William Greene 543(e.g., Revelt and Train, 1998, and, especially, McFadden and Train, 2000) for this
model. The choice-specific constants,αji, and the elements ofβiare distributed
randomly across individuals with fixed means. A refinement of the model is to
allow the means of the parameter distributions to be heterogeneous with observed
datazi(which does not include a constant). This would be a set of choice invariant
characteristics that produce individual heterogeneity in the means of the randomly
distributed coefficients so that:
βki=βk+z′iδk+σkvki,and likewise for the constants. The model is not limited to the normal distribution.
One important variation is the log-normal model,
βki=exp(βk+z′iδk+σkvki).Thevkis are individual and choice specific, unobserved random disturbances – the
source of the heterogeneity. Thus, as stated above, in the population, if the random
terms are normally distributed:
βki∼Normal or Lognormal[βk+z′iδk,σk^2 ].(Other distributions may be specified.) For the full vector ofKrandom coefficients
in the model, we may write the full set of random parameters as:
βi=β+ zi+vi,whereis a diagonal matrix which containsσkon its diagonal. For convenience at
this point, we will simply gather the parameters or choice specific constants under
the subscript “k.”
Greene and Hensher (2006) have developed a counterpart to the random effects
model that essentially generalizes the mixed logit model to a stochastic form of
the nested logit model. The general notation is fairly cumbersome, but an example
suffices to develop the model structure. Consider a four outcome-choice set: Air,
Train, Bus, Car. The utility functions in an MNL or mixed logit model could be:
Uit,Air =αAir +x′it,Airβi +εit,Air +θ 1 Ei,Private
Uit,Train =αTrain +x′it,Trainβii +εit,Train +θ 2 Ei,Public
Uit,Bus =αBus +x′it,Busβii +εit,Bus +θ 2 Ei,Public
Uit,Car =x′it,Carβi +εit,Car +θ 1 Ei,Private,where the componentsEi,PrivateandEi,Publicare independent, normally distributed
random elements of the utility functions. Thus this is a two-level nested logit
model.
The probabilities defined above are conditioned on the random terms,vi, and the
error components,Ei. The unconditional probabilities are obtained by integrating
vikandEimout of the conditional probabilities:Pj=Ev,E[P(j|vi,Ei)]. This is a
multiple integral which does not exist in closed form. The integral is approximated
by simulation (see Greene and Hensher, 2006, and Greene, 2007a, for discussion).
Parameters are estimated by maximizing the simulated log-likelihood.