Practical note on specification of discrete choice model. Toshiyuki Yamamoto Nagoya University, Japan

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1 Practical note on specification of discrete choice model Toshiyuki Yamamoto Nagoya University, Japan 1

2 Contents Comparison between bary logit model and bary probit model Comparison between multomial logit model and nested logit model Comparison between nested logit model and mixed logit model 2

3 Comparison between bary logit model and bary probit model 3

4 Random utility models Random utility U jn = V jn + ε jn V jn : determistic part of utility ε jn : stochastic part of utility Conventional lear utility function V jn = β X jn X jn : vector of lanatory variables β : vector of coefficients 4

5 Bary choice models When the choice set contas only two alternatives Probability for dividual n to choose alternative i P = Prob(U > U jn ) = Prob(V + ε > V jn + ε jn ) = Prob(ε jn ε < V V jn ) If ε jn and ε follow normal distribution, ε jn ε also follows normal distribution -> Bary probit model If ε jn and ε follow iid Gumbel distribution, ε jn ε follows logistic distribution -> Bary logit model 5

6 Gumbel distribution: G(η, µ) Probability density function f ( ε ) = µ { µ ( ε η) } [ { µ ( ε η) }] Mode = η, Mean = η + r/µ, variance = π 2 /6µ 2, where r (Euler s constant) Cumulative density function ( ) = ( ) F ε { µ ε η } 6

7 Bary logit model If ε and ε jn follow G(η i, µ) and G(η j, µ) respectively, ε jn ε = ε n follows logistic distribution as below ( ) F ε n 1 = 1 + Assumg η i = η j = 0, probability to choose i is ( ε ε ) ( ) P = Pr < V V = F V V jn jn jn 1 = = 1+ { µ ( V )} Vjn { µη ( )} j ηi εn ( µ V ) ( µ V ) + ( µ V ) jn 7

8 Normal distribution: N(m, σ 2 ) Probability density function ( ε ) 1 1 = 2πσ 2 Mode = Mean = m, variance = σ 2 f ε m σ 2 Cumulative density function F ε ( ε ) ( ) = e= f e de 8

9 Bary probit model ε jn ε = ε n is assumed to follow N(0, σ 2 ) where m = 0 ( ε ε ) P = Pr < V V jn jn V V 1 jn 1 = εn = 2πσ 2 V V jn = Φ σ ε n σ 2 dε n If V(ε ) = V(ε jn ) and COV(ε, ε jn ) = 0 (i.i.d.), V(ε ) = V(ε jn ) = σ 2 /2 9

10 Identifiability of parameters Bary logit model: P ( V ) ( ) ( ) Bary probit model: ( X ) µ µβ = = µ V + µ V µβ X + µβ X ( ) ( ) jn jn V Vjn βx βx jn β β P =Φ =Φ =Φ X X jn σ σ σ σ µ and σ are always connected with β Thus, µ and σ cannot be identified 10

11 Standardization Bary logit model: µ = 1 V(ε ) = π 2 /6 P ( µ V ) ( µ ) ( µ ) ( V ) = = V + V V + V ( ) ( ) jn jn Bary probit model: σ = 1 V V P =Φ =Φ V V σ ( ) jn jn V(ε ) = 1/2 when V(ε ) = V(ε jn ) and COV(ε, ε jn ) = 0 (i.i.d.) Estimates of V jn = β X jn have different sizes Also applies when comparg multomial logit and probit models 11

12 Comparison between multomial logit model and nested logit model 12

13 Multomial logit model P = = J j= 1 J j= 1 j= 1 J µ V ( ) ( µ V ) µβ X jn ( ) ( µβ X ) β X ( ) jn ( β X ) jn where ε follows G(0, µ) µ Is always connected with β Thus, µ cannot be identified standardized by µ = 1 V(ε ) = π 2 /6 13

14 Nested logit model Jot choice of trip destation and mode Destation d = {I, T}, mode m = {A, R} Utility function: U dm = V d + V m + V dm +ε d + ε dm Tree structure Ise (I) Takayama (T) V d : utility specific to destation d V m : utility specific to mode m V dm : utility specific to combation of destation d and mode m (such as travel time) ε d : stochastic utility specific to destation d ε dm : stochastic utility specific to combation of destation d and mode m Car (A) Car (A) Tra (R) Tra (R) Alternative1 {I, A} 2 {I, R} 3 {T, A} 4 {T, R} 14

15 Identifiability of parameters (, ) Pdm { } = { AR} m', { µ dm ( Vm + Vdm )} µ ( V + V ) { dm m' dm' } µ µ Vd + ln { µ dm ( Vm + Vdm )} µ dm m { AR, } µ µ V + ln µ V + V { } { ( )} d' dm ' m dm ' d' IT, µ dm ' m AR, ε dm follows G(0, µ dm ) and ε d + ε dm follows G(0, µ) which means µ µ dm One of µ and µ dm can be identified, and the other should be fixed 15

16 µ dm = 1 (, ) Pdm Two ways of standardization 0 µ 1 V(ε d + ε dm ) π 2 /6 µ Vd + µ ln ( Vm + Vdm ) ( Vm + Vdm ) m { AR, } = ( Vm ' + Vdm' ) m' { AR, } µ Vd' + µ ln ( Vm+ Vdm ' ) d' { IT, } m { AR, } µ = 1 1 µ dm V(ε d + ε dm ) = π 2 /6 (, ) Pdm 1 Vd + ln { ( )} { ( )} µ dm Vm + Vdm µ dm Vm + Vdm µ dm m { AR, } = { µ dm ( Vm ' + Vdm' )} 1 m' { AR, } Vd' + ln { µ dm ' ( Vm+ Vdm ' )} d' { IT, } µ dm ' m { AR, } µ = 1 is recommended to keep the size of β comparable with multomial logit model 16

17 Comparison between nested logit model and mixed logit model 17

18 Stochastic terms of nested logit model and mixed logit model Nested logit model Tree structure U dm = V d + V m + V dm +ε d + ε dm ε dm follows G(0, µ dm ) ε d + ε dm follows G(0, µ) Ise (I) Takayama (T) Mixed logit model Car (A) Car (A) Tra (R) Tra (R) U dm = V d + V m + V dm +ε d + ε dm ε dm follows G(0, µ dm ) ε d follows N(0, σ d2 ) Alternative1 {I, A} 2 {I, R} 3 {T, A} 4 {T, R} 18

19 Stochastic terms of nested logit model Nested logit model and mixed logit model U dm = V d + V m + V dm +ε d + ε dm ε dm follows G(0, µ dm ) ε d + ε dm follows G(0, µ) Mixed logit model U dm = V d + V m + V dm +ε d + ε dm ε dm follows G(0, µ dm ) ε d follows N(0, σ d2 ) Different probability distributions are mixed Distributions other than normal can be used, but normal is often used Standardized by µ dm = 1, V(ε d + ε dm ) = σ d2 + π 2 /6 Size of β becomes different from nested logit model 19

20 Nested logit model U dm = V d + V m + V dm +ε d + ε dm ε dm follows G(0, µ dm ) ε d + ε dm follows G(0, µ) Utility function for each alternative Tree structure Ise (I) Takayama (T) 1. U IA = V I + V A + V IA +ε I + ε IA 2. U IR = V I + V R + V IR +ε I + ε IR Car (A) Car (A) Tra (R) Tra (R) 3. U TA = V T + V A + V TA +ε T + ε TA 4. U TR = V T + V R + V TR +ε T + ε TR Alternative1 {I, A} 2 {I, R} 3 {T, A} 4 {T, R} 20

21 Nested logit model U dm = V d + V m + V dm +ε d + ε dm ε dm follows G(0, µ dm ) ε d + ε dm follows G(0, µ) Utility function for each alternative 1. U IA = V I + V A + V IA +ε I + ε IA 2. U IR = V I + V R + V IR +ε I + ε IR 3. U TA = V T + V A + V TA +ε T + ε TA 4. U TR = V T + V R + V TR +ε T + ε TR ε I is common for alt. 1 & 2, so V(ε IA ) = V(ε IR ) ε T is common for alt. 3 & 4, so V(ε TA ) = V(ε TR ) However, V(ε I ) and V(ε T ) can be different It means µ dm and µ d m can be different 21

22 (, ε, ε ) Pdm I Mixed logit model U dm = V d + V m + V dm +ε d + ε dm ε dm follows G(0, 1) T = ε d follows N(0, σ d2 ) { IA IR TA TR} d ' m',,, ( Vd + Vm + Vdm + ε d ) ( V + V + V + ε ) d' m' dm ' ' d' (, ) (,, ) 1 ε 1 I ε T Pdm = PdmεI εt φ φ dεidε ε T I = εt = σi σi σt σt Numerical tegration is needed for 2 dimensions 22

23 Identifiability of parameters U dm = V d + V m + V dm +ε d + ε dm ε dm follows G(0, 1) ε d follows N(0, σ d2 ) Utility function for each alternative 1. U IA = V I + V A + V IA +ε I + ε IA 2. U IR = V I + V R + V IR +ε I + ε IR 3. U TA = V T + V A + V TA +ε T + ε TA Different from nested logit model, σ I2 and σ 2 T cannot be estimated together Considerg [only difference utility matters], settg ε I = ε I - ε T and ε T = 0 gives the same β 4. U TR = V T + V R + V TR +ε T + ε TR Then, why can both be estimated nested logit model? 23

24 Reference Carrasco, J.A. and Ortuzar, J. de D. (2002) Review and assessment of the nested logit model, Transport Reviews, Vol. 22(2), pp Walker, J.L., Ben-Akiva, M. and Bolduc, D. (2007) Identification of parameters normal error component logit-mixture (NECLM) models, Journal of Applied Econometrics, Vol. 22, pp

Lecture-20: Discrete Choice Modeling-I

Lecture-20: Discrete Choice Modeling-I 1 In Today s Class Introduction to discrete choice models General formulation Binary choice models Specification Model estimation Application Case Study 2 Discrete