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Modeling zero response data from willingness to pay surveys
A semi-parametric estimation
a ,
*
b cSeung-Hoon Yoo
, Tai-Yoo Kim , Jai-Ki Lee
a
Institute of Economic Research, Korea University, 5-1 Anam-Dong, Sungbuk-Ku, Seoul, 136-701, South Korea
b
Techno-Economics and Policy Program, Seoul National University, 56-1 San, Shinrim-Dong, Kwanak-Ku,
Seoul, 151-742, South Korea
c
Department of International Area Studies, Hoseo University, 29-1 Sechul-Ri, Baebang-Myun,
Asan Chungnam, 336-795, South Korea Received 19 April 2000; accepted 11 October 2000
Abstract
This paper models zero response data from willingness to pay surveys by employing parametric and semi-parametric estimation methods. The result of the specification test indicates the semi-parametric estimation outperforms the parametric estimation significantly. 2001 Published by Elsevier Science B.V.
Keywords: Zero response; Willingness to pay; Symmetrically trimmed least squares
JEL classification: C24
1. Introduction
A typical characteristic of willingness to pay (WTP) for an item is that many respondents would not be willing to pay anything for it. In our data on household WTP for the educational near video-on-demand (ENVOD) service, this is the case for 26.2% of all observations. In such cases, least squares estimates will be inconsistent. To account for this, one type of limited-dependent variable model, the familiar Tobit model (Tobin, 1958), has been widely used.
The Tobit estimator, based on the maximum likelihood estimation (MLE) method, assumes the homoscedasticity and normality on the distribution of the error term. If the assumptions are not satisfied, it is again inconsistent (e.g. see Robinson (1982)). Since economic theory generally yields no restrictions concerning the form of the error distribution or homoscedasticity of the residuals, the
*Corresponding author. Tel.: 182-2-3290-2715; fax:182-2-928-4948.
E-mail address: [email protected] (S.-H. Yoo).
sensitivity of likelihood-based procedures to such assumptions is a serious concern. The test results, explained in Section 3, imply that the hypotheses of homoscedasticity and normality are clearly rejected at the 1% level.
Recently, the econometric theory of semi-parametric estimation methods for the Tobit model has gained much attention. However, empirical applications of the methods remain lacking. This paper, therefore, has two major goals. The first is to analyze some determinants of WTP for the ENVOD service. The second goal is to explore the use of a consistent and robust estimator when estimating a WTP equation using zero response data from the WTP survey. To this end, we propose a systematic approach of testing the parametric assumptions on the error term distribution of the Tobit model, a semi-parametric re-estimation of the model, and specification test of the parametric estimation vs. the semi-parametric estimation.
2. A model of WTP
An individual’s optimal WTP can be derived within the constrained utility maximization framework. Assuming the utility function is continuous and quasi-concave, then the optimal WTP can be expressed as a function of the respondent’s tastes or personal characteristics. Denote these determinants of WTP as a vector x and assume a linear functional form for the WTP equation. Then,
*
for individual i51, . . . , T, the optimal WTP, y , can be written as:i
*
9
yi 5xib1ui (1)
where b is a vector of parameters, and u is a random error. In reality, an individual’s choice isi subject to non-negativity constraints, and, therefore, a corner solution could result. One representative way to accommodate corner solutions is to use the Tobit model, in which case observed WTP, denoted
*
y , relates to the latent WTP yi i such that:
*
yi5maxhy , 0i j (2)
where u is assumed to be distributed as normal with mean zero and standard deviationi s.
3. Data and pretests
The data on household WTP for the ENVOD service, and characteristics used in this analysis come from a 1998 survey of 427 households in Pusan, Korea. The sample is censored, with 112 households (26.2%) reporting zero WTP. The variables in the model are described in Table 1.
We test for the existence of heteroscedasticity and non-normality in the error term of the Tobit
2 2
model. Firstly, we consider the heteroscedastic Tobit model in which we specify that si 5s
9
exp(xia). The null hypothesis of homoscedasticity is a50. The Lagrange multiplier (LM) statistic was calculated as 35.10. This is asymptotically distributed as chi-squared with 10 degrees of freedom
2
under the null hypothesis. Given that x0.01(10)523.21, the hypothesis can be rejected at the 1% level.
Table 1
Description of variables in model
Variables Definition
WTP Willingness to pay for educational near video-on-demand service
a
(Unit: 10 000 Korean won )
INTEREST Degree of respondent’s interest in educating his / her child
(From 15very little to 55very much)
SATISFACTION Degree of how much respondent is satisfied with current education in school
(From 15very little to 55very much)
NEED Opinion about how necessary extracurricular work is for respondent’s child
(From 15very little to 55very much)
HELP Opinion about how helpful the proposed educational video-on-demand is for respondent’s child (From 15very little to 55very much)
EXTRA Dummy for respondent’s child being engaged in extracurricular work
(05no; 15yes)
AGE Age of the respondent
CATV Dummy for respondent’s watching cable television (05no; 15yes)
EBS Dummy for respondent’s child watching the Educational Broadcasting System
(05no; 15yes)
EDUCATION Education level
(from 15lowest to 95highest)
INCOME Monthly household total income after tax deduction
a
(Unit: 10 000 Korean won )
a
At the time of the survey, US$1 was approximately equal to 1400 Korean won.
and Irish (1987). The test is equivalent to testing the null hypothesis ofg15g250 in a modification
2 3
of the normal cumulative distribution function (cdf), Pr(ut,t)5F(t)5F(t1g01g1t 1g2t ), where
1 2
F(?) is the univariate standard normal cdf. The LM statistic follows asymptotically ax distribution with two degrees of freedom under the null hypothesis of censored normality. The test statistic was computed as 412.89, which is large enough to reject the hypothesis at the 1% level, given that
2
x0.01(2)59.21. Thus, it appears clear from the test results given above that the assumptions required to use the Tobit model are too strong to be satisfied.
1
4. A semi-parametric estimation
One could estimate the heteroscedastic Tobit models and use an alternative distribution to rectify heteroscedasticity and non-normality of the error term in the Tobit MLE model. However, this method does not necessarily solve the problem, and may make it worse. Accordingly, devising a new estimator that is robust to these two problems requiring fewer assumptions is more appealing.
As an alternative to the Tobit model, we use symmetrically trimmed least squares (STLS) estimator proposed by Powell (1986), which is consistent and asymptotically normal for a wide class of error
2
distributions with heteroscedasticity of unknown form and a censored dependent variable. For a
ˆ
sample of size T, the STLS estimator, bSTLS, is defined as:
T
2
ˆ
9
9
9
bSTLS5arg min
O
I(xib.0) min( y , 2xf
i ib)2xibg
(3)b[B i51
where B denotes the relevant parameter space for b, and the indicator function, I(?), takes the value of one if the argument is true, and zero otherwise. See Powell (1986) for an iterative procedure to
ˆ ˆ
obtain bSTLS, proof of consistency of bSTLS, and its asymptotic covariance.
Rather than estimate the asymptotic covariance for the STLS estimator, we use a bootstrap
ˆ
estimator of the covariance, V, given by
R
where bSTLS5(1 /R)oj51bSTLS and R is the number of bootstrap replications. This bootstrap
ˆ
procedure results in a consistent estimator of the covariance ofbSTLS, which is robust to violations of the assumption that the residuals are identically distributed.
5. Estimation results and specification test
To obtain the STLS estimator, convergence occurs when the maximum change in any parameter 25
estimate is less than 10 in two consecutive iterations. The STLS estimator converged in 48 iterations. The value of R is set to 5000. Table 2 shows the coefficients of our basic equation estimated by Tobit MLE and STLS, respectively.
To compare the parametric and semi-parametric estimations more generally, we conduct Hausman’s (1978) type specification test given in Melenberg and van Soest (1996). The test requires an estimator that is consistent and efficient under the null hypothesis but inconsistent under the alternative (the Tobit estimator), and an estimator that is consistent under both hypotheses, but inefficient under the null hypothesis (the STLS estimator). We can construct the Wald statistic as:
2
Table 2
Estimation results of WTP equations
a
The variables are defined in Table 1. The numbers in parentheses below the estimates are standard errors. The standard errors of Tobit estimates are computed using the analytic second derivatives of the log-likelihood. Those of STLS estimates are calculated by the use of the bootstrap method with 5000 replications.
21
This is a x statistic with K degrees of freedom under the null hypothesis, where K is rank of L. Thus, the null hypothesis for W is that the parametric Tobit model estimates would be consistent. The alternative hypothesis is that the model is semi-parametric. The calculated statistic for the STLS
2
estimator is 40.22. Given that x0.01(11)524.72, we can reject the null hypothesis at the 1% level.
3 ˆ ˆ
This is rather dramatic evidence of the impact of changing estimator in a situation of heteroscedastici-ty or the failure of the normally distributed error term as related to censoring.
6. Concluding remarks
The parametric estimation technique for dealing with zero WTP data, the MLE of the Tobit model, is vulnerable to existence of heteroscedasticity and non-normal error structures. The STLS estimator is robust under those stresses and can deal with zero response data. In the application reported here, the STLS estimation outperformed the parametric estimation significantly, reducing the implicit restric-tions involved in the parametric model. Even if this semi-parametric estimation method can be easily calculated with any commercial computer packages such as GAUSS, it has not been popular because of the difficulty of the literature to people accustomed to MLE. However, judging from our application it is a practical and theoretically promising way of modeling zero WTP data.
References
Chesher, A., Irish, M., 1987. Residual analysis in the grouped data and censored normal linear model. Journal of Econometrics 34, 33–62.
Hausman, J., 1978. Specification test in econometrics. Econometrica 46, 1251–1271.
Melenberg, B., van Soest, A., 1996. Parametric and semi-parametric modeling of vacation expenditures. Journal of Applied Econometrics 11, 59–76.
Powell, J.L., 1984. Least absolute deviations for censored regression model. Journal of Econometrics 25, 303–325. Powell, J.L., 1986. Symmetrically trimmed least squares estimation for tobit models. Econometrica 54, 1435–1460. Robinson, P.M., 1982. On the asymptotic properties of estimators of models containing limited dependent variables.
Econometrica 50, 27–41.
Tobin, J., 1958. Estimation of relationships for limited dependent variables. Econometrica 26, 24–36.
