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LATENT THRESHOLDS ANALYSIS OF CHOICE DATA UNDER VALUE UNCERTAINTY
MIMAKO KOBAYASHI, KLAUS MOELTNER, AND KIMBERLY ROLLINS
In many non-market valuation settings stakeholders will be uncertain as to their exact willingnessto-pay for a proposed environmental amenity. It then makes sense for the analyst to treat this value as a random variable with distribution only known to the respondent. In stated preference settings, researchers have used elicitation formats with multiple bids and uncertain-response options to learn about individual value distributions. Past efforts have focused on inference involving the expectation of individual densities. This requires stringent and likely unrealistic assumptions regarding the shape or moments of individual value distributions. We propose a Latent Thresholds Estimator that focuses instead on the range, i.e. minimum and maximum willingness-to-pay of individual respondents. The estimator efficiently exploits correlation patterns in individual responses and does not require any restrictive assumptions on underlying values. It also nests some of the existing approaches, which are not statistically supported for our empirical application. Key words:stated preference,multiple bounded elicitation,polychotomous choice,Bayesian estimation, value uncertainty. JEL codes: C11, C15, C35, C52, Q51.

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The maintained assumption for this study is that individuals do not hold a specific, fixed value for an environmental amenity or proposed policy, but rather treat this value, call it vi , as a random variable with smooth, continuous probability density function (pdf ) gi (vi ). This pdf is known to the individual, but not to the researcher. As discussed in Opaluch and Segerson (1989), Ready, Whitehead, and Blomquist (1995), Shaikh, Sun, and vanKooten (2007), Akter, Bennett, and Akhter (2008),Akter et al. (2009), and Kobayashi, Rollins, and Evans (2010) there are numerous reasons that can contribute to such “value uncertainty”, such as uncertain stream of future income, uncertain location of residence at the time of policy implementation, and uncertainty over the

Klaus Moeltner is an Associate Professor at the Department of Agricultural and Resource Economics, Virginia Tech. Mimako Kobayashi is a Research Assistant Professor and Kimberly Rollins an Associate Professor in the Department of Economics, University of Nevada, Reno. We thank Douglass Shaw, J. Scott Shonkwiler, seminar participants at the 2010 W2133 Regional Meetings in Tucson, AZ, and two anonymous reviewers for helpful comments. Part of this research was conducted while Moeltner was a Visiting Scholar at the Luskin Center for Innovation, University of California, Los Angeles.

future availability of substitutes.1 We believe that accounting for value uncertainty is a prudent research strategy in many stated preference settings, especially when the time frame for policy implementation is moved relatively far into the future compared to the timing of value elicitation or when respondents lack experience with or knowledge of the amenity or policy in question. A common approach to capture value uncertainty in stated preference surveys is to combine a bid-based elicitation format with discrete response categories representing varying degrees of uncertainty. These uncertainty measures can take the form of a numerical scale (e.g. Li and Mattsson 1995; Champ et al. 1997; Ekstrand and Loomis 1998) or a discrete set of qualitative response options (“definitely yes”, “probably yes”, “not sure”, etc) (e.g. Ready, Whitehead, and Blomquist 1995; Wang 1997; Ready, Navrud, and Dubourg 2001). Ready, Whitehead, and Blomquist

1 Opaluch and Segerson (1989) also argue that in some cases individuals might also be uncertain as to the exact shape of gi (vi ). In this study we abstract from this type of “ambiguity”. In the same vein, we also assume stability of gi (vi ) over time, at least in the short run.

Amer. J. Agr. Econ. 94(1): 189–208; doi: 10.1093/ajae/aar129 Received November 24 2011 © The Author (2012). Published by Oxford University Press on behalf of the Agricultural and Applied Economics Association. All rights reserved. For permissions, please e-mail: [email protected]

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(1995) deem this elicitation approach the “polychotomous choice” (PC) format. Originally, these uncertainty scales were employed in combination with a single referendum question. Welsh and Poe (1998) extend this framework by adding additional bid amounts, each of which is paired with a polychotomous response choice. The authors label this approach the “Multiple Bounded Discrete Choice” format. It is also referred to as the “Multiple Bounded Uncertainty Choice” format (Evans, Flores, and Boyle 2003), and the “Multiple Bounded Polychotomous Choice” format (Alberini, Boyle, and Welsh 2003) in subsequent applications. We adopt Evans, Flores, and Boyle’s acronym “MBUC” throughout this text. The main focus of the bulk of these contributions was not to learn about gi (vi ) per se, but rather to circumvent some of the practical hurdles, such as non-responses and “protestzeros”, that often plague generic Dichotomous Choice (DC) formats (e.g. Ready, Whitehead, and Blomquist 1995). A second motivation was to gain insights into the reliability and interpretation of responses when participants are constrained to choose between a simple “yes” and “no” (e.g. Ready, Whitehead, and Blomquist 1995; Champ et al. 1997; Welsh and Poe 1998; Ready, Navrud, and Dubourg 2001).2 Noteworthy exceptions, to be discussed below in more detail, are Li and Mattsson (1995), Wang (1997), Alberini, Boyle, and Welsh (2003), and Evans, Flores, and Boyle (2003), who all use a PC or MBUC format to gain inference on properties of gi (vi ). However, we argue that these existing routes are based on relatively restrictive assumptions on gi (vi ), without fully exploiting the information content flowing from MBUC elicitation. We propose a new estimator that focuses on the simultaneous estimation of the multiple decision thresholds that separate the polychotomous response categories rather than the expectation of an underlying individual value distribution. Our Latent Thresholds Estimator (LTE) recovers the range and thus the minimum and maximum of vi for each respondent in the sample, and the expectation of these statistics for the underlying population at large. This can be achieved without resorting to any assumptions on the shape or moments of gi (vi ).
2 For example,Welsh and Poe (1998) find that unsure individuals are likely to respond affirmatively in a simple DC setting, and that welfare estimates associated with open-ended and payment card techniques are more consistent with higher levels of certainty.

In addition, the LTE allows for inference on the effects of observable respondent characteristics on the central tendency and spread of this distribution. It also nests several existing MBUC models and thus allows for a rigorous examination of the underlying assumptions associated with these alternative specifications. We believe that our approach broadens the applicability of the MBUC method as tool for both welfare estimation and guidance in bid and survey design. We illustrate our framework using data from the first field implementation of the MBUC format in the 1994 Glen Canyon Pilot Study (Welsh et al. 1995). In the next section we discuss existing models of uncertain responses and highlight their strength and shortcomings. The third section introduces the econometric framework for the Latent Thresholds Estimator (LTE). This is followed by an empirical section that introduces the data and discusses estimation results. The last section concludes.

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Modeling Uncertain Responses To motivate the discussion of the existing literature,consider figure 1. It depicts g(vi ) for three individuals. We purposefully let these densities differ vastly in spread and shape. Each can be assumed to have its own expectation and variance. Person one’s pdf (solid line) is unimodal and relatively symmetric. Person two (dashed line) has a bi-modal value function (perhaps due to equal probabilities of lower or higher future income),and g3 (v3 ) (dashed-dotted line) is skewed to the right. The figure also shows the outermost thresholds for each individual. For example, in a five-tiered polychotomous response format, the left-most threshold would correspond to the value of vi at which a respondent would switch from “definitely yes” (DY) to “probably yes” (PY). Analogously, the rightmost threshold indicates the switching point from “probably no” (PN) to “definitey no” (DN). Following the bulk of the literature we interpret these points as minimum and maximum willingness-to-pay (WTP), respectively. Our ultimate goal is to gain inference on the population distribution of these extrema. For example, as drawn in the figure and observed for our empirical application, the leftmost threshold exhibits lower population variance than its rightmost counterpart. For ease of exposition the figure does not depict any inner thresholds. i.e. the points at

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Figure 1. Examples for Value Distributions and Outer Thresholds which respondents switch from PY to NS, and from NS to PN. However, we do estimate these in our application. Their role is twofold. They aid in the identification of the outer thresholds via error correlations (see below),and they allow for an examination of marginal effects of personal characteristics along the entire range of the value distribution, as discussed below in more detail. As indicated in the figure, our model allows all individual thresholds to differ from one another. It does not impose any restrictions on threshold location relative to the (unknown) expectation of vi , nor does it impose common interpretability of the area under the value curve to the right or left of a specific bid amount. For example, consider the specific dollar amount marked in the figure as bid bj . Clearly, the value of the cumulative distribution function (cdf ) at this point, i.e prob(vi < bj ) is vastly different for each individual. Yet, all existing studies that treat vi as a random variable, and that use a PC or MBUC format to model value uncertainty, build either on this “common area” assumption or impose proportionality of threshold locations vis-à-vis the expectation of vi . Table 1 summarizes the key features of these contributions. All of them offer one or several bids bj to a given respondent, each of which triggers a response ri (bj ). As indicated in column two of the table, for a numerical scale, such as in Li and Mattsson (1995), ri (bj ) might be “80% certain”, while for a polychotomous scale, used in Wang (1997), Alberini, Boyle, and Welsh (2003), and Evans, Flores, and Boyle (2003), ri (bj ) takes the form of a qualitative answer, such as “probably yes” and “definitely no”. Li and Mattsson (1995) and Evans, Flores, and Boyle (2003) interpret an observed response directly as a numerical cumulative probability, i.e. ri (bj ) → prob(vi < bj ) = Gi (bj ) = αm , where Gi (.) is the cdf for vi and αm is some numerical value chosen by the analyst for response category m. As captured in column three of table 1, in Li and Mattsson (1995) this is possible by assigning a normal density to vi , ∀i, with common variance across all individuals. Evans, Flores, and Boyle (2003) leave the density family for gi (vi ) unspecified, but assume a common probabilistic interpretation of Gi (bj )|ri (bj ) for all respondents with pre-assigned numerical values. For example, a response of “probably yes” (PY) to bid bj is translated as prob(vi < bj ) = 0.25 for any respondent. In contrast to these probability-based estimators, Wang (1997) and Alberini, Boyle, and Welsh (2003) abstract from any numerical interpretation of Gi (bj ). Instead, they specify a normal density for the expectation E(vi ) at the population level, and use the observed responses ri (bj ) to relate the location of E(vi ) vis-à-vis the decision thresholds that separate the discrete response categories. They label this approach the “Random Valuation Model” (RVM). For example, let t1i be the threshold dollar value that separates a “definitely yes” (DY)

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Table 1. Existing Key Contributions for Bid-based Elicitation of Uncertain Values
Source Elicitation Format Assumptions on vi Interpretation of ri (bj ) Comments

PROBABILITY-BASED ESTIMATORS Li & Mattsson single bid, vi ∼ normal with (1995) numerical common uncertainty variance, scale in 5% E(vi ) ∼ normal increments in the population Evans et al. multiple bids, pre-determined (2003) qualitative fixed values for uncertainty Gi (bj ) for a scale: DY, PY, given response; NS, PN, DN unspecified but common Gi (.)∀i; “realized” vi ∼ normal in population, with common variance THRESHOLD-BASED ESTIMATORS Wang (1997) single bid, no assumptions on qualitative gi (vi ); E(vi ) ∼ uncertainty normal in the scale: Y, DK, N population Alberini et al. (2003) multiple bids, qualitative uncertainty scale: DY, PY, NS, PN, DN E(vi ) ∼ normal in the population

ri (bj ) translates into exact value for Gi (bj )

forces common variance, normality of vi ∀i forces common numerical interpretation of Gi (bj ) ∀i; gi (vi ) undefined for repeated PY, NS, or PN over two or more bids (“RIA” problem)

ri (bj ) translates into exact value for Gi (bj )

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ri (bj ) informs location of E(vi ) vis-à-vis decision threshold

ri (bj ) informs location of E(vi ) vis-à-vis decision threshold

requires restrictions on distance between E(vi ) and decision thresholds requires restrictions on distance between E(vi ) and decision thresholds; multiple responses by same i over bids treated as independent (implies: E(vi ) changes over bids)

Note: vi = value held by person i for non-market good; ri (bj ) = person i’s response to bid j; E = expectation operator Y = yes, N = no, DK = don’t know, DY = definitely yes, PY = probably yes, NS = not sure, PN = probably no, DN = definitely no.

response from a “probably yes” (PY) answer, and t2i be the threshold dollar value that separates a “probably yes” (PY) response from a “not sure” (NS) answer for person i.3 Also, let d1i be the distance between t1i and E(vi ), and d2i be the distance between t2i and E(vi ). This situation is depicted in figure 2. In Alberini,

3 We follow the bulk of existing PC and MBUC studies and interpret NS as a response that describes an underlying value segment that is wedged between the segments for PY and PN.

Boyle, and Welsh (2003), an observed response of NS in reaction to bid bj would be interpreted as the probability that the offered bid falls between the two thresholds, or - equivalently as the probability that E(vi ) is located between bj + d2,i and bj + d1,i . Similar boundary expressions can be derived for all other response categories. Adding restrictions on the relative magnitude of distance parameters di1 through di4 then allows for the estimation of E(vi ). All of these existing approaches provide estimates of the population expectation of E(vi ).

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Figure 2. Response Categories, Decision Thresholds and Distance Parameters for the Random Valuation Model However, as should be clear from an inspection of figures 1 and 2 this “payoff” comes at the prohibitive cost of imposing highly restrictive assumptions on either the distribution of vi itself, as is the case for Li and Mattsson (1995) and Evans, Flores, and Boyle (2003), or the location of E(vi ) relative to the decision thresholds, as in Wang (1997) and Alberini, Boyle, and Welsh (2003). Given the plethora of unobserved, heterogeneous forces that can shape a person’s valuation density, it is unlikely that these densities have much in common across individuals, let alone identical density families, variances, or cdf values. Along the same vein, it is rather unrealistic to assume that the exact same relationship holds for all respondents between the location of the expectation of vi and decision thresholds, as implied in the RVM approach. Consider, for example, the symmetry assumption imposed by Wang (1997) and Alberini, Boyle, and Welsh (2003) for some of their sub-models, i.e. d3i = d2i (see figure 2). This implies t2i = E(vi ) − d2i , t3i = E(vi ) + d2i , or, equivalently, d2i = (t3i − t2i )/2. This, in turn, restricts E(vi ) to lie precisely between the second and third threshold, and always in the “NS” segment for all individuals. This strikes us as a rather strong assumption in any application. An additional problem in Evans, Flores, and Boyle (2003) arises under “repeated inframarginal answers” (RIAs). For example, it is quite possible (and frequently observed in practice) that a given subject issues the same non-certainty response for a series of sequential bids. In fact, this situation is virtually unavoidable whenever the number of bids (say J) exceeds the number of response categories (M). This is the case in most MBUC applications, with M ranging from 3 to 5, and J usually lying in the 10-20 range. In Evans, Flores, and Boyle (2003) such RIA responses imply gaps with zero density for interior regions of gi (vi ). This is difficult to rationalize for any individual.4 The MBUC model in Alberini, Boyle, and Welsh (2003) circumvents the RIA problem by assuming that bid-specific responses are independent for a given individual. However, this essentially implies that E(vi ), which is modeled as a normal random variate, can take different values for the same respondent over the bid range. At best, this is an unlikely occurrence. At worst, this would imply serious formatting flaws in the MBUC design. As is evident from figure 1 our Latent Thresholds Estimator (LTE) abstracts from any of these assumptions. Instead, it exploits the notion that decision thresholds are likely correlated across respondents - if a relatively higher bid is required to make a given individual switch from DY to PY, then the same is likely to hold for all other decision thresholds. While this forgoes the estimation of the population expectation of E(vi ), it still allows for substantial learning about gi (vi ) and underlying consumer preferences. As an added benefit, the LTE nests conventional re-mapping estimators, which are based on re-coding uncertain responses into “yes” or “no”, and estimating a single, fixed value of vi (e.g. Champ et al. 1997; Welsh and Poe 1998; Ready, Navrud, and Dubourg 2001). Within the LTE framework this would be akin to setting threshold correlations to zero. The LTE also allows for a formal examination
4 This “density gap” dilemma could only be circumvented by adopting an ad hoc rule of admitting only the response to the highest bid in a sequence of identical responses, and interpreting resulting value estimates as upper bounds.

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of threshold restrictions, as imposed in the RVM approach of Wang (1997) and Alberini, Boyle, and Welsh (2003). We find strong empirical evidence against both the independent thresholds-assumption and threshold symmetry. Econometric Framework Consider an MBUC format with j = 1 . . . J bid levels and m = 1 . . . M response categories. This implies that there are S = M − 1 decision thresholds, one for each category transition. Thresholds will be individual-specific, given (i) the expected heterogeneity in the range and shape of gi (vi ), and (ii) personal preferences in where to switch to a different response category along this distribution. The latter point is an important distinction from Evans, Flores, and Boyle (2003) who assign equal numerical interpretation of response categories as cdf values over all individuals, as discussed in the previous section. We do, however, share with existing MBUC contributions the assumption that the outer thresholds t1i and tSi indicate the cutoff points for the full range of gi (vi ), i.e. min(vi ) and max(vi ). Over the entire population of individuals, we model each threshold as a simple linear function of observables and a normally distributed additive error term, i.e. (1) tsi = xsi βs +
si 2 ∼ n(0, σs ), si ,

across the threshold equations, the associated attribute primarily increases the variance of gi (vi ), while (potentially) inducing little change in the expectation. It is important to note that our model still allows each individual value distribution to have its own expectation and variance. The variance of gi (vi ) is closely related to its spread, which, in turn, is delineated by the two outermost thresholds t1i and tSi . Since these are allowed to take on individual-specific values, the individual-specific variance of gi (vi ) remains completely unrestricted. For example, consider again figure 1. If these three individuals constituted the full population (or each represented a population segment with similarly-spread densities),this would translate into a lower population variance for the first threshold compared to the last threshold, i.e. in 2 terms of equation 1, σ1 would be small relative 2 to σS . At the individual (panel) level the full model with correlated thresholds can be written as shown in equation (2), (2) ti = Xi β + i where ⎡ ⎤ ⎡ t1i x1i 0 ⎢t2i ⎥ ⎢ 0 x2i ti = ⎢ . ⎥ , Xi = ⎢ ⎣.⎦ ⎣0 0 . 0 0 tSi ⎡ ⎤ ⎡ ⎤ β1 1i ⎢β2 ⎥ ⎢ 2i ⎥ β=⎢ . ⎥, =⎢ . ⎥, ⎣.⎦ ⎣ . ⎦ . . βS Si ∼ n(0, ) where is an ex-ante unrestricted covariance matrix. If individuals have a tendency to switch response categories at relatively higher values for all categories, we would expect the offdiagonal elements of to be positive. If, in contrast,there is a population tendency to switch at a higher value of vi between two response categories whenever the switch between two other categories occurs at a lower value, the corresponding covariance term in would emerge as negative. In our application we find strong, positive correlation between all thresholds. If thresholds were observed, equation 2 would describe a basic Seemingly Unrelated Regression (SUR) model. However, instead of the actual thresholds we only observe a series of bid/response combinations from each survey

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0 0 .. . 0

⎤ 0 0⎥ ⎥, 0⎦ xSi

i = 1 . . . N, s = 1 . . . S,

where xis is a vector of individual characteristics. This allows for individual-and-thresholdspecific means, and threshold-specific coefficients and variance. One noteworthy advantage of our framework over Alberini, Boyle, and Welsh’s (2003) Random Valuation Model is that in theory all threshold functions are fully identified by the exogenous bids, such that the contents of xis can remain unchanged, and marginal effects can be allowed to vary across thresholds. This, in turn allows for a closer examination of how individual attributes affect the value distribution gi (vi ). For example, if a specific coefficient βsk , s = 1 . . . S, exhibits the same sign in all S equations, the corresponding observed characteristic moves all thresholds to the right, and thus unambiguously shift the entire range of the value distribution. In contrast, if βsk changes from pronouncedly negative to strongly positive

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participant. Let yij be the observed response by individual i when confronted with bid j. If we sort the response options from most affirmative (usually “DY”) to most disapproving (usually “DN”) and code them as increasing integers we have yij ∈ {1, 2, . . . , M}. We observe yij |bij = m if the cdf of the underlying valuation vi at bid bij falls between the cdf of the preceding and the following threshold, i.e. Gi (tm−1,i ) < Gi (bij ) < Gi (tm,i ). Since we do not wish to explicitly specify the cumulative value distribution, we re-express this condition directly in terms of the location of bij vis-à-vis the two adjacent thresholds, i.e. (3) yij = m if tm−1,i < bij < tm,i

For the most affirmative response, the left bound in (3) will generally be negative infinity. It could also be zero if a given sample includes only confirmed program supporters, i.e. individuals who would always answer DY to a bid of zero. This will be the case for our empirical application. For yij = M the upper bound of (3) will usually be infinity. Using the entire series of J observed responses for person i, collected in vector yi , it is then straightforward to express observed bid-response combinations as a joint probability for a set of S location restrictions for the thresholds, as shown in equation (4). (4) prob(yi |β, , Xi ) ⎡ ⎤ b1i,l < t1i < b1i,u ⎢ b2i,l < t2i < b2i,u ⎥ ⎥ = prob ⎢ . ⎣ ⎦ . . bSi,l < tSi < bSi,u = (Xi β, ; Ri ),

where with slight abuse of notation, (.) denotes the cdf of the truncated multivariate normal density with mean Xi β, variance matrix , and truncation region Ri implicitly defined by the S boundary conditions. Vector β comprises all S sets of threshold coefficients. Bids bsi,l and bsi,u denote, respectively, the relevant lower and upper bounds for a given threshold.5

The relevant bounds are the bids closest to a given threshold. Due to the possibility of repeated infra-marginal answers (RIAs, see above), not all bids offered to a given respondents will be relevant. Furthermore, it is possible for several thresholds to share one or both bounds if the observed answer pattern for a given individual does not traverse the entire M-dimensional response space, i.e. if one or more response categories are skipped. For those cases we impose the general ranking restriction t1i < t2i < . . . < tSi in our estimation framework, as described in detail in Appendix B. Examples for the identification of relevant bounds under different response scenarios are given in Appendix A. Equation (4) also describes the likelihood contribution by person i. In theory, the model parameters β and could be estimated via Maximum Likelihood techniques. We prefer a Bayesian approach for the following two reasons: (i) The effective sample size for the identification of some of the covariance terms in is relatively small in our application, preempting the interpretation of estimation results in the light of classical asymptotic theory, and (ii) it would be computationally challenging to impose the simultaneous threshold boundary and ranking conditions in an MLE framework. Our Bayesian Gibbs Sampler can handle these restrictions in a straightforward fashion. A Bayesian approach requires the specification of priors for all model parameters. We choose the standard multivariate normal priors for β and an inverse Wishart (IW) prior for the elements of , i.e. β ∼ n(μ0 , V0 ), ∼ IW(ν0 , S0 ), where ν0 and S0 are the degrees of freedom and scale matrix, respectively. The IW density is parameterized such that E( ) = (ν0 − S − 1)−1 S0 , where S is the number of thresholds and thus the dimension of . When combined with the likelihood function, these priors yield tractable conditional posterior densities. We further facilitate the implementation of our posterior simulator (Gibbs Sampler) by augmenting the model with draws of the unknown thresholds. A general discussion of the merits of this technique of data augmentation is given in Tanner and Wong (1987) and vanDyk and Meng (2001). The augmented posterior distribution will thus be

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5 This model differs form the Ordered Probit (OP) framework underlying Wang’s (1997) and Alberini, Boyle, and Welsh’s (2003) Random Valuation (RV) Model in two important aspects. First, the OP model is based on a single latent variable (the population distribution of E(vi ) in the RV case) while in our case we have a joint i.e. multivariate - distribution of S latent thresholds. Second, while both frameworks require the response thresholds (or threshold distances in the RV case) to be estimated along with all other model

parameters, the RV model loses an additional degree of freedom by attempting to also estimate the population mean of E(vi ). This makes it necessary to restrict one of the threshold distances to be proportional to others. Such additional identifying restrictions are not required in our case.

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proportional to the priors times the augmented likelihood, i.e. (5) p(β, , {t}N |y, X) ∝ p(β)p( )∗ i=1 p({t}N |β, , X)p(y|{t}N ) i=1 i=1 The Gibbs Sampler draws consecutively and repeatedly from the conditional posterior distributions p(β| , {t}N , X), p( |β, {t}N , X) and i=1 i=1 p({t}N |β, , y, X). Posterior inference is based i=1 on the marginals of the joint posterior distribution p(β, |y, X). The detailed steps of the posterior simulator are given in Appendix B. Empirical Application We examine the performance of our estimator using the Glen Canyon (GC) Pilot Study described in Welsh et al. (1995) and considered in Evans, Flores, and Boyle (2003).6 The general aim of the GC non-market valuation project, implemented in 1994 and 1995, was to elicit stakeholders’ WTP for reducing environmentally harmful fluctuations of water levels of the Colorado River due to varying discharges from Glen Canyon Dam. The pilot study used an MBUC format with 13 bid amounts ($0.1, $0.5, $1, $5, $10, $20, $30, $40, $50, $75, $100, $150, and $200) and a response scale of DY, PY, NS, PN, and DN.7 It was implemented via several versions that differed in fluctuation scenario, target population, and the type of information provided to respondents. We combine all data associated with a “seasonally adjusted steady flow” scenario (versions 3, 5, 6, 8, and 9), yielding an original sample size of 384 individuals. After eliminating observations with missing bid-responses or other key variables we retain 370 observations, for a total of 370 × 13 = 4810 observed bid/response combinations. Following the original study, all of these individuals had been screened to support the environmental program at no cost. Thus our general lower bound for all thresholds

6 We thank Mary Evans and V. Kerry Smith for providing this data set and all accompanying documentation. 7 The exact wording of the MBUC question was: “How would you vote on this proposal if passage of the proposal would cost your household these amounts every year for the foreseeable future? (CIRCLE ONE LETTER FOR EACH DOLLAR AMOUNT TO SHOW HOW YOU WOULD VOTE)”. This question is followed by a table that shows bid amounts, sorted from smallest to largest, in rows, and response categories, sorted from most to least affirmative, as column headings. Each row then displays the letters A through E, one per response category. See Welsh and Poe (1998) for details.

is zero. This bound becomes active for cases where the first threshold is not bounded from below by one of the offered bids, i.e. where a respondent’s answer to the lowest bid is not DY. A subset (28%) of the data stems from residents living in the market area served by GC-generated electrical power. The remaining observations flow from a nation-wide sample. It was hypothesized that market participants would exhibit lower WTP for the environmental benefits of reduced flow fluctuations, as they would be exposed to resulting higher energy prices. We capture this sub-segment with an indicator variable “market”. Another subset of participants (20%) from the national sample received a truncated information brochure that omitted a listing of the economic drawbacks of flow regulation for some stakeholders. We identify this group via the binary indicator “empathy”, taking a value of “1” if costs to others were dropped from the information pamphlet. Contrary to expectations, the original authors found that WTP decreased significantly for this sub-group. Our richer analysis sheds additional light on this issue, as shown below. The remaining variables in our GC model are as follows: (i) a standardized knowledge score (“knowledge”) based on respondents’ answers to a short quiz at the beginning of the main survey instrument that covered the contents of the information brochure, (ii) a standardized factor-analytical score summarizing participant’s relative preference of economic security over environmental protection (“econ”), and (iii) annual household income in $1000s. Knowledge scores range from −3.53 (poor score on the introductory quiz) to 0.77 (close to perfect score). Economic security scores cover a range of −1.18 (strong preference for environmental health over economic security) to 2.47 (strong preference for economic security over environmental health). Both variables have standardized means of zero. The average household income (in 1994 dollars) lies at $55,000, with a standard deviation of $31,000. Table 2 captures the response statistics for our sample. From an econometric perspective, the most important characteristic of the GC data is that all thresholds and threshold covariances are fully identified via reasonably large sub-samples. For example, there are 276 cases (out of 370) that provide both DY and PY answers, thus identifying the first threshold and its variance. Similarly, there are 204 individuals

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Table 2. Response Statistics for the GC Application
Response Category DY PY NS PN DN Total Identification Obs. 2349 638 469 458 896 4810 Share of Total 0.49 0.13 0.10 0.10 0.19 1
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Threshold t1 t2 t3 t4

Obs. 276 204 173 152

Covariances t1, t2 t1, t3 t1, t4 t2, t3 t2, t4 t3, t4

Obs. 204 302 304 147 254 122

Note: Obs. = number of observations.

that exhibit the full triplet of DY, PY, and NS answers, thus jointly identifying the first two thresholds and their covariance (last column of Table 2). This limits cases where thresholds share common bounds, and thus enhances the efficiency of our estimator. A more detailed discussion of relevant bounds for each threshold is given in Appendix A and summarized in Table A1. Perhaps the most striking feature of our sample with respect to relevant bounds and threshold identification is the wide spread of observations across virtually the entire set of relevant bounds for all four thresholds. For example, focusing on t1 , a given individual might switch from DY to PY at virtually any point between $0.1, and $200, although the bulk of switches occur in the $1–$30 range. Similar patterns can be observed for t2 through t4 . This highlights the pronounced heterogeneity in the range (and likely shape) of the underlying value distribution across individuals. From an estimation perspective this is both desirable, as it aids in the identification of marginal effects of regressors, and problematic, as the unobservable effects in the threshold equations (1), and therefore threshold variances, can be expected to be large given our rather sparse set of observables. The second most important feature of Table A1 is the relatively large share of observations that fall into the $200 - to - infinity category, especially for t3 and t4 (96 and 164 cases, respectively). As our results will show this both inflates threshold variances and decreases the posterior efficiency for parameters related to these thresholds. In retrospect, a few additional bids at the upper end of the spectrum would have likely been extremely beneficial for this application. As we will show below, our LTE estimator can also be used for guidance in bid design.

Estimation Results We estimate the LTE model using the following vague but proper parameter settings for our priors: μ0 = 0, V0 = diag(10), ν0 = S + 2, and S0 = IS .8 We also estimate a model version with independent thresholds, where is restricted to a diagonal matrix. For these cases we specify inverse-gamma (ig) priors for the S variance terms with shape and scale parameters set to 1/2. The estimates flowing from the independent model can be interpreted as those that would be obtained from four separate dichotomous choice models, each of which treats all responses to the left of a given threshold as “DY”, and those to the right as “DN” (e.g. Ready, Whitehead, and Blomquist 1995; Welsh and Poe 1998; Ready, Navrud, and Dubourg 2001). We first test all models using simulated data to assure the accuracy of our computational algorithm. For all actual estimation runs we discard the first 2000 draws generated by the Gibbs Sampler as “burn-ins”, and retain the following 10000 draws for posterior inference. We evaluate the performance of the posterior simulator using Geweke’s (1992) convergence diagnostics (CD), and inefficiency (IEF) scores as described in Chib (2001). The CD scores clearly indicate convergence for all our models. The IEF scores, which convey the degree of (undesirable) autocorrelation in the series of posterior draws, range from the single digits (i.e. near-independence) for most slope coefficients and variance terms to 10-20 for the
8 “Proper”prior distributions are those that integrate to one over their entire range. This characteristic is required for the derivation of Bayes Factors for model comparison, an important consideration in our case. “Vague” refers to the fact that the distribution has a relatively large variance, which preempts substantial prior density mass for any specific segment of the distribution range. This reflects the absence of any existing information to aid in the construction of priors.

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Table 3. Estimation Results
Full Model Coefficients Constant Empathy (1 = no costs to others mentioned in pamphlet) Market (1 = respondent lives in electricity market served by GC Power) Knowledge (quiz score) Econ (high score = economic security outweighs environmental concerns) Income ($1000s) Constant Empathy (1 = no costs to others mentioned in pamphlet) Market (1 = respondent lives in electricity market served by GC Power) Knowledge (quiz score) Econ (high score = economic security outweighs environmental concerns) Income ($1000s) SD (t1 ) Corr (t1 , t2 ) SD (t2 ) Corr (t1 , t3 ) Corr (t2 , t3 ) SD (t3 ) Corr (t1 , t4 ) Corr (t2 , t4 ) Corr (t3 , t4 ) SD (t4 ) Mean 0.28 −6.07 −2.67 (SD) Indep. Model Mean (SD) (4.71) (5.82) (5.44) Full Model Mean 3.06 −8.14 −2.49 (SD) Indep. Model Mean (SD) (5.60) (6.69) (6.30)

Threshold 1 (3.22) 20.27 (3.69) −11.63 (3.42) −7.26

Threshold 2 (4.17) 33.85 (4.51) −9.98 (4.24) −4.55

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2.22 −5.81

(2.01) (2.35)

5.97 −17.90

(2.77) (3.35)

4.70 −8.73

(2.94) (3.34)

8.12 −24.20

(3.39) (4.12)

0.76 5.84 −1.84 −0.83

(0.06)

0.53

(0.07) (6.59) (7.52) (7.15)

1.32 17.35 5.76 3.97

(0.10)

0.74

(0.08) (8.32) (8.74) (8.41)

Threshold 3 (5.51) 45.57 (5.71) −4.70 (5.52) −2.23

Threshold 4 (8.29) 46.60 (8.18) 2.15 (8.05) 4.96

4.46 −11.07

(4.32) (4.75)

5.86 −27.51

(4.16) (4.97)

6.40 −17.13

(6.85) (7.45)

4.21 −28.17

(5.82) (6.79)

2.11 57.36 0.96 96.72 0.91 0.97 151.37 0.87 0.95 0.98 242.29

(0.16) (2.42) (0.01) (4.73) (0.01) (0.00) (9.26) (0.02) (0.01) (0.00) (18.57)

1.06 52.48 – 66.38 – – 83.56 – – – 126.09

(0.11) (2.06) – (2.66) – – (3.62) – – – (7.32)

3.42

(0.29)

2.03

(0.18)

Note: Indep. = Independent (error correlations set to zero), SD = standard deviation, Corr = correlation.

somewhat less clearly identified variances and covariances associated with higher thresholds. Table 3 summarizes the estimation results for both models. The table reports the posterior mean and standard deviation for the coefficients in all four threshold equations, as well as the elements of the error covariance matrix . The latter are given in terms of standard deviations and correlations (for the full model) for ease of interpretation.

We immediately observe that the error terms in the full model exhibit close to perfect correlation for all six threshold pairs. Moreover, these correlation terms exhibit high posterior precision as indicated by the negligible magnitudes of the respective posterior standard deviations. This casts serious doubt on the legitimacy of the independent model. To allow for a more rigorous comparison we compute the marginal likelihood for each model using the

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simulation method outlined in Chib (1995). The difference between the marginal likelihood of the full model (-3373) and the independent model (-4133) yields a logged Bayes Factor (BF), i.e. the logged odds of the full model over the independent model, of 760.4. Using the interpretation thresholds for BFs given in Kass and Raftery (1995) this result provides “decisive” evidence in favor of the full model. The efficiency gains from the full model compared to the independent version are evident from the table via the smaller posterior standard deviations for most coefficients, especially for the first three thresholds. More importantly, because of the truncated nature of the threshold equations in the likelihood function, the independent model, by ignoring error correlations, also produces inconsistent estimates of all model parameters.This will become obvious when examining threshold predictions below. Not surprisingly given our discussion of bid ranges and relevant bounds above the posterior means of estimated threshold standard deviations increase from the lowest threshold (57.36) to the highest (244.29). The posterior standard deviations for threshold coefficients follow the same pattern, suggesting a general trend of decreasing posterior precision as one moves from the lowest to the highest threshold. However, some slope coefficients, such as those corresponding to income and the economic security score (econ) are estimated with relatively high precision for all four thresholds. Learning about Individual Values In this section we discuss how our results aid in gaining a better understanding of the underlying individual valuation distributions gi (vi ). We begin with a closer examination of trends in marginal effects across the four threshold equations. As mentioned above, an important advantage of the LTE framework compared to the RVM (Alberini, Boyle, and Welsh 2003) is that the same set of regressors can be used in all S threshold equations, and that the marginal effects of these explanatory variables can differ across thresholds. This allows for the identification of variables that exert an even shift on the entire value distribution, and thus unambiguously affect its expectation in the same direction for all individuals. We can also identify regressors that “pull” the outer thresholds further apart, or “push” them inward, thus primarily affecting the variance of gi (vi ).

As is evident from table 3 virtually all of our explanatory variables exhibit noticeable changes in marginal effects across thresholds, as measured via the posterior mean of the corresponding coefficient. For example, the empathy indicator shifts the lower thresholds to the left and the highest threshold to the right by comparable magnitude. Thus, omitting reminders of costs to others in a survey version does not necessarily lead to lower expected WTP (the puzzling conclusion reached by Welsh et al. (1995)), but primarily increases the spread of the value distribution. This subtle but important difference in inference becomes only apparent when all four thresholds are estimated. A similar finding holds for the market indicator - participants from the market area served by GC power have lower estimates for t1 and t2 (i.e. switch from DY to PY and from PY to NS at lower bids),but are also more reluctant to enter the DN category. Thus, market participants exhibit a wider spread in underlying valuation than non-market respondents. In contrast, the direction of marginal effects remains unchanged across thresholds for the knowledge score (knowledge), the economic security score (econ) and income. A better understanding of GC power generation and related environmental issues, as measured by knowledge, shifts the entire value distribution to the right, with a slightly increasing trend across thresholds. On the other hand, individuals with high preference for economic security over environmental conservation, i.e. a high econ score, can be associated with both a strong leftward shift of their value distribution, and a tighter overall distribution. The latter insight stems from the fact that the leftward shift for the highest threshold (-17.13) is substantially larger than the leftward shift for the lowest threshold (-5.81). This reduces the overall range of the underlying valuation. As expected, income exhibits an efficiently estimated positive effect on all thresholds. Since its marginal effect also increases from lowest to highest threshold, we can infer that higher income also translates into higher variability of underlying valuation. A second avenue of learning about underlying WTP afforded by the LTE framework is via the posterior predictive distributions (PPDs) of thresholds for a given setting of regressors. That is, we can examine the posterior distribution of thresholds for a specific setting of regressors, controlling for parameter uncertainty. This is especially informative for the outermost ones that determine the range of

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gi (vi ), or, alternatively put, the minimum and maximum of vi . Formally, this PPD is given via (6) p(t|xp ; X, y) = p(t|θ, xp )p(θ|X, y)dθ,

θ

where t is the S-dimensional threshold vector, θ comprises all model parameters (i.e. the elements of β and ), xp is a vector with specific settings for explanatory variables, and X and y denote the sample data. The second function in the integral is simply the joint posterior distribution. Thus, we can conveniently obtain draws from p(t|xp ; X, y) by drawing t from the S-fold system of latent threshold equations p(t|θ, xp ) (see equation (2)) for each draw of θ from the Gibbs Sampler.9 To illustrate, we derive PPDs for four settings of xp , corresponding to the three feasible combinations of the market and empathy indicators. The remaining regressors are set to their respective sample mean, except for the last version, where the economic security score econ is set to its 75th percentile. Table 4 captures the posterior means and standard deviations for these PPDs for both the full and independent model. The most important result captured in table 4 is the pronounced difference in posterior means between the full and the independent model. Specifically, the independent model tends to over-predict the lowest threshold and severely under-predict all remaining thresholds. The latter shortcoming could have serious implications in policy applications where a conservative estimate of welfare losses is sought, which would logically shift the inferential focus to the upper end of the value distribution. Focusing on the outer threshold predictions for the full model, we can thus infer that vi for the typical out-of-market stakeholder can be expected to range between $47.45 and $216.53. Similarly, the expected minimum and maximum values of vi for an in-market resident are $45.78 and $221.34, respectively. Therefore, any policy that stays below a per-stakeholder cost of $45 is assured to generate positive net benefits. Conversely, if per-capita costs exceed $220, the policy is guaranteed to impose net societal losses. The full-model PPDs for all four thresholds for the first stakeholder type from table 4
9 Note that these threshold draws are not conditioned on any observed bounds, other than the “natural” leftmost bound of zero, if we continue to focus on a population of strict program supporters.

are given in figure 3, with expectations super-imposed as dotted lines. The spread of these densities captures two sources of variation: (i) unobserved individual heterogeneity, as modeled via the error term in the threshold equations (1), and remaining parameter uncertainty, as introduced to the model via our priors in (5). Since all thresholds are estimated under equal prior settings, the figure indicates that individual heterogeneity increases over thresholds. There appears to be considerably less variability amongst stakeholders as to the minimum they would be willing to pay to support the Glen Canyon project compared to the maximum.10 To reiterate, knowledge of the population distribution of minimum and maximum WTP suffices for policy decisions based on societal net-benefits if per-household costs either fall below the population expectation of t1 (guaranteed positive net benefits) or exceed the population expectation of t4 (guaranteed net losses). For per-capita costs between these bounds the net welfare effect is ambiguous, given our exclusive focus on the extrema of the individual vi distributions. However, further insight into equity implications of policy implementation under a specific per-unit cost can be gained via inspection of the cdf or, equivalently, the survival function (1 − cdf ) of the outer thresholds. Figure 4 shows the survival function for t1 (i.e. minimum WTP, top graph) and the cdf for t4 (maximum WTP, bottom graph). For each dollar amount on the x-axis the top graph shows the population share with minimum WTP exceeding that amount. In contrast, the bottom graph depicts the population share with maximum WTP falling below a given dollar amount. The zero, one, and 0.5 probability levels are superimposed as dotted lines. For example, the top graph shows that
10 To some extent this increasing loss in posterior precision going from the lowest to the highest threshold is also related to the increasing monetary intervals over the bid range (see table A1). In addition, the lower thresholds “benefit” the most from the natural lower bound on vi of zero (since the sample only includes declared zero-cost supporters), while such a natural bound is absent for the upper end of the value distribution. We repeated our analysis with a counterfactual upper bound of $1000. This did not affect our results, which suggests that such a “natural upper bound”, if it existed, would have to be located close to the highest bid to induce a tightening of the distribution for the upper threshold. A uniform bid range and expanding the sample to include non-supporters (i.e. dropping the lower natural bound of zero) would be necessary to fully disentangle the individual heterogeneity effect from formatting effects. Alternatively put, the figure also hints at the potential benefits of tighter bid intervals for the posterior precision of upper thresholds. This should be a key consideration for research geared towards improved MBUC designs.

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Table 4. Predictive Threshold Moments for Different Population Types
Full Model Variable Settings Empathy, no market, mean knowledge, mean econ, mean income Empathy, market, mean knowledge, mean econ, mean income No empathy, no market, mean knowledge, mean econ, mean income Empathy, no market, mean knowledge, high econ, mean income t1 t2 t3 t4 t1 t2 t3 t4 t1 t2 t3 t4 t1 t2 t3 t4 Mean 47.45 84.42 132.04 216.53 45.78 83.14 131.94 221.34 44.15 79.95 134.05 226.68 47.11 83.81 132.06 215.72 SD (31.38) (50.67) (81.07) (134.51) (31.09) (51.09) (82.19) (136.34) (30.53) (50.33) (81.95) (137.32) (31.54) (51.34) (82.41) (136.87) Independent Model Mean 50.75 68.71 87.75 121.00 47.82 66.31 87.18 124.83 45.91 63.17 85.56 121.16 47.02 62.73 80.51 115.87 SD (35.57) (47.10) (59.20) (85.31) (34.41) (45.85) (59.90) (87.13) (33.42) (45.52) (58.82) (86.06) (34.31) (44.59) (56.40) (84.04)

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Note: SD = standard deviation, market = resident of affected electricity market, knowledge = knowledge score from survey quiz, econ = economic security score.

T1 (DY / PY)
0.015

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mean=$47.45 0.010

mean=$84.42

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Figure 3. Posterior Predictive Densities for Baseline Type

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prob[max(wtp)<x]

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median=$195.3

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Figure 4. Predicted Population Share with Minimum WTP Exceeding a Given Amount (top) and Maximum WTP Falling below a Given Amount (bottom) approximately 80% of typical out-of-market stakeholders would be willing to pay, with certainty, at least $25 to see that proposed policy implemented. The 50% level for minimum WTP lies at $42.6. At $80 only about 20% can be expected to have a minimum WTP that is guaranteed to exceed this level. Consider, for example,a proposed river flow policy that came at a per-capita cost of $45. This would generate positive net benefits overall for this population segment, given an expected minimum value of $47.45 for this group (see figure 3). Moreover, as is evident from figure 4, approximately 40% of stakeholders are guaranteed to experience net benefits, as their minimum WTP exceeds this amount. Turning to maximum WTP, from the bottom graph of figure 4 we can infer that 20% of stakeholders are certain they would not pay $100 or more, 50% are certain to have a WTP that falls below $200, and 90–95% are guaranteed to hold a maximum value of $500 or less. Thus, if the proposed flow scenario generated per-capita costs of $500 or higher, it would lead to guaranteed losses for the bulk of stakeholders. Naturally, it would also lead to guaranteed overall net losses, as this amount exceeds the expected maximum WTP of $216.53 (see figure 3). Now consider a cost of $180. This amount lies between expected minimum WTP and expected maximum WTP, so the effect on societal net benefits cannot be inferred with certainty from our analysis. However, it is clear from figure 4 that at that cost about 40% of stakeholders are guaranteed to experience losses. Thus, such an intervention may be politically unpalatable based on equity considerations. A Test for Threshold Symmetry Our modeling framework also allows for an examination of the maintained assumption in Alberini, Boyle, and Welsh’s (2003) primary Random Valuation Model that both inner and outer thresholds are equidistant to the expectation of the value distribution. We can cast this assumption as a linear model restriction, i.e E((t4i − t3i ) − (t2i − t4i )) = 0 or, equivalently, a linear parameter restriction, i.e.

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((β4 − β3 ) − (β2 − β4 )) = 0. Based on the results in table 4, such dual symmetry seems unlikely. We also perform a formal model comparison, which produces a logged Bayes Factor of 104.7 in favor of the unrestricted specification.11 Therefore, this dual-symmetry assumption is not supported by the observed data for our application. Any valuation model based on such an assumption would be mis-specified. Conclusion We propose a new estimator for MBUC data that utilizes all observed response patterns for the simultaneous estimation of the full set of underlying decision thresholds. Our Latent Threshold Estimator has several advantages over existing approaches that process MBUC data, or - more generally - data from value elicitation with uncertain response options. It does not require stringent ex ante restrictions on value distributions or threshold locations, and handles repeated infra-marginal responses in a straightforward manner. By allowing thresholds to be fully correlated our framework exploits linkages across observed response patterns. This turns out to be crucial for the consistent estimation of thresholds, and thus the range and extrema of vi . In addition, the LTE framework provides insights into the marginal effect of regressors that would remain undiscovered using other estimation strategies. It clearly highlights if a given regressor primarily shifts the entire value distribution (and thus its expectation), or if it also affects the spread of the distribution. This can be exploited to examine the impact of changes in survey format, such as the provision of additional information, or altering the scope or scale of a proposed policy intervention. Furthermore, by returning the full distribution of the outer thresholds, the LTE approach provides clear and explicit guidance as to the expected variability in the range of underlying values that can be expected for a given stakeholder population. It also generates population estimates of the expected minimum and maximum willingness-to-pay. In many cases, this may suffice to guide policy decisions. Learning about the distribution of boundary

thresholds can also be helpful in devising efficient bid designs. Our results strongly suggest that tighter intervals at the upper end of the bid spectrum could greatly enhance the estimation precision for the highest threshold.This,in turn, could broaden the applicability of the MBUC format to environmental policy scenarios with a primary focus on loss prevention or damage assessment. Naturally, important caveats remain. Our modeling framework is fully anchored in the assumption that individuals are truly unable to assign a point value estimate to a given nonmarket amenity or service due to latent and potentially permanent uncertain factors. Thus, we rule out the possibility that respondents are unwilling to exert sufficient effort to zoom in on a single value, a concern raised by Alberini, Boyle, and Welsh (2003), or choose an uncertain response because they perceive that, given the increased action space compared to a simple yes/no context, they are expected to do so (Vossler and McKee 2006). Neither do we address the issue of bid ordering effects on measured values examined in Alberini, Boyle, and Welsh (2003), although our LTE framework could potentially be useful in identifying and controlling for such effects. In general, our modeling assumptions appear reasonable and are certainly less stringent than those required for alternative estimation strategies. We believe that the LTE approach is a natural and to date overlooked extension of the MBUC framework. It has the potential to substantially broaden the applicability of this elicitation approach. We also conclude that the common practice of using MBUC data solely to derive an estimate for a single threshold, interpreted as point estimate of WTP, leaves useful information untapped and can produce misleading results if thresholds are highly correlated.

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References Akter, S., J. Bennett, and S. Akhter. 2008. Preference Uncertainty in Contingent Valuation. Ecological Economics 67: 345–351. Akter, S., R. Brouwer, L. Brander, and P. van Beukering. 2009. Respondent Uncertainty in a Contingent Market for Carbon Offsets. Ecological Economics 68: 1858–1863. Alberini, A., K. Boyle, and M. Welsh. 2003. Analysis of Contingent Valuation Data with Multiple Bids and Response Options Allowing Respondents to Express

11 We derive this Bayes Factor via the Savage-Dickey Density Ration (SDDR). Details of these computations are available upon request.

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Moeltner,K.,R. Johnston,R. Rosenberger,and J. Duke. 2009. Benefit Transfer from Multiple Contingent Experiments: A Flexible Two-Step Model Combining Individual Choice Data with Community Characteristics. American Journal of Agricultural Economics 91(5): 1335–42. Opaluch, J., and K. Segerson. 1989. Rational Roots of “Irrational” Behavior: New Theories of Economic Decision-Making. Northeastern Journal of Agricultural and Resource Economics 18(2): 81–95. Ready, R., S. Navrud, and W. Dubourg. 2001. How do Respondents with Uncertain Willingness to Pay Answer Contingent Valuation Questions? Land Economics 77: 315–326. Ready, R., J. Whitehead, and G. Blomquist. 1995. Contingent Valuation when Respondents are Ambivalent. Journal of Environmental Economics and Management 29: 181–196. Shaikh, S., L. Sun, and G. vanKooten. 2007. Treating Respondent Uncertainty in Contingent Valuation: A Comparison of Empirical Treatments. Ecological Economics 62: 115–125. Tanner, M., and W. Wong. 1987. The Calculation of Posterior Distributions by Data Augmentation. Journal of the American Statistical Association 82: 528–550. vanDyk, D., and X. L. Meng. 2001. The Art of Data Augmentation. Journal of Computational and Graphical Statistics 10(1): 1–50. Vossler, C., and M. McKee. 2006. Inducedvalue Tests of Contingent Valuation Elicitation Mechanisms. Environmental and Resource Economics 35: 137–168. Wang, H. 1997. Treatment of “Don’t Know” Responses in Contingent Valuation Surveys:A Random Valuation Model. Journal of Environmental Economics and Management 32: 219–232. Welsh, M., R. Bishop, M. Phillips, and R. Baumgartner. 1995. GCES Non-Use Value Study. U.S. Bureau of Reclamation, Glen Canyon Environmental Studies Non-Use Value Committee. Welsh, M., and G. Poe. 1998. Elicitation Effects in Contingent Valuation: Comparison to a Multiple Bounded Discrete Choice Approach. Journal of Environmental Economics and Management 36: 170–185.

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Appendix A: Relevant and shared bounds in the LTE framework Case 1: Full threshold identification This marks the ideal scenario in which all possible answer categories are observed for a given respondent. It leads to finite and unique lower and upper bounds for each threshold. For example, consider the following response pattern: (A.1) b1 → DY, b2 → PY, b3 → PY, b4 → PY, b5 → NS, b6 → PN, b7 → DN This implies the following boundary conditions for thresholds: (A.2) b1 < t1 < b2 , b4 < t2 < b5 , b5 < t3 < b6 , b6 < t4 < b7 Bid b3 contributes no new information to the identification of threshold locations and becomes irrelevant. Also, the general ranking condition t1 < t2 < t3 < t4 is automatically assured through the increasing bid amounts. Case 2: Partial threshold identification Now consider the following response pattern: (A.3) b1 → DY, b2 → PY, b3 → PY, b4 → PY, b5 → PN, b6 → PN, b7 → DN Compared to the previous case, the NS category has been skipped. The resulting boundary conditions are: (A.4) b1 < t1 < b2 , b4 < t2 < b5 , b4 < t3 < b5 , b6 < t4 < b7 Thresholds two and three share the same upper and lower bounds. In our computational algorithm we handle this case by drawing such partially identified thresholds simultaneously from their shared interval, and imposing the relevant ranking condition ex post. Table A1 provides a closer look at the distribution of observations across relevant bounds for each threshold for the GC application. The first three columns list the lower bound, upper bound, and inter-bound range for each set of relevant bounds observed in the data. Each possible consecutive bid interval figures as relevant bound for the first three thresholds. For

the fourth threshold (t4 ) there are no observations that falls within the lowest two bid segments. As has been standard practice in MBUC applications, inter-bid ranges increase substantially over the entire set of bids, here from $0.1 for the lowest bracket to $50 for the highest two brackets. For each threshold and relevant bound the Table depicts the number of fully identified cases (“fi” column), the number of partially identified cases (“pi”), and the total number of observations for which a given pair of bids forms the relevant bound. The “pi” cases include both observations for which a given bound is shared by two or more other thresholds for the same individual, and observations for which the upper bound is infinite. For example, looking at the first row in the t1 triplet of columns, there are 22 individuals in our sample that respond with DY to a bid of $0 and PY to a bid of $0.1. Consequently, the $0 / $0.1 pair becomes the relevant set of bounds for t1 for these cases. For three individuals in this group other thresholds in addition to t1 also fall into the same relevant bracket. This implies that these individuals skipped the PY and perhaps additional higher response categories, i.e. they answered DY to $0, and NS, PN, or DN to $0.1. As a result, t1 , t2 and perhaps even higher thresholds all fall within the $0 / $0.1 relevant bracket. The total number of fully identified cases for each threshold also corresponds to the respective entry in the third-to-last column of table 2.

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Appendix B: Gibbs Sampler Details Building on (4) the likelihood for the full sample can be written as
N

(A.5) p(y|β, , X) =
i=1

(Xi β, ; Ri )

The priors for our main parameters of interest are explicitly given as follows: (A.6) p(β) = (2π)−k/2 |V0 |−1/2
1 ∗ exp − 2 (β − μ0 ) V−1 (β − μ0 ) 0

p( ) =
N −1 ν0 +1−i 2 i=1

2

ν0 S/2

π

S(S−1)/4

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Table A1. Relevant Bounds and Threshold Identification for the GC Application
Lower 0 0.1 0.5 1 5 10 20 30 40 50 75 100 150 200 Bounds ($) Upper Range 0.1 0.4 0.5 4 5 10 10 10 10 25 25 50 50 Inf Fi 19 10 10 26 28 42 34 32 13 25 15 19 3 0 276 75% t1 Pi 3 0 1 11 4 11 11 3 3 9 3 8 0 27 94 25% Total 22 10 11 37 32 53 45 35 16 34 18 27 3 27 370 100% Fi 0 1 4 11 12 19 29 15 28 32 21 22 10 0 204 55% t2 Pi 3 0 2 17 6 16 18 7 6 14 10 10 2 55 166 45% Total 3 1 6 28 18 35 47 22 34 46 31 32 12 55 370 100% Fi 0 1 0 2 3 15 22 13 16 26 35 25 15 0 173 47% t3 Pi 1 0 1 13 5 13 14 7 7 13 10 12 5 96 197 53% Total 1 1 1 15 8 28 36 20 23 39 45 37 20 96 370 100% Fi 0 0 0 0 2 4 9 22 14 20 31 32 18 0 152 41% t4 Pi 0 0 1 9 4 8 5 4 3 5 4 8 3 164 218 59% Total 0 0 1 9 6 12 14 26 17 25 35 40 21 164 370 100%

0.1 0.5 1 5 10 20 30 40 50 75 100 150 200 Inf Column total % of sample

Note: Fi = fully identified (threshold does not share bounding bid amounts with other thresholds), Pi = partially identified (threshold shares one or both bounding bids with other thresholds).

Amer. J. Agr. Econ.

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Kobayashi, Moeltner, and Rollins

Latent Thresholds

207

∗ |S0 |ν/2 | |−(ν0 +S+1)/2
1 ∗ exp − 2 tr(S0 −1

)

The augmented likelihood, in turn, can be written as
N

with k denoting the length of vector β and S indicating the number of thresholds. As indicated in (5) the full posterior kernel, augmented with draws of the threshold vectors t1 , t2 , . . . , tN is conceptually given as: (A.7) p(β, , {t}N |y, X) i=1 ∝ p(β)p( ) ∗ p({t}N |β, , X)p(y|{t}N ) i=1 i=1 The last two component in (A.7) can be interpreted, respectively, as the parameterconditioned prior density of the augmented components and the augmented likelihood. Recall that outcome vector y represents the series of observed bid/response pairs. Naturally, if thresholds were observed, responses to a set of bids would become deterministic. Thus, the augmented likelihood does not depend on either parameters or explanatory data. This preempts the need to evaluate the multi-dimensional integrals implicit in the (.) component of the likelihood function. The augmented prior, in turn, does not depend on observed responses. If thresholds were observed, they would simply follow an unrestricted S-variate normal density, as indicated in (2). For each individual threshold this is analogous to the unrestricted normal density for the latent variable underlying a Probit or Ordered Probit specification. For the entire set of thresholds this latent system is analogous to the unrestricted system of random utilities underlying a Multinomial Probit or related specification based on utility differences or rankings (for a Bayesian treatment of such models see e.g. McCulloch, Rossi, and Polson (2000), Layton and Levine (2003), Layton and Levine (2005), and Moeltner et al. (2009)). For the full sample the augmented prior takes the following explicit form: (A.8) p({t}N |β, , X) i=1
N

(A.9)

p(y|{t}N ) = i=1
i=1

(I(bil < t < biu )

+ (1 − I(bil < t < biu ))) ∗ (t1i < t2i < . . . < tSi ) The right-hand side is a deterministic product of index functions that simply state that the thresholds are either fully bounded by relevant bids (first index) or, if only partially identified by bid-bounds, stipulated to follow a ranking condition. The explicit form of the augmented posterior kernel thus emerges as: (A.10) p(β, , {t}N |y, X) i=1
1 ∝ exp − 2 (β − μ0 ) V−1 (β − μ0 ) 0 1 ∗ | |−(ν0 +S+1)/2 exp − 2 tr(S0 −1

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)

∗ | |−N/2
1 ∗ exp − 2 (ti − Xi β) N −1

(ti − Xi β)

∗
i=1

(I(bil < t < biu )

+ (1 − I(bil < t < biu ))) ∗ (t1i < t2i < . . . < tSi ) To derive the conditional posterior kernel for we need to consider all components of the full posterior kernel that are nor multiplicatively separable from expressions that include this parameter, i.e. (A.11) p(β| , {ti }N , X) i=1 ∝ exp
N

−

1 2

(β − μ0 ) V−1 (β − μ0 ) 0
−1

=
i=1

(2π)−S/2 | |−1/2
−1

+ (ti − Xi β) (ti − Xi β)
i=1

(ti − Xi β)

(ti − Xi β)

1 ∗ exp − 2 (ti − Xi β)

= (2π)

−SN/2

| |

−N/2 −1

1 ∗ exp − 2 (ti − Xi β)

This corresponds to the conditional kernel for a generalized regression model, yielding the usual multivariate normal conditional

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Amer. J. Agr. Econ.

posterior: (A.12) β| , {ti }N , X ∼ n(μ1 , V1 ) with i=1
N −1

The third and final step of the Gibbs Sampler involves the draws of the latent thresholds. Collecting again all relevant components from the full posterior kernel, we obtain (A.15) ,
−1

V1 = V−1 + 0
i=1

Xi
N

−1

Xi

‘p({ti }N |β, , y, X) i=1 ∝ | |−N/2
1 ∗ exp − 2 (ti − Xi β) N −1

μ1 = V1 V−1 μ0 + 0
i=1

Xi

ti

(ti − Xi β)
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Similarly, for draws of (A.13) p |β, {ti }N , X i=1

we obtain

∗
i=1

(I (bil < t < biu )

+ (1 − I (bil < t < biu ))) ∗ (t1i < t2i < . . . < tSi )
−1

∝ | |−(ν0 +S+1+N)/2 ∗ exp
N

−

1 2

tr S0

+
i=1

(ti − Xi β)

−1

(ti − Xi β)

which implies (A.14) |β, {ti }N , X ∼ IW(ν1 , S1 ) i=1 ν1 = ν0 + N,
N

with

S1 = S0 +
i=1

(ti − Xi β)(ti − Xi β)

For each individual this stipulates draws from the S-variate normal density restricted to the sphere outlined by the boundary and ranking conditions. We use a Gibbs-within-Gibbs routine as described in Layton and Levine (2003) and Layton and Levine (2005) to obtain these draws. For partially identified thresholds, i.e. where two or more thresholds share the same bid-bounds, we proceed as follows: We first draw each partially identified threshold from the bid-bounded interval, conditional on all other thresholds, then impose the ranking condition ex post. This proved less costly than rejecting all draws that violate the ranking condition.