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

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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

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Comparison between binary logit model and binary probit model

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Random utility models

• Random utility

Ujn = Vjn + εjn Vjn : deterministic part of utility εjn : stochastic part of utility

• Conventional linear utility function

Vjn = β Xjn Xjn : vector of explanatory variables β : vector of coefficients

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Binary choice models

When the choice set contains only two alternatives

• Probability for individual n to choose alternative i

Pin = Prob(Uin> Ujn) = Prob(Vin + εin > Vjn + εjn) = Prob(εjn− εin < Vin − Vjn)

• If εjn and εin follow normal distribution, εjn− εin

also follows normal distribution -> Binary probit model

• If εjn and εin follow iid Gumbel distribution, εjn−

εin follows logistic distribution -> Binary logit model

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Gumbel distribution: G(η, µ)

• Probability density function

– Mode = η, Mean = η + r/µ, variance = π2/6µ2, where r ≈ 0.577 (Euler’s constant)

• Cumulative density function

( ) ( ) { } ( ) { }

[ ]

η ε µ η ε µ µ ε − − − − − = exp exp exp f

( ) ( )

{ }

exp exp F ε µ ε η   = − − −  

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Binary logit model

• If εin and εjn follow G(ηi, µ) and G(ηj, µ)

respectively, εjn − εin = εn follows logistic distribution as below

• Assuming ηi = ηj = 0, probability to choose i is

( ) ( ) ( )

{ }

( ) ( )

( )

Pr exp 1 exp exp 1 exp

in jn in in jn in jn in in jn in jn

P V V F V V V V V V V ε ε µ µ µ µ = − < − = − = = + + − −

( )

( )

{ }

1 1 exp

n j i n

F ε µ η η ε = + − −

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Normal distribution: N(m, σ2)

• Probability density function

– Mode = Mean = m, variance = σ2

• Cumulative density function

( )

2

1 1 exp 2 2 m f ε ε σ πσ   −   = −          

( ) ( )

e

F f e de

ε

ε

=−∞

= ∫

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Binary probit model

• εjn − εin = εn is assumed to follow N(0, σ2)

where m = 0

( )

2

Pr 1 1 exp 2 2

in jn n

in jn in in jn V V n n in jn

P V V d V V

ε

ε ε ε ε σ πσ σ

− =−∞

= − < −     = −           −   = Φ   

∫

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If V(εin) = V(εjn) and COV(εin, εjn) = 0 (i.i.d.), V(εin) = V(εjn) = σ2/2

Identifiability of parameters

• Binary logit model:
• Binary probit model:

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( ) ( )

( )

( ) ( )

( )

exp exp exp exp exp exp

in in in in jn in jn

V X P V V X X µ µβ µ µ µβ µβ = = + +

in jn in jn in in jn

V V X X P X X β β β β σ σ σ σ − −       = Φ = Φ = Φ −            

µ and σ are always connected with β Thus, µ and σ cannot be identified

Standardization

• Binary logit model: µ = 1
• Binary probit model: σ = 1

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( ) ( )

( )

( ) ( )

( )

exp exp exp exp exp exp

in in in in jn in jn

V V P V V V V µ µ µ = = + +

( )

in jn in in jn

V V P V V σ −   = Φ = Φ −    

V(εin) = π2/6 V(εin) = 1/2

when V(εin) = V(εjn) and COV(εin, εjn) = 0 (i.i.d.)

Estimates of Vjn = β Xjn have different sizes

Also applies when comparing multinomial logit and probit models

Comparison between multinomial logit model and nested logit model

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Multinomial logit model

where εin follows G(0, µ)

( )

( )

( )

( )

( )

( )

1 1 1

exp exp exp exp exp exp

in in J jn j in J jn j in J jn j

V P V X X X X µ µ µβ µβ β β

= = =

= = →

∑ ∑ ∑

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µ Is always connected with β Thus, µ cannot be identified standardized by µ = 1 V(εin) = π2/6

Nested logit model

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• Tree structure

Ise (I) Takayama (T) Car (A) Car (A) Train (R) Train (R) Alternative1 {I, A} 2 {I, R} 3 {T, A} 4 {T, R}

• Joint choice of trip destination and

mode

• Destination d = {I, T}, mode m = {A, R}
• Utility function:

Udm = Vd + Vm + Vdm +εd + εdm

• Vd: utility specific to destination d
• Vm: utility specific to mode m
• Vdm: utility specific to combination of

destination d and mode m (such as travel time)

• εd: stochastic utility specific to

destination d

• εdm: stochastic utility specific to

combination of destination d and mode m

Identifiability of parameters

• εdm follows G(0, µdm) and εd + εdm follows G(0, µ)

which means µ ≤ µdm

• One of µ and µdm can be identified, and the other

should be fixed

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( ) ( )

{ }

( )

{ }

{ }

( )

{ }

{ }

( )

{ }

{ } { }

' ' ' , , ' ' ' ' , , '

exp , exp exp ln exp exp ln exp

dm m dm dm m dm m A R d dm m dm m A R dm d d m m d m d I T m A R d m

V V P d m V V V V V V V V µ µ µ µ µ µ µ µ µ µ

∈ ∈ ∈ ∈

+ = +     + +       ×     + +      

∑ ∑ ∑ ∑

Two ways of standardization

• µdm = 1

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( ) ( ) ( )

{ }

( )

{ }

( )

{ } { }

, ' ' ' , ' ' ' , ,

exp ln exp exp , exp exp ln exp

d m dm m A R m dm m dm m A R d m d m d I T m A R

V V V V V P d m V V V V V µ µ µ µ

∈ ∈ ∈ ∈

    + +   +     = × +     + +      

∑ ∑ ∑ ∑

0 ≤ µ ≤ 1 V(εd + εdm) ≥ π2/6

• µ = 1

1 ≤ µdm V(εd + εdm) = π2/6

( ) ( )

{ }

( )

{ }

{ }

( )

{ }

{ }

( )

{ }

{ } { }

, ' ' ' , ' ' ' ' , , '

1 exp ln exp exp , exp 1 exp ln exp

d dm m dm m A R dm dm m dm dm m dm m A R d d m m d m d I T m A R d m

V V V V V P d m V V V V V µ µ µ µ µ µ

∈ ∈ ∈ ∈

    + +   +     = × +     + +      

∑ ∑ ∑ ∑

µ = 1 is recommended to keep the size of β comparable with multinomial logit model

Comparison between nested logit model and mixed logit model

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Stochastic terms of nested logit model and mixed logit model

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• Tree structure

Ise (I) Takayama (T) Car (A) Car (A) Train (R) Train (R) Alternative1 {I, A} 2 {I, R} 3 {T, A} 4 {T, R}

Nested logit model

• Udm = Vd + Vm + Vdm +εd + εdm
• εdm follows G(0, µdm)
• εd + εdm follows G(0, µ)

Mixed logit model

• Udm = Vd + Vm + Vdm +εd + εdm
• εdm follows G(0, µdm)
• εd follows N(0, σd

2)

Stochastic terms of nested logit model and mixed logit model

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Nested logit model

• Udm = Vd + Vm + Vdm +εd + εdm
• εdm follows G(0, µdm)
• εd + εdm follows G(0, µ)

Mixed logit model

• Udm = Vd + Vm + Vdm +εd + εdm
• εdm follows G(0, µdm)
• εd follows N(0, σd

2)

• Different probability

distributions are mixed

• Distributions other than

normal can be used, but normal is often used

• Standardized by µdm = 1,

V(εd + εdm) = σd

2 + π2/6

• Size of β becomes different

from nested logit model

Nested logit model

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• Tree structure

Ise (I) Takayama (T) Car (A) Car (A) Train (R) Train (R) Alternative1 {I, A} 2 {I, R} 3 {T, A} 4 {T, R}

• Udm = Vd + Vm + Vdm +εd + εdm
• εdm follows G(0, µdm)
• εd + εdm follows G(0, µ)

Utility function for each alternative

• 1. UIA = VI + VA + VIA +εI + εIA
• 2. UIR = VI + VR + VIR +εI + εIR
• 3. UTA = VT + VA + VTA +εT + εTA
• 4. UTR = VT + VR + VTR +εT + εTR

Nested logit model

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• Udm = Vd + Vm + Vdm +εd + εdm
• εdm follows G(0, µdm)
• εd + εdm follows G(0, µ)

Utility function for each alternative

• 1. UIA = VI + VA + VIA +εI + εIA
• 2. UIR = VI + VR + VIR +εI + εIR
• 3. UTA = VT + VA + VTA +εT + εTA
• 4. UTR = VT + VR + VTR +εT + εTR
• εI is common for alt. 1 & 2,

so V(εIA) = V(εIR)

• εT is common for alt. 3 & 4,

so V(εTA) = V(εTR)

• However, V(εI) and V(εT)

can be different

• It means µdm and µd’m can

be different

Mixed logit model

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( )

( ) 1

1 , , ,

I T

I T I T I T I I T T

P d m P d m d d

ε ε

ε ε ε ε φ φ ε ε σ σ σ σ

∞ ∞ =−∞ =−∞

    =        

∫ ∫

( )

( ) ( )

{ }

' ' ' ' ' ' ' , , ,

exp , , exp

d m dm d I T d m d m d d m IA IR TA TR

V V V P d m V V V ε ε ε ε

∈

+ + + = + + +

∑

• Udm = Vd + Vm + Vdm +εd + εdm
• εdm follows G(0, 1)
• εd follows N(0, σd

2)

Numerical integration is needed for 2 dimensions

Identifiability of parameters

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• Different from nested

logit model, σI

2 and σT 2

cannot be estimated together

• Considering [only

difference in utility matters], setting εI’ = εI - εT and εT’ = 0 gives the same β

• Udm = Vd + Vm + Vdm +εd + εdm
• εdm follows G(0, 1)
• εd follows N(0, σd

2)

Utility function for each alternative

• 1. UIA = VI + VA + VIA +εI + εIA
• 2. UIR = VI + VR + VIR +εI + εIR
• 3. UTA = VT + VA + VTA +εT + εTA
• 4. UTR = VT + VR + VTR +εT + εTR

Then, why can both be estimated in nested logit model?

Reference

• Carrasco, J.A. and Ortuzar, J. de D. (2002)

Review and assessment of the nested logit model, Transport Reviews, Vol. 22(2), pp. 197- 218.

• Walker, J.L., Ben-Akiva, M. and Bolduc, D.

(2007) Identification of parameters in normal error component logit-mixture (NECLM) models, Journal of Applied Econometrics, Vol. 22, pp. 1095-1125.

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