Bayesian Estimation of Willingness-to-pay WhereRespondents Mis-report Their Preferences*

K. Balcombe�, A. Bailey�, A. Chalak� and I. Fraser�

�Department of Agricultural and Food Economics, University of Reading, Reading, UK(e-mail: k.g.balcombe@reading.ac.uk)�AEBM Group, Kent Business School, University of Kent, Wye, Kent, UK(e-mail: alastair.bailey@imperial.ac.uk; a.chalak@imperial.ac.uk; i.fraser@imperial.ac.uk)

Abstract

We introduce a modified conditional logit model that takes account of uncertaintyassociated with mis-reporting in revealed preference experiments estimating will-ingness-to-pay (WTP). Like Hausman et al. [Journal of Econometrics (1988) Vol.87, pp. 239–269], our model captures the extent and direction of uncertainty byrespondents. Using a Bayesian methodology, we apply our model to a choicemodelling (CM) data set examining UK consumer preferences for non-pesticidefood. We compare the results of our model with the Hausman model. WTP estimatesare produced for different groups of consumers and we find that modified estimatesof WTP, that take account of mis-reporting, are substantially revised downwards. Wefind a significant proportion of respondents mis-reporting in favour of the non-pesticide option. Finally, with this data set, Bayes factors suggest that our model ispreferred to the Hausman model.

I. Introduction

Stated preferences are now commonly used in conjunction with the choice modelling(CM) framework in environmental valuation (e.g. Bennett and Blamey, 2001;Hanley, Mourato and Wright, 2001). Naturally, an assumption underpinning suchstudies is that agents are able to answer in a way that reflects their preferences, aswould be revealed by a ‘true’ choice. Part of the rationale for preferring CM tocontingent valuation (CV) is that by setting out a number of attributes, and varying

*We are grateful for the comments from seminar participants at Deakin University and the UniversityNewcastle, Australia. This research was partly funded by UK DEFRA grant number: PS2302.JEL Classification numbers: C25, C11, Q51.

OXFORD BULLETIN OF ECONOMICS AND STATISTICS, 69, 3 (2007) 0305-9049doi: 10.1111/j.1468-0084.2006.00198.x

413� Blackwell Publishing Ltd and the Department of Economics, University of Oxford, 2006. Published by Blackwell Publishing Ltd,9600 Garsington Road, Oxford OX4 2DQ, UK and 350 Main Street, Malden, MA 02148, USA.

those attributes, the outcomes of a well-designed experiment should closely reflectthe preferences of respondents (Hanley et al., 2001). Yet, arguably, even in a well-designed experiment, many respondents may be uncertain that their stated(hypothetical) choices would be in accordance with their actual choices, and thereis also the possibility that some respondents deliberately state preferences they knowto be false, or probably false.

Unlike some other studies in the environmental valuation literature (e.g. Welshand Poe, 1998; Evans, Flores and Boyle, 2004; Vossler et al., 2003), which haveaddressed this problem, this paper introduces an approach that does not requireexplicit acknowledgement by respondents about their uncertainty. In common withother CM studies, the essential idea behind our approach is that individuals areassigned a linear utility function, with differences in choices being due to a randomresidual. In addition, each respondent is assigned a probability that they will respondin a manner not consistent with this utility function (which we term mis-reporting).Moreover, each option within the choice set is assigned a probability that,should respondents mis-report their preferences, is in favour of that option.These probabilities are then estimated along with the usual coefficients of the logitmodel.

The model we develop is similar to that introduced in Hausman, Abrevaya andScott-Morton (1998) (henceforth referred to as the Hausman model or Hausmanlogit). As in the Hausman model, our approach assumes that respondents potentiallygive inaccurate responses, deliberately or otherwise, and the propensity for them todo so is explicitly estimated within the model (by generalizing the likelihoodfunction). However, whereas the Hausman model specifies conditional probabilitiesfor mis-reporting within the likelihood that depend on the nature of the ‘true’ choice,we specify an unconditional probability that there is mis-reporting. Consequently,henceforth we refer to our model as the unconditional probability of mis-reporting(UPMR) logit to distinguish it from the Hausman model which we refer to as theconditional probability of mis-reporting (CPMR) logit. The UPMR and CPMRapproaches to the treatment of mis-reporting produce models that are non-nested.However, the UPMR generalizes more easily into the multinomial logit, and is moreparsimonious than the CPMR model except in the binomial case when the number ofparameters in each model are equal.

Within the environmental valuation literature, the approach proposed in this paperis indirectly related to Welsh and Poe (1998), Alberini, Boyle and Walsh (2003),Evans et al. (2004), Vossler et al. (2003) and Vossler and Poe (2004), among others.In this literature, researchers have explicitly allowed survey participants as part of thesurvey elicitation format to indicate qualitative and/or quantitative levels ofuncertainty associated with particular choices in non-market CV studies. This richersurvey design has led to a number of modifications in how WTP is estimated such asthe dual-uncertainty decision estimator (DUDE) of Evans et al. (2004). In addition,it has allowed calibration of responses to ‘deflate’ WTP estimates (e.g. Vossler et al.,2003).

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The approach we propose in this paper is fundamentally different. First, themethods are applied to stated preference data collected as part of a CM experimentand not CV. Secondly, the mis-reporting logit models can estimate the proportionof respondents replying in a way that does not reflect their true preferences.Thirdly, biases in responses can be estimated. Fourthly, the modified logit modelsyield modified WTP estimates without the need to adjust the CM experiment.Finally, in this paper, as the framework is developed in the context of a conditionallogit model, the approach could potentially be extended to the probit, nested logit,or perhaps the random parameter (mixed) versions of either the logit or probitmodels.

In this study we use a Bayesian approach to estimation, which builds on the workof Koop and Poirier (1993), except that, as with Lahiri and Gao (2002), we use aMetropolis–Hastings (M–H) approach to estimation rather than importance samp-ling. Among the advantages of a Bayesian approach relative to the classical approachis that non-identification of parameters is less problematic (see Bauwens, Lubranoand Richard, 1999, p. 41), and inequality restrictions are also easily implementedwhen using an acceptance–rejection step within an M–H algorithm (Chib andGreenberg, 1995). As will be described, both non-identification and inequalityrestrictions are features of both UPMR and CPMR models. Likewise, a ‘test’ for thevalidity of the standard logit model would require a restriction on a parameter at apoint on the edge of the parameter space. This also poses problems within theclassical framework, but creates no particular difficulties for the calculation of Bayesfactors in the manner of Chib and Jeliazkov (2001). As we discuss in section III,Bayesian methods do not require differentiability of the likelihood, and the existenceof a non-singular Hessian has no bearing on the theoretical or practical constructionof the marginal likelihood used to ‘test’ the hypotheses. Using the samemethodology the non-nested alternatives can also be ‘tested’, enabling the UPMRand CPMR models to be compared.

This paper adds to a small literature of non-market studies employing Bayesianmethods to estimate WTP (e.g. Fernandez et al., 2004; Arana and Leon, 2005).These applications estimate WTP using various probit specifications to captureheterogeneity. However, in both cases sample data are generated as part of CVexercises, a related but different form of stated preference experiment compared withCM. By way of illustration, we apply our new model to a choice data set thatexamines WTP for a basket of food produced using ‘non-pesticide’ technologies. Wecalculate and examine the distributions for WTP by grouped respondents for both thenormal and generalized logit models.

The structure of this paper is as follows. In section II, we briefly review therelevant literature on CM methods and existing studies that have estimated the WTPfor pesticide avoidance. In section III, the UPMR and CPMR logits are outlinedalong with the (Bayesian) methodology used to estimate and test these models. Insection IV, we outline the CM survey, present estimates derived from the standardand mis-reporting models and discuss the results. Section V concludes.

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II. Literature review

The CM methodology is based on random utility theory (RUT) (McFadden,1974). RUT assumes that individual consumers select products because they yieldthe highest level of utility and as such the probability of selecting a productincreases as the utility associated with it increases. The idea behind CM is thatconsumers can choose between alternative options that contain a number ofattributes, set at different levels. CM provides a means of analysing the trade-offsthat are made by consumers when presented with products that offer alternative,new or improved commodity attributes.

In general, a CM experiment consists of the following elements:

• A set of fixed-choice options that may have explicit names or labels.• A set of attributes that describe the potential differences in the choice optionswhich, it is hypothesized, play an important role in the choice behaviour ofinterest.

• A set of levels (values) assigned to each attribute of each choice that representvariation in an attribute appropriate to the research objectives.

• A sample of subjects (survey participants) who evaluate all or a subset of thechoice sets, selecting one of a possible number of options presented in eachset.

Importantly, when one of the options in the attribute set is monetary, and whenthe coefficient of the monetary option divides the coefficient of the other attributes,the resulting quantity represents an estimate of the WTP for the attribute in question.

Choice modelling has been used extensively to deliver WTP estimates thatinform decision makers and in public policy analysis (see Bennett and Blamey,2001; Hanley et al., 2001, for examples). However, CM, along with relatedmethods such as CV frequently produces WTP estimates that exceed expecta-tions. There is a large literature on the tendency of expressed or stated preferencetechniques to find overvalued WTP figures. Garrod and Willis (1999, Ch. 5)review the literature on the tendency for CV to inflate WTP measures. Forexample, Bishop, Heberlein and Kealy (1983) found that CV values couldoverestimate WTP measures by 50%, and Bohm (1994) and Neil et al. (1994)found inflated WTP measures relative to revealed auction methods. Vossler et al.(2003) note that respondents will frequently say ‘yes’ to a hypothetical choice butwhen faced with the actual choice they decline. The tendency to overstate WTPhas also been noted recently in Adamowicz (2004). The literature is fullof possible explanations of this behaviour (e.g. yea-saying, strategic bias) becauseit is important. Biased valuation estimates that grossly exceed policy makers’expectations will result in a distrust of the valuation estimates provided as well asthe methods used.

The approach we propose in this paper is related to these studies in that wecalibrate the results via estimation of WTP. This is different from most other

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approaches in the literature that attempt to mitigate excessive WTP estimates viasurvey design, calibration of responses, and by combining revealed and statedpreference data. Modifications to survey design constitute a significant body ofresearch. However, it is apparent that the cost of conducting a survey that satisfiesthe many requirements needed to avoid the cause of inflated WTP gives rise tosignificant costs of implementation. It is also the case that even when extreme careand attention are given to survey design, delivery and analysis, the inherent driversof inflated WTP need not be resolved. Delivering realistic WTP measuresundoubtedly requires appropriate survey design and implementation, but thesemay be complemented by improved statistical/modelling methods that are capable ofdealing with deliberate or accidental mis-reporting by respondents. A framework fordoing this is therefore suggested in section III.

There exist several non-market studies of consumer WTP to reduce pesticide use.Travisi, Florax and Nijkamp (2004) provide a meta-analysis of this literature. Theyobserve that the justification used to motivate non-market valuation of pesticides iscomplex. In addition, there are significant differences in the magnitude and scope ofthe WTP estimates generated. For example, they find a mean WTP of US$ 262 perannum (year 2000) to reduce the impact of pesticides on farmers, US$ 289 perannum to reduce the impact on the aquatic environment, US$ 246 per annum toreduce the impact on terrestrial ecosystems, and US$ 42 per annum to reduceimpacts on consumers’ health. In summary, previous estimates of WTP to avoidpesticides have been large as well as diverse.

Within this literature, there is only one study that estimates WTP for thereduction of pesticide use in the UK. Foster and Mourato (2000) estimated WTPby employing a contingent ranking methodology. They focused on very specificimpacts of pesticide use: biodiversity proxied by declining farmland bird species;and the impact on farmers’ health. Foster and Mourato acknowledge that thesetwo aspects of pesticide impact are only a subset of the many potential impactsof pesticide use. However, to include all potential impacts of pesticide use wouldincrease the complexity of the choice task faced by survey respondents. Theyestimated that UK consumers are WTP £1.15 (or 191% extra) for a green loaf ofbread in order to reduce the number of cases of ill health per year and thenumber of declining farmland bird species to zero. The results we provide in thispaper, therefore, add to a very small but important body of research in an area ofpolicy design and implementation that has proven complex as well as politicallysensitive.

III. The logit with potential mis-reporting

As with the standard conditional logit model, each individual (i) is assumed to deriveutility from a basket of goods x0ij (a 1 · k vector). The utility function is of the linearform

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Uij ¼ x0ijbi þ eij ð1Þ

where b0i are the preference parameters for the ith individual and eij is Gumbel-distributed. In addition, we observe z0i, a 1 · p vector of characteristics, for the ithindividual, and the preference parameters are conditioned on these characteristics asin equation (2):

bi ¼ ðIk � z0iÞh ð2Þwhere h is a pk · 1 vector of parameters. Accordingly, the utility function can beexpressed in terms of parameters which do not vary over individuals as inequation (3):

Uij ¼ Z 0ijhþ eij

Z 0ij ¼ x0ijðIk � z0iÞ:ð3Þ

If each individual is given the option of J choices, then the vector which records thechoices of the ith individual is:

yi ¼ ðyi1; yi2; . . . ; yiJ Þ ð4Þwhere yij ¼ 1 if the ith individual makes the jth choice (Uij > Uik for all k 6¼ j) andzero otherwise. Under the assumptions above:

Prðyij ¼ 1Þ ¼ eZ0ijh

ðPJ

j¼1 eZ 0ijhÞ

0@

1A ¼ pij: ð5Þ

The associated likelihood, where n individual choice are observed, is therefore:1

L0ðhÞ ¼Yn

i¼1

YJ

j¼1ðpijÞyij : ð6Þ

Equations (5) and (6) together specify the standard logit model.

The UPMR logit

This model can be modified by introducing the probability that an individual mis-reports their preferences. Denote the indicator variable

vi ¼ 1 if individual i correctly reports their preferences0 otherwise.

nDefining the probability of correct reporting as:

Prðvi ¼ 1Þ ¼ p; ð7Þthe probability of the ith individual choosing the jth option is therefore:

Prðyij ¼ 1Þ ¼ Prðyij ¼1 j vi ¼1Þpþ Prðyij ¼ 1 j vi ¼ 0Þð1� pÞ: ð8Þ

1Obviously, this logistic probability could be replaced by some other depending on the underlyingassumptions about the distributions of eij.

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Further assigning a probability that the ith individual will mis-report in favour of thejth option (if they mis-report) as

Prðyij ¼1 j vi ¼ 0Þ ¼ kj ð9Þwhere XJ

j¼1kj ¼ 1: ð10Þ

We can define

f ð1Þij ¼ Prðyij ¼ 1Þ ¼ pijpþ kjð1� pÞ ð11Þ

and the associated log-likelihood is:

lnðL1ðH1ÞÞ ¼Xn

i¼1

XJ

j¼1yij lnðpijpþ kjð1� pÞÞ

!¼Xn

i¼1

XJ

j¼1yij ln f ð1Þij

!: ð12Þ

The adjustment of the likelihood as in equation (12) based on the definitions inequations (7) to (11) define what we term to be the UPMR logit. The parameter setfor the UPMR logit is:

H1 ¼ ðh0; p;K0Þ where K0 ¼ ðk1; . . . ; kJ Þ: ð13ÞFor priors, we use the special case of the conjugate priors (for the standard logit) inKoop and Poirier (1993):

fhðh j r; ZÞ ¼Yn

i¼1

XJ

j¼1expð _Z 0ijhÞ

" #�r

_Z0ij ¼ ðZ 0ij � �Z 0:jÞ

ð14Þ

becoming neutral (non-informative) if r is a small number.2 The priors for the otherparameters (p, K0) are, in this paper, taken to be uniform over the interval (0, 1).3

Therefore, denoting fU(p; a, b) to be a uniform prior on the interval [a, b]:

fpðpÞ ¼ fUðp; 0; 1Þ and fKðKÞ ¼YJ�1j¼1

fUðkj; 0; 1Þ: ð15Þ

The posterior distribution f1(Q1 | Y), (Q1 ¼ (h0, p, K0)), for the UPMR logit istherefore proportional to h1(Q1 | Y)

f1ðH1 j Y Þ / h1ðH1 j Y Þ ¼ L1ðH1Þ � fhðh j r; ZÞ � fKðKÞ � fpðpÞ: ð16ÞThe first two components on the right-hand side are (after dividing by the integrationconstant) the posterior of the standard logit, and could be mapped using importance

2For details on the value of r employed herein, see our discussion later.3We take these uniform priors to be non-informative in the current context. Other priors are possible of

course. However, we do not explore them in this paper.

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sampling. However, with the introduction of the second components, the priors areno longer conjugate. Nevertheless, they can be simulated using the M–H algorithm.4

The UPMR model requires the assumption that people mis-report in a way thatdoes not depend on their preferences for particular options. The plausibility of thisassumption can justified in at least two ways. First, people may simply reply infavour of one option without truly assessing their preference (perhaps because oftime constraints or the inability to understand the choices). Alternatively, they mightassess which option they prefer, but their preferences have no bearing on how theymis-report (perhaps because of yea-saying and strategic responses).

CPMR logit

In the simple binomial choice case, the UPMR is similar to that introduced byHausman et al. (1998), used by Caudill and Mixon (2005), which we termthe CPMR logit. The Hausman specification (in the binomial case) has twoparameters p1|2, the probability of mis-classifying choice one as choice two, and theprobability of the counter-case p2|1. Here, we have an unconditional probability ofmis-reporting p, and then a ‘directional’ set of mis-reporting probabilities K. Thoughthe CPMR can be generalized beyond the binomial case, it rapidly increases thenumber of parameters required, with a multinomial logit of order J requiring J(J ) 1)(non-redundant) conditional probabilities to be specified. The UPMR approachoutlined in this paper extends more naturally into the multinomial choice case witheach additional choice requiring the addition of one further parameter (as it requiresthe specification of J non-redundant probabilities).

That the CPMR amd UPMR logits are not equivalent can be seen in thefollowing. Let y� relate to the reported choice, and y to the observed choice foran individual (dropping the subscript ‘i’ for the moment). Define an event,conditional on a set of (exogenous) variables x, with the probabilities

PkjðxÞ ¼ Prðy� ¼ k; y ¼ jÞ

Pk�ðxÞ ¼ Prðy� ¼ kÞ ¼X

j

PkjðxÞ ð17Þ

and

P�jðxÞ ¼ Prðy ¼ jÞ ¼X

k

PkjðxÞ:

Defining the conditional probabilities:

pkjjðxÞ ¼ Prðy� ¼ k j y ¼ jÞ ¼ PkjðxÞP�jðxÞ

; ð18Þ

4For a general discussion of the M–H algorithm, readers are again referred to Chib and Greenberg (1995).Details of the algorithm used are given in section IV.

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the (generalized) CPMR approach has the parameter sets P (the conditionalprobabilities {pi|j}) and, as before, the logit parameters h. Defining Q2 ¼ (h, P), thelikelihood is:

L2ðH2Þ ¼Xn

i¼1

XJ

j¼1yij ln f ð2Þij

!

f ð2Þij ðH2Þ ¼ pjjjpij þXk 6¼j

pjjkpik

! ð19Þ

subject to the summation conditionsPJ

k¼1 pkjj ¼ 1. The parameters in the CPMRlogit can be compared with the UPMR parameters defined in equations (11) and (12)[allowing them to vary with (x)] which are:

pðxÞ ¼X

k

pkjkðxÞP�kðxÞ

kkðxÞ ¼P

j 6¼k pkjjðxÞP�jðxÞPk

Pj 6¼k pkjjðxÞP�jðxÞ

:

ð20Þ

As the values of P•i(x) vary, one cannot simultaneously fix the pi|j(x) along with p(x)and ki(x). Whereas the CPMR model fixes the conditional probabilities pi|j(x) ¼ pi|j,the UPMR logit fixes p(x) ¼ p and ki(x) ¼ ki. In the multinomial case, the UPMRlogit has fewer parameters and delivers results which, we would argue, are more easilyinterpreted. The CPMR may appear to be ‘richer’ because it arguably embodies theidea that people will mis-report, depending on their preferences for the options thatthey are presented with. Thus, the ‘type’ of individual can determine the nature of theirmis-reporting. However, importantly, the CPMR cannot can be viewed as moregeneral than the UPMR (or vice versa) because, as we show in this section, neithermodel nests the other. The more parsimonious nature of the UPMR in the multinomialcase is not an unambiguously good property. However, parsimony does bringadvantages in terms of computational practicality. Ultimately, discriminating betweenthe models requires comparing the models’ performance, and this can be conducted ina Bayesian manner through the computation of the Bayes factors.

In order to estimate the CPMR logit, the priors in Koop and Poirier (1993) canagain be specified for h. For the priors on the conditional mis-reporting probabilities,uniform priors are again specified as:

fPðPÞ ¼YJðJ�1Þ

j¼1fUðPj; 0; 1Þ ð21Þ

where P is minimal set of conditional probabilities with redundant probabilities(implied by the summation conditions) removed. The CPMR logit therefore has theposterior [at a point Q2 ¼ (h0, P)]

f2ðH2 j Y Þ / h2ðH2 j Y Þ ¼ L2ðH2Þ � fhðh j r; ZÞ � fPðPÞ: ð22Þ

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Identification issues

Within a given model, a parameter is said to be non-identified if the data are notinformative about its value. If a model has many parameters, then a given parametermay not be identified when other parameters take particular values. The problem ofnon-identification has received considerable attention from the classical perspective.Ramalho (2002, p. 17) sets out the generalized conditions for identification for theextended CPMR logit. The concern of classical statisticians with models that containpotentially non-identified parameters is not only because of the fact the data may notbe informative about the value of a parameter, but also because non-identificationcan complicate classical methods of estimation and inference. Using maximumlikelihood, potential non-identification is highly problematic because of the possiblesingularity of the likelihoods’ Hessian matrix and multiple maxima of the likelihoodfunction. Non-identification can prevent finding a maximum, and can preclude theuse of the inverse Hessian in producing standard errors for parameters.

Equally, the UPMR logit also has identification conditions. In this paper, we donot attempt to formally outline the identification conditions within our model. This isbecause non-identification does not have quite the same implications when usingBayesian methods. It is not our purpose to suggest that non-identification ofparameters is somehow irrelevant to Bayesian analysis. Clearly, if the data arenon-informative about a given set of parameters, their posterior distributions willonly reflect the nature of the prior distributions. Thus, in Bayesian analysis,non-identification of a given set of parameters will result in a failure of the posteriorsto diverge from the priors. However, using diffuse priors, the diffuse nature of theresulting posteriors for parameters which are not identified accurately reflectsthe non-informative nature of the data. It is equally important to recognize that theexistence of a well-behaved posterior distribution does not hinge on the existence ofa unique maximum point for the likelihood function, the existence of first or secondderivatives for that likelihood, or the non-singularity of the Hessian. Moreover, themethod of construction or interpretation of Bayes factors is not affected byidentification issues.

The use of any proper prior, however diffuse, inevitably imposes priorinformation on the model. However, prior information can be extremely ‘weak’ orresult from conditions that are intrinsic to the construction of the model. Thus, for anon-identified parameter a diffuse prior will lead to a diffuse posterior, correctlyreflecting the non-informative nature of the data. For example, within our model, ifp ¼ 1 (no mis-reporting), then the parameters K are not identified. The likelihoodwill be flat over all values of {ki} lying within the unit intervals [0, 1]. However,estimating the model in the Bayesian framework only requires prior information thatthese probabilities lie between zero and one. Should the ‘true’ data generatingprocess involve no-mis-reporting (p ¼ 1), then the resulting estimates of theposterior distributions of ki will be correspondingly ‘flat’ over the interval 0 and 1,when using uniform priors. Thus non-identification in no way precludes estimation

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and the posterior distributions will correctly reflect the fact that we know little ornothing about the nature of mis-reporting, should p be equal to one. Within aBayesian framework, we ‘test’ the assumption that p ¼ 1 by estimating the standard(non-mis-reporting model) and construct the Bayes factors for the mis-reportingmodel and standard model.

Likewise, if p ¼ 0, the coefficients h will not be identified because this impliesthat respondents ignore all values of attributes in the choice sets when (mis)-reporting their preferences. In this circumstance, a wide set of values of h will beequally compatible with the data (in the sense that the posterior will be relatively flatover the domain of h). Again, diffuse information is used in the setting of the priorsof Koop and Poirier (1993). However, when this prior is set diffusely (as in thispaper), the resulting estimates of the h parameters will be accordingly diffuse if p iszero, correctly reflecting that we have learned little or nothing about the values of hfrom the data.

Points of clarification

A few points are worth noting with regard to the specifications above that relate tothe nature of CM studies. It is important to note the fact that in many survey designsthere is nothing ‘intrinsic’ about the options in a choice set, and no consistentordering of the options (j ¼ 1, 2, . . . , k), as they are offered to respondents.Importantly, this does not undermine the rationale for employing the modelsdescribed for the following reasons:

• The models above nest the standard logit (setting p ¼ 1, or p1|2 ¼ p2|1 ¼ 0).However, these models do not require that people mis-report, or imply thatpeople will have a tendency to mis-report in any particular direction. Forexample, if k1 ¼ k2 ¼ � � � ¼ kJ, then mis-reporting is equally probable overeach of the alternatives. Generally, we would expect the direction of mis-reporting to be highly ‘asymmetric’ in the sense that people do not have thesame tendency to mis-report in favour of each of the options. Within thecontingent valuation literature it has been demonstrated that various types ofasymmetry exist such as ‘framing effects’ whereby equivalent choices arestated differently leading to quite different responses on the part of individuals(Tversky and Kahneman, 1981). Likewise, ‘status quo’ bias and strategicbiases that are shared by a significant proportion of the population are likely toinduce ‘asymmetry’ in mis-reporting. In the CPMR model, there would be noreason to expect that people who actually prefer option 1, mis-report to thesame degree as those who prefer option 2, etc. In the UPMR model we wouldnot expect values of k to be equal (in a sense, a symmetrical mis-reportingresponse). A symmetric outcome would reflect an almost random tendency tomis-report. Such an outcome would be consistent with the case where asubgroup of respondents could not either understand or would not invest the

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time to understand the choices and therefore selected options with equalprobability. Alternatively, different groups might respond strategically by mis-reporting in a way that coincidentally ‘cancels out’ the other group leading toapproximately the same level of mis-reporting for different options. Asym-metry in the type of mis-reporting (in either model) would be in accordancewith the idea that people generally identified either one of the options as beingthe right, or easiest option to choose, even though it was not in accordancewith their preferences.

• It is possible that something as simple as the sequence in which the options arepresented to the respondent will determine the direction of their mis-reporting.However, it is not the ordering of the options as they are presented to respondentsthat (probably) matters. Pre-estimation re-ordering of options can be performedto give meaning to the sequence. Some survey designs do have an intrinsicquality to some of the options, and also an ordering. For example, it is commonpractice inmany CM studies to include a common ‘status quo’ option as the firstoption (Bennett andBlamey, 2001).However, from an econometric perspective,there is nothing that requires a common ‘status quo’ option to come first orappear at a consistent point in the option-ordering within each choice set,because the options can be re-ordered so as to make the common option alwaysappear as the jth option, prior to estimation.

• The mis-reporting framework does not require that there are common options(e.g. the status quo) appearing in all choice sets. If respondents identifyelements in some of the options that make them predisposed to mis-report infavour of that option, and if pre-estimation re-ordering of the options reflectsthat ordering, the model will identify mis-reporting and the direction of thatmis-reporting. For example, if respondents tended to mis-report in favour ofthe option with the lowest (highest) payment, and if pre-estimation pre-ordering of options was done according the level of payment option, then thedirection of mis-reporting would be identified. Another example is in studieswhere one or more of the attributes is environmental. In such cases, mis-reporting may be in favour of the option which has the best (or worst)environmental outcome. It is in these circumstances that upwardly (down-wardly) biased estimates of WTP will arise.5 An appropriate pre-estimationre-ordering of the options according to the environmental outcome wouldenable the framework above to identify the existence and direction of the mis-reporting.

• Pre-estimation re-ordering may be performed in multiple ways. Indeed,multiple estimation with multiple re-orderings may be a useful way to test therobustness of the results.

5We make this claim on the basis of Monte Carlo studies performed using maximum likelihood estimation,though we do not report those results here. Of course, the issue of bias is generally recognized (e.g. inHausman et al., 1998).

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Finally, we reiterate that we are not advocating that mis-reporting models arean alternative to good experimental designs that aim to identify uncertain choicesor, through the use of multiple choice sets, aim to identify inconsistent choices.What we imply is that they are tools to be employed in conjunction withinnovative survey design and delivery.

Model assessment and Bayes factors

The evidence in favour of mis-reporting can be partially assessed by examiningthe posterior distributions of p and K or alternatively, the p0ijjs. For the UPMRmodel, if there is no mis-reporting, then the mass of the posterior of p will, for areasonably sized sample, be concentrated towards one, and the posteriors of Kwill tend to be uniformly distributed over the unit interval (because where p ¼ 1,K is not identified). For the CPMR model, the posteriors for each of the p0ijjswould be concentrated towards zero if no-misreporting was taking place.However, in order to properly test and compare the models, Bayes factors can beconstructed in a straightforward manner, using the methods in Chib and Jeliazkov(2001).

If all the parameters are treated as one block, then the marginal likelihood of themodel can be estimated without estimation of the integration constants, using

ln miðyÞ ¼ ln hiðHi j Y Þ � ln fiðHi j Y Þ ð23Þ

evaluated at a high density point H, such as the mean of the posterior. Using aone-block M–H algorithm, the second component (ln fi(Q | Y)) can be estimatedusing equation (9) of Chib and Jeliazkov (2001). Herein, we are interested in thetwo models. The standard logit model, which we will denote as model zero, hasthe posterior proportional to:

f0ðH0 j Y Þ / h0ðH0 j Y Þ ¼ L0ðH0Þ � fhðh0 j r;ZÞ ð24Þ

where H0 ¼ ðh00; 1; 00Þ and h00 is the estimated mean of the posterior when usingmodel 0 (the standard model), whereby for the UPMR and CPMR models:

f1ðH1 j Y Þ / h1ðH1 j Y Þ ¼ L1ðH1Þ � fhðh1 j r; ZÞ � fKðK1Þ � fpðp1Þ ð25Þ

and

f2ðH2 j Y Þ / h1ðH2 j Y Þ ¼ L2ðH2Þ � fhðh2 j r; ZÞ � fPðP2Þ

where H1 ¼ ðh01; p1; K01Þ or H2 ¼ ðh02; P02Þ are the estimated posterior mean

parameter vectors for the UPMR and CPMR models, respectively. Noting thatbecause the two models contain the same priors on h with the same integratingconstant c, and uniform priors on, p, K and P, the model probabilities can becomputed up to (plus or minus) the same constant:

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ln m2ðyÞ ¼ ln cþ ln h2ðH2 j Y Þ � ln f2ðH2 j Y Þln m1ðyÞ ¼ ln cþ ln h1ðH1 j Y Þ � ln f1ðH1 j Y Þln m0ðyÞ ¼ ln cþ ln h0ðH0 j Y Þ � ln f0ðH0 j Y Þ

ð26Þ

where by ln f2ðH2 j Y Þ, ln f1ðH1 j Y Þ and ln f0ðH0 j Y Þ are simulated at therespective posterior means H2, H1 and H0 (as estimated in the M–H algorithm) as inequation (9) of Chib and Jeliazkov (2001),6 and c is redundant from the point ofview of comparison. Consequently, the Bayes factor is then estimated as (for anyvalue of c)

BFij ¼ expðln miðyÞ � ln mjðyÞÞ: ð27Þ

IV. Empirical section

The survey

Our CM experiment was conducted as part of larger survey designed to establishsocially acceptable levels of pesticide impacts on non-target species, and ultimately,environmentally acceptable concentrations (EACs) of pesticide applications. Thesurvey was conducted by MORI. The survey polled a representative sample of 2,049adults, aged 15 years or more, at 201 sampling points across Great Britain. Indesigning and testing the survey, we held a number of focus groups and well asreceived valuable input from MORI.

The CM experiment was designed to yield monetary values (WTP estimates) forconsuming a basket of food items (i.e. bread, meat, milk, fruits and vegetables)produced using a non-pesticide technology. The payment vehicle chosen for thisexperiment was the weekly purchase of the basket of food items. The NationalOceanic and Atmosphere Administration (NOAA) panel (Arrow et al., 1993)recommended that reminders of substitute goods and budget constraints be includedwithin a survey instrument. Our CM experiment explicitly included an alternativeoption in the form of leisure goods as well as a hypothetical budget constraint.

Specifically, survey participants faced two options. Buy the basket of goods onwhich pesticides have been used at a constant price of £12 per week. Alternatively,buy another basket of goods labelled ‘No Pesticides Have Been Used’. This basketof goods was offered at three different prices (£14, £17 or £23 per week) that whererandomized over participants. The survey participants then indicated which optionthey preferred.

6In this paper, we used a random-walk M–H algorithm with a normal innovation. In estimating theposterior for the Bayes factor, we also used a normal candidate density, with a variances based on the varianceestimated from the M–H output. The variance was inflated and reduced around this level to ensure that theestimates were robust.

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The hypothetical budget constraint made the trade-off faced explicit when optingto purchase the non-pesticide food. We randomized the available budget at £30 and£50 per week. In this CM design, participants cannot violate their budget constraintsdirectly. Thus, if the resulting estimates of WTP conform to these budget constraintsthen, theoretically, we should see WTP measures (as estimated from the logit) at amaximum of £18 for the £30 budget group and £38 for the £50 budget group.Otherwise, these will exceed the maximum additional payments that could beafforded over and above the £12 paid for the standard basket of goods. In addition, tothe choice data we also collected individual-specific socioeconomic data on genderand age (less than or greater than 45). We also asked individuals to indicate if theyconsidered themselves to be price-sensitive, food safety-sensitive or environmentalsafety-sensitive.

Importantly, our survey design is simple in that the choice sets do not link thechoice of food production technology with any of the associated negative impacts ofpesticides. We employed this design because of the significant scientific uncertaintythat surrounds the impact of pesticides which makes the provision of concise andunambiguous information to survey participants almost impossible. We concludedthat it is inappropriate to include (potentially inaccurate) information in choice sets.As a result, our WTP estimates reflect public perception of pesticides as opposed topotentially value-laden information included as part of the survey. It was also thecase that MORI imposed limits on the size and cognitive difficulty of the questionswe asked survey participants which meant that we had to keep our CM experimentshort and to the point.

Issues in estimation and inference

As outlined in previous sections, an M–H algorithm is employed to simulate theposterior distributions of the standard and mis-reporting logit models outlined insection III. The M–H algorithm used a random-walk step for the parameters withnormal innovations and a common variance across the innovations. The variancewas calibrated so as to give an acceptance rate of around 25%. Prior to estimating themodel using the survey data, we tested our procedures by using Monte Carlo-generated data, with various degrees of mis-reporting (including none at all). Wefound that with Monte Carlo data, the procedures worked extremely well, clearlyidentifying if mis-reporting was taking place, and accurate estimates of p and Kwhere mis-reporting was occurring.

When conducting the empirical study, two inequality conditions were placed onthe parameters of both models. The first required that the monetary coefficient wasthe correct sign (positive as we entered payment as a negative payment). Next, itshould be recalled that the maximum WTP estimates that are in accordance with thestated budget constraints given to consumers are £18 and £38 pounds for the lower(£30) and upper (£50) budget groups, respectively. When estimating the models, weplaced bounds on the WTP of £50 and )£50, which greatly exceeded the budget

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constraints given to respondents. As WTP is calculated by dividing the parameters ofother attributes by the payment coefficient, small payment coefficients can inducevery large WTP estimates which, we would argue, are not a priori credible.Furthermore, the moments of the WTP distribution will not exist. Other methods canbe employed to deal with this problem (e.g. by the transformation of the coefficientinto one which is also positive, or by setting some lower positive bound on thepayment coefficient). However, a priori, we do not see that these are preferential tothe truncation of the WTP distribution. Both sets of inequality constraints wereenforced within the M–H algorithm by specifying them as additional conditions forthe acceptance of a step.

With regard to the hyper-parameters used in the prior distributions (r) weexperimented, by setting r to 0, 10)3, 10)2 and 10)1 (when using the pesticide data).However, we only produce the results for the most diffuse, but non-zero, prior(r ¼ 10)3). We found that the results (in terms of WTP, p and k parameters) were notsubstantially affected by choosing between r ¼ 0, 10)3 or 10)2. However, adoptingr ¼ 10)1 had a large impact on our results. Although 10)3 delivered posteriorestimates that were very close to maximum likelihood estimates (in the standard logitformulation), setting r ¼ 10)1 caused a substantive downward revision in ourparameter estimates (and an upward revision in WTP). When using this informativeprior, the differences between the standard and generalized models became smaller,with the parameter p becoming closer to one. In addition, whereas the Bayes factorsfor the non-informative priors strongly favoured the general model, the Bayes factorscalculated using the informative prior actually (slightly) favoured the standardmodel. On the other hand, using r ¼ 10)3 seemed to give better (faster) convergencein the M–H algorithms than when using r ¼ 0.

Convergence was monitored as follows. First, we noted the degree of serialcorrelation in the sampled parameters. Although, as noted above, the sampler wascalibrated so that it had about a 25% acceptance rate, the degree of dependence(which we estimated using the first-order autocorrelations for each of the parameters)was very high. This was broadly consistent with the performance of the samplerwhen using Monte Carlo data. However, we found that the degree of dependencewhen using our survey data was more acute. Accordingly, we recorded only every1,000th draw from the sampler. Even so, we found that serial correlation between therecorded draws, though less (with first order autocorrelation coefficients of around0.1), was significant perhaps up to an order of 16 or more. Therefore, we used a verylarge burn-in of 106 draws, with a further 20 · 106 draws (of which we sampled20,000) which we used to map the posterior distributions.

Secondly, we conducted t-tests for the difference between the mean of the firstand second half of the sample values by resampling every 20th (approximatelyindependent) draw from the first and the second 10,000 recorded values. We alsoconducted t-tests and F-tests between the means of these sub-samples (each of 500).These were (all) insignificantly different at the 5% level. The overall mean values forthe entire first and second sub-samples were very similar (to two decimal places). We

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also experimented with initializing the sampler from different points in smaller runs,with no significant differences in the final results.

Finally, the posterior distributions for the WTP for each of the groupings, wasproduced by simulation after the M–H algorithm ended. Prior to the estimation of thedistributions using kernal density methods, each of the simulated parameters {hg}from the sampler were drawn along with a random draw from the population ofvectors of characteristics {zi} in order to produce fbg

i g as in equation (2), subjectto the requirement that the vector of characteristics conformed to a prespecifiedcondition (e.g. that it should be male or female, etc.). Accordingly, the resultingposteriors will reflect the underlying frequencies with which other characteristicsoccurred within our sample. For the budget, age and gender characteristics, thesewere approximately 50%. For the rest, approximately 63% of people declaredthemselves to be food safety-conscious, around 8% declared themselves to beenvironmental safety-conscious, with the remaining 29% declaring themselves to beprice-conscious.

Results

The results for each of the models are presented in this section. We do not discuss, indetail, the parameters (h) for each model (the mean and standard deviations of whichare presented in Table 2) because, in themselves, they are not of particular interest. Itis only the ratio of these parameters which are of interest to us here because theseproduce the WTP estimates which are our main concern. With regard to theseparameters (h), we only note that the two mis-reporting models have similar valuesthat differ sharply from the standard logit.

Table 1 and Figures 1 and 2 present the WTP results for the standard and themis-reporting models. Table 1 presents the posterior mean and standard deviationsfor WTP for both models by various groups (24 groups given our survey design)that result from the specific socioeconomic data. The WTPs, by group, have beenrank-ordered according to the estimates delivered by the standard logit. Plots ofkernal density estimates for posterior distributions of WTP, by group, arepresented for the standard logit in Figure 1a–d, the UPMR model in Figure 1e–h,and for the CPMR model in Figure 1i–l. The results in Table 1 give thenumerical values that underpin the bar graph in Figure 2e.

As can be seen from Table 1 and Figure 2, WTP estimates for the three modelsare reasonably consistent in terms of rankings. The WTP for male, young and price-sensitive respondents under a budget of £30 per week ranked the lowest, while thosefor female, old and environmentally sensitive ranked the highest.

There are several important points in our results that merit further comment. First,we observe that the WTP estimates from the mis-reporting logits are considerablydownsized compared with the standard logit model by an average factor of morethan 30%. Secondly, the WTP for the UPMR model is very similar to that of theCPMR model, with the CPMR model delivering slightly smaller WTPs. Thirdly,

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respondents were, in the survey, randomly assigned two notional budgets. Given thatthe WTP measures in Table 1 purport to be the additional amounts that people areWTP for the non-pesticide budget (over and above £12), it is evident that thestandard logit model generates WTP measures that grossly exceed the budgetconstraint for groups 15, 17, 19, 21 and 23 (which have maximum possible WTPs of£18). It is also clear from Table 1 that the generalized models lead to less violationsof the budget constraint in comparison with the standard model, and these violationsremain minimal (groups 19 and 23 have estimates of just over £18 for the UPMR

TABLE 1

Willingness-to-pay estimates by group

Group Budget Gender Age Sens

Standard UPMR CPMR

Mean SD Mean SD Mean SD

All — — — — 17.83 9.71 11.62 9.45 10.64 8.56— — — 18.67 9.67 13.13 9.91 12.07 8.90— — — 16.99 9.67 9.94 8.62 8.96 7.78

— Female — — 21.95 9.75 15.02 9.83 13.77 8.76— Male — — 13.59 7.67 7.84 7.20 7.26 6.77— — Old — 19.75 9.01 13.88 9.08 12.75 8.17— — Young — 15.92 10.03 9.21 9.20 8.15 8.26— — — Food 22.84 7.68 15.60 7.79 14.38 6.69— — — Env 18.85 5.89 16.72 7.00 15.66 5.94— — — Price 7.08 4.14 1.66 4.05 1.06 3.68

1 30 Male Young Price 2.04 2.48 )1.59 3.35 )2.06 3.192 50 Male Young Price 4.16 1.81 )0.12 3.28 )0.63 3.183 30 Male Old Price 5.27 1.63 0.64 3.06 0.07 2.924 50 Male Old Price 7.12 1.53 2.14 3.08 1.59 2.765 30 Female Young Price 7.58 2.26 1.39 3.75 0.75 3.146 50 Female Young Price 9.76 2.68 3.19 3.69 2.52 2.997 30 Female Old Price 10.86 3.20 3.94 3.99 3.12 3.118 50 Female Old Price 12.70 3.60 5.76 4.00 4.95 3.129 30 Male Young Env 15.13 4.30 11.06 4.46 10.26 3.7710 50 Male Young Food 15.70 4.67 7.64 4.60 6.70 3.9811 30 Male Young Env 16.19 4.51 13.67 5.02 12.96 4.3712 50 Male Old Env 16.46 4.26 13.25 4.61 12.49 3.8713 30 Male Old Env 17.44 4.48 15.84 5.21 15.17 4.5514 50 Male Young Food 17.69 5.24 11.02 4.71 10.22 4.2315 30 Male Old Food 18.07 4.61 10.94 4.45 10.07 3.7216 50 Male Old Food 19.78 5.05 14.39 4.84 13.66 4.2117 30 Female Young Env 19.81 6.20 16.38 6.64 15.11 5.3118 50 Female Young Env 20.79 6.27 19.47 7.15 18.19 5.8119 30 Female Old Env 20.97 6.06 18.59 6.81 17.34 5.5420 50 Female Old Env 21.89 6.19 21.69 7.37 20.40 6.0521 30 Female Young Food 24.88 7.37 14.99 6.99 13.43 5.4522 50 Female Young Food 26.53 7.30 19.51 7.27 17.93 5.8623 30 Female Old Food 26.62 6.94 18.34 7.25 16.76 5.7324 50 Female Old Food 28.11 7.03 22.90 7.46 21.21 5.99

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logit and are less than £18 for the CPMR logit).7 In percentage terms, the downsizingof WTP estimates is more pronounced under the £30 per week budget as opposed tothe £50 per week budget. In absolute terms, however, the reductions of the estimatedWTPs from the UPMR and CPMR logits relative to the standard logit are fairlysimilar across the board (between £5 and £8), except for environmental safety-sensitive respondents, which had only a small revision downward (a little over £2or £3). Fourthly, in both the standard and mis-reporting logits, there is a clearbimodality in the distributions according to budget, age and gender. It is also evidentthat from Figure 1d, h and i that this bimodality reflects very different behaviour bythose identifying themselves as price-sensitive as opposed to those identifyingthemselves as food safety- or environmental safety-sensitive. It can be argued thatthe bimodality displayed in the posterior WTP distributions across the budget,gender and age categories in all models (Figure 1a–c, 1e–g and 1i–k, respectively)can be captured in the posterior WTP distributions across the price/food safety/

TABLE 2

Mean and standard deviations of parameter distributions

Standard UPMR CPMR

Mean SD Mean SD Mean SD

Payment 0.97 0.30 2.35 0.85 2.48 0.79Payment * budget 0.03 0.23 )0.11 0.42 )0.11 0.42Payment * gender )0.11 0.24 )0.21 0.43 )0.17 0.43Payment * age 0.07 0.23 0.04 0.40 0.05 0.41Payment * food )0.19 0.27 )0.93 0.77 )1.03 0.76Payment * env 0.53 0.52 )0.36 0.88 )0.46 0.86Pesticide 0.25 0.22 )0.31 0.54 )0.42 0.53Pesticide * budget 0.19 0.18 0.32 0.31 0.35 0.32Pesticide * gender 0.38 0.18 0.57 0.32 0.61 0.33Pesticide * age 0.32 0.18 0.46 0.31 0.48 0.32Pesticide * food 0.89 0.20 1.34 0.49 1.37 0.50Pesticide * env 1.86 0.43 2.32 0.56 2.35 0.56p — — 0.60 0.12 — —k — — 0.79 0.14 — —p1|2 — — — — 0.36 0.08p2|1 — — — — 0.08 0.04Marginal likelihoods — )1,118.04 — )1,115.60 — )1,116.60Bayes factor relative tostandard logit

— — — 11.47 — 4.22

Bayes factor relative toUPMR

— 0.09 — — — 0.37

7We could have enforced the budget constraints more directly. Both models were estimated enforcing theseconstraints. However, unsurprisingly, the results for the standard model yielded posterior WTP distributionswhere the mass of the £30 budget group was concentrated at the upper end of the constraint (e.g. near £18),and a mean of close to £18. The generalized model estimates, however, remained fairly similar.

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environmental safety categories (Figure 1d, h and l). This suggests quite differentpatterns of behaviour and preferences between price-sensitive and other respondents.

Overall, the mis-reporting models resulted in a substantial downward revision ofWTP estimates that better conformed with theoretical budget constraints given torespondents. Moreover, the resulting WTP estimates from the mis-reporting modelsconform more closely to those previously reported in the literature (e.g. Foster andMourato, 2000). This aside, the resulting estimates for the generalized modelssuggested that certain groups such as older females, who classified themselves as

30

Male

Young

Old

Price

Food

Env

Female

50 50

50

30 30

Male Male

Female Female

Young Young

Old Old

Env Env

Price Price

Food Food

Standard logit D

ensi

tyD

ensi

tyD

ensi

tyD

ensi

tyUPMR logit CPMR logit

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)(l)

(k)

(j)

(i)

Figure 1. Posterior distributions for WTP by major group

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either food safety- or environmental safety-sensitive were willing to pay more than150% more for non-pesticide foods (an additional £18 pounds for non-pesticideversion of a £12 basket of goods). At the other end of the spectrum, the resultssuggested that young, male, price-sensitive respondents were not WTP anyadditional money for non-pesticide options, and perhaps even needed to becompensated. The average for all groups suggested that people were WTP around80–100% more for the non-pesticide option. The standard deviations for the WTPdistributions were high, however, giving us little confidence in using the posteriormeans as reliable point estimates.

(e)

(a) (b)

(d) (c)

UPMR mis-reporting parameters

CPMR mis-reporting parameters

WTP by group

Den

sity

Den

sity

Den

sity

–5.00

0.00

5.00

10.00

15.00

20.00

25.00

30.00

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Standard logit UPMR logit CPMR logit

Figure 2. Mean WTP by subgroups and mis-reporting parameters

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To assess the robustness of our model specifications and the level of supportfor mis-reporting, we calculated the Bayes factors (outlined in section III). TheUPMR is preferred to that standard logit given the Bayes factor of approximately11.5 (second bottom row of Table 2). This indicates a strong support for theUPMR logit compared with the standard logit. Likewise, the CPMR logit (Bayesfactor of 4.22) is also preferred to the standard logit, but is not preferred to theUPMR logit (with a Bayes factor of 0.37 at the bottom of Table 2). We wouldemphasize that these Bayes factors also constitute a Bayesian ‘test’ for theexistence of mis-reporting. Thus, the support of both the UPMR and the CPMRmodels over the standard model suggests that the data support the existence ofmis-reporting.

Estimates for the mis-reporting parameters are presented in the centre rows ofTable 2, and their posterior distributions are in Figures 2a–d. For the UPMR, meanvalues for p (0.59) and k (0.79) indicate that a significant proportion of respondentsmis-report their preferences. This concurs with our own and other researchers’ beliefsthat, respondents in stated preference surveys are likely to exaggerate their WTP.According to these results, only 59% of respondents reported their preferencesaccurately. In the majority of these cases, mis-reporting was in the expected direction,i.e. 79% people stating that they would choose the ‘no pesticide’ food in a real market,given that they are mis-reporting. According the CPMR, the values for p1|2 (0.36) andp2|1 (0.08) indicate that around 36% of respondents who prefer the pesticide optionfalsely reported that they preferred the ‘no pesticide’ option. However, only 8% ofthose who preferred the pesticide option mis-reported in favour of the alternative.

The findings of the CPMR and UPMR model with regard to mis-reportinggenerally agree and they both suggest that the direction of mis-reporting shows ahigh degree of asymmetry. The UPMR suggests that around 59% of people reportcorrectly. The CPMR suggests that 64% report correctly if they like the pesticideoption, but 92% of respondents report correctly if they prefer the non-pesticideoption. Therefore, UPMR estimates a larger proportion of respondents to be mis-reporting than the CPMR. However, in examining the posterior distributions inFigure 2, it is evident that the posterior for correct reporting (p) in the UPMR model(Window a of Figure 2) has more than a 25% of its mass above 0.75 (75%)suggesting that the results from the two models do not substantively disagree. Bothmodels clearly suggest that the tendency to mis-report is in favour of the non-pesticide option. In the UPMR this asymmetry is embodied in the value of lambda,which indicates that 79% or all mis-reporting is in favour of the non-pesticide option.In the CPMR, this asymmetry is embodied in the fact that the proportion mis-reporting in favour of the non-pesticide option (36%) is a much larger proportionthan that mis-reporting in favour of the pesticide option (8%). Thus, the resultsconsistently suggest that the majority of people mis-reported in favour of the ‘nopesticide’ option. It is also the case that our WTP estimates and posteriordistributions segregate better in the mis-reporting logit models than in the standardlogit model.

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Finally, some qualifying remarks concerning our results are warranted. Werecognize that the logit framework is restrictive, particularly with regard to thewell-recognized assumption of the ‘independence or irrelevant alternatives’ (seep. 109, McFadden, 1974) which underpins all the estimated models. Therefore, werecognize that a richer model structure may deal with (what we perceive to be) theproblems of overvalued WTP estimates for non-pesticide food, without the need toemploy the idea that people are mis-reporting. Likewise, the finding of mis-reportingmay not be evident in models which are less restrictive in other ways. However, insupport of our approach, we would argue that the idea that people report in anuncertain manner, and that some may reply in a deliberately misleading way, ishardly controversial. The real question is how to inhibit mis-reporting, or to mitigateits effects. We would argue that any models which do not attempt to deal with mis-reporting are open to the challenge of being unrealistic. Moreover, as part of ourwork, but which we have not presented here, we have also examined other modelssuch as the mixed logit (or random-parameter logit) using both classical and Bayesianmethods. We found that these other generalizations did not significantly downwardlyrevise the WTP estimates, unless explicit restrictions on WTP measures wereimposed. Therefore, we believe that there is some circumstantial evidence that thefinding of mis-reporting is not an artifact of the simple logit structure which we haveemployed. These comments aside, similar approaches, as to the ones being employedhere, may be embedded into ‘richer’ models and this might be explored in the future.

V. Conclusions

This paper presents a modification of the standard logit approach that deals withpotential mis-reporting of respondents in stated preference CM studies in a mannersimilar to Hausman et al. (1998). It is not our intention to offer this model as apanacea for mis-reporting by respondents in stated preference CM. However, webelieve that in conjunction with other models/methods and careful survey design, theintegration of mis-reporting parameters into models may increase the credibility ofWTP studies. We have, in this paper, offered several suggestions about how theframework suggested here could be developed in the future. However, one furthersuggestion might be to examine the integration of survey information regardingresponse uncertainty into the current framework, and therefore modelling hetero-geneity in the parameters relating to mis-reporting.

A Bayesian methodology was described and employed for estimation.We considerthat this offered valuable advantages over classical estimation, particularly in view ofthe potential lack of identification of some of the parameters in the mis-reportingmodels. When applying the UPMR and CPMR logits to CM data for non-pesticidefoods, Bayes factors preferred the use of the UPMR logit developed in this paper to theCPMR logit of Hausman et al. (1998). The findings suggested that around 40% ofrespondents were mis-reporting their preferences, with around 80% of mis-reportingbeing in favour of the non-pesticide basket over a standard basket of goods. In practical

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terms, both the mis-reporting models delivered similar results. Their applicationresulted in a considerable downsizing ofWTP estimates, which nevertheless suggestedthat most consumers have high WTP for non-pesticide foods.

Final Manuscript Received: February 2006

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