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Discrete Choice Modeling
Hybrid Choice Models
Discrete Choice Modeling
William Greene
Stern School of Business
New York University
0 Introduction1 Summary2 Binary Choice3 Panel Data4 Bivariate Probit5 Ordered Choice6 Count Data7 Multinomial Choice8 Nested Logit9 Heterogeneity10 Latent Class11 Mixed Logit12 Stated Preference13 Hybrid Choice
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Discrete Choice Modeling
Hybrid Choice Models
What is a hybrid choice model?
Incorporates latent variables in choice model
Extends development of discrete choice model to incorporate other aspects of preference structure of the chooser
Develops endogeneity of the preference structure.
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Discrete Choice Modeling
Hybrid Choice Models
Endogeneity "Recent Progress on Endogeneity in Choice Modeling" with Jordan Louviere
& Kenneth Train & Moshe Ben-Akiva & Chandra Bhat & David Brownstone & Trudy Cameron & Richard Carson & J. Deshazo & Denzil Fiebig & William Greene & David Hensher & Donald Waldman, 2005. Marketing Letters Springer, vol. 16(3), pages 255-265, December.
Narrow view: U(i,j) = b’x(i,j) + (i,j), x(i,j) correlated with (i,j)(Berry, Levinsohn, Pakes, brand choice for cars, endogenous price attribute.) Implications for estimators that assume it is.
Broader view: Sounds like heterogeneity. Preference structure: RUM vs. RRM Heterogeneity in choice strategy – e.g., omitted attribute models Heterogeneity in taste parameters: location and scaling Heterogeneity in functional form: Possibly nonlinear utility functions
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Discrete Choice Modeling
Hybrid Choice Models
Heterogeneity Narrow view: Random variation in marginal
utilities and scale RPM, LCM Scaling model Generalized Mixed model
Broader view: Heterogeneity in preference weights RPM and LCM with exogenous variables Scaling models with exogenous variables in
variances Looks like hierarchical models
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Discrete Choice Modeling
Hybrid Choice Models
Heterogeneity and the MNL Model
j ij
J(i)
j ijj=1
exp(α + ' )P[choice j | i] =
exp(α + ' )
β x
β x
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Discrete Choice Modeling
Hybrid Choice Models
Observable Heterogeneity in Preference Weights
U
ij j i ij j i ij
x z
i i
i,k k k i
Hierarchical model - Interaction terms
= +
Parameter heterogeneity is observable.
Each parameter β = β +
Prob[choi
β β Φh
φ h
i
j ij i
J
j ij ij=1
exp(α + + )ce j | i] =
exp(α + + )
i j
i
β x γ z
β x γ z
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Discrete Choice Modeling
Hybrid Choice Models
‘Quantifiable’ Heterogeneity in Scaling
Uij j i ij j i ij
x z2 2 2
ij j i 1
Var[ε ] = σ exp( ), σ = π / 6jδ w
wi = observable characteristics: age, sex, income, etc.
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Discrete Choice Modeling
Hybrid Choice Models
Unobserved Heterogeneity in Scaling
1
HEV formulation: U (1 / )
Generalized model with = 1 and = [ ].
Produces a scaled multinomial logit model with
exp( )Prob(choice = j) = , 1,..., , 1,...,
exp( )i
ij ij i ij
i i
i iji iJ
i ijj
i N j J
x
0
x
x
2
2
The random variation in the scaling is
exp( / 2 )
The variation across individuals may also be observed, so that
exp( / 2 )
i i
i i i
w
w
z
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Discrete Choice Modeling
Hybrid Choice Models
Generalized Mixed Logit Model i i,j i,j
i i i i i i
i i
i
U(i, j) = Common effects + ε
Random Parameters
= σ [ + ]+[γ +σ (1- γ)]
=
is a lower triangular matrix
with 1s on the diagonal (Cholesky matrix)
is a
β x
β β Δh Γ v
Γ ΛΣ
Λ
Σ
k k i
2i i i i i
i i
diagonal matrix with φ exp( )
Overall preference scaling
σ = exp(-τ / 2+τ w + ]
τ = exp( )
0 < γ < 1
ψ h
θ h
λ r
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Discrete Choice Modeling
Hybrid Choice Models
A helpful way to view hybrid choice models
Adding attitude variables to the choice model
In some formulations, it makes them look like mixed parameter models
“Interactions” is a less useful way to interpret
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Discrete Choice Modeling
Hybrid Choice Models
Observable Heterogeneity in Utility Levels
Uij j i ij j i ij
x z
j ij i
J(i)
j ij ij=1
exp(α + + )Prob[choice j | i] =
exp(α + + )
jβ'x γ z
β'x γ z
Choice, e.g., among brands of cars
xitj = attributes: price, features
zit = observable characteristics: age, sex, income
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Discrete Choice Modeling
Hybrid Choice Models
Unbservable heterogeneity in utility levels and other preference indicators
Multinomial Choice Model
U
Indicators (Measurement) Model(s)
Outco
i i
ij j i ij j i ij
x z
t
i
j ij i
J (i)
j ij ij=1
exp(α + + )Prob[choice j | i] =
exp(α + + )
j
z b w
β'x γ z
β'x γ z
mes = f ( ,v )m imim iy z
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Discrete Choice Modeling
Hybrid Choice Models
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Discrete Choice Modeling
Hybrid Choice Models
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Discrete Choice Modeling
Hybrid Choice Models
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Discrete Choice Modeling
Hybrid Choice Models
*1 1 1 1
*2 2 2 2
*3 3 3 3
*1 1 1 1
*2 2 1 1
* *3 3 1 2 1
*4 4 2 2
*5 5 2 2
*6 6 3 3
*7 7 3 3
( , )
( , )
( , , )
( , )
( , )
( , )
( , )
z u
z u
z u
y g z
y g z
y g z z
y g z
y g z
y g z
y g z
h
h
h
Observed Latent Observed x z* y
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Discrete Choice Modeling
Hybrid Choice Models
MIMIC ModelMultiple Causes and Multiple Indicators
X z* Y1 1 1
2 2 2* *+w z ... ... ...
M M M
y
yz
y
β x
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Discrete Choice Modeling
Hybrid Choice Models
should be kl ikx
Note. Alternative i, Individual j.
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Discrete Choice Modeling
Hybrid Choice Models
*
*
U =
ij k ik kl ik jl ijk l k
k ik kl jl ik ijk k l
k ik kl jl ik ijk k l
k ik kj ik ijk k
k kj ik ijk
x x
x x
x x
x x
x
This is a mixed logit model. The interesting extension is the source of the individual heterogeneity in the random parameters.
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Discrete Choice Modeling
Hybrid Choice Models
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Discrete Choice Modeling
Hybrid Choice Models
“Integrated Model” Incorporate attitude measures in preference
structure
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Discrete Choice Modeling
Hybrid Choice Models
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Discrete Choice Modeling
Hybrid Choice Models
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Discrete Choice Modeling
Hybrid Choice Models
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Discrete Choice Modeling
Hybrid Choice Models
Hybrid choice Equations of the MIMIC Model
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Discrete Choice Modeling
Hybrid Choice Models
Identification Problems
Identification of latent variable models with cross sections
How to distinguish between different latent variable models. How many latent variables are there? More than 0. Less than or equal to the number of indicators.
Parametric point identification
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Discrete Choice Modeling
Hybrid Choice Models
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Discrete Choice Modeling
Hybrid Choice Models
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Discrete Choice Modeling
Hybrid Choice Models
Caution
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Discrete Choice Modeling
Hybrid Choice Models
Swait, J., “A Structural Equation Model of Latent Segmentation and Product Choice for Cross Sectional Revealed Preference Choice Data,” Journal of Retailing and Consumer Services, 1994
Bahamonde-Birke and Ortuzar, J., “On the Variabiity of Hybrid Discrete Choice Models, Transportmetrica, 2012
Vij, A. and J. Walker, “Preference Endogeneity in Discrete Choice Models,” TRB, 2013
Sener, I., M. Pendalaya, R., C. Bhat, “Accommodating Spatial Correlation Across Choice Alternatives in Discrete Choice Models: An Application to Modeling Residential Location Choice Behavior,” Journal of Transport Geography, 2011
Palma, D., Ortuzar, J., G. Casaubon, L. Rizzi, Agosin, E., “Measuring Consumer Preferences Using Hybrid Discrete Choice Models,” 2013
Daly, A., Hess, S., Patruni, B., Potoglu, D., Rohr, C., “Using Ordered Attitudinal Indicators in a Latent Variable Choice Model: A Study of the Impact of Security on Rail Travel Behavior”