part 24: stated choice [1/117] econometric analysis of panel data william greene department of...

Part 24: Stated Choice [1/117]

Econometric Analysis of Panel Data

William Greene

Department of Economics

Stern School of Business


Econometric Analysis of Panel Data

24. Multinomial Choice and Stated Choice Experiments


A Microeconomics Platform Consumers Maximize Utility (!!!) Fundamental Choice Problem: Maximize U(x1,x2,

…) subject to prices and budget constraints A Crucial Result for the Classical Problem:

Indirect Utility Function: V = V(p,I) Demand System of Continuous Choices

Observed data usually consist of choices, prices, income

The Integrability Problem: Utility is not revealed by demands

j*

j

V( ,I)/ px = -

V( ,I)/ I

p

p


Implications for Discrete Choice Models Theory is silent about discrete choices Translation of utilities to discrete choice requires:

Well defined utility indexes: Completeness of rankings Rationality: Utility maximization Axioms of revealed preferences

Consumers often act to simplify choice situations This allows us to build “models.”

What common elements can be assumed? How can we account for heterogeneity?

However, revealed choices do not reveal utility, only rankings which are scale invariant.


Multinomial Choice Among J Alternatives

• Random Utility Basis Uitj = ij + i’xitj + ijzit + ijt

i = 1,…,N; j = 1,…,J(i,t); t = 1,…,T(i)

N individuals studied, J(i,t) alternatives in the choice set, T(i) [usually 1] choice situations examined.

• Maximum Utility Assumption Individual i will Choose alternative j in choice setting t if and only if

Uitj > Uitk for all k j.

• Underlying assumptions Smoothness of utilities Axioms of utility maximization: Transitive, Complete, Monotonic


Features of Utility Functions The linearity assumption Uitj = ij + ixitj + jzit + ijt

To be relaxed later: Uitj = V(xitj,zit,i) + ijt

The choice set: Individual (i) and situation (t) specific Unordered alternatives j = 1,…,J(i,t)

Deterministic (x,z,j) and random components (ij,i,ijt)

Attributes of choices, xitj and characteristics of the chooser, zit. Alternative specific constants ij may vary by individual

Preference weights, i may vary by individual

Individual components, j typically vary by choice, not by person

Scaling parameters, σij = Var[εijt], subject to much modeling


Unordered Choices of 210 Travelers


Data on Multinomial Discrete Choices


The Multinomial Logit (MNL) Model Independent extreme value (Gumbel):

F(itj) = Exp(-Exp(-itj)) (random part of each utility) Independence across utility functions Identical variances (means absorbed in constants) Same parameters for all individuals (temporary)

Implied probabilities for observed outcomes

],

itj it i,t,j i,t,k

j itj j it

J(i,t)

j itj j itj=1

P[choice = j | , ,i,t] = Prob[U U k = 1,...,J(i,t)

exp(α + + ' ) =

exp(α + ' + ' )

x z

β'x γ z

β x γ z


Multinomial Choice Models

Conditional logit model depends on attrib

Multinomial logit model depends on characteristics

utes

j j itit J(i,t)

j j itj=1

jitj

exp(α + ' ) P[choice = j | ,i, t] =

exp(α + ' )

exp(α + P[choice = j | ,i, t] =

γ zz

γ z

β'xx

THE multinomial logit model accommodates both.

There is no meaningful distinction.

itj

J(i,t)

j itjj=1

j itj j ititj it J(i,t)

j itj j itj=1

)

exp(α + ' )

exp(α + + ' ) P[choice = j | , ,i, t] =

exp(α + ' + ' )

β x

β'x γ zx z

β x γ z


Specifying the Probabilities• Choice specific attributes (X) vary by choices, multiply by generic

coefficients. E.g., TTME=terminal time, GC=generalized cost of travel mode

• Generic characteristics (Income, constants) must be interacted with

choice specific constants.

• Estimation by maximum likelihood; dij = 1 if person i chooses j],

itj it i,t,j i,t,k

j itj j it

J(i,t)

j itj j itj=1

N J(i)

iji=1 j=1

P[choice = j | , ,i, t] = Prob[U U k = 1,...,J(i,t)

exp(α + + ' ) =

exp(α + ' + ' )

logL = d lo

x z

β'x γ z

β x γ z

ijgP


Willingness to PayGenerally a ratio of coefficients

β(Attribute Level) WTP =

β(Income)

Use negative of cost coefficient as a proxu for MU of income

negative β(Attribute Level) WTP =

β(cost)

Measurable using model parameters

Ratios of possibly random parameters can produce wild and

unreasonable values. We will consider a different approach later.


An Estimated MNL Model-----------------------------------------------------------Discrete choice (multinomial logit) modelDependent variable ChoiceLog likelihood function -199.97662Estimation based on N = 210, K = 5Information Criteria: Normalization=1/N Normalized UnnormalizedAIC 1.95216 409.95325Fin.Smpl.AIC 1.95356 410.24736Bayes IC 2.03185 426.68878Hannan Quinn 1.98438 416.71880R2=1-LogL/LogL* Log-L fncn R-sqrd R2AdjConstants only -283.7588 .2953 .2896Chi-squared[ 2] = 167.56429Prob [ chi squared > value ] = .00000Response data are given as ind. choicesNumber of obs.= 210, skipped 0 obs--------+--------------------------------------------------Variable| Coefficient Standard Error b/St.Er. P[|Z|>z]--------+-------------------------------------------------- GC| -.01578*** .00438 -3.601 .0003 TTME| -.09709*** .01044 -9.304 .0000 A_AIR| 5.77636*** .65592 8.807 .0000 A_TRAIN| 3.92300*** .44199 8.876 .0000 A_BUS| 3.21073*** .44965 7.140 .0000--------+--------------------------------------------------


-----------------------------------------------------------Discrete choice (multinomial logit) modelDependent variable ChoiceLog likelihood function -199.97662Estimation based on N = 210, K = 5Information Criteria: Normalization=1/N Normalized UnnormalizedAIC 1.95216 409.95325Fin.Smpl.AIC 1.95356 410.24736Bayes IC 2.03185 426.68878Hannan Quinn 1.98438 416.71880R2=1-LogL/LogL* Log-L fncn R-sqrd R2AdjConstants only -283.7588 .2953 .2896Chi-squared[ 2] = 167.56429Prob [ chi squared > value ] = .00000Response data are given as ind. choicesNumber of obs.= 210, skipped 0 obs--------+--------------------------------------------------Variable| Coefficient Standard Error b/St.Er. P[|Z|>z]--------+-------------------------------------------------- GC| -.01578*** .00438 -3.601 .0003 TTME| -.09709*** .01044 -9.304 .0000 A_AIR| 5.77636*** .65592 8.807 .0000 A_TRAIN| 3.92300*** .44199 8.876 .0000 A_BUS| 3.21073*** .44965 7.140 .0000--------+--------------------------------------------------

Estimated MNL Model


Estimated MNL Model-----------------------------------------------------------Discrete choice (multinomial logit) modelDependent variable ChoiceLog likelihood function -199.97662Estimation based on N = 210, K = 5Information Criteria: Normalization=1/N Normalized UnnormalizedAIC 1.95216 409.95325Fin.Smpl.AIC 1.95356 410.24736Bayes IC 2.03185 426.68878Hannan Quinn 1.98438 416.71880R2=1-LogL/LogL* Log-L fncn R-sqrd R2AdjConstants only -283.7588 .2953 .2896Chi-squared[ 2] = 167.56429Prob [ chi squared > value ] = .00000Response data are given as ind. choicesNumber of obs.= 210, skipped 0 obs--------+--------------------------------------------------Variable| Coefficient Standard Error b/St.Er. P[|Z|>z]--------+-------------------------------------------------- GC| -.01578*** .00438 -3.601 .0003 TTME| -.09709*** .01044 -9.304 .0000 A_AIR| 5.77636*** .65592 8.807 .0000 A_TRAIN| 3.92300*** .44199 8.876 .0000 A_BUS| 3.21073*** .44965 7.140 .0000--------+--------------------------------------------------

2

0

2

0

logPseudo R = 1- .

log

N(J-1) logAdjusted Pseudo R =1- .

N(J-1)-K log

L

L

L

L


jj m k

m,k m,k

Partial effects :

Change in attribute "k" of alternative "m" on the

probability that the individual makes choice "j"

PProb(j) = =P [ (j = m) -P ]β

x x1

m = Car

j = Train

k = Price


jj j k

j,k j,k

jj m k

m,k m,k

Partial effects :

Own effects :

PProb(j) = =P [1-P ]β

x x

Cross effects :

PProb(j) = = -PP β

x x

m = Carj = Train

k = Price

j = Train


j m,kj m k

m,k m,k j

m m,k k

Elasticities for proportional changes :

logP xlogProb(j)= = P [ (j = m) -P ]β

logx logx P

= [ (j = m) -P ] x β

Note the elasticity is the same for all j. This i

1

1

s a

consequence of the IIA assumption in the model

specification made at the outset.


Own effect

Cross effects

+---------------------------------------------------+| Elasticity averaged over observations.|| Attribute is INVT in choice AIR || Mean St.Dev || * Choice=AIR -.2055 .0666 || Choice=TRAIN .0903 .0681 || Choice=BUS .0903 .0681 || Choice=CAR .0903 .0681 |+---------------------------------------------------+| Attribute is INVT in choice TRAIN || Choice=AIR .3568 .1231 || * Choice=TRAIN -.9892 .5217 || Choice=BUS .3568 .1231 || Choice=CAR .3568 .1231 |+---------------------------------------------------+| Attribute is INVT in choice BUS || Choice=AIR .1889 .0743 || Choice=TRAIN .1889 .0743 || * Choice=BUS -1.2040 .4803 || Choice=CAR .1889 .0743 |+---------------------------------------------------+| Attribute is INVT in choice CAR || Choice=AIR .3174 .1195 || Choice=TRAIN .3174 .1195 || Choice=BUS .3174 .1195 || * Choice=CAR -.9510 .5504 |+---------------------------------------------------+| Effects on probabilities of all choices in model: || * = Direct Elasticity effect of the attribute. |+---------------------------------------------------+

Note the effect of IIA on the cross effects.

Elasticities are computed for each observation; the mean and standard deviation are then computed across the sample observations.


A Multinomial Logit Common Effects Model

How to handle unobserved effects in other nonlinear models? Single index models such as probit, Poisson,

tobit, etc. that are functions of an xit'β can be modified to be functions of xit'β + ci.

Other models – not at all obvious. Rarely found in the literature.

Dealing with fixed and random effects? Dynamics makes things much worse.


A Multinomial Logit Model

i,t i,t i,t

The multinomial logit model for unordered choices

U (j) ( j) ( j), j = 1,...,J (choice set)

t = 1,...,T (choice situations)

i =

x

i,t

i,t i,t i,t i,t i,t i,t

1,...,N (individuals)

( j) ~ I.I.D. Type 1 extreme value.

j * = j = index of choice such that U ( j *) U (k) for j * k.

How to modify the model to include common (random or

fixed) effects?


A Heterogeneous Multinomial Logit Model

i,t i,t i,t i

i,t i,t i,t i

i,t i,t i,t i

The multinomial logit model for unordered choices

U (1) (1) (1) u (1),

U (2) (2) (2) u (2)

...

U (J) (J) (J) u (J)

t = 1,...,T (choice situations)

i = 1,...,N (individua

x

x

x

i,t i,t i,t i,t i,t i,t

i,t

i i i i

ls)

j * = j = index of choice such that U ( j *) U (k) for j * k

( j) ~ I.I.D. Type 1 extreme value, j=1,...,J.

[u (1),u (2),...,u (J)] = J common individual effects.

[A dynamic version

u

of this model in Gong, et al., "Mobility in the

Urban Labor Market" IZA Working Paper 213, Bonn, 2000]


Common Effects Multinomial Logit

T i,t i,t i i,ti i Jt 1

j 1 i,t i

Fixed Effects is complicated. Needs N sets of J

dummy variable coefficients (that sum to zero across choices).

Random Effects:

exp[ ( j *) u ( j *)]L |

exp[ ( j) u ( j)]

xu

x

i

T

t 1

T i,t i,t i i,ti i iJt 1

j 1 i,t i

Prob[choice made | (i)]

Unconditional contribution to the log likelihood for person i is

exp[ ( j *) u ( j *)]logL log f( )d

exp[ ( j) u ( j)]u

u

xu u

x


Simulation Based Estimation

i

i

T i,t i,t i i,ti i iJt 1

j 1 i,t i

TN i,t i,t i i,ti iJi 1 t 1

j 1 i,t i

N

i 1


exp[ ( j) u ( j)]


exp[ ( j) u ( j)]

1SimulatedLogL log

R

u

u

xu u

x

xu u

x

TR i,t i,t j* i,r i,t

Jr 1 t 1j 1 i,t j i,r

i,r

1 J

exp[ ( j *) v ( j *)]

exp[ ( j) v ( j)]

where v ( j) are random draws from the assumed population.

This function is maximized over and ,...,

x

x


Application Shoe Brand Choice

Simulated Data: Stated Choice, N=400 respondents, T=8 choice situations, 3,200 observations

3 choice/attributes + NONE J=4 Fashion = High / Low Quality = High / Low Price = 25/50/75,100 coded 1,2,3,4; and Price2

Heterogeneity: Sex, Age (<25, 25-39, 40+)

Underlying data generated by a 3 class latent class process (100, 200, 100 in classes)

Thanks to www.statisticalinnovations.com (Latent Gold)


Application

2i,t 1 i,t 2 i,t 3 i,t 4 i,t

S,1 i Y,1 i O,1 i i,t i

2i,t 1 i,t 2 i,t 3 i,t 4 i,t

S,2 i Y,2 i O,2 i i,t i

i

U (1) F Q P P

Sex Young Older (1) u (1)

U (2) F Q P P


U

2,t 1 i,t 2 i,t 3 i,t 4 i,t

S,3 i Y,3 i O,3 i i,t i

i,t i,t

(3) F Q P P


U (none) (none)


No Common Effects

+---------------------------------------------+| Start values obtained using MNL model || Log likelihood function -4119.500 |+---------------------------------------------++--------+--------------+----------------+--------+--------+|Variable| Coefficient | Standard Error |b/St.Er.|P[|Z|>z]|+--------+--------------+----------------+--------+--------+ FASH | 1.45964424 .07748860 18.837 .0000 QUAL | 1.10637961 .07153725 15.466 .0000 PRICE | 2.31763951 3.98732636 .581 .5611 PRICESQ | -55.5527148 13.8684229 -4.006 .0001 ASC4 | .64637513 .24440240 2.645 .0082 B1_MAL1 | -.16751621 .10552035 -1.588 .1124 B1_YNG1 | -.58118337 .11969068 -4.856 .0000 B1_OLD1 | -.02600079 .14091863 -.185 .8536 B2_MAL2 | -.05966758 .10055110 -.593 .5529 B2_YNG2 | -.14991404 .11180414 -1.341 .1800 B2_OLD2 | -.15128297 .14133889 -1.070 .2845 B3_MAL3 | -.12076085 .09301010 -1.298 .1942 B3_YNG3 | -.12265952 .10419547 -1.177 .2391 B3_OLD3 | -.04753400 .12950649 -.367 .7136


Revealed and Stated Preference Data

Pure RP Data Market (ex-post, e.g., supermarket scanner data) Individual observations

Pure SP Data Contingent valuation (?) Validity

Combined (Enriched) RP/SP Mixed data Expanded choice sets


Revealed Preference Data

Advantage: Actual observations on actual behavior

Disadvantage: Limited range of choice sets and attributes – does not allow analysis of switching behavior.


Stated Preference Data

Pure hypothetical – does the subject take it seriously?

No necessary anchor to real market situations

Vast heterogeneity across individuals


Pooling RP and SP Data Sets - 1 Enrich the attribute set by replicating choices E.g.:

RP: Bus,Car,Train (actual) SP: Bus(1),Car(1),Train(1)

Bus(2),Car(2),Train(2),… How to combine?


Each person makes four choices from a choice set that includes either 2 or 4 alternatives.

The first choice is the RP between two of the 4 RP alternatives

The second-fourth are the SP among four of the 6 SP alternatives.

There are 10 alternatives in total.

A Stated Choice Experiment with Variable Choice Sets


Enriched Data Set – Vehicle Choice

Choosing between Conventional, Electric and LPG/CNG Vehicles in Single-Vehicle Households

David A. Hensher William H. Greene Institute of Transport Studies Department of Economics School of Business Stern School of Business The University of Sydney New York University NSW 2006 Australia New York USA

September 2000


Fuel Types Study

Conventional, Electric, Alternative 1,400 Sydney Households Automobile choice survey RP + 3 SP fuel classes Nested logit – 2 level approach – to handle

the scaling issue


Attribute Space: Conventional


Attribute Space: Electric


Attribute Space: Alternative


Mixed Logit Approaches Pivot SP choices around an RP outcome. Scaling is handled directly in the model Continuity across choice situations is handled by

random elements of the choice structure that are constant through time Preference weights – coefficients Scaling parameters

Variances of random parameters Overall scaling of utility functions


Application

Survey sample of 2,688 trips, 2 or 4 choices per situationSample consists of 672 individualsChoice based sample

Revealed/Stated choice experiment: Revealed: Drive,ShortRail,Bus,Train Hypothetical: Drive,ShortRail,Bus,Train,LightRail,ExpressBus

Attributes: Cost –Fuel or fare Transit time Parking cost Access and Egress time


Nested Logit Approach

Car Train Bus SPCar SPTrain SPBus

RP

Mode

Use a two level nested model, and constrain three SP IV parameters to be equal.


Each person makes four choices from a choice set that includes either 2 or 4 alternatives.

The first choice is the RP between two of the 4 RP alternatives

The second-fourth are the SP among four of the 6 SP alternatives.

There are 10 alternatives in total.

A Stated Choice Experiment with Variable Choice Sets


Customers’ Choice of Energy Supplier California, Stated Preference Survey 361 customers presented with 8-12 choice situations

each Supplier attributes:

Fixed price: cents per kWh Length of contract Local utility Well-known company Time-of-day rates (11¢ in day, 5¢ at night) Seasonal rates (10¢ in summer, 8¢ in winter, 6¢ in

spring/fall)

(TrainCalUtilitySurvey.lpj)


Population Distributions

Normal for: Contract length Local utility Well-known company

Log-normal for: Time-of-day rates Seasonal rates

Price coefficient held fixed


Estimated Model Estimate Std errorPrice -.883 0.050Contract mean -.213 0.026 std dev .386 0.028Local mean 2.23 0.127 std dev 1.75 0.137Known mean 1.59 0.100 std dev .962 0.098TOD mean* 2.13 0.054 std dev* .411 0.040Seasonal mean* 2.16 0.051 std dev* .281 0.022*Parameters of underlying normal.


Distribution of Brand Value

Brand value of local utility

Standard deviation10% dislike local utility

00 2.5¢2.5¢

=2.0¢


Random Parameter Distributions


Time of Day Rates (Customers do not like – lognormal coefficient. Multiply variable by -1.)


Estimating Individual Parameters Model estimates = structural parameters, α, β, ρ, Δ, Σ, Γ Objective, a model of individual specific parameters, βi

Can individual specific parameters be estimated? Not quite – βi is a single realization of a random

process; one random draw. We estimate E[βi | all information about i] (This is also true of Bayesian treatments, despite

claims to the contrary.)


Expected Preferences of Each Customer

Customer likes long-term contract, local utility, and non-fixed rates.

Local utility can retain and make profit from this customer by offering a long-term contract with time-of-day or seasonal rates.


Posterior Estimation of i

ˆ

i

i

i i i i i

T

i i i i i i t=1

T

i i i i i t=1

=E | , , , , ,z

P(choice j | X , )g( | , , ,etc., ) d

=

P(choice j | X , )g(β | , , ,etc., ) d

β

β

β β β Δ Γ y X

β β β β Δ Γ z β

β β Δ Γ z β

ˆ ˆ ˆ ˆ ˆ ˆ

ˆ

ˆ ˆ ˆ ˆ ˆ

ˆ ˆ ˆ ˆ

TR

ir i i i ir=1 t=1

i TR

i i i ir=1 t=1

ir i ir

1β P(choice j | X , )g( | , , ,etc.,z )

R= ,

1 P(choice j | X , )g( | , , ,etc.,z )R

= + z + w

β β β Δ Γ

β

β β β Δ Γ

β β Δ Γ

Estimate by simulation


Application: Shoe Brand Choice

Simulated Data: Stated Choice, 400 respondents, 8 choice situations, 3,200 observations

3 choice/attributes + NONE Fashion = High / Low Quality = High / Low Price = 25/50/75,100 coded 1,2,3,4

Heterogeneity: Sex (Male=1), Age (<25, 25-39, 40+)



Stated Choice Experiment: Unlabeled Alternatives, One Observation

t=1

t=2

t=3

t=4

t=5

t=6

t=7

t=8


Random Parameters Logit Model

,

,

,

n s

n s

n s

1,n 1,n,s 2 1,n,s 3 1,n,s Brand1,n,s

1,n 2,n,s 2 2,n,s 3 2,n,s Brand2,n,s

1,n 3,n,s 2 3,n,s 3 3,n,s Bra

U(brand1) = β Fashion +β Quality +β Price +ε



,n s

nd3,n,s

4 No Brand,n,s

1,n 1 11 12 13 1 n1

U(None) = β + ε

β β +δ Sex +δ Age2539+δ Age40+η z

Individual parameters


Panel Data

Repeated Choice Situations Typically RP/SP constructions (experimental) Accommodating “panel data”

Multinomial Probit [marginal, impractical] Latent Class Mixed Logit


Customers’ Choice of Energy Supplier California, Stated Preference Survey 361 customers presented with 8-12 choice

situations each Supplier attributes:

Fixed price: cents per kWh Length of contract Local utility Well-known company Time-of-day rates (11¢ in day, 5¢ at night) Seasonal rates (10¢ in summer, 8¢ in winter, 6¢ in

spring/fall)


Application: Shoe Brand Choice

Simulated Data: Stated Choice, 400 respondents, 8 choice situations, 3,200 observations

3 choice/attributes + NONE Fashion = High / Low Quality = High / Low Price = 25/50/75,100 coded 1,2,3,4

Heterogeneity: Sex (Male=1), Age (<25, 25-39, 40+)



Stated Choice Experiment: Unlabeled Alternatives, One Observation

t=1

t=2

t=3

t=4

t=5

t=6

t=7

t=8


This an unlabelled choice experiment: Compare Choice = (Air, Train, Bus, Car)To Choice = (Brand 1, Brand 2, Brand 3, None) Brand 1 is only Brand 1 because it is first in the list.

What does it mean to substitute Brand 1 for Brand 2?

What does the own elasticity for Brand 1 mean?

Unlabeled Choice Experiments


Aggregate Data and Multinomial Choice:

The Model of Berry, Levinsohn and Pakes


ResourcesAutomobile 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/Nevo-BLP.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/Lee-BLP.pdf

Elasticities of Market Shares and Social Health Insurance Choice in Germany: A Dynamic Panel Data Approach, M. Tamm et al., Health Economics, 16, 2007, 243-256.

http://people.stern.nyu.edu/wgreene/Econometrics/Tamm.pdf


Aggregate Data and Multinomial Choice:

The Model of Berry, Levinsohn and Pakes


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.


BLP Automobile Market

t

Jt


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 w i 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


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( )


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.


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.


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.


Instrumental Variables(ξ and ω are our h and f.)


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


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


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


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


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


ABLP Iteration

ξt is our ft. is our(β,)

No superscript is our (M); superscript 0 is our (M-1).


Side Results


ABLP Iterative Estimator


BLP Design Data


Exogenous price and nonrandom parameters


IV Estimation


Full Model


Some Elasticities


Fixed Effects Multinomial Logit:

Application of Minimum Distance Estimation


Binary Logit Conditional Probabiities

i

i

1 1 2 2

1 1

1

T

S1 1

All

Prob( 1| ) .1

Prob , , ,

exp exp

exp exp

i it

i it

i i

i i

i i

t i

i

t i

it it

i i i i iT iT

T T

it it it itt t

T T

it it it i

T

itt

td St t

ey

e

Y y Y y Y y

y

d d

y

y

x

xx

x x

x x

β

β

i

t i

different

iT

it t=1 i

ways that

can equal S

.

Denominator is summed over all the different combinations of T valuesof y that sum to the same sum as the observed . If S is this sum,

there are

it

it

d

y

i

terms. May be a huge number. An algorithm by KrailoS

and Pike makes it simple.

T


1 7

1 2 7

*| 1

Prob[ = (1,0,0,0,1,1,1)| ]=

exp( ) exp( )1...

1 exp( ) 1 exp( ) 1 exp( )

There are 35 different sequences of y (permutations) that sum to 4.

For example, y might b

i

i i

i i i

it

it p

y X

x x

x x x

717

1 35 7 *1 |1

e (1,1,1,1,0,0,0). Etc.

exp yProb[y=(1,0,0,0,1,1,1)| , y =7] =

exp y

t it it

i t it

t it p itp

xX

x

Example: SevenPeriod Binary Logit


12 14

With T = 50, the number of permutations of sequences of

y ranging from sum = 0 to sum = 50 ranges from 1 for 0 and 50,

to 2.3 x 10 for 15 or 35 up to a maximum of 1.3 x 10 for sum =25.

These are the numbers of terms that must be summed for a model

with T = 50. In the application below, the sum ranges from 15 to 35.


The sample is 200 individuals each observed 50 times.


The data are generated from a probit process with b1 = b2 = .5. But, it is fit as a logit model. The coefficients obey the familiar relationship, 1.6*probit.


1

0 01

= 1 if individual i makes choice j in period t

Prob( 1| ) , 1,..., .1

1Prob( 1| ) .

1

The probability attached to the sequence of choices i

ij itj

im itm

im itm

itj

itj itj Jm

it it Jm

y

ey j J

e

ye

x

x

x

x

x

i

ij

t

1 11 1

T different

1 1 S1 1

All ways that

s remarkably complicated.

exp exp

exp exp

i i

i i

t itj ij

it

T TJ J

itj itj itj itjj jt t

T TJ J

it it itj itjj d S jt td

y y

d d

x x

x x

β

β

i

ij

iT

itj t=1 ij

ij

can equal S

.

Denominator is summed over all the different combinations of T valuesof y that sum to the same sum as the observed . If S is this sum,

there are terms. May bS

j

ityT

e a huge number. Larger yet by summing over choices.

Multinomial Logit Model: J+1 choices including a base choice.


Estimation Strategy

Conditional ML of the full MNL model. Impressively complicated.

A Minimum Distance (MDE) Strategy Each alternative treated as a binary choice vs.

the base provides an estimator of Select subsample that chose either option j or the

base Estimate using this binary choice setting This provides J different estimators of the same

Optimally combine the different estimators of


Minimum Distance Estimationˆ ˆThere are J estimators of the same parameter vector, .

Each estimator is consistent and asymptotically normal.

ˆEstimated covariance matrices . How to combine the estimators?

ˆMDE: Minimize wrt

j

jV

β β

1 * 1 *

2 * 2 **

* *

ˆ ˆ ˆ ˆ

ˆ ˆ ˆ ˆ q =

ˆ ˆ ˆ ˆ

What to use for the weighting matrix ? Any positive definite matrix will do.J J

W

W

β β β β

β β β β

β β β β


MDE Estimation

1 * 1 *

2 * 2 **

* *

ˆEstimated covariance matrices . How to combine the estimators?

ˆ ˆ ˆ ˆ

ˆ ˆ ˆ ˆˆMDE: Minimize wrt q = . Propose a GLS approach

ˆ ˆ ˆ ˆ

j

J J

V

W

β β β β

β β β ββ

β β β β1

1

1 2

ˆ 0 0

ˆ0 0

ˆ0 0 J

V

V W = A

V


MDE Estimation

1

1 * 1 *1

22 * 2 **

* *

11 1 1 1 1

* 1 2 1 1 2

ˆ ˆ ˆ ˆˆ 0 0

ˆ ˆ ˆ ˆˆ0 0ˆMDE: Minimize wrt q = .

ˆˆ ˆ ˆ ˆ0 0

ˆ ˆ ˆˆ ˆ ˆ ˆ ˆThe solution is ...

JJ J

J

V

V

V

V V V V V

β β β β

β β β ββ

β β β β

β β 12

11 1

1 1

1 1

ˆˆ...

ˆˆ ˆ =

ˆ = where

J J

J J

j j jj j

J J

j j jj j

V

V V

H H I

β β

β

β


Why a 500 fold increase in speed?

MDE is much faster Not using Krailo and Pike, or not using

efficiently Numerical derivatives for an extremely

messy function (increase the number of function evaluations by at least 5 times)

part 24: stated choice [1/117] econometric analysis of panel data william greene department of...

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