Simulation Based Estimation of Discrete Choice...
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Simulation Based Estimation of Discrete Choice Models
William GreeneDepartment of EconomicsStern School of Business
New York University
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Simulation Based Estimation
• History:– 1981: Lerman and Manski, Multivariate normal probabilities– 1984: GHK simulator and the multinomial probit model (vastly
oversold)– 1996 Gourieroux and Monfort: Simulation Based Econometrics– 1996 – present: Trivedi, Train, Greene, et. al. applications– 2003: Train, Discrete Choice Methods with Simulation
• Wave of the future? – A census of the JAE– Outside Economics?
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Journal of Applied Econometrics
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Agenda
• Methodology• Maximum simulated likelihood• Contrast to Bayesian MCMC estimation• Mechanics of MSL, MCMC and ML• Applications - several
– Panel model for counts: GEE (whiz)– Health care – binary choice, counts– Random parameters multinomial logit – Logit kernels and bank distress
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Simulation Based Estimation
• Range of applications– Microeconometric applications – Typically discrete choice – No limitation as such
• Typical implementation – modeling unobserved heterogeneity
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The Conditional Density
• Data Structure– Panel: yi = (yi1,…,yi,Ti)'– Multivariate: yi = (si, ri, hi)'– Both…
• Unobserved heterogeneity: wi.• Conditional density
– fi(y) = f(yi | Xi, wi,β) conditioned on wi
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Log Likelihood
=
=
= =
=
∑
∑ ∫i
N
i i i ii 1
N
i i ii 1
i
logL | {( ), i 1,...,N} log f( | , , )
logL( ) log f( | , , )p( | )d
To be maximized over and (parameters of marginal distribution of ).
i iw
Conditional
w y X w β
Unconditional
β,θ y X w β w θ w
β θw
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Simulated Log Likelihood
=
=
=
⎡ ⎤⎣ ⎦
∑ ∫∑
i iw
w
β,θ y X w β w θ w
y X w βi
i
N
i i ii 1
N
i i ii 1
The unconditional log likelihood is an expected value
logL( ) log f( | , , )p( | )d
= logE f( | , , )
The expected value can be estimated by random sampling
logL= =
= ⎡ ⎤⎣ ⎦
⎯⎯→
∑ ∑β,θ y X w β
β,θ β,θ
N RSi i iri 1 r 1
pS
1( ) log f( | , , )
RBy the law of large numbers,
logL ( ) logL( )Sample of draws from the population wi.
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Properties of the MSLE
• Assume– Likelihood is “regular”– MLE achieves the usual desirable properties– Smoothness with respect to the heterogeneity
• Simulation (For useful details, see M&T)– Rate of increase in draws is faster than N1/2 (to be
revisited below)• Same asymptotic properties as MLE (See
Munkin and Trivedi (1999) – cited below.)
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A Random Parameters Model
=
= +
= =
⊥
=
i
0i
y X w β y X β
β β w
w ∆z w Σ ΓΓ'w X
z x
i i i i i
i
i i i
i i
i i
Random parameters model( | , , ) p( | , )
Partially observed heterogeneityE[ ] , Var[ ]=
It is assumed that . Perhaps a Mundlak
correction in a panel,
Structural para
f
= +w ∆z Γv v 0 v Ii i i i i
meters where E[ ]= , Var[ ] =
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Simulated Log Likelihood for a Panel
= =
= = =
= ∑ ∑
∑ ∑ ∏
0
0i ir
β ,∆,Γ y X w β
x β +∆z +Γv
v
i
N RSi i iri 1 r 1
TN R
it iti 1 r 1 t 1
ir
Simulated Log Likelihood1
logL ( ) log f( | , , )R1
= log f[y | ,( )]R
Simulation is over draws of . Standard Normal, or other...
logL 0β ,∆,ΓS ( ) is maximized using conventional gradient methods
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RP Binomial Logit Model
= = =
′−=
′+ −
∑ ∑ ∏ iTN RSit iti 1 r 1 t 1
it itit it
it it
1logL ( ) = log f[y | ,( )]
Rexp[(2y 1) ( )]
f[y | ,( )]1 exp[(2y 1) ( )]
0 0i ir
00 i ir
i ir 0i ir
β ,∆,Γ x β +∆z +Γv
x β +∆z +Γvx β +∆z +Γv
x β +∆z +Γv
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An ApplicationData Source: Regina T. Riphahn, Achim Wambach, and Andreas Million, "Incentive Effects in the Demand for Health Care: A Bivariate Panel Count Data Estimation", Journal of Applied Econometrics, Vol. 18, No. 4, 2003, pp. 387-405.
German Health Care Usage Data (JAE Data Archive)
N = 7,293 Individuals,
Unbalanced PanelTi : 1=1525, 2=2158, 3=825, 4=926, 5=1051, 6=1000, 7=987
There are altogether 27,326 observations
Variables in the file areHHNINC = household nominal monthly net income in German marks / 10000.
(4 observations with income=0 were dropped)HHKIDS = children under age 16 in the household = 1; otherwise = 0EDUC = years of schooling MARRIED = marital StatusDOCTOR = Number of doctor visits > 0.
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Estimated Random Parameters Logit Model for DOCTOR
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Fixed RandomMeans
RandomStd.Devs.
Constant 1.239 (.068) 1.185 (.057) 1.504 (.020)HHNINC -.191 (.075) .045 (.063) .441 (.038)HHKIDS -.429 (.027) -.369 (.022) .298 (.024)EDUC -.058 (.006) -.057 (.005) .013 (.001)MARRIED .251 (.031) .191 (.026) .045 (.017)
Log Likelihood -17804.1 -16408.7
Note: In this model, Γ = I and ∆ = 0.
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Bayesian EstimationRandom Parameters? Not "randomly distributed across observations."What is "randomness" in this context?
The posterior mean is a conditional meanL(Data| )Prior( )
Posterior density for = p(Data)
β ββ
∫
∫∫
L(Data| )Prior( ) =
L(Data| )Prior( )d
Bayesian estimator is the posterior mean:
L(Data| )Prior( )d =
L(Data| )Prior( )d
β
β
β
β ββ β β
β β β β
β β β
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Bayesian Estimation
= =∫
∫
Posterior density via Bayes Theoremp(Data| )Prior( ) p(Data, ) p(Data, )
p( )= p(Data) p(Data) p(Data, )d
L(Data| )Prior( ) = (Note: Fn of L and Prior)
L(Data| )Prior( )d
Posteri
β
β
β β β ββ|Dataβ β
β ββ β β
==
⎯⎯→
∫∫
∑R
r 1
P
or mean
L(Data| )Prior( )dˆ=
L(Data| )Prior( )d
MCMC (Gibbs sampling) estimator1ˆ |posterior densityR
ˆBy the law of large numbers, E[ ]
β
β
MCMC r
MCMC
β β β ββ
β β β
β β
β β|Data
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MCMC vs. MSL
=∫ ∫∫ ∫
Neglecting the interpretative difference - in purely numerical terms,
With noninformative (flat) priors, the posterior mean is
L(Data| )Prior( )d L(Data| )dˆ =
L(Data| )Prior( )d L(Data|β β
β β
β β β β β β ββ
β β β β=
•
•
∫ p( Data)d)d
The Bayesian estimator is the mean of the population of vectors whose distribution is defined by the conditional (on the data) likelihood.
The MLE is the vector which sits
ββ β| β
β
at the mode of the same likelihood.
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Why do MCMC and MSL differ numerically?
• Pure sampling variability – they are different statistics• Asymmetry of the log likelihood function; mean ≠ mode• Strength of the prior: Posterior ≠ Likelihood• … only… by and large, they don’t differ.• A Theorem of Von-Mises: Under weak priors, the
“estimators converge” to the same thing, and the posterior variance converges to the inverse of the information matrix.
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Application: Panel (RE) ProbitHarris, M., Rogers, M., Siouclis, A., "Modelling Firm Innovation Using PanelProbit Estimators," Melbourne Institute of Applied Economic and SocialResearch, WP 20/01, 2002.Sample = 3,757 Australian firms, 3 years:Random Effects Probit Model: Innovation reported in the survey year ML MCMCConstant -1.646 (.052) -1.646 (.052)Labor 0.075 (.017) 0.075 (.017)Lagged Profit Margin -0.063 (.056) -0.063 (.056)Business Plan 0.485 (.042) 0.485 (.042)Network 0.410 (.044) 0.409 (.043)Export 0.072 (.050) 0.071 (.051)Startup Firm 0.005 (.081) 0.001 (.082)R&D 1.613 (.061) 1.613 (.063)Correlation 0.380 (.020) 0.381 (.053)(ML obtained by Hermite quadrature - Butler and Moffitt method.)
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The Simulations
• An arms race in the MCMC literature: How many draws for the Gibbs sampler? 5,000? 50,000? There is no theory to define R.
• MSL – Considerations for good properties– R/N1/2 →∞ Many draws needed to invoke LLN– High quality random number generators with long
periods. (Unclear why these are important.)– It’s a moot point
• Randomness is not the issue. Even coverage of the range of integration is what is needed.
• Halton draws and “intelligent” draws reduce the number of draws needed by a large percent. 90 if K=1 and N is large.
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Quadrature and Simulation
ii i i
K 2k=1 i,k
General Problem: How to compute
f( | , , )p( | )d
Thus far, we have considered simulation and Gibbs sampling.
If p( | ) = Exp(- w ) (Multivariate spherical normal)
then Cartesian Hermite
Σ
∫ i iw
i
y X w β w θ w
w θ
i
K K 1 1 1 K KK K 1 1
i i i
H H H
h h h i i h h 1 hh 1 h 1 h 1
K
quadrature is a third possibility.
f( | , , )p( | )d
w w ... w f( | ,z ,...z , z , )
Note: H function evaluations per observation per iteration,...−−
−= = =≈
∫∑ ∑ ∑
i iwy X w β w θ w
y X β
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Discussion
Rabe-Heketh, S., Skrondal, A., Pickles, A., “Maximum Likelihood Estimation of Limited and Discrete Dependent Variable Models with Nested Random Effects,” Journal of Econometrics, 128, 2, October, 2005, pp. 301-323.
Develops an “Adaptive Quadrature” method for these problems.
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On Adaptive Quadrature
• “Performs well for large cluster sizes”• “Gives empirical Bayes predictions for cluster or
individual specific random effects and their standard errors” (We will return to this claim.) (A claim usually made by Bayesians, typically followed by an untrue assertion that classical methods cannot do this.)
(Page 320)
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On Alternatives
“Computer intensivealternatives to adaptive quadrature include simulation based approaches such as MCMC and MSL.”
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Rabe-Hesketh on MCMC• “The hierarchical structure of multilevel models lends
itself naturally to MCMC using Gibbs sampling. If vague priors are specified, the method essentially yields maximum likelihood estimates. (We knew that.)
• Quibbles with MCMC estimation– How to ensure that a truly stationary distribution has been
obtained? (MCMC always works, even when it shouldn’t)– There is no diagnostic for assessing empirical identification.
(Same issue. How do you know if a problem is not identified?)
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Rabe-Hesketh on MSL• “Regarding simulated maximum likelihood, a
merit is that conditional independence specifications implicit in standard multilevel models may be relaxed….
• “Futhermore, unlike methods based on quadrature, simulation methods allow analysis of the approximation error” (See Munkin and Trivedi)
• (No shortcomings are cited)
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Quadrature
• Virtues– Quite accurate for relatively few nodes
• Vices– Excruciatingly slow if K > 2 (Compare hours
vs. seconds for simulation or Gibbs sampling)– Torturously complicated– Limited to normal densities for integrands– Limited to independent effects
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Why Consider Quadrature?
• The last paragraph of the 21 page paper states “Maximum likelihood estimation and empirical Bayes predictions for all of these models using adaptive quadrature is implemented in gllamm which runs in Stata”
• Stata does not contain a simulation estimator nor any Gibbs sampling routines.
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Parlor Tricks: MSL Estimation of a Panel Poisson Model
1 2
How to construct a bivariate Poisson model?(1) Traditional: Observed Y1 = Z1 + U Observed Y2 = Z2 + U Z1, Z2, U all Poisson with means , ,
A bit clumsy and only al
λ λ µ
lows positive covariance (Not good for Y1 = doctor visits, Y2 = hospital visits )(2) Munkin, M. and Trivedi, P., "Simulated Maximum Likelihood Estimation of Multivariate Mixed Poisson Regres
µ
sion Models, with Application," The Econometrics Journal, 2, 1, 1999, pp. 29-48.
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Bivariate Poisson1 2y y
1 1 2 21 1 2 2
1 2
1 1 1 1 2 2 2 2
1 2
exp( ) exp( )P(y | x ) , P(y | x )
(y 1) (y 1)
exp( x ), exp( x )
( , ) have a bivariate distribution with nonzero covariance
that can be simul
−λ λ −λ λ= =
Γ + Γ +′ ′λ = β + ε λ = β + ε
ε ε
ated. M&T assume joint normality.
The likelihood does not have a closed form, but the conditionallikelihood can be easily simulated. The model is estimated by MSL
Application: Health care utilization: y1 = emergency room visits y2 = hospitalizations
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A Panel Poisson Model
ityit it it it
it it itit
it it it it i
i1 i2 i7
German Health Care Data: Unbalanced panel, up to 7 periods
exp[ ( | )]( | )P(y | x , ) ,
(y 1)
( | ) exp( ), t = 1,...,T
( , ,..., ) ~ 7 variate normal w
− λ ε λ εε =
Γ +′λ ε = + ε
ε ε ε
β x
its
it it it i1 it1 i2 it2 i3 it3 i4 it4 i5 it5 i6 it6 i7 it7
0i i
ith free correlation matrix
There are 7 specific dates in the data set. d a set of time dummies
( | ) exp( d d d d d d d )
=′λ ε = + δ + δ + δ + δ + δ + δ + δ
= +
β x
δ δ Γv
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Estimated ModelsFixed
ParametersRandom
ConstantsUncorrelated
EffectsRandom
ConstantsCorrelated
Effects
Means Means Standard Deviations
Means Standard Deviations
HHNINC -0.683 (.024) -0.633 (.008) -0.322 (.011)
HHKIDS -0.363 (.007) -0.369 (.002) -0.117 (.003)
EDUC -0.056 (.002) -0.065 (.001) -0.098 (.001)
MARRIED 0.142 (.009) 0.117 (.003) 0.074 (.004)
Period 1=1984 2.006 (.022) 1.772 (.010) 1.376 (.006) 2.347 (.012) 1.696
Period 2=1985 1.984 (.022) 1.892 (.009) 1.052 (.005) 2.395 (.012) 1.379
Period 3=1986 2.121 (.022) 1.920 (.0090) 1.046 (.005) 2.541 (.012) 1.590
Period 4=1987 2.064 (.022) 2.208 (.008) 0.909 (.005) 2.426 (.012) 1.404
Period 5=1988 1.936 (.022) 1.889 (.008) 0.910 (.004) 2.482 (.012) 1.191
Period 6=1991 1.933 (.022) 1.897 (.009) 0.761 (.004) 2.350 (.012) 1/045
Period 7=1994 2.285 (.022) 2.293 (.008) 0.707 (.004) 2.449 (.012) 1.316
Log Likelihood -105,520.20 -82,897.02 -72,467.9
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Cross Period Correlation Matrix
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Multinomial Logit Model
• Workhorse for empirical analysis of discrete choice among unordered alternatives
• Numerous extensions have been developed for different situations and microeconomic assumptions
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A Random Utility ModelRandom Utility Model for Discrete Choice Among J alternatives at time t by person i.Uitj = αj + β′xitj + εijt
αj = Choice specific constant
xitj = Attributes of choice presented to person(Information processing strategy. Not allattributes will be evaluated. E.g., lexicographicutility functions over certain attributes.)
β = ‘Taste weights,’ ‘Part worths,’ marginal utilities
εijt = Unobserved random component of utility
Mean=E[εijt] = 0; Variance=Var[εijt] = σ2
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The Multinomial Logit Model
Independent type 1 extreme value (Gumbel):– F(εitj) = 1 – Exp(-Exp(εitj)) – Independence across utility functions– Identical variances, σ2 = π2/6– Same taste parameters for all individuals
Model extensions build around IIA assumptions.Not our interest here.
∑ t
j itjJ (i)
j itjj=1
exp(α +β'x )Prob[choice j | i,t] =
exp(α +β'x )
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Building Individual Effects into Utilities for the MNL Model
i,t,j itj itj i
A counterpart to individual specific, random effects?U + u
Same effect in all utility functions is undesirableChoice specific effect is not the objective.
′= + ε σ
Repeated Choice Situations
β x
i,1 i,M
i,t,j itj itj 1,1 1 i,1 1,2 2 i,2 1,M M i,M
i,t,j itj itj 2,1 1 i,1 2,2 2 i,2 2,M M i,M
i,t,j itj itj J,1 1 i,
Define a set of M J "kernels" u ,...,u
U + C u C u ... C u
U + C u C u ... C u
...U + C u
≤
′= + ε σ + σ + + σ
′= + ε σ + σ + + σ
′= + ε σ
β x
β x
β x 1 J,2 2 i,2 J,M M i,M
j,m
C u ... C u
C dummy variable for placing kernel m in utility j
+ σ + + σ
=
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The Kernel Logit Model• Individual effects in utility functions, not tied to the choice
structure• Provides a stochastic specification that produces the
nested logit model – simpler to estimate and always satisfies utility maximization without parameter constraints
• Estimable as a random parameters multinomial logit model with MSL.
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Application: Australian Banks• Hensher, D., Jones, S., Greene, W., “Corporate
Bankruptcy and Insolvency Risk in Australia: A Kernel Logit Analysis,” Department of Accounting, University of Sydney, WP, 2006.
• Application– Nonfailed; Insolvent; Distressed; Bankrupt and in receivership– Unordered choice – MNL model.– Quibles: Are these choices ordered? How should this outcome
be modeled?
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A Complicated Application of MSL:MNL for Shoe Brand Choice
• Simulated Data: Stated Choice, 400 respondents, 8 choice situations
• 3 choice/attributes + NONE– Fashion = High / Low– Quality = High / Low– Price = 25/50/75,100 coded 1,2,3,4
• Heterogeneity: Sex, Age (<25, 25-39, 40+)• Underlying data actually generated by a 3 class latent class
process (100, 200, 100 in classes) (True process is known.)• Thanks to www.statisticalinnovations.com (Latent Gold and Jordan
Louviere)
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A Discrete (4 Brand) Choice Model
i,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 P
U =β Fashion +β Quality +β Price +λ K +ε
U =β Fashion +β Quality +β Price +λ K +ε
U =β Fashion +β Quality +β ,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 i
Price +λ K +ε
U =α +λ K +ε
β =β +δ Sex +[σ exp(γ AgeL25 +γ Age2539 )] w ; w ~N[0,1]
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]
K ~N[0,1]
K ~N[0,1]
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Random Parameters Model
i,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 P
U =β Fashion +β Quality +β Price +λ K +ε
U =β Fashion +β Quality +β Price +λ K +ε
U =β Fashion +β Quality +β ,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
Price +λ K +ε
U =α +λ K +ε
β =β +δ Sex +[σ exp(γ AgeL25 +γ Age2539 )] w ; w ~N[0,1]
β =β +δ Sexi P P1 i P2 i P,i P,i
Brand,i
NONE,i
+[σ exp(γ AgeL25 +γ Age2539 )] w ; w ~N[0,1]
K ~N[0,1]
K ~N[0,1]
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Heterogeneous (in the Means) Random Parameters Model
i,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 P
U =β Fashion +β Quality +β Price +λ K +ε
U =β Fashion +β Quality +β Price +λ K +ε
U =β Fashion +β Quality +β ,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
Price +λ K +ε
U =α +λ K +ε
β =β +δ Sex +[σ exp(γ AgeL25 +γ Age2539 )] w ; w ~N[0,1]
β =β +δ Sexi P P1 i P2 i P,i P,i
Brand,i
NONE,i
+[σ exp(γ AgeL25 +γ Age2539 )] w ; w ~N[0,1]
K ~N[0,1]
K ~N[0,1]
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Heterogeneity in Both Means and Variances
i,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 P
U =β Fashion +β Quality +β Price +λ K +ε
U =β Fashion +β Quality +β Price +λ K +ε
U =β Fashion +β Quality +β ,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
Price +λ K +ε
U =α +λ K +ε
β =β +δ Sex +[σ exp(γ AgeL25 +γ Age2539 )] w ; w ~N[0,1]
β =β +δ Sexi P P1 i P2 i P,i P,i
Brand,i
NONE,i
+[σ exp(γ AgeL25 +γ Age2539 )] w ; w ~N[0,1]
K ~N[0,1]
K ~N[0,1]
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Individual (Kernel) Effects Model
i,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 P
U =β Fashion +β Quality +β Price +λ K +ε
U =β Fashion +β Quality +β Price +λ K +ε
U =β Fashion +β Quality +β ,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
Price +λ K +ε
U =α +λ K +ε
β =β +δ Sex +[σ exp(γ AgeL25 +γ Age2539 )] w ; w ~N[0,1]
β =β +δ Sexi P P1 i P2 i P,i P,i
Brand,i
NONE,i
+[σ exp(γ AgeL25 +γ Age2539 )] w ; w ~N[0,1]
K ~N[0,1]
K ~N[0,1]
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Estimated Brand Choice Models
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“Estimating Individual Effects”
µ
i i
i
0
0
f(y |parameters) = g(data, )
p( ) = h( ) (Typically normal)
p( ) = Normal( , )
p( ) = Inverse Wishart(a , ) Individual Estimate? Posterior mean is an estima
0
0
Hierarchical Bayesβ
β β,Σ
β Σ
Σ A
= + +
i i
i i
i i i
i
i i
te of E[ |Data ]
f(y |parameters) = g(data, )
density of is specified...
Individual Estimate? We can estimate E[ |Data ]
0
βClassical Random Parameters
β
β β ∆z Γvv
βNeither is an "e i
1 N
i
x
x (x ,..., x ).
i
stimate" of β any more than is an estimate ofin a sample It is an estimate of the conditional mean
of the population from which β is a draw.
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Simulation Estimation of E[βi|Datai]=
=
=
=∫ ∫i i
i i i i
i i
i i i i i
i i i i i
p(Data | ) Individual contribution to the likelihood, L(Data | )
Marginal p( ) Specified in the RP model = g( | )
p(Data ) L(Data | )g( | )
Marginal p(Data )= p(Data )d L(Data |
i
i
β β
β ββ β θ, z
,β β β θ, z
,β β
=
=
∫
∫∫
i
i
i
i i i
i i ii i
i i i i
i i i i i
i ii i i i
)g( | ) d
Now, use Bayes TheoremL(Data | )g( | )
p( |Data )L(Data | )g( | ) d
The Conditional Mean Estimator
L(Data | )g( | )dE( |Data )
L(Data | )g( | ) d
i
i
iβ
iβ
iβ
β β θ, z β
β β θ, zβ
β β θ, z β
β β β θ, z ββ
β β θ, z β
We can also estimate the conditional variance this way.This is precisely the counterpart to the hierarchical Bayes estimator
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RP Estimator for MNL
= + + =
+ +=∫
i
i
i i
i i i i i
i i
i i
Likelihood is formed for the Multinomial Logit Model |
Marginal density for is formed from that of
where ~ N[ , ] f( )
The estimator is
( ) ME[ |Data ]
0
0
v
ββ v
β β ∆z Γv v 0 I v
β ∆z Γvβ
+ +
+ +∫i
i i i i i
i i i i i
NL[Data | ( )]f( )d
MNL[Data | ( )]f( )d
0
0
v
β ∆z Γv v v
β ∆z Γv v v
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Simulation Estimator
=
=
+ + + +=
+ +
∑
∑
R
i ir i i ir 1
i iR
i i irr 1
i i
1 ( )MNL[Data | ( )]RE[ |Data ]
1 MNL[Data | ( )]R
We do likewise for the expected square and then compute anestimate of Var[ |Data ]
A Caveat: The "plug-in" estim
0 0
0
β ∆z Γv β ∆z Γvβ
β ∆z Γv
β
iator uses the MLEs of , , .
Statistical properties may be questionable. The same caveat applies to the hierarchical Bayes estimators.
0β ∆z Γ
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Individual E[βi|datai] Estimates
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Estimated Upper and Lower Bounds for BPrice(i)
PERSON
-25
-15
-5
5
15
-3581 161 240 320 4001
B_P
rice
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Analyzing Willingness to Pay
i
R
i ir i i ir 1
i iR
i i irr 1
Extending the idea of estimating E[ |Data ]
Simulation estimator for the mean is1 ˆ ˆˆ ˆ ˆ ˆ ( )MNL[Data | ( )]RE[ |Data ]
1 ˆ ˆ ˆMNL[Data | ( )]R
Extend this to a functi
=
=
+ + + +=
+ +
∑
∑
i
0 0
0
β
β ∆z Γv β ∆z Γvβ
β ∆z Γv
R
i ir i i ir 1
i iR
i i irr 1
on of the parameters1 ˆ ˆˆ ˆ ˆ ˆ g( )MNL[Data | ( )]RE[g( )|Data ]
1 ˆ ˆ ˆMNL[Data | ( )]R
=
=
+ + + +=
+ +
∑
∑
0 0
0
β ∆z Γv β ∆z Γvβ
β ∆z Γv
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WTP
Attribute Attribute
Price Income
R Attribute,ii irr 1
Price,i
R
ir 1
Willingness to pay measures in a MNL model:ˆ ˆ
Usually: - . Where no price is available, ˆ ˆ
ˆ1 ˆ MNL[Data | ( )]ˆR
Estimation :1 MNL[DataR
=
=
β β
β β
∑
∑
ββ
β
irˆ| ( )]β
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WTP Application
Greene, W., Hensher, D., Rose, J., “Using Classical Simulation Based Estimators to Estimate Individual WTP Values,” in R. Scarpa and A. Alberini, eds., Applications of Simulation Methods in Environmental and Resource Economics, Springer, 2005
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Stated Choice Experiment• Survey Data• 223 Commuting Trips in Northwest Sydney• T = 10 repetitions (scenarios)• Choices:
– Existing: Car, bus, train, busway– Proposed: light rail, heavy rail, new busway
• Demographics: Age, Hours worked per week, Income, Household size, Number of kids, Gender
• Attributes: Fare, In vehicle time, Waiting time, Access Mode/Time, Egress Time (various)
• Experimental design in several levels for each.
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Mixed Logit Model
0k,i k k ,i k,i
0k
k,i
Simulation is tailored to the application:
(1 v ) where v has a "tent" distribution from -1 to +1.
is free.
v a transformation of a standard uniform U(0,1). Si
β = β +
β
=
k,i i
k,i i
mulation is trivial.
v 2u - 1 if u .5
v 1 - 2(1 u ) if u > .5
= ≤
= −
0 β 2β
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Value of Travel Time Saved
Willingness to Pay Attribute VTTS Mean
VTTS Range
Main mode in vehicle time – car 28.80 6.37 - 39.80Egress time – car 31.61 15.80 - 75.60Main mode in vehicle time – public transport 15.71 2.60 - 31.50Waiting time – all bus 19.10 9.90 - 40.30Access time – all bus 17.10 8.70 - 31.70Access plus wait time – all rail 10.50 9.60 - 22.90Egress time – all public transport 3.40 2.00 - 7.40
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Observations
• MSL is a fast, flexible method of handling different forms of heterogeneity.
• MCMC (current practice) and MSL are alternative approaches. Statistical “aspects” are a red herring
• Individual effects are equally straightforward with either approach.
Camp Econometrics. Saratoga Springs, 2006