Final Mixed Logit Model

Table 7.16 shows the final ML model that was achieved following the step-by step m ethodology described in the previous section. Each variable was added at a time, and it was kept if the distribution o f the random coefficients as tested improved the likelihood o f the base model.

It is found that the mixed logit model performs better in explaining preferences for quiet, considering the significance o f the random coefficients pointing out for individuals’ intrinsic tastes across the already modelled groups based on observed attributes. Also, the log likelihood at convergence o f the mixed logit is -2448.513 in comparison to the value o f - 2834.390 for the MNL-INT model, and following a likelihood ratio test this represented a significant improvement considering for the additional 7 degrees o f freedom.

Following the ML specification reported in Table 7.16, it is possible to retrieve more information on the factors that influence the variation o f householders’ tastes:

1 and 2 - Deteriorations versus Improvements in the levels o f quiet: the best specification for the coefficient for deteriorations was a fixed one. The standard deviation o f the random component for all distributions was not statistically significant. It had the expected sign but it was lower than the corresponding coefficient for improvements (base). However, the best specification for the coefficient for improvements was a random parameters one, when a normal distribution was assumed. The standard deviation o f the random coefficient was highly significant (8.61). This finding seems to indicate quite rigid behaviour o f individuals when concerning losses in quiet in comparison to improvements in quiet (loss aversion). The proportion o f individuals with wrong sign is 2%.

3 - Interaction o f quiet with general flat exposure: householders located at the quieter facade (back) have a higher marginal value o f quiet in comparison to those fronting the main road.

Considering the normally distributed coefficient, it can be seen that the standard deviation o f this coefficient is highly significant (6.47) due to householders’ heterogeneity. The proportion o f individuals with wrong sign is 11%.

Table 7.16: Combined M NL-INT M odels with Random Parameters. 1 QDET: Deteriorations in quiet levels (base) 0.0473

(2.54)

"

2 QIMP: Improvements in quiet levels (base) ♦ 0.0439 (2.49) 6 QFEM: Interaction o f quiet with gender ♦ 0.02698

(2.977)

* Functional form: Quiet*(Quiet-Base Level in the status quo)2

* * Functional form: Housing Service Charge/Income per person 0 5

♦ Normal Distribution. ♦ ♦ Log-normal distribution. Note that estimation gives the log (coefficient estimate).

4 - Interaction o f quiet with num ber o f years living at the site (dummy variable for number o f years > 5): the best specification was found when this coefficient was allowed to vary across individuals following a normal distribution. The standard deviation o f this coefficient is statistically significant at the usual 5% level o f confidence. The proportion o f individuals with wrong sign is 14%.

5 - Interaction o f quiet with familiarity to the SP choice context (lot): The best specification for this coefficient was found when it was allowed to vary across individuals following a normal distribution. The standard deviation o f this coefficient (5.56) confirms the importance o f random variation across observed heterogeneity. The mean value was not statistically significant at the 5% level o f confidence, and this fact indicates that tastes tend to balance out w ith respect to the effect o f familiarity o f choice context on the marginal values o f quiet.

6 - Interaction o f quiet with gender (dummy variable for females'): The best specification for the interaction o f this dummy coefficient with quiet was when it was allowed to vary across individuals following a normal distribution. The standard deviation o f the mean coefficient is highly statistically significant (4.07), reflecting the importance o f unobserved random variation intrinsic to each individual (female) case. All individuals had the right sign for this parameter.

7 - Interaction o f quiet with size o f quiet changes relative to the base: the coefficient in Table 7.19 is rescaled (/1000) in order to get elements o f the Hessian o f the same order of magnitude. The best specification for this coefficient was a fixed one. In this case, considering the functional form for this interaction term, it can be said that it has a fixed (same) effect on each household type (i.e. with same base level o f quiet and facing same size o f changes in levels). The proportion o f individuals with wrong sign for this param eter is

1%.

8 - Interaction o f quiet with dummy for floor num ber greater equal than four: the best specification was a fixed effects one, and it shall be noted that this coefficient is only statistically significant at a low level o f significance, below 5%. The proportion o f individuals with wrong sign for this parameter is 8%.

9 - Interaction o f housing service charge with current payment (/10(’): the presence o f reference effects with respect to the base payment (housing service charge) follows a fixed effect specification. This specification is imposed in order to obtain marginal values o f quiet.

This is because the distribution o f the ratio coefficient that involves the computation o f the marginal value o f quiet (ratio o f two partial derivatives in the conditional indirect utility function) is not easy tractable in all cases, depending on the distribution o f the random param eters in the numerator (quiet) and denominator (cost). All individuals had the right sign for this parameter.

10 - Interaction o f housing service charge with adiusted income per person (considering household composition): This cost coefficient is kept fixed, as in 9 in order to be able to derive tractable marginal values o f quiet. It shall be noted that the ratio ol two random coefficients e.g. normally distributed over log-normal (this would be the case it this cost variable would have been specified following a log-normal distribution, as it can be expected that it is always negative for all householders), is not as easy to treat because the resultant density distribution o f the coefficient ratio (marginal value o f quiet) will have a non closed form. All individuals had the right sign for this parameter.

11 _ Interaction o f housing service charge with missing income: for the same reasons as in 9 and 10, this cost coefficient is kept fixed. All individuals had the right sign for this parameter.

12 - View: the best distribution for this coefficient was log-normal, and its mean and standard deviation are highly statistically significant. It shall be noted that GAUSS gives the log (view coefficient estimate), and this justifies the negative sign o f the mean o f the log (view coefficient). Following the log-normal distribution, the mean and standard deviation o!

the view coefficient have to be calculated. It shall be noted that GAUSS gives as output the logarithm o f the view coefficient. Therefore the mean o f the view and standard deviation o f the view coefficient need to be computed as: mean is exp (m+s~/2) and its standaid deviation is equal to: exp (m+s2/2)*sqrt (exp (s2 -1)), being m and 5 respectively the mean and standard deviation o f the logarithm of the view coefficient. Following the log-normal distribution by definition, all individuals had the right sign lor this parameter.

13 - Sun Exposure: the best specification for this coefficient was when it was allowed to vary following a normal distribution. It can be seen that the mean and standard deviation of this coefficient has an high statistical significance, as in 12 confirming the importance ol dealing with the issue o f householders’ taste variation lor qualitative variables. 1 he proportion o f individuals with wrong sign for this param eter was 24%, and this higher value is consistent with the fact that for some individuals more sunlight is considered as a bad, w hilst for others is a good.