Discrete Choice Modeling
Aggregate Share Data - BLP
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Discrete Choice Modeling
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Introduction
Summary
Binary Choice
Panel Data
Bivariate Probit
Ordered Choice
Count Data
Multinomial Choice
Nested Logit
Heterogeneity
Latent Class
Mixed Logit
Stated Preference
Hybrid Choice
Spatial Data
Aggregate Market Data
William Greene
Stern School of Business
New York University
Discrete Choice Modeling
Aggregate Share Data - BLP
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Aggregate Data and Multinomial Choice:
The Model of Berry, Levinsohn and Pakes
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Aggregate Share Data - BLP
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Resources

Automobile Prices in Market Equilibrium, S. Berry, J. Levinsohn,
A. Pakes, Econometrica, 63, 4, 1995, 841-890. (BLP)
http://people.stern.nyu.edu/wgreene/Econometrics/BLP.pdf

A Practitioner’s Guide to Estimation of Random-Coefficients Logit
Models of Demand, A. Nevo, Journal of Economics and
Management Strategy, 9, 4, 2000, 513-548
http://people.stern.nyu.edu/wgreene/Econometrics/NevoBLP.pdf

A New Computational Algorithm for Random Coefficients Model
with Aggregate-level Data, Jinyoung Lee, UCLA Economics,
Dissertation, 2011
http://people.stern.nyu.edu/wgreene/Econometrics/LeeBLP.pdf
Discrete Choice Modeling
Aggregate Share Data - BLP
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Theoretical Foundation



Consumer market for J differentiated brands of a good
 j =1,…, Jt brands or types
 i = 1,…, N consumers
 t = i,…,T “markets” (like panel data)
Consumer i’s utility for brand j (in market t) depends on
 p = price
 x = observable attributes
 f = unobserved attributes
 w = unobserved heterogeneity across consumers
 ε = idiosyncratic aspects of consumer preferences
Observed data consist of aggregate choices, prices and
features of the brands.
Discrete Choice Modeling
Aggregate Share Data - BLP
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BLP Automobile Market
Jt
t
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Random Utility Model

Utility: Uijt=U(wi,pjt,xjt,fjt|), i = 1,…,(large)N,
j=1,…,J




wi = individual heterogeneity; time (market) invariant. w has a
continuous distribution across the population.
pjt, xjt, fjt, = price, observed attributes, unobserved features of
brand j; all may vary through time (across markets)
Revealed Preference: Choice j provides maximum
utility
Across the population, given market t, set of prices pt
and features (Xt,ft), there is a set of values of wi that
induces choice j, for each j=1,…,Jt; then, sj(pt,Xt,ft|)
J
is the market share of brand
X t ,market
ft | θ) < 1t.
 j=1 s j (pjt ,in
There is an outside good that attracts a nonnegligible
market share, j=0. Therefore,
t

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Functional Form
(Assume one market for now so drop “’t.”)
Uij=U(wi,pj,xj,fj|)= xj'β – αpj + fj + εij
= δj + εij
 Econsumers i[εij] = 0, δj is E[Utility].

Market Share j  E q j Prob( j  q  )



Will assume logit form to make integration
unnecessary. The expectation has a
closed form.
Discrete Choice Modeling
Aggregate Share Data - BLP
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Heterogeneity
Assumptions so far imply IIA. Cross price
elasticities depend only on market shares.
 Individual heterogeneity: Random
parameters
 Uij=U(wi,pj,xj,fj|i)= xj'βi – αpj + fj + εij
βik = βk + σkvik.
 The mixed model only imposes IIA for a
particular consumer, but not for the
market as a whole.

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Endogenous Prices: Demand side
Uij=U(wi,pj,xj,fj|)= xj'βi – αpj + fj + εij
 fj is unobserved
 Utility responds to the unobserved fj
 Price pj is partly determined by features fj.
 In a choice model based on observables,
price is correlated with the unobservables
that determine the observed choices.

Discrete Choice Modeling
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Endogenous Price: Supply Side





There are a small number of competitors in this market
Price is determined by firms that maximize profits given the
features of its products and its competitors.
mcj = g(observed cost characteristics c,
unobserved cost characteristics h)
At equilibrium, for a profit maximizing firm that produces
one product,
sj + (pj-mcj)sj/pj = 0
Market share depends on unobserved cost characteristics as
well as unobserved demand characteristics, and price is
correlated with both.
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Instrumental Variables
(ξ and ω are our h and f.)
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Econometrics: Essential Components
Uijt  x jti  fjt  ijt
Ui0t  i0t (Outside good)
i    v i ,   diagonal(1 ,...)
ijt ~ Type I extreme value, IID across all choices
Market shares: s j ( X t , ft : i ) 
exp( x jti  fjt )
1   m1 exp( x mti  fmt )
J
, j  1,..., Jt
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Econometrics
Market Shares: s j ( X t , ft : i ) 
exp( x jti  fjt )
1   m1 exp( x mti  fmt )
Expected Share: E[s j ( X t , ft :  )] 
J

i
, j  1,..., Jt
exp( x jti  fjt )
1   m1 exp( x mti  fmt )
J
Expected Shares are estimated using simulation:
exp[x jt   v ir )  fjt ]
1 R
ŝ j ( X t , ft :  )   r 1
J
R
1   m1 exp[x mt   v ir )  fmt ]
dF(i )
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GMM Estimation Strategy - 1
exp[x jt   v ir )  fjt ]
1 R
ŝ jt ( X t , ft :  )   r 1
J
R
1   m1 exp[x mt   v ir )  fmt ]
We have instruments z jt such that
E[fjt ( )z jt ]  0
fjt is obtained from an inverse mapping by equating the
fitted market shares, ˆ
s t , to the observed market shares, S t .
ˆ
s t ( X t , ft :  )  S t so ˆft  ˆ
s t 1 ( X t , S t :  ).
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GMM Estimation Strategy - 2
We have instruments z jt such that
E[fjt ( )z jt ]  0
ˆ
s t ( X t , ft :  )  S t so ˆft  ˆ
s t 1 ( X t , S t :  ).
1
ˆ
Define gt =
Jt

Jt
j1
ˆf z
jt jt
ˆ ( )  g
ˆ Wg
ˆ
GMM Criterion would be Q
t
t
t
where W = the weighting matrix for mi nimum distance estimation.
For the entire sample, the GMM estimator is built on
ˆ = 1 T 1
g
T t 1 Jt

Jt
j 1
ˆf z and Q( )=g
ˆWg
ˆ
jt jt
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BLP Iteration
Begin with starting values for ft  ˆft(0) and starting values for
structural parameters  and .
1)
ˆ(M1) , 
ˆ (M1) ).
Compute predicted shares ˆ
s (M
( X t , ˆft(M1) : 
t
INNER (Contraction Mapping) Find a fixed point for
1)
ˆf (M)  ˆf (M1)  log(S )  log[ˆ
ˆ(M1) , 
ˆ(M 1) , 
ˆ (M1) )]  ˆft(M) (ˆft(M1) , 
ˆ(M 1) )
s (M
( X t , ˆft(M1) : 
t
t
t
t
ˆ(M) , 
ˆ (M) .
OUTER (GMM Step) With ˆft(M) in hand, use GMM to (re)estimate 
Return to INNER step or exit if ˆft(M) - ˆft(M1) is sufficiently small.
GMM step is straightforward - concave function (quadratic form) of a
concave function (logit probability).
Solving the INNER step is time consuming and very complicated.
Recent research has produced several alternative algorithms.
Overall complication: The estimates ˆft(M) can diverge.
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ABLP Iteration
ξt is
our ft.
 is our
(β,)
No superscript is
our (M);
superscript 0 is
our (M-1).
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Side Results
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ABLP Iterative Estimator
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BLP Design Data
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Exogenous price and nonrandom parameters
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IV Estimation
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Full Model
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Some Elasticities