Multinomial Logit Models with Continuous and Discrete Individual Heterogeneity in R : The gmnl Package

Abstract

This paper introduces the package gmnl in R for estimation of multinomial logit models with unobserved heterogeneity across individuals for cross-sectional and panel (longitudinal) data. Unobserved heterogeneity is modeled by allowing the parameters to vary randomly over individuals according to a continuous, discrete, or mixture distribution, which must be chosen a priori by the researcher. In particular, the models supported by gmnl are the multinomial or conditional logit, the mixed multinomial logit, the scale heterogeneity multinomial logit, the generalized multinomial logit, the latent class logit, and the mixed-mixed multinomial logit. These models are estimated using either the Maximum Likelihood Estimator or the Maximum Simulated Likelihood Estimator. This article describes and illustrates with real databases all functionalities of gmnl, including the derivation of individual conditional estimates of both the random parameters and willingness-to-pay measures.

Full text

Multinomial Logit Models with Continuous and
Discrete Individual Heterogeneity in R: The gmnl
Package
Mauricio Sarrias
Universidad Cat´olica del Norte
Ricardo A. Daziano
Cornell University
Abstract
This paper introduces the package gmnl in Rfor estimation of multinomial logit models
with unobserved heterogeneity across individuals for cross-sectional and panel (longitudi-
nal) data. Unobserved heterogeneity is modeled by allowing the parameters to vary ran-
domly over individuals according to a continuous, discrete, or discrete-continuous mixture
distribution, which must be chosen a priori by the researcher. In particular, the models
supported by gmnl are the multinomial or conditional logit, the mixed multinomial logit,
the scale heterogeneity multinomial logit, the generalized multinomial logit, the latent
class logit, and the mixed-mixed multinomial logit. These models are estimated using
either the Maximum Likelihood Estimator or the Maximum Simulated Likelihood Es-
timator. This article describes and illustrates with real databases all functionalities of
gmnl, including the derivation of individual conditional estimates of both the random
parameters and willingness-to-pay measures.
Keywords: latent class, mixed multinomial logit, random parameters, preference heterogeneity,
R.
1. Introduction
Modeling individual choices has been a very important avenue of research in diverse fields
such as marketing, transportation, political science, and environmental, health, and urban
economics. In all these areas the most widely used method to model choice among mutually
exclusive alternatives has been the Conditional or Multinomial Logit model (MNL) (McFad-
den 1974), which belongs to the family of Random Utility Maximization (RUM) models. The
main advantage of the MNL model has been its simplicity in terms of both estimation and
interpretation of the resulting choice probabilities and elasticities. On the one hand, the MNL
has a closed-form choice probability and a likelihood function that is globally concave (for
a complete overview of MNL and its properties, see Train 2009). MNL estimation is thus
straightforward using the Maximum Likelihood Estimator (MLE). On the other hand, it has
been recognized that MNL not only imposes constant competition across alternatives — as
a consequence of the independence of irrelevant alternatives (IIA) property — but also lacks
the flexibility to allow for individual-specific preferences.
With the advent of more powerful computers and the improvement of simulation-aided infer-
ence in the last decades, researchers are no longer constrained to use models with closed-form

2gmnl Package in R
solutions that may lead to unrealistic behavioral specifications. In fact, much of recent work
in choice modeling focuses on extending MNL to allow for random-parameter models that
accommodate unobserved preference heterogeneity.
The most popular MNL extension is the Mixed Logit Model (MIXL). MIXL allows parameters
to vary randomly over individuals by assuming some continuous heterogeneity distribution a
priori while keeping the MNL assumption that the error term is independent and identically
distributed (i.i.d) extreme value type 1 (McFadden and Train 2000;Train 2009;Hensher and
Greene 2003). MIXL is a very flexible model (MIXL properties are discussed in McFadden and
Train 2000) that does not exhibit pure IIA substitution (cf. Dotson, Brazell, Howell, Lenk,
Otter, MacEachern, and Allenby 2015). Furthermore, using the parametric heterogeneity
distribution that describes how preferences vary in the population it is possible to derive
conditional estimates of the parameters at the individual-level (Train 2009).
Below we present a brief example to introduce the idea of unobserved heterogeneity across
individuals, and how gmnl works. This example uses microdata about individual choice
among four transportation modes: air, train, bus and car. Both the data itself and its correct
formatting — which follows the mlogit data frame— as well as the gmnl syntax are explained
further in the following sections.
R> library("gmnl")
R> data("TravelMode", package = "AER")
R> library("mlogit")
R> TM <- mlogit.data(TravelMode, choice = "choice", shape = "long",
+ alt.levels = c("air", "train", "bus", "car"))
R> mixl <- gmnl(choice ~ vcost + travel + wait | 1 ,
+ data = TM,
+ model = "mixl",
+ ranp = c(travel = "n"),
+ R = 50)
Estimating MIXL model
In the estimated model, choice is a discrete (multinomial response) dependent variable that
indicates which of the four alternative modes was actually chosen by each individual. There
are three explanatory variables that have values that are alternative-specific (vcost: in-vehicle
cost, travel: in-vehicle time, and wait: waiting time). The model also includes alternative-
specific constants (| 1). Random parameters are considered just for the in-vehicle time
component, making this example an application of a Mixed Logit Model (model = "mixl").
The ‘marginal disutility’ of travel time1is assumed to be normally distributed in the line of
code that identifies the random parameters (ranp = c(travel = "n")). The assumption of
normality is an example of representing variation in preferences according to a parametric
continuous distribution. All other parameters (the alternative-specific constants, wait and
vcost) are assumed fixed. Finally, R = 50 evaluation points are used to simulate the likelihood
function.
1In a discrete choice model, the parameters of interest are interpreted as marginal utilities, as choice is
modeled as selecting the alternative that maximizes utility, which is a function of the explanatory variables.
Some attributes cause a reduction in utility: travelers desire to reduce travel time, and hence its associated
parameter is expected to be negative.

Mauricio Sarrias, Ricardo Daziano 3
R> summary(mixl)
Model estimated on: Wed Mar 02 10:35:12 2016
Call:
gmnl(formula = choice ~ vcost + travel + wait | 1, data = TM,
model = "mixl", ranp = c(travel = "n"), R = 50, method = "bfgs")
Frequencies of categories:
air train bus car
0.276 0.300 0.143 0.281
The estimation took: 0h:0m:9s
Coefficients:
Estimate Std. Error z-value Pr(>|z|)
train:(intercept) 0.36131 1.01014 0.36 0.72058
bus:(intercept) -0.44685 1.03936 -0.43 0.66725
car:(intercept) -4.90453 1.09319 -4.49 7.2e-06 ***
vcost -0.02611 0.00954 -2.74 0.00621 **
wait -0.11642 0.01484 -7.84 4.4e-15 ***
travel -0.00808 0.00236 -3.42 0.00062 ***
sd.travel 0.00531 0.00204 2.60 0.00927 **
---
Signif. codes: 0 '***'0.001 '**'0.01 '*'0.05 '.'0.1 ' ' 1
Optimization of log-likelihood by BFGS maximization
Log Likelihood: -189
Number of observations: 210
Number of iterations: 156
Exit of MLE: successful convergence
Simulation based on 50 draws
The output of calling the gmnl function includes the point estimates, standard errors, z-value,
and p-value of the parameters of each explanatory variable. Note that for the normally
distributed parameter, both the population mean and standard deviation are estimated:
travel=-0.008 is the point estimate of the mean and sd.travel=0.005 is the point esti-
mate of the standard deviation of the normal distribution that represents how preferences for
in-vehicle travel time reductions are distributed in the population. Note that, we can infer
the profile of individuals’ preferences for travel time by looking at the shape of the parameter,
which is distributed in this sample as N(−0.008,0.0052).
In addition to MIXL, there are other models that can represent unobserved preference hetero-
geneity. Latent Class (LC) discrete choice models offer an alternative to MIXL by replacing
the continuous distribution assumption with a discrete distribution in which preference het-
erogeneity is captured by membership in distinct classes or segments (Boxall and Adamowicz
2002;Greene and Hensher 2003;Shen 2009). The standard LC specification with class-specific

4gmnl Package in R
multinomial logit models of choice (LC-MNL) is useful if the assumption of preference homo-
geneity holds within segments. In effect, in an LC-MNL all individuals in a given class have
the same parameters (fixed parameters within a class), but the parameters vary across classes
(heterogeneity across classes).
Rossi, Allenby, and McCulloch (2005) in a Bayesian setting, and more recently Bujosa, Riera,
and Hicks (2010) and Greene and Hensher (2013) in the frequentist context, have derived
models with discrete-continuous mixing distributions of unobserved heterogeneity in the form
of a (finite) mixture of normals. This model is also known as Mixed-Mixed Logit (MM-MNL)
(Keane and Wasi 2013).
Other researchers have focused on MNL extensions that allow for a more flexible representa-
tion of heteroskedasticity. For example, Fiebig, Keane, Louviere, and Wasi (2010) proposed
two new models, namely the Scale Heterogeneity (S-MNL) model and the Generalized Multi-
nomial Logit (G-MNL) model. S-MNL extends the MNL by letting the scale of errors vary
across individuals (via a parametric specification of heteroskedasticity), whereas the G-MNL
nests the S-MNL, MIXL, and MNL models. For a discussion of confounding effects between
scale and preference heterogeneity, see Hess and Rose (2012) and Hess and Stathopoulos
(2013).
In addition to gmnl, there exist different packages in R(RCore Team 2015) that estimate
models with multinomial responses. Some packages for estimating Multinomial Logit models
with fixed parameters are mlogit (Croissant 2012), RSGHB (Dumont, Keller, and Carpenter
2014), mnlogit (Hasan, Zhiyu, and Mahani 2015), the function multinom function from the
nnet package (Venables and Ripley 2002), VGAM (Yee 2010), and bayesm (Rossi. 2012). The
Multinomial Probit (MNP) model is fitted in bayesm,MNP (Imai and Dyk 2005) and mlogit.
Models with random parameters are supported by mlogit,mclogit2,bayesm,ChoiceModelR3
and RSGHB. In terms of models with latent classes RSGHB,flexmix (Leisch 2004), and
poLCA (Linzer and Lewis 2011) offer alternative estimation procedures. gmnl is the only
package to date that handles G-MNL specifications, and the only package that implements
the maximum simulated likelihood estimator for MM-MNL. MM-MNL models are also im-
plemented using a Bayes estimator in bayesm. However, gmnl allows the researcher to specify
covariates that explain assignment to classes, whereas bayesm assumes constant weights for
the continuous components of the double mixture.4
Among all the packages mentioned above, mlogit is one of the most complete and user-friendly
Rpackages for the estimation of models with multinomial responses. For this reason, we have
adopted most of the mlogit syntax in gmnl. Table 1presents a more complete overview of
the models supported by each package and the estimation procedure used to estimate the
parameters.
The gmnl package (Sarrias and Daziano 2015) is intended to consolidate in a single Rpackage
the whole range of discrete choice models with random parameters for the use of researchers
and practitioners. It shares similar functionalities with mlogit and mnlogit in terms of the
formula interface. Furthermore, since gmnl is able to estimate G-MNL models, it also allows
the user to estimate models in willingness-to-pay space with a minimal extra reformulation.
2Only random intercepts are allowed in the current version.
3Both bayesm and ChoiceModelR only allow normally distributed parameters.
4One of the advantages of the Bayes estimator and its implementation in bayesm is the possibility of letting
the number of discrete components of the double mixture free. This is achieved by assuming a Dirichlet Process
prior for the heterogeneity distribution.

Mauricio Sarrias, Ricardo Daziano 5
Model Package Estimation procedure
MNL
gmnl Maximum likelihood
mclogit Maximum likelihood
mlogit Maximum likelihood
MNL Maximum likelihood
mnlogit Maximum likelihood
multinom function (nnet) Maximum likelihood
VGAM Maximum likelihood
bayesm Bayesian inference
MCMCpack Bayesian inference
RSGHB Bayesian inference
MNP mlogit Maximum simulated likelihood
bayesm Bayesian inference
MNP Bayesian inference
MIXL
gmnl Maximum simulated likelihood
mlogit Maximum simulated likelihood
mclogit Penalized quasi-likelihood
bayesm Bayesian inference
ChoiceModelR Bayesian inference
RSGHB Bayesian inference
G-MNL gmnl Maximum simulated likelihood
S-MNL gmnl Maximum simulated likelihood
LC-MNL
gmnl Maximum likelihood
flexmix Expectation-Maximization
poLCA Expectation-Maximization
RSGHB Bayesian inference
MM-MNL gmnl Maximum simulated likelihood
bayesm Bayesian inference
Table 1: Packages available in Rfor models with multinomial response.
Our package also provides the ability of constructing the conditional estimates for the individ-
ual parameters and willingness-to-pay. gmnl is available from the Comprehensive RArchive
Network (CRAN) at http://cran.r-project.org/package=gmnl.
The paper is organized as follows: Section 2presents a brief overview of the models supported
by gmnl. Section 3discusses the functionalities of the package. Section 4explains some com-
putational issues that may arise when estimating random parameter models using maximum
simulated likelihood. Finally, Section 5concludes the paper.
2. Models
2.1. Mixed and latent class logit models
MIXL generalizes the MNL model by allowing the preference or taste parameters to be differ-

6gmnl Package in R
ent for each individual (McFadden and Train 2000;Train 2009). MIXL is basically a random
parameter logit model with continuous heterogeneity distributions. The random utility of
person ifor alternative jand for choice occasion tis:
Uijt =x>
ijtβi+ijt i= 1, ..., N ;j= 1, ..., J, t = 1, ..., Ti,(1)
where xijt is a K×1 vector of observed alternative attributes; ijt is the idiosyncratic error
term or taste shock, and is i.i.d. extreme value type 1; the parameter vector βiis unobserved
for each iand is assumed to vary in the population following the continuous density f(βi|θ),
where θare the parameters of this distribution. This mixing distribution can in principle take
any shape. For example, when assuming that the parameters are distributed multivariate
normal, βi∼MVN(β,Σ), the vector βican be re-written as:
βi=β+Lηi,
where ηi∼N(0,I), and Lis the lower-triangular Cholesky factor of Σsuch that LL>=
VAR(βi) = Σ. If the off-diagonal elements of Lare zero, then the parameters are indepen-
dently normally distributed. Observed heterogeneity (deterministic taste variations) can also
be accommodated in the random parameters by including individual-specific covariates (see
for example Greene 2012). Specifically, the vector of random coefficients is:
βi=β+Πzi+Lηi,(2)
where ziis a set of Mcharacteristics of individual ithat influence the mean of the preference
parameters; and Πis a K×Mis a matrix of additional parameters.
Unlike the MIXL model, LC uses a discrete mixing distribution, where individual ibelongs
to class qwith probability wiq, i.e.,:
βi=βqwith probability wiq for q= 1, ..., Q,
where Pqwiq = 1 and wiq >0. The discrete mixing distribution (or class assignment prob-
ability) is unknown to the analyst. The most widely used formulation for wiq is the semi-
parametric Multinomial Logit format (Greene and Hensher 2003;Shen 2009):
wiq =exp h>
iγq
PQ
q=1 exp h>
iγq;q= 1, ..., Q, γ1=0,
where hidenotes a set of socio-economic characteristics that determine assignment to classes.
The parameters of the first class are normalized to zero for identification of the model. Note
that one could omit any socio-economic covariate as a determinant of the class assignment
probability. Under this scenario, the class probabilities simply become:
wiq =exp (γq)
PQ
q=1 exp (γq);q= 1, ..., Q, γ1= 0,
where γqis a constant (Scarpa and Thiene 2005).

Mauricio Sarrias, Ricardo Daziano 7
Let yijt = 1 if individual ichooses jon occasion t, and 0 otherwise. Then, the unconditional
probabilities of the sequence of choices by individual ifor MIXL and LC are respectively given
by:
Pi(θ) = Z


T
Y
t
J
Y
j
exp x>
ijtβi
PJ
j=1 exp x>
ijtβi

yijt 

f(βi)dβi
Pi(θ) =
Q
X
q
wiq 


T
Y
t
J
Y
j
exp x>
ijtβq
PJ
j=1 exp x>
ijtβq

yijt 

.
Both MIXL and LC are widely used in practice to accommodate preference heterogeneity
across respondents. As discussed above, in the MIXL approach parameters are assumed to
vary across the population according to some prespecified statistical distribution that con-
tinuously represents preferences. In the LC model a discrete number of separate classes or
segments, each with different fixed parameters, recover preference heterogeneity. In addition
to differentiation in terms of continuous versus discrete consumer segments, there exist fur-
ther differences between MIXL and LC. For example, compared with the MIXL approach,
the LC model has the advantage of being “relatively simple, reasonably plausible and statis-
tically testable” (Shen 2009). In addition, because LC is a semiparametric specification that
depends only on the prespecified number of classes, it avoids misspecification problems in the
distribution of individual heterogeneity. In fact, the main disadvantage of MIXL is that the
researcher has to choose the distribution of the random parameters a priori. Nevertheless,
LC is less flexible than MIXL precisely because the parameters in each class are fixed. An-
other important difference between these two models is the estimation procedure. The MIXL
requires the use of the maximum simulated likelihood estimator – which can be very costly
in terms of computational time – but no simulation is required for LC.5gmnl implements the
Maximum Likelihood Estimator for both LC and MIXL, using analytical expressions for the
appropriate gradient.
2.2. Mixed-Mixed Logit model
To take advantage of the benefits of both MIXL and LC, recent empirical papers have derived
a mixture of both models. This double-mixture is known as the ‘Mixed-Mixed’ Logit model
(MM-MNL) (Keane and Wasi 2013).6Bujosa et al. (2010), and Greene and Hensher (2013)
developed this MM-MNL model by extending the LC model to allow for random parameters
within each class.
Consider the case where the heterogeneity distribution is generalized to a discrete mixture of
multivariate normal distributions. In this case we have:
βi∼N(βq,Σq) with probability wiq for q= 1, ..., Q. (3)
5For an empirical comparison between these two models, see for example Greene and Hensher (2003), Shen
(2009) and Hess, Ben-Akiva, Gopinath, and Walker (2011).
6Train (2008) refers to this model as ‘discrete mixture of continuous distributions’, whereas Greene and
Hensher (2013) label it ‘LC-MIXL’.

8gmnl Package in R
The appeal of using a Gaussian mixture for the heterogeneity distribution is that any contin-
uous distribution can be approximated by a discrete mixture of normal distributions (Train
2008). Note that the MM-MNL with only one class is equivalent to the MIXL model. Fur-
thermore, if Σq→0 for all q, the model in Equation 3becomes a LC-MNL model (Bujosa
et al. 2010;Keane and Wasi 2013). Thus, MM-MNL nests both MIXL and LC.
The choice probabilities for the MM-MNL are given by:
Pi(θ) =
Q
X
q
wiq Z


T
Y
t
J
Y
j
exp x>
ijtβi
PJ
j=1 exp x>
ijtβi

yijt 

fq(βi)dβi,
where fq(βi) = N(βq,Σq). Due to the complex expression of the probability, gmnl imple-
ments the maximum likelihood estimator for the MM-MNL parameters with the Monte-Carlo
approximation of this choice probability and the analytical expression of the gradient.
2.3. Generalized multinomial logit model
Fiebig et al. (2010) proposed a general version of the MIXL model, which they called the
G-MNL model, where the parameters vary across individuals according to:
βi=σiβ+ [γ+σi(1 −γ)] Lηi,(4)
where σiis the individual-specific scale of the idiosyncratic error term, and γis a scalar
parameter that controls how the variance of residual taste heterogeneity Lηivaries with
scale. To better understand this specification, it is useful to note that differing sub-models
arise when some structural parameters in the G-MNL model are constrained:
•G-MNL-I: If γ= 1, then βi=σiβ+Lηi. In this model, the residual taste heterogeneity
is independent of the scaling of β.
•G-MNL-II: If γ= 0, then βi=σi(β+Lηi). In this model, the residual taste hetero-
geneity is proportional to σi.
•S-MNL: If VAR(ηi) = 0, then βi=σiβ. As pointed out by Fiebig et al. (2010), this
model is observationally equivalent to the particular type of heterogeneity in which the
parameters increase or decrease proportionally across individuals by the scaling factor
σi. S-MNL provides a more parsimonious representation of continuous heterogeneity
than MIXL, because βσihas fewer parameters than β+Lηi(Fiebig et al. 2010).
•MIXL: βi=β+Lηi, if σi= 1.
•MNL: βi=β, if σi= 1 and VAR(ηi) = 0.
Fiebig et al. (2010) note that some restrictions need to be considered to estimate the G-MNL
model. First, the domain of σishould be the positive real line. A positive scale parameter is
ensured by assuming that σiis distributed log-normal with standard deviation τand mean ¯σ
Fiebig et al. (2010):

Mauricio Sarrias, Ricardo Daziano 9
σi= exp(¯σ+τ υi),
where υ∼N(0,1). Fiebig et al. (2010) also note that when τis too large, numerical problems
arise for extreme draws of υi. To avoid this numerical issue, the authors suggest to use a
truncated normal distribution for υiwith truncation at ±2, so that υ∼T N [−2,+2]. Greene
and Hensher (2010) found that constraining υiat −1.96 and +1.96 maintains the smoothness
of the estimator. Specifically, the authors used υir = Φ−1(0.025 + 0.95uir), where uir is a
draw from the standard uniform distribution. gmnl allows the user to choose between these
two ways of drawing from υi, using the argument typeR (see Section 3.3).
Note that the parameters ¯σ,τ, and βare not separately identified. Fiebig et al. (2010) suggest
that one can normalize the mean ¯σby setting:
¯σ=−log "1
N
N
X
i=1
exp (τυi)#.
Another important issue in G-MNL is the domain of γ. Initially, Fiebig et al. (2010) imposed
γ∈[0,1]. To constrain γin this interval, the authors used the logistic transformation:
γ=exp(γ∗)
1 + exp(γ∗),
and estimated γ∗. However, Keane and Wasi (2013) pointed out that both γ < 0 and γ > 1
still have meaningful behavioral interpretations. Thus, these authors estimate γdirectly.
gmnl allows to estimate γusing both procedures.
Finally, one can allow the mean of the scale to differ across individuals by including individual-
specific characteristics. In this case the scale parameter can be written as:
σi= exp(¯σ+δsi+τ υi),
where siis a vector of attributes of individual i.
In terms of computation, all models, except for the LC and the MNL model, are estimated
in gmnl using the maximum simulated likelihood estimator (MSLE) and maxLik function
from maxLik package (Henningsen and Toomet 2011). All models are estimated using the
analytical gradient (instead of the numerical gradient). The MNL is estimated using also the
analytical Hessian.
For a complete derivation of the asymptotic properties of the MSLE and a more comprehensive
review of how to implement this estimator, see Train (2009), Lee (1992), Gourieroux and
Monfort (1997)orHajivassiliou and Ruud (1986).
3. The gmnl package
3.1. Format of data

10 gmnl Package in R
The function mlogit.data from mlogit is very useful to handle multinomial data formats.
gmnl thus uses the same class of data for estimation. If the user forgets to set the data in the
mlogit.data format, gmnl will give an error message and the estimation process will stop.
For illustration purposes, we use the Travel Mode data from the AER package (Kleiber and
Zeileis 2008). As quickly presented in the introduction, the TravelMode data contains actual
individual choices7among four transportation modes (air,train,bus and car) for travel
between the cities of Sydney and Melbourne in Australia.
R> data("TravelMode", package = "AER")
R> with(TravelMode, prop.table(table(mode[choice == "yes"])))
air train bus car
0.276 0.300 0.143 0.281
Each mode is characterized by four alternative-specific variables (wait,vcost,travel,gcost),8
and two individual-specific variables (income,size). The observed shares of each mode are
27.62% (air), 30% (train), 14.29% (bus), and 28.1% (car). More details about and examples
using this dataset can be found in Kleiber and Zeileis (2008).
R> head(TravelMode)
individual mode choice wait vcost travel gcost income size
1 1 air no 69 59 100 70 35 1
2 1 train no 34 31 372 71 35 1
3 1 bus no 35 25 417 70 35 1
4 1 car yes 0 10 180 30 35 1
5 2 air no 64 58 68 68 30 2
6 2 train no 44 31 354 84 30 2
As can be seen above, the data is in a “long”format (one row per available mode) and can be
transformed into the structure needed by gmnl using the mlogit.data in the following way:
R> library("mlogit")
R> TM <- mlogit.data(TravelMode, choice = "choice", shape = "long",
+ alt.levels = c("air", "train", "bus", "car"))
The argument choice indicates the choice made by the individuals; shape specifies the original
format of the data; and alt.levels is a character vector that contains the name of the
alternatives. We show how to transform other kinds of data in the examples below. For a
more complete treatment of the data using mlogit.data function see Croissant (2012).
Before formal modeling, it is useful to summarize the (unconditional) relationship between
the travel mode and the regressors. In the next example, we reshape the data from long to
wide format:
7In the choice modeling literature, microdata that represents real choices is known as ‘revealed preference’
(RP) data.
8vcost denotes travel cost, whereas gcost represents a generalized cost that combines both vcost and
travel – which is in-vehicle travel time – using an exogenous value of time. wait is for waiting time.

Mauricio Sarrias, Ricardo Daziano 11
R> wide_TM <- reshape(TravelMode,
+ idvar = c("individual", "income", "size"),
+ timevar = "mode" ,
+ direction = "wide")
R> wide_TM$chosen_mode[wide_TM$choice.air == "yes"] <- "air"
R> wide_TM$chosen_mode[wide_TM$choice.car == "yes"] <- "car"
R> wide_TM$chosen_mode[wide_TM$choice.train == "yes"] <- "train"
R> wide_TM$chosen_mode[wide_TM$choice.bus == "yes"] <- "bus"
For the case-specific income variable, we get the following summary
R> library("plyr")
R> ddply(wide_TM, ~ chosen_mode,
+ summarize,
+ mean.income = mean(income))
chosen_mode mean.income
1 air 41.7
2 bus 29.7
3 car 42.2
4 train 23.1
On average, those individuals choosing train have the lowest income and those choosing car
have the highest. The relationship between the chosen travel mode and the alternative-specific
regressor vcost is summarized as follows (a similar analysis can be done for the rest of the
alternative-specific variables):
R> ddply(wide_TM, ~ chosen_mode, summarize,
+ mean.air = mean(vcost.air),
+ mean.car = mean(vcost.car),
+ mean.train = mean(vcost.train),
+ mean.bus = mean(vcost.bus))
chosen_mode mean.air mean.car mean.train mean.bus
1 air 97.6 23.4 58.5 34.3
2 bus 89.3 26.8 62.3 33.7
3 car 76.3 15.6 53.6 33.8
4 train 80.4 21.1 37.5 32.3
Note that these figures show that the chosen mode is not determined solely by travel cost.
The purpose of the choice model is precisely to determine the tradeoffs across attributes that
help to explain choices.
3.2. Formula interface
The specification of Multinomial Logit models using gmnl is similar to that of mlogit and
mnlogit. In particular, we use the Rpackage Formula (Zeileis and Croissant 2010), which is
able to handle multi-part formulae.

12 gmnl Package in R
Consider the TravelMode data and suppose that we want to estimate a Multinomial Logit
model where the variables wait and vcost are alternative-specific variables with a generic co-
efficient β;income is an individual-specific variable with an alternative specific coefficient γj;
and the variable travel is alternative-specific variables with an alternative-specific coefficient
δj. This is done using the following 3-part formula:
R> f1 <- choice ~ wait + vcost | income | travel
By default, the alternative-specific constants (ASC) for each alternative are included. They
can be omitted by adding +0 or -1 in the second part of the formula. For example:
R> f2 <- choice ~ wait + vcost | income + 0 | travel
R> f2 <- choice ~ wait + vcost | income - 1 | travel
Some parts may be omitted when there is no ambiguity. For instance, a model with only
individual specific variables can be specified as follows:
R> f3 <- choice ~ 0 | income + size | 0
R> f3 <- choice ~ 0 | income + size | 1
Similarly, a Conditional Logit model, that is, a model with alternative-specific variables with
a generic coefficient β, can be specified using either of the following formula objects:
R> f4 <- choice ~ wait + vcost | 0
R> f4 <- choice ~ wait + vcost | 0 | 0
R> f4 <- choice ~ wait + vcost | -1 | 0
For other models, such as the MIXL, S-MNL, LC-MNL and MM-MNL model, we require
to use the fourth and fifth part of the formula. As explained in Section 2.1,gmnl allows
incorporating observed heterogeneity in the mean of the random parameters. This can be
achieved by including individual-specific characteristics (income and size) in the fourth part
of the formula:
R> f5 <- choice ~ wait + vcost | 0 | 0 | income + size - 1
and then use the mvar argument to indicate how these two variables modify the mean of the
random parameters. For a more complete example see Section 3.4.
The fifth part of the formula is reserved for either models with heterogeneity in the scale
parameter or models with latent classes. For example, an S-MNL or G-MNL model where the
scale varies across individuals by individual-specific characteristics can be specified as follows:
R> f6 <- choice ~ wait + vcost | 1 | 0 | 0 | income + size - 1
The same formulation can be used if a model with latent classes is estimated and both income
and size determine the class assignment.

Mauricio Sarrias, Ricardo Daziano 13
3.3. Estimating S-MNL models
In this example, we estimate an S-MNL model using the TravelMode data where the ASCs
are fixed and not scaled. Fiebig et al. (2010) found that in a model where all attributes are
scaled — including the ASCs — the estimates often show a explosive behavior and the model
actually produces a worse fit. The basic syntax for estimation is the following:
R> library("gmnl")
R> smnl.nh <- gmnl(choice ~ wait + vcost + travel | 1,
+ data = TM,
+ model = "smnl",
+ R = 30,
+ notscale = c(1, 1, 1, rep(0, 3)))
The following variables are not scaled:
[1] "train:(intercept)" "bus:(intercept)" "car:(intercept)"
Estimating SMNL model
The component | 1 in the formula means that the model is fitted using ASCs for the J−1
alternatives. The main argument in the model is model = "smnl", which indicates to the
function that the user wants to estimate the S-MNL model (without random parameters).
The rest of the models allowed by gmnl are given in Table 2.R = 30 indicates that 30 Halton
draws are used to simulate the probabilities. Another important argument in this example is
notscale. This is a vector that indicates which variables will not be scaled (1 = not scaled
and 0 = scaled). Since the ASCs are always the first variables entering in the model (if they
are specified using | 1 in the second part of formula) and only J−1 = 3 ASCs are created,
notscale = c(1, 1, 1, rep(0, 3)) implies that the constants will not be scaled.
Options for "model" Model
"mnl" Multinomial Logit Model
"mixl" Mixed Logit Model
"smnl" Scaled Multinomial Logit Model
"gmnl" Generalized Multinomial Logit Model
"lc" Latent Class Multinomial Logit Model
"mm" Mixed-Mixed Multinomial Logit Model
Table 2: Models supported by gmnl.
R> summary(smnl.nh)
Model estimated on: Wed Mar 02 10:35:15 2016
Call:
gmnl(formula = choice ~ wait + vcost + travel | 1, data = TM,
model = "smnl", R = 30, notscale = c(1, 1, 1, rep(0, 3)),
method = "bfgs")

14 gmnl Package in R
Frequencies of categories:
air train bus car
0.276 0.300 0.143 0.281
The estimation took: 0h:0m:3s
Coefficients:
Estimate Std. Error z-value Pr(>|z|)
train:(intercept) -1.25946 0.52091 -2.42 0.01561 *
bus:(intercept) -2.02267 0.64669 -3.13 0.00176 **
car:(intercept) -7.19922 1.42002 -5.07 4.0e-07 ***
wait -0.14085 0.02777 -5.07 3.9e-07 ***
vcost -0.02198 0.01196 -1.84 0.06615 .
travel -0.00481 0.00127 -3.78 0.00016 ***
tau 0.54412 0.16422 3.31 0.00092 ***
---
Signif. codes: 0 '***'0.001 '**'0.01 '*'0.05 '.'0.1 ' ' 1
Optimization of log-likelihood by BFGS maximization
Log Likelihood: -188
Number of observations: 210
Number of iterations: 54
Exit of MLE: successful convergence
Simulation based on 30 draws
The results report the point estimates for each variable. Note that all preference parameters
are statistically significant and negative,9meaning that cost and the time components create
a ‘disutility’ to the traveler, so that people prefer modes that are both cheaper and faster.
τ, which represents the standard deviation of σi, is also significant supporting the presence
of heterogeneous preferences by means of different error variances for each individual. The
output also gives additional information about estimation. The model is estimated using the
BFGS procedure. Other optimization procedures such as the BHHH and Newton Raphson
(NR) can be called using the argument method passed to the maxLik function.
Another important point is that the number of observations reported by gmnl corresponds
to N/J if cross-sectional data is used, or N×T /J if panel data (repeated choice situations)
is used. Finally, it is always important to check all the details in the estimation output.
In our example, the output informs us that the convergence was achieved successfully. If
convergence fails, the analyst needs to revise identification of the model, starting values,
estimation procedure, and measurement scale of the attributes. For further details about
potential convergence problems see Section 4.
In the next example, we allow the scale to differ across individuals according to their income.
Basically, we assume that:
9vcost is significant at the 10% level.

Mauricio Sarrias, Ricardo Daziano 15
σi= exp ( ¯σ+δincomeincomei+τ υi).
The syntax is very similar to our previous example, with minor changes in the formula
argument:
R> smnl.het <- gmnl(choice ~ wait + vcost + travel | 1 |
+ 0 | 0 | income - 1,
+ data = TM,
+ model = "smnl",
+ R = 30,
+ notscale = c(1, 1, 1, 0, 0, 0),
+ typeR = FALSE)
The following variables are not scaled:
[1] "train:(intercept)" "bus:(intercept)" "car:(intercept)"
Estimating SMNL model
The fifth part of the formula is reserved for individual-specific variables that affect scale. In
this example, we specify that the variable income and no constant are included in σi.
R> summary(smnl.het)
Model estimated on: Wed Mar 02 10:35:19 2016
Call:
gmnl(formula = choice ~ wait + vcost + travel | 1 | 0 | 0 | income -
1, data = TM, model = "smnl", R = 30, notscale = c(1, 1,
1, 0, 0, 0), typeR = FALSE, method = "bfgs")
Frequencies of categories:
air train bus car
0.276 0.300 0.143 0.281
The estimation took: 0h:0m:4s
Coefficients:
Estimate Std. Error z-value Pr(>|z|)
train:(intercept) -0.99226 0.49290 -2.01 0.04410 *
bus:(intercept) -1.69842 0.61673 -2.75 0.00589 **
car:(intercept) -6.92476 1.22665 -5.65 1.6e-08 ***
wait -0.11795 0.02253 -5.23 1.7e-07 ***
vcost -0.01489 0.01019 -1.46 0.14412
travel -0.00439 0.00111 -3.95 7.8e-05 ***
tau 0.54198 0.14234 3.81 0.00014 ***
het.income 0.00564 0.00339 1.66 0.09635 .

16 gmnl Package in R
---
Signif. codes: 0 '***'0.001 '**'0.01 '*'0.05 '.'0.1 ' ' 1
Optimization of log-likelihood by BFGS maximization
Log Likelihood: -185
Number of observations: 210
Number of iterations: 66
Exit of MLE: successful convergence
Simulation based on 30 draws
The results are similar to those of the previous example. However, vcost is no longer statis-
tically significant. All the parameters for the variables that enter in the scale are preceded by
the string het. Thus, the coefficient het.income corresponds to δincome, and is significant at
the 10% level. This result indicates that the variability of the error term for each individual
depends on their income. Finally, the argument typeR determines the type of draws used for
the scale parameter. If TRUE, truncated normal draws are used for the scale parameter. In this
case, the function rtruncnorm of truncnorm (Trautmann, Steuer, Mersmann, and Bornkamp
2014) is used. If typeR = FALSE, as in this example, the procedure suggested by Greene and
Hensher (2010) is used. See Section 2.3 for more details.
Suppose now that we want to test the null hypothesis H0:δincome = 0. This test can
be performed using the function waldtest or lrtest from the package lmtest (Zeileis and
Hothorn 2002):
R> library("lmtest")
R> waldtest(smnl.nh, smnl.het)
Wald test
Model 1: choice ~ wait + vcost + travel | 1
Model 2: choice ~ wait + vcost + travel | 1 | 0 | 0 | income - 1
Res.Df Df Chisq Pr(>Chisq)
1 203
2 202 1 2.76 0.096 .
---
Signif. codes: 0 '***'0.001 '**'0.01 '*'0.05 '.'0.1 ' ' 1
R> lrtest(smnl.nh, smnl.het)
Likelihood ratio test
Model 1: choice ~ wait + vcost + travel | 1
Model 2: choice ~ wait + vcost + travel | 1 | 0 | 0 | income - 1
#Df LogLik Df Chisq Pr(>Chisq)
1 7 -188
2 8 -185 1 5.21 0.022 *
---
Signif. codes: 0 '***'0.001 '**'0.01 '*'0.05 '.'0.1 ' ' 1

Mauricio Sarrias, Ricardo Daziano 17
Both tests are asymptotically equivalent, but they can provide different results for finite
samples. In this case, both reject the null hypothesis, but at different significance levels.
In some cases, one will need to decide among non-nested models. In such cases the LR, Wald
and Lagrange multiplier tests cannot be applied. However, one can use either the AIC or BIC
criteria to measure the relative quality of models. In general, given a set of candidate models
for the data, the preferred model is the one with the minimum AIC or BIC value. We can
obtain the AIC and BIC criteria by typing:
R> AIC(smnl.nh)
[1] 390
R> AIC(smnl.het)
[1] 387
R> BIC(smnl.nh)
[1] 414
R> BIC(smnl.het)
[1] 414
The results show that AIC favors the smnl.het model, whereas the BIC is not able to dis-
criminate between both models.
3.4. Estimating MIXL models
In the following examples we show how to estimate MIXL models using gmnl. The package
mlogit is very efficient in estimating MIXL models. However, one advantage of using gmnl is
the inclusion of individual-specific variables to explain the mean of the random parameters
(see Equation 2). Other important expansions include the possibility of producing point and
interval estimates at the individual level, and the consideration of Johnson Sbheterogeneity
distributions.
If we assume that the coefficients of travel and wait vary across individuals according to:
βtravel,i =β1+π11income +π12 size +σ1η1i
βwait,i =β2+π21income +σ2η2i,
where η1iis triangular and η2i∼N(0,1), the corresponding MIXL model is estimated by
typing:

18 gmnl Package in R
R> mixl.hier <- gmnl(choice ~ vcost + travel + wait | 1 |
+ 0 | income + size - 1,
+ data = TM,
+ model = "mixl",
+ ranp = c(travel = "t", wait = "n"),
+ mvar = list(travel = c("income","size"),
+ wait = c("income")),
+ R = 50,
+ haltons = list("primes" = c(2, 17),
+ "drop" = rep(19, 2)))
Estimating MIXL model
The argument model = "mixl" indicates that the MIXL model will be estimated. The dis-
tribution of the random coefficients are specified by the argument ranp. The distributions
supported by gmnl are presented in Table 3.10 Note also that the fourth part of the formula
is reserved for all the variables that enter the mean of the random parameters. The argument
mvar indicates which variables enter each specific random parameter. For example, travel
= c("income","size") indicates that the mean of the travel coefficient varies according
to income and size. Finally, haltons is relevant if ranp is not NULL. If haltons = NULL,
pseudo-random draws are used instead of Halton sequences. If haltons = NA, the first K
primes are used to generate the Halton draws, where Kis the number of random parameters,
and 15 of the initial sequence of elements are dropped. Otherwise, haltons should be a list
with elements prime and drop. In this example we use the prime numbers 2 and 17, and we
drop the first 19 elements for each series. For a further explanation of Halton draws see Train
(2009).
Shorthands Distributions
"n" Normal distribution
"ln" Log-normal distribution
"cn" Truncated (at zero) normal distribution
"t" Triangular distribution
"u" Uniform distribution
"sb" Johnson Sbdistribution
Table 3: Continuous distributions supported by gmnl.
R> summary(mixl.hier)
Model estimated on: Wed Mar 02 10:36:37 2016
10It is worth mentioning that given how the random parameters of the G-MNL model are constructed
(see Equation 4), the distributions allowed when model = "gmnl" are the normal, uniform, and triangular.
Similarly, when the model is estimated with correlated random parameters, only the normal distribution and
its transformations —log-normal and truncated normal— are allowed.

Mauricio Sarrias, Ricardo Daziano 19
Call:
gmnl(formula = choice ~ vcost + travel + wait | 1 | 0 | income +
size - 1, data = TM, model = "mixl", ranp = c(travel = "t",
wait = "n"), R = 50, haltons = list(primes = c(2, 17), drop = rep(19,
2)), mvar = list(travel = c("income", "size"), wait = c("income")),
method = "bfgs")
Frequencies of categories:
air train bus car
0.276 0.300 0.143 0.281
The estimation took: 0h:1m:18s
Coefficients:
Estimate Std. Error z-value Pr(>|z|)
train:(intercept) -1.70e-01 1.13e+00 -0.15 0.88005
bus:(intercept) -1.14e+00 1.23e+00 -0.93 0.35120
car:(intercept) -9.00e+00 2.52e+00 -3.57 0.00035 ***
vcost -3.07e-02 1.52e-02 -2.02 0.04288 *
travel -7.75e-03 3.29e-03 -2.35 0.01861 *
wait -1.53e-01 4.48e-02 -3.42 0.00062 ***
travel.income -1.59e-04 6.42e-05 -2.48 0.01324 *
travel.size 3.51e-03 1.37e-03 2.56 0.01057 *
wait.income -1.19e-03 6.48e-04 -1.84 0.06568 .
sd.travel 5.48e-03 4.02e-03 1.36 0.17288
sd.wait 8.41e-02 3.48e-02 2.41 0.01582 *
---
Signif. codes: 0 '***'0.001 '**'0.01 '*'0.05 '.'0.1 ' ' 1
Optimization of log-likelihood by BFGS maximization
Log Likelihood: -165
Number of observations: 210
Number of iterations: 357
Exit of MLE: successful convergence
Simulation based on 50 draws
The output shows the estimates in the following order: fixed parameters, mean of the ran-
dom parameters, effect of the variables that affect the mean of the random parameters, and
finally the standard deviation/spread of the random parameters. Note that travel.income
corresponds to π11,travel.size corresponds to π12, and wait.income corresponds to π21 .
The parameters have the expected sign. Note that individuals with higher income are more
sensitive to in-vehicle time (due to the significant and negative travel.income). Whereas
both components of time were assumed random, only waiting time appears as having signifi-
cant variation in how it is perceived across travelers. This result may be due to income and
size explaining taste variations (observed heterogeneity).
We now estimate a correlated random parameter model. For this example, we will use the

20 gmnl Package in R
Electricity data from the mlogit package, which is a panel dataset. There are 4,308 obser-
vations in this microdata set, but only 361 individuals. The analyst designed 12 hypothetical
choice scenarios according to a discrete choice experiment,11 where four hypothetical elec-
tricity suppliers were described in terms of price – which could be fixed (pf), time-of-day
rate (tod), or seasonal rate (seas); length of contract (cl); and being local (loc) or ‘well-
known’ (wk). The experimental design considered unlabeled alternatives, which means that
alternative-specific constants can be set at zero. This microdata is in a “wide format” (one
row describes all alternatives in a given choice situation). Given time compilation restrictions,
in this example we will use just a subsample of this database (subset = 1:3000). The user
may want to use the whole sample to reproduce this case study.
R> data("Electricity", package = "mlogit")
R> Electr <- mlogit.data(Electricity, id.var = "id", choice = "choice",
+ varying = 3:26, shape = "wide", sep = "")
In this example, two arguments are especially relevant in the gmnl function. First, panel =
TRUE indicates that the data is a panel. When using panel data, the user needs to specify a
variable in the id.var argument of the mlogit.data function that identifies the individual.
Second, to estimate correlated random parameters correlation = TRUE needs to be indicated
in the gmnl function. The syntax is the following:
R> Elec.cor <- gmnl(choice ~ pf + cl + loc + wk + tod + seas | 0,
+ data = Electr,
+ subset = 1:3000,
+ model = 'mixl',
+ R = 50,
+ panel = TRUE,
+ ranp = c(cl = "n", loc = "n", wk = "n",
+ tod = "n", seas = "n"),
+ correlation = TRUE)
Estimating MIXL model
R> summary(Elec.cor)
Model estimated on: Wed Mar 02 10:36:53 2016
Call:
gmnl(formula = choice ~ pf + cl + loc + wk + tod + seas | 0,
data = Electr, subset = 1:3000, model = "mixl", ranp = c(cl = "n",
loc = "n", wk = "n", tod = "n", seas = "n"), R = 50,
correlation = TRUE, panel = TRUE, method = "bfgs")
Frequencies of categories:
11Discrete choice experiments collect ‘stated preference’ (SP) data, where choices reflect intended behavior.
In this example, some respondents did not provide their choices for all 12 choice situations.

Mauricio Sarrias, Ricardo Daziano 21
1234
0.215 0.303 0.217 0.265
The estimation took: 0h:0m:16s
Coefficients:
Estimate Std. Error z-value Pr(>|z|)
pf -0.8702 0.0786 -11.07 < 2e-16 ***
cl -0.1765 0.0430 -4.11 4.0e-05 ***
loc 2.3822 0.3053 7.80 6.0e-15 ***
wk 1.9447 0.2493 7.80 6.2e-15 ***
tod -8.5026 0.7423 -11.45 < 2e-16 ***
seas -8.6456 0.7803 -11.08 < 2e-16 ***
sd.cl.cl 0.3919 0.0420 9.33 < 2e-16 ***
sd.cl.loc 0.4921 0.1983 2.48 0.01311 *
sd.cl.wk 0.5514 0.2131 2.59 0.00966 **
sd.cl.tod -0.9834 0.2802 -3.51 0.00045 ***
sd.cl.seas -0.1470 0.2297 -0.64 0.52206
sd.loc.loc 2.5925 0.4226 6.14 8.5e-10 ***
sd.loc.wk 1.9311 0.3610 5.35 8.8e-08 ***
sd.loc.tod 1.0198 0.5651 1.80 0.07114 .
sd.loc.seas 0.0941 0.4579 0.21 0.83723
sd.wk.wk -0.3330 0.2212 -1.51 0.13226
sd.wk.tod 1.9341 0.3208 6.03 1.7e-09 ***
sd.wk.seas 0.7349 0.3030 2.43 0.01529 *
sd.tod.tod 2.0635 0.3301 6.25 4.1e-10 ***
sd.tod.seas 1.1689 0.2539 4.60 4.2e-06 ***
sd.seas.seas 1.7034 0.2533 6.72 1.8e-11 ***
---
Signif. codes: 0 '***'0.001 '**'0.01 '*'0.05 '.'0.1 ' ' 1
Optimization of log-likelihood by BFGS maximization
Log Likelihood: -692
Number of observations: 750
Number of iterations: 97
Exit of MLE: successful convergence
Simulation based on 50 draws
Note that the estimates for the population means indicate that on average individuals prefer
contracts that are cheaper and shorter, and companies that are local and well-known. But,
there is great variation in preferences. The estimates from sd.cl.cl to sd.seas.seas are the
elements of the lower triangular matrix L. If the user is interested in the standard errors of the
variance-covariance matrix of the random parameters LL>=Σor the standard deviations,
the S3 function vcov can be used for finding these elements. The syntax for both cases is the
following:12
12To compute the standard errors, gmnl uses the deltamethod function from the msm package (Jackson

22 gmnl Package in R
R> vcov(Elec.cor, what = 'ranp', type = 'cov', se = 'true')
Elements of the variance-covariance matrix
Estimate Std. Error z-value Pr(>|z|)
v.cl.cl 0.1536 0.0329 4.67 3.1e-06 ***
v.cl.loc 0.1928 0.0816 2.36 0.0181 *
v.cl.wk 0.2161 0.0917 2.36 0.0185 *
v.cl.tod -0.3854 0.1290 -2.99 0.0028 **
v.cl.seas -0.0576 0.0906 -0.64 0.5249
v.loc.loc 6.9630 2.2065 3.16 0.0016 **
v.loc.wk 5.2776 1.6637 3.17 0.0015 **
v.loc.tod 2.1599 1.3323 1.62 0.1050
v.loc.seas 0.1715 1.1222 0.15 0.8785
v.wk.wk 4.1440 1.3293 3.12 0.0018 **
v.wk.tod 0.7832 0.8530 0.92 0.3585
v.wk.seas -0.1441 0.7525 -0.19 0.8481
v.tod.tod 10.0058 3.4763 2.88 0.0040 **
v.tod.seas 4.0739 1.5217 2.68 0.0074 **
v.seas.seas 4.8384 1.1851 4.08 4.5e-05 ***
---
Signif. codes: 0 '***'0.001 '**'0.01 '*'0.05 '.'0.1 ' ' 1
R> vcov(Elec.cor, what = 'ranp', type = 'sd', se = 'true')
Standard deviations of the random parameters
Estimate Std. Error z-value Pr(>|z|)
cl 0.392 0.042 9.33 < 2e-16 ***
loc 2.639 0.418 6.31 2.8e-10 ***
wk 2.036 0.327 6.23 4.5e-10 ***
tod 3.163 0.549 5.76 8.6e-09 ***
seas 2.200 0.269 8.17 2.2e-16 ***
---
Signif. codes: 0 '***'0.001 '**'0.01 '*'0.05 '.'0.1 ' ' 1
The correlation matrix of the random parameters can be recovered using the following syntax:
R> vcov(Elec.cor, what = 'ranp', type = 'cor')
cl loc wk tod seas
cl 1.0000 0.1865 0.2708 -0.311 -0.0668
loc 0.1865 1.0000 0.9825 0.259 0.0295
wk 0.2708 0.9825 1.0000 0.122 -0.0322
tod -0.3109 0.2588 0.1216 1.000 0.5855
seas -0.0668 0.0295 -0.0322 0.586 1.0000
2011).

Mauricio Sarrias, Ricardo Daziano 23
3.5. Estimating G-MNL models
In the following examples we show how to estimate G-MNL models in gmnl. Although we
do not need to specify constants in an unlabeled experiment (alternatives are fully defined
by the experimental attributes), just for illustrative purposes we will assume that the ASCs
are random. Using the formula to create the ASCs produces problems in the ranp argument
due to the way the constants are labeled. So, we first create the ASCs by hand :
R> Electr$asc2 <- as.numeric(Electr$alt == 2)
R> Electr$asc3 <- as.numeric(Electr$alt == 3)
R> Electr$asc4 <- as.numeric(Electr$alt == 4)
The G-MNL model is estimated using model = "gmnl":
R> Elec.gmnl <- gmnl(choice ~ pf + cl + loc + wk + tod + seas +
+ asc2 + asc3 + asc4 | 0,
+ data = Electr,
+ subset = 1:3000,
+ model = 'gmnl',
+ R = 50,
+ panel = TRUE,
+ notscale = c(rep(0, 6), 1, 1, 1),
+ ranp = c(cl = "n", loc = "n", wk = "n",
+ tod = "n", seas = "n",
+ asc2 = "n", asc3 = "n", asc4 = "n"))
The following variables are not scaled:
[1] "asc2" "asc3" "asc4"
Estimating GMNL model
R> summary(Elec.gmnl)
Model estimated on: Wed Mar 02 10:37:15 2016
Call:
gmnl(formula = choice ~ pf + cl + loc + wk + tod + seas + asc2 +
asc3 + asc4 | 0, data = Electr, subset = 1:3000, model = "gmnl",
ranp = c(cl = "n", loc = "n", wk = "n", tod = "n", seas = "n",
asc2 = "n", asc3 = "n", asc4 = "n"), R = 50, panel = TRUE,
notscale = c(rep(0, 6), 1, 1, 1), method = "bfgs")
Frequencies of categories:
1234
0.215 0.303 0.217 0.265
The estimation took: 0h:0m:22s

24 gmnl Package in R
Coefficients:
Estimate Std. Error z-value Pr(>|z|)
pf -0.8733 0.1066 -8.19 2.2e-16 ***
cl -0.1718 0.0422 -4.07 4.7e-05 ***
loc 1.8081 0.2289 7.90 2.9e-15 ***
wk 1.7543 0.2222 7.89 2.9e-15 ***
tod -8.5960 0.9865 -8.71 < 2e-16 ***
seas -8.8653 1.0128 -8.75 < 2e-16 ***
asc2 0.3044 0.1539 1.98 0.0479 *
asc3 0.1563 0.1598 0.98 0.3279
asc4 0.1133 0.1568 0.72 0.4698
sd.cl 0.3643 0.0442 8.25 2.2e-16 ***
sd.loc 1.1015 0.2738 4.02 5.7e-05 ***
sd.wk 1.2053 0.2493 4.84 1.3e-06 ***
sd.tod 1.4655 0.2336 6.28 3.5e-10 ***
sd.seas 1.8110 0.2958 6.12 9.3e-10 ***
sd.asc2 0.5264 0.1791 2.94 0.0033 **
sd.asc3 0.0849 0.2246 0.38 0.7055
sd.asc4 0.2407 0.1881 1.28 0.2007
tau 0.6777 0.1490 4.55 5.4e-06 ***
gamma 0.3625 0.1747 2.08 0.0380 *
---
Signif. codes: 0 '***'0.001 '**'0.01 '*'0.05 '.'0.1 ' ' 1
Optimization of log-likelihood by BFGS maximization
Log Likelihood: -729
Number of observations: 750
Number of iterations: 98
Exit of MLE: successful convergence
Simulation based on 50 draws
Since we are including the ASCs as additional variables, the second part of the formula does
not include the ASCs (| 0). Note also that even though the ASCs are random, they are not
scaled: notscale = c(rep(0, 6), 1, 1, 1) indicates that the last three variables in the
first part of the formula (asc2,asc3, and asc4) are not scaled.
Another important issue is that gmnl estimates γdirectly by default as suggested by Keane
and Wasi (2013). However, one can estimate γ∗, where γ= exp(γ∗)/(1+exp(γ∗)) as suggested
by Fiebig et al. (2010), by specifying hgamma = "indirect". Thus, hgamma = "direct" is
the default setting.
The G-MNL estimation code is also very convenient when one wants to estimate S-MNL
models with random effects (Keane and Wasi 2013). In this case, the user can fix γand use
model = "gmnl".
R> Elec.smnl.re <- gmnl(choice ~ pf + cl + loc + wk + tod + seas +
+ asc2 + asc3 + asc4 | 0,

Mauricio Sarrias, Ricardo Daziano 25
+ data = Electr,
+ subset = 1:3000,
+ model = 'gmnl',
+ R = 50,
+ panel = TRUE,
+ print.init = TRUE,
+ notscale = c(rep(0, 6), 1, 1, 1),
+ ranp = c(asc2 = "n", asc3 = "n", asc4 = "n"),
+ init.gamma = 0,
+ fixed = c(rep(FALSE, 16), TRUE),
+ correlation = TRUE)
The following variables are not scaled:
[1] "asc2" "asc3" "asc4"
Starting Values:
pf cl loc wk tod
-0.6018 -0.1350 1.2223 1.0387 -5.3686
seas asc2 asc3 asc4 sd.asc2.asc2
-5.5623 0.2097 0.0811 0.1065 0.1000
sd.asc2.asc3 sd.asc2.asc4 sd.asc3.asc3 sd.asc3.asc4 sd.asc4.asc4
0.1000 0.1000 0.1000 0.1000 0.1000
tau gamma
0.1000 0.0000
Estimating GMNL model
R> summary(Elec.smnl.re)
Model estimated on: Wed Mar 02 10:37:31 2016
Call:
gmnl(formula = choice ~ pf + cl + loc + wk + tod + seas + asc2 +
asc3 + asc4 | 0, data = Electr, subset = 1:3000, model = "gmnl",
ranp = c(asc2 = "n", asc3 = "n", asc4 = "n"), R = 50, correlation = TRUE,
panel = TRUE, init.gamma = 0, notscale = c(rep(0, 6), 1,
1, 1), print.init = TRUE, fixed = c(rep(FALSE, 16), TRUE),
method = "bfgs")
Frequencies of categories:
1234
0.215 0.303 0.217 0.265
The estimation took: 0h:0m:16s
Coefficients:
Estimate Std. Error z-value Pr(>|z|)

26 gmnl Package in R
pf -0.6299 0.1148 -5.49 4.1e-08 ***
cl -0.1377 0.0328 -4.20 2.7e-05 ***
loc 1.2749 0.2200 5.80 6.8e-09 ***
wk 1.1119 0.1929 5.76 8.2e-09 ***
tod -6.2008 1.0450 -5.93 3.0e-09 ***
seas -6.3681 1.0802 -5.90 3.7e-09 ***
asc2 0.2124 0.1442 1.47 0.1409
asc3 0.2295 0.1351 1.70 0.0894 .
asc4 0.1536 0.1310 1.17 0.2410
sd.asc2.asc2 0.5694 0.2184 2.61 0.0091 **
sd.asc2.asc3 0.3066 0.1813 1.69 0.0908 .
sd.asc2.asc4 0.1508 0.1995 0.76 0.4497
sd.asc3.asc3 0.0445 0.2192 0.20 0.8390
sd.asc3.asc4 0.0893 0.2188 0.41 0.6832
sd.asc4.asc4 -0.0406 0.2034 -0.20 0.8419
tau 1.1009 0.1852 5.94 2.8e-09 ***
---
Signif. codes: 0 '***'0.001 '**'0.01 '*'0.05 '.'0.1 ' ' 1
Optimization of log-likelihood by BFGS maximization
Log Likelihood: -841
Number of observations: 750
Number of iterations: 76
Exit of MLE: successful convergence
Simulation based on 50 draws
The argument init.gamma indicates the initial value for γ. In this case we set it at zero.
The next step is to set the parameters that are fixed by using the argument fixed, which
is passed to the maxLik function. Note that the user needs to be careful with the order of
the parameters. We encourage the user to estimate first a model where all the parameters
are freely estimated with the argument print.init = TRUE. This argument will display the
initial values and the order used by gmnl. Generally, γis the last parameter that enters
the likelihood specification. So, by typing fixed = c(rep(FALSE, 16), TRUE) we are only
holding γfixed at zero, and the rest of the coefficients are freely estimated.
By default, the initial values for the mean of the random parameters come from an MNL, and
the standard deviations or spread are set at 0.1. However, the starting values from an MNL
model may not be the best guess, since the G-MNL model is not globally concave. The best
starting values for a G-MNL model with correlated parameters might be: 1) G-MNL with
uncorrelated parameters, 2) MIXL with correlated parameters, or 3) GMNL with correlated
parameters with γfixed at 0. One can first get these initial parameters and then use the
start argument of gmnl to indicate the vector of appropriate starting values (see Section 3.7
for an example of how to use the start argument).
3.6. Estimating LC and MM-MNL models
The next example shows how an LC model with two classes can be estimated:

Mauricio Sarrias, Ricardo Daziano 27
R> Elec.lc <- gmnl(choice ~ pf + cl + loc + wk + tod + seas | 0 |
+ 0|0|1,
+ data = Electr,
+ subset = 1:3000,
+ model = 'lc',
+ panel = TRUE,
+ Q = 2)
Estimating LC model
Note that for the LC model, one needs to specify at least a constant in the fifth part of the
formula. If the class assignment wiq is also determined by socio-economic characteristics,
those covariates can also be included in the fifth part. The LC model is estimated by typing
model = "lc", and the prespecified number of classes is indicated with the argument Q.
R> summary(Elec.lc)
Model estimated on: Wed Mar 02 10:37:32 2016
Call:
gmnl(formula = choice ~ pf + cl + loc + wk + tod + seas | 0 |
0 | 0 | 1, data = Electr, subset = 1:3000, model = "lc",
Q = 2, panel = TRUE, method = "bfgs")
Frequencies of categories:
1234
0.215 0.303 0.217 0.265
The estimation took: 0h:0m:0s
Coefficients:
Estimate Std. Error z-value Pr(>|z|)
class.1.pf -0.4458 0.0876 -5.09 3.6e-07 ***
class.1.cl -0.1847 0.0301 -6.14 8.3e-10 ***
class.1.loc 1.2144 0.1618 7.50 6.2e-14 ***
class.1.wk 0.9641 0.1429 6.75 1.5e-11 ***
class.1.tod -3.2184 0.6880 -4.68 2.9e-06 ***
class.1.seas -3.4865 0.6929 -5.03 4.9e-07 ***
class.2.pf -0.8431 0.0968 -8.71 < 2e-16 ***
class.2.cl -0.1242 0.0453 -2.74 0.0061 **
class.2.loc 1.6445 0.2689 6.12 9.6e-10 ***
class.2.wk 1.4139 0.2120 6.67 2.6e-11 ***
class.2.tod -9.3732 0.8676 -10.80 < 2e-16 ***
class.2.seas -9.2647 0.8847 -10.47 < 2e-16 ***
(class)2 -0.2200 0.0788 -2.79 0.0052 **
---

28 gmnl Package in R
Signif. codes: 0 '***'0.001 '**'0.01 '*'0.05 '.'0.1 ' ' 1
Optimization of log-likelihood by BFGS maximization
Log Likelihood: -793
Number of observations: 750
Number of iterations: 77
Exit of MLE: successful convergence
Note that the underlying assumption in this example is that there are two types of cus-
tomers. For both types, the parameters have expected signs and are statistically significant.
Probability of class assignment is assumed constant in this case, but in a dataset with sociode-
mographics, assignment to classes can vary for each individual using those sodiodemographics
as explanatory variables of the class assignment probability.
The following example estimates an MM-MNL with a mixture of two normal distributions:
R> Elec.mm <- gmnl(choice ~ pf + cl + loc + wk + tod + seas | 0 |
+ 0|0|1,
+ data = Electr,
+ subset = 1:3000,
+ model = 'mm',
+ R = 50,
+ panel = TRUE,
+ ranp = c(pf = "n", cl = "n", loc = "n",
+ wk = "n", tod = "n", seas = "n"),
+ Q = 2,
+ iterlim = 500)
Estimating MM-MNL model
R> summary(Elec.mm)
Model estimated on: Wed Mar 02 10:38:09 2016
Call:
gmnl(formula = choice ~ pf + cl + loc + wk + tod + seas | 0 |
0 | 0 | 1, data = Electr, subset = 1:3000, model = "mm",
ranp = c(pf = "n", cl = "n", loc = "n", wk = "n", tod = "n",
seas = "n"), R = 50, Q = 2, panel = TRUE, iterlim = 500,
method = "bfgs")
Frequencies of categories:
1234
0.215 0.303 0.217 0.265
The estimation took: 0h:0m:37s

Mauricio Sarrias, Ricardo Daziano 29
Coefficients:
Estimate Std. Error z-value Pr(>|z|)
class.1.pf -1.28036 0.12279 -10.43 < 2e-16 ***
class.1.cl -0.49715 0.08070 -6.16 7.3e-10 ***
class.1.loc 0.64445 0.24097 2.67 0.00749 **
class.1.wk 0.71241 0.21823 3.26 0.00110 **
class.1.tod -11.62474 1.02748 -11.31 < 2e-16 ***
class.1.seas -12.65698 1.13957 -11.11 < 2e-16 ***
class.2.pf -0.43575 0.11588 -3.76 0.00017 ***
class.2.cl 0.08464 0.08288 1.02 0.30713
class.2.loc 3.38611 0.38188 8.87 < 2e-16 ***
class.2.wk 2.72095 0.32688 8.32 < 2e-16 ***
class.2.tod -4.71637 1.05414 -4.47 7.7e-06 ***
class.2.seas -4.64753 0.96000 -4.84 1.3e-06 ***
class.1.sd.pf 0.10526 0.03938 2.67 0.00752 **
class.1.sd.cl 0.26906 0.05835 4.61 4.0e-06 ***
class.1.sd.loc 0.00764 0.26552 0.03 0.97704
class.1.sd.wk 0.11414 0.58053 0.20 0.84413
class.1.sd.tod 2.20533 0.45035 4.90 9.7e-07 ***
class.1.sd.seas 2.32243 0.45203 5.14 2.8e-07 ***
class.2.sd.pf 0.19696 0.03371 5.84 5.2e-09 ***
class.2.sd.cl 0.35523 0.07491 4.74 2.1e-06 ***
class.2.sd.loc 0.58717 0.28699 2.05 0.04076 *
class.2.sd.wk 1.10025 0.30323 3.63 0.00029 ***
class.2.sd.tod 1.40225 0.51875 2.70 0.00687 **
class.2.sd.seas 0.07309 0.25655 0.28 0.77572
(class)2 -0.13714 0.07844 -1.75 0.08039 .
---
Signif. codes: 0 '***'0.001 '**'0.01 '*'0.05 '.'0.1 ' ' 1
Optimization of log-likelihood by BFGS maximization
Log Likelihood: -672
Number of observations: 750
Number of iterations: 125
Exit of MLE: successful convergence
Simulation based on 50 draws
The specification is similar to that of the LC model, but we now allow the parameters in each
class to be normally distributed using the argument ranp. It is worth mentioning that the
number of iterations required for this model is greater than that for previous models.13 For
13Another important issue is weak identification. Large magnitudes of the coefficients or standard errors,
or even models with slow convergence, might be a sign of weak identification, especially in complex models
such as the MM-Logit model (Ruud 2007). Users should remain suspicious and further investigate such cases
by analyzing the Hessian matrix of second-order partial derivatives. If the Hessian matrix is positive definite
(e.g., all its eigenvalues are positive), the model is said to be locally identified (see Wedel and Kamakura 2012,
pag. 91). The Hessian can be obtained by typing my_model$logLik$hessian after estimating a model of class

30 gmnl Package in R
that reason we have set the maximum of iterations at 500 using the argument iterlim. The
example below adds the consideration of correlated parameters, which is the model typically
used in Bayesian treatments.
R> Elec.mm.c <- gmnl(choice ~ pf + cl + loc + wk + tod + seas | 0 |
+ 0|0|1,
+ data = Electr,
+ subset = 1:3000,
+ model = 'mm',
+ R = 50,
+ panel = TRUE,
+ ranp = c(pf = "n", cl = "n", loc = "n",
+ wk = "n", tod = "n", seas = "n"),
+ Q = 2,
+ iterlim = 500,
+ correlation = TRUE)
Estimating MM-MNL model
R> summary(Elec.mm.c)
Model estimated on: Wed Mar 02 10:39:42 2016
Call:
gmnl(formula = choice ~ pf + cl + loc + wk + tod + seas | 0 |
0 | 0 | 1, data = Electr, subset = 1:3000, model = "mm",
ranp = c(pf = "n", cl = "n", loc = "n", wk = "n", tod = "n",
seas = "n"), R = 50, Q = 2, correlation = TRUE, panel = TRUE,
iterlim = 500, method = "bfgs")
Frequencies of categories:
1234
0.215 0.303 0.217 0.265
The estimation took: 0h:1m:33s
Coefficients:
Estimate Std. Error z-value Pr(>|z|)
class.1.pf -0.9996 0.1311 -7.63 2.4e-14 ***
class.1.cl -0.2168 0.0545 -3.98 7.0e-05 ***
class.1.loc 2.7749 0.4689 5.92 3.3e-09 ***
class.1.wk 2.3225 0.3165 7.34 2.2e-13 ***
class.1.tod -9.8537 1.1690 -8.43 < 2e-16 ***
class.1.seas -9.5464 1.0975 -8.70 < 2e-16 ***
gmnl.

Mauricio Sarrias, Ricardo Daziano 31
class.2.pf -1.9182 0.4702 -4.08 4.5e-05 ***
class.2.cl -0.8815 0.1673 -5.27 1.4e-07 ***
class.2.loc 3.7753 1.0432 3.62 0.00030 ***
class.2.wk 3.0521 0.8006 3.81 0.00014 ***
class.2.tod -13.4974 3.6628 -3.68 0.00023 ***
class.2.seas -15.0237 3.6817 -4.08 4.5e-05 ***
class.1.sd.pf.pf 0.6324 0.1245 5.08 3.8e-07 ***
class.1.sd.pf.cl 0.2018 0.0522 3.87 0.00011 ***
class.1.sd.pf.loc 1.4214 0.3932 3.62 0.00030 ***
class.1.sd.pf.wk 1.0639 0.2577 4.13 3.6e-05 ***
class.1.sd.pf.tod 5.5303 1.0575 5.23 1.7e-07 ***
class.1.sd.pf.seas 4.3941 1.0112 4.35 1.4e-05 ***
class.1.sd.cl.cl 0.2116 0.0638 3.32 0.00091 ***
class.1.sd.cl.loc -1.0122 0.3917 -2.58 0.00976 **
class.1.sd.cl.wk -0.6071 0.2838 -2.14 0.03239 *
class.1.sd.cl.tod 2.0922 0.4038 5.18 2.2e-07 ***
class.1.sd.cl.seas 1.5087 0.3366 4.48 7.4e-06 ***
class.1.sd.loc.loc 1.3945 0.5937 2.35 0.01883 *
class.1.sd.loc.wk 0.8431 0.3562 2.37 0.01793 *
class.1.sd.loc.tod 0.5368 0.3483 1.54 0.12325
class.1.sd.loc.seas 0.6143 0.4411 1.39 0.16371
class.1.sd.wk.wk 0.5915 0.1638 3.61 0.00030 ***
class.1.sd.wk.tod 0.2274 0.2398 0.95 0.34301
class.1.sd.wk.seas 0.7723 0.2370 3.26 0.00112 **
class.1.sd.tod.tod 1.6883 0.5447 3.10 0.00194 **
class.1.sd.tod.seas 1.9974 0.4498 4.44 9.0e-06 ***
class.1.sd.seas.seas 0.0275 0.4979 0.06 0.95597
class.2.sd.pf.pf -1.0278 0.2735 -3.76 0.00017 ***
class.2.sd.pf.cl 0.3755 0.1017 3.69 0.00022 ***
class.2.sd.pf.loc 1.5795 0.5169 3.06 0.00224 **
class.2.sd.pf.wk 1.9678 0.5497 3.58 0.00034 ***
class.2.sd.pf.tod -7.1137 2.0080 -3.54 0.00040 ***
class.2.sd.pf.seas -6.0019 2.0065 -2.99 0.00278 **
class.2.sd.cl.cl 1.2556 0.2070 6.07 1.3e-09 ***
class.2.sd.cl.loc 0.0671 0.5375 0.12 0.90058
class.2.sd.cl.wk -0.5454 0.4364 -1.25 0.21141
class.2.sd.cl.tod -1.0686 0.5221 -2.05 0.04068 *
class.2.sd.cl.seas 4.0996 0.8738 4.69 2.7e-06 ***
class.2.sd.loc.loc 3.8107 0.9585 3.98 7.0e-05 ***
class.2.sd.loc.wk 3.3262 0.7860 4.23 2.3e-05 ***
class.2.sd.loc.tod 0.3255 0.6085 0.53 0.59279
class.2.sd.loc.seas 2.8477 0.9473 3.01 0.00265 **
class.2.sd.wk.wk -0.7910 0.3234 -2.45 0.01446 *
class.2.sd.wk.tod 0.7507 0.3089 2.43 0.01508 *
class.2.sd.wk.seas -1.6942 0.6594 -2.57 0.01019 *
class.2.sd.tod.tod -2.6490 0.6178 -4.29 1.8e-05 ***
class.2.sd.tod.seas -0.4250 0.6364 -0.67 0.50425

32 gmnl Package in R
class.2.sd.seas.seas 1.4412 0.4171 3.46 0.00055 ***
(class)2 -1.0232 0.0983 -10.40 < 2e-16 ***
---
Signif. codes: 0 '***'0.001 '**'0.01 '*'0.05 '.'0.1 ' ' 1
Optimization of log-likelihood by BFGS maximization
Log Likelihood: -640
Number of observations: 750
Number of iterations: 292
Exit of MLE: successful convergence
Simulation based on 50 draws
The standard deviations of the random parameters and their standard errors for each class
can be obtained using the vcov function in the following way
R> vcov(Elec.mm.c, what = "ranp", Q = 1, type = 'sd', se = TRUE)
Standard deviations of the random parameters
Estimate Std. Error z-value Pr(>|z|)
pf 0.6324 0.1245 5.08 3.8e-07 ***
cl 0.2924 0.0514 5.69 1.2e-08 ***
loc 2.2337 0.4054 5.51 3.6e-08 ***
wk 1.6004 0.3159 5.07 4.1e-07 ***
tod 6.1767 0.9809 6.30 3.0e-10 ***
seas 5.1525 0.9761 5.28 1.3e-07 ***
---
Signif. codes: 0 '***'0.001 '**'0.01 '*'0.05 '.'0.1 ' ' 1
R> vcov(Elec.mm.c, what = "ranp", Q = 2, type = 'sd', se = TRUE)
Standard deviations of the random parameters
Estimate Std. Error z-value Pr(>|z|)
pf 1.028 0.273 3.76 0.00017 ***
cl 1.311 0.209 6.26 3.8e-10 ***
loc 4.126 1.017 4.06 4.9e-05 ***
wk 3.982 0.885 4.50 6.8e-06 ***
tod 7.709 1.960 3.93 8.4e-05 ***
seas 8.128 1.853 4.39 1.2e-05 ***
---
Signif. codes: 0 '***'0.001 '**'0.01 '*'0.05 '.'0.1 ' ' 1
3.7. Willingness-to-pay space

Mauricio Sarrias, Ricardo Daziano 33
Willingness-to-pay space models reparameterize the parameter space in such a way that the
marginal WTP for each attribute (and the parameters of its heterogeneity distribution for a
random parameter model) is directly estimated rather than the marginal utility (preference
parameters). To motivate the WTP space model, consider the following latent utility
Uijt =−αpijt +x>
ijtβ+ijt,(5)
where pijt is the price coefficient. This model is known as the model in preference space. The
utility in WTP-space is obtained by dividing the attribute’s coefficients by the price coefficient
in the following way
Uijt =−αpijt +x>
ijt −αβ
α+ijt
=−αpijt +x>
ijt (−αγ) + ijt,
where γis the WTP parameter vector, and αis fixed and equal to 1. Although both,
preference and WTP-space are behaviorally equivalent, the latter approach is useful when
allowing for random heterogeneity in γ.
In effect, the WTP-space approach is very appealing because it allows the analyst to specify
and estimate the distributions of WTP directly, rather than deriving them indirectly from
distributions of coefficients in preference space model (Scarpa, Thiene, and Train 2008). In the
preference space model, the distribution of WTP is derived from the distribution of the ratio of
both αand β. However this ratio may not result in a well-specified distribution. For example,
if both αand βare normally distributed, then the ratio produces a Cauchy distribution with
no finite moments (Daly, Hess, and Train 2011). Motivated by this problem, Train and Weeks
(2005) and Sonnier, Ainslie, and Otter (2007) extended the WTP-space approach by allowing
γto follow any distribution and thus to avoid the problem of non-finite moments for the
distribution of WTP.
To illustrate the concept of WTP-space, and how it can be estimated using gmnl, we will first
show the case without random parameters. The standard procedure to derive willingness-to-
pay measures is to start with a model in preference space, and then make inference on the
appropriate ratio that represents the marginal rate of substitution between a given attribute
and price. For example, consider the simple conditional logit model,
R> clogit <- gmnl(choice ~ pf + cl + loc + wk + tod + seas | 0,
+ data = Electr,
+ subset = 1:3000)
R> summary(clogit)
Model estimated on: Wed Mar 02 10:39:43 2016
Call:
gmnl(formula = choice ~ pf + cl + loc + wk + tod + seas | 0,
data = Electr, subset = 1:3000, method = "nr")
Frequencies of categories:

34 gmnl Package in R
1234
0.215 0.303 0.217 0.265
The estimation took: 0h:0m:0s
Coefficients:
Estimate Std. Error z-value Pr(>|z|)
pf -0.6113 0.0548 -11.15 < 2e-16 ***
cl -0.1398 0.0204 -6.85 7.2e-12 ***
loc 1.1986 0.1197 10.01 < 2e-16 ***
wk 1.0304 0.1063 9.69 < 2e-16 ***
tod -5.4540 0.4341 -12.56 < 2e-16 ***
seas -5.6648 0.4419 -12.82 < 2e-16 ***
---
Signif. codes: 0 '***'0.001 '**'0.01 '*'0.05 '.'0.1 ' ' 1
Optimization of log-likelihood by Newton-Raphson maximisation
Log Likelihood: -870
Number of observations: 750
Number of iterations: 4
Exit of MLE: gradient close to zero
To estimate the willingness to pay for each attribute, one needs to divide each attribute
parameter by that of price pf. This ratio can be easily retrieved using the function wtp.gmnl:
R> wtp.gmnl(clogit, wrt = "pf")
Willigness-to-pay respect to: pf
Estimate Std. Error t-value Pr(>|t|)
cl 0.2287 0.0358 6.38 1.8e-10 ***
loc -1.9610 0.2304 -8.51 < 2e-16 ***
wk -1.6858 0.1949 -8.65 < 2e-16 ***
tod 8.9226 0.2025 44.07 < 2e-16 ***
seas 9.2675 0.2164 42.83 < 2e-16 ***
---
Signif. codes: 0 '***'0.001 '**'0.01 '*'0.05 '.'0.1 ' ' 1
The argument wrt = "pf" indicates that all the parameters should be divided by the pa-
rameter of the attribute pf.14 Using the estimated ratios, we can say, for example, that a
individual with average price and contract lenght (cl) is willing to pay ≈0.23, or almost
one-fifth a cent per kWh extra, to have a contract that is one year shorter.
14In the current version of gmnl, the standard error of the WTP estimates are calculated using the delta
method, which only works well when there are no problems of weak identification in the ratio.

Mauricio Sarrias, Ricardo Daziano 35
Another way to estimate the same WTP coefficients is to use the S-MNL model to derive
a specification in WTP-space. To do so, we need first to compute the negative of the price
attribute using the mlogit.data function:
R> ElectrO <- mlogit.data(Electricity, id = "id", choice = "choice",
+ varying = 3:26, shape = "wide", sep = "",
+ opposite = c("pf"))
Next, we need to set the values for the price parameter and τat 1 and 0, respectively. The
fixed argument is used to set these values.
R> start <- c(1, 0, 0, 0, 0, 0, 0, 0)
R> wtps <- gmnl(choice ~ pf + cl + loc + wk + tod + seas | 0 |
+ 0|0|1,
+ data = ElectrO,
+ model = "smnl",
+ subset = 1:3000,
+ R = 1,
+ fixed = c(TRUE, FALSE, FALSE, FALSE, FALSE,
+ FALSE, TRUE, FALSE),
+ panel = TRUE,
+ start = start,
+ method = "bhhh",
+ iterlim = 500)
Estimating SMNL model
Note that we fitted the S-MNL model with a constant in the scale. This constant, after a
proper transformation, will represent the price parameter. Since we are working with a fixed
parameter model, the number of draws is set equal to 1.
R> summary(wtps)
Model estimated on: Wed Mar 02 10:39:43 2016
Call:
gmnl(formula = choice ~ pf + cl + loc + wk + tod + seas | 0 |
0 | 0 | 1, data = ElectrO, subset = 1:3000, model = "smnl",
start = start, R = 1, panel = TRUE, fixed = c(TRUE, FALSE,
FALSE, FALSE, FALSE, FALSE, TRUE, FALSE), method = "bhhh",
iterlim = 500)
Frequencies of categories:
1234
0.215 0.303 0.217 0.265

36 gmnl Package in R
The estimation took: 0h:0m:1s
Coefficients:
Estimate Std. Error z-value Pr(>|z|)
cl -0.2287 0.0361 -6.34 2.4e-10 ***
loc 1.9609 0.2284 8.59 < 2e-16 ***
wk 1.6857 0.1915 8.80 < 2e-16 ***
tod -8.9226 0.2025 -44.06 < 2e-16 ***
seas -9.2675 0.2166 -42.79 < 2e-16 ***
het.(Intercept) -0.4922 0.0917 -5.37 7.9e-08 ***
---
Signif. codes: 0 '***'0.001 '**'0.01 '*'0.05 '.'0.1 ' ' 1
Optimization of log-likelihood by BHHH maximisation
Log Likelihood: -870
Number of observations: 750
Number of iterations: 14
Exit of MLE: successive function values within tolerance limit
Simulation based on 1 draws
Each value in the output represents the WTP estimates for each respective attribute. Note
that these WTP estimates are the same as those obtained using the wtp.gmnl function. The
price coefficient can be obtained using the following transformation:
R> -exp(coef(wtps)["het.(Intercept)"])
het.(Intercept)
-0.611
If one requires the standard error for the price coefficient the deltamethod function from the
msm (Jackson 2011) package can be used in the following way:
R> library("msm")
R> estmean <- coef(wtps)
R> estvar <- vcov(wtps)
R> se <- deltamethod(~ -exp(x6), estmean, estvar, ses = TRUE)
R> se
[1] 0.056
Using the same idea, one can let the WTP vary across individuals. To do so, we can estimate
a G-MNL where the parameter of price and γare fixed as in the previous example:
R> start2 <- c(1, coef(wtps), rep(0.1, 5), 0.1, 0)
R> wtps2 <- gmnl(choice ~ pf + cl + loc + wk + tod + seas | 0 | 0 | 0 | 1,

Mauricio Sarrias, Ricardo Daziano 37
+ data = ElectrO,
+ subset = 1:3000,
+ model = "gmnl",
+ R = 50,
+ fixed = c(TRUE, rep(FALSE, 12), TRUE),
+ panel = TRUE,
+ start = start2,
+ ranp = c(cl = "n", loc = "n", wk = "n", tod = "n", seas = "n"))
Estimating GMNL model
R> summary(wtps2)
Model estimated on: Wed Mar 02 10:40:10 2016
Call:
gmnl(formula = choice ~ pf + cl + loc + wk + tod + seas | 0 |
0 | 0 | 1, data = ElectrO, subset = 1:3000, model = "gmnl",
start = start2, ranp = c(cl = "n", loc = "n", wk = "n", tod = "n",
seas = "n"), R = 50, panel = TRUE, fixed = c(TRUE, rep(FALSE,
12), TRUE), method = "bfgs")
Frequencies of categories:
1234
0.215 0.303 0.217 0.265
The estimation took: 0h:0m:26s
Coefficients:
Estimate Std. Error z-value Pr(>|z|)
cl -0.2727 0.0518 -5.26 1.4e-07 ***
loc 2.1631 0.2452 8.82 < 2e-16 ***
wk 1.9424 0.1935 10.04 < 2e-16 ***
tod -9.6782 0.2933 -32.99 < 2e-16 ***
seas -9.8866 0.2772 -35.67 < 2e-16 ***
het.(Intercept) 0.1142 0.1401 0.82 0.41
sd.cl 0.4115 0.0520 7.91 2.7e-15 ***
sd.loc 1.7850 0.2515 7.10 1.3e-12 ***
sd.wk 1.2865 0.2213 5.81 6.1e-09 ***
sd.tod 1.7174 0.2478 6.93 4.2e-12 ***
sd.seas 2.2146 0.3723 5.95 2.7e-09 ***
tau 0.6904 0.1394 4.95 7.3e-07 ***
---
Signif. codes: 0 '***'0.001 '**'0.01 '*'0.05 '.'0.1 ' ' 1
Optimization of log-likelihood by BFGS maximization

38 gmnl Package in R
Log Likelihood: -737
Number of observations: 750
Number of iterations: 143
Exit of MLE: successful convergence
Simulation based on 50 draws
Note that the model recast in WTP-space that is implemented in the G-MNL specification
above allows the researcher to specify directly the heterogeneity distribution of WTP mea-
sures (Sonnier et al. 2007). When working in preference space, and then deriving WTP as
a parameter ratio, normally distributed WTP measures can only be derived if the price pa-
rameter is fixed (i.e., no unobserved heterogeneity in the marginal utility of income, which is
a strong assumption) and the rest of the attributes are assumed to be normally distributed.
The problem with all normals in preference space is that the ratio of two normally distributed
parameters has a distribution with very long tails and without moments, which leads to
unexpected individual-level predictions.
Finally, a WTP-space model with correlated random parameters can be estimated in the
following way:
R> n_ran <- 5
R> start3 <- c(1, coef(wtps), rep(0.1, .5 * n_ran * (n_ran + 1)), 0.1, 0)
R> wtps3 <- gmnl(choice ~ pf + cl + loc + wk + tod + seas | 0 | 0 | 0 | 1,
+ data = ElectrO,
+ subset = 1:3000,
+ model = "gmnl",
+ R = 50,
+ fixed = c(TRUE, rep(FALSE, 22), TRUE),
+ panel = TRUE,
+ start = start3,
+ ranp = c(cl = "n", loc = "n", wk = "n", tod = "n", seas = "n"),
+ correlation = TRUE)
Estimating GMNL model
R> summary(wtps3)
Model estimated on: Wed Mar 02 10:40:50 2016
Call:
gmnl(formula = choice ~ pf + cl + loc + wk + tod + seas | 0 |
0 | 0 | 1, data = ElectrO, subset = 1:3000, model = "gmnl",
start = start3, ranp = c(cl = "n", loc = "n", wk = "n", tod = "n",
seas = "n"), R = 50, correlation = TRUE, panel = TRUE,
fixed = c(TRUE, rep(FALSE, 22), TRUE), method = "bfgs")
Frequencies of categories:

Mauricio Sarrias, Ricardo Daziano 39
1234
0.215 0.303 0.217 0.265
The estimation took: 0h:0m:40s
Coefficients:
Estimate Std. Error z-value Pr(>|z|)
cl -0.1968 0.0588 -3.35 0.00082 ***
loc 2.1682 0.3228 6.72 1.9e-11 ***
wk 1.8273 0.2523 7.24 4.4e-13 ***
tod -9.9662 0.3736 -26.68 < 2e-16 ***
seas -9.9928 0.3535 -28.27 < 2e-16 ***
het.(Intercept) -0.1385 0.0945 -1.47 0.14268
sd.cl.cl 0.4741 0.0625 7.59 3.2e-14 ***
sd.cl.loc 0.3903 0.2560 1.52 0.12730
sd.cl.wk 0.6118 0.2444 2.50 0.01231 *
sd.cl.tod -1.3139 0.2727 -4.82 1.5e-06 ***
sd.cl.seas -0.2438 0.2655 -0.92 0.35840
sd.loc.loc 2.5819 0.4937 5.23 1.7e-07 ***
sd.loc.wk 1.6391 0.4104 3.99 6.5e-05 ***
sd.loc.tod 1.9078 0.4725 4.04 5.4e-05 ***
sd.loc.seas 0.9679 0.3765 2.57 0.01014 *
sd.wk.wk -0.7259 0.2745 -2.64 0.00818 **
sd.wk.tod 2.2373 0.3745 5.97 2.3e-09 ***
sd.wk.seas 1.5985 0.3521 4.54 5.6e-06 ***
sd.tod.tod 2.5361 0.4169 6.08 1.2e-09 ***
sd.tod.seas 1.7418 0.3636 4.79 1.7e-06 ***
sd.seas.seas 2.0034 0.2872 6.98 3.0e-12 ***
tau -0.2557 0.1219 -2.10 0.03595 *
---
Signif. codes: 0 '***'0.001 '**'0.01 '*'0.05 '.'0.1 ' ' 1
Optimization of log-likelihood by BFGS maximization
Log Likelihood: -691
Number of observations: 750
Number of iterations: 174
Exit of MLE: successful convergence
Simulation based on 50 draws
Note that n_ran is the number of random coefficients, which is used to compute the number
of initial values for the Lmatrix in the start3 vector of initial parameters: let this number
be Ka, then the number of elements is equal to (1/2) ·Ka·(Ka+ 1).
3.8. Individual parameters
Similarly to the Rchoice package (Sarrias 2015), gmnl also allows the analyst to get the
conditional estimates for each individual in the sample (see for example Train 2009;Greene

40 gmnl Package in R
2012). Using Bayes’ theorem we obtain
f(βi|yi,Xi,θ) = f(yi|Xi,βi)g(βi|θ)
Rβif(yi|Xi,βi)g(βi|θ)dβi
,
where f(βi|yi,Xi,θ) is the distribution of the individual parameters βiconditional on the
observed sequence of choices, and g(βi|θ) is the unconditional distribution. The conditional
expectation of βiis thus given by:
E[βi|yi,Xi,θ] = Rβiβif(yi|Xi,βi)g(βi|θ)dβi
Rβif(yi|Xi,βi)g(βi|θ)dβi
.(6)
The expectation in Equation 6gives us the conditional mean of the distribution of the random
parameters, which can also be interpreted as the posterior distribution of the individual pa-
rameters. Simulators for this conditional expectation are presented below for the continuous,
discrete and mixture cases, respectively:
b¯
βi=b
E[βi|yi,Xi,θ] =
1
RPR
r=1 b
βir Qtf(yit|xit,b
βir,b
θ)
1
RPR
r=1 Qtf(yit|xit,b
βir,b
θ)
b¯
βi=b
E[βi|yi,Xi,θq] = PQ
q=1 b
βqbwiq Qtf(yit|xit,b
βiq,b
θq)
PQ
q=1 bwiq Qtf(yit|xit,b
βiq,b
θq)
b¯
βi=b
E[βi|yi,Xi,θq] = PQ
q=1 bwiq 1
RPR
r=1 b
βiqr Qtf(yit|xit,b
βirq ,b
θq)
PQ
q=1 bwiq 1
RPR
r=1 Qtf(yit|xit,b
βiqr ,b
θq)
In order to construct the confidence interval for b¯
βi, we can derive an estimator of the condi-
tional variance from the point estimates as follows (Greene 2012, chap. 15):
b
Vi=b
Eβ2
i|yi,Xi,θ−b
E[βi|yi,Xi,θ]2.(7)
An approximate normal-based 95% confidence interval can be then constructed as b¯
βi±1.96×
b
V1/2
i. The gmnl package uses these formulae to compute the individual parameters along with
their 95% confidence interval. However, it is worth mentioning that there are two shortcomings
with the procedure describe above for computing the conditional variance of βi. First, the
estimator in Equation 7is an estimator of the variance of the conditional distribution of
¯
βi, and not an estimator of the sampling variance of the estimator of the expected value.
The estimated conditional variance will approach the estimated variance in the population
as the number of choice situations faced by each person increases without bound (Hensher,
Greene, and Rose 2006). Second, it does not take into account the sampling variability of the
parameter estimates.15
As an illustration, we can plot the kernel density of the individuals’ conditional mean for the
loc parameter using Elec.cor model by typing the following:
15Under the Bayesian framework, the estimation of the individual-level estimates fully accounts for uncer-
tainty in the population-level parameters in the estimation routine (see for example Daziano and Achtnicht
2014).

Mauricio Sarrias, Ricardo Daziano 41
R> plot(Elec.cor, par = "loc", effect = "ce", type = "density",
+ col = "grey")
Figure 1displays the distribution of the individuals’ conditional mean for the parameter of
loc. The gray area gives us the proportion of individuals with a positive conditional mean.
−202468
0.00 0.05 0.10 0.15
Conditional Distribution for loc
E(βi
^)
Density
Figure 1: Kernel density of the individuals’ conditional mean.
The 95% confidence interval of the conditional mean for the first 30 individuals is shown in
Figure 2, which was plotted using the following syntax:16
R> plot(Elec.cor, par = "loc", effect = "ce", ind = TRUE, id = 1:30)
Another important function in gmnl is effect.gmnl. This function allows the users to get the
individuals’ conditional mean of both the preference parameters and the willingness-to-pay
measures.
For example, one can get the individual conditional mean and standard errors plotted in
Figure 2by typing:
16gmnl uses plotrix package (Lemon 2006) to create the confidence interval graph.

42 gmnl Package in R
0 5 10 15 20 25 30
−2 0 2 4 6 8
95% Probability Intervals for loc
Individuals
E(βi
^)
Figure 2: 95% confident interval for the conditional means.
R> bi.loc <- effect.gmnl(Elec.cor, par = "loc", effect = "ce")
R> summary(bi.loc$mean)
Min. 1st Qu. Median Mean 3rd Qu. Max.
-0.79 0.42 2.04 2.12 3.46 7.13
R> summary(bi.loc$sd.est)
Min. 1st Qu. Median Mean 3rd Qu. Max.
0.113 0.564 0.795 0.866 1.130 1.860
The conditional mean of the willingness to pay for“loc” (wtp = βi,loc /βpf ) for all individuals
in the sample can be obtained using:
R> wtp.loc <- effect.gmnl(Elec.cor, par = "loc", effect = "wtp", wrt = "pf")
Note that the argument par is the variable whose parameter goes in the numerator, and the
argument wrt is a string indicating which parameter goes in the denominator.

Mauricio Sarrias, Ricardo Daziano 43
R> summary(wtp.loc$mean)
Min. 1st Qu. Median Mean 3rd Qu. Max.
-8.19 -3.98 -2.35 -2.44 -0.48 0.91
R> summary(wtp.loc$sd.est)
Min. 1st Qu. Median Mean 3rd Qu. Max.
0.130 0.648 0.914 0.996 1.300 2.130
4. Computational issues
There are some issues about computation and convergence of maximum likelihood worth
mentioning before concluding this paper. Regarding models estimated by maximum simulated
likelihood, there are at least four factors influencing the estimation of the parameters. First,
if the draws used in the estimation are pseudo-random draws, instead of Halton draws, then
the parameters might change if the seed is changed. Second, the number of draws used in
the simulations is very important in order to have a good approximation of the likelihood.
In this paper, we used just a few draws due to time restrictions. Nevertheless, in applied
work researchers must use a greater number of draws, especially if pseudo-random draws are
used. The ‘standard rule’ is to increase the number of draws in each run until the estimates
stabilize. For a comprehensive review of the impact of the number of draws see for example
Bhat (2001) and S´andor and Train (2004).
Another important factor affecting convergence and estimates is the starting values. It is
important to stress that the likelihood of the models reviewed in this article are complex and
are not globally concave. Thus, poor choice of initial values might lead to local maxima instead
of global maxima, or getting stuck in a flat region of the loglikelihood due to numeric overruns
in the Hessian. If this is the case, the user of gmnl will receive the following message: Warning
message: In sqrt(diag(vcov(object))): NaNs produced. We encourage users to try
different initial values using the argument start.
Finally, the algorithm used for optimization of the MSL is another important factor that
users should consider. gmnl uses the function maxLik to maximize the log-likelihood func-
tion, which implements the Newton-Raphson (NR), BGFS and Berndt-Hall-Hall-Hausman
(BHHH) procedures. As default, all models using simulation are estimated using the BFGS
algorithm. But if the estimation does not converge, users should try a different algorithm. As
a caution note, gmnl uses the numerical Hessian if the NR algorithm is used. Thus, it can be
very slow compared to the other methods. BHHH is generally faster, but it might fail if the
variables have very different scale. The larger the ratio between the largest standard deviation
and the smallest standard deviation of the variables, the more problems the user will have
with the estimation procedure. Therefore, users should check the variables and re-scale them
if necessary, and always look at the output message regarding convergence. It is good practice
to use the argument print.level = 2 to trace the optimization procedure in real time. For
more information about the arguments for optimization, type help(maxLik).
5. Conclusions

44 gmnl Package in R
The package gmnl implements the maximum likelihood estimator of random parameter logit
models with heterogeneity distributions that can be continuous, discrete, or discrete-continuous
mixtures. In this paper we have shown how gmnl can fit several extensions to the standard
multinomial logit model, including the recently derived mixed-mixed multinomial logit (MM-
MNL). To our knowledge there is no other widely available statistical package that has imple-
mented the maximum simulated likelihood estimator of MM-MNL, and we want to highlight
that gmnl makes use of analytical expressions of the gradient. gmnl is also the first imple-
mentation in Rof the estimator of the scale heterogeneity multinomial logit (S-MNL), the
generalized multinomial logit (G-MNL), and the latent class logit (LC). Whereas there are
other packages in Rfor the estimation of MIXL, gmnl allows for the inclusion of individual-
specific variables to explain the mean of the random parameters for a mixture of deterministic
taste variations and unobserved preference heterogeneity. In addition, gmnl also implements
Johnson Sbheterogeneity distributions.
Another key post-estimation functionality of gmnl that we have illustrated in this paper is
the derivation of conditional point and interval estimates of either the random parameters
or willingness-to-pay measures at the individual level. Random parameter models can be
used to make inference on the preference parameters of each individual in the sample, but
most packages that estimate MIXL models lack a command to produce individual-level esti-
mates. gmnl is able to compute individual parameters for all generalized logit models that
are implemented in the package, including G-MNL, MIXL, and LC.
Additional functionalities that we expect to incorporate in the future are the consideration of
different choice sets for each individual and the implementation of different methods for the
construction of confidence intervals of willingness-to-pay measures.
Acknowledgments
We would like to express our gratitude to the two anonymous referees and the editor whose
comments greatly improved this paper and the package. We are also very grateful to all the
users of this package who have helped us to improve it through their questions and suggestions.
References
Bhat CR (2001). “Quasi-Random Maximum Simulated Likelihood Estimation of the Mixed
Multinomial Logit Model.” Transportation Research Part B: Methodological,35(7), 677–
693. doi:http://dx.doi.org/10.1016/S0191-2615(00)00014-X.
Boxall PC, Adamowicz WL (2002). “Understanding Heterogeneous Preferences in Random
Utility Models: A Latent Class Approach.” Environmental and Resource Economics,23(4),
421–446. doi:10.1023/A:1021351721619.
Bujosa A, Riera A, Hicks RL (2010). “Combining Discrete and Continuous Representations
of Preference Heterogeneity: A Latent Class Approach.” Environmental and Resource
Economics,47(4), 477–493. doi:10.1007/s10640-010-9389-y.
Croissant Y (2012). “Estimation of Multinomial Logit Models in R: The mlogit Packages.”

Mauricio Sarrias, Ricardo Daziano 45
Rpackage version 0.2-2. URL http://cran.r-project.org/web/packages/mlogit/
vignettes/mlogit.pdf.
Daly A, Hess S, Train K (2011). “Assuring Finite Moments for Willingness to Pay in
Random Coefficient Models.” Transportation,39(1), 19–31. ISSN 1572-9435. doi:
10.1007/s11116-011-9331-3. URL http://dx.doi.org/10.1007/s11116-011-9331-3.
Daziano RA, Achtnicht M (2014). “Accounting for Uncertainty in Willingness to Pay for
Environmental Benefits.” Energy Economics,44, 166 – 177. ISSN 0140-9883. doi:http:
//dx.doi.org/10.1016/j.eneco.2014.03.023. URL http://www.sciencedirect.com/
science/article/pii/S014098831400070X.
Dotson J, Brazell J, Howell J, Lenk P, Otter T, MacEachern S, Allenby G (2015). “A Probit
Model with Structured Covariance for Similarity Effects and Source of Volume Calcula-
tions.” Available at SSRN: http://ssrn.com/abstract=1396232.
Dumont J, Keller J, Carpenter C (2014). RSGHB: Functions for Hierarchical Bayesian
Estimation: A Flexible Approach.Rpackage version 1.0.2, URL http://CRAN.R-project.
org/package=RSGHB.
Fiebig DG, Keane MP, Louviere J, Wasi N (2010). “The Generalized Multinomial Logit
Model: Accounting for Scale and Coefficient Heterogeneity.” Marketing Science,29(3),
393–421. doi:http://dx.doi.org/10.1287/mksc.1090.0508.
Gourieroux C, Monfort A (1997). Simulation-based Econometric Methods. Oxford University
Press.
Greene WH (2012). Econometric Analysis. 7th edition. Prentice Hall.
Greene WH, Hensher DA (2003). “A Latent Class Model for Discrete Choice Analysis: Con-
trasts With Mixed Logit.” Transportation Research Part B: Methodological,37(8), 681–698.
doi:10.1016/S0191-2615(02)00046-2.
Greene WH, Hensher DA (2010). “Does Scale Heterogeneity Across Individuals Matter? An
Empirical Assessment of Alternative Logit Models.” Transportation,37(3), 413–428. doi:
10.1007/s11116-010-9259-z.
Greene WH, Hensher DA (2013). “Revealing Additional Dimensions of Preference Hetero-
geneity in a Latent Class Mixed Multinomial Logit Model.” Applied Economics,45(14),
1897–1902. doi:10.1080/00036846.2011.650325.
Hajivassiliou VA, Ruud PA (1986). “Classical Estimation Methods for LDV Models Using
Simulation.” In RF Engle, D McFadden (eds.), Handbook of Econometrics, volume 4 of
Handbook of Econometrics, chapter 40, pp. 2383–2441. Elsevier.
Hasan A, Zhiyu W, Mahani AS (2015). mnlogit: Multinomial Logit Model.Rpackage version
1.2.1, URL http://CRAN.R-project.org/package=mnlogit.
Henningsen A, Toomet O (2011). “maxLik: A Package for Maximum Likelihood Estimation
in R.” Computational Statistics,26(3), 443–458. doi:10.1007/s00180-010-0217-1.

46 gmnl Package in R
Hensher DA, Greene WH (2003). “The Mixed Logit Model: The State of Practice.” Trans-
portation,30(2), 133–176. doi:10.1023/A:1022558715350.
Hensher DA, Greene WH, Rose JM (2006). “Deriving Willingness-to-pay Estimates of Travel-
time Savings from Individual-based Parameters.” Environment and Planning A,38(12),
2365. doi:10.1068/a37395.
Hess S, Ben-Akiva M, Gopinath D, Walker J (2011). “Advantages of Latent Class Over
Continuous Mixture of Logit Models.” Working paper, Institute for Transport Studies,
University of Leeds.
Hess S, Rose JM (2012). “Can Scale and Coefficient Heterogeneity be Separated in
Random Coefficients Models?” Transportation,39(6), 1225–1239. doi:10.1007/
s11116-012-9394-9.
Hess S, Stathopoulos A (2013). “Linking Response Quality to Survey Engagement: A Com-
bined Random Scale and Latent Variable Approach.” Journal of Choice Modelling,7, 1–12.
doi:10.1016/j.jocm.2013.03.005.
Imai K, Dyk DAV (2005). “MNP:RPackage for Fitting the Multinomial Probit Model.”
Journal of Statistical Software,14(3), 1–32. doi:10.18637/jss.v014.i03. URL http:
//www.jstatsoft.org/v14/i03.
Jackson CH (2011). “Multi-State Models for Panel Data: The msm Package for R.” Journal
of Statistical Software,38(8), 1–29. doi:10.18637/jss.v038.i08. URL http://www.
jstatsoft.org/v38/i08/.
Keane M, Wasi N (2013). “Comparing Alternative Models of Heterogeneity in Consumer
Choice Behavior.” Journal of Applied Econometrics,28(6), 1018–1045. doi:10.1002/jae.
2304.
Kleiber C, Zeileis A (2008). Applied Econometrics with R. Springer-Verlag, New York. URL
http://CRAN.R-project.org/package=AER.
Lee LF (1992). “On Efficiency of Methods of Simulated Moments and Maximum Simulated
Likelihood Estimation of Discrete Response Models.” Econometric Theory,8(04), 518–552.
Leisch F (2004). “FlexMix: A General Framework for Finite Mixture Models and Latent
Class Regression in R.” Journal of Statistical Software,11(8), 1–18. doi:10.18637/jss.
v011.i08. URL http://www.jstatsoft.org/v11/i08/.
Lemon J (2006). “Plotrix: A Package in the Red Light District of R.” R-News,6(4), 8–12.
Linzer DA, Lewis JB (2011). “poLCA: An RPackage for Polytomous Variable Latent Class
Analysis.” Journal of Statistical Software,42(10), 1–29. doi:10.18637/jss.v042.i10.
URL http://www.jstatsoft.org/v42/i10/.
McFadden D (1974). “Conditional Logit Analysis of Qualitative Choice Behavior.”In P Zarem-
bka (ed.), Frontiers in Econometrics, pp. 105–142. Academic Press, New York.
McFadden D, Train K (2000). “Mixed MNL Models for Discrete Response.” Journal of
Applied Econometrics,15(5), 447–470. doi:10.1002/1099-1255(200009/10)15:5<447::
AID-JAE570>3.0.CO;2-1.

Mauricio Sarrias, Ricardo Daziano 47
RCore Team (2015). R: A Language and Environment for Statistical Computing.RFounda-
tion for Statistical Computing, Vienna, Austria. URL http://www.R-project.org.
Rossi P (2012). bayesm: Bayesian Inference for Marketing/Micro-econometrics.Rpackage
version 2.2-5, URL http://CRAN.R-project.org/package=bayesm.
Rossi P, Allenby G, McCulloch R (2005). Bayesian Statistics and Marketing. Wiley, Hoboken,
NJ.
Ruud PA (2007). “Estimating Mixtures of Discrete Choice Model.” Technical report.
S´andor Z, Train K (2004). “Quasi-Random Simulation of Discrete Choice Models.” Trans-
portation Research Part B: Methodological,38(4), 313–327. doi:http://dx.doi.org/10.
1016/S0191-2615(03)00014-6.
Sarrias M (2015). Rchoice: Discrete Choice (Binary, Poisson and Ordered) Models with
Random Parameters.Rpackage version 0.3, URL http://CRAN.R-project.org/package=
Rchoice.
Sarrias M, Daziano R (2015). gmnl: Multinomial Logit Models with Random Parameters.R
package version 1.1, URL http://CRAN.R-project.org/package=gmnl.
Scarpa R, Thiene M (2005). “Destination Choice Models for Rock Climbing in the Northeast-
ern Alps: A Latent-class Approach Based on Intensity of Preferences.” Land Economics,
81(3), 426–444. doi:10.3368/le.81.3.426.
Scarpa R, Thiene M, Train K (2008). “Utility in Willingness to Pay Space: A Tool to
Address Confounding Random Scale Effects in Destination Choice to the Alps.” American
Journal of Agricultural Economics,90(4), 994–1010. doi:http://dx.doi.org/10.1111/
j.1467-8276.2008.01155.x.
Shen J (2009). “Latent Class Model or Mixed Logit Model? A Comparison by Transport Mode
Choice Data.” Applied Economics,41(22), 2915–2924. doi:10.1080/00036840801964633.
Sonnier G, Ainslie A, Otter T (2007). “Heterogeneity Distributions of Willingness-to-Pay in
Choice Models.” Quantitative Marketing and Economics,5(3), 313–331. doi:10.1007/
s11129-007-9024-6.
Train K (2009). Discrete Choice Methods with Simulation. 2nd edition. Cambridge University
Press.
Train K, Weeks M (2005). “Discrete Choice Models in Preference Space and Willingness-
to-Pay Space.” In R Scarpa, A Alberini (eds.), Applications of Simulation Methods in
Environmental and Resource Economics, volume 6 of The Economics of Non-Market Goods
and Resources, pp. 1–16. Springer Netherlands. ISBN 978-1-4020-3683-5. doi:10.1007/
1-4020-3684-1_1. URL http://dx.doi.org/10.1007/1-4020-3684-1_1.
Train KE (2008). “EM Algorithms for Nonparametric Estimation of Mixing Distributions.”
Journal of Choice Modelling,1(1), 40–69. doi:10.1016/S1755-5345(13)70022-8.
Trautmann H, Steuer D, Mersmann O, Bornkamp B (2014). truncnorm: Truncated Nor-
mal Distribution.Rpackage version 1.0-7, URL http://CRAN.R-project.org/package=
truncnorm.

48 gmnl Package in R
Venables WN, Ripley BD (2002). Modern Applied Statistics with S. Fourth edition. Springer-
Verlag, New York. ISBN 0-387-95457-0, URL http://www.stats.ox.ac.uk/pub/MASS4.
Wedel M, Kamakura WA (2012). Market Segmentation: Conceptual and Methodological Foun-
dations, volume 8. Springer Science & Business Media.
Yee TW (2010). “The VGAM Package for Categorical Data Analysis.” Journal of Statistical
Software,32(10), 1–34. doi:10.18637/jss.v032.i10.
Zeileis A, Croissant Y (2010). “Extended Model Formulas in R: Multiple Parts and Multiple
Responses.” Journal of Statistical Software,34(1), 1–13. URL http://www.jstatsoft.
org/v34/i01/.
Zeileis A, Hothorn T (2002). “Diagnostic Checking in Regression Relationships.” RNews,
2(3), 7–10. URL http://CRAN.R-project.org/doc/Rnews/.
Affiliation:
Mauricio Sarrias
Department of Economics
Universidad Cat´olica del Norte
0610 Avenida Angamos, Antofagasta, Chile
E-mail: msarrias86@gmail.com
URL: https://msarrias.weebly.com
Ricardo A. Daziano
School of Civil and Environmental Engineering
Cornell University
305 Hollister Hall, Ithaca, NY 14850, USA
E-mail : daziano@cornell.edu
URL: http://www.cee.cornell.edu/research/groups/daziano/