exhibiting a covariance matrix Σ = μσψμ + σ21, with μ as K × 1 vector of zeros
and ones that create the correlation structure.
Another drawback of the conditional logit model (3) is that it does not allow
for taste variation, meaning that any household specific deviation from the mean-
sample taste would enter into the unobserved part of utility eij. In the present
application, this would preclude the possibility that households exhibit different
responses to the determinants of retrofitting alternatives. An appropriate method
to deal with such heterogeneity in adoption behavior is to allow for household
specific coefficients βi =(β + ui), with ui as a household specific deviation from
the sample mean β , such that β exhibits a distribution across the sample of
households. This gives rise to the random-parameter logit model:
(7)
Pi(j)=
( ∖
eVij(βi)
Ç eVk ( βi )
f(β)dβ.
Equation (7) is a generalization of Equation (3) as it estimates not only the mean
coefficient but the parameters of the underlying distribution for those coefficients
that are specified as random (Train 2003). For example, if a random parameter
β is assumed be normally distributed in the population, the random-parameter
logit model estimates the mean and standard deviation of β . The coefficients can
thus vary across observations, thereby accounting for taste variations with respect
to the attributes of the available retrofitting alternatives. In this way, some parts
of the unobserved heterogeneity inherent in the conditional logit model can be
removed (Hensher and Greene 2003).
The random-parameter logit fully relaxes the IIA property and additionally
allows for any correlation structure between the utility of different alternatives.
If the representation of a particular correlation pattern is deemed important,
the random-parameter logit can also be specified using the error components de-
scribed above. As discussed by Koppelman and Bhat (2006), this more flexible
approach captures both heterogeneous preferences and complex correlation pat-
10
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