Dealing with censored data from contingent valuation surveys: symmetrically-trimmed least squares estimation.

Russell, Clifford S.

I. Introduction

The purpose of this paper is to explore the use of a consistent and robust estimator when estimating a willingness to pay (WTP) equation using the censored data collected by a contingent valuation survey.(1)

Mitchell and Carson's payment card format [11; 12] strongly encourages the respondent to give a WTP value that is zero or greater, because payment cards do not normally include any negative values. But some respondents who answer with a zero WTP may in fact have negative WTP. As a simple example, if people are asked about improving a salt water pond's water quality to the point that shellfish taken from it would be edible, there may be individuals in the sample who use the pond for other recreational activities that would be hindered by the presence of people shellfishing. In addition, anyone who enjoys quietness around and on the pond also might want to be paid a certain amount to allow the ponds to be made shellfishable. In brief, not every "public effect" on net is a good to every affected person, but anticipating the varieties of reasons for negative valuation is at least difficult if not impossible. This was true in our research aimed at estimating the willingness to pay for improving the level of security of tap water quality in Seoul, Korea, as we discuss in section II.

This censoring of the WTP responses becomes problematic when one seeks to estimate a willingness to pay equation as a check on internal consistency of the study results and potentially as a basis for total benefit projection. Thus, with WTP censored at zero, OLS parameter estimates will be inconsistent. One popular method that has been used in such cases is the Tobit model [1; 17]. However, there are two possible pitfalls to this approach. First, the Tobit estimator becomes inconsistent when heteroskedasticity occurs in the errors [2; 9]. Moreover, when the normality assumption on the distribution of the error term is not satisfied, it is again inconsistent [2]. We have tested the hypothesis of an i.i.d. censored normality by employing the test procedure of Nelson [13]. We can reject the hypothesis at the 1% level. The assumptions required to use the Tobit model are, therefore, too strong to be satisfied.

As an alternative to the Tobit estimation, we consider here the symmetrically trimmed (censored) least squares estimation (STLS) method proposed by Powell [16]. This method is based on symmetric censoring of the upper tail of the distribution of the dependent variable. This semi-parametric estimator is consistent and asymptotically normal for a wide class of distributions of the error term and is robust to unknown heteroskedasticity.

In this paper, we apply the STLS method to estimating a WTP equation and compare the results with those from OLS and the Tobit estimation. As noted, this estimator is robust under quite general conditions, so differences between coefficients estimated in STLS compared to OLS or Tobit can be interpreted as evidence of problems of inappropriate clustering at zero, or violation of assumption of homoskedasticity and normality. The paper proceeds as follows. Section II describes the WTP model to be estimated. Section III explains the estimation method. A discussion of results appears in section IV. Conclusions will be found in the final section.

II. A Model of WTP and the Censoring Problem

Our theoretical model for explaining individuals' WTP comes from the income compensating function [18]. When we take WTP as the desired benefit measure, the income compensating function is referred to as the WTP function, and we hypothesize that the arguments are elements of a vector of the respondent's tastes or personal characteristics as well as variables representing both the respondent's environmental and economic situations. Thus:

WTP([q.sub.1], [q.sub.0]) = f([P.sub.0], [q.sub.1], [q.sub.0], [Q.sub.0], [Y.sub.0], T), (1)

where [P.sub.0] is the price level of private goods, the [q.sub.i] are tap water "quality" descriptions, [Q.sub.0] is other environmental goods, [Y.sub.0] is income, and T is a vector of the respondents' tastes or characteristics.

In our study, [q.sub.0] was the current situation as related to the security of Seoul's tap water quality, while [q.sub.1] represented the improvement goal described in the survey. Both the current situation and the policy goal were described to respondents by relating them to incidents that occurred in a large river south of Seoul in 1991, less than a year before our survey. These incidents were two accidental industrial spills of phenolic compounds that went undiscovered until they had contaminated the water supply of a major city that used the river (the Nak-dong) as its raw water source. Routine chlorination made the effect worse in one sense because the water gave off an even worse smell after treatment, but it is possible that the terrible smell was a major cause of so little reported health damage.

The base condition, [q.sub.0], described to our Seoul respondents was vulnerability to a Nak-dong-like spill incident. We let respondents in effect define their own subjective probabilities of such an event occurring.(2) The goal, [q.sub.1], ascribed to the government was "to reduce the probability that, in your life, you will experience an accident similar to the phenol accident in the Nak-dong river to zero or very close to zero."

Prices and other aspects of environmental quality were held constant for respondents, and we asked for their WTP for security of tapwater quality. In our survey we also asked about actions taken to change drinking water quality for the household (buying bottled water, traveling to a spring for water, or installing a filter system); about the respondent's attitude toward current tap water quality and her assessment of the number of incidents her household might face without a policy change; and about several socio-economic characteristics of the household.

The resulting model is then (for variable definitions, see appendix):

[WTP.sup.*] = [a.sub.0] + [a.sub.1]ATT + [a.sub.2]FILT + [a.sub.3]BOTL + [a.sub.4]TSPW + [a.sub.5]NAC + [a.sub.6]AGE + [a.sub.7]EDU + [a.sub.8]NCHD + [a.sub.9]YRS + [a.sub.10]BILL + [a.sub.11]PINC + U (2)

where [WTP.sup.*] = Household willingness to pay per month (Unit = 1000 won), and the observed willingness to pay is given by:

WTP = [WTP.sup.*] if [WTP.sup.*] [greater than] 0

WTP = 0 if [WTP.sup.*] [less than or equal to] 0.

We asked the respondents who gave a zero WTP (28 of 298 observations) for their reasons. Examples of the responses that suggest possibly negative WTP include the following. One person said she was the wife of a bottled water salesman; another two said that their households depended on the income from the vending of coffee, toast, and simple breakfast or lunch on the way to a popular spring.(3) If tap water quality were to be improved, the number of people visiting the springs would be reduced. The families' expected loss of income might actually exceed their personal benefits from improving public tap water quality. These respondents felt the government should compensate them for the loss of income. Some other respondents who gave zero WTP answers said that they did not want the security of tap water quality improved in spite of believing that current tap water quality was not good. Unfortunately, the interviewer could not get them to specify their reasons. We guess that some of those respondents may also have faced some loss of income due to any improvement of the security of tap water quality. Because we had no reason to anticipate negative values, however, the interviewers were not prepared to seek them systematically in the interview process. So, the censuring problem is obviously involved in our data.

A summary of the survey data and an indication of the signs of the coefficients we expect in the WTP equation(4) are provided in Table I. The size of the usable sample was 298 households. For a description of the sampling plan and some limited comparisons between sample characteristics and what little is known (publicly at least) about the population of Seoul, see Kwak [10].

Table I. A Summary of the Survey Data, Independent Variables

 Mean Standard Deviation Expected Sign of Coefficient

ATT 3.597 0.769 +
FILT 1.571 3.601 ?
BOTL 1.859 5.520 ?
TSPW 0.356 0.480 +
NAC 3.873 3.145 +
AGE 37.775 9.430 +
EDU 11.456 3.222 +
NCHD 0.923 0.875 +
YRS 19.523 13.078 -
BILL 4.816 2.328 +
PINC 274.900 185.18 +

III. Pretests and Estimation

As discussed in the introduction, the Tobit estimator is sensitive to the error specification. Specifically when heteroskedasticity occurs in the error term or when the normality assumption is violated, the estimator is inconsistent. This problem is actually found in our sample data. When the Lagrangian multiplier (LM) test of Breusch-Pagan [4] is employed, the null hypothesis of homoskedasticity of the error terms from the Tobit estimation is rejected at the 1 percent significance level. (The LM statistic which follows a chi-square distribution was computed as 662.8.) Furthermore the assumption of normality of the error terms is not supported for either the OLS estimation or the MLE estimations when the Bera-Jarque [3] tests are used.

To examine more precisely the problem of violating the assumptions required to use the Tobit model, we also employ a formal test procedure of Nelson [13]. We test the maintained hypothesis of the i.i.d. censored normality of the Tobit MLE model. This method uses the maximum likelihood estimates and the method of moments estimates for certain population moments. The test is known to have decent power and correct size of the test under the null hypothesis. The test statistic, not presented here for reasons of space, is given in Nelson [13]. It follows asymptotically a chi-square distribution with degrees of freedom equal to the number of regressors. The test statistic is applied to our case using the Tobit estimates, and is computed as 225.8, which is quite large enough to reject the null hypothesis of the i.i.d. censored normality (rejection at 1% level requires 26.22 in this case; d.f. = 12). As noted by Nelson, it would be difficult to distinguish between sources of bias without a priori speculation about particular sources. However, it appears clear from the test results given above that the Tobit estimates contain two problematic sources of inconsistency.(5)

As an alternative estimation technique, we use the method of symmetrically trimmed (censored) least squares proposed by Powell [16]. This method is based on censoring of the upper tail of the distribution of the dependent variable in a way symmetric with the implied censoring of the lower tail. The resulting semiparametric estimator is known to be consistent and asymptotically normally distributed for an error specification of non-normality and/or heteroskedasticity of unknown form. It is also quite easy computationally.(6)

The idea of the STLS is intuitive. First, we express the true underlying regression equation as:

[Mathematical Expression Omitted]. (3)

A censored regression model applies to a sample for which the independent variables [x.sub.t] and dependent variable [y.sub.t] are observed in such a way that [y.sub.t] = 0 if [Mathematical Expression Omitted] and [y.sub.t] = [x[prime].sub.t][[Beta].sub.0] + [u.sub.t] if [Mathematical Expression Omitted]. Thus some of the dependent variable values are not truly observed if [Mathematical Expression Omitted]. This induces asymmetry in the distribution of the error terms. The STLS estimator restores symmetry of the error distribution by symmetric trimming in such a way that the uncensored observations in the upper tail are replaced by their estimated symmetrically censored values, and the observations with [x[prime].sub.t][[Beta].sub.0] [less than or equal to] 0 are trimmed. This amounts to replacing the dependent variable with min{[y.sub.t], 2[x[prime].sub.t][[Beta].sub.0]}.(7)

The symmetrically censored least squares estimator [Mathematical Expression Omitted] is then obtained by

[Mathematical Expression Omitted] (4)

where [Mathematical Expression Omitted] stands for an indicator function, which is one if [Mathematical Expression Omitted] and zero otherwise. See Powell [16] for the proof of consistency of [Mathematical Expression Omitted] and its asymptotic variance.

The STLS estimator is computed using equation [4] as a recursion formula, with maximum likelihood estimates as starting values. Our iteration was terminated when the maximum change in any parameter estimate was less than 0.00001. The estimator converged in 45 iterations. The corresponding t-statistics are then calculated.

IV. The Results

In Table II the first three columns show the coefficients of our basic equation estimated by OLS, Tobit, and STLS respectively. (In the OLS and Tobit versions, not all coefficients are significantly different from zero, while using STLS they are.) Thus, as respondents have a worse opinion about current tap water quality, so their WTP to protect tap water quality is higher. As a housewife's reported expenditures on a water filtration system or on bottled water are bigger, her WTP is higher. The households that made trips to a spring for drinking water are willing to pay more than others. The housewife's subjective estimate of the number of drinking water accidents that might occur in the next five years, if the government takes no action, also has a positive relation to her expressed WTP. Older respondents are willing to pay more than younger respondents. More highly educated respondents are also willing to pay more than less well educated respondents. The number of children in the household under 13 years of age has a positive relation to the WTP. However, the number of years a respondent has been a resident of Seoul affects her willingness to pay negatively. A household's monthly water bill is positively related to WTP. Finally, the household income per family member is also strongly positively related to WTP.

Table II. Regression Results Using OLS, Tobit, and STLS Estimation

Variable OLS Tobit STLS

CONST. -3.063 -3.748 -11.452
 (-2.182)(*) (-2.523)(*) (-4.023)(**)

ATT 0.315 0.344 0.829
 (1.644) (1.682) (2.800)(**)

FILT 0.183 0.190 0.145
 (4.274)(**) (4.246)(**) (2.475)(*)

BOTL 0.152 0.158 0.114
 (5.548)(**) (5.530)(**) (3.456)(**)

TSPW 0.676 0.886 1.206
 (2.277)(*) (2.833)(**) (3.419)(**)

NAC 0.163 0.187 0.280
 (3.548)(**) (3.871)(**) (3.420)(**)

AGE 0.009 0.010 0.070
 (0.434) (0.456) (2.411)(*)

EDU 0.090 0.106 0.257
 (1.605) (1.757) (2.567)(*)

NCHD 0.315 0.370 0.707
 (1.682) (1.877) (3.415)(**)

YRS -0.027 -0.033 -0.052
 (-2.051)(*) (-2.422)(*) (-2.873)(**)

BILL 0.122 0.133 0.182
 (1.925) (1.995)(*) (2.904)(**)

PINC 0.005 0.005 0.007
 (5.860)(**) (5.712)(**) (5.645)(**)

N = 298

t-statistics are in parentheses below coefficient estimates.

* Significant at the level of five percent

** Significant at the level of one percent

The most interesting part of the Table II, however, is the pattern of coefficient change across the estimators. For each variable except FILT and BOTL the coefficients increase monotonically in absolute value between the OLS and the STLS version. In addition the number of statistically [TABULAR DATA FOR TABLE III OMITTED] significant (at least the 5 percent level) coefficients increases from 7 of 12 in OLS version, to 8 of 12 in the Tobit version, and 12 of 12 in the STLS version. This is rather dramatic evidence of the impact of changing estimators in a situation of heteroskedasticity and failure of normally distributed error term related to censoring.

As a final exercise, we show in Table III the WTPs calculated for the average household in our sample for each equation in Table II. Based on the STLS estimator, the average household's willingness to pay in our sample for the automatic monitoring system and the described 'goal' is 2,560 Won (US $3.28) per month. This amount is smaller than the average WTP (2603 Won) of the raw data. Because some zero WTPs almost certainly represent negative values, the average WTP of the raw data should be an overestimate of the correct value.

V. Conclusion

Willingness to pay values obtained from surveys that allow respondents to state a value are normally censored at zero. While survey technique may in principle be adjusted to avoid this, in practice it may prove difficult to anticipate the motivations and amounts involved in potential negative values. This can make elicitation difficult in many cases. Conventional econometric technique for dealing with data with a mass of observations at zero - the Tobit - is vulnerable to heteroskedasticity and non-normal error structures. The symmetric trimming proposed by Powell is robust under those stresses. In the application reported here STLS performed very well, improving the significance of the WTP equation coefficients and, in most cases, increasing the size of the coefficients, consistent with what we know about OLS bias. STLS is not computationally difficult, and so is a practical as well as a theoretically promising way of dealing with such data.

Appendix: Variable Definitions

ATT: The respondent's attitude toward current tap water quality

1 = Very good 2 = Good 3 = Average 4 = Bad 5 = Very bad

FILT: Monthly expenditure for a tap water filtration system (unit = 1,000 won)

BOTL: Monthly expenditure for bottled water (unit = 1,000 won)

TSPW: Dummy for having taken a trip to obtain spring water to use for drinking during last 5 years

1 = Yes 0 = No

NAC: Subjective estimate of the number of drinking water contamination accidents that might occur in the next five years if the government takes no action.

AGE: Age of the respondent

EDU: Education level of the respondent in years from 0 = no education to 18 = post graduate

NCHD: Number of children in the respondent's household under 13 years of age

YRS: Number of years respondent has been a resident of Seoul

BILL: Monthly combined bill for water and sewerage service

PINC: Monthly total household income divided by number in the family living in the household (Unit = 1,000 won)

The authors gratefully acknowledge the advice and assistance of their colleague, Professor Georgine Pion; and the financial support of the Korean Environmental Protection Agency. They also appreciate the constructive comments of a referee. The usual disclaimer, of course, applies.

1. While the major problems in the application of the contingent valuation method are in questionnaire design and application [14] even in such reasonably well controlled situations, econometric challenges still exist. See, for example, on outliers Desvousges, Smith, and Fisher [6] and on dealing with referendum (take-it-or-leave-it) type data, Cameron and James [5].

2. We recognize that even quite sophisticated respondents can have problems estimating probabilities of unlikely events [7]. We had no objective information to offer, however, beyond reminding them of events, which had been widely reported in Korea.

3. In Korea, people often exercise and have a small social hour around trips to obtain spring water in the early morning.

4. Signs are not indicated for the coefficients of BOTL and FILT. Concern with everyday water quality (taste, color, background contaminants) drives spending on bottled water and in-home filtration systems. Concern with possibility of rare events that will lead to unpleasant smells and substantial inconvenience (getting water from tank trucks in the street for several days) drives WTP for the monitoring and reservoir system to prevent such effects. Thus, though both concerns have to do with drinking water quality, they involve different probability regimes (certain average daily quality vs. very improbable events) and different facets of the supply situation. It is therefore not obvious apriori what the relationship should look like, though intuition suggests this will be positive.

5. One could estimate the Tobit model with heteroskedasticity. We have estimated several versions of the heteroskedastic Tobit model, where the heteroskedasticity is assumed to depend on various different sets of independent variables. The results (not reported here) vary, depending on which set of variables are chosen. Furthermore, some coefficients become insignificant, contrary to our expectation. An arbitrary selection of variables to incorporate the heteroskedasticity problem may invite additional problems. Thus, a method requiring fewer assumption is more appealing.

6. One might consider a least absolute deviations estimator of Powell [15] which is also robust to the non-Gaussian and heteroskedastic error specification. However, its main drawback is its computational complexity [1; 8], and it is not used in this paper.

7. The error term of the censored model has the form [Mathematical Expression Omitted]. Therefore, symmetric censoring would replace [Mathematical Expression Omitted] with [Mathematical Expression Omitted] whenever [x[prime].sub.t][[Beta].sub.T] [greater than] 0, and would censor the observation otherwise.

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