Theoretical Validity

In contingent valuation studies, theoretical validity is usually assessed by estimating a bid function which seeks to explain the variation in willingness to pay across respondents in terms o f their underlying socio-economic characteristics. Owing to the statistical specification of the logit model it is not possible to incorporate socio­ economic regressors directly into the utility index. Instead, the impact of socio-

However, panel data approaches such as the Random Parameters Logit (Train, 1998), in their present form, do not incorporate information on the full ranking of alternatives, only catering for data on the most preferred alternative. Hence, for the purposes of this and the following chapter, in order to avoid correlation between individual error terms, the econometric analysis presented is based on the use of a single randomly selected four-way ranking set per respondent (note that Ben Akiva et a l, 1991 and Lareau and Rae, 1985 only assigned one four-way ramking set per respondent). While ensuring the independence of observations, this approach throws away information by not utilising all available data. Alternative panel data approaches to modelling ranking data will be investigated in Chapter 8.

The reason for this is that, as illustrated in equation (6.2) above, the random utility framework is based on differencing the values of attributes across different alternatives. Thus any attribute, such as a socio-economic characteristic of a respondent, which does not vary across alternatives would simply drop out of the model as a result of this process.

economic factors on willingness to pay can be gauged by one of two alternative methods. The first entails interacting socio-economic variables with the attributes of the alternative loaves/^ and examining the signs and significance o f the resulting interacted variables. The second involves estimating the model on sub-samples of the data differentiated by socio-economic factors, and comparing the resulting willingness to pay across sub-samples. Each of these two approaches will be considered in turn. An important prediction of economic theory is that willingness to pay will be an increasing function of income. A natural way of testing this hypothesis in the present context is to incorporate an additional interaction variable which expresses price as a function o f individual income (fi), thereby modifying the utility index along the lines indicated in equation (6.13). The predicted willingness to pay then becomes a function o f income level as shown in equation (6.14). Because of the form o f the interaction, a positive relationship between income and willingness to pay will show-up as a negative coefficient on the price-income interaction variable (Jb^.

Uij = bpPij + by{Pij / YÎ) + bhHiJ + bbBij (6.13) dUij

dHij bh

The results o f applying this approach are illustrated in Table 6.3, where the MPA model with the income interaction variable is compared to the baseline specification with no interaction. Given that the income variable was missing for one third o f the sample, the baseline MPA model was re-estimated on the sub-sample for which the income variable is available, in order to facilitate comparisons between the two specifications.^^

Interacting the socio-economic characteristics with the attributes of the alternatives circumvents the earlier problem because the interacted variable both contains individual-specific socio-economic information and varies across alternatives in the choice set.

In order to check for the presence of sample selectivity bias brought about as a result of income non­ response, the log-likelihood fimction for tiie whole sample is compared with the sum of the log- likelihood fimctions estimated for the sub-samples that include and exclude income observations. This likelihood ratio procedure, described by Ben-Akiva and Lerman (1985), cannot reject the null hypothesis of equality between the coefficients estimated on these exhaustive and mutually exclusive sub-samples. Moreover, the willingness to pay estimates resulting fi-om the two sub-samples are not significantly different.

Table 6.3: Comparison of MPA models with and without the income interaction

Baseline Missing Income Observation with Model with

Model Observations

Only

Income Only Income

Interaction Price* -3.515 -3.768 -3.087 -2.564 (0.372) (0.471) (0.609) (0.559) -9.443 -7.994 -5.070 -4.586 Human health* -2.391 -2.399 -2.391 -2.474 (0.246) (0.311) (0.405) (0.314) -9.703 -7.721 -5.903 -7.886 Bio-diversity* -1.837 -1.850 -1.829 -1.903 (0.223) (0.278) (0.374) (0.281) -8.249 -6.656 -4.894 -6.779 Price-income — — — -7.130 interaction* _(1.923) -3.709 Log-likelihood -621.942 -400.408 -220.667 -392.834 No. of 501 324 177 324 observations WTP for 0.681 0.642 0.791 0.778 human health** (0.062) (0.073) (0.138) (0.109) WTP for bio­ 5.252 4.933 6.045 5.981 diversity* _{(0.574) (0.655) (1.134)} ---:--- :---—______ (0.933)

units of pence per loaf of bread.

coefficient, standard error and t-ratio. The WTP is expressed in

The income interaction variable proves to be highly significant, with the expected negative sign. The price variable on its own also remains highly significant, though less so than in the baseline specification. This demonstrates that willingness to pay for the ‘health’ and ‘birds’ attributes increases with income level, as predicted by economic theory. A likelihood ratio test strongly rejects the exclusion o f the income interaction variable, with a test statistic of 15.15 well above the 95% critical value of 3.84.

However, the ultimate willingness to pay estimates for the interaction model based on the average income level lie within the 95% confidence interval of the baseline MPA model estimated on the same sample. A very similar pattern of results is obtained from estimating the same models on the basis of the RD specification.

The second approach to testing for theoretical validity relies on estimating willingness to pay for a series of exhaustive and mutually exclusive groupings of the population. Clearly, there are a large number o f ways in which the sample could be segmented. Here, the general approach is illustrated by examining cuts according to gender, degree of interest in environmental issues (as measured by an attitudinal variable included in the introductory section of the survey), educational level, and bird- watching status.

The resulting models for each of the four pairs of cuts are summarised in Table 6.4, which indicates substantial variations in coefficients across population segments, particularly as regards the price coefficients. In order to test whether these differences are statistically significant overall, Ben-Akiva and Lerman (1985) propose a likelihood ratio test based on comparing the log-likelihood function for the model estimated on the pooled sample, where coefficients across all segments of the population are implicitly restricted to be equal {logL^, and the sum of the log- likelihoods estimated across a series of exhaustive and mutually exclusive segments of the population where coefficients are allowed to vary {ZulogLij). The test statistic, which is given in equation (6.15), is distributed as a chi-squared variable with degrees o f freedom equal to the difference in the number of coefficients estimated across the restricted and unrestricted models.

S = 2 [ ( ^ log Lu) - log Lj,] (6.15)

U

The test statistics for this likelihood ratio test are also reported in Table 6.4, and can be compared against a critical value o f 7.82 at the 95% confidence level. This suggests that only degree o f interest in environmental issues and bird-watching status have a material impact on the coefficient estimates.

This finding is corroborated by a comparison of the willingness to pay estimates produced by the Krinksy and Robb procedure and their associated standard errors. These indicate that while women exhibit a higher willingness to pay than men, and those with tertiary education exhibit a higher willingness to pay than those with only secondary education, these differences are not statistically significant. However, those with a high degree of interest in environmental issues are willing to pay around three times as much to preserve human health and bio-diversity as those who do not. While birdwatchers are willing to pay nearly twice as much to protect bird species as those who do not undertake this pastime. This pattern o f results conforms to a priori theoretical expectations and provides further evidence o f the theoretical validity o f the results. Once again, repeating this estimation procedure with the RD model instead of the MPA model does not materially affect the conclusions drawn.

Table 6.4: Comparison of MPA models estimated on different population segments

Gender Concern for environment Educational level Birdwatcher

Males Females High Low Secondary Tertiary Yes No

Price* -3.771 -3.313 -3.396 -4.495 -3.392 -4.269 -3.825 -3.431 (0.561) (0.499) (0.421) (0.967) (0.403) (1.002) (0.858) (0.416) -6.726 -6.641 -8.067 -4.648 -8.420 -4.262 -4.456 -8.242 Human health* -2.372 -2.446 -2.919 -0.808 -2.256 -3.114 -3.052 -2.240 (0.356) (0.345) (0.282) (0.588) (0.270) (0.623) (0.566) (0.276) -6.658 -7.100 -10.347 -1.374 -8.357 -4.998 -5.390 -8.109 Bio-diversity* -2.029 -1.668 -2.222 -0.894 -1.688 -2.614 -2.994 -1.518 (0.332) (0.302) (0.257) (0.520) (0.243) (0.579) (0.518) (0.249) -6.115 -5.529 -8.643 -1.719 -6.952 -4.519 -5.781 -6.099 Log-likelihood -306.891 -314.520 -480.505 -112.827 -511.746 -108.424 -123.315 -492.518 No. of observations 247 254 405 96 408 93 111 390 LR test statistic 1.062 57.220 3.544 12.218 WTP for human 0.633 0.752 0.867 0.174 0.672 0.738 0.826 0.656 health ** (0.079) (0.100) (0.090) (0.124) (0.074) (0.143) (0.176) (0.070) WTP for bio­ 5.449 5.064 6.561 1.959 4.966 6.245 8.032 4.440 diversity* *____ :---- r----—---:— (0.685) (0.842) (0.719) ■ r ' (1.095) (0.610) (1.294) (1.596) (0.591) 182