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Nychka 7North Carolina State University  Y E?V. Kerry Smith*ă D?Duke University >December 21, 1995!0*0*0*  Y xB Abstract ă The purpose of this research is to use measures of willingness to pay derived from random utility models as the basis for evaluating the properties of parametric and nonparametric estimators with binary choice data. Monte Carlo simulations are conducted to evaluate two parametric methods and a nonparametric method. The simulation results indicate that simple parametric methods outperform the nonparametric method for a range of underlying preference structures.  Y  Key Words :` ` !X )Estimator Selecting Criteria, Discrete Choice Methods, Loss Function, Nonparametric Regression, Cubic Smoothing Splines.(#  Yd  JEL: ` ` ! )2C25, Q20d0*0*0*  Y     V? 1. Introduction ă Increasing recognition of the effects of parametric modeling assumptions for the properties of Hicksian surplus estimates, (see Hayes and PorterHudak [1987] and Kling [1988]),  Yw has focused attention on nonparametric alternatives.X1Í.X1Í.YwXThere have also been numerous comparisons of demand models with count data for situations that resemble travel cost recreation demand applications. Hayes and PorterHudak [1987] report results similar to Hausman [1981] in that they describe measures of deadweight loss as random variables and illustrate their properties.Y Hausman and Newey's [forthcoming] recent comparison of three kernel estimators with a conventional parametric approach is one such example. They find that parametric methods' estimates of consumer surplus and of dead weight loss (due to taxes) are very sensitive to changes in the specification selected for the demand function. By contrast, the nonparametric estimates of these measures are quite stable across alternative modeling assumptions. The sensitivity of Hicksian welfare estimates from discrete choice models appears to be even more pronounced (Kling [1988]). To date, however, there have been no studies comparing nonparametric to parametric estimators based on the  Y properties of the welfare measures derived from them.fwThere have been numerous studies proposing distribution free semiparametric (i.e. specifying the nonstochastic component of the model in parametric terms but using a distribution free model for the error) and semiparametric methods with nonparametric estimation of expectations, and distribution free nonparametric methods for binary choice models (for further discussion see Ahn [1995]). To the extent, the research has attempted to evaluate the small sample properties of the estimators, they have generally followed a format similar to Horowitz [1992]. His sampling experiments were intended to be illustrative, selected very simple models and focused on the estimates of a key parameter not the integral of the estimated distribution function. His results indicate clear support both in terms of MSE and bias for the logit estimator over both the maximum score and the smoothed maximum score estimator in all cases except those with significant heteroscedasticity.f The purpose of this paper is to report the first such evaluation, comparing probit, logit, and cubic smoothing spline. Our analysis is based on a wide range of simulated choice data derived from constrained optimization models with each of three different preference specifications. These models vary the importance of the restrictions linking commodities assumed available on an "all or nothing" basis (i.e., through the discrete choice decisions) with goods that can be consumed in varying amounts at fixed prices per unit. Our results suggest that a cubic smoothing spline is inferior to either probit or logit, based on the normalized root mean squared error (NRMSE) for the Hicksian willingness to pay (WTP) for the commodity rationed through "take it or leave it" choices. While the overall size of the error in the WTP estimates varies with the behavioral model assumed responsible for the choices, support for parametric;&0*((aa models is clear and relatively insensitive to the structure of the preferences considered in these experiments. This finding contrasts with Hausman and Newey's conclusion. There are, however, important differences in the type of information available to their nonparametric estimator for the case of a continuous demand model, as well as a distinction in the criteria used in their evaluation versus this one. Our findings are based on models where the "true" WTP is known. The estimators can be compared based on how well each performs in recovering this true value from the simulated choice data. Their empirical evaluation uses actual data and is a comparison of results, not an evaluation of sampling properties of parametric versus nonparametric methods. It is important to acknowledge that cubic smoothing splines are not the only approach for estimating binary choice models and some of these other methods have been subjected to limited sampling evaluations. While the results are limited, where comparisons have been made they conform to Manski and Thompson [1986] and Horowitz [1992], the nonparametric approach (or in their case the maximum score method) is inferior to a conventional logit, based on the mean squared error and bias in estimates of a model's parameters. However, with appreciable heteroscedasticity, the alternative methods are often superior. These findings provide another reason supporting the design of our simulation models. Because errors are introduced into the behavioral model as an additive term to the indirect utility function, they induce true willingness to pay values that are heteroscedastic (for two of the three preference specifications) and these values provide the basis for the choices represented in the samples used to evaluate each estimator.:&0*((aaԌSection two describes the estimation of WTP using parametric binary choice models and the cubic smoothing spline. Section three outlines the design of sampling experiments. Section four summarizes the simulation results. The implications of the results are presented in Section five.  Y h ) 2. Parametric and Nonparametric Discrete Choice Estimators ă Binary response data have received increasing attention in economics as individual choice data have become available for analysis. The models used to define welfare measures rely on these outcomes being the result of a constrained optimization process. Hanemann [1984] first demonstrated how a welfaretheoretic interpretation could be developed from McFadden's [1974] random utility model. The logic underlying this framework links the probability of a choice to parameters describing the circumstances of that choice. For example, equation (1) describes the probability a commodity is selected (i.e. purchased at a cost of t). t defines a lower bound for that individual's WTP. If we assume WTP is a random variable, then the last term in equation (1) connects t to the distribution for WTP.  W" (1)` ` !Prob(purchase) = Prob(WTP  t) = 1 GWTP(t)  Y where GWTP(.) = distribution function for WTP. The distribution GWTP(.) can be derived in a variety of ways, from the specification of indirect utility functions, as in Hanemann [1984], or via direct specification of the Hicksian compensating surplus function (see Cameron [1988], and McConnell [1990]).<&0*((aaԌConventional practice uses these choices, together with an assumed distribution for the  Y errors to induce GWTP(.) and to define a likelihood function based on it. Probit (or logit) follows this logic. As a rule some functional structure must be maintained for the indirect utility or compensation function. As is in the case of the demand functions considered in the HausmanNewey comparison, we can expect the assumptions about this specification will influence the properties of the parametric methods' estimates for WTP. This is easily verified from the definition of the mean WTP as in equation (2). ! ddddddddxXS(2)~~~~~~~~~~ E(WTP)~=~INT_0 ^{inf}~ (1~~G_WTP (s))ds~~ INT_{inf}^0 ~G_WTP (s)ds(Y2)pE(+WTP)Q!A!xx>xx10 ( 1 n Gxx uWTP (1 s} ) )ds Axx>0xxd1xx1|GxxuWTP(?s)ds$(#(# (#(#!!'#$  Y4 If the underlying distributional assumptions do not restrict GWTP(.) to the first quadrant, then the specifications used for the nonstochastic function describing choices will influence the importance of the second term in equation (2). This has been one of the persistent issues identified with parametric models (see Hanemann and Kanninen [1996]). Nonparametric estimation of E(WTP) reduces the restrictions imposed on the modeling structure. A cubic smoothing spline, for example, focuses on the smoothness of the choice function. To combine the smoothness criterion and the maximum likelihood criterion, a penalized likelihood function (O'Sullivan et al. (1986)) is used. With a logistic link function, this penalized likelihood function can be written as equation (3). ~A#x(dddddEddj x(3)~~~~~~~~~max from {f IN W_2 ^2[a,b]} ~~ SUM from {i=1} to n (y_i`` f_i~~log(1+e^{f_i}))~~ lambda INT from a to b (f''(x))^2dx,Xw PE37XPXw PE37XPXw PE37XP9(Z939)}9maxdd 2dd/2;Y[Y,Y]qt19(} 9log} 9( 91O 9) 9)9(9(v9)9)29,YfYWgYaYb6nti@9yi9f0 i 9e fdd3 iGbLa89f9x?9dx>Y!t 9$ 9-9IxL9~$(#(#"(#(#!A'#$@#'0*((aa!'#!('#,A@Ԍ Y where f is the estimator of the choice function and is assumed to belong to the second order  Y Sobolev space, W2    2 $ [a,b]. W2    2 $ [a,b] is a function space that all functions on [a,b] in the space have the first derivative absolutely continuous and the second derivative square integrable. The first summation in equation (3) is the usual loglikelihood function for logit model expressed in terms of this general function, f. The integral in the second part of the penalized likelihood function is the roughness function constraining smoothness of choice function in terms  Y of the stated cost, described here as x. The smaller the value of this term, the smoother is f.  Y  is the smoothing parameter controlling the relative importance of the two terms in (3). As  is allowed to become arbitrarily large, the second term must approach zero. This implies a zero second derivative and a linear fit. Thus, the optimization problem in (3) includes a linear specification as a special case. The penalized likelihood function is similar to the optimization problem that defines ridge regression and can be regarded as a generalized ridge regression model.  Y O'Sullivan et al. (1986) demonstrated that for a given value of , the solution to (3) exists and it is a natural cubic smoothing spline function. This is a piecewise cubic polynomial with two continuous derivatives within the data range and a linear function outside the data  Y4 range. The optimal value of the smoothing parameter  can be determined by data driven methods such as generalized cross validation (Craven and Wahba (1979), O'Sullivan et al.  Y (1986)). Thus, the selection of  is equivalent to choosing the best spline estimator among a class of alternatives.  Y$ The WTP estimator derived from a cubic smoothing spline can be written in terms of f,  YZ& the nonparametric estimator of the choice function by substituting the estimated logisticZ&0*((aa  Y probability distribution function into 1GWTP(.) in equation (2) as in (4): a#x7dddddddx(4)~~~~~~~~~W(f)~=~ INT from 0 to INF {e^{f(x)}} over {1+e^{f(x)}}dx~~ INT from { INF } to 0 1 over {1+e^{f(x)}}dx ~=~INT from 0 to INF`pi(f(x))dx~~INT from {INF} to 0`[1pi(f(x))]dx.Xw PE37XPXw PE37XPXw PE37XP'#(j#4#)#(K#)60[()]d1(0 ) 10 1& d1 ( )B60=#(#(T#)#)10u#[#1#(#(0#)s#)#]#.)#W#fe6fx9defxw #dx dem f x@#dx#f#x#dx\#f#x#dx#1d #\ 6 6 dU#;1#66#LCRm L R%LL#!#!$(#(#(#(#!a'#$  Y5 The estimated expected willingness to pay involves integrating nonlinear function of f(x) using the estimated cubic smoothing spline function. The primary large sample property of this nonparametric WTP estimator can be described as follows:  W Theorem 1: Let W(f) be defined as equation (4). Let f0 be the true function and fn be  WS the unique maximizer of (4). If W(f) is finite, then lim Prob( W(fn)W(f0)  )=1, where  is  W% a small constant.      Theorem 1 ensures the consistency of the WTP estimator derived from the nonparametric  Y discrete choice method. The proof of the theorem is given in the appendix.w Y ԍXOne issue raised in the theorem is the existence of W(n). It is possible that the integrals from  to  in (4) are not finite. Given that the WTP values are usually  Y positive, the integrals can be truncated in a range of positive values, [a,b] to ensure existence of the expected WTP. However, the truncated distribution of WTP must be modified to satisfy the basic properties of a distribution function. A common practice with parametric models has involved the use of linear specifications, linking the choice function to conditional indirect utility functions or to a CameronMcConnell  Y compensation function. For linear models, such as (5a), with v the utility difference (i.e., with and without the object of choice), t the onetime price, and the error specification for e leading to a probit or logit, the expected value for willingness to pay is a simple function of the estimated parameters, as in (5b).0#0*((aa7'# a0Ԍ#xaddddd dd> xX0(5a)~~~~~~~~~~ DELTA v~=~ alpha ~~ beta t~+~ e B(YB5Ba!B)BYBv BBBzBBty BI Beߓ$(#(#(#(#!'#$ s  ddddddL dd xX((5b)~~~~~~~~~~E(WTP)~=~{alpha over beta}(Y5b!)EL(WTP)BBs$(#(#(#(#!'#$ Thus, with parametric models the expression for E(WTP) depends on the restrictions imposed  Y on GWTP(.) through distributional assumptions or the choice function (see Hanemann [1989] and Hanemann and Kanninen [1996]). There are two important aspects of the WTP distribution implied by each model of the choice process. Most parametric models estimates of E(WTP) are conditional on other determinants specified to influence people's decisions. Thus, the hypothesized "true" value is expected to vary with these other variables. For the simple cubic smoothing spline, choices are hypothesized to be related to the one time fee, t. Therefore, the estimate of E(WTP) is constant. A second difference arises in the computation of estimates for E(WTP) with the parametric and nonparametric methods. For the former, there will generally be a closed form  Y7 expression while the latter is a numerical integral based on the estimate of GWTP(.). As discussed earlier, the mean square error is a common criterion for evaluating  Y estimators. The MSE of the WTP estimator for individual i can be written as the expected value  Y! of the square distance between Wi !   * !  and Wi J 0, E[(Wi !   * ! өWi   0)2], where Wi !  { * !  is the estimated expected  Y# WTPi derived from the methods being evaluated, and Wi  0 is the true expected value for WTPi. Substituting the expressions for the cubic smoothing spline's estimated WTP and the true WTP  Y/' into the definition of MSE, it can be written as equation (6), with the subscript i omitted for@/' 0*((aa!a'#~  '#  @ simplicity. #x3ddddddd x(6)~~~~~~~~~MSE~=~ E from {f_{n lambda}} `` LEFT [ ` INT from a to b `{e^{f_ {n lambda} (x)}} over {1+e^{f_ {n lambda} (x)}}` dx~~LEFT (` INT from 0 to INF `{e^{f_0(x)}} over {1+e^{f_0(x)}}`dx~~ INT from { INF } to 0 `1 over {1+e^{f_0(x)}}`dx `RIGHT ) `RIGHT ] ^{`2}.Xw PE37XPXw PE37XPXw PE37XP((Z(6() (Z )F1a ( )} ;0dd0M()F1dd0(")60+1DF1dd0(I)2[(.(MSE(E{fddQn6b;ae^ fdd n xa Fe fdd n x% (dxefyxFedfx(dx Fefx(dx(F: (v 6F(d;;FddQdd dd# vxSxxw}S~LkW h j Sj j iyoyqySqyqyp` L5WuL5*Wߔ$(#(#(#(#!'#$  Y1 The first part of equation (6) is the estimated expected WTP that is truncated at [a,b] and the expression in the parenthesis is the true expected WTP. Equation (6) can be considered an integrated loss function in that it evaluates the overall fit of the probability curvenot the  Y estimator of f itself. Hence, the estimation methods used to generate WTP measures are in fact evaluated by their global performance. Because the WTP measure varies across individuals in each generated sample (and depends on individual characteristics), a normalized estimate of MSE relative to the true WTP (labeled NRMSE) will enable the comparison of WTP estimators across individuals. Hence, our criterion is the square root of the estimated MSE relative to the true WTP for each individual. Since the above expectation is unobserved, an estimator of NRMSE replaces the expectation by the average of replicates from a Monte Carlo simulation. It provides the basis for our evaluation of each of three estimatorsprobit, logit, and the cubic smoothing spline. This is given in equation (7). #xD&ddddddd| M{(7)~~~~~~~~~ STACK { AND # NRMSE_i}~=~{sqrt { SMALLSUM from {j=1} to m ( W_{ij}^* W_{i0})^2}} over W_{i0} ~~~~i=1,..,n``,Xw PE37XPXw PE37XPXw PE37XP(b7)J t1 !(` !) o21u,.. ,,< D#T#S#BS#UR!!NRMSEi mtj !Ww ijQ !W i0 vW 6i0ai=n)t  !M$(#(#(#(#v!'#$ where m is the number of replicates in each simulation, and n the number of individuals in each@+' 0*((aa!3'#{ D&'#,+ @ sample.  Y B 4 3. Design of the Sampling Experiments ă Three specifications were used to represent the behavioral models underlying how people make choices. The preference specifications, including the direct and indirect utility function as well as the implied WTP function, are given in Table 1. Each individual is assumed to  Y consume two goods: a good, x1, linked to the environmental quality, and a composite good, x2.  Y R represents the level of environmental quality. Rm is some minimum essential level of R and  Y is assumed fixed across observations. x1 could be interpreted as the number of beach trips using  Yp the environmental resource with R as some aspect of the quality of the beaches being used. p1  YF is the price of x1 normalized by the price of x2. I is the normalized individual income. 1 and  Y  are parameters in the utility function. The model specifications and parameter values were selected to correspond to the characteristics generally encountered in applying these models to applications involving nonmarket valuation of environmental resources. More specifically, the variable intended to  Yh represent environmental quality enters all specifications "linked" to x1 as well as in a separable  Y: component. The former is intended to reflect use (i.e., through increasing consumption of x1 values) and the later reflects nonuse values (i.e., independent of consumption of any commodity). Model 2 introduces interactions between the marketed goods so that both commodities are dependent on environmental quality in different ways. The last model (3) adopts the Cameron [1992] specification using a quadratic utility function and allows quality to modify its parameters.T& 0*((aaԌ Y As noted earlier, an error (e) is added to the indirect utility function V(.) implied by each specification to introduce the randomness of choices originating from each behavioral model. Choices arise based on the comparison of two states: the original condition, hypothesized to be  Yz without a change in environmental quality (R=R0) and a new state, with R1 (R1 > R0), offered.  YP If V(R0, Ii, p1i)+ei0  V(R1, Iiĩti, p1i )+ei1, i.e., the offer of paying ti in exchange for the higher  Y& environmental quality will be accepted by the observation i, then the value of the choice variable  Y assumed to be observed is coded 1. If V(R0, Ii, p1i)+ei0  V(R1, Iiĩti, p1i)+ei1, then choice is  Y coded as 0. In general, the true WTP varies across individuals and is conditional on individual income, the price of the quality related good, and the parameters of the preference function. For example, the true expected WTP from model 1 is a linear function of income and price of  YH x1 while it is nonlinear in the other two models. Because the error is introduced additively to V(.), the induced error in WTP can be expected to be heteroscedastic for models 2 and 3. Specification errors seem likely to be more important with complexity of the models involved. With logit and probit models, the utility difference function is assumed to be linear in  Y income and/or price of x1. For the cubic smoothing spline estimator, the choice function is  Yj assumed to be a function of t only. In this case, the use of a single explanatory variable is common practice to avoid the dimensionality problems arising with nonparametric methods. A variety of sampling experiments were performed with different sets of values for the  Y parameters, 1 and , as well as the error variance. Six values of 1 combined with two values of  represent twelve parameterizations for each of the three underlying behavior models. Two error properties, e N(0,1) and e N(0, 0.36), are assumed. The parameter values were selected to vary the importance of the use and nonuse sources of value and to increase theZ& 0*((aa importance of the random error in influencing the generated choices. Each sample includes 200 observations (intended to represent different individuals). The  Y income and price of x1 are assumed to follow normal and uniform distributions, respectively. One of twenty values of t is randomly assigned to each of the 200 observations. The characteristics of these independent variables describing the sample are fixed in the 100 replications used for each experiment. Considering the three estimators, three models, variations in parameters and error properties, and 200 possible estimates of WTP in each sample, there are 34,200 estimates of the normalized root mean squared error. Table 2 presents a tree diagram to outline the key differences in the experimental design used for the sampling experiments.   Y4 L< 4. Simulation Results ă Table 3 reports response surface models to summarize how the estimator, attributes of the experiment, model, and sample characteristics influence the normalized root mean square. Four models are reported. The first considers each effect as independent and additive. The remaining three specifications evaluate whether the characteristics of the model, true parameter values, or sample characteristics influenced the severity of specification error associated with using simple models where the choice is hypothesized to be a function only of t. A smaller error variance and larger relative size of nonuse to use values reduce the error in all the estimators' measures of expected WTP. The mix of values selected for the income and  Y" price of the market good (i.e. x1) influence the relative size of the true WTP and how it varies across the observations in a given sample. Both are significant influences on our performance measure. As expected, the level of the proposed fee (t) is not. Models with interactions among<& 0*((aa the market goods, link the takeitorleave choice to the market goods. These links increase the error in the WTP measure, for all estimators. Most importantly, given our objectives, the nonparametric method identified in the response surface by the cubic smoothing spline (CSS) dummy variable is a negative and significant determinant of the proportionate error in the estimate for the WTP. This implies that CSS is always inferior to either probit or logit. Moreover, probit and logit are not significantly different in their WTP estimates. Our other response surface models indicate that the cubic smoothing spline's performance is also sensitive to the model considered and sample characteristics. Overall it is best when the choices are more likely to arise because the "takeitorleaveit" good makes a contribution to  Yb preferences that is independent of x1 and x2 (a feature that would mimic presence of nonuse values for environmental resources). In these cases, the specification errors associated with  Y omitting p1 and income are less important.  Y A> 5. Implications ă Nonparametric cubic smoothing splines do not offer estimators with superior properties to conventional probit or logit methods in estimating willingness to pay from discrete choice responses. Given growing interest in the use of these flexible models for commodity demand and cost functions, this finding may seem surprising. One potential reason for it follows from the criterion used in evaluating the estimatorsthe expected WTP. The maintained hypothesis underlying the cubic smoothing spline for these applications was a constant expected WTP. Thus, all the models generating our choice data introduced specification errors from the respective of CSS. That is, by considering only one determinant (the fee, t), the specification<&0*((aa requires the analyst to assume a constant WTP across individuals, when the true models imply differing WTP values with prices and income. The more important the link between the "takeitorleaveit" good (here used to represent nonmarketed environmental quality) and the other goods rationed by prices, the larger are these specification errors. Indeed a plot of the shape  YH of the true underlying distribution function, GWTP(.), versus WTP suggests a limited curvature that becomes more pronounced with income. Thus, the reduced relative error for all models (and marginally so for the cubic smoothing spline) with income reflects this connection. This feature of the true choice function implies our conclusions may not be relevant to the use of cubic smoothing splines in other contexts where there is greater curvature to this distribution function. Nonetheless, because our model specifications were selected to represent conditions often used to characterize individual preferences for environmental resources, the findings suggest that the parametric models seem to offer preferred alternatives for estimating WTP under a range of conditions.0*((aa  Y  8  8  X  6 Table 1: Three Behavioral Models ă 2  Y  Model 1:  (Madariaga and McConnell (1987)) #xddddddd  alignl{~~~~~U(x_{1i},~x_{2i},~R;~R_m)~=~ alpha _1(RR_m)x_{1i}~+~ln(x_{2i})~+~e^{ beta (RR_m)}} # {~~~} #alignl{~~~~~V(p_{1i},~I_i,~R;~R_m)~=~{ alpha _1 (RR_m)I_i} over p_{1i}~+~ln LEFT (p_{1i} over { alpha _1(RR_m)} RIGHT )~+~e^ {beta (RR_m)}~~1} # {~~~} #alignl{~~~~~WTP_i~=~{(R_1R_0)I_i} over {R_1R_m}~+~p_{1i} over { alpha _1 (R_1R_m)} LEFT [ln LEFT ({R_0R_m} over {R_1 R_m} RIGHT )~+~e^{beta(R_1R_m)}~~e^{beta(R_0R_m)} RIGHT ] }Xw PE37XPXw PE37XPXw PE37XP3=U=xYY'i=xYYiw=R'=RYY}m^=R=RYYUm=xYY)i =xYY i =eYY RYY RCCQ lm3VpYY iIYYi9RRYY?m=RRYY4mIYYivEpYYi4 pYY i ER ERYY m eYY( RYY RCC m3WTPYYBihR6RIYYiMROMRYY#mpYYiiMRMRYY#m< R RYY` m< MR MRYY` #m eYYE RYY RCC meYYRYYLRCCm=(YY1@=,YY2=,=;=)YY12=(=)YY1 =lnn =(YY 2 =)YY (YY )(YY19,,;)YY1(t)YY1lnYYw 1YY 1 E(W E)YY (YY )1<(YY1YY0)YY#1YY<1YYu#1M(YY#12M)lnYY 0YY #1YY' (CC{ 1YY= )YY(CC0YY)#==|= =YY 6q E YY^ VMvLM  M YY  YY=YYR o EYY MYY YYfnz( h( j( i o q p] *9 pvpxpw}~hji o q p*  $(#(#(#(#_!'#$  X  Model 2:  c!#x #ddddd ddP  alignl{~~~~~U(x_{1i},~x_{2i},~R;~R_m)~=~ alpha _1(RR_m)x_{1i}x_{2i}~+~x_{2i}~+~e^{beta(RR_m)}} # {~~~} #alignl{~~~~~V(p_{1i},~I_i,~R;~R_m)~=~ {(alpha_1(RR_m)I_i+p_{1i})^2} over {4 alpha_1(RR_m)p_{1i}}~+~e^{ beta(RR_m)}} # {~~~} #alignl{~~~~~WTP_i~=~{Bsqrt{B^24AC}} over {2A}} # {~~~} #alignl{~~~~~~~~~where~~A~=~ alpha _1 ^3(R_0R_m)(R_1R_m)^2} # {~~~} #alignl{~~~~`~~~~~~~~~~~~B~=~2 alpha _1^2(R_1R_m)^2I_i~~2 alpha _1(R_1R_m)p_{1i}} # {~~~} #alignl{~~~~~~~~~~~~~~~~C~=~ alpha _1(R_0R_m) ( alpha _1(R_1R_m)I_i+p_{1i} )^2~~ alpha _1(R_1R_m) ( alpha _1(R_0R_m)I_i+p_{1i} )^2} # {~~~} #alignc{~~~~~~+~4 alpha _1 ^2(R_0R_m)(R_1R_m)p_{1i}(e^{ beta (R_1R_m)}e^{ beta (R_0R_m)})}Xw PE37XPXw PE37XPXw PE37XP3m Um xYY'C im xYYC iwm R'm RYY}C m^m Rm RYYUC mm xYY)C iBm xYYC i m xYY C i m eYYl RYY RCC m3VpYY iIYYi9RRYY?mi3R 3RYY` m3IYY if3pYY iaRaRYY7mQapYY7i eYY RYY) RCCd m3WTPYYBiBwBACSA&where&A&Rl&RYYmZ&R(&RYY~mBRRYYBmIYY i R RYY mm pYY iCRSRYYumRRYY um IYY ui pYY uis RA RYY umRRYYumpIYYui pYYzuiwNRENRYY$m3NR NRYYW $m NpYY3 $ix NeYY RYY RCC }m~ NeYY RYY RCC }mm (YYC 1@m ,YYC 2m ,m ;m )YYC 12m (m )YYC 1YY}C 2YY C 2YYN (YYB )(YY19,,;)3(YY 1=3(3)YY 13)YY g2a4YY71a(%a)YY71YY (YY )YY92Q4sS2YYEm3YYE1r&(YY0&).&(YY1&)YYZ2*2YY 2YY1(YYo1)YY 22YY 1 (YY. 1A )YY 1YY,u1Y(YYu0)(YYu1(YYCu1V )YY_ u1 )YY 2YY u1G (YY u1 )(YYu1(YY1u0D)YYMu1)YY2N4YY2YY,1KN(YY$0N)N(YY$1 N)YY $1L N(YY (CCN }1YY )YY (CCT }0YY)4N)#m m m B m YY 33?a YY <c&&&P_[ Jp 8  ^NNNYYp . NYYv m YY 3=aYYe &m, A /NYY YY z*iwi-Pc$(#(# (#(# !!'#$  Y  Model 3:  (Cameron [1992]) CA3x dddddhS dd  0salignl{~~~~~U(x_{1i},~x_{2i},~R;R_m)~=~ alpha _1(RR_m)x_{1i} ~+~a_2x_{2i}~+~ alpha _2(RR_m)x_{1i}^2~+~a_4x_{1i}x_{2i}~+~ a_5x_{2i}^2~+~e^{ beta (RR_m)}} # {~~~} #alignl{~~~~~V(p_{1i},~I_i,~R; R_m)~=~D~+EI_i~+~FI_i^2~+~e^{ beta (RR_m)}} # {~~~~} #alignl{~~~~~~~~~~where~~D~=~D(R)~=~{(a_2 p_{1i} alpha _1(RR_m))^2} over {2G}} # {~~~} #alignl{~~~~~~~~~~~~~~~~~~E~=~E(R)~=~ {alpha_1(RR_m)(2a_5p_{1i}a_4)+a_2(2 alpha_2(RR_m)a_4p_{1i})} over G} # {~~~} #alignl{~~~~~~~~~~~~~~~~~~F~=~F(R)~=~{4 alpha _2(RR_m)a_5a_4^2} over {2G}} # {~~~} #alignl{~~~~~~~~~~~~~~~~~~G~=~G(R)~=~2a_5p_{1i}^2~~2a_4p_{1i}~+~2 alpha _2(RR_m)}Xw PE37XPXw PE37XPXw PE37XP3UxYY'ixYYiwRRYYCm$RRYYmxYYia< xYY i2 R RYY) m xYY i aJxYYixYY3iaxYYieYYeRYYRCC m3\V\pYY 2i\IYY2i9\R\RYY2m5\D\EIYY2i~\FIYY:i\eYY RYY RCC> mXwhereD7DR"a "pYY}i "R8 "RYY m+ PG,EAERKRnKRYY!m Ka KpYY !i Ka Kad KRKRYY[!mKaKpYY!i| yG,FAFRLRLRYY "ms La3 LazG,MGPMGMR`MaMpYY@+i` Ma MpYY@ #i MR MRYY #m(YY1@,YY2,;)YY1([)YY1YY 2YYw 2YY 2 (i )YY 2YY 1YY4YY1YY2YYS5YY2YY2YYG(YY;)\(YY219\,\,\;E\)YY2YYz (YYn )()q"(YY2YYP1YY> 1k "( ") ")YY& V2P2()YYt!1K( K)0 K(\ K2YY !5YYR !1YY+ !4X K)YY !2D K(p K2YY !28 K(K)YYZ!4YY!1K)()L4YY"2L(G L)YY "5YY 2YYv *4z2M(-M)M2YY#5YY2YY+1 M2YY #4YY #1 M2YY #2 M(H M)uB  P ;YY\\\n\YY z!"" "uK K K KKuaL LMMM Mb Ms YY YY? "K K_L` M  C$(#(#(#(# !A'#$ a#x!(dddddYdd$  C Dalignl{~~~~~WTP_i~=~{E_1+2F_1I_i+sqrt{(E_1+2F_1I_i)^24F_1^2 LEFT [D_1D_0+e^{ beta (R_1R_m)}e^{ beta (R_0R_m)}+(E_1E_0)I_i+(F_1F_0)I_i^2 RIGHT ]}} over {2F_1^2}} # {~~~} #alignl{~~~~~~~~~~where~~D_0~=~D(R_0);~~E_0~=~E(R_0); ~~F_0~=~F(R_0)} # {~~~} #alignl{~~~~~~~~~~~`~~~~~~D_1~=~D(R_1);~~E_1~=~E(R_1); ~~F_0~=~F(R_1)}Xw PE37XPXw PE37XPXw PE37XP3(WTPYYBi<EMFIYYgiEFjIYYgiFE D" D eYY RYY, RCCg m eYY RYY2RCCmm7EEIYYgiuFCFIYYoi3 FXrwhererDdrDrR4rEvrErR7 rFy rF rRMDQMDMR!MEcMEMR$ MFf MF MR(Y   YY  YY rr rMM MO*GpRpeRQ~* Rt~RR{YYg1 2YYg1(YY,g12YY=g1)YY2]4YY2YYo1YY g1YY g0YY (CC 1YY )YY (CC 0YY) (YYg1YYVg0)I(YYg1YYg0)YY%2 2YY 2YY u1YYsH0r(YYAH0nr)r;YYH0r(YYD H0q r) r;YY H0 r(YYG H0t r)YY`#1M(YY.#1[M)M;YYr#1M(YY1 #1^ M) M;YYu #0 M(YY4 #1a M)YYF YYL C $(#(##(#(#_!a'#$2`)-,,55A'##'#a!'#I(A('#;.a`  c    2 Table 3: Fitted Response Surfaces for  NRMSE ă 20  xP #c PE37 P# 44 % - (1)<<6> (2)FDDO (3)W_ (4) 20  xPs Intercept % -12.1210***>11.6923***DDO11.5349***_11.9844***  xP; 44 % -(15.6007)a>(14.2031)DDO(13.0749)_(14.0167)  xP Income44 % -1.3510***>1.3510***DDO1.2727***_1.2724*** 44 % -(13.4013)>(13.4052)DDO(10.8404)_(10.8349)  zP# p144 % - 7.9263***> 7.9263***DDO 7.1219***_ 7.1208*** 44 % -(10.4779)>(10.4810)DDO(8.0827)_(8.0803)  zP} t (proposed fee) % -0.2901<<6> 0.2901FDDO 0.2901W_ 0.3175 44 % -(0.7831)>(0.7833)DDO(0.7832)_(0.7357)  zP V=f(fee) Dummy -0.8547***>0.5093FDDO0.6432**_0.8547*** 44 % -(2.7151)>(1.5142)DDO(1.9659)_(2.7151) Probit Dummy % - 0.0255<<6> 0.0255FDDO 0.0255W_ 0.0255 44 % -(0.0993)>(0.0993)DDO(0.0993)_(0.0993)  xP Cubic SS Dummy - 1.9838***> 2.8408***DDO 3.7079**_ 2.5029* 44 % -(5.4703)>(2.9973)DDO(2.3710)_(1.7826)  zP 144 % -94.1486***>90.4797***DDO91.0139***_94.1486*** 44 % -(14.2803)>(11.8630)DDO(11.9385)_(14.2806)  zP; 44 % -52.6495***>45.2084***DDO47.3204***_52.6495*** 44 % -(15.3022)>(10.6428)DDO(11.3811)_(15.3026)  zP e N(0,.36) Dummy -2.0510***>1.6337***DDO2.0335***_2.0510*** 44 % -(4.8118)>(3.1533)DDO(4.7702)_(4.8119)  xP True Specification -3.3350***>3.1492***DDO3.2602***_3.3350*** Dummy % -(5.7032)>(5.3690)DDO(5.5659)_(5.7033)  xPG Model2 Dummy % - 1.4491***> 1.2158***DDO 1.4491***_ 1.4491*** 44 % - (4.9223)>(3.6148)DDO(4.9228)_(4.9225)  xP Model3 Dummy % - 7.1129***> 6.5181***DDO 7.1129***_ 7.1129*** 44 % -(24.1621)>(19.3793)DDO(24.1645)_(24.1628) CSS*Income % -<<6>FDDO0.2974W_0.2985 44 % -<<6>FDDO(1.3003)_LLh(1.3040)  xPO% CSS*p144 % -<<6>FDDO 3.0566*_ 3.0610* 44 % -<<6>FDDO(1.7798)_(1.7819) CSS*t44 % -<<6>FDDOW_0.1041 44 % -<<6>FDDOW_(0.1237)  xP) CSS*144 % -<<6>14.1007DDO12.0478 44 % -<<6>(0.9559)DDO(0.8187)*-,,55Ԍ xP ԙCSS*44 % -<<6>22.5427***DDO15.3313** 44 % -<<6>(3.1064)DDO(2.2497) CSS*e44 % -<<6>0.9935 44 % -<<6>(1.1138) CSS*Model2 % -<<6> 0.9330 44 % -<<6>(1.4323)  xP CSS*Model3 % -<<6> 2.3792*** 44 % -<<6>(3.6525)  xP` R244 % - 0.04198> 0.04268DDO 0.04228_LLh 0.04212 F Statistic % -124.8364> 89.6480DDO 94.3129_100.1976 Degrees of Freedom - 34187<<6> 34182FDDO 34183W_ 34184 20  xP ***Significant at the .01 level.  xPH **Significant at the .05 level.  xP *Significant at the .1 level.  xP aNumbers in parentheses are the ratio of the estimated parameter to the estimated standard error.-,,55  Y # Xw PE37XP#C REFERENCES ă X44Ahn, Hyungtaik, 1995, "Nonparametric Twostage Estimation of Conditional Choice  Yx Probabilities in a Binary Choice Model Under Uncertainty," Journal of Econometrics, 67, (June): 337378.(#4 X44Cameron, Trudy A., 1988, "A New Paradigm for Valuing NonMarket Goods Using Referendum Data: Maximum Likelihood Estimation by Censored Logistic  Y Regression," Journal of Environmental Economics and Management, 15: 355379.(#4 X44Cameron, Trudy A., 1992, "Combining Contingent Valuation and Travel Cost Data for ""{the  Yl Valuation of NonMarket Goods," Land Economics, 68: 302317.(#4 X44Chen, Heng Z.: "SemiNonparametric Estimation of Constrained Qualitative Response Models With Application to Contingent Valuation of Environmental Resources," unpublished, Dept. of Agricultural Economics, Ohio State University, August 7 (1992).(#4 X44Cox, Dennis D., and Finbarr O'Sullivan, 1990, "Asymptotic Analysis of Penalized  Y\ Likelihood and Related Estimators," The Annals of Statistics, 18: 16761695.(#4 X44Craven, P., and G. 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Raynor, Jr., 1986, "Automatic  Y Smoothing of Regression Functions in Generalized Linear Models," Journal of  Y American Statistical Association, 81, (March): 96103.(#4-,,55  Y   B ENDNOTES ă *XAssistant Professor, Department of Economics, East Carolina University; Professor, Department of Statistics, North Carolina State University; Arts and Sciences Professor of Environmental Economics, Duke University and University Fellow Resources for the Future, respectively. Huang and Smith acknowledge partial support from the UNC Sea Grant Program under Grant #RMD/R23.(# II-,,55  Y B APPENDIX ă As a rule we can appeal to the Slutsky Theorem to establish that any continuous function of a consistent estimator is consistent. The Theorem developed below (due to Cox and O'Sullivan [1990]) describes the specific norm that assures consistency for the cubic smoothing spline. We summarize their Theorem and the implications below.  W Theorem A: Suppose m2 and f0W2    mp $ [0,1], where 3/(2m)lzz'  zz zz zz. ,#zzR zzzzuuy T$(#(# (#(#v!'#$where   2 = <,> is the squared norm of W2   6 mb $  with the inner product defined as  Y  #xAddddd{3dd x?&alignl{ (AI)~~~~```~~~~~~ LANGLE theta , zeta RANGLE _{W_2 ^m}~=~ LANGLE theta , zeta RANGLE _{L_2}~+~ LANGLE theta ^{(m)}, zeta ^{(m)} RANGLE _{L_2}} # {~~~} #alignl{~~~~~~~~~~~~~~~~~```~~~=~~ INT~ theta zeta~ dx~+~ INT~ theta '' zeta ''~dx}Xw PE37XPXw PE37XPXw PE37XPm(m)'m,\\J2@m,\\A 2zz (zz ) m,zzC (zz )\\\ 2RmAIzz/W\\frmzz3Lzz0 mzzl mzz 3L zdx zdxmmmm} m mL= LmUm mnm  m m lzz  z z m mz zzza zz ?߼$(#(# (#(#!'#$  f0 is the true function that maximizes L(f), the limiting function of Ln. 5d   Yo #xIdddddddbx&y(A2)~~~~~~~~~L(f)~=~lim from {n  INF }~L_n(f)~=~ INT from a to b ~LEFT [pi(f_0(x))f(x)log(1+e^{f(x)}) RIGHT ] g(x)dxXw PE37XPXw PE37XPXw PE37XP ( ) ( )P lim= ( )Y (zz 0 ( ) )A ( )z loge ( 1zzS(zzOS)x )= ( )R A2/ L fzzbmn Lzznz fzz bzz 2a fC x f~ xl ezzSfzzSx gz x dx zzmzzm:     L t{ !ߔ$(#(#o(#(#!'#$According to the theorem, as the smoothing parameter goes to zero and n is large, the  Y distance between the solution to (3) and the true function f0 is less than a small number with  Y" probability one. A heuristic description is as follows. Suppose that fn is the unique maximizer  Y~$ of (3). The difference between the unique maximizer of L, f0, and fn is called the estimation error. The estimation error consists of a systematic error and a stochastic error. The two error components represent the bias and the sampling variabilities, respectively, as reflected in the following equation.P)-,,551'#A'#'#P#PԌ Y ԙ(A3)` ` ! fn f0 = (f f0) + (fn f),  Y where f is the maximizer of L(f)((f.)2dx. The difference between f and f0 is attributable to the addition of the penalty function; hence, it is a systematic error and is independent of the  YP sampling. The difference between fn and f is a random error due to sampling. Cox and O'Sullivan (1990) analyzed both of these errors and the overall accuracy is the content of Theorem 2. With some additional assumptions, Theorem A can be used to show the consistency of the welfare measure developed from these estimates. Considering the absolute difference  Yn between W(fn) and W(f0), #xydddddddxo&.(A4)~~~~~~~~~ LINE W(f_{n lambda })W(f_0) LINE ~=~ LEFT LINE ~INT from 0 to INF ~LEFT ({e^{f_{n lambda }(x)}} over {1+e^{f_{n lambda }(x)}} {e^{f_0 (x)}} over {1+e^{f_0 (x)}} RIGHT ) dx~+~ INT from { INF } to 0 ~LEFT ( 1 over {1+e^{f_{n lambda} (x)}} 1 over {1+e^{f_0(x)}} RIGHT ) dx ~RIGHT LINE Xw PE37XPXw PE37XPXw PE37XP ( ) ( )K (zz0 )zz 40zz] (zz ) @1zz (zz! )\\0zz(zzn) @1\\F0zzt(zz)zz;0Er1T@1zz (zzh)r1?@1\\0zz(zz)R A4x WM fzzn W fr rezz f\\ nzz x @ezz9 f\\[ nzz x]rezzfzz8x@ezz$fzzxV dx@ezzf\\nzz2x @ezzkfzzx dx/  E 5   zz i @c T@S zz4zzZ4@ @zz\\$ \\ \\ 5   ] 5] L ~h 6j j i~o6qqpA 8 8!L6hji\6o\q\pA<8'8o$(#(#D(#(#!'#$  Y Formally, the expectation of WTP requires integrating the probability curve from  to . In fact, a welfare measure is usually finite and positive (with rare cases that are negative) in that its distribution is truncated at some finite values. If the welfare measure is assumed to be in a  Y finite range [éN,M] where N and M are positive finite numbers, then the righthand side of (9)  Y has finite integrals from éN to zero and zero to M. This is a plausible assumption and it enables us to apply Theorem A.  Y# Since both ef/(1+ef) and 1/(1+ef) are monotonic functions, there exists a function  Yq% f  *(x)=f0(x)+(1)fn(x), 01 and =(x) such that equation (9) with the assumption of  YG' finite range [éN,M] can be rewritten as follows.  3x5ddddd(dd6x&(A5)~~~~~~~~~ LINE W(f_{n lambda })~~W(f_0) LINE ~=~ LEFT LINE ~INT from 0 to M {e^{f_0}} over {(1+e^{f_0})^2}(f^*f_0) dx ~~ INT from {N} to 0 {e^{f_0}} over {(1+e^{f_0})^2}(f^*f_0)dx~ RIGHT LINE Xw PE37XPXw PE37XPXw PE37XP ( ) ( ) (zz[0 )zz 40\\h 0# @(` @1\\ 0 @)zz 2o (zz0 )zz=0\\0@( @1\\[0@)zz2 (zz0 )R A5x WM fzznS W( fzz M rezzF f* @ezz f f fT dxzzX4Nrezzf@ezz9fY fT f dx/     n  @zz U9 Q zz4i@zzU zz, 5, , , ], 5] L/ 8#L/8߃  Y* Notice that equation (10) is derived by linearizing the logistic function at f0 and is not based on@*-,,55!y'#m5'#) @ $(#(#(#(#!'#$the intermediate mean value theorem. By the CauchySchwartz inequality, B!3xf ddddd-ddx& alignl{ (A6)~ LINE W(f_{n lambda })W(f_0) LINE ~ <= ~SQRT { INT from 0 to M ({e^f_0} over {(1+e^{f_0})^2} )^2 dx~ INT from 0 to M (f^* f_0) ^2 dx}} # {~~~~} #{~~~~~~~~~~~~~~~~~~~~~~+~SQRT{ INT from { N } to 0 ({e^{f_0}} over {(1+e^{f_0})^2})^2dx~ INT from { N } to 0 (f^*f_0)^2dx}}`.Xw PE37XPXw PE37XPXw PE37XPb(b)b(b)b(zz7(0tb)zz0b(\\ 0(1\\ 0: )zzw 2 b)zz 2zz^ 0 b(zz!(0^b)zz2zz0 (\\ 0_ @( @1\\ 0 @)zzU 2 )zz U2zzP 0 (zz'0d )zzU2 .RbA6bWbfzz(n/bWbfzzZM6 ezz /f ezz fG bdxzzJ ZM bf bfbdxzz4N rezz ff @ezz f% dxzzk 4N f f dxb bb Jb zz7  b zza4 @zz" 4zz=U zz9(9 ccTSSSCSjRL/iD L aTS~S6SSRuL/G 86 LB$(#(#1(#(#1!!'#$  Y4 If the expected willingness to pay, W(f0), is finite, (ef/(1+ef)dx will be finite. Hence,  Y  ((ef/(1+ef))2dx is finite and can be set to be some constant. It is seen that A 3x5dddddzdd x&(A7)~~ LINE W(f_{n lambda })~~W(f_0) LINE ~ <= ~C_1 SQRT{ INT from 0 to M (f^*f_0)^2dx}~+~C_2 SQRT{ INT from { N } to 0 (f^* f_0)^2 dx } # {~~~} #alignr{ <= ~C_1 SQRT{ INT from { N} to M (f^*f_0)^2 dx}~+~C_2 SQRT{ INT from { N} to M (f^*f_0)^2 dx }} # {~~~} #alignc{``~~~~ =~ C~ SQRT{ INT from { N} to M (f^*f_0)^2 dx}} # {~~~} #alignc{~~~~~<= ~C~ SQRT{ INT from { N} to M (f_{n lambda }f_0)^2dx}} # {~~~~} #alignr{~ (BECAUSE~ f^*~=~ beta f_0~+~(1 beta )f_{n lambda },~~0 <= beta <= 1)} # {~~~} #alignc{~~~~~~~~`` =~C~ DLINE~ f_{n lambda }~~f_0~ DLINE _{L_2[N,M]}} # {~~~} #alignr{~~~=~C~DLINE~f_{n lambda}~~f_0~DLINE_{W_2^{bm}[N,M]}~~~``with~ b=0,}Xw PE37XPXw PE37XPXw PE37XP()()(zz(0e)zzs1zzm E0 (zz0 0m )zz f2zz2zz*0(zz0>)zz{f2zzK 1 (zz0 0m )zz  2zz 2 (zz 0> )zz{ 2 t (zzX : 0 t )zz 2 (zz 0 )zz g2G(zz 0 ( 1T)L,01()zzE 0\\_ 2zz [zzP ,zz ]zzD U0\\ *2zz Q[zz Q,zzCQ]07,RA7EWfzzMn WfCzzY M f f dx CzzEENffdx CzzE Mzzt N f f dx Czz MzzE N f f dxt Czzm l Mzz N* t f% t f t dxCzzj Mzz FN' fzzZ nP f: dxB fV ffzzn@CI @fzz| n @fzz Lzz Nzzo MCH fzz{ Un fzz QW\\ bmzzm QNzz QMwithb b ;zzF f  zzEzzf` zz+ zzF    zz zz ` (t zzS zzn  t %zzP F ezz   m w^(@@T @ @zz 'S  zz$ Qmzzzz   zzzz zz US !!TSSISS(RS L!Y!TYSYSYISYSY(RL+  T S< S S S R? c L Y TY SY< SY SY SY Rc LS w w T/ S S SW S~Rg  L!P ""TSSJSS)Rd Lߕ  Y( where C1, C2, and C are some fixed positive constants. Thus,  W(fn)W(f0)  is less than or  Y equal to the norm of fnĩf0. That is, by setting b=0 in Theorem A, when the number of  Y observations is large and the smoothing parameter is very small,  W(fn)W(f0)  is less than some  Y" small number   with probability equal to one. The consistency of the welfare measure with finite range derived from the penalized likelihood function is verified. #Xx PE37XP#