A Practitioner's Guide To Estimation Of Random Coefficients Logit S Demand BLP

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A Practitioner’s Guide to
Estimation of Random-Coef cients
Logit Models of Demand
Aviv Nevo
University of California–Berkeley, Berkeley,
CA 94720-388 0 and
NBER

Estimation of demand is at the heart of many recent studies that examine questions of market power, mergers, innovation, and valuation of new
brands in differentiated-products markets. This paper focuses on one of the
main methods for estimating demand for differentiated products: randomcoefcients logit models. The paper carefully discusses the latest innovations
in these methods with the hope of increasing the understanding, and therefore the trust among researchers who have never used them, and reducing
the difculty of their use, thereby aiding in realizing their full potential.

1. Introduction
Estimation of demand has been a key part of many recent studies
examining questions regarding market power, mergers, innovation,
and valuation of new brands in differentiated-product industries.1
This paper explains the random-coefcients (or mixed) logit methodology for estimating demand in differentiated-product markets using
An earlier version of this paper circulated under the title “A Research Assistant’s Guide
to Random Coefcients Discrete Choice Model of Demand.” I wish to thank Steve Berry,
Iain Cockburn, Bronwyn Hall, Ariel Pakes, various lecture and seminar participants,
and an anonymous referee for comments, discussions, and suggestions. Financial support from the UC Berkeley Committee on Research Junior Faculty Grant is gratefully
acknowledged.
1. Just to mention some examples, Bresnahan (1987) studies the 1955 price war in the
automobile industry; Gasmi et al. (1992) empirically study collusive behavior in a softdrink market; Hausman et al. (1994) study the beer industry; Berry et al. (1995, 1999)
examine equilibrium in the automobile industry and its implications for voluntary trade
restrictions; Goldberg (1995) uses estimates of the demand for automobiles to investigate trade policy issues; Hausman (1996) studies the welfare gains generated by a new
brand of cereal; Berry et al. (1996) study hubs in the airline industry; Bresnahan et al.
(1997) study rents from innovation in the computer industry; Nevo (2000a, b) examines
price competition and mergers in the ready-to-eat cereal industry; Davis (1998) studies
spatial competition in movie theaters; and Petrin (1999) studies the welfare gains from
the introduction of the minivan.
© 2000 Massachusetts Institute of Technology.
Journal of Economics & Management Strategy, Volume 9, Number 4, Winter 2000, 513–548

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Journal of Economics & Management Strategy

market-level data. This methodology can be used to estimate the demand for a large number of products using market data and allowing
for the endogenity of price. While this method retains the benets of
alternative discrete-choice models, it produces more realistic demand
elasticities. With better estimates of demand, we can, for example, better judge market power, simulate the effects of mergers, measure the
benets from new goods, or formulate innovation and competition
policy. This paper carefully discusses the recent innovations in these
methods with the intent of reducing the barriers to entry and increasing the trust in these methods among researchers who are not familiar
with them.
Probably the most straightforward approach to specifying demand for a set of closely related but not identical products is to specify
a system of demand equations, one for each product. Each equation
species the demand for a product as a function of its own price, the
price of other products, and other variables. An example of such a
system is the linear expenditure model (Stone, 1954), in which quantities are linear functions of all prices. Subsequent work has focused
on specifying the relation between prices and quantities in a way that
is both exible (i.e., allows for general substitution patterns) and consistent with economic theory.2
Estimating demand for differentiated products adds two additional nontrivial concerns. The rst is the large number of products,
and hence the large number of parameters to be estimated. Consider, for example, a constant-elasticity or log–log demand system,
in which logarithms of quantities are linear functions of logarithms of
all prices. Suppose we have 100 differentiated products; then without
additional restrictions this implies estimating at least 10,000 parameters (100 demand equations, one for each product, with 100 prices
in each). Even if we impose symmetry and adding up restrictions,
implied by economic theory, the number of parameters will still be
too large to estimate them. The problem becomes even harder if we
want to allow for more general substitution patterns.
An additional problem, introduced when estimating demand for
differentiated products, is the heterogeneity in consumer tastes: If all
consumers are identical, then we would not observe the level of differentiation we see in the marketplace. One could assume that preferences are of the right form [the Gorman form: see Gorman (1959)],
so that an aggregate, or average, consumer exists and has a demand
function that satises the conditions specied by economic theory.3
2. Examples include the Rotterdam model (Theil, 1965; Barten, 1966), the translog
model (Christensen et al., 1975), and the almost ideal demand system (Deaton and
Muellbauer, 1980).
3. For an example of a representative consumer approach to demand for differentiated products see Dixit and Stiglitz (1977) or Spence (1976).

Random-Coefcients Logit Models of Demand

515

However, the required assumptions are strong and for many applications seem to be empirically false. The difference between an aggregate
model and a model that explicitly reects individual heterogeneity
can have profound affects on economic and policy conclusions.
The logit demand model (McFadden, 1973)4 solves the dimensionality problem by projecting the products onto a space of characteristics, making the relevant size the dimension of this space and not
the square of the number of products. A problem with this model
is the strong implication of some of the assumptions made. Due to
the restrictive way in which heterogeneity is modeled, substitution
between products is driven completely by market shares and not by
how similar the products are. Extensions of the basic logit model relax
these restrictive assumptions, while maintaining the advantage of the
logit model in dealing with the dimensionality problem. The essential idea is to explicitly model heterogeneity in the population and
estimate the unknown parameters governing the distribution of this
heterogeneity. These models have been estimated using both marketand individual-level data.5 The problem with the estimation is that it
treats the regressors, including price, as exogenously determined. This
is especially problematic when aggregate data is used to estimate the
model.
This paper describes recent developments in methods for estimating random-coefcients discrete-choice models of demand [Berry,
1994; Berry et al., 1995 (henceforth BLP)]. The new method maintains
the advantage of the logit model in handling a large number of products. It is superior to prior methods because (1) the model can be
estimated using only market-level price and quantity data, (2) it deals
with the endogeneity of prices, and (3) it produces demand elasticities
that are more realistic—for example, cross-price elasticities are larger
for products that are closer together in terms of their characteristics.
The rest of the paper is organized as follows. Section 2 describes
a model that encompasses, with slight alterations, the models previously used in the literature. The focus is on the various modeling
assumptions and their implications for estimation and the results. In
Section 3 I discuss estimation, including the data required, an outline
of the algorithm, and instrumental variables. Many of the nitty-gritty
details of estimation are described in an appendix (available from
4. A related literature is the characteristic s approach to demand, or the address
approach (Lancaster, 1966, 1971; Quandt, 1968; Rosen, 1974). For a recent exposition of
it and a proof of its equivalence to the discrete choice approach see Anderson et al.
5. For example, the generalized extreme-value model (McFadden, 1978) and the
random-coefcients logit model (Cardell and Dunbar, 1980; Boyd and Mellman, 1980;
Tardiff, 1980; Cardell, 1989; and references therein). The random-coefcients model is
often called the hedonic demand model in this earlier literature; it should not be confused with the hedonic price model (Court, 1939; Griliches, 1961).

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Journal of Economics & Management Strategy

http://elsa.berkeley.edu/~ nevo). Section 4 provides a brief example
of the type of results the estimation can produce. Section 5 concludes
and discusses various extensions and alternatives to the method described here.

2. The Model
In this section I discuss the model with an emphasis on the various modeling assumptions and their implications. In the next section
I discuss the estimation details. However, for now I want to stress
two points. First, the method I discuss here uses (market-level) price
and quantity data for each product, in a series of markets, to estimate
the model. Some information regarding the distribution of consumer
characteristics might be available, but a key benet of this methodology is that we do not need to observe individual consumer purchase
decisions to estimate the demand parameters.6
Second, the estimation allows prices to be correlated with the
econometric error term. This will be modeled in the following way.
A product will be dened by a set of characteristics. Producers and
consumers are assumed to observe all product characteristics. The
researcher, on the other hand, is assumed to observe only some of the
product characteristics. Each product will be assumed to have a characteristic that inuences demand but that either is not observed by the
researcher or cannot be quantied into a variable that can be included
in the analysis. Examples are provided below. The unobserved characteristics will be captured by the econometric error term. Since the producers know these characteristics and take them into account when
setting prices, this introduces the econometric problem of endogenous
prices.7 The contribution of the estimation method presented below
is to transform the model in such a way that instrumental-variable
methods can be used.

2.1

The Setup

Assume we observe t 5 1, . . . , T markets, each with i 5 1, . . . , It
consumers. For each such market we observe aggregate quantities,
6. If individual decisions are observed, the method of analysis differs somewhat
from the one presented here. For clarity of presentation I defer discussion of this case
to Section 5.
7. The assumption that when setting prices rms take account of the unobserved
(to the econometrician) characteristic s is just one way to generate correlation between
prices and these unobserved variables. For example, correlation can also result from
the mechanics of the consumer’s optimizatio n problem (Kennan, 1989).

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Random-Coefcients Logit Models of Demand

average prices, and product characteristics for J products.8 The denition of a market will depend on the structure of the data. BLP use
annual automobile sales over a period of twenty years, and therefore
dene a market as the national market for year t, where t 5 1, . . . , 20.
On the other hand, Nevo (2000a) observes data in a cross section of
cities over twenty quarters, and denes a market t as a city–quarter
combination, with t 5 1, . . . , 1124. Yet a different example is given
by Das et al. (1994), who observe sales for different income groups,
and dene a market as the annual sales to consumers of a certain
income level.
The indirect utility of consumer i from consuming product j in
market t, U (x j t , »j t , p j t , ¿i ; h ), 9 is a function of observed and unobserved (by the researcher) product characteristics, x j t and »j t respectively;
price, p j t ; individual characteristics, ¿i ; and unknown parameters, h .
I focus on a particular specication of this indirect utility,10
u ijt 5

a i (yi
i5

pjt ) 1

xjt b

1, . . . , I t ,

j 5

»j t 1
e

ij t ,

1, . . . , J ,

1, . . . , T ,

(1)

where yi is the income of consumer i, p j t is the price of product j in
market t, x j t is a K-dimensional (row) vector of observable characteristics of product j , »j t is the unobserved (by the econometrician) product characteristic, and e ij t is a mean-zero stochastic term. Finally, a i is
consumer i’s marginal utility from income, and b i is a K-dimensional
(column) vector of individual-specic taste coefcients.
Observed characteristics vary with the product being considered.
BLP examine the demand for cars, and include as observed characteristics horsepower, size and air conditioning. In estimating demand
for ready-to-eat cereal Nevo (2000a) observes calories, sodium, and
ber content. Unobserved characteristics, for example, can include
the impact of unobserved promotional activity, unquantiable factors
(brand equity), or systematic shocks to demand. Depending on the
structure of the data, some components of the unobserved characteristics can be captured by dummy variables. For example, we can model
»j t 5 »j 1 »t 1 D »j t and capture »j and »t by brand- and market-specic
dummy variables.
Implicit in the specication given by equation (1) are three
things. First, this form of the indirect utility can be derived from a
8. For ease of exposition I have assumed that all products are offered to all consumers in all markets. The methods described below can easily deal with the case
where the choice set differs between markets and also with different choice sets for
different consumers.
9. This is sometimes called the conditional indirect utility, i.e., the indirect utility
conditional on choosing this option.
10. The methods discussed here are general and with minor adjustments can deal
with different functional forms.

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Journal of Economics & Management Strategy

quasilinear utility function, which is free of wealth effects. For some
products (for example, ready-to-eat cereals) this is a reasonable assumption, but for other products (for example, cars) it is an unreasonable one. Including wealth effects alters the way the term yi p j t enters
equation (1). For example, BLP build on a Cobb-Douglas utility function to derive an indirect utility that is a function of log(y i p j t ). In
principle, we can include f (yi p j t ), where f ( ? ) is a exible functional
form (Petrin, 1999).
Second, equation (1) species that the unobserved characteristic,
which among other things captures the elements of vertical product
differentiation, is identical for all consumers. Since the coefcient on
price is allowed to vary among individuals, this is consistent with
the theoretical literature of vertical product differentiation. An alternative is to model the distribution of the valuation of the unobserved
characteristics, as in Das et al. (1994). As long as we have not made
any distributional assumptions on consumer-specic components (i.e.,
anything with subscript i), their model is not more general. Once we
make such assumptions, their model has slightly different implications for some of the normalizations usually made. An exact discussion of these implications is beyond the scope of this paper.
Finally, the specication in equation (1) assumes that all consumers face the same product characteristics. In particular, all consumers are offered the same price. Depending on the data, if different
consumers face different prices, using either a list or average transaction price will lead to measurement error bias. This just leads to
another reason why prices might be correlated with the error term
and motivates the instrumental-variable procedure discussed below.11
The next component of the model describes how consumer preferences vary as a function of the individual characteristics, ¿i . In the
context of equation (1) this amounts to modeling the distribution of
consumer taste parameters. The individual characteristics consist of
two components: demographics, which I refer to as observed, and
additional characteristics, which I refer to as unobserved, denoted Di
and v i respectively. Given that no individual data is observed, neither
component of the individual characteristics is directly observed in the
choice data set. The distinction between them is that even though we
do not observe individual data, we know something about the distribution of the demographics, Di , while for the additional characteristics, vi , we have no such information. Examples of demographics are
income, age, family size, race, and education. Examples of the type of
11. However, as noted by Berry (1994), the method proposed below can deal with
measurement error only if the variable measured with error enters in a restrictive way.
Namely, it only enters the part of utility that is common to all consumers, d j in equation
(3) below.

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Random-Coefcients Logit Models of Demand

information we might have is a large sample we can use to estimate
some feature of the distribution (e.g., we could use Census data to
estimate the mean and standard deviation of income). Alternatively,
we might have a sample from the joint distribution of several demographic variables (e.g., the Current Population Survey might tell us
about the joint distribution of income, education, and age in different cities in the US). The additional characteristics, m i , might include
things like whether the individual owns a dog, a characteristic that
might be important in the decision of which car to buy, yet even very
detailed survey data will usually not include this fact.
Formally, this will be modeled as
a

5
b

a
1

ÕDi 1

åm i ,
m

Pm * (m ) ,

Di ~

PÃD* (D),

(2)

where Di is a d ´ 1 vector of demographic variables, m i captures the
additional characteristics discussed in the previous paragraph, Pm * (? ) is
a parametric distribution, PÃ D* (? ) is either a nonparametric distribution
known from other data sources or a parametric distribution with the
parameters estimated elsewhere, Õ is a (K 1 1) ´ d matrix of coefcients
that measure how the taste characteristics vary with demographics,
and å is a ( K 1 1) ´ (K 1 1) matrix of parameters.12 If we assume
that Pm * (? ) is a standard multivariate normal distribution, as I do in
the example below, then the matrix å allows each component of m i
to have a different variance and allows for correlation between these
characteristics. For simplicity I assume that m i and Di are independent. Equation (2) assumes that demographics affect the distribution
of the coefcients in a fairly restrictive linear way. For those coefcients that are most important to the analysis (e.g., the coefcients on
price), relaxing the linearity assumption could have important implications [for example, see the results reported in Nevo (2000a, b)].
As we will see below, the way we model heterogeneity has
strong implications for the results. The advantage of letting the taste
parameters vary with the observed demographics, Di is twofold. First,
it allows us to include additional information, about the distribution
of demographics, in the analysis. Furthermore, it reduces the reliance
on parametric assumptions. Therefore, instead of letting a key element
of the method, the distribution of the random coefcients, be determined by potentially arbitrary distributional assumptions, we bring
in additional information.
12. To simplify notation I assume that all characteristics have random coefcients.
This need not be the case. I return to this in the appendix, when I discuss the details
of estimation.

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Journal of Economics & Management Strategy

The specication of the demand system is completed with the
introduction of an outside good: the consumers may decide not to purchase any of the brands. Without this allowance, a homogenous price
increase (relative to other sectors) of all the products does not change
quantities purchased. The indirect utility from this outside
option is
u i0t 5

a i yi 1

»0t 1

¼0 Di 1

¾0 v i0 1
e

i0t .

The mean utility from the outside good, »0t , is not identied (without either making more assumptions or normalizing one of the inside
goods). Also, the coefcients ¼0 and ¾0 are not identied separately
from coefcients on an individual-specic constant term in equation
(1). The standard practice is to set »0t , ¼0 , and ¾0 to zero, and since the
term a i yi will eventually vanish (because it is common to all products), this is equivalent to normalizing the utility from the outside
good to zero.
Let h 5 (h 1 , h 2 ) be a vector containing all the parameters of the
model. The vector h 1 5 (a , b ) contains the linear parameters, and the
vector h 2 5 (Õ, å) the nonlinear parameters. 13 Combining equations
(1) and (2), we have
u ijt 5 a i yi 1
d

5 xjt b

j t (x j t ,

a pjt 1

p j t , »j t ; h 1 ) 1
»j t ,

u ijt 5

u ijt ( x j t , p j t , v i , Di ; h 2 ) 1
[ p j t , x j t ] ( ÕDi 1

åm i ),

ijt ,

(3)

where [ p j t , x j t ] is a 1 ´ (K 1 1) (row) vector. The indirect utility
is now expressed as a sum of three (or four) terms. The rst term,
a i yi , is given only for consistency with equation (1) and will vanish, as we will see below. The second term, d j t , which is referred
to as the mean utility, is common to all consumers. Finally, the last
two terms, l ijt 1 e ijt , represent a mean-zero heteroskedastic deviation from the mean utility that captures the effects of the random
coefcients.
Consumers are assumed to purchase one unit of the good that
gives the highest utility.14 Since in this model an individual is de13. The reasons for the names will become apparent below.
14. A comment is in place here about the realism of the assumption that consumers
choose no more than one good. We know that many households own more than one
car, that many of us buy more than one brand of cereal, and so forth. We note that even
though many of us buy more than one brand at a time, less actually consume more
than one at a time. Therefore, the discreteness of choice can be sometimes defended by
dening the choice period appropriately. In some cases this will still not be enough, in
which case the researcher has one of two options: either claim that the above model
is an approximation, or reduced-form, to the true choice model, or model the choice
of multiple products, or continuous quantities, explicitly [as in Dubin and McFadden
(1984) or Hendel (1999)].

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Random-Coefcients Logit Models of Demand

ned as a vector of demographics and product-specic shocks,
( Di , m i , e i 0t , . . . , e iJ t ), this implicitly denes the set of individual
attributes that lead to the choice of good j . Formally, let this set be
A j t (x ? t , p ? t , d

? t;

h 2) 5

(Di , v i , e
" l5

i 0t ,

...,e

iJ t )|u ij t

u ilt

0, 1, . . . , J ,

where x ? t 5 ( x lt , . . . , x J t ) ¢ , p ? t 5 (p lt , . . . , p J t ) ¢ , and d ? t 5 (d lt , . . . , d J t ) ¢
are observed characteristics, prices, and mean utilities of all brands,
respectively. The set A j t denes the individuals who choose brand j in
market t. Assuming ties occur with zero probability, the market share
of the j th product is just an integral over the mass of consumers in
the region A j t . Formally, it is given by
s j t ( x? t , p ? t , d

? t;

h 2) 5
5
5

dP * (D, v, e )

dP * (e |D, m ) dP * (m |D) dPD* (D)
dPe * (e ) dPm * (m ) d PÃD* (D) ,

(4)

where P * ( ? ) denotes population distribution functions. The second
equality is a direct application of Bayes’ rule, while the last is a consequence of the independence assumptions previously made.
Given assumptions on the distribution of the (unobserved) individual attributes, we can compute the integral in equation (4), either
analytically or numerically. Therefore, for a given set of parameters
equation (4) predicts of the market share of each product in each
market, as a function of product characteristics, prices, and unknown
parameters. One possible estimation strategy is to choose parameters that minimize the distance (in some metric) between the market shares predicted by equation (4) and the observed shares. This
estimation strategy will yield estimates of the parameters that determine the distribution of individual attributes, but it does not account
for the correlation between prices and the unobserved product characteristics. The method proposed by Berry (1994) and BLP, which is
presented in detail in the Section 3, accounts for this correlation.

2.2

Distributional Assumptions

The assumptions on the distribution of individual attributes made in
order to compute the integral in equation (4) have important implications for the own- and cross-price elasticities of demand. In this
section I discuss some possible assumptions and their implications.

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Journal of Economics & Management Strategy

Possibly the simplest distributional assumption one can make in
order to evaluate the integral in equation (4) is that consumer heterogeneity enters the model only through the separable additive random
shock, e ij t . In our model this implies h 2 5 0, or b i 5 b and a i 5 a for
all i, and equation (1) becomes
u ijt 5

pjt ) 1

a (yi
i5

xjt b 1

1, . . . , It ,

»j t 1
j5

ijt ,

1, . . . , J ,

1, . . . , T .

(5)

At this point, before we specify the distribution of e ij t , the model
described by equation (5) is as general as the model given in
equation (1).15 Once we assume that e ij t is i.i.d., then the implied substitution patterns are severely restricted, as we will see below. If we
also assume that e ijt are distributed according to a Type I extremevalue distribution, this is the (aggregate) logit model. The market
share of brand j in market t, dened by equation (4), is
sj t 5

a pjt 1

exp(x j t b
11

J
k5 1

exp(x kt b

»j t )
a p kt 1

»kt )

(6)

Note that income drops out of this equation, since it is common to all
options.
Although the model implied by equation (5) and the extremevalue distribution assumption is appealing due to its tractability, it
restricts the substitution patterns to depend only on the market shares.
The price elasticities of the market shares dened by equation (6) are
´j kt 5

@sj t p kt
@p kt sj t

(

a p j t (1
a p kt skt

sj t )

if j 5 k,
otherwise.

There are two problems with these elasticities. First, since in
most cases the market shares are small, the factor a (1 s j t ) is nearly
constant; hence, the own-price elasticities are proportional to own
price. Therefore, the lower the price, the lower the elasticity (in absolute value), which implies that a standard pricing model predicts a
higher markup for the lower-priced brands. This is possible only if the
marginal cost of a cheaper brand is lower (not just in absolute value,
but as a percentage of price) than that of a more expensive product.
For some products this will not be true. Note that this problem is
a direct implication of the functional form in price. If, for example,
indirect utility was a function of the logarithm of price, rather than
price, then the implied elasticity would be roughly constant. In other
15. To see this compare equation (5) with equation (3).

Random-Coefcients Logit Models of Demand

523

words, the functional form directly determines the patterns of ownprice elasticity.
An additional problem, which has been stressed in the literature,
is with the cross-price elasticities. For example, in the context of RTE
cereals the cross-price elasticities imply that if Quaker CapN Crunch
(a childern’s cereal) and Post Grape Nuts (a wholesome simple nutrition cereal) have similar market shares, then the substitution from
General Mills Lucky Charms (a children’s cereal) toward either of
them will be the same. Intuitively, if the price of one children’s cereal
goes up, we would expect more consumers to substitute to another
children’s cereal than to a nutrition cereal. Yet, the logit model restricts
consumers to substitute towards other brands in proportion to market
shares, regardless of characteristics.
The problem in the cross-price elasticities comes from the i.i.d.
structure of the random shock. In order to understand why this is the
case, examine equation (3). A consumer will choose a product either
because the mean utility from the product, d j t , is high or because the
consumer-specic shock, l ijt 1 e ij t , is high. The distinction becomes
important when we consider a change in the environment. Consider,
for example, the increase in the price of Lucky Charms discussed in
the previous paragraph. For some consumers who previously consumed Lucky Charms, the utility from this product decreases enough
so that the utility from what was the second choice is now higher.
In the logit model different consumers will have different rankings of
the products, but this difference is due only to the i.i.d. shock. Therefore, the proportion of these consumers who rank each brand as their
second choice is equal to the average in the population, which is just
the market share of the each product.
In order to get around this problem we need the shocks to utility to be correlated across brands. By generating correlation we predict that the second choice of consumers that decide to no longer
buy Lucky Charms will be different than that of the average consumer. In particular, they will be more likely to buy a product with
a shock that was positively correlated to Lucky Charms, for example
CapN Crunch. As we can see in equation (3), this correlation can be
generated either through the additive separable term e ij t or through
the term l ij t , which captures the effect the demographics, Di and v i .
Appropriately dening the distributions of either of these terms can
yield the exact same results. The difference is only in modeling convenience. I now consider models of the two types.
Models are available that induce correlation among options by
allowing e ij t to be correlated across products rather than independently distributed, are available (see the generalized extreme-value
model, McFadden, 1978). One such example is the nested logit model,

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Journal of Economics & Management Strategy

in which all brands are grouped into predetermined exhaustive and
mutually exclusive sets, and e ijt is decomposed into an i.i.d. shock
plus a group-specic component.16 This implies that correlation
between brands within a group is higher than across groups; thus, in
the example given above, if the price of Lucky Charms goes up, consumers are more likely to rank CapN Crunch as their second choice.
Therefore, consumers that currently consume Lucky Charms are more
likely to substitute towards Grape Nuts than the average consumer.
Within the group the substitution is still driven by market shares, i.e.,
if some children’s cereals are closer substitutes for Lucky Charms than
others, this will not be captured by the simple grouping.17 The main
advantage of the nested logit model is that, like the logit model, it
implies a closed form for the integral in equation (4). As we will see
in Section 3, this simplies the computation.
This nested logit can t into the model described by equation (1)
in one of two ways: by assuming a certain distribution of e ij t , as was
motivated in the previous paragraph, or by assuming one of the characteristics of the product is a segment-specic dummy variable and
assuming a particular distribution on the random coefcient of that
characteristic. Given that we have shown that the model described by
equations (1) and (2) can also be described by equation (3), it should
not be surprising that these two ways of describing the nested logit
are equivalent. Cardell (1997) shows the distributional assumptions
required for this equivalence to hold.
The nested logit model allows for somewhat more exible substitution patterns. However, in many cases the a priori division of
products into groups, and the assumption of i.i.d. shocks within a
group, will not be reasonable, either because the division of segments
is not clear or because the segmentation does not fully account for
the substitution patterns. Furthermore, the nested logit does not help
with the problem of own-price elasticities. This is usually handled
by assuming some “nice” functional form (i.e., yield patterns that are
consistent with some prior), but that does not solve the problem of
having the elasticities driven by the functional-form assumption.
In some industries the segmentation of the market will be multilayered. For example, computers can be divided into branded versus generic and into frontier versus nonfrontier technology. It turns
16. For a formal presentation of the nested logit model in the context of the model
presented here, see Berry (1994) or Stern (1995).
17. Of course, one does not have to stop at one level of nesting. For example, we
could group all children’s cereals into family-acceptabl e and not acceptable. For an
example of such grouping for automobiles see Goldberg (1995).

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Random-Coefcients Logit Models of Demand

out that in the nested logit specication the order of the nests matters.18 For this reason Bresnahan et al. (1997) build on a the general
extreme-value model (McFadden 1978) to construct what they call the
principles-of-differentiation general extreme-value (PD GEV) model of
demand for computers. In their model they are able to use two dimensions of differentiation, without ordering them. With the exception of
dealing with the problem of ordering the nests, this model retains all
the advantages and disadvantages of the nested logit. In particular it
implies a closed-form expression for the integral in equation (4).
In principle one could consider estimating an unrestricted
variance-covariance matrix of the shock, e ijt . This, however, reintroduces the dimensionality problem discussed in the Introduction.19 If
in the full model, described by equations (1) and (2), we maintain
the i.i.d. extreme-value distribution assumption on e ijt . Correlation
between choices is obtained through the term l ij t . The correlation
will be a function of both product and consumer characteristics: the
correlation will be between products with similar characteristics, and
consumers with similar demographics will have similar rankings of
products and therefore similar substitution patterns. Therefore, rather
than having to estimate a large number of parameters, corresponding to an unrestricted variance-covariance matrix, we only have to
estimate a smaller number.
The price elasticities of the market shares, sj t , dened by
equation (4) are
´j kt 5

@sj t p kt
@p kt sj t

(

pj t
p kt
sj t

sj t

i s ij t (1

sij t ) d PÃ D* (D) dPm * (v)

Ã*
i s ijt s ikt d PD (D)

dPm (v)
*

if j 5

otherwise,

where sij t 5 exp(d j t 1 l ijt )/ [11 S Kk5 1 exp(d kt 1 l ikt )] is the probability of
individual i purchasing product j . Now the own-price elasticity will
not necessary be driven by the functional form. The partial derivative
of the market shares will no longer be determined by a single parameter, a . Instead, each individual will have a different price sensitivity,
which will be averaged to a mean price sensitivity using the individual specic probabilities of purchase as weights. The price sensitivity
18. So, for example, classifying computers rst into branded/nonbranded and then
into frontier/nonfrontier technology implies different substitution patterns than classifying rst into frontier/nonfrontier technology and then into branded/nonbranded,
even if the classication of products does not change.
19. See Hausman and Wise (1978) for an example of such a model with a small
number of products.

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will be different for different brands. So if, for example, consumers of
Kellogg’s Corn Flakes have high price sensitivity, then the own-price
elasticity of Kellogg’s Corn Flakes will be high despite the low prices
and the fact that prices enter linearly. Therefore, substitution patterns
are not driven by functional form, but by the differences in the price
sensitivity, or the marginal utility from income, between consumers
that purchase the various products.
The full model also allows for exible substitution patterns,
which are not constrained by a priori segmentation of the market (yet
at the same time can take advantage of this segmentation by including
a segment dummy variable as a product characteristic). The composite random shock, l ij t 1 e ijt , is then no longer independent of the
product and consumer characteristics. Thus, if the price of a brand
goes up, consumers are more likely to switch to brands with similar
characteristics rather than to the most popular brand.
Unfortunately, these advantages do not come without cost. Estimation of the model specied in equation (3) is not as simple as that
of the logit, nested logit, or GEV models. There are two immediate
problems. First, equation (4) no longer has an analytic closed form
[like that given in equation (6) for the Logit case]. Furthermore, the
computation of the integral in equation (4) is difcult. This problem
is solved using simulation methods, as described below. Second, we
now require information about the distribution of consumer heterogeneity in order to compute the market shares. This could come in
the form of a parametric assumption on the functional form of the
distribution or by using additional data sources. Demographics of the
population, for example, can be obtained by sampling from the CPS.

3. Estimation
3.1

The Data

The market-level data required to consistently estimate the model previously described consists of the following variables: market shares
and prices in each market, and brand characteristics. In addition
information on the distribution of demographics, PÃ D* , is useful, 20 as
are marketing mix variables (such as advertising expenditures or the
availability of coupons or promotional activity). In principle, some
of the parameters of the model are identied even with data on one
20. Recall that we divided the demographic variables into two types. The rst were
those variables for which we had some information regarding the distribution. If such
information is not available we are left with only the second type, i.e., variables for
which we assume a parametric distribution.

Random-Coefcients Logit Models of Demand

527

market. However, it is highly recommended to gather data on several markets with variation in relative prices of the products and/or
products offered.
Market shares are dened using a quantity variable, which
depends on the context and should be determined by the specics
of the problem. BLP use the number of automobiles sold, while Nevo
(2000a, b) converts pounds of cereal into servings. Probably the most
important consideration in choosing the quantity variable is the need
to dene a market share for the outside good. This share will rarely
be observed directly, and will usually be dened as the total size of
the market minus the shares of the inside goods. The total size of the
market is assumed according to the context. So, for example, Nevo
(2000a, b) assumes the size of the market for ready-to-eat cereal to be
one serving of cereal per capita per day. Bresnahan et al. (1997), in
estimating demand for computers, take the potential market to be the
total number of ofce-based employees.
In general I found the following rules useful when dening the
market size. You want to make sure to dene the market large enough
to allow for a nonzero share of the outside good. When looking at historical data one can use eventual growth to learn about the potential
market size. One should check the sensitivity of the results to the
market denition; if the results are sensitive, consider an alternative.
There are two parts to dening the market size: choosing the variable
to which the market size is proportional, and choosing the proportionality factor. For example, one can assume that the market size is
proportional to the size of the population with the proportionality
factor equal to a constant factor, which can be estimated (Berry et al.,
1996). From my own (somewhat limited) experience, getting the right
variable from which to make things proportional is the harder, and
more important, component of this process.
An important part of any data set required to implement the
models described in Section 2 consists of the product characteristics.
These can include physical product characteristics and market segmentation information. They can be collected from manufacturer’s
descriptions of the product, the trade press, or the researcher’s prior.
In collecting product characteristics we recall the two roles they play
in the analysis: explaining the mean utility level d (? ) in equation (3),
and driving the substitution patterns through the term l (? ) in
equation (3). Ideally, these two roles should be kept separate. If the
number of markets is large enough relative to the number of products, the mean utility can be explained by including product dummy
variables in the regression. These variables will absorb any product
characteristics that are constant across markets. A discussion of the
issues arising from including brand dummy variables is given below.

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Relying on product dummy variables to guide substitution patterns is equivalent to estimating an unrestricted variance-covariance
matrix of the random shock e ijt in equation (1). Both imply estimating J (J 1)/ 2 parameters. Since part of our original motivation was
to reduce the number of parameters to be estimated, this is usually
not a feasible option. The substitution patterns are explained by the
product characteristics, and in deciding which attributes to collect the
researcher should keep this in mind.
The last component of the data is information regarding the
demographics of the consumers in different markets. Unlike market
shares, prices, or product characteristics, this estimation can proceed
without demographic information. In this case the estimation will rely
on assumed distributional assumptions rather than empirical distributions. The Current Population Survey (CPS) is a good, widely available source for demographic information.

3.2

Identi cation

This section discusses, informally, some of the identication issues.
There are several layers to the argument. First, I discuss how in general a discrete choice model helps us identify substitution patterns,
using aggregate data from (potentially) a small number of markets.
Second, I ask what in the data helps us distinguish between different
discrete choice models—for example, how we can distinguish between
the logit and the random-coefcients logit.
A useful starting point is to ask how one would approach the
problem of estimating price elasticities if a controlled experiment
could be conducted. The answer is to expose different consumers to
randomly assigned prices and record their purchasing patterns. Furthermore, one could relate these purchasing patterns to individual
characteristics. If individual purchases could not be observed, the
experiment could still be run by comparing the different aggregate
purchases of different groups. Once again, in principle, these patterns
could be related to the difference in individual characteristics between
the groups.
There are two potential problems with mapping the data described in the previous section into the data that arises from the ideal
controlled experiment, described in the previous paragraph. First,
prices are not randomly assigned; rather, they are set by prot-maximizing rms that take into account information that, due to inferior
knowledge, the researcher has to include in the error term. This problem can be solved, at least in principle, by using instrumental
variables.
The second, somewhat more conceptual difculty arises because
discrete choice models, for example the logit model, can be estimated

Random-Coefcients Logit Models of Demand

529

using data from just one market; hence, we are not mimicking the
experiment previously described. Instead, in our experiment we ask
consumers to choose between products, which are perceived as bundles of attributes. We then reveal the preferences for these attributes,
one of which is price. The data from each market should not be seen
as one observation of purchases when faced with a particular price
vector; rather, it is an observation on the relative likelihood of purchasing J different bundles of attributes. The discrete choice model
ties these probabilities to a utility model that allows us to compute
price elasticities. The identifying power of this experiment increases
as more markets are included with variation both in the characteristics
of products and in the choice set. The same (informal) identication
argument holds for the nested logit, GEV, and random-coefcients
models, which are generalized forms of the logit model.
There are two caveats to the informal argument previously
given. If one wants to tie demographic variables to observed purchases [i.e., allow for ÕDi in equation (2)], several markets, with
variation in the distribution of demographics, have to be observed.
Second, if not all the product characteristics are observed and these
unobserved attributes are correlated with some of the observed characteristics, then we are faced with an endogeneity problem. The problem can be solved by using instrumental variables, but we note that
the formal requirements from these instrumental variables depend
on what we believe goes into the error term. In particular, if brandspecic dummy variables are included, we will need the instrumental
variables to satisfy different requirements. I return to this point in
Section 3.4.
A different question is: What makes the random-coefcients logit
respond differently to product characteristics? In other words, what
pins down the substitution patterns? The answer goes back to the difference in the predictions of the two models and can be best explained
with an example. Suppose we observe three products: A, B, and C.
Products A and B are very similar in their characteristics, while products B and C have the same market shares. Suppose we observe market shares and prices in two periods, and suppose the only change
is that the price of product A increases. The logit model predicts that
the market shares of both products B and C should increase by the
same amount. On the other hand, the random-coefcients logit allows
for the possibility that the market share of product B, the one more
similar to product A, will increase by more. By observing the actual
relative change in the market shares of products B and C we can distinguish between the two models. Furthermore, the degree of change
will allow us to identify the parameters that govern the distribution
of the random coefcients.

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This argument suggests that having data from more markets
helps identify the parameters that govern the distribution of the random coefcients. Furthermore, observing the change in market shares
as new products enter or as characteristics of existing products change
provides variation that is helpful in the estimation.

3.3

The Estimation Algorithm

In this subsection I outline how the parameters of the models described in Section 2 can be consistently estimated using the data described in Section 3.1. Following Berry’s (1994) suggestion, a GMM
estimator is constructed. Given a value of the unknown parameters,
the implied error term is computed and interacted with the instruments to form the GMM objective function. Next, a search is performed over all the possible parameter values to nd those values
that minimize the objective function. In this subsection I discuss what
the error term is, how it can be computed, and some computational
details. Discussion of the instrumental variables is deferred to the next
section.
As previously pointed out, a straightforward approach to the
estimation is to solve
Min d s(x, p, d (x, p, »; h 1 ) ; h 2 )
h

Sd ,

(7)

where s(? ) are the market shares given by equation (4), and S are the
observed market shares. However, this approach is usually not taken,
for several reasons. First, all the parameters enter the minimization
in equation (7) in a nonlinear fashion. In some applications the inclusion of brand and time dummy variables results in a large number of
parameters and a costly nonlinear minimization problem. The estimation procedure suggested by Berry (1994), which is described below,
avoids this problem by transforming the minimization problem so that
some (or all) of the parameters enter the objective function linearly.
Fundamentally, though, the main contribution of the estimation
method proposed by Berry (1994) is that it allows one to deal with
correlation between the (structural) error term and prices (or other
variables that inuence demand). As we saw in Section 2.1, there are
several variables that are unobserved by the researcher. These include
the individual-level characteristics, denoted (Di , v i , e i ), as well as the
unobserved product characteristics, »j . As we saw in equation (4),
the unobserved individual attributes ( Di , vi , e i ) were integrated over.
Therefore, the econometric error term will be the unobserved product
characteristics, »j t . Since it is likely that prices are correlated with this
term, the econometric estimation will have to take account of this. The

Random-Coefcients Logit Models of Demand

531

standard nonlinear simultaneous-equations model [see, for example,
Amemiya (1985, Chapter 8)] allows both parameters and variables to
enter in a nonlinear way, but requires a separable additive error term.
Equation (7) does not meet this requirement. The estimation method
proposed by Berry (1994), and described below, shows how to adapt
the model described in the previous section to t into the standard
(linear or) nonlinear simultaneous-equations model.
Formally, let Z 5 [z 1 , . . . , z M ] be a set of instruments such that
E[Z m x (h * ) ] 5

1, . . . , M ,

(8)

where x , a function of the model parameters, is an error term dened
below, and h * denotes the “true” values of the parameters. The GMM
estimate is
hÃ 5

argmin x (h ) ¢ ZF

Z ¢ x (h ),

(9)

where F is a consistent estimate of E[Z¢ x x ¢ Z]. The logic driving this
estimate is simple enough. At the true parameter value, h * , the population moment, dened by equation (8), is equal to zero. So we
choose our estimate such that it sets the sample analog of the moments
dened in equation (8), i.e., Z ¢ x Ã , to zero. If there are more independent moment equations than parameters [i.e., dim(Z) > dim(h )], we
cannot set all the sample analogs exactly to zero and will have to set
them as close to zero as possible. The weight matrix, F 1 , denes the
metric by which we measure how close to zero we are. By using the
inverse of the variance-covariance matrix of the moments, we give
less weight to those moments (equations) that have a higher variance.
Following Berry (1994), the error term is not dened as the difference between the observed and predicted market shares, as in
equation (7); rather it is dened as the structural error, »j t . The advantage of working with a structural error is that the link to economic
theory is tighter, allowing us to think of economic theories that would
justify various instrumental variables.
In order to use equation (9) we need to express the error term
as an explicit function of the parameters of the model and the data.
The key insight, which can be seen in equation (3), is that the error
term, »j t , only enters the mean utility level, d (? ). Furthermore, the
mean utility level is a linear function of »j t ; thus, in order to obtain an
expression for the error term we need to express the mean utility as a

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Journal of Economics & Management Strategy

linear function of the variables and parameters of the model. In order
to do this, we solve for each market the implicit system of equations
s(d ? t ; h 2 ) 5

S? t ,

1, . . . , T ,

(10)

where s(? ) are the market shares given by equation (4), and S are the
observed market shares.
The intuition for why we want to do this is given below. In
solving this system of equations we have two steps. First, we need a
way to compute the left-hand side of equation (10), which is dened
by equation (4). For some special cases of the general model (e.g., logit,
nested logit, and PD GEV) the market-share equation has an analytic
formula. For the full random-coefcients model the integral dening
the market shares has to be computed by simulation. There are several
ways to do this. Probably the most common is to approximate the
integral given by equation (4) by
sj t (p ? t , x ? t , d
5

? t,

Pns ; h 2 )

1 ns
s 5
ns i 5 1 j ti
´

1
ns

exp d
11

J
m5 1

ns
i5 1

exp d

x jkt (¾k v ik 1

K
k5 1
mt

K
k5 1

¼k1 Di 1 1

k
x mt
(¾k v ik

? ? ? 1

¼k1 Di 1 1

¼kd Di d )
? ? ? 1

¼kd Di d )

(11)
where (m 1i , . . . , m Ki ) and (Di1 , . . . , Did ), i 5 1, . . . , ns, are draws from
PÃ m * (v) and PD* (D) , respectively, while x jkt , k 5 1, . . . , K, are the variables
that have random slope coefcients. Note the we use the extreme-value
distribution Pe * (e ), to integrate the e ¢ s analytically. Issues regarding
sampling from Pm * v and PÃ D* (D), alternative methods to approximate
the market shares, and their advantages are discussed in detail in the
appendix (available from http://elsa.berkeley.edu/~ nevo).
Second, using the computation of the market share, we invert the
system of equations. For the simplest special case, the logit model,
this inversion can be computed analytically by d j t 5 ln Sj t ln S0t ,
where S0t is the market share of the outside good. Note that it is the
observed market shares that enter this equation. This inversion can
also be computed analytically in the nested logit model (Berry, 1994)
and the PD GEV model (Bresnahan et al., 1997).
For the full random-coefcients model the system of equations
(10) is nonlinear and is solved numerically. It can be solved by using

Random-Coefcients Logit Models of Demand

533

the contraction mapping suggested by BLP (see there for a proof of
convergence), which amounts to computing the series
5

h1 1
?t

h
?t

ln S? t
t5

ln S(p ? t , x ? t , d ?ht , Pns ; h 2 ),

1, . . . , T,

0, . . . , H ,

(12)

where s(? ) are the predicted market shares computed in the rst step,
H is the smallest integer such that ||d H? t d ?Ht 1 || is smaller than some
tolerance level, and d H? t is the approximation to d ? t .
Once the inversion has been computed, either analytically or
numerically, the error term is dened as
x

5
d

j t ( S? t ;

h 2)

(x j t b 1

a pj t ) º

»j t .

(13)

Note that it is the observed market shares, S, that enter this equation.
Also, we can now see the reason for distinguishing between h 1 and
h 2 : h 1 enters this term, and the GMM objective, in a linear fashion,
while h 2 enters nonlinearly.
The intuition to the denition is as follows. For given values of
the nonlinear parameters h 2 , we solve for the mean utility levels d ? t (? ),
that set the predicted market shares equal to the observed market
shares. We dene the residual as the difference between this valuation
and the one predicted by the linear parameters a and b . The estimator,
dened by equation (9), is the one the minimizes the distance between
these different predictions.
Usually,21 the error term, as dened by equation (13), is the
unobserved product characteristic, »j t . However, if enough markets
are observed, then brand-specic dummy variables can be included as
product characteristics. The coefcients on these dummy variable capture both the mean quality of observed characteristics that do not vary
over markets, b x j , and the overall mean of the unobserved characteristics, »j . Thus, the error term is the market-specic deviation from the
main valuation, i.e., D »j t º »j t »j . The inclusion of brand dummy
variables introduces a challenge in estimating the taste parameters, b ,
which is dealt with below.
In the logit and nested logit models, with the appropriate choice
of a weight matrix,22 this procedure simplies to two-stage least
squares. In the full random-coefcients model, both the computation
of the market shares and the inversion in order to get d j t (? ) have to
be done numerically. The value of the estimate in equation (9) is then
computed using a nonlinear search. This search is simplied by noting
21. See for example Berry (1994), BLP, Berry et al. (1996), and Bresnahan et al. (1997).
22. That is, F 5 Z¢ Z, which is the “optimal” weight matrix under the assumption
of homoskedastic errors.

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Journal of Economics & Management Strategy

that the rst-order conditions of the minimization problem dened in
equation (9) with respect to h 1 are linear in these parameters. Therefore, these linear parameters can be solved for (as a function of the
other parameters) and plugged into the rest of the rst-order conditions, limiting the nonlinear search to the nonlinear parameters only.
The details of the computation are given in the appendix.

3.4

Instruments

The identifying assumption in the algorithm previously given is
equation (8), which requires a set of exogenous instrumental variables.
As is the case with many hard problems, there is no global solution
that applies to all industries and data sets. Listed below are some of
the solutions offered in the literature. A precise discussion of how
appropriate each set of assumptions has to be done on a case-by-case
basis, but several advantages and problems are mentioned below.
The rst set of variables that comes to mind are the instrumental variables dened by ordinary (or nonlinear) least squares, namely
the regressors (or more generally the derivative of the moment function with respect to the parameters). As previously discussed, there
are several reasons why these are invalid. For example, a variety of
differentiated-products pricing models predict that prices are a function of marginal cost and a markup term. The markup term is a function of the unobserved product characteristic, which is also the error
term in the demand equation. Therefore, prices will be correlated with
the error term, and the estimate of the price sensitivity will be biased.
A standard place to start the search for demand-side instrumental variables is to look for variables that shift cost and are uncorrelated with the demand shock. These are the textbook instrumental
variables, which work quite well when estimating demand for homogeneous products. The problem with the approach is that we rarely
observe cost data ne enough that the cost shifters will vary by brand.
A restricted version of this approach uses whatever cost information is
available in combination with some restrictions on the demand specication [for example, see the cost variables used in Nevo (2000a)].
Even the restricted version is rarely feasible, due to lack of any
cost data.
The most popular identifying assumption used to deal with the
above endogeneity problem is to assume that the location of products
in the characteristics space is exogenous, or at least determined prior
to the revelation of the consumers’ valuation of the unobserved product characteristics. This assumption can be combined with a specic
model of competition and functional-form assumptions to generate
an implicit set of instrumental variables (as in Bresnahan, 1981, 1987).

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535

BLP derive a slightly more explicit set of instrumental variables, which
build on a similar economic assumption. They use the observed product characteristics (excluding price and other potentially endogenous
variables), the sums of the values of the same characteristics of other
products offered by that rm (if the rm produces more than one
product), and the sums of the values of the same characteristics of
products offered by other rms.23
Instrumental variables of this type have been quite successful
in the study of many industries, including automobiles, computers,
and pharmaceutical drugs. One advantage of this approach is that the
instrumental variables vary by brand. The main problem is that in
some cases the assumption that observed characteristics are uncorrelated with the unobserved components is not valid. One example is
when certain types of products are better characterized by observed
attributes. Another example is if the time required to change the
observed characteristics is short and therefore changes in characteristics could be reacting to the same sort of shocks as prices. Finally,
once a brand dummy variable is introduced, a problem arises with
these instrumental variables: unless there is variation in the products
offered in different markets, there is no variation between markets in
these instruments.
The last set of instrumental variables I discuss here was introduced by Hausman et al. (1994) and Hausman (1996) and was used in
the context of the model described here by Nevo (2000a, b). The essential ideal is to exploit the panel structure of the data. This argument
is best demonstrated by an example. Nevo (2000a, b) observes quantities and prices for ready-to-eat cereal in a cross section of cities over
twenty quarters. Following Hausman (1996), the identifying assumption made is that, controlling for brand-specic intercepts and demographics, the city-specic valuations of the product, D »j t 5 »j t »j ,
are independent across cities but are allowed to be correlated within
a city over time. Given this assumption, the prices of the brand in
other cities are valid instruments; prices of brand j in two cities will
be correlated due to the common marginal cost, but due to the independence assumption will be uncorrelated with the market-specic
valuation of the product.
There are several plausible situations in which the independence
assumption will not hold. Suppose there is a national (or regional)
demand shock, for example, discovery that ber may reduce the risk
23. Just to be sure, suppose the product has two characteristics : horsepower (HP)
and size (S), and assume there are two rms producing three products each. Then we
have six instrumental variables: The values of HP and S for each product, the sum of
HP and S for the rm’s other two products, and the sum of HP and S for the three
products produced by the competition.

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Journal of Economics & Management Strategy

of cancer. This discovery will increase the unobserved valuation of
all ber-intensive cereal brands in all cities, and the independence
assumption will be violated. Alternatively, suppose one believes that
local advertising and promotions are coordinated across city borders
and that these activities inuence demand. Then the independence
assumption will be violated.
The extent to which the assumptions needed to support any
of the above instrumental variables are valid in any given situation
is an empirical issue. Resolving this issue beyond any reasonable
doubt is difcult and requires comparing results from several sets
of instrumental variables, combing additional data sources, and using
the researcher’s knowledge of the industry.

3.5

Brand-Speci c Dummy Variables

As previously pointed out, I believe that brand-specic xed effects
should be used whenever possible. There are at least two good reasons
to include these dummy variables. First, in any case where we are
unsure that the observed characteristics capture the true factors that
determine utility, xed effects should be included in order to improve
the t of the model. We note that this helps t the mean utility level
d j while substitution patterns are driven by observed characteristics
(either physical characteristics or market segmentation), as is the case
if we do not include a brand xed effect.
Furthermore, the major motivation (Berry, 1994) for the estimation scheme previously described is the need to instrument for the
correlation between prices and the unobserved quality of the product, »j t . A brand-specic dummy variable captures the characteristics that do not vary by market and the product-specic mean of
unobserved components, namely, x j b 1 »j . Therefore, the correlation
between prices and the brand-specic mean of unobserved quality is
fully accounted for and does not require an instrument. In order to
introduce a brand dummy variable we require observations on more
than one market. However, even without brand dummy variables, tting the model using observations from a single market is difcult
(see BLP, footnote 30).
Once brand dummy variables are introduced, the error term is
no longer the unobserved characteristics. Rather, it is the marketspecic deviation from this unobserved mean. This additional variance was not introduced by the dummy variables; it is present in all
models that use observations from more than one market. The use of
brand dummy variables forces the researcher to discuss this additional
variance explicitly.

Random-Coefcients Logit Models of Demand

537

There are two potential objections to the use of brand dummy
variables. First, as previously mentioned, a major difculty in estimating demand in differentiated product markets is that the number
of parameters increases proportionally to the square of the number
of products. The main motivation for the use of discrete choice models was to reduce this dimensionality problem. Does the introduction
of parameters that increase in proportion to the number of brands
defeat the whole purpose? No. The number of parameters increases
only with J (the number of brands) and not J 2 . Furthermore, the brand
dummy variables are linear parameters and do not increase the computational difculty. If the number of brands is large, the size of the
design matrix might be problematic, but given the computing power
required to run the full model, this is unlikely to be a serious difculty.
A more serious objection to the use of brand dummy variables
is that the taste coefcients b cannot be identied. Fortunately, this is
not true. The taste parameters can be retrieved by using a minimumdistance procedure (as in Chamberlain, 1982). Let d 5 ( d 1 , . . . , d j ) ¢
denote the J ´ 1 vector of brand dummy coefcients, X be the J ´
K (K < J ) matrix of product characteristics that are xed across markets, and » 5 (»1 , . . . , »j ) ¢ be the J ´ 1 vector of unobserved product
qualities. Then from equation (1),
d5

Xb 1

».

If we assume that E[» | X] 5
bÃ 5

X ¢ Vd 1 X

X ¢ Vd 1 d,Ã

0, 24 then the estimates of b and » are
»Ã 5

dÃ

Xb Ã ,

where dÃ is the vector of coefcients estimated from the procedure
described in the previous section, and Vd is the variance-covariance
matrix of these estimates. This is simply a GLS regression where the
independent variable consists of the estimated brand effects, estimated
using the GMM procedure previously described and the full sample. The number of “observations” in this regression is the number
of brands. The correlation in the values of the dependent variable
is treated by weighting the regression by the estimated covariance
matrix, Vd , which is the estimate of this correlation. The coefcients
on the brand dummy variables provide an unrestricted estimate of
the mean utility. The minimum-distance procedure project these estimate onto a lower-dimensional space, which is implied by a restricted
24. Note that this is the assumption required to justify the use of observed product
characteristics as instrumental variables. Here, however, this assumption is only used
to recover the taste parameters. If one is unwilling to make it, the price sensitivity can
still be recovered using the other assumptions discussed in the previous section.

538

Journal of Economics & Management Strategy

model that sets » to zero. Chamberlain (1982) provides a chi-square
test to evaluate these restrictions.

4. AN APPLICATION
In this section I briey present the type of estimates one can obtain
from the random-coefcients Logit model. The data used for this
demonstration were motivated by real scanner data (from the readyto-eat industry). However, the data is not real and should not be used
for any analysis. The focus is on providing a data set that can easily be used to learn the method. Therefore, I estimate a somewhat
restricted version of the model (with a limited amount of data). For a
more detailed and realistic use of the models presented here see either
BLP or Nevo (2000a, b). The data set used to generate the results and
the Matlab code used to perform the computation, is available from
http://elsa.berkeley.edu/~ nevo.
The data used for the analysis below consists of quantity and
prices for 24 brands of a differentiated product in 47 cities over 2 quarters. The data was generated from a model of demand and supply.25
The marginal cost and the parameters required to simulate this model
were motivated by the estimates of Nevo (2000b). I use two product characteristics: Sugar, which measures sugar content, and Mushy,
a dummy variable equal to one if the product gets soggy in milk.
Demographics were drawn from the Current Population Survey. They
include the log of income (Income), the log of income squared,
(Income Sq), Age, and Child, a dummy variable equal to one if the
individual is less than sixteen. The unobserved demographics, vi , were
drawn from a standard normal distribution. For each market I draw
20 individuals [i.e., ns 5 20 in equation (11)].
The results of the estimation can be found in Table I. The means
of the distribution of marginal utilities (b ¢ s) are estimated by a
minimum-distance procedure described above and presented in the
rst column. The results suggest that for the average consumer more
sugar increases the utility from the product. Estimates of heterogeneity around these means are presented in the next few columns. The
column labeled “Standard Deviations” captures the effects of the
unobserved demographics. The effects are insignicant, both economically and statistically. I return to this below. The last four columns
present the effect of demographics on the slope parameters. The point
estimates are economically signicant; I return to statistical signicance below. The estimates suggest that while the average consumer
25. The demand model of Section 2 was used. Supply was modeled using a standard
multiproduct-rms differentiated-product Bertrand model.

32.433
(7.743)

1.841a
(0.258)

0.148a
(0.258)

0.788a
(0.013)

Price

Constant

Sugar

Mushy

Standard

0.081
(0.205)

0.004
(0.012)

0.377
(0.129)

1.848
(1.075)

Deviations
¾

14.9
(7)

1.468
(0.697)

0.193
(0.005)

3.089
(1.213)

16.598
(172.334)

Income

—

0.659
(8.955)

IncomeSq

1.514
(1.103)

0.029
(0.036)

1.186
(1.016)

—

Age

Interactions with Demographic Variables

—

11.625
(5.207)

Child

Based on 2256 observations. Except where noted, parameters are GMM estimates. All regressions include brand and time dummy variables. Asymptotically robust standard
errors are given in parentheses.
a Estimates from a minimum-distance procedure.

GMM objective
(Degrees of freedom)

Variable

Means

Results from the Full Model

TABLE I.

Random-Coefcients Logit Models of Demand
539

540

Journal of Economics & Management Strategy

might like a soggy cereal, the marginal valuation of sogginess decreases with age and income. In other words, adults are less sensitive
to the crispness of a cereal, as are the wealthier consumers. The distribution of the Mushy coefcient can be seen in Figure 1. Most of the
consumers value sogginess in a positive way, but approximately 31%
of consumers actually prefer a crunchy cereal.
The mean price coefcient is negative. Coefcients on the interaction of price with demographics are economically signicant, while
the estimate of the standard deviation suggests that most of the heterogeneity is explained by the demographics (an issue we shall return
to below). Children and above-average-income consumers tend to be
less price-sensitive. The distribution of the individual price sensitivity
can be seen in Figure 2. It does not seem to be normal, which is a
result of the empirical distribution of demographics. In principle, the
tail of the distribution can reach positive values—implying that the
higher the price, the higher the utility. However, this is not the case
for these results.
Most of the coefcients are not statistically signicant. This is
due for the most part to the simplications I made in order to make
this example more accessible. If the focus were on the real use of
these estimates, their efciency could be greatly improved by using
more data, increasing the number of simulations (ns), improving the
simulation methods (see the appendix on the web), or adding a supply
side (see Section 5). For now I focus on the economic signicance.

FIGURE 1. FREQUENCY
SOGGINESS

DISTRIBUTION

TASTE

FOR

Random-Coefcients Logit Models of Demand

541

FIGURE 2. FREQUENCY DISTRIBUTION OF PRICE COEFFICIENT
As noted above, all the estimates of the standard deviations
are economically insignicant,26 suggesting that the heterogeneity in
the coefcients is mostly explained by the included demographics.
A measure of the relative importance of the demographics and random shocks can be obtained from the ratios of the variance explained
by the demographics to the total variation in the distribution of the
estimated coefcients; these are over 90%. This result is somewhat
at odds with previous work.27 The results here do not suggest that
observed demographics can explain all heterogeneity; they only suggest that the data rejects the assumed normal distribution.
An alternative explanation for this result has to do with the
structure of the data used here. Unlike other work (for example, BLP),
by construction I have no variation across markets in the choice set. As
I mentioned in Section 3.2, this sort of variation in the choice set helps
identify the variance of the random shocks. This explanation, however, does not explain why the point estimates are low (as opposed
to the standard errors being high) and why the effect of demographic
variables is signicant.
Table II presents a sample of estimated own- and cross-price
elasticities. Each entry i, j , where i indexes row and j column, gives
the elasticity of brand i with respect to a change in the price of j . Since
the model does not imply a constant elasticity, this matrix will depend
26. Unlike the interactions with demographics, even after taking measures to
improve the efciency of the estimates, they will still stay statisticall y insignicant.
27. Rossi et al. (1996) nd that using previous purchasing history helps explain
heterogeneity above and beyond what is explained by demographics alone. Berry et al.
(1998) reach a similar conclusion using second-choice data.

2
18
4
3
12

14
3
4
14
1

11
4
3
13
13

16
10
3
20
7

14
6
12
0

1
2
3
4
5

6
7
8
9
10

11
12
13
14
15

16
17
18
19
20

21
22
23
24

0
0
0
0

1
0
0
1
0

0
1
1
0
0

0
1
0
0
0

1
1
1
0
0

Mushy

0.0303

0.0773
0.1500
0.0854
0.2249

0.1430
0.1138
0.1900
0.0944
0.1420

0.0967
0.2711
0.2827
0.0834
0.0889

0.0699
0.2828
0.1743
0.0735
0.2109

2.1689
0.1119
0.2715
0.1868
0.0965

Brand

0.2433

0.1706
0.0950
0.1561
0.0548

0.2282
0.1399
0.0717
0.2691
0.0999

0.1556
0.0691
0.0597
0.1825
0.1763

0.1642
0.0606
0.0789
0.1704
0.0643

0.0562
3.4505
0.0674
0.0736
0.1540

Brand

0.0396

0.0741
0.1283
0.0815
0.1731

0.1523
0.1005
0.1542
0.1130
0.1270

0.0890
0.2339
0.2390
0.0786
0.0814

0.0698
0.2390
0.1476
0.0717
0.1682

0.2429
0.1302
2.9208
0.1501
0.0947

Brand

0.0552

0.1160
0.1602
0.1307
0.1619

0.0873
0.1466
0.1656
0.0771
0.1546

0.1377
0.1121
0.0983
0.1247
0.1256

0.1123
0.1035
0.1622
0.1128
0.1665

0.1161
0.0924
0.1083
3.2709
0.1326

Brand

0.2279

0.2534
0.1797
0.2559
0.1110

0.1447
0.2172
0.1362
0.2163
0.1791

0.2426
0.0735
0.0632
0.2697
0.2491

0.2518
0.0648
0.1524
0.2596
0.1224

0.0645
0.1905
0.0709
0.1470
5.5116

Brand

Elasticity

0.3349

0.3573
0.2096
0.3484
0.1044

0.1442
0.2867
0.1508
0.2618
0.2074

0.3322
0.0662
0.0554
3.8393
0.3542

0.3628
0.0563
0.1620
0.3586
0.1199

0.0596
0.2305
0.0627
0.1557
0.2852

Brand

0.3716

0.3794
0.2085
0.3753
0.1114

0.1702
0.2912
0.1578
0.3107
0.2080

0.3427
0.0704
0.0582
0.3922
4.3982

0.3971
0.0598
0.1656
0.3966
0.1298

0.0611
0.2606
0.0684
0.1564
0.3085

Brand

0.3896

0.3530
0.1530
0.3154
0.0733

0.3270
0.2498
0.1026
3.3335
0.1628

0.2899
0.0985
0.0866
0.3380
0.3383

0.3513
0.0865
0.1180
0.3443
0.0817

0.0759
0.4380
0.0944
0.1037
0.2988

Brand

0.0399

0.1416
0.2311
0.1608
4.1215

0.1264
0.1875
0.2660
0.1013
0.2251

0.1769
0.2260
0.2189
0.1600
0.1582

0.1369
0.2225
0.2555
0.1333
0.2809

0.2375
0.1165
0.2237
0.2670
0.1715

Brand

Cell entries i, j , where i indexes row and j column, give the percent change in market share of brand i with a one-percent change in price of j . Each entry represents the median of the elasticities
from the 94 markets.

Outside good

Sugar

Brand

Characteristics

TABLE II.

Median Own- and Cross-Price Elasticities

542
Journal of Economics & Management Strategy

Random-Coefcients Logit Models of Demand

543

on the values of the variables used to evaluate it. Rather than choosing
a particular value (say the average, or a value at a particular market),
I present the median of each entry over the 94 markets in the sample.
The results demonstrate how the substitution patterns are determined
in this model. Products with similar characteristics will have larger
substitution patterns, all else equal. For example, brands 14 and 15
have identical observed characteristics, and therefore their cross-price
elasticities are essentially identical.
A diagnostic of how far the results are from the restrictive form
imposed by the logit model is given by examining the variation in
the cross-price elasticities in each column. As discussed in Section 2,
the logit model restricts all elasticities within a column to be equal.
Therefore, an indicator of how well the model has overcome these
restrictions is to examine the variation in the estimated elasticities.
One such measure is given by examining the ratio of the maximum
to the minimum cross-price elasticity within a column (the logit model
implies that all cross-price elasticities within a column are equal and
therefore a ratio of one). This ratio varies from 9 to 3. Not only does
this tell us the results have overcome the logit restrictions, but more
importantly it suggests for which brands the characteristics do not
seem strong enough to overcome the restrictions. This test therefore
suggests which characteristics we might want to add.28

5. CONCLUDING REMARKS
This paper has carefully discussed recent developments in methods
of estimating random-coefcients (mixed) logit models. The emphasis
was on simplifying the exposition, and as a result several possible
extensions were not discussed. I briey mention these now.

5.1

Supply Side

In the above presentation the supply side was used only in order
to motivate the instrumental variables; it was not fully specied and
estimated. In some cases we will want to fully specify a supply relationship and estimate it jointly with the demand-side equations (for
example, see BLP). This ts into the above model easily by adding
moment conditions to the GMM objective function. The increase in
computational and programming complexity is small for standard
static supply-side models. As usual, estimating demand and supply
28. A formal specication test of the logit model [in the spirit of Hausman and
McFadden (1984)] is the test of the hypothesis that all the nonlinear parameters are
jointly zero. This hypothesis is easily rejected.

544

Journal of Economics & Management Strategy

jointly has the advantage of increasing the efciency of the estimates,
at the cost of requiring more structure. The cost and benets are specic to each application and data set.

5.2

Consumer-Level Data

This paper has assumed that the researcher does not observe the purchase decisions of individuals. There are many cases where this is not
true. In cases where only consumer data is observed, usually estimation is conducted using either maximum likelihood or the simulated
method of moments [for recent examples and details see Goldberg,
(1995), Rossi et al. (1996), McFadden and Train (2000), or Shum (1999)].
The method discussed here can be applied in such cases by using the
consumer-level data to estimate the mean utility level d j t . The estimated mean utility levels can now be treated in a similar way to the
mean utility levels computed from the inversion of the aggregate market shares. Care has to be taken when computing the standard errors,
since the mean utility levels are now measured with error.
In most studies that use consumer-level data, the correlation
between the regressors and the error term, which was the main motivation for the method discussed here, is usually ignored [one notable
exception is Villas-Boas and Winer (1999)]. This correlation might still
be present, for at least two reasons. First, even though consumers take
prices and other product characteristics as given, their optimal choice
from a menu of offerings could imply that econometric endogeneity
might still exist (Kennan, 1989). Second, unless enough control variables are included, common unobserved characteristics, »j t , could still
bias the estimates. The method proposed here could, in principle, deal
with the latter problem.
Potentially, one could observe both consumer and aggregate
data. In such cases the analysis proposed here could be enriched.
Petrin (1999) observes, in addition to the aggregate market shares
of automobile models, the probability of purchase by consumers of
different demographic groups. He uses this information in the form
of additional moment restriction (thus forcing the estimated probabilities of purchase to predict the observed probabilities). Although
technically somewhat different, the idea is similar to using multiple
observations on the same product in different markets (i.e., different
demographic groups). Berry et al. (1998) generalize this strategy by tting three sets of moments to their sample counterparts: (1) the market
shares, as above, (2) the covariance of the product characteristics and
the observed demographics, and (3) the covariance of rst and second
choice (they have a survey that describes what the consumer’s second
choice was). As Berry et al. (1998) point out, the algorithm they use is

Random-Coefcients Logit Models of Demand

545

very similar to the one they introduced in BLP, which was the basis
for the discussion above.

5.3

Alternative Methods

An alternative to the discrete-choice methods discussed here is a multilevel demand model. The essential idea is to use aggregation and
separability assumptions to justify different levels of demand [see
Gorman (1959, 1971), or Deaton and Muellbauer (1980) and references therein]. Originally these methods were developed to deal with
demand for broad categories like food, clothing, and shelter. Recently,
however, they have been adapted to demand for differentiated products [see Hausman et al. (1994) or Hausman (1996)]. The top level
is the overall demand for the product category (for example, RTE
cereal). Intermediate levels of the demand system model substitution
between various market segments (e.g., between children’s and natural cereals). The bottom level is the choice of a brand within a segment.
Each level of the demand system can be estimated using a exible
functional form. This segmentation of the market reduces the number of parameters in inverse proportion to the number of segments.
Therefore, with either a small number of brands or a large number
of (a priori) reasonable segments, this methods can use exible functional forms [for example, the almost ideal demand system of Deaton
and Muellbauer (1980)] to give good rst-order approximations to any
demand system. However, as the number of brands in each segment
increases beyond a handful, this method becomes less feasible. For a
comparison between the methods described below and these multilevel models see Nevo (1997, Chapter 6).

5.4

Dynamics

The model presented here is static. However, it has close links to several dynamic models. The rst class of dynamic models are models of
dynamic rm behavior. The links are twofold. The model used here
can feed into the dynamic model as in Pakes and McGuire (1994).
On the other hand, the dynamic model can be used to characterize
the endogenous choice of product characteristics, therefore supplying
more general identifying conditions.
An alternative class of dynamic models examine demand-side
dynamics (Erdem and Keane, 1996; Ackerberg, 1996). These models
generalize the demand model described here and are estimated using
consumer-level data. Although in principle these models could also
be estimated using aggregate (high-frequency) data, consumer-level
data is better suited for the task.

546

5.5

Journal of Economics & Management Strategy

Instruments and Additional Applications

As was mentioned in Sections 3.2–3.4, the identication of parameters
in these models relies heavily on having an adequate set of exogenous instrumental variables. Finding such instrumental variables is
crucial for any consistent estimation of demand parameters, and in
models of demand for differentiated products this problem is further
complicated by the fact that cost data are rarely observed and proxies
for cost will rarely exhibit much cross-brand variation. Some of the
solutions available in the literature have been presented, yet all suffer from potential drawbacks. It is important not to get carried away
in the technical reworks and to remember this most basic, yet very
difcult, identication problem.
This paper has surveyed some of the growing literature that uses
the methods described here. The scope of application and potential of
use are far from exhausted. Of course, there are many more potential
applications within the study of industrial economics, both in studying new industries and in answering different questions. However,
the full scope of these methods is not limited to industrial organization. It is my hope that this paper will facilitate further application of
these methods.

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Title                           : A Practitioner's Guide to Estimation of Random-Coefficients Logit Models of Demand
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A Practitioner's Guide To Estimation Of Random Coefficients Logit S Demand BLP

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