Discrete Choice Modeling

Transcription

1 [Part 13] 1/30 0 Introducton 1 Summary 2 Bnary Choce 3 Panel Data 4 Bvarate Probt 5 Ordered Choce 6 Count Data 7 Multnomal Choce 8 Nested Logt 9 Heterogenety 10 Latent Class 11 Mxed Logt 12 Stated Preference 13 Hybrd Choce Wllam Greene Stern School of Busness New York Unversty

2 [Part 13] 2/30 What s a hybrd choce model? Incorporates latent varables n choce model Extends development of dscrete choce model to ncorporate other aspects of preference structure of the chooser Develops endogenety of the preference structure.

3 [Part 13] 3/30 Endogenety "Recent Progress on Endogenety n Choce Modelng" wth Jordan Louvere & Kenneth Tran & Moshe Ben-Akva & Chandra Bhat & Davd Brownstone & Trudy Cameron & Rchard Carson & J. Deshazo & Denzl Febg & Wllam Greene & Davd Hensher & Donald Waldman, Marketng Letters Sprnger, vol. 16(3), pages , December. Narrow vew: U(,j) = b x(,j) + (,j), x(,j) correlated wth (,j) (Berry, Levnsohn, Pakes, brand choce for cars, endogenous prce attrbute.) Implcatons for estmators that assume t s. Broader vew: Sounds lke heterogenety. Preference structure: RUM vs. RRM Heterogenety n choce strategy e.g., omtted attrbute models Heterogenety n taste parameters: locaton and scalng Heterogenety n functonal form: Possbly nonlnear utlty functons

4 [Part 13] 4/30 Heterogenety Narrow vew: Random varaton n margnal utltes and scale RPM, LCM Scalng model Generalzed Mxed model Broader vew: Heterogenety n preference weghts RPM and LCM wth exogenous varables Scalng models wth exogenous varables n varances Looks lke herarchcal models

5 [Part 13] 5/30 Heterogenety and the MNL Model P[choce j ] = exp(α + βx ' ) J() j=1 j exp(α + βx ' ) j j j

6 [Part 13] 6/30 Observable Heterogenety n Preference Weghts Herarchcal model - Interacton terms U j j x j j z j β Each parameter β = β + φh = β + Φh Parameter heterogenety s observable. Prob[choce j ] =,k k k exp(α + βx + γz ) J j=1 j j j exp(α + βx + γz ) j j

7 [Part 13] 7/30 Quantfable Heterogenety n Scalng U j j x j j z j Var[ε ] = σ exp( δw ), σ = π / 6 j j j 1 w = observable characterstcs: age, sex, ncome, etc.

8 [Part 13] 8/30 Unobserved Heterogenety n Scalng HEV formulaton: U x (1/ ) j j j Generalzed model wth = 1 and = [ 0]. Produces a scaled multnomal logt model wth exp( xj ) Prob(choce = j) =, 1,..., N, j 1,..., J J exp( x ) j1 The random varaton n the scalng s 2 exp( / 2 w) 2 exp( / 2 w ) j The varaton across ndvduals may also be observed, so that z

9 [Part 13] 9/30 Generalzed Mxed Logt Model U(, j) = βx Common effects + ε,j,j Random Parameters β = σ [ β + Δh ]+[γ +σ (1- γ)] Γ v Γ = ΛΣ Λ s a lower trangular matrx wth 1s on the dagonal (Cholesky matrx) Σ s a dagonal matrx wth φ exp( ψh ) Overall preference scalng 2 σ = exp(-τ / 2+ τ w + θh ] τ = exp( λr ) 0 < γ < 1 k k

10 [Part 13] 10/30 A helpful way to vew hybrd choce models Addng atttude varables to the choce model In some formulatons, t makes them look lke mxed parameter models Interactons s a less useful way to nterpret

11 [Part 13] 11/30 Observable Heterogenety n Utlty Levels U j j x j j z j Prob[choce j ] = exp(α + β'x + γz ) J() j=1 j j j exp(α + β'x + γz ) j j Choce, e.g., among brands of cars x tj = attrbutes: prce, features z t = observable characterstcs: age, sex, ncome

12 [Part 13] 12/30 Unbservable heterogenety n utlty levels and other preference ndcators Multnomal Choce Model U x z j j j j j Prob[choce j ] = Indcators (Measurement) Model(s) Outcomes y = f ( z,v ) m m bw exp(α + β'x + γz ) J () t j=1 m z j j j exp(α + β'x + γz ) j j

13 [Part 13] 13/30

14 [Part 13] 14/30

15 [Part 13] 15/30

16 [Part 13] 16/30 Observed Latent Observed x z* y z z z h u * h u * h u * y g ( z, ) * y g ( z, ) * y g ( z, z, ) * * y g ( z, ) * y g ( z, ) * y g ( z, ) * y g ( z, ) *

17 [Part 13] 17/30 MIMIC Model Multple Causes and Multple Indcators X z* Y y1 1 1 y * 2 2 * 2 βx +w z z ym M M

18 [Part 13] 18/30 should be kl xk Note. Alternatve, Indvdual j.

19 [Part 13] 19/30 U = j k k kl k jl j k l k k k kl jl k j k k l k k kl jl k j k k l k k x x x x x x x x * k k kj k j k x * k kj k j Ths s a mxed logt model. The nterestng extenson s the source of the ndvdual heterogenety n the random parameters.

20 [Part 13] 20/30

21 [Part 13] 21/30 Integrated Model Incorporate atttude measures n preference structure

22 [Part 13] 22/30

23 [Part 13] 23/30

24 [Part 13] 24/30

25 [Part 13] 25/30 Hybrd choce Equatons of the MIMIC Model

26 [Part 13] 26/30 Identfcaton Problems Identfcaton of latent varable models wth cross sectons How to dstngush between dfferent latent varable models. How many latent varables are there? More than 0. Less than or equal to the number of ndcators. Parametrc pont dentfcaton

27 [Part 13] 27/30

28 [Part 13] 28/30

29 [Part 13] 29/30 Cauton

30 [Part 13] 30/30 Swat, J., A Structural Equaton Model of Latent Segmentaton and Product Choce for Cross Sectonal Revealed Preference Choce Data, Journal of Retalng and Consumer Servces, 1994 Bahamonde-Brke and Ortuzar, J., On the Varabty of Hybrd Dscrete Choce Models, Transportmetrca, 2012 Vj, A. and J. Walker, Preference Endogenety n Dscrete Choce Models, TRB, 2013 Sener, I., M. Pendalaya, R., C. Bhat, Accommodatng Spatal Correlaton Across Choce Alternatves n Dscrete Choce Models: An Applcaton to Modelng Resdental Locaton Choce Behavor, Journal of Transport Geography, 2011 Palma, D., Ortuzar, J., G. Casaubon, L. Rzz, Agosn, E., Measurng Consumer Preferences Usng Hybrd Dscrete Choce Models, 2013 Daly, A., Hess, S., Patrun, B., Potoglu, D., Rohr, C., Usng Ordered Atttudnal Indcators n a Latent Varable Choce Model: A Study of the Impact of Securty on Ral Travel Behavor