Nested Logit Models

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. 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 Vbusas 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 probabilitiesfor 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

27. Swissmetro example (2) • NL: model 1

28. Swissmetro example (3) • NL: model 2

29. NL in Biogeme • Specify nesting structure [NLNests] // Name paramvalue LowerBound UpperBound status list of alt Existing 1.0 1 10 0 1 3 Future 1.0 1 10 1 2 • Select model [Model] $NL • Nesting constraints [ConstraintNestCoef] // (CarNest = BusNest)