![Page 1: William Greene Department of Economics Stern School of Business](https://reader030.vdocument.in/reader030/viewer/2022032607/56813062550346895d9633a6/html5/thumbnails/1.jpg)
Empirical Methods for Microeconomic Applications
University of Lugano, SwitzerlandMay 27-31, 2013
William Greene
Department of Economics
Stern School of Business
![Page 2: William Greene Department of Economics Stern School of Business](https://reader030.vdocument.in/reader030/viewer/2022032607/56813062550346895d9633a6/html5/thumbnails/2.jpg)
1A. Descriptive Tools, Regression, Panel Data
![Page 3: William Greene Department of Economics Stern School of Business](https://reader030.vdocument.in/reader030/viewer/2022032607/56813062550346895d9633a6/html5/thumbnails/3.jpg)
Agenda• Day 1
• A. Descriptive Tools, Regression, Models, Panel Data, Nonlinear Models
• B. Binary choice and nonlinear modeling, panel data• C. Ordered Choice, endogeneity, control functions,
Robust inference, bootstrapping• Day 2
• A. Models for count data, censoring, inflation models• B. Latent class, mixed models• C. Multinomial Choice
• Day 3• A. Stated Preference
![Page 4: William Greene Department of Economics Stern School of Business](https://reader030.vdocument.in/reader030/viewer/2022032607/56813062550346895d9633a6/html5/thumbnails/4.jpg)
Agenda for 1A
• Models and Parameterization
• Descriptive Statistics• Regression
• Functional Form• Partial Effects• Hypothesis Tests• Robust Estimation
• Bootstrapping• Panel Data• Nonlinear Models
![Page 5: William Greene Department of Economics Stern School of Business](https://reader030.vdocument.in/reader030/viewer/2022032607/56813062550346895d9633a6/html5/thumbnails/5.jpg)
Cornwell and Rupert Panel DataCornwell and Rupert Returns to Schooling Data, 595 Individuals, 7 YearsVariables in the file are
EXP = work experienceWKS = weeks workedOCC = occupation, 1 if blue collar, IND = 1 if manufacturing industrySOUTH = 1 if resides in southSMSA = 1 if resides in a city (SMSA)MS = 1 if marriedFEM = 1 if femaleUNION = 1 if wage set by union contractED = years of educationBLK = 1 if individual is blackLWAGE = log of wage = dependent variable in regressions
These data were analyzed in Cornwell, C. and Rupert, P., "Efficient Estimation with Panel Data: An Empirical Comparison of Instrumental Variable Estimators," Journal of Applied Econometrics, 3, 1988, pp. 149-155.
![Page 6: William Greene Department of Economics Stern School of Business](https://reader030.vdocument.in/reader030/viewer/2022032607/56813062550346895d9633a6/html5/thumbnails/6.jpg)
![Page 7: William Greene Department of Economics Stern School of Business](https://reader030.vdocument.in/reader030/viewer/2022032607/56813062550346895d9633a6/html5/thumbnails/7.jpg)
Model Building in Econometrics
• Parameterizing the model• Nonparametric analysis• Semiparametric analysis• Parametric analysis
• Sharpness of inferences follows from the strength of the assumptions
A Model Relating (Log)Wage to Gender and Experience
![Page 8: William Greene Department of Economics Stern School of Business](https://reader030.vdocument.in/reader030/viewer/2022032607/56813062550346895d9633a6/html5/thumbnails/8.jpg)
Nonparametric RegressionKernel regression of y on x
Semiparametric Regression: Least absolute deviations regression of y on x
Parametric Regression: Least squares – maximum likelihood – regression of y on x
Application: Is there a relationship between Log(wage) and Education?
![Page 9: William Greene Department of Economics Stern School of Business](https://reader030.vdocument.in/reader030/viewer/2022032607/56813062550346895d9633a6/html5/thumbnails/9.jpg)
A First Look at the DataDescriptive Statistics
• Basic Measures of Location and Dispersion
• Graphical Devices• Box Plots• Histogram• Kernel Density Estimator
![Page 10: William Greene Department of Economics Stern School of Business](https://reader030.vdocument.in/reader030/viewer/2022032607/56813062550346895d9633a6/html5/thumbnails/10.jpg)
![Page 11: William Greene Department of Economics Stern School of Business](https://reader030.vdocument.in/reader030/viewer/2022032607/56813062550346895d9633a6/html5/thumbnails/11.jpg)
Box Plots
![Page 12: William Greene Department of Economics Stern School of Business](https://reader030.vdocument.in/reader030/viewer/2022032607/56813062550346895d9633a6/html5/thumbnails/12.jpg)
From Jones and Schurer (2011)
![Page 13: William Greene Department of Economics Stern School of Business](https://reader030.vdocument.in/reader030/viewer/2022032607/56813062550346895d9633a6/html5/thumbnails/13.jpg)
Histogram for LWAGE
![Page 14: William Greene Department of Economics Stern School of Business](https://reader030.vdocument.in/reader030/viewer/2022032607/56813062550346895d9633a6/html5/thumbnails/14.jpg)
![Page 15: William Greene Department of Economics Stern School of Business](https://reader030.vdocument.in/reader030/viewer/2022032607/56813062550346895d9633a6/html5/thumbnails/15.jpg)
The kernel density estimator is ahistogram (of sorts).
n i mm mi 1
** *x x1 1
f̂(x ) K , for a set of points xn B B
B "bandwidth" chosen by the analyst
K the kernel function, such as the normal
or logistic pdf (or one of several others)
x* the point at which the density is approximated.
This is essentially a histogram with small bins.
![Page 16: William Greene Department of Economics Stern School of Business](https://reader030.vdocument.in/reader030/viewer/2022032607/56813062550346895d9633a6/html5/thumbnails/16.jpg)
Kernel Density Estimator
n i mm mi 1
** *x x1 1
f̂(x ) K , for a set of points xn B B
B "bandwidth"
K the kernel function
x* the point at which the density is approximated.
f̂(x*) is an estimator of f(x*)
1
The curse of dimensionality
n
ii 1
3/5
Q(x | x*) Q(x*). n
1 1But, Var[Q(x*)] Something. Rather, Var[Q(x*)] * Something
N NˆI.e.,f(x*) does not converge to f(x*) at the same rate as a mean
converges to a population mean.
![Page 17: William Greene Department of Economics Stern School of Business](https://reader030.vdocument.in/reader030/viewer/2022032607/56813062550346895d9633a6/html5/thumbnails/17.jpg)
Kernel Estimator for LWAGE
![Page 18: William Greene Department of Economics Stern School of Business](https://reader030.vdocument.in/reader030/viewer/2022032607/56813062550346895d9633a6/html5/thumbnails/18.jpg)
From Jones and Schurer (2011)
![Page 19: William Greene Department of Economics Stern School of Business](https://reader030.vdocument.in/reader030/viewer/2022032607/56813062550346895d9633a6/html5/thumbnails/19.jpg)
Objective: Impact of Education on (log) Wage
• Specification: What is the right model to use to analyze this association?
• Estimation• Inference• Analysis
![Page 20: William Greene Department of Economics Stern School of Business](https://reader030.vdocument.in/reader030/viewer/2022032607/56813062550346895d9633a6/html5/thumbnails/20.jpg)
Simple Linear RegressionLWAGE = 5.8388 + 0.0652*ED
![Page 21: William Greene Department of Economics Stern School of Business](https://reader030.vdocument.in/reader030/viewer/2022032607/56813062550346895d9633a6/html5/thumbnails/21.jpg)
Multiple Regression
![Page 22: William Greene Department of Economics Stern School of Business](https://reader030.vdocument.in/reader030/viewer/2022032607/56813062550346895d9633a6/html5/thumbnails/22.jpg)
Specification: Quadratic Effect of Experience
![Page 23: William Greene Department of Economics Stern School of Business](https://reader030.vdocument.in/reader030/viewer/2022032607/56813062550346895d9633a6/html5/thumbnails/23.jpg)
Partial Effects
![Page 24: William Greene Department of Economics Stern School of Business](https://reader030.vdocument.in/reader030/viewer/2022032607/56813062550346895d9633a6/html5/thumbnails/24.jpg)
Model Implication: Effect of Experience and Male vs. Female
![Page 25: William Greene Department of Economics Stern School of Business](https://reader030.vdocument.in/reader030/viewer/2022032607/56813062550346895d9633a6/html5/thumbnails/25.jpg)
Hypothesis Test About Coefficients
• Hypothesis• Null: Restriction on β: Rβ – q = 0• Alternative: Not the null
• Approaches• Fitting Criterion: R2 decrease under the null?• Wald: Rb – q close to 0 under the
alternative?
![Page 26: William Greene Department of Economics Stern School of Business](https://reader030.vdocument.in/reader030/viewer/2022032607/56813062550346895d9633a6/html5/thumbnails/26.jpg)
Hypotheses
All Coefficients = 0?
R = [ 0 | I ] q = [0]
ED Coefficient = 0?
R = 0,1,0,0,0,0,0,0,0,0,0,0
q = 0
No Experience effect?
R = 0,0,1,0,0,0,0,0,0,0,0,0 0,0,0,1,0,0,0,0,0,0,0,0
q = 0 0
![Page 27: William Greene Department of Economics Stern School of Business](https://reader030.vdocument.in/reader030/viewer/2022032607/56813062550346895d9633a6/html5/thumbnails/27.jpg)
Hypothesis Test Statistics
2
2 21 0
121 1
Subscript 0 = the model under the null hypothesis
Subscript 1 = the model under the alternative hypothesis
1. Based on the Fitting Criterion R
(R -R ) / J F = =F[J,N-K ]
(1-R ) / (N-K )
2. Bas
-1
2 -11 1
ed on the Wald Distance : Note, for linear models, W = JF.
Chi Squared = ( - ) s ( ) ( - )Rb q R X X R Rb q
![Page 28: William Greene Department of Economics Stern School of Business](https://reader030.vdocument.in/reader030/viewer/2022032607/56813062550346895d9633a6/html5/thumbnails/28.jpg)
Hypothesis: All Coefficients Equal Zero
All Coefficients = 0?
R = [0 | I] q = [0]
R12 = .42645
R02 = .00000
F = 280.7 with [11,4153]
Wald = b2-12[V2-12]-1b2-12
= 3087.83355
Note that Wald = JF = 11(280.7)
![Page 29: William Greene Department of Economics Stern School of Business](https://reader030.vdocument.in/reader030/viewer/2022032607/56813062550346895d9633a6/html5/thumbnails/29.jpg)
Hypothesis: Education Effect = 0
ED Coefficient = 0?
R = 0,1,0,0,0,0,0,0,0,0,0,0
q = 0
R12 = .42645
R02 = .36355 (not shown)
F = 455.396
Wald = (.05544-0)2/(.0026)2
= 455.396
Note F = t2 and Wald = F
For a single hypothesis about 1 coefficient.
![Page 30: William Greene Department of Economics Stern School of Business](https://reader030.vdocument.in/reader030/viewer/2022032607/56813062550346895d9633a6/html5/thumbnails/30.jpg)
Hypothesis: Experience Effect = 0
No Experience effect?
R = 0,0,1,0,0,0,0,0,0,0,0,0 0,0,0,1,0,0,0,0,0,0,0,0
q = 0 0R0
2 = .34101, R12 = .42645
F = 309.33
Wald = 618.601 (W* = 5.99)
![Page 31: William Greene Department of Economics Stern School of Business](https://reader030.vdocument.in/reader030/viewer/2022032607/56813062550346895d9633a6/html5/thumbnails/31.jpg)
Built In Test
![Page 32: William Greene Department of Economics Stern School of Business](https://reader030.vdocument.in/reader030/viewer/2022032607/56813062550346895d9633a6/html5/thumbnails/32.jpg)
Robust Covariance Matrix
• What does robustness mean?• Robust to: Heteroscedasticty• Not robust to:
• Autocorrelation• Individual heterogeneity• The wrong model specification
• ‘Robust inference’
-1 2 -1i i ii
The White Estimator
Est.Var[ ] = ( ) e ( )b X X x x X X
![Page 33: William Greene Department of Economics Stern School of Business](https://reader030.vdocument.in/reader030/viewer/2022032607/56813062550346895d9633a6/html5/thumbnails/33.jpg)
Robust Covariance Matrix
Uncorrected
![Page 34: William Greene Department of Economics Stern School of Business](https://reader030.vdocument.in/reader030/viewer/2022032607/56813062550346895d9633a6/html5/thumbnails/34.jpg)
Bootstrapping and Quantile Regresion
![Page 35: William Greene Department of Economics Stern School of Business](https://reader030.vdocument.in/reader030/viewer/2022032607/56813062550346895d9633a6/html5/thumbnails/35.jpg)
Estimating the Asymptotic Variance of an Estimator
• Known form of asymptotic variance: Compute from known results
• Unknown form, known generalities about properties: Use bootstrapping• Root N consistency• Sampling conditions amenable to central limit
theorems• Compute by resampling mechanism within the
sample.
![Page 36: William Greene Department of Economics Stern School of Business](https://reader030.vdocument.in/reader030/viewer/2022032607/56813062550346895d9633a6/html5/thumbnails/36.jpg)
Bootstrapping
Method:1. Estimate parameters using full sample: b2. Repeat R times:
Draw n observations from the n, with replacement
Estimate with b(r). 3. Estimate variance with
V = (1/R)r [b(r) - b][b(r) - b]’
(Some use mean of replications instead of b. Advocated (without motivation) by original designers of the method.)
![Page 37: William Greene Department of Economics Stern School of Business](https://reader030.vdocument.in/reader030/viewer/2022032607/56813062550346895d9633a6/html5/thumbnails/37.jpg)
Application: Correlation between Age and Education
![Page 38: William Greene Department of Economics Stern School of Business](https://reader030.vdocument.in/reader030/viewer/2022032607/56813062550346895d9633a6/html5/thumbnails/38.jpg)
Bootstrap Regression - Replications
namelist;x=one,y,pg$ Define Xregress;lhs=g;rhs=x$ Compute and
display bproc Define
procedureregress;quietly;lhs=g;rhs=x$ … Regression
(silent)endproc Ends
procedureexecute;n=20;bootstrap=b$ 20 bootstrap repsmatrix;list;bootstrp $ Display replications
![Page 39: William Greene Department of Economics Stern School of Business](https://reader030.vdocument.in/reader030/viewer/2022032607/56813062550346895d9633a6/html5/thumbnails/39.jpg)
--------+-------------------------------------------------------------Variable| Coefficient Standard Error t-ratio P[|T|>t] Mean of X--------+-------------------------------------------------------------Constant| -79.7535*** 8.67255 -9.196 .0000 Y| .03692*** .00132 28.022 .0000 9232.86 PG| -15.1224*** 1.88034 -8.042 .0000 2.31661--------+-------------------------------------------------------------Completed 20 bootstrap iterations.----------------------------------------------------------------------Results of bootstrap estimation of model.Model has been reestimated 20 times.Means shown below are the means of thebootstrap estimates. Coefficients shownbelow are the original estimates basedon the full sample.bootstrap samples have 36 observations.--------+-------------------------------------------------------------Variable| Coefficient Standard Error b/St.Er. P[|Z|>z] Mean of X--------+------------------------------------------------------------- B001| -79.7535*** 8.35512 -9.545 .0000 -79.5329 B002| .03692*** .00133 27.773 .0000 .03682 B003| -15.1224*** 2.03503 -7.431 .0000 -14.7654--------+-------------------------------------------------------------
Results of Bootstrap Procedure
![Page 40: William Greene Department of Economics Stern School of Business](https://reader030.vdocument.in/reader030/viewer/2022032607/56813062550346895d9633a6/html5/thumbnails/40.jpg)
Bootstrap Replications
Full sample result
Bootstrapped sample results
![Page 41: William Greene Department of Economics Stern School of Business](https://reader030.vdocument.in/reader030/viewer/2022032607/56813062550346895d9633a6/html5/thumbnails/41.jpg)
Quantile Regression
• Q(y|x,) = x, = quantile• Estimated by linear programming• Q(y|x,.50) = x, .50 median regression• Median regression estimated by LAD (estimates
same parameters as mean regression if symmetric conditional distribution)
• Why use quantile (median) regression?• Semiparametric• Robust to some extensions (heteroscedasticity?)• Complete characterization of conditional distribution
![Page 42: William Greene Department of Economics Stern School of Business](https://reader030.vdocument.in/reader030/viewer/2022032607/56813062550346895d9633a6/html5/thumbnails/42.jpg)
Estimated Variance for Quantile Regression
• Asymptotic Theory
• Bootstrap – an ideal application
![Page 43: William Greene Department of Economics Stern School of Business](https://reader030.vdocument.in/reader030/viewer/2022032607/56813062550346895d9633a6/html5/thumbnails/43.jpg)
1 1
Model : , ( | , ) , [ , ] 0
ˆˆResiduals: u
1Asymptotic Variance:
= E[f (0) ] Estimated by
Asymptotic Theory Based Estimator of Variance of Q - REG
x | x
A C A
A xx
i i i i i i i i
i i i
u
y u Q y Q u
y
N
βx βx
-βx
1
.2
1 1 1ˆ1 | | B
B 2 Bandwidth B can be Silverman's Rule of Thumb:
ˆ ˆ( | .75) ( | .25)1.06 ,
1.349
(1- )(1- ) [ ] Estimated by
x x
C = xx
N
i i ii
i iu
uN
Q u Q uMin s
N
EN
12For =.5 and normally distributed u, this all simplifies to .2
But, this is an ideal application for bootstrapping
X
X
.
X
Xus
![Page 44: William Greene Department of Economics Stern School of Business](https://reader030.vdocument.in/reader030/viewer/2022032607/56813062550346895d9633a6/html5/thumbnails/44.jpg)
= .25
= .50
= .75
![Page 45: William Greene Department of Economics Stern School of Business](https://reader030.vdocument.in/reader030/viewer/2022032607/56813062550346895d9633a6/html5/thumbnails/45.jpg)
OLS vs. Least Absolute Deviations----------------------------------------------------------------------Least absolute deviations estimator...............Residuals Sum of squares = 1537.58603 Standard error of e = 6.82594Fit R-squared = .98284 Adjusted R-squared = .98180Sum of absolute deviations = 189.3973484--------+-------------------------------------------------------------Variable| Coefficient Standard Error b/St.Er. P[|Z|>z] Mean of X--------+------------------------------------------------------------- |Covariance matrix based on 50 replications.Constant| -84.0258*** 16.08614 -5.223 .0000 Y| .03784*** .00271 13.952 .0000 9232.86 PG| -17.0990*** 4.37160 -3.911 .0001 2.31661--------+-------------------------------------------------------------Ordinary least squares regression ............Residuals Sum of squares = 1472.79834 Standard error of e = 6.68059 Standard errors are based onFit R-squared = .98356 50 bootstrap replications Adjusted R-squared = .98256--------+-------------------------------------------------------------Variable| Coefficient Standard Error t-ratio P[|T|>t] Mean of X--------+-------------------------------------------------------------Constant| -79.7535*** 8.67255 -9.196 .0000 Y| .03692*** .00132 28.022 .0000 9232.86 PG| -15.1224*** 1.88034 -8.042 .0000 2.31661--------+-------------------------------------------------------------
![Page 46: William Greene Department of Economics Stern School of Business](https://reader030.vdocument.in/reader030/viewer/2022032607/56813062550346895d9633a6/html5/thumbnails/46.jpg)
Benefits of Panel Data
• Time and individual variation in behavior unobservable in cross sections or aggregate time series
• Observable and unobservable individual heterogeneity
• Rich hierarchical structures• More complicated models• Features that cannot be modeled with only
cross section or aggregate time series data alone
• Dynamics in economic behavior
![Page 47: William Greene Department of Economics Stern School of Business](https://reader030.vdocument.in/reader030/viewer/2022032607/56813062550346895d9633a6/html5/thumbnails/47.jpg)
![Page 48: William Greene Department of Economics Stern School of Business](https://reader030.vdocument.in/reader030/viewer/2022032607/56813062550346895d9633a6/html5/thumbnails/48.jpg)
![Page 49: William Greene Department of Economics Stern School of Business](https://reader030.vdocument.in/reader030/viewer/2022032607/56813062550346895d9633a6/html5/thumbnails/49.jpg)
![Page 50: William Greene Department of Economics Stern School of Business](https://reader030.vdocument.in/reader030/viewer/2022032607/56813062550346895d9633a6/html5/thumbnails/50.jpg)
![Page 51: William Greene Department of Economics Stern School of Business](https://reader030.vdocument.in/reader030/viewer/2022032607/56813062550346895d9633a6/html5/thumbnails/51.jpg)
![Page 52: William Greene Department of Economics Stern School of Business](https://reader030.vdocument.in/reader030/viewer/2022032607/56813062550346895d9633a6/html5/thumbnails/52.jpg)
![Page 53: William Greene Department of Economics Stern School of Business](https://reader030.vdocument.in/reader030/viewer/2022032607/56813062550346895d9633a6/html5/thumbnails/53.jpg)
![Page 54: William Greene Department of Economics Stern School of Business](https://reader030.vdocument.in/reader030/viewer/2022032607/56813062550346895d9633a6/html5/thumbnails/54.jpg)
![Page 55: William Greene Department of Economics Stern School of Business](https://reader030.vdocument.in/reader030/viewer/2022032607/56813062550346895d9633a6/html5/thumbnails/55.jpg)
![Page 56: William Greene Department of Economics Stern School of Business](https://reader030.vdocument.in/reader030/viewer/2022032607/56813062550346895d9633a6/html5/thumbnails/56.jpg)
![Page 57: William Greene Department of Economics Stern School of Business](https://reader030.vdocument.in/reader030/viewer/2022032607/56813062550346895d9633a6/html5/thumbnails/57.jpg)
Application: Health Care UsageGerman Health Care Usage Data, 7,293 Individuals, Varying Numbers of PeriodsThis is an unbalanced panel with 7,293 individuals. There are altogether 27,326 observations. The number of observations ranges from 1 to 7. Frequencies are: 1=1525, 2=2158, 3=825, 4=926, 5=1051, 6=1000, 7=987. Downloaded from the JAE Archive.Variables in the file include DOCTOR = 1(Number of doctor visits > 0) HOSPITAL = 1(Number of hospital visits > 0) HSAT = health satisfaction, coded 0 (low) - 10 (high) DOCVIS = number of doctor visits in last three months HOSPVIS = number of hospital visits in last calendar year PUBLIC = insured in public health insurance = 1; otherwise = 0 ADDON = insured by add-on insurance = 1; otherswise = 0 INCOME = household nominal monthly net income in German marks / 10000. (4 observations with income=0 will sometimes be dropped) HHKIDS = children under age 16 in the household = 1; otherwise = 0 EDUC = years of schooling AGE = age in years MARRIED = marital status
![Page 58: William Greene Department of Economics Stern School of Business](https://reader030.vdocument.in/reader030/viewer/2022032607/56813062550346895d9633a6/html5/thumbnails/58.jpg)
Balanced and Unbalanced Panels• Distinction: Balanced vs. Unbalanced
Panels• A notation to help with mechanics
zi,t, i = 1,…,N; t = 1,…,Ti
• The role of the assumption • Mathematical and notational convenience:
Balanced, n=NT Unbalanced:
• Is the fixed Ti assumption ever necessary? Almost never.
• Is unbalancedness due to nonrandom attrition from an otherwise balanced panel? This would require special considerations.
N
ii=1n T
![Page 59: William Greene Department of Economics Stern School of Business](https://reader030.vdocument.in/reader030/viewer/2022032607/56813062550346895d9633a6/html5/thumbnails/59.jpg)
An Unbalanced Panel: RWM’s GSOEP Data on Health Care
![Page 60: William Greene Department of Economics Stern School of Business](https://reader030.vdocument.in/reader030/viewer/2022032607/56813062550346895d9633a6/html5/thumbnails/60.jpg)
Nonlinear Models
• Specifying the model• Multinomial Choice
• How do the covariates relate to the outcome of interest
• What are the implications of the estimated model?
![Page 61: William Greene Department of Economics Stern School of Business](https://reader030.vdocument.in/reader030/viewer/2022032607/56813062550346895d9633a6/html5/thumbnails/61.jpg)
![Page 62: William Greene Department of Economics Stern School of Business](https://reader030.vdocument.in/reader030/viewer/2022032607/56813062550346895d9633a6/html5/thumbnails/62.jpg)
Unordered Choices of 210 Travelers
![Page 63: William Greene Department of Economics Stern School of Business](https://reader030.vdocument.in/reader030/viewer/2022032607/56813062550346895d9633a6/html5/thumbnails/63.jpg)
Data on Discrete Choices
![Page 64: William Greene Department of Economics Stern School of Business](https://reader030.vdocument.in/reader030/viewer/2022032607/56813062550346895d9633a6/html5/thumbnails/64.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],
itj it i,t,j i,t,k
j itj j it
J(i,t)
j itj j itj=1
N J(i)
iji=1 j=1
P[choice = j | , ,i, t] = Prob[U U k = 1,...,J(i,t)
exp(α + + ' ) =
exp(α + ' + ' )
logL = d lo
x z
β'x γ z
β x γ z
ijgP
![Page 65: William Greene Department of Economics Stern School of Business](https://reader030.vdocument.in/reader030/viewer/2022032607/56813062550346895d9633a6/html5/thumbnails/65.jpg)
Estimated MNL Model
],
itj it i,t,j i,t,k
j itj j it
J(i,t)
j itj j itj=1
P[choice = j | , ,i, t] = Prob[U U k = 1,...,J(i,t)
exp(α + + ' ) =
exp(α + ' + ' )
x z
β'x γ z
β x γ z