1. Nested Logit Models

2. GEV
GEV models have the advantage that the choice probabilities usually take a closed form

3. The most widely used member of the GEV family is the Nested Logit

8. This covariance violates the assumption underlying the MNL model
The Problem
The common error components creates a covariance between the total error for bus and LRT
This covariance violates the assumption underlying the MNL model

9. The choice probabilities for alternative

12. Decomposing into two Logits

15. Nested logit model Group similar alternatives in nests
Two-level choice:
Choice of nest
Choice of alternative within nest

16. Lower level model Conditional probability
Choice between bus alternatives
Conditional on choice of the nest

17. Upper level model Choice between car and bus
represents both bus alternatives
Nest systematic utility
Expected value of maximum utility
Define Vbus as the expected maximum utility of red bus and blue bus

18. Expected maximum utility
For i.i.d Gumbel errors
Inclusive value
Where µb is the scale parameter for the MNL associated with the choice between red bus and blue bus

19. Choice probabilities

20. Choice probabilities for µb = 1, µ is normalized to 1

21. Variance-covariance structure
MNL NL

22. Simultaneous estimation
Sequential estimation:
Estimation of NL decomposed into two estimations of MNL
Estimator is consistent but not efficient
Simultaneous estimation:
Log-likelihood function is generally non concave
No guarantee of global maximum
Estimator asymptotically efficient

23. Simultaneous estimation

24. Example: Mode Choice (Correlated Alternatives)

25. Tree Representation of Nested Logit

26. Swissmetro example MNL Parameter Value t-Stat Constant_Train -0.668
-18.7
Constant_SM
-0.52
b_TravelTime
-21.7
b_TravelCost
-30.0

27. Swissmetro example (2) NL: model 1 Parameter Value t-Stat
Constant_Train
-0.557
-5.8
Constant_SM
0.26
b_TravelTime
-21.3
b_TravelCost
-26.2
Sigma
1.082
1.13 (vs. 1)

28. Swissmetro example (3) NL: model 2 Parameter Value t-Stat
Constant_Train
-0.372
-18.7
Constant_SM
0.05
b_TravelTime
-20.0
b_TravelCost
-22.5
Sigma
2.051
11.1 (vs. 1)

29. NL in Biogeme Specify nesting structure Select model
[NLNests]
// Name paramvalue LowerBound UpperBound status list of alt
Existing
Future
Select model
[Model]
$NL
Nesting constraints
[ConstraintNestCoef]
// (CarNest = BusNest)