Heterogeneity in Hybrid Choice Models

Discrete Choice Modeling Hybrid Choice Models [Part 13] 1/30 Discrete Choice Modeling 0 1 2 3 4 5 6 7 8 9 10 Latent Class 11 Mixed Logit 12 Stated Preference 13 Hybrid Choice Introduction Summary Binary Choice Panel Data Bivariate Probit Ordered Choice Count Data Multinomial Choice Nested Logit Heterogeneity William Greene Stern School of Business New York University

Discrete Choice Modeling Hybrid Choice Models [Part 13] 2/30 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.

Discrete Choice Modeling Hybrid Choice Models [Part 13] 3/30 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

Discrete Choice Modeling Hybrid Choice Models [Part 13] 4/30 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

Discrete Choice Modeling Hybrid Choice Models [Part 13] 5/30 Heterogeneity and the MNL Model exp( + ' x ) j ij P[choice j|i]= J(i) exp( + ' x ) j ij j=1

Discrete Choice Modeling Hybrid Choice Models [Part 13] 6/30 Observable Heterogeneity in Preference Weights Hierarchical model - Interaction terms i = + = + + x z U Parameter heterogeneity is observable. Each parameter = + ij j ij j i ij + h i i k x h i,k k i i j exp( + + z ) j ij i Prob[choi ce j|i]= J i exp( + x + z ) i j ij i j=1

Discrete Choice Modeling Hybrid Choice Models [Part 13] 7/30 Quantifiable Heterogeneity in Scaling i = + + + x z Uij j ij j i ij 2 j 2 1 2 Var[ ]= exp( j w ), = /6 ij i wi = observable characteristics: age, sex, income, etc.

Discrete Choice Modeling Hybrid Choice Models [Part 13] 8/30 Unobserved Heterogeneity in Scaling HEV formulation: U ij ij = + x (1/ ) i ij 0 Generalized model with = 1 and = [ ]. Produces a scaled multinomial logit model with exp( Prob(choice = j) = i i J j = The random variation in the scaling is exp( The variation across individuals may also be observed, so that exp( / 2 i = = i i i x ) ij = = , 1,..., , 1,..., i N j J i i x exp( ) ij 1 = + 2 / 2 ) w i i + + 2 z ) w i i

Discrete Choice Modeling Hybrid Choice Models [Part 13] 9/30 Generalized Mixed Logit Model i + U(i,j) = Random Parameters = [ + = is a lower triangular matrix with 1s on the diagonal (Cholesky matrix) is a diagonal matrix with exp( Overall preference scaling =exp(- /2+ w + =exp( ) 0 < < 1 Common effects + x i,j i,j h ]+[ + (1- )] v i i i i i i i i k h ) i k i 2 i h ] i i i i r i i

Discrete Choice Modeling Hybrid Choice Models [Part 13] 10/30 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

Discrete Choice Modeling Hybrid Choice Models [Part 13] 11/30 Observable Heterogeneity in Utility Levels = + exp( + + + x z Uij j i ij j i ij j 'x + z ) j ij i Prob[choice j|i]= J(i) exp( + 'x + z ) j ij i j=1 Choice, e.g., among brands of cars xitj = attributes: price, features zit = observable characteristics: age, sex, income

Discrete Choice Modeling Hybrid Choice Models [Part 13] 12/30 Unbservable heterogeneity in utility levels and other preference indicators Multinomial Choice Model U ij j i ij j i = + + + x z exp( + Prob[choice j|i]= = + z b w i i i ij j 'x + z ) j ij i J (i) exp( + 'x + z ) t j ij i j=1 Indicators (Measurement) Model(s) Outco mes = f ( ,v ) im y z i m im

Discrete Choice Modeling Hybrid Choice Models [Part 13] 13/30

Discrete Choice Modeling Hybrid Choice Models [Part 13] 14/30

Discrete Choice Modeling Hybrid Choice Models [Part 13] 15/30

Discrete Choice Modeling Hybrid Choice Models [Part 13] 16/30 Observed Latent Observed x z* 1 1 h = = = = = = = = = = + + + * 1 * 2 * 3 h h z u 1 y z u 2 2 2 z u 3 3 3 * 1 ( , ) g z y 1 1 1 * 1 * 1 ( , ) g z y 2 2 1 * 2 , ) ( , y g z z 3 3 1 * 2 * 2 ( , ) y g z 4 4 2 ( , ) y g z 5 5 2 * 3 ( , ) y g z 6 6 3 * 3 ( , ) y g z 7 7 3

Discrete Choice Modeling Hybrid Choice Models [Part 13] 17/30 MIMIC Model Multiple Causes and Multiple Indicators X z* Y y y 1 1 1 2 2 2 = + * * +w z z x ... y ... ... M M M

Discrete Choice Modeling Hybrid Choice Models [Part 13] 18/30 should be ik x kl Note. Alternative i, Individual j.

Discrete Choice Modeling Hybrid Choice Models [Part 13] 19/30 + + U = ij x x k ik kl ik jl ij k l k = + + x x k ik kl jl ik ij k k l = + + x x k ik kl jl ik ij k k l ( ) = + + * kj x x k ik ik ij k k ( ) = + + * kj x k ik ij k This is a mixed logit model. The interesting extension is the source of the individual heterogeneity in the random parameters.

Discrete Choice Modeling Hybrid Choice Models [Part 13] 20/30

Discrete Choice Modeling Hybrid Choice Models [Part 13] 21/30 Integrated Model Incorporate attitude measures in preference structure

Discrete Choice Modeling Hybrid Choice Models [Part 13] 22/30

Discrete Choice Modeling Hybrid Choice Models [Part 13] 23/30

Discrete Choice Modeling Hybrid Choice Models [Part 13] 24/30

Discrete Choice Modeling Hybrid Choice Models [Part 13] 25/30 Hybrid choice Equations of the MIMIC Model

Discrete Choice Modeling Hybrid Choice Models [Part 13] 26/30 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

Discrete Choice Modeling Hybrid Choice Models [Part 13] 27/30

Discrete Choice Modeling Hybrid Choice Models [Part 13] 28/30

Discrete Choice Modeling Hybrid Choice Models [Part 13] 29/30 Caution

Discrete Choice Modeling Hybrid Choice Models [Part 13] 30/30 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