Ranked Data Model

As noted above, although the RD model makes use of a richer set o f information about respondent preferences, it is significantly more restrictive than the MPA model in the sense that it assumes the same choice process governs the top rank as governs each successive rank. Hausman and Ruud (1987) note the possibility that individuals might pay much more attention to identifying their first best choice than to ranking the remaining alternatives, and show that such an outcome would render the use o f the RD model illegitimate. Ben-Akiva et a l (1991) also find significant differences among choice models run for different ranks, hence rejecting the RD formulation. Using similar four-way rankings to the ones used in this chapter, these authors find that choice data from the third rank does not provide any additional estimation accuracy over and above the first two choices. The implication is that the decision criteria adopted for the top ranks may well be different from that o f the low ranks. The notion that choices should be consistent across ranks suggests a natural likelihood ratio based specification test which involves comparing the outcome of the RD model, ■with the outcome of estimating a series of separate MPA models for the selection of each successive rank from the choice set which remains after the previously most highly ranked alternative has been deleted (Ben-Akiva et al., 1991). The test is based on the recognition that the log-likelihood for the RD model can be broken down into the sum o f the log-likelihoods for each of the series of MPA models just described.

This is shown in equation (6.11). In the RD specification the coefficients governing each successive choice are constrained to be equal, whereas separate estimation of the underlying MPA models (logLupAj) allows them to vary across ranks. Under the null hypothesis, the log-likelihoods associated with each o f these two estimation procedures should be equal. The test statistic is stated in equation (6.12). The degrees of freedom for the test are given by the difference between the number of coefficients estimated in each case.

logZ = x t l o g [ ^ ^ ^ e ^ ] (6.11)

' '^Qxpibxh)

k=j

^ e x p (6 % * ) ' ^ ^ ^exp(6% ik) ' ^exp(6% & )

; j=\

*=] *=2 *=3

J - \

s = 2[( Yj^ogLMPAj) - (6.12)

7=1

Table 6.2 contains the results of estimating the series o f separate MPA models described above. The first column presents the results of modelling the first choice out of the four alternatives (this corresponds to the standard MPA specification); the second column corresponds to modelling the best choice out of the remaining three alternatives, after the first ranked alternative is removed; the third column corresponds to the choice o f the best out o f the two remaining alternatives, after the first and second ranked alternatives are eliminated from the choice set; finally, the last column presents the standard RD model.*® The application o f the likelihood ratio test given by (6.12) to this data set yields a test statistic of 66.99 in comparison with a critical value

For the purposes of estimation, the attributes levels were re-scaled in the following way: price is expressed in pounds per loaf; the ‘health’ attribute was divided by 100; and the ‘birds’ attribute was divided by 10. This affects the interpretation of the coefficients. The re-scaling is removed during the process of calculating willingness to pay.

for the chi-squared distribution with six degrees of freedom o f 12.59. This constitutes a strong rejection of the RD specification over and against the MPA model.

These results confirm the findings of Ben Akiva et al. (1991) and Hausman and Ruud (1987) for a data set that conforms to the IIA assumption and show that there can be potential bias in coefficient estimates when using the full RD specification. Chapter 7 provides further evidence on this point.

6.4.3 Comparison of W TP Across Both Specifications

Hausman and Ruud (1987) maintain that while use of a misspecified RD model may lead to significant differences in the coefficients estimated relative to the MPA model, this will not necessarily make any material difference to the ultimate willingness to pay estimates which are ratios of those coefficients. In order to ascertain whether this is the case in the present context, Table 6.2 also presents the willingness to pay values associated with coefficient estimates for the MPA and RD models (first and last columns respectively).

The results show that the coefficients for both models have the expected negative signs and are highly statistically significant. Relative to the MPA model the RD model has a very similar coefficient for human health and smaller coefficients for price and bio-diversity. In order to examine whether these differences were statistically significant a Hausman test for overall equality o f the coefficients was applied, comparable to that defined by equation (6.10). The resulting test statistic of 43.90 lies well above 7.82, which is the 95% critical value for the chi-squared distribution 'with

" The equality of individual parameters across the three MPA models in Table 6.2 can also be tested according to the following asymptotically normal test statistic (Ben-Akiva et al, 1991):

1 2

test statistic = ^

yvar(6 )+var(6 )

The results show that the price coefficient varies significantly (at the 5% level) across all MPA specifications while the health coefficient is only different between the second and third choice models. The bio-diversity coefficient is not significantly different across models. The differences in the price attribute explain the wildly different WTP values obtained across specifications.

three degrees o f freedom. This rejection is consistent with the results o f the likelihood ratio test reported above.

Table 6.2: Comparison of MPA and RD specifications

MPA Models RD

Model P* Choice 2““ Choice 3'^“ Choice

Price* -3.515 -1.792 -0.447 -2.015 (0.372) (0.380) (0.500) (0.077) -9.443 -4.711 -0.893 -26.295 Human health* -2.391 -3.087 -1.016 -2.346 (0.246) (0.314) (0.407) (0.103) -9.703 -9.838 -2.494 -22.853 Bio-diversity* -1.837 -1.260 -1.393 -1.496 (0.223) (0.217) (0.294) (0 .1 0 2) -8.249 -5.811 -4.737 -14.699 Log-likelihood -621.942 -483.588 -319.176 -1458.2 No. of observations 501 501 501 501

WTP for human health 0.681 1.798 2.039 1.168

(0.062) (0.344) (31.164) (0.083)

WTP for bio diversity* 5.252 7.325 21.259 7.444

i_— ... (0.574) (1.614) (500.53) (0.639)

are coefficient, standard error and t-ratio. The WTP is expressed in units of pence per loaf of bread.

In line with prior expectations, the standard errors on the coefficients o f the RD model are substantially smaller than those obtained for the MPA model, leading to t-ratios two to three times greater. This higher degree o f efficiency is attributable to the fact that the RD model is making use of all the available information about consumer rankings as opposed to just the first best choice.

As noted in equation (6.9), the implicit values of willingness to pay are non-linear functions o f the coefficient estimates. As was already mentioned in Chapter 5, Krinsky and Robb (1986) show that, under these circumstances, classical linear

A test for differences in individual coefficients found a significant difference, at the 5% level, between the price coefficient of the MPA and the RD specifications but not for the other attributes.

approximations are unlikely to provide accurate estimates o f the variance of willingness to pay. A superior approach is to use simulation techniques to establish the empirical distribution o f the welfare measure, based on N random draws from the multivariate normal distribution defined by the coefficients and covariance matrix estimated firom the logit model. All willingness to pay estimates and associated standard errors reported in this chapter are estimated using this technique, with simulations based on 1,000 repetitions.

Willingness to pay is found to be substantially larger under the RD model than under the MPA model, 1.2 pence per loaf as against 0.7 pence per loaf to prevent each case of human ill-health and 7.4 pence per loaf as against 5.3 pence to reverse the decline of each species o f farmland birds. Another way of expressing these results is that on average respondents were only willing to tolerate between 6-8 cases of human ill- health to save an entire species of farmland birds. This indicates the importance that the sample attached to the preservation of human health.

In order to ascertain whether these differences are statistically significant, it is necessary to examine the magnitude of the standard errors generated by the Krinsky and Robb (1986) procedure. These indicate that the observed differences in willingness to pay are statistically significant in the case o f human health but not in the case of bio-diversity. Thus this finding provides only partial support for Hausman and Ruud’s (1987) suggestion that misspecification need not be a problem if ultimate interest lies in willingness to pay rather than in individual coefficient estimates.

Overall, the results indicate that the greater efficiency and reduced sample sizes that are possible when ranks data is used come at the expense o f a considerable degree of misspecification. On the strength o f these conclusions the MPA model will be used as the basis of analysis throughout the remainder of this chapter. However, for comparison, the results of the RD specification will also be reported where they show any interesting divergence from the conclusions o f the MPA model. In this way, the consequences of misspecification can be fully explored.

It should be noted that the contingent ranking data that was available for this chapter is effectively a panel with multiple rankings per individual. Consequently, using pooled cross-sectional estimation techniques such as the MPA or the RD models, as is commonly done in the literature (Adamowicz et a l, 1994; Hanley et a l, 1998), could be expected to lead to biased results in as much as the error terms on the choice sets ranked by a particular individual could be expected to correlate across each other.