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Discrete Choice Modeling

Hybrid Choice Models


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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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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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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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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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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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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‘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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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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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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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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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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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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*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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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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should be kl ikx

Note. Alternative i, Individual j.

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*

*

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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“Integrated Model” Incorporate attitude measures in preference

structure

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Hybrid choice Equations of the MIMIC Model

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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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Caution

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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”

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