A Simple Method for Bounding Willingness to Pay Using a Probit or Logit Model. August 13, 1997

Transcription

1 A Smple Method for Boundng Wllngness to Pay Usng a Probt or Logt Model August 3, 997 Tmothy C. Haab Department of Economcs East Carolna Unversty Greenvlle, NC (99) haabt@mal.ecu.edu Kenneth E. McConnell Department of Agrcultural and Resource Economcs Unversty of Maryland College Park, MD (30) tedm@arec.umd.edu Abstract The purpose of ths note s to present a workable model of referendum responses that mposes upper and lower bounds on wllngness to pay, and can be estmated by canned programs such as SAS or LIMDEP wthout further programmng work. The framework descrbed here allows the researcher to ncorporate bounds on wllngness to pay usng ordnary logt or probt models. The model takes the focus off of fnessng the dstrbuton of wllngness to pay and nstead allows the researcher to focus on the nformaton contaned n the referendum responses. In contrast to some of the more complcated models, the model has a smple closed form soluton for medan wllngness to pay and subsequently for covarate effects on medan wllngness to pay. A straghtforward approxmaton for mean wllngness to pay s also presented.

2 Introducton Recent research provdes strong support for modelng dscrete responses to contngent valuaton questons wth reasonable bounds on wllngness to pay. Whle the model of utlty dfferences wth lnear utlty functon and addtve error s ntutvely appealng, t typcally mples that wllngness to pay les between plus and mnus nfnty. Yet n most cases wllngness to pay s bounded below by zero and above by ncome. Incorporatng these bounds leads to new estmators for mean wllngness to pay and the effects of covarates on mean wllngness to pay. A smple and parsmonous way of estmatng mean wllngness to pay s to use the Turnbull expected lower bound suggested by Carson et al. (see also Haab and McConnell (997); Krstrom has a smlar estmator). The Turnbull, however, does not lend tself to the easy estmaton of the effects of covarates. Instead, one must turn to more complcated estmaton of models such as the beta or pnched logt (Haab and McConnell forthcomng, Ready and Hu), and use often complcated maxmum lkelhood programs. The purpose of ths note s to present a workable model that mposes natural upper and lower bounds on wllngness to pay, and can be estmated by canned programs such as SAS or LIMDEP wthout further programmng work. The advantages of the approach are clear: t allows the researcher to ncorporate bounds usng ordnary logt or probt models. Further, the model takes the focus off of fnessng the dstrbuton of wllngness to pay and nstead allows the researcher to focus on the nformaton contaned n the referendum responses. In contrast to some of the more complcated models, the model has a smple closed form soluton for medan wllngness to pay and subsequently for covarate effects on medan wllngness to pay. Despte the smple nature of our model, t has two dsadvantages. It requres some approxmaton to estmate mean wllngness to pay and the effects of covarates. Second s the nflexblty of the model. It assumes a specfc functonal form for wllngness to pay as a functon of covarates, and ths form must be mantaned for emprcal tractablty. There seems to be no way around these problems. Recognzng that wllngness to pay s bounded rrevocably makes the econometrcs more complcated. One can pursue models that are more complcated to estmate, such as the beta (see Hanemann and Kannnen, or Haab and McConnell, forthcomng), but whch are flexble and gve means whch are easy to calculate, or one can estmate off the shelf

3 models such as the one we outlne below, but whch requre some approxmaton for calculaton of the mean and dervatves. A Workable Model Suppose that WTP, wllngness to pay, les between zero and some upper bound, A, whch could be ncome or even some proporton of ncome (for example, after tax ncome). In order to explan the outcomes of referendum-type questons, we need a model whch specfes wllngness to pay as a stochastc functon of covarates but whch constrans WTP to le between 0 and A. Let WTP for ndvdual be specfed: A () WTP = X β+ ε + e, 2 where ε ~ N( 0, σ ) and X β s the nner product of the J covarates (X =X X J ) and a vector of coeffcents (β ) and A s a known constant for ndvdual, such as ncome, whch s assumed to be a reasonable upper bound on wllngness to pay. Note that for a gven value of ε, as the value of X β becomes large and negatve, WTP approaches A, and f X β becomes very large, WTP approaches zero. The dervatves of WTP wth respect to covarates have the form (2) WTP X j Xβ+ ε β β ε ja e A e = X = β Z ε β ε j + ( X + ) where Z β+ ε ( + e ) ( + e ) 2 2 X < 0. (The number Z has no partcular meanng but s useful for computaton.) There are two thngs to note about the dervatves. Frst, the sgn of the dervatve of wllngness to pay of a covarate s the opposte of the sgn of the coeffcent. Second, the dervatve s a functon of all covarates and the error, whch are contaned n the term Z (ε ), the same for all dervatves. Now suppose that the th respondent s asked Would you pay t for a proposed hypothetcal servce? The probablty of a no response s the probablty that wllngness to pay would be less than t. In ths model ths probablty becomes A P WTP t = P X + e (3) ( ) β+ ε X t = P ε β ln σ σ A t ( t ) The last expresson on the rght hand sde s the contrbuton to the lkelhood functon for a standard probt model, where the probablty of a no response s modeled wth the covarates X. 2

4 and ln A t. For a sample of I respondents, where the th respondent wth upper bound A t and covarates X responds yes or no to the queston Would you pay t for a proposed hypothetcal servce?, the log lkelhood functon s (4) I ( L) = ( y ) ( P( no to queston) ) + y ( P( no to queston) ) ln ln ( ) ln = A t A t ( ) X ln t ( t ) β I X ln β = ( y ) Φ y + Φ = σ σ σ σ where y = f the respondent answers yes to the queston. Ths s a standard probt wth X and ln A t t as covarates. The parameter estmates on X wll be an estmate of β σ and the parameter estmate for ln A t t wll be an estmate of σ. The estmates of β and have the σ σ usual propertes of maxmum lkelhood estmates. Recovery of β occurs by dvdng the A t coeffcents on X through by the coeffcent on the composte term ln. t The smple procedure for estmatng a bound probt (or logt) s as follows 2 : A t a) Create a new varable X J + = X = $ ln. Where A s the assumed upper bound on the t th ndvdual s wllngness to pay, and t s the bd offered to the th ndvdual. b) Usng any standard econometrc package, run a smple probt (or logt) of the 0/ no/yes responses on the vector of covarates (ncludng a constant) X and the new varable X $. c) Recover estmates of β by dvdng the coeffcents on X ( σ β ) by the coeffcent on X ( σ ). Mean and Medan Wllngness to Pay Models estmated from dscrete responses should provde a central tendency measure of wllngness to pay and the effects of margnal changes n covarates on that central measure, typcally mean and medan wllngness to pay. Mean wllngness to pay has been the tradtonal measure of central tendency, but medan wllngness to pay s less senstve to the shape of the tals of the assumed dstrbuton. The bound probt/logt model presented here has a smple closed 3

5 form soluton for medan wllngness to pay that allows for the straghtforward calculaton of covarate effects on a central tendency measure of wllngness to pay. Mean wllngness to pay on the other hand s more complcated because of the non-lnear nature of the error dsturbance n the assumed form for wllngness to pay. However, a straghtforward approxmaton can be derved whch allows for the calculaton of mean wllngness to pay. Medan Wllngness to Pay Medan wllngness to pay (M(WTP )) represents the ffteth percentle of the dstrbuton of wllngness to pay, or n general the soluton to the non-lnear equaton P(WTP <MWTP )=.5. Observng that for any WTP functonal form wth a mean zero symmetrc error term, medan wllngness to pay wll occur when the error takes on a value of zero. Pluggng n a zero for the error n () yelds: A (5) M (WTP ) = X β + e. Smply pluggng the estmated parameters from the probt or logt nto the WTP functon from () and excludng the error term yelds the estmate of medan wllngness to pay for the bound probt/logt 3. The sample medan could be calculated by fndng the value MWTP whch dvdes the array of ndvdual sample medans n half. Margnal covarate effects on medan wllngness to pay are found by dfferentatng (5) wth respect to X : (6) M(WTP ) = X j β A e j X β 2 X β ( + e ) = β Z (0). j Agan, smple substtuton of parameter estmates yelds estmates of the covarate effects on medan wllngness to pay. Mean Wllngness to Pay Because of ts smplcty, medan wllngness to pay s a very attractve measure of central tendency for the bound probt/logt model. In addton, medan wllngness to pay appears to be less senstve to dstrbutonal msspecfcaton n a referendum framework. However, falure of medan wllngness to pay to take nto account the tals of the dstrbuton 4 has left many researchers reluctant to abandon mean wllngness to pay as a central tendency measure. But, for 4

6 the bound probt, the nonlnear form of the error term n WTP yelds an estmate of mean WTP wth no closed form soluton. The general form of expected wllngness to pay s gven by (7) ( ) = ( ) ( ) E WTP WTP X β, ε f ε dε. By substtutng the form of WTP from (), and assumng ether the standard normal or logstc dstrbuton for ε, nto (7) t s apparent that the ntegral does not have an obvous closed form soluton. But that does not preclude the use of mean wllngness to pay n ths nstance. The ntegral n (7) can be approxmated by k (8) E( WTP ) WTP( X β, ε )( ε ε ) n k= σ φ ε σ k k k where φ( ) s the standard normal pdf, ε k are ponts on the dstrbutonal support of ε and n s large enough so that the approxmaton s smooth. The trouble wth a large n s that t requres one to go to the computer after the model s estmated. Our goal s to make t feasble to calculate expected WTP wthout resort to the computer, gven an estmated model. An approxmaton to mean wllngness to pay s gven by (9) E( WTP ) m σ k= k φ ε WTP( X β, ε k ) σ m j φ ε. σ j= σ Ths approxmaton represents a more dscrete approxmaton to the ntegral n (7) than does (8). The numerator n (9) evaluates the ntegrand from (7) at m dscrete ponts over the range of ntegraton. The denomnator normalzes the dscrete approxmaton n the ntegral to account for the area under the densty of WTP mssed by the dscrete approxmaton. The dea n applyng (9) s to pck m small enough, say n the range of 2 to 2, to calculate expected wllngness to pay by hand. If m s chosen small enough, expected wllngness to pay can then be approxmated by performng the followng calculatons. ) Defne θ=ε/σ, where θ s N(0,) and can be pcked from a standard normal densty table. Pck the lowest value of θ, θ such that φ(θ ) 0. Thnk of θ as the number of standard devatons of the N(0,) dstrbuton over whch the approxmaton wll be performed. Because the normal and logstc dstrbutons are symmetrc, we can defne the hghest value of θ, θ m as θ m =-θ 5

7 For example, suppose a bound probt model s estmated resultng n parameter estmates such that Xβ=, and σ =.. The frst step to calculatng mean wllngness to pay s to choose a value for θ. We want to choose θ so that φ(θ ) 0 and we cover the entre support of the dstrbuton of WTP. Suppose we choose θ =-3 (φ(θ )=.0044). Snce the normal dstrbuton s symmetrc, the hghest value for θ s θ m =-θ =3. ) Pck a value of m and dvde the range of θ nto m- equal ncrements, where s defned by = (θ m -θ )/(m-) = -2θ /(m-). The next step s to pck a number of ponts (m) along the support of WTP where we wll calculate WTP. Ths number should be small to mnmze the number of calculatons needed, but large enough to closely approxmate the curvature of the densty. Contnung the example suppose we choose m=4. Accordng to ) the ncrement for the range of θ s =-2θ /(m-)=(-2*-3)/(4-)=2. ) Calculate expresson (9) for the chosen values of m and θ and the estmated parameters β and σ usng the relatonshps θ k = θ + for k> and ε k = σθ k Usng defned n step ) the ponts at whch WTP wll be evaluated are θ =-3, θ 2 =-, θ 3 =, θ 4 =3, and the correspondng values of ε k = σθ k are ε = -.3, ε 2 = -., ε 3 =., and ε 4 =.3. a) To calculate WTP (equaton 9) frst calculate N= φ( θ ) b) Then calculate D= φ( θ ). j= c) Then WTP N/D. Fnshng the example: m j m k= WTP( X β, ε ). k k N=(.0044*.332)+(.2420*.289)+(.2420*.250)+(.0004*.24)=.328 D=( )=.4928 WTP=N/D=.270 Ths approxmaton to WTP can be nterpreted as a percentage of the upper bound: WTP s 27% of the upper bound. The exact value of WTP (found by calculatng the approxmaton n (9) usng m=00,000 and θ =-5), s WTP=.270. In ths case, the approxmaton s accurate to the thrd decmal place. Ths wll not always be the case. Some care and good luck s needed n pckng m and θ. For example f θ s pcked to be qute low, say -4, and m small, say 3, then the curvature of the densty wll not be approxmated well. In general, f m s low, t s better to have smaller range for θ. We wll show some senstvty results for the choce of m and θ below. 6

8 by Usng (7), the margnal expected wllngness to pay wth respect to covarate X j s gven E WTP X j β j Z ε f ε dε where Z ( ε) A e Xβ+ ε ( + e ) (0) ( ) / = ( ) ( ) X β+ ε The chef reason for estmatng a bound model such as we are proposng s to understand the effects of covarates. Otherwse one can use a smple non-parametrc approach such as the Turnbull to estmate the mean wllngness to pay. Smlar to E(WTP ), the dervatve can be approxmated by ( ) E WTP / X j m σ k= φ ε k σ β jz ( ε k ) m σ j= φ ε j σ where the steps would be the same as -v above, except that we would substtute m N = ε σφ k jz k k= σ β ( ε ) for N. Note that E(WTP) and ts dervatve are proportonal to A, the bound on WTP. Hence we can normalze these numbers and smply multply them by the upper bound to complete the calculatons. Table shows the percent errors n approxmated wllngness to pay for varous values of m (the number of dvsons of the normal dstrbuton), σ and Xβ, whch are the estmated parameters, and the standard devaton of the normal table, θ n step ) above. These errors are defned as 00*(true - calculated)/true. Also, for each σ and Xβ, the true value of the WTP s gven. Ths s calculated by dvdng the normal densty nto 00,000 ncrements, so that the approxmaton s almost perfect. Ths table can be used to determne approxmately how many ncrements would be needed to get an accurate estmate of WTP. The shaded cells of Table represents errors below 0% of true wllngness to pay. Naturally when the estmated varance (σ) s small, not many ncrements are needed. For example n the submatrces for σ =., the approxmaton errors for even two ncrements are qute small, less than 0% n all cases and less than % n most cases 5. Table also gves the correct value of F = Z ( ε) f( ε) dε, the factor 2. 7

9 whch s multpled by β j to gve the partal dervatve ( ) / j = j ( ) ( ) E WTP X β Z ε f ε dε =β j F. And n Table 2, the approxmatons for F are gven. These errors are small f the correct choce of m and θ are made. To llustrate the workngs of Table, suppose we estmate a model wth σ = and Xβ = 5. Consder a person wth ncome of $25,000, whch would be the exogenously gven A, the upper bound of WTP. Then the true WTP, gven σ = and Xβ = 5, s $25,000*.0 = $275. If ths s approxmated usng 2 standard devatons (θ = -2) and 4 ncrements (m=4) the calculated WTP s $26, wth an error of -5.7%. The approxmaton of partal dervatves works smlarly, usng Table 2. Suppose that a model wth σ =2 and Xβ = has been estmated. Further, consder an observaton wth A = $30,000, the upper bound on WTP. The correct dervatve, gven σ =2 and Xβ =, s β j *.4*30,000. If ths s approxmated by dvdng 3 standard devatons nto 4 ncrements (θ = -3, m = 4) the calculated dervatve would be.05*β j *.4*30,000, wth an error of.05%. An Example wth Econometrcs We llustrate the bound probt model by estmatng t wth data from a contngent valuaton study measurng the wllngness to pay of Kentucky resdents to prevent the mnng of western Kentucky wetlands 6 (see Blomqust and Whtehead for a full descrpton of the survey). For comparson, three other models were estmated: a beta, a log-probt, and a probt. The beta accounts for bounds correctly, the log probt bounds WTP from below at zero, and the probt allows WTP to be unbounded. The data contans nformaton on the respondents GENDER (male=; mean=.478), AGE (mean=48.82), EDUCATION (mean=2.66), number of CHILDREN (mean=.68), INCOME (mean=$24,660) and whether the respondent s a MEMBER of an envronmental organzaton (yes=; mean =.9). Tables 3-6 report the estmaton results for the four models on the dchotomous choce wllngness to pay responses. In addton to the demographc covarates a seres of control dummes (VERB, VERC, VERD, and YES6DUM) were ncluded n the estmaton to control for dfferences n questonnare formats among respondents. These 8

10 questonnare dfferences were the focus of Blomqust and Whtehead. Few conclusons can be drawn from the parameters shown n tables 3-6. Snce the estmated models dffer n the functonal form and/or dstrbuton assumed for wllngness to pay, the parameter estmates are not comparable across models. For comparson, mean and medan wllngness to pay are calculated for each of the models along wth the correspondng covarate effects (where applcable). Medan and mean WTP are calculated at the mean values of AGE, EDUCATION and CHILDREN. For these calculatons the dummes ndcatng gender, and envronmental organzaton membershp are set to zero. The questonnare dummes are set to zero and covarate effects for these control dummes are not reported (for brevty). Table 7 reports the results. For the bound probt, two sets of calculatons are presented for mean wllngness to pay: An approxmaton based on dvdng the range of WTP n 0,000 ntervals (a smooth approxmaton), and the same approxmaton based on 2 dvsons, and 4 standard devatons. Once agan, the approxmaton performs well. Medan wllngness to pay s smlar across the bound probt, beta, and log-probt models. It s dffcult to obtan covarate effects on medan wllngness to pay from the beta model. Medan WTP s found by solvng the non-lnear equaton that determnes the 50 th percentle of the beta dstrbuton. Total dfferentaton of the non-lnear equaton nvolves the dfferentaton of the ncomplete beta functon. Snce the bound probt provdes a closed form soluton for medan wllngness to pay, the beta appears to be unnecessarly complcated. The log-probt and the bound probt provde smlar covarate effects on medan wllngness to pay, once agan supportng the result that medan wllngness to pay s robust across dstrbutonal assumptons. The probt model provdes covarate effects wth the same sgn as the bound probt and logprobt, but they dffer by an order of magntude. As expected, the two models that bound wllngness to pay by ncome (the bound probt and the beta) provde mean WTP estmates lower than the unbounded (from above) log-probt. The log-probt suffers from a fat upper tal snce the dstrbuton of wllngness to pay s allowed to range from zero to nfnty. Integratng over ths full range provdes a large estmate of mean wllngness to pay. The probt model estmates mean WTP to be negatve. The goods valued n ths survey are local wetlands and should not nduce reductons n utlty, so that WTP should be nonnegatve. 9

11 The bound probt and beta models provde covarate effects on mean WTP that dffer substantally n some cases. Whle not necessarly surprsng, ths result s somewhat troublng as t s dffcult to determne whch of these two models s approprate. Snce both models consst of ad hoc dstrbutonal assumptons, the choce of the approprate model s left to the researcher. Horowtz (983) provdes a method for rankng non-nested maxmum lkelhood models. The smplest of hs rankngs bols down to a smple comparson of lkelhood functon values f the number of parameters estmated are dentcal and the number of choces gven to each ndvdual are dentcal (as s the case here). In ths case, the bound probt appears to provde a better ft than does the beta. It should be noted however that Horowtz rankng s not a method for statstcally dstngushng between models. The bound probt and beta models restrct wllngness to pay to conform to reasonable bounds. Usng a smple rankng of models based on lkelhood functon values t appears that the bound probt provdes a better ft to the data than does the beta. In addton the bound probt provdes a closed form soluton for medan wllngness to pay and a smple approxmaton procedure for mean WTP. Conclusons The current approaches to estmatng WTP from dscrete choce referendum models allows three possbltes. The smplest s to use a dstrbuton-free model such as the Turnbull to estmate mean WTP. Ths approach makes the estmaton of covarate effects qute dffcult. It also reles on a carefully chosen set of bds. The most common approach s to estmate the standard dfference n utlty model, whch often requres truncatng estmated WTP at zero or some upper bound. The approprate choce would nvolve estmatng a complcated model such as the beta or the pnched logt. Ths paper offers an alternatve model that naturally mposes bounds of zero and an ndvdual-specfc upper bound but whch can be estmated wth canned econometrc packages. The estmaton of ths model allows the calculaton of mean and medan WTP as well as the covarate effects on mean and medan WTP. Medan WTP calculatons are qute smple, but all calculatons for mean, medan and partal dervatves can be done wth a hand calculator. 0

12 Table Errors n Approxmatng Wllngness to Pay Standard Devatons Standard Devatons Standard Devatons Xβ= σ=. Xβ=5 σ=. Xβ=0 σ=. WTP =.269 F=-.96 WTP=.007 F=-.007 WTP= F= % 0.00% -0.03% -0.% 0.00% 0.00% -0.08% -0.32% 0.00% -0.0% -0.08% -0.33% % 0.00% -0.03% -0.% 0.00% 0.00% -0.07% -0.32% 0.00% 0.00% -0.08% -0.33% % 0.00% -0.02% -0.% 0.00% 0.00% -0.06% -0.3% 0.00% 0.00% -0.07% -0.32% % 0.00% -0.02% -0.0% 0.0% 0.00% -0.05% -0.30% 0.0% 0.00% -0.05% -0.30% 4 0.3% 0.02% -0.0% -0.09% 0.39% 0.07% -0.02% -0.26% 0.39% 0.07% -0.02% -0.27% %.33% 0.50% 0.00% 7.4% 3.93%.47% 0.00% 7.57% 4.0%.50% 0.00% Xβ= σ= Xβ=5 σ= Xβ=0 σ= WTP=..303 F=-.78 WTP=.0 F=-.0 WTP= F= % -0.05% -.7% -6.63% 0.07% -0.69% -8.32% % 0.0% -0.85% -9.2% % % -0.03% -.06% -6.50% 0.08% -0.54% -7.76% % 0.% -0.68% -8.54% -28.3% 8 0.0% -0.02% -0.89% -6.29% 0.08% -0.34% -6.9% % 0.2% -0.44% -7.65% % % 0.00% -0.63% -5.93% 0.24% -0.0% -5.47% % 0.28% -0.5% -6.2% % % 2.92% -0.6% -5.08% 24.66% 3.09% -2.58% % 24.45% 3.02% -3.04% % % 48.9% 28.36% 2.09% 47.% % 24.04% -5.7% % 50.98% 28.45% -6.28% Xβ= σ=2 Mu=5 σ=2 Mu=0 σ=2 WTP=.352 f=-.4 WTP=.032 F=-.025 WTP=.0003 F= % -0.04% -.5% -0.38% 0.07% -.% % %.28% -7.3% % % 0 0.0% -0.03% -.02% -0.09% 0.0% -0.82% -9.08% -6.99%.44% -6.6% % % % -0.0% -0.84% -9.63% 0.0% -0.46% -6.8% -6.06%.62% -4.49% % % 6 4.9% 0.56% -0.56% -8.8% -0.80% -0.03% -2.84% %.8% -.93% -3.06% -72.0% %.05%.5% -6.99% 39.00% -5.93% -4.9% -55.9% 5.82%.37% % % % 4.2% 36.6% 0.50% % % 37.54% % % % % % Xβ= σ=4 Xβ=5 σ=4 Xβ=0 σ=4

13 WTP=..40 F=0.089 WTP=.30 F=-.047 WTP=.0 F= % 0.00% -0.64% -7.96% -0.25% -0.20% -8.43% % 0.75% -3.5% % -97.2% 0.57% 0.9% -0.57% -7.63% 2.4% -0.55% -7.48% % 0.23% -2.47% % % %.27% -0.42% -7.4% -3.5%.75% -6.08% % -5.88% -2.72% % % % 5.87% 0.54% -6.30% -9.89% -6.6% -3.32% -6.4% 30.88% -0.70% % % % 7.9% 8.07% -4.35% 29.69%.9% -5.78% -52.7% % % % % % 22.07% 2.97% 7.0% 294.0% % % 6.02% 426.4% 37.30% 45.80% % Xβ=5 σ=8 Xβ=0 σ=8 Xβ=20 σ=8 WTP=.27 F=-.040 WTP=. F=-.023 WTP=.007 F= %.29% -2.67% -3.73% -7.45% 2.64% -9.87% % 4.2% -3.82% % % % -2.98% -.56% % 6.55% -6.66% -0.05% % % -9.9% % % % -.0% -0.85% -28.2% 0.54%.47% -4.93% % 3.50% % % % % 3.36% -0.64% % % -9.9% -3.70% % 53.8% % % % % 75.90% 8.62% -2.35% 96.84% % -3.66% -78.9% % 9.90% % % % 84.49% 84.49% 75.74% % % % % % % 22.05% % Xβ=5 σ=20 Xβ=0 σ=20 Xβ=20 σ=20 WTP=.402 F=-.09 WTP=.309 F=-.07 WTP=.60 F= % 4.9% -4.7% -8.99% % -5.55% 2.33% % 29.0% -3.40% -2.75% -79.8% % 4.52% -3.60% -8.9% -6.47% -6.93% 2.9% -2.4%.86% 4.95% 4.96% -76.4% % 22.35% 6.20% -0.04% 37.48% -9.72% -9.63% % % 7.42% % % % 24.39% 20.95% -2.38% 6.3% 46.27% % -0.24% -72.7% % 24.33% % % 24.48% 24.45% 2.43% 6.68% 6.68% 56.92% % 22.84% 59.44% % -38.8% % 24.48% 24.48% 24.48% 6.68% 6.68% 6.68% 6.68% 23.24% 23.24% 23.24% 56.62% 2

14 Table 2 Errors n Approxmatng F, Used n Computng the Covarate Effects Standard Devatons Standard Devatons Standard Devatons Xβ= σ=. Xβ=5 σ=. Xβ=0 σ=. WTP=.269 F=-.96 WTP=.007 F=-.007 WTP= F= % 0.00% 0.0% 0.06% 0.00% 0.00% -0.08% -0.32% 0.00% -0.0% -0.08% -0.33% % 0.00% 0.0% 0.06% 0.00% 0.00% -0.07% -0.3% 0.00% 0.00% -0.08% -0.33% % 0.00% 0.0% 0.06% 0.00% 0.00% -0.06% -0.3% 0.00% 0.00% -0.07% -0.32% % 0.00% 0.0% 0.05% 0.0% 0.00% -0.05% -0.29% 0.0% 0.00% -0.05% -0.30% % -0.0% 0.00% 0.05% 0.38% 0.07% -0.02% -0.26% 0.39% 0.07% -0.02% -0.27% % -0.73% -0.27% 0.00% 7.25% 3.85%.44% 0.00% 7.57% 4.0%.50% 0.00% Xβ= σ= Xβ=5 σ= Xβ=0 σ= WTP=.303 F=-.78 WTP=.0 F=-.00 WTP= F= % 0.07%.49% 6.95% 0.05% -0.55% -7.57% % 0.0% -0.85% -9.% % % 0.05%.36% 6.84% 0.05% -0.43% -7.04% % 0.% -0.68% -8.53% -28.2% 8 0.0% 0.03%.6% 6.67% 0.06% -0.26% -6.24% -25.2% 0.2% -0.44% -7.64% % 6 0.0% 0.04% 0.84% 6.35% 0.2% -0.07% -4.87% -24.6% 0.28% -0.5% -6.2% -26.6% % -.43% 0.39% 5.6% 24.84% 3.6% -2.8% -2.89% 24.45% 3.02% -3.03% % % % % -0.26% % 40.63% 9.49% -4.3% % 50.54% 28.42% -6.27% Xβ= σ=2 Xβ=5 σ=2 Xβ=0 σ=2 WTP=.352 F=-.40 WTP=.032 F=-.025 WTP=.0003 F= % 0.09% 2.69% 2.56% -0.0% -0.08% -0.48% %.09% -6.92% % -74.2% 0 0.2% 0.07% 2.40% 2.04% 0.00% -0.04% -9.29% %.22% -5.80% % % % 0.9%.97% 20.22% 0.35% 0.02% -7.5% -5.69%.33% -4.7% % -73.0% %.04%.40% 8.75% -4.90% 0.84% -4.59% %.42% -.69% % -7.77% % -5.46%.22% 5.32% 60.42% -3.39% 0.47% % 4.97%.34% -20.8% % % -97.3% -8.56% -3.96% -0.55% % 289.5% -8.75% 696.% % % -47.0% 3

15 Xβ= σ=4 Xβ=5 σ=4 Xβ=0 σ=4 WTP=.40 F=-,089 WTP=.27 F= WTP=.0 F= % 0.62% 2.90% 35.48% 3.5% -0.2% 2.66% % -.9% 0.2% % % %.48% 2.65% 34.07%.95% 0.8% 2.43% % 6.70%.53% -37.0% % % 0.97% 2.8% 3.92% % 4.33%.26% % -8.36% 3.9% -30.8% % % -4.24% 2.96% 28.30% 20.32% -27.2% 7.4% -6.78% 34.7% -3.27% -8.48% % 4-9.8% -7.35% % 22.45% 56.70% 04.0% -9.86% -2.03% -3.98% % 6.08% % % % % % % % % 07.82% -8.8% % % -8.80% Xβ=5 σ=8 Xβ=0 σ=8 Xβ=20 σ=8 WTP=.27 F=-.040 WTP=. F=-.023 WTP=.007 F= % -5.5% 4.23% 36.4% 2.76% -2.75% 5.7% % 20.08% 8.45% % % % -3.39% 9.22% 34.79% 43.58% -6.44% 2.56% % 7.70% 2.49% -8.34% % % 2.06% % 32.20% -82.8% 60.57% -5.0% % -76.3% -25.9% -77.0% % % 45.2% -5.94% 36.02% % % 65.55% % % % % % % % 57.52% -7.02% % 22.93% -82.6% -0.44% % % % % % % % % % % % 27.02% % % % % Xβ=5 σ=20 Xβ=0 σ=20 Xβ=20 σ=20 WTP=.40 F=-.09 WTP=.309 F=-.07 WTP=.60 F= % 58.83% 25.58% 46.95% -0.84% % 5.66% 38.59% 58.92% % 0.52% 40.60% % 74.68% 3.66% 5.20% 24.08% % % 33.72% % % % 59.85% % % 55.98% -6.9% 244.% 77.30% % 33.5% % % -3.44% 94.05% % -98.% % 07.5% % 39.25% 78.70% 6.03% % % % 59.20% % % %.38% % % -8.95% % % 94.56% -95.5% % % % % % % % % % % % % 933.4% 4

16 Table 3 Bound Probt Regresson Results Kentucky Wetlands Data Parameters Estmates Std. err. Est./s.e XHAT B VERB VERC VERD GENDER AGE EDUCATION CHILDREN MEMBER YES6DUM Mean log-lkelhood Number of cases 379 Table 4 Beta Regresson Results Kentucky Wetlands Data Parameters Estmates Std. err LN(P) Q VERB VERC VERD GENDER AGE EDUCATION CHILDREN MEMBER YES6DUM Mean log-lkelhood Number of cases 379 5

17 Table 5 Log-Probt Regresson Results Kentucky Wetlands Data Parameters Estmates Std. err. Est./s.e LN(TAX) B VERB VERC VERD GENDER AGE EDUCATIO CHILDREN MEMBER YES6DUM Mean log-lkelhood Number of cases 379 Table 6 Probt Regresson Results Kentucky Wetlands Data Parameters Estmates Std. err. Est./s.e TAX B VERB VERC VERD GENDER AGE EDUCATION CHILDREN MEMBER YES6DUM Mean log-lkelhood Number of cases 379 6

18 Table 7 Wllngness to Pay and Margnal Effects from Probt Models on Kentucky Wetlands Data Bound Probt Beta Log-Probt Probt Medan WTP Margnal Effect on Medan WTP Gender (=male) Age (Years) Educaton (Years) Number of Chldren Member of Envronmental Organzaton? (Yes=) Iteratons for Approxmaton 4 Standard Devatons 0,000 2 Mean WTP Margnal Effects on Mean WTP Gender (=male) Age (Years) Educaton (Years) Number of Chldren Member of Envronmental Organzaton? (Yes =) Xβ = σ =

19 References Blomqust, G. C. and J. C. Whtehead (forthcomng) Resource Qualty Informaton and Valdty of Wllngness to Pay n Contngnent Valuaton. Resource and Energy Economcs. Carson, R., W. M. Hanemann, R. Kopp, J. Krosnck, R. Mtchell, S. Presser, P. Ruud, and V.K. Smth (994) Prospectve Interm Lost Use Value Due to DDT and PCB Contamnaton n the Southern Calforna Bght. NOAA Contract No. DGNC Haab, T.C. and K.E. McConnell forthcomng Referendum Models and Economc Values: Theoretcal, Intutve and Practcal Bounds on Wllngness to Pay Land Economcs Haab, T.C. and K.E. McConnell (997) Referendum Models and Negatve Wllngness to Pay: Alternatve Solutons. Journal of Envronmental Economcs and Management. Hanemann, W.M. and B. Kannnen (996) The Statstcal Analyss of Dscrete-Response Data. Department of Agrcultural and Resource Economcs Workng Paper No Unversty of Calforna, Berkeley. Horowtz, Joel. (983) Statstcal Comparson of Non-Nested Probablstc Dscrete Choce Models. Transportaton Scence. 7: Kannnen, B.J. (995) Bas n Dscrete Response Contngent Valuaton. Journal of Envronmental Economcs and Management 28:4-25. Krnsky, I. and A. Robb (986) On Approxmatng the Statstcal Propertes of Elastctes. Revew of Economcs and Statstcs 86:75-9. Krstrom, B. (990) A Non-parametrc Approach to the Estmaton of Welfare Measures n Dscrete Response Valuaton Studes. Land Economcs 66: Ready, R.C. and D. Hu (995) Statstcal Approaches to the Fat Tal Problem for Dchotomous Choce Contngent Valuaton. Land Economcs 7: Turnbull, B. (976) The Emprcal Dstrbuton Functon wth Arbtrarly Grouped, Censored, and Truncated Data. Journal of the Royal Statstcal Socety, Seres B. 38:

20 Endnotes For the sake of clarty n notaton we wll assume the error s normally dstrbuted throughout. As wll be seen, ths results n the dervaton of a probt type model. By smply changng the error dstrbuton to a logstc dstrbuton and followng the same procedure, an dentcal logt verson of the model emerges. 2 Care must be taken f the bound probt procedure s estmated usng SAS. Because SAS reverses the probabltes of 0 and observatons, the coeffcent sgns wll be the opposte of those estmated usng other packages and those expected from the dervaton here. Smply reversng the sgns of the SAS estmated coeffcents resolves the problem. 3 The actual calculaton of medan wllngness to pay requres the researcher to decde how to handle the ndvdual specfc covarates. An ndvdual specfc medan wllngness to pay can be calculated for each ndvdual by substtutng the covarates nto (7). Alternatvely, sample average wllngness to pay estmates can be obtaned by averagng the ndvdual specfc WTP s over the sample, or by substtutng average covarate values nto (7) and calculatng one medan wllngness to pay measure for the sample. The relatve strengths and weaknesses of these methods are beyond the scope of ths paper and are relegated to future research. 4 Medan wllngness to pay s not a suffcent statstc. 5 The authors have avalable detaled tables (n Mcrosoft Excel format) that show the exact wllngness to pay factors and covarate factors for values of Xβ=(-20, 20) n ntervals of., and values of σ=(., 50) n ntervals of.. The tables each consst of 205,000 possble combnatons of Xβ and σ, and allow for the drect calculaton of mean WTP wth covarate effects wthout resortng to the approxmaton procedure presented here. The tables are avalable upon request. 6 We would lke to thank John Whtehead for makng ths survey data avalable to us. 9