Discrete Choice Modeling

1
Discrete Choice Modeling

• William Greene
• Stern School of Business
• New York University

2
Part 8

• Multinomial Logit Models

3
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
• The Integrability Problem Utility is not
  revealed by demands

4
Theory for Discrete Choice

• Theory is silent about discrete choices
• Translation to discrete choice
• Existence of well defined utility indexes
  Completeness of rankings
• Rationality Utility maximization
• Axioms of revealed preferences
• Choice sets and consideration sets consumers
  simplify choice situations
• Implication for choice among a set of discrete
  alternatives
• Commonalities and uniqueness
• Does this allow us to build models?
• What common elements can be assumed?
• How can we account for heterogeneity?
• Revealed choices do not reveal utility, only
  rankings which are scale invariant

5
Multinomial Choice Among J Alternatives

• Random Utility Basis
• Uitj ?ij ?i xitj ?izit ?ijt
• i 1,,N j 1,,J(i) t 1,,T(i)
• Maximum Utility Assumption
• Individual i will Choose alternative j in
  choice setting t iff Uitj gt Uitk for all k ? j.
• Underlying assumptions
• Smoothness of utilities
• Axioms Transitive, Complete, Monotonic

6
Utility Functions

• The linearity assumption and curvature
• The choice set
• Deterministic and random components The model
• Generic vs. alternative specific components
• Attributes and characteristics
• Coefficients
• Part worths
• Alternative specific constants
• Scaling

7
The Multinomial Logit (MNL) Model

• Independent extreme value (Gumbel)
• F(?itj) 1 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

8
Specifying Probabilities

• Choice specific attributes (X) vary by choices,
  multiply by generic
• coefficients. E.g., TTME, GC
• Generic characteristics (Income, constants) must
  be interacted with
• choice specific constants. (Else they fall out
  of the probability)
• Estimation by maximum likelihood dij 1 if
  person i chooses j

9
Consumer Surplus

• General
• Utility and marginal utility are unobservable
• For the MNL model (only)
• C constant of integration (utility is
  unobservable)
• Logsum Inclusive value (denominator of Probj
  a useful coincidence)

10
Valuing a Policy Change

• Individual change in consumer surplus Price
  change, income change, change in choice set.
  Compare equilibria
• Approximate MUI by coefficient on income if
  available or the negative of the coefficient on a
  price variable
• Aggregate over individuals (add them up).

11
Willingness to Pay

• Generally ratio of coefficients
• ?cost or price / ?attribute level
• Sampling distribution
• Ratio of asymptotic normals?
• (Later) Ratio of random parameters?

12
Observed Data

• Types of Data
• Individual choice
• Market shares
• Frequencies
• Ranks
• Attributes and Characteristics
• Choice Settings
• Cross section
• Repeated measurement (panel data)

13
Data on Discrete Choices

• Line MODE TRAVEL INVC INVT
  TTME GC HINC
• 1 AIR .00000 59.000 100.00
  69.000 70.000 35.000
• 2 TRAIN .00000 31.000 372.00
  34.000 71.000 35.000
• 3 BUS .00000 25.000 417.00
  35.000 70.000 35.000
• 4 CAR 1.0000 10.000 180.00
  .00000 30.000 35.000
• 5 AIR .00000 58.000 68.000
  64.000 68.000 30.000
• 6 TRAIN .00000 31.000 354.00
  44.000 84.000 30.000
• 7 BUS .00000 25.000 399.00
  53.000 85.000 30.000
• 8 CAR 1.0000 11.000 255.00
  .00000 50.000 30.000
• 321 AIR .00000 127.00 193.00
  69.000 148.00 60.000
• 322 TRAIN .00000 109.00 888.00
  34.000 205.00 60.000
• 323 BUS 1.0000 52.000 1025.0
  60.000 163.00 60.000
• 324 CAR .00000 50.000 892.00
  .00000 147.00 60.000
• 325 AIR .00000 44.000 100.00
  64.000 59.000 70.000
• 326 TRAIN .00000 25.000 351.00
  44.000 78.000 70.000
• 327 BUS .00000 20.000 361.00
  53.000 75.000 70.000
• 328 CAR 1.0000 5.0000 180.00
  .00000 32.000 70.000

14
Estimated MNL Model
---------------------------------------------
Discrete choice (multinomial logit) model
Maximum Likelihood Estimates
Model estimated Jan 20, 2004 at 030511PM.
Dependent variable Choice
Weighting variable None
Number of observations 210
Iterations completed 6
Log likelihood function -199.9766
R21-LogL/LogL Log-L fncn R-sqrd RsqAdj
Constants only -283.7588 .29526 .28962
Chi-squared 2 167.56429
Prob chi squared gt value .00000
Response data are given as ind. choice.
Number of obs. 210, skipped 0 bad obs.
---------------------------------------------
----------------------------------------------
---------- Variable Coefficient Standard
Error b/St.Er.PZgtz ---------------------
----------------------------------- GC
-.01578375 .00438279 -3.601 .0003
TTME -.09709052 .01043509 -9.304
.0000 A_AIR 5.77635888 .65591872
8.807 .0000 A_TRAIN 3.92300124
.44199360 8.876 .0000 A_BUS
3.21073471 .44965283 7.140 .0000
15
Estimated MNL Model
---------------------------------------------
Discrete choice (multinomial logit) model
Maximum Likelihood Estimates
Model estimated Jan 20, 2004 at 030511PM.
Dependent variable Choice
Weighting variable None
Number of observations 210
Iterations completed 6
Log likelihood function -199.9766
R21-LogL/LogL Log-L fncn R-sqrd RsqAdj
Constants only -283.7588 .29526 .28962
Chi-squared 2 167.56429
Prob chi squared gt value .00000
Response data are given as ind. choice.
Number of obs. 210, skipped 0 bad obs.
---------------------------------------------
----------------------------------------------
---------- Variable Coefficient Standard
Error b/St.Er.PZgtz ---------------------
----------------------------------- GC
-.01578375 .00438279 -3.601 .0003
TTME -.09709052 .01043509 -9.304
.0000 A_AIR 5.77635888 .65591872
8.807 .0000 A_TRAIN 3.92300124
.44199360 8.876 .0000 A_BUS
3.21073471 .44965283 7.140 .0000
16
Estimated MNL Model
---------------------------------------------
Discrete choice (multinomial logit) model
Maximum Likelihood Estimates
Model estimated Jan 20, 2004 at 030511PM.
Dependent variable Choice
Weighting variable None
Number of observations 210
Iterations completed 6
Log likelihood function -199.9766
R21-LogL/LogL Log-L fncn R-sqrd RsqAdj
Constants only -283.7588 .29526 .28962
Chi-squared 2 167.56429
Prob chi squared gt value .00000
Response data are given as ind. choice.
Number of obs. 210, skipped 0 bad obs.
---------------------------------------------
----------------------------------------------
---------- Variable Coefficient Standard
Error b/St.Er.PZgtz ---------------------
----------------------------------- GC
-.01578375 .00438279 -3.601 .0003
TTME -.09709052 .01043509 -9.304
.0000 A_AIR 5.77635888 .65591872
8.807 .0000 A_TRAIN 3.92300124
.44199360 8.876 .0000 A_BUS
3.21073471 .44965283 7.140 .0000
17
Estimated MNL Model
---------------------------------------------
Discrete choice (multinomial logit) model
Maximum Likelihood Estimates
Model estimated Jan 20, 2004 at 030511PM.
Dependent variable Choice
Weighting variable None
Number of observations 210
Iterations completed 6
Log likelihood function -199.9766
R21-LogL/LogL Log-L fncn R-sqrd RsqAdj
Constants only -283.7588 .29526 .28962
Chi-squared 2 167.56429
Prob chi squared gt value .00000
Response data are given as ind. choice.
Number of obs. 210, skipped 0 bad obs.
---------------------------------------------
----------------------------------------------
---------- Variable Coefficient Standard
Error b/St.Er.PZgtz ---------------------
----------------------------------- GC
-.01578375 .00438279 -3.601 .0003
TTME -.09709052 .01043509 -9.304
.0000 A_AIR 5.77635888 .65591872
8.807 .0000 A_TRAIN 3.92300124
.44199360 8.876 .0000 A_BUS
3.21073471 .44965283 7.140 .0000
18
Estimated MNL Model
---------------------------------------------
Discrete choice (multinomial logit) model
Maximum Likelihood Estimates
Model estimated Jan 20, 2004 at 030511PM.
Dependent variable Choice
Weighting variable None
Number of observations 210
Iterations completed 6
Log likelihood function -199.9766
R21-LogL/LogL Log-L fncn R-sqrd RsqAdj
Constants only -283.7588 .29526 .28962
Chi-squared 2 167.56429
Prob chi squared gt value .00000
Response data are given as ind. choice.
Number of obs. 210, skipped 0 bad obs.
---------------------------------------------
----------------------------------------------
---------- Variable Coefficient Standard
Error b/St.Er.PZgtz ---------------------
----------------------------------- GC
-.01578375 .00438279 -3.601 .0003
TTME -.09709052 .01043509 -9.304
.0000 A_AIR 5.77635888 .65591872
8.807 .0000 A_TRAIN 3.92300124
.44199360 8.876 .0000 A_BUS
3.21073471 .44965283 7.140 .0000
19
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)
• Estimated model Logit probabilities
• Compute log likelihood
• Measure improvement in log likelihood with
  R-squared 1 LogL/LogL0 (Adjusted for number
  of parameters in the model.)
• NOT A MEASURE OF FIT!

20
Fit the Model with Only ASCs
Iterations completed 1
Log likelihood function -283.7588
R21-LogL/LogL Log-L fncn R-sqrd RsqAdj
Constants only -283.7588 .00000 -.00478
--------------------------------------------
------------ Variable Coefficient
Standard Error b/St.Er.PZgtz
--------------------------------------------
------------ A_AIR -.01709443
.18490682 -.092 .9263 A_TRAIN
.06559728 .18116889 .362 .7173 A_BUS
-.67634006 .22423757 -3.016
.0026 Log likelihood function -199.9766
R21-LogL/LogL Log-L fncn R-sqrd RsqAdj
Constants only -283.7588 .29526 .28962
Chi-squared 2 167.56429
Prob chi squared gt value .00000
---------------------------------------------
----------------------------------------------
---------- Variable Coefficient Standard
Error b/St.Er.PZgtz ---------------------
----------------------------------- GC
-.01578375 .00438279 -3.601 .0003
TTME -.09709052 .01043509 -9.304
.0000 A_AIR 5.77635888 .65591872
8.807 .0000 A_TRAIN 3.92300124
.44199360 8.876 .0000 A_BUS
3.21073471 .44965283 7.140 .0000
21
Descriptive Statistics
-------------------------------------------------
------------------------
Descriptive Statistics for Alternative AIR
Utility Function
58.0 observs.
Coefficient All 210.0
obs.that chose AIR Name Value
Variable Mean Std. Dev.Mean Std.
Dev. ------------------- --------
-------------------------------------- GC
-.0158 GC 102.648 30.575
113.552 33.198 TTME -.0971 TTME
61.010 15.719 46.534 24.389
A_AIR 5.7764 ONE 1.000
.000 1.000 .000 ----------------------
--------------------------------------------------
- ----------------------------------------------
---------------------------
Descriptive Statistics for Alternative TRAIN
Utility Function
63.0 observs.
Coefficient All 210.0
obs.that chose TRAIN Name Value
Variable Mean Std. Dev.Mean Std.
Dev. ------------------- --------
-------------------------------------- GC
-.0158 GC 130.200 58.235
106.619 49.601 TTME -.0971 TTME
35.690 12.279 28.524 19.354
A_TRAIN 3.9230 ONE 1.000
.000 1.000 .000 ----------------------
--------------------------------------------------
-
22
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
• Njk ? ?i dij ? Predicted P(i,k)
• Request by adding CROSSTAB to command.

23
CLOGIT Fit Measures

• Based on the log likelihood

Values in parentheses below show the number of
correct predictions by a model with only choice
specific constants.
---------------------------------------------
Log likelihood function -172.9437
Log-L for Choice model -172.9437
R21-LogL/LogL Log-L fncn R-sqrd RsqAdj
No coefficients -291.1218 .40594 .39636
Constants only -283.7588 .39053 .38070
Chi-squared 7 221.63022
Significance for chi-squared 1.00000
---------------------------------------------
Based on the model predictions
------------------------------------------------
------ Cross tabulation of actual vs.
predicted choices. Row indicator is
actual, column is predicted.
Predicted total is F(k,j,i)Sum(i1,...,N)
P(k,j,i). Column totals may be subject to
rounding error. -------------------------
----------------------------- Matrix Crosstab
has 5 rows and 5 columns. AIR
TRAIN BUS CAR Total
------------------------------------------
---------------------------- AIR 35.0000
(16) 7.0000 4.0000 13.0000
58.0000 TRAIN 7.0000 41.0000 (19)
4.0000 11.0000 63.0000 BUS
5.0000 4.0000 16.0000 (4) 4.0000
30.0000 CAR 11.0000 11.0000
6.0000 31.0000 (17) 59.0000 Total
58.0000 63.0000 30.0000
59.0000 210.0000
24
Effects of Changes in Attributes on Probabilities

• Partial Effects Effect of a change in attribute
  k of alternative m on the probability that
  choice j will be made is
• Proportional changes Elasticities

Note the elasticity is the same for all choices
j. (IIA)
25
Elasticities for CLOGIT

• Request Effects attribute (choices where
  changes occur )
• Effects INVT() (INVT changes in all choices)

-------------------------------------------------
---------------- Elasticity
Averaged over observations.
Effects on probabilities of all choices in the
model indicates direct
Elasticity effect of the attribute.
Trunk Limb Branch
Choice Effect Attribute is INVT in
choice AIR
ChoiceAIR .000 .000 .000 -1.336
-1.336 ChoiceTRAIN .000 .000
.000 .535 .535 ChoiceBUS
.000 .000 .000 .535 .535
ChoiceCAR .000 .000 .000 .535
.535 Attribute is INVT in choice
TRAIN
ChoiceAIR .000 .000 .000 2.215
2.215 ChoiceTRAIN .000 .000
.000 -6.298 -6.298 ChoiceBUS
.000 .000 .000 2.215 2.215
ChoiceCAR .000 .000 .000 2.215
2.215 Attribute is INVT in choice
BUS
ChoiceAIR .000 .000 .000 1.194
1.194 ChoiceTRAIN .000 .000
.000 1.194 1.194 ChoiceBUS
.000 .000 .000 -7.615 -7.615
ChoiceCAR .000 .000 .000 1.194
1.194 Attribute is INVT in choice
CAR
ChoiceAIR .000 .000 .000 2.085
2.085 ChoiceTRAIN .000 .000
.000 2.085 2.085 ChoiceBUS
.000 .000 .000 2.085 2.085
ChoiceCAR .000 .000 .000 -5.937
-5.937 -----------------------------------
------------------------------
Own effect Cross effects
Note the effect of IIA on the cross effects.
26
Analyzing Behavior of Market Shares

• Scenario What happens to the number of people
  how 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

27
Model Simulation
Generalized cost of CAR rises by 25
-------------------------------------------------
------ 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
16.105 34 -11.514 -24 TRAIN
30.000 63 19.515 41 -10.485 -22
BUS 14.286 30 8.299 17
-5.987 -13 CAR 28.095 59
56.081 118 27.986 59 Total
100.000 210 100.000 210 .000 0
--------------------------------------------
------------
Changes in the predicted market shares when
GC_CAR changes
28
More Complicated Model Simulation
Generalized cost of CAR falls by 10 Market is
limited to ground (Train, Bus, Car)
NLOGIT Lhs Mode Choices
Air,Train,Bus,Car Rhs TTME,INVC,INVT,GC
Rh2 One ,Hinc Simulation
TRAIN,BUS,CAR Scenario GC(car).9
29
Model Simulation
-------------------------------------------------
----- 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
.900 ---------------------------------------------
---------------------------- 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 ------------------------------------
-------------------- TRAIN 37.321 78
43.523 91 6.202 13 BUS
19.805 42 24.365 51 4.560 9
CAR 42.874 90 32.112 67
-10.762 -23 Total 100.000 210
100.000 209 .000 -1
--------------------------------------------
------------
30
Compound Scenario GC(Car) falls by 10, TTME
(Air,Train) rises by 25 (at the same time).
-------------------------------------------------
----- 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
.900 TTME AIR TRAIN
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
19.888 42 -7.731 -16 TRAIN
30.000 63 26.851 56 -3.149 -7
BUS 14.286 30 23.579 50
9.293 20 CAR 28.095 59
29.682 62 1.587 3 Total
100.000 210 100.000 210 .000 0
--------------------------------------------
------------
31
Choice Based Sampling

• Over/Underrepresenting alternatives in the data
  set
• Biases in parameter estimates? (Constants only?)
• Biases in estimated variances
• Weighted log likelihood, weight ?j / Fj for all
  i.
• Fixup of covariance matrix
• Choices list of names / list of true
  proportions

32
Choice Based Sampling Estimators
----------------------------------------------
---------- Variable Coefficient Standard
Error b/St.Er.PZgtz ---------------------
-----------------------------------
Unweighted GC .7577656131E-01
.18331991E-01 4.134 .0000 TTME
-.1028868983 .11087157E-01 -9.280 .0000
INVT -.1399485532E-01 .26709164E-02 -5.240
.0000 INVC -.8043945139E-01 .19950713E-01
-4.032 .0001 A_AIR 4.370346415
1.0573353 4.133 .0000 AIRxHIN1
.4275438233E-02 .13061691E-01 .327 .7434
A_TRAIN 5.914073895 .68992964 8.572
.0000 TRAxHIN2 -.5907284040E-01 .14709175E-01
-4.016 .0001 A_BUS 4.462691316
.72332545 6.170 .0000 BUSxHIN3
-.2295037775E-01 .15917353E-01 -1.442
.1493 ----------------------------------------
---------------- Weighted GC
.1022492766 .22662522E-01 4.512 .0000
TTME -.1361098346 .19321208E-01 -7.045
.0000 INVT -.1772099171E-01 .33128059E-02
-5.349 .0000 INVC -.1035114747
.23306867E-01 -4.441 .0000 A_AIR
4.525045167 1.2865721 3.517 .0004
AIRxHIN1 .7458987986E-02 .13402559E-01 .557
.5778 A_TRAIN 5.532288683 .71701137
7.716 .0000 TRAxHIN2 -.6026155867E-01
.17377917E-01 -3.468 .0005 A_BUS
4.365784894 .78651423 5.551 .0000
BUSxHIN3 -.1956868658E-01 .17288002E-01 -1.132
.2577
33
Changes in Estimated Elasticities
-------------------------------------------------
---------------- Elasticity
Averaged over observations.
Attribute is GC in choice CAR
Effects on probabilities of all
choices in the model indicates
direct Elasticity effect of the attribute.
Unweighted
ChoiceAIR
.000 .000 .000 -1.922 -1.922
ChoiceTRAIN .000 .000 .000 -1.922
-1.922 ChoiceBUS .000 .000
.000 -1.922 -1.922 ChoiceCAR
.000 .000 .000 5.308 5.308
-----------------------------------------------
------------------ Weighted

ChoiceAIR .000 .000 .000 -4.482
-4.482 ChoiceTRAIN .000 .000
.000 -4.482 -4.482 ChoiceBUS
.000 .000 .000 -4.482 -4.482
ChoiceCAR .000 .000 .000 5.274
5.274 -----------------------------------
------------------------------
34
The I.I.D Assumption

• Uitj ?ij ?i xitj ?izit ?ijt
• F(?itj) 1 Exp(-Exp(?itj)) (random part of
  each utility)
• Independence across utility functions
• Identical variances (means absorbed in constants)
• Restriction on scaling
• Correlation across alternatives?
• Implication for cross elasticities (we saw
  earlier)
• Behavioral assumption, independence from
  irrelevant alternatives (IIA)

35
A Test for IIA?

• Estimate full model with irrelevant
  alternatives
• Estimate 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?
• Hausman test
• Chi-squared, d.f. Number of parameters estimated
• Fit the model, then again with
• IAS the irrelevant
  alternative(s)

36
Is AIR Irrelevant to Mode Choice?
---------------------------------------------
Log likelihood function -244.1342
Log-L for Choice model -244.1342
R21-LogL/LogL Log-L fncn R-sqrd RsqAdj
No coefficients -291.1218 .16140 .15604
Constants only -283.7588 .13964 .13414
Response data are given as ind. choice.
Number of obs. 210, skipped 0 bad obs.
---------------------------------------------
----------------------------------------------
---------- Variable Coefficient Standard
Error b/St.Er.PZgtz ---------------------
----------------------------------- GC
.3182945727E-01 .13728561E-01 2.318 .0204
TTME -.3480666872E-01 .46939661E-02 -7.415
.0000 INVT -.6344728477E-02 .18416761E-02
-3.445 .0006 INVC -.2242963034E-01
.14354086E-01 -1.563 .1181 ------------------
--------------------------- Log likelihood
function -96.34853 Number of obs.
210, skipped 58 bad obs. Hausman test for
IIA. Excluded choices are AIR
ChiSqrd 4
62.9330, Pr(Cgtc) .000000 ------------------
--------------------------- -------------------
------------------------------------- Variabl
e Coefficient Standard Error
b/St.Er.PZgtz --------------------------
------------------------------ GC
.4311545877 .13318989 3.237 .0012
TTME -.2241209049E-02 .71428544E-02 -.314
.7537 INVT -.7723644784E-01 .19351682E-01
-3.991 .0001 INVC -.4351129783
.13277876 -3.277 .0010
The Hausman statistic is large. Reject IIA.
37
Behavioral Issue Omitted Attributes

• Do all consumers evaluate all attributes?
• Information processing strategy minimize
  processing cost
• Lexicographic preferences some attributes are
  irrelevant.
• Do we know which attributes are evaluated?
• How to incorporate omitted attributes information
  in the model
• Zero fill in the data. Zero is not a valid
  PRICE.
• Change the equation True zeros in index
  functions

38
Modeling Attribute Choice

• Conventional Uijt ?'xijt. For ignored
  attributes, set xk,ijt 0. Eliminates xk,ijt
  from utility function
• Price 0 is not a reasonable datum. Distorts
  choice probabilities
• Appropriate Formally set ?k 0
• Requires a person specific model
• Accommodate as part of model estimation
• (Work in progress) Stochastic determination of
  attribution choices

39
Choice Strategy Heterogeneity

• Methodologically, a rather minor point
  construct appropriate likelihood given known
  information
• Not a latent class model. Classes are not latent.
• Not the variable selection issue (the worst
  form of stepwise modeling)
• Familiar strategy gives the wrong answer.

40
Application Sydney Commuters Route Choice

• Stated Preferences
• Multinomial and Mixed Logit
• Consumers included data on which attributes were
  ignored.
• (Ignored attributes coded -888 in NLOGIT are
  automatically treated correctly in model
  estimation.)

41
Application of Information Strategy

• Stated/Revealed preference study, Sydney car
  commuters. 500 surveyed, about 10 choice
  situations for each.
• Existing route vs. 3 proposed alternatives.
• Attribute design
• Original respondents presented with 3, 4, 5, or
  6 attributes
• Attributes four level design.
• Free flow time
• Slowed down time
• Stop/start time
• Trip time variability
• Toll cost
• Running cost
• Final respondents use only some attributes and
  indicate when surveyed which ones they ignored

42
Stated Preference Survey
43
(No Transcript)
44
Estimated Models
45
Estimated Elasticities and WTP
46
Discrete Choice Model Extensions

• Heteroscedasticity and other forms of
  heterogeneity
• Across individuals
• Across alternatives
• Panel data (Repeated measures)
• Random and fixed effects models
• Building into a multinomial logit model
• The IIA assumption
• The nested logit model
• Latent class model
• Mixed logit, error components and multinomial
  probit models
• Combining revealed and stated preference data