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[Part 15] 1/24
Discrete Choice Modeling
Aggregate Share Data - BLP
Discrete Choice Modeling
William Greene
Stern School of Business
New York University
0 Introduction1 Summary2 Binary Choice3 Panel Data4 Bivariate Probit5 Ordered Choice6 Count Data7 Multinomial Choice8 Nested Logit9 Heterogeneity10 Latent Class11 Mixed Logit12 Stated Preference13 Hybrid Choice14. Spatial Data15. Aggregate Market Data
[Part 15] 2/24
Discrete Choice Modeling
Aggregate Share Data - BLP
Aggregate Data and Multinomial Choice:
The Model of Berry, Levinsohn and Pakes
[Part 15] 3/24
Discrete Choice Modeling
Aggregate Share Data - BLP
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, 513-548http://people.stern.nyu.edu/wgreene/Econometrics/Nevo-
BLP.pdf
A New Computational Algorithm for Random Coefficients Model with Aggregate-level Data, Jinyoung Lee, UCLA Economics, Dissertation, 2011http://people.stern.nyu.edu/wgreene/Econometrics/Lee-BLP.pdf
[Part 15] 4/24
Discrete Choice Modeling
Aggregate Share Data - BLP
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.
[Part 15] 5/24
Discrete Choice Modeling
Aggregate Share Data - BLP
BLP Automobile Market
t
Jt
[Part 15] 6/24
Discrete Choice Modeling
Aggregate Share Data - BLP
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|) is the market share of brand j in market t.
There is an outside good that attracts a nonnegligible market share, j=0. Therefore,
< j t t t tJ
j=1s ( , , | ) 1p X f θ
[Part 15] 7/24
Discrete Choice Modeling
Aggregate Share Data - BLP
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].
Will assume logit form to make integration unnecessary. The expectation has a closed form.
j j qq j
Market Share E Prob( )
[Part 15] 8/24
Discrete Choice Modeling
Aggregate Share Data - BLP
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.
[Part 15] 9/24
Discrete Choice Modeling
Aggregate Share Data - BLP
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.
[Part 15] 10/24
Discrete Choice Modeling
Aggregate Share Data - BLP
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.
[Part 15] 11/24
Discrete Choice Modeling
Aggregate Share Data - BLP
Instrumental Variables(ξ and ω are our h and f.)
[Part 15] 12/24
Discrete Choice Modeling
Aggregate Share Data - BLP
Econometrics: Essential Components
ijt jt i jt ijt
i0t i0t
i i 1
ijt
jt i jtj t t i J
mt i mtm 1
U f
U (Outside good)
v , diagonal( ,...)
~ Type I extreme value, IID across all choices
exp( f )Market shares: s ( , : ) , j 1,...,
1 exp( f )
x
xX f
xtJ
[Part 15] 13/24
Discrete Choice Modeling
Aggregate Share Data - BLP
Econometrics
i
jt i jtj t t i tJ
mt i mtm 1
jt i jtj t t iJ
mt i mtm 1
exp( f )Market Shares: s ( , : ) , j 1,..., J
1 exp( f )
exp( f )Expected Share: E[s ( , : )] dF( )
1 exp( f )
Expected Shares are estimated using simulati
xX f
x
xX f
x
R jt ir jtj t t Jr 1
mt ir mtm 1
on:
exp[ v ) f ]1s ( , : )
R 1 exp[ v ) f ]
xX f
x
[Part 15] 14/24
Discrete Choice Modeling
Aggregate Share Data - BLP
GMM Estimation Strategy - 1
R jt ir jtjt t t Jr 1
mt ir mtm 1
jt
jt jt
jt
exp[ v ) f ]1s ( , : )
R 1 exp[ v ) f ]
We have instruments such that
E[f ( ) ] 0
f is obtained from an inverse mapping by equating the
ˆfitted market shares,
xX f
x
z
z
s
t
1t t t t t
, to the observed market shares, .
ˆˆ ˆ( , : ) so ( , : ).t
S
s X f S f s X S
t
t t
[Part 15] 15/24
Discrete Choice Modeling
Aggregate Share Data - BLP
GMM Estimation Strategy - 2
t
jt
jt jt
1t t t t t
J
t jt jtj 1t
t t t
We have instruments such that
E[f ( ) ] 0
ˆˆ ˆ( , : ) so ( , : ).
1 ˆˆDefine = fJ
ˆ ˆ ˆGMM Criterion would be Q ( )
where = the weighting matrix for mi
t
z
z
s X f S f s X S
g z
g Wg
W
t t
tT J
jt jtt 1 j 1t
nimum distance estimation.
For the entire sample, the GMM estimator is built on
1 1 ˆˆ ˆ ˆ = f and Q( )=T J
g z gWg
[Part 15] 16/24
Discrete Choice Modeling
Aggregate Share Data - BLP
BLP Iteration
(0)t t
(M 1) (M 1) (M 1) (M 1)t t t
ˆBegin with starting values for and starting values for
structural parameters and .
ˆ ˆ ˆˆCompute predicted shares ( , : , ).
Find a fixedINNER (Contraction Mapping)
ff
s X f
(M) (M 1) (M 1) (M 1) (M 1) (M 1) (M) (M 1) (M 1) (M 1)t t t t t t t t
(M) (M) (M)t
point for
ˆ ˆ ˆ ˆ ˆˆ ˆˆ ˆˆlog( ) log[ ( , : , )] ( , , )
ˆ ˆ ˆ With in hand, use GMM to (re)estimate , .
Return to
OUTER (GMM Step)
ff S s
IN
X ff
f
NER
f
(M) (M 1)t t
ˆ ˆstep or exit if - is sufficiently small.
step is straightforward - concave function (quadratic form) of a
concave function (logit probability).
Solving the step is time consuming INNER
f
GMM
f
(M)t
and very complicated.
Recent research has produced several alternative algorithms.
ˆOverall complication: The estimates can diverge.f
[Part 15] 17/24
Discrete Choice Modeling
Aggregate Share Data - BLP
ABLP Iteration
ξt is our ft. is our(β,)
No superscript is our (M); superscript 0 is our (M-1).
[Part 15] 18/24
Discrete Choice Modeling
Aggregate Share Data - BLP
Side Results
[Part 15] 19/24
Discrete Choice Modeling
Aggregate Share Data - BLP
ABLP Iterative Estimator
[Part 15] 20/24
Discrete Choice Modeling
Aggregate Share Data - BLP
BLP Design Data
[Part 15] 21/24
Discrete Choice Modeling
Aggregate Share Data - BLP
Exogenous price and nonrandom parameters
[Part 15] 22/24
Discrete Choice Modeling
Aggregate Share Data - BLP
IV Estimation
[Part 15] 23/24
Discrete Choice Modeling
Aggregate Share Data - BLP
Full Model
[Part 15] 24/24
Discrete Choice Modeling
Aggregate Share Data - BLP
Some Elasticities