Binary Choice Matthieu de Lapparent matthieu.delapparent@epfl.ch - PowerPoint PPT Presentation

Binary Choice Matthieu de Lapparent matthieu.delapparent@epfl.ch Transport and Mobility Laboratory, School of Architecture, Civil and Environmental Engineering, Ecole Polytechnique F´ ed´ erale de Lausanne Transport and Mobility Laboratory Binary Choice 1 / 53

Outline Model specification 1 Applying the model 2 Maximum likelihood estimation 3 Estimation output 4 Back to the scale 5 Transport and Mobility Laboratory Binary Choice 2 / 53

Model specification Example Data Unit of analysis: travelers (simulated observations) Choice set: choice of car (C) or transit (T) Independent variable: travel time Ben-Akiva & Lerman (1985) Discrete Choice Analysis: Theory and Applications to Travel Demand, MIT Press (p.88) Transport and Mobility Laboratory Binary Choice 3 / 53

Model specification Example Data from 21 decision makers Time Time Time Time # auto transit Choice # auto transit Choice 1 52.9 4.4 T 11 99.1 8.4 T 2 4.1 28.5 T 12 18.5 84.0 C 3 4.1 86.9 C 13 82.0 38.0 C 4 56.2 31.6 T 14 8.6 1.6 T 5 51.8 20.2 T 15 22.5 74.1 C 6 0.2 91.2 C 16 51.4 83.8 C 7 27.6 79.7 C 17 81.0 19.2 T 8 89.9 2.2 T 18 51.0 85.0 C 9 41.5 24.5 T 19 62.2 90.1 C 10 95.0 43.5 T 20 95.1 22.2 T 21 41.6 91.5 C Transport and Mobility Laboratory Binary Choice 4 / 53

Model specification Binary choice model Specification of utility functions U C = β 1 T C + ε C U T = β 1 T T + ε T where T C is the travel time by car (min) and T T the travel time by transit (min). Choice model P ( C |{ C , T } ) = P ( U C ≥ U T ) = P ( β 1 T C + ε C ≥ β 1 T T + ε T ) = P ( β 1 T C − β 1 T T ≥ ε T − ε C ) = P ( ε ≤ β 1 ( T C − T T )) where ε = ε T − ε C . Transport and Mobility Laboratory Binary Choice 5 / 53

Model specification Error term Three questions about the random variables ε T and ε C 1 What’s their distribution? 2 What’s their moments: Mean? 1 Variance? 2 Note For binary choice it is sufficient to make assumptions about ε = ε T − ε C Transport and Mobility Laboratory Binary Choice 6 / 53

Model specification First-order moment: mean Note Adding the same constant µ to all utility functions does not affect the choice model Pr( U C ≥ U T ) = Pr( U C + µ ≥ U T + µ ) ∀ µ ∈ R . Why? An utility function is defined up to a monotone increasing transformation. Transport and Mobility Laboratory Binary Choice 7 / 53

Model specification First-order moment: mean, cont. Change of variables Assume that E[ ε C ] = β C and E[ ε T ] = β T . Define ε ′ C = ε C − β C and ε ′ T = ε T − β T , so that E[ ε ′ C ] = E[ ε ′ T ] = 0. Choice model P ( C |{ C , T } ) = Pr( β 1 ( T C − T T ) ≥ ε T − ε C ) = ε ′ T + β T − ε ′ Pr( β 1 ( T C − T T ) ≥ C − β C ) = ε ′ T − ε ′ Pr( β 1 ( T C − T T ) + ( β C − β T ) ≥ C ) = ε ′ ) Pr( β 1 ( T C − T T ) + β 0 ≥ where β 0 = β C − β T and ε ′ = ε ′ T − ε ′ C . Transport and Mobility Laboratory Binary Choice 8 / 53

Model specification First-order moment: mean, cont. Mean The mean of ε can be included as a parameter of the deterministic part of utility Only the mean of the difference of the error terms is meaningful Alternative Specific Constant (ASC) U C = β 1 T C + ε T or U C + ε C = β 1 T C − β 0 + ε C U T = β 1 T T + β 0 U T = β 1 T T + ε T In practice, one needs to associate an ASC with all alternatives but one: exclusion constraint to define a one-to-one mapping between vector of parameters and choice probabilities Transport and Mobility Laboratory Binary Choice 9 / 53

Model specification Second-order moment: the variance Utility is ordinal Utilities can be scaled up or down without changing the choice probability Pr( U C ≥ U T ) = Pr( α U C ≥ α U T ) ∀ α > 0 Repeat once more! A utility function is defined up to a monotone increasing transformation. Link with the variance α 2 Var( U C ) Var( α U C ) = α 2 Var( U T ) Var( α U T ) = Variance is not identified As any α can be selected arbitrarily, any variance can be assumed. No way to identify the variance of the error terms from data. Transport and Mobility Laboratory Binary Choice 10 / 53 The scale has to be arbitrarily defined: normalization constraint to

Model specification Practical summary Only difference in levels of utility matters It is not possible to estimate all ASC but only their differences. Choose arbitrarily one of the ASCs as reference and fix it to 0: estimated differences of ASCs are wrt to this reference Scale is arbitrary It means for a linear utility function that the values of the parameters are not sensible. Transport and Mobility Laboratory Binary Choice 11 / 53

Model specification The normal distribution Assumption 1 ε T and ε C are the sum of many r.v. capturing unobservable attributes (e.g. mood, experience), measurement and specification errors. Central-limit theorem The sum of many i.i.d. random variables approximately follows a normal distribution: N ( µ, σ 2 ). Assumed distribution ε C ∼ N (0 , 1) , ε T ∼ N (0 , 1) , ε C ⊥ ε T Transport and Mobility Laboratory Binary Choice 12 / 53

Model specification The normal distribution, cont. 0.4 c * exp(-x*x/2.0) 0.35 Probability density function 0.3 0.25 (pdf): Utility 0.2 0.15 e − ( t − µ )2 1 f ( t ) = √ 0.1 2 σ 2 σ 2 π 0.05 0 -3 -2 -1 0 1 2 3 Time Cumulative distribution function 1 norm(x) 0.9 (CDF) 0.8 0.7 � c 0.6 P ( c ≥ ε ) = F ( c ) = f ( t ) dt Utility 0.5 0.4 −∞ 0.3 0.2 No closed form 0.1 0 -3 -2 -1 0 1 2 3 Time Transport and Mobility Laboratory Binary Choice 13 / 53

Model specification The normal distribution, cont. ε = ε T − ε C From the properties of the normal distribution, we have ε C ∼ N (0 , 1) ε T ∼ N (0 , 1) ε = ε T − ε C ∼ N (0 , 2) As the variance is arbitrary, we may also assume ε C ∼ N (0 , 0 . 5) ε T ∼ N (0 , 0 . 5) ε = ε T − ε C ∼ N (0 , 1) Transport and Mobility Laboratory Binary Choice 14 / 53

Model specification The binary probit model Choice model P ( C |{ C , T } ) = Pr( β 1 ( T C − T T ) + β 0 ≥ ε ) = F ε ( β 1 ( T C − T T ) + β 0 ) The binary probit model � β 1 ( T C − T T ) − β 0 1 e − 1 2 t 2 dt P ( C |{ C , T } ) = √ 2 π −∞ Not a closed form expression Transport and Mobility Laboratory Binary Choice 15 / 53

Model specification The binary probit model The distribution If the error terms are assumed to follow a normal distribution, the corresponding model is called Probability Unit Model or Probit Model. Transport and Mobility Laboratory Binary Choice 16 / 53

Model specification The Gumbel distribution Assumption 2 ε T and ε C are the maximum of many r.v. capturing unobservable attributes (e.g. mood, experience), measurement and specification errors. Gumbel theorem The maximum of many i.i.d. random variables approximately follows an Extreme Value distribution: EV( η, µ ). Assumed distribution ε C ∼ EV(0 , 1) , ε T ∼ EV(0 , 1) , ε C ⊥ ε T Transport and Mobility Laboratory Binary Choice 17 / 53

Model specification The type 1 Extreme Value distribution EV1( η, µ ) Probability density function (pdf) f ( t ) = µ e − µ ( t − η ) e − e − µ ( t − η ) Cumulative distribution function (CDF) � c P ( c ≥ ε ) = F ( c ) = f ( t ) dt −∞ e − e − µ ( c − η ) = Transport and Mobility Laboratory Binary Choice 18 / 53

Model specification The type 1 Extreme Value distribution 0.4 Gumbel PDF mu=0 sigma=1 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 -3 -2 -1 0 1 2 3 1 Gumbel CDF mu=0 sigma=1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 -3 -2 -1 0 1 2 3 Transport and Mobility Laboratory Binary Choice 19 / 53

Model specification The type 1 Extreme Value distribution Properties If ε ∼ EV( η, µ ) then Var[ ε ] = π 2 E[ ε ] = η + γ and 6 µ 2 µ where γ is Euler’s constant. Euler’s constant � ∞ k 1 e − x ln xdx ≈ 0 . 5772 � γ = lim i − ln k = − k →∞ 0 i =1 Transport and Mobility Laboratory Binary Choice 20 / 53

Model specification Difference of independent type 1 Extreme Value distributions ε = ε T − ε C From the properties of the extreme value distribution, we have ε C ∼ EV(0 , 1) ε T ∼ EV(0 , 1) ε ∼ Logistic(0 , 1) Transport and Mobility Laboratory Binary Choice 21 / 53

Model specification The Logistic distribution: Logistic( η , µ ) Probability density function (pdf) µ e − µ ( t − η ) f ( t ) = (1 + e − µ ( t − η ) ) 2 Cumulative distribution function (CDF) � c 1 P ( c ≥ ε ) = F ( c ) = f ( t ) dt = 1 + e − µ ( c − η ) −∞ with µ > 0. Transport and Mobility Laboratory Binary Choice 22 / 53

Model specification The binary logit model Choice model P ( C |{ C , T } ) = Pr( β 1 ( T C − T T ) + β 0 ≥ ε ) = F ε ( β 1 ( T C − T T ) + β 0 ) The binary logit model e β 1 T C + β 0 1 P ( C |{ C , T } ) = 1 + e − ( β 1 ( T C − T T )+ β 0 ) = e β 1 T C + β 0 + e β 1 T T The binary logit model e V C P ( C |{ C , T } ) = e V C + e V T Transport and Mobility Laboratory Binary Choice 23 / 53

Model specification Logit curve P n (i) 1 logit curve for non- limiting cases V in -V jn 0 Transport and Mobility Laboratory Binary Choice 24 / 53