logitr Fast Estimation of Multinomial and Mixed Logit Models with Preference Space and Willingness to Pay Space Utility Parameterizations

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JSS Journal of Statistical Software

MMMMMM YYYY, Volume VV, Issue II. doi: 10.18637/jss.v000.i00

logitr: Fast Estimation of Multinomial and Mixed Logit

Models with Preference Space and Willingness to Pay

Space Utility Parameterizations

John Paul Helveston

George Washington University

Abstract

This paper introduces the logitr Rpackage for fast maximum likelihood estimation of multi-

nomial logit and mixed logit models with unobserved heterogeneity across individuals, which

is modeled by allowing parameters to vary randomly over individuals according to a chosen

distribution. The package is faster than other similar packages such as mlogit,gmnl,mixl,

and apollo, and it supports utility models speciﬁed with “preference space” or “willingness to

pay (WTP) space” parameterizations, allowing for the direct estimation of marginal WTP. The

typical procedure of computing WTP post-estimation using a preference space model can lead

to unreasonable distributions of WTP across the population in mixed logit models. The paper

provides a discussion of some of the implications of each utility parameterization for WTP esti-

mates. It also highlights some of the design features that enable logitr’s performant estimation

speed and includes a benchmarking exercise with similar packages. Finally, the paper highlights

additional features that are designed speciﬁcally for WTP space models, including a consistent

user interface for specifying models in either space and a parallelized multi-start optimization

loop, which is particularly useful for searching the solution space for diﬀerent local minima when

estimating models with non-convex log-likelihood functions.

Keywords: logit, utility, preference, willingness to pay, discrete choice models, R, maximum likeli-

hood estimation.

1. Introduction

Choice modeling is a well-established statistical method for assessing consumer preferences across

a wide variety of ﬁelds. One of the most common approaches for modeling choice is the maximum

likelihood estimation of multinomial logit models (McFadden 1974), which is rooted in the theory

of random utility models (Louviere, Hensher, and Swait 2000;Train 2009). The central assumption

of these models is that individual consumers make choices that maximize an underlying random

arXiv:2210.10875v1 [stat.ME] 19 Oct 2022

2logitr: Preference and WTP Space Logit Models in R

utility model, which can be parameterized as a function of a product’s observed attributes and

a random variable representing the portion of utility unobservable to the modeler. These models

produce estimates of the marginal utility for changes in each attribute relative to one another.

In many applications, modelers are interested in estimating marginal “willingness to pay” (WTP)

for changes in product attributes. The typical procedure to obtain these estimates is to divide the

estimated parameters of a “preference space” utility model by the negative of the price parameter.

Despite this common practice, it can yield unreasonable distributions of WTP across the population

in heterogeneous random parameter (or “mixed logit”) models (Train and Weeks 2005;Sonnier,

Ainslie, and Otter 2007;Helveston, Feit, and Michalek 2018). For example, if the parameters for

the price attribute and another non-price attribute are both assumed to be normally distributed

across the population, then the resulting WTP estimate follows a Cauchy distribution, implying

that WTP has an inﬁnite variance across the population.

An alternative approach is to re-parameterize the utility model into the “WTP space” prior to

estimation. Estimating a WTP space model allows the modeler to directly specify assumptions

of how WTP is distributed, which has been found to yield more reasonable estimates of WTP

(Train and Weeks 2005;Sonnier et al. 2007;Daly, Hess, and Train 2012). WTP space models have

also been found to be more consistent with respondent’s true underlying preferences (Crastesa,

Beaumaisb, Mahieud, Martinez-Camblore, and Scarpa 2014). Finally, since WTP estimates are

independent of error scaling, they can be conveniently compared across diﬀerent models estimated

on diﬀerent data.

Several statistical packages support the estimation of multinomial and mixed logit models with

WTP space utility parameterizations. One of the most common approaches involves an adaptation

of the generalized multinomial logit (GMNL) model (Fiebig, Keane, Louviere, and Wasi 2010) to ﬁt

WTP space models via an implementation of the scaled multinomial logit (SMNL) model, though

this requires that the price parameter estimate and standard error be calculated post-estimation.

Estimation of WTP space models via GMNL has been implemented in R with the gmnl package

(Sarrias, Daziano et al. 2017) and in STATA with the gmnl package (Gu, Hole, and Knox 2013).

WTP space models can also be estimated using the apollo (Hess and Palma 2019) and mixl (Molloy,

Becker, Schmid, and Axhausen 2021) R packages as they allow the user to hand-specify any valid

utility model. Finally, Professor Arne Rise Hole developed two STATA packages that share a

common syntax for estimating mixed logit models in the preference space (mixlogit) and WTP

space (mixlogitwtp) (Hole 2007). Many other packages exist for estimating a wider variety of logit

models, but they are limited to preference space models. Of these, package mlogit (Croissant 2020)

is perhaps the most complete and widely used for estimating multinomial logit and mixed logit

models in R via maximum likelihood estimation.

The logitr package is designed speciﬁcally to support the estimation of multinomial logit and mixed

logit models models with either preference space or WTP space utility parameterizations. While

logitr is less general in scope compared to more ﬂexible packages like mixl and apollo, it oﬀers

other functionality that is particularly useful for estimating WTP space models and conveniently

switching between preference and WTP space models. For example, given their non-linear utility

speciﬁcation, WTP space models often diverge during estimation and can be sensitive to starting

parameters. To address this, the package includes a parallelized multi-start optimization loop

to search for diﬀerent local minima from diﬀerent random starting points when minimizing the

negative log-likelihood. The user interface is also more streamlined and simpliﬁed for estimating

models in either space.

Package logitr is also computationally eﬃcient and faster than other similar packages, including the

Journal of Statistical Software 3

mixl package which uses high performance C++ code to compile the log-likelihood function (Molloy

et al. 2021) (though mixl can be accelerated considerably via multi-core processing). The perfor-

mance gains are the result of a combination of design features, including how the choice probabilities

are speciﬁed, avoiding redundant computation by pre-computing constant intermediate variables,

and the use of analytic gradients that are optimized for eﬃciency.

The rest of the article is organized as follows: Section 2 provides an overview of the models sup-

ported by logitr, including multinomial and mixed logit models with preference space and WTP

space utility parameterizations. Section 3 discusses several important implications of preference

versus WTP space utility parameterizations on WTP estimates. Section 4 describes the software

architecture and performance. Section 5 then introduces the logitr package, including examples

of estimating multinomial and mixed logit models in both preference and WTP spaces as well as

additional functionality for estimating weighted models and making predictions. Section 6 explains

some limitations of WTP space models. Finally, Section 7 concludes the paper.

2. Models

2.1. The random utility model in two spaces

Random utility models assume that consumers choose the alternative jfrom a set of alternatives

that has the greatest utility uj. Utility is a random variable that is modeled as uj=vj+εj, where

vjis the “observed utility” (a function of the observed attributes such that vj=f(xj)) and εjis a

random variable representing the portion of utility unobservable to the modeler.

Adopting the same notation as in Helveston et al. (2018), consider the following utility model:

u∗

j=β∗>xj+α∗pj+ε∗

j, ε∗

j∼Gumbel 0, σ2π2

6!(1)

where β∗is the vector of coeﬃcients for non-price attributes xj,α∗is the coeﬃcient for price pj,

and the error term, ε∗

j, is an IID random variable with a Gumbel extreme value distribution of

mean zero and variance σ2(π2/6).

This model is not identiﬁed since there exists an inﬁnite set of combinations of values for β∗,α∗,

and σthat will produce the same choice probabilities. In order to specify an identiﬁable model,

Equation 1must be normalized. One approach is to normalize the scale of the error term by

dividing Equation 1by σ, producing the “preference space” utility speciﬁcation (Train and Weeks

2005):

u∗

σ!=β∗

σ>

xj+α∗

σpj+ ε∗

σ!, ε∗

σ!∼Gumbel 0,π2

6!(2)

The typical preference space parameterization of the multinomial logit model can then be written

by rewriting Equation 2with uj= (u∗

j/σ),β= (β∗/σ),α= (α∗/σ), and εj= (ε∗

j/σ):

uj=β>xj+αpj+εjεj∼Gumbel 0,π2

6!(3)

The vector βin Equation 3represents the marginal utility for changes in each non-price attribute

(relative to the standardized scale of the error term), and αrepresents the marginal utility obtained

4logitr: Preference and WTP Space Logit Models in R

from changes in price (relative to the standardized scale of the error term). The coeﬃcients βand

αare only relative values rather than absolute and do not have units. Using this model, estimates

of the marginal WTP for changes in each non-price attribute could be computed by dividing ˆ

βby

−ˆα, where the “hat” symbol indicates a parameter estimate.

An alternative approach to normalizing Equation 1is to divide by −α∗instead of σ, resulting in

the “WTP space” utility parameterization:

u∗

−α∗!=β∗

−α∗>

xj+α∗

−α∗pj+ ε∗

−α∗!, ε∗

−α∗!∼Gumbel 0,σ2

(−α∗)2

π2

6!(4)

Since the error term in Equation 4is scaled by λ2=σ2/(−α∗)2, it can be rewritten by multiplying

both sides by λ= (−α∗/σ) and renaming uj= (λu∗

j/−α∗),ω= (β∗/−α∗), and εj= (λε∗

j/−α∗):

uj=λω>xj−pj+εjεj∼Gumbel 0,π2

6!(5)

The vector ωin Equation 5represents the marginal WTP for changes in each non-price attribute,

and λrepresents the scale of the deterministic portion of utility relative to the standardized scale

of the random error term (also called the “scale parameter”). In contrast to the βcoeﬃcients

from the preference space model in Equation 3, the ωcoeﬃcients have absolute value with units

of currency.

The logitr package can ﬁt logit models with either utility parameterization, and it contains functions

that facilitate the comparison of WTP estimates between models from the two model spaces.

2.2. Multinomial and mixed logit probabilities

By assuming that the error term in Equations 3and 5follows a Gumbel extreme value distribution,

the probability that a consumer will choose alternative jin choice situation nfollows a convenient,

closed form expression, cf. Train (2009):

Pnj =exp (vnj )

kexp (vnk),(6)

where vnj is the deterministic portion of the utility model and Jis the number of alternatives

in choice situation n. The multinomial logit model assumes homogeneous preferences across the

population and possess the independence of irrelevant alternatives (IIA) property, which means that

the ratio of any two probabilities is independent of the functions determining any other outcome

since

Pnj

Pnk

=exp (vnj )

exp (vnk),(7)

To relax this assumption and allow for heterogeneity of preferences across the population, the

multinomial logit model can be extended to the random coeﬃcients “mixed” logit model (McFadden

and Train 2000) where probabilities are the integrals of standard logit probabilities over a density

of parameters across people:

Pnj =Z exp (vnj )

kexp (vnk)!f(θ)dθ,(8)

Journal of Statistical Software 5

where f(θ)is a density function and θcontains the parameters in the deterministic portion of the

utility model, which are βand αfor preference space models (Equation 3) and ωand λfor WTP

space models (Equation 5). The mixed logit probability can be interpreted as a weighted average

of the multinomial logit probability with weights given by the density f(θ).

Modelers often specify diﬀerent mixing distributions for parameters in θdepending on assumptions

of how preferences might be distributed across the population. For example, modelers may assume

αfollows a log-normal or zero-censored normal distribution to force the price coeﬃcient to remain

positive—an assumption based on the logic that most people prefer price decreases rather than

increases. Likewise, parameters in βare often assumed to follow a normal distribution if it is

unclear whether the utility parameters for attributes xjshould be positive or negative.

2.3. Maximum likelihood estimation

Parameters in the preference or WTP space utility models can be estimated by maximizing the

log-likelihood function. For the multinomial logit model, the log-likelihood is given by:

ynj ln Pnj (9)

where ynj = 1 if alternative jis chosen in situation nand 0otherwise, Nis the number of choice

situations, Jis the number of alternatives in choice situation n, and the probabilities Pnj are given

by Equation 6.

For mixed logit models, the log-likelihood can be estimated using simulation to obtain estimates

of Pnj in Equation 8(Train 2009). Over a series of iterations, parameters are drawn from f(θ)

and used to compute the logit probability in Equation 6. The average probabilities over all of

the iterations, ˆ

Pnj , are then used in place of Pnj in Equation 9to compute the simulated log-

likelihood. Should the data contain a panel structure where multiple observations come from the

same individual, the product of the logit probabilities in Equation 6over all trials for each individual

must ﬁrst computed and then averaged over the draws of each parameter drawn from f(θ)(Train

2009).

McFadden (1974) shows that the log-likelihood function is globally concave for linear-in-parameters

utility models with ﬁxed parameters. This implies that optimization algorithms should always arrive

at a global solution when minimizing the negative log-likelihood for preference space models with

ﬁxed parameters. In contrast, WTP space utility models (as well as mixed logit models with either

utility parameterization) have non-convex log-likelihood functions and thus are not guaranteed

to arrive at a global solution. For these models, diﬀerent optimization strategies should be used

to minimize the negative log-likelihood, such as using a multi-start loop where the optimization

algorithm is run multiple times from diﬀerent random starting points to search for multiple local

minima.

3. Implications of preference versus WTP space utility parameterizations

WTP estimates can be obtained from both preference and WTP space utility parameterizations. In

the preference space utility model given by Equation 3, WTP is estimated as ˆ

β/−ˆα; in the WTP

space model given by Equation 5, WTP is simply ˆ

ω. The choice of which approach to use can

have important implications for estimates of WTP, and modelers should consider which outcomes

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JSSJournalofStatisticalSoftwareMMMMMMYYYY,VolumeVV,IssueII.doi:10.18637/jss.v000.i00logitr:FastEstimationofMultinomialandMixedLogitModelswithPreferenceSpaceandWillingnesstoPaySpaceUtilityParameterizationsJohnPaulHelvestonGeorgeWashingtonUniversityAbstractThispaperintroducesthelogitrRpackageforfastma...

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