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\begin{center} {\Large {\bf The BLP Method of Demand Curve Estimation in
Industrial Organization}  } \\





 \bigskip


%15 March 2006 \\

14 December 2007 (revised as of January 15, 2020)\\


\bigskip

Eric Rasmusen \\



\bigskip

{\it Abstract}
  \end{center}

\begin{small}

 This is an exposition of the BLP method of structural demand
estimation  using the random-coefficients logit model. It is published in 
Japanese  in Gendai Keizaigaku 1, mikuro-bunseki,  edited by Isao Miura
and Tohru Naito, Tokyo: Keiso shobo. The English version will stay
posted on the web and most likely  be revised later.

\bigskip

\end{small}

\noindent {\small \hspace*{20pt}   
Department  of Business Economics and Public Policy, Kelley School of
Business,
Indiana University,  1309 E. 10th Street, Bloomington, Indiana,
47405-
1701. Office: (812) 855-9219. Fax: 812-855-3354. Erasmuse@indiana.edu.
\url{ http://www.rasmusen.org.} Copies of this paper can be found at
 \url{ http://www.rasmusen.org/papers/blp-rasmusen.pdf.   }}

 {\small  I thank Mishal Ahmed, Alberto Gaggero, Timothy Havel, Zhong Liu,    Fei Lu, and Ilia Sorvachev    for
their comments. }

%%-----------------------------%----------------

\newpage

\bigskip \noindent


%\begin{raggedright}

%\baselineskip 12pt

\parindent 24pt

 \noindent
 \textbf{I. Introduction}

 I came to know Professor Moriki Hosoe when he led the project of
translating
the first edition of my book, {\it Games and Information}, into Japanese
in 1990
and visited him at Kyushu University.   He realized, as I did, that a
great
many  useful new tools had been developed for economists, and that
explaining
the tools to applied modellers would be extraordinarily useful because
of the
difficulty of reading the original expositions in journal articles.
Here, my
objective is similar: to try to explain a new technique, but in this
case the
statistical one of the ``BLP Method'' of econometric estimation, named
after
Berry,    Levinsohn  \&   Pakes (1995). I hope it will be a useful
contribution
to the festschrift in honor of Professor Hosoe. I, alas, do not speak
Japanese,
so I thank xxx for their translation of this paper. I know from  editing
the
1997 {\it Public Policy and Economic Analysis} with Professor  Hosoe how
time-
consuming it can be to edit a collection of articles, especially with
authors of
mixed linguistic backgrounds, and so I thank Isao Miura and Tohru Naito
for
their editing of this volume.

 The BLP Method   is a way to estimate demand curves, a way that lends
itself to
testing theories of industrial organization. It   combines  a variety of
new
econometric techniques of the 1980's and  90's.  Philosophically, it is
in the
style of structural modelling, in which empirical work starts with a
rigorous
theoretical model in which players maximize utility and profit
functions, and
everything, including the disturbance terms, has an economic
interpretation, the
style for which McFadden won the  Nobel Prize.  That is in contrast to
the
older, reduced-form,  approach, in which the economist essentially looks
for
conditional correlations consistent with his theory. Both approaches
remain
useful. What we get with the structural approach is the assurance that
we do
have a self-consistent theory  and the ability to test much finer
hypotheses
about economic behavior. What we lose is simplicity and robustness to
specification error.

  Other people have written  explanations of the BLP Method. Nevo (2000)
is
justly famous for doing this, an interesting case of a commentary on a
paper
being a leading article in the literature itself. (Recall, though,
Bertrand's
1883 comment on Cournot  or Hicks's 1937 ``Mr Keynes and the
Classics.'')    The
present paper will be something of a commentary on a commentary, because
I will
use Nevo's article  as my foundation. I will use his notation  and
equation
numbering, and point out typos in his article as I go along.  I hope
that this
paper, starting from the most basic problem of demand estimation, will
be useful
to those, who like myself, are trying to understand modern structural
estimation.




\bigskip

\noindent
 \textbf{II. The Problem }

  Suppose we are trying to estimate a market demand curve. Our data is
the
quantity sold of a product, $q_{t}$, and the price, $p_{t}$ in twenty
towns $t =
1, ...20$. Our theory is that demand is linear, with this equation:
\begin{equation} \label{e01}
 q_{t} (p_{t})  = \alpha - \beta p_{t} + \varepsilon_{t}.
\end{equation}


 Let's start with an industry subject to price controls. A mad dictator
sets the
price in each town, changing it from town to town for entirely whimsical
reasons.  The result will be data that looks like Figure 1a. That  data
nicely
fits our model in equation (\ref{e01}), as Figure 1b shows.

\vspace*{24pt}
 \begin{center}
\begin{minipage}{1.0\textwidth}
 \begin{center}
  {\sc Figure 1: Supply and Demand with Price Controls  }

%\vspace{-22pt}

    \includegraphics[width=150mm]{blp1.png}
\end{center}
%\vspace*{-98pt}
\end{minipage}
\end{center}
 

Next, suppose we do not have price controls. Instead, we have a
situation of
normal supply and demand.  The problem is that now we might observe data
like
that in Figure 2a. Quantity rises with price; the demand curve seems to
slope
the wrong way, if we use ordinary least squares (OLS) as in Figure 2b.

\vspace*{24pt}
\begin{center}
\begin{minipage}{1.0\textwidth}
\begin{center}
 {\sc Figure 2:  Supply and Demand without  Price Controls  }
\vspace*{12pt}

    \includegraphics[width=150mm]{blp2.png}
\end{center}
%\vspace*{-98pt}
\end{minipage}
 \end{center}

 The solution to the paradox is shown in Figure 2c: OLS estimates the
supply
curve, not the demand curve. This is what Working (1927) pointed out.

 It could be that the unobservable variables $\varepsilon_{t}$ are what
are
shifting the demand curve in Figure 2. Or, it could be that it is some
observable variable that we have left out of our theory. So perhaps we
should
add income, $y_t$:
\begin{equation} \label{e02}
 q_{t} (p_{t})  = \alpha - \beta p_{t} + \gamma y_t  + \varepsilon_{t}.
\end{equation}

This  will help make the estimation of $\beta$ more accurate if $p_t$
and $y_t$ are correlated, but it does not help our basic problem of
untangling supply shifts from demand shifts. We need  to include some
observed
variable that shifts supply.


  Another approach would be to try to use a different kind of data. What
if
we could get data on individual purchases of product $j$?  Our theory
for the
individual is different from our theory of the market. The demand curve
looks
much the same, except now we have a subscript for consumer $i$.
\begin{equation} \label{e03}
 q_{it} (p_{t})  = \alpha_i - \beta_i p_{t} + \gamma_i y_{it}  +
\varepsilon_{it}.
\end{equation}
  But what about the supply curve? From the point of view of any one
small
consumer, the supply curve is flat. He has  only a trivial  influence on
the
quantity supplied and the market price, an influence we ignore in
theories of
perfect competition, where the consumer is a price-taker. Thus, we are
back to
something like Figure 1.




There is indeed one important difference. Now,  all we've done is
estimate the demand curve for one consumer.  That is enough, if we are
willing
to simplify our theory to assume that all consumers have identical
demand
functions:
\begin{equation} \label{e04}
 q_{it} (p_{t})  = \alpha  - \beta  p_{t} + \gamma  y_{it}  +
\varepsilon_{it}.
\end{equation}
This is not the same as assuming that all consumers have the same
quantity
demanded, since they will still differ in income $y_{it}$ and unobserved
influences, $\varepsilon_{it}$, but it does   say  that the effect of an
increase in price on quantity is the same for all consumers. If we are
willing
to accept that, however, we can estimate the demand curve for one
consumer, and
we can use our estimate  $\hat{\beta}$ for the market demand curve. Or,
we can
use data on $n$ different consumers, to get more variance in income, and
estimate  $\hat{\beta}$ that way.

      But we would not  have to use the theory  that consumers are
identical.
One alternative is to use the demand of one or $n$ consumers  anyway,
arriving
at the same estimate  $\hat{\beta}$ that we would under the theory (\ref
{e01}).
The interpretation would be different, though--it would be that we have
estimated the {\it average} value of $\beta$, and interpreting the
standard
errors would be harder, since they would be affected by the amount of
heterogeneity in $\beta_i$ as well as in $\varepsilon_{it}$. (The
estimates
would be unbiased, though, unlike the estimates I criticize in Rasmusen
(1989a,
b), since   $p_t$ is not under the control of the consumer.) Or, we
could
estimate the $\beta_i$ for all $n$ consumers and then average {\it the
$n$
estimates} to get a market $\beta$, as opposed to running one big
regression.
Under either alternative, if we had reason to believe that the $n$
consumers in
our sample were not representative of the entire population,  we would
want to
weight each consumer by the likely  frequency of his   type of
preferences  in
the population.

 Individual consumer data, however, is no panacea. For one thing, it is
hard to
get-- especially since it is important to get a representative sample of
the
population.  For another, its freedom from the endogeneity problem is
deceptive.
Recall that we assumed that each individual's demand had no effect on
the market
price. That is not literally correct, of course--- every one of   900,
000 buyers
of  toothbrushes has   some positive if trivial effect on market sales.
If one
of them decides not to buy a toothbrush, sales fall to 899,999. That
effect is
so small that the one consumer can ignore it, and the econometrician
could not
possibly estimate it given even a small amount of noise in the data. The
problem
is that changes in individuals' demand are unlikely to be  statistically
independent of each other. When the unobservable variable
$\epsilon_{900000t}$
for  consumer $i=900,000$ is unusually negative, so too in all
likelihood is the
unobservable variable $\epsilon_{899999t}$ for  consumer $i=899,999$.
Thus, they
will both reduce  their purchases at the same time, which will move the
equilibrium to a new point on the supply curve, reducing the market
price. Price
is endogenous for the purposes of estimation, even though it is
exogenous from
the point of view of any one consumer.

 So we are left with a big problem---identification--- for demand
estimation.
The analyst needs to use instrumental variables, finding some variable
that is
correlated with the price but not with anything else in the demand
equation, or
else he must find a situation like our initial price control example
where
prices are exogenous.

In fact,  even   price controls might not lead to exogenous prices. A
mad
dictator is much more satisfactory, at least  if he is truly mad.
Suppose we
have a political process setting the price controls,  either a
democratic one
or a sane dictator who is making decisions with an eye to everything in
the
economy and public opinion too. When is politics going to result in a
higher
regulated price? Probably when demand is stronger and  quantity is
greater. If
the supply curve would naturally slope up,  both buyers and sellers will
complain more if the demand curve shifts out and the regulated price
does not
change. Thus, even the regulated price will have a supply curve.

All of these problems arise in any method used to estimate a demand
curve,
whether it be the reduced-form methods just described or the BLP
structural
method. One  thing about  structural methods is that they force us to
think more
about the econometric problems. Your structural model will say what the
demand
disturbance is--- unobserved variables that influence demand. If you
must build
a maximizing model of where regulated prices come from, you will realize
that
they might depend on those unobserved variables.

What does all this have to do with industrial organization? Isn't it
just price
theory, or even just consumer theory?    Where are the firms? The reason
this is
so important in industrial organization is that price theory underlies
it.
Production starts because somebody demands something. An entrepreneur
then
discovers supply. That entrepreneur needs to organize the supply, and so
we have
the firm. Other entrepreneurs compete, and we have an industry. How
consumers
react to price changes is fundamental to this.   Natural extensions to
this
problem bring in most of how firms behave. Demand for a product depends
on the
prices of all firms in the industry, and so we bring in the theory of
oligopoly.
Demand depends on product characteristics, and so we have monopolistic
competition and location theory. Demand depends on consumer information,
and so
we have search theory, advertising, adverse selection, and moral hazard.
As we
have seen, estimating demand inevitably brings in supply. Or, if you
like, you
could think of starting with the problem of estimating the supply curve
in a
perfectly competitive industry, a problem that can be approached in the
same way
as we approach demand here.


\bigskip

\noindent
 \textbf{III. The Structural Approach  }

 Let us now start again, but with a structural approach. We will not
begin with
a demand curve this time. Instead, we will start with consumer utility
functions. The standard approach in microeconomic theory  is to start
with
the primitives of utility functions (or even preference orderings) and
production functions and then see how maximizing choices of consumers
and firms
result in observed behavior. Or, in game theory terms, we  begin with
players,
actions, information, and payoffs, and see what equilibrium strategies
result
from  players choosing actions to try to maximize their payoffs given
their
information.


Suppose we are trying to estimate a demand elasticity--- how quantity
demanded
responds to price.   We   have observations from  20 towns        of
cereal market data, the same 50 brands
of
cereal for each town (50 ``products''), which makes a total of 1,000
data points. We also
have data on 6 characteristics
of each cereal brand  and we have demographic data on how  4  consumer
characteristics   are distributed  in the population in each town.

   The  older structural approach is to use a  model of consumer
preferences over the products and estimate 50 interconnected demand
curves, as in the ``linear expenditure model'' of Stone (1954) (for the
descendants of that approach, see the references in
Hosken, Daniel,
Daniel O’Brien,
David Scheffman \&
Michael Vita [2002] and Gould [2006]).
 A problem with that approach    is that if there are 50 demand curves,
and demand for each product depends on the prices of the others  there
are 2,500 parameters in the model, and we only have 1,000 data points
with which to estimate them. This is the ``curse of dimensionality.''
The symmetry of the cross-elasticities and the adding-up restrictions
required by the common budget constraint reduce the number of  free
parameters, but estimation is still impractical. Even if we had more
observations, we also would need sufficient variation in the data to
sort out all the different interactions--- we would need sufficient
differences in the patterns of all 50 prices, for example.   Thus, we
will use a different approach, making consumer utility a function of
product characteristics  instead of the products themselves, and making
the  individual consumer's problem one of the probability of buying a
particular  product rather than how much to buy.



Each type of consumer will decide  which product  to buy, buying either
one
or zero units. We do not observe individual decisions, but we will model
them
so we can aggregate them to obtain the   product market shares that we
do observe.   The fact that  the frequency of different consumer types
is different in different towns   is the variance in the data that
allows us to estimate how each of the 4 consumer characteristics affects
demand. We can see if more sweet cereal is bought in Smallville, with
its many children, than in Littleton, with its aging population.


At this point, we could decide to   estimate the elasticity of demand
for each product and all the cross-elasticities directly, but with 50
products that would require estimating 2,500 numbers.   Instead, we will
focus on the product characteristics. There are only 6 of these, and
there would only be 6 even if there were 500 brands of cereal instead of
50. In effect, we will be estimating cross-elasticities between cereal
characteristics, but if know   those numbers, we can calculate the
cross-elasticities between products  by  combining the characteristic
effect parameter estimates with the characteristic level for each
product.


\bigskip

\noindent
 {\bf  The Consumer Decision}



The utility of consumer $i$  if he were to buy product $j$ in town $t$
is given by the following equation, denoted equation (1n) because it is
equation (1) in Nevo (2000):
 $$
\begin{array}{lll}
   &  u_{ijt}  &  = \alpha_i (y_{i} - p_{jt}) + {\bf x_{jt}\bbeta_i}  +
\xi_{jt}
+ \epsilon_{ijt}\\
& &  \\
  & & i = 1, ..., 400, \;\;  j = 1, ..., 50, \;\; t = 1, ..., 20, \;\;\\
\end{array}\eqno(1n)
$$
 where $y_{i}$ is the income of consumer $i$ (which is unobserved and
which we will assume does not vary across time), $ p_{jt}$ is the
observed  price of product $j$ in town $t$, $ {\bf x_{jt}}$   is a $6$-
dimensional vector of observed characteristics of product $j$ in town
$t$, $\xi_{jt}$ (the letter ``xi'') is a disturbance scalar summarizing
unobserved  characteristics of product $j$ in  town $t$, and
$\epsilon_{ijt}$ is the usual unobserved  disturbance with mean zero.
The parameters to be estimated are  consumer $i$'s  marginal utility of
income,   $\alpha_i$, and his marginal utility of product
characteristics, the $6$-vector {\bf $\bbeta_i$}.   I have boldfaced the
symbols for vectors and matrices above, and will do so  throughout the
article.


Consumer $i$ also has the choice to not buy any product at all. We will
model this outside option as buying ``product 0'' and normalize by
setting  the $j=0$    parameters equal to zero (or, if you like, by
assuming that it  has a zero price and zero values of the
characteristics):
\begin{equation} \label{e05a}
u_{i0t} \equiv  \alpha_i  y_{i}   + \epsilon_{i0t}
\end{equation}


 Equation (1n) is an indirect utility function, depending on income
$y_i$ and price $p_{jt}$ as well as on the real variables $x_{jt}$,
$\xi_{jt}$, and
$\epsilon_{ijt}$. It is easily derived from a quasilinear utility
function, however,  in which the consumer's utility is the utility  from
his consumption of one (or zero) of the 50 products, plus utility which
is linear in his consumption of the outside good. Our  consumers are
solving the problem,
\begin{equation} \label{e05b}
\begin{array}{ll}
   \stackrel {Maximize }{q_0, q_1, ...q_{50}}&  \left\{ q_0 + u_1 (q_1)+
u_2 (q_2)+ \cdots u_{50} (q_{50}) \right\}  \\
 & \\
   & {\rm  such \; that} \\
 & \\
& (a)\;\; \sum_{j=0}^{50} \left\{q_j p_j \right\} \leq y\\
 & \\
  &  (b)\;\; \sum_{j=0}^{50} \left\{q_j\right\} =1\\
 \end{array}
\end{equation}
  Constraint (a) is the usual budget constraint. Constraint (b) requires
that only one unit of one of the 50 goods, or no units of any of them (
so $q_0=1$ instead), be bought.



   Quasilinear utility is linear, not concave, in income, so it lacks
income effects.As a result, we do not need to observe consumer incomes
and we can denote income as $y_{i}$, constant across towns,     rather
than $y_{it}$, varying across towns.  If income effects  are important,
they can be modelled by indirect utility that  is a function not   of
$  (y_{i} - p_{jt})$ but of some concave function such as
$ log (y_{it} - p_{jt})$, as in BLP (1995).  Consumer $i$'s income does
appear, but it  does not change across towns, which may seem strange.
That is because  we are implicitly assuming that utility is linear in
money, so different income levels would make no difference to choices.



 A consumer's utility  depends  on a product's characteristics (
${\bf x_{jt}} $),  directly on the product in a way that is the same for
all consumers ($\xi_{jt}$), and on unobserved effects special to that
consumer, product, and town ($\epsilon_{ijt}$). Why is there no fixed
consumer effect $v_{it}$ analogous to the fixed  product effect
$\xi_{jt}$--- an effect    special to   consumer  $i$ in   town $t$, but
independent of  product $j$? We could add a $v_{it}$ variable without
harming the model, but it would have no effect, because when a consumer
comes to compare the utilities from different products  that variable
would be the same for all of them and make no difference. The
Smallsville consumer may be an unusually  happy person, but that does
not affect his choice between cereals.


 Consumer characteristics do not appear directly  in the utility
function.   They play a role later in the model, in determining
$\bbeta_i$, the marginal utility of product characteristics. Note that
consumer income could appear there if we observe it varying across
towns, as affecting the utility of different characteristics even if we
assume that utility is linear in money and the price elasticity does not
depend on income.



We will assume that  $\epsilon_{ijt}$ follows the Type I extreme-value
distribution,  which if it has mean zero and scale
parameter one  has the  density and cumulative distribution
\begin{equation} \label{e05}
f(x) = e^{-x} e^{-e^{-x}}  \;\;\;\;\;\;\;\;   F(x)  =   e^{-e^{-x}}.
\end{equation}
  This is the limiting  distribution of the maximum value of a   series
of draws of independent identically distributed random variables with
support over the entire real line.  The  more draws, the bigger the
expected maximum,  which is why the mean and  scale parameter can take
various values.
 Figure 3 illustrates the density, which is not dissimilar to the normal
distribution, except that it is slightly asymmetric and has thicker
tails. Note that it has infinite support, so the  unobserved effect
$\epsilon_{ijt}$ always has some probability of outweighing product
characteristics, price, and everything else.
 This distribution is standardly used in logit models because its
cumulative distribution is related to the probability   of  $x$ being
larger than any other of a  number of draws, which is like the random
utility from one choice being higher than that from a number of other
choices. This   leads to a convenient formula for   the probability that
a consumer makes a particular choice, and thus for a product's market
share, as we will see below.

 
\vspace*{24pt}
\begin{center}
\begin{minipage}{1.0\textwidth}

  {\sc Figure 3: The Type I Extreme-Value Distribution} 

\includegraphics[width=100mm]{blp3.png}

%\vspace*{-28pt}
\end{minipage}
\end{center}

 



 Consumer $i$ buys product $j$ in town $t$  if it yields him the highest
utility of any product or of not buying at all.     What we observe,
though, is not consumer $i$'s decision, but the market share of product
$j$. Also, though we do not  observe $\bbeta_i$, consumer $i$'s marginal
utility of product characteristics, we do observe  a sample of
observable characteristics. Even if we did observe his decision, we
would still have to choose between regular logit and  BLP's random-
coefficients logit, depending on whether we assumed that every consumer
had the same marginal utility of characteristics $\bbeta$ or whether
$\bbeta_i$ depended on consumer characteristics.





\bigskip

\noindent
{\bf  Standard Multinomial Logit: Identical Consumers}

   One way to proceed would be to assume that all consumers are
identical in
their taste parameters; i.e., that $\alpha_i = \alpha$ and {\bf
$\bbeta_i=
\bbeta$}, and that the $\epsilon_{ijt}$ disturbances are uncorrelated
across
$i$'s. Then we have the standard multinomial
logit model (multinomial because there are multiple choices, not just
two).     The utility function   reduces to the following form, which is
(1n) except that the parameters are no longer $i$-specific.
$$
\begin{array}{lll}
   &  u_{ijt}  &  = \alpha  (y_i - p_{jt}) + {\bf x_{jt}\bbeta }  +
\xi_{jt}
+ \epsilon_{ijt}\;\; \\
 & & \\
 & &  i = 1, ..., 400, \;\;  j = 1, ..., 50, \;\; t = 1, ..., 20. \\
\end{array}\eqno(5n)
$$
  Now the coefficients are the same for all consumers, but     incomes
differ. We can aggregate by adding up the incomes, however,   since the
coefficient on each consumer has the same value, $\alpha$.  Thus we
obtain an aggregate utility function,
\begin{equation} \label{e06}
     u_{ jt}    = \alpha  (y  - p_{jt}) + {\bf x_{jt}\bbeta }  +  \xi_{
jt}
+ \epsilon_{jt},  \;\;  j = 1, ..., 50, \;\; t = 1, ..., 20.
\end{equation}
  If we assume that  $\epsilon_{jt}$ follows the Type I extreme-value
distribution, then this is the multinomial logit model.

Since $\epsilon_{jt}$ follows the Type I extreme value distribution by
assumption,    the market share of product $j$ under our utility
function  is
$$
  {\displaystyle   s_{jt}   =  \frac{e^{ {\bf  x_{jt} \bbeta} -\alpha p_
{jt} +
\xi_{jt} }
}
{1+  \sum_{k=1}^{50}  e^{ {\bf  x_{kt} \bbeta} -\alpha p_{kt} + \xi_{kt}
}} }
\eqno(6n)
$$
    Equation (6n) is by no means obvious.  The market share of product
$j$ is the probability that $j$ has the highest utility, which occurs if
$\epsilon_{jt}$ is high enough relative to the other disturbances. The
probability that product 1 has a higher utility than the other 49
products and the outside good (which has a utility normalized to zero)
is thus
\begin{equation} \label{e06a}
\begin{array}{l}
      Prob(   \alpha  (y  - p_{1t}) + {\bf x_{1t}\bbeta }  +  \xi_{1t}
+ \epsilon_{1t} > \alpha  (y  - p_{2t}) + {\bf x_{2t}\bbeta }  +  \xi_{
2t}
+ \epsilon_{2t} ) * \\
Prob(   \alpha  (y  - p_{1t}) + {\bf x_{1t}\bbeta }  +  \xi_{1t}
+ \epsilon_{1t} > \alpha  (y  - p_{3t}) + {\bf x_{3t}\bbeta }  +  \xi_{
3t}
+ \epsilon_{3t} ) *  \cdots\\
  *   Prob(   \alpha  (y  - p_{1t}) + {\bf x_{1t}\bbeta }  +  \xi_{1t}
+ \epsilon_{1t} > \alpha  (y  - p_{50t}) + {\bf x_{50t}\bbeta }  +  \xi_
{50t}
+ \epsilon_{50t} )* \\
  Prob(   \alpha  (y  - p_{1t}) + {\bf x_{1t}\bbeta }  +  \xi_{1t}
+ \epsilon_{1t} >  \alpha  y      + \epsilon_{0t})\\
\end{array}
\end{equation}
   Substituting  for the Type I extreme value distribution into equation
(\ref{e06a}) and solving this out yields, after much algebra, equation (
6n). (For a step-by-step derivation,    see chapter 3.1 and its appendix
in Train (2003) on the Web.)  Since  $\alpha y$ appears on both sides of
each inequality, it drops out.  The  1 in equation (6n)  appears because
$e^0=1$ and    the outside good  adds  0  to utility.

\bigskip

\noindent
 {\bf Elasticities of Demand }




To  find  the elasticity of demand, we need to calculate
$\frac{\partial  s_{jt}}{\partial  p_{kt}  }$ for products
$k = 1, ..., 50$.  It
is  helpful  to rewrite equation (6n)
by defining $M_j$ as
\begin{equation} \label{e07}
  {\displaystyle  M_j  \equiv   e^{ {\bf  x_{jt} \bbeta} -\alpha p_{jt}
+
\xi_{jt}}.  }
\end{equation}
so
\begin{equation} \label{e08}
 {\displaystyle   s_{jt}      =  \frac{M_j } {1+  \sum_{k=1}^{50}   M_k}
.}
\end{equation}
 Then
\begin{equation} \label{e09}
  {\displaystyle  \frac{\partial  s_{jt}}{\partial  p_{kt}  }      =
\frac{\frac{\partial  M_j}{\partial  p_{kt}  }  } {1+  \sum_{k=1}^{50}
M_k}
+   \left( \frac{-M_j}{ (1+  \sum_{k=1}^{50}   M_k)^2 }\right) \left(
\frac{\partial   M_k}{\partial  p_{kt}  }  \right) }
\end{equation}

 First, suppose $k\neq j$. Then
\begin{equation} \label{e10}
\begin{array}{lll}
  &  \frac{\partial  s_{jt}}{\partial  p_{kt}  }    &  {\displaystyle=
\frac{0 }
{1+  \sum_{k=1}^{50}   M_k} +   \left( \frac{-M_j}{ (1+  \sum_{k=1}^{50}
M_k)^2 }  \right) \left(  - \alpha M_k       \right)}  \\
 & & \\
 & &  {\displaystyle=   \alpha   \left( \frac{ M_j}{  1+
\sum_{k=1}^{50}
M_k   }\right)
    \left(  \frac{ M_k}{  1+  \sum_{k=1}^{50}   M_k   }       \right) }
\\
 & & \\
 & & =   \alpha    s_{jt} s_{kt}  \\
\end{array}
\end{equation}




Second, suppose $k = j$. Then
\begin{equation} \label{e11}
\begin{array}{lll}
   &\frac{\partial  s_{jt}}{\partial  p_{jt}  }    &  {\displaystyle  =
\frac{
-\alpha M_j  } {1+  \sum_{k=1}^{50}   M_k} +   \left( \frac{-M_j}{ (1+
\sum_{k=1}^{50}   M_k)^2 }\right) \left(   -\alpha M_j    \right) } \\
 & & \\
& &  =   -\alpha s_{jt}   +   \alpha s_{jt}^2  \\
  & & \\
& &  =   -\alpha s_{jt}  (1-s_{jt}  )    \\
\end{array}
\end{equation}

We  can now calculate the elasticity of the market share:  the
percentage change
in the market share of product $j$ when the price of product $k$ goes
up:
\begin{equation} \label{e12}
\begin{array}{lll}
   & {\displaystyle \eta_{jkt} \equiv  \frac{\% \Delta s_{jt}} {\%
\Delta
p_{kt}}     = \frac{\partial  s_{jt}}{\partial  p_{kt}  }   \cdot \frac{
p_{kt}
}{s_{jt}}      =  \left\{
\begin{array}{ll }
  -\alpha p_{jt}(1-s_{jt}) & if \; j=k\\
 \;\; \;  \alpha p_{kt} s_{kt}  & {\rm otherwise}. \\
\end{array}
   \right.} \\
\end{array}
\end{equation}






\bigskip
\noindent
 {\bf   Why Multinomial Logit Is Unsatisfactory for Demand Estimation}

The theoretical structure of the
elasticities in equation (\ref{e12}) is unrealistic in two ways.

\noindent
1. If market shares are small, as is frequently the case, then
$\alpha (1-
s_{jt})$ is close to $\alpha$, so that own-price elasticities are close
to
$-\alpha p_{jt} $. This says that if the price is lower, demand is less
elastic,
less responsive to price, which in turn implies that the seller will
charge a
higher  markup on products with low marginal cost. There is no
particular reason
why we want to assume this, and in reality we often see that markups are
higher
on products with higher marginal cost, e.g. luxury cars compared to
cheap cars.

\noindent
2. The cross-price elasticity  of product $j$ with respect to the price
of  product $k$ is $ \alpha p_{kt} s_{kt}$, which only depends on
features   of product $k$--- its price and market share. If product $k$
raises its price, it  loses customers equally to each other product.
\footnote{An apparent third feature, that a high price of product $k$
means it has higher cross-elasticities with every other product, is
actually the same  problem  as problem (1). }   This is a standard
defect of multinomial logit, which  McFadden's famous  red-bus/blue-bus
example illustrates. If you can choose among   going to work by riding
the red bus,   the blue bus, or a bicycle,  and the price of riding the
red bus rises, are you equally likely to shift to the blue bus and  to
riding a bicycle?  Of course not, but the   multinomial logit model says
that you will.




   Another way to proceed is to use nested logit. In the red-bus/blue-
bus  example, you would first decide whether the blue bus or the red bus
had the highest utility, and then decide whether the best bus's utility
was greater than the bicycle's. In such a model, if the price of the red
bus rose, you might switch to the blue bus, but you would not switch to
the bicycle. A problem with nested logit, however, is that you need to
use prior information to decide how to construct the nesting.  In the
case of automobiles, we might want to make the first choice to be
between a large car and small car, but it is not clear that this makes
more sense than making the first choice to be between a high-quality car
and a low-quality car, especially if we are forcing all consumers to use
the same nesting.



\bigskip

\noindent
{\bf  Random Coefficients Logit: Heterogeneous Consumers}



An alternative to simple logit or nested logit is  to assume that the
parameters--- the marginal utilities of the product characteristics---
are different across  consumers, and  are determined by  the
consumer characteristics. Random coefficients is the name used for this
approach, though it is a somewhat misleading name. The approach does not
say that the consumer behaves randomly. Rather,  each consumer has fixed
coefficients in his utility function, but these coefficients are a
function both of fixed parameters that multiply his observed
characteristics and on unobservable characteristics that might as well
be random. Thus,
``random coefficients'' really means   ``individual coefficients''.
 We will  denote the average values of the parameters $\alpha_i$ and
$\bbeta_i$
across consumers as $\alpha$ and $\bbeta$,  and assume the following
specification (where the second step depends on  $\Sigma$ being diagonal): 



 $$
\begin{array}{ll}
   \left(
 \begin{array}{l}
 \alpha_i\\
  \bbeta_i  \\
  \end{array}
 \right)
 & =     \left(
 \begin{array}{l}
 \alpha \\
  \bbeta   \\
  \end{array}
 \right)
   +   {\bf   \Pi D_i  + \Sigma \bnu_i }   \\
  & \\
  &  =  \left(
 \begin{array}{l}
 \alpha \\
 {\bf \bbeta } \\
  \end{array}
 \right)
 +
  \left(
 \begin{array}{l}
 {\bf \Pi_{\alpha}} \\
 {\bf \Pi_{\bbeta} } \\
  \end{array}
 \right)  {\bf  D_i }
   +
 \left(
 \begin{array}{l}
 {\bf\Sigma_{\alpha}} \\
 {\bf \Sigma_{\bbeta} } \\
  \end{array}
 \right)
  \left(
   \bnu_{i \alpha}, \;\;
   \bnu_{i \bbeta}
 \right)
   \\
\end{array}\eqno(2n)
$$
 where  $ {\bf D_i}$ is a $4 \times 1$ vector of consumer $i$'s
observable characteristics,   $   \bnu_i $ is a $7 \times 1$ vector of
the effect of consumer $i$'s unobservable  characteristics on his
$\alpha_i$ and $\bbeta_i$ parameters; ${\bf \Pi}$ is a $7 \times 4$
matrix of how  parameters (the $\alpha_i$ and the 6 elements of
$\bbeta_i$ ) depend on consumer observables, ${\bf \Sigma}$ is a
$7 \times 7$ matrix of how those 7  parameters depend on the
unobservables, and   $(\bnu_{i \alpha}, \;\; {\bf \bnu_{i \beta} })$,  (
${\bf \Pi_{\alpha}}, {\bf \Pi_{\beta})}$  and (
${\bf \Sigma_{\alpha},  \Sigma_{\beta})}$ just split each  vector or
matrix into two parts.

We will denote the distributions of ${\bf D}$ and  $\bnu$ by
 ${\bf P^*_D (D)}$  and ${\bf P^*_{\bnu}(\bnu)}$.  Since we'll be
estimating the distribution of the consumer characteristics ${\bf D}$,
you will see the notation ${\bf \hat{P}^*_D(D) }$  show up too. We will
assume that
${\bf P^*_{\bnu}(\bnu)}$ is multivariate normal.\footnote{ Nevo
variously uses   ${\bf P^*_D ( D})$, ${\bf \hat{P}^*_D ( D})$,
${\bf P^*_{\bnu}(  \bnu})$, and ${\bf \hat{P}^*_{\bnu} ( \bnu})$ in his
exposition. I  have tried to follow him, but I may simply have
misunderstood what he is doing.}

\bigskip

\noindent
{\bf Utility in the Random Coefficients Logit Model}


Equation (1n) becomes\footnote{There is a   typographical error in the
Nevo paper on p. 520 in
equation (3n): $u$ instead of $\mu$ in each line. }
\begin{equation} \label{e13}
\begin{array}{lll}
   u_{ijt}  &  =&  \alpha_i (y_i - p_{jt}) + {\bf x_{jt}\bbeta_i}  +
\xi_{jt} +
\epsilon_{ijt}\\
& &  \\
        & =&  {\displaystyle   \alpha_i  y_i -  ( \alpha +  {\bf \Pi_{
\alpha}}
{\bf  D_i }+  {\bf   \Sigma_{\alpha}}  \bnu_{i \alpha}  ) p_{jt}
  + {\bf x_{jt} ( \bbeta + {\bf \Pi_{\beta} } {\bf D_i} + {\bf \Sigma_{
\bbeta} }
{\bf \bnu_{i \bbeta} }     ) }  +    \xi_{jt} + \epsilon_{ijt}}\\
& &  \\
          & =& {\displaystyle  \alpha_i  y_i + \left(- \alpha   p_{jt} +
{\bf
x_{jt}\bbeta } +  \xi_{jt}  \right)    - \left( {\bf \Pi_{\alpha}} {\bf
D_i} +
{\bf \Sigma_{\alpha}}  \bnu_{i \alpha}  ) p_{jt}  \right. \left.  + {\bf
x_{jt} (
{\bf \Pi_{\bbeta} } D_i + {\bf \Sigma_{\bbeta} }  {\bf \bnu_{i \bbeta} }
}
\right) + \epsilon_{ijt}}\\
& &  \\
        & =&  {\displaystyle  \alpha_i  y_i + \left(- \alpha   p_{jt} +
{\bf
x_{jt}\bbeta } +  \xi_{jt}  \right)    +
 \left( -p_{jt},  {\bf x_{jt}} \right)  \left(
 {\bf \Pi    D_i + \Sigma  \bnu_i }
  \right)
    + \epsilon_{ijt}} \\
& &  \\
        & =&    \alpha_i  y_i +  \delta_{jt} +  \mu_{ijt}
   + \epsilon_{ijt}\\
& &  \\
  & &      j = 1, ..., 50, \;\; t = 1, ..., 20.  \\
\end{array}
\end{equation}

  What I have done above is to reorganize the terms to separate them
into four
parts. First, there is the utility from income,  $\alpha_i  y_i$. This
plays no
part in  the consumer's choice, so it will drop out.

 Second, there is the ``mean utility'',
$\delta_{jt}$, which is the component of utility from  a consumer's
choice of product $j$ that is the
same across all consumers.
\begin{equation} \label{e14}
    {\displaystyle  \delta_{jt} \equiv - \alpha   p_{jt} + {\bf
x_{jt}\bbeta } +  \xi_{jt} }
\end{equation}
 Third and fourth, there is  a heteroskedastic  disturbance,
$\mu_{ijt}$, and a homoskedastic i.i.d. disturbance, $\epsilon_{ijt}$.
\begin{equation} \label{e15}
    {\displaystyle  \mu_{ijt}  \equiv  \left( -p_{jt},  {\bf x_{jt}}
\right)
\left(
 {\bf \Pi    D_i + \Sigma  \bnu_i }
  \right)  }
\end{equation}




\bigskip

\noindent
{\bf Market Shares and Elasticities in the BLP Model}


If we use the Type I extreme value distribution for $\epsilon_{ijt}
$, then the market share of  product $j$ for a consumer  of type $i$
turns out to be
\begin{equation} \label{e16}
\begin{array}{lll}
   &  s_{ijt} & {\displaystyle  =  \frac{ e^{\delta_{jt}   + \mu_{ijt}}
}
{1+  \sum_{k=1}^{50}  e^{\delta_{kt}   + \mu_{ikt}} }.} \\
\end{array}
\end{equation}

Recall that we  denote the distributions of ${\bf D}$ and  ${\bf \bnu}$
by
${\bf P^*_D ( D})$  and $P^*_{\bnu}( {\bf \bnu})$.  Since we will  be
estimating the
distribution of the consumer characteristics ${\bf D}$, you will see the
notation  ${\bf \hat{P}^*_D ( D})$  show up too.

The overall market share of product $j$ in town $t$ is found by
integrating the     market shares picked by each consumer in equation (
\ref{e16}) across the
individual types, weighting each type by its probability in the
population:
\begin{equation} \label{e17}
\begin{array}{lll}
 &  s_{jt} &  {\displaystyle =  \int_{\bnu}   \int_{\bf D} s_{ijt}{\bf
d\hat{P}^*_D ( D )
dP^*_{\bnu}(   \bnu})}\\
 & & \\
 & & =  {\displaystyle \int_{\bnu}   \int_{\bf D} \left[
  \frac{ e^{\delta_{jt}   + \mu_{ijt}} }
{1+  \sum_{k=1}^{50}  e^{\delta_{kt}   + \mu_{ikt}} }  \right]  {\bf   d
\hat{P}^*_D
(D ) dP^*_{\bnu}(   \bnu )}}\\
\end{array}
\end{equation}
     Equation (\ref{e17})  adds up  the market shares of different types
$i$ of consumers based on how common that type $i$ is in town $t$.

The price elasticity of  the market share of product $j$ with respect to
the price
of product $k$ is
\begin{equation} \label{e18}
\begin{array}{lll}
  &  {\displaystyle \eta_{jkt} \equiv  \frac{\partial s_{jt}}{\partial
p_{kt}
} \cdot  \frac{  p_{kt} }{s_{jt}} }  & \\
 & & \\
 & & =    \left\{
\begin{array}{ll}
 {\displaystyle   -\frac{p_{jt}} {s_{jt}}\int_{\bnu}   \int_{\bf D}
\alpha_i s_{ijt} (1-s_{ijt})
{\bf d\hat{P}^*_D ( D}) {\bf  dP^*_{\bnu}}( {\bf \bnu})} & if \; j=k \\
 & \\
   {\displaystyle \; \;\; \frac{ p_{kt}} { s_{jt}} \int_{\bnu}   \int_{
\bf D}  \alpha_i s_{ijt}
s_{ikt}{\bf d \hat{P}^*_D ( D}) {\bf dP^*_{\bnu}( \bnu}) } & {\rm
otherwise}. \\
\end{array}
 \right.\\
\end{array}
\end{equation}


   This is harder to estimate than the ordinary logit model, whose
analog of equation (\ref{e17}) is equation (6n), repeated below.
$$
  {\displaystyle   s_{jt}   =  \frac{e^{ {\bf  x_{jt} \bbeta} -\alpha p_
{jt} +
\xi_{jt} }
}
{1+  \sum_{k=1}^{50}  e^{ {\bf  x_{kt} \bbeta} -\alpha p_{kt} + \xi_{kt}
}} }
\eqno(6n)
$$
  The difficulty comes from the
integrals in  (\ref{e17}). Usually these need to be calculated by
simulation,  starting with our real-world knowledge of the distribution
of consumer  types $i$ in a given town $t$ and the characteristics of
product $j$, and combining that with estimates of how different consumer
types value different product characteristics. This suggests that we
might begin  with an initial set of parameter estimates, calculate what
market shares that   generates for each town, see how those match the
observed market shares, and then pick a new set of parameter estimates
to try to get a closer match.

   What is special about random-coefficients logit is not that it allows
for interactions between product and consumer characteristics, but that
it does so in a structural model.   One  non-structural approach would
have been to use  ordinary least squares to estimate the following
equation, including product dummies to account for the $\xi_{jt}$
product fixed  effects.
\begin{equation} \label{e18a}
       s_{jt} = {\bf  x_{jt} \bbeta }   - \alpha p_{jt}+ \xi_{jt}
\end{equation}
    Like simple logit, the method of equation (\ref{e18a})  implies that
the market share depends on the product characteristics and prices  but
not on any interaction between those things and consumer
characteristics. We can incorporate consumer characteristics by creating
new variables in a vector  { \bf  $d_{t}$} that represents the mean
value of the 4 consumer characteristics in town $t$, and then
interacting the  4  consumer variables in  ${\bf   d_{t} }$ with the 6
product variables in ${\bf   x_{t} }$ to create a $1 \times 24  $
variable ${\bf   w_{t}}$. Then we could use least squares with product
dummies to estimate
\begin{equation} \label{e18b}
       s_{jt} = {\bf  x_{jt} \bbeta }  - \alpha p_{jt}+ {\bf  d_{t}
\btheta_1} +  {\bf  w_{t} \btheta_w} +  \xi_{jt}
\end{equation}
 where  $\btheta_1$ is the coefficient vector for the 4  direct effects
of the
consumer variables and $\btheta_w$ is the coefficient vector for the  24
interactions between consumer and product variables.

 Equation (\ref{e18b}) adds
  market shares  directly  from the 4 consumer characteristics and less
directly via the 24 consumer-product characteristic  interactions.
Thus, it has some of the flexibility of the BLP model. It is not the
result of a consistent theoretical model. Even aside from whether
rational consumer behavior could result in a reduced form like (\ref{
e18a}) or (\ref{e18b}), those equations do not take account of the
relationships between the market shares of the different products---
the sum of all the predicted  market shares should not add up to more
than one, for example.  The random-coefficient logit model avoids this
kind of internal contradition.


 Before going on to estimate the random coefficients model, however,
recall that regardless of how we specify the utility function--- logit,
nested logit, random-coefficients logit, or something else---   we face
the  basic simultaneity problem  of estimating demand and supply
functions.
Market shares
depend on prices and disturbance terms, but prices will also depend on
the
disturbance terms.  If demand is unusually strong for a product because
of some unobservable variable, the price  of that product will be higher
too.
This calls for some kind of instrumental variables estimation. What  we
will use  here is the generalized method of moments.




\bigskip




\noindent
{\bf The Generalized  Method of Moments }

  The generalized method of moments (GMM) of Hansen (1982)  will
combine aspects of  overidentified instrumental variables,
generalized least squares, and a nonlinear specification  in our demand
setting. It helps to separate these things out. We will go one step at a
time, before circling back to the BLP method's use of GMM.


 Suppose we want to estimate
\begin{equation}
  y = x_1 \beta_1 + x_2 \beta_2 +  \epsilon,
\end{equation}
 where we observe $y$, $x_1$, and $x_2$, but not $\epsilon$, though we
know that $\epsilon$ has a mean of zero.
  We assume that the $x$'s and  the unobservable disturbances
$\bepsilon$ are uncorrelated, which is the definition of the $x$'s
being exogenous (if we included a constant term, it would be
uncorrelated with $\bepsilon$ too--- another way to include the
assumption that $\bepsilon$ has a mean of zero). This lack of
correlation makes up the   two ``moment conditions'' that we can write
as
\begin{equation}
 M_1:  E {\bf (x_1'   \bepsilon)}=0, \;\;\;\;\;\;   M_2: E ({\bf x_2'
\bepsilon})=0.
\end{equation}
 or
\begin{equation}
   E (M_1) =0,  \;\;\;\;\;\;   E (M_2) =0.
\end{equation}
   Note that if there are $T$ observations,   ${\bf x_1}$  is a
$T \times  1$ vector, but  $M_1 ={\bf  x_1'   \bepsilon}$ is
$1 \times 1$.

In the  generalized method of moments, we choose $\hat{\beta}$ to make a
weighted sum of squares of   the sample moment expressions (which are
the $M$'s that equal zero in the moment condition) as small as possible.
Let's put aside the question of what   weights to use for now.
The sum of squares of the moment expressions  is
\begin{equation}
  (M_1 \;\; M_2)'(M_1 \;\; M_2)
\end{equation}
  Think of $M_1$ as a random variable, since it incorporates those $T$
random variables $\bepsilon$ in the $T$ observations.  The expected
value  of $M_1$ is zero, by assumption, but in our sample   its
realization  might be   positive or negative, because its variance is
not zero.  The matrix $(M_1 \;\; M_2)$ is  $2 \times 1$, so the sum of
squared moment expressons  is $ 1 \times 1$.

 Another way to write the problem  is  to choose $\hat{\bbeta}$ to
minimize   ${\bf M'M}$.  If  ${\bf M= X' \bepsilon}$,   we will find
\begin{equation} \label{e39}
\begin{array}{lll}
   &  \hat{  \bbeta}  & {\displaystyle = \stackrel{argmin}{  \bbeta } {
\bf
\hat{\bepsilon}'   X     X' \hat{\bepsilon}  } } \\
\end{array}
\end{equation}
 Thus, we minimize the function $f(\hat{\bbeta})$:
 \begin{equation} \label{e40}
\begin{array}{lll}
 f(\hat{\bbeta}) &= &
   {\bf
\hat{\bepsilon}'  {\bf  X      X}' \hat{\bepsilon}} \\
  &  & \\
  & =&  ({\bf y - X \hat{\bbeta})'   X      X' (y - X \hat{\bbeta}) } \\
  &  & \\
 & =&  {\bf y'X X'y }   -   \hat{\bbeta}'{\bf X'   X      X'  y}    -  {
\bf y'X
X'  X}
\hat{\bbeta}   +  \hat{\bbeta}'{\bf X'   X      X'  X} \hat{\bbeta}  \\
 \end{array}
\end{equation}
 We can differentiate equation (\ref{e40}) with respect to
$\hat{\bbeta}$ to get the first order
condition,
\begin{equation} \label{e41}
\begin{array}{lll}
 f'(\hat{\bbeta})   & =&     -    {\bf X'   X      X'  y}   -    {\bf
y'X X'  X}
+
2\hat{\bbeta}'  {\bf X'   X      X'  X}     \\
 &  & \\
 & =&     -    2  {\bf X'   X      X'  y}        +  2\hat{\bbeta}'  {\bf
X'   X
X'  X}
\\
 &  & \\
 & =&          2  {\bf X'   X}       (-  {\bf X'  y}        +  \hat{
\bbeta}'
{\bf X'  X})  =0   \\
 \end{array}
 \end{equation}
 in which case
\begin{equation}
 \hat{\bbeta}      =          ( {\bf X'   X})^{-1}        {\bf X'  y}
\end{equation}
 and we  have the  OLS estimator.

 \bigskip

  We  might also know that the $x$'s and disturbances are {\it
independent}:
\begin{equation}
   E (\epsilon|x_1, x_2)=0.
\end{equation}
 This is different from being uncorrelated. For example, suppose
$\epsilon$'s value is either  $-1$ or $+1$ with equal probability, and
$x_1$ equals  0 if $\epsilon=-1$ and    $3$ or $-3$ with equal
probability if $\epsilon = 1$. Then     $  E ({\bf \bepsilon'x_1 })=0$,
but $E (\epsilon|x_1=0)=-1$.  Moreover, various functions of ${\bf x_1}$
are correlated with $\epsilon$ even though   ${\bf x_1}$ is not.
$E ( {\bf  \bepsilon'(x_1^2) })= .5 (-1)(0) + .5(.5* 3^2*1) +
.5*(-3)^2*1 = 4.5, $. Thus, $\epsilon$ and $x_1$ are    uncorrelated but
not independent.


  We want to use all available information, for efficient estimation, so
we would like to use that independence information.  It will turn out to
be useful information if  the variance depends on $ {\bf X}$, though not
otherwise. (That ``not otherwise'' will show up as that if the variance
does not depend on $ {\bf X}$, then the moment variance covariance
matrix entry is zero, and if we know  ex ante  the information that it
equals zero then we should use that information.)

Independence gives us lots of other potential moment conditions. Here
are a couple:
\begin{equation}
E {\bf    ((x_1^2)'   \bepsilon)} = E(M_3) =0,   \;\;\;\;\;\;    E  {\bf
((x_2*x_1)'   \bepsilon)} = E(M_4)=0.
\end{equation}

Some of these conditions  are more reliable than others. So we'd like to
weight them  when we use them. Since $M_3$ and $M_4$ are random
variables, they have variances. So let's weight them by the the inverse
of their variances--- more precisely, by the   inverse of their
variance-covariance matrix, since they have cross-correlations.  Call
the variance-covariance matrix of all the moment conditions
${\bf  \bPhi(M).}$ We can estimate that matrix consistently by running a
preliminary  consistent regression such as OLS and making use of the
residuals.


The GMM estimator uses the inverse of the variance-covariance matrix of
the moment conditions, ${\bf  \bPhi(M)^{-1} }$,  to weight them in the
estimation, a weighting scheme that  has been shown to be optimal (see
Hansen [1982]). We minimize the weighted square of the moment conditions
by choice of the parameters $\hat{\bbeta}$.
\begin{equation}
  (M_1 \;\; M_2 \;\; M_3 \;\; M_4)' {\bf (\bPhi(M)^{-1})}  (M_1 \;\; M_2
\;\; M_3 \;\; M_4)
\end{equation}

 The weighting matrix is crucial. OLS   uses  the most obviously useful
information.    We can throw in lots and lots of other moment conditions
using the independence assumption, but they will contain less and less
new information. Adding extra information is always good in itself, but
in finite samples, the new information, the result of random chance,
could well cause more harm than good.    In such a case, we wouldn't
want to weight the less important moment conditions, which might have
higher variance, as much as the basic exogeneity ones.   Consider the
moment condition $M_5$:
\begin{equation}
 E  {\bf ((x_2^3*x_1^5)'   \bepsilon) } = E (  M_5) =0.
\end{equation}
 That moment condition doesn't add a lot of information, and it could
have a big variance not reflected in the consistent estimate of
${\bf  \bPhi(M)}$ that we happen to obtain from our finite sample.

   We have now gotten something like  generalized least squares, GLS,
from the generalized method of moments.   I did not demonstrate it, but
 ${\bf   \bPhi(M)}$ will turn out to be  an estimate of the  variance
covariance matrix   of    $\bepsilon$.  It is not the same as other
estimates used in GLS, because it depends on exactly which moment
conditions are used,  but it is consistent.  We have a correction for
heteroskedasticity, which is something we need for estimation of the BLP
problem.    Notice that this means that GMM can be useful even though:

(a) This is a linear estimation problem, not nonlinear.

(b) No   explanatory variables are endogenous, so this is not an
instrumental variables problem.

\bigskip

    There are other ways to correct for heteroskedasticity, but the GMM
estimator shows its true strength when used as a form of instrumental
variables estimation.\footnote{The following exposition is based on
Hall(1996).}
 Suppose  that one of our basic moment conditions fails.
$  E {\bf (x_2 \bepsilon) }\neq 0$, because ${\bf x_2}$ is endogenous,
and we have lost our moment conditions $M_2$ and $M_4$.    What we need
is a  new basic moment condition that will enable us to estimate
$\beta_2$--- that is, we need an instrument correlated with ${\bf x_2}$
but not with $\bepsilon$. Suppose we do have a  number of such
conditions, a set of variables ${\bf z_1}$ and  ${\bf z_2}$.  We can use
our old conditions $M_1$ and $M_3$, and we'll add a couple others too,
ending up with this set:
\begin{equation}
 E ({\bf x_1} \bepsilon) =  0  \;\;\;  E(({\bf x_1^2})'   \bepsilon) =
0,   \;\;\;   E ({\bf z_1} \bepsilon)   =0   \;\;\;     E ({\bf z_2}
\bepsilon)   =0  \;\;\;     E (({\bf z_1*x_1})'   \bepsilon)= 0 \;\;\;
E (({\bf z_1*z_2})'   \bepsilon)= 0.
\end{equation}
    We will abbreviate these six moment conditions as
\begin{equation}
 E {\bf (Z'\bepsilon) }=  E {\bf  (M) } =0,
\end{equation}
  where the matrix   ${\bf Z}$  includes separate columns for the
original variable ${\bf x_1}$, the simple instruments ${\bf z_1}$ and
${\bf z_2}$, and the interaction instruments,
${\bf z_1*x_1}$ and ${\bf z_1*z_2}$.


 Let's suppose also, for the moment, that we have the ex ante
information that the disturbances are independent of each other and
${\bf Z}$, so there is no heteroskedasticity. Then we can derive the
weighting matrix thus (noting that a variance is calculated using the
deviation from the mean, which here is zero):
\begin{equation}
{\bf \bPhi(M) =   Var (M)  =    Var (Z'\bepsilon) =}  E {\bf ( Z'
\bepsilon \bepsilon' Z)}  - E {\bf (Z'\bepsilon)}  E{\bf (\bepsilon'Z)}
= E {\bf ( Z'  (I \sigma^2)  Z)  - 0^2 = \sigma^2 Z'Z.}
\end{equation}

The GMM estimator uses that variance-covariance matrix in the weighting
matrix that puts different weight on the six moment conditions. It
solves the problem of choosing the parameters
${\bf \hat{\bbeta}_{2SLS}}$  to  minimize
 \begin{equation} \label{e40aa}
\begin{array}{lll}
 f({\bf \hat{\bbeta}_{2SLS}}) &= &
   {\bf
\hat{\bepsilon}_{2SLS}'  {\bf  Z  (\sigma^2 Z'Z)^{-1}     Z}' \hat{
\bepsilon}_{2SLS}} \\
  &  & \\
  & =&  ({\bf y - X \hat{\bbeta}_{2SLS})'  {\bf  Z  (\sigma^2 Z'Z)^{-1}
Z}' (y - X \hat{\bbeta}_{2SLS}) }
 \end{array}
\end{equation}
 We can differentiate this with respect to ${\bf \hat{\bbeta}_{2SLS}}$
to get the first order
condition
\begin{equation} \label{e41aa}
\begin{array}{lll}
 f'( {\bf \hat{\bbeta}_{2SLS}) }  & =&    {\bf   - X'    {\bf  Z  (
\sigma^2 Z'Z)^{-1}     Z}' (y - X \hat{\bbeta}_{2SLS}) }=0,
 \end{array}
 \end{equation}
which solves to
\begin{equation}
 {\bf \hat{\bbeta}_{2SLS}    =   [ X'Z   ( Z'Z)^{-1}  Z'X]^{-1} X'Z  (
Z'Z)^{-1} Z' y}
\end{equation}

   This   estimator is both the GMM estimator and the 2SLS (two-stage
least squares) estimator. They are equivalent when the disturbances are
independently distributed, though if there were heteroskedasticity they
would become different  because GMM would use the weighting matrix
${\bf (\bPhi(M))^{-1}}$, which would not be the same as
${\bf (Z'Z)^{-1}}$. 2SLS could   be improved upon with
heteroskedasticity corrections, however,    in the same way that OLS can
be improved.

  Notice that this is the 2SLS estimator, rather than the simpler
instrumental variables (IV) estimator that is computed by calculating IV
directly:
\begin{equation}\label{e100}
 {\bf \hat{\bbeta}_{IV}   =   ( X'Z)^{-1} Z' y}
\end{equation}
   Two-stage least squares and IV are the same if the number of
instruments is the same as the number of parameters to be estimated, but
otherwise the formula in equation (\ref{e100}) cannot be used, because
when ${\bf X}$ is $ T   \times J$ and ${\bf Z}$ is $ T  \times K$,
${\bf X'Z}$ is  $J \times K$, which is not square and cannot be inverted
.   What  2SLS is doing differently from IV is  projecting   ${\bf X}$
onto   ${\bf Z}$  with  the projection matrix,
${\bf Z   ( Z'Z)^{-1}  Z'}$,  to generate a square matrix that can be
inverted.    GMM does something similar, but with    $ {\bf \bPhi(M)}$
instead of   ${\bf Z   ( Z'Z)^{-1}  Z'}$.

\bigskip

   We have so far solved for {\bf $\hat{\bbeta}$} analytically, but that
is not an essential
part of GMM. The parameters {\bf $\bbeta$} might enter the problem
nonlinearly, in which case  minimizing the moment expression could be
done using some kind of search algorithm. For example, suppose our
theory is that
 \begin{equation}\label{e101}
y   =   x_1^{\beta_1} + \beta_1 \beta_2 x_2 + \epsilon,
\end{equation}
and our moment conditions are
 \begin{equation}\label{e102}
   E  {\bf (x_1'\bepsilon)} =M_1=0,  \;\;\;\;  E   {\bf
(x_2'\bepsilon)} =M_2=0 \;\;\;\; E   {\bf (x_1*x_2'\bepsilon)}=M_3=0.
\end{equation}
  We could then search over values of $\beta_1$ and $\beta_2$ to
minimize the moment expression,
 \begin{equation}\label{e102a}
   {\bf   (y  -   x_1^{\beta_1} + \beta_1*\beta_2*x_2)'M (\bPhi(M))^{-1}
M' (y  -   x_1^{\beta_1} + \beta_1*\beta_2*x_2)},
\end{equation}
 where we would have to also estimate   ${\bf \bPhi(M)}$  during some
part of the search.

\bigskip


  \noindent
{\bf Combining Logit and GMM   }

 \noindent
Now let us return to  random coefficients logit.  If our
assumption on the population is that\footnote{I think there is a typo in
Nevo
here, on page 531, and $z_m'$ should replace $Z_m$ in equation (8n).}
 $$
\begin{array}{lll}
  & {\bf  E  (Z_m  \bomega(\btheta^*)) }= 0,      & m = 1, \ldots, M,
\\
\end{array}\eqno(8n)
$$
 then  the GMM  estimator is
$$
\begin{array}{lll}
   &  \hat{\bf \btheta} & {\displaystyle  = \stackrel{argmin}{\bf
\btheta }   {\bf
\bomega(\btheta)'   Z  \bPhi^{-1}     Z' \bomega(\btheta),}  } \\
\end{array}\eqno(9n)
$$
 where ${\bf \bPhi}$ is a consistent estimator of
${\bf E ( Z'  \bomega
\bomega'
Z})$. The  instrument matrix ${\bf Z}$  consists both of   single-
variable instruments and multi-variable instruments consisting of
squares and interactions of the single variables.


As in ordinary least squares,but unlike in maximum
likelihood, does not require us to know the distribution of the
disturbances. In
our demand estimation, though, we will still have to use the assumption
that the
$\epsilon_{ijt}$ follow the extreme value distribution, because we need
it to
calculate the   market shares aggregated across consumer types, whether
by plain logit or random-coefficients logit.

In more detail, here is how the estimation proceeds. First, the modeller
must set up his theoretical model, which at a minimum means choosing
lists of   observation units (towns, or months, or town/months, etc.),
products,  product characteristics, and  consumer  characteristics.  He
may also choose to include additional variables besides the prices which
are common to all consumers:  advertising, hot weather, a time trend,
and so forth (in which case the parameter $\alpha$ would become a vector
of parameters).  Following the Appendix to Nevo (2000), denote by
${\bf X_1}$ the matrix of  explanatory variables common to all
consumers, including not only the prices, advertising, and so forth but
a dummy variable for each product. These are separated out because  they
are the only variables that enter into the simpler, linear part of the
estimation needed to obtain the average parameter values,
$(\alpha,   {\bf \bbeta }). $
 Denote by ${\bf X_2}$  the matrix of directly observed  explanatory
variables  that   enter into the nonlinear part of the estimation, the
estimation of ${\bf ( \Pi, \Sigma})$ for the random   coefficients (i.e.
the individuals' deviations from the average coefficients):   the prices
and any other common variables which affect consumers with different
characteristics differently, and the product characteristics, but not
product dummies. Also, neither ${\bf X_1}$ nor ${\bf X_2}$ includes the
consumer characteristics, since they enter the estimation separately, at
the point where the market shares are estimated by aggregating across
consumers.  The modeller must also
  choose  instruments for price and perhaps for other common variables.
All this data must be collected, as must the market shares for each
product and   a distribution or actual  sample of consumers with their
characteristics. The   estimation then
  take the following steps (the labelling of the steps  is adapted  from
Nevo [2000], Appendix, p. 1).


\noindent
 (-1)  Select  arbitrary values for $    \bdelta $  and
$(  {\bf  \Pi, \Sigma})$ (for step 1)     and for
$(\alpha,   {\bf \bbeta })$  (for step  (3))
as
starting points.       Recall that   ${\bf \bdelta}$ from (\ref{e14}) is
the  vector of  the mean utility from each of the  products,   that
$({\bf \Pi, \Sigma})$ is the matrix  of parameters  showing how observed
and unobserved  consumer characteristics and product characteristics
interact to generate utility, and that  $(\alpha,   {\bf \bbeta })$  is
the average value of  the parameters across consumers.


\noindent
(0)  Draw random values for $({\bf \bnu_i, D_i  })$ for $i = 1, ...n_s$
from the
distributions ${\bf  P^*_{\bnu} (  \bnu)}$ and
${\bf \hat{P}^*_{D} (  D}  )$ for a
sample of size $n_s$, where the bigger  you pick $n_s$  the more
accurate your
estimate will
be.



\noindent
(1)   Using the starting values and the random values,  and using the
assumption
that the $\epsilon_{ijt}$ follow the extreme-value distribution,
approximate the
integral for market share that results from aggregating
across $i$ by
    $$
\begin{array}{lll}
   &  s_{jt} &  {\displaystyle =\left(  \frac{1}{n_s} \right) \sum_{i=1}
^{n_s}
s_{ijt}} \\
 & & \\
 &   &  {\displaystyle = \left(  \frac{1}{n_s} \right) \sum_{i=1}^{n_s}
\left[
 \frac{
e^{[\delta_{jt} + \Sigma_{k=1}^{6}  x^k_{jt} ( \sigma_k \nu^k_i + \pi_{
k1}
D_{i1} + \cdots + \pi_{k4} D_{i4})]    }
 }
{
1+  \sum_{m=1}^{50}  e^{ [ \delta_{mt} + \sum_{k=1}^{6}  x^k_{mt}  (
\sigma_k \nu^k_i + \pi_{k1} D_{i1} + \cdots + \pi_{k4} D_{i4})  ] }
}
   \right],  } \\
\end{array}\eqno(11n)
$$
where $(\nu_i^1, \ldots, \nu_i^6)$ and $(D_{i1}, \ldots, D_{i4})$ for
$i = 1,
\ldots n_s$ are  those  random draws from the previous step.

Thus, in step (1)  we obtain      predicted market shares for given
values of the individual consumer parameters   $(  {\bf \Pi, \Sigma})$
and for given values of the mean utilities $\bdelta$.




\noindent
(2)    Use  the following  contraction mapping, which, a bit
surprisingly,
converges.  Keeping $(  {\bf   \Pi, \Sigma})$ fixed at their starting
points,
find  values of ${\bf  \bdelta}$ by the following iterative process.
  $$
\begin{array}{lll}
   &  \bdelta^{h+1}_{\cdot t} &  {\displaystyle = \bdelta^{h }_{\cdot t}
+
{\bf(ln (
S_{\cdot t}) - ln (s_{\cdot t})}), }  \\
\end{array}\eqno(12n)
$$
where ${\bf S_{\cdot t}} $ is the  observed market share. and
${\bf s_{\cdot t}}$ is the
predicted market share from step (1) that uses $\bdelta^{h+1}_{\cdot t}$
as its
starting point.   Start with the arbitrary $\bdelta^0$ of step (-1).

 If the observed and predicted market shares are equal, then
$\bdelta^{h+1}
_{\cdot t}   = \bdelta^{h }_{\cdot t}$ and the series has converged.  In
practice, keep iterating until
${\bf (ln (S_{\cdot t}) - ln (s_{\cdot t}))}$ is small
enough for you to be satisfied with its accuracy.



Thus, in step (2) we come out  with  values for $\bdelta$.


\noindent
(3) Figure out the value  of the  moment expression,  using  the
starting values for $(\alpha,   {\bf \bbeta }) $ from step (0)  and the
$\bdelta$ estimate from step (2).




\noindent
(3a) Calculate  the error term $\bomega_{jt}$.\footnote{In my original version, and in Nevo's equation (13), there is a mistake: the minus sign in front of $\alpha$ is omitted. This expression comes from equation (3) in Nevo and equation (\ref{e14}) of the present paper. Equation (13n)below has the mistake fixed now---November 18, 2011.} 
 $$
\begin{array}{lll}
  &   \bomega_{jt}  &  {\displaystyle =  \bdelta_{jt} - ({\bf - \alpha
p_{jt} +x_{jt}} \bbeta
})  \\
\end{array}\eqno(13n)
$$


\noindent
(3b)  Calculate the value of the moment expression,
\begin{equation} \label{e45}
\begin{array}{lll}
  &     & {\bf \bomega'Z \bPhi^{-1} Z' \bomega}   \\
\end{array}
\end{equation}

You need a  weighting matrix $\bPhi^{-1}$ to do this, which ideally is
\begin{equation} \label{e45b}
 \bPhi^{-1}  =    (E {\bf ( Z' \bomega \bomega' Z)  })^{-1}.
\end{equation}
 In practice,  we use a consistent estimator of  $\bPhi^{-1}$.  Until
  step (4c),      just use
$\bPhi^{-1} ={\bf (Z'Z)^{-1}}$ as a starting point.



\noindent
 (4) Compute better estimates of all the parameters:the common
parameters  $(\alpha,   {\bf \bbeta }) $, the individual parameters
${\bf ( \Pi, \Sigma})$, and the weighting matrix $\bPhi$.

\noindent
(4a) Find an estimate of the parameters  that are common to all
consumers,  $(\alpha,   {\bf \bbeta }) $,    using the GMM estimator,
\begin{equation} \label{e45a}
 (\hat{\alpha},    \hat{ \bbeta })  =    {\bf (X'Z \bPhi^{-1} Z'X)^{-1}
X' Z   \bPhi^{-1} Z' \bdelta }
\end{equation}

   This is a linear estimator that can be found analytically  by
multiplying   matrices without any need for  numerically minimizing
something
with search algorithm.  Separating out the parameters that can be
linearly  estimated from the parameters that require a search algorithm
is  why we  use all these steps instead of simply setting up the moment
expression and then using a minimization algorithm to find parameter
values that minimize it. Searching takes the computer  longer than
multiplying matrices, and is less reliable in finding the true minimum,
or, indeed, in converging to any solution.

\noindent
(4b) Estimate the value of the error term in  (13n), $\hat{\bomega}$
and then the  moment expression in   (\ref{e45}), using the improved
estimates of $(\alpha,   {\bf \bbeta }) $ from equation (\ref{e45a}).

\noindent
(4c) Estimate the value of the weighting matrix
${\bf \bPhi = Z'\bomega \bomega' Z}$ using the  $\hat{\bomega}$ just
calculated.\footnote{The mistake in equation (13n) above carried over to equation (\ref{e45b}).   I omitted a  minus sign in front of $\alpha$, but I have now included it--- Nov. 18, 2011.}
\begin{equation} \label{e45b}
\begin{array}{lll}
  &   \hat{\bomega}_{jt}   &  {\displaystyle =  \bdelta_{jt} - ({\bf
\hat {-\alpha}
p_{jt} +x_{jt}} \hat{ \bbeta  }
})  \\
\end{array}
\end{equation}



 \noindent
(4d)  Use a search algorithm to find  new values for
$({\bf  \Pi, \Sigma})$. Take the new values and return to step (1). Keep
iterating, searching for parameter estimates that
  minimize  the moment expression (\ref{e45}),
 until the value of the moment expression is close
enough to zero.



Nevo notes that you could then iterate between estimating parameters (
step 4a)  and estimating the weighting matrix (step 4c). Both methods
are consistent, and neither has more attractive theoretical properties,
so it is acceptable to skip over step (4c) after the first iteration.

\bigskip


\noindent
{\bf Conclusion}

     Now that we have gone through the entire procedure, it may be
helpful to list some of the ideas we have used.

\begin{enumerate}

 \item
 {\it  Instrumental variables}. We use instruments to  correct for the
endogeneity of prices, the classic problem in estimating supply and
demand.

\item
 {\it  Product characteristics}. We look at the effect of
characteristics on demand, and then build up to products that have
particular levels of the characteristics.    Going from 50 products to 6
characteristics   drastically  reduces  the number of parameters to be
estimated.

\item
{\it Consumer and product characteristics interact.}  This is what is
going on when consumer marginal utilities are allowed to depend on
consumer characteristics. This makes the pattern of consumer  purchases
substituting from     one product to another more sensible.

\item
  {\it Structural estimation.}  We do not just look at conditional
correlations of relevant variables with a disturbance term tacked on to
account for the imperfect fit of the regression equation. Instead, we
start with a model in which individuals maximize their payoffs by choice
of actions, and the model includes  the  disturbance term which will
later show up in the regression.

\item
   {\it The contraction mapping.} A contraction mapping is used to
estimate the parameters that are averaged  across consumers, an
otherwise difficult optimization problem.

\item
{\it Separating linear and nonlinear estimation problems.} The
estimation is divided into one part that uses a search algorithm to
numerically estimate parameters that enter nonlinearly  and a second
part that uses an analytic formula to estimate the parameters that enter
linearly.

\item
 {\it The generalized method of moments.} The generalized method of
moments is used to estimate the other parameters.

\end{enumerate}

 Not all of these are special to the BLP method. Ideas (1), (2), and (3)
can all be used with least squares (which itself is a simplified version
of (7)). Idea (4) is used in standard logit.  Ideas (5) and (6)   are
special to BLP, but of course BLP is a combination of all seven ideas,
which is why it is so complex  to explain.



    The BLP method has been widely used because   it is general enough
to use for a variety of estimation
problems in industrial organization, not just for simple demand
problems. It is attractive compared to older methods
because it imposes relatively little structure on the theoretical model,
and so
allows many different kinds of  firm and consumer behavior to be tested.
This
flexibility, however, is achieved at the cost of considerable intricacy.
The BLP
method is made up of a modelling part and an estimation part. The
modelling part
is a logit model of  a maximizing consumer's  choice of product
depending on
consumer and product characteristics. This is a structural model, and
really any
structural model of maximizing choice, by consumer, government, or firm,
could
be used in its place. The estimation part estimates the importance of
the
product characteristics, consumer characteristics, and prices using the
generalized method of moments. This is a highly flexible method,
requiring
weaker assumptions than maximium likelihood but like that procedure
requiring a
large number of observations and much computing power. I hope in this
summary I
have made clearer how the economist would go about combining this
modelling and
estimation that forms the BLP model.



\newpage

\noindent
{\bf References}


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\end{document}


