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William Greene Department of Economics Stern School of Business. Topics in Microeconometrics University of Queensland Brisbane, QLD July 7-9, 2010. Multinomial Choice. Multinomial Unordered Choice. Observed Data. Types of Data Individual choice Market shares – consumer markets

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William greene department of economics stern school of business l.jpg

William Greene

Department of Economics

Stern School of Business

Topics in MicroeconometricsUniversity of QueenslandBrisbane, QLDJuly 7-9, 2010




Observed data l.jpg
Observed 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


Multinomial choice among j alternatives l.jpg
Multinomial Choice Among J Alternatives

• Random Utility Basis

Uitj = ij+i’xitj+ ijzit+ ijt

i = 1,…,N; j = 1,…,J(i,t); t = 1,…,T(i)

N individuals studied, J(i,t) alternatives in the choice set, T(i) [usually 1] choice situations examined.

• Maximum Utility Assumption

Individual i will Choose alternative j in choice setting t iff Uitj > Uitk for all k  j.

•Underlying assumptions

Smoothness of utilities

Axioms: Transitive, Complete, Monotonic


The multinomial logit mnl model l.jpg
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


Specifying the probabilities l.jpg
Specifying the Probabilities

  • •Choice specific attributes (X) vary by choices, multiply by generic

  • coefficients. E.g., TTME=terminal time, GC=generalized cost of travel mode

  • Generic characteristics (Income, constants) must be interacted with

  • choice specific constants.

  • • Estimation by maximum likelihood; dij = 1 if person i chooses j



Example data on discrete choices n 210 commuters between sydney and melbourne l.jpg
Example Data on Discrete Choices: N = 210Commuters Between Sydney and Melbourne

CHOICE ATTRIBUTES CHARACTERISTIC

MODE TRAVEL INVC INVT TTME GC HINC

AIR .00000 59.000 100.00 69.000 70.000 35.000

TRAIN .00000 31.000 372.00 34.000 71.000 35.000

BUS .00000 25.000 417.00 35.000 70.000 35.000

CAR 1.0000 10.000 180.00 .00000 30.000 35.000

AIR .00000 58.000 68.000 64.000 68.000 30.000

TRAIN .00000 31.000 354.00 44.000 84.000 30.000

BUS .00000 25.000 399.00 53.000 85.000 30.000

CAR 1.0000 11.000 255.00 .00000 50.000 30.000

AIR .00000 127.00 193.00 69.000 148.00 60.000

TRAIN .00000 109.00 888.00 34.000 205.00 60.000

BUS 1.0000 52.000 1025.0 60.000 163.00 60.000

CAR .00000 50.000 892.00 .00000 147.00 60.000

AIR .00000 44.000 100.00 64.000 59.000 70.000

TRAIN .00000 25.000 351.00 44.000 78.000 70.000

BUS .00000 20.000 361.00 53.000 75.000 70.000

CAR 1.0000 5.0000 180.00 .00000 32.000 70.000


Estimated mnl model l.jpg
Estimated MNL Model

-----------------------------------------------------------

Discrete choice (multinomial logit) model

Dependent variable Choice

Log likelihood function -199.97662

Estimation based on N = 210, K = 5

Information Criteria: Normalization=1/N

Normalized Unnormalized

AIC 1.95216 409.95325

Fin.Smpl.AIC 1.95356 410.24736

Bayes IC 2.03185 426.68878

Hannan Quinn 1.98438 416.71880

R2=1-LogL/LogL* Log-L fncn R-sqrd R2Adj

Constants only -283.7588 .2953 .2896

Chi-squared[ 2] = 167.56429

Prob [ chi squared > value ] = .00000

Response data are given as ind. choices

Number of obs.= 210, skipped 0 obs

--------+--------------------------------------------------

Variable| Coefficient Standard Error b/St.Er. P[|Z|>z]

--------+--------------------------------------------------

GC| -.01578*** .00438 -3.601 .0003

TTME| -.09709*** .01044 -9.304 .0000

A_AIR| 5.77636*** .65592 8.807 .0000

A_TRAIN| 3.92300*** .44199 8.876 .0000

A_BUS| 3.21073*** .44965 7.140 .0000

--------+--------------------------------------------------


Slide11 l.jpg

Estimated MNL ModelWhat do the Coefficients Mean?

-----------------------------------------------------------

Discrete choice (multinomial logit) model

Dependent variable Choice

Log likelihood function -199.97662

Estimation based on N = 210, K = 5

Information Criteria: Normalization=1/N

Normalized Unnormalized

AIC 1.95216 409.95325

Fin.Smpl.AIC 1.95356 410.24736

Bayes IC 2.03185 426.68878

Hannan Quinn 1.98438 416.71880

R2=1-LogL/LogL* Log-L fncn R-sqrd R2Adj

Constants only -283.7588 .2953 .2896

Chi-squared[ 2] = 167.56429

Prob [ chi squared > value ] = .00000

Response data are given as ind. choices

Number of obs.= 210, skipped 0 obs

--------+--------------------------------------------------

Variable| Coefficient Standard Error b/St.Er. P[|Z|>z]

--------+--------------------------------------------------

GC| -.01578*** .00438 -3.601 .0003

TTME| -.09709*** .01044 -9.304 .0000

A_AIR| 5.77636*** .65592 8.807 .0000

A_TRAIN| 3.92300*** .44199 8.876 .0000

A_BUS| 3.21073*** .44965 7.140 .0000

--------+--------------------------------------------------



Elasticities for clogit l.jpg
Elasticities for CLOGIT

+---------------------------------------------------+

| Elasticity averaged over observations.|

| Attribute is INVT in choice AIR |

| Mean St.Dev |

| * Choice=AIR -.2055 .0666 |

| Choice=TRAIN .0903 .0681 |

| Choice=BUS .0903 .0681 |

| Choice=CAR .0903 .0681 |

+---------------------------------------------------+

| Attribute is INVT in choice TRAIN |

| Choice=AIR .3568 .1231 |

| * Choice=TRAIN -.9892 .5217 |

| Choice=BUS .3568 .1231 |

| Choice=CAR .3568 .1231 |

+---------------------------------------------------+

| Attribute is INVT in choice BUS |

| Choice=AIR .1889 .0743 |

| Choice=TRAIN .1889 .0743 |

| * Choice=BUS -1.2040 .4803 |

| Choice=CAR .1889 .0743 |

+---------------------------------------------------+

| Attribute is INVT in choice CAR |

| Choice=AIR .3174 .1195 |

| Choice=TRAIN .3174 .1195 |

| Choice=BUS .3174 .1195 |

| * Choice=CAR -.9510 .5504 |

+---------------------------------------------------+

| Effects on probabilities of all choices in model: |

| * = Direct Elasticity effect of the attribute. |

+---------------------------------------------------+

Note the effect of IIA on the cross effects.

Own effect

Cross effects

Elasticities are computed for each observation; the mean and standard deviation are then computed across the sample observations.


Model simulation l.jpg
Model Simulation

+---------------------------------------------+

| Discrete Choice (One Level) Model |

| Model Simulation Using Previous Estimates |

| Number of observations 210 |

+---------------------------------------------+

+------------------------------------------------------+

|Simulations of Probability Model |

|Model: Discrete Choice (One Level) Model |

|Simulated choice set may be a subset of the choices. |

|Number of individuals is the probability times the |

|number of observations in the simulated sample. |

|Column totals may be affected by rounding error. |

|The model used was simulated with 210 observations.|

+------------------------------------------------------+

-------------------------------------------------------------------------

Specification of scenario 1 is:

Attribute Alternatives affected Change type Value

--------- ------------------------------- ------------------- ---------

GC CAR Scale base by value 1.250

-------------------------------------------------------------------------

The simulator located 209 observations for this scenario.

Simulated Probabilities (shares) for this scenario:

+----------+--------------+--------------+------------------+

|Choice | Base | Scenario | Scenario - Base |

| |%Share Number |%Share Number |ChgShare ChgNumber|

+----------+--------------+--------------+------------------+

|AIR | 27.619 58 | 29.592 62 | 1.973% 4 |

|TRAIN | 30.000 63 | 31.748 67 | 1.748% 4 |

|BUS | 14.286 30 | 15.189 32 | .903% 2 |

|CAR | 28.095 59 | 23.472 49 | -4.624% -10 |

|Total |100.000 210 |100.000 210 | .000% 0 |

+----------+--------------+--------------+------------------+

Changes in the predicted market shares when GC_CAR increases by 25%.


Slide15 l.jpg

Model Simulation Step: 10% Fall in INVC of CAR

+---------------------------------------------+

| 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

--------- ------------------------------- ------------------- ---------

INVC CAR Scale base by value .900

-------------------------------------------------------------------------

The simulator located 210 observations for this scenario.

Simulated Probabilities (shares) for this scenario:

+----------+--------------+--------------+------------------+

|Choice | Base | Scenario | Scenario - Base |

| |%Share Number |%Share Number |ChgShare ChgNumber|

+----------+--------------+--------------+------------------+

|TRAIN | 37.321 78 | 35.854 75 | -1.467% -3 |

|BUS | 19.805 42 | 18.641 39 | -1.164% -3 |

|CAR | 42.874 90 | 45.506 96 | 2.632% 6 |

|Total |100.000 210 |100.000 210 | .000% 0 |

+----------+--------------+--------------+------------------+


Model fit based on log likelihood l.jpg
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!”


Model fit based on predictions l.jpg
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)


Using the most probable cell l.jpg
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 Multinomial Choice 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 |

+-------------+-------------+-------------+-------------+-------------+


Fit measures based on crosstabulation l.jpg
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 |

+-------------+-------------+-------------+-------------+-------------+


Measuring willingness to pay l.jpg
Measuring Willingness to Pay

U(alt) = aj + bINCOME*INCOME + bAttribute*Attribute + …

WTP = MU(Attribute)/MU(Income)

When MU(Income) is not available, an approximationoften 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 l.jpg
WTP from CLOGIT Model

-----------------------------------------------------------

Discrete choice (multinomial logit) model

Dependent variable Choice

--------+--------------------------------------------------

Variable| Coefficient Standard Error b/St.Er. P[|Z|>z]

--------+--------------------------------------------------

GC| -.00286 .00610 -.469 .6390

INVT| -.00349*** .00115 -3.037 .0024

TTME| -.09746*** .01035 -9.414 .0000

AASC| 4.05405*** .83662 4.846 .0000

TASC| 3.64460*** .44276 8.232 .0000

BASC| 3.19579*** .45194 7.071 .0000

--------+--------------------------------------------------

WALD ; fn1=WTP_INVT=b_invt/b_gc ; fn2=WTP_TTME=b_ttme/b_gc$

-----------------------------------------------------------

WALD procedure.

--------+--------------------------------------------------

Variable| Coefficient Standard Error b/St.Er. P[|Z|>z]

--------+--------------------------------------------------

WTP_INVT| 1.22006 2.88619 .423 .6725

WTP_TTME| 34.0771 73.07097 .466 .6410

--------+--------------------------------------------------



The i i d assumption l.jpg
The I.I.D Assumption

Uitj = ij+’xitj+ ’zit+ ijt

F(itj) = 1 – 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 independence from irrelevant alternatives is unreasonable (IIA). If an alternative is removed, probability is spread equally across the remaining alternatives.


A hausman test for iia l.jpg
A Hausman 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?

Use a Hausman test:

Chi-squared, d.f. Number of parameters estimated

Practicalities: Fit the model, then again with

;IAS = the irrelevant alternative(s)


Iia test for choice air l.jpg
IIA Test for Choice AIR

+--------+--------------+----------------+--------+--------+

|Variable| Coefficient | Standard Error |b/St.Er.|P[|Z|>z]|

+--------+--------------+----------------+--------+--------+

GC | .06929537 .01743306 3.975 .0001

TTME | -.10364955 .01093815 -9.476 .0000

INVC | -.08493182 .01938251 -4.382 .0000

INVT | -.01333220 .00251698 -5.297 .0000

AASC | 5.20474275 .90521312 5.750 .0000

TASC | 4.36060457 .51066543 8.539 .0000

BASC | 3.76323447 .50625946 7.433 .0000

+--------+--------------+----------------+--------+--------+

GC | .53961173 .14654681 3.682 .0002

TTME | -.06847037 .01674719 -4.088 .0000

INVC | -.58715772 .14955000 -3.926 .0001

INVT | -.09100015 .02158271 -4.216 .0000

TASC | 4.62957401 .81841212 5.657 .0000

BASC | 3.27415138 .76403628 4.285 .0000

Matrix IIATEST has 1 rows and 1 columns.

1

+--------------

1| 33.78445 Test statistic

+------------------------------------+

| Listed Calculator Results |

+------------------------------------+

Result = 9.487729 Critical value


Discrete choice model extensions l.jpg
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 nested logit model

Latent class model

Mixed logit, error components and multinomial probit models

A Generalized Mixed Logit Model – The frontier

Combining revealed and stated preference data


Several types of heterogeneity l.jpg
Several Types of Heterogeneity

  • Observational: Observable differences across choice makers

  • Choice strategy: How consumers make decisions. (E.g., omitted attributes)

  • Structure: Model frameworks

  • Preferences: Model ‘parameters’


Heterogeneity in choice strategy l.jpg
Heterogeneity in Choice Strategy

  • Consumers avoid ‘complexity’

    • Lexicographic preferences eliminate certain choices  choice set may be endogenously determined

    • Simplification strategies may eliminate certain attributes

  • Information processing strategy is a source of heterogeneity in the model.


Structural heterogeneity l.jpg
Structural Heterogeneity

  • Marketing literature

  • Latent class structures

    • Yang/Allenby - latent class random parameters models

    • Kamkura et al – latent class nested logit models with fixed parameters


Accommodating heterogeneity l.jpg
Accommodating Heterogeneity

  • Observed? Enter in the model in familiar (and unfamiliar) ways.

  • Unobserved? Takes the form of randomness in the model.


Heterogeneity and the mnl model l.jpg
Heterogeneity and the MNL Model

  • Limitations of the MNL Model:

    • IID  IIA

    • Fundamental tastes are the same across all individuals

  • How to adjust the model to allow variation across individuals?

    • Full random variation

    • Latent clustering – allow some variation


Observable heterogeneity in utility levels l.jpg
Observable Heterogeneity in Utility Levels

Choice, e.g., among brands of cars

xitj = attributes: price, features

zit = observable characteristics: age, sex, income


Observable heterogeneity in preference weights l.jpg
Observable Heterogeneity in Preference Weights


Heteroscedasticity in the mnl model l.jpg
Heteroscedasticity in the MNL Model

•Motivation: Scaling in utility functions

• If ignored, distorts coefficients

• Random utility basis

Uij = j + ’xij + ’zi+ jij

i = 1,…,N; j = 1,…,J(i)

F(ij) = Exp(-Exp(-ij)) now scaled

• Extensions: Relaxes IIA

Allows heteroscedasticity across choices and across individuals


Quantifiable heterogeneity in scaling l.jpg
‘Quantifiable’ Heterogeneity in Scaling

wit = observable characteristics: age, sex, income, etc.


Modeling unobserved heterogeneity l.jpg
Modeling Unobserved Heterogeneity

  • Modeling individual heterogeneity

    • Latent class – Discrete approximation

    • Mixed logit – Continuous

    • The mixed logit model (generalities)

  • Data structure – RP and SP data

    • Induces heterogeneity

    • Induces heteroscedasticity – the scaling problem


Random parameters models l.jpg
Random Parameters Models

  • 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 = i’xitj + i’zit+ ijt

  • Random parameters in multinomial logit model

    • i = random parameters that may vary across individuals and across time

  • Maintain I.I.D. assumption for ijt(given )


Continuous random variation in preference weights l.jpg
Continuous Random Variation in Preference Weights


Random parameters logit model l.jpg
Random Parameters Logit Model

Multiple choice situations: Independent conditioned on the individual specific parameters


Customers choice of energy supplier l.jpg
Customers’ Choice of Energy Supplier

  • California, Stated Preference Survey

  • 361 customers presented with 8-12 choice situations each

  • Supplier attributes:

    • Fixed price: cents per kWh

    • Length of contract

    • Local utility

    • Well-known company

    • Time-of-day rates (11¢ in day, 5¢ at night)

    • Seasonal rates (10¢ in summer, 8¢ in winter, 6¢ in spring/fall)


Population distributions l.jpg
Population Distributions

  • Normal for:

    • Contract length

    • Local utility

    • Well-known company

  • Log-normal for:

    • Time-of-day rates

    • Seasonal rates

  • Price coefficient held fixed


Estimated model l.jpg
Estimated Model

Estimate Std error

Price -.883 0.050

Contract mean -.213 0.026

std dev .386 0.028

Local mean 2.23 0.127

std dev 1.75 0.137

Known mean 1.59 0.100

std dev .962 0.098

TOD mean* 2.13 0.054

std dev* .411 0.040

Seasonal mean* 2.16 0.051

std dev* .281 0.022

*Parameters of underlying normal.


Distribution of brand value of the local utility l.jpg
Distribution of Brand Value of the Local Utility

Standard deviation

10% dislike local utility

=2.0¢

0

2.5¢


Slide44 l.jpg

29%

Contract LengthMean: -.24Standard Deviation: .55

-0.24¢

0

Local UtilityMean: 2.5Standard Deviation: 2.0

10%

0

2.5¢

Well known companyMean 1.8Standard Deviation: 1.1

5%

0

1.8¢


Time of day rates customers do not like l.jpg
Time of Day Rates (Customers do not like.)

Strictly negative (or positive) sign was built into the model

Time-of-day Rates

0

-10.4

Seasonal Rates

-10.2

0


Expected preferences of each customer l.jpg
Expected Preferences of Each Customer

Customer likes long-term contract, local utility, and non-fixed rates.

Local utility can retain and make profit from this customer by offering a long-term contract with time-of-day or seasonal rates.


Estimating individual preferences l.jpg
Estimating Individual Preferences

  • Model estimates = structural parameters, α, β, ρ, Δ, Σ, Γ

  • Objective, a model of individual specific parameters, βi

  • Can individual specific parameters be estimated?

    • Not quite – βi is a single realization of a random process; one random draw.

    • We estimate E[βi | all information about i]

    • (This is also true of Bayesian treatments, despite claims to the contrary.)


Estimating individual distributions l.jpg
Estimating Individual Distributions

  • Form posterior estimates of E[i|datai]

  • Use the same methodology to estimate E[i2|datai] and Var[i|datai]

  • Plot individual “confidence intervals” (assuming near normality)

  • Sample from the distribution and plot kernel density estimates


Posterior estimation of i l.jpg
Posterior Estimation of i

Expected value of βi given the sample information

Estimate by simulation


Application shoe brand choice l.jpg
Application: Shoe Brand Choice

  • Simulated Data: Stated Choice, 400 respondents, 8 choice situations, 3,200 observations

  • 3 choice/attributes + NONE

    • Fashion = High / Low

    • Quality = High / Low

    • Price = 25/50/75,100 coded 1,2,3,4

  • Heterogeneity: Sex, Age (<25, 25-39, 40+)

  • Underlying data generated by a 3 class latent class process (100, 200, 100 in classes)

  • Thanks to www.statisticalinnovations.com (Latent Gold)


An error components logit model l.jpg
An Error Components Logit Model

  • Alternative approach to building cross choice correlation

  • Common (random) “effects”



Error components logit model l.jpg
Error Components Logit Model

-----------------------------------------------------------

Error Components (Random Effects) model

Dependent variable CHOICE

Log likelihood function -4158.45044

Estimation based on N = 3200, K = 5

Response data are given as ind. choices

Replications for simulated probs. = 50

Halton sequences used for simulations

ECM model with panel has 400 groups

Fixed number of obsrvs./group= 8

Number of obs.= 3200, skipped 0 obs

--------+--------------------------------------------------

Variable| Coefficient Standard Error b/St.Er. P[|Z|>z]

--------+--------------------------------------------------

|Nonrandom parameters in utility functions

FASH| 1.47913*** .06971 21.218 .0000

QUAL| 1.01385*** .06580 15.409 .0000

PRICE| -11.8052*** .86019 -13.724 .0000

ASC4| .03363 .07441 .452 .6513

SigmaE01| .09585*** .02529 3.791 .0002

--------+--------------------------------------------------

Random Effects Logit Model

Appearance of Latent Random

Effects in Utilities

Alternative E01

+-------------+---+

| BRAND1 | * |

+-------------+---+

| BRAND2 | * |

+-------------+---+

| BRAND3 | * |

+-------------+---+

| NONE | |

+-------------+---+

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


A heterogeneous heteroscedastic random parameters model with error components l.jpg
A Heterogeneous, Heteroscedastic Random Parameters Model with Error Components


Individual e i data i estimates l.jpg
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 l.jpg
WTP Application (Value of Time Saved)

Estimating Willingness to Pay for Increments to an Attribute in a Discrete Choice Model

Random


Extending the rp model to wtp l.jpg
Extending the RP Model to WTP

  • Use the model to estimate conditional distributions for any function of parameters

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

  • Use simulation method


Panel data l.jpg
Panel Data

  • Repeated Choice Situations

  • Typically RP/SP constructions (experimental)

  • Accommodating “panel data”

    • Multinomial Probit [Marginal, impractical]

    • Latent Class

    • Mixed Logit


Revealed and stated preference data l.jpg
Revealed and Stated Preference Data

  • Pure RP Data

    • Market (ex-post, e.g., supermarket scanner data)

    • Individual observations

  • Pure SP Data

    • Contingent valuation

    • (?) Validity

  • Combined (Enriched) RP/SP

    • Mixed data

    • Expanded choice sets


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Revealed Preference Data

  • Advantage: Actual observations on actual behavior

  • Disadvantage: Limited range of choice sets and attributes – does not allow analysis of switching behavior.


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Mixed Logit Approaches

  • Pivot SP choices around an RP outcome.

  • Scaling is handled directly in the model

  • Continuity across choice situations is handled by random elements of the choice structure that are constant through time

    • Preference weights – coefficients

    • Scaling parameters

      • Variances of random parameters

      • Overall scaling of utility functions


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A Generalized Mixed Logit Model

Uij = j + i′xij + ′zi + ij.

Uijt = j + i′xitj + ′zit + ijt.

i =  + vi

i = exp(-2/2 + wi)

i = i + [ + i(1 - )]vi


Application l.jpg
Application

Survey sample of 2,688 trips, 2 or 4 choices per situation

Sample consists of 672 individuals

Choice based sample

Revealed/Stated choice experiment:

Revealed: Drive,ShortRail,Bus,Train

Hypothetical: Drive,ShortRail,Bus,Train,LightRail,ExpressBus

Attributes:

Cost –Fuel or fare

Transit time

Parking cost

Access and Egress time


Slide64 l.jpg

Each person makes four choices from a choice set that includes either two or four alternatives.

The first choice is the RP between two of the RP alternatives

The second-fourth are the SP among four of the six SP alternatives.

There are ten alternatives in total.


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A Random Parameters Approach includes either two or four alternatives.

NLOGIT

;lhs=chosen,cset,altij

;choices=RPDA,RPRS,RPBS,RPTN,SPDA,SPRS,SPBS,SPTN,SPLR,SPBW

/.592,.208,.089,.111,1.0,1.0,1.0,1.0,1.0,1.0

; rpl

; pds=4

; halton ; pts=25

; fcn=invc(n)

; model:

U(RPDA) = rdasc+ invc*fcost+tmrs*autotime ?+prkda*vehprkct+

+ pinc*pincome+CAVDA*CARAV/

U(RPRS) = rrsasc + invc*fcost+tmrs*autotime/?+ ?egt*autoegtm+prk*vehprkct+

U(RPBS) = rbsasc + invc*mptrfare+mtpt*mptrtime/?+acegt*rpacegtm/

U(RPTN) = cstrs*mptrfare+mtpt*mptrtime/?+acegt*rpacegtm/

U(SPDA) = sdasc + invc*fueld + tmrs*time+cavda*carav ?+prkda*parking

+ pinc*pincome/

U(SPRS) = srsasc + invc*fueld + tmrs*time/? cavrs*carav/

U(SPBS) = invc*fared + mtpt*time +acegt*spacegtm/

U(SPTN) = stnasc + invc*fared + mtpt*time+acegt*spacegtm/

U(SPLR) = slrasc + invc*fared + mtpt*time+acegt*spacegtm/

U(SPBW) = sbwasc + invc*fared + mtpt*time+acegt*spacegtm$


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Connecting Choice includes either two or four alternatives.Situations through RPs

--------+--------------------------------------------------

Variable| Coefficient Standard Error b/St.Er. P[|Z|>z]

--------+--------------------------------------------------

|Random parameters in utility functions

INVC| -.58944*** .03922 -15.028 .0000

|Nonrandom parameters in utility functions

RDASC| -.75327 .56534 -1.332 .1827

TMRS| -.05443*** .00789 -6.902 .0000

PINC| .00482 .00451 1.068 .2857

CAVDA| .35750*** .13103 2.728 .0064

RRSASC| -2.18901*** .54995 -3.980 .0001

RBSASC| -1.90658*** .53953 -3.534 .0004

MTPT| -.04884*** .00741 -6.591 .0000

CSTRS| -1.57564*** .23695 -6.650 .0000

SDASC| -.13612 .27616 -.493 .6221

SRSASC| -.10172 .18943 -.537 .5913

ACEGT| -.02943*** .00384 -7.663 .0000

STNASC| .13402 .11475 1.168 .2428

SLRASC| .27250** .11017 2.473 .0134

SBWASC| -.00685 .09861 -.070 .9446

|Distns. of RPs. Std.Devs or limits of triangular

NsINVC| .45285*** .05615 8.064 .0000

--------+--------------------------------------------------


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