Empirical Methods for Microeconomic Applications University of Lugano, Switzerland May 27-31, 2019 William Greene Department of Economics Stern School.

Empirical Methods for Microeconomic Applications University of Lugano, Switzerland May 27-31, 2019
William Greene Department of Economics Stern School of Business New York University

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

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 CHARACTERISTIC MODE TRAVEL INVC INVT TTME GC HINC AIR TRAIN BUS CAR AIR TRAIN BUS CAR AIR TRAIN BUS CAR AIR TRAIN BUS CAR

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

Multinomial Choice Models

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

Using the Model to Measure Consumer Surplus

Measuring the Change in Consumer Surplus

Willingness to Pay

Observed Choice3 Data Types of Data Attributes and Characteristics
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 CHARACTERISTIC MODE TRAVEL INVC INVT TTME GC HINC AIR TRAIN BUS CAR AIR TRAIN BUS CAR AIR TRAIN BUS CAR AIR TRAIN BUS CAR 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.

A Stated Choice Experiment with Variable Choice Sets
Each person makes four choices from a choice set that includes either two or four alternatives. The first choice is the RP between two of the RP alternatives The second-fourth are the SP among four of the six SP alternatives. There are ten alternatives in total. A Stated Choice Experiment with Variable Choice Sets

Stated Choice Experiment: Unlabeled Alternatives, One Observation

An Estimated MNL Model for Travel
Discrete choice (multinomial logit) model Dependent variable Choice Log likelihood function Estimation based on N = , K = 5 Information Criteria: Normalization=1/N Normalized Unnormalized AIC Fin.Smpl.AIC Bayes IC Hannan Quinn R2=1-LogL/LogL* Log-L fncn R-sqrd R2Adj Constants only Chi-squared[ 2] = Prob [ chi squared > value ] = 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| *** TTME| *** A_AIR| *** A_TRAIN| *** A_BUS| ***

Estimated MNL Model Discrete choice (multinomial logit) model Dependent variable Choice Log likelihood function Estimation based on N = , K = 5 Information Criteria: Normalization=1/N Normalized Unnormalized AIC Fin.Smpl.AIC Bayes IC Hannan Quinn R2=1-LogL/LogL* Log-L fncn R-sqrd R2Adj Constants only Chi-squared[ 2] = Prob [ chi squared > value ] = 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| *** TTME| *** A_AIR| *** A_TRAIN| *** A_BUS| ***

Model Fit Based on Log Likelihood
Three sets of predicted probabilities No model: Pij = 1/J (.25) Constants only: Pij = (1/N)i dij (58,63,30,59)/210=.286,.300,.143,.281 Constants only model matches sample shares Estimated model: Logit probabilities Compute log likelihood Measure improvements in log likelihood with pseudo R-squared = 1 – LogL/LogL0 (“Adjusted” for number of parameters in the model.)

Fit the Model with Only ASCs
If the choice set varies across observations, this is the only way to obtain the restricted log likelihood. Discrete choice (multinomial logit) model Dependent variable Choice Log likelihood function Estimation based on N = , K = 3 Information Criteria: Normalization=1/N Normalized Unnormalized AIC Fin.Smpl.AIC Bayes IC Hannan Quinn R2=1-LogL/LogL* Log-L fncn R-sqrd R2Adj Constants only Response data are given as ind. choices Number of obs.= 210, skipped 0 obs Variable| Coefficient Standard Error b/St.Er. P[|Z|>z] A_AIR| A_TRAIN| A_BUS| ***

Estimated MNL Model Discrete choice (multinomial logit) model Dependent variable Choice Log likelihood function Estimation based on N = , K = 5 Information Criteria: Normalization=1/N Normalized Unnormalized AIC Fin.Smpl.AIC Bayes IC Hannan Quinn R2=1-LogL/LogL* Log-L fncn R-sqrd R2Adj Constants only Chi-squared[ 2] = Prob [ chi squared > value ] = 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| *** TTME| *** A_AIR| *** A_TRAIN| *** A_BUS| ***

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)

Effects of Changes in Attributes on Probabilities
j = Train m = Car k = Price

k = Price j = Train j = Train m = Car

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%

More Complicated Model Simulation
In vehicle cost of CAR falls by 10% Market is limited to ground (Train, Bus, Car) CLOGIT ; Lhs = Mode ; Choices = Air,Train,Bus,Car ; Rhs = TTME,INVC,INVT,GC ; Rh2 = One ,Hinc ; Simulation = TRAIN,BUS,CAR ; Scenario: GC(car)=[*].9$

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

Estimation in WTP Space

The I.I.D Assumption Uitj = ij + ’xitj + ’zit + ijt
F(itj) = Exp(-Exp(-itj)) (random part of each utility) Independence across utility functions Identical variances (means absorbed in constants) Restriction on equal scaling may be inappropriate Correlation across alternatives may be suppressed Equal cross elasticities is a substantive restriction Behavioral implication of IID is independence from irrelevant alternatives.. If an alternative is removed, probability is spread equally across the remaining alternatives. This is unreasonable

IIA Implication of IID

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

The Multinomial Probit Model

Warning about Stata Multinomial Probit (mprobit)

Multinomial Choice Models: Where to From Here?
Panel data (Repeated measures) Random and fixed effects models Building into a multinomial logit model The nested logit model Latent class model Mixed logit, error components and multinomial probit models A generalized mixed logit model – The frontier Combining revealed and stated preference data

Random Parameters Model
Allow model parameters as well as constants to be random Allow multiple observations with persistent effects Allow a hierarchical structure for parameters – not completely random Uitj = 1’xi1tj + 2i’xi2tj + zit + ijt Random parameters in multinomial logit model 1 = nonrandom (fixed) parameters 2i = random parameters that may vary across individuals and across time Maintain I.I.D. assumption for ijt (given )

Continuous Random Variation in Preference Weights

The Random Parameters Logit Model
Multiple choice situations: Independent conditioned on the individual specific parameters

Modeling Variations Parameter specification Stochastic 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 Denote by 1 all “fixed” parameters
Denote by 2i all random and hierarchical parameters

Estimating the RPL Model
Estimation: 1 2it = 2 + Δhi + Γvi,t Uncorrelated: Γ is diagonal Autocorrelated: vi,t = Rvi,t-1 + ui,t (1) Estimate “structural parameters” (2) Estimate individual specific utility parameters (3) Estimate elasticities, etc.

Classical Estimation Platform: The Likelihood
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.)

Maximum Simulated Likelihood
True log likelihood 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

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

Error Components Logit Modeling
Alternative approach to building cross choice correlation Common ‘effects.’ Wi is a ‘random individual effect.’

Implied Covariance Matrix Nested Logit Formulation

Error Components Logit Model
Error Components (Random Effects) model Dependent variable CHOICE Log likelihood function Estimation based on N = , K = 5 Response data are given as ind. choices Replications for simulated probs. = 50 Halton sequences used for simulations ECM model with panel has groups Fixed number of obsrvs./group= Number of obs.= 3200, skipped 0 obs Variable| Coefficient Standard Error b/St.Er. P[|Z|>z] |Nonrandom parameters in utility functions FASH| *** QUAL| *** PRICE| *** ASC4| SigmaE01| *** Random Effects Logit Model Appearance of Latent Random Effects in Utilities Alternative E01 | BRAND1 | * | | BRAND2 | * | | BRAND3 | * | | NONE | | Correlation = { / [ ]}1/2 =

Extended MNL Model Utility Functions

Extending the Basic MNL Model
Random Utility

Random Parameters Model

Heterogeneous (in the Means) Random Parameters Model

Heterogeneity in Both Means and Variances

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Random Parms/Error Comps. Logit Model Dependent variable CHOICE Log likelihood function ( for MNL) Restricted log likelihood (Chi squared = 278.5) Chi squared [ 12 d.f.] Significance level McFadden Pseudo R-squared Estimation based on N = , K = 12 Information Criteria: Normalization=1/N Normalized Unnormalized AIC Fin.Smpl.AIC Bayes IC Hannan Quinn R2=1-LogL/LogL* Log-L fncn R-sqrd R2Adj No coefficients Constants only At start values Response data are given as ind. choices Replications for simulated probs. = 50 Halton sequences used for simulations RPL model with panel has groups Fixed number of obsrvs./group= Hessian is not PD. Using BHHH estimator Number of obs.= 3200, skipped 0 obs

Estimating Individual Distributions
Form posterior estimates of E[i|datai] Use the same methodology to estimate E[i2|datai] and Var[i|datai] Plot individual “confidence intervals” (assuming near normality) Sample from the distribution and plot kernel density estimates

What is the ‘Individual Estimate?’
Point estimate of mean, variance and range of random variable i | datai. Value is NOT an estimate of i ; it is an estimate of E[i | datai] This would be the best estimate of the actual realization i|datai An interval estimate would account for the sampling ‘variation’ in the estimator of Ω. Bayesian counterpart to the preceding: Posterior mean and variance. Same kind of plot could be done.

Individual E[i|datai] Estimates*
The random parameters model is uncovering the latent class feature of the data. *The intervals could be made wider to account for the sampling variability of the underlying (classical) parameter estimators.

WTP Application (Value of Time Saved)
Estimating Willingness to Pay for Increments to an Attribute in a Discrete Choice Model Random

Extending the RP Model to WTP
Use the model to estimate conditional distributions for any function of parameters Willingness to pay = -i,time / i,cost Use simulation method

Sumulation of WTP from i

A Generalized Mixed Logit Model

Estimation in Willingness to Pay Space
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 LIMB Travel BRANCH Private Public TWIG Air Car Train Bus

Degenerate Branches Shoe Choice Choice Situation Purchase Opt Out
Choose Brand Brand None Brand1 Brand2 Brand3

Correlation Structure for a Two Level Model
Within a branch Identical variances (IIA applies) Covariance (all same) = variance at higher level Branches have different variances (scale factors) Nested logit probabilities: Generalized Extreme Value Prob[Alt,Branch] = Prob(branch) * Prob(Alt|Branch)

Probabilities for a Nested Logit Model

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

Model Structure Tree Structure Specified for the Nested Logit Model
Sample proportions are marginal, not conditional. Choices marked with * are excluded for the IIA test. Trunk (prop.)|Limb (prop.)|Branch (prop.)|Choice (prop.)|Weight|IIA Trunk{1} |TRAVEL |PRIVATE |AIR | 1.000| | | |CAR | 1.000| | |PUBLIC |TRAIN | 1.000| | | |BUS | 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 |

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 = LogL (MNL) = Chi-squared with 2 d.f. = 2( ( )) = The critical value is 5.99 (95%) The MNL is rejected (as usual)

Higher Level Trees E.g., Location (Neighborhood)
Housing Type (Rent, Buy, House, Apt) Housing (# Bedrooms)

Degenerate Branches LIMB Travel BRANCH Fly Ground TWIG Air Train Car
Bus

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 $

Using Degenerate Branches to Reveal Scaling
LIMB Travel BRANCH Fly Rail Drive GrndPblc TWIG Air Train Car Bus

An Error Components Model

Empirical Methods for Microeconomic Applications University of Lugano, Switzerland May 27-31, 2019 William Greene Department of Economics Stern School.

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Empirical Methods for Microeconomic Applications University of Lugano, Switzerland May 27-31, 2019 William Greene Department of Economics Stern School.

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