1/122: Topic 3.3 – Discrete Choice; The Multinomial Logit Model
Microeconometric Modeling
William Greene
Stern School of Business
New York University
New York NY USA
3.3 Discrete Choice; The
Multinomial Logit
Model
2/122: Topic 3.3 – Discrete Choice; The Multinomial Logit Model
Concepts
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•
•
•
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•
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•
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•
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Random Utility
Multinomial Choice
Attributes and Characteristics
Stated and Revealed Preference
Unlabeled Choice
IID and IIA
Consumer Surplus
Choice Based Sampling
Alternative Specific Constants
Partial Effects and Elasticities
Simulation
WTP Space
Hausman – McFadden Test for IIA
Random Regret
Minimum Distance Estimator
Scaled MNL Model
Nonlinear Utility Functions
Models
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Multinomial Logit Model
Conditional Logit
Fixed Effects Multinomial Logit
Best/Worst
Nested Logit
Exploded Logit
Heteroscedastic Extreme Value (HEV)
Generalized Mixed Logit
3/122: Topic 3.3 – Discrete Choice; The Multinomial Logit Model
A Microeconomics Platform



Consumers Maximize Utility (!!!)
Fundamental Choice Problem: Maximize U(x1,x2,…)
subject to prices and budget constraints
A Crucial Result for the Classical Problem:


Indirect Utility Function: V = V(p,I)
Demand System of Continuous Choices
*
j
x =

V(p,I)/p j
V(p,I)/I
Observed data usually consist of choices, prices, income
The Integrability Problem: Utility is not revealed by
demands
4/122: Topic 3.3 – Discrete Choice; The Multinomial Logit Model
Implications for Discrete Choice Models


Theory is silent about discrete choices
Translation of utilities to discrete choice requires:





Consumers often act to simplify choice situations
This allows us to build “models.”



Well defined utility indexes: Completeness of rankings
Rationality: Utility maximization
Axioms of revealed preferences
What common elements can be assumed?
How can we account for heterogeneity?
However, revealed choices do not reveal utility, only rankings
which are scale invariant.
5/122: Topic 3.3 – Discrete Choice; The Multinomial Logit Model
Multinomial Choice
Among J Alternatives
• Random Utility Basis
Uitj = ij + i’xitj + ijzit + ijt
i = 1,…,N; j = 1,…,J(i,t); t = 1,…,T(i)
N individuals studied, J(i,t) alternatives in the choice
set, T(i) [usually 1] choice situations examined.
• Maximum Utility Assumption
Individual i will Choose alternative j in choice setting t if and only if
Uitj > Uitk for all k  j.
• Underlying assumptions


Smoothness of utilities
Axioms of utility maximization: Transitive, Complete, Monotonic
6/122: Topic 3.3 – Discrete Choice; The Multinomial Logit Model
Features of Utility Functions


The linearity assumption Uitj = ij + ixitj + jzit + ijt
To be relaxed later:
Uitj = V(xitj,zit,i) + ijt
The choice set:




Individual (i) and situation (t) specific
Unordered alternatives j = 1,…,J(i,t)
Deterministic (x,z,j) and random components (ij,i,ijt)
Attributes of choices, xitj and characteristics of the chooser, zit.




Alternative specific constants ij may vary by individual
Preference weights, i may vary by individual
Individual components, j typically vary by choice, not by person
Scaling parameters, σij = Var[εijt], subject to much modeling
7/122: Topic 3.3 – Discrete Choice; The Multinomial Logit Model
Unordered Choices of 210 Travelers
8/122: Topic 3.3 – Discrete Choice; The Multinomial Logit Model
Data on Multinomial Discrete Choices
9/122: Topic 3.3 – Discrete Choice; The Multinomial Logit Model
Each person makes four choices
from a choice set that includes either
two or four alternatives.
The first choice is the RP between
two of the RP alternatives
The second-fourth are the SP among
four of the six SP alternatives.
There are ten alternatives in total.
A Stated Choice Experiment with Variable Choice Sets
10/122: Topic 3.3 – Discrete Choice; The Multinomial Logit Model
Stated Choice Experiment: Unlabeled Alternatives, One Observation
t=1
t=2
t=3
t=4
t=5
t=6
t=7
t=8
11/122: Topic 3.3 – Discrete Choice; The Multinomial Logit Model
Unlabeled Choice Experiments
This an unlabelled choice experiment: Compare
Choice = (Air, Train, Bus, Car)
To
Choice = (Brand 1, Brand 2, Brand 3, None)
Brand 1 is only Brand 1 because it is first in
the list.
What does it mean to substitute Brand 1 for
Brand 2?
What does the own elasticity for Brand 1 mean?
12/122: Topic 3.3 – Discrete Choice; The Multinomial Logit Model
The Multinomial Logit (MNL) Model

Independent extreme value (Gumbel):





F(itj) = Exp(-Exp(-itj)) (random part of each utility)
Independence across utility functions
Identical variances (means absorbed in constants)
Same parameters for all individuals (temporary)
Implied probabilities for observed outcomes
P[choice = j | xitj , zit ,i,t] = Prob[Ui,t,j  Ui,t,k ], k = 1,...,J(i,t)
=
exp(α j + β'xitj + γ j'zit )

J(i,t)
j=1
exp(α j + β'xitj + γ j'zit )
13/122: Topic 3.3 – Discrete Choice; The Multinomial Logit Model
Multinomial Choice Models
Multinomial logit model depends on characteristics
P[choice = j | zit ,i,t] =
exp(α j + γ j'zit )

J(i,t)
j=1
exp(α j + γ j'zit )
Conditional logit model depends on attributes
P[choice = j | x itj,i,t] =
exp(α j + β'x itj )

J(i,t)
j=1
exp(α j + β'x itj )
THE multinomial logit model accommodates both.
P[choice = j | x itj, zit ,i,t] =
exp(α j + β'x itj + γ j'zit )

J(i,t)
j=1
exp(α j + β'x itj + γ j'z it )
There is no meaningful distinction.
14/122: Topic 3.3 – Discrete Choice; The Multinomial Logit Model
Specifying the Probabilities
• Choice specific attributes (X) vary by choices, multiply by generic
coefficients. E.g., TTME=terminal time, GC=generalized cost of travel mode
• Generic characteristics (Income, constants) must be interacted with
choice specific constants.
• Estimation by maximum likelihood; dij = 1 if person i chooses j
P[choice = j | x itj , zit ,i,t] = Prob[Ui,t,j  Ui,t,k ], k = 1,...,J(i,t)
=
exp(α j + β'x itj + γ j'zit )

J(i,t)
j=1
logL =  i=1
N
exp(α j + β'x itj + γ j'zit )

J(i)
j=1
dijlogPij
15/122: Topic 3.3 – Discrete Choice; The Multinomial Logit Model
Using the Model to Measure Consumer
Surplus
Maximum j (Uj )
Consumer Surplus =
Marginal Utility of Income
Utility and marginal utility are not observable
For the multinomial logit model (only),
1
E[CS] =
log
MUI

J(i,t)
j=1

exp(α j + β'x itj + γ j'z it ) + C
Where Uj = the utility of the indicated alternative and C
is the constant of integration.
The log sum is the "inclusive value."
16/122: Topic 3.3 – Discrete Choice; The Multinomial Logit Model
Measuring the Change in Consumer Surplus

1
E[CS | Scenario B] =
log  
MU
E[CS | Scenario A] =
1
log
MUI
I
J(i,t)
j=1
J(i,t)
j=1

+ γ 'z ) | B  + C
exp(α j + β'x itj + γ j'zit ) | A + C
exp(α j + β'x itj
j
it
MUI and the constant of integration do not change under scenarios.
Change in expected consumer surplus from a policy (scenario) change
E[CS | Scenario A] - E[CS | Scenario B]
J(i,t)

exp(α j + β'xitj + γ j'zit ) | A 

1 
j=1
=
log J(i,t)
MUI 
 j=1 exp(α j + β'xitj + γ j'zit ) | B 

17/122: Topic 3.3 – Discrete Choice; The Multinomial Logit Model
Willingness to Pay
Generally a ratio of coefficients
WTP =
β(Attribute Level)
β(Income)
Use negative of cost coefficient as a proxu for MU of income
WTP =
negative β(Attribute Level)
β(cost)
Measurable using model parameters
Ratios of possibly random parameters can produce wild and
unreasonable values. We will consider a different approach later.
18/122: Topic 3.3 – Discrete Choice; The Multinomial Logit Model
Observed Data

Types of Data





Attributes and Characteristics



Individual choice
Market shares – consumer markets
Frequencies – vote counts
Ranks – contests, preference rankings
Attributes are features of the choices such as price
Characteristics are features of the chooser such as age, gender and
income.
Choice Settings


Cross section
Repeated measurement (panel data)


Stated choice experiments
Repeated observations – THE scanner data on consumer choices
19/122: Topic 3.3 – Discrete Choice; The Multinomial Logit Model
Choice Based Sampling



Over/Underrepresenting alternatives in the data set
Choice
Air
Train
Bus
Car
True
0.14
0.13
0.09
0.64
Sample
0.28
0.30
0.14
0.28
May cause biases in parameter estimates. (Possibly constants only)
Certainly causes biases in estimated variances


Weighted log likelihood, weight = j / Fj for all i.
Fixup of covariance matrix – use “sandwich” estimator. Using weighted
Hessian and weighted BHHH in the center of the sandwich
20/122: Topic 3.3 – Discrete Choice; The Multinomial Logit Model
An Estimated MNL Model
----------------------------------------------------------Discrete choice (multinomial logit) model
Dependent variable
Choice
Log likelihood function
-199.97662
Estimation based on N =
210, K =
5
Information Criteria: Normalization=1/N
Normalized
Unnormalized
AIC
1.95216
409.95325
Fin.Smpl.AIC
1.95356
410.24736
Bayes IC
2.03185
426.68878
Hannan Quinn
1.98438
416.71880
R2=1-LogL/LogL* Log-L fncn R-sqrd R2Adj
Constants only
-283.7588 .2953 .2896
Chi-squared[ 2]
=
167.56429
Prob [ chi squared > value ] =
.00000
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|
-.01578***
.00438
-3.601
.0003
TTME|
-.09709***
.01044
-9.304
.0000
A_AIR|
5.77636***
.65592
8.807
.0000
A_TRAIN|
3.92300***
.44199
8.876
.0000
A_BUS|
3.21073***
.44965
7.140
.0000
--------+--------------------------------------------------
21/122: Topic 3.3 – Discrete Choice; The Multinomial Logit Model
Estimated MNL Model
----------------------------------------------------------Discrete choice (multinomial logit) model
Dependent variable
Choice
Log likelihood function
-199.97662
Estimation based on N =
210, K =
5
Information Criteria: Normalization=1/N
Normalized
Unnormalized
AIC
1.95216
409.95325
Fin.Smpl.AIC
1.95356
410.24736
Bayes IC
2.03185
426.68878
Hannan Quinn
1.98438
416.71880
R2=1-LogL/LogL* Log-L fncn R-sqrd R2Adj
Constants only
-283.7588 .2953 .2896
Chi-squared[ 2]
=
167.56429
Prob [ chi squared > value ] =
.00000
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|
-.01578***
.00438
-3.601
.0003
TTME|
-.09709***
.01044
-9.304
.0000
A_AIR|
5.77636***
.65592
8.807
.0000
A_TRAIN|
3.92300***
.44199
8.876
.0000
A_BUS|
3.21073***
.44965
7.140
.0000
--------+--------------------------------------------------
22/122: Topic 3.3 – Discrete Choice; The Multinomial Logit Model
Model Fit Based on Log Likelihood

Three sets of predicted probabilities





No model:
Pij = 1/J (.25)
Constants only: Pij = (1/N)i dij
(58,63,30,59)/210=.286,.300,.143,.281
Constants only model matches sample shares
Estimated model: Logit probabilities
Compute log likelihood
Measure improvement in log likelihood with
Pseudo R-squared = 1 – LogL/LogL0
(“Adjusted” for number of parameters in the
model.)
23/122: Topic 3.3 – Discrete Choice; The Multinomial Logit Model
Fit the Model with Only ASCs
If the choice set varies across observations, this is
the only way to obtain the restricted log likelihood.
----------------------------------------------------------Discrete choice (multinomial logit) model
Dependent variable
Choice
If the choice set is fixed at
Log likelihood function
-283.75877
Estimation based on N =
210, K =
3
 Nj 
J
Information Criteria: Normalization=1/N
logL =
N log 

j 1 l
Normalized
Unnormalized
N


AIC
2.73104
573.51754
J
Fin.Smpl.AIC
2.73159
573.63404

N log Pj
Bayes IC
2.77885
583.55886
j 1 l
Hannan Quinn
2.75037
577.57687
R2=1-LogL/LogL* Log-L fncn R-sqrd R2Adj
Constants only
-283.7588 .0000-.0048
Response data are given as ind. choices
Number of obs.=
210, skipped
0 obs
--------+-------------------------------------------------Variable| Coefficient
Standard Error b/St.Er. P[|Z|>z]
--------+-------------------------------------------------A_AIR|
-.01709
.18491
-.092
.9263
A_TRAIN|
.06560
.18117
.362
.7173
A_BUS|
-.67634***
.22424
-3.016
.0026
--------+--------------------------------------------------


J, then
24/122: Topic 3.3 – Discrete Choice; The Multinomial Logit Model
Estimated MNL Model
----------------------------------------------------------Discrete choice (multinomial logit) model
Dependent variable
Choice
Log likelihood function
-199.97662
Estimation based on N =
210, K =
5
Information Criteria: Normalization=1/N
log L
Pseudo R 2 = 1.
Normalized
Unnormalized
log L0
AIC
1.95216
409.95325
Fin.Smpl.AIC
1.95356
410.24736

Adjusted Pseudo R 2 =1- 
Bayes IC
2.03185
426.68878

Hannan Quinn
1.98438
416.71880
R2=1-LogL/LogL* Log-L fncn R-sqrd R2Adj
Constants only
-283.7588 .2953 .2896
Chi-squared[ 2]
=
167.56429
Prob [ chi squared > value ] =
.00000
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|
-.01578***
.00438
-3.601
.0003
TTME|
-.09709***
.01044
-9.304
.0000
A_AIR|
5.77636***
.65592
8.807
.0000
A_TRAIN|
3.92300***
.44199
8.876
.0000
A_BUS|
3.21073***
.44965
7.140
.0000
--------+--------------------------------------------------
N(J-1)   log L 
.

N(J-1)-K   log L0 
25/122: Topic 3.3 – Discrete Choice; The Multinomial Logit Model
Model Fit Based on Predictions



Nj = actual number of choosers of “j.”
Nfitj = i Predicted Probabilities for “j”
Cross tabulate:
Predicted vs. Actual, cell prediction is cell probability
Predicted vs. Actual, cell prediction is the cell
with the largest probability
Njk = i dij  Predicted P(i,k)
26/122: Topic 3.3 – Discrete Choice; The Multinomial Logit Model
Fit Measures Based on Crosstabulation
AIR
TRAIN
BUS
CAR
Total
AIR
TRAIN
BUS
CAR
Total
+-------------------------------------------------------+
| Cross tabulation of actual choice vs. predicted P(j) |
| Row indicator is actual, column is predicted.
|
| Predicted total is F(k,j,i)=Sum(i=1,...,N) P(k,j,i). |
| Column totals may be subject to rounding error.
|
+-------------------------------------------------------+
NLOGIT Cross Tabulation for 4 outcome Multinomial Choice Model
AIR
TRAIN
BUS
CAR
Total
+-------------+-------------+-------------+-------------+-------------+
|
32
|
8
|
5
|
13
|
58
|
|
8
|
37
|
5
|
14
|
63
|
|
3
|
5
|
15
|
6
|
30
|
|
15
|
13
|
6
|
26
|
59
|
+-------------+-------------+-------------+-------------+-------------+
|
58
|
63
|
30
|
59
|
210
|
+-------------+-------------+-------------+-------------+-------------+
NLOGIT Cross Tabulation for 4 outcome Constants Only Choice Model
AIR
TRAIN
BUS
CAR
Total
+-------------+-------------+-------------+-------------+-------------+
|
16
|
17
|
8
|
16
|
58
|
|
17
|
19
|
9
|
18
|
63
|
|
8
|
9
|
4
|
8
|
30
|
|
16
|
18
|
8
|
17
|
59
|
+-------------+-------------+-------------+-------------+-------------+
|
58
|
63
|
30
|
59
|
210
|
+-------------+-------------+-------------+-------------+-------------+
27/122: Topic 3.3 – Discrete Choice; The Multinomial Logit Model
Partial effects :
Change in attribute "k" of alternative "m" on the
probability that the individual makes choice "j"
j = Train
Prob(j) Pj
=
= Pj [1(j = m) - Pm ]βk
xm,k
xm,k
m = Car
k = Price
28/122: Topic 3.3 – Discrete Choice; The Multinomial Logit Model
Partial effects :
k = Price
Own effects :
j = Train
Prob(j) Pj
=
= Pj [1- Pj ]βk
x j,k
x j,k
Cross effects : j = Train
m = Car
Prob(j) Pj
=
= -PP
j m βk
x m,k
x m,k
29/122: Topic 3.3 – Discrete Choice; The Multinomial Logit Model
Elasticities for proportional changes :
xm,k
logProb(j) logPj
=
=
Pj [1(j = m) - Pm ]βk
logxm,k
logx m,k
Pj
= [1(j = m) - Pm ] x m,k βk
Note the elasticity is the same for all j. This is a
consequence of the IIA assumption in the model
specification made at the outset.
30/122: Topic 3.3 – Discrete Choice; The Multinomial Logit Model
+---------------------------------------------------+
| Elasticity
averaged over observations.|
| Attribute is INVT
in choice AIR
|
|
Mean
St.Dev
|
| *
Choice=AIR
-.2055
.0666
|
|
Choice=TRAIN
.0903
.0681
|
|
Choice=BUS
.0903
.0681
|
|
Choice=CAR
.0903
.0681
|
+---------------------------------------------------+
| Attribute is INVT
in choice TRAIN
|
|
Choice=AIR
.3568
.1231
|
| *
Choice=TRAIN
-.9892
.5217
|
|
Choice=BUS
.3568
.1231
|
|
Choice=CAR
.3568
.1231
|
+---------------------------------------------------+
| Attribute is INVT
in choice BUS
|
|
Choice=AIR
.1889
.0743
|
|
Choice=TRAIN
.1889
.0743
|
| *
Choice=BUS
-1.2040
.4803
|
|
Choice=CAR
.1889
.0743
|
+---------------------------------------------------+
| Attribute is INVT
in choice CAR
|
|
Choice=AIR
.3174
.1195
|
|
Choice=TRAIN
.3174
.1195
|
|
Choice=BUS
.3174
.1195
|
| *
Choice=CAR
-.9510
.5504
|
+---------------------------------------------------+
| Effects on probabilities of all choices in model: |
| * = Direct Elasticity effect of the attribute.
|
+---------------------------------------------------+
Note the effect of IIA on
the cross effects.
Own effect
Cross effects
Elasticities are computed
for each observation; the
mean and standard
deviation are then
computed across the
sample observations.
31/122: Topic 3.3 – Discrete Choice; The Multinomial Logit Model
Use Krinsky and Robb to compute standard errors for Elasticities
32/122: Topic 3.3 – Discrete Choice; The Multinomial Logit Model
Analyzing the Behavior of Market
Shares to Examine Discrete Effects

Scenario: What happens to the number of people who make
specific choices if a particular attribute changes in a
specified way?

Fit the model first, then using the identical model setup, add
; Simulation = list of choices to be analyzed
; Scenario = Attribute (in choices) = type of change

For the CLOGIT application
; Simulation = *
? This is ALL choices
; Scenario: GC(car)=[*]1.25$ Car_GC rises by 25%
33/122: Topic 3.3 – Discrete Choice; The Multinomial Logit Model
Model Simulation
+---------------------------------------------+
| Discrete Choice (One Level) Model
|
| Model Simulation Using Previous Estimates
|
| Number of observations
210
|
+---------------------------------------------+
+------------------------------------------------------+
|Simulations of Probability Model
|
|Model: Discrete Choice (One Level) Model
|
|Simulated choice set may be a subset of the choices. |
|Number of individuals is the probability times the
|
|number of observations in the simulated sample.
|
|Column totals may be affected by rounding error.
|
|The model used was simulated with
210 observations.|
+------------------------------------------------------+
------------------------------------------------------------------------Specification of scenario 1 is:
Attribute Alternatives affected
Change type
Value
--------- ------------------------------- ------------------- --------GC
CAR
Scale base by value
1.250
------------------------------------------------------------------------The simulator located
209 observations for this scenario.
Simulated Probabilities (shares) for this scenario:
+----------+--------------+--------------+------------------+
|Choice
|
Base
|
Scenario
| Scenario - Base |
|
|%Share Number |%Share Number |ChgShare ChgNumber|
+----------+--------------+--------------+------------------+
|AIR
| 27.619
58 | 29.592
62 | 1.973%
4 |
|TRAIN
| 30.000
63 | 31.748
67 | 1.748%
4 |
|BUS
| 14.286
30 | 15.189
32 |
.903%
2 |
|CAR
| 28.095
59 | 23.472
49 | -4.624%
-10 |
|Total
|100.000
210 |100.000
210 |
.000%
0 |
+----------+--------------+--------------+------------------+
Changes in the predicted
market shares when GC_CAR
increases by 25%.
34/122: Topic 3.3 – Discrete Choice; The Multinomial Logit Model
More Complicated Model Simulation
In vehicle cost of CAR falls by 10%
Market is limited to ground (Train, Bus, Car)
CLOGIT ; Lhs = Mode
; Choices = Air,Train,Bus,Car
; Rhs = TTME,INVC,INVT,GC
; Rh2 = One ,Hinc
; Simulation = TRAIN,BUS,CAR
; Scenario: GC(car)=[*].9$
35/122: Topic 3.3 – Discrete Choice; The Multinomial Logit Model
Model Estimation Step
----------------------------------------------------------Discrete choice (multinomial logit) model
Dependent variable
Choice
Log likelihood function
-172.94366
Estimation based on N =
210, K = 10
R2=1-LogL/LogL* Log-L fncn R-sqrd R2Adj
Constants only
-283.7588 .3905 .3807
Chi-squared[ 7]
=
221.63022
Prob [ chi squared > value ] =
.00000
Response data are given as ind. choices
Number of obs.=
210, skipped
0 obs
--------+-------------------------------------------------Variable| Coefficient
Standard Error b/St.Er. P[|Z|>z]
--------+-------------------------------------------------TTME|
-.10289***
.01109
-9.280
.0000
INVC|
-.08044***
.01995
-4.032
.0001
INVT|
-.01399***
.00267
-5.240
.0000
GC|
.07578***
.01833
4.134
.0000
A_AIR|
4.37035***
1.05734
4.133
.0000
AIR_HIN1|
.00428
.01306
.327
.7434
A_TRAIN|
5.91407***
.68993
8.572
.0000
TRA_HIN2|
-.05907***
.01471
-4.016
.0001
A_BUS|
4.46269***
.72333
6.170
.0000
BUS_HIN3|
-.02295
.01592
-1.442
.1493
--------+--------------------------------------------------
Alternative specific
constants and interactions
of ASCs and Household
Income
36/122: Topic 3.3 – Discrete Choice; The Multinomial Logit Model
+---------------------------------------------+
| Discrete Choice (One Level) Model
|
| Model Simulation Using Previous Estimates
|
| Number of observations
210
|
+---------------------------------------------+
+------------------------------------------------------+
|Simulations of Probability Model
|
|Model: Discrete Choice (One Level) Model
|
|Simulated choice set may be a subset of the choices. |
|Number of individuals is the probability times the
|
|number of observations in the simulated sample.
|
|The model used was simulated with
210 observations.|
+------------------------------------------------------+
------------------------------------------------------------------------Specification of scenario 1 is:
Attribute Alternatives affected
Change type
Value
--------- ------------------------------- ------------------- --------INVC
CAR
Scale base by value
.900
------------------------------------------------------------------------The simulator located
210 observations for this scenario.
Simulated Probabilities (shares) for this scenario:
+----------+--------------+--------------+------------------+
|Choice
|
Base
|
Scenario
| Scenario - Base |
|
|%Share Number |%Share Number |ChgShare ChgNumber|
+----------+--------------+--------------+------------------+
|TRAIN
| 37.321
78 | 35.854
75 | -1.467%
-3 |
|BUS
| 19.805
42 | 18.641
39 | -1.164%
-3 |
|CAR
| 42.874
90 | 45.506
96 | 2.632%
6 |
|Total
|100.000
210 |100.000
210 |
.000%
0 |
+----------+--------------+--------------+------------------+
Model Simulation Step
37/122: Topic 3.3 – Discrete Choice; The Multinomial Logit Model
Willingness to Pay
U(alt) = aj + bINCOME*INCOME + bAttribute*Attribute + …
WTP = MU(Attribute)/MU(Income)
When MU(Income) is not available, an approximation
often used is –MU(Cost).
U(Air,Train,Bus,Car)
= αalt + βcost Cost + βINVT INVT + βTTME TTME + εalt
WTP for less in vehicle time = -βINVT / βCOST
WTP for less terminal time = -βTIME / βCOST
38/122: Topic 3.3 – Discrete Choice; The Multinomial Logit Model
WTP from CLOGIT Model
----------------------------------------------------------Discrete choice (multinomial logit) model
Dependent variable
Choice
--------+-------------------------------------------------Variable| Coefficient
Standard Error b/St.Er. P[|Z|>z]
--------+-------------------------------------------------GC|
-.00286
.00610
-.469
.6390
INVT|
-.00349***
.00115
-3.037
.0024
TTME|
-.09746***
.01035
-9.414
.0000
AASC|
4.05405***
.83662
4.846
.0000
TASC|
3.64460***
.44276
8.232
.0000
BASC|
3.19579***
.45194
7.071
.0000
--------+-------------------------------------------------WALD ; fn1=WTP_INVT=b_invt/b_gc ; fn2=WTP_TTME=b_ttme/b_gc$
----------------------------------------------------------WALD procedure.
--------+-------------------------------------------------Variable| Coefficient
Standard Error b/St.Er. P[|Z|>z]
--------+-------------------------------------------------WTP_INVT|
1.22006
2.88619
.423
.6725
WTP_TTME|
34.0771
73.07097
.466
.6410
--------+--------------------------------------------------
Very different estimates suggests this might not be a very good model.
39/122: Topic 3.3 – Discrete Choice; The Multinomial Logit Model
Estimation in WTP Space
Problem with WTP calculation : Ratio of two estimates that
are asymptotically normally distributed may have infinite variance.
Sample point estimates may be reasonable
Inference - confidence intervals - may not be possible.
WTP estimates often become unreasonable in random parameter
models in which parameters vary across individuals.
Estimation in WTP Space
U(Air) = α + βCOST COST + βTIME TIME + βattr Attr + ε


β
β
= α + βCOST COST + TIME TIME + attr Attr  + ε
βCOST
βCOST


= α + βCOST COST + θTIME TIME + θattr Attr  + ε
For a simple MNL the transformation is 1:1. Results will be identical
to the original model. In more elaborate, RP models, results change.
40/122: Topic 3.3 – Discrete Choice; The Multinomial Logit Model
The I.I.D Assumption
Uitj = ij + ’xitj + ’zit + ijt
F(itj) = Exp(-Exp(-itj)) (random part of each utility)
Independence across utility functions
Identical variances (means absorbed in constants)
Restriction on equal scaling may be inappropriate
Correlation across alternatives may be suppressed
Equal cross elasticities is a substantive restriction
Behavioral implication of IID is independence from irrelevant
alternatives. If an alternative is removed, probability is
spread equally across the remaining alternatives. This is
unreasonable
41/122: Topic 3.3 – Discrete Choice; The Multinomial Logit Model
IIA Implication of IID
exp[U (train)]
exp[U (air )]  exp[U (train)]  exp[U (bus )]  exp[U (car )]
exp[U (train)]
Prob(train|train,bus,car) =
exp[U (train)]  exp[U (bus )]  exp[U (car )]
Air is in the choice set, probabilities are independent from air if air is
Prob(train) =
not in the condition. This is a testable behavioral assumption.
42/122: Topic 3.3 – Discrete Choice; The Multinomial Logit Model
+---------------------------------------------------+
| Elasticity
averaged over observations.|
| Attribute is INVT
in choice AIR
|
|
Mean
St.Dev
|
| *
Choice=AIR
-.2055
.0666
|
|
Choice=TRAIN
.0903
.0681
|
|
Choice=BUS
.0903
.0681
|
|
Choice=CAR
.0903
.0681
|
+---------------------------------------------------+
| Attribute is INVT
in choice TRAIN
|
|
Choice=AIR
.3568
.1231
|
| *
Choice=TRAIN
-.9892
.5217
|
|
Choice=BUS
.3568
.1231
|
|
Choice=CAR
.3568
.1231
|
+---------------------------------------------------+
| Attribute is INVT
in choice BUS
|
|
Choice=AIR
.1889
.0743
|
|
Choice=TRAIN
.1889
.0743
|
| *
Choice=BUS
-1.2040
.4803
|
|
Choice=CAR
.1889
.0743
|
+---------------------------------------------------+
| Attribute is INVT
in choice CAR
|
|
Choice=AIR
.3174
.1195
|
|
Choice=TRAIN
.3174
.1195
|
|
Choice=BUS
.3174
.1195
|
| *
Choice=CAR
-.9510
.5504
|
+---------------------------------------------------+
| Effects on probabilities of all choices in model: |
| * = Direct Elasticity effect of the attribute.
|
+---------------------------------------------------+
Behavioral
Implication of IIA
Own effect
Cross effects
Note the effect of IIA on
the cross effects.
Elasticities are computed
for each observation; the
mean and standard
deviation are then
computed across the
sample observations.
43/122: Topic 3.3 – Discrete Choice; The Multinomial Logit Model
A Hausman and McFadden Test for IIA


Estimate full model with “irrelevant alternatives”
Estimate the short model eliminating the irrelevant alternatives



Eliminate individuals who chose the irrelevant alternatives
Drop attributes that are constant in the surviving choice set.
Do the coefficients change? Under the IIA assumption, they
should not.


Use a Hausman test:
Chi-squared, d.f. Number of parameters estimated
H = bshort - b full  '  Vshort - Vfull  bshort - b full 
-1
44/122: Topic 3.3 – Discrete Choice; The Multinomial Logit Model
IIA Test for Choice AIR
+--------+--------------+----------------+--------+--------+
|Variable| Coefficient | Standard Error |b/St.Er.|P[|Z|>z]|
+--------+--------------+----------------+--------+--------+
GC
|
.06929537
.01743306
3.975
.0001
TTME
|
-.10364955
.01093815
-9.476
.0000
INVC
|
-.08493182
.01938251
-4.382
.0000
INVT
|
-.01333220
.00251698
-5.297
.0000
AASC
|
5.20474275
.90521312
5.750
.0000
TASC
|
4.36060457
.51066543
8.539
.0000
BASC
|
3.76323447
.50625946
7.433
.0000
+--------+--------------+----------------+--------+--------+
GC
|
.53961173
.14654681
3.682
.0002
TTME
|
-.06847037
.01674719
-4.088
.0000
INVC
|
-.58715772
.14955000
-3.926
.0001
INVT
|
-.09100015
.02158271
-4.216
.0000
TASC
|
4.62957401
.81841212
5.657
.0000
BASC
|
3.27415138
.76403628
4.285
.0000
Matrix IIATEST has 1 rows and 1 columns.
1
+-------------1|
33.78445
Test statistic
+------------------------------------+
| Listed Calculator Results
|
IIA is rejected
+------------------------------------+
Result =
9.487729
Critical value
45/122: Topic 3.3 – Discrete Choice; The Multinomial Logit Model
Alternative to Utility Maximization (!)
Minimizing Random Regret
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RUM vs. Random Regret
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53/122: Topic 3.3 – Discrete Choice; The Multinomial Logit Model
Fixed Effects Multinomial
Logit:
Application of Minimum
Distance Estimation
54/122: Topic 3.3 – Discrete Choice; The Multinomial Logit Model
Binary Logit Conditional
Probabiities
ei  xit 
Prob( yit  1| xit ) 
.
1  ei  xit 
Ti


Prob  Yi1  yi1 , Yi 2  yi 2 , , YiTi  yiTi  yit 
t 1


Ti


 Ti

exp   yit xit  
exp   yit xit β 
 t 1

 t 1



.
Ti
 Ti



dit xit    All  Ti  different ways that exp   dit xit β 
 t dit Si exp  
 Si 
 t 1

 t 1

 t dit can equal Si
Denominator is summed over all the different combinations of Ti values
of yit that sum to the same sum as the observed  Tt=1i yit . If Si is this sum,
T 
there are   terms. May be a huge number. An algorithm by Krailo
 Si 
and Pike makes it simple.
55/122: Topic 3.3 – Discrete Choice; The Multinomial Logit Model
Example: Seven Period Binary Logit
Prob[y = (1,0,0,0,1,1,1)|Xi ]=
exp( i  x1 )
exp( i  x 7 )
1

 ... 
1  exp( i  x1 ) 1  exp( i  x 2 )
1  exp( i  x 7 )
There are 35 different sequences of y it (permutations) that sum to 4.
For example, y*it| p1 might be (1,1,1,1,0,0,0). Etc.
Prob[y=(1,0,0,0,1,1,1)|Xi ,t71y it =7] =

exp t71 yit xit 
7
*



exp


y
t

1
it | p x it 

p 1
35
56/122: Topic 3.3 – Discrete Choice; The Multinomial Logit Model
57/122: Topic 3.3 – Discrete Choice; The Multinomial Logit Model
With T = 50, the number of permutations of sequences of
y ranging from sum = 0 to sum = 50 ranges from 1 for 0 and 50,
to 2.3 x 1012 for 15 or 35 up to a maximum of 1.3 x 1014 for sum =25.
These are the numbers of terms that must be summed for a model
with T = 50. In the application below, the sum ranges from 15 to 35.
58/122: Topic 3.3 – Discrete Choice; The Multinomial Logit Model
The sample is 200 individuals each observed 50 times.
59/122: Topic 3.3 – Discrete Choice; The Multinomial Logit Model
The data are generated from a probit process with b1 = b2 = .5. But, it is fit as a
logit model. The coefficients obey the familiar relationship, 1.6*probit.
60/122: Topic 3.3 – Discrete Choice; The Multinomial Logit Model
Multinomial Logit Model: J+1 choices including a base choice.
yitj = 1 if individual i makes choice j in period t
  x 
e ij itj
Prob( yitj  1| xitj ) 
  , j  1,..., J .
1   mJ 1eim  xitm
Prob( yit 0  1| xit 0 ) 
1
  .
 im  xitm
J
1   m 1e
The probability attached to the sequence of choices is remarkably complicated.
 Ti


exp
y
x

 j 1  
itj itj 
t

1




Ti


J
J

exp
d
x

 j 1  t ditj Sij  
 j 1
it it 
 t 1

J
 Ti


exp
y
x
β
 j 1  
itj itj 
t

1


.
Ti


ditj xitj β 
 All  STiji  different ways that exp  
 t 1

 t ditj can equal Sij
J
Denominator is summed over all the different combinations of Ti values
of yitj that sum to the same sum as the observed  Tt=1i yit . If Sij is this sum,
T 
there are   terms. May be a huge number. Larger yet by summing over choices.
 Sij 
61/122: Topic 3.3 – Discrete Choice; The Multinomial Logit Model
Estimation Strategy
Conditional ML of the full MNL model.
Impressively complicated.
 A Minimum Distance (MDE) Strategy


Each alternative treated as a binary choice vs.
the base provides an estimator of 




Select subsample that chose either option j or the
base
Estimate  using this binary choice setting
This provides J different estimators of the same 
Optimally combine the different estimators of 
62/122: Topic 3.3 – Discrete Choice; The Multinomial Logit Model
Minimum Distance Estimation
There are J estimators βˆ j of the same parameter vector, βˆ .
Each estimator is consistent and asymptotically normal.
ˆ . How to combine the estimators?
Estimated covariance matrices V
j
 βˆ 1  βˆ *    βˆ 1  βˆ * 




 βˆ 2  βˆ * 
 βˆ 2  βˆ * 
ˆ
MDE: Minimize wrt * q = 
 W





ˆ
ˆ
ˆ 
ˆ 
β

β
β

β
*
*
 J
 J
What to use for the weighting matrix W? Any positive definite matrix will do.
63/122: Topic 3.3 – Discrete Choice; The Multinomial Logit Model
MDE Estimation
ˆ . How to combine the estimators?
Estimated covariance matrices V
j
 βˆ 1  βˆ *   βˆ 1  βˆ * 




ˆ
ˆ
ˆ
ˆ
β 2  β* 
β  β* 
W
MDE: Minimize wrt βˆ * q =  2
 . Propose a GLS approach






ˆ
ˆ
ˆ 
ˆ 
β J  β* 
β J  β* 
ˆ
V
1

0
W = A 1  

0

0
ˆ
V
2
0
0

0


ˆ 
V
J
1
64/122: Topic 3.3 – Discrete Choice; The Multinomial Logit Model
MDE Estimation
ˆ
 βˆ 1  βˆ *   V
  1

 βˆ 2  βˆ *   0
ˆ
MDE: Minimize wrt β* q = 
 
 

ˆ
ˆ  
β J  β*   0
0
ˆ
V
2
0
0

0


ˆ 
V
J
1
 βˆ 1  βˆ * 


ˆ
ˆ
 β 2  β* 
.



ˆ
ˆ 
β J  β* 
1
1
1
1
ˆ
ˆ 1βˆ 
ˆ 1βˆ  ...  V
ˆ 1βˆ  V
ˆ
ˆ
ˆ


The solution is β*   V1  V2  ...  VJ   V
J
J
2
2
1
1


1
J
ˆ 1βˆ 
ˆ 1    J V
=   j 1 V
j
  j 1 j j 

J
J
=  j 1 H j βˆ j where  j 1 H j  I
65/122: Topic 3.3 – Discrete Choice; The Multinomial Logit Model
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73/122: Topic 3.3 – Discrete Choice; The Multinomial Logit Model
Why a 500 fold increase in speed?
MDE is much faster
 Not using Krailo and Pike, or not using
efficiently
 Numerical derivatives for an extremely
messy function (increase the number of
function evaluations by at least 5 times)

74/122: Topic 3.3 – Discrete Choice; The Multinomial Logit Model
Rank Data and Best/Worst
75/122: Topic 3.3 – Discrete Choice; The Multinomial Logit Model
76/122: Topic 3.3 – Discrete Choice; The Multinomial Logit Model
Rank Data and Exploded Logit
Alt 1 is the best overall
Alt 3 is the best among
remaining alts 2,3,4,5
Alt 5 is the best among
remaining alts 2,4,5
Alt 2 is the best among
remaining alts 2,4
Alt 4 is the worst.
77/122: Topic 3.3 – Discrete Choice; The Multinomial Logit Model
Exploded Logit
U[j] = jth favorite alternative among 5 alternatives
U[1] = the choice made if the individual indicates only the favorite
Prob{j = [1],[2],[3],[4],[5]} = Prob{[1]|choice set = [1]...[5]} 
Prob{[2]|choice set = [2]...[5]} 
Prob{[3]|choice set = [3]...[5]} 
Prob{[4]|choice set = [4],[5]} 
1
78/122: Topic 3.3 – Discrete Choice; The Multinomial Logit Model
Exploded Logit
U[j] = jth favorite alternative among 5 alternatives
U[1] = the choice made if the individual indicates only the favorite
Individual ranked the alternatives 1,3,5,2,4
Prob{This set of ranks}
exp(x3 )
exp(x1 )
=


 j 1,2,3,4,5 exp(x j )  j 2,3,4,5 exp(x j )

exp(x5 )
j  2,4,5
exp(x j )


exp(x 2 )
j  2,4
exp(x j )
1
79/122: Topic 3.3 – Discrete Choice; The Multinomial Logit Model
Best Worst



Individual simultaneously ranks best and worst
alternatives.
Prob(alt j) = best = exp[U(j)] / mexp[U(m)]
Prob(alt k) = worst = exp[-U(k)] / mexp[-U(m)]
80/122: Topic 3.3 – Discrete Choice; The Multinomial Logit Model
81/122: Topic 3.3 – Discrete Choice; The Multinomial Logit Model
Choices
82/122: Topic 3.3 – Discrete Choice; The Multinomial Logit Model
Best
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Worst
84/122: Topic 3.3 – Discrete Choice; The Multinomial Logit Model
85/122: Topic 3.3 – Discrete Choice; The Multinomial Logit Model
Uses the result that if U(i,j) is the lowest utility, -U(i,j) is the highest.
86/122: Topic 3.3 – Discrete Choice; The Multinomial Logit Model
Uses the result that if U(i,j) is the lowest utility, -U(i,j) is the highest.
87/122: Topic 3.3 – Discrete Choice; The Multinomial Logit Model
88/122: Topic 3.3 – Discrete Choice; The Multinomial Logit Model
89/122: Topic 3.3 – Discrete Choice; The Multinomial Logit Model
90/122: Topic 3.3 – Discrete Choice; The Multinomial Logit Model
Nonlinear Utility Functions
Generalized (in functional form) multinomial logit model
U(i, j) = Vj (x ij,zi,  ) + ε ij (Utility function may vary by choice.)
F(ε ij ) = exp(-exp(-(ε ij )) - the standard IID assumptions for MNL
Prob(i, j) =

exp  Vj (xij ,zi ,β)
J
m=1
exp  Vm (x im ,zi,β)
Estimation problem is more complicated in practical terms
Large increase in model flexibility.
Note : Coefficients are no longer generic.
WTP(i, k | j) = -
Vj (xij ,zi ,β) / xi,j (k)
Vj (xij ,zi ,β) / Cost
91/122: Topic 3.3 – Discrete Choice; The Multinomial Logit Model
Assessing Prospect Theoretic Functional
Forms and Risk in a Nonlinear Logit
Framework: Valuing Reliability Embedded
Travel Time Savings
David Hensher
The University of Sydney, ITLS
William Greene
Stern School of Business, New York University
8th Annual Advances in Econometrics Conference
Louisiana State University
Baton Rouge, LA
November 6-8, 2009
Hensher, D., Greene, W., “Embedding Risk Attitude and Decisions Weights in Non-linear Logit to Accommodate Time
Variability in the Value of Expected Travel Time Savings,” Transportation Research Part B
92/122: Topic 3.3 – Discrete Choice; The Multinomial Logit Model
Prospect Theory

Marginal value function for an attribute (outcome)
v(xm) = subjective value of attribute

Decision weight w(pm) = impact of a probability on
utility of a prospect

Value function V(xm,pm) = v(xm)w(pm) = value of a prospect that
delivers outcome xm with probability pm

We explore functional forms for w(pm) with
implications for decisions
93/122: Topic 3.3 – Discrete Choice; The Multinomial Logit Model
An Application of Valuing Reliability (due to Ken
Small)
late
late
94/122: Topic 3.3 – Discrete Choice; The Multinomial Logit Model
Stated Choice Survey

Trip Attributes in Stated Choice Design












Routes A and B
Free flow travel time
Slowed down travel time
Stop/start/crawling travel time
Minutes arriving earlier than expected
Minutes arriving later than expected
Probability of arriving earlier than expected
Probability of arriving at the time expected
Probability of arriving later than expected
Running cost
Toll Cost
Individual Characteristics: Age, Income, Gender
95/122: Topic 3.3 – Discrete Choice; The Multinomial Logit Model
Value and Weighting Functions
x1-α
Value Function : V(x) =
1- α
Weighting Functions :
Model 1 =
pmγ
1
γ γ
[pmγ + (1- pm ) ]
τPmγ
Model 2 =
[τPmγ + (1- pm )γ ]
Model 3 = exp(-τ(-lnpm )γ ) Model 4 = exp(-(-lnpm )γ )
96/122: Topic 3.3 – Discrete Choice; The Multinomial Logit Model
Choice Model
U(j) = βref + βcostCost + βAgeAge + βTollTollASC
+ βcurr
w(pcurr)v(tcurr)
+ βlate
w(plate) v(tlate)
+ βearly
w(pearly)v(tearly) + εj
Constraint: βcurr = βlate = βearly
U(j) = βref + βcostCost + βAgeAge + βTollTollASC
+ β[w(pcurr)v(tcurr) + w(plate)v(tlate) + w(pearly)v(tearly)]
+ εj
97/122: Topic 3.3 – Discrete Choice; The Multinomial Logit Model
Application

2008 study undertaken in Australia


toll vs. free roads
stated choice (SC) experiment involving two
SC alternatives (i.e., route A and route B)
pivoted around the knowledge base of
travellers (i.e., the current trip).
280 Individuals
 32 Choice Situations (2 blocks of 16)

98/122: Topic 3.3 – Discrete Choice; The Multinomial Logit Model
Data
99/122: Topic 3.3 – Discrete Choice; The Multinomial Logit Model
100/122: Topic 3.3 – Discrete Choice; The Multinomial Logit Model
Reliability Embedded Value of Travel Time Savings in Au$/hr
$4.50
101/122: Topic 3.3 – Discrete Choice; The Multinomial Logit Model
SCALING IN CHOICE MODELS
102/122: Topic 3.3 – Discrete Choice; The Multinomial Logit Model
Using Degenerate Branches to Reveal
Scaling
LIMB
BRANCH
TWIG
Travel
Fly
Air
Rail
Train
Drive
Car
GrndPblc
Bus
103/122: Topic 3.3 – Discrete Choice; The Multinomial Logit Model
Scaling in Transport Modes
----------------------------------------------------------FIML Nested Multinomial Logit Model
Dependent variable
MODE
Log likelihood function
-182.42834
The model has 2 levels.
Nested Logit form:IVparms=Taub|l,r,Sl|r
& Fr.No normalizations imposed a priori
Number of obs.=
210, skipped
0 obs
--------+-------------------------------------------------Variable| Coefficient
Standard Error b/St.Er. P[|Z|>z]
--------+-------------------------------------------------|Attributes in the Utility Functions (beta)
GC|
.09622**
.03875
2.483
.0130
TTME|
-.08331***
.02697
-3.089
.0020
INVT|
-.01888***
.00684
-2.760
.0058
INVC|
-.10904***
.03677
-2.966
.0030
A_AIR|
4.50827***
1.33062
3.388
.0007
A_TRAIN|
3.35580***
.90490
3.708
.0002
A_BUS|
3.11885**
1.33138
2.343
.0192
|IV parameters, tau(b|l,r),sigma(l|r),phi(r)
FLY|
1.65512**
.79212
2.089
.0367
RAIL|
.92758***
.11822
7.846
.0000
LOCLMASS|
1.00787***
.15131
6.661
.0000
DRIVE|
1.00000
......(Fixed Parameter)......
--------+--------------------------------------------------
NLOGIT ; Lhs=mode
; Rhs=gc,ttme,invt,invc,one
; Choices=air,train,bus,car
; Tree=Fly(Air),
Rail(train),
LoclMass(bus),
Drive(Car)
; ivset:(drive)=[1]$
104/122: Topic 3.3 – Discrete Choice; The Multinomial Logit Model
A Model with Choice
Heteroscedasticity
U(i,t, j)  α j + β'x itj + γ j'zit + σ jε i,t,j
F(ε i,t,j ) = exp(-exp(-ε i,t,j ))
IID after scaling by a choice specific scale parameter
P[choice = j | x itj , zit ,i,t] = Prob[Ui,t,j  Ui,t,k ], k = 1,...,J(i,t)
=

exp (α j + β'x itj + γ j'zit ) /  j 
J(i,t)
exp(α j + β'x itj + γ j'zit ) /  j 
Normalization required as only ratios can be estimated;
j=1
σ j = 1 for one of the alternatives
(Remember the integrability problem - scale is not identified.)
105/122: Topic 3.3 – Discrete Choice; The Multinomial Logit Model
Heteroscedastic Extreme Value Model (1)
+---------------------------------------------+
| Start values obtained using MNL model
|
| Maximum Likelihood Estimates
|
| Log likelihood function
-184.5067
|
| Dependent variable
Choice
|
| Response data are given as ind. choice.
|
| Number of obs.=
210, skipped
0 bad obs. |
+---------------------------------------------+
+--------+--------------+----------------+--------+--------+
|Variable| Coefficient | Standard Error |b/St.Er.|P[|Z|>z]|
+--------+--------------+----------------+--------+--------+
GC
|
.06929537
.01743306
3.975
.0001
TTME
|
-.10364955
.01093815
-9.476
.0000
INVC
|
-.08493182
.01938251
-4.382
.0000
INVT
|
-.01333220
.00251698
-5.297
.0000
AASC
|
5.20474275
.90521312
5.750
.0000
TASC
|
4.36060457
.51066543
8.539
.0000
BASC
|
3.76323447
.50625946
7.433
.0000
106/122: Topic 3.3 – Discrete Choice; The Multinomial Logit Model
Heteroscedastic Extreme Value Model (2)
+---------------------------------------------+
| Heteroskedastic Extreme Value Model
|
| Log likelihood function
-182.4440
| (MNL logL was -184.5067)
| Number of parameters
10
|
| Restricted log likelihood
-291.1218
|
+---------------------------------------------+
+--------+--------------+----------------+--------+--------+
|Variable| Coefficient | Standard Error |b/St.Er.|P[|Z|>z]|
+--------+--------------+----------------+--------+--------+
---------+Attributes in the Utility Functions (beta)
GC
|
.11903513
.06402510
1.859
.0630
TTME
|
-.11525581
.05721397
-2.014
.0440
INVC
|
-.15515877
.07928045
-1.957
.0503
INVT
|
-.02276939
.01122762
-2.028
.0426
AASC
|
4.69411460
2.48091789
1.892
.0585
TASC
|
5.15629868
2.05743764
2.506
.0122
BASC
|
5.03046595
1.98259353
2.537
.0112
---------+Scale Parameters of Extreme Value Distns Minus 1.0
s_AIR
|
-.57864278
.21991837
-2.631
.0085
Normalized for estimation
s_TRAIN |
-.45878559
.34971034
-1.312
.1896
s_BUS
|
.26094835
.94582863
.276
.7826
s_CAR
|
.000000
......(Fixed Parameter).......
---------+Std.Dev=pi/(theta*sqr(6)) for H.E.V. distribution.
s_AIR
|
3.04385384
1.58867426
1.916
.0554
Structural parameters
s_TRAIN |
2.36976283
1.53124258
1.548
.1217
s_BUS
|
1.01713111
.76294300
1.333
.1825
s_CAR
|
1.28254980
......(Fixed Parameter).......
107/122: Topic 3.3 – Discrete Choice; The Multinomial Logit Model
HEV Model - Elasticities
+---------------------------------------------------+
| Elasticity
averaged over observations.|
| Attribute is INVC
in choice AIR
|
| Effects on probabilities of all choices in model: |
| * = Direct Elasticity effect of the attribute.
|
|
Mean
St.Dev
|
| *
Choice=AIR
-4.2604
1.6745
|
|
Choice=TRAIN
1.5828
1.9918
|
|
Choice=BUS
3.2158
4.4589
|
|
Choice=CAR
2.6644
4.0479
|
| Attribute is INVC
in choice TRAIN
|
|
Choice=AIR
.7306
.5171
|
| *
Choice=TRAIN
-3.6725
4.2167
|
|
Choice=BUS
2.4322
2.9464
|
|
Choice=CAR
1.6659
1.3707
|
| Attribute is INVC
in choice BUS
|
|
Choice=AIR
.3698
.5522
|
|
Choice=TRAIN
.5949
1.5410
|
| *
Choice=BUS
-6.5309
5.0374
|
|
Choice=CAR
2.1039
8.8085
|
| Attribute is INVC
in choice CAR
|
|
Choice=AIR
.3401
.3078
|
|
Choice=TRAIN
.4681
.4794
|
|
Choice=BUS
1.4723
1.6322
|
| *
Choice=CAR
-3.5584
9.3057
|
+---------------------------------------------------+
Multinomial Logit
+---------------------------+
| INVC
in AIR
|
|
Mean
St.Dev
|
| *
-5.0216
2.3881
|
|
2.2191
2.6025
|
|
2.2191
2.6025
|
|
2.2191
2.6025
|
| INVC
in TRAIN
|
|
1.0066
.8801
|
| *
-3.3536
2.4168
|
|
1.0066
.8801
|
|
1.0066
.8801
|
| INVC
in BUS
|
|
.4057
.6339
|
|
.4057
.6339
|
| *
-2.4359
1.1237
|
|
.4057
.6339
|
| INVC
in CAR
|
|
.3944
.3589
|
|
.3944
.3589
|
|
.3944
.3589
|
| *
-1.3888
1.2161
|
+---------------------------+
108/122: Topic 3.3 – Discrete Choice; The Multinomial Logit Model
Variance Heterogeneity in MNL
We extend the HEV model by allowing variances to differ
across individuals
U(i,t, j) = α j + β'x itj + γ j'zit + σ ijε i,t,j
σ ij = exp(  j + δw i ). δ = 0 returns the HEV model
F(ε i,t,j ) = exp(-exp(-ε i,t,j ))
 j = 0 for one of the alternatives
Scaling now differs both across alternatives and across individuals
109/122: Topic 3.3 – Discrete Choice; The Multinomial Logit Model
Application: Shoe Brand Choice

Simulated Data: Stated Choice, 400
respondents, 8 choice situations, 3,200
observations

3 choice/attributes + NONE



Fashion = High / Low
Quality = High / Low
Price = 25/50/75,100 coded 1,2,3,4
Heterogeneity: Sex, Age (<25, 25-39, 40+)
 Underlying data generated by a 3 class latent

class process (100, 200, 100 in classes)
110/122: Topic 3.3 – Discrete Choice; The Multinomial Logit Model
Multinomial Logit Baseline Values
+---------------------------------------------+
| Discrete choice (multinomial logit) model
|
| Number of observations
3200
|
| Log likelihood function
-4158.503
|
| Number of obs.= 3200, skipped
0 bad obs. |
+---------------------------------------------+
+--------+--------------+----------------+--------+--------+
|Variable| Coefficient | Standard Error |b/St.Er.|P[|Z|>z]|
+--------+--------------+----------------+--------+--------+
FASH
|
1.47890473
.06776814
21.823
.0000
QUAL
|
1.01372755
.06444532
15.730
.0000
PRICE
|
-11.8023376
.80406103
-14.678
.0000
ASC4
|
.03679254
.07176387
.513
.6082
111/122: Topic 3.3 – Discrete Choice; The Multinomial Logit Model
Multinomial Logit Elasticities
+---------------------------------------------------+
| Elasticity
averaged over observations.|
| Attribute is PRICE
in choice BRAND1
|
| Effects on probabilities of all choices in model: |
| * = Direct Elasticity effect of the attribute.
|
|
Mean
St.Dev
|
| *
Choice=BRAND1
-.8895
.3647
|
|
Choice=BRAND2
.2907
.2631
|
|
Choice=BRAND3
.2907
.2631
|
|
Choice=NONE
.2907
.2631
|
| Attribute is PRICE
in choice BRAND2
|
|
Choice=BRAND1
.3127
.1371
|
| *
Choice=BRAND2
-1.2216
.3135
|
|
Choice=BRAND3
.3127
.1371
|
|
Choice=NONE
.3127
.1371
|
| Attribute is PRICE
in choice BRAND3
|
|
Choice=BRAND1
.3664
.2233
|
|
Choice=BRAND2
.3664
.2233
|
| *
Choice=BRAND3
-.7548
.3363
|
|
Choice=NONE
.3664
.2233
|
+---------------------------------------------------+
112/122: Topic 3.3 – Discrete Choice; The Multinomial Logit Model
HEV Model without Heterogeneity
+---------------------------------------------+
| Heteroskedastic Extreme Value Model
|
| Dependent variable
CHOICE
|
| Number of observations
3200
|
| Log likelihood function
-4151.611
|
| Response data are given as ind. choice.
|
+---------------------------------------------+
+--------+--------------+----------------+--------+--------+
|Variable| Coefficient | Standard Error |b/St.Er.|P[|Z|>z]|
+--------+--------------+----------------+--------+--------+
---------+Attributes in the Utility Functions (beta)
FASH
|
1.57473345
.31427031
5.011
.0000
QUAL
|
1.09208463
.22895113
4.770
.0000
PRICE
|
-13.3740754
2.61275111
-5.119
.0000
ASC4
|
-.01128916
.22484607
-.050
.9600
---------+Scale Parameters of Extreme Value Distns Minus 1.0
s_BRAND1|
.03779175
.22077461
.171
.8641
s_BRAND2|
-.12843300
.17939207
-.716
.4740
s_BRAND3|
.01149458
.22724947
.051
.9597
s_NONE |
.000000
......(Fixed Parameter).......
---------+Std.Dev=pi/(theta*sqr(6)) for H.E.V. distribution.
s_BRAND1|
1.23584505
.26290748
4.701
.0000
s_BRAND2|
1.47154471
.30288372
4.858
.0000
s_BRAND3|
1.26797496
.28487215
4.451
.0000
s_NONE |
1.28254980
......(Fixed Parameter).......
Essentially no
differences in
variances
across choices
113/122: Topic 3.3 – Discrete Choice; The Multinomial Logit Model
Homogeneous HEV Elasticities
Multinomial Logit
+---------------------------------------------------+
| Attribute is PRICE
in choice BRAND1
|
|
Mean
St.Dev
|
| *
Choice=BRAND1
-1.0585
.4526
|
|
Choice=BRAND2
.2801
.2573
|
|
Choice=BRAND3
.3270
.3004
|
|
Choice=NONE
.3232
.2969
|
| Attribute is PRICE
in choice BRAND2
|
|
Choice=BRAND1
.3576
.1481
|
| *
Choice=BRAND2
-1.2122
.3142
|
|
Choice=BRAND3
.3466
.1426
|
|
Choice=NONE
.3429
.1411
|
| Attribute is PRICE
in choice BRAND3
|
|
Choice=BRAND1
.4332
.2532
|
|
Choice=BRAND2
.3610
.2116
|
| *
Choice=BRAND3
-.8648
.4015
|
|
Choice=NONE
.4156
.2436
|
+---------------------------------------------------+
| Elasticity
averaged over observations.|
| Effects on probabilities of all choices in model: |
| * = Direct Elasticity effect of the attribute.
|
+---------------------------------------------------+
+--------------------------+
| PRICE
in choice BRAND1|
|
Mean
St.Dev
|
| *
-.8895
.3647
|
|
.2907
.2631
|
|
.2907
.2631
|
|
.2907
.2631
|
| PRICE
in choice BRAND2|
|
.3127
.1371
|
| *
-1.2216
.3135
|
|
.3127
.1371
|
|
.3127
.1371
|
| PRICE
in choice BRAND3|
|
.3664
.2233
|
|
.3664
.2233
|
| *
-.7548
.3363
|
|
.3664
.2233
|
+--------------------------+
114/122: Topic 3.3 – Discrete Choice; The Multinomial Logit Model
Heteroscedasticity Across Individuals
+---------------------------------------------+
| Heteroskedastic Extreme Value Model
| Homog-HEV
| Log likelihood function
-4129.518[10] | -4151.611[7]
+---------------------------------------------+
+--------+--------------+----------------+--------+--------+
|Variable| Coefficient | Standard Error |b/St.Er.|P[|Z|>z]|
+--------+--------------+----------------+--------+--------+
---------+Attributes in the Utility Functions (beta)
FASH
|
1.01640726
.20261573
5.016
.0000
QUAL
|
.55668491
.11604080
4.797
.0000
PRICE
|
-7.44758292
1.52664112
-4.878
.0000
ASC4
|
.18300524
.09678571
1.891
.0586
---------+Scale Parameters of Extreme Value Distributions
s_BRAND1|
.81114924
.10099174
8.032
.0000
s_BRAND2|
.72713522
.08931110
8.142
.0000
s_BRAND3|
.80084114
.10316939
7.762
.0000
s_NONE |
1.00000000
......(Fixed Parameter).......
---------+Heterogeneity in Scales of Ext.Value Distns.
MALE
|
.21512161
.09359521
2.298
.0215
AGE25
|
.79346679
.13687581
5.797
.0000
AGE39
|
.38284617
.16129109
2.374
.0176
MNL
-4158.503[4]
115/122: Topic 3.3 – Discrete Choice; The Multinomial Logit Model
Variance Heterogeneity Elasticities
Multinomial Logit
+---------------------------------------------------+
| Attribute is PRICE
in choice BRAND1
|
|
Mean
St.Dev
|
| *
Choice=BRAND1
-.8978
.5162
|
|
Choice=BRAND2
.2269
.2595
|
|
Choice=BRAND3
.2507
.2884
|
|
Choice=NONE
.3116
.3587
|
| Attribute is PRICE
in choice BRAND2
|
|
Choice=BRAND1
.2853
.1776
|
| *
Choice=BRAND2
-1.0757
.5030
|
|
Choice=BRAND3
.2779
.1669
|
|
Choice=NONE
.3404
.2045
|
| Attribute is PRICE
in choice BRAND3
|
|
Choice=BRAND1
.3328
.2477
|
|
Choice=BRAND2
.2974
.2227
|
| *
Choice=BRAND3
-.7458
.4468
|
|
Choice=NONE
.4056
.3025
|
+---------------------------------------------------+
+--------------------------+
| PRICE
in choice BRAND1|
|
Mean
St.Dev
|
| *
-.8895
.3647
|
|
.2907
.2631
|
|
.2907
.2631
|
|
.2907
.2631
|
| PRICE
in choice BRAND2|
|
.3127
.1371
|
| *
-1.2216
.3135
|
|
.3127
.1371
|
|
.3127
.1371
|
| PRICE
in choice BRAND3|
|
.3664
.2233
|
|
.3664
.2233
|
| *
-.7548
.3363
|
|
.3664
.2233
|
+--------------------------+
116/122: Topic 3.3 – Discrete Choice; The Multinomial Logit Model
117/122: Topic 3.3 – Discrete Choice; The Multinomial Logit Model
118/122: Topic 3.3 – Discrete Choice; The Multinomial Logit Model
Generalized Mixed Logit Model
U(i,t, j) = βi xi,t,j  Common effects + εi,t,j
Random Parameters
βi = σi [β + Δhi ] + [γ + σi (1- γ)]Γivi
Γi = ΛΣi
Λ is a lower triangular matrix
with 1s on the diagonal (Cholesky matrix)
Σi is a diagonal matrix with φk exp(ψkhi )
Overall preference scaling
σi = exp(-τi2 / 2 + τi w i + θhi ]
τi = exp( λri )
0 < γ <1
119/122: Topic 3.3 – Discrete Choice; The Multinomial Logit Model
Unobserved Heterogeneity in Scaling
HEV formulation: U it , j  x itj  (1 / i )it , j
Generalized model with  = 1 and  = [0].
Produces a scaled multinomial logit model with i  i
exp(i x itj )
Prob(choiceit = j) =
, i  1,..., N , j  1,..., J it , t  1,..., Ti
J it
 j 1 exp(i xitj )
The random variation in the scaling is
i  exp( 2 / 2  wi )
The variation across individuals may also be observed, so that
i  exp( 2 / 2  wi  zi )
120/122: Topic 3.3 – Discrete Choice; The Multinomial Logit Model
Scaled MNL
121/122: Topic 3.3 – Discrete Choice; The Multinomial Logit Model
Observed and Unobserved Heterogeneity
122/122: Topic 3.3 – Discrete Choice; The Multinomial Logit Model
Price Elasticities
123/122: Topic 3.3 – Discrete Choice; The Multinomial Logit Model
Data on Discrete Choices
CHOICE
MODE
TRAVEL
AIR
.00000
TRAIN
.00000
BUS
.00000
CAR
1.0000
AIR
.00000
TRAIN
.00000
BUS
.00000
CAR
1.0000
AIR
.00000
TRAIN
.00000
BUS
1.0000
CAR
.00000
AIR
.00000
TRAIN
.00000
BUS
.00000
CAR
1.0000
INVC
59.000
31.000
25.000
10.000
58.000
31.000
25.000
11.000
127.00
109.00
52.000
50.000
44.000
25.000
20.000
5.0000
ATTRIBUTES
INVT
TTME
100.00
69.000
372.00
34.000
417.00
35.000
180.00
.00000
68.000
64.000
354.00
44.000
399.00
53.000
255.00
.00000
193.00
69.000
888.00
34.000
1025.0
60.000
892.00
.00000
100.00
64.000
351.00
44.000
361.00
53.000
180.00
.00000
GC
70.000
71.000
70.000
30.000
68.000
84.000
85.000
50.000
148.00
205.00
163.00
147.00
59.000
78.000
75.000
32.000
CHARACTERISTIC
HINC
35.000
35.000
35.000
35.000
30.000
30.000
30.000
30.000
60.000
60.000
60.000
60.000
70.000
70.000
70.000
70.000
This is the ‘long form.’ In the ‘wide form,’ all data for the individual appear on a single ‘line’. The ‘wide
form’ is unmanageable for models of any complexity and for stated preference applications.