Practical note on specification of discrete choice model Toshiyuki - PowerPoint PPT Presentation

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

Contents • Comparison between binary logit model and binary probit model • Comparison between multinomial logit model and nested logit model • Comparison between nested logit model and mixed logit model 2

Comparison between binary logit model and binary probit model 3

Random utility models • Random utility U jn = V jn + ε jn V jn : deterministic part of utility ε jn : stochastic part of utility • Conventional linear utility function V jn = β X jn X jn : vector of explanatory variables β : vector of coefficients 4

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

Gumbel distribution: G( η , µ ) • Probability density function [ ] ( ) { ( ) } { ( ) } ε = µ − µ ε − η − − µ ε − η f exp exp exp – Mode = η , Mean = η + r/ µ , variance = π 2 /6 µ 2 , where r ≈ 0.577 (Euler’s constant) • Cumulative density function { } ( ) ( )   F ε = − − µ ε − η exp exp   6

Binary logit model • If ε in and ε jn follow G( η i , µ ) and G( η j , µ ) respectively, ε jn − ε in = ε n follows logistic distribution as below 1 ( ) F ε = { } ( ) n + µ η − η − ε 1 exp j i n • Assuming η i = η j = 0, probability to choose i is ( ) ( ) = ε − ε < − = − P Pr V V F V V in jn in in jn in jn ( ) µ exp V 1 = = in { } ( ) ( ) ( ) µ + µ + − µ − exp V exp V 1 exp V V in jn in jn 7

Normal distribution: N(m, σ 2 ) • Probability density function   ε − 2   1 1 m ( ) ε = −     f exp σ πσ     2   2 – Mode = Mean = m, variance = σ 2 • Cumulative density function = ∫ ε ( ) ( ) ε F f e de =−∞ e 8

Binary probit model • ε jn − ε in = ε n is assumed to follow N(0, σ 2 ) where m = 0 ( ) = ε − ε < − P Pr V V in jn in in jn   ε 2   − 1 1 ∫ V V = − ε in jn   n   exp d σ πσ n   ε =−∞   2 2   n −   V V = Φ in jn  σ   If V( ε in ) = V( ε jn ) and COV( ε in , ε jn ) = 0 (i.i.d.), V( ε in ) = V( ε jn ) = σ 2 /2 9

Identifiability of parameters • Binary logit model: ( ) ( ) µ µβ exp V exp X = = in in P ( ) ( ) ( ) ( ) µ + µ µβ + µβ in exp V exp V exp X exp X in jn in jn • Binary probit model: − β − β     β β   V V X X = Φ = Φ = Φ − in jn in jn       P X X σ σ σ σ in in jn       µ and σ are always connected with β Thus, µ and σ cannot be identified 10

Standardization • Binary logit model: µ = 1 V( ε in ) = π 2 /6 ( ) ( ) µ exp V exp V = = in in P ( ) ( ) ( ) ( ) µ + µ + in exp V exp V exp V exp V in jn in jn V( ε in ) = 1 /2 • Binary probit model: σ = 1 −   when V( ε in ) = V( ε jn ) and ( ) V V = Φ = Φ − in jn   COV( ε in , ε jn ) = 0 (i.i.d.) P V V σ in in jn   Estimates of V jn = β X jn have different sizes Also applies when comparing multinomial logit and probit models 11

Comparison between multinomial logit model and nested logit model 12

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

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

Identifiability of parameters { } ( ) µ + exp V V ( ) = dm m dm P d m , { } ∑ ( ) µ + exp V V dm m ' dm ' { } ∈ m ' A R ,   µ   { } ∑ ( ) µ + µ +   exp V ln exp V V µ d dm m dm     { } ∈ m A R , × dm   µ   { } ∑ ∑ ( ) µ + µ +   exp V ln exp V V µ d ' d m ' m d m '     { } { } ∈ ∈ d ' I T , m A R , d m ' • 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

Two ways of standardization V( ε d + ε dm ) ≥ π 2 /6 • µ dm = 1 0 ≤ µ ≤ 1     ∑ ( ) µ + µ +   exp V ln exp V V ( ) + d m dm     exp V V { } ( ) ∈ = × m A R , m dm P d m , ∑ ( ) +     exp V V ∑ ∑ ( ) µ + µ + m ' dm '   exp V ln exp V V { } ∈ m ' A R , d ' m d m '     { } { } ∈ ∈ d ' I T , m A R , V( ε d + ε dm ) = π 2 /6 • µ = 1 1 ≤ µ dm     { } 1 ∑ ( ) + µ +   exp V ln exp V V { } ( ) µ + µ d dm m dm   exp V V   { } ( ) ∈ m A R , = × dm m dm dm P d m , { } ∑ ( ) µ +     exp V V { } 1 ∑ ∑ ( ) + µ + dm m ' dm '   exp V ln exp V V { } ∈ m ' A R , µ d ' d m ' m d m '     { } { } ∈ ∈ d ' I T , m A R , d m ' µ = 1 is recommended to keep the size of β comparable with multinomial logit model 16

Comparison between nested logit model and mixed logit model 17

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

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

Nested logit model • U dm = V d + V m + V dm + ε d + ε dm • Tree structure • ε dm follows G(0, µ dm ) • ε d + ε dm follows G(0, µ ) Ise (I) Takayama (T) Utility function for each alternative 1. U IA = V I + V A + V IA + ε I + ε IA Car (A) Car (A) Train (R) Train (R) 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 Alternative1 2 3 4 4. U TR = V T + V R + V TR + ε T + ε TR {I, A} {I, R} {T, A} {T, R} 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 ) • ε I is common for alt. 1 & 2, so V( ε IA ) = V( ε IR ) Utility function for each alternative • ε T is common for alt. 3 & 4, 1. U IA = V I + V A + V IA + ε I + ε IA so V( ε TA ) = V( ε TR ) 2. U IR = V I + V R + V IR + ε I + ε IR • However, V( ε I ) and V( ε T ) 3. U TA = V T + V A + V TA + ε T + ε TA can be different • It means µ dm and µ d’m can 4. U TR = V T + V R + V TR + ε T + ε TR be different 21

Mixed logit model • U dm = V d + V m + V dm + ε d + ε dm • ε dm follows G(0, 1 ) ( ) + + + ε exp V V V ( ) ε ε = d m dm d P d m , , ∑ ( ) + + + ε I T exp V V V d ' m ' d m ' ' d ' { } ∈ d m ' ' IA IR TA TR , , , • ε d follows N(0, σ d 2 )     ε ε ( ) 1 ∞ ∞ 1 ( ) ∫ ∫ = ε ε φ φ ε ε     I T P d m , P d m , , d d σ σ σ σ I T I T ε =−∞ ε =−∞     I T I I T T Numerical integration is needed for 2 dimensions 22