Modeling Consumer Decision Making and Discrete Choice Behavior

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(No Transcript)
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A Masterclass Discrete Choice ModelingFrontier
Modeling and Efficiency EstimationProfessor
William GreeneStern School of BusinessNew York
UniversitySeptember 2-4, 2007
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Multinomial Choice Modeling

• Session 1
• Random Utility Models
• Binary Choice Modeling
• Extensions Heterogeneity, Semiparametrics, Panel
  Data
• Session 2
• Multinomial Choice
• Multinomial Logit, Nested Logit, Scaling
• Models for Heterogeneity
• Session 3
• Mixed Logit and Error Components
• Stated Preferences and Choice Experiments, Panel
  Data
• Student Presentation Peter Silvey
• Session 4 Computer Exercises and Applications

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Random Indirect Utility Functions
U(i,t,j) ?(i,j) ?(i)x(i,t,j) ?(i)z(i,t)
?(i,t,j) V(i,t,j) ?(i,t,j)
?(i,j) Choice specific constant x(i,t,j)
Attributes of choice presented to person,
such as Price ?(i) Person specific taste
weights z(i,t) Characteristics of the
person (age,income) ?(i) Weights on person
specific characteristics ?(i,t,j) Unobserved
random component of utility
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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

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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

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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)

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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).

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Willingness to Pay

• Generally ratio of coefficients
• -?cost or price / ?attribute level
• When the negative of a direct price coefficient
  is unavailable, might be able to use a
  coefficient on income

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Value of Time

• We can also compute the value of time as
• If the direct cost measure is unavailable, use
  the negative of the income coefficient.
  (Numerator will generally be negative.)

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Observed Data

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

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Data on Discrete Choices

• Line MODE TRAVEL INVC INVT
  TTME GC HINC
• i1
• 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
• i2
• 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
• i81
• 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
• i82
• 325 AIR .00000 44.000 100.00
  64.000 59.000 70.000

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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
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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
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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
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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
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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
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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!

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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
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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 ----------------------
--------------------------------------------------
-
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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)

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LOGIT 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
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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)
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Elasticities for CLOGIT
-------------------------------------------------
---------------- 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 IID on the cross effects.
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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

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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
Evidently something wrong with this model.
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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

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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
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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 -----------------------------------
------------------------------
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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). (Red Bus-Blue Bus
  Problem)

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The Red Bus-Blue Bus Problem

• Suppose the relative probability of choosing the
  bus over a car is 21 (this is the odds ratio).
  Initially only a red bus is available.
• Then another bus comes available, which is blue,
  but otherwise identical to the red bus. It seems
  sensible that the relative probability of
  choosing any bus over a car is 21, so the
  relative probability of choosing each bus over
  the car should be 11

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Red Bus-Blue Bus Problem (contd)

• But for the logit, the relative probability Pi/Pk
  is unchanged when a new alternative become
  available, so the red bus still has 21 odds
  ratio over a car.
• Therefore, the blue bus also has a 21 odds of
  being chosen (as it is identical to the red).
• So the relative probability of taking any bus
  over a car becomes 41 when the blue bus is added.

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Red bus-blue bus problem (contd)

• Why does this problem with the logit arise?
• Because we have assumed that the random terms in
  the utility functions are independent and
  identically distributed.
• This contrasts with the multinomial probit model,
  where it is only assumed that the differences in
  random terms are normally distributed, and these
  errors are correlated
• Solution Any model that does not make the IID
  assumption will escape the IIA problem.
  Nonindependence or heteroscedasticity suffice.

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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

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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.
36
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

37
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

38
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.

39
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.)

40
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

41
Stated Preference Survey
42
(No Transcript)
43
Estimated Models
44
Estimated Elasticities and WTP
45
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

46
Whats Wrong with the MNL Model?

• I.I.D. ? IIA (Independence from irrelevant
  alternatives)
• Peculiar behavioral assumption
• Leads to skewed, implausible empirical results
• Functional forms, e.g., nested logit, avoid IIA
• IIA will be a nonissue in what follows.
• Insufficiently heterogeneous
• economists are often more interested in
  aggregate effects and regard heterogeneity as a
  statistical nuisance parameter problem which must
  be addressed but not emphasized. Econometricians
  frequently employ methods which do not allow for
  the estimation of individual level parameters.
  (Allenby and Rossi, Journal of Econometrics, 1999)

47
Variance Heterogeneity in MNL

• We extend the MNL model
• Uij ?j ?xij ?zi ?j?ij, i
  1,,N j 1,,J(i)
• ?j exp(?j). Only variance ratios can be
  estimated.
• dJ 0. Var(?ij)
  p2/6
• F(?ij) 1 Exp(-Exp(?ij)) is now scaled
  by choice

48
Heteroscedastic Extreme Value Model (1)
---------------------------------------------
Start values obtained using nonnested model
Maximum Likelihood Estimates
Dependent variable Choice
Weighting variable None
Number of observations 210
Iterations completed 6
Log likelihood function -184.5067
Log-L for Choice model -184.5067
R21-LogL/LogL Log-L fncn R-sqrd RsqAdj
No coefficients -291.1218 .36622 .35910
Constants only -283.7588 .34978 .34247
Chi-squared 4 198.50415
Significance for chi-squared 1.00000
Response data are given as ind. choice.
Number of obs. 210, skipped 0 bad obs.
---------------------------------------------
----------------------------------------------
---------- Variable Coefficient Standard
Error b/St.Er.PZgtz ---------------------
----------------------------------- TTME
-.1036495471 .10938147E-01 -9.476 .0000
INVC -.8493181748E-01 .19382513E-01 -4.382
.0000 INVT -.1333219613E-01 .25169842E-02
-5.297 .0000 GC .6929537303E-01
.17433063E-01 3.975 .0001 A_AIR
5.204742747 .90521312 5.750 .0000
A_TRAIN 4.360604565 .51066543 8.539
.0000 A_BUS 3.763234465 .50625946
7.433 .0000
49
Heteroscedastic Extreme Value Model (2)
---------------------------------------------
Heteroskedastic Extreme Value Model
Maximum Likelihood Estimates
Dependent variable MODE
Weighting variable None
Number of observations 840
Iterations completed 33
Log likelihood function -182.4440
Restricted log likelihood -291.1218
Chi-squared 217.3557
Degrees of freedom 10
Significance level .0000000
R21-LogL/LogL Log-L fncn R-sqrd RsqAdj
No coefficients -291.1218 .37331 .36320
Constants only -283.7588 .35705 .34668
At start values -218.6505 .16559 .15213
Response data are given as ind. choice.
---------------------------------------------
----------------------------------------------
---------- Variable Coefficient Standard
Error b/St.Er.PZgtz ---------------------
-----------------------------------
Attributes in the Utility Functions (beta) TTME
-.1152558142 .57213967E-01 -2.014
.0440 INVC -.1551587735 .79280449E-01
-1.957 .0503 INVT -.2276938838E-01
.11227621E-01 -2.028 .0426 GC
.1190351260 .64025102E-01 1.859 .0630
A_AIR 4.694114603 2.4809179 1.892
.0585 A_TRAIN 5.156298676 2.0574376
2.506 .0122 A_BUS 5.030465948
1.9825935 2.537 .0112 Scale
Parameters of Extreme Value Distns. s_AIR
.4213572229 .21991837 1.916 .0554
s_TRAIN .5412144144 .34971034 1.548
.1217 s_BUS 1.260948354 .94582863
1.333 .1825 s_CAR 1.000000000
........(Fixed Parameter)........
Std.Devpi/(sigmasqr(6)) for H.E.V.
distribution. s_AIR 3.043853838
1.5886743 1.916 .0554 s_TRAIN
2.369762826 1.5312426 1.548 .1217
s_BUS 1.017131111 .76294300 1.333
.1825 s_CAR 1.282549800 ........(Fixed
Parameter)........
Normalized for estimation
Structural parameters
50
HEV Model - Elasticities
------------------------------------------------
------------------------- Elasticity
Averaged over observations. MNL
Model indicates direct Elasticity effect of
the attribute. Attribute is
INVT in choice AIR
ChoiceAIR .000 .000
.000 -.998 -.998 (-1.270)
ChoiceTRAIN .000 .000 .000 .358
.358 ( .503) ChoiceBUS .000
.000 .000 .727 .727 ( .503)
ChoiceCAR .000 .000 .000 .609
.609 ( .503) Attribute is INVT in
choice TRAIN
ChoiceAIR .000 .000 .000
1.425 1.425 ( 2.044) ChoiceTRAIN
.000 .000 .000 -6.293 -6.293
(-6.066) ChoiceBUS .000 .000
.000 4.957 4.957 ( 2.044)
ChoiceCAR .000 .000 .000 3.401
3.401 ( 2.044) Attribute is INVT in
choice BUS
ChoiceAIR .000 .000 .000
1.013 1.013 ( 1.148) ChoiceTRAIN
.000 .000 .000 1.594 1.594 (
1.148) ChoiceBUS .000 .000
.000 -18.278 -18.278 (-7.245)
ChoiceCAR .000 .000 .000 5.338
5.338 ( 1.148) Attribute is INVT in
choice CAR
ChoiceAIR .000 .000 .000
1.671 1.671 ( 2.040) ChoiceTRAIN
.000 .000 .000 2.447 2.447 (
2.040) ChoiceBUS .000 .000
.000 7.432 7.432 ( 2.040)
ChoiceCAR .000 .000 .000 -12.254
-12.254 (-5.602) ------------------------------
-------------------------------------------
51
Variance Heterogeneity in MNL

• We extend the HEV model
• Uij ?j ?xij ?zi ?ij?ij
• i 1,,N j 1,,J(i)
• ?ij exp(?j ?wi)
• F(?ij) 1 Exp(-Exp(?ij)) is now scaled
  by choice and person

52
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 (lt25, 25-39, 40)
• Underlying data generated by a 3 class latent
  class process (100, 200, 100 in classes)
• Thanks to www.statisticalinnovations.com (Latent
  Gold)

53
NLOGIT Commands for HEV Model
Nlogit lhschoice choicesBrand1,Brand2,Br
and3,None1,2,3,4 Rhs F1,F2,F3,F4,Q1,Q2,Q3,Q4
,P1,P2,P3,P4,N1,N2,N3,N4 AttrFashion,Quality,Pr
ice,NONE_ASC heteroscedasticity hfnmale,agel2
5,age2539
54
MNL
55
Heteroscedasticity Across Choices
Essentially no differences in variances across
choices
56
Variance Heterogeneity
57
The Multinomial Probit Model

• Uj ?j ?xj ?jz ?j
• Correlated normal with free variances
• Implied probabilities for observed outcomes, J
  variate normal CDF (J-1 relative values)
• Probability requires simulation

58
Multinomial Probit Model

----------------------------------------------
---------- Variable Coefficient Standard
Error b/St.Er.PZgtz ---------------------
-----------------------------------
Attributes in the Utility Functions (beta) GC
-.1723478613E-01 .77369459E-02 -2.228
.0259 TTME -.6079153929E-01 .23260331E-01
-2.614 .0090 INVT -.4128962447E-02
.13121951E-02 -3.147 .0017 HINCA
.5395491283E-01 .32252594E-01 1.673 .0943
A_AIR -1.291786620 1.5713258 -.822
.4110 A_TRAIN 3.250173238 1.0013602
3.246 .0012 A_BUS 2.836341831
.93447229 3.035 .0024 Std. Devs.
of the Normal Distribution. sAIR
4.290644619 1.6562711 2.591 .0096
sTRAIN 1.255252723 .70455295 1.782
.0748 sBUS 1.000000000 ........(Fixed
Parameter)........ sCAR 1.000000000
........(Fixed Parameter)........
Correlations in the Normal Distribution rAIR,TRA
.6198374877 .35576890 1.742 .0815
rAIR,BUS .7365723924 .34029323 2.165
.0304 rTRA,BUS .5973376260 .31285316
1.909 .0562 rAIR,CAR .0000000000
........(Fixed Parameter)........ rTRA,CAR
.0000000000 ........(Fixed Parameter)........
rBUS,CAR .0000000000 ........(Fixed
Parameter)........
59
MNP Elasticities

• -------------------------------------------
  ----------------------
• Attribute is INVT in choice AIR
  
• ChoiceAIR .000 .000
  .000 -.185 -.185
• ChoiceTRAIN .000 .000
  .000 .108 .108
• ChoiceBUS .000 .000
  .000 .139 .139
• ChoiceCAR .000 .000
  .000 .040 .040
• Attribute is INVT in choice TRAIN
  
• ChoiceAIR .000 .000
  .000 .370 .370
• ChoiceTRAIN .000 .000
  .000 -3.024 -3.024
• ChoiceBUS .000 .000
  .000 2.362 2.362
• ChoiceCAR .000 .000
  .000 .942 .942
• Attribute is INVT in choice BUS
  
• ChoiceAIR .000 .000
  .000 .230 .230
• ChoiceTRAIN .000 .000
  .000 1.112 1.112
• ChoiceBUS .000 .000
  .000 -5.514 -5.514
• ChoiceCAR .000 .000
  .000 .748 .748
• Attribute is INVT in choice CAR
  
• ChoiceAIR .000 .000
  .000 .155 .155
• ChoiceTRAIN .000 .000
  .000 1.114 1.114

60
Extended Formulation

• Clusters of similar alternatives
• Compound Utility U(Alt)U(AltBranch)U(branch)
• Behavioral implications Correlations across
  branches

Travel
LIMB
Private
Public
BRANCH
TWIG
Air
Car
Train
Bus
61
The Nested Logit Tree Structure
The NL model is often presented as a tree
structure, where the connections at each level
define the error variances within the model
Trunks (Level 4)
Limbs (Level 3)
Branches (Level 2)
Alternatives or twigs (Level 1)
Note Respondents are exposed to alternatives in
Level 1 only.
62
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
• ProbAlt,Branch Prob(branch)
  Prob(AltBranch)

63
Probabilities for a Nested Logit Model

• Utility functions notation (drop observation
  i,.)
• Twig level k j denotes alt given branch
• ?kj ?xkj
• Branch level ?yj
• Twig level probability
• Inclusive value for the branch IV(j)
• Branch probability

?j 1 for all branches returns the MNL model.
64
Estimation Strategy

• Familiar two step estimation
• For each branch, just fit MNL
• Loses efficiency replicates coefficients
• Does not insure consistency with utility
  maximization
• For branch level, fit separate model, just
  including y and the inclusive values
• Again loses efficiency
• Definitely not consistent with utility
  maximization Note form of branch probability
• Full information ML. Just fit the entire model
  all at once, imposing all restrictions

65
Estimates of a Nested Logit Model
NLOGIT lhsmode rhsgc,ttme,invt,invc
rh2one,hinc choicesair,train,bus,ca
r treeTravelPrivate(Air,Car),
Public(Train,Bus) show tree
effectsgc() Describe ru1 Selects
the branch normalization
66
Model Structure
Tree Structure Specified for the Nested Logit
Model Sample proportions are marginal,
not conditional. Choice Based
Choices marked with are excluded for the IIA
test. Sampling -----------------------------
-------------------------------------------- T
runk (prop.)Limb (prop.)Branch
(prop.)Choice (prop.)WeightIIA --------------
-----------------------------------------------
------------ Trunk1 1.00000TRAVEL
1.00000PRIVATE .55714AIR .27619
1.000
CAR .28095 1.000
PUBLIC .44286TRAIN
.30000 1.000
BUS .14286
1.000 -----------------------------------------
-------------------------------- Model
Specification Utility Functions for
Alternatives Table entry is the attribute that
multiplies the indicated parameter.
Parameter Row 1 GC TTME INVT
INVC A_AIR AIRxHIN1 A_TRAIN Row 2
TRAxHIN3 A_BUS BUSxHIN4 Choice AIR 1
GC TTME INVT INVC Constant
HINC 2 CAR 1 GC TTME
INVT INVC 2 TRAIN 1 GC
TTME INVT INVC
Constant 2 HINC BUS 1 GC
TTME INVT INVC 2
Constant HINC
67
MNL Starting Values
---------------------------------------------
Discrete choice (multinomial logit) model
Maximum Likelihood Estimates
Dependent variable Choice
Weighting variable None
Number of observations 210
Iterations completed 6
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
Response data are given as ind. choice.
Number of obs. 210, skipped 0 bad obs.
---------------------------------------------
----------------------------------------------
---------- Variable Coefficient Standard
Error b/St.Er.PZgtz ---------------------
----------------------------------- 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 TRAxHIN3 -.5907284040E-01
.14709175E-01 -4.016 .0001 A_BUS
4.462691316 .72332545 6.170 .0000
BUSxHIN4 -.2295037775E-01 .15917353E-01 -1.442
.1493
68
FIML Parameter Estimates
---------------------------------------------
FIML Nested Multinomial Logit Model
Log likelihood function -166.6483
Restricted log likelihood -291.1218
Chi-squared 248.9469
Degrees of freedom 12
Significance level .0000000
R21-LogL/LogL Log-L fncn R-sqrd RsqAdj
No coefficients -291.1218 .42756 .41645
Constants only -283.7588 .41271 .40131
At start values -172.9437 .03640 .01769
---------------------------------------------
FIML Nested Multinomial Logit Model
The model has 2 levels.
Random Utility Form 1IV parms are lmdaji
Number of obs. 210, skipped 0 bad obs.
---------------------------------------------
----------------------------------------------
---------- Variable Coefficient Standard
Error b/St.Er.PZgtz ---------------------
-----------------------------------
Attributes in the Utility Functions (beta) GC
.6579426257E-01 .18775111E-01 3.504
.0005 TTME -.7738173146E-01 .12171535E-01
-6.358 .0000 INVT -.1335082674E-01
.26981706E-02 -4.948 .0000 INVC
-.7045959659E-01 .20521563E-01 -3.433 .0006
A_AIR 2.493635860 1.0108408 2.467
.0136 AIRxHIN1 .3567490789E-02 .10571112E-01
.337 .7358 A_TRAIN 3.498674921
.80634347 4.339 .0000 TRAxHIN3
-.3580829277E-01 .13788318E-01 -2.597 .0094
A_BUS 2.301421632 .81284159 2.831
.0046 BUSxHIN4 -.1128147854E-01 .14593705E-01
-.773 .4395 IV parameters,
lambda(ji),gamma(i) PRIVATE 2.160948513
.47192642 4.579 .0000 PUBLIC
1.562945071 .34500462 4.530 .0000
Underlying standard deviation
pi/(IVparmsqr(6)) PRIVATE .5935124287
.12961632 4.579 .0000 PUBLIC
.8205981284 .18113890 4.530 .0000
69
------------------------------------------------
-------------------------
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
.1151 GC 102.648 30.575
113.552 33.198 TTME -.1341 TTME
61.010 15.719 46.534 24.389
INVT -.0226 INVT 133.710
48.521 124.828 50.288 INVC
-.1257 INVC 85.252 27.409 97.569
31.733 A_AIR 3.9713 ONE
1.000 .000 1.000 .000 AIRxHIN1
.0151 HINC 34.548 19.711
41.724 19.115 ------------------------------
-------------------------------------------
Descriptive Statistics for Alternative
CAR Utility Function
59.0 observs.
Coefficient All
210.0 obs.that chose CAR Name
Value Variable Mean Std. Dev.Mean
Std. Dev. ------------------- --------
-------------------------------------- GC
.1151 GC 95.414 46.827
89.085 49.833 TTME -.1341 TTME
.000 .000 .000 .000
INVT -.0226 INVT 573.205
274.855 527.373 301.131 INVC
-.1257 INVC 20.995 14.678 15.644
9.629 --------------------------------------
-----------------------------------
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
.1151 GC 130.200 58.235
106.619 49.601 TTME -.1341 TTME
35.690 12.279 28.524 19.354
INVT -.0226 INVT 608.286
251.797 532.667 249.360 INVC
-.1257 INVC 51.338 27.032 37.460
20.676 A_TRAIN 7.7651 ONE
1.000 .000 1.000 .000 TRAxHIN3
-.0589 HINC 34.548 19.711
23.063 17.287 ------------------------------
-------------------------------------------
Descriptive Statistics for Alternative
BUS Utility Function
30.0 observs.
Coefficient All
210.0 obs.that chose BUS Name
Value Variable Mean Std. Dev.Mean
Std. Dev. ------------------- --------
-------------------------------------- GC
.1151 GC 115.257 44.934
108.133 43.244 TTME -.1341 TTME
41.657 12.077 25.200 14.919
INVT -.0226 INVT 629.462
235.408 618.833 273.610 INVC
-.1257 INVC 33.457 12.591 33.733
11.023 A_BUS 6.0530 ONE
1.000 .000 1.000 .000 BUSxHIN4
-.0163 HINC 34.548 19.711
29.700 16.851 ------------------------------
-------------------------------------------
70
Estimated Elasticities
-------------------------------------------------
---------------- Elasticity
Averaged over observations.
Effects on probabilities of all choices in the
model indicates direct
Elasticity effect of the attribute.
Decomposition of Effect
Total Trunk
Limb Branch Choice Effect Attribute is
GC in choice AIR
BranchPRIVATE
ChoiceAIR
.000 .000 2.724 3.523 6.246
ChoiceTRAIN .000 .000 2.724 -3.231
-.507 BranchPUBLIC
ChoiceBUS
.000 .000 -4.258 .000 -4.258
ChoiceCAR .000 .000 -4.258 .000
-4.258 Attribute is GC in choice
TRAIN
BranchPRIVATE
ChoiceAIR .000
.000 3.180 -3.074 .105
ChoiceTRAIN .000 .000 3.180 3.203
6.383 BranchPUBLIC
ChoiceBUS
.000 .000 -3.464 .000 -3.464
ChoiceCAR .000 .000 -3.464 .000
-3.464 Attribute is GC in choice
BUS
BranchPRIVATE
ChoiceAIR .000
.000 -3.357 .000 -3.357
ChoiceTRAIN .000 .000 -3.357 .000
-3.357 BranchPUBLIC
ChoiceBUS
.000 .000 4.747 3.382 8.128
ChoiceCAR .000 .000 4.747 -5.185
-.438 Attribute is GC in choice
CAR indicates
direct Elasticity effect of the attribute.
BranchPRIVATE
ChoiceAIR
.000 .000 -1.620 .000 -1.620
ChoiceTRAIN .000 .000 -1.620 .000
-1.620 BranchPUBLIC
ChoiceBUS
.000 .000 2.737 -2.788 -.051
ChoiceCAR .000 .000 2.737 4.795
7.532 -----------------------------------
------------------------------
Note the decomposition
71
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 -166.6843
• LogL (MNL) -172.9437
• Chi-squared with 2 d.f. 2(-166.6843-(-172.9437))
  12.5188
• The MNL is rejected

72
Moving Scaling Down to Twig Level
Use RU2 for this normalization
µj 1 for all branches returns the MNL model.
73
Utility Maximization

• µj 1 within branch equal correlation
• If 0 lt µj 1, probabilities consistent with
  utility maximization for all xij
• If µj gt 1, probabilities are consistent with
  utility maximization for some xij.
• If µj 0, probabilities not consistent with
  utility maximization
• NLOGIT allows µj exp(dzi) covariance
  heterogeneity.

74
Higher Level Trees
E.g., Location (Neighborhood) Housing
Type (Rent, Buy, House, Apt) Housing (
Bedrooms)
75
Degenerate Branches
Travel
LIMB
Fly
Ground
BRANCH
Air
TWIG
Train
Bus
Car
76
NL Model with Degenerate Branch
----------------------------------------------
---------- Variable Coefficient Standard
Error b/St.Er.PZgtz ---------------------
-----------------------------------
Attributes in the Utility Functions (beta) GC
.6527311416E-01 .17874605E-01 3.652
.0003 TTME -.6114254564E-01 .11185422E-01
-5.466 .0000 INVT -.1231199082E-01
.28277279E-02 -4.354 .0000 INVC
-.7018419777E-01 .19512209E-01 -3.597 .0003
A_AIR 1.225446380 .87245443 1.405
.1601 AIRxHIN1 .1501487772E-01 .12257418E-01
1.225 .2206 A_TRAIN 3.444084262
.68388215 5.036 .0000 TRAxHIN2
-.2822807019E-01 .85245516E-02 -3.311 .0009
A_BUS 2.583999103 .63247372 4.086
.0000 BUSxHIN4 -.7261929758E-02 .10749879E-01
-.676 .4993 IV parameters, RU2
form mu(ji),gamma(i) FLY 1.000000000
........(Fixed Parameter)........ GROUND
2.093001135 .46033753 4.547 .0000
Underlying standard deviation
pi/(IVparmsqr(6)) FLY 1.282549800
........(Fixed Parameter)........ GROUND
.6127802696 .13477573 4.547 .0000
77
Estimates of a Nested Logit Model
NLOGIT lhsmode rhsgc,ttme,invt,invc
rh2one,hinc choicesair,train,bus,ca
r treeTravelFly(Air),
Ground(Train,Car,Bus) show tree
effectsgc() Describe ru2 (This
is RANDOM UTILITY FORM 2. The different
normalization shows the effect of the degenerate
branch.)
78
Using Degenerate Branches to Reveal Scaling
Travel
LIMB
Fly
Rail
BRANCH
Drive
GrndPblc
TWIG
Air
Car
Train
Bus
79
Scaling in Transport Modes
----------------------------------------------
---------- Variable Coefficient Standard
Error b/St.Er.PZgtz ---------------------
-----------------------------------
Attributes in the Utility Functions (beta) GC
.9622486614E-01 .38754476E-01 2.483
.0130 TTME -.8330878028E-01 .26973001E-01
-3.089 .0020 INVC -.1090428879
.36769818E-01 -2.966 .0030 INVT
-.1888151144E-01 .68412845E-02 -2.760 .0058
A_AIR 4.508271115 1.3306195 3.388
.0007 A_TRAIN 3.355795339 .90489560
3.708 .0002 A_BUS 3.118848372
1.3313845 2.343 .019