1/110: Topic 4.3 – Mixed Models and Random Parameters Microeconometric Modeling William Greene Stern School of Business New York University New York NY

1/110: Topic 4.3 – Mixed Models and Random Parameters

Microeconometric Modeling

William GreeneStern School of BusinessNew York UniversityNew York NY USA

4.3 Mixed Models and Random Parameters


Concepts

• Random Effects• Simulation• Random Parameters• Maximum Simulated Likelihood• Cholesky Decomposition• Heterogeneity• Hierarchical Model• Conditional Means• Population Distribution• Nested Logit• Willingness to Pay (WTP)• Random Parameters and WTP• WTP Space• Endogeneity• Market Share Data

Models

• Random Parameters• RP Logit• Error Components Logit• Generalized Mixed Logit• Berry-Levinsohn-Pakes Model• Hybrid Choice• MIMIC Model


A Recast Random Effects Model

i

1 2 ,1

+ + , ~ [0, ]

T = observations on individual i

For each period, 1[ 0] (given u )

Joint probability for T observations is

Prob( , ,... | ) ( )

Write

x

x

i

it i it i u

i i

i

it it

i

T

i i i it i itt

it u u N

u

y U

y y u F

U

y

u u

1 , u1

1

= , ~ [0,1],

log | ,... log ( ( ) )

It is not possible to maximize log | ,... because of

the unobserved random effects embedded in .

xi

i i i i i

N T

N it i iti i t

N

i

u v v N v

L v v F y v

L v v


A Computable Log Likelihood

1 1

u

[log | ] log ( , )

Maximize this function with respect to , , . ( )

How to compute the in

The unobserved heterogeneity is averaged out of

tegral?

(

l |

1

og

i

i

TN

it i it i ii t

i u i

E L F y f d

v

L

v

v

v x

) Analytically? No, no formula exists.

(2) Approximately, using Gauss-Hermite quadrature

(3) Approximately using Monte Carlo simulation


Simulation

iT

it i i i

2i

N

ii 1

N

ii 1

N

i 1

i

i

tt 1

i

logL log d

= log d

This ]

The expected value of the function

F(y , )

g(

of can be ap

equals

proxim

-1

at

log

ed

by

exp2

dr

a

)

g(

win

2

g

E[

ra

)

R

x

iTN R

S it u ir iti 1 r 1 t

i

ir u ir

1

r

1logL log F(y

ndom draws v from the population N[0,1] and

averaging the R functio

,( v ) )R

(We did this previously for

ns of

the r

v . We

andom e

ma

ff

ximi

ect

z

r

e

s p

x

obit model.)


Random Effects Model: Simulation----------------------------------------------------------------------Random Coefficients Probit ModelDependent variable DOCTOR (Quadrature Based)

Log likelihood function -16296.68110 (-16290.72192) Restricted log likelihood -17701.08500Chi squared [ 1 d.f.] 2808.80780Simulation based on 50 Halton draws--------+-------------------------------------------------Variable| Coefficient Standard Error b/St.Er. P[|Z|>z]--------+------------------------------------------------- |Nonrandom parameters AGE| .02226*** .00081 27.365 .0000 ( .02232) EDUC| -.03285*** .00391 -8.407 .0000 (-.03307) HHNINC| .00673 .05105 .132 .8952 ( .00660) |Means for random parametersConstant| -.11873** .05950 -1.995 .0460 (-.11819) |Scale parameters for dists. of random parametersConstant| .90453*** .01128 80.180 .0000--------+-------------------------------------------------------------Implied from these estimates is .904542/(1+.904532) = .449998.


The Entire Parameter Vector is Random

i i 1

1 2 ,1

1

+ ,

, ~ [ , ( ,..., )]

Joint probability for observations is

Prob( , ,... | ) ( )

For convenience, write = , ~ [0,1],

log | ,..

i

i it it

i K

i

T

i i i it i itt

i

ik k ik ik ik k ik

t

k

N diag

T

y y u F y

u v v N v

U

L v

x

u u 0

x

,1

1

. log ( )

It is not possible to maximize log | ,... because of

the unobserved random effects embedded in .

iTN

N it i iti i t

N

i

v F y

L

x

v v


Maximum Simulated Likelihood

+ i

i

i

i

i

T

i i i i i it=1

T

i i i i i it=1β

N

i i i i ii=1 β

L ( | data ) = f(data | )

L ( | data ) = f(data | )f( | )d

logL = log L ( | data )f( | )d

β β β β u

Ω β β Ω β

β β Ω β

True log likelihood

+

ˆ

ir N R

S i ir i iri=1 r=1

S

1logL = log L ( | data , )

R

= argmax(logL )

β Ω β β u

Ω

Simulated log likelihood



S

M

= ( )( ) where the diagonal

elements of equal 1, and is the diagonal

matrix with free positive elements. (Cholesky values)

i i

LL MS MS

M S

u MSv


MSSM


Modeling Parameter Heterogeneity

i,k

of the parameters

+

E[ | , ]

of the parameters

Var[u | da

i i i

i i i

β =β Δz u

u X z

Individual heterogeneity in the means

Heterogeneity in the variances

i kta] exp( )

Estimation by maximum simulated likelihoodi khδ


A Hierarchical Probit Model

Uit = 1i + 2iAgeit + 3iEducit + 4iIncomeit + it. 1i=1+11 Femalei + 12 Marriedi + u1i

2i=2+21 Femalei + 22 Marriedi + u2i



Yit = 1[Uit > 0]

All random variables normally distributed.



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.)


Estimating i

+

Is it possible to compute ? No. That requires the unobserved .

The would be

E[ ] = .

It is possible to do better than this. We observe (

i i i

i i

i i

unconditional esti

β =β Δz u

β u

β β Δz

mator

y

i i

i i i i

i

i i

, ). What is

E[ |( , )] = E[ + | ( , )]

(For convenience, embed and any other exogenous data in the

subscript i.) We use Bayes Theorem

f( | ) = Likelihood | f

i i i

i i

X

β y X β Δz u y X

X

y β β

i

i

T

itt 1

T

i i itt 1

i ii

i

(y | )

p( ) = Normal( , ) (or some other known distribution)

f( ) = f( | )p( ) f(y | ) p( )

f( ) f( ) f( ) =

f( ) f(

i

i i

i i i i i

i ii

β

β β Δz

y ,β y β β β β

y ,β y ,ββ | y

y

i

i

i

i

T

itt 1

Ti itt 1

T

itt 1

i i T

itt 1

f(y | ) p( )

)d f(y | ) p( )d

f(y | ) p( )d E[ ] f( )d

f(y | ) p( )d

i i

i

i

i

i i

i i i i iβ β

i i i iβi i i iβ

i i iβ

β β

y ,β β β β β

β β β ββ | y β β | y β

β β β


i

i

T

itt 1

i i T

itt 1

i i i

R R

r 1 r 1

f(y | ) p( )dE[ ] f( | )d

f(y | ) p( )d

How to estimate this using simulation.

ˆ ˆ ˆ ˆ(1)

1 1ˆ ˆ(2) To estimate h( )d use h( ) h(R R

i

i

i

i

i i i iβi i i iβ

i i iβ

i i irβ

β β β ββ | y β β y β

β β β

β β Δz u

β β β β

ir i ir ir

)

ˆ ˆ ˆ ˆ ˆˆ ˆ . = draw from N[ , ].

ir

β β Δz C v CC v 0 I

Conditional Estimate of i


i

i

i

i

T

itt 1

i i T

itt 1

TR

i ir it i irr 1 t 1

iTR

itr 1 t 1

f(y | ) p( )dE[ ] f( | )d

f(y | ) p( )d

How to estimate this using simulation.

1 ˆ ˆˆ ˆˆ ˆf(y | )RE[ ]

1f(y |

R

i

i

i

i i i iβi i i iβ

i i iβ

i

β β β ββ | y β β y β

β β β

β Δz C v β Δz C vβ | y

i ir

R

i ir ir 1

R

ir 1

R Riir ir ir irRr 1 r 1

ir 1

ˆ ˆˆ )

1 ˆ ˆˆˆ Likelihood( | )R =

1 ˆLikelihood( | )R

ˆLikelihood( | )ˆˆ = w where w , w 1ˆLikelihood( | )

ir

ir

ir

ir

β Δz C v

β Δz C v y β

y β

y β β

y β

Conditional Estimate of i





“Individual Coefficients”


The Random Parameters Logit Model

t

j i itj j,i iti J (i)

j i itj j.i itj=1

exp(α + + )Prob[choice j | i, t, ] =

exp(α + + )

β x γ zβ

β x γ z

t

i i

T(i) j i itj j,i it

J (i)t=1j i itj j,i itj=1

Prob[choice j | i, t =1,...,T, ] =

exp(α + + )

exp(α + + )

β

β x γ z

β x γ z

Multiple choice situations: Independent conditioned on the individual specific parameters


Continuous Random Variation in Preference Weights

i

ijt j i itj j it ijt

i i i

i,k k k i i,k

i

eterogeneity arises from continuous variation

in across individuals. (Note Classical and Bayesian)

U = α + + +ε

= + +

β = β + + w

Most treatments set = = +

Hβ

β x γ z

β β Δh w

δ h

Δ 0, β β w

t

i

j i itj j iti J (i)

j i itj j itj=1

exp(α + + ) Prob[choice j | i, t, ] =

exp(α + + )

β x γ zβ

β x γ z


Random Parameters Model Allow model parameters as well as constants to be random Allow multiple observations with persistent effects Allow a hierarchical structure for parameters – not completely

random

Uitj = 1’xi1tj + 2i’xi2tj + i’zit + ijt

Random parameters in multinomial logit model 1 = nonrandom (fixed) parameters 2i = random parameters that may vary across

individuals and across time Maintain I.I.D. assumption for ijt (given )


Modeling Variations Parameter specification

“Nonrandom” – variance = 0 Correlation across parameters – random parts correlated Fixed mean – not to be estimated. Free variance Fixed range – mean estimated, triangular from 0 to 2 Hierarchical structure - ik = k + k’zi

Stochastic specification Normal, uniform, triangular (tent) distributions Strictly positive – lognormal parameters (e.g., on income) Autoregressive: v(i,t,k) = u(i,t,k) + r(k)v(i,t-1,k) [this picks up

time effects in multiple choice situations, e.g., fatigue.]


Model Extensions

AR(1): wi,k,t = ρkwi,k,t-1 + vi,k,t

Dynamic effects in the model Restricting sign – lognormal distribution:

Restricting Range and Sign: Using triangular distribution and range = 0 to 2.

Heteroscedasticity and heterogeneity

i,k k k i k iβ = exp(μ + + )δ z γ w

i i i= + +β β Δz Γw

k,i k iσ = σ exp( )θ h


Estimating the Model

j,i i itj j,i it

J(i)

j,i i itj j,i itj=1

i i i i i

exp(α + + )P[choice j | i,t] =

exp(α + + )

α , , = functions of underlying [α, , , , , , ]

β x γ z

β x γ z

β γ β Δ Γ ρ z v

Denote by 1 all “fixed” parameters in the model

Denote by 2i,t all random and hierarchical parameters in the model


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

0 2.5¢

=2.0¢


Random Parameter Distributions


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


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


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.


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+)

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


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




Error Components Logit Modeling

Alternative approach to building cross choice correlation Common ‘effects.’ Wi is a ‘random individual effect.’

i 1 1,i 2 1,i 3 1,i Brand1,i i

i 1 2,i 2 2,i 3 2,i Brand2,i i

i 1 3,i 2 3,i 3 3,i Brand3,i i

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



U(None) 4 No Brand,i= β + ε


Implied Covariance MatrixNested Logit Formulation

2

2 2 2Brand1

2 2 2Brand2

2 2 2Brand3

NONE

2 2

Var[ε] = π / 6 =1.6449

Var[W] =1

ε +σW 1.6449+σ σ σ 0

ε +σW σ 1.6449+σ σ 0= Var =

ε +σW σ σ 1.6449+σ 0

ε 0 0 0 1.6449

Cross Brand Correlation = σ / [1.6449+σ ]


Error Components Logit Model

Correlation = {0.09592 / [1.6449 + 0.09592]}1/2 = 0.0954

-----------------------------------------------------------Error Components (Random Effects) modelDependent variable CHOICELog likelihood function -4158.45044Estimation based on N = 3200, K = 5Response data are given as ind. choicesReplications for simulated probs. = 50Halton sequences used for simulationsECM model with panel has 400 groupsFixed number of obsrvs./group= 8Number of obs.= 3200, skipped 0 obs--------+--------------------------------------------------Variable| Coefficient Standard Error b/St.Er. P[|Z|>z]--------+-------------------------------------------------- |Nonrandom parameters in utility functions FASH| 1.47913*** .06971 21.218 .0000 QUAL| 1.01385*** .06580 15.409 .0000 PRICE| -11.8052*** .86019 -13.724 .0000 ASC4| .03363 .07441 .452 .6513SigmaE01| .09585*** .02529 3.791 .0002--------+--------------------------------------------------

Random Effects Logit ModelAppearance of Latent Random Effects in Utilities Alternative E01+-------------+---+| BRAND1 | * |+-------------+---+| BRAND2 | * |+-------------+---+| BRAND3 | * |+-------------+---+| NONE | |+-------------+---+


Extended MNL Modeli,1,t F,i i,1,t Q i,1,t P,i i,1,t Brand i,Brand i,1,t

i,2,t F,i i,2,t Q i,2,t P,i i,2,t Brand i,Brand i,2,t

i,3,t F,i i,3,t Q i,3,t

U =β Fashion +β Quality +β Price +λ W +ε


U =β Fashion +β Quality + P,i i,3,t Brand i,Brand i,3,t

i,NONE,t NONE NONE i,NONE i,NONE,t

F,i F F i F F1 i F2 i F,i F,i

P,i P P

β Price +λ W +ε

U =α +λ W +ε

β =β +δ Sex +[σ exp(γ AgeL25 +γ Age2539)] w ; w ~N[0,1]

β =β +δ S i P P1 i P2 i P,i P,i

Brand,i

NONE,i

ex +[σ exp(γ AgeL25 +γ Age2539)] w ; w ~N[0,1]

W ~N[0,1]

W ~N[0,1]

Utility Functions


Extending the Basic MNL Model



i,3,t F,i i,3,t Q i,3,t



U =β Fashion +β Quality + P,i i,3,t Brand i,Brand i,3,t



P,i P P

β Price +λ W +ε

U =α +λ W +ε


β =β +δ S i P P1 i P2 i P,i P,i

Brand,i

NONE,i

ex +[σ exp(γ AgeL25 +γ Age2539)] w ; w ~N[0,1]

W ~N[0,1]

W ~N[0,1]

Random Utility


Error Components Logit Model



i,3,t F,i i,3,t Q i,3,t P



U =β Fashion +β Quality +β ,i i,3,t Brand i,Brand i,3,t



P,i P P i

Price +λ W +ε

U =α +λ W +ε


β =β +δ Sex P P1 i P2 i P,i P,i

Brand,i

NONE,i

+[σ exp(γ AgeL25 +γ Age2539)] w ; w ~N[0,1]

W ~N[0,1]

W ~N[0,1]

Error Components


Random Parameters Model









P,i P P

Price +λ W +ε

U =α +λ W +ε


β =β +δ Sexi P P1 i P2 i P,i P,i

Brand,i

NONE,i


W ~N[0,1]

W ~N[0,1]Random Parameters


Heterogeneous (in the Means) Random Parameters Model









P,i P P

Price +λ W +ε

U =α +λ W +ε



Brand,i

NONE,i


W ~N[0,1]

W ~N[0,1]

Heterogeneity in Means


Heterogeneity in Both Means and Variances









P,i P P

Price +λ W +ε

U =α +λ W +ε



Brand,i

NONE,i


W ~N[0,1]

W ~N[0,1]

Heteroscedasticity


---------------------------------------------------------------Random Parms/Error Comps. Logit ModelDependent variable CHOICELog likelihood function -4019.23544 (-4158.50286 for MNL)Restricted log likelihood -4436.14196 (Chi squared = 278.5)Chi squared [ 12 d.f.] 833.81303Significance level .00000McFadden Pseudo R-squared .0939795Estimation based on N = 3200, K = 12Information Criteria: Normalization=1/N Normalized UnnormalizedAIC 2.51952 8062.47089R2=1-LogL/LogL* Log-L fncn R-sqrd R2AdjNo coefficients -4436.1420 .0940 .0928Constants only -4391.1804 .0847 .0836At start values -4158.5029 .0335 .0323Response data are given as ind. choicesReplications for simulated probs. = 50Halton sequences used for simulationsRPL model with panel has 400 groupsFixed number of obsrvs./group= 8Number of obs.= 3200, skipped 0 obs---------------------------------------------------------------


--------+--------------------------------------------------Variable| Coefficient Standard Error b/St.Er. P[|Z|>z]--------+-------------------------------------------------- |Random parameters in utility functions FASH| .62768*** .13498 4.650 .0000 PRICE| -7.60651*** 1.08418 -7.016 .0000 |Nonrandom parameters in utility functions QUAL| 1.07127*** .06732 15.913 .0000 ASC4| .03874 .09017 .430 .6675 |Heterogeneity in mean, Parameter:VariableFASH:AGE| 1.73176*** .15372 11.266 .0000FAS0:AGE| .71872*** .18592 3.866 .0001PRIC:AGE| -9.38055*** 1.07578 -8.720 .0000PRI0:AGE| -4.33586*** 1.20681 -3.593 .0003 |Distns. of RPs. Std.Devs or limits of triangular NsFASH| .88760*** .07976 11.128 .0000 NsPRICE| 1.23440 1.95780 .631 .5284 |Heterogeneity in standard deviations |(cF1, cF2, cP1, cP2 omitted...) |Standard deviations of latent random effectsSigmaE01| .23165 .40495 .572 .5673SigmaE02| .51260** .23002 2.228 .0258--------+--------------------------------------------------Note: ***, **, * = Significance at 1%, 5%, 10% level.-----------------------------------------------------------

Random Effects Logit Model Appearance of Latent Random Effects in Utilities Alternative E01 E02+-------------+---+---+| BRAND1 | * | |+-------------+---+---+| BRAND2 | * | |+-------------+---+---+| BRAND3 | * | |+-------------+---+---+| NONE | | * |+-------------+---+---+

Heterogeneity in Means.Delta: 2 rows, 2 cols. AGE25 AGE39FASH 1.73176 .71872PRICE -9.38055 -4.33586

Estimated RP/ECL Model


Estimated Elasticities

+---------------------------------------------------+| Elasticity averaged over observations.|| Attribute is PRICE in choice BRAND1 || Effects on probabilities of all choices in model: || * = Direct Elasticity effect of the attribute. || Mean St.Dev || * Choice=BRAND1 -.9210 .4661 || Choice=BRAND2 .2773 .3053 || Choice=BRAND3 .2971 .3370 || Choice=NONE .2781 .2804 |+---------------------------------------------------+| Attribute is PRICE in choice BRAND2 || Choice=BRAND1 .3055 .1911 || * Choice=BRAND2 -1.2692 .6179 || Choice=BRAND3 .3195 .2127 || Choice=NONE .2934 .1711 |+---------------------------------------------------+| Attribute is PRICE in choice BRAND3 || Choice=BRAND1 .3737 .2939 || Choice=BRAND2 .3881 .3047 || * Choice=BRAND3 -.7549 .4015 || Choice=NONE .3488 .2670 |+---------------------------------------------------+

+--------------------------+| Effects on probabilities || * = Direct effect te. || Mean St.Dev || PRICE in choice BRAND1 || * BRAND1 -.8895 .3647 || BRAND2 .2907 .2631 || BRAND3 .2907 .2631 || NONE .2907 .2631 |+--------------------------+| PRICE in choice BRAND2 || BRAND1 .3127 .1371 || * BRAND2 -1.2216 .3135 || BRAND3 .3127 .1371 || NONE .3127 .1371 |+--------------------------+| PRICE in choice BRAND3 || BRAND1 .3664 .2233 || BRAND2 .3664 .2233 || * BRAND3 -.7548 .3363 || NONE .3664 .2233 |+--------------------------+

Multinomial Logit


Estimating Individual Distributions Form posterior estimates of E[i|datai]

Use the same methodology to estimate E[i2|

datai] and Var[i|datai] Plot individual “confidence intervals”

(assuming near normality) Sample from the distribution and plot kernel

density estimates


What is the ‘Individual Estimate?’

Point estimate of mean, variance and range of random variable i | datai.

Value is NOT an estimate of i ; it is an estimate of E[i | datai]

This would be the best estimate of the actual realization i|datai

An interval estimate would account for the sampling ‘variation’ in the estimator of Ω.

Bayesian counterpart to the preceding: Posterior mean and variance. Same kind of plot could be done.


Individual E[i|datai] Estimates*

The random parameters model is uncovering the latent class feature of the data.

*The intervals could be made wider to account for the sampling variability of the underlying (classical) parameter estimators.


WTP Application (Value of Time Saved)

Estimating Willingness to Pay for Increments to an Attribute in a Discrete Choice Model

WTP = MU(attribute) / MU(Income)

attribute,i

cost

βWTP = -

β

Random


Extending the RP Model to WTPUse the model to estimate conditional

distributions for any function of parameters

Willingness to pay = -i,time / i,cost

Use simulation methodˆ ˆ

ˆˆ ˆ

ˆ

R Tr=1 ir t=1 ijt ir it

i i R Tr=1 t=1 ijt ir it

R

i,r irr=1

(1/ R)Σ WTP Π P (β |Ω,data )E[WTP | data ] =

(1/ R)Σ Π P (β |Ω,data )

1 = w WTP

R


Simulation of WTP from i

ˆ

i

i

i,Attributei i i i

i,Cost

Ti,Attribute

i i i i i i i βt=1i,Cost

T

i i i i i i i βt=1

i,

i

-βWTP =E | , , , , ,

β

-βP(choice j | , )g( | , , , , , ) d

β =

P(choice j | , )g( | , , , , , ) d

-β1R

WTP =

β Δ Γ y X z

X β β β Δ Γ y X z β

X β β β Δ Γ y X z β

ˆˆ

ˆ ˆ ˆ ˆ

ˆ

TRAttribute

i irr=1 t=1i,Cost

ir i irTR

i irr=1 t=1

P(choice j | , )β

, = + +1

P(choice j | X , )R

X β

β β Δz Γw

β



WTP


A Generalized Mixed Logit Model

i i,t,j i,t,j

i i i i i i

i i

U(i,t, j) = Common effects + ε

Random Parameters

= σ [ + ]+[γ +σ (1- γ)]

=

is a lower triangular matrix

with 1s on the diagonal (Cholesky matrix)

β x

β β Δz Γ v

Γ ΛΣ

Λ

Σ

i k k i

2 2i i i i i

i i

is a diagonal matrix with φ exp( )

Overall preference scaling

σ = σexp(-τ / 2+τ w + ]

τ = exp( )

0 < γ < 1

ψ h

θ h

λ r


Generalized Multinomial Choice Model









P,i P P

Price +λ W +ε

U =α +λ W +ε



Brand,i

NONE,i


W ~N[0,1]

W ~N[0,1]


Estimation in Willingness to Pay Space

θ θ

θ θ

θ θ

i,1,t P,i F,i i,1,t Q i,1,t i,1,t Brand i,Brand i,1,t

i,2,t P,i F,i i,2,t Q i,2,t i,2,t Brand i,Brand i,2,t

i,3,t P,i F,i i,3,t Q i,3

U =β Fashion + Quality +Price +λ W +ε

U =β Fashion + Quality +Price +λ W +ε

U =β Fashion + Quality ,t i,3,t Brand i,Brand i,3,t


Brand,i NONE,i

+Price +λ W +ε

U =α +λ W +ε

W ~N[0,1] W ~N[0,1

0[ (1 )] F

i iPF P

θ θF,i F,i F,iF F i

P,i P,i P,iP P i

]

w w ~N[0,1]+δ Sex

β w w ~N[0,1]β +δ Sex

Both parameters in the WTP calculation are random.


--------+--------------------------------------------------Variable| Coefficient Standard Error b/St.Er. P[|Z|>z]--------+-------------------------------------------------- |Random parameters in utility functions QUAL| -.32668*** .04302 -7.593 .0000 1.01373 renormalized PRICE| 1.00000 ......(Fixed Parameter)...... -11.80230 renormalized |Nonrandom parameters in utility functions FASH| 1.14527*** .05788 19.787 .0000 1.4789 not rescaled ASC4| .84364*** .05554 15.189 .0000 .0368 not rescaled |Heterogeneity in mean, Parameter:VariableQUAL:AGE| .05843 .04836 1.208 .2270 interaction termsQUA0:AGE| -.11620 .13911 -.835 .4035PRIC:AGE| .23958 .25730 .931 .3518PRI0:AGE| 1.13921 .76279 1.493 .1353 |Diagonal values in Cholesky matrix, L. NsQUAL| .13234*** .04125 3.208 .0013 correlated parameters CsPRICE| .000 ......(Fixed Parameter)...... but coefficient is fixed |Below diagonal values in L matrix. V = L*LtPRIC:QUA| .000 ......(Fixed Parameter)...... |Heteroscedasticity in GMX scale factor sdMALE| .23110 .14685 1.574 .1156 heteroscedasticity |Variance parameter tau in GMX scale parameterTauScale| 1.71455*** .19047 9.002 .0000 overall scaling, tau |Weighting parameter gamma in GMX modelGammaMXL| .000 ......(Fixed Parameter)...... |Coefficient on PRICE in WTP space formBeta0WTP| -3.71641*** .55428 -6.705 .0000 new price coefficientS_b0_WTP| .03926 .40549 .097 .9229 standard deviation | Sample Mean Sample Std.Dev.Sigma(i)| .70246 1.11141 .632 .5274 overall scaling |Standard deviations of parameter distributions sdQUAL| .13234*** .04125 3.208 .0013 sdPRICE| .000 ......(Fixed Parameter)......--------+--------------------------------------------------

Estimated Model for WTP











Aggregate Data and Multinomial Choice:

The Model of Berry, Levinsohn and Pakes


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/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


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 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


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











Latent Class Mixed Logit





Appendix: Maximum Simulated Likelihood

TN

iti=1 t 1

K 1 K

logL( )=

log f(y | , )h( | , )d

,..., , ,..., ,

i

it i i i iβ

1

θ,Ω

x ,β θ β z Ω β

Ω=β,Δ, δ δ Γ


Monte Carlo Integration

range of v

(1) Integral is of the form

K = g(v|data, ) f(v| ) dv

where f(v) is the density of random variable v

possibly conditioned on a set of parameters

and g(v|data, ) is a function of data and

β Ω

Ω

β

r

parameters.

(2) By construction, K( ) = E[g(v|data, )]

(3) Strategy:

a. Sample R values from the population

of v using a random number generator.

b. Compute average K = (1/R) g(v |dat

Ω β

R

r=1a, )

By the law of large numbers, plim K = K.

β


Monte Carlo Integration

ii

RP

ir i i i u iur=1

1f(u ) f(u )g(u )du =E [f(u )]

R

(Certain smoothness conditions must be met.)

ir

ir ir ir

-1 2ir ir

Drawing u by 'random sampling'

u = t(v ), v ~ U[0,1]

E.g., u = σΦ (v )+μ for N[μ,σ ]

Requires many draws, typically hundreds or thousands


Example: Monte Carlo Integral

2

1 2 3

1 2 3

exp( v / 2)(x .9v) (x .9v) (x .9v) dv

2where is the standard normal CDF and

x = .5, x = -.2, x = .3.

The weighting function for v is the standard normal.

Strategy: Draw R (say 1000) standard

r

1 r 2 r 3 r

normal random

draws, v . Compute the 1000 functions

(x .9v ) (x .9v ) (x .9v ) and average them.

(Based on 100, 1000, 10000, I get .28746, .28437, .27242)


Simulated Log Likelihood for a Mixed Probit Model

i

i

it it

TN

iti=1 t 1

TN RSiti=1 r 1 t 1

Random parameters probit model

f(y | ) [(2y 1) ]

~N[ , ]

LogL( )= log [(2y 1) ] N[ , ]d

1LogL log [(2y 1) ( )]

RWe now m

i

it i it i

i i

2i

2it i iβ

it ir

x ,β x β

β β+u

u 0 ΓΛ Γ'

β,Γ,Λ x β β ΓΛ Γ' β

x β+ΓΛv

aximize this function with respect to ( ).β,Γ,Λ


Generating Random DrawsMost common approach is the "inverse probability transform"

Let u = a random draw from the standard uniform (0,1).

Let x = the desired population to draw from

Assume the CDF of x is F(x).

The random -1 draw is then x = F (u).

Example: exponential, . f(x)= exp(- x), F(x)=1-exp(- x)

Equate u to F(x), x = -(1/ )log(1-u).

Example: Normal( , ). Inverse function does not exist in

closed form. There are good polynomial approxi-

mations to produce a draw from N[0,1] from a U(0,1).

Then x = + v.

This leaves the question of how to draw the U(0,1).


Drawing Uniform Random Numbers

Computer generated random numbers are not random; they

are Markov chains that look random.

The Original IBM SSP Random Number Generator for 32 bit computers.

SEED originates at some large odd number

d3 = 2147483647.0

d2 = 2147483655.0

d1=16807.0

SEED=Mod(d1*SEED,d3) ! MOD(a,p) = a - INT(a/p) * p

X=SEED/d2 is a pseudo-random value between 0 and 1.

Problems:

(1) Short period. Base 31d on 32 bits, so recycles after 2 1 values

(2) Evidently not very close to random. (Recent tests have

discredited this RNG)


Quasi-Monte Carlo Integration Based on Halton Sequences

0

Coverage of the unit interval is the objective,

not randomness of the set of draws.

Halton sequences --- Markov chain

p = a prime number,

r= the sequence of integers, decomposed as

H(r|p)

I iii

b p

b 1

0, ,...1 r = r (e.g., 10,11,12,...)

I iiip

For example, using base p=5, the integer r=37 has b0 =

2, b1 = 2, and b2 = 1; (37=1x52 + 2x51 + 2x50). Then

H(37|5) = 25-1 + 25-2 + 15-3 = 0.448.


Halton Sequences vs. Random Draws

Requires far fewer draws – for one dimension, about 1/10. Accelerates estimation by a factor of 5 to 10.


Estimating the RPL Model

Estimation: 1

2it = 2 + Δzi + Γvi,t

Uncorrelated: Γ is diagonal

Autocorrelated: vi,t = Rvi,t-1 + ui,t

(1) Estimate “structural parameters”

(2) Estimate individual specific utility parameters

(3) Estimate elasticities, etc.


Classical Estimation Platform: The Likelihood

ˆ

ˆ

i

i

i

i i iβ

i

Marginal : f( | data, )

Population Mean =E[ | data, ]

= f( | )d

= = a subvector of

= Argmax L( ,i =1,...,N| data, )

Estimator =

β Ω

β Ω

β β Ω β

β Ω

Ω β Ω

β

Expected value over all possible realizations of i. I.e., over all possible samples.


When Var[ ] is not a diagonal matrix. How to estimate a positive

definite matrix, .

~ N[ , ]

= where is upper triangular

with ~ N[ , ]

Convenient Refinement:

i

i

i i i

Cholesky Decomposi

u

u 0

LL L

u Lv v

i

0

o :

I

t n

= ( )( ) where the diagonal

elements of equal 1, and is the diagonal

matrix with free positive eleme

LL MS MS

M S

nts. (Cholesky values)

returns the original uncorrelated case.

We used the Cholesky decomposition in developing the Krinsky and Robb

method for standard errors for partial effects, in Part 3.

i iu MSv

M I


Simulation Based Estimation

Probability = P[data |(1,2,Δ,Γ,R,vi,t)] Need to integrate out the unobserved random term E{P[data | (1,2,Δ,Γ,R,vi,t)]}

= P[…|vi,t]f(vi,t)dvi,t

Integration is done by simulation Draw values of v and compute then probabilities Average many draws Maximize the sum of the logs of the averages (See Train[Cambridge, 2003] on simulation methods.)

v


WinBUGS: MCMC User specifies the model – constructs the Gibbs Sampler/Metropolis Hastings

MLWin: Linear and some nonlinear – logit, Poisson, etc. Uses MCMC for MLE (noninformative priors)

SAS: Proc Mixed. Classical Uses primarily a kind of GLS/GMM (method of moments algorithm for loglinear models)

Stata: Classical Several loglinear models – GLAMM. Mixing done by quadrature. Maximum simulated likelihood for multinomial choice (Arne Hole, user provided)

LIMDEP/NLOGIT Classical Mixing done by Monte Carlo integration – maximum simulated likelihood Numerous linear, nonlinear, loglinear models

Ken Train’s Gauss Code, miscellaneous freelance R and Matlab code Monte Carlo integration Mixed Logit (mixed multinomial logit) model only (but free!)

Biogeme Multinomial choice models Many experimental models (developer’s hobby)

Programs differ on the models fitted, the algorithms, the paradigm, and the extensions provided to the simplest RPM, i = +wi.

Documents

1/110: Topic 4.3 – Mixed Models and Random Parameters Microeconometric Modeling William Greene Stern School of Business New York University New York NY