William Greene Department of Economics Stern School of Business

Empirical Methods for Microeconomic Applications

University of Lugano, SwitzerlandMay 27-31, 2013

William GreeneDepartment of EconomicsStern 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

• Observed data usually consist of choices, prices, income• The Integrability Problem: Utility is not revealed

by demands

j*

j

V( ,I)/ px = -

V( ,I)/ Ipp

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 ATTRIBUTES CHARACTERISTICMODE TRAVEL INVC INVT TTME GC HINCAIR .00000 59.000 100.00 69.000 70.000 35.000TRAIN .00000 31.000 372.00 34.000 71.000 35.000BUS .00000 25.000 417.00 35.000 70.000 35.000CAR 1.0000 10.000 180.00 .00000 30.000 35.000AIR .00000 58.000 68.000 64.000 68.000 30.000TRAIN .00000 31.000 354.00 44.000 84.000 30.000BUS .00000 25.000 399.00 53.000 85.000 30.000CAR 1.0000 11.000 255.00 .00000 50.000 30.000AIR .00000 127.00 193.00 69.000 148.00 60.000TRAIN .00000 109.00 888.00 34.000 205.00 60.000BUS 1.0000 52.000 1025.0 60.000 163.00 60.000CAR .00000 50.000 892.00 .00000 147.00 60.000AIR .00000 44.000 100.00 64.000 59.000 70.000TRAIN .00000 25.000 351.00 44.000 78.000 70.000BUS .00000 20.000 361.00 53.000 75.000 70.000CAR 1.0000 5.0000 180.00 .00000 32.000 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

],

itj i i,t,j i,t,k

j itj j iJ(i,t)

j itj j ij=1

P[choice = j | , ,i, t] = Prob[U U k = 1,...,J(i,t)

exp(α + + ' ) =

exp(α + ' + ' )

x zβ'x γ z

β x γ z

Multinomial Choice Models

Conditional logit model depends on at

Multinomial logit model

tribut

depends on characteri i

s

t cs

e

s

j j ii J(i)

j j tj=1

j itjitj

exp(α + ' ) P[choice = j | ,i] =

exp(α + ' )

exp(α + ) P[choice = j | ,i,t] =

exp

γ zz

γ z

β'xx

THE multinomial logit model accommodates both.

There is no meaningful distinction.

J(i,t)j itjj=1

j itj j iitj i J(i,t)

j itj j ij=1

(α + ' )

exp(α + + ' ) P[choice = j | , ,i,t] =

exp(α + ' + ' )

β x

β'x γ zx z

β x γ z

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

],

itj i i,t,j i,t,k

j itj j iJ(i,t)

j itj j ij=1

N J(i,t)iji=1 j=1

P[choice = j | , ,i,t] = Prob[U U k = 1,...,J(i,t)

exp(α + + ' ) =

exp(α + ' + ' )

logL = d log

x zβ'x γ z

β x γ z

ijP

Using the Model to Measure Consumer Surplus

J(i,t)j itj j

j j

itj=1

Maximum (U ) Consumer Surplus =

Marginal Utility of IncomeUtility and marginal utility are not observableFor the multinomial logit model (only),

exp(α + ' + ' )β x γ zI

1E[CS]= log +MUjWhere U = the utility of the indicated alternative and C

is the constant of integration. The log sum is the "inclusive value." (The sum is thedenominator of the probability.)

C

Measuring the Change in Consumer Surplus

J(i,t)j itj j ij=1

J(i,t)j itj=

I

j ij 1

MU and the constant of integration

exp(α + ' + ' ) | A

exp(α +

do not change under

' +

scenarios.

' ) |B

β x γ z

β x γ z

I

I

1E[CS| Scenario A]= log +A ConstantMU1E[CS| Scenario B]= log +A ConstantMU

J(i,t)j itj j ij=1

J(i,t)j itj j ij=1

Change in expected consumer surplus from a polic

exp(α + ' + ' ) | A

exp(α + ' + ' ) |B

y (scenario) change

β x γ z

β x γ zI

E[CS| Scenario A] - E[CS| Scenario B]

1 = logMU

Willingness to PayGenerally a ratio of coefficients

β(Attribute Level) WTP = β(Income)

Use negative of cost coefficient as a proxu for MU of income

negative β(Attribute Level) WTP = 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

• Individual choice• Market shares – consumer markets• Frequencies – vote counts• Ranks – contests, preference rankings

• Attributes and Characteristics• 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 ATTRIBUTES CHARACTERISTICMODE TRAVEL INVC INVT TTME GC HINCAIR .00000 59.000 100.00 69.000 70.000 35.000TRAIN .00000 31.000 372.00 34.000 71.000 35.000BUS .00000 25.000 417.00 35.000 70.000 35.000CAR 1.0000 10.000 180.00 .00000 30.000 35.000AIR .00000 58.000 68.000 64.000 68.000 30.000TRAIN .00000 31.000 354.00 44.000 84.000 30.000BUS .00000 25.000 399.00 53.000 85.000 30.000CAR 1.0000 11.000 255.00 .00000 50.000 30.000AIR .00000 127.00 193.00 69.000 148.00 60.000TRAIN .00000 109.00 888.00 34.000 205.00 60.000BUS 1.0000 52.000 1025.0 60.000 163.00 60.000CAR .00000 50.000 892.00 .00000 147.00 60.000AIR .00000 44.000 100.00 64.000 59.000 70.000TRAIN .00000 25.000 351.00 44.000 78.000 70.000BUS .00000 20.000 361.00 53.000 75.000 70.000CAR 1.0000 5.0000 180.00 .00000 32.000 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 alternativesThe 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) modelDependent variable ChoiceLog likelihood function -199.97662Estimation based on N = 210, K = 5Information Criteria: Normalization=1/N Normalized UnnormalizedAIC 1.95216 409.95325Fin.Smpl.AIC 1.95356 410.24736Bayes IC 2.03185 426.68878Hannan Quinn 1.98438 416.71880R2=1-LogL/LogL* Log-L fncn R-sqrd R2AdjConstants only -283.7588 .2953 .2896Chi-squared[ 2] = 167.56429Prob [ chi squared > value ] = .00000Response data are given as ind. choicesNumber 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--------+--------------------------------------------------

-----------------------------------------------------------Discrete choice (multinomial logit) modelDependent variable ChoiceLog likelihood function -199.97662Estimation based on N = 210, K = 5Information Criteria: Normalization=1/N Normalized UnnormalizedAIC 1.95216 409.95325Fin.Smpl.AIC 1.95356 410.24736Bayes IC 2.03185 426.68878Hannan Quinn 1.98438 416.71880R2=1-LogL/LogL* Log-L fncn R-sqrd R2AdjConstants only -283.7588 .2953 .2896Chi-squared[ 2] = 167.56429Prob [ chi squared > value ] = .00000Response data are given as ind. choicesNumber 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) modelDependent variable ChoiceLog likelihood function -199.97662Estimation based on N = 210, K = 5Information Criteria: Normalization=1/N Normalized UnnormalizedAIC 1.95216 409.95325Fin.Smpl.AIC 1.95356 410.24736Bayes IC 2.03185 426.68878Hannan Quinn 1.98438 416.71880R2=1-LogL/LogL* Log-L fncn R-sqrd R2AdjConstants only -283.7588 .2953 .2896Chi-squared[ 2] = 167.56429Prob [ chi squared > value ] = .00000Response data are given as ind. choicesNumber 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

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

-----------------------------------------------------------Discrete choice (multinomial logit) modelDependent variable ChoiceLog likelihood function -283.75877Estimation based on N = 210, K = 3Information Criteria: Normalization=1/N Normalized UnnormalizedAIC 2.73104 573.51754Fin.Smpl.AIC 2.73159 573.63404Bayes IC 2.77885 583.55886Hannan Quinn 2.75037 577.57687R2=1-LogL/LogL* Log-L fncn R-sqrd R2AdjConstants only -283.7588 .0000-.0048Response data are given as ind. choicesNumber 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--------+--------------------------------------------------

If the choice set varies across observations, this is the only way to obtain the restricted log likelihood.

1

1

If the choice set is fixed at J, then

logL = log

log

J jlj

Jl jj

NN

N

N P

Estimated MNL Model-----------------------------------------------------------Discrete choice (multinomial logit) modelDependent variable ChoiceLog likelihood function -199.97662Estimation based on N = 210, K = 5Information Criteria: Normalization=1/N Normalized UnnormalizedAIC 1.95216 409.95325Fin.Smpl.AIC 1.95356 410.24736Bayes IC 2.03185 426.68878Hannan Quinn 1.98438 416.71880R2=1-LogL/LogL* Log-L fncn R-sqrd R2AdjConstants only -283.7588 .2953 .2896Chi-squared[ 2] = 167.56429Prob [ chi squared > value ] = .00000Response data are given as ind. choicesNumber 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--------+--------------------------------------------------

2

0

2

0

logPseudo R = 1- . log

N(J-1) logAdjusted Pseudo R =1- . N(J-1)-K log

LL

LL

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 +-------------------------------------------------------+ | 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 +-------------+-------------+-------------+-------------+-------------+AIR | 32 | 8 | 5 | 13 | 58 |TRAIN | 8 | 37 | 5 | 14 | 63 |BUS | 3 | 5 | 15 | 6 | 30 |CAR | 15 | 13 | 6 | 26 | 59 | +-------------+-------------+-------------+-------------+-------------+Total | 58 | 63 | 30 | 59 | 210 | +-------------+-------------+-------------+-------------+-------------+ NLOGIT Cross Tabulation for 4 outcome Constants Only Choice Model AIR TRAIN BUS CAR Total +-------------+-------------+-------------+-------------+-------------+AIR | 16 | 17 | 8 | 16 | 58 |TRAIN | 17 | 19 | 9 | 18 | 63 |BUS | 8 | 9 | 4 | 8 | 30 |CAR | 16 | 18 | 8 | 17 | 59 | +-------------+-------------+-------------+-------------+-------------+Total | 58 | 63 | 30 | 59 | 210 | +-------------+-------------+-------------+-------------+-------------+

Using the Most Probable Cell +-------------------------------------------------------+ | 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 +-------------+-------------+-------------+-------------+-------------+AIR | 40 | 3 | 0 | 15 | 58 |TRAIN | 4 | 45 | 0 | 14 | 63 |BUS | 0 | 3 | 23 | 4 | 30 |CAR | 7 | 14 | 0 | 38 | 59 | +-------------+-------------+-------------+-------------+-------------+Total | 51 | 65 | 23 | 71 | 210 | +-------------+-------------+-------------+-------------+-------------+ NLOGIT Cross Tabulation for 4 outcome Constants only Model AIR TRAIN BUS CAR Total +-------------+-------------+-------------+-------------+-------------+AIR | 0 | 58 | 0 | 0 | 58 |TRAIN | 0 | 63 | 0 | 0 | 63 |BUS | 0 | 30 | 0 | 0 | 30 |CAR | 0 | 59 | 0 | 0 | 59 | +-------------+-------------+-------------+-------------+-------------+Total | 0 | 210 | 0 | 0 | 210 | +-------------+-------------+-------------+-------------+-------------+

Effects of Changes in Attributes on Probabilities

jj m k

m,k m,k

Partial effects : Change in attribute "k" of alternative "m" on the probability that the individual makes choice "j"

PProb(j) = =P [ (j = m) -P ]βx x

1

m = Car

j = Train

k = Price


jj j k

j,k j,k

jj m k

m,k m,k

Partial effects : Own effects :

PProb(j) = =P [1-P ]βx x

Cross effects :PProb(j) = = -PP β

x x

m = Carj = Train

k = Price

j = Train


j m,kj m k

m,k m,k j

m m,k k

Elasticities for proportional changes :logP xlogProb(j) = = P [ (j = m) -P ]β

logx logx P

= [ (j = m) -P ] x βNote the elasticity is the same for all j. T

1

1his is a

consequence of the IIA assumption in the model specification made at the outset.

Elasticities for CLOGIT

Own effectCross effects

+---------------------------------------------------+| 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.

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 SimulationIn 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) modelDependent variable ChoiceLog likelihood function -172.94366Estimation based on N = 210, K = 10R2=1-LogL/LogL* Log-L fncn R-sqrd R2AdjConstants only -283.7588 .3905 .3807Chi-squared[ 7] = 221.63022Prob [ chi squared > value ] = .00000Response data are given as ind. choicesNumber 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 .0000AIR_HIN1| .00428 .01306 .327 .7434 A_TRAIN| 5.91407*** .68993 8.572 .0000TRA_HIN2| -.05907*** .01471 -4.016 .0001 A_BUS| 4.46269*** .72333 6.170 .0000BUS_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) modelDependent 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 .6725WTP_TTME| 34.0771 73.07097 .466 .6410--------+--------------------------------------------------

Very different estimates suggests this might not be a very good model.

Estimation in WTP SpaceProblem with WTP calculation : Ratio of two estimates thatare asymptotically normally distributed may have infinite variance. Sample point estimates may be reasonable Inference - confidence

COST TI

intervals - may not be possible.WTP estimates often become unreasonable in random parametermodels in which parameters vary across individuals.Estimation in WTP Space U(Air) = α+β COST + β

ME attr

attrTIMECOST

COST COST

COST TIME attr

TIME + β Attr + ε

ββ = α+β COST + TIME + Attr + ε β β

= α+β COST + θ TIME + θ Attr + ε

For a simple MNL the transformation is 1:1. Results will be identicalto 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 IIDexp[ ( )]Prob(train) =

exp[ ( )] exp[ ( )] exp[ ( )] exp[ ( )]exp[ ( )]Prob(train|train,bus,car) =

exp[ ( )] exp[ ( )] exp[ ( )]Air is in the choice set, probabilities are in

U trainU air U train U bus U car

U trainU train U bus U car

dependent from air if air is

not in the condition. This is a testable behavioral assumption.

Behavioral Implication of IIA

Own effectCross effects

+---------------------------------------------------+| 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.

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

-1short full short full short fullH = - ' - -b b V V b b

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 .0000Matrix IIATEST has 1 rows and 1 columns. 1 +-------------- 1| 33.78445 Test statistic+------------------------------------+| Listed Calculator Results |+------------------------------------+ Result = 9.487729 Critical value

IIA is rejected

The Multinomial Probit Model

j ij j i i,j

1 2 J

U(i, j) α + + ' +ε

[ε ,ε ,...,ε ] ~ Multivariate Normal[ , ] Correlation across choicesHeteroscedasticitySome restrictions for identification

Relaxes the IID assumptions, therefore, does not

β'x

a

γ z0 Σ

ssume IIA.

* * * ... * ** * * ... * ** * * ... * *

* * * 1 *0 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. s[AIR] | 3.60695794 1.42963795 2.523 .0116 s[TRAIN]| 1.59318892 .81711159 1.950 .0512 s[BUS] | 1.00000000 ......(Fixed Parameter)....... s[CAR] | 1.00000000 ......(Fixed Parameter).......---------+Correlations in the Normal Distribution rAIR,TRA| .30491746 .49357120 .618 .5367 rAIR,BUS| .40383018 .63548534 .635 .5251 rTRA,BUS| .36973127 .42310789 .874 .3822 rAIR,CAR| .000000 ......(Fixed Parameter)....... rTRA,CAR| .000000 ......(Fixed Parameter)....... rBUS,CAR| .000000 ......(Fixed Parameter).......

Correlation Matrix for Air, Train, Bus, Car

1 .305 .404 0.305 1 .370 0.404 .370 1 0

0 0 0 1

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 |+---------------------------------------------------+

+---------------------------+| 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 |+---------------------------+

Multinomial Logit

Warning about Stata Multinomial Probit (mprobit)

* * * ... * ** * * ... * ** * * ... * *

* * * 1 *0 0 0 ... 0 1

1 0 0 ... 0 00 1 0 ... 0 00 0 1 ... 0 0

0 0 0 1 00 0 0 ... 0 1

j ij j i i,j

1 2 J

U(i, j) α + + ' +ε

[ε ,ε ,...,ε ] ~ Multivariate Normal[ , ] Correlation across choicesHeteroscedasticitySome restrictions for identification

β'x γ z0 Σ

j ij j i i,j

1 2 J

U(i, j) α + + ' +ε

[ε ,ε ,...,ε ] ~ Multivariate Normal[ , ] No correlation across choicesNo heteroscedasticityThis model retains the IID assump

β'x γ

ti

z0 I

ons.

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 modelLatent class modelMixed logit, error components and multinomial probit modelsA generalized mixed logit model – The frontierCombining 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

i

ijt j i itj j i ijt

i i i

i,k k k i i,k

i i

eterogeneity arises from continuous variationin across individuals. (Note Classical and Bayesian) U = α + + +ε

= + + β = β + + w

Most treatments set = = +

Hβ

β x γ zβ β Δh w

δ h

Δ 0, β β w

t

j i itj j ii J (i)

j i itj j ij=1

exp(α + + )

Prob[choice j | i, t, ] =exp(α + + )

β x γ zβ

β x γ z

The Random Parameters Logit Model

t

j i itj j ii J (i)

j i itj j ij=1

exp(α + + )Prob[choice j | i, t, ] =

exp(α + + )

β x γ zβ

β x γ z

t

i i

T(i) j i itj j iJ (i)t=1

j i itj j ij=1

Prob[choice j | i, t =1,...,T, ] =exp(α + + )

exp(α + + )

ββ x γ z

β x γ z

Multiple choice situations: Independent conditioned on the individual specific parameters

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

i

j,i i itj j i

J(i)j,i i itj j ij=1

j,i i i

exp(α + + )P[choice j | i, t] =

exp(α + + )

α , = functions of underlying [α, , , , , ]

β x γ zβ x γ z

β β Δ Γ ρ h ,v

Denote by 1 all “fixed” parametersDenote by 2i all random and hierarchical parameters

Estimating the RPL ModelEstimation: 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

ˆ

ˆ

i

i

i

i i iβ

i

Marginal : f( | data, )Population Mean =E[ | data, ]

= f( | )d

= = a subvector of

= Argmax L( ,i =1,...,N| data, )

Estimator =

β Ωβ Ωβ β Ω β

β Ω

Ω β Ω

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)]} = 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.)

v

Maximum Simulated Likelihood

i

i

i

i

Ti i i i it=1

Ti i i i i it=1β

Ni i i i ii=1 β

L ( | data ) = f(data | )

L ( | data ) = f(data | )f( | )d

logL = log L ( | data )f( | )d

β β

Ω β β Ω β

β β Ω β

True log likelihood

ˆ

N RS i iR ii=1 r=1

S

1logL = log L ( | data , )R

= argmax(logL )

β Ω

Ω

Simulated log likelihood

Model Extensions

• AR(1): wi,k,t = ρkwi,k,t-1 + vi,k,t

Dynamic effects in the model• Restricting sign – lognormal distribution:

• Restricting Range and Sign: Using triangular distribution and range = 0 to 2.

• Heteroscedasticity and heterogeneity

i,k k k i k iβ = exp(μ + + )δ h γ w

i i i= + +β β Δh Γw

k,i k iσ = σ exp( )θ h

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 (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.’

it 1 1,it 2 1,it 3 1,it Brand1,it i



U(brand1) = β Fashion +β Quality +β Price +ε +σ W



U( 4 No Brand,itNone) = β + ε

Implied Covariance MatrixNested Logit Formulation

2

2 2 2Brand1

2 2 2Brand2

2 2 2Brand3

NONE

2 2

Var[ε] = π / 6 =1.6449Var[W] =1

ε +σW 1.6449 +σ σ σ 0ε +σW σ 1.6449+σ σ 0

= Var =ε +σW σ σ 1.6449+σ 0

ε 0 0 0 1.6449

Cross Brand Correlation = σ / [1.6449+σ ]

Error Components Logit Model

Correlation = {0.09592 / [1.6449 + 0.09592]}1/2 = 0.0954

-----------------------------------------------------------Error Components (Random Effects) modelDependent variable CHOICELog likelihood function -4158.45044Estimation based on N = 3200, K = 5Response data are given as ind. choicesReplications for simulated probs. = 50Halton sequences used for simulationsECM model with panel has 400 groupsFixed number of obsrvs./group= 8Number 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 .6513SigmaE01| .09585*** .02529 3.791 .0002--------+--------------------------------------------------

Random Effects Logit ModelAppearance of Latent Random Effects in Utilities Alternative E01+-------------+---+| BRAND1 | * |+-------------+---+| BRAND2 | * |+-------------+---+| BRAND3 | * |+-------------+---+| NONE | |+-------------+---+

Extended MNL Model

i,1,t F,i i,1,t Q i,1,t P,i i,1,t Brand i,Brand i,1,t


i,3,t F,i i,3,t Q i,3,t

U =β Fashion +β Quality +β Price +λ W +εU =β Fashion +β Quality +β Price +λ W +εU =β Fashion +β Quality + P,i i,3,t Brand i,Brand i,3,t

i,NONE,t NONE NONE i,NONE i,NONE,t

F,i F F i F F1 i F2 i F,i F,i

P,i P P

β Price +λ W +εU =α +λ W +εβ =β +δ Sex +[σ exp(γ AgeL25 +γ Age2539)] w ; w ~N[0,1]β =β +δ S i P P1 i P2 i P,i P,i

Brand,i

NONE,i

ex +[σ exp(γ AgeL25 +γ Age2539)] w ; w ~N[0,1]W ~N[0,1]W ~N[0,1]

Utility Functions

Extending the Basic MNL Model



i,3,t F,i i,3,t Q i,3,t

U =β Fashion +β Quality +β Price +λ W +εU =β Fashion +β Quality +β Price +λ W +εU =β Fashion +β Quality + P,i i,3,t Brand i,Brand i,3,t



P,i P P

β Price +λ W +εU =α +λ W +εβ =β +δ Sex +[σ exp(γ AgeL25 +γ Age2539)] w ; w ~N[0,1]β =β +δ S i P P1 i P2 i P,i P,i

Brand,i

NONE,i

ex +[σ exp(γ AgeL25 +γ Age2539)] w ; w ~N[0,1]W ~N[0,1]W ~N[0,1]

Random Utility

Error Components Logit Model



i,3,t F,i i,3,t Q i,3,t P

U =β Fashion +β Quality +β Price +λ W +εU =β Fashion +β Quality +β Price +λ W +εU =β Fashion +β Quality +β ,i i,3,t Brand i,Brand i,3,t



P,i P P i

Price +λ W +εU =α +λ W +εβ =β +δ Sex +[σ exp(γ AgeL25 +γ Age2539)] w ; w ~N[0,1]β =β +δ Sex P P1 i P2 i P,i P,i

Brand,i

NONE,i

+[σ exp(γ AgeL25 +γ Age2539)] w ; w ~N[0,1]W ~N[0,1]W ~N[0,1]

Error Components

Random Parameters Model







P,i P P

Price +λ W +εU =α +λ W +εβ =β +δ Sex +[σ exp(γ AgeL25 +γ Age2539)] w ; w ~N[0,1]β =β +δ Sexi P P1 i P2 i P,i P,i

Brand,i

NONE,i


Heterogeneous (in the Means) Random Parameters Model







P,i P P


Brand,i

NONE,i


Heterogeneity in Both Means and Variances







P,i P P


Brand,i

NONE,i


-----------------------------------------------------------Random Parms/Error Comps. Logit ModelDependent variable CHOICELog likelihood function -4019.23544 (-4158.50286 for MNL)Restricted log likelihood -4436.14196 (Chi squared = 278.5)Chi squared [ 12 d.f.] 833.81303Significance level .00000McFadden Pseudo R-squared .0939795Estimation based on N = 3200, K = 12Information Criteria: Normalization=1/N Normalized UnnormalizedAIC 2.51952 8062.47089Fin.Smpl.AIC 2.51955 8062.56878Bayes IC 2.54229 8135.32176Hannan Quinn 2.52768 8088.58926R2=1-LogL/LogL* Log-L fncn R-sqrd R2AdjNo coefficients -4436.1420 .0940 .0928Constants only -4391.1804 .0847 .0836At start values -4158.5029 .0335 .0323Response data are given as ind. choicesReplications for simulated probs. = 50Halton sequences used for simulationsRPL model with panel has 400 groupsFixed number of obsrvs./group= 8Hessian is not PD. Using BHHH estimatorNumber of obs.= 3200, skipped 0 obs--------+--------------------------------------------------

--------+--------------------------------------------------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:VariableFASH:AGE| 1.73176*** .15372 11.266 .0000FAS0:AGE| .71872*** .18592 3.866 .0001PRIC:AGE| -9.38055*** 1.07578 -8.720 .0000PRI0: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 effectsSigmaE01| .23165 .40495 .572 .5673SigmaE02| .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 AGE39FASH 1.73176 .71872PRICE -9.38055 -4.33586

Estimated RP/ECL Model

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 |+---------------------------------------------------+

+--------------------------+| 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 |+--------------------------+

Multinomial Logit

Estimating Individual Distributions• Form posterior estimates of E[i|datai]• Use the same methodology to estimate E[i

2|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 forIncrements to an Attribute in a

Discrete Choice Model

attribute,i

cost

βWTP = -

β

Random

Extending the RP Model to WTPUse the model to estimate conditional

distributions for any function of parameters

Willingness to pay = -i,time / i,cost

Use simulation methodˆ ˆ

ˆˆ ˆ

ˆ

R Tr=1 ir t=1 ijt ir it

i i R Tr=1 t=1 ijt ir it

Ri,r irr=1

(1/ R)Σ WTP Π P (β |Ω,data )E[WTP | data ] =

(1/ R)Σ Π P (β |Ω,data )

1 = w WTPR

Sumulation of WTP from i

i

i

i,Attributei i i i i

i,Cost

Ti,Attribute

i i i i i i i i βt=1i,Cost

T

i i i i i i i i βt=1

-βWTP =E | , , , , , ,

β

-βP(choice j | , )g( | , , , , , , ) d

β =

P(choice j | , )g( | , , , , , , ) d

WTP

β Δ Γ y X h z

X β β β Δ Γ y X h z β

X β β β Δ Γ y X h z β

ˆˆ

ˆˆ ˆ ˆ ˆ

ˆ

TRi,Attribute

i irr=1 t=1i,Cost

i ir i irTR

i irr=1 t=1

-β1 P(choice j | , )R β

= , = + +1 P(choice j | , )R

X ββ β Δh Γw

X β

A Generalized Mixed Logit Model i i,t,j i,t,j

i i i i i i

i i

U(i,t, j) = Common effects + ε

Random Parameters= σ [ + ]+[γ +σ (1- γ)]=

is a lower triangular matrix with 1s on the diagonal (Cholesky matrix)

β x

β β Δh Γ v Γ ΛΣ

Λ

Σ

i k k i

2 2i i i i i

i i

is a diagonal matrix with φ exp( )Overall preference scaling

σ = σexp(-τ / 2+τ w + ]

τ = exp( ) 0 < γ < 1

ψ h

θ hλ r

Estimation in Willingness to Pay Space

θ θ

θ θ

θ θ

i,1,t P,i F,i i,1,t Q i,1,t i,1,t Brand i,Brand i,1,t

i,2,t P,i F,i i,2,t Q i,2,t i,2,t Brand i,Brand i,2,t

i,3,t P,i F,i i,3,t Q i,3

U =β Fashion + Quality +Price +λ W +εU =β Fashion + Quality +Price +λ W +εU =β Fashion + Quality ,t i,3,t Brand i,Brand i,3,t


Brand,i NONE,i

+Price +λ W +εU =α +λ W +ε W ~N[0,1] W ~N[0,1

0[ (1 )] F

i iPF P

θ θF,i F,i F,iF F i

P,i P,i P,iP P i

]w w ~N[0,1]+δ Sex

β w w ~N[0,1]β +δ Sex

Both parameters in the WTP calculation are random.

Extended Formulation of the MNL Groups of similar alternatives

Compound Utility: U(Alt)=U(Alt|Branch)+U(branch) Behavioral implications – Correlations across branches

Travel

Private

Public

Air Car Train Bus

LIMB

BRANCH

TWIG

Degenerate Branches

Choice Situation

Opt Out

Choose Brand

None Brand2

Brand1

Brand3

Purchase

Brand

Shoe Choice

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

k|j k|j

j

Utility functions; (Drop observation indicator, i.) Twig level : k | j denotes alternative k in branch j U(k | j) = α +

Branch level U(j) = y

Twig level proba

β x

( )

( )

( )

k|j k|j

k|j K|jm|j m|jm=1

K|jm=1 m|j m|j

j j

b

exp α +bility : P(k | j) = P =

exp α +

Inclusive value for branch j = IV(j) = log Σ exp α +

exp λ γ'y +IV(j)Branch level probability : P(j) =

exp λ

β xβ x

β x

Bbb=1

j

γ'y +IV(b)

λ = 1 for all branches returns the original MNL model

Estimation Strategy for Nested Logit Models• 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 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

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 |+---------------------------------------------------------------+

Model Structure

MNL Starting Values-----------------------------------------------------------Discrete choice (multinomial logit) modelDependent variable ChoiceLog likelihood function -172.94366Estimation based on N = 210, K = 10R2=1-LogL/LogL* Log-L fncn R-sqrd R2AdjConstants only -283.7588 .3905 .3787Chi-squared[ 7] = 221.63022Prob [ chi squared > value ] = .00000Response data are given as ind. choicesNumber 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 .0000AIR_HIN1| .00428 .01306 .327 .7434 A_TRAIN| 5.91407*** .68993 8.572 .0000TRA_HIN3| -.05907*** .01471 -4.016 .0001 A_BUS| 4.46269*** .72333 6.170 .0000BUS_HIN4| -.02295 .01592 -1.442 .1493--------+--------------------------------------------------

FIML Parameter Estimates-----------------------------------------------------------FIML Nested Multinomial Logit ModelDependent variable MODELog likelihood function -166.64835The model has 2 levels.Random Utility Form 1:IVparms = LMDAb|lNumber 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 .0136AIR_HIN1| .00357 .01057 .337 .7358 A_TRAIN| 3.49867*** .80634 4.339 .0000TRA_HIN3| -.03581*** .01379 -2.597 .0094 A_BUS| 2.30142*** .81284 2.831 .0046BUS_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--------+--------------------------------------------------

Estimated Elasticities – Note Decomposition

| 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 |

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

Travel

Fly Ground

Air CarTrain Bus

BRANCH

TWIG

LIMB

NL Model with Degenerate Branch-----------------------------------------------------------FIML Nested Multinomial Logit ModelDependent variable MODELog 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 .0005AIR_HIN1| .02163 .01326 1.631 .1028 A_TRAIN| 6.50129*** 1.01147 6.428 .0000TRA_HIN2| -.06427*** .01768 -3.635 .0003 A_BUS| 4.52963*** .99877 4.535 .0000BUS_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 ModelDependent variable MODELog 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 .1601AIR_HIN1| .01501 .01226 1.225 .2206 A_TRAIN| 3.44408*** .68388 5.036 .0000TRA_HIN2| -.02823*** .00852 -3.311 .0009 A_BUS| 2.58400*** .63247 4.086 .0000BUS_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

Travel

Fly Rail

Air CarTrain Bus

LIMB

BRANCH

TWIG

Drive GrndPblc

Scaling in Transport Modes-----------------------------------------------------------FIML Nested Multinomial Logit ModelDependent variable MODELog likelihood function -182.42834The model has 2 levels.Nested Logit form:IVparms=Taub|l,r,Sl|r& Fr.No normalizations imposed a prioriNumber 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 .0000LOCLMASS| 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 ModelNLOGIT ; 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 1.500Simulated 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

AIR 1 i,AIR i,AIR i,1

TRAIN 1 i,TRAIN i,TRAIN i,1

BUS 1 i,BUS

Random terms in utility functions share random componentsU(Air,i) = α +β INVC +...+ ε + w

U(Train,i) = α +β INVC +...+ ε + w

U(Bus,i) = α +β INVC

i,BUS i,2

1 i,CAR i,CAR i,2

2 2 2ε 1 1

2 2 21 ε 1

2 2 2ε 2 2

2 2 22 ε 2

+...+ ε + w

U(Car,i) = β INVC +...+ ε + w

Air σ +θ θ 0 0Train θ σ +θ 0 0

Cov =Bus 0 0 σ +θ θCar 0 0 θ σ +θ

This model is estimated by maximum simulated likelihood.

Error Components Logit Model-----------------------------------------------------------Error Components (Random Effects) modelDependent variable MODELog likelihood function -182.27368Response data are given as ind. choicesReplications for simulated probs. = 25Halton sequences used for simulationsECM model with panel has 70 groupsFixed number of obsrvs./group= 3Hessian is not PD. Using BHHH estimatorNumber 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 effectsSigmaE01| -.27336 3.25167 -.084 .9330SigmaE02| 1.21988 .94292 1.294 .1958--------+--------------------------------------------------

William Greene Department of Economics Stern School of Business

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