Analysis of CVM data

After collecting WTP responses from the sampled households in a CVM study, there are a number of steps to statistically compute mean WTP of the sample. These technical steps are needed because of three main reasons (Bateman et al. 2002). Firstly, the elicitation question formats used in the CVM do not always produce precise figures for each household’s WTP. It

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was mentioned that only the OE format produces precise WTP statements. Other formats such as DC and PC produce only WTP statements as interval data. Secondly, only the sa mpled, not all, households are interviewed for their stated WTP, which makes it necessary to determine the level of confidence with which the sample means can be extrapolated to the whole population in the sample frame. For these reasons, certain statistical methods are needed to estimate the mean WTP which can then be multiplied by the total number of households in the population to arrive at the social value of the proposed environmental project. Thirdly, a statistical estimation must also be employed to find the determinants of WTP. Information on the determinants helps researchers to identify the characteristics of households that obtain more or less benefits from the environmental project. The information on which variables have an impact on people’s WTP statements can also be used for the validation of WTP statements.

For the OE format which produces continuous WTP data, the calculation of mean WTP is rather straightforward. No parametric models are required because when the OE format is used, respondents are asked to state the exact amount of money that they would be willing to pay for the program considered. The obtained WTP statements can simply be averaged to produce the mean WTP of the survey sample. However, the result is the mean WTP of the sample households but not of the population. The two are not necessarily identical because sampling procedures naturally incur errors. To infer the population mean using the rather unreliable sample mean, a confidence interval must be calculated. The confidence interval indicates that range of values that the estimate of interest will assume with a specified level of certainty. As general practices, researchers employ certainty levels of 90%, 95%, or 99%. For example, the 95% confidence interval implies that from 100 times the sampling is repeated, there are 95 times that the true parameter of the population falls in that interval. The following formula is used: 95% confidence interval of 𝜇 = 𝑥̅ ± 1.96 (𝑆.𝐷.

√𝑁), where 𝜇 represents the mean

of the population, 𝑥̅ denotes the sample’s mean, S.D. is the standard deviation of the mean, and √𝑁 is the square root of the sample size (Berry and Lindgren 1996).

Estimating the determinants of WTP for OE data is equally straightforward. Simple regression techniques, usually Ordinary Least Squares regression (OLS), can be used. The stated WTP is taken as the dependent variable and the socio-economic and attitudinal characteristics are taken as the independent variables. But as OE datasets usually consist of a high proportion of zero responses and no nonnegative WTP statements, the use of simple of OLS will produce biased parameters. As a consequence, it is more correct to use parametric

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models that can deal with censored datasets, such as the tobit regression model. The topic which will be discussed at the end of this section.

For other elicitation question formats like the DC and the PC, the estimation of the mean WTP and its determinants is a little more challenging. Both single- and double-bounded DC formats produce binary data indicating whether or not respondents accept or reject the proposed bids. Without further calculations, we will only know that the WTP of a respondent is more or is less than some specified amount of money. Unlike the OE question, the WTP that respondents had in their mind when answering DC questions remains unrevealed. To estimate the mean WTP from the yes/no responses, the idea is to model the underlying utility difference problem that is solved by survey participants when accepting or rejecting the proposed bid. The aim here is to take accurate account of important factors determining the yes/no decisions of households including the underlying WTP itself. This is the basic idea of the utility difference approach which is a dominating approach to estimate respondents’ WTP from DC data (Hanemann 1984). The details of this approach are as follows.

As the DC format produces binary data indicating whether or not an individual household ℎ, ℎ ∈ [1,2, … , 𝐻] with certain demographic and socio-economic characteristics 𝑠_ℎaccepts a bid amount, a decision model based on the characteristics of that household is needed. It is therefore assumed that household ℎ has the following indirect utility function

𝑣ℎ = 𝑣ℎ(𝐼ℎ, 𝑧𝑖, 𝑠ℎ), (2-21)

where 𝑣_ℎ(𝐼_ℎ, 𝑧𝑖_{, 𝑠}

ℎ) represents the indirect utility function of household ℎ. It represents the

maximum obtainable level of utility that the household can obtain given the level of environmental goods 𝑧𝑖 (𝑖 = 0 refers to the situation before the implementation of the project, and 𝑖 = 1 is the situation after), the disposable income of the household 𝐼_ℎ, and the vector of the households’ socio-economic and attitudinal characteristics 𝑠_ℎ.

When a “yes” response from household ℎ, to a proposed bid 𝑡ℎ in a CVM survey is

received, it can be expected that the utility of household ℎ, after making the payment of 𝑡_ℎ , is still at least as high as the utility in the status quo (otherwise household ℎ would have answered “no” to the required payment). That means for those households who accept the required payment it holds that

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The weak inequality in (2-22) can be inferred from any “yes” response to a DC question in a CVM survey as long as the sampled households aim to maximize their utility, which this approach assumes. Nonetheless, if the household answers “yes” for other reasons, e.g. to please interviewers, the inequality (2-22) will not be applicable anymore. The design of a CVM study must ensure that the “yes” response can be explained by (2-22). Further, the inequality (2-22) implies that for the “yes” response to be meaningful, household ℎ must have a clear idea about the change of its welfare between situations 0 and 1. If not, the WTP response, as well as its statistical estimate, will not represent the actual utility changes that the household expects from the project. In the next steps, the parametric version of the weak inequality in (2-22) shall be introduced.

The weak inequality in (2-22) is of course only an illustration of what might happen in respondents’ heads when giving a “yes” answer to the DC question format. The true rationales for this answer are unobservable by nature. This means that the true form of 𝑣_ℎ(𝐼_ℎ, 𝑧𝑖_{, 𝑠}

ℎ) is

unknown to the analyst. So the aim of this approach is the approximation of the real form of 𝑣_ℎ(𝐼_ℎ, 𝑧𝑖_{, 𝑠}

ℎ). This leads to the core idea underpinning the framework of the Random Utility

Model (RUM) as developed by McFadden (1974). According to the RUM, the true indirect utility function of household ℎ consists of two parts: the deterministic term 𝑣̅_ℎ(𝐼_ℎ, 𝑧𝑖, 𝑠_ℎ). representing the approximation of the real indirect utility function made by the analyst and the stochastic term 𝜀_ℎ𝑖 which refers to the part of the true indirect utility function that is unobservable for the analyst and thus can only be taken into account implicitly. Household ℎ’s level of utility can thus be expressed as:

𝑣_ℎ(𝐼_ℎ, 𝑧𝑖_{, 𝑠}

ℎ). = 𝑣̅ℎ(𝐼ℎ, 𝑧𝑖, 𝑠ℎ). +𝜀ℎ𝑖. (2-23)

The framework of the RUM shown in (2-23) will be used as a building block for the statistical estimation of mean WTP and its determinants. Equation (2-22) will be transformed according to the RUM framework. Further, the analyst can only investigate the WTP responses in terms of probability. The probability of a respondent to accept the proposed bid 𝑡ℎ reads:

𝑃𝑟 {𝑦𝑒𝑠ℎ} = 𝑃𝑟 {𝑣̅ℎ(𝐼ℎ, 𝑧0, 𝑠ℎ) + 𝜀ℎ0 ≤ 𝑣̅ℎ (𝐼ℎ− 𝑡ℎ, 𝑧1, 𝑠ℎ) + 𝜀ℎ1}

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From equation (2-24), it can be seen that the probability of household ℎ to accept the required payment of 𝑡_ℎ (and to say “yes” to the DC question) depends on the deterministic part of the indirect utility function, i.e. the part that is based on the observable characteristics of households, and its stochastic part, i.e. the part that is based on the unobservable characteristics of households or measurement errors. The larger the deterministic utility difference the higher the probability that it will exceed the stochastic utility difference and the higher the probability that household ℎ will not experience a utility loss from contributing to the environmental project and therefore tend to say “yes” to WTP question. Given that 𝜀_ℎ0_{− 𝜀}

ℎ1= 𝜀ℎ and that

𝑣̅_ℎ(𝐼_ℎ− 𝑡_ℎ, 𝑧1, 𝑠_ℎ) − 𝑣̅_ℎ(𝐼_ℎ, 𝑧0, 𝑠_ℎ) = ∆𝑣̅_ℎ, the probability that the random variable 𝜀_ℎ is less than ∆𝑣_ℎcan be expressed in terms of the cumulative distribution function of 𝜀, 𝐹_𝜀( . ). Equation (2-24) can be rearranged into

𝑃𝑟 {𝑦𝑒𝑠ℎ} = 𝐹𝜀( ∆𝑣̅ℎ ). (2-25)

If the binary response obtained from the DC format is to be interpreted as the utility maximizing choice, it must be expressed by equation (2-25). Thus, this equation provides a criterion for investigating whether a given statistical model for estimating DC responses is consistent with the economic theory of utility maximization. It also provides a starting point for specifying the functional forms of statistical models so that the related variables can be estimated (Hanemann 1984). Both ∆𝑣̅_ℎ and 𝐹_𝜀( . ) are only general terms representing variations of functions. They have to be specified. For ∆𝑣̅_ℎ, the form of utility function that is often employed is linear in income and other observable characteristics of the household 𝑠_ℎ (Bateman et al. 2002). These other observable characteristics of the household will now be aggregated in 𝛼 but a model to explicitly take them into account will be introduced below. The linear utility function can thus be expressed as

𝑣̅_ℎ = 𝛼 + 𝛽𝐼_ℎ, (2-26)

where 𝛽 represents marginal utility of income. After the form of utility function is specified, the deterministic utility difference ∆𝑣̅ℎ can now be expressed as:

∆𝑣̅_ℎ = 𝑣̅_ℎ1− 𝑣̅_ℎ0= (𝛼1+ 𝛽(𝐼_ℎ− 𝑡_ℎ)) - ( 𝛼0+ 𝛽𝐼_ℎ)

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where 𝛼 = 𝛼1_{- 𝛼}0_{. It should be noted that other forms of indirect utility functions are available}

(Bateman et al. 2002). However, as the focus of this study is not on examining the influences of differences in the parametric specifications of utility on the estimation of WTP but on the methodological improvement of the CVM, it seems justified to employ this version of the indirect utility function.

After the form of utility function is specified, the only task left is to assume the form of the distribution of the error term 𝜀_ℎ𝑖. For the purpose of simplification, 𝜀_ℎ𝑖 is assumed to be independently and identically distributed with mean zero. Further, 𝜀_ℎ𝑖 is assumed to follow a normal distribution. With these specifications, 𝐹_𝜀( ∆𝑣̅_ℎ ) becomes 𝛷( ∆𝑣̅_ℎ ), with 𝛷( . ) being the standard normal cumulative distribution function. The result is the probit model for WTP estimation.2 _{The error term of the probit model is assumed to be normally distributed with a} mean of 0 and variance of 1, that is 𝜀_ℎ𝑖 ~ 𝑁 (0,1). Since the error term 𝜀_ℎ𝑖 is distributed with 𝑁 (0, 𝜎2), the parameters α and 𝛽 have to be normalized to α

𝜎 and 𝛽 𝜎 , 𝛷( ∆𝑣̅ℎ ) thus becomes 𝛷( 𝛼 𝜎− 𝛽

𝜎 𝑡ℎ ), and equation (2-25) reads:

𝐹𝜀( ∆𝑣̅ℎ ) = 𝛷( 𝛼 𝜎−

𝛽

𝜎 𝑡ℎ ) (2-28)

After all functions are specified, the initial task of calculating the WTP can now be approached. First related variables that are necessary for the calculation of the mean WTP have to be identified. Afterwards, it is explained how they can be estimated. Considering equation (2 -27), by assuming that the proposed bid tℎ is exactly equal to households’ WTP, it follows that the

utility difference of household ℎ between the initial state of the environment and the final state

of the environment is zero, i.e. ∆𝑣̅ℎ = 0. Equation (2-27) now reads: 0 = 𝛼 − 𝛽𝑊𝑇𝑃ℎ. So that

𝑊𝑇𝑃ℎ = 𝛼

𝛽. (2-29)

Because mean WTP is to be calculated, the aim is to find α and 𝛽 that best represent those of all individual households. The task then boils down to deriving α and 𝛽 from the “yes” and “no”

2 _{An alternative distributional assumption that is often made with respect to 𝜀}

ℎ𝑖 and should be mentioned here is

the standard logistic distribution. This leads to the so-called logit model for WTP estimation. Both the probit and the logit model have dominated the estimation of the WTP datasets obtained from the binary choice format. There is minute difference between the two: the standard logistic distribution assumes a higher probability density at the tails of the distribution than the standard normal distribution.

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responses in the survey dataset. The maximum likelihood method (MLM) is used for this purpose. The MLM begins with the construction of a likelihood function, which models the occurrence of all observations. In this case, the observations of interest are the “yes” and “no” responses. The likelihood function is maximized by finding those parameters 𝛼 and 𝛽 that maximize the likelihood of the model to the pattern of responses that was actually obtained.

With the help of the parametric model specified in (2-28), the specific likelihood function can be developed. For the single-bounded DC format, the likelihood function using the probit specification reads: 𝐿 (𝛼, 𝛽|𝑡ℎ) = ∏ [𝛷( 𝛼 𝜎− 𝛽 𝜎 𝑡ℎ )] 𝑦𝑒𝑠ℎ . 𝐻 ℎ=1 [1 − 𝛷( 𝛼 𝜎 − 𝛽 𝜎 𝑡ℎ )] 1−𝑦𝑒𝑠ℎ . (2-30)

Equation (2-30) represents the probability of the occurrence of all the “yes” and “no” responses. Now, to fit the model to the dataset it is necessary to fill in variables 𝑦𝑒𝑠_ℎ and 𝑡_ℎ. If the household answers “yes” to the proposed bid, 𝑦𝑒𝑠ℎ is 1 and (2-30) collapses leaving only the

first term. If the household’s response is “no,” 𝑦𝑒𝑠ℎ is 0 and only the latter term of (2-30)

remains. For 𝑡_ℎ the amounts of the proposed bid are filled in. With this process, there will be the multiplication of H terms as the right hand side of equation (2-30). As it is easier to work with summation rather than multiplication, the likelihood function is often transformed to the log-likelihood function 𝑙𝑛 𝐿 (𝛼, 𝛽|𝑡_ℎ) = ∑ 𝑦𝑒𝑠_ℎ. 𝑙𝑛 [𝛷 ( 𝛼 𝜎 − 𝛽 𝜎 𝑡ℎ )] 𝐻 ℎ=1 +(1 − 𝑦𝑒𝑠ℎ). 𝑙𝑛 [1 − 𝛷( 𝛼 𝜎− 𝛽 𝜎 𝑡ℎ )] (2-31)

The next step of the MLM is to find the values of α and 𝛽 that produce the greatest possibility that all observations will occur simultaneously, i.e. that maximizes equation (2-31). The estimation of parameters using the MLM depends on an iteration process. In the calculation, certain starting values for the parameters 𝛼 and 𝛽 are chosen and fed into the model resulting in the likelihood estimate according to (2-31). This process is repeated and continues until it is not possible to find other values of the parameter estimates that bring about a greater likelihood

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function. The actual calculation can be done using a statistical software package such as LIMDEP or STATA. For the double-bounded DC format the likelihood function becomes:

𝑙𝑛 𝐿 (𝛼, 𝛽|𝑡_ℎ𝑙𝑜𝑤, 𝑡_ℎ, 𝑡_ℎ𝑢𝑝) = ∑ 𝑌𝑒𝑠𝑌𝑒𝑠_ℎ. 𝑙𝑛 [1 −( 𝛼 𝜎− 𝛽 𝜎 𝑡ℎ 𝑢𝑝 )] 𝐻 ℎ=1 + 𝑌𝑒𝑠𝑁𝑜_ℎ. 𝑙𝑛 [𝛷 ( 𝛼 𝜎− 𝛽 𝜎 𝑡ℎ 𝑢𝑝 ) − 𝛷 ( 𝛼 𝜎− 𝛽 𝜎 𝑡ℎ)] + 𝑁𝑜𝑌𝑒𝑠ℎ. 𝑙𝑛 [𝛷 ( 𝛼 𝜎 − 𝛽 𝜎 𝑡ℎ) − 𝛷 ( 𝛼 𝜎− 𝛽 𝜎 𝑡ℎ 𝑙𝑜𝑤_)] + 𝑁𝑜𝑁𝑜_ℎ. 𝑙𝑛 [ 𝛷 ( 𝛼 𝜎− 𝛽 𝜎 𝑡ℎ 𝑙𝑜𝑤_)] (2-32)

where 𝑡ℎrefers to the first proposed bid, 𝑡ℎ𝑙𝑜𝑤 refers to the second lower bid, and 𝑡ℎ 𝑢𝑝

refers to the second upper bid. The estimated parameters 𝛼 and 𝛽 obtained from the maximum likelihood method can be used to compute mean WTP according to (2-29). Since mean WTP is calculated from the ratio of two parameter estimates each with its own standard error, the calculation of the 95% confidence interval is not as straightforward as in the case of OE data. The bootstrapping method developed by Park et al. (1991) is recommended. This method utilizes data obtained from the estimated coefficients, together with variance and covariance matrices to randomly estimate mean WTP 1,000 times. To calculate a 95% confidence interval of mean WTP the first and last 25 from these 1,000 WTP estimates have to be eliminated. This will result in 950 stated WTP estimates the first and last of which are the boundaries of the 95% confidence interval of the mean WTP.

It has been mentioned that another aim of the CVM is the identification of determinants of WTP. With regard to this issue, WTP determinants can be investigated using the extended version of the linear utility model presented in (2-27):

∆𝑣̅_ℎ = 𝛼 − 𝛽𝑡_ℎ+ 𝛾_𝑗𝑠_ℎ𝑗, (2-33)

where the vector 𝑠_ℎ𝑗, j = (1,2,…,J) consists of 𝐽 socio-economic, demographic, or attitudinal characteristics of the households, 𝛾𝑗 refers to the parameter vector of the 𝐽 observed variables

of the vector 𝑠ℎ𝑗. WTP determinants are those variables of the vector 𝑠ℎ𝑗 that have significant

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For the DC dataset of this study, parameters 𝛼, 𝛽, and 𝛾_𝑗 will be estimated based on the probit model. For the PC dataset, two estimation methods for the calculation of mean WTP are possible. First, the mean WTP can be assessed using interval data. In this case, any bid interval in the payment card can be described as (𝑡_ℎ𝑙𝑜𝑤, 𝑡_ℎ𝑢𝑝). When a payment card is selected by respondents it can be interpreted using the logic of the DC approach in that they accept 𝑡_ℎ𝑙𝑜𝑤 and reject 𝑡_ℎ𝑢𝑝. The log-likelihood function for the PC dataset therefore reads:

𝑙𝑛 𝐿 (𝛼, 𝛽|𝑡_ℎ𝑙𝑜𝑤, 𝑡_ℎ𝑢𝑝) = ∑ 𝑙𝑛 [𝛷 ( 𝛼 𝜎− 𝛽 𝜎 𝑡ℎ 𝑢𝑝 ) − 𝛷 ( 𝛼 𝜎− 𝛽 𝜎 𝑡ℎ 𝑙𝑜𝑤_)] 𝐻 ℎ=1 (2-34)

Alternatively, the analyst could also calculate the midpoints of each response interval and directly calculate mean WTP based on these midpoints. In this case, there is no need for any parametric estimation technique. The computation of mean WTP based on midpoints frees researchers from the uncertainty regarding the correct distribution for dataset. It also releases