alberini analysis of contingent valuation data with multiple

Journal of Environmental Economics and Management 45 (2003) 40–62

Analysis of contingent valuation data with multiple bids andresponse options allowing respondents to express uncertainty

Anna Alberini,a,* Kevin Boyle,b and Michael Welshc

aDepartment of Agricultural and Resource Economics, University of Maryland, 2200 Symons Hall,

College Park, MD 20742-5535, USAbLibra Professor of Environmental Economics, University of Maine, Orono, ME, USA

cChristensen Consulting, Madison, WI, USA

Abstract

The NOAA Panel on Contingent Valuation advocated a ‘‘no answer’’ response option to dichotomous-choice payment questions, but did not give guidance as to how this additional response should beinterpreted conceptually or analytically. We investigate the econometric modeling and response effectsassociated with multiple-bounded, polychotomous-choice payment question. We find that using multiplebids with responses to each bid can increase the efficiency of welfare estimates, but this approach is not freefrom bid design effects. Moreover, in our application, explicitly modeling uncertain responses can increasewelfare estimates by over 100%.r 2003 Elsevier Science (USA). All rights reserved.

1. Introduction

Contingent-valuation questions usually follow one of several possible formats. The NOAABlue Ribbon Panel on Contingent Valuation [19] sanctioned the use of dichotomous-choicequestions for eliciting nonuse values when the contingent market is framed as a referendum.1 TheNOAA Panel also advocated a ‘‘would not vote’’ response option in addition to the traditional’’‘‘vote for/vote against’’ options. However, no guidance was provided on the conceptual orempirical interpretation of ‘‘would not vote’’ or ‘‘no answer’’ responses.One interpretation of the ‘‘no answer’’ response is that it identifies people who are not in the

market for the good being valued at the particular bid amount [9]. An alternative interpretation is

*Corresponding author.

E-mail address: [email protected] (A. Alberini).1By analogy, and the fact that the application of contingent valuation to estimate use values was not called into

question, we presume that this endorsement also applies to the estimation of use values.

0095-0696/03/$ - see front matter r 2003 Elsevier Science (USA). All rights reserved.

PII: S 0 0 9 5 - 0 6 9 6 ( 0 2 ) 0 0 0 1 0 - 4

that ‘‘no answer’’ is equivalent to ‘‘don’t know,’’ i.e., it identifies people who have notmade up their minds. Following the voting literature and excluding ‘‘don’t know’’ responsesfrom empirical analysis carries the implicit assumption that a sample-selection problem is notcreated.Subsequent to the NOAA Panel report, a number of investigators have implemented response

formats that allow for expressions of uncertainty [18,10,21,23], and have implicitly or explicitlyimplied that uncertainty is another interpretation of ‘‘no answer’’ responses. Recently, Welsh andPoe [24] have devised a ‘‘multiple-bounded’’ question format with uncertain response options.The multiple-bounded format differs from single- and double-bounded formats in that it lists apanel of k bid amounts and respondents are asked whether they would pay each bid amount. Themultiple bounded format refines the interval where latent willingness to pay resides and can beimplemented with traditional ‘‘yes’’/’’no’’ responses to each of the k bid amounts. However,Welsh and Poe utilized polychotomous responses of ‘‘definitely’’ and ‘‘probably’’ yes, ‘‘unsure’’,and ‘‘definitely’’ and ‘‘probably’’ no, similar to what was done with a single-bounded question byReady et al. In this paper, we will refer the Welsh–Poe question as the multiple-bounded,polychotomous-choice elicitation approach.While some researchers have implied that uncertainty is one way to interpret the NOAA

Panel’s ‘‘don’t know’’ recommendation, we do not imply that uncertainty is the app-ropriate interpretation. Uncertainty, however, may well be one possible reason why aperson might respond by checking the no-answer category. We also do not imply thatthe no-answer category is conceptually or methodologically the same as the three levels ofuncertain response options that Welsh and Poe provided for respondents. However, given thatWelsh and Poe and others have begun to use uncertain response options, there are a number ofconceptual and methodological issues that must be investigated before this type of responseformat can be considered valid or invalid. In this paper we begin to investigate some of theseissues.None of the studies cited above present a conceptual model to support the use and analysis of

uncertain response options. Furthermore, Welsh and Poe and Ready et al. assume away theuncertainty revealed by the polychotomous responses through recoding all responses to ‘‘yes’’ or‘‘no’’. Moreover, Welsh and Poe assume that responses to the multiple bids were driven by aunique willingness to pay (WTP) for each respondent and that all information about WTP isrevealed by the bid intervals where responses switch from ‘‘yes’’ to ‘‘no’’.The purpose of this paper is three-fold. First, we investigate relaxing the restrictive assumption

employed by Welsh and Poe that responses by an individual to all bids are driven by a uniqueWTP amount. We also propose a conceptual framework that supports the ‘‘random valuation’’model proposed by Wang for analyzing uncertain responses and implement this model to avoidrecoding uncertain responses to certain responses.Second, our experimental design administers a multiple-bounded question with the uncertainty

scale used by Welsh and Poe to one sample, and a multiple-bounded question with the same bidlevels where respondents simply answer ‘‘yes’’ or ‘‘no’’ to another sample. This allows us toinvestigate the effects of allowing people to express their uncertainty.Third, since it has been shown that the magnitude of bid amounts can affect responses to single-

bounded and double-bounded questions [5,16]; we investigate whether the ordering of the panel ofbids (ascending or descending) influences responses.

A. Alberini et al. / Journal of Environmental Economics and Management 45 (2003) 40–62 41

2. Why use a multiple-bounded question?

Single-bounded, dichotomous-choice questions are notoriously imprecise as the onlyinformation revealed is whether WTP resides above or below the threshold provided by a singlebid. Double-bounded questions reduce the range of latent WTP to one of three intervals, whichincreases the efficiency of the resulting welfare estimates [15]. The multiple-bounded question,utilized by Welsh and Poe, increases the number of possible intervals to k þ 1 (where k is thenumber of bids shown to a respondent), which further increases the efficiency of the welfareestimate. They find that their multiple-bounded question with 13 bids (14 intervals) reduces theconfidence bounds around estimates of WTP by over 60% relative to a single-bounded questionwith the same bid design.In their econometric analysis Welsh and Poe assumed that responses to all of their 13 bids

were driven by a single WTP amount. This assumption implies that the only relevant statisticalinformation is the interval between the highest bid that a respondent answer ‘‘yes’’ to andthe next higher bid. This assumption implies that a multiple-bounded question is not needed andthe same information could be obtained using a payment card where respondents circle thehighest amount they would pay. If the payment card has the same bid amounts as the multiple-bounded question, and there are no differing response effects that arise from the two responseformats, then both approaches reveal the same statistical information (this is a testablepresumption).We relax Welsh and Poe’s restrictive assumption of complete dependence and estimate a Probit

model that utilizes all k bids in the multiple-bounded panel, assuming separate draws from thedistribution of WTP. We estimate the model with random effects to allow for correlation betweenthe multiple answers provided by each respondent.2 Assuming a new draw from the distributionof WTP for each of the k bids should increase the efficiency of the estimate of central tendency forWTP as the sample has been expanded by a factor of k:Both of the above efficiency improvements arise from the multiple-bounded question format

and are not related to the use of uncertain response options. Thus, the most obvious reason forusing a multiple-bounded question is to increase the efficiency of the resulting welfare estimate.Another potential advantage of the multiple-bounded format is that the range of bids may be

useful when limited information is available to develop an optimal bid design [2].3 The panel ofbids may also reduce the potential for respondents to anchor their responses on bid amounts asoccurs in single and double-bounded questions. However, other questions arise such as whetherthe range of dollar amounts covered by the bids, the intervals between bids, the number of bids orother design features affect welfare estimates [20,22]. Here we investigate one potential issue,whether the order (ascending or descending) of the panel of bids affects welfare estimates, as hasbeen done by DeShazo [11] for double-bounded questions.

2Cameron et al. [8] also analyze data from a multiple-bounded question and utilize all k bids in the analysis, but

appear to assume independence of the k responses from each individual.3The statistical efficiency of welfare estimates can be low if most bid values are too far away from the center of the

distribution, while placing all bid values too close to the center can make the researcher unable to get a good feel for the

dispersion of the welfare measure [17,2].

A. Alberini et al. / Journal of Environmental Economics and Management 45 (2003) 40–6242

3. Why use uncertain response options?

While there is a considerable literature regarding uncertainty in the estimation of nonmarketvalues, all of this literature assumes people know their preferences with certainty. Uncertainty inresponses to valuation questions is presumed to arise from unknown conditions in demand (e.g.,future income) or supply (e.g., future quality of the item being valued) [13]. Specifically, it isassumed that people use expected utility to formulate an option price under conditions ofuncertainty and it is this value that each person uses to evaluate the bids presented in valuationquestions. While respondents are uncertain about future conditions, the resolution of the expectedutility problem implies that respondents have values that are firm and known to them. Thisfurther implies that, when faced with a dichotomous-choice or multiple-bounded question,respondents should be able to answer without any uncertainty.There are reasons to believe that this conventional wisdom may be flawed, even for market

goods. Consider the purchase of a market good such as running shoes. No runner knows how theshoes will fit until they actually run in them. Thus, purchase decisions are made using the bestavailable information at the time of the sale, which may include experience running in the brandand model purchased. All uncertainty is not removed because the model design changes or thedesired model may no longer be stocked. Thus, there is reason to believe that uncertainty remainsat the time of the purchase that the shoes may ultimately not fit well. Due to this uncertainty, anumber of stores that sell running shoes will accept returns with little wear if they do not fit wellfor the purchaser. This gives the purchaser some assurance that they will not be penalized formaking a wrong decision with limited information. A similar situation arises with ‘‘lemon laws’’to protect purchasers of motor vehicles.There is every reason to believe that more uncertainty exists in responses to stated-preference

questions. Respondents do not have prior purchase experience and cannot use brand and modelsas signals of quality, nor do they have prior experience with the researcher that implies thevaluation exercise is credible and they will have some recourse for making a bad decision. Inaddition, respondents are restricted to making a decision based on the scant information availablein the scenario that describes the item to be valued.If we accept that people may be less sure about their responses to a stated-preference question

and do not have the recourse for a bad decision as they do for some market situations, thequestion becomes one of how respondents answer in the face of this uncertainty. One way toconsider this is to postulate that people who answer definitely yes will only do so if their utility foranswering yes, plus some error factor, exceeds their utility from answering no. That is, arespondent will answer yes to any given bid amount in a single, double or multiple-boundedquestion if:

Vðw; I � BÞ þ x > Vðw=o; IÞ; ð1Þ

or

Vðw; I � BÞ � Vðw=o; IÞ > �x; ð2Þ

where Vð�Þ is an indirect utility function, w and w=o indicate being with and without the itembeing valued, I is income, B is the bid and x is a negative error function (the loss from making anerror). To make the point perfectly clear, it is worth stating that x is a loss function that is known


to respondents and is not the random econometric error that arises from the empiricalinvestigator’s inability to observe all relevant arguments. That is, the larger an individual’sperceived error, the larger will be |x|. This error function is reduced for a market purchase by priorpurchase experience with the brand and model, and recourse for a mistake in the purchasedecision. Thus, x may be minuscule for the purchase of a commodity such as ‘‘Cheerios’’ andcould be quite large for a stated-preference question eliciting nonuse values for an obscureenvironmental amenity. This conceptualization of a threshold for expressing preferences providesan interpretation of the random-valuation model Wang used to analyze response options of‘‘yes’’, ‘‘no’’ and ‘‘don’t know’’. As done by Wang, it is possible to parameterize x in terms ofrespondent characteristics.Thus, the reason for implementing uncertain response options in stated-preference questions is

that the commodity description may not be complete or clear, or respondents may haveinsufficient experience with the item being valued to provide clear, fixed responses. This we referto as ‘‘true’’ uncertainty. Uncertain response options can also reveal what we refer to as ‘‘false’’uncertainty. False uncertainty arises when polychotomous responses encourage people to be lazyin resolving their answers to the valuation question, provide an expectation that respondentsshould use all response options, or allow people to indicate some support for the item beingvalued even if they would not pay the specified bid. In this paper we do not address this issue of‘‘true’’ and ‘‘false’’ uncertainty, we simply take the uncertainty expressed in respondents answersat face value in the modeling. It will be important for future investigations to address this issue of‘‘true’’ versus ‘‘false’’ uncertainty if this question format is to have conceptual and empiricalcredibility.While the focus of multiple bids is on the efficiency of welfare estimates, there is no reason to

believe that allowing uncertain response options will affect efficiency of welfare estimates. Thereason for examining how people answer questions with uncertain responses is to develop a betterunderstanding of the effects on estimates of central tendency.

4. Statistical models for valuation questions with multiple bids and polychotomous responses

4.1. Models for valuation questions with multiple bids

Our first model for multiple-bounded, polychotomous-choice responses follows Welsh and Poefor purposes of comparison. In this model, which is based on the same five response categories asWelsh and Poe, we code ‘‘definitely’’ and ‘‘probably yes’’ as ‘‘yes’’ and any other response as‘‘no’’.4 The analysis of the recoded data models the intervals where respondents switch from a‘‘yes’’ to a ‘‘no’’ response. With this model, all uncertainty in the responses is assumed away andthe bid intervals where respondents switch from ‘‘yes’’ to ‘‘not yes’’ are assumed to convey all theinformation required to recover the distribution of respondents’ latent values. Clearly, thismodeling approach (Welsh–Poe model hereafter) assumes that one value for each respondentdrives responses to all bids. Thus, the sole reason for using the multiple-bounded framework is to

4Welsh and Poe show that different recodings of the polychotomous responses scale value estimates up as more

categories are recoded as a definite ‘‘yes’’, and using alternative recodings of our data has the same result.


narrow the interval within which the latent value resides. The recoding may imply that there is noneed for polychotomous responses if the same statistical information would be provided by apayment-card question.Next, we estimate a Probit model that relaxes the assumption of complete dependence among

the responses implicit in the Welsh–Poe model. ‘‘Definitely’’ and ‘‘probably yes’’ are again codedas ‘‘yes,’’ and all other responses are coded as ‘‘no,’’ which continues to assume away uncertaintyin the responses. All k responses for each respondent (one for each bid in the panel) are used. Themodel we fit is a random-effect Probit, which treats responses within an individual as correlated,but individuals as independent of one another.The empirical question is whether the Welsh and Poe model, which models the switching

interval, provides welfare estimates with smaller standard errors, or does our random-effectProbit model with k responses per person provide more efficient welfare estimates?

4.2. Inclusion of uncertain response categories

The next model uses the information provided by the uncertain, polychotomous responses.In these models, uncertainty is not assumed away by recoding to certain responses. Ourmodels follow the approach introduced by Wang [23], estimating the thresholds whererespondents switch from ‘‘definitely yes’’ to more uncertain response categories (‘‘probablyyes,’’ ‘‘unsure’’ and ‘‘probably no’’) and to ‘‘definitely no’’ as the magnitude of the bidincreases (random-valuation models hereafter). In the first of these two models, the thresholds areconstants, while in the second they are functions of respondent characteristics. A modelwhere the thresholds are considered to be functions of angler characteristics could be motivatedby angler heterogeneity where more experienced, dedicated anglers might have smaller range ofuncertainty.A summary of the assumptions and properties of all models (Welsh–Poe, random-effects

Probit and random-valuation) is shown in Table 1. The likelihood functions for each of themodels, plus the identification conditions for the random-valuation model, are reported in anappendix.5

Another issue that merits investigation is how respondents answer payment questions with andwithout uncertain response categories. To answer this question, two independent samples receivedquestionnaires that were identical in all respects, except for the possible response categories to thepayment question. The first sample received a panel of 14 bids arranged in ascending order from$1 to $2000, with response options of ‘‘definitely’’ and ‘‘probably’’ yes, ‘‘unsure’’, and ‘‘definitely’’and ‘‘probably’’ no. The second sample received a similar payment question, but the only tworesponse categories were ‘‘definitely yes’’ and ‘‘definitely no’’.To ascertain the effect of the inclusion of uncertain response categories we first compare the

distribution of responses across the questionnaire versions. Next, we estimate mean WTP andcompare the welfare estimates from the two samples.

5To estimate the models of this paper, we programmed directly the respective likelihood functions using the GAUSS

Maxlik optimization module. The random-valuation model can also be estimated using an ordered Probit packaged

routine, in which case estimates of the regression coefficients, standard deviation of WTP, and thresholds are obtained

in the usual fashion for a discrete choice model of WTP responses, exploiting the identifying restriction discussed in the

appendix. For a small number of bid amounts per individual, a random effects Probit routine is available in LIMDEP.


4.3. Presentation of the bids

Finally, we wish to investigate whether the multiple-bounded approach can create undesirableresponse effects. For instance, is it possible that respondents are influenced by the order in whichthe bids are presented to them?To answer this question, a third independent sample received the 14 bids arranged in

descending order from $2000 to $1. The response options were, once again, ‘‘definitely’’and ‘‘probably’’ yes, ‘‘unsure’’, and ‘‘definitely’’ and ‘‘probably’’ no. To determine the effectof the presentation of the bid we compare the raw distribution of responses for the first andthird samples, as well as the respective welfare estimates produced by each of our econometricmodels.

5. The data

The data used to demonstrate the econometric models and investigate the response effects werecollected from a random sample of 5000 anglers who held a fishing license in Maine during 1994.The sample was randomly stratified into three subsamples of 1666, 1667 and 1667 anglers; eachformat of the valuation question was contained in a different version of the survey. Surveys weremailed in October 1994, after the close of the 1994 open-water, fishing season. Anglers were askedto value their fishing for the 1994 open-water, fishing season (April 1, 1994–September 30, 1994).

Table 1

Summary of statistical models employed to analyze data from the multiple-bid questions with polychotomous responses

that allow for uncertainty

Model Respondents allowed to

make revisions as they

answer the bid panel?

Correlation between

response to bids for each

respondent?

Is information about

uncertainty used?

Welsh–Poe No, one underlying WTP

for all bids

Yes, equal to 1 (perfect

correlation)

No

Random effects Probit Yes, one underlying WTP

for each bid

Yes, correlation between

any two WTP amounts is

the same. Simplifies to the

independent Probit model

if the correlation between

the error terms of the

WTP amounts within a

person is equal to zero

No

Random-valuation models

I and II

Yes, one underlying WTP

for each bid

No, independent Yes, to estimate thresholds

that must be crossed

before respondents

provide certain responses


Of the 5000 surveys sent to anglers, the US Postal Service returned 192 as undeliverable and 2982completed surveys were returned for an effective response rate of 60%.6

After eliciting total, trip expenses for the 1994 open-water fishing season, respondents were told:‘‘We would like to know whether you would have gone open-water fishing in Maine during the1994 season if your expenditures were more than the total you just reported in Question 4. Pleasetell us if you would have gone fishing at all at each of the increased costs listed below. (Definitely

yes means ‘I would have still gone fishing at least once.’ Definitely no means ‘I would not havegone fishing at all.’). It is very important that you respond to all dollar amounts’’.The amounts listed immediately following this question were $1, $5, $10, $25, $50, $75, $100,

$200, $300, $400, $500, $1000, $1500 and $2000. The polychotomous response options were‘‘definitely yes’’, ‘‘probably yes’’, ‘‘not sure’’, ‘‘probably no’’, and ‘‘definitely no’’. The binaryresponse version used the ascending bid panel, but only had responses of ‘‘definitely yes’’ and‘‘definitely no’’. We will refer to these versions as ascending bids with polychotomous responses(ABPR) and ascending bids with binary responses (ABBR). The version with the descending bidpanel—hereafter labeled as DBPR for descending bids with polychotomous responses—was thesame as ABPR with the exception that the bids were presented in reverse order, $2000 to $1.The valuation question elicits consumer surplus experienced in the recent fishing season.

Formally, denote with the respondent’s indirect utility function as Vðp; y;ZÞ; where p denotes theprice per fishing trip, y is income, and Z is a set of individual characteristics influencing utility.Willingness to pay, or Hicksian surplus, is defined as WTP� that solves:

Vðp; y � WTP�;ZÞ ¼ Vðpc; y;ZÞ; ð3Þ

where p is the price per trip actually faced by the respondent in the recent fishing season, and pc isthe choke price. Eq. (3) suggests that respondent surplus should depend on price per trip, income,plus individual characteristics:

WTP� ¼ WTP�ðp; y;ZÞ: ð4Þ

In this paper, we model WTP directly, following the approach presented in Cameron and James[7]). The respondents answer ‘‘yes’’ to a bid if surplus exceeds the bid.The regressors in our econometric equations are average expenditure per trip (p),7 income and

various demographics, and dummies capturing fishing taste and habits, such as whether the anglerparticipates in ice fishing (ICEFISH) and whether the angler participates in marine fishing(MARINEF). Descriptive statistics for all variables used in our statistical models are presented inTable 2.

6The return rate by questionnaire version were 60.56%, 58.46% and 59.93%. The three subsamples are very similar

in terms of sociodemographics, implying that any difference in the response patterns to the panel of bids should be

attributed to the experimental treatment.7Average expenditure per trip is a different variable from the bid level. The bid level is used to compute the

contributions to the likelihood, but does not appear explicitly in the expression for latent WTP. Average expenditure

per trip is included in the expression for latent WTP because Eq. (4) posits that WTP may depend on this factor, among

others.


6. Results

6.1. Distribution of responses

The response pattern for the ABPR version is shown in Fig. 1. Only 50 respondents (1.7% ofthe sample) always answered ‘‘definitely yes’’, and 21 respondents (0.7% of the sample) alwaysanswered ‘‘definitely no’’. Most respondents checked each response category at least once.The percentage of respondents who gave ‘‘definitely yes’’ responses is very high (92.5%) for a

bid level equal to $1 and declines sharply as the bid level increases. When the additional costper season is $2000, only 1.8% would definitely pay this amount. The percentage of ‘‘definitelyno’’ responses is very low at the low bid values and rises in a regular fashion with the bid. Eighty-seven percent of the respondents would definitely not pay $2000. A Pearson chi-square test easilyrejects the null hypothesis that the responses are independent of the bid levels at the 1% level ofsignificance.8

Further examination of the data reveals that about 31.4% of the respondents checked the ‘‘notsure’’ response option at least once. This indicates that uncertainty is an important component of theresponse distribution, and the different modeling frameworks proposed here are relevant for analyzingthese data. The percentage of respondents providing ‘‘not sure’’ responses is generally low at thelowest and highest bid amounts, and peaks at the middle amounts, where it is as high as 18.3%.9

Table 2

Descriptive statistics

Variable and description Sample average

DBPR

version

ABPR

version

ABBR

version

ICEFISH (dummy=1 if respondent engages in ice fishing) 0.38 0.34 0.38

MARINEF 0.23 0.25 0.23

(dummy=1 if respondent engages in marine fishing)

Age (years) 42.61 42.32 43.12

Income (household income) (thou. $) 44.11 43.36 42.44

MALE (dummy=1 if the respondent is a male) 0.86 0.84 0.86

PRICE (previous season’s expenditures on fishing trips, divided by

price per trip) ($)

101.33 86.15 94.15

8 If the choice of a response category is truly independent of the bid level, the frequencies along the rows of a

contingency table crossing the bid levels against the response categories should remain approximately the same. The test

statistic is w2 ¼P

ðnij � mijÞ2=mij ; where nij are the observed frequencies, and mij are the frequencies predicted by the

independence model [1]. In this particular case, since all of the n respondents are confronted with the complete list of

payment levels, mij ¼ npþj ; where pþj is the marginal probability of each response category. Here, the test is distributed

as chi-square with 52 degrees of freedom. The test statistic was computed to be 16,148.51, which falls in the rejection for

the null hypothesis of independence at the 1% level or better.9While the response data presented here suggest that people center their responses in the middle of the bid panel,

Roach and Boyle [20] have shown that this centering pattern is not maintained when the bid panels is skewed to high

bid amounts.


To look for associations between the uncertainty of the response and individual characteristics, wefit a multinomial logit (MNL) model, assuming that for each bid level people choose the responsecategory that gives the highest utility. The advantage of a MNL model over Wang’s randomvaluation model is that the former helps find associations between respondent characteristics and theselected response categories without imposing any assumption about the ordering of the responses.10

The regressors include the bid, price of a trip, income and other individual characteristicspotentially related to tastes for fishing (gender, age, and dummies denoting participation in icefishing and marine fishing). We find that the probability of selecting each response category isrelated to the bid level, as well as to several of the angler characteristics. Wald tests imply that‘‘not sure’’ responses should be interpreted as distinct from both the ‘‘probably yes’’ and the‘‘probably no’’ responses. Similarly, ‘‘probably yes’’ and ‘‘probably no’’ should not be combinedwith the ‘‘definitely’’ responses. These results contrast with those reported in Alberini and Champ[3] and Carson et al. [9] who found that the characteristics of people who answered ‘‘unsure’’ weremore similar to those who answered ‘‘no’’ than they were to people who answered ‘‘yes’’.

6.2. Models for multiple-bounded responses

The results of the alternative statistical models seeking to estimate WTP are reported in Table 3for the ABPR version. All models assume that respondents’ latent values are normally distributed,

$1 $5 $10 $25 $50 $75 $100 $200 $300 $400 $500 $1,000 $1,500 $2,000

def.yes

notsure

def.no0

10

20

30

40

50

60

70

80

90

100

Per

cent

Fig. 1. Response distribution for version ABPR.

10 Implicit in the MNL model is the assumption that the utility level is observed with an error, that the error terms

associated with each response options are independent of one another, and that they follow a standard type-I extreme

value distribution.


and that the welfare statistic of interest is mean WTP. All reported results for the Welsh–Poemodel assume a switch from ‘‘probably yes’’ to ‘‘not sure’’ identifies the upper and lower boundsfor the latent variable. ‘‘Definitely’’ and ‘‘probably yes’’ are reclassified as ‘‘yes’’ for the Probitmodels.Comparing the Welsh–Poe model with the random-effects Probit model shows that the

coefficients on MARINEF and MALE are of comparable magnitude: they differ by only 5% and3%, respectively.11 The coefficient of income changes by about 24%: its magnitude implies that a$1000 change in household income will change mean WTP by $2.42 to $3.26. ICEFISH, age andprice per trip are insignificant in the Welsh–Poe model, but are statistically significant, and ofgreater absolute magnitude, in the random-effects Probit model. The t-statistics are generallylarger in the Probit model than in the Welsh–Poe model, showing that employing multipleobservations from each respondent (as the random-effects Probit model effectively does) morethan compensates for the fact that the intervals around WTP are much broader. The estimates ofs; the standard deviations of WTP are similar, in the two models.Estimated mean WTP is quite similar and only differs by 14%: $207 for the Welsh–Poe model

and $236 in the Probit model. While Welsh and Poe found that their multiple-bounded questionmodel increased the efficiency of the welfare estimate by over 60% relative to a dichotomous-choicequestion, we find that assuming that the response to each of the 14 bids is a new draw from thedistribution of WTP further increases the welfare estimate by more 50% (B$15 against B$7).12

Fitting the random-effects Probit model proved to be very instructive. The coefficient ofcorrelation between responses from the same individual, which measures the fraction of the totalvariance of the error terms explained by the individual-specific component of the error, is onlyabout 0.06. This coefficient is statistically significant, but it is so close to zero that it effectivelyleaves the point estimates of all coefficients and their standard errors very similar to those that areobtained by fitting a Probit that treats all observations as independent (by restricting thecorrelation coefficient to be zero).13 Because of these findings, we believe it is acceptable to fitrandom-valuation models based on the assumption of independent draws, as we do next.

11The coefficients reported in the various columns of Table 4 are the regression coefficients of WTP on the various

regressors, i.e., bj ¼ @EðWTPÞ=@xj : Rather than presenting the coefficient of the bid variable, as is frequently done in

much CV empirical work, we report directly (the negative of) its reciprocal, s; which is the standard deviation of WTP.

The standard deviation of WTP is identified from the change in the probability of a positive response as the bid

changes.12Mean WTP is computed as fitted WTP at the means of the explanatory variables ½ð1=nÞ

Pni¼1 x

0i#b ¼ %x0 #b; and its

standard error as the square root of %x0 Covð #bÞ %x; where Covð #bÞ is the covariance matrix of the estimated regression

coefficients.13The statistical literature (see Fahrmeir and Tutz [12]) informs us that, if random effects are present, the point

estimates of the coefficients of the random effects Probit should be close to those of a Probit model that assumes

independent observations. The standard errors of the estimates, however, can be quite different, depending on the

correlation between the unobserved WTP amounts by the same respondent. We were surprised to obtain such a low

estimate of the correlation coefficient, r; which we had expected to be substantially higher. Perhaps this result reflects

the routine’s difficulty in identifying r: If r is less than one, one would expect to observe some ‘‘reversals’’ in the

responses (e.g., at least a few respondents might say that they are willing to pay $25, but not, say, $15. This apparently

inconsistent answer can be explained as arising from different draws from the distribution of WTP.) In this study, no

inconsistencies were observed, which corroborated our expectation to find that r is close to one. This expectation was

not borne out in the estimation results.


6.3. Models for polychotomous choice responses

The first type of random-valuation model is one that holds the thresholds constant acrossindividuals. We first fit this model imposing the fewest restrictions necessary for identification ofthe welfare measure. Specifically, we only need to impose the constraint that two of the thresholds

Table 3

Estimation results for ascending bids with polychotomous responses (version ABPR). (t-statistics in parentheses)

(N ¼ 8540) (642 respondents)

(A) Welsh–

Poe; Switch

away from

prob. Yes

(B) Random-

effects Probit

(Def. and

prob. Yes=1

all else=0)

(C) Random

valuation; a

and b are

constants

(D) Random

valuation; a

and b are

constants

(E) Random

valuation; a

and b linear

in ICEFISH

and

MARINEF

Constant �0.62 �12.03 36.42 79.13 80.35

(�0.10) (�0.46) (58.02) (2.58) (2.70)

ICEFISH �5.24 28.08 58.02

(�0.16) (2.06) (3.51)

MARINEF 92.15 88.38 130.17

(2.626) (5.84) (7.15)

AGE 0.10 1.20 1.17 0.97 0.92

(0.10) (2.67) (2.25) (1.79) (1.74)

MALE 58.98 63.20 95.90 132.84 134.53

(1.48) (3.66) (4.58) (6.38) (6.52)

INCOME in thou. $ 3.26 2.42 3.26 3.08 3.08

(5.40) (9.01) (9.98) (9.70) (9.74)

PRICE PER TRIP �0.07 0.08 0.16 0.10 0.10

(0.79) (1.93) (3.14) (2.09) (1.99)

s 362.40 354.30 576.29 580.62 580.27

(31.42) (34.83) (54.44) (54.34) (54.34)

Threshold coefficients:

g10 [constant] 307.90 308.25 297.27

(46.72) (46.60) (38.30)

g11 [MARINEF] 20.92

(1.74)

g12 [ICEFISH] 17.09

(1.58)

g20 [constant] 100.51 100.61 96.64

(28.41) (28.38) (21.98)

g21 [MARINEF] 5.31

(0.66)

g22 [ICESFISH] 7.84

(1.06)

Welfare measure:

Mean WTP 207 236 373 373 374

Std. Error around mean WTP 14.97 6.89 7.68 7.69 7.69


be symmetric around mean WTP, while the other two are estimated as free parameters. Weexperimented with first setting the two ‘‘inner’’ thresholds (those that separate ‘‘not sure’’ and‘‘yes’’ or ‘‘no’’ responses) equal to one another and then setting the ‘‘outer’’ thresholds (those thatseparate the ‘‘probably’’ from the ‘‘definitely’’ responses) equal to one another.However, likelihood ratio tests that compared each of these models with the fully constrained

model (where the two pairs of thresholds are constrained to be symmetric with one another) didnot reject the null hypothesis of full symmetry.14 Accordingly, we report the results of random-valuation models with both pairs of thresholds restricted to be symmetric with one another incolumns (C) and (D) of Table 3.Comparison of the random-valuation model (column (C)) with the Welsh–Poe model and the

random-effects model shows that, with the only exceptions of the coefficients of age and income,all coefficients are dramatically different. Several coefficients double in size when moving fromcolumn (B) to column (C), and even the standard deviation of WTP, s; increases by 62%. Thethresholds are large and statistically significant: They imply that a respondent’s WTP amountmust be at least $308 greater than the bid before a ‘‘definitely yes’’ answer is provided.Respondents whose WTP exceeds the bid by an amount between $100 and $308 will select the‘‘probably yes’’ response category, while persons with WTP within a $100 of the bid will declarethat they are ‘‘not sure’’.

Table 4

Estimation results for ascending bids with binary responses (version ABBR). (t-statistics in parentheses) (N ¼ 9019)

(A) Welsh–Poe; Switch away from

def. yes

(B) Random-effects Probit (Def.

yes=1 all else=0)

Constant 56.53 33.67

(1.04) (1.23)

ICEFISH 73.13 94.88

(2.46) (6.93)

MARINEF 60.64 57.41

(1.83) (3.77)

AGE 0.63 0.33

(0.50) (0.77)

MALE �16.17 �6.02(�0.40) (�0.33)

INCOME in thou. $ 2.89 3.07

(5.14) (11.58)

PRICE PER TRIP �0.06 �0.03(�0.82) (�0.81)

s 344.02 347.73

(34.35) (36.08)

Mean WTP 229 221

Std. error around mean WTP 13.54 6.42

14The likelihood ratio statistics are 0.68 (when the two ‘‘inner’’ thresholds are forced to be symmetric around mean

WTP) and 0.69 (when the two ‘‘outer’’ thresholds are forced to be symmetric). In neither case is the fully restricted

model rejected at the usual significance levels.


At $374, the estimated mean is larger than in the Welsh–Poe model and the random-effectsProbit (an 80% and 63% increase, respectively). The standard error of the mean, at 7.69, is notmuch larger than that of the random-effects Probit. Thus, modeling the uncertain responses has adramatic effect on the magnitude of the welfare measure, but no substantial effect on theefficiency of the welfare estimate.In specification (D), we omit two regressors, the dummies for participation in marine fishing

and ice fishing, and reserve them for later use as instruments to predict the thresholds that must becrossed before respondents feel more (or less) confident that they would make the payment.15

Omitting these variables results in relatively large changes in the intercept and the gendercoefficient, but has minor effects on the other coefficients. The standard deviation of WTP, meanWTP and the standard error around mean WTP are virtually the same as in specification (C).In practice, despite our prior that fishing taste and experience would be important in

determining the thresholds, column (E) shows that there is little evidence in favor of the dummies(MARINEF and ICEFISH) we picked to explain the thresholds of the random-valuation model.The coefficients of the variables entering in the thresholds are largely insignificant. Even moreimportant, the estimated coefficients, their standard errors and mean WTP remain virtually thesame as in column (D), where the thresholds are the same across individuals.16

6.4. How do people respond in the absence of uncertain response options?

The percentages of ‘‘yes’’ responses for the ABBR version are approximately equal to the sumof the percentages of ‘‘definitely yes’’ and ‘‘probably yes’’ responses to the ABPR version. In thecurrent application this implies that when respondents are forced to give either a firm ‘‘yes’’ or afirm ‘‘no’’ and they are leaning toward answering ‘‘yes’’ (‘‘definitely’’ or ‘‘probably yes’’), they willtend to answer ‘‘yes.’’The models fitted to the data from ABBR version (Table 4) are qualitatively similar to the

comparable models for the ABPR version (Table 3), and give estimated means that are similar.The estimated means from the Probit models are statistically undistinguishable across versionsABPR and ABBR for both the Welsh–Poe and the random-effects Probit models. Theappropriate Wald statistics are 1.11 (p ¼ 0:29), and 2.52 (p ¼ 0:13), respectively. Wald tests alsoindicate that the estimate of s is not statistically different across subsamples (Wald statistics 1.45for the Welsh–Poe model and 0.22 for the random effects Probit model, p-values 0.23 and 0.64).

15We avoid having overlapping variables between the determinants of mean WTP and the determinants of the

thresholds to ensure identification of all parameters (see Sections A.1 and A.3 of Appendix A).16The assumption that the observations within a respondent are independent requires some care in interpreting the

estimates of the standard errors around the point estimates of the mean. For good measure, we also estimated these

models using a single response, randomly selected among the responses provided to the panel of 14 bids, for each

person, and averaged the estimates of the coefficients and of the welfare measure over 100 replications. This procedure

is robust to the lack or presence of correlation among the responses (as correlation within an individual, if any, is no

longer relevant), and allows one to check for possible order effects or abnormalities in the response patterns as the bid

values change. The results obtained when the random-valuation models were re-estimated using a single response per

person are very similar to those reported for each of these models in Table 3, except, of course, for the fact that the

standard errors are larger. Since estimation with random draws is robust to the presence or absence of correlation

among the responses, this confirms that our assumption of independence for the models reported in Table 3 does not

bias the point estimates.


This clearly indicates that, when faced with the choice between firm ‘‘yes’’ and ‘‘no’’ responses,persons who were uncertain, but leaning toward answering yes, chose ‘‘yes’’.

6.5. Is there an order bias?

We found some interesting differences in the responses to the ABPR and the DBPR versions.The general shape of the distribution of the responses from the DBPR version is similar to thatfrom the ABPR version shown in Fig. 1. That is, the proportion of ‘‘definitely yes’’ responsesdeclines with the bid, the percentage of ‘‘definitely no’’ responses increases with the bid, and ‘‘notsure’’ responses are more common at middle bid values. However, as shown in Fig. 2, thefrequency of ‘‘definitely yes’’ responses decline more slowly as the bid amount increases in DBPRversion than in the ABPR version. In fact, the percentages of ‘‘definitely yes’’ responses are higherat every bid level in the DBPR version than in the ABPR version, and the percentages of‘‘definitely no’’ responses are lower at every bid level.Comparisons of the frequencies of ‘‘not sure’’ responses reveal that at all bid levels the people

responding to the ABPR version were slightly more likely to indicate that they were unsure thanwere respondents to the DBPR version. The highest percentages of ‘‘not sure’’ responses are forthe $200 bid in the ABPR version (18.3%) and for the $300 bid in the DBPR version (11.2%).To formally test for respondent sensitivity to the order of the bids, we estimate multinomial

logit (MNL) models using the bid level as the only regressor. For the null hypothesis that theMNL coefficients are the same in both versions of the survey, the likelihood ratio test is x ¼�2½logLR � logLU ; where R denotes ‘‘restricted’’ (one model for both versions) and U stands for‘‘unrestricted’’ (separate models for each version). Under the null hypothesis of no difference, thestatistic x is distributed as chi-square with 8 degrees of freedom. The test statistic is equal to 721,falling in the rejection region of the null hypothesis, and suggesting that there is an order effect.We may further speculate that the DBPR version is likely to yield higher welfare estimates thanthe ABPR version due to the differing patterns of ‘‘definitely yes’’ and ‘‘definitely no’’ responses.The data suggests that starting the panel of bids with the highest bid leads respondents to be morelikely to answer yes to all subsequent bid amounts relative to a panel where the first bid is the

0

10

20

30

40

50

60

70

80

90

100

$0 $500 $1,000 $1,500 $2,000

Bid Amount

Pe

rc

en

t

Pct. Def Yes (ABPR) Pct. Def Yes (DBPR)

$2,500

Fig. 2. Percent ‘‘Definitely Yes’’ responses in ABPR and DBPR versions.


lowest dollar amount. This suggests a starting-point issue such as has been identified for single-bounded [8] and double-bounded [16] questions. This may also be evidence of uncertainty inpreferences if respondents anchor their responses on the first bid provided: Well-definedpreferences should not lead to anchoring.Finally, the results from fitting the Welsh–Poe, random-effects Probit, and random-valuation

models to data from the DBPR version are not surprising given the distributions of responsespresented in the figures. The estimated coefficients for the descending bids (Table 5) tend to bemuch larger than their comparable estimates for the ascending bids (Table 3). The samplepresented with descending bids provides larger estimates of mean WTP (over a 100% increase forthe Welsh–Poe and random-effect Probit models), and larger standard errors around mean WTPfor all models. Overall, the qualitative differences between the results of all models fitted to thedata from the DBPR version are similar to those between the results of all models for the samplepresented with the ABPR version. Using responses to all bids, and treating them as separatedraws from the distribution of WTP, increases the efficiency of the welfare estimate and modelingthe uncertain responses substantially increases the welfare estimate.We checked whether the tendency of persons given the DBPR to reveal higher WTP values was

specific to certain groups of respondents (e.g., more highly educated subjects vs. less educatedsubjects, anglers with ice fishing experience and without, etc.), but found that the differences inresponses between these groups for the ABPR and DBPR treatments were roughly the same as thedifferences for the sample as a whole.Having established that presenting the bids in descending order results in larger mean WTP, we

wished to check whether such effect was due to a uniform increase in all regression coefficients, orwas restricted to the coefficients of a few key regressors. We present in Table 6 the results of suchan investigation when we fit a random-valuation model with constant thresholds. Column (A) ofTable 5 reports the intercept plus the estimated coefficients on the usual five regressors (ICEFISH,MARINEF, age, household income, and expenditure per trip in the past fishing season), a dummyfor version DBPR, and interaction terms between each of the five regressors and the dummy forversion DBPR.The coefficient of the DBPR dummy, therefore, when summed with the intercept, gives the

intercept for respondents who were given the DBPR questionnaire. Similarly, the coefficients ofthe interaction terms tell us by how much the DBPR slope coefficients differ from their ABPRcounterparts. Column (A) shows clearly that the impact of the order in which the bids werepresented is limited to the intercept and the coefficient of ICEFISH, which are much larger in theDBPR version, while the coefficients of all of the other regressors are roughly the same.In column (B) of Table 6 we report the results of fitting a similar model with pooled samples

and a single set of coefficients, confirming once again that while pooling the data introducesdramatic bias in the estimates of the intercept and ICEFISH, the other coefficients are much lessseverely affected.17

17The results of column (B), Table 6, can also be used, along with those displayed in columns (D) of Tables 3 and 5,

to formally test the null hypothesis that all parameters differ across the ABPR and DBPR subsamples. The likelihood

ratio statistic is 478.2, and the null hypothesis is, therefore, soundly rejected. When this exercise is repeated for the

independent Probit and the Welsh–Poe model, the resulting likelihood ratio statistics are 396.18 and 190.65. In both

cases, the null of identical parameters is rejected, the test statistic falling in the rejection region of the chi-square with 8

degrees of freedom.


Table 5

Estimation results for descending bids with polychotomous responses (version DBPR). (t-statistics in parentheses)

(N ¼ 8483) (642 respondents)

(A) Welsh–Poe;

Switch away

from prob. Yes

(B) Random-

effects Probit

(Def. and prob.

Yes=1 all

else=0)

(C) Random

valuation; a and

b are constants

(D) Random

valuation; a and

b are constants

(E) Random

valuation; a and

b linear in

ICEFISH and

MARINEF

Constant 67.65 165.63 289.56 416.16 416.44

(0.81) (3.79) (8.05) (10.51) (13.03)

ICEFISH 152.69 184.23 163.29

(3.22) (8.16) (8.70)

MARINEF 104.29 90.92 116.45

(1.95) (6.61) (5.60)

AGE 3.29 1.98 1.52 0.43 0.41

(2.25) (2.76) (2.60) (0.56) (0.87)

MALE 0.08 30.75 62.50 99.05 100.87

(0.12) (1.06) (2.56) (4.08) (4.14)

INCOME in

thou. $

2.60 2.98 2.70 2.35 2.36

(4.06) (7.05) (7.65) (6.60) (6.67)

PRICE PER

TRIP

0.01 0.13 0.11 0.05 0.04

(0.23) (2.65) (2.89) (1.36) (1.19)

s 543.13 518.72 632.61 640.63 640.24

(33.42) (40.72) (57.69) (57.47) (57.44)

Threshold

coefficients

g10 [constant] 311.35 311.23 299.13

(6.18) (45.97) (36.44)

g11[MARINEF]

28.87

(2.10)

g12 [ICEFISH] 15.13

(1.26)

g20 [constant] 91.52 91.39 91.87

(5.13) (25.10) (19.13)

g21[MARINEF]

13.62

(1.61)

g22 [ICEFISH] �9.35(�1.27)

Welfare

measure:

Mean 448 514 627 626 627

Std. error

around mean

WTP

21.65 10.98 9.60 9.50 10.42


Table 6

Comparison of random-valuation models for ascending bids with polychotomous responses (version ABPR) and

descending bids with polychotomous responses (version DBPR). (t-statistics in parentheses)

(A) Pooled data model

with shift terms

(B) Pooled data model

with no shift terms

Constant 28.08 142.71

(0.58) (5.86)

ICEFISH 60.38 123.27

(3.48) (9.74)

MARINEF 134.07 114.00

(7.02) (8.17)

AGE 1.22 1.32

(1.86) (3.23)

MALE 98.34 94.64

(4.37) (5.84)

INCOME in thou. $ 3.36 3.04

(10.03) (12.65)

PRICE PER TRIP 0.17 0.15

(3.17) (4.93)

Version DBPR dummy 262.02

(5.20)

ICEFISH�DBPR dummy 99.42

(3.98)

MARINEF�DBPR dummy �20.02(�0.72)

AGE�DBPR dummy 0.25

(0.81)

MALE�DBPR dummy �37.08(11.18)

INCOME�DBPR dummy �0.72(�1.48)

PRICE PER TRIP�DBPR

dummy

�0.07

(�0.95)

s 606.26 618.86

(79.56) (78.94)

Threshold coefficients

g10 311.55 310.80

(65.99) (65.33)

g20 97.11 96.55

(38.06) (37.89)

Log likelihood �19,052.82 �19,279.74N 17023 17023


7. Conclusions

Four important lessons emerge from our analysis. First, interpreting responses to all bids asseparate draws from the distribution of WTP results in comparable welfare estimates, but smallerstandard errors, when compared to the Welsh–Poe model that models the interval whereresponses switch from ‘‘yes’’ to ‘‘no.’’ Thus, while multiple bids can reduce the interval withinwhich the latent variable lies, using the individual responses to all bids as separate draws appearsto further increase the statistical efficiency of the resulting welfare estimate. In addition, the errorcorrelation within individual responses is only 0.06. While this may be specific to our individualapplication, it does suggest that researchers should not immediately abandon the treatment ofmultiple responses as independent draws.Second, respondents do utilize the response categories that imply some degree of uncertainty.

Moreover, in our application, attempting to explicitly include uncertain responses in theeconometric analyses substantially increases welfare estimates.18 These results are in sharpcontrast with those by Li and Mattsson [18], who find that mean WTP decreases after oneaccounts for uncertainty in the responses. This divergence of findings, and the dramatic increase inwelfare estimates we observed, suggest that more research is needed to understand why peoplechoose to use uncertain responses to contingent-valuation questions, before the validity of usingpolychotomous responses can be established.Third, comparisons of the response distributions and welfare estimates for our ABPR and

ABBR versions suggest that in the absence of uncertain response options people who would haveanswered ‘‘definitely’’ or ‘‘probably yes’’ will chose to answer ‘‘yes,’’ which is the recoding weused. Thus, our results do not support the finding of Welsh and Poe that people who areuncertain, but not leaning toward answering ‘‘no,’’ will answer ‘‘yes,’’ i.e., those who answer‘‘unsure’’ will not answer ‘‘yes.’’ Our results also do not support the Carson et al. [9] finding thatall uncertain responses would be no responses in a binary yes/no response choice.Fourth, the order of presenting the bid panel does affect the responses and the magnitude of the

welfare estimates. This is against some researchers’ heuristic argument that the panel of bidsreduces the focus on a single bid or on two sequential bids and thereby avoids anchoring effects.Roach and Boyle [20] show that truncating the bid range also affects welfare estimates frommultiple-bounded questions, which is similar to the result that Rowe et al. [22] found when therange of a payment card is truncated. DeShazo [11] discusses reasons for differences in theresponses between ascending and descending sequences where bids are administered sequentially,but it is unclear whether these possible explanations carry over to the multiple-bounded context,where all bid amounts are shown to the respondent at the same time. The intuitive analogy here isthat the magnitude of the starting, or first, bid in the panel still appears to provide an anchor thataffects welfare estimates.The bottom line with respect to using multiple bids is that the efficiency of welfare estimates is

increased, but this approach does not resolve bid-design problems. This leaves the researcher with

18The results agree with the evidence from another study which used the multiple-bounded, polychotomous-choice

approach to obtain the value of increases in the populations of grassland birds associated with USDA’s Conservation

Reserve Program [4]. The random valuation modeling framework always gives higher mean WTP amounts than the

Welsh–Poe and Probit models.


the age-old tradeoff between efficiency and bias. In our application, welfare estimates increasedramatically when uncertain responses are explicitly modeled. Our findings, and the contrastbetween our findings and those of previous studies that have allowed uncertain responses, suggestthat more consideration needs to be given to the framing of the questions and responses formatsthat allow for uncertainty, and the reasons people choose to give responses that indicateuncertainty.

Appendix A. Econometric models employed

A.1. Welsh–Poe model

If a person checks ‘‘probably yes’’ at $5, but ‘‘not sure’’ at $10, one way to interpret thisresponse, for a recoding of ‘‘definitely’’ and ‘‘probably yes’’ to yes’’, is to treat the respondent asbeing willing to pay $5, but not $10. Hence, willingness to pay lies within the interval between $5and $10, and this person’s contribution to the likelihood function is the probability that $5 and$10 bracket this subject’s WTP amount. The log likelihood function is

logL ¼Xn

i¼1log PrðWTPo$X H

i Þ � PrðWTPo$X Li Þ

� �; ðA:1Þ

where $X Li is the highest bid at which respondent i answered ‘‘probably yes,’’ and $X H

i is the bid

level at which the subject switched to a ‘‘not sure’’ response. Welsh and Poe [24] assume that WTPis a logistic, but other distributions are possible; here, we assume normality. The log likelihoodfunction is thus

logL ¼Xn

i¼1log F

X Hi

s� xib

s

� �� F

X Li

s� xib

s

� �� ; ðA:2Þ

where xi is a 1� k vector of regressors, b is a k � 1 vector of parameters, s is the standarddeviation of WTP, and F is the standard normal cdf.

A.2. Random-effects Probit model

The recoded yes/no indicators are stacked, resulting in 14 observations for each respondent(one response for each bid in the panel). It is assumed that respondents potentially revise theirWTP amounts when answering the payment question. Hence, the response at each dollar amountis motivated by WTPij ¼ xibþ eij; where i indexes the respondent, j indexes the bids, and the eijs

are error terms. Each revision, however, remains unobserved to the researcher. A ‘‘yes’’ responseis observed if WTP�

ijXBidj:

It is further assumed that the error term e can be broken down into two components: eij ¼ni þ Zij: This decomposition assumes that when answering payment question j; respondent i bases

his choice of response category on WTP��ij ¼ xibþ ½ni þ Zij: The WTP amounts underlying the

response at each bid levels, WTP��ij ; are, therefore, correlated, and the correlation coefficient

between the WTP amounts underlying responses at any two bid levels is constant.


The independent variables in the Probit model include the intercept, individual characteristicsof the respondent ðxiÞ; and the bid value; a ‘‘yes’’ response is coded as a one for the purpose offitting the random-effects Probit model. Denote the estimated Probit coefficients for xi as #a and

for the bid variable as #d:If latent WTP is normally distributed around mean xib and has variance s2; it can be shown [7]

that #b ¼ #a=#d; #s ¼ �1=#d:The standard errors of #b and #s are derived from the covariance matrix of the Probit coefficients

using the delta method [6]. Specifically, the covariance matrix of #b and #s2 is computed as g0Vg;where:

g ¼

1=d

1=d

y

1=d2

266664

377775 ðA:3Þ

and V is the block of the covariance matrix of the estimates that refers to #a and #d:

A.3. Random-valuation models

The random valuation model developed by Wang [23] is well suited for this interpretation ofhow subjects answer polychotomous-choice payment questions. Wang [23] suggests that arespondent answers ‘‘yes’’ only if latent WTP amount is sufficiently large relative to the bid, ‘‘no’’only if latent WTP amount is sufficiently small relative to the bid, and ‘‘don’t know’’ if latentWTP amount lies in between. Assuming that WTP is normally distributed, the log likelihoodfunction is for three response categories and a single-shot payment question:

logL ¼XiAyes

log 1� Fti þ ai � xib

s

� �� þ

XiAno

log Fti � bi � xib

s

� ��

þX

iADK

log Fti þ ai � xib

s

� �� F

ti � bi � xibs

� �� ; ðA:4Þ

where ti is the bid level assigned to respondent i: If ai and bi are constants (ai � a and bi � b for alli’s), then Eq. (A.4) is effectively an ordered Probit model. This is easily seen if the contributions tothe likelihood in Eq. (A.4) are compared for those of an ordered Probit model [14]. The orderedProbit model produces one less estimated coefficient than the number of parameters in the model,requiring that one identification restriction be imposed. To illustrate, if (A.4) is estimated as anordered Probit model, the two constants produced by the estimation routine are a1 ¼ ða � b0Þ=sand a2 ¼ ðb � b0Þ=s; where b0 is the intercept in the expression for WTP (and mean WTP if noother regressors are included in WTP equation). Examples of possible identification restrictionsare (i) setting a and b symmetric around mean WTP (i.e., a ¼ �b), or (ii) setting a=b equal to aconstant.To better capture the fact that individual may have different search costs, the thresholds ai and

bi may be specified as linear functions of a set of individual characteristics: ai ¼ zig1 and bi ¼ zig2:


If the thresholds are linear functions of regressors and parameters, the parameters b and g areidentified only if z and x do not include any overlapping variables, or the ratio of ai to bi is set to aspecified constant.We adapt the Wang model to the situation with five response categories, assuming that

respondents can revise their WTP amounts, and the error terms are independent. Specifically, weintroduce four threshold levels, a; b; c; and d: One identifying restriction is needed to separatelyidentify all coefficients and thresholds, since the ordered Probit routine produces one fewercoefficients than the number of parameters of the model. One can set d ¼ �a; or c ¼ �b:Here, we assume that c ¼ �b; and d ¼ �a: A respondent answers ‘‘definitely yes’’ to the

question if WTP > Bid þ a; ‘‘probably yes’’ if Bid þ boWTPoBid þ a; ‘‘not sure’’ if Bid �boWTPoBid þ b; ‘‘probably not’’ if Bid � aoWTPoBid � b; and ‘‘definitely not’’ ifWTPoBid � b: The log likelihood function is:

logL ¼Xn

i¼1

XjAdef :yes

log 1� Ftj þ ai � xib

s

� �� (

þX

jAprob:yes

log Ftj þ ai � xib

s

� �� F

tj þ bi � xibs

� ��

þX

jAnot sure

log Ftj þ bi � xib

s

� �� F

tj � bi � xibs

� ��

þX

jAprob:no

log Ftj � bi � xib

s

� �� þ

XjADK

log Ftj � ai � xib

s

� �� ); ðA:5Þ

where j indexes the payment question within the questionnaire.In another variant of Eq. (A.5) we allow a; b; c; and d to vary with the respondent, and to be a

function of respondent characteristics, mirroring the notion that different people incur differentsearch costs and have different ability to answer with confidence the payment questions.

References

[1] A. Agresti, An Introduction to Categorical Data Analysis, Wiley, New York, 1996.

[2] A. Alberini, Optimal designs for discrete choice contingent valuation surveys: single-bound, double-bound and

bivariate models, J. Environ. Econom. Manage. 28 (1995) 187–306.

[3] A. Alberini, P. Champ, An approach for dealing with uncertain responses to a contingent valuation question, in:

P.M. Jakus (Ed.), W-133 Benefits and Costs of Resource Policies Affecting Public and Private Land, Eleventh

Interim Report, University of Tennessee, Knoxville, 1998.

[4] K.J. Boyle, R.C. Bishop, D. Hellerstein, M.P. Welsh, M.C. Ahearn, A. Laughland, J. Charbonneau, R. O’Connor,

Tests of Scope in Contingent-Valuation Studies: Are the Numbers for the Birds?, Unpublished paper, Department

of Resource Economics and Policy, University of Maine, 2000.

[5] K.J. Boyle, F.R. Johnson, D.W. McCollum, Anchoring and adjustment in single-bounded, dichotomous-choice

questions, Am. J. Agric. Econom. 79 (1997) 1495–1500.

[6] T.A. Cameron, Interval estimates of non-market resource values from referendum contingent valuation surveys,

Land Econom. 67 (1991) 413–421.


[7] T.A. Cameron, M.D. James, Efficient estimation methods for ‘closed-ended’ contingent valuation Surveys, Rev.

Econom. Statist. 69 (1987) 269–276.

[8] T.A. Cameron, G.L. Poe, R.G. Ethier, W.D. Schulze, Alternative nonmarket value-elicitation methods: are

underlying preferences the same? J. Environ. Econom. Manage. 44 (2002), 391–425.

[9] R.T. Carson, W.M. Hanemann, R.J. Kopp, J.A. Krosnick, R.C. Mitchell, S. Presser, P.A. Ruud, V.K. Smith,

Referendum design and contingent valuation: the NOAA panel’s no-vote recommendation, Rev. Econom. Statist.

80 (3) (1998) 484–487.

[10] P.A. Champ, R.C. Bishop, T.C. Brown, D.W. McCollum, Using donation mechanisms to value nonuse benefits of

public goods, J. Environ. Econom. Manage. 33 (1997) 151–162.

[11] J.R. DeShazo, Designing transactions without framing effects in iterative question formats, J. Environ. Econom.

Manage., forthcoming.

[12] L. Fahrmeir, G. Tutz, Multivariate Statistical Modeling Based on Generalized Linear Models, Springer, New

York, 1994.

[13] A.M. Freeman, III, The Measurement of Environmental and Resource Values: Theory and Methods, Resources

for the Future, Washington, DC, 1993.

[14] W.H. Greene, Econometric Analysis, 4th ed., Prentice-Hall, Upper Saddle River, NJ, 2000.

[15] W.M. Hanemann, J. Loomis, B. Kanninen, Statistical efficiency of double-bounded dichotomous choice

contingent valuation, Am. J. Agric. Econom. 73 (1) (1991) 1255–1263.

[16] J.A. Herriges, J.F. Shogren, Starting point bias in dichotomous choice valuation with follow-up questioning,

J. Environ. Econom. Manage. 30 (1996) 112–131.

[17] B.J. Kanninen, Optimal experimental design for contingent valuation surveys, Ph.D. Dissertation, University of

California Berkeley, 1991.

[18] C.Z. Li, L. Mattsson, Discrete choice under preference uncertainty: an improved structural model for contingent

valuation, J. Environ. Econom. Manage. 28 (1995) 256–269.

[19] NOAA Blue Ribbon Panel on Contingent Valuation, Natural resource damage assessments under the oil Pollution

Act of 1990, Federal Reg. 58 (1993) 460–4614.

[20] B. Roach, K.J. Boyle, Testing bid design effects in multiple-bounded contingent valuation, Land Econom., under

review.

[21] R.C. Ready, J.C. Whitehead, G.C. Bloomquist, Contingent valuation when respondents are ambivalent,


[22] R.D. Rowe, W.D. Schulze, W.S. Breffle, A test for payment card biases, J. Environ. Econom. Manage. 31 (1996)

178–185.

[23] H. Wang, Treatment of ‘don’t know’ responses in contingent valuation survey: a random valuation model,


[24] M.P. Welsh, G.L. Poe, Elicitation effects in contingent valuation: comparisons to a multiple bounded discrete

choice approach, J. Environ. Econom. Manage. 36 (1998) 170–185.


alberini analysis of contingent valuation data with multiple

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