*Corresponding author. Tel: (603) 862-3279; fax: (603) 862-3383.
E-mail addresses:jchuang@cisunix.unh.edu (J.-C. Huang), nychka@ucar.edu (D.W. Nychka).
A nonparametric multiple choice method
with-in the random utility framework
Ju-Chin Huang
!,
*, Douglas W. Nychka
",#
!Department of Economics, University of New Hampshire, McConnell Hall, Durham 03824, USA
"National Center for Atmospheric Research, Climate and Global Dynamics Division, Boulder, CO80307, USA
#Department of Statistics, North Carolina State University, Raleigh, NC27695, USA Received 1 August 1997; received in revised form 28 July 1999; accepted 1 August 1999
Abstract
Many researchers use categorical data analysis to recover individual consumption preferences, but the standard discrete choice models require restrictive assumptions. To improve the #exibility of discrete choice data analysis, we propose a nonparametric multiple choice model that applies the penalized likelihood method within the random utility framework. We show that the deterministic component of the random utility function in the model is a cubic smoothing spline function. The method subsumes the conventional conditional logit model (McFadden, 1973, in: Zarembka, P., (Ed.), Fron-tiers in Econometrics) as a special case. In this paper, we present the model, describe the estimator, provide the computational algorithm of the model, and demonstrate the model by applying it to nonmarket valuation of recreation sites. ( 2000 Elsevier Science S.A. All rights reserved.
JEL classixcation: C14
Keywords: Polychotomous choices; Cubic smoothing splines; Random utility model;
Welfare measurement; Nonmarket valuation
1Through December, 1998, the seminal paper by McFadden (1973) has generated over 1000 citations according to the Social Science Citation Index and the Science Citation Index. The total number of citations on the sequence of papers by McFadden on this subject is over 1600. 1. Introduction
Following the development of the random utility model (RUM, McFadden, 1973) and given the increased availability of detailed micro-data that describe people's consumption choices, categorical data analysis has become an impor-tant tool for recovering individual preferences1. In particular, multiple choice analysis based on RUM is now used to analyze market data for the study of individual choice decisions in "nance, marketing research, transportation re-search, occupational choices, and many other"elds. In the past twenty years, multiple choice methods have also been applied to nonmarket valuation to examine individuals' consumption choices of nonmarket goods. For example, researchers and policy makers may want to estimate the recreational value of improving quality of a sport"shing site or to derive the welfare loss of closing a recreation site for commercial uses (e.g., Morey, 1981; Bockstael et al., 1991; Morey et al., 1991; Kaoru et al., 1995; Kling and Thomson, 1996). Under certain assumptions, these bene"ts or losses can be estimated through individuals'
recreational choice decisions.
The basic assumption of RUM is that the utility function is unknown but it can be partially recovered by relating an individual's consumption choices to the individual's socio-economic characteristics, the relative prices that he or she faces, and the characteristics of the goods available for consumption. A part of the utility function, however, cannot be recovered by the researcher because of unknown factors about individuals. Hence, the utility function in RUM contains two components: a deterministic component that can be explained by known factors and a random error component that indicates the unknown variation across individuals. Conventional discrete choice models cast within the random utility framework assume a parametric speci"cation for the deterministic com-ponent and a distributional assumption for the error comcom-ponent of the random utility function. Since the true preference structure cannot be actually observed, any model entails some degree of misspeci"cation. Because the conventional models impose these parametric assumptions, they tend to increase the degree of misspeci"cation, which frequently leads to biased estimates of marginal values (Hanemann, 1984; Ozuna et al., 1993).
been applied to binary choice, multiple choice, sequential choice, and contingent ranking models (e.g., Manski, 1975,1985; Cosslett, 1983; Gallant and Nychka, 1987; Han, 1987; Ichimura and Lee, 1991; Horowitz, 1992; Klein and Spady, 1993; Lee, 1995). Relaxing the distributional assumption on the random com-ponent of RUM helps provide a better connection between choices and underly-ing preference structure, although misspeci"cation of the deterministic component of the utility function can still be present.
The second group of estimation methods generalizes the discrete choice models by relaxing the parametric speci"cation assumptions of the deterministic component of RUM. Most of the studies in this group focus on the binary choice data analysis (Matzkin, 1992; Manski, 1991; Ahn and Manski, 1993; Huang et al., 1999). The two multiple choice applications in this group are Matzkin (1991) and Abe (1999). Ahn (1995) combines the approaches applied in these two groups and proposes a nonparametric binary choice model that uses both a distribution free error structure and a nonparametric deterministic utility function.
In general, the theory of nonparametric binary choice methods is studied more extensively. Although there are a few existing applications, little theory on nonparametric multiple choice methods has been explicitly addressed. In this paper, we examine a nonparametric multiple choice model that assumes a smooth nonparametric deterministic component of the utility function to allow more#exibility in describing the impact of a particular variable on utility. The proposed model"ts into the second group of nonparametric generalization of the discrete choice methods. The objective function of this model is a penal-ized likelihood function that contains a likelihood function and a roughness penalty function to control the smoothness of the nonparametric utility func-tion. When the penalized likelihood function is maximized over the second-order Sobolev function space, we show that the deterministic component of the random utility function is a cubic smoothing spline function. As the penalty of roughness increases, the cubic smoothing spline function approaches linearity. Hence, the proposed method subsumes the conventional conditional logit model as a special case and allows for more#exible"tting of the utility function in analyzing multiple choice data. The method extends the use of the penalized likelihood method in binary choice data analysis by O'Sullivan et al. (1986) and Cox and O'Sullivan (1990). This paper provides the basic theory behind the applications of spline"tting to multiple choice data. In addition, we apply the method to nonmarket valuation of recreation site choices for its need for more
#exible estimation of the marginal utility of income.
2This expression is in fact a conditional indirect utility function given that one of theJgoods is consumed. It is, however, unconditional comparing to the expression of the indirect utility function in (2). In this paper, we assume that one and only one of theJgoods is consumed. For an analysis that incorporates no consumption of allJgoods, see Morey et al. (1991).
(WTP) for maintaining the access to a recreation site. We discuss the WTP estimators implied by both the conventional conditional logit model and the proposed nonparametric multiple choice method. Section 5 provides a case study that estimates the WTP for maintaining access to a recreation site in the Monongahela River Basin in Pennsylvania. Section 6 gives concluding remarks and suggests future research.
2. Review of multiple choice random utility model
Consider a utility maximization for the individual i subject to his budget constraint.
max
yij ;(y
i1,2,yiJ,qi1,2,qiJ,zi,ai)# J
+
j/1
y
ijeij
s.t. I
i" J
+
j/1
p
ijyij#zi (1)
where;()) is the utility function.y
ij's indicate the consumption of a group of
Jgoods that y
ij"1 if the jth good is consumed and yij"0, otherwise. The
consumption of these J goods is assumed to be mutually exclusive; that is,
y
ilyik"0 for alllOk.qij is the quality attribute corresponding toyij.zi is the
numeraire good and a
i is a vector of exogenous variables. eij's are random
variables. The randomness of the utility function represents the unknown characteristics of an individual.p
ij is the price or cost of consumingyij.
It is assumed that the partial derivative of the direct utility function with respect to q
ij is zero if the jth good is not consumed. This is the weak
complementarity assumption introduced by MaKler (1974). Under this assump-tion, an individual does not care about the quality attributes of a good if it is not consumed. Hence, the indirect utility function conditional on the consumption of thejth good is
;ij"<(q
ij,Ii,pij,ai)#eij"<ij#eij. (2)
For the jth good to be chosen for consumption, the utility derived from consuming thejth good must be the highest. Consequently the unconditional indirect utility function for the individualiis the maximum of theJconditional indirect utility functions, ;
3The i.i.d. extreme value distributional assumption of errors implies the hypothesis of indepen-dence of irrelevant alternatives (IIA). Under IIA, the ratio of two probabilities (e.g., p
*1/p*2) is independent of the other goods available. When two or more of the J alternatives are close substitutes, the IIA assumption is not plausible. The IIA hypothesis can be lifted by assuming a generalized extreme value distribution for the errors, which allows interdependence across choices. The relaxation of the i.i.d. and IIA assumptions will be considered in future research.
(nij) that the jth good is consumed by the individual i, given that one of the J goods is consumed, is equivalent to the probability that maxM<
ik#eik, fork"1,2,JN"<ij#eij";ij. Hence,nij is the probability
that the Jth order statistic of (;
i1,;i2,2,;iJ) is ;ij, for any value of ;ij.
Assume that e
ij's are independent and identically distributed (i.i.d.) and each
follows a type I extreme value distribution with location and scale parameters being zero and one, respectively3. It can be shown that probability of consuming goodjby the individualiis
nij"Pr(;
(J)";ij)"eVij
N
+J k/1eVik (3)
where;
(J) is theJth order statistic of (;i1,;i2,2,;iJ). Once the functional
form of the utility function is speci"ed, we can apply the maximum likelihood estimation method to estimate the probabilities and the parameters in the utility function.
max¸"+n
i/1
(y
i1log(ni1)#yi2log(ni2)#2#yiJlog(niJ)) (4)
where¸is the log likelihood function. Notice that the expression ofn
ij in (3) is
invariant to an additive constant of all V's. Hence, the conditional indirect utility function can only be uniquely estimated up to an additive constant. Nevertheless, the welfare measures derived from this model are uniquely de"ned because the Hicksian measures of welfare changes are invariant to monotonic transformations of the utility function (Mishan, 1977).
3. Nonparametric multiple choice method
method to incorporate the theory of splines into the conditional logit model. In this section we show the characterization and representation of the estimator, discuss the determination of smoothness of the utility function, and provide the computational algorithm.
3.1. Smoothing splines in analyzing multiple choice data
The penalized likelihood method is initially proposed by Good and Gaskins (1971). Anderson and Blair (1982) apply the estimation method to logistic regression that examines impact of individual characteristics on binary choices. In this paper, we examine a nonparametric multiple choice model within the random utility framework that considers the e!ect of a choice speci"c variable on choice decisions. The following objective function is used to derive the optimal nonstochastic component of the utility function in RUM based on multiple choice data.
Assume that<, the deterministic component of the random utility function, is a function of a choice speci"c variable,x. The double subscript ofxemphasizes that the value of x varies with both individuals (i) and choices (j), and
a)x
ij)b. It is assumed that the values of xij's are distinct. The penalized
likelihood functionQ(<) consists of two terms. The"rst term,¸()), is the log
likelihood function for a discrete response model. The second term, :"!(VA(x))2dx, is the integral of the squared second derivative of <(x). This is commonly called the roughness penalty function that measures the roughness of the"tted<(x) curve. The measure increases with the roughness of<.
Instead of assuming a parametric function, <is assumed to belong to the function space=2
4=
that any function in the space has absolutely continuous "rst derivative and square integrable second derivative on the interval [a,b].jis called the smooth-ing parameter for its ability to determine the degree of smoothness. The value of j controls the relative weights of the two terms in the penalized likelihood function. Ifjis set very large, it imposes a high penalty to roughness of the"tted function. Asjgoes to in"nity, to maximize the penalized likelihood function the second term must be set to zero, which implies a zero second derivative and a linear function for <. Thus, the proposed nonparametric multiple choice model subsumes the linearity of<as a special case.
There are various presentations of the solution to (5). A well-known repres-entation employs reproducing kernel Hilbert space theory. Suppose that =2
ais the left end point of the range. With the inner product being de"ned as
Sg
It should be noted that while reproducing kernels are associated with a function space, the veri"cation of the formula in (6) can be established simply by integration by parts twice. In particular, the inner products of representors on =2
2,BC associated with the x's are piecewise cubic polynomial functions of the
5The theorem proceeds under the assumption that a maximizer exists. Due to the highly nonlinear forms in the likelihood, the conditions to ensure a maximizer are not obvious. Also, in conditional logit models, certain restriction must be imposed in order forVto be identi"able. The identi"cation condition could be setd
0,0. The restriction imposed in the estimation is described in9. 6In de"ning the set ofoxij, it is necessary that allMx
ijNbe distinct. For de"ning the basis, repeated values should be dropped. For example, letMx
iNdenote the unique set of values in the covariates Mx
ijN, thens(x)"d0#d1#+cioxi(x).
7Cox and O'Sullivan (1990) provide an error analysis of the binary choice case for the penalized likelihood estimation. Under suitable conditions onjand sample size they establish existence of the estimator and its consistency. The extension of their results to the multiple choice case should be straight forward but beyond the scope of this paper.
With this form for g, one can then maximize (5) over the linear portion involvingd
0 andd1. The following theorem suggests a representation of the
solution to (5).
Theorem. If there exists a maximizer to (5), then it must have the form
s(x)"d
s(x) is a natural cubic smoothing spline function. That is,s(x) is a piecewise cubic polynomial with continuous derivatives up to order 2 at x (the knot) within the data range and is linear beyond the data range. The theorem says that the estimator of <(x), for any value of x, has the form described in (8). The proof of the Theorem is in the Appendix. Hence, for a"xedj, the maximizer of the penalized likelihood function in (5) is a natural cubic smooth-ing spline function. The basic approach of continuous spline smoothsmooth-ing is expounded by Wahba (1990). Our model extends her work to discrete choice data analysis.7
Lets(x
hl) be the estimator of<evaluated atxhl. Applying the formula in (6), we
have: s(x
just identi"ed by thenJobservations plus the boundary conditions that restrict the"tted function to be linear outside the data range. The boundary conditions
sA(a)"s@@@(a)"sA(b)"s@@@(b)"0 can be represented by ¹Tc"0 (¹&transpose' c"0), which adds two constraints to the estimation: (¹
This can be veri"ed by taking the second and the third derivatives ofs(x) with respect to x and setting them equal to zero. Consequently, the optimization problem in (5) is equivalent to the maximization over a"nite collection of basis functions [¹,K] to derive [d,c] inS(x).
3.2. Computation and smoothness of the penalized likelihood estimate
It is assumed that there is one preference structure to characterize individuals'
choice decisions. Hence, there is only one nonparametric function to be esti-mated in this multiple choice setting. In estimation allx
ij's are stacked"rst by
choices (j) then by individuals (i) to create the value vector of the explanatory variable. The estimation of < is based on nJ observed values of <(x
ij)
with choice probabilities summed to one for each individual. x
ij can be a
characteristic or the price of the good j, or any economic variable that varies across choices. For simplicity, we will use<
ijfor<(xij) in the rest of this
paper.
For computational purpose, we want to rewrite the maximization problem into a minimization problem by examining the "rst-order conditions of the likelihood function, ¸. The partial derivatives of the log-likelihood function with respect to <
wherenijis the probability that the individualiselects thejth choice. By taking the"rst order Taylor series expansion at some initial value<0
Letz The expression in (10) is equivalent to the "rst derivative of
!+
i+j(zij!<ij)2wij that can be written in a matrix form:
!(Z!S(x))T=(Z!S(x)), whereZis the vector ofMz
ijNnJC1,=is thenJ]nJ
matrix with the diagonal elementsMw
ijNnJC1 and 0 elsewhere, andS(x) is the
estimator of the vector of M<
ijNnJC1. Further, substituting the estimator of
M<
ijNnJC1,S(x)"¹d#Kc, into the roughness penalty function, it can be easily
veri"ed that:[SA(x)]2dx":[(+ni/1+Jj/1c
ijoAxij(x))]2"cTKc(Green and
Silver-man, 1994,pp. 24}25). Recall thatK is annJ]nJmatrix of inner products of representors associated with thex's andcis thenJ]1 vector ofMc
ijN.
Substitu-ting [¹d#Kc] forM<
ijNnJC1andcTKcfor the roughness penalty function, the
maximizer of the optimization problem in (5) can be approximated by the minimizer of the following optimization problem (expressed in a matrix form).
min The expression in (12) can be viewed as a generalization of the ridge regression model. Hence, the smoothing spline estimator is a generalized ridge estimator that belongs to the family of shrinkage estimators. After simple manipulation, the normal equations for minimizing (12) can be presented as follows.
¹d#(j=~1#K)c"Z,
¹Tc"0. (13)
Notice that for a"xed value ofj, there arenJ#2 equations to solvenJ#2 unknowns. As discussed in Section 3.1, equations¹Tc"0 imply the boundary conditions.
The method can be extended to account for repeatedxvalues, it is computa-tionally more complex due to an additional step in updating zand w in the estimation. Nonetheless, the basic model is the same.
8The theoretical background for using GCV function to select the optimaljvalue in the binary choice model can be found in O'Sullivan et al. (1986).
9Notice that V cannot be uniquely identi"ed in the conditional logit models. In estimation a constraint must be imposed to hold the values ofVwithin a range. We restrict the mean of the estimated<
ijoverjto be zero in each iteration. The constraint dictates the values of the"tted utility function. However, the scaling does not a!ect the probability and welfare estimation since they are invariant to monotonic transformations of the utility function. Hastie and Tibshirani (1990) suggest a similar restriction in estimating the generalized additive models to resolve the identi"cation issue they called concurvity.
popularity, we choose to use the generalized cross-validation (GCV) as the criterion to determine the optimalk}a data based method proposed by Craven and Wahba (1979). LetFK be the vector of estimatesFK"¹dK#Kc("H(j)Zfor a givenj.H(j) is the weight matrix commonly called the hat matrix. The GCV function for selectingjin the proposed multiple choice model can be formulated as follows.8
GC<(j)"+ni/1+Jj/1(yij!nHij)2/wHij
[n!Tr(H(j))]2 , (14)
wherenHijis the estimated probability;wHijis the estimated variance; and Tr(H(j)) is the trace of the hat matrix. The optimaljis determined by minimizing the GCV function with respect toj.
To compute the estimates for <(x), we suggest a computational algorithm that is equivalent to a combination of the iteratively reweighted ridge regression procedure (O'Sullivan et al., 1986) plus an embedded probability constraint to incorporate the restriction that choice probabilities for each individual must sum to 1. We de"neconst
ij"+kEjeVikNnij"eVij/(eVij#constij). The variable
const
ij is used to impose the constraint in the estimation/iterations that
const
ij#eVij is a constant across j for the individual i. The computational
algorithm to estimatedandcis summarized as follows.
(i) Read in (x
ij,yij) and stack the data by choices (j) then by observations (i).
(ii) Create a grid ofjvalues. (iii) Create pseudo data z
ij"(yij!nij)/(nij(1!nij))#<ij and
w
ij"nij(1!nij)/2.
(iv) Estimatedandcfor a given value ofj9. (v) Calculate predictedconst
ij andnij; calculate predictedzij andwij.
(vi) Repeat steps (iv) and (v) with z
ij and wij updated each time until nij's
converge; calculate Tr(H(j)) andGC<. (vii) Changejvalue; repeat steps (iv), (v), and (vi).
(viii) Determine the optimaljaccording to the minimumGC<.
(ix) Repeat steps (iv), (v), and (vi) withj"optimalj; calculate predicted<(x
ij)
andn
10If the conditional indirect utility function is not quasi-linear in income, the loss (WTA or compensating variation) of eliminating an alternative and the gain (WTP or equivalent variation) of maintaining an alternative will di!er. The WTP for maintaining alternativetis derived by solving
J + j/1
eV(pj,qj,I~WTP)"+J jj/1Et
eV(pj,qj,I),
and the WTA for losing alternativetis solved from J
+ j/1 jEt
eV(pj,qj,I)"+J j/1
eV(pj,qj,I`WTA).
4. Willingness to pay for the accessibility to a good
Multiple choice models have been widely applied in describing the relation-ship between recreation sites and the site characteristics. Individual choices of recreation sites can be used to recover his value for a particular site. Consider an individual who wants to go to a beach withJpossible sites from which he can choose. Once he decides on a site to visit, he cannot visit the other sites simultaneously and it is reasonable to assume that he does not care about the quality attributes of the other sites. If the random errors are assumed to follow the i.i.d. extreme value distributions, it can be shown that the expected utility of the individual i who faces J site choices is E(max(<
i1#ei1,
<
i2#ei2,2,<iJ#eiJ))"ln(+Jj/1eVij)#k, where k is the Eulers constant
+0.57 (Bockstael et al., 1991). Based on the conditional utility functions for all site choices, we can derive the willingness to pay (WTP) for reserving the access (or willingness to accept (WTA) for the loss of access) to a recreation site by equating the expected utilities from two states: one state with access to the site and one without. If the conditional utility function is assumed quasi-linear in income, the welfare measure of the availability of site t has a convenient analytical solution as follows (Bockstael et al., 1991).
CH
i"
ln(+Jj/1eVij)!ln(+Jj/1,jEteVij)
c (15)
11The income information is not available in the data set for the application. The assumption we make is that the derivative ofVwith respect to price at a given price indicates the negative marginal utility of income at that particular price since the price of a trip could be thought as a reduction of income.
12Alternatively the marginal utility of income at sitetfor the individualican be estimated by the weighted average of!<@
ijover allJsites. Two possible measures of average marginal utility of income are considered in Huang (1994). The"rst one is the negative sum of<@
ij weighted by the probabilities,!+Jj
/1npij<@ij. Bothnijand<@ijcan be estimated from the nonparametric multiple choice model. This measure is also suggested by one of the reviewers. Since survey respondents took multiple trips to the same site within the given time period, the second measure of the average marginal utility of income is to use the actual proportions of trips taken to various sites as weights: !(+Jj
/1nij<@(xij))/(+Jj/1nij), wherenijis the number of trips taken to sitej. A comparison of the nonparametric welfare measures based on alternative estimates of marginal utility of income can be found in Huang (1994).
13A Monte-Carlo simulation is conducted to compare the two welfare measures implied by the nonparametric and parametric multiple choice models, respectively. Under three assumed true utility functions}two simple random utility models (taken from Morey et al., 1993) and one share model (based on Morey, 1981)}the simulation results show no signi"cant statistical di!erence between the two welfare estimators. The details of sampling experiments and response surface analysis of mean squared errors of the welfare estimates are available upon request from the"rst author.
a modi"cation in the denominator, the formula in (15) can be used as an approximation of the WTP for a site choice from a nonparametric utility function. Letx
ijbe the cost (price) of going to sitejfor the individuali.11Let the
sitetbe the site being measured. If the price (x
it) of vising the sitetis thought as
a reduction of income, the marginal utility of income for going to sitetcan be measured as the negative derivative of<(x) evaluated atx
it:!<@it. Hence, the
following formula is used to approximate the WTP for maintaining the recre-ation sitetbased on the estimated nonparametric multiple choice model.12
CHH
i "
ln(+Jj/1eVij)!ln(+Jj
/1,jEteVij)
!<@
it
. (16)
It must be emphasized that since the welfare measure in (16) is an approxima-tion, not an exact measure, the error of approximation is embedded in the nonparametric welfare estimates and cannot be separated from the estimation errors.13
5. An example of nonparametric nonmarket valuation
14It is common, although not always correct, to assume that the decision of each of the multiple trips taken by the same individual/household is independent of other trips. In this application, we maintain the assumption of independent decisions of multiple trips by the same household.
Monongahela River Basin in Pennsylvania in 1981, concerns households' recre-ational decisions in the Basin. A total of 397 households was randomly selected for in-person interviews. The number of fully completed interviews is 303. A subset of 112 households that are users of seven selected recreation sites is used in this application. The seven sites are: Kittanning, Keystone Dam, Lake Arthur in Moraine State Park, Ohiopyle State Park, North Park Lake, Youghiogheny River Lake Reservoir, and Pittsburgh (The Point, Smiths"eld Bridge). Multiple trips to the recreation sites by the same household in the season will be stacked and treated as independent visits. The total number of trips to all sites used in estimation is 945. The data set provides information on costs of visiting various sites,"sh catch rates at the sites, number of trips taken by individuals, and socio-economic characteristics of the interviewed indi-viduals.14
Table 1
Point estimates and bootstrapped results of the average per trip WTP Logit Spline
opt. log(j)# log(j)"5 log(j)"!1 log(j)"!7
Point estimate 0.3722 0.2238 0.3721 0.3678 0.2166
Mean! 0.3725 0.2227 0.3725 0.3722 0.2568
STD 0.0083 0.7436 0.0085 0.0093 0.0098
Median 0.3722 0.2160 0.3724 0.3720 0.2561
Lower bound" 0.3874 0.2602 0.3865 0.3882 0.2728
Upper bound 0.3600 0.0805 0.3584 0.3581 0.2412
!It is the mean of the 500 bootstrapped average bene"t estimates.
"The lower and upper bounds are created based on Efron's (1982) percentile method. They form a 90% con"dence interval for each of the bene"t estimates.
#The`logahere is the natural logarithm. The optimal log(j) selected by the GCV function for the point estimate is!11.jvaries according to the GCV criterium in the bootstrap when deriving the con"dence interval. However, thejvalue is"xed in resamplings in the next three columns.
to the di!erence between the parametric and nonparametric welfare measures. Further investigation is needed. Nonetheless, this simple exercise demonstrates that the proposed nonparametric multiple choice model allows more#exibility in estimating WTP and provides a test for linear approximation of the utility function.
6. Summary and remarks
special case so it can be used to re-examine and select the most plausible parametric model to derive the important parameter estimates that in#uence policy decisions.
The model can be extended to include multiple covariates. For example, with additional assumptions on the relationship of covariates, methods such as generalized additive models (Hastie and Tibshirani, 1987,1990) and semiparametric generalized linear models (Green and Silverman, 1994) can be incorporated to handle multivariate situations. Based on our results, the theory and derivation of the estimators for these models should be straight forward.
The i.i.d. assumption of errors and the IIA hypothesis can be relaxed by exploring more generalized statistical distributions such as a generalized ex-treme value distribution that allows nonconstant variances and interdependence across errors. There are other extensions. We may, for example, examine through sampling experiments the performance of the nonparametric discrete choice method under highly nonlinear behavioral models. We can also extend the method to multivariate cases as well as explore the asymptotic and small sample properties of the nonparametric welfare estimators. Nonparametric functions can also be used to analyze more complex choice settings such as nested and ranked data that are now commonly elicited in surveys. Indeed, the potential of nonparametric multiple choice methods is unlimited. The properties of the estimators and economic implications of these models must be further explored so the #exibility of nonparametric function estimation can be fully appreciated.
Appendix A
Proof of Theorem: As seen in Section 3.1, by the properties of reproducing kernels,l
ij(<)"Soxij,<T"f(<ij). The optimization in (5) can be presented in
the following simple form.
max
V|W2 2
Q(<),max
V|W2 2
¸(l
11(<),2,lij(<),2,lnJ(<))!j
P
ba
(<A(x))2dx. (A.1)
The Hilbert space projection theorem (Taylor and Lay, 1986) starts with the assumption thatMis a closed subspace of the Hilbert space. IfM3H(in our case,H,=2
2), de"ne the space perpendicular toMbyMM"Mw3H,Sw,zT"0
∀z3MN. For anyx3H x, can be decomposed uniquely asx"z#wsuch that
z3M,w3MM. Based on the projection theorem and the decomposition of =2
2 to =0 and =22,BC, let the maximizer of (5) be of the form
<K"s#l"d
0#d1x#+ni/1+Jj/1cijoxij(x)#l, wheresis as de"ned in (8) and
l is some element in =2
is,(l,oxij*0 and (l,d
0#d1x*:10d2(d0#d1x)/dx2l@@dt"0. Substituting
<K into (A1), we have Q(<K )"¸(l
11(<K),2,lnJ(<K ))!j
P
b a(<K A)2dx
"¸(l
11(<K ),2,lnJ(<K))!jS<K ,<KT
"¸(l11(s#l),2,l
nJ(s#l)!jSs#l,s#lT
"¸(So
11,s#lT,2,SonJ,s#lT)!j[Ss,sT#2Ss,lT#Sl,lT]
"¸(So
11,sT#So11,lT,2,SonJ,sT#SonJ,lT)
!jSs,sT!2jSs,lT!jSl,lT "¸(So
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References
Abe, M., 1999. A generalized additive model for discrete-choice data. Journal of Business and Economic Statistics 17, 271}284.
Ahn, H., Manski, C.F., 1993. Distribution theory for the analysis of binary choice under uncertainty with nonparametric estimation of expectations. Journal of Econometrics 56, 291}321. Ahn, H., 1995. Nonparametric two-stage estimation of conditional choice probabilities in a binary
choice model under uncertainty. Journal of Econometrics 67, 337}378.
Anderson, J.A., Blair, V., 1982. Penalized maximum likelihood estimation in logistic regression and discrimination. Biometrika 69, 123}136.
Bockstael, N.E., McConnell, K.E., Strand, I.E., 1991. Recreation. In: Braden, J.B., Kolstad, C.D. (Eds.), Measuring the Demand for Environmental Quality. Elsevier Science, New York, NY. Cosslett, S.R., 1983. Distribution-free maximum likelihood estimator of the binary choice model.
Econometrica 51, 765}782.
Cox, D.D., O'Sullivan, F., 1990. Asymptotic analysis of penalized likelihood and related estimators. Annals of Statistics 18, 1676}1695.
Craven, P., Wahba, G., 1979. Smoothing noisy data with spline functions: estimating the correct degree of smoothing by the method of generalized cross-validation. Numerische Mathematik 31, 377}403.
Efron, B., 1982. The Jackknife, the Bootstrap, and other resampling plans. CBMS-NSF Regional Conference Series in Applied Mathematics, Monograph 38. Society for Industrial and Applied Mathematics, Philadelphia.
Good, I.J., Gaskins, R.A., 1971. Non-parametric roughness penalties for probability densities. Biometrika 58, 255}277.
Green, P.J., Silverman, B.W., 1994. Nonparametric Regression and Generalized Linear Models. Chapman and Hall, New York, NY.
Han, A.K., 1987. Non-parametric analysis of a generalized regression model: the maximum rank correlation estimator. Journal of Econometrics 35, 303}316.
Hanemann, W.M., 1984. Statistical issues in discrete-response contingent valuation studies. Califor-nia Agr. Exp. Sta., Giannini Foundation of Agr. Econ., Working paper No. 322.
Hastie, T.J., Tibshirani, R.J., 1987. Non-parametric logistic and proportional odds regression. Applied Statistics 36 (2), 260}276.
Hastie, T.J., Tibshirani, R.J., 1990. Generalized Additive Models, Monographs on Statistics and Applied Probability, Vol. 43. Chapman & Hall, New York.
Horowitz, J.L., 1992. A smooth maximum score estimator for the binary response model. Econo-metrica 60, 505}531.
Huang, J.-C., 1994. New discrete choice methods for valuing environmental amenities: theory and evaluation. Ph.D. Dissertation, North Carolina State University.
Huang, J.-C., Nychka, D.W., Smith, V.K., 1999. Semi-parametric Discrete choice measures of willingness to pay. Working paper, University of New Hampshire.
Ichimura, H., Lee, L.F., 1991. Semiparametric estimation of multiple index models: single equation estimation. In: Barnett, W.A., Powell, J., Tauchen, G. (Eds.), Nonparametric and Semiparametric Methods in Econometrics and Statistic. Cambridge University Press, New York, NY (Chapter 1). Kaoru, Y., Smith, V.K., Liu, J.L., 1995. Using random utility models to estimate the recreational
value of estuarine resources. American Journal of Agricultural Economics 77, 141}151. Klein, R.W., Spady, R.H., 1993. An e$cient semiparametric estimator for binary response models.
Econometrica 61, 387}421.
Kling, C.L., Thomson, C.J., 1996. The implications of model speci"cation for welfare estimation in nested logit models. American Journal of Agricultural Economics 78, 103}114.
Lee, L.F., 1995. Semiparametric maximum likelihood estimation of polychotomous and sequential choice models. Journal of Econometrics 65, 381}428.
MaKler, K.G., 1974. Environmental Economics: A Theoretical Inquiry. Johns Hopkins University Press, Baltimore.
Manski, C.F., 1975. Maximum score estimation of the stochastic utility model of choice. Journal of Econometrics 3, 205}228.
Manski, C.F., 1985. Semiparametric analysis of discrete response: asymptotic properties of the maximum score estimator. Journal of Econometrics 27, 313}333.
Manski, C.F., 1991. Nonparametric estimation of expectations in the analysis of discrete choice under uncertainty. In: Barnett, W., Powell, J., Tauchen, G. (Eds.), Nonparametric and Semiparametric Methods in Econometrics and Statistics. Cambridge University Press, Cambridge.
Matzkin, R.L., 1991. Semiparametric estimation of monotone and concave utility functions for polychotomous choice models. Econometrica 59, 1315}1327.
Matzkin, R.L., 1992. Nonparametric and distribution-free estimation of the binary threshold crossing and the binary choice models. Econometrica 60, 239}270.
McFadden, D., 1973. Conditional logit analysis of discrete choice behavior. In: Zarembka, P. (Ed.), Frontiers in Econometrics. Academic Press, New York, NY.
Morey, E.R., 1981. The demand for site-speci"c recreational activities. a characteristics approach. Journal of Environmental Economics and Management 8, 345}371.
Morey, E.R., Shaw, W.D., Rowe, R.D., 1991. A discrete-choice model of recreational participation, site choice, and activity valuation when complete trip data are not available. Journal of Environmental Economics and Management 20, 181}201.
O'Sullivan, F., Yandell, B.S., Raynor Jr., W.J., 1986. Automatic smoothing of regression functions in generalized linear models. Journal of American Statistical Association 81, 96}103.
Ozuna Jr., T., Jang, K.Y., Stoll, J.R., 1993. Testing intervals for misspeci"cation in the referendum contingent valuation approach. American Journal of Agricultural Economics 75, 332}338. Smith, V.K., Desvousges, W.H., 1986. Measuring Water Quality Bene"ts. Kluwer, Boston, MA. Taylor, A.E., Lay, D.C., 1986. Introduction to Functional Analysis. Krieger, Melbourne, FL. Wahba, G., 1990. Spline Models for Observational Data. Society for Industrial and Applied
