Random
PMC method
This
technique approximates the choice probabilities by computing the integrand in equation
(15) at randomly chosen values for each 
. Since the terms are independent
across individuals and variables, and are distributed standard normal, we
generate a matrix s of standard normal random numbers with Q*K elements (one element for
each variable and individual combination) and compute the corresponding
individual choice probabilities for a given value of the parameter vector to be
estimated. This process is repeated R
times for the given value of the parameter vector and the integrand is
approximated by averaging over the computed choice probabilities in the
different draws. This results in an unbiased estimator of the actual individual
choice probabilities. The variance of ‹ decreases as R increases also has the appealing
properties of being smooth (i.e., twice
differentiable) and being
strictly positive for any realization of the finite R draws. The parameter vector is estimated as the vector value that
maximizes the simulated log-likelihood function. Under rather weak regularity
conditions, the PMC estimator is consistent, asymptotically efficient, and
asymptotically normal. However, the estimator will generally be a biased
simulation of the maximum likelihood (ML) estimator because of the logarithmic
transformation of the choice probabilities in the log-likelihood function. The
bias decreases with the variance of the probability simulator; that is, it
decreases as the number of repetitions increase.
. Since the terms are independent
across individuals and variables, and are distributed standard normal, we
generate a matrix s of standard normal random numbers with Q*K elements (one element for
each variable and individual combination) and compute the corresponding
individual choice probabilities for a given value of the parameter vector to be
estimated. This process is repeated R
times for the given value of the parameter vector and the integrand is
approximated by averaging over the computed choice probabilities in the
different draws. This results in an unbiased estimator of the actual individual
choice probabilities. The variance of ‹ decreases as R increases also has the appealing
properties of being smooth (i.e., twice
differentiable) and being
strictly positive for any realization of the finite R draws. The parameter vector is estimated as the vector value that
maximizes the simulated log-likelihood function. Under rather weak regularity
conditions, the PMC estimator is consistent, asymptotically efficient, and
asymptotically normal. However, the estimator will generally be a biased
simulation of the maximum likelihood (ML) estimator because of the logarithmic
transformation of the choice probabilities in the log-likelihood function. The
bias decreases with the variance of the probability simulator; that is, it
decreases as the number of repetitions increase.
Random
QMC method
The
quasi-random Halton sequence is designed to span the domain of the S-dimensional unit cube uniformly and
efficiently (the interval of each dimension of the unit cube is between 0 and
1). In one dimension, the Halton sequence is generated by choosing a prime
number r 
and expanding the
sequence of integers 0, 1, 2, …g, ...G in terms of the base r:
and expanding the
sequence of integers 0, 1, 2, …g, ...G in terms of the base r:
. (18)
Thus,
g (g = 1, 2, ...,
G) can be represented by the r-adic
integer string 
. The Halton sequence in the prime base r is obtained by taking the radical inverse of g (g = 1, 2, ..., G) to the base r by reflecting through the radical point:
. The Halton sequence in the prime base r is obtained by taking the radical inverse of g (g = 1, 2, ..., G) to the base r by reflecting through the radical point: 
. (19)
The
sequence above is very uniformly distributed in the interval (0,1) for each
prime r. The Halton sequence in K dimensions is obtained by pairing K one-dimensional sequences based on K pairwise relatively prime integers
(usually the first K primes):
. (20)
The
Halton sequence is generated number-theoretically rather than randomly and so
successive points at any stage “know” how to fill in the gaps left by earlier
points, leading to a uniform distribution within the domain of integration.
The
simulation technique to evaluate the integral in the log-likelihood function of
equation (15) involves generating the K-dimensional
Halton sequence for a specified number of “draws” R for each individual. To avoid correlation in simulation errors
across individuals, separate independent draws of R Halton numbers in K
dimensions are taken for each individual. This is achieved by generating a
Halton “matrix” Y of size G x K,
where G = R*Q+10 (Q is the total number of individuals in
the sample). The first ten terms in each dimension are then discarded because
the integrand may be sensitive to the starting point of the Halton sequence.
This leaves a (R*Q) x K Halton matrix which is partitioned
into Q sub-matrices of size R x K,
each sub-matrix representing the R
Halton draws in K dimensions for each
individual (thus, the first R rows of
the Halton matrix Y are assigned to
the first individual, the second R
rows to the second individual, and so on).
The
Halton sequence is uniformly distributed over the multi-dimensional cube. To
obtain the corresponding multivariate normal points over the multi-dimensional
domain of the real line, the inverse standard normal distribution transformation
of Y is taken. By the integral
transform result, 
provides the Halton
points for the multi-variate normal distribution (see Fang and Wang, 1994;
Chapter 4). The integrand in equation (15) is computed at the resulting points
in the columns of the matrix X for
each of the R draws for each
individual and then the simulated likelihood function is developed in the usual
manner as the average of the values of the integrand across the R draws.
provides the Halton
points for the multi-variate normal distribution (see Fang and Wang, 1994;
Chapter 4). The integrand in equation (15) is computed at the resulting points
in the columns of the matrix X for
each of the R draws for each
individual and then the simulated likelihood function is developed in the usual
manner as the average of the values of the integrand across the R draws.
Comparison
of the estimation techniques
Bhat
(1999a) recently proposed and introduced the use of the Halton sequence for
estimating the mixed logit model and conducted Monte Carlo simulation
experiments to study the performance of this QMC simulation method vis-à-vis the cubature and PMC
simulation methods (this study, to the author’s knowledge, is the first attempt
at employing the QMC simulation method in discrete choice). Bhat’s results
indicate that the QMC method out-performs the polynomial-cubature and PMC
methods for mixed logit model estimation. In 
dimensions, simulation
estimation with as few as 75 Halton draws provides considerably better accuracy
than with 2000 pseudo-random draws. In higher dimensions (4 or 5), 100 Halton
draws provide about the same level of accuracy as 2000 pseudo-random draws and
125 Halton draws provides much better accuracy in about one-tenth the time
required for convergence using 2000 pseudo-random draws. Bhat also noted that
this substantial reduction in computational cost has the potential to
dramatically influence the use of the mixed logit model in practice.
Specifically, given the flexibility of the mixed logit model to accommodate
very general patterns of competition among alternatives and/or random
coefficients, the use of the QMC simulation method of estimation should
facilitate the application of behaviorally rich structures for discrete choice
modeling. An application of the Halton sequence may be found in Bhat (1999b) in
the context of a spatial model of work mode choice. A subsequent study by Train
(1999) confirmed the substantial reduction in computational time for mixed
logit estimation using the QMC method. More recently, Hensher (1999) has investigated
Halton sequences involving draws of 10, 25, 50, 100, 150 and 200, and compared
the findings with random draws for mixed logit model estimation. He noted that
the data fit and parameter values of the mixed logit model remain almost the
same beyond 50 Halton draws. He concluded that the QMC method “is a phenomenal
development in the estimation of complex choice models”.
dimensions, simulation
estimation with as few as 75 Halton draws provides considerably better accuracy
than with 2000 pseudo-random draws. In higher dimensions (4 or 5), 100 Halton
draws provide about the same level of accuracy as 2000 pseudo-random draws and
125 Halton draws provides much better accuracy in about one-tenth the time
required for convergence using 2000 pseudo-random draws. Bhat also noted that
this substantial reduction in computational cost has the potential to
dramatically influence the use of the mixed logit model in practice.
Specifically, given the flexibility of the mixed logit model to accommodate
very general patterns of competition among alternatives and/or random
coefficients, the use of the QMC simulation method of estimation should
facilitate the application of behaviorally rich structures for discrete choice
modeling. An application of the Halton sequence may be found in Bhat (1999b) in
the context of a spatial model of work mode choice. A subsequent study by Train
(1999) confirmed the substantial reduction in computational time for mixed
logit estimation using the QMC method. More recently, Hensher (1999) has investigated
Halton sequences involving draws of 10, 25, 50, 100, 150 and 200, and compared
the findings with random draws for mixed logit model estimation. He noted that
the data fit and parameter values of the mixed logit model remain almost the
same beyond 50 Halton draws. He concluded that the QMC method “is a phenomenal
development in the estimation of complex choice models”.
a). Transport Applications
The transport
applications of the MMNL model are discussed under two headings:
error-components applications and random-coefficients applications.
Error-components
applications
Brownstone and Train
(1999) applied an error-components MMNL structure to model households’ choices
among gas, methanol, compressed natural gas (CNG), and electric vehicles. Their
specification allows non-electric vehicles to share an unobserved random
component, thereby increasing the sensitivity of non-electric vehicles to one
another compared to an electric vehicle. Similarly, a non-CNG error component
is introduced. Two additional error components
related to the size of the
vehicle are also introduced: one is a normal deviate multiplied by the size of
the vehicle, and the second is a normal deviate multiplied by the luggage
space. All these error components are statistically significant, indicating
non-IIA competitive patterns.
Bhat (1998a) applied
the MMNL model to a multidimensional choice situation. Specifically, his
application accommodates unobserved correlation across both dimensions in a
two-dimensional choice context. The model is applied to an analysis of travel-mode
and departure-time choice for home-based social-recreational trips using data
drawn from the 1990 San Francisco Bay area household survey. The empirical
results underscore the need to capture unobserved attributes along both the
mode and departure-time dimensions, both for improved data fit and for more
realistic policy evaluations of transportation control measures.
Random-coefficients
applications
There
have been several applications of the MMNL model motivated from a
random-coefficients perspective. Bhat (1998b) estimated a model of intercity
travel-mode choice that accommodates variations in responsiveness to
level-of-service measures due to both observed and unobserved individual
characteristics. The model is applied to examine the impact of improved rail
service on weekday business travel in the Toronto-Montreal corridor. The
empirical results show that not accounting adequately for variations in
responsiveness across individuals leads to a statistically inferior data fit
and also to inappropriate evaluations of policy actions aimed at improving
inter-city transportation services.
Bhat (2000) formulated a MMNL model of
multi-day urban travel-mode choice that accommodates variations in mode
preferences and responsiveness to level-of-service. The model is applied to
examine the travel-mode choice of workers in the San Francisco Bay area. Bhat’s
empirical results indicate significant unobserved variation (across
individuals) in intrinsic mode preferences and level-of-service responsiveness.
A comparison of the average response coefficients (across individuals in the
sample) among the fixed-coefficient and random-coefficient models shows that
the random-coefficients model implies substantially higher monetary values of
time than the fixed-coefficient model. Overall, the empirical results emphasize
the need to accommodate observed and unobserved heterogeneity across
individuals in urban mode-choice modeling.
Train
(1998) used a random-coefficients specification to examine the factors
influencing anglers’ choice of fishing sites.
Explanatory variables in the model include fish stock (measured in fish
per 1000 feet of river), esthetics rating of fishing site, size of each site,
number of camp grounds and recreation access at site, number of restricted
species at the site, and the travel cost to the site (including the money value
of travel time). The empirical results indicate highly significant taste
variation across anglers in the sensitivity to almost all the factors listed
above. In this study, as well as the one by Bhat (2000), there was a very
dramatic increase in data fit after including random variation in coefficients.
Mehndiratta
(1997) proposed and formulated a theory to accommodate variations in the
resource value of time in time-of-day choice for intercity travel. Mehndiratta
then proceeded to implement his theoretical model using a random-coefficients
specification for the resource value of disruption of leisure and sleep. He
uses a stated choice sample in his analysis.
Hensher
(2000) undertook a stated-choice analysis of the
valuation of non-business travel time savings for car drivers undertaking long
distance trips (up to 3 hours) between major urban areas in New Zealand. Hensher disaggregates
overall travel time into several different components, including free flow
travel time, slowed-down time, and stop time. The coefficients of each of these
attributes are allowed to vary randomly across individuals in the population.
The study showed significant taste heterogeneity to the various components of travel
time, and adds to the accumulating evidence that the restrictive travel time
response homogeneity assumption undervalues the mean value of travel-time
savings.
2). Conclusions
The
structure, estimation techniques, and transport applications of two classes of
discrete choice models — heteroscedastic models and flexible structure
models—have been described, and within each class alternative formulations have
been discussed. The formulations presented are quite flexible (this is especially
the case with the flexible structure models), although estimation using the
maximum likelihood technique requires the evaluation of one-dimensional
integrals (in the HEV model) or multi-dimensional integrals (in the MMNL model).
However, these integrals can be approximated using Gaussian quadrature
techniques or simulation techniques. The advent of fast computers and the
development of increasingly more efficient sequences for simulation has now
made the estimation of such analytically intractable model formulations very
practical. In this regard, the recent use of QMC simulation techniques seems to
be particularly effective, since it needs very few draws for accurate and
efficient model estimation.
A
note of caution before closing. It is important for the analyst to continue to
think carefully about model-specification issues rather than to use the
(relatively) advanced model formulations presented in this chapter as a panacea
for all systematic specification ills. The flexible structures presented here should
be viewed as formulations that recognize the inevitable presence of unobserved
heterogeneity in individual responsiveness across individuals and/or of
interactions among unobserved components affecting the utility of alternatives
(because it is impossible to identify, or collect data on, all factors
affecting choice decisions). The flexible structures are not, however, a
substitute for careful identification of systematic variations in the
population. The analyst must always explore alternative and improved ways to
incorporate systematic effects in a model. The flexible structures can then be
superimposed on models that have attributed as much heterogeneity to
systematic variations as possible. Another important issue in using flexible
structure models is that the specification adopted should be easy to interpret;
the analyst would do well to retain as simple a specification as possible while
attempting to capture the salient interaction patterns in the empirical context
under study. The MMNL model is particularly appealing in this regard since it “forces”
the analyst to think structurally during model specification.
The confluence of continued careful structural
specification with the ability to accommodate very flexible substitution
patterns or unobserved heterogeneity should facilitate the application of
behaviorally rich structures in transportation-related discrete
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