Empirical Methods for
Microeconomic Applications
University of Lugano, Switzerland
May 27-31, 2013
William Greene
Department of Economics
Stern School of Business
2C. Multinomial Choice
Agenda for 2C
•
•
•
•
•
•
•
•
•
•
•
Random Utility
The Multinomial Logit Model
Choice Data
Estimating the MNL
Model Fit
Elasticities
Willingness to Pay
The IIA Assumption(s)
Multinomial Probit
Mixed Logit (Random Parameters)
Nested Logit
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
Multinomial Choice Among J Alternatives
• Random Utility Basis
Uitj = ij + i’xitj + ijzit + ijt
i = 1,…,N; j = 1,…,J(i,t); t = 1,…,Ti
N individuals studied, J(i,t) alternatives in the choice
set, Ti [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.
Mode Choices of 210 Travelers
Features of Utility Functions
•
•
The linearity assumption Uitj = ij + i xitj + j zi + 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, zi.
•
•
•
•
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
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
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, zi ,i,t] = Prob[Ui,t,j  Ui,t,k ], k = 1,...,J(i,t)
=
exp(α j + β'xitj + γ j'zi )

J(i,t)
j=1
exp(α j + β'xitj + γ j'zi )
Multinomial Choice Models
Multinomial logit model depends on characteristics
P[choice = j | z i ,i] =
exp(α j + γ j'zi )

J(i)
j=1
exp(α j + γ j'z t )
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 , zi ,i,t] =
exp(α j + β'x itj + γ j'zi )

J(i,t)
j=1
exp(α j + β'x itj + γ j'z i )
There is no meaningful distinction.
Specifying the Probabilities
• Choice specific attributes (X) vary by choices, multiply by generic
coefficients. E.g., INVT=In vehicle time, INVC=In vehicle cost
• Generic characteristics, HINC=Household income, must be interacted with
choice specific constants.
• Estimation by maximum likelihood; dij = 1 if person i chooses j
P[choice = j | xitj, zi ,i,t] = Prob[Ui,t,j  Ui,t,k ], k = 1,...,J(i,t)
=
exp(α j + β'xitj + γ j'zi )

J(i,t)
j=1
logL =  i=1
N
exp(α j + β'xitj + γ j'zi )

J(i,t)
j=1
dijlogPij
Using the Model to Measure Consumer Surplus
Consumer Surplus =
Maximum j (Uj )
Marginal Utility of Income
Utility and marginal utility are not observable
For the multinomial logit model (only),
1
J(i,t)
E[CS] =
log  j=1 exp(α j + β'x itj + γ j'z it ) + C
MUI


Where Uj = the utility of the indicated alternative and C
is the constant of integration.
The log sum is the "inclusive value." (The sum is the
denominator of the probability.)
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  + A Constant
exp(α j + β'x itj + γ j'zi ) | A + A Constant
exp(α j + β'x itj
j
i
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'zi ) | A 

1 
j=1
=
log J(i,t)

MUI
exp(α j + β'x itj + γ j'zi ) | B 


j=1


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.
Observed Choice3 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
Individual 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.
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
Stated Choice Experiment: Unlabeled Alternatives, One Observation
t=1
t=2
t=3
t=4
t=5
t=6
t=7
t=8
An Estimated MNL Model for Travel
----------------------------------------------------------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
--------+--------------------------------------------------
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
--------+--------------------------------------------------
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
--------+--------------------------------------------------
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 improvements in log likelihood with
pseudo R-squared = 1 – LogL/LogL0
(“Adjusted” for number of parameters in the
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
Fin.Smpl.AIC
2.73159
573.63404
J
 j 1 Nl log Pj
Bayes IC
2.77885
583.55886
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
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
 N(J-1)   log L
Adjusted Pseudo R 2 =1- 
Bayes IC
2.03185
426.68878

N(J-1)-K

  log L0
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
--------+--------------------------------------------------

.

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)
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
|
+-------------+-------------+-------------+-------------+-------------+
Using the Most Probable Cell
AIR
TRAIN
BUS
CAR
Total
AIR
TRAIN
BUS
CAR
Total
+-------------------------------------------------------+
| Cross tabulation of actual y(ij) vs. predicted y(ij) |
| Row indicator is actual, column is predicted.
|
| Predicted total is N(k,j,i)=Sum(i=1,...,N) Y(k,j,i). |
| Predicted y(ij)=1 is the j with largest probability. |
+-------------------------------------------------------+
NLOGIT Cross Tabulation for 4 outcome Multinomial Choice Model
AIR
TRAIN
BUS
CAR
Total
+-------------+-------------+-------------+-------------+-------------+
|
40
|
3
|
0
|
15
|
58
|
|
4
|
45
|
0
|
14
|
63
|
|
0
|
3
|
23
|
4
|
30
|
|
7
|
14
|
0
|
38
|
59
|
+-------------+-------------+-------------+-------------+-------------+
|
51
|
65
|
23
|
71
|
210
|
+-------------+-------------+-------------+-------------+-------------+
NLOGIT Cross Tabulation for 4 outcome Constants only Model
AIR
TRAIN
BUS
CAR
Total
+-------------+-------------+-------------+-------------+-------------+
|
0
|
58
|
0
|
0
|
58
|
|
0
|
63
|
0
|
0
|
63
|
|
0
|
30
|
0
|
0
|
30
|
|
0
|
59
|
0
|
0
|
59
|
+-------------+-------------+-------------+-------------+-------------+
|
0
|
210
|
0
|
0
|
210
|
+-------------+-------------+-------------+-------------+-------------+
Effects of Changes in
Attributes on Probabilities
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
Effects of Changes in Attributes on Probabilities
Partial effects :
Own effects :
k = Price
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 mβk
xm,k
xm,k
Effects of Changes in Attributes on Probabilities
Elasticities for proportional changes :
xm,k
logProb(j) logPj
=
=
Pj [1(j = m) - Pm ]βk
logx m,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.
Elasticities for CLOGIT
+---------------------------------------------------+
| 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.
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%
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 of
CAR increases by 25%.
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$
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
Model Simulation Step
+------------------------------------------------------+
|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 |
+----------+--------------+--------------+------------------+
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
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.
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


βattr
βTIME
= α + βCOST COST +
TIME +
Attr  + ε
β
β
COST
COST


= 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.
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
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
not in the condition. This is a testable behavioral assumption.
Prob(train) =
Behavioral Implication of IIA
+---------------------------------------------------+
| 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.
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  b short - b full 
-1
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
The Multinomial Probit Model
U(i, j)  α j + β'x ij + γ j'zi + ε i,j
[ε1,ε 2 ,...,ε J ] ~ Multivariate Normal[0, Σ]
Correlation across choices
Heteroscedasticity
Some restrictions for identification
Relaxes the IID assumptions,
therefore, does not assume IIA.
*
*

*


*

0
* * ... * * 
* * ... * * 
* * ... * * 


* *
1 *

0 0 ... 0 1 
Multinomial Probit Model
+---------------------------------------------+
| Multinomial Probit Model
|
| Dependent variable
MODE
|
| Number of observations
210
|
| Iterations completed
30
|
| Log likelihood function
-184.7619
| Not comparable to MNL
| Response data are given as ind. choice.
|
+---------------------------------------------+
+--------+--------------+----------------+--------+--------+
|Variable| Coefficient | Standard Error |b/St.Er.|P[|Z|>z]|
+--------+--------------+----------------+--------+--------+
---------+Attributes in the Utility Functions (beta)
GC
|
.10822534
.04339733
2.494
.0126
TTME
|
-.08973122
.03381432
-2.654
.0080
INVC
|
-.13787970
.05010551
-2.752
.0059
INVT
|
-.02113622
.00727190
-2.907
.0037
AASC
|
3.24244623
1.57715164
2.056
.0398
TASC
|
4.55063845
1.46158257
3.114
.0018
BASC
|
4.02415398
1.28282031
3.137
.0017
---------+Std. Devs. of the Normal Distribution.
Correlation Matrix for
s[AIR] |
3.60695794
1.42963795
2.523
.0116
s[TRAIN]|
1.59318892
.81711159
1.950
.0512
Air, Train, Bus, Car
s[BUS] |
1.00000000
......(Fixed Parameter).......
s[CAR] |
1.00000000
......(Fixed Parameter).......
.305 .404 0 
 1
---------+Correlations in the Normal Distribution
.305

rAIR,TRA|
.30491746
.49357120
.618
.5367
1
.370
0
rAIR,BUS|
.40383018
.63548534
.635
.5251


rTRA,BUS|
.36973127
.42310789
.874
.3822
.404 .370
1
0
rAIR,CAR|
.000000
......(Fixed Parameter).......


rTRA,CAR|
.000000
......(Fixed Parameter).......
0
0
0
1


rBUS,CAR|
.000000
......(Fixed Parameter).......
Multinomial Probit 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
-4.2785
1.7182
|
|
Choice=TRAIN
1.9910
1.6765
|
|
Choice=BUS
2.6722
1.8376
|
|
Choice=CAR
1.4169
1.3250
|
| Attribute is INVC
in choice TRAIN
|
|
Choice=AIR
.8827
.8711
|
| *
Choice=TRAIN
-6.3979
5.8973
|
|
Choice=BUS
3.6442
2.6279
|
|
Choice=CAR
1.9185
1.5209
|
| Attribute is INVC
in choice BUS
|
|
Choice=AIR
.3879
.6303
|
|
Choice=TRAIN
1.2804
2.1632
|
| *
Choice=BUS
-7.4014
4.5056
|
|
Choice=CAR
1.5053
2.5220
|
| Attribute is INVC
in choice CAR
|
|
Choice=AIR
.2593
.2529
|
|
Choice=TRAIN
.8457
.8093
|
|
Choice=BUS
1.7532
1.3878
|
| *
Choice=CAR
-2.6657
3.0418
|
+---------------------------------------------------+
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
|
+---------------------------+
Warning about Stata Multinomial Probit (mprobit)
*
*

*


*

0
* * ... * * 
* * ... * * 
* * ... * * 


* *
1 *

0 0 ... 0 1 
1
0

0


0

0
0 0 ... 0 0 
1 0 ... 0 0 
0 1 ... 0 0 


0 0
1 0

0 0 ... 0 1 
U(i, j)  α j + β'xij + γ j'zi + ε i,j
U(i, j)  α j + β'x ij + γ j'zi + ε i,j
[ε1,ε 2 ,...,ε J ] ~ Multivariate Normal[0, Σ]
[ε1,ε 2 ,...,ε J ] ~ Multivariate Normal[0, I]
Correlation across choices
Heteroscedasticity
No correlation across choices
Some restrictions for identification
This model retains the IID assumptions.
No heteroscedasticity
Multinomial Choice Models: Where to From Here?
Panel data (Repeated measures)
•
•
Random and fixed effects models
Building into a multinomial logit model
The nested logit model
Latent class model
Mixed logit, error components and multinomial probit models
A generalized mixed logit model – The frontier
Combining revealed and stated preference data
Random Parameters Model
•
•
•
Allow model parameters as well as constants to be random
Allow multiple observations with persistent effects
Allow a hierarchical structure for parameters – not completely
random
Uitj = 1’xi1tj + 2i’xi2tj + zit + ijt
•
•
Random parameters in multinomial logit model
•
1 = nonrandom (fixed) parameters
•
2i = random parameters that may vary across
individuals and across time
Maintain I.I.D. assumption for ijt (given )
Continuous Random Variation
in Preference Weights
Heterogeneity arises from continuous variation
in βi across individuals. (Note Classical and Bayesian)
Uijt = α j + βi x itj + γj zi + ε ijt
βi = β + Δhi + w i
βi,k = βk + δkhi + w i,k
Most treatments set Δ = 0, βi = β + w i
Prob[choice j | i,t,βi ] =
exp(α j + βi x itj + γj zi )

Jt (i)
j=1
exp(α j + βi x itj + γj zi )
The Random Parameters Logit Model
Prob[choice j | i,t,βi ] =
exp(α j + βi xitj + γj zi )

Jt (i)
j=1
exp(α j + βi xitj + γj zi )
Multiple choice situations: Independent conditioned
on the individual specific parameters
Prob[choice j | i,t =1,...,T,i βi ] =
exp(α j + βi x itj + γj zi )
T(i)

t=1

Jt (i)
j=1
exp(α j + βi xitj + γj zi )
Modeling Variations
•
Parameter specification
•
“Nonrandom” – variance = 0
Correlation across parameters – random parts correlated
Fixed mean – not to be estimated. Free variance
•
Fixed range – mean estimated, triangular from 0 to 2
•
Hierarchical structure - ik = k + k’hi
•
•
•
Stochastic specification
•
•
•
Normal, uniform, triangular (tent) distributions
Strictly positive – lognormal parameters (e.g., on income)
Autoregressive: v(i,t,k) = u(i,t,k) + r(k)v(i,t-1,k) [this picks up
time effects in multiple choice situations, e.g., fatigue.]
Estimating the Model
P[choice j | i,t] =
exp(α j,i + βi x itj + γj zi )

J(i)
j=1
exp(α j,i + βi x itj + γj zi )
α j,i ,βi  = functions of underlying [α,β, Δ, Γ, ρ, hi ,vi ]
Denote by 1 all “fixed” parameters
Denote by 2i all random and hierarchical parameters
Estimating the RPL Model
Estimation: 1
2it = 2 + Δhi + Γvi,t
Uncorrelated: Γ is diagonal
Autocorrelated: vi,t = Rvi,t-1 + ui,t
(1) Estimate “structural parameters”
(2) Estimate individual specific utility parameters
(3) Estimate elasticities, etc.
Classical Estimation Platform:
The Likelihood
Marginal : f(βi | data, Ω)
Population Mean = E[βi | data, Ω]
=  βi f(βi | Ω)dβi
βi
= β = a subvector of Ω
ˆ = Argmax L(β ,i = 1,...,N | data, Ω)
Ω
i
ˆ
Estimator = β
Expected value over all possible realizations of i (according to the
estimated asymptotic distribution). I.e., over all possible samples.
Simulation Based Estimation
•
•
•
•
Choice probability = P[data |(1,2,Δ,Γ,R,hi,vi,t)]
Need to integrate out the unobserved random term
E{P[data | (1,2,Δ,Γ,R,hi,vi,t)]}
= v P[…|vi,t]f(vi,t)dvi,t
Integration is done by simulation
•
•
•
•
Draw values of v and compute  then probabilities
Average many draws
Maximize the sum of the logs of the averages
(See Train[Cambridge, 2003] on simulation methods.)
Maximum Simulated Likelihood
True log likelihood
i
Li (βi | datai ) =  t=1
f(datai | βi )
T

Li (Ω | datai ) = 
βi
Ti
t=1
f(datai | βi )f(βi | Ω)dβi
=  i=1 log Li (βi | datai )f(βi | Ω)dβi
N
logL
βi
Simulated log likelihood
logLS =  i=1 log
N
1 R
Li (βiR | datai, Ω)

r=1
R
ˆ = argmax(logL )
Ω
S
Model Extensions
βi = β + Δhi + Γw i
•
•
AR(1): wi,k,t = ρkwi,k,t-1 + vi,k,t
Dynamic effects in the model
Restricting sign – lognormal distribution:
βi,k = exp(μk + δkhi + γk w i )
•
•
Restricting Range and Sign: Using triangular
distribution and range = 0 to 2.
Heteroscedasticity and heterogeneity
σ k,i = σ k exp(θhi )
Application: Shoe Brand Choice
•
Simulated Data: Stated Choice,
•
•
•
3 choice/attributes + NONE
•
•
•
•
•
400 respondents,
8 choice situations, 3,200 observations
Fashion = High / Low
Quality = High / Low
Price = 25/50/75,100 coded 1,2,3,4
Heterogeneity: Sex (Male=1), Age (<25, 25-39, 40+)
Underlying data generated by a 3 class latent class
process (100, 200, 100 in classes)
Stated Choice Experiment: Unlabeled Alternatives, One Observation
t=1
t=2
t=3
t=4
t=5
t=6
t=7
t=8
Error Components Logit Modeling
•
•
Alternative approach to building cross choice correlation
Common ‘effects.’ Wi is a ‘random individual effect.’
U(brand1)it = β1Fashion1,it + β2Quality1,it + β3Price1,it + ε Brand1,it + σ Wi
U(brand2)it = β1Fashion2,it + β2Quality 2,it + β3Price 2,it + ε Brand2,it + σ Wi
U(brand3)it = β1Fashion3,it + β2Quality 3,it + β3Price3,it + ε Brand3,it + σ Wi
U(None)
= β4
+ εNo Brand,it
Implied Covariance Matrix
Nested Logit Formulation
Var[ε] = π 2 / 6 = 1.6449
Var[W] = 1
 εBrand1 + σW  1.6449 + σ 2
σ2
σ2
0 


ε

2
2
2
+
σW
σ
1.6449
+
σ
σ
0
Brand2

 = Var ε + σW  =  σ 2
2
2
σ
1.6449 + σ
0 
Brand3


 
0
0
0
1.6449 
 εNONE  
Cross Brand Correlation = σ 2 / [1.6449 + σ 2 ]
Error Components Logit Model
----------------------------------------------------------Error Components (Random Effects) model
Dependent variable
CHOICE
Log likelihood function
-4158.45044
Estimation based on N =
3200, K =
5
Response data are given as ind. choices
Replications for simulated probs. = 50
Halton sequences used for simulations
ECM model with panel has
400 groups
Fixed number of obsrvs./group=
8
Number of obs.= 3200, skipped
0 obs
--------+-------------------------------------------------Variable| Coefficient
Standard Error b/St.Er. P[|Z|>z]
--------+-------------------------------------------------|Nonrandom parameters in utility functions
FASH|
1.47913***
.06971
21.218
.0000
QUAL|
1.01385***
.06580
15.409
.0000
PRICE|
-11.8052***
.86019
-13.724
.0000
ASC4|
.03363
.07441
.452
.6513
SigmaE01|
.09585***
.02529
3.791
.0002
--------+--------------------------------------------------
Random Effects Logit Model
Appearance of Latent Random
Effects in Utilities
Alternative
E01
+-------------+---+
| BRAND1
| * |
+-------------+---+
| BRAND2
| * |
+-------------+---+
| BRAND3
| * |
+-------------+---+
| NONE
|
|
+-------------+---+
Correlation = {0.09592 / [1.6449 + 0.09592]}1/2 = 0.0954
Extended MNL Model
Utility Functions
Ui,1,t = βF,i Fashioni,1,t +β Q Quality i,1,t +βP,i Pricei,1,t + λ Brand Wi,Brand + ε i,1,t
Ui,2,t = βF,i Fashioni,2,t +β Q Quality i,2,t +βP,i Pricei,2,t + λ Brand Wi,Brand + ε i,2,t
Ui,3,t = βF,i Fashioni,3,t +β Q Quality i,3,t + βP,i Pricei,3,t + λ Brand Wi,Brand + εi,3,t
Ui,NONE,t = αNONE
+ λ NONE Wi,NONE + εi,NONE,t
βF,i = βF + δF Sex i +[σ F exp(γ F1 AgeL25i + γ F2 Age2539i )] w F,i ; w F,i ~ N[0,1]
βP,i = βP + δPSex i +[σP exp(γ P1 AgeL25i + γ P2 Age2539i )] w P,i ; w P,i ~ N[0,1]
WBrand,i ~ N[0,1]
WNONE,i ~ N[0,1]
Extending the Basic MNL Model
Random Utility
Ui,1,t = βF,i Fashioni,1,t +β Q Quality i,1,t +βP,i Pricei,1,t + λ Brand Wi,Brand + ε i,1,t
Ui,2,t = βF,i Fashioni,2,t +β Q Quality i,2,t +βP,i Pricei,2,t + λ Brand Wi,Brand + ε i,2,t
Ui,3,t = βF,i Fashioni,3,t +β Q Quality i,3,t + βP,i Pricei,3,t + λ Brand Wi,Brand + εi,3,t
Ui,NONE,t = αNONE
+ λ NONE Wi,NONE + εi,NONE,t
βF,i = βF + δF Sex i +[σ F exp(γ F1 AgeL25i + γ F2 Age2539i )] w F,i ; w F,i ~ N[0,1]
βP,i = βP + δPSex i +[σP exp(γ P1 AgeL25i + γ P2 Age2539i )] w P,i ; w P,i ~ N[0,1]
WBrand,i ~ N[0,1]
WNONE,i ~ N[0,1]
Error Components Logit Model
Error Components
Ui,1,t = βF,i Fashioni,1,t +β Q Quality i,1,t +βP,i Pricei,1,t + λ Brand Wi,Brand + ε i,1,t
Ui,2,t = βF,i Fashioni,2,t +β Q Quality i,2,t +βP,i Pricei,2,t + λ Brand Wi,Brand + ε i,2,t
Ui,3,t = βF,i Fashioni,3,t +β Q Quality i,3,t +βP,i Pricei,3,t + λBrand Wi,Brand + εi,3,t
Ui,NONE,t = αNONE
+ λ NONE Wi,NONE + εi,NONE,t
βF,i = βF + δF Sex i +[σ F exp(γ F1 AgeL25i + γ F2 Age2539i )] w F,i ; w F,i ~ N[0,1]
βP,i = βP + δPSex i +[σ P exp(γ P1 AgeL25i + γ P2 Age2539i )] w P,i ; w P,i ~ N[0,1]
WBrand,i ~ N[0,1]
WNONE,i ~ N[0,1]
Random Parameters Model
Ui,1,t = βF,i Fashioni,1,t +β Q Quality i,1,t +βP,i Pricei,1,t + λ Brand Wi,Brand + ε i,1,t
Ui,2,t = βF,i Fashioni,2,t +β Q Quality i,2,t +βP,i Pricei,2,t + λ Brand Wi,Brand + ε i,2,t
Ui,3,t = βF,i Fashioni,3,t +β Q Quality i,3,t +βP,i Pricei,3,t + λBrand Wi,Brand + εi,3,t
Ui,NONE,t = αNONE
+ λ NONE Wi,NONE + εi,NONE,t
βF,i = βF + δF Sex i +[σ F exp(γ F1 AgeL25i + γ F2 Age2539i )] w F,i ; w F,i ~ N[0,1]
βP,i = βP + δPSex i +[σP exp(γ P1 AgeL25i + γ P2 Age2539i )] w P,i ; w P,i ~ N[0,1]
WBrand,i ~ N[0,1]
WNONE,i ~ N[0,1]
Heterogeneous (in the Means)
Random Parameters Model
Ui,1,t = βF,i Fashioni,1,t +β Q Quality i,1,t +βP,i Pricei,1,t + λ Brand Wi,Brand + ε i,1,t
Ui,2,t = βF,i Fashioni,2,t +β Q Quality i,2,t +βP,i Pricei,2,t + λ Brand Wi,Brand + ε i,2,t
Ui,3,t = βF,i Fashioni,3,t +β Q Quality i,3,t +βP,i Pricei,3,t + λBrand Wi,Brand + εi,3,t
Ui,NONE,t = αNONE
+ λ NONE Wi,NONE + εi,NONE,t
βF,i = βF + δF Sex i +[σ F exp(γ F1 AgeL25i + γ F2 Age2539i )] w F,i ; w F,i ~ N[0,1]
βP,i = βP + δPSex i +[σP exp(γ P1 AgeL25i + γ P2 Age2539i )] w P,i ; w P,i ~ N[0,1]
WBrand,i ~ N[0,1]
WNONE,i ~ N[0,1]
Heterogeneity in Both
Means and Variances
Ui,1,t = βF,i Fashioni,1,t +β Q Quality i,1,t +βP,i Pricei,1,t + λ Brand Wi,Brand + ε i,1,t
Ui,2,t = βF,i Fashioni,2,t +β Q Quality i,2,t +βP,i Pricei,2,t + λ Brand Wi,Brand + ε i,2,t
Ui,3,t = βF,i Fashioni,3,t +β Q Quality i,3,t +βP,i Pricei,3,t + λBrand Wi,Brand + εi,3,t
Ui,NONE,t = αNONE
+ λ NONE Wi,NONE + εi,NONE,t
βF,i = βF + δF Sex i +[σ F exp(γ F1 AgeL25i + γ F2 Age2539i )] w F,i ; w F,i ~ N[0,1]
βP,i = βP + δPSex i +[σP exp(γ P1 AgeL25i + γ P2 Age2539i )] w P,i ; w P,i ~ N[0,1]
WBrand,i ~ N[0,1]
WNONE,i ~ N[0,1]
----------------------------------------------------------Random Parms/Error Comps. Logit Model
Dependent variable
CHOICE
Log likelihood function
-4019.23544 (-4158.50286 for MNL)
Restricted log likelihood
-4436.14196 (Chi squared = 278.5)
Chi squared [ 12 d.f.]
833.81303
Significance level
.00000
McFadden Pseudo R-squared
.0939795
Estimation based on N =
3200, K = 12
Information Criteria: Normalization=1/N
Normalized
Unnormalized
AIC
2.51952
8062.47089
Fin.Smpl.AIC
2.51955
8062.56878
Bayes IC
2.54229
8135.32176
Hannan Quinn
2.52768
8088.58926
R2=1-LogL/LogL* Log-L fncn R-sqrd R2Adj
No coefficients -4436.1420 .0940 .0928
Constants only -4391.1804 .0847 .0836
At start values -4158.5029 .0335 .0323
Response data are given as ind. choices
Replications for simulated probs. = 50
Halton sequences used for simulations
RPL model with panel has
400 groups
Fixed number of obsrvs./group=
8
Hessian is not PD. Using BHHH estimator
Number of obs.= 3200, skipped
0 obs
--------+--------------------------------------------------
Estimated RP/ECL Model
--------+-------------------------------------------------Variable| Coefficient
Standard Error b/St.Er. P[|Z|>z]
--------+-------------------------------------------------|Random parameters in utility functions
FASH|
.62768***
.13498
4.650
.0000
PRICE|
-7.60651***
1.08418
-7.016
.0000
|Nonrandom parameters in utility functions
QUAL|
1.07127***
.06732
15.913
.0000
ASC4|
.03874
.09017
.430
.6675
|Heterogeneity in mean, Parameter:Variable
FASH:AGE|
1.73176***
.15372
11.266
.0000
FAS0:AGE|
.71872***
.18592
3.866
.0001
PRIC:AGE|
-9.38055***
1.07578
-8.720
.0000
PRI0:AGE|
-4.33586***
1.20681
-3.593
.0003
|Distns. of RPs. Std.Devs or limits of triangular
NsFASH|
.88760***
.07976
11.128
.0000
NsPRICE|
1.23440
1.95780
.631
.5284
|Heterogeneity in standard deviations
|(cF1, cF2, cP1, cP2 omitted...)
|Standard deviations of latent random effects
SigmaE01|
.23165
.40495
.572
.5673
SigmaE02|
.51260**
.23002
2.228
.0258
--------+-------------------------------------------------Note: ***, **, * = Significance at 1%, 5%, 10% level.
-----------------------------------------------------------
Random Effects Logit
Model Appearance of
Latent Random Effects
in Utilities
Alternative
E01 E02
+-------------+---+---+
| BRAND1
| * |
|
+-------------+---+---+
| BRAND2
| * |
|
+-------------+---+---+
| BRAND3
| * |
|
+-------------+---+---+
| NONE
|
| * |
+-------------+---+---+
Heterogeneity in Means.
Delta:
2 rows, 2 cols.
AGE25
AGE39
FASH
1.73176
.71872
PRICE -9.38055 -4.33586
Estimated 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
-.9210
.4661
|
|
Choice=BRAND2
.2773
.3053
|
|
Choice=BRAND3
.2971
.3370
|
|
Choice=NONE
.2781
.2804
|
+---------------------------------------------------+
| Attribute is PRICE
in choice BRAND2
|
|
Choice=BRAND1
.3055
.1911
|
| *
Choice=BRAND2
-1.2692
.6179
|
|
Choice=BRAND3
.3195
.2127
|
|
Choice=NONE
.2934
.1711
|
+---------------------------------------------------+
| Attribute is PRICE
in choice BRAND3
|
|
Choice=BRAND1
.3737
.2939
|
|
Choice=BRAND2
.3881
.3047
|
| *
Choice=BRAND3
-.7549
.4015
|
|
Choice=NONE
.3488
.2670
|
+---------------------------------------------------+
Multinomial Logit
+--------------------------+
| Effects on probabilities |
| * = Direct effect te.
|
|
Mean St.Dev |
| PRICE in choice BRAND1 |
| * BRAND1
-.8895 .3647 |
|
BRAND2
.2907 .2631 |
|
BRAND3
.2907 .2631 |
|
NONE
.2907 .2631 |
+--------------------------+
| PRICE in choice BRAND2 |
|
BRAND1
.3127 .1371 |
| * BRAND2 -1.2216 .3135 |
|
BRAND3
.3127 .1371 |
|
NONE
.3127 .1371 |
+--------------------------+
| PRICE in choice BRAND3 |
|
BRAND1
.3664 .2233 |
|
BRAND2
.3664 .2233 |
| * BRAND3
-.7548 .3363 |
|
NONE
.3664 .2233 |
+--------------------------+
Estimating Individual Distributions
•
•
•
•
Form posterior estimates of E[i|datai]
Use the same methodology to estimate
E[i2|datai] and Var[i|datai]
Plot individual “confidence intervals”
(assuming near normality)
Sample from the distribution and plot kernel
density estimates
What is the ‘Individual Estimate?’
Point estimate of mean, variance and range of
random variable i | datai.
Value is NOT an estimate of i ; it is an estimate
of E[i | datai]
This would be the best estimate of the actual
realization i|datai
An interval estimate would account for the sampling
‘variation’ in the estimator of Ω.
Bayesian counterpart to the preceding: Posterior
mean and variance. Same kind of plot could be done.
Individual E[i|datai] Estimates*
The random parameters model is uncovering the latent class feature of the data.
*The intervals could be made wider to account for the sampling
variability of the underlying (classical) parameter estimators.
WTP Application (Value of Time Saved)
Estimating Willingness to Pay for
Increments to an Attribute in a
Discrete Choice Model
βattribute,i
WTP = βcost
Random
Extending the RP Model to WTP
Use the model to estimate conditional
distributions for any function of
parameters
Willingness to pay = -i,time / i,cost
Use simulation method
R
T
ˆ | Ω,data
ˆ
(1/
R)Σ
WTP
Π
P
(β
r=1
ir
t=1
ijt
ir
it )
ˆ
E[WTPi | datai ] =
ˆ
(1/ R)ΣR
ΠT P (βˆ | Ω,data
)
r=1
1 R
ˆ i,r WTPir
=  r=1 w
R
t=1 ijt
ir
it
Sumulation of WTP from i
 -βi,Attribute

WTPi = E 
| β, Δ,Γ, yi, X i,hi, zi 
 βi,Cost

-βi,Attribute  T

P(choice j | X i ,βi )g(βi | β, Δ,Γ, yi, X i,hi, zi )dβi
 βi βi,Cost 
t=1

=
 T

P(choice
j
|
X
,
β
)g(
β
|
β
,
Δ
,
Γ
,
y
,
X
,
h
,
z
)
i
i
i
i
i
i
i dβi
 βi 
t=1

1 R -βˆ i,Attribute  T
ˆ )
P(choice
j
|
X
,
β


i
ir 
R r=1 βˆ i,Cost  t=1
 ˆ
ˆ + Γw
ˆ
WTPi =
, βir = βˆ + Δh
i
ir
T
R

1 
P(choice j | X i ,βˆ ir )



R r=1  t=1

A Generalized Mixed Logit Model
U(i,t, j) = βi xi,t,j  Common effects + εi,t,j
Random Parameters
βi = σ i [β + Δhi ] +[γ + σ i (1- γ)]Γi vi
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
σ i = σexp(-τi2 / 2 + τ i2 w i + θhi ]
τi = exp( λri )
0 < γ <1
Estimation in Willingness to Pay Space
Both parameters in the WTP calculation are
random.
Ui,1,t = βP,i  θF,i Fashioni,1,t + θQ Quality i,1,t +Price i,1,t  + λ BrandWi,Brand + ε i,1,t
Ui,2,t = βP,i  θF,i Fashioni,2,t + θQ Quality i,2,t +Price i,2,t  + λ BrandWi,Brand + ε i,2,t
Ui,3,t = βP,i  θF,i Fashioni,3,t + θQ Quality i,3,t +Pricei,3,t  + λ Brand Wi,Brand + εi,3,t
Ui,NONE,t = αNONE
+ λNONE Wi,NONE + εi,NONE,t
WBrand,i ~ N[0,1] WNONE,i ~ N[0,1]
 θF + δF Sex i 
 θF,i 
 F
  [i   (1   )] 
   i 
β
 PF
βP + δP Sex i 
 P,i 
0   wF,i 


 P   w P,i 
w F,i ~ N[0,1]
w P,i ~ N[0,1]
Extended Formulation of the MNL
Groups of similar alternatives
LIMB
Travel
BRANCH
TWIG
Private
Air
Public
Car
Train
Bus
Compound Utility: U(Alt)=U(Alt|Branch)+U(branch)
Behavioral implications – Correlations across branches
Degenerate Branches
Shoe Choice
Purchase
Brand
Choice Situation
Opt Out
None
Choose Brand
Brand1
Brand2
Brand3
Correlation Structure for a Two Level Model
•
Within a branch
•
•
•
•
Identical variances (IIA 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)
Probabilities for a Nested Logit Model
Utility functions; (Drop observation indicator, i.)
Twig level : k | j denotes alternative k in branch j
U(k | j) = αk|j + βxk|j
Branch level U(j)
= y j
Twig level probability : P(k | j) = Pk|j =
exp(αk|j + βx k|j )

K|j
m=1
exp(αm|j + βxm|j )


Inclusive value for branch j = IV(j) = log ΣK|j
m=1exp(αm|j + β x m|j ) 
exp  λ j  γ'y j +IV(j)  
Branch level probability :
P(j) = B
 b=1 exp λb  γ'yb +IV(b)
λ j = 1 for all branches returns the original MNL model
Estimation Strategy for Nested Logit Models
•
Two step estimation
•
For each branch, just fit MNL


•
For branch level, fit separate model, just including y and the
inclusive values


•
Loses efficiency – replicates coefficients
Does not insure consistency with utility maximization
Again loses efficiency
Not consistent with utility maximization – note the form of the
branch probability
Full information ML
Fit the entire model at once, imposing all restrictions
Estimates of a Nested Logit Model
NLOGIT ; Lhs=mode
; Rhs=gc,ttme,invt,invc
; Rh2=one,hinc
; Choices=air,train,bus,car
; Tree=Travel[Private(Air,Car),
Public(Train,Bus)]
; Show tree
; Effects: invc(*)
; Describe
; RU1 $ Selects branch normalization
Model Structure
Tree Structure Specified for the Nested Logit Model
Sample proportions are marginal, not conditional.
Choices marked with * are excluded for the IIA test.
----------------+----------------+----------------+----------------+------+--Trunk
(prop.)|Limb
(prop.)|Branch
(prop.)|Choice
(prop.)|Weight|IIA
----------------+----------------+----------------+----------------+------+--Trunk{1} 1.00000|TRAVEL
1.00000|PRIVATE
.55714|AIR
.27619| 1.000|
|
|
|CAR
.28095| 1.000|
|
|PUBLIC
.44286|TRAIN
.30000| 1.000|
|
|
|BUS
.14286| 1.000|
----------------+----------------+----------------+----------------+------+--+---------------------------------------------------------------+
| Model Specification: Table entry is the attribute that
|
| multiplies the indicated parameter.
|
+--------+------+-----------------------------------------------+
| Choice |******| Parameter
|
|
|Row 1| GC
TTME
INVT
INVC
A_AIR
|
|
|Row 2| AIR_HIN1 A_TRAIN TRA_HIN3 A_BUS
BUS_HIN4 |
+--------+------+-----------------------------------------------+
|AIR
|
1| GC
TTME
INVT
INVC
Constant |
|
|
2| HINC
none
none
none
none
|
|CAR
|
1| GC
TTME
INVT
INVC
none
|
|
|
2| none
none
none
none
none
|
|TRAIN
|
1| GC
TTME
INVT
INVC
none
|
|
|
2| none
Constant HINC
none
none
|
|BUS
|
1| GC
TTME
INVT
INVC
none
|
|
|
2| none
none
none
Constant HINC
|
+---------------------------------------------------------------+
MNL Starting Values
----------------------------------------------------------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 .3787
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]
--------+-------------------------------------------------GC|
.07578***
.01833
4.134
.0000
TTME|
-.10289***
.01109
-9.280
.0000
INVT|
-.01399***
.00267
-5.240
.0000
INVC|
-.08044***
.01995
-4.032
.0001
A_AIR|
4.37035***
1.05734
4.133
.0000
AIR_HIN1|
.00428
.01306
.327
.7434
A_TRAIN|
5.91407***
.68993
8.572
.0000
TRA_HIN3|
-.05907***
.01471
-4.016
.0001
A_BUS|
4.46269***
.72333
6.170
.0000
BUS_HIN4|
-.02295
.01592
-1.442
.1493
--------+--------------------------------------------------
FIML Parameter Estimates
----------------------------------------------------------FIML Nested Multinomial Logit Model
Dependent variable
MODE
Log likelihood function
-166.64835
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|
.06579***
.01878
3.504
.0005
TTME|
-.07738***
.01217
-6.358
.0000
INVT|
-.01335***
.00270
-4.948
.0000
INVC|
-.07046***
.02052
-3.433
.0006
A_AIR|
2.49364**
1.01084
2.467
.0136
AIR_HIN1|
.00357
.01057
.337
.7358
A_TRAIN|
3.49867***
.80634
4.339
.0000
TRA_HIN3|
-.03581***
.01379
-2.597
.0094
A_BUS|
2.30142***
.81284
2.831
.0046
BUS_HIN4|
-.01128
.01459
-.773
.4395
|IV parameters, lambda(b|l),gamma(l)
PRIVATE|
2.16095***
.47193
4.579
.0000
PUBLIC|
1.56295***
.34500
4.530
.0000
|Underlying standard deviation = pi/(IVparm*sqr(6)
PRIVATE|
.59351***
.12962
4.579
.0000
PUBLIC|
.82060***
.18114
4.530
.0000
--------+--------------------------------------------------
| 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
.000
.000 -2.456 -3.091
-5.547 3.525 |
|
Choice=CAR
.000
.000 -2.456
2.916
.460 3.178 |
|
Branch=PUBLIC
|
|
Choice=TRAIN
.000
.000
3.846
.000
3.846 4.865 |
|
Choice=BUS
.000
.000
3.846
.000
3.846 4.865 |
+-----------------------------------------------------------------------+
| Attribute is INVC
in choice CAR
|
|
Branch=PRIVATE
|
|
Choice=AIR
.000
.000
-.757
.650
-.107
.589 |
| *
Choice=CAR
.000
.000
-.757
-.830
-1.587 1.292 |
|
Branch=PUBLIC
|
|
Choice=TRAIN
.000
.000
.647
.000
.647
.605 |
|
Choice=BUS
.000
.000
.647
.000
.647
.605 |
+-----------------------------------------------------------------------+
| Attribute is INVC
in choice TRAIN
|
|
Branch=PRIVATE
|
|
Choice=AIR
.000
.000
1.340
.000
1.340 1.475 |
|
Choice=CAR
.000
.000
1.340
.000
1.340 1.475 |
|
Branch=PUBLIC
|
| *
Choice=TRAIN
.000
.000 -1.986 -1.490
-3.475 2.539 |
|
Choice=BUS
.000
.000 -1.986
2.128
.142 1.321 |
+-----------------------------------------------------------------------+
| Attribute is INVC
in choice BUS
|
|
Branch=PRIVATE
|
|
Choice=AIR
.000
.000
.547
.000
.547
.871 |
|
Choice=CAR
.000
.000
.547
.000
.547
.871 |
|
Branch=PUBLIC
|
|
Choice=TRAIN
.000
.000
-.841
.888
.047
.678 |
| *
Choice=BUS
.000
.000
-.841 -1.469
-2.310 1.119 |
Estimated
Elasticities –
Note
Decomposition
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
= -166.68435
LogL (MNL) = -172.94366
Chi-squared with 2 d.f. = 2(-166.68435-(-172.94366))
= 12.51862
The critical value is 5.99 (95%)
The MNL is rejected (as usual)
Higher Level Trees
E.g., Location (Neighborhood)
Housing Type (Rent, Buy, House, Apt)
Housing (# Bedrooms)
Degenerate Branches
LIMB
BRANCH
TWIG
Travel
Fly
Air
Ground
Train
Car
Bus
NL Model with Degenerate Branch
----------------------------------------------------------FIML Nested Multinomial Logit Model
Dependent variable
MODE
Log likelihood function
-148.63860
--------+-------------------------------------------------Variable| Coefficient
Standard Error b/St.Er. P[|Z|>z]
--------+-------------------------------------------------|Attributes in the Utility Functions (beta)
GC|
.44230***
.11318
3.908
.0001
TTME|
-.10199***
.01598
-6.382
.0000
INVT|
-.07469***
.01666
-4.483
.0000
INVC|
-.44283***
.11437
-3.872
.0001
A_AIR|
3.97654***
1.13637
3.499
.0005
AIR_HIN1|
.02163
.01326
1.631
.1028
A_TRAIN|
6.50129***
1.01147
6.428
.0000
TRA_HIN2|
-.06427***
.01768
-3.635
.0003
A_BUS|
4.52963***
.99877
4.535
.0000
BUS_HIN3|
-.01596
.02000
-.798
.4248
|IV parameters, lambda(b|l),gamma(l)
FLY|
.86489***
.18345
4.715
.0000
GROUND|
.24364***
.05338
4.564
.0000
|Underlying standard deviation = pi/(IVparm*sqr(6)
FLY|
1.48291***
.31454
4.715
.0000
GROUND|
5.26413***
1.15331
4.564
.0000
--------+--------------------------------------------------
Estimates of a Nested Logit Model
NLOGIT
;
;
;
;
;
lhs=mode
rhs=gc,ttme,invt,invc
rh2=one,hinc
choices=air,train,bus,car
tree=Travel[Fly(Air),
Ground(Train,Car,Bus)]
; show tree
; effects:gc(*)
; Describe $
Nested Logit Model
----------------------------------------------------------FIML Nested Multinomial Logit Model
Dependent variable
MODE
Log likelihood function
-168.81283 (-148.63860 with RU1)
--------+-------------------------------------------------Variable| Coefficient
Standard Error b/St.Er. P[|Z|>z]
--------+-------------------------------------------------|Attributes in the Utility Functions (beta)
GC|
.06527***
.01787
3.652
.0003
TTME|
-.06114***
.01119
-5.466
.0000
INVT|
-.01231***
.00283
-4.354
.0000
INVC|
-.07018***
.01951
-3.597
.0003
A_AIR|
1.22545
.87245
1.405
.1601
AIR_HIN1|
.01501
.01226
1.225
.2206
A_TRAIN|
3.44408***
.68388
5.036
.0000
TRA_HIN2|
-.02823***
.00852
-3.311
.0009
A_BUS|
2.58400***
.63247
4.086
.0000
BUS_HIN3|
-.00726
.01075
-.676
.4993
|IV parameters, RU2 form = mu(b|l),gamma(l)
FLY|
1.00000
......(Fixed Parameter)......
GROUND|
.47778***
.10508
4.547
.0000
|Underlying standard deviation = pi/(IVparm*sqr(6)
FLY|
1.28255
......(Fixed Parameter)......
GROUND|
2.68438***
.59041
4.547
.0000
--------+--------------------------------------------------
Using Degenerate Branches to Reveal Scaling
LIMB
BRANCH
TWIG
Travel
Fly
Air
Rail
Train
Drive
Car
GrndPblc
Bus
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]$
Simulating the Nested Logit Model
NLOGIT
; lhs=mode;rhs=gc,ttme,invt,invc ; rh2=one,hinc
; choices=air,train,bus,car
; tree=Travel[Private(Air,Car),Public(Train,Bus)]
; simulation = *
; scenario:gc(car)=[*]1.5
+------------------------------------------------------+
|Simulations of Probability Model
|
|Model: FIML: Nested Multinomial Logit Model
|
|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
1.500
Simulated Probabilities (shares) for this scenario:
+----------+--------------+--------------+------------------+
|Choice
|
Base
|
Scenario
| Scenario - Base |
|
|%Share Number |%Share Number |ChgShare ChgNumber|
+----------+--------------+--------------+------------------+
|AIR
| 26.515
56 | 8.854
19 |-17.661%
-37 |
|TRAIN
| 29.782
63 | 12.487
26 |-17.296%
-37 |
|BUS
| 14.504
30 | 71.824
151 | 57.320%
121 |
|CAR
| 29.200
61 | 6.836
14 |-22.364%
-47 |
|Total
|100.000
210 |100.000
210 |
.000%
0 |
+----------+--------------+--------------+------------------+
An Error Components Model
Random terms in utility functions share random components
U(Air,i)
= αAIR + β1INVCi,AIR +...+ ε i,AIR
+ w i,1
U(Train,i) = αTRAIN + β1INVCi,TRAIN +...+ ε i,TRAIN + w i,1
U(Bus,i) = αBUS + β1INVCi,BUS +...+ ε i,BUS +
w i,2
β1INVCi,CAR +...+ ε i,CAR +
w i,2
U(Car,i)
=
θ12
0
0 
 Air  σ 2ε + θ12


 
2
2
2
Train
θ
σ
+
θ
0
0
1
ε
1

=
Cov 
 Bus   0
0
σ 2ε + θ22
θ22 

 
2
2
2
0
θ2
σ ε + θ2 
 Car   0
This model is estimated by maximum simulated likelihood.
Error Components Logit Model
----------------------------------------------------------Error Components (Random Effects) model
Dependent variable
MODE
Log likelihood function
-182.27368
Response data are given as ind. choices
Replications for simulated probs. = 25
Halton sequences used for simulations
ECM model with panel has
70 groups
Fixed number of obsrvs./group=
3
Hessian is not PD. Using BHHH estimator
Number of obs.=
210, skipped
0 obs
--------+-------------------------------------------------Variable| Coefficient
Standard Error b/St.Er. P[|Z|>z]
--------+-------------------------------------------------|Nonrandom parameters in utility functions
GC|
.07293***
.01978
3.687
.0002
TTME|
-.10597***
.01116
-9.499
.0000
INVT|
-.01402***
.00293
-4.787
.0000
INVC|
-.08825***
.02206
-4.000
.0001
A_AIR|
5.31987***
.90145
5.901
.0000
A_TRAIN|
4.46048***
.59820
7.457
.0000
A_BUS|
3.86918***
.67674
5.717
.0000
|Standard deviations of latent random effects
SigmaE01|
-.27336
3.25167
-.084
.9330
SigmaE02|
1.21988
.94292
1.294
.1958
--------+--------------------------------------------------