Results of the tobit regression

The Tobit model (from Tobin, 1958) considers the amounts declared by respondents as if they were truncated at 0. Bids can not be smaller than zero and are thus cut down at zero.

Indeed, respondents have at least a willingness to pay equal to zero and their value cannot be negative. This truncation results in a high concentration of bids equal to zero, which is

common in contingent valuation studies. Tobit models are adapted to take this particular feature into account. The structure of the model is given by following equations for the individual i :

i i

i X

y^* = ^'α+ε ,

ε

_i∼ N(0,

σ

²)

⎩⎨

= ⎧

else 0

0 if ^*

* i f

i i

y y y

Where states for the latent dependant variable (unobserved), the observed variable (declared wtp), a vector of explanatory variables for the individual i and

*

yi y_i

Xi

α

the vector of

coefficients. We assume that the unobserved continuous dependant variable y_i* is the respondent’s true WTP for the chosen scenario. Moreover we assume that the underlying distribution of y* is a normal distribution and that y* is conditional on a vector of explanatory variables : X. The Tobit model assumes that residuals follow a Gaussian distribution, with a zero mean and standard deviation equal to 1. In this model, the variable defining truncation is the same than the one which is observed when there is no truncation. The model does not explicitly model truncation. We assume that the decision to participate and the amount are defined by the same variables.

Different regression models were explored, i.e. with or without protest answers, with or without logarithms. For comparisons, the base regression exclude protest answers and log amounts and quantitative independent variables: age, income, number of persons living in the household. Dichotomous variables remain unchanged. For zero amounts, one alternative is to assume that log (0) = 0; the other alternative is to add one to all WTP amounts/values, using log (wtp+1) as dependent variable. This second alternative is presented in this section, since the hypothesis seems to be quite strong. Nevertheless, results are very similar between both alternatives.

Table 19. Results of the Tobit regression explaining respondent’s WTP values for groundwater quality improvement¹⁶

Log-log without protest Log-log with protest Linear without protest Perception and link to water

Good quality

observations 346 (including 145 left truncated) 427 (including 226 left

truncated) 346 (including 145 left truncated)

In total, 346 interviews were used to calculate the coefficients presented in Table 14. Among them, 145 are truncated at zero. None are truncated on the right since there is no upper limit for contributions. The only limit would be the budget constraint that is unknown and might not have been relevant.

For the model without protest and logged variables, the Pseudo-R² is equal to 0.1719 which is relatively high for a contingent valuation survey. But the largest proportion of contribution is still unexplained by measured variables. Nevertheless, the Pseudo-R² has its limits. This is why it is interesting to estimate also the correlation between predicted variable and the actual

16 Note : Reported marginal effects, which can be interpreted as elasticity, are calculated at the means of explanatory variables.

dependant variable. IN the model presented above, it is equal to 0.3. Nevertheless, the model is globally significant and better explains the values given by respondents than a constant alone. The likelihood ratio test states that the likelihood with all variables included is significantly higher than the one with only the constant.

a. Quantitative variables

Since quantitative variables and contribution amount are logged, coefficients can be interpreted in term of elasticity. A 1% change in the explanatory variable will result in a αj% change in the dependant variable (α_j being the parameter estimated). Due to the use of the maximum likelihood method, it is the marginal effects which have to be considered. So holding other things constant, respondents who are 10% older will be willing to pay 6.9%

less. Income elasticity is equal to 0.78 which means that shallow groundwater quality is a normal good - its elasticity is smaller than 1 and higher than 0 - confirming results of Cho (2005) about conservation easements. Its demand increases with income but not as fast as income increases. When income increases by 10%, the proposed contribution will be 7.8%

higher.

b. Dichotomous variables

For dichotomous variables, coefficients can not be interpreted as elasticity. They can not be logged. Commonly, a one unit change is considered as conducting to a 100*α_j unit change in the contribution. Nevertheless, Halvorsen and Palmquist (1980) showed that the change is in fact of 100*exp(

α

j−1)%. According to Kennedy (1981) estimate of variance ( ) should be taken into account so that the change is of

^

walk often along watercourses, they accept to pay 67% more. For dummy variables, the fact that good groundwater quality is considered as important has the stronger effect on the proposed financial contribution since it is significant at the 1% level with an estimated coefficient equal to 2.14. Thinking that it is important to have good quality shallow groundwater in Riga leads to WTP values higher by 303%. The confidence in the groundwater improvement programmes is also an important explanation of the willingness to pay values, with a positive and significant coefficient.

Citing environment problems among the more important problems for the region increases the contribution amounts. This coefficient, however, is only signfiicant at the 10% level.

c. Without Protest

When including protests, the goodness of fit is not as good as when they are excluded, 0.1453 only versus 0.1719 when excluding protest. Indeed, some answers are not the right one since they are not revealed. The significant variables are however the same in both regressions, except for environmental problems which no more explain the amounts people are willing to pay for groundwater improvement programmes.

d. Without Log Transformation

The log transformation increases the Pseudo-R2 significantly as the linear regression goodness of fit is equal to only 0.0645. The interpretation of coefficients also differ as coefficients are interpretable in terms of units and no more in terms of percentage. Having a well is now significant and increases people’s willingness to pay by 17 euros. Indeed, well owners and users are directly connected to groundwater as they use it or may even drink it.

Citing environmental problems is once again not significant showing that this variable might not clearly represent an important aspect in people’s perception and views for the method used.

e. Chow Test

The choice of scenarios does not influence the willingness to pay amounts. It could however influence the way the amount level is chosen. The stability of coefficients is tested through the chow test for Scenarios 2 and 3. A separate regression could not be done for Scenario 1 because of the limited number of respondents who chose this first (basic) scenario. The following hypothesis was tested : H0, ∀j, αj scenario2 = αj scenario3 = αj scenario2 and scenario3 together.

This hypothesis is accepted. Thus, coefficients are equal, whether regressions are made separately (one for each scenario) or respondents grouped independently of their scenario choice.

Dans le document ACTeon Innovation, policy, environment (Page 56-60)