# Random Utility Models with Skewed Random Components:
the Smallest versus Largest Extreme Value Distribution

Richard T. Carson, Derrick H. Sun, and Yixiao Sun Department of Economics, UC
San Diego. Email<EMAIL_ADDRESS>of Mathematics and College of
Computing, Data Science, and Society, UC Berkeley. Email:
<EMAIL_ADDRESS>of Economics, UC San Diego. Email<EMAIL_ADDRESS>
###### Abstract

At the core of most random utility models (RUMs) involving multinomial choice
behavior is an individual agent with a random utility component following a
_largest_ extreme value Type I (LEVI) distribution. What if, instead, the
random component follows its mirror image — the _smallest_ extreme value Type
I (SEVI) distribution? Differences between these specifications, closely tied
to the random component’s skewness, can be quite profound. For the same
preference parameters, the two RUMs, equivalent with only two choice
alternatives, diverge progressively as the number of alternatives increases,
resulting in substantially different estimates and predictions for key
measures, such as elasticities and market shares.

The LEVI model imposes the well-known independence-of-irrelevant-alternatives
property, while SEVI does not. Instead, the SEVI choice probability for a
particular option involves enumerating all subsets that contain this option.
The SEVI model, though more complex to estimate, is shown to have
computationally tractable closed-form choice probabilities. Much of the paper
delves into explicating the properties of the SEVI model and exploring
implications of the random component’s skewness, including offering new
insights into multinomial probit models. SEVI-based counterparts exist for
most LEVI-based generalizations of the conditional logit model such as mixed
logit, and the LEVI versus SEVI issue is largely orthogonal to the issues
these models are intended to address.

Conceptually, the difference between the LEVI and SEVI models centers on
whether information, known only to the agent, is more likely to increase or
decrease the systematic utility parameterized using observed attributes. LEVI
does the former; SEVI the latter. An immediate implication is that if choice
is characterized by SEVI random components, then the observed choice is more
likely to correspond to the systematic-utility-maximizing choice than if
characterized by LEVI. Examining standard empirical examples from different
applied areas, we find that the SEVI model outperforms the LEVI model,
suggesting the relevance of its inclusion in applied researchers’ toolkits.

Keywords: Conditional Logit, Gumbel Distribution, Independence of Irrelevant
Alternatives, Multinomial Logit, Reverse Gumbel Distribution, Largest Extreme
Value Type I Distribution, Smallest Extreme Value Type I Distribution.

JEL codes: C10, C18, C25

##  1 Introduction

Random utility models (RUM) are a cornerstone of applied microeconomics and
other fields such as marketing, health policy, and transportation research.
Since the seminal work of McFadden, (1973), the vast majority of RUMs have
been formulated as conditional or multinomial logit models, or their
generalizations (Hensher et al., (2015) and Train, (2009)).111Conditional
logit typically refers to choices between alternatives with different
observable attributes. In contrast, multinomial logit focuses on choices where
the decision-makers themselves differ in characteristics. These two models can
be combined by interacting alternative attributes with agent characteristics.
Here, we primarily use the conditional logit representation for simplicity.
These models posit that the total utility of an alternative comprises two
components: a systematic component parameterized by observable attributes, and
an idiosyncratic component known only to the agent (Manski, (1977)). The
latter is often referred to as the random component, since it is unknown to
the econometrician, who generally assumes it follows a specific distribution.
The agent chooses the alternative with the highest total utility, making
choice behavior appear random from the econometrician’s perspective, even
though it may be regarded as deterministic from the agent’s viewpoint.

The conditional logit model and its standard generalizations assume that the
random component follows the Largest Extreme Value Type I (LEVI) distribution,
also known as the Gumbel or double exponential distribution (Gumbel, (1958)).
This distribution is continuous, has a single mode, is asymmetric and skewed
with a long _right_ tail. Mirroring the LEVI distribution is the Smallest
Extreme Value Type I (SEVI) distribution, also known as the Reverse Gumbel. It
is also asymmetric but with a long _left_ tail. This paper presents a
systematic study of RUMs based on the SEVI distribution, highlighting their
differences from conventional logit models and practical implications.

The natural question at this point for a choice modeler is: does the choice
between LEVI and SEVI random components matter empirically? Two important bits
of information suggest it should not. First, the difference between two
(standard) LEVI random variables or two (standard) SEVI random variables is
the same (standard) logistic distribution, and thus, RUMs based on LEVI and
SEVI random components are equivalent when there are only two alternatives.
Second, there is a long standing belief (Hausman and Wise, (1977); Horowitz,
(1982)) that a multinomial probit (MNP) model produces essentially the same
results as the conditional/multinomial logit model.222As succinctly summarized
in the popular graduate econometric text by Greene, (2018): “An MNP model that
replaces a normal distribution with $\Sigma=I$ [zero covariance between the
random components] will yield virtually identical results (probabilities and
elasticities) compared to the multinomial logit model.” This belief has been
taken to suggest that it is the iid assumption imposed on the random
components that drives the well-known independence of irrelevant alternatives
(IIA) substitution pattern. However, insights from the case with only two
alternatives do not extend to the case with more than two alternatives. In
this paper, we show that as the number of alternatives increases, the choice
probabilities from a LEVI-based conditional logit and a SEVI-based conditional
logit model increasingly diverge from each other. We also show that the IIA
property does not follow from the iid assumption. Even though the random
components are still iid, a SEVI-based conditional logit model with more than
two alternatives exhibits a distinctly non-IIA substitution pattern.

Let’s consider a simple stylized example. An agent (individual $i$) enters a
store with the intention of purchasing a pair of running shoes, and there are
five different alternatives $(j\in\\{1,2,3,4,5\\}$ and $J=5)$ to choose from.
Before trying them on, the individual rates each of the five pairs based on
observable features like brand and price. These ratings capture the systematic
components, $\left\\{V_{j}\right\\}$, of a random utility model. For
illustrative purposes, assume that the $V_{j}$’s are known to be
$\\{0.25,0.50,0.75,1.50,$ and $2.00\\},$ so there is no need to specify the
shoe attributes and the associated preference parameters.333The vector of
systematic utilities can vary among different individuals. Here, for
simplicity, we assume that $V_{j}$’s are the same across all individuals.
Next, the individual tries on each of the five pairs of shoes, making an
idiosyncratic judgement on comfort and fit. This provides the random
component, $\varepsilon_{ij}$, to add to the systematic component, $V_{j}$, to
get the total utility level, $U_{ij}=V_{j}+\varepsilon_{ij},$ for each pair of
shoes. The chosen pair for purchase is the one with the highest $U_{ij}$.

Now let the store’s customers comprise of a large number of individuals
($i=1,\ldots,n$), with identical and known systematic components
$\left\\{V_{j}\right\\}_{j=1}^{5}$, but different random components
$\varepsilon_{ij}.$ They try on the shoes, therefore observing their
individual random components, then make their decision on which of the five
pairs of shoes to purchase. The question we posed can now be framed solely in
terms of: what are the choice probabilities cast in percentage terms (or
equivalently, market shares) for each of the five pairs of shoes, if the
$\varepsilon_{ij}$ are drawn from the following distributions: (a) the
standard LEVI distribution, (b) the standard SEVI distribution, (c)
N$(0,\pi^{2}/6)$ (hereafter referred to as NORM, a normal distribution
normalized to have the same variance as the standard LEVI and SEVI
distributions)?

The answer to this question is displayed in Figure 1. The results are quite
striking, particularly for the alternatives with the smallest and largest
systematic utilities. With a LEVI error component, the alternative with the
smallest systematic utility has a market share 137% higher than its SEVI
counterpart and 25% higher than its independent normal counterpart. This
pattern continues for the alternatives with the second smallest and third
smallest systematic utilities, with LEVI giving rise to market shares 73% and
37% higher than SEVI, respectively. However, this pattern of a higher LEVI
market share reverses around a 20% market share (i.e., $1/J$ or equal share
for all alternatives), where the SEVI distribution leads to a higher market
share than the LEVI distribution. For the alternative with the second-largest
systematic utility, the SEVI market share is 11% higher than its LEVI
counterpart and 5% higher than that under the NORM distribution. The pattern
of a higher market share under SEVI intensifies for the alternative with the
largest systematic utility. Here, SEVI leads to a market share 21% higher than
LEVI and 16% higher than NORM.

Figure 1: Market shares for all five alternatives in an RUM with identical
systematic utilities but different distributions of the random utility
components

The magnitude of all of these differences is likely to be important in
empirical applications. For instance, if Alternative 1 were a new product, use
of a LEVI-based prediction when the true error component was SEVI would have
led to an overestimation of demand by more than double (7.6% vs. 3.2% market
shares). The potential for the random component to be SEVI rather than LEVI
provides one possible explanation for why most new products perform much worse
than predicted by marketing researchers. On the other hand, a LEVI-based
prediction for the market leader, when the true error component is SEVI, could
substantially underestimate its market share (43.7% vs. 52.7%). It is easy to
see how either of these faulty predictions could be disastrous from both
production and profit perspectives. Antitrust implications are also obvious.

Why do these differences in choice probabilities arise, despite these three
simple models sharing the same (known) systematic components and random
components from iid draws from simple, single-peaked distributions with the
same finite variance? The answer lies in the skewness of the random
components. From the econometrician’s viewpoint, the fundamental question in
our stylized example is: for a randomly chosen agent, is the overall utility
from a randomly chosen pair of shoes (a) more likely to move up, (b) stay the
same, or (c) move down? Surprisingly, this fundamental property of the random
component of a RUM seems to have never been raised in the voluminous
literature on multinomial choice models and serves as the driving force behind
the present paper.

If the total utility $U_{ij}$ of an arbitrarily chosen agent $i$ for an
arbitrarily chosen alternative $j$ is more likely to move upward relative to
the systematic utility $V_{ij}$ after receiving the private signal
$\varepsilon_{ij}$ representing the random component, then the random
component $\varepsilon_{ij}$ is right-skewed. The classic exemplar in this
scenario is the LEVI distribution, and the canonical model is the conditional
logit. If the total utility for this agent is just as likely to move up as
move down, then the random component is symmetric. The normal distribution is
the exemplar, and the multinomial probit is the canonical model. If this
agent’s total utility is more likely to move down, then the random component
is left-skewed. There is not a similar exemplar distribution or canonical
model for this left-skewed case. We propose employing the SEVI distribution
and its corresponding model for the left-skewed scenario, serving as the
natural counterpart to the standard LEVI formulation.

Once the potential importance of the skewness of the random component is
acknowledged, an entirely new line of inquiry is opened up. Are there
conditions (e.g., in-store vs. online purchases) or deep personality traits
(e.g., risk averse vs. risk loving) that influence the random component’s
skewness? We don’t provide answers to this question but hope that we provide
tools that can help researchers explore these issues.

Fitting an RUM based on LEVI random components to data generated with SEVI
random components results in substantially biased estimates of the preference
parameters and vice versa. For the same preference parameters in the RUM’s
systematic components, SEVI-based random components result in substantively
different (compared to LEVI-based random components) estimates for quantities
such as elasticities and average partial derivatives routinely relied upon in
decision making.444We identify one specific scenario: a ratio estimator, such
as those used in many willingness-to-pay calculations, where the LEVI-based
estimator remains consistent even if the true error component is SEVI,
provided that an important set of restrictions on the attribute matrix is met.
Standard generalizations of the multinomial/conditional logit do not address
issues that arise when a LEVI random component is assumed, incorrectly, when
SEVI is appropriate. Instead, SEVI-based analogues of those models exist to
deal with the specific deviations from the standard multinomial/conditional
logit that those models are intended to address.

While we make no claim that the SEVI assumption is always or even generally
preferable to that of LEVI, the empirical relevance of the SEVI model is
easily demonstrated. When examining a set of six datasets taken from standard
econometric textbooks and well-known papers, we find that the SEVI
distributional assumption is consistently preferred over LEVI and is favored
over the normal distribution in five out of the six datasets. For any
particular application, the skewness of the random component is obviously an
empirical question, and a priori, nothing rules out mixtures of random
component skewness types, an issue we examine near the end of this paper (see
Section 6.1).

The bulk of the paper is devoted to studying the theoretical properties of a
SEVI model and delving into aspects of the estimation, inference, and
prediction within the SEVI framework. Here, we offer a preview of some
intriguing properties of the SEVI model.

First, both the choice probabilities and the surplus function in a SEVI model
have intuitive and closed-form representations, allowing us to explore new
avenues of discrete choice analysis. Using the strict convexity of the social
surplus function, we show that a SEVI model is identified under the same
conditions that ensure the identification of a LEVI model.

Second, the structure of the choice probability for any given alternative in a
SEVI model is considerably richer than in the LEVI model, where its dependence
on other alternatives is solely through the sum of the exponentiated
systematic components for each alternative. In the SEVI model, the choice
probability for a given alternative can be seen as an aggregation involving
all choice subsets in which the alternative occurs and the overall choice set.
This dependence of the choice probability on all subsets of alternatives is a
distinctive feature of the SEVI model that differentiates it from the LEVI
model. As a consequence of such dependence, IIA does not hold in the SEVI
model.

Third, on its surface, the all-subsets representation of SEVI might seem
computationally challenging when the number of alternatives $J$ is of any
appreciable size. We overcome this issue by employing Gosper’s hack (Knuth,
(2011)) from the computer science literature to efficiently enumerate all
possible subsets containing a target alternative. For a moderately large $J$
$\left(J\leq 15\right)$ with typical sample sizes and attribute numbers in
economic analysis, the computation time for a SEVI model is somewhat higher
than its standard logit LEVI counterpart but much lower than that of the MNP
with iid normal errors. In the case of $J=12,$ the computation time for a SEVI
model can be less than 2% of that of the corresponding independent MNP
model.555It is an open issue how SEVI’s enumeration of subsets of alternatives
interacts with the computational intensity inherent in the generalizations of
the conditional logit model, where long run times are often the norm.

Fourth, we explore the regions where the SEVI model is likely to produce
results similar to the LEVI model and where the results are most likely to
diverge. We develop a Vuong-type test, showing its effectiveness in
distinguishing between the SEVI and LEVI models (Vuong, (1989)). We also show
that AIC/BIC can be employed to help guide appropriate model choice.

Fifth, we explore the implications of the skewness of the random component as
the number of alternatives increases, going beyond the case with only two
alternatives. One immediate implication is that in an RUM with SEVI random
components, the observed choice and the choice that maximizes the systematic
utility are more likely to correspond than in the case of LEVI. This is
because SEVI error components have less of an impact on determining the
selected alternative compared to LEVI ones. In contrast, SEVI random
components play a larger role than their LEVI counterparts in determining what
is not chosen.666One way to conceptualize the role of the random components,
consistent with this observation, is that an agent with SEVI random components
seeks idiosyncratic reasons to reject alternatives, whereas under LEVI, the
agent aims to find a high-quality idiosyncratic match component to accept an
alternative.

The rest of the paper is organized as follows. Section 2 provides a literature
overview, largely from a historical perspective, that hints at why the
importance of the skewness of the random component was overlooked. Section 3
studies the structure of the choice probabilities in a SEVI model, contrasting
its properties with those of the conventional LEVI model. This section shows
that while Luce’s choice axiom holds for the SEVI model when $J=2$, this
restriction does not hold when $J>2$. This section also provides a closed-form
expression for the surplus function and shows how it can be used to compute
the compensating variation and establish the identification of the SEVI model.
Section 4 considers estimating the SEVI model by MLE and QMLE, studies the
consistency of the QMLE up to a scale normalization, and describes the use of
AIC/BIC and the Vuong test for selecting between the SEVI and LEVI models.
Section 5 presents a variety of simulation results, exploring different
aspects of the SEVI model. Section 6 introduces a mixed model that
accommodates individuals of either LEVI or SEVI type in a population. It also
contains a straightforward extension to discrete choice problems where agents
aim to minimize, rather than maximize, an objective function, such as cost and
regret. Section 7 contains a set of empirical applications drawn from
textbooks and well-known papers, allowing us to compare the goodness of fit
using LEVI, SEVI, and independent MNP with the same error variance. The final
section offers concluding remarks and outlines some future research directions
based on generalizing the SEVI model. Proofs are provided in the appendix, and
an online supplementary appendix contains additional materials and figures.

##  2 Literature Review: a Historical Perspective

Early empirical research on individual-level choices was hindered by the lack
of data and appropriate statistical models. Operationally, the study of choice
requires a conceptualization involving a systematic component and a random
component. The psychologist Thurstone, (1927) is generally credited with
pioneering this concept and proposing a variant of the now-familiar binary
probit model. The binary logit model soon emerged from biology. Statistical
advancements related to the random component, whether logistic or normal,
largely took place in the realm of biometrics (Cox, (1969)).

The transition from the binary choice case to the multinomial choice proved
not to be easy. While significant advancements occurred within biometrics
(e.g., Cox, (1969)), the pendulum swung back to psychology and, in particular,
to the seminal work of Luce, (1959). This effort quickly spilled over into
economics (e.g., Block and Marschak, (1960)). Formal requirements for choice
models to meet the rationality requirements of utility maximization were
explored, and patterns of choice behavior, such as the IIA property in the
multinomial logit model, were examined. The multinomial logit model and its
statistically equivalent flipped version, known as the conditional logit
model, came together in its current form with the tour-de-force synthesis of
McFadden, (1973). The work of McFadden and those following in his footsteps
spawned a truly massive empirical enterprise in economics and other social
sciences (see McFadden, (2001)). This involves the construction and
application of various random utility models, almost all based on the LEVI
kernel of the conditional logit.

There are many reasons why the LEVI distribution was originally chosen for
RUMs and has become popular in applied research.

First, utility is inherently unobservable and hence is typically represented
as a latent variable. The assumption of continuity makes economic theory and
applied work much more tractable. This ruled out discrete distributions.

Second, in the case of a binary response, there had been a long-running debate
over tacking on an error term, first from a normal and later from a logistic
distribution, in bioassay experiments. In these experiments, the error
component was perceived as coming from an unobserved tolerance distribution.
When response data were plotted, bell-shaped tolerance distributions seemed
more appropriate than either a uniform or bimodal distribution. Eventually,
the binary logit model prevailed over its probit counterpart because of its
computational advantages and the early literature’s observation of largely
indistinguishable empirical results across the two models with sample sizes
available at the time (Chambers and Cox, (1967)). This outcome also influenced
the adoption of the binary logit model in the social sciences; see Chapter 9
of Cramer, (2003) for a historical account.

Third, there was a long-standing belief that the multinomial logit and the
multinomial probit with iid normal unobserved utilities were very close
approximations to each other.777Hausman and Wise, (1977) state: “We might
expect the independent probit and the logit models to have similar properties
[including inducing the IIA red bus/blue bus property via the iid random
component assumption] and, in fact, they lead to almost identical empirical
results.” Horowitz, (1982) proposes a Lagrange multiplier test of the
unrestricted MNP model, assuming that the MNL estimates adequately approximate
the restricted MNP results. The belief may have led to the misconception that
there is no need to go beyond the multinomial logit if the random utility
components are iid. However, this belief turns out not to be generally true.
The misconception likely stems from limitations in computational techniques
and resources, restricting early literature to examine only cases with a small
number of alternatives. When the number of alternatives, $J,$ is small,
particularly when $J=2$ and $J=3$, the estimates from an independent
multinomial probit with a common variance normalization tend to be very close
to those from a LEVI multinomial logit. Nevertheless, as shown in Figure 1,
the multinomial logit and probit models can exhibit significant differences
for even a moderate $J,$ such as $J=5.$

These, however, are not reasons for not exploring the implications of other
distributions, as Manski, (1977) had urged. This is particularly relevant in
light of the increased computational power and insights from behavioral
economics, which suggest frequent violations of the IIA property.

The “non-IIA” property of a SEVI model bears some resemblance to that of the
mother/universal logit (McFadden et al., (1978)) and the random regret
minimization model (e.g., Mai et al., (2017)). However, the mechanisms of
removing the IIA property are fundamentally different. By definition, a SEVI
model considered here is a utility maximization model, and hence the resulting
choice probability function can be rationalized by a probability distribution
over linear preference orderings (Block and Marschak, (1960)). The absence of
the IIA property is due solely to the distribution of the error component. In
contrast, most implementations of the mother/universal logit and the random
regret minimization models are inconsistent with random utility maximization,
and the lack of the IIA property in these models is a result of the
specification that the difference in the systematic (not random) component of
utility between any two alternatives depends on the presence or absence of
another alternative.

A significant portion of the history of this field revolves around building
models utilizing the LEVI kernel to relax the IIA constraint. This is achieved
by permitting agents to have different preferences or scale parameters or by
allowing subsets of alternatives to be correlated in some fashion, while still
maintaining computational feasibility (Hensher et al., (2015); Train, (2009);
Greene, (2018)). Among these models, random-parameter (mixed) logit, latent-
class multinomial logit, scale-heterogeneity logit, generalized multinomial
logit, and nested logit stand out as the most prominent examples. All these
more general specifications can be constructed using a SEVI rather than a LEVI
random component. Conceptually, specifying the random component’s skewness
should precede relaxing restrictive assumptions of the conditional logit
model, such as the independence of random components and the homogeneity of
preferences across individuals.

Our discovery of the LEVI vs. SEVI distinction came about through an
inadvertent sign flip while generating draws from a standard LEVI
distribution. In doing so, a SEVI random component was generated instead. This
would likely have gone unnoticed if our objective had not been to study
whether the behavior of agents facing choice sets of varying sizes differed.
In a baseline simulation, we observed sizeable differences in the conditional
logit model estimates of preference parameters and other statistics, as the
number of alternatives increased. These unexpected results led us to realize
that we had fit a LEVI-based logit model to data generated by a SEVI logit. A
search of general and specialized econometrics texts revealed no discussion of
LEVI vs. SEVI or the broader issue of the role of the skewness of the random
component in choice behavior.

We eventually found one short and largely ignored paper in the literature (Hu,
(2005)) which points out that the likelihood functions of SEVI and LEVI-based
choice models are different. However, that paper does not delve into any of
the other results presented here. It asserts that the SEVI model does not
appear to have a closed-form solution and hence has to be estimated using a
simulation approach. In Theorem 1, we show this not to be the case. SEVI has a
mathematically elegant and intuitive closed-form solution that facilitates
direct comparisons between the properties of LEVI vs. SEVI choice models. In
an empirical marketing application with three alternatives involving types of
bread, Hu, (2005) shows that the SEVI model fits marginally better than the
LEVI model, but the difference is smaller than the variation due to different
smoothing parameter values used in the simulation approach. To our knowledge,
the SEVI model has not been used in subsequent empirical applications.

##  3 The Model: Choice Probability and Surplus Function

###  3.1 The SEVI-based RUM

We consider a probabilistic choice model based on utility maximization. We
assume that the utility of individual $i$ from alternative $j$ is given by

$U_{ij}\left(\beta_{0}\right)=V_{ij}\left(\beta_{0}\right)+\varepsilon_{ij},\text{
}j=1,2,\ldots,J,$ (1)

where $J$ is the number of alternatives, $V_{ij}\left(\beta_{0}\right)$ is the
observable utility that individual $i$ obtains from choosing alternative $j$,
$\varepsilon_{ij}$ captures the utility component unobservable to the
econometrician, and $\\{\varepsilon_{ij}\\}$ is independent of
$\left\\{V_{ij}\left(\beta_{0}\right)\right\\}$. As is customary in the
literature, we refer to $V_{ij}\left(\beta_{0}\right)$ as the _systematic_
utility and $\varepsilon_{ij}$ as the _random/stochastic_ utility. A common
parametrization of the systematic utility is
$V_{ij}\left(\beta_{0}\right)=Z_{i}\beta_{0,z,j}+X_{ij}\beta_{0,x}$ where
$X_{ij}$ is a vector of attributes of alternative $j$ as perceived by
individual $i$ and $Z_{i}$ is a vector of characteristics of individual $i$.

Individuals choose the alternative that maximizes their total utility. Let
$Y_{i}$ denote the choice of individual $i;$ then

$Y_{i}=\arg\max_{j}\left\\{U_{i1}\left(\beta_{0}\right),\ldots,U_{iJ}\left(\beta_{0}\right)\right\\}.$
(2)

One of our goals is to estimate the model parameter $\beta_{0}$ based on
individual choice data.

For the stochastic utility, we assume that $\varepsilon_{ij}$ is iid over
$j=1,2,\ldots,J$ and follows the extreme value type I distribution based on
the _smallest_ extreme value. The CDF of the (standard) smallest extreme value
type I (SEVI) distribution is

$F(a)=1-\exp\left(-\exp(a)\right),$ (3)

and the corresponding pdf is

$f(a)=F^{\prime}(a)=\exp\left(a-\exp(a)\right).$ (4)

Our distributional assumption is in contrast with that in standard logit
models, where the latent utility is given by
$\tilde{U}_{ij}\left(\beta_{0}\right)=V_{ij}\left(\beta_{0}\right)+\tilde{\varepsilon}_{ij}$,
and the stochastic utility $\tilde{\varepsilon}_{ij}$ follows the (standard)
extreme value Type I distribution based on the _largest_ extreme value (i.e.,
the _largest_ extreme value type I (LEVI) distribution). These two forms of
extreme value distributions are related in that
$\left(\varepsilon_{i1},\ldots,\varepsilon_{iJ}\right)^{\prime}$ and
$-\left(\tilde{\varepsilon}_{i1},\ldots,\tilde{\varepsilon}_{iJ}\right)^{\prime}$
have the same distribution, so they can be regarded as mirror images of each
other. For easy reference, we call a stochastic utility following the SEVI
distribution a SEVI error and the resulting RUM as the SEVI model. The same
nomenclature applies to the LEVI case.888Throughout the paper, the SEVI is the
standard SEVI with pdf given in (4) and variance $\pi^{2}/6.$ A similar
comment applies to the LEVI case.

Figure 2: Probability density functions of the LEVI, NORM, and SEVI
distributions

Figure 2 illustrates the pdfs of the LEVI and SEVI distributions. The LEVI
distribution exhibits a strong right-skewness with a heavier upper tail in
contrast to its lower tail, while SEVI exhibits the opposite (mirror image)
behavior. As a symmetric benchmark, the normal pdf with the same variance is
also plotted. The SEVI distribution is appropriate when there is a prior
belief that the random utility component is more likely to have a negative
rather than a positive or neutral effect on the total utility. Such a belief
is plausible, and we have been unable to find a strong _a priori_ argument
against using a SEVI random component or a mixture of SEVI and LEVI
components, as we discuss in Section 6.1.

This notion of random component skewness has to be seen as conditional on what
alternative attributes and agent characteristics are observed by the
econometrician. To see this, note that in the case of a linear latent utility
model, $\varepsilon_{ij}$ consists of
$(\tilde{X}_{ij}-E\tilde{X}_{ij})\tilde{\beta}_{0,x}$ for a scalar attribute
$\tilde{X}_{ij}$ known to individual $i$ but not observable to the
econometrician. If $\tilde{X}_{ij}$ has a negative skewness and
$\tilde{\beta}_{0,x}>0$, then
$(\tilde{X}_{ij}-E\tilde{X}_{ij})\tilde{\beta}_{0,x}$ will also have a
negative skewness, implying that $\varepsilon_{ij}$ may exhibit left-skewness
and follow the SEVI distribution. Thus, the inclusion or exclusion of a
particular covariate may change the skewness of the stochastic utility
component. More generally, the system utility may be misspecified, with the
resulting specification error effectively becoming part of the stochastic
utility. Given that specification errors can be manifest as left-skewed,
symmetric, or right-skewed, the stochastic utility can display any of these
forms of skewness as a consequence. This argument lends another support to the
notion that the SEVI distribution of the stochastic utility is at least as
plausible as the LEVI distribution.

A logical question at this point is whether the SEVI distribution belongs to
the class of Generalized Extreme Value (GEV) distributions with the CDF given
by $F_{\mathrm{GEV}}(a;\xi)=\exp(-\left(1+\xi a\right)^{-1/\xi})$ for some
parameter $\xi.$ The SEVI distribution, however, does not align with a GEV
distribution.999When $\xi\neq 0,$ the GEV distribution is not supported on the
whole real line. When $\xi=0,$ the GEV distribution is right-skewed. Nor does
it correspond to the marginal distribution of a multivariate extreme value
(MEV) distribution with CDF:

$F_{\mathrm{MEV}}(a_{1},\ldots,a_{J})=\exp(-\mathbb{G}\left(\exp\left(-a_{1}\right),\ldots,\exp\left(-a_{J}\right)\right),$
(5)

for some function $\mathbb{G}$. In both cases, the marginal distribution is
right-skewed instead of being left-skewed like a SEVI distribution. See, for
example, Chapter 4 of Train, (2009). In principle, exploring an extension of
the GEV or MEV family by using the SEVI as the marginal distribution,
potentially encompassing the SEVI as a special case within this broader
framework, could be considered for future research.

###  3.2 Choice Probabilities in the SEVI-based RUM

In a SEVI-based RUM, similar to a LEVI-based RUM, the choice probabilities
also have closed-form expressions, albeit a bit more complex. To simplify the
notation, we suppress the dependence of $V_{ij}\left(\beta_{0}\right)$ and
$V_{i}\left(\beta_{0}\right):=(V_{i1}\left(\beta_{0}\right),\ldots,V_{iJ}\left(\beta_{0}\right))^{\prime}$
on $\beta_{0}$, and we write $U_{ij}:=U_{ij}\left(\beta_{0}\right),$
$V_{ij}:=V_{ij}\left(\beta_{0}\right),$ and
$V_{i}:=V_{i}\left(\beta_{0}\right)$. We adopt the same convention for the
realized values $v_{ij}\left(\beta_{0}\right)$ and
$v_{i}\left(\beta_{0}\right)$ of $V_{ij}\left(\beta_{0}\right)$ and
$V_{i}\left(\beta_{0}\right)$ in the theorem below.

###### Theorem 1

Assume that $U_{ij}=V_{ij}+\varepsilon_{ij}$, $j=1,2,\ldots,J$ where
$\varepsilon_{i}:=\left(\varepsilon_{i1},\ldots,\varepsilon_{i,J}\right)^{\prime}$
is independent of $V_{i}$ and the elements of $\varepsilon_{i}$ follow
independent and identical SEVI distributions. Then, under utility
maximization, the probability of individual $i$ choosing alternative $j$ is

$\displaystyle P^{\mathrm{SEVI}}\left(Y_{i}=j|v_{i}\right)$
$\displaystyle:=P^{\mathrm{SEVI}}\left(Y_{i}=j|V_{i}=v_{i}\right)$
$\displaystyle=1+\sum_{\ell=1}^{J-1}\left(-1\right)^{\ell}\sum_{k_{1}<\ldots<k_{\ell}:\left\\{k_{1},\ldots,k_{\ell}\right\\}\subseteq\mathcal{J}_{-j}}\frac{\exp\left(-v_{ij}\right)}{\exp\left(-v_{ij}\right)+\sum_{k\in\left\\{k_{1},\ldots,k_{\ell}\right\\}}\exp\left(-v_{ik}\right)}$
(6)
$\displaystyle=1+\sum_{\ell=1}^{J-1}\left(-1\right)^{\ell}\sum_{k_{1}<\ldots<k_{\ell}:\left\\{k_{1},\ldots,k_{\ell}\right\\}\subseteq\mathcal{J}_{-j}}\frac{1}{1+\sum_{k\in\left\\{k_{1},\ldots,k_{\ell}\right\\}}\exp\left(-v_{ik}+v_{ij}\right)},$

where, for $\mathcal{J}=\left\\{1,\ldots,J\right\\},$
$\mathcal{J}_{-j}=\mathcal{J}\backslash\left\\{j\right\\}$ is the subset of
alternatives leaving the $j$-th alternative out.

###### Remark 1

In a standard LEVI conditional logit model, the choice probability is

$P^{\mathrm{LEVI}}\left(Y_{i}=j|v_{i}\right)=\frac{\exp(v_{ij})}{\sum_{k=1}^{J}\exp\left(v_{ik}\right)}.$
(7)

We can compare the choice probabilities across the SEVI and LEVI models using
a few examples. When there are only two alternatives so that $J=2,$ we have,
using Theorem 1,

$\displaystyle P^{\mathrm{SEVI}}\left(Y_{i}=1|v_{i}\right)$
$\displaystyle=1-\frac{\exp\left(-v_{i1}\right)}{\exp\left(-v_{i1}\right)+\exp\left(-v_{i2}\right)}$
$\displaystyle=\frac{\exp(-v_{i2})}{\exp(-v_{i1})+\exp(-v_{i2})}=\frac{\exp(v_{i1})}{\exp(v_{i1})+\exp(v_{i2})}.$

The choice probability is exactly the same as what we obtain in a standard
LEVI model. Therefore, in the presence of only two alternatives, it does not
matter whether the stochastic utility follows the LEVI or SEVI distribution.
This is because the difference between two independent LEVI or SEVI draws
follows the same logistic distribution.101010By Lemma 5 of Yellott, (1977),
the condition that the difference of two iid random components is logistic is
necessary for the IIA to hold. Our results presented later can be seen as
showing the existence of a simple distribution, SEVI, which meets this
necessary condition, yet IIA does not hold.

When there are more than two alternatives so that $J>2,$ the choice
probabilities in a SEVI model are different from those in a LEVI model.
Consider the case with $J=3.$ It follows from Theorem 1 that

$\displaystyle P^{\mathrm{SEVI}}\left(Y_{i}=1|v_{i}\right)$
$\displaystyle=1-\frac{\exp\left(-v_{i1}\right)}{\exp\left(-v_{i1}\right)+\exp\left(-v_{i2}\right)}-\frac{\exp\left(-v_{i1}\right)}{\exp\left(-v_{i1}\right)+\exp\left(-v_{i3}\right)}$
(8)
$\displaystyle+\frac{\exp\left(-v_{i1}\right)}{\exp\left(-v_{i1}\right)+\exp\left(-v_{i2}\right)+\exp\left(-v_{i3}\right)}.$

This is clearly different from the LEVI choice probability, which is equal to

$P^{\mathrm{LEVI}}\left(Y_{i}=1|v_{i}\right):=\frac{\exp\left(v_{i1}\right)}{\exp\left(v_{i1}\right)+\exp\left(v_{i2}\right)+\exp\left(v_{i3}\right)}.$

To understand the formula for $P^{\mathrm{SEVI}}\left(Y_{i}=1|v_{i}\right),$
we can examine $1-P^{\mathrm{SEVI}}\left(Y_{i}=1|v_{i}\right),$ which
represents the probability that alternative 1 is _not_ chosen. The latter
event can be expressed as a union of two events $\left\\{U_{i1}\leq
U_{i2}\right\\}\cup\left\\{U_{i1}\leq U_{i3}\right\\}$, indicating that
alternative 1 is dominated by either alternative 2 or alternative 3. But the
probability of this union is

$\displaystyle\Pr\left[\left\\{U_{i1}\leq
U_{i2}\right\\}\cup\left\\{U_{i1}\leq U_{i3}\right\\}\right]$
$\displaystyle=\Pr\left[U_{i1}\leq U_{i2}\right]+\Pr\left[U_{i1}\leq
U_{i3}\right]-\Pr\left[\left\\{U_{i1}\leq
U_{i2}\right\\}\cap\left\\{U_{i1}\leq U_{i3}\right\\}\right]$
$\displaystyle=\Pr\left[-U_{i1}\geq-U_{i2}\right]+\Pr\left[-U_{i1}\geq-
U_{i3}\right]-\Pr\left[\left\\{-U_{i1}\geq-
U_{i2}\right\\}\cap\left\\{-U_{i1}\geq-U_{i3}\right\\}\right].$

Each of the three probabilities above corresponds to a choice probability in a
conventional LEVI model. Substituting the familiar LEVI formulas yields
Equation (8), and Theorem 1 is a generalization of these arguments.

###### Remark 2

When $J=4,$ Theorem 1 reveals that

$\displaystyle P^{\mathrm{SEVI}}\left(Y_{i}=1|v_{i}\right)$
$\displaystyle=1-\frac{\exp\left(-v_{i1}\right)}{\exp\left(-v_{i1}\right)+\exp\left(-v_{i2}\right)}-\frac{\exp\left(-v_{i1}\right)}{\exp\left(-v_{i1}\right)+\exp\left(-v_{i3}\right)}-\frac{\exp\left(-v_{i1}\right)}{\exp\left(-v_{i1}\right)+\exp\left(-v_{i4}\right)}$
$\displaystyle+\frac{\exp\left(-v_{i1}\right)}{\exp\left(-v_{i1}\right)+\exp\left(-v_{i2}\right)+\exp\left(-v_{i3}\right)}+\frac{\exp\left(-v_{i1}\right)}{\exp\left(-v_{i1}\right)+\exp\left(-v_{i2}\right)+\exp\left(-v_{i4}\right)}$
$\displaystyle+\frac{\exp\left(-v_{i1}\right)}{\exp\left(-v_{i1}\right)+\exp\left(-v_{i3}\right)+\exp\left(-v_{i4}\right)}$
$\displaystyle-\frac{\exp\left(-v_{i1}\right)}{\exp\left(-v_{i1}\right)+\exp\left(-v_{i2}\right)+\exp\left(-v_{i3}\right)+\exp\left(-v_{i4}\right)},$

which involves enumerating all choice subsets that contain the first
alternative. In contrast, the corresponding choice probability in a LEVI model
is

$P^{\mathrm{LEVI}}\left(Y_{i}=1|v_{i}\right)=\frac{\exp\left(v_{i1}\right)}{\left[\exp\left(v_{i1}\right)+\exp\left(v_{i2}\right)+\exp\left(v_{i3}\right)+\exp\left(v_{i4}\right)\right]},$

which involves only the largest choice set. As shown in Figure 1, and later,
when $J$ becomes larger than 2, the choice probabilities in a SEVI model can
be different from those in a LEVI model in an economically meaningful way.
This has practical implications, underscoring the need for careful
consideration when formulating distribution assumptions in discrete choice
modeling.

###### Remark 3

When $v_{ij}=v_{ik}$ for all $j$ and $k,$ we have

$\displaystyle P^{\mathrm{SEVI}}\left(Y_{i}=j|v_{i}\right)$
$\displaystyle=1+\sum_{\ell=1}^{J-1}\left(-1\right)^{\ell}\sum_{k_{1}<\ldots<k_{\ell}:\left\\{k_{1},\ldots,k_{\ell}\right\\}\subseteq\mathcal{J}_{-j}}\frac{1}{\ell+1}$
$\displaystyle=\sum_{\ell=0}^{J-1}\left(-1\right)^{\ell}\binom{J-1}{\ell}\frac{1}{\ell+1}=\int_{0}^{1}\sum_{\ell=0}^{J-1}\left(-1\right)^{\ell}\binom{J-1}{\ell}s^{\ell}ds$
$\displaystyle=\int_{0}^{1}\left(1-s\right)^{J-1}ds=\frac{1}{J}.$

This result can also be obtained by using a symmetric argument. Since all
alternatives have the same systematic utility, they are indistinguishable and
hence have an equal chance of being chosen. The result is the same as that
obtained under the LEVI assumption. On the other hand, when $v_{ij}$ increases
and dominates all other $v_{ik}$ for $k\neq j,$
$P^{\mathrm{SEVI}}\left(Y_{i}=j|v_{i}\right)$ approaches 1. The choice
probability $P^{\mathrm{SEVI}}\left(Y_{i}=j|v_{i}\right)$ can, therefore, be
regarded as a type of softmax function that maps $\mathbb{R}^{J}$ into
$\left(0,1\right)^{J}.$ For example, it maps $\left(1,2,8\right)$ into
$(4.24\times 10^{-4},$ $2.29\times 10^{-3},0.997),$ which amounts to assigning
almost all probability mass to the element with maximal systematic utility.
This result is qualitatively similar to that obtained under the LEVI
assumption.

###### Remark 4

The probability of choosing alternative $j$ in a SEVI model involves a
summation over all subsets of $\mathcal{J}_{-j},$ resulting in an iteration
over a total of $2^{J-1}$ subsets of alternatives. The choice probability in a
LEVI model takes a simpler, more convenient form. While the choice probability
in a SEVI model is considerably richer than that in a LEVI model, the SEVI
model has a computational disadvantage when $J$ is large. To mitigate this, we
employ Gosper’s hack, a technique rooted in bitwise operations, to streamline
the iteration process efficiently. Additionally, by leveraging the arguments
following Proposition 4.4 in Ross, (2013), it can be shown that for any $M<J$:

$\left|P^{\mathrm{SEVI}}\left(Y_{i}=j|v_{i}\right)-P_{M}^{\mathrm{SEVI}}\left(Y_{i}=j|v_{i}\right)\right|\leq\sum_{k_{1}<\ldots<k_{M}:\left\\{k_{1},\ldots,k_{M}\right\\}\subseteq\mathcal{J}_{-j}}\frac{1}{1+\sum_{k\in\left\\{k_{1},\ldots,k_{\ell}\right\\}}\exp\left(-v_{ik}+v_{ij}\right)}$
(9)

where

$P_{M}^{\mathrm{SEVI}}\left(Y_{i}=j|v_{i}\right)=1+\sum_{\ell=1}^{M-1}\left(-1\right)^{\ell}\sum_{k_{1}<\ldots<k_{\ell}:\left\\{k_{1},\ldots,k_{\ell}\right\\}\subseteq\mathcal{J}_{-j}}\frac{1}{1+\sum_{k\in\left\\{k_{1},\ldots,k_{\ell}\right\\}}\exp\left(-v_{ik}+v_{ij}\right)}.$

This allows us to approximate $P^{\mathrm{SEVI}}\left(Y_{i}=j|v_{i}\right)$ by
$P_{M}^{\mathrm{SEVI}}\left(Y_{i}=j|v_{i}\right)$ with a well-controlled
approximation error. Note that
$P_{M}^{\mathrm{SEVI}}\left(Y_{i}=j|v_{i}\right)$ takes the same form as
$P^{\mathrm{SEVI}}\left(Y_{i}=j|v_{i}\right)$ but involves subsets of at most
$M-1$ alternatives, and so it is computationally less intensive when $M$ is
smaller than $J.$ To achieve a desired level of precision, we can choose $M$
large enough and stop iterating over subsets of more than $M-1$ alternatives.
Combining Gosper’s hack with an early stopping makes it computationally easy
to estimate a SEVI model for $J$ as large as 14. Section 5.3 reports and
compares the computation times for various models using simulated data with
typical sample size and attribute numbers. Section 7 contains a comparison of
computation times for real data applications.111111By leveraging multiple
processors running in parallel to enumerate choice subsets, the feasible size
of $J$ can be significantly increased. Unlike the sequential approach required
for integration in the multinomial probit model, this can be accomplished
simultaneously.

###### Remark 5

For the SEVI model with a linear parametrization of
$V_{ij}\left(\beta_{0}\right),$ if $Z_{i}$ is not present so that
$V_{ij}\left(\beta_{0}\right)=X_{ij}\beta_{0,x},$ then the model is called a
conditional SEVI (logit) model. As an extension of the basic model, we can
allow the coefficient on $X_{ij}$ to be alternative specific and so we may
have $V_{ij}\left(\beta_{0}\right)=X_{ij}\beta_{0,x,j}.$ On the other hand, if
$X_{ij}$ is not present so that
$V_{ij}\left(\beta_{0}\right)=Z_{i}\beta_{0,z,j},$ then the model is called a
multinomial SEVI (logit) model. These forms can be combined so that
$V_{ij}\left(\beta_{0}\right)=\alpha_{j}+Z_{i}\beta_{0,z,j}+X_{ij}\beta_{0,x}+W_{ij}\beta_{0,w,j},$
which provides the same flexibility as in a standard LEVI (logit) model. As in
a standard multinomial LEVI (logit) model, not all of $\beta_{0,z,j}$’s are
identified, and we need to normalize one of them, say $\beta_{0,z,J},$ to be
zero.

To illustrate the difference between the SEVI and LEVI models, we simulate the
choice probabilities when $V_{ij}\left(\beta_{0}\right)=X_{ij}\beta_{0,x}$
(there is no individual-specific characteristic or alternative-specific
constant), and the stochastic utility is iid SEVI or LEVI. We set the number
of covariates $L$ in $X_{ij}$ to be $3$ (there are three attributes for each
alternative) and let $\beta_{0}=\beta_{0,x}=(1,2,1)^{\prime}.$ We consider two
data generating processes (DGP) for the attributes $\left\\{X_{ij}\right\\}$.
In the first DGP, for each $i$ and $j,$ we let $X_{ij,\ell}$ be iid
$N(0,\pi^{2}\omega_{j}^{2}/36)$ over $\ell=1,2,3$ where

$\omega_{j}=\frac{j-\left(J+1\right)/2}{\sqrt{\left(J^{2}-1\right)/12}},\text{
}j=1,\ldots,J.\text{ }$ (10)

Here, $\left(\omega_{1},\ldots,\omega_{J}\right)$ is a just standardized
version of the sequence ($1,2,\ldots,J$). In this case, each attribute
$X_{ij,\ell}$ becomes more dispersed when the alternative label $j$ is closer
to $1$ or $J$. As a result, the mean choice probabilities (i.e.,
$E\Pr\left(Y_{i}=j|V_{i}\right)$) are not all the same across the
alternatives. In the second DGP, for each $i$ and $j,$ we let $X_{ij,\ell}$ be
iid $N(0,\pi^{2}/36)$ over $\ell=1,2,3.$ In this case, the attributes have the
same distribution across the alternatives, and the mean choice probabilities
are the same across all alternatives. Under these two DGPs for
$\left\\{X_{ij}\right\\},$ the variances of the systematic utility
$V_{ij}\left(\beta_{0}\right)=X_{ij}\beta_{0}$ are $\pi^{2}\omega_{j}^{2}/6$
and $\pi^{2}/6,$ respectively. These variances are comparable to $\pi^{2}/6,$
the variance of the stochastic utility.

Figure 3: Simulated choice probabilities and the average of theoretical choice
probabilities against alternative labels $(j=1,\ldots,J)$ for a sample of
$n=10,000$ individuals when $X_{ij,\ell}$ is iid
$N(0,\pi^{2}\omega_{j}^{2}/36)$

For each model, we compute the choice probabilities according to the
theoretical formulae (i.e., (6) under the SEVI and (7) under the LEVI), and
take the average of these choice probabilities over 10,000 individuals. We
also simulate the decisions of 10,000 independent individuals and compute the
proportion of individuals choosing each alternative. Figure 3 reports results
for various $J$ values when $X_{ij,\ell}$ $\thicksim$ iid
$N(0,\pi^{2}\omega_{j}^{2}/36).$ Not surprisingly, since the sample size is
large, the average theoretical choice probabilities match the simulated choice
proportions well for both the SEVI and LEVI models. When $J=2,$ the
theoretical choice probabilities are the same across the two models. With only
two alternatives, the simulated choice shares are very close across the two
models, but not identical due to simulation errors. When $J>2,$ for
alternatives with relatively higher choice probabilities under the LEVI, the
SEVI model assigns a higher choice probability than the LEVI model. On the
other hand, when $J>2,$ for alternatives with a relatively lower choice
probability under the LEVI, the SEVI model assigns a lower choice probability
than the LEVI model. The difference between the choice probabilities across
the two models grows with $J.$ The choice probability in a SEVI model is
closer to the “arg max” function than that in a LEVI model. In this context, a
SEVI model leans more towards embodying the concept of “the winner takes it
all” than a LEVI model. More specifically, given the same set of systematic
utilities, the probability of choosing the alternative with the maximum
systematic utility in a SEVI model tends to be larger than that in a LEVI
model.

Next, we report the results when $X_{ij,\ell}$ is iid $N(0,\pi^{2}/36)$, but
we use a different type of plot. Figure 4 provides a scatter plot of the ratio
of the choice probabilities in the SEVI and LEVI models (i.e.,
$P^{\mathrm{SEVI}}/P^{\mathrm{LEVI}})$ against $P^{\mathrm{LEVI}}$. Instead of
examining the average choice probabilities as done in Figure 3, we consider
the choice probability of choosing a specific alternative (here, we use
alternative 1 without loss of generality) for each realization of the
systematic utilities. Figure 5 provides a supplementary illustration, offering
a closer examination of certain details.

Figure 4: Scatter plot of the ratio of choice probabilities in the SEVI and
LEVI models ($P^{\mathrm{SEVI}}/P^{\mathrm{LEVI}})$ against the choice
probability in the LEVI model ($P^{\mathrm{LEVI}})$ with identical systematic
utilities when $X_{ij,\ell}$ is iid $N(0,\pi^{2}/36)$

In both Figure 4 and Figure 5, the systematic utilities are held constant for
each data point with either LEVI errors or SEVI errors. We note from these two
figures that when $J>2$ and $P^{\mathrm{LEVI}}$ is low, $P^{\mathrm{SEVI}}$
can be more than 50 percent lower. For example, when there are five
alternatives ($J=5$) and $P^{\mathrm{LEVI}}=0.1,$ $P^{\mathrm{SEVI}}$ can be
as low as $0.05$. The LEVI probability can be roughly twice as large as the
SEVI probability. Thus, from a practical point of view, a choice between the
two models can have an enormous impact on predicting a lift for a minor brand
with a market share between 5% and 10%. On the other hand, when $J>2$ and
$P^{\mathrm{LEVI}}$ is high, $P^{\mathrm{SEVI}}$ tends to be higher. For
example, when $J=5$ and $P^{\mathrm{LEVI}}=0.4,$ $P^{\mathrm{SEVI}}$ can be 20
percent higher; when $J=11$ and $P^{\mathrm{LEVI}}=0.3,$ $P^{\mathrm{SEVI}}$
can be 50 percent higher. The patterns we observe here are similar to those in
Figure 1. The difference in the choice probabilities across the two models can
be important in various contexts, ranging from antitrust actions to production
planning.

Figure 5: Scatter plot of the choice probability in the SEVI model versus that
in the LEVI model with left and right panels having different ranges for the
horizontal axis ($X_{ij,\ell}$ is iid $N(0,\pi^{2}/36))$

Figure 5 shows that when $J=5$ and the LEVI probability is below 0.20, the
corresponding SEVI probability is lower. When the LEVI probability is above
0.20, the corresponding SEVI probability is higher. Similar patterns are found
for other values of $J.$ These qualitative patterns are consistent with what
we find in Figure 3. The vector of choice probabilities in the SEVI model
(i.e., $[P^{\mathrm{SEVI}}(Y=1),\ldots,P^{\mathrm{SEVI}}\left(Y=J\right)]$) is
closer to the vertices of the probability simplex than that in the LEVI model.
In other words,
$[P^{\mathrm{SEVI}}(Y=1),\ldots,P^{\mathrm{SEVI}}\left(Y=J\right)]$ is closer
to $\arg\max\left(v_{1},\ldots,v_{J}\right):=(0,\ldots,1,\ldots,0),$ a $0$-$1$
vector with the $j$-th element equal to 1 if and only if $v_{j}$ is the
maximum value of $\left(v_{1},\ldots,v_{J}\right).$121212For ease of
discussion, we assume that there is a unique maximum among
$v_{1},\ldots,v_{J}.$ The 0-1 vector representation of $\arg\max$ is the so-
called one-hot encoding of $\arg\max.$

We now offer insights into why the probability vector
$[P^{\mathrm{SEVI}}(Y=1),\ldots,P^{\mathrm{SEVI}}\left(Y=J\right)]$ is closer
to the one-hot encoding of $\arg\max\left(v_{1},\ldots,v_{J}\right)$ than
$[P^{\mathrm{LEVI}}(Y=1),\ldots,P^{\mathrm{LEVI}}\left(Y=J\right)].$ When
$J\geq 3,$ the top two values among $J$ independent draws from the stochastic
utility distribution are more likely to be above or equal to the median of
this distribution rather than below it. For a draw from the LEVI distribution,
conditional on being above the median, the mean is 1.545 and the standard
deviation is 1.079. In contrast, for the SEVI case, these values are 0.391 and
0.500, respectively. Hence, the absolute difference between any two above-the-
median draws from the LEVI distribution is expected to be larger than that for
the SEVI distribution. In fact, Monte Carlo simulations show that the former
first-order stochastically dominates the latter (see Figure S.1). This implies
that two above-the-median draws from the SEVI distribution are more likely to
be closer to each other than those from the LEVI distribution. Therefore, with
more than two alternatives, it is less probable for the stochastic utility
components under the SEVI to make a difference in determining which
alternative has the highest total utility. As a result, in the presence of
more than two alternatives, the choice probabilities under the SEVI are better
aligned with the systematic utilities than those under the LEVI. In other
words, in a SEVI model with $J\geq 3$, the observed choice is more likely to
correspond to the choice that maximizes the systematic utility than in a LEVI
model. See Figure S.2 for an additional illustration.

###  3.3 Algebraic Relation between SEVI and LEVI Choice Probabilities

To relate the choice probabilities under SEVI and LEVI assumptions, we
consider a general latent utility model $U_{ij}=V_{ij}+\varepsilon_{ij},$
$j=1,\ldots,J$ where $\varepsilon_{ij}$ is iid across $j=1,\ldots,J$ and may
follow either SEVI or LEVI,
$\varepsilon_{i}=\left(\varepsilon_{i1},\ldots,\varepsilon_{iJ}\right)$ is
independent of $V_{i}=(V_{i1},\ldots,V_{iJ})$. For any alternative $j$ and any
subset of alternatives $\mathcal{S}_{\ell}$ that does not include alternative
$j,$ we let

$\displaystyle P^{\mathrm{SEVI}}(Y_{i}$
$\displaystyle=j|\mathcal{S}_{\ell}\cup
j,V_{i}=v_{i})=\Pr(Y_{i}=j|\mathcal{S}_{\ell}\cup
j,V_{i}=v_{i},\varepsilon_{im}\thicksim iid\text{ }\mathrm{SEVI}\text{ over
}m\in\mathcal{J}),$ $\displaystyle P^{\mathrm{LEVI}}(Y_{i}$
$\displaystyle=j|\mathcal{S}_{\ell}\cup
j,V_{i}=v_{i})=\Pr(Y_{i}=j|\mathcal{S}_{\ell}\cup
j,V_{i}=v_{i},\varepsilon_{im}\thicksim iid\text{ }\mathrm{LEVI}\text{ over
}m\in\mathcal{J}),$

which are the probabilities of individual $i$ choosing alternative $j$ from
the subset $\mathcal{S}_{\ell}\cup j:=$
$\mathcal{S}_{\ell}\cup\left\\{j\right\\}$ under SEVI and LEVI errors,
respectively. It is important to point out that in the above definition, the
probabilities depend on the choice subset under consideration.

###### Proposition 2

The choice probabilities under the SEVI and LEVI satisfy the following
relations:

$\displaystyle P^{\mathrm{SEVI}}(Y_{i}$
$\displaystyle=j|\mathcal{J},V_{i}=v_{i})$
$\displaystyle=1+\sum_{\ell=1}^{J-1}\left(-1\right)^{\ell}\sum_{k_{1}<\ldots<k_{\ell}:\text{
}\mathcal{S}_{\ell}=\left\\{k_{1},\ldots,k_{\ell}\right\\}\subseteq\mathcal{J}_{-j}}P^{\mathrm{LEVI}}\left(Y_{i}=j|\mathcal{S}_{\ell}\cup
j,V_{i}=-v_{i}\right);$ $\displaystyle P^{\mathrm{LEVI}}(Y_{i}$
$\displaystyle=j|\mathcal{J},V_{i}=v_{i})$
$\displaystyle=1+\sum_{\ell=1}^{J-1}\left(-1\right)^{\ell}\sum_{k_{1}<\ldots<k_{\ell}:\text{
}\mathcal{S}_{\ell}=\left\\{k_{1},\ldots,k_{\ell}\right\\}\subseteq\mathcal{J}_{-j}}P^{\mathrm{SEVI}}\left(Y_{i}=j|\mathcal{S}_{\ell}\cup
j,V_{i}=-v_{i}\right).$

The proposition shows an intrinsic connection between the choice probabilities
under the two assumptions of the stochastic utility. Note that the right-hand
side of each relation involves the probabilities of choosing alternative $j$
over all subsets of alternatives that contain alternative $j$ but under a
“reflected” model for $V_{i}$ ($V_{i}=v_{i}$ on the left-hand side becomes
$V_{i}=-v_{i}$ on the right-hand side).

By taking the difference between the two equations in Proposition 2, we have

$\displaystyle P^{\mathrm{SEVI}}(Y_{i}$
$\displaystyle=j|\mathcal{J},V_{i}=v_{i})-P^{\mathrm{LEVI}}(Y_{i}=j|\mathcal{J},V_{i}=v_{i})$
$\displaystyle=\sum_{\ell=2}^{J-1}\left(-1\right)^{\ell-1}\sum_{k_{1}<\ldots<k_{\ell}:\text{
}\mathcal{S}_{\ell}=\left\\{k_{1},\ldots,k_{\ell}\right\\}\subseteq\mathcal{J}_{-j}}\left[P^{\Delta}\left(Y_{i}=j|\mathcal{S}_{\ell}\cup
j,V_{i}=-v_{i}\right)\right],$

where

$P^{\Delta}\left(Y_{i}=j|\mathcal{S}_{\ell}\cup
j,V_{i}=-v_{i}\right)=P^{\mathrm{SEVI}}\left(Y_{i}=j|\mathcal{S}_{\ell}\cup
j,V_{i}=-v_{i}\right)-P^{\mathrm{LEVI}}\left(Y_{i}=j|\mathcal{S}_{\ell}\cup
j,V_{i}=-v_{i}\right).$

So, the difference in the SEVI and LEVI probabilities of choosing alternative
$j$ from the entire choice set $\mathcal{J}$ is an aggregation of the
probability differences from choosing alternative $j$ from all subsets of
alternatives that contain $j$ but with flipped systematic utilities. This is a
recursive relationship, as each probability difference on the right-hand side
can be represented in the same fashion as the left-hand side. An intriguing
question emerges: can the difference in choice probabilities be fully captured
by a simple index that measures the concentration of the systematic utilities?
Given the intricate recursive relationship described above and the highly
nonlinear dependence of the probability difference on systematic utilities, as
shown in Figure S.3, it appears unlikely that commonly used measures like the
Gini coefficient or the Herfindahl-Hirschman index would be sufficient for
this task. However, in empirical practice, particularly for moderately sized
$J$, these measures are likely to prove useful in predicting differences in
choice probabilities.

The second relation in Proposition 2 can be presented equivalently as follows:

$\displaystyle P^{\mathrm{LEVI}}(Y_{i}$
$\displaystyle=j|\mathcal{J},V_{i}=-v_{i})$
$\displaystyle=1+\sum_{\ell=1}^{J-1}\left(-1\right)^{\ell}\sum_{k_{1}<\ldots<k_{\ell}:\mathcal{S}_{\ell}:=\left\\{k_{1},\ldots,k_{\ell}\right\\}\subseteq\mathcal{J}_{-j}}P^{\mathrm{SEVI}}\left(Y_{i}=j|\mathcal{S}_{\ell}\cup
j,V_{i}=v_{i}\right).$ (11)

Since $P^{\mathrm{LEVI}}(Y_{i}=j|\mathcal{J},V_{i}=-v_{i})$ is nonnegative,
the sum in (11) is also nonnegative. This sum is a special Block-Marschak
polynomial (Block and Marschak, (1960)). In the presence of a finite number of
alternatives, it is well-known in the stochastic choice theory that the
nonnegativity of Block-Marschak polynomials is both necessary and sufficient
for a random choice rule to have an additive random utility representation.
For a detailed discussion, see, for example, Chapters 1 and 2 of the recent
book by Strzalecki, (2023). In the present setting, the choice probabilities
under SEVI are derived from random utility maximization and as such, they
trivially have an additive random utility representation. The significance of
(11) is that it provides a simple characterization of a special Block-Marschak
polynomial.

More generally, let $j$ be a target alternative of interest and

$\mathcal{D}_{j}:=\left\\{\left(d_{1},\ldots,d_{\rho},j\right):d_{1}<\ldots<d_{\rho}\text{
and }\left(d_{1}<\ldots<d_{\rho}\right)\subseteq\mathcal{J}_{-j}\right\\}$

be a subset of alternatives that _includes_ alternative $j.$ Then, the
(general) Block-Marschak polynomial for our probability choice rule under SEVI
errors is defined to be

$\displaystyle\mathcal{K}^{\mathrm{SEVI}}(j,\mathcal{D}_{j})$
$\displaystyle:=\Pr\left(Y_{i}=j|\mathcal{D}_{j},V_{i}=v_{i},\varepsilon_{im}\thicksim
iid\text{ }\mathrm{SEVI}\text{ over }m\in\mathcal{J}\right)$
$\displaystyle+\sum_{\ell=1}^{J-\left|\mathcal{D}_{j}\right|}\left(-1\right)^{\ell}\sum_{k_{1}<\ldots<k_{\ell}:\mathcal{S}_{\ell}:=\left\\{k_{1},\ldots,k_{\ell}\right\\}\subseteq\mathcal{J}\backslash\mathcal{D}_{j}}P^{\mathrm{SEVI}}\left(Y_{i}=j|\mathcal{S}_{\ell}\cup\mathcal{D}_{j},V_{i}=v_{i}\right).$

Note that when $\mathcal{D}_{j}=\left\\{j\right\\},$ the first term in
$\mathcal{K}^{\mathrm{SEVI}}(j,\mathcal{D}_{j})$ becomes $1$ and
$\mathcal{K}^{\mathrm{SEVI}}(j,\mathcal{D}_{j})$ reduces to the special Block-
Marschak polynomial in (11). We can show that
$\mathcal{K}^{\mathrm{SEVI}}(j,\mathcal{D}_{j})$ is the probability that the
$j$-th alternative dominates all other alternatives in $\mathcal{D}_{j}$ but
is dominated by the alternatives that are not in $\mathcal{D}_{j}.$ See, for
example, Barberá and Pattanaik, (1986). Hence,
$\mathcal{K}^{\mathrm{SEVI}}(j,\mathcal{D}_{j})\geq 0$ for all
$\left(j,\mathcal{D}_{j}\right),$ which is consistent with the nonnegativity
of Block-Marschak polynomials for any additive RUM model.

The Block-Marschak polynomial $\mathcal{K}^{\mathrm{SEVI}}(j,\mathcal{D}_{j})$
is precisely the probability relevant for rank-ordered data. While using the
SEVI model for rank-ordered data necessitates additional research, the formula
for $\mathcal{K}^{\mathrm{SEVI}}(j,\mathcal{D}_{j})$ provides a good starting
point.

###  3.4 Patterns of Substitution and the Lack of IIA

In this subsection, we compute $\partial
P^{\mathrm{SEVI}}\left(Y_{i}=k|v_{i}\right)/\partial v_{ij}$, the effect of an
infinitesimal change in alternative $j$’s systematic utility (as perceived by
individual $i$) on the probability of individual $i$ choosing alternative $k.$
This computation is essential for determining $\partial
P^{\mathrm{SEVI}}\left(Y_{i}=k|v_{i}\right)/\partial X_{ij},$ the effect of
changing alternative $j$’s attributes on the probability of choosing
alternative $k.$

For notational economy, we let
$P_{ik}^{\mathrm{SEVI}}:=P^{\mathrm{SEVI}}\left(Y_{i}=k|v_{i}\right).$ We
have, using Proposition 2 and some simple calculations,

$\frac{\partial P_{ij}^{\mathrm{SEVI}}}{\partial
v_{ij}}=-\sum_{\ell=1}^{J-1}\left(-1\right)^{\ell}\sum_{k_{1}<\ldots<k_{\ell}:\mathcal{S}_{\ell}=\left\\{k_{1},\ldots,k_{\ell}\right\\}\subseteq\mathcal{J}_{-j}}P^{\mathrm{LEVI}}(Y_{i}=j|\mathcal{S}_{\ell}\cup
j,-v_{i})P^{\mathrm{LEVI}}(Y_{i}\neq j|\mathcal{S}_{\ell}\cup j,-v_{i}),$ (12)

where for
$\mathcal{S}_{\ell-1}=\left\\{k_{1},\ldots,k_{\ell-1}\right\\}\subseteq\mathcal{J}_{-j},$

$\displaystyle P^{\mathrm{LEVI}}(Y_{i}$
$\displaystyle=j|\mathcal{S}_{\ell}\cup
j,-v_{i})=\frac{\exp(-v_{ij})}{\sum_{m\in\mathcal{S}_{\ell}\cup
j}\exp\left(-v_{im}\right)},$ $\displaystyle P^{\mathrm{LEVI}}(Y_{i}$
$\displaystyle\neq j|\mathcal{S}_{\ell}\cup
j,-v_{i})=1-P^{\mathrm{LEVI}}(Y_{i}=j|\mathcal{S}_{\ell}\cup j,-v_{i}).$

Furthermore, for $k\neq j,$

$\displaystyle\frac{\partial P_{ij}^{\mathrm{SEVI}}}{\partial v_{ik}}$
$\displaystyle=\sum_{\ell=1}^{J-1}\left(-1\right)^{\ell}\sum_{k_{1}<\ldots<k_{\ell-1}:\mathcal{S}_{\ell-1}=\left\\{k_{1},\ldots,k_{\ell-1}\right\\}\subseteq\mathcal{J}_{-\left(j,k\right)}}$
$\displaystyle
P^{\mathrm{LEVI}}(Y_{i}\overset{}{=}j|\mathcal{S}_{\ell-1}\cup\left\\{j,k\right\\},-v_{i})\cdot
P^{\mathrm{LEVI}}(Y_{i}\overset{}{=}k|\mathcal{S}_{\ell-1}\cup\left\\{j,k\right\\},-v_{i}),$
(13)

where for
$\mathcal{S}_{\ell-1}=\left\\{k_{1},\ldots,k_{\ell-1}\right\\}\subseteq\mathcal{J}_{-\left(j,k\right)},$
a subset of $\mathcal{J}$ excluding $j$ and $k,$

$\displaystyle P^{\mathrm{LEVI}}(Y_{i}$
$\displaystyle=j|\mathcal{S}_{\ell}\cup\left\\{j,k\right\\},-v_{i})=\frac{\exp(-v_{ij})}{\sum_{m\in\mathcal{S}_{\ell}\cup\left\\{j,k\right\\}}\exp\left(-v_{im}\right)},$
$\displaystyle P^{\mathrm{LEVI}}(Y_{i}$
$\displaystyle=k|\mathcal{S}_{\ell}\cup\left\\{j,k\right\\},-v_{i})=\frac{\exp(-v_{ik})}{\sum_{m\in\mathcal{S}_{\ell}\cup\left\\{j,k\right\\}}\exp\left(-v_{im}\right)}.$

In equation (13), the first term in the summation (i.e., when $\ell=1)$ is
understood to be

$-\frac{\exp(-v_{ij})}{\left[\exp(-v_{ij})+\exp\left(-v_{ik}\right)\right]}\frac{\exp(-v_{ik})}{\left[\exp(-v_{ij})+\exp\left(-v_{ik}\right)\right]}.$

Equation (13) reveals that

$\frac{\partial P_{ij}^{\mathrm{SEVI}}}{\partial v_{ik}}=\frac{\partial
P_{ik}^{\mathrm{SEVI}}}{\partial v_{ij}}$ (14)

for any $k$ and $j.$ That is, the choice probabilities satisfy a symmetry
property analogous to Slutsky symmetry in continuous choice models.

The expression of $\partial P_{ij}^{\mathrm{SEVI}}/\partial v_{ik}$ can be
used to derive the score function, which is essential for finding the MLE and
developing the asymptotic theory. It can also be employed to examine whether
the IIA property holds in a SEVI model when $J\geq 3$. For the IIA to hold,
the cross (semi)elasticity $\partial\log P_{ij}^{\mathrm{SEVI}}/\partial
v_{ik}$ should be the same for all $j\neq k.$ That is,

$\frac{\partial\log P_{ij_{1}}^{\mathrm{SEVI}}}{\partial
v_{ik}}=\frac{\partial\log P_{ij_{2}}^{\mathrm{SEVI}}}{\partial v_{ik}}$

for all $j_{1}\neq j_{2}\neq k.$ However, this condition is not met, as is
clear from (13). Alternatively, for the IIA to hold, the probability ratio
$P_{ij_{1}}^{\mathrm{SEVI}}/P_{ij_{2}}^{\mathrm{SEVI}}$ should not depend on
the systematic utilities of alternatives other than alternatives $j_{1}$ and
$j_{2}.$ However, it is evident that the probability ratio _does_ depend on
the systematic utilities of all alternatives, indicating that the IIA does not
hold.

To illustrate the violation of the IIA in a SEVI model with $J\geq 3$, we
start with a scenario featuring two initial alternatives: individuals must
choose between alternative 1 and alternative 2. In this context, the choice
between the SEVI or LEVI does not matter. For a fixed scale value $s_{1}$ (we
set $v_{1}$ to $\log 1$ so that $s_{1}=\exp(v_{1})=1$),131313Following
Yellott, (1977)), we refer to the exponentiated systematic utility for
alternative $j$ $\left(\text{i.e.},\text{ }\exp\left(v_{j}\right)\right)$ as
the scale value ($s_{j}$) for this alternative. we examine a range of possible
values for $s_{2}:=\exp\left(v_{2}\right)\in\left\\{0.01:0.01:3\right\\},$ a
grid of values from 0.01 to 3 with increment 0.01. Each combination of
$\left(s_{1},s_{2}\right)$ corresponds to a probability ratio:

$\frac{P(Y=2|v_{1},v_{2})}{P(Y=1|v_{1},v_{2})}=\frac{P^{\mathrm{SEVI}}(Y=2|v_{1},v_{2})}{P^{\mathrm{SEVI}}(Y=1|v_{1},v_{2})}=\frac{P^{\mathrm{LEVI}}(Y=2|v_{1},v_{2})}{P^{\mathrm{LEVI}}(Y=1|v_{1},v_{2})}=\frac{\exp(v_{2})}{\exp(v_{1})}=s_{2}.$

Next, we introduce a third alternative with its scale value
$s_{3}=\exp\left(v_{3}\right)\in\left\\{0.01:0.01:3\right\\}$, the same grid
for $s_{2}.$ We investigate how the probability ratio of choosing alternative
2 versus alternative 1 changes. This ratio now depends on whether the
stochastic utilities follow the SEVI or LEVI distribution.

The difference of the ratios under SEVI and LEVI models, denoted as
$D_{\mathrm{PR}},$ is

$\displaystyle D_{\mathrm{PR}}$
$\displaystyle=\frac{P^{\mathrm{SEVI}}(Y=2|v_{1},v_{2},v_{3})}{P^{\mathrm{SEVI}}(Y=1|v_{1},v_{2},v_{3})}-\frac{P^{\mathrm{LEVI}}(Y=2|v_{1},v_{2},v_{3})}{P^{\mathrm{LEVI}}(Y=1|v_{1},v_{2},v_{3})}$
$\displaystyle=\frac{P^{\mathrm{SEVI}}(Y=2|v_{1},v_{2},v_{3})}{P^{\mathrm{SEVI}}(Y=1|v_{1},v_{2},v_{3})}-\frac{P^{\mathrm{LEVI}}(Y=2|v_{1},v_{2})}{P^{\mathrm{LEVI}}(Y=1|v_{1},v_{2})}$
$\displaystyle=\frac{P^{\mathrm{SEVI}}(Y=2|v_{1},v_{2},v_{3})}{P^{\mathrm{SEVI}}(Y=1|v_{1},v_{2},v_{3})}-\frac{P^{\mathrm{SEVI}}(Y=2|v_{1},v_{2})}{P^{\mathrm{SEVI}}(Y=1|v_{1},v_{2})}$
$\displaystyle=\frac{P^{\mathrm{SEVI}}(Y=2|v_{1},v_{2},v_{3})}{P^{\mathrm{SEVI}}(Y=1|v_{1},v_{2},v_{3})}-s_{2}.$

Here we have used properties of the LEVI probabilities. By the third equality
above, $D_{\mathrm{PR}}$ also gauges the extent of IIA violation. With $s_{1}$
set to 1, $D_{\mathrm{PR}}$ is a function of $s_{2}$ and $s_{3}$, so we can
write it as $D_{\mathrm{PR}}(s_{2},s_{3}).$ We present a contour plot of this
function in Figure 6. To aid interpretation, we label the horizontal axis as
$P(Y=2|v_{1},v_{2})/P(Y=1|v_{1},v_{2})=\exp(v_{2})=s_{2},$ representing the
initial probability ratio with only two alternatives available.

It is clear from Figure 6 that $D_{\mathrm{PR}}(s_{2},s_{3})$ is not a zero
function. After the introduction of the third alternative, the market shares
of alternatives 1 and 2 do not decrease proportionally in the SEVI model as
what happens in the LEVI model, indicating a violation of the IIA.

Figure 6 also shows that there are two scenarios where
$D_{\mathrm{PR}}(s_{2},s_{3})$ equals zero or becomes approximately zero.
First, $D_{\mathrm{PR}}(s_{2},s_{3})$ equals zero when alternatives 1 and 2
have the same initial market share (i.e.,
$P(Y=2|v_{1},v_{2})/P(Y=1|v_{1},v_{2})=s_{2}=1)$. This is expected, as in this
case, the first two alternatives are indistinguishable, and their market
shares reduce proportionally in both the LEVI and SEVI models when the third
alternative is introduced. Second, $D_{\mathrm{PR}}(s_{2},s_{3})$ approaches
zero when one of the alternatives becomes almost completely dominant in the
market. This corresponds to a scenario where either $s_{2}$ is small and
$s_{3}$ is very large, or $s_{3}$ is small and $s_{2}$ is very large. Hence,
the difference between a SEVI model and a LEVI model may not be large when
there is clearly a dominant player in the market who commands almost the
entire market, and their disparities become more apparent when no alternative
has nearly all market share.

Figure 6: Contour plot of the difference of the choice probability ratios
$D_{\mathrm{PR}}(s_{2},s_{3})$ between the SEVI and LEVI models against
$\left[s_{2},s_{3}\right]$ when $s_{1}=1$

In an additively separable RUM with $U_{ij}=V_{ij}+\varepsilon_{ij},$ as
considered here, it is well known that the IIA property holds for any $J\geq
3$ if and only if the stochastic utility $\varepsilon_{ij}$ follows the LEVI
distribution (cf., Yellott, (1977)). Under certain conditions that permit an
infinite $J,$ it can be shown that in a random utility model with possibly
nonseparable utility functions, the IIA holds for any $J\geq 3$ only if the
latent utility $U_{ij}$ can be represented as
$H\left(V_{ij}+\varepsilon_{ij}\right)$ for some strictly increasing function
$H\left(\cdot\right)$ and LEVI-distributed $\varepsilon_{ij}.$ This result is
a corollary to Theorem 1 of Dagsvik, (2016); see also Lindberg and Smith,
(2017) and the references therein.141414When $J$ is finite, the latent utility
presentation of $H\left(V_{ij}+\varepsilon_{ij}\right)$ is not necessary for
IIA to hold, but the counterexample given in Lindberg and Smith, (2017) is
unlikely to be relevant to most empirical applications. The SEVI model does
not have the required representation, and this provides another perspective on
the absence of the IIA property in the SEVI model when $J\geq 3.$

One consequence of the lack of IIA is that we cannot consistently estimate the
preference parameters in a SEVI model using the random sampling approach. In
this approach, for each decision maker, we sample a random subset of
alternatives that contains the chosen alternative and treat this subset as if
it were the choice set faced by this decision maker. In the simplest case with
three total alternatives $(J=3)$, the reason for the inconsistency is that

$\frac{P^{\mathrm{SEVI}}(Y=1|v_{1},v_{2},v_{3})}{P^{\mathrm{SEVI}}(Y=1|v_{1},v_{2},v_{3})+P^{\mathrm{SEVI}}(Y=2|v_{1},v_{2},v_{3})}\neq
P^{\mathrm{SEVI}}(Y=1|v_{1},v_{2}).$

This contrasts with a LEVI-based logit, where such a random sampling procedure
can be used to reduce the number of alternatives without inducing
inconsistency. For a detailed discussion of the LEVI case, see page 64 of
Train, (2009).

We note in passing that IIA also does not hold for standard multinomial probit
models, i.e., the multinomial probit with iid normal random utility
components. See Section S.D in the supplementary appendix for details.

###  3.5 The Surplus Function: Compensating Variation and Identification

A fundamental concept in the discrete choice analysis is the surplus function,
defined as:

$W(v)=E\left[\max_{j\in\mathcal{J}}\left\\{v_{j}+\varepsilon_{j}\right\\}\right]-E\max_{j\in\mathcal{J}}\left\\{\varepsilon_{j}\right\\},$

where the expectations are taken with respect to the distribution of
$\left\\{\varepsilon_{1},\ldots,\varepsilon_{J}\right\\}.$ In the above, the
first term is the expected value of the maximum utility achieved, given the
vector of the systematic utilities
$v=\left(v_{1},\ldots,v_{J}\right)^{\prime}$ and the second term serves as a
constant benchmark.151515If these terms are not well defined, $W(v)$ can
always be expressed as
$E\left[\max_{j\in\mathcal{J}}\left\\{v_{j}+a_{j}\right\\}-\max_{j\in\mathcal{J}}\left\\{a_{j}\right\\}\right],$
which is well defined.  See, for example, McFadden, (1981). Among other uses,
the surplus function can be used to obtain the choice probabilities. More
specifically, the Williams-Daly-Zachary theorem161616It was termed the WDZ
theorem in Section 5.8 of McFadden, (1981). McFadden refers to Daly and
Zachery, (1978) and Williams, (1977) as forerunners of the result. says that

$P_{ij}=\frac{\partial W(v_{i1},\ldots,v_{iJ})}{\partial v_{ij}}.$

The specific form of the surplus function $W(v)$ depends on the distribution
of $\varepsilon_{1},\ldots,\varepsilon_{J}.$ When $\varepsilon_{j}\thicksim
iid$ LEVI, it is well-known that

$W^{\mathrm{LEVI}}(v)=\ln\left[\frac{1}{J}\sum_{j=1}^{J}\exp\left(v_{j}\right)\right],$

taking the form of a “log-sum”. Here the superscript “$\mathrm{LEVI}$”
signifies that the surplus function is based on the assumption that
$\varepsilon_{j}\thicksim iid$ LEVI.

The next proposition gives the form of the surplus function when
$\varepsilon_{j}\thicksim iid$ SEVI.

###### Proposition 3

If $\varepsilon_{j}\thicksim iid$ $\mathrm{SEVI}$ across $j=1,\ldots,J,$ then
the surplus function, denoted by $W^{\mathrm{SEVI}}(v),$ is

$\displaystyle W^{\mathrm{SEVI}}(v)$
$\displaystyle=\sum_{\ell=1}^{J}\left(-1\right)^{\ell}\sum_{k_{1}<k_{2}<\ldots<k_{\ell}:\left\\{k_{1},\ldots,k_{\ell}\right\\}\subseteq\mathcal{J}}\ln\left[\frac{1}{\ell}\sum_{k\in\left\\{k_{1},\ldots,k_{\ell}\right\\}}\exp\left(-v_{k}\right)\right]$
$\displaystyle=\sum_{\ell=1}^{J}\left(-1\right)^{\ell}\sum_{k_{1}<k_{2}<\ldots<k_{\ell}:\left\\{k_{1},\ldots,k_{\ell}\right\\}\subseteq\mathcal{J}}W^{\mathrm{LEVI}}(-\left(v_{k_{1}},\ldots,v_{k_{\ell}}\right)),$
(15)

where

$W^{\mathrm{LEVI}}(-\left(v_{k_{1}},\ldots,v_{k_{\ell}}\right))=\ln\left[\frac{1}{\ell}\sum_{k\in\left\\{k_{1},\ldots,k_{\ell}\right\\}}\exp\left(-v_{k}\right)\right].$

Furthermore, $W^{\mathrm{SEVI}}(v)$ is strictly convex.

Although the surplus function still contains “log-sum”, $W^{\mathrm{SEVI}}(v)$
is more complex than $W^{\mathrm{LEVI}}(v)$. Nevertheless, Proposition 3 gives
us a closed-form expression for $W^{\mathrm{SEVI}}(v)$ and shows that it is
strictly convex. The second representation given in (15) shows how
$W^{\mathrm{SEVI}}(v)$ is related to the log-sum over all subsets of choice
alternatives but with the systematic utilities flipped.

As the first application of Proposition 3, consider a SEVI model of the form:

$U_{ij}=V_{ij}+\varepsilon_{ij}\text{,
}V_{ij}=\tilde{V}_{ij}+\left(\mathcal{I}_{i}-K_{j}\right)\lambda,$ (16)

where $\varepsilon_{ij}\thicksim iid$ $\mathrm{SEVI}$ for $j=1,2,\ldots,J.$
Here we have introduced an additional covariate $\mathcal{I}_{i}-K_{j}$ as
part of the systematic utility, where $\mathcal{I}_{i}$ is the income of
individual $i$ and $K_{j}$ is the cost/price of alternative $j.$ The
coefficient $\lambda$ on this covariate represents the marginal utility of
income. In this model, the systematic utility that individual $i$ obtains from
alternative $j$ consists of two components: the utility
$\left(\mathcal{I}_{i}-K_{j}\right)\lambda$ derived from the expenditure on
the numeraire and the utility $\tilde{V}_{ij}$ derived from the
characteristics of alternative $j$. Such a formulation is common in applied
welfare analysis using discrete choice models.171717In this formulation,
income does not influence the choice probabilities. This assumption may be
reasonable when costs associated with the alternatives are negligible compared
to income or where measurement errors in income outweigh the costs. While it
is possible to account for income effects, we do not explore this extension
here.

Now suppose we change the cost/price for the $m$-th alternative from $K_{m}$
to $K_{m}+\Delta_{m}$ while keeping all else the same as before. For each
individual $i,$ the compensating variation $\mathrm{CV}$ is defined implicitly
via the equation below:

$\displaystyle\max_{j\in\mathcal{J}}\left\\{\tilde{V}_{ij}+\left[\mathcal{I}_{i}+\mathrm{CV}-\left(K_{j}+1\left\\{m=j\right\\}\Delta_{m}\right)\right]\lambda+\varepsilon_{ij}\right\\}$
$\displaystyle=\max_{j\in\mathcal{J}}\left\\{\tilde{V}_{ij}+\left(\mathcal{I}_{i}-K_{j}\right)\lambda+\varepsilon_{ij}\right\\},$

where $1\left\\{\cdot\right\\}$ is the indicator function. That is,
$\mathrm{CV}$ is the additional income that individual $i$ would need to
maintain the same utility when the cost of alternative $m$ changes from
$K_{m}$ to $K_{m}+\Delta_{m}.$ Solving for $\mathrm{CV}$ yields

$\mathrm{CV}=\frac{1}{\lambda}\left[\max\left\\{\tilde{V}_{ij}+\left(\mathcal{I}_{i}-K_{j}\right)\lambda+\varepsilon_{ij}\right\\}-\max_{j}\left\\{\tilde{V}_{ij}+\left[\mathcal{I}_{i}-\left(K_{j}+1\left\\{m=j\right\\}\Delta_{m}\right)\right]\lambda+\varepsilon_{ij}\right\\}\right].$

Using Proposition 3, we can find that the expected value of $\mathrm{CV}$
conditional on $V$ and $\Delta_{m}$ is

$E\left(\mathrm{CV}|V,\Delta_{m}\right)=\frac{1}{\lambda}\left[W^{\mathrm{SEVI}}(V-e_{m}\Delta_{m}\lambda)-W^{\mathrm{SEVI}}(V)\right],$
(17)

where $e_{m}$ is the unit vector with a value of 1 at the $m$-th element and
$0$ elsewhere. We can average $E\left(\mathrm{CV}|V,\Delta_{m}\right)$ over
the distribution of $V$ to get the expected compensating variation over the
population:

$E\left(\mathrm{CV}|\Delta_{m}\right):=E\left[E\left(\mathrm{CV}|V,\Delta_{m}\right)|\Delta_{m}\right].$

In empirical applications, we can estimate
$E\left(\mathrm{CV}|V,\Delta_{m}\right)$ and
$E\left(\mathrm{CV}|\Delta_{m}\right)$ using sample averages after plugging in
parameter estimates.

As a second and closely related application, we can employ the surplus
function to define the compensating variation when an alternative is removed
from the choice set. Using the same model as in (16), we investigate the
minimum compensation necessary for individual $i$ so that they would not be
worse off if alternative $k$ were eliminated from their choice set. Denote
this compensation as $\mathrm{CV}\left(k\right)$. Then,
$\mathrm{CV}\left(k\right)$ solves

$\max_{j\in\mathcal{J}\backslash
k}\left\\{\tilde{V}_{ij}+\left[\mathcal{I}_{i}+\mathrm{CV}\left(k\right)-K_{j}\right]\lambda+\varepsilon_{ij}\right\\}=\max_{j\in\mathcal{J}}\left\\{\tilde{V}_{ij}+\left(\mathcal{I}_{i}-K_{j}\right)\lambda+\varepsilon_{ij}\right\\}.$

By a similar argument used to derive (17), we find

$\displaystyle\mathrm{CV}\left(k\right)$
$\displaystyle=\frac{1}{\lambda}\left[\max_{j\in\mathcal{J}}\left\\{\tilde{V}_{ij}+\left(\mathcal{I}_{i}-K_{j}\right)\lambda+\varepsilon_{ij}\right\\}-\max_{j\in\mathcal{J}\backslash
k}\left\\{\tilde{V}_{ij}+\left[\mathcal{I}_{i}-K_{j}\right]\lambda+\varepsilon_{ij}\right\\}\right]$
$\displaystyle=\frac{1}{\lambda}\left[W^{\mathrm{SEVI}}(V)-W^{\mathrm{SEVI}}(V_{-k})+W^{\mathrm{SEVI}}(0_{J})-W^{\mathrm{SEVI}}(0_{J-1})\right],$

where

$W^{\mathrm{SEVI}}(0_{J})=E\left[\max_{j\in\mathcal{J}}\varepsilon_{ij}\right],W^{\mathrm{SEVI}}(0_{J-1})=E\left[\max_{j\in\mathcal{J}\backslash
k}\varepsilon_{ij}\right]\text{ for }\varepsilon_{ij}\thicksim
iid\mathrm{SEVI.}$

By definition, $\mathrm{CV}\left(k\right)$ represents compensation for an
individual with systematic utility $V.$ To emphasize the dependence of
$\mathrm{CV}\left(k\right)$ on $V$ and the distribution of random utilities,
we denote it as $\mathrm{CV}^{\mathrm{SEVI}}\left(k,V\right)$ hereafter.

We can define $\mathrm{CV}^{\mathrm{LEVI}}\left(k,V\right)$ similarly under
the LEVI model. For each value of $V,$ the ratio
$\mathrm{CV}^{\mathrm{SEVI}}\left(k,V\right)/\mathrm{CV}^{\mathrm{LEVI}}\left(k,V\right)$
quantifies the relative magnitude of the (conditional) compensating variations
across the SEVI and LEVI models. Figure 7 plots this ratio against
$\mathrm{CV}^{\mathrm{LEVI}}\left(k,V\right)$ when $V$ follows the same data
generating process as that in Figure 3, with one of the $X$’s regarded as the
price/cost. Each point in the figure represents the ratio for a particular
instance of $k$ and $V.$ Qualitatively, the figure resembles Figure 4, where
the general patterns are applicable to the DGP in Figure 3. Specifically, when
the CV under the LEVI model is low, the CV under the SEVI model tends to be
even lower, and conversely, when the CV under the LEVI model is high, the CV
under the SEVI model tends to be higher. These results are intuitive: removing
an alternative that is chosen more (or less) often should have a larger (or
smaller) welfare implication.

A drawback of Figure 7 is that each point in the figure does not clearly
indicate which alternative is excluded from the choice set. To address this,
we simulate the expected CVs, $E[\mathrm{CV}^{\mathrm{SEVI}}\left(k,V\right)]$
and $E[\mathrm{CV}^{\mathrm{LEVI}}\left(k,V\right)],$ for each alternative $k$
excluded, with respect to the distribution of $V$. Figure S.6 in the
supplementary appendix plots the ratio of these two expected CVs against the
label of the excluded alternative ($k$). From the figure, it can be observed
that the expected CV under the LEVI appears to be larger than that under the
SEVI, regardless of which alternative is excluded. In addition, the percentage
difference between the two expected CVs appears to grow with $J,$ the number
of alternatives.

Figure 7: Scatter plot of the ratio of compensating variations in the SEVI and
LEVI models
$(\mathrm{CV}^{\mathrm{SEVI}}(k,V)/\mathrm{CV}^{\mathrm{LEVI}}(k,V))$ against
the compensating variation in the LEVI model
$(\mathrm{CV}^{\mathrm{LEVI}}(k,V))$ with the same DGPs as in Figure 3
($X_{ij,\ell}$ is iid $N(0,\pi^{2}\omega_{j}^{2}/36))$

As a third application, we use Proposition 3 to establish the identification
of a SEVI model. Note that the model under the vector of systematic utilities
$V=(V_{1},\ldots,V_{J})$ is equivalent to that under
$V^{o}=(V_{1}-V_{J},\ldots,V_{J-1}-V_{J},0)$, as the choice problem depends
only on utility differences. For the purpose of studying identification, we
normalize the systematic utility of the last alternative to zero and maintain
this normalization in the rest of this subsection. Each vector of systematic
utilities $V\in\mathbb{R}_{o}^{J}:=\mathbb{R}^{J-1}\otimes\left\\{0\right\\}$
maps into a vector of choice probabilities
$P\in\mathbb{S}^{J-1}:=\\{P=\left(P_{1},\ldots,P_{J-1}\right):P_{j}\geq 0$ and
$\sum_{j=1}^{J-1}P_{j}\leq 1\\}.$ The question remains whether we can deduce
the systematic utilities from the choice probabilities.

Define the conjugate surplus function:

$W_{\ast}^{\mathrm{SEVI}}\left(P\right):=\sup_{v\in\mathbb{R}_{o}^{J}}\left[\sum_{j=1}^{J-1}v_{j}P_{j}-W^{\mathrm{SEVI}}(v)\right]\text{
for any }P\in\mathbb{S}^{J-1}.$

Since $W^{\mathrm{SEVI}}(v)$ is known and given in Proposition 3, it follows
by construction that all of $W_{\ast}^{\mathrm{SEVI}}\left(\cdot\right):$
$\mathbb{S}^{J}\rightarrow\mathbb{R}$ and $\left\\{\partial
W_{\ast}^{\mathrm{SEVI}}\left(\cdot\right)/\partial
P_{j}:\mathbb{S}^{J}\rightarrow\mathbb{R}\right\\}_{j=1}^{J-1}$ are known
functions. Using Theorem 7 of Sørensen and Fosgerau, (2022), we
have:181818This may be of independent interest in IO/marketing applications if
the objective is to derive the systematic utilities from the observed market
shares, but here we focus on identification.

$V_{j}=\left.\frac{\partial W_{\ast}^{\mathrm{SEVI}}\left(P\right)}{\partial
P_{j}}\right|_{P=\left[P^{\mathrm{SEVI}}(Y=1|V),\ldots,P^{\mathrm{SEVI}}\left(Y=J-1|V\right)\right]}$

for $j=1,\ldots,J-1.$ That is, we can plug the vector of choice probabilities
into $\left\\{\partial W_{\ast}^{\mathrm{SEVI}}\left(\cdot\right)/\partial
P_{j}\right\\}_{j=1}^{J-1}$ to recover the vector of systematic utilities.

Given that the vector of choice probabilities is identified, the vector of
systematic utilities is also identified. The problem of identifying the model
parameter $\beta_{0}$ in a SEVI model reduces to the problem of identifying
$\beta_{0}$ from the (joint) distribution of
$V^{\ast}\left(\beta_{0}\right):=(V_{1}\left(\beta_{0}\right)-V_{J}\left(\beta_{0}\right),\ldots,V_{J-1}\left(\beta_{0}\right)-V_{J}\left(\beta_{0}\right))\in\mathbb{R}^{J-1}.$
Intuitively, if we have an infinite number of independent draws from the
distribution of $V^{\ast}\left(\beta_{0}\right),$ can we pin down $\beta_{0}?$
Mathematically, if

$\Pr(V^{\ast}\left(\beta_{1}\right)\neq
V^{\ast}\left(\beta_{2}\right))>0\text{ for any }\beta_{1}\neq\beta_{2},$ (18)

then $\beta_{0}$ is identified. This identification condition requires that,
for two different model parameters, there is a positive probability that the
resulting vectors of (normalized) systematic utilities are different. For
example, in the linear case where
$V_{i}^{\ast}\left(\beta\right)=\left[\left(X_{i1}-X_{iJ}\right)\beta,\ldots,\left(X_{iJ-1}-X_{iJ}\right)\beta\right],$
this requires that there is no perfect multicollinearity among the variables
in $X_{ij}-X_{iJ}$ for at least one $j\in\left\\{1,\ldots,J-1\right\\}.$

The identification condition in (18) does not involve the distribution of
$\varepsilon_{ij}.$ Therefore, it is applicable to any RUM model, provided
that the unobserved random components
$\left(\varepsilon_{i1},\ldots,\varepsilon_{iJ}\right)$ is independent of the
systematic utility $\left(V_{i1},\ldots,V_{iJ}\right)$ and is absolutely
continuous with full support $\mathbb{R}^{J}.$ See Corollary 1 of Sørensen and
Fosgerau, (2022) for more discussions. In particular, this condition is
applicable to a LEVI model. Hence, a SEVI model is identified if and only if
the corresponding LEVI model is identified. No additional identification
challenges arise in a SEVI model.

##  4 The MLE and QMLE of a SEVI Model

###  4.1 The Estimators

Consider the following conditional SEVI model

$U_{ij}\left(\beta_{0}\right)=V_{ij}\left(\beta_{0}\right)+\varepsilon_{ij}\text{
and }V_{ij}\left(\beta_{0}\right)=X_{ij}\beta_{0}$

for $j=1,\ldots,J$ where $X_{ij}\in\mathbb{R}^{L}$ and
$\varepsilon_{ij}\thicksim iid$ $\mathrm{SEVI.}$ All other models in Remark 5
can be reformulated to take the above form. For example, if we want to include
alternative-specific constants, we can introduce a dummy variable for each
alternative and include the dummies for $\left(J-1\right)$ alternatives as
part of $X_{ij}$. This dummy-variable method can also be used to allow for
alternative-specific coefficients for any covariate, including individual-
specific characteristics.

Given a simple random sample
$\left\\{Y_{i},X_{i1},\ldots,X_{iJ}\right\\}_{i=1}^{n}$ from the above model,
we consider estimating $\beta_{0}$ by MLE and QMLE. The MLE is the maximum
likelihood estimator when the likelihood function is based on a correctly
specified SEVI model, and the QMLE is the maximum likelihood estimator based
on an incorrectly specified LEVI model.

The negative log-likelihood function of the model under the correct SEVI
specification is

$\ell_{\mathrm{SEVI}}\left(\beta\right)=\sum_{i=1}^{n}\ell_{i}^{\mathrm{SEVI}}\left(\beta\right)\text{
for
}\ell_{i}^{\mathrm{SEVI}}\left(\beta\right)=-\sum_{k=1}^{J}Y_{ik}\log\left(P_{ik}^{\mathrm{SEVI}}\left(\beta\right)\right),$

where $Y_{ik}:=1\left\\{Y_{i}=k\right\\}$ and
$P_{ik}^{\mathrm{SEVI}}\left(\beta\right):=P^{\mathrm{SEVI}}\left(Y_{i}=k|v_{i}\left(\beta\right)\right)$
is defined as in (6) with $v_{i}$ replaced by $v_{i}\left(\beta\right).$ By
minimizing the above negative log-likelihood function
$\ell_{\mathrm{SEVI}}\left(\beta\right),$ we obtain the MLE:

$\hat{\beta}_{\mathrm{MLE}}=\arg\min_{\beta\in\mathcal{B}}\ell_{\mathrm{SEVI}}\left(\beta\right),$

where $\mathcal{B}$ is a large enough compact set in $\mathbb{R}^{L}.$

To compute the MLE, we can use a gradient-based algorithm. The gradient
function is

$\frac{\partial\ell_{\mathrm{SEVI}}\left(\beta\right)}{\partial\beta}=\sum_{i=1}^{n}\frac{\partial\ell_{i}^{\mathrm{SEVI}}\left(\beta\right)}{\partial\beta}:=\sum_{i=1}^{n}S_{i}^{\mathrm{SEVI}}\left(\beta\right),$

where

$S_{i}^{\mathrm{SEVI}}\left(\beta\right)=\sum_{j=1}^{J}\frac{\partial\ell_{i}^{\mathrm{SEVI}}\left(\beta\right)}{\partial
v_{ij}\left(\beta\right)}\frac{\partial
v_{ij}\left(\beta\right)}{\partial\beta}=-\sum_{j=1}^{J}\sum_{k=1}^{J}\frac{Y_{ik}}{P_{ik}^{\mathrm{SEVI}}\left(\beta\right)}\frac{\partial
P_{ik}^{\mathrm{SEVI}}\left(\beta\right)}{\partial
v_{ij}\left(\beta\right)}X_{ij}^{\prime}.$

In the above, the expression of the derivative $\frac{\partial
P_{ik}^{\mathrm{SEVI}}\left(\beta\right)}{\partial v_{ij}\left(\beta\right)}$
is given in (12) or (13) after plugging in the parameter $\beta.$ The closed-
form expression for the gradient can be supplied to a gradient-based
optimization algorithm.

Given the score $S_{i}^{\mathrm{SEVI}}\left(\beta\right)$ and the estimator
$\hat{\beta}_{\mathrm{MLE}},$ we can estimate the asymptotic variance of
$\hat{\beta}_{\mathrm{MLE}}$ by

$\hat{\Omega}=\left[\frac{1}{n}\sum_{i=1}^{n}S_{i}^{\mathrm{SEVI}}(\hat{\beta}_{\mathrm{MLE}})S_{i}^{\mathrm{SEVI}}(\hat{\beta}_{\mathrm{MLE}})^{\prime}\right]^{-1}.$

Following standard arguments, we can show that
$\hat{\Omega}^{-1/2}\sqrt{n}(\hat{\beta}_{\mathrm{MLE}}-\beta_{0})\rightarrow^{d}N\left(0,I_{L}\right).$
Details are omitted here, as the theory is standard; See, for example, Chapter
13 of Wooldridge, (2002).

If we want to test $H_{0}:R\beta_{0}=r$ against $H_{1}:R\beta_{0}\neq r$ for
some $q\times L$ matrix $R$ with row rank $q$ and $q\times 1$ vector $r,$ we
can form the Wald statistic and show that it is asymptotically chi-squared
under the null:

$n\left(R\hat{\beta}_{\mathrm{MLE}}-r\right)\left(R\hat{\Omega}R^{\prime}\right)^{-1}\left(R\hat{\beta}_{\mathrm{MLE}}-r\right)\rightarrow^{d}\chi_{q}^{2}.$

In particular, we can use the square root of the diagonal elements of
$\left[\sum_{i=1}^{n}S_{i}^{\mathrm{SEVI}}(\hat{\beta}_{\mathrm{MLE}})S_{i}^{\mathrm{SEVI}}(\hat{\beta}_{\mathrm{MLE}})^{\prime}\right]^{-1}$,
denoted by $\hat{\sigma}_{\ell},\ell=1,\ldots,L,$ as the standard errors for
the elements of $\hat{\beta}_{\mathrm{MLE}}.$ We can construct a 95%
confidence interval as
$[\hat{\beta}_{\mathrm{MLE},\ell}-1.96\hat{\sigma}_{\ell},\hat{\beta}_{\mathrm{MLE},\ell}+1.96\hat{\sigma}_{\ell}].$

If the stochastic utilities follow SEVI, but we employ the maximum likelihood
method under the incorrect assumption that they follow LEVI, we obtain the
quasi-MLE (QMLE):191919Here, we focus on the QMLE based on the LEVI
likelihood. Other QMLEs may be considered. For instance, we might explore the
QMLE derived from the specification that the random utility components follow
an MEV distribution (see (5)). As long as the marginal distribution is
incorrectly specified, such an alternative QMLE will generally be inconsistent
for the preference parameters.

$\displaystyle\hat{\beta}_{\mathrm{QMLE}}$
$\displaystyle=\arg\min_{\beta\in\mathcal{B}}\ell_{\mathrm{LEVI}}\left(\beta\right)\text{
where
}\ell_{\mathrm{LEVI}}\left(\beta\right)=\sum_{i=1}^{n}\ell_{i}^{\mathrm{LEVI}}\left(\beta\right)\text{
for }$ (19) $\displaystyle\ell_{i}^{\mathrm{LEVI}}\left(\beta\right)$
$\displaystyle=-\sum_{k=1}^{J}Y_{ik}\log\left(P_{ik}^{\mathrm{LEVI}}\left(\beta\right)\right)=-\sum_{k=1}^{J}Y_{ik}\left[X_{ik}\beta\right]-\log\left\\{\sum_{j=1}^{J}\exp\left[X_{ij}\beta\right]\right\\}.$

The asymptotic properties of the QMLE are also standard results. In
particular, we use the sandwich form for estimating the asymptotic variance of
$\hat{\beta}_{\mathrm{QMLE}}$ when we know that the LEVI model is not the
correct model.

###  4.2 On the Estimation of Parameter Ratios

Under some standard regularity conditions, $\hat{\beta}_{\mathrm{MLE}}$ is
consistent for $\beta_{0},$ but in general, $\hat{\beta}_{\mathrm{QMLE}}$ is
not. However, the ratio of the individual elements of
$\hat{\beta}_{\mathrm{QMLE}}$, say
$\hat{\beta}_{\mathrm{QMLE},2}/\hat{\beta}_{\mathrm{QMLE},1},$ may still be
consistent for the true ratio, namely $\beta_{0,2}/\beta_{0,1},$ for a nonzero
$\beta_{0,1}.$ In this subsection, we provide sufficient conditions for the
consistency of the ratio estimator, even though the distribution of the random
utility component is misspecified.

We consider an RUM with $V_{ij}\left(\beta_{0}\right)=X_{ij}\beta_{0}$ and
_specify_ $\varepsilon_{ij}\thicksim iid\mathrm{Q}$ for some distribution
$\mathrm{Q}.$ This includes LEVI as an example but $\mathrm{Q}$ may be more
general and incorrectly specified. Denote the choice probability derived under
$\mathrm{Q}$ by $P^{\mathrm{Q}}(Y_{i}=k|V_{i}\left(\beta\right))$. Let
$\beta_{\mathrm{Q}}^{\ast}$ be the pseudo-true value defined by

$\beta_{\mathrm{Q}}^{\ast}=\arg\min_{\beta\in\mathcal{B}}-E\sum_{k=1}^{J}Y_{ik}\log
P^{\mathrm{Q}}(Y_{i}=k|V_{i}\left(\beta\right)),$

which is also the probability limit of
$\hat{\beta}_{\mathrm{Q}}:=\arg\min_{\beta\in\mathcal{B}}-n^{-1}\sum_{i=1}^{n}\sum_{k=1}^{J}Y_{ik}\log
P^{\mathrm{Q}}(Y_{i}=k|V_{i}\left(\beta\right)).$

We assume that $\beta_{\mathrm{Q}}^{\ast}$ is an interior point of the compact
set $\mathcal{B}$ so that $\beta_{\mathrm{Q}}^{\ast}$ satisfies the first
order zero-derivative conditions. The derivative of the population objective
function with respect to $\beta$ is

$-E\sum_{k=1}^{J}Y_{ik}\frac{\partial\log
P^{\mathrm{Q}}(Y_{i}=k|\mathcal{J},V_{i}\left(\beta\right))}{\partial\beta}=-EX_{i}^{\prime}\sum_{k=1}^{J}Y_{ik}\lambda_{ik}^{\mathrm{Q}}\left(X_{i}\beta\right)=-EX_{i}^{\prime}\Lambda_{i}^{\mathrm{Q}}\left(X_{i}\beta\right)\overrightarrow{Y}_{i},$

where

$\displaystyle X_{i}$
$\displaystyle=(X_{i1}^{\prime},\ldots,X_{iJ}^{\prime})^{\prime}\in\mathbb{R}^{J\times
L},$ $\displaystyle\lambda_{ik}^{\mathrm{Q}}\left(X_{i}\beta\right)$
$\displaystyle=\left.\frac{\partial\log
P^{\mathrm{Q}}(Y_{i}=k|\mathcal{J},V_{i}=v_{i})}{\partial
v_{i}}\right|_{v_{i}=X_{i}\beta}\in\mathbb{R}^{J\times 1},$
$\displaystyle\Lambda_{i}^{\mathrm{Q}}\left(X_{i}\beta\right)$
$\displaystyle=\left[\lambda_{i1}^{\mathrm{Q}}\left(X_{i}\beta\right),\ldots,\lambda_{iJ}^{\mathrm{Q}}\left(X_{i}\beta\right)\right]\in\mathbb{R}^{J\times
J},\text{
}\overrightarrow{Y}_{i}=(Y_{i1},\ldots,Y_{iJ})^{\prime}\in\mathbb{R}^{J\times
1}.$

So, $\beta_{\mathrm{Q}}^{\ast}$ satisfies
$E\left[X_{i}^{\prime}\Lambda_{i}^{\mathrm{Q}}(X_{i}\beta_{\mathrm{Q}}^{\ast})\overrightarrow{Y}_{i}\right]=0.$

###### Proposition 4

Assume that (i)
$\hat{\beta}_{\mathrm{Q}}\rightarrow^{p}\beta_{\mathrm{Q}}^{\ast}\in
Int(\mathcal{B)}$, and $\beta_{\mathrm{Q}}^{\ast}$ is the unique solution to
$E\left[X_{i}^{\prime}\Lambda_{i}^{\mathrm{Q}}\left(X_{i}\beta\right)\overrightarrow{Y}_{i}\right]=0;$

(ii) $\beta_{0}\neq 0$, and there exists a nonzero scalar $\delta_{1}^{\circ}$
such that
$E\left[X_{i}^{\prime}\Lambda_{i}^{\mathrm{Q}}\left(X_{i}\beta_{0}\delta_{1}^{\circ}\right)\overrightarrow{Y}_{i}\right]=0;$

(iii)
$E\left[X_{i,j,\ell}|X_{i1}\beta_{0},X_{i2}\beta_{0},\ldots,X_{iJ}\beta_{0}\right]=\Theta_{0,\ell}+\left(X_{ij}\beta_{0}\right)\Theta_{1,\ell}$
for some scalars $\Theta_{0,\ell}$ and $\Theta_{1,\ell}$ and for
$\ell=2,\ldots,L.$

Then: $\beta_{\mathrm{Q}}^{\ast}=\delta_{1}^{\circ}\beta_{0}.$

Proposition 4 shows that while the QMLE $\hat{\beta}_{\mathrm{Q}}$ may not be
consistent for $\beta_{0}$, it converges to $\beta_{0}$ up to a multiplicative
factor. In particular, if $\beta_{0,m_{1}}\neq 0,$ then

$\frac{\hat{\beta}_{\mathrm{Q,}m_{2}}}{\hat{\beta}_{\mathrm{Q,}m_{1}}}\rightarrow^{p}\frac{\beta_{0,m_{2}}}{\beta_{0,m_{1}}}.$

This says that the plug-in ratio estimator is consistent for the true ratio.

Assumption (iii) in Proposition 4 holds if $X_{i1},\ldots,X_{iJ}$ are
independent and each follows a multivariate normal or elliptical distribution.
The latter assumption is sufficient but not necessary. There may exist other
distributions of $X_{i1},\ldots,X_{iJ}$ or specific ways to relax the
independence assumption such that the proportionality result holds. In
empirical applications, the ratio estimator could still be close to the true
ratio even though Assumption (iii) does not hold.

Proposition 4 is proved along the lines of the proof of Theorem 1 in Ruud,
(1986), which does not cover multinomial choice models. Some adjustments to
the proof in Ruud, (1986) are needed to account for two main differences:
there is no constant term in a multinomial choice model and the covariate
$X_{i}$ in this model is a matrix rather than a vector.

###  4.3 The Choice between SEVI and LEVI

In empirical applications, it is often the case that neither the SEVI nor the
LEVI is correctly specified. However, we still have to decide which model to
use. To aid in this decision, we can employ the Akaike Information Criterion
(AIC) or the Bayesian Information Criterion (BIC). Since both models have the
same number of parameters, we can compare their empirical (quasi) log-
likelihoods, which leads to the likelihood ratio statistic given by

$\displaystyle\mathrm{LR}$
$\displaystyle:=\ell_{\mathrm{SEVI}}(\hat{\beta}_{\mathrm{SEVI}})-\ell_{\mathrm{LEVI}}(\hat{\beta}_{\mathrm{LEVI}})$
$\displaystyle=\sum_{i=1}^{n}\left[\ell_{i}^{\mathrm{SEVI}}\left(\hat{\beta}_{\mathrm{SEVI}}\right)-\ell_{i}^{\mathrm{LEVI}}\left(\hat{\beta}_{\mathrm{LEVI}}\right)\right].$

Note that in the previous section, $\hat{\beta}_{\mathrm{SEVI}}$ and
$\hat{\beta}_{\mathrm{LEVI}}$ were denoted as $\hat{\beta}_{\mathrm{MLE}}$ and
$\hat{\beta}_{\mathrm{QMLE}},$ respectively. However, in this section, both
the SEVI model and the LEVI model can be misspecified and so both
$\hat{\beta}_{\mathrm{SEVI}}$ and $\hat{\beta}_{\mathrm{LEVI}}$ can be quasi-
maximum likelihood estimators.

In empirical applications, the choice between the SEVI model and the LEVI
model can be based on the likelihood ratio. If $\mathrm{LR}>0$, then the LEVI
model is preferred.202020Note that $\ell_{\mathrm{SEVI}}\left(\cdot\right)$
and $\ell_{\mathrm{LEVI}}\left(\cdot\right)$ are the negative log-likelihood
functions so when $LR>0,$ we may conclude that the LEVI model fits the data
better. Conversely, if $\mathrm{LR}<0$, then the SEVI model is preferred. This
simple empirical rule of thumb is derived from the empirical goodness of fit
and serves as a practical guide for selecting between the two models. It is
worth pointing out again that this rule is also the model selection rule based
on the AIC or BIC, as the two models have the same number of parameters.

Another method to determine which model to use is by employing Vuong’s test
(Vuong, (1989)), which formally assesses the null hypothesis that the SEVI
model and the LEVI model provide equally good fits to the data. Let
$\beta_{\mathrm{SEVI}}^{\ast}$ and $\beta_{\mathrm{LEVI}}^{\ast}$ be the
respective limits of $\hat{\beta}_{\mathrm{SEVI}}$ and
$\hat{\beta}_{\mathrm{LEVI}}$. In the present setting, the null hypothesis of
Vuong’s test can be expressed as:

$H_{0}:E\left[\ell_{\mathrm{SEVI}}\left(\beta_{\mathrm{SEVI}}^{\ast}\right)|V\right]=E\left[\ell_{\mathrm{LEVI}}\left(\beta_{\mathrm{LEVI}}^{\ast}\right)|V\right]$

almost surely, where $V$ is the vector collecting the systematic utilities.
Note that

$E\left[\ell_{\mathrm{SEVI}}\left(\beta_{\mathrm{SEVI}}^{\ast}\right)|V\right]=-\sum_{k=1}^{J}P_{ik}^{o}\log\left(P_{ik}^{\mathrm{SEVI}}\left(\beta_{\mathrm{SEVI}}^{\ast}\right)\right)$

where $P_{ik}^{o}:=\Pr(Y_{i}=k|V_{i})$ is the true choice probability. The
right-hand side of the above equation is the cross entropy between
$\left(P_{i1}^{0},\ldots,P_{iJ}^{o}\right)$ and
($P_{i1}^{\mathrm{SEVI}}\left((\beta_{\mathrm{SEVI}}^{\ast}\right),\ldots,P_{iJ}^{\mathrm{SEVI}}\left(\beta_{\mathrm{SEVI}}^{\ast}\right)$),
which quantifies the difference between these two distributions. By adding a
constant $\sum_{k=1}^{J}P_{ik}^{o}\log\left(P_{ik}^{o}\right)$ to both sides,
we have

$E\left[\ell_{\mathrm{SEVI}}\left(\beta_{\mathrm{SEVI}}^{\ast}\right)|V\right]+\sum_{k=1}^{J}P_{ik}^{o}\log\left(P_{ik}^{o}\right)=-\sum_{k=1}^{J}P_{ik}^{o}\log\left(\frac{P_{ik}^{\mathrm{SEVI}}\left(\beta_{\mathrm{SEVI}}^{\ast}\right)}{P_{ik}^{o}}\right),$

where the right-hand side is the KL distance between the two distributions
$\left(P_{i1}^{0},\ldots,P_{iJ}^{o}\right)$ and
($P_{i1}^{\mathrm{SEVI}}\left(\beta_{\mathrm{SEVI}}^{\ast}\right),\ldots,P_{iJ}^{\mathrm{SEVI}}\left(\beta_{\mathrm{SEVI}}^{\ast}\right)$)
conditional on $V.$ Similar calculations can be performed for
$E\left[\ell_{\mathrm{LEVI}}\left(\beta_{\mathrm{LEVI}}^{\ast}\right)|V\right].$
Therefore, the null of Vuong’s test is that the SEVI and LEVI models are
equidistant from the true data generating process, with the “distance”
measured by either the cross-entropy or the KL distance.

Since the SEVI model and the LEVI model are nonnested,
$n^{-1/2}\left[\ell_{\mathrm{SEVI}}(\hat{\beta}_{\mathrm{SEVI}})-\ell_{\mathrm{LEVI}}(\hat{\beta}_{\mathrm{LEVI}})\right]$
is asymptotically normal. We can standardize the LR statistic to obtain the
$\mathcal{V}_{n}$ statistic of Vuong, (1989):

$\mathcal{V}_{n}=\frac{\sum_{i=1}^{n}\left[\ell_{i}^{\mathrm{SEVI}}\left(\hat{\beta}_{\mathrm{SEVI}}\right)-\ell_{i}^{\mathrm{LEVI}}\left(\hat{\beta}_{\mathrm{LEVI}}\right)\right]}{\left\\{\sum_{i=1}^{n}\left[\ell_{i}^{\mathrm{SEVI}}\left(\hat{\beta}_{\mathrm{SEVI}}\right)-\ell_{i}^{\mathrm{LEVI}}\left(\hat{\beta}_{\mathrm{LEVI}}\right)-n^{-1}\mathrm{LR}\right]^{2}\right\\}^{1/2}}.$

Following the arguments in Vuong, (1989), we can show that under the null
$\mathcal{V}_{n}\rightarrow^{d}N(0,1).$ This serves as the basis for both one-
sided tests and two-sided tests. For example, if
$\left|\mathcal{V}_{n}\right|>1.96,$ then we may reject the null that the SEVI
and LEVI models fit the data equally well at the 5% significance level. On the
other hand, if $\mathcal{V}_{n}>1.645,$ we may reject the null that the SEVI
model fits the data better at the 5% significance level. If
$\mathcal{V}_{n}<-1.645,$ we may reject the null that the LEVI model fits the
data better at the 5% significance level.

##  5 Simulation Evidence

We generate the data from a conditional SEVI model. More specifically, we
assume that

$Y_{i}=\arg\max_{j=1,\ldots,J}\left\\{X_{ij}\beta+\varepsilon_{ij}\right\\}$

where $\varepsilon_{ij}\thicksim iid$ SEVI over $j$ and $X_{ij,\ell}\thicksim
iidN(0,\pi^{2}\omega_{j}^{2}/36)$ or $X_{ij,\ell}\thicksim iidN(0,\pi^{2}/36)$
over $\ell=1,2$ and $3$. The DGPs are the same as the SEVI DGPs in Figures
3–5. Given a simple random sample
$\left\\{Y_{i},X_{i1},\ldots,X_{iJ}\right\\}_{i=1}^{n},$ we use the MLE and
QMLE in Section 4 to estimate $\beta_{0}=\left(1,2,1\right)^{\prime}.$ To
reiterate, the MLE is based on the correct specification that
$\varepsilon_{ij}\thicksim iid$ SEVI, and the QMLE is based on the incorrect
specification that $\varepsilon_{ij}$ $\thicksim iid$ LEVI.

###  5.1 Simulation with a Large Sample Size

We first consider the case that the sample size is large, and there is only
one simulation replication. The purpose of this exercise is to see how large
the misspecification bias is. Table 1 reports the estimates when the sample
size $n$ is $10,000.$ When $J=2,$ the MLE and QMLE give the same estimate of
$\beta_{0}.$ This is consistent with the result that the SEVI model and the
LEVI model are the same when there are only two alternatives. When $J>2,$ the
MLE and QMLE are quite different. While the MLE is close to the true parameter
vector for all values of $J$ considered, the QMLE grows with $J.$ For example,
when $J=8$ and $X_{ij,\ell}\thicksim iidN(0,\pi^{2}\omega_{j}^{2}/36),$ the
QMLE is $\left(1.48,2.99,1.49\right)$, each element of which is higher than
the corresponding true value by about 50%. The misspecification bias of the
QMLE is substantial. In line with Proposition 4, the ratio of the estimates of
any two elements of $\beta_{0}$, say
$\hat{\beta}_{\mathrm{QMLE},2}/\hat{\beta}_{\mathrm{QMLE},1},$ is close to the
true ratio $\beta_{0,2}/\beta_{0,1}.$ This is true regardless of whether the
attributes have the same variance across the alternatives or not. Proposition
4 permits variance heterogeneity across the alternatives.

| True |  | $J=2$ |  | $J=5$ |  | $J=8$ |  | $J=11$
---|---|---|---|---|---|---|---|---|---
| $(\boldsymbol{\beta}_{\boldsymbol{0}})$ |  | MLE | QMLE |  | MLE | QMLE |  | MLE | QMLE |  | MLE | QMLE
|  |  | $X_{ij,\ell}\thicksim iidN(0,\pi^{2}\omega_{j}^{2}/36)$
$\beta_{1}$ | 1.00 |  | 1.01 | 1.01 |  | 1.00 | 1.37 |  | 0.98 | 1.48 |  | 1.02 | 1.64
$\beta_{2}$ | 2.00 |  | 1.95 | 1.95 |  | 2.01 | 2.75 |  | 1.99 | 2.99 |  | 1.98 | 3.17
$\beta_{3}$ | 1.00 |  | 0.99 | 0.99 |  | 1.00 | 1.37 |  | 1.01 | 1.49 |  | 0.98 | 1.57
|  |  | $X_{ij,\ell}\thicksim iidN(0,\pi^{2}/36)$
$\beta_{1}$ | 1.00 |  | 1.01 | 1.01 |  | 1.01 | 1.35 |  | 0.99 | 1.51 |  | 1.00 | 1.64
$\beta_{2}$ | 2.00 |  | 2.03 | 2.03 |  | 2.05 | 2.80 |  | 1.98 | 3.09 |  | 1.98 | 3.37
$\beta_{3}$ | 1.00 |  | 1.03 | 1.03 |  | 1.03 | 1.38 |  | 1.03 | 1.56 |  | 1.00 | 1.64
Table 1: MLE and QMLE of the model parameters for various J

In a nonlinear model, instead of focusing on the parameter values, it can be
more informative to examine the (empirical) average partial effects, defined
by

$\displaystyle APE(j;\beta,\mathrm{SEVI})$
$\displaystyle=\frac{1}{n}\sum_{i=1}^{n}\frac{\partial
P_{ij}^{\mathrm{SEVI}}\left(\beta\right)}{\partial
X_{ij,1}}=\frac{1}{n}\sum_{i=1}^{n}\frac{\partial
P_{ij}^{\mathrm{SEVI}}\left(\beta\right)}{\partial
v_{ij}\left(\beta\right)}\beta_{1},$ $\displaystyle
APE(j;\beta,\mathrm{LEVI})$
$\displaystyle=\frac{1}{n}\sum_{i=1}^{n}\frac{\partial
P_{ij}^{\mathrm{LEVI}}\left(\beta\right)}{\partial
X_{ij,1}}=\frac{1}{n}\sum_{i=1}^{n}P_{ij}^{\mathrm{LEVI}}\left(\beta\right)\left(1-P_{ij}^{\mathrm{LEVI}}\left(\beta\right)\right)\beta_{1}.$

In the above, $P_{ij}^{\mathrm{SEVI}}$ and $P_{ij}^{\mathrm{LEVI}}$ are the
choice probabilities under the SEVI and LEVI specifications, respectively.
$APE(j;\beta,\mathrm{SEVI})$ and $APE(j;\beta,\mathrm{LEVI})$ are the
corresponding empirical APEs that measure the effect on the probability of
choosing alternative $j$ when a marginal change is made to one of alternative
$j$’s attributes. We focus on the first attribute $X_{ij,1}$ and plot the
ratios:

$\frac{APE(j;\hat{\beta}_{\mathrm{MLE}},\mathrm{SEVI})}{APE(j;\beta_{0},\mathrm{SEVI})}\text{
and
}\frac{APE(j;\hat{\beta}_{\mathrm{QMLE}},\mathrm{LEVI})}{APE(j;\beta_{0},\mathrm{SEVI})}$

against the label of each alternative $j=1,\ldots,J.$ Figure 8 reports the
results for the case that $X_{ij,\ell}\thicksim
iidN(0,\pi^{2}\omega_{j}^{2}/36)$ and makes it clear that the empirical APE is
not close to the target APE (i.e., $APE(j;\beta_{0},\mathrm{SEVI})$) when the
model is misspecified. This shows that the difference in the parameter
estimates due to model misspecification has substantial effects on APE
estimation.

Figure 8: Plots of the empirical APE under the correct SEVI specification and
the incorrect LEVI specification against alternative labels for a sample of
$n=10,000$ individuals

To further illustrate the difference between the SEVI and LEVI specifications,
we conduct an out-of-sample prediction exercise. Let
$\hat{\beta}_{\mathrm{MLE}}^{\left(J\right)}$ and
$\hat{\beta}_{\mathrm{QMLE}}^{\left(J\right)}$ be the MLE and QMLE of
$\beta_{0}$, respectively, when there are $J$ alternatives. We use
$\hat{\beta}_{\mathrm{MLE}}^{\left(J\right)}$ and
$\hat{\beta}_{\mathrm{QMLE}}^{\left(J\right)}$ to compute out-of-sample choice
probabilities for a certain choice problem. More specifically, we assume that
an out-of-sample individual chooses between the first two alternatives only,
and the rest $J-2$ alternatives are not available. In this case, the out-of-
sample choice probabilities are

$\hat{P}_{i_{o},\mathrm{MLE}}:=P\left(Y_{i_{o}}=1|X_{i_{o}};\hat{\beta}_{\mathrm{MLE}}^{\left(J\right)}\right)\text{
and
}\hat{P}_{i_{o},\mathrm{QMLE}}=P\left(Y_{i_{o}}=1|X_{i_{o}};\hat{\beta}_{\mathrm{QMLE}}^{\left(J\right)}\right)$

where

$P\left(Y_{i_{o}}=1|X_{i_{o}};\beta\right)=\frac{\exp(X_{i_{o},1}\beta)}{\exp(X_{i_{o},1}\beta)+\exp(X_{i_{o},2}\beta)}$

and $i_{o}\in\left\\{n+1,\ldots,2n\right\\}$ represents an out-of-sample
individual whose observation $(X_{i_{o}},Y_{i_{o}})$ was not used in
constructing the MLE or QMLE, but $\left(X_{i_{o}},Y_{i_{o}}\right)$ is
generated in the same way as any in-sample observation. Note that the number
of “out-of-sample” individuals is the same as the number of “in-sample”
individuals and both are equal to 10,000.

When there are only two alternatives in the out-of-sample prediction exercise,
the formulae for the choice probabilities are the same under SEVI or LEVI.
Hence, as given above, $\hat{P}_{i_{o},\mathrm{MLE}}$ and
$\hat{P}_{i_{o},\mathrm{QMLE}}$ are different only because they are based on
different estimated parameters.

Figure 9(a&b) reports the kernel density estimators (KDE) of
$\left\\{P\left(Y_{i_{o}}=1|X_{i_{o}};\beta\right)\right\\}_{i=n+1}^{2n}$ for
$\beta=\beta_{0},\hat{\beta}_{\mathrm{MLE}}^{\left(J\right)}$ and
$\hat{\beta}_{\mathrm{QMLE}}^{\left(J\right)}$ when $J=5$ and
$X_{ij,\ell}\thicksim iidN(0,\pi^{2}\omega_{j}^{2}/36).$ It is not surprising
that the KDE under $\hat{\beta}_{\mathrm{MLE}}^{\left(J\right)}$ is very close
to the target KDE, that is, the KDE under $\beta_{0}.$ When the sample size is
large, $\hat{\beta}_{\mathrm{MLE}}^{\left(J\right)}$ is very close to
$\beta_{0}$, as demonstrated in Table 1. As a result,
$P(Y_{i_{o}}=1|X_{i_{o}};\hat{\beta}_{\mathrm{MLE}}^{\left(J\right)})$ is
close to $P\left(Y_{i_{o}}=1|X_{i_{o}};\beta_{0}\right)$ and therefore, the
corresponding KDEs are close to each other. On the other hand, the KDE under
$\hat{\beta}_{\mathrm{QMLE}}^{\left(J\right)}$ is quite different from the
target KDE. This demonstrates that the model misspecification introduces not
only a significant bias in the parameter estimator but also a substantial
discrepancy in the predicted choice probabilities.

The KDE figure for the case that $J=5$ and $X_{ij,\ell}\thicksim
iidN(0,\pi^{2}/36)$ is qualitatively similar to Figure 9(a&b) and is thus
omitted. The KDE figure for the case that $J=11$ and $X_{ij,\ell}\thicksim
iidN(0,\pi^{2}\omega_{j}^{2}/36)$ is reported in Figure 9(c&d). The difference
between the KDE under $\hat{\beta}_{\mathrm{QMLE}}^{(J)}$ and the target KDE
appears to increase with $J.$

Figure 9: Kernel density estimation of the out-of-sample predictive
probabilities when $n=10,000,J=5,11$ and $X_{ij,\ell}\thicksim
iidN(0,\pi^{2}\omega_{j}^{2}/36)$

As final evidence that it matters whether the stochastic utility follows the
SEVI or LEVI distribution, we compute the out-of-sample predicted choice
probabilities when there are four alternatives $(J=4)$ for both the in-sample
estimation and out-of-sample prediction. For the latter, we compute
$\hat{P}_{i_{o},\mathrm{MLE}}^{\mathrm{SEVI}}(j):=P(Y_{i_{o}}=j|X_{i_{o}};\mathrm{SEVI},$
$\hat{\beta}_{\mathrm{MLE}}^{\left(4\right)})$ and
$\hat{P}_{i_{o},\mathrm{QMLE}}^{\mathrm{LEVI}}\left(j\right):=P(Y_{i_{o}}=j|X_{i_{o}};\mathrm{LEVI},\hat{\beta}_{\mathrm{QMLE}}^{\left(4\right)})$
where we have emphasized the dependence of the choice probability on the
distributional assumption imposed on the stochastic utility. We plot the
difference
$\hat{P}_{i_{o},\mathrm{MLE}}^{\mathrm{SEVI}}(j)-\hat{P}_{i_{o},\mathrm{QMLE}}^{\mathrm{LEVI}}(j)$
against the true choice probability
$P_{i_{o},0}^{\mathrm{SEVI}}(j):=P(Y_{i_{o}}=j|X_{i_{o}};\mathrm{SEVI},$
$\beta_{0})$ as a scatter plot. We omit the figure here, but it is available
as Figure S.7 in the supplementary appendix. The figure clearly shows that,
relative to the true choice probability, the difference can be substantial.

###  5.2 Simulation with Smaller Sample Sizes

In this subsection, we study the performances of MLE and QMLE in finite
samples. Using the same DGP as in the previous subsection, we consider the
sample size $n=500$, $1000$ and the number of alternatives $J=2,5,8,11.$ The
number of simulation replications is 5000. For each estimator, we compute the
bias, the empirical standard derivation, and the average of the standard
errors across the simulation replications. Furthermore, we construct a 95%
confidence interval for each element of $\beta_{0}$ following the standard
procedure of adding and subtracting 1.96 times the standard error and report
the empirical coverage of the CI (i.e., the proportion of times the so-
constructed confidence interval contains the true parameter value).

Table 2 reports the results when $X_{ij,\ell}\thicksim
iidN(0,\pi^{2}\omega_{j}^{2}/36)$ and $n=500$. The reported results are
representative of other cases with a different DGP for $X_{ij,\ell}$ or sample
size. A few patterns emerge. First, the bias of the MLE is small and decreases
with $J.$ On the other hand, the bias of the QMLE is large and increases with
$J.$ This is consistent with our large sample result when $n=10,000$ and there
is only one simulation replication. Here, the finite sample biases (with
$n=500$) are averaged over 5000 simulation replications. Second, for both the
MLE and QMLE, the empirical standard deviation decreases with $J.$ A larger
$J$ can be regarded as having more observations and, hence, smaller sampling
noises. Third, for both estimators, the average of the standard errors across
the simulation replications matches with the corresponding standard deviation
very well. This indicates that the standard errors are reliable estimates of
the standard deviation. Finally, for the MLE, the empirical coverage of the
95% confidence intervals is very close to 95%. This shows that the asymptotic
normal approximation is reliable. On the other hand, for the QMLE, the
empirical coverage of the 95% confidence intervals is much smaller than 95%.
The only exception is the case when $J=2$, where MLE and QMLE are identical.
In this case, there is a small and nearly invisible difference in the
empirical coverages, because the standard error for the MLE is based on the
non-robust asymptotic variance estimator while the standard error for the QMLE
is based on the robust sandwich asymptotic variance estimator. Both variance
estimators are consistent when $J=2.$ When $J>2,$ the under-coverage of the
CIs based on the QMLE is due to the bias in the QMLE. The larger the number of
alternatives $J$ is, the larger the bias is, and the smaller the empirical
coverage is.

| MLE |  | QMLE
---|---|---|---
| Empirical | Empirical | Average of | Empirical |  | Empirical | Empirical | Average of | Empirical
| Bias | StD | SE’s | Coverage |  | Bias | StD | SE’s | Coverage
| $J=2$
$\beta_{1}$ | 0.009 | 0.211 | 0.210 | 0.951 |  | 0.009 | 0.211 | 0.209 | 0.948
$\beta_{2}$ | 0.023 | 0.240 | 0.241 | 0.950 |  | 0.023 | 0.240 | 0.239 | 0.950
$\beta_{3}$ | 0.010 | 0.209 | 0.210 | 0.950 |  | 0.010 | 0.209 | 0.209 | 0.950
| $J=5$
$\beta_{1}$ | 0.003 | 0.105 | 0.107 | 0.957 |  | 0.366 | 0.141 | 0.142 | 0.261
$\beta_{2}$ | 0.010 | 0.136 | 0.137 | 0.950 |  | 0.738 | 0.175 | 0.173 | 0.004
$\beta_{3}$ | 0.004 | 0.106 | 0.107 | 0.958 |  | 0.367 | 0.142 | 0.142 | 0.266
| $J=8$
$\beta_{1}$ | 0.006 | 0.087 | 0.088 | 0.952 |  | 0.514 | 0.130 | 0.129 | 0.017
$\beta_{2}$ | 0.011 | 0.117 | 0.116 | 0.951 |  | 1.026 | 0.159 | 0.158 | 0.000
$\beta_{3}$ | 0.006 | 0.088 | 0.088 | 0.951 |  | 0.513 | 0.129 | 0.129 | 0.017
| $J=11$
$\beta_{1}$ | 0.004 | 0.080 | 0.080 | 0.950 |  | 0.600 | 0.124 | 0.123 | 0.000
$\beta_{2}$ | 0.007 | 0.105 | 0.107 | 0.955 |  | 1.201 | 0.149 | 0.150 | 0.000
$\beta_{3}$ | 0.007 | 0.081 | 0.080 | 0.946 |  | 0.605 | 0.126 | 0.123 | 0.001
Table 2: Finite sample performances of the MLE and QMLE

Next, we consider testing IIA using the case when $X_{ij,\ell}\thicksim
iidN(0,\pi^{2}/36),\ell=1,2,3$ and $n=500$ as an example. We follow the
standard procedure to perform the Hausman-McFadden test (cf., Hausman and
McFadden, (1984)). For each simulation replication, we first estimate the
model under the (incorrect) LEVI specification using the full sample. We then
carry out the same estimation procedure but use only the subsample of
individuals who choose either alternative 1 or 2. We compare these two
estimates using the Wald statistic based on the 95% critical value from the
chi-squared distribution with three degrees of freedom ($\chi^{2}(3)$). The
empirical rates of rejection over 5000 simulation replications for $J=5,8,$
and $11$ are 30.30%, 40.18%, and 47.52%, respectively. This indicates a fairly
good chance of the IIA being rejected. Following an IIA rejection, the
standard practice is to employ discrete choice models that do not exhibit the
IIA property, such as the mixed logit. However, in the present setting, such a
practice would lead to a wrong characterization of agents’ preferences.

We note that the LEVI model is misspecified for both the full sample and the
subsample of the individuals who choose either alternative 1 or 2. The
misspecification of the LEVI model for the full sample is obvious as the
stochastic utility follows the SEVI distribution rather than the LEVI
distribution. For the subsample with the two alternatives selected by the
econometrician, the LEVI model is still misspecified. From the point of view
of the econometrician, the conditional probability of choosing alternative 1
given that either alternative 1 or alternative 2 is chosen is

$\frac{\Pr(Y=1|\mathcal{J},V=v,\varepsilon_{j}\thicksim iid\text{
}\mathrm{SEVI}\text{ over
}j\in\mathcal{J})}{\Pr(Y=1|\mathcal{J},V=v,\varepsilon_{j}\thicksim iid\text{
}\mathrm{SEVI}\text{ over
}j\in\mathcal{J})+\Pr(Y=2|\mathcal{J},V=v,\varepsilon_{j}\thicksim iid\text{
}\mathrm{SEVI}\text{ over }j\in\mathcal{J})}.$ (20)

This is not equal to $\exp(v_{1})/[\exp(v_{1})+\exp(v_{2})]$ unless
$\mathcal{J}=\left\\{1,2\right\\}.$ However, the Hausman–McFadden test
requires at least three alternatives in total, as otherwise the test statistic
equals zero and the IIA null is never rejected. So, whenever the Hausman-
McFadden test is meaningful, the LEVI model is misspecified for any subsample
with at least two alternatives.

Finally, we consider using Vuong’s test and AIC/BIC to decide which model fits
the data better. When the SEVI is the correct model, $X_{ij,\ell}\thicksim
iidN(0,\pi^{2}\omega_{j}^{2}/36)$ and $n=500$, the fractions of the test
statistic $\mathcal{V}_{n}$ less than $-1.645$ across 5000 simulation
replications are 47.08%, 69.08, and 80.82%, respectively, for $J=5,8,$ and
$11$. The corresponding fractions of the test statistic less than $0$ are
92.56%, 97.86%, and 99.32%, respectively. These results underscore the
reasonable power of Vuong’s test and the high accuracy of the AIC/BIC in
selecting the correct SEVI model. Results are similar when
$X_{ij,\ell}\thicksim iidN(0,\pi^{2}/36)$ and for other sample sizes. On the
other hand, when the LEVI is the correct model, $X_{ij,\ell}\thicksim
iidN(0,\pi^{2}\omega_{j}^{2}/36)$ and $n=500$, we observe that the test
statistic $\mathcal{V}_{n}$ exceeds 1.645 in 30.58%, 48.22%, and 60.72% of the
cases across 5000 simulation replications for $J=5,8,$ and $11$, respectively.
The corresponding percentages for $\mathcal{V}_{n}$ greater than zero are
87.56%, 94.94%, and 97.18% for $J=5,8,$ and $11$, respectively. This affirms
the power of Vuong’s test and the effectiveness of using the AIC/BIC in
selecting the correct LEVI model. Therefore, regardless of whether the SEVI or
LEVI model is the true model, Vuong’s test has good power and the AIC and BIC
perform very well in correctly identifying the true model.

###  5.3 Computation Time Comparison

In this subsection, we compare the computation times required for estimating
the LEVI and SEVI models. Additionally, we include the multinomial probit
model with iid normal errors with variance $\pi^{2}/6$ in our comparison. For
the sake of uniformity with the nomenclature used for the other two models, we
will henceforth refer to this restricted multinomial probit model as the NORM
model.

The data are generated according to
$Y_{i}=\max_{j=1}^{J}\left\\{X_{ij}\beta+\varepsilon_{ij}\right\\}$ where
$X_{ij,\ell}\thicksim iidN(0,\pi^{2}\omega_{j}^{2}/36)$ over $\ell=1,\ldots,L$
(cf., equation (10)) and $\varepsilon_{ij}\thicksim iid$ $\mathrm{SEVI}$ over
$j.$ We fit the LEVI, SEVI, NORM models to the so-generated data. All models
have the same correct linear specification
$V_{ij}\left(\beta\right)=X_{ij}\beta$ for the systematic utility but differ
with respect to the distributional specification of the random components.

We use a sample size of 1000, explore different numbers of alternatives $(J)$
ranging from 2 to 16, and consider two different numbers of attributes $(L=3$
and $L=10)$. To mimic real data analysis, we simulate only one sample with
model parameter $\beta_{0}$ drawn randomly from a multivariate standard normal
distribution.

We document the time needed to obtain the ML/QML estimate of the model
parameters for each of the three models. The time represents the seconds spent
by our MATLAB program to find each point estimate. All computations are
performed on a desktop PC equipped with an Intel i7-13700F processor and 32GB
of RAM. While no parallel method is employed,212121As noted earlier, this
represents one potential direction to explore for improving the computational
speed of the SEVI model for larger values of $J.$ we strive to vectorize all
operations as much as possible. For the SEVI model, the choice probabilities
are computed using the formula in Theorem 1. In the case of the NORM model,
the choice probabilities are obtained using the Geweke–Hajivassiliou–Keane
(GHK) simulator with 500 simulations (Geweke and Keane, (2001); Hajivassiliou
and McFadden, (1998); Keane and Wolpin, (1994)).

Figure 10: Comparison of computation times for the LEVI, SEVI, and NORM (MNP
with $\varepsilon_{ij}\thicksim iidN(0,\pi^{2}/6))$ models when
$V_{ij}=X_{ij}\beta$

Figure 10 plots the computation time against the number of alternatives. On
one hand, when $J\leq 12,$ the computation time for the SEVI model is
comparable to that for the LEVI model. When $J>12$, the computation time for
the SEVI model starts to increase with $J$ and becomes significantly larger
than that for the LEVI model, especially when $J$ reaches $16.$ On the other
hand, when $J\leq 12,$ the NORM model exhibits a clear computational
disadvantage compared to the LEVI and SEVI models. While the computation time
for each of the latter two models is less than 27 seconds when the number of
attributes is 10 and the number of alternatives is 12, the corresponding
computation time for the NORM model is more than 1000 seconds. In our
experiment, the SEVI model starts to become computationally as costly as the
NORM model only when $J=16.$

While the first DGP contains no alternative specific constant, the second
group of DGPs we consider does. We examine two scenarios: one with only
alternative-specific constants where $V_{ij}=Z_{i}\beta_{zj}$ for $Z_{i}=1$
(hereafter, ASC-only) and another with alternative-specific constants
$\left(Z_{i}\right)$ and three attributes $X_{ij,\ell}\thicksim
iidN(0,\pi^{2}\omega_{j}^{2}/36),\ell=1,2,3$. In the latter case,
$V_{ij}=Z_{i}\beta_{zj}+X_{ij,1}\beta_{x1}+X_{ij,2}\beta_{x2}+X_{ij,3}\beta_{x3}$
where $Z_{i}=1.$ In both cases, $\varepsilon_{ij}\thicksim iid$
$\mathrm{SEVI}$ over $j=1,\ldots,J.$ For each dataset, we fit the LEVI, SEVI,
and NORM models with correctly specified system utility. Figure 11 reports the
computation times. The general patterns observed in Figure 10 carry over to
this figure. When $J\leq 12,$ the SEVI and LEVI models require less than 65
seconds for estimation, with much shorter times for a smaller $J.$ In
contrast, the time needed for the NORM model grows rapidly as $J$ increases
from 2 to 12. For example, when $J=10$ and $12,$ the required times for the
NORM model for the ASC-only case are 2274 and 5159 seconds, respectively. In
other words, estimating a standard multinomial probit model takes between half
an hour and two hours when $J$ is about 10 or 12, whereas the corresponding
time for a SEVI model is just about 1 minute. As before, the SEVI model
becomes computationally less competitive for $J>12$ and has a computational
cost starting to approach the NORM model when $J$ reaches 16.

Figure 11: Comparison of computation times for the LEVI, SEVI, and NORM
(standard MNP) models when $V_{ij}=Z_{i}\beta_{zj}$ or
$V_{ij}=Z_{i}\beta_{zj}+X_{ij}\beta_{x}$

##  6 Some Extensions

###  6.1 Mixed LEVI-SEVI Model

One way to understand the random components in an RUM is to conceptualize them
as capturing private signals exclusive to the individual agent making the
choice, yet undisclosed to the econometrician (cf., Manski, (1977)). These
signals may exhibit left or right skewness, varying across individuals. This
variability suggests the possibility of two distinct groups of individuals:
one characterized by the LEVI type and the other by the SEVI type. While the
group membership of a specific individual remains unobservable, it is still
useful to estimate the relative sizes of these two groups for the purpose of
prediction and policy analysis.

Assume that individuals in the population have identical preference parameters
but may differ in terms of random utility components. We postulate that with a
probability of $\rho_{0},$ $\varepsilon_{ij}\thicksim iid$ $\mathrm{SEVI}$
over $j=1,\ldots,J$ and with a probability of $1-\rho_{0},$
$\varepsilon_{ij}\thicksim iid$ $\mathrm{LEVI}$ over $j=1,\ldots,J.$ In such a
mixed LEVI-SEVI model, the probability of a randomly chosen individual, say
individual $i,$ selecting alternative $j,$ given the vector of systematic
utilities $v_{i},$ is

$\Pr\left(Y_{i}=j|V_{i}=v_{i}\right)=\rho_{0}P^{\mathrm{SEVI}}\left(Y_{i}=j|V_{i}=v_{i}\right)+\left(1-\rho_{0}\right)P^{\mathrm{LEVI}}\left(Y_{i}=j|V_{i}=v_{i}\right).$