or heard about the threatened and endangered fish in the Colorado River, and 0 otherwise. Also included in Table III is the chi-squared test for goodness of fit. This test looks at the overall significance of the model by comparing the calculated test statistic, Equation (22), to a critical chi-squared statistic. c = −2(LLR − LLU )

(22)

LLR and LLU are the Log likelihoods for the restricted and unrestricted models respectively. The critical chi-squared statistic is 14.07 for 5% level of significance. In all cases, this test indicates significance of the models. The coefficient on BID in Table III are negative, indicating that the higher the bid amount, the less likely it is for a respondent to vote in favor of the program. Education and Income have positive coefficients, showing evidence that respondents with higher levels of education and income would be more likely to vote yes for the program. Projob and Protect have the expected opposing signs, which suggest that respondents agreeing with Projob questions would be less likely to vote in favor of the program, and those agreeing with the Protect questions would be more likely to vote in favor of the program. Finally, the positive coefficient on Knowfish implies that respondents who have read or heard about the threatened and endangered fish in the Colorado River would be more likely to vote in favor of the program that protects them.5 The willingness to pay results shown in Table IV are quite similar across the models. As expected, the coefficient of variation for the SNPDF model is higher than for the parametric models. It is higher still for the Turnbull, as the SNPDF functional form is more flexible than the parametric models, but in this small sample case, is less flexible than the nonparametric Turnbull. Note that the Turnbull.2 method is somewhat more efficient than Turnbull.1. Turnbull.1 uses lower bound estimate of WTP, namely, the triangles that form the trapezoids in Turnbull.2 are left out. Turnbull.2 is a trapezoidal estimator (triangles added). The first row of Table IV presents the corresponding point estimates of mean WTP calculated using equation (21). The sixth row gives the standard errors associated with these point estimates, derived via the jackknife method with 1,000 repetitions in each case. The empirical 95% confidence intervals for median WTP based on the jackknife output are shown in the fourth and fifth rows. The second and third rows gives Efron’s (1987) Bias Corrected Accelerated (BCa) 95% confidence intervals, which adjust the jackknife output for potential non-normalities. The statistical techniques described above lead to the estimates in Table IV. The mean WTP from the log-logit model could not be calculated because of the BID coefficient in the log-logit model is greater than −1. This may indicate that the non-

−0.006 (−4.91) 0.048 (1.19) −0.003 (−4.89) 0.030 (1.23) −0.387 (−5.87) 0.037 (0.90) −0.004 (−4.60) 0.046 (1.82) −0.003 (−3.83) −0.006 (−0.07)

Education 6.2e-6 (2.17) 3.6e-6 (2.17) 6.5e-6 (2.26) 3.8e-6 (2.18) 5.1e-6 (2.15)

Income −0.218 (−3.94) −0.134 (−4.17) −0.222 (−4.00) −0.102 (−2.95) −0.131 (−3.99)

Projob 0.509 (8.91) 0.296 (9.14) 0.512 (8.88) 0.366 (9.20) 0.306 (9.25)

Protect

$164.19 143.88–185.15 140.94–183.61 10.53 0.065 15.1

Logit 165.75 145.12–187.03 142.05–185.76 10.69 0.065 15.2

Probit

186.26 165.32–207.88 166.17–209.66 10.86 0.058 17.1

Weibull

173.05 150.16–196.68 148.12–195.52 11.87 0.069 15.9

SNPDF

151.75 128.51–175.75 122.11–170.19 12.05 0.082 13.9

Turbull 1

192.79 192.02 155.27 195.51 218.08

Chi-squared test

162.23 139.55–185.62 137.29–183.90 11.75 0.073 14.9

Turnbull 2

0.299 (1.46) 0.170 (1.41) 0.293 (1.45) 0.130 (1.01) 0.084 (1.58)

Knowfish

Table IV. Mean WTP point estimates (0 < WTP < maximum bid) and associated statistics (US$).

0.448 (0.60) 0.198 (0.45) 1.473 (1.91) −0.109 (−0.12) 0.714 (0.54)

E(WTP) 95% BCa C.I. for WTP 95% empir. C.I. for WTP S. error Ceof. of var National WTP ($billion)

Logit Probit Log-logit Weibull SNPDF

Technique Coefficients and (t-statistics) Constant BID

Table III. Results from model regressions.


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Table V. Hypothesis tests of equality of WTP – evaluated by pairwise comparisons of benefit estimates from each model (1 = equality accepted; 0 = equality not accepted, ∝ = 0.05). Logit Probit Weibull SNPDF Turnbull 1 Turnbull 2 Logit Probit Weibull SNPDF Turnbull 1 Turnbull 2

– 1 1 1 0 0

1 – 1 1 0 0

1 1 – 1 1 1

1 1 1 – 0 0

1 0 1 0 – 1

0 0 1 0 0 –

Notes: Lower triangle is the hypothesis test based on BCa transformations of the bootstrap results. Upper triangle is the hypothesis tests based on empirical confidence intervals of the bootstrap results.

negativity constraint is to restrictive. For brevity, only the final SNPDF results are presented here; the original parametric results that went into the SNPDF are available from the authors. Since the parametric logit model is nested in the SNPDF, likelihood ratio tests, i.e., λLR = 2[1nL − lnLR ] with critical value χ 2 (2,0.05) = 14.07, can be used to compare the models. Note in particular that two models are statistically different from a few others at the 95% level. As expected, the logit and probit parametric models are quite similar. When testing the Hypothesis (19) from the estimation techniques, there is an indication that only the Turnbull estimates are significantly different from the logit estimations, logit being the most frequently used statistical model for dichotomous choice CVM. Table V presents the results of the pairwise hypothesis test Ho: WTPi − WTPj = 0, where the subscripts reference the approaches. Each of the 1,000 simulated data sets was saved, thereby allowing each data set to be analysed by each approach and hence, allowing pairwise comparisons of the WTP estimates across these approaches. The tests are based on 95% confidence intervals (CIs) from the empirical interval as well as 95% CIs from the BCa transformation. In most cases except for pairwise comparisons with the Turnbull, the null hypothesis is not rejected at the 10% level of significance. 5.2. COSTS OF ENDANGERED FISH RECOVERY PROGRAM The US Fish and Wildfish Service has estimated the construction and water flow acquisition costs of the endangered fish recovery effort at $120 to 224 million (US Fish and Wildlife Service 1996). To this, $18.5 million in annual maintenance must be added, yielding a total cost of $138.5 to $242.5 million (Cleymaet 1998). In order to make an informal benefit-cost comparison, our per household benefit estimates from each model must be aggregated upwards to the national benefits. We use national benefits for two reasons. First, this endangered species


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program is a federal program, paid for with federal tax dollars. Because of this, we sampled nationally and believe we had a fairly representative sample from the United States. Our relatively good response rate of about 54% and comparability to the US population suggests we can expand our per household values upward by 91,993,582 households (US Census Bureau 1994). Doing so for six estimated benefit levels is shown in Table IV. As is clear, all of the benefit estimates exceed the costs, and frequently by a factor of ten. Thus, while different statistical analysis approaches may yield sizable differences in per household WTP, for this particular policy analysis, the endangered species recovery program is economically justified regardless of estimation used.

6. Conclusions The contingent valuation method is one of the only available methods to estimate passive use values. This method is controversial for many reasons, including the techniques used to estimate willingness to pay from closed-ended, single bounded responses. This paper shows evidence that using the jackknife approach the WTP estimates are not significantly different between various parametric and semi-nonparametric modeling techniques with the exception of Turnbull technique estimates. Turnbull estimates have been shown to be useful in bid design, but a number of researchers have stated that they are not reliable for final benefit estimation (Duffield and Patterson 1991). The distributional assumptions may contribute to the significant differences between the Turnbull and other estimation techniques. The main distributional assumption in Turnbull and Weibull is the restriction of the WTP to fall only in the nonnegative region. Even with the non-negativity constraint on the Weibull technique, it shows no evidence of being significantly different from any of the estimates. Other estimation methods outlined in this study allow for negative WTP amounts. The log-logit approach also forces the model to lie in the positive region, and is sensitive to outlying observations. The WTP and the confidence intervals could not be estimated using the log-logit method in this study because the coefficient on the bid amount was greater than −1, indicating an undefined WTP amount. Because the data is forced into the positive region, WTP estimates will be relatively larger than that of the unrestricted techniques. Nonparametric estimation techniques have advantages in the fact that they do not impose a distribution onto the data, but they can lead to inconsistent results depending on the maximum bid amount, and the size of the sample. These techniques do not allow for the inclusion of socioeconomic characteristics. Again, Duffield and Patterson (1991) suggest that the non-parametric models should be used in survey bid design, not necessarily as a replacement for the parameter approaches to estimating the final WTP. Resource managers and policy analysts need to use caution when interpreting WTP estimates. They should be made aware of the estimation technique, and the

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assumptions associated with it. All techniques have advantages and disadvantages, so the authors suggest that sensitivity analysis should be performed if the benefitcost ratio is close to one to ensure economic efficiency recommendations are robust in the statistical estimator used. Notes 1. In addition to appending Xβ to the Fourier series in equation (11), 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. 2. With 14 unique bid values in our data set, our specification permits a max(j) = 5 to avoid singularity in the regression. For our data, since increasing J to values above 1 yielded little change in the regression results, J = 1 appears to proved the best balance in the trade-off between bias and efficiency. 3. The program to perform the maximization is written in GAUSS and is available from the authors. 4. Six likert-scale questions were asked on a scale of 1 to 5, with 1 being strongly agree, 3 being neutral, and 5 being strongly disagree. i) Businessess should be allowed to extract natural resources from Federal lands; ii) All species endangered due to human activities should be protected from extinction whether or not they appear important to human well being; iii) Plants and animals have as much right as humans to exist; iv) I am glad that the endangered species in the Four Corners Region are protected even if I never see them; v) If any jobs are lost, the cost of protecting a Threatened or Endangered Species is too large; and vi) Protection of Threatened and Endangered Species is a responsibility I am willing to pay for. Questions i) and v) were added together and multipled by −1 to form PROJOB. Questions ii), iii), iv), and vi) were added together and multiplied by −1/2 to form PROTECT. 5. This variable is not significant at the 95% confidence level, but it is the 90% confidence level for most estimation techniques. The variable remained in the regression to avoid omitted variable bias.

References Bishop, R. and T. Heberlein (1979), ‘Measuring Values of Extra Market Goods: Are Indirect Measures Biased?’, American Journal of Agricultural Economics 61, 926–930. Chen, H. Z. and A. Randall (1997), ‘Semi-Parametric Estimation of Binary Response Models with an Application to Natural Resource Valuation’, Journal of Econometrics 76(1–2), 323–340. Cleymaet, R. (1998), Use of CVM in Cost-Benefit Analysis of Nine Threatened and Endangered Fish Species in the Colorado River Basin. Technical Paper, Colorado State University, Department of Agricultural and Resource Economics. Cooper, J. (2000), ‘Nonparametric and Semi-Parametric Travel Cost Methods’, American Journal of Agricultural Economics (May) 82, 245–259. Cooper, J. and D. Hellerstein (1989), CVM: A Set of CVM Estimators. Washington, DC: USDS/ERS. http://rpbcam.econ.ag.gov/GOGRBL Creel, M. and J. Loomis (1997), ‘Semi-Nonparamentric, Distribution Free Dichotomous Choice Contingent Valuation’, Journal of Environmental Economics and Management 32(3), 341–358.


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Creel, M. (1995), ‘A Semi-nonparametric, Distribution Free Estimator for Binary Discrete Responses’, Revision of Working Paper 267.94. Department of Economics and Economic History, University Autonoma de Barcelona. Cummings, R., G. Harrison and E. E. Rutstrom (1995), ‘Homegrown Values and Hypothetical Surveys: Is Dichotomous Choice Approach Incentive Compatible?’, American Economic Review 85, 260–266. Diamond, P. and J. Hausman (1994), ‘Contingent Valuation: Is Some Number Better Than No Number?’, Journal of Economic Perspectives 8(4), 45–64. Dilman, D. (1978), Mail and Telephone Surveys. New York: John Wiley and Sons. Duffield, J. and D. Patterson (1991), ‘Inference and Optimal Design for a Welfare Measure in Dichotomous Choice Contingent Valuation’, Land Economics 67(2), 225–239. Efron, B. (1987), ‘Better Bootstrap Confidence Intervals’, Journal of the American Statistical Association 82(397), 171–185. Gallant, A. (1982), ‘Unbiased Determination of Production Technologies’, Journal of Econometrics 20, 285–323. Flores, N. (1996), ‘Reconsidering the Use of Hicks Neutrality to Recover Total Values’, Journal of Environmental Economics and Management 31(1), 49–64. GAUSS for Windows NT/95 Version 3.2.32 (Dec 19 1997). Aptech Systems, Inc. Maple Valley, WA. Haab, T. and K. McConnell (1997), ‘Referendum Models and Negative Willingness to Pay: Alternative Solutions’, Land Economics 32(1), 251–270. Hanemann, M. and Kanninen (1999), ‘Valuing Environemntal Preferences: Theory and Practice of the Contingent Valuation Method in the US, EU, and Developing Countries’, in I. Bateman and K. Willis, eds., Chapter 10 Statistical Considerations in CVM. New York: Oxford University Press. Hanemann, M. (1989), ‘Welfare Evaluations in Contingent Valuation Experiments with Discrete Response Data: Reply’, American Journal of Agricultural Economics 71(4), 1057–1061. Hanemann, M. (1984), ‘Welfare Evaluations in Contingent Valuation Experiments with Discrete Responses’, American Journal of Agricultural Economics 66(3), 332–341. Hellerstein, D. (1998), GRBL: A Non-Linear Econometric Package. Washington, DC: USDS/ERS. http://rpbcam.econ.ag.gov/GOGRBL Hoehn, J. and A. Randall (1987), ‘A Stisfactory Benefit Cost Indicator from Contingetn Valuation’, Journal of Environmental Economics and Management 13(3), 226–247. Krinsky, I. and A. Robb (1986), ‘Approximating the Statistical Properties of Elasticities’, Review of Economics and Statistics (Nov) 68, 715–719. Kristrom, B. (1990), ‘A Non-Parametric Approach to the Estimation of Welfare Measures in Discrete Response Valuation Studies’, Land Economics 66(2), 135–139. Mitchell, R. and R. Carson (1989), Using Surveys to Value Public Goods: The Contingent Valuation Method. Washington, DC: Resources for the Future. Neill, H., R. Cummings, P. Garderton, G. Harrison and T. McGuckin (1994), ‘Hypothetical Surveys and Real Economics Commitments’, Land Economics 70(1), 145–154. Park, T., J. Loomis, and M. Creel (1991), ‘Confidence Intervals for Evaluating Benefits from Dichotomous Choice Contingent Valuation Studies’, Land Economics 67(1), 64–73. Poe, G., E. Severance-Lossin and M. Welsh (1994), ‘Measuring the Difference (X-Y) of Simulated Distributions: A Convolutions Approach’, American Journal of Agricultural Economics 76(4), 904–915. Studenmend, A. H. (1992), Using Econometrics: A Practical Guide, 2nd edn. Harper Collins Publisher, pp. 518–526. Turnbull, B. (1976), ‘The Empirical Distribution Function with Arbitrarily Grouped, Censored and Truncated Data’, Journal of the Royal Statistics Ser. B 38, 290–295. U.S. Census Bureau (1994), Current Population Survey. Washington, DC: Government Printing Office.

346

GIRAUD ET AL.

U.S. Department of Commerce (1993), ‘Arrow, Kenneth, Robert Solow, Edward Leamer, Roy Rander and Howard Schuman. Report of the NOAA Panel on Contingent Valuation’, Federal Register 58(10), 4602–4614. U.S. Fish and Wildlife Service (1996), Recovery Implementation Program for Endangered Fishes of the Upper Colorado River Basin. Recovery Program for the Endangered Fishes of the Upper Colorado, March 18–March 22.