[Part 15] 1/24 Discrete Choice Modeling Aggregate Share Data - BLP Discrete Choice Modeling William Greene Stern School of Business New York University.

[Part 15] 1/24

Discrete Choice Modeling

Aggregate Share Data - BLP


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

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Aggregate Data and Multinomial Choice:

The Model of Berry, Levinsohn and Pakes

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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, 513-548http://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

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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.

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BLP Automobile Market

t

Jt

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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|) 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

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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].

Will assume logit form to make integration unnecessary. The expectation has a closed form.

j j qq j

Market Share E Prob( )

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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.

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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

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

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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

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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

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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

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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

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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

[Part 15] 1/24 Discrete Choice Modeling Aggregate Share Data - BLP Discrete Choice Modeling William Greene Stern School of Business New York University.

Documents

choice j

x j p j f j ij

good j

j t brands

x j i p j f j ij ik

market share of brand

s j p t

aggregate market data