模拟离散选择方法：Logit与混合模型详解

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《离散选择方法与模拟》是一本于2009年出版的专业书籍，由Kenneth Train撰写，主要探讨了在经济学和运输规划等领域广泛应用的离散选择模型（Discrete Choice Model）。该模型是研究个体在有限选项中做出选择的一种统计分析工具，特别关注如Logit、Probit和Mixed Logit等模型，这些模型广泛用于预测消费者行为、交通出行模式分析以及政策评估。离散选择方法的核心思想是通过估计每个选项的效用，来解释个体如何在多个可能的选择中进行决策。Logit模型是最常见的离散选择模型之一，它基于最大似然估计，假设个体的选择遵循伯努利分布，使得模型易于处理多属性和不确定性的决策问题。Probit模型则基于正态分布，适合于那些连续变量的效应更为显著的情况。而Mixed Logit模型则引入了随机参数，允许不同个体对同一选项的偏好存在差异，增加了模型的灵活性。书中不仅详细介绍了理论基础，还涵盖了如何通过模拟技术来实施这些模型，这对于理解和应用这些复杂的方法至关重要。作者Kenneth Train不仅是该领域的专家，还与Daniel McFadden共同工作，后者在离散选择模型的发展中也扮演了重要角色。这本书是对离散选择理论的重要贡献，提供了深入理解个体决策过程的实用工具，并且自出版以来已被引用4,616次，显示出其在学术界和实践中的影响力。此外，书中还包含了关于数据可靠性（如CV响应的可靠性）的讨论，强调了实际应用中数据质量对模型结果的重要性。该书的修订和增强版本反映了作者对模型持续的研究兴趣和改进，以及与同行们在相关项目上的合作。《离散选择方法与模拟》是一本深度解析离散选择模型并强调实证应用的著作，对于经济学者、政策分析师和运输工程师来说，它提供了一个不可或缺的参考资源，帮助他们更好地理解和应用离散选择理论来解决实际问题。

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6 Introduction

As we will see in the chapters to follow, this simulator, while easy to

understand, has some unfortunate properties. Choice probabilities can

often be expressed as averages of other statistics, rather than the average

of an indicator function. The simulators based on these other statistics

are calculated analogously, by taking draws from the density, calculating

the statistic, and averaging the results. Probit (in Chapter 5) is the most

prominent example of a model estimated by complete simulation. Vari-

ous methods of simulating the probit probabilities have been developed

based on averages of various statistics over various (related) densities.

1.2.3. Partial Simulation, Partial Closed Form

So far we have provided two polar extremes: either solve the

integral analytically or simulate it. In many situations, it is possible to

do some of both.

Suppose the random terms can be decomposed into two parts labeled

and ε

. Let the joint density of ε

and ε

be f (ε) = f (ε

,ε

). The

joint density can be expressed as the product of a marginal and a condi-

tional density: f (ε

,ε

) = f (ε

|ε

) · f (ε

). With this decomposition,

the probability in equation (1.1) can be expressed as

P(y |x) =



I [h(x,ε) = y] f (ε) dε







I [h(x,ε

,ε

) = y] f (ε

|ε

) dε



f (ε

) dε

Now suppose that a closed form exists for the integral in large brackets.

Label this formula g(ε

) ≡



I [h(x,ε

,ε

) = y] f (ε

|ε

) dε

, which

is conditional on the value of ε

. The probability then becomes

P(y |x) =



g(ε

) f (ε

) dε

. If a closed-form solution does not ex-

ist for this integral, then it is approximated through simulation. Note

that it is simply the average of g over the marginal density of ε

. The

probability is simulated by taking draws from f (ε

), calculating g(ε

)

for each draw, and averaging the results.

This procedure is called convenient error partitioning (Train, 1995).

The integral over ε

given ε

is calculated exactly, while the integral over

is simulated. There are clear advantages to this approach over com-

plete simulation. Analytic integrals are both more accurate and easier to

calculate than simulated integrals. It is useful, therefore, when possible,

to decompose the random terms so that some of them can be integrated

analytically, even if the rest must be simulated. Mixed logit (in Chap-

ter 6) is a prominent example of a model that uses this decomposition

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Introduction 7

effectively. Other examples include Gourieroux and Monfort’s (1993)

binary probit model on panel data and Bhat’s (1999) analysis of ordered

responses.

1.3 Outline of Book

Discrete choice analysis consists of two interrelated tasks: speciﬁcation

of the behavioral model and estimation of the parameters of that model.

Simulation plays a part in both tasks. Simulation allows the researcher to

approximate the choice probabilities that arise in the behavioral model.

As we have stated, the ability to use simulation frees the researcher

to specify models without the constraint that the resulting probabilities

must have a closed form. Simulation also enters the estimation task.

The properties of an estimator, such as maximum likelihood, can change

when simulated probabilities are used instead of the actual probabilities.

Understanding these changes, and mitigating any ill effects, is important

for a researcher. In some cases, such as with Bayesian procedures, the

estimator itself is an integral over a density (as opposed to the choice

probability being an integral). Simulation allows these estimators to be

implemented even when the integral that deﬁnes the estimator does not

take a closed form.

The book is organized around these two tasks. Part I describes be-

havioral models that have been proposed to describe the choice process.

The chapters in this section move from the simplest model, logit, to

progressively more general and consequently more complex models. A

chapter is devoted to each of the following: logit, the family of gener-

alized extreme value models (whose most prominent member is nested

logit), probit, and mixed logit. This part of the book ends with a chapter

titled “Variations on a Theme,” which covers a variety of models that

build upon the concepts in the previous chapters. The point of this chap-

ter is more than simply to introduce various new models. The chapter

illustrates the underlying concept of the book, namely, that researchers

need not rely on the few common speciﬁcations that have been pro-

grammed into software but can design models that reﬂect the unique

setting, data, and goals of their project, writing their own software and

using simulation as needed.

Part II describes estimation of the behavioral models. Numerical max-

imization is covered ﬁrst, since most estimation procedures involve

maximization of some function, such as the log-likelihood function.

We then describe procedures for taking draws from various kinds of

densities, which are the basis for simulation. This chapter also describes

different kinds of draws, including antithetic variants and quasi-random

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8 Introduction

sequences, that can provide greater simulation accuracy than indepen-

dent random draws. We then turn to simulation-assisted estimation, look-

ing ﬁrst at classical procedures, including maximum simulated likeli-

hood, method of simulated moments, and method of simulated scores.

Finally, we examine Bayesian estimation procedures, which use simula-

tion to approximate moments of the posterior distribution. The Bayesian

estimator can be interpreted from either a Bayesian or classical perspec-

tive and has the advantage of avoiding some of the numerical difﬁculties

associated with classical estimators. The power that simulation provides

when coupled with Bayesian procedures makes this chapter a ﬁtting

ﬁnale for the book.

1.4 Topics Not Covered

I feel it is useful to say a few words about what the book does not cover.

There are several topics that could logically be included but are not.

One is the branch of empirical industrial organization that involves esti-

mation of discrete choice models of consumer demand on market-level

data. Customer-level demand is speciﬁed by a discrete choice model,

such as logit or mixed logit. This formula for customer-level demand is

aggregated over consumers to obtain market-level demand functions that

relate prices to shares. Market equilibrium prices are determined as the

interaction of these demand functions with supply, based on marginal

costs and the game that the ﬁrms are assumed to play. Berry (1994)

and Berry et al. (1995) developed methods for estimating the demand

parameters when the customer-level model takes a ﬂexible form such as

mixed logit. The procedure has been implemented in numerous markets

for differentiated goods, such as ready-to-eat cereals (Nevo, 2001).

I have decided not to cover these procedures, despite their importance

because doing so would involve introducing the literature on market-

level models, which we are not otherwise considering in this book. For

market demand, price is typically endogenous, determined by the in-

teraction of demand and supply. The methods cited previously were

developed to deal with this endogeneity, which is probably the central

issue with market-level demand models. This issue does not automati-

cally arise in customer-level models. Prices are not endogenous in the

traditional sense, since the demand of the customer does not usually

affect market price. Covering the topic is therefore not necessary for our

analysis of customers’ choices.

It is important to note, however, that various forms of endogeneity

can indeed arise in customer-level models, even if the traditional type of

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Introduction 9

endogeneity does not. For example, suppose a desirable attribute of prod-

ucts is omitted from the analysis, perhaps because no measure of it exists.

Price can be expected to be higher for products that have high levels of

this attribute. Price therefore becomes correlated with the unobserved

components of demand, even at the customer level: the unobserved part

of demand is high (due to a high level of the omitted attribute) when

the price is high. Estimation without regard to this correlation is incon-

sistent. The procedures cited above can be applied to customer-level

models to correct for this type of endogeneity, even though they were

originally developed for market-level data. For researchers who are con-

cerned about the possibility of endogeneity in customer-level models,

Petrin and Train (2002) provide a useful discussion and application of

the methods.

A second area that this book does not cover is discrete–continuous

models. These models arise when a regression equation for a continuous

variable is related in any of several ways to a discrete choice. The most

prominent situations are the following.

1. The continuous variable depends on a discrete explanatory

variable that is determined endogenously with the dependent

variable. For example, consider an analysis of the impact of job-

training programs on wages. A regression equation is speciﬁed

with wages as the dependent variable and a dummy variable for

whether the person participated in a job-training program. The

coefﬁcient of the participation dummy indicates the impact of

the program on wages. The situation is complicated, however, by

the fact that participation is voluntary: people choose whether to

participate in job-training programs. The decision to participate

is at least partially determined by factors that also affect the per-

son’s wage, such as the innate drive, or “go-for-it” attitude, of the

person. Estimation of the regression by ordinary least squares is

biased in this situation, since the program-participation dummy

is correlated with the errors in the wage equation.

2. A regression equation is estimated on a sample of observations

that are selected on the basis of a discrete choice that is de-

termined endogenously with the dependent variable. For exam-

ple, a researcher might want to estimate the effect of weather

on peak energy load (that is, consumption during the highest-

demand hour of the day). Data on energy loads by time of day are

available only for households that have chosen time-of-use rates.

However, the households’ choice of rate plan can be expected

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10 Introduction

to be related to their energy consumption, with customers who

have high peak loads tending not to choose time-of-use rates,

since those rates charge high prices in the peak. Estimation of

the regression equation on this self-selected sample is biased

unless the endogeneity of the sample is allowed for.

3. The continuous dependent variable is truncated. For example,

consumption of goods by households is necessarily positive.

Stated statistically, consumption is truncated below at zero, and

for many goods (such as opera tickets) observed consumption

is at this truncation point for a large share of the population.

Estimation of the regression without regard to the truncation

can cause bias.

The initial concepts regarding appropriate treatment of discrete–

continuous models were developed by Heckman (1978, 1979) and Dubin

and McFadden (1984). These early concepts are covered in my earlier

book (Train, 1986, Chapter 5). Since then, the ﬁeld has expanded tremen-

dously. An adequate discussion of the issues and procedures would take

a book in itself. Moreover, the ﬁeld has not reached (at least in my view)

the same type of juncture that discrete choice modeling has reached.

Many fundamental concepts are still being hotly debated, and poten-

tially valuable new procedures have been introduced so recently that

there has not been an opportunity for researchers to test them in a vari-

ety of settings. The ﬁeld is still expanding more than it is coalescing.

There are several ongoing directions of research in this area. The

early procedures were highly dependent on distributional assumptions

that are hard to verify. Researchers have been developing semi- and

nonparametric procedures that are hopefully more robust. The special

1986 issue of the Journal of Econometrics provides a set of important

articles on the topic. Papers by Lewbel and Linton (2002) and Levy

(2001) describe more recent developments. Another important devel-

opment concerns the representation of behavior in these settings. The

relation between the discrete and continuous variables has been gen-

eralized beyond the fairly simple representation that the early methods

assumed. For example, in the context of job training, it is likely that

the impact of the training differs over people and that people choose to

participate in the training program on the basis of the impact it will have

on them. Stated in econometric terms: the coefﬁcient of the participation

dummy in the wage equation varies over people and affects the value of

the dummy. The dummy is correlated with its own coefﬁcient, as well

as with the unobserved variables that enter the error of the regression.

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