1. William Greene Stern School of Business New York University
1 Introduction
2 Binary Choice
3 Panel Data
4 Ordered Choice
5 Multinomial Choice
6 Heterogeneity
7 Latent Class
8 Mixed Logit
9 Stated Preference
William Greene
Stern School of Business
New York University

2. Extended Formulation of the MNL
Sets of similar alternatives Compound Utility: U(Alt)=U(Alt|Branch)+U(branch) Behavioral implications – Correlations within branches
LIMB
Travel
BRANCH
Private
Public
TWIG
Air
Car
Train
Bus

3. Correlation Structure for a Two Level Model
Within a branch
Identical variances (IIA (MNL) applies)
Covariance (all same) = variance at higher level
Branches have different variances (scale factors)
Nested logit probabilities: Generalized Extreme Value
Prob[Alt,Branch] = Prob(branch) * Prob(Alt|Branch)

4. Probabilities for a Nested Logit Model

5. Model Form RU1

6. Moving Scaling Down to the Twig Level

7. Higher Level Trees E.g., Location (Neighborhood)
Housing Type (Rent, Buy, House, Apt)
Housing (# Bedrooms)

8. Estimation Strategy for Nested Logit Models
Two step estimation (ca. 1980s)
For each branch, just fit MNL
Loses efficiency – replicates coefficients
For branch level, fit separate model, just including y and the inclusive values in the branch level utility function
Again loses efficiency
Full information ML (current) Fit the entire model at once, imposing all restrictions

9. -----------------------------------------------------------
Discrete choice (multinomial logit) model
Dependent variable Choice
Log likelihood function
Estimation based on N = , K = 10
R2=1-LogL/LogL* Log-L fncn R-sqrd R2Adj
Constants only
Chi-squared[ 7] =
Prob [ chi squared > value ] =
Response data are given as ind. choices
Number of obs.= 210, skipped 0 obs
Variable| Coefficient Standard Error b/St.Er. P[|Z|>z]
GC| ***
TTME| ***
INVT| ***
INVC| ***
A_AIR| ***
AIR_HIN1|
A_TRAIN| ***
TRA_HIN3| ***
A_BUS| ***
BUS_HIN4|
MNL Baseline

10. FIML Parameter Estimates
FIML Nested Multinomial Logit Model
Dependent variable MODE
Log likelihood function
The model has 2 levels.
Random Utility Form 1:IVparms = LMDAb|l
Number of obs.= 210, skipped 0 obs
Variable| Coefficient Standard Error b/St.Er. P[|Z|>z]
|Attributes in the Utility Functions (beta)
GC| ***
TTME| ***
INVT| ***
INVC| ***
A_AIR| **
AIR_HIN1|
A_TRAIN| ***
TRA_HIN3| ***
A_BUS| ***
BUS_HIN4|
|IV parameters, lambda(b|l),gamma(l)
PRIVATE| ***
PUBLIC| ***
FIML Parameter Estimates

11. Elasticities Decompose Additively

12. +-----------------------------------------------------------------------+
| Elasticity averaged over observations |
| Attribute is INVC in choice AIR |
| Decomposition of Effect if Nest Total Effect|
| Trunk Limb Branch Choice Mean St.Dev|
| Branch=PRIVATE |
| * Choice=AIR |
| Choice=CAR |
| Branch=PUBLIC |
| Choice=TRAIN |
| Choice=BUS |
| Attribute is INVC in choice CAR |
| Choice=AIR |
| * Choice=CAR |
| Choice=TRAIN |
| Choice=BUS |
| Attribute is INVC in choice TRAIN |
| Choice=AIR |
| Choice=CAR |
| * Choice=TRAIN |
| Choice=BUS |
| * indicates direct Elasticity effect of the attribute |

13. Testing vs. the MNL Log likelihood for the NL model
Constrain IV parameters to equal 1 with ; IVSET(list of branches)=[1]
Use likelihood ratio test
For the example:
LogL (NL) =
LogL (MNL) =
Chi-squared with 2 d.f. = 2( ( )) =
The critical value is 5.99 (95%)
The MNL (and a fortiori, IIA) is rejected

14. Degenerate Branches LIMB Travel BRANCH Fly Ground TWIG Air Train Car
Bus

15. NL Model with a Degenerate Branch
FIML Nested Multinomial Logit Model
Dependent variable MODE
Log likelihood function
Variable| Coefficient Standard Error b/St.Er. P[|Z|>z]
|Attributes in the Utility Functions (beta)
GC| ***
TTME| ***
INVT| ***
INVC| ***
A_AIR| ***
AIR_HIN1|
A_TRAIN| ***
TRA_HIN2| ***
A_BUS| ***
BUS_HIN3|
|IV parameters, lambda(b|l),gamma(l)
FLY| ***
GROUND| ***

16. Using Degenerate Branches to Reveal Scaling

17. Heterogeneity and the MNL Model
Limitations of the MNL Model:
IID  IIA
Fundamental tastes are the same across all individuals
How to adjust the model to allow variation across individuals?
Full random variation
Latent grouping – allow some variation

18. Accommodating Heterogeneity
Observed? Enter in the model in familiar (and unfamiliar) ways.
Unobserved? Takes the form of randomness in the model.

19. Observable Heterogeneity in Utility Levels
Choice, e.g., among brands of cars
xitj = attributes: price, features
zit = observable characteristics: age, sex, income

20. Observable Heterogeneity in Preference Weights

21. Modeling Unobserved Heterogeneity
Latent class – Discrete approximation
Mixed logit – Continuous
Many extensions and blends of LC and RP

22. Latent Class Models

24. Latent Classes
Population contains a mixture of individuals of different types
Common form of the generating mechanism within the classes
Observed outcome y is governed by the common process F(y|x,j )
Classes are distinguished by the parameters, j.

25. The “Finite Mixture Model”
An unknown parametric model governs an outcome y
F(y|x,)
This is the model
We approximate F(y|x,) with a weighted sum of specified (e.g., normal) densities:
F(y|x,)  j j G(y|x,)
This is a search for functional form. With a sufficient number of (normal) components, we can approximate any density to any desired degree of accuracy. (McLachlan and Peel (2000))
There is no “mixing” process at work

26. Density. Note significant mass below zero
Density? Note significant mass below zero. Not a gamma or lognormal or any other familiar density.

27. ML Mixture of Two Normal Densities

28. Mixing probabilities .715 and .285

29. The actual process is a mix of chi squared(5) and normal(3,2) with mixing probabilities .7 and .3.

30. Approximation
Actual Distribution

31. The Latent Class “Model”
Parametric Model:
F(y|x,)
E.g., y ~ N[x, 2], y ~ Poisson[=exp(x)], etc.
Density F(y|x,)  j j F(y|x,j ),  = [1, 2,…, J, 1, 2,…, J]
j j = 1
Generating mechanism for an individual drawn at random from the mixed population is F(y|x,).
Class probabilities relate to a stable process governing the mixture of types in the population

34. RANDOM Parameter Models

35. A Recast Random Effects Model

36. A Computable Log Likelihood

37. Simulation

38. Random Effects Model: Simulation
Random Coefficients Probit Model
Dependent variable DOCTOR (Quadrature Based)
Log likelihood function ( )
Restricted log likelihood
Chi squared [ 1 d.f.]
Simulation based on 50 Halton draws
Variable| Coefficient Standard Error b/St.Er. P[|Z|>z]
|Nonrandom parameters
AGE| *** ( )
EDUC| *** ( )
HHNINC| ( )
|Means for random parameters
Constant| ** ( )
|Scale parameters for dists. of random parameters
Constant| ***
Implied  from these estimates is /( ) =

39. The Entire Parameter Vector is Random

40. Maximum Simulated Likelihood
True log likelihood
Simulated log likelihood

42. S M

43. MSSM

44. A Hierarchical Probit Model
Uit = 1i + 2iAgeit + 3iEducit + 4iIncomeit + it.
1i=1+11 Femalei + 12 Marriedi + u1i
2i=2+21 Femalei + 22 Marriedi + u2i
3i=3+31 Femalei + 32 Marriedi + u3i
4i=4+41 Femalei + 42 Marriedi + u4i
Yit = 1[Uit > 0]
All random variables normally distributed.

46. Simulating Conditional Means for Individual Parameters
Posterior estimates of E[parameters(i) | Data(i)]

50. “Individual Coefficients”

53. Generalized Mixed Logit Model

54. Scaled MNL

55. Observed and Unobserved Heterogeneity

56. Price Elasticities