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Discrete Choice Modeling
William GreeneStern School of BusinessIFS at UCLFebruary 11-13, 2004
Discrete Choice Modeling
Econometric Methodology Binary Choice Models Multinomial Choice
Model Building Specification Estimation Analysis Applications
NLOGIT Software
Our Agenda1. Methodology2. Discrete Choice Models3. Binary Choice Models4. Panel Data Models for Binary Choice5. Introduction to NLOGIT6. Discrete Choice Settings7. The Multinomial Logit Model8. Heteroscedasticity in Utility Functions9. Nested Logit Modeling10. Latent Class Models11. Mixed Logit Models and Simulation Based Estimation12. Revealed and Stated Preference Data Sets
Part 1
Methodology
Measurement as Observation
Population Measurement Theory
CharacteristicsBehavior Patterns
Individual Behavioral Modeling Assumptions about behavior
Common elements across individuals Unique elements
Prediction Population aggregates Individual behavior
Modeling Choice
Activity as choices Preferences Behavioral axioms Choice as utility maximization
Inference
Population Measurement Econometrics
CharacteristicsBehavior PatternsChoices
Econometric Frameworks
Nonparametric Parametric
Classical (Sampling Theory) Bayesian
Likelihood Based Inference Methods
Behavioral TheoryStatistical Theory
Observed Measurement
LikelihoodFunction
The likelihood function embodies the theoretical description of the population. Characteristics of the population are inferred from the characteristics of the likelihood function. (Bayesian and Classical)
Modeling Discrete Choice Theoretical foundations Econometric methodology
Models Statistical bases Econometric methods
Estimation with econometric software Applications
Part 2
Basics of Discrete Choice Modeling
Modeling Consumer Choice:Continuous Measurement
• What do we measure?
• What is revealed by the data?
• What is the underlying model?
• What are the empirical tools?
Example: Travel expenditure based on price and income
Expenditure
Income
Low price
High price
Discrete Choice Observed outcomes
Inherently discrete: number of occurrences (e.g., family size; considered separately)
Implicitly continuous: the observed data are discrete by construction (e.g., revealed preferences; our main subject)
Implications For model building For analysis and prediction of behavior
Two Fundamental Building Blocks Underlying Behavioral Theory: Random
utility model
The link between underlying behavior and observed data
Empirical Tool: Stochastic, parametric model for binary choice
A platform for models of discrete choice
Random Utility A Theoretical Proposition About Behavior
Consumer making a choice among several alternatives
Example, brand choice (car, food) Choice setting for a consumer: Notation
Consumer i, i = 1, …, N
Choice setting t, t = 1, …, Ti (may be one)
Choice set j, j = 1,…, Ji (may be fixed)
Behavioral Assumptions
Preferences are transitive and complete wrt choice situations
Utility is defined over alternatives: Uijt
Utility maximization assumption
If Ui1t > Ui2t, consumer chooses alternative 1, not alternative 2.
Revealed preference (duality) If the consumer chooses alternative 1 and not
alternative 2, then Ui1t > Ui2t.
Random Utility Functions
Uitj = j + i ’xitj + i’zit + ijt
j = Choice specific constant
xitj = Attributes of choice presented to person i = Person specific taste weights
zit = Characteristics of the person
i = Weights on person specific characteristics
ijt = Unobserved random component of utility
Mean: E[ijt] = 0, Var[ijt] = 1
Part 3
Modeling Binary Choice
A Model for Binary Choice Yes or No decision (Buy/Not buy) Example, choose to fly or not to fly to a destination
when there are alternatives. Model: Net utility of flying Ufly = +1Cost + 2Time + Income + Choose to fly if net utility is positive Data: X = [1,cost,terminal time] Z = [income]
y = 1 if choose fly, Ufly > 0, 0 if not.
What Can Be Learned from the Data? (A Sample of Consumers, i = 1,…,N)
• Are the attributes “relevant?”
• Predicting behavior
- Individual
- Aggregate
• Analyze changes in behavior when
attributes change
Application 210 Commuters Between Sydney and
Melbourne Available modes = Air, Train, Bus, Car Observed:
Choice Attributes: Cost, terminal time, other Characteristics: Household income
First application: Fly or other
Binary Choice Data
Choose Air Gen.Cost Term Time Income1.0000 86.000 25.000 70.000.00000 67.000 69.000 60.000.00000 77.000 64.000 20.000.00000 69.000 69.000 15.000.00000 77.000 64.000 30.000.00000 71.000 64.000 26.000.00000 58.000 64.000 35.000.00000 71.000 69.000 12.000.00000 100.00 64.000 70.0001.0000 158.00 30.000 50.0001.0000 136.00 45.000 40.0001.0000 103.00 30.000 70.000.00000 77.000 69.000 10.0001.0000 197.00 45.000 26.000.00000 129.00 64.000 50.000.00000 123.00 64.000 70.000
An Econometric Model Choose to fly iff UFLY > 0
Ufly = +1Cost + 2Time + Income + Ufly > 0
> -(+1Cost + 2Time + Income) Probability model: For any person observed by the
analyst, Prob(fly) = Prob[ > -(+1Cost + 2Time + Income)]
Note the relationship between the unobserved and the outcome
A Regression - Like Model
INDEX
.2
.4
.6
.8
1.0
.0-1.8 -.6 .6 1.8 3.0-3.0
Pr[
Fly
]
+1Cost + 2TTime + Income
Econometrics How to estimate , 1, 2, ?
It’s not regression The technique of maximum likelihood
Prob[y=1] =
Prob[ > -(+1Cost + 2Time + Income)] Prob[y=0] = 1 - Prob[y=1]
Requires a model for the probability
0 1Prob[ 0] Prob[ 1]
y yL y y
Completing the Model: F() The distribution
Normal: PROBIT, natural for behavior Logistic: LOGIT, allows “thicker tails” Gompertz: EXTREME VALUE, asymmetric, underlies
the basic logit model for multiple choice Does it matter?
Yes, large difference in estimates Not much, quantities of interest are more stable.
Estimated Binary Choice Models
LOGIT PROBIT EXTREME VALUE
Variable Estimate t-ratio Estimate t-ratio Estimate t-ratio
Constant 1.78458 1.40591 0.438772 0.702406 1.45189 1.34775
GC 0.0214688 3.15342 0.012563 3.41314 0.0177719 3.14153
TTME -0.098467 -5.9612 -0.0477826 -6.65089 -0.0868632 -5.91658
HINC 0.0223234 2.16781 0.0144224 2.51264 0.0176815 2.02876
Log-L -80.9658 -84.0917 -76.5422
Log-L(0) -123.757 -123.757 -123.757
A Regression - Like Model
INDEX
.2
.4
.6
.8
1.0
.0-1.8 -.6 .6 1.8 3.0-3.0
Pr[
Fly
]
+1Cost + 2Time + (Income+1)
Effect on predicted probability of an increase in income
( is positive)
Marginal Effects in Probability Models Prob[Outcome] = some F(+1Cost…) “Partial effect” = F(+1Cost…) / ”x”
(derivative) Partial effects are derivatives Result varies with model
Logit: F(+1Cost…) / x = Prob * (1-Prob) * Probit: F(+1Cost…) / x = Normal density
Scaling usually erases model differences
The Delta Method
ˆ ˆ ,
ˆ ˆ ˆˆ. . , ,
ˆˆ . .
ˆ ,ˆ ,
ˆ
f
Est AsyVar
Est AsyVar
f
x
G x V G x
V =
xG x
Marginal Effects for Binary Choice Logit
Probit
ˆ ˆ ˆ[ | ] exp / 1 exp
[ | ]ˆ ˆ ˆ ˆ1
ˆ ˆ ˆ ˆ1 1 2
y
y
x x x x
x x xx
G x x I x x
ˆ[ | ]
[ | ]ˆ ˆ ˆ
ˆ ˆ ˆ
y
y
x x
x xx
G x I x x
Estimated Marginal Effects
Estimate t-ratio Estimate t-ratio Estimate t-ratio
GC .003721 3.267 .003954 3.466 .003393 3.354
TTME -.017065 -5.042 -.015039 -5.754 -.016582 -4.871
HINC .003869 2.193 .004539 2.532 .033753 2.064
Logit Probit Extreme Value
Marginal Effect for a Dummy Variable
Prob[yi = 1|xi,di] = F(’xi+di)
=conditional mean Marginal effect of d
Prob[yi = 1|xi,di=1]=Prob[yi= 1|xi,di=0] Logit: ˆ ˆˆ( )id x x
1 1 0 0
ˆ ˆ(1 ) (1 )
ˆ 0
g
(Marginal) Effect – Dummy Variable
HighIncm = 1(Income > 50)+-------------------------------------------+| Partial derivatives of probabilities with || respect to the vector of characteristics. || They are computed at the means of the Xs. || Observations used are All Obs. |+-------------------------------------------++---------+--------------+----------------+--------+---------+----------+|Variable | Coefficient | Standard Error |b/St.Er.|P[|Z|>z] | Mean of X|+---------+--------------+----------------+--------+---------+----------+ Characteristics in numerator of Prob[Y = 1] Constant .4750039483 .23727762 2.002 .0453 GC .3598131572E-02 .11354298E-02 3.169 .0015 102.64762 TTME -.1759234212E-01 .34866343E-02 -5.046 .0000 61.009524
Marginal effect for dummy variable is P|1 - P|0. HIGHINCM .8565367181E-01 .99346656E-01 .862 .3886 .18571429
(Autodetected)
Computing Effects Compute at the data means?
Simple Inference is well defined
Average individual effects More appropriate? Asymptotic standard errors. (Not done correctly
in the literature – terms are correlated!)
Elasticities
Elasticity =
How to compute standard errors? Delta method Bootstrap
Bootstrap the individual elasticities? (Will neglect variation in parameter estimates.)
Bootstrap model estimation?
ˆlog [ 1| ]ˆ
ˆlog [ 1| ]k
k kk
xP y
x P y
x
x
Estimated Income Elasticity for Air Choice Model
+------------------------------------------+| Results of bootstrap estimation of model.|| Model has been reestimated 25 times. || Statistics shown below are centered || around the original estimate based on || the original full sample of observations.|| Result is ETA = .71183 || bootstrap samples have 840 observations.|| Estimate RtMnSqDev Skewness Kurtosis || .712 .266 -.779 2.258 || Minimum = .125 Maximum = 1.135 |+------------------------------------------+
Mean Income = 34.55, Mean P = .2716, Estimated ME = .004539, Estimated Elasticity=0.5774.
Odds Ratio – Logit Model Only Effect Measure? “Effect of a unit change in
the odds ratio.”Prob[ 1| , ]
exp[ ]Prob[ 0 | , ]
Prob[ 1| , 1]Prob[ 0 | , 1] exp(Prob[ 1| , ]
Prob[ 0 | , ]
y zz
y z
y zy z
y zy z
xx
x
xx
xx
Inference for Odds Ratios Logit coefficient = , estimate = b Coefficient = exp(), estimate = exp(b) Standard error = exp(b) times se(b) t ratio is the same
How Well Does the Model Fit? There is no R squared “Fit measures” computed from log L
“pseudo R squared = 1 – logL0/logL Others… - these do not measure fit.
Direct assessment of the effectiveness of the model at predicting the outcome
Fit Measures for Binary Choice Likelihood Ratio Index
Bounded by 0 and 1 Rises when the model is expanded
Cramer (and others)ˆ ˆ ˆ F | = 1 - F | = 0
=
Mean y Mean y reward for correct predictions minus
penalty for incorrect predictions
Fit Measures for the Logit Model+----------------------------------------+| Fit Measures for Binomial Choice Model || Probit model for variable MODE |+----------------------------------------+| Proportions P0= .723810 P1= .276190 || N = 210 N0= 152 N1= 58 || LogL = -84.09172 LogL0 = -123.7570 || Estrella = 1-(L/L0)^(-2L0/n) = .36583 |+----------------------------------------+| Efron | McFadden | Ben./Lerman || .45620 | .32051 | .75897 || Cramer | Veall/Zim. | Rsqrd_ML || .40834 | .50682 | .31461 |+----------------------------------------+| Information Akaike I.C. Schwarz I.C. || Criteria .83897 189.57187 |+----------------------------------------+
Predicting the Outcome
Predicted probabilities
P = F(a + b1Cost + b2Time + cIncome) Predicting outcomes
Predict y=1 if P is large Use 0.5 for “large” (more likely than not)
Count successes and failures
Individual Predictions from a Logit Model
Observation Observed Y Predicted Y Residual x(i)b Pr[Y=1]
81 .00000 .00000 .0000 -3.3944 .0325
85 .00000 .00000 .0000 -2.1901 .1006
89 1.0000 .00000 1.0000 -2.6766 .0644
93 1.0000 1.0000 .0000 .8113 .6924
97 1.0000 1.0000 .0000 2.6845 .9361
101 1.0000 1.0000 .0000 2.4457 .9202
105 1.0000 .00000 1.0000 -3.2204 .0384
109 1.0000 1.0000 .0000 .0311 .5078
113 .00000 .00000 .0000 -2.1704 .1024
117 .00000 .00000 .0000 -3.3729 .0332
445 .00000 1.0000 -1.0000 .0295 .5074
Note two types of errors and two types of successes.
Predictions in Binary Choice Predict y = 1 if P > P*
Success depends on the assumed P*
ROC Curve Plot %Y=1 correctly predicted vs. %y=1
incorrectly predicted 450 is no fit. Curvature implies fit. Area under the curve compares models
Aggregate PredictionsFrequencies of actual & predicted outcomes
Predicted outcome has maximum probability.
Threshold value for predicting Y=1 = .5000
Predicted
------ ---------- + -----
Actual 0 1 | Total
------ ---------- + -----
0 151 1 | 152
1 20 38 | 58
------ ---------- + -----
Total 171 39 | 210
Analyzing PredictionsFrequencies of actual & predicted outcomes
Predicted outcome has maximum probability.
Threshold value for predicting Y=1 is P* .5000.
(This table can be computed with any P*.)
Predicted
------ -------------------- + -----
Actual 0 1 | Total
------ ----------------------+-------
0 N(a0,p0) N(a0,p1) | N(a0)
1 N(a1,p0) N(a1,p1) | N(a1)
------ ----------------------+ -----
Total N(p0) N(p1) | N
Analyzing Predictions - Success Sensitivity = % actual 1s correctly predicted = 100N(a1,p1)/N(a1) % [100(38/58)=65.5%]
Specificity = % actual 0s correctly predicted = 100N(a0,p0)/N(a0) % [100(151/152)=99.3%]
Positive predictive value = % predicted 1s that were actual 1s = 100N(a1,p1)/N(p1) % [100(38/39)=97.4%]
Negative predictive value = % predicted 0s that were actual 0s = 100N(a0,p0)/N(p0) % [100(151/171)=88.3%]
Correct prediction = %actual 1s and 0s correctly predicted = 100[N(a1,p1)+N(a0,p0)]/N [100(151+38)/210=90.0%]
Analyzing Predictions - Failures False positive for true negative = %actual 0s predicted as 1s =
100N(a0,p1)/N(a0) % [100(1/152)=0.668%]
False negative for true positive = %actual 1s predicted as 0s = 100N(a1,p0)/N(a1) % [100(20/258)=34.5%]
False positive for predicted positive = % predicted 1s that were actual 0s = 100N(a0,p1)/N(p1) % [100(1/39)=2/56%]
False negative for predicted negative = % predicted 0s that were actual 1s = 100N(a1,p0)/N(p0) % [100(20/171)=11.7%]
False predictions = %actual 1s and 0s incorrectly predicted = 100[N(a0,p1)+N(a1,p0)]/N [100(1+20)/210=10.0%]
Aggregate Prediction is a Useful Way to Assess the Importance of a Variable
Frequencies of actual & predicted outcomes. Predicted outcome has maximum probability. Threshold value for predicting Y=1 = .5000
Predicted
------ ---------- + -----
Actual 0 1 | Total
------ ---------- + -----
0 145 7 | 152
1 48 10 | 58
------ ---------- + -----
Total 193 17 | 210
Predicted
------ ---------- + -----
Actual 0 1 | Total
------ ---------- + -----
0 151 1 | 152
1 20 38 | 58
------ ---------- + -----
Total 171 39 | 210
Model fit without TTME
Model fit with TTME
Simulating the Model to Examine Changes in Market Shares
Suppose TTME increased by 25% for everyone.
Before increase After increase
Predicted
------ ---------- + -----
Actual 0 1 | Total
------ ---------- + -----
0 151 1 | 152
1 20 38 | 58
------ ---------- + -----
Total 171 39 | 210
Predicted
------ ---------- + -----
Actual 0 1 | Total
------ ---------- + -----
0 152 0 | 152
1 29 29 | 58
------ ---------- + -----
Total 181 29 | 210
• The model predicts 10 fewer people would fly
• NOTE: The same model used for both sets of predictions.
ScalingUitj = j + i ’xitj + i’zit + ijt
ijt = Unobserved random component of utility
Mean: E[ijt] = 0, Var[ijt] = 1 Why assume variance = 1? What if there are subgroups with different variances?
Cost of ignoring the between group variation? Specifically modeling
More general heterogeneity across people Cost of the homogeneity assumption Modeling issues
Choice Between Two Alternatives By way of example: Automobile type
Choices (1) SUV or (2) Sedan, Ji = 2 One choice situation: Ti = 1 Attribute: xij = price, perhaps others Characteristic: zi = income No variation in taste parameters, i =
What do revealed choices tell us?
Modeling the Binary Choice Ui,suv = suv + Psuv + suvIncome + i,suv Ui,sed
= sed + Psed + sedIncome + i,sed
Chooses SUV: Ui,suv > Ui,sed Ui,suv - Ui,sed > 0
(SUV-SED) + (PSUV-PSED) + (SUV-sed)Income
+ i,suv - i,sed > 0
i > -[ + (PSUV-PSED) + Income]
Probability Model for Choice Between Two Alternatives
i > -[ + (PSUV-PSED) + Income]
Individual vs. Grouped Data Proportions and Frequencies Likelihood is the same
Yji may be 1s and 0s, proportions, or frequencies for the two outcomes.
0 1
1Prob[ 0] Prob[ 1]i i
N y y
iL y y
Weighting and Choice Based Sampling Weighted log likelihood for all data types
Endogenous weights for individual data
“Biased” sampling – “Choice Based”
1
log y0 log Prob[ 0 | ] 1 log Prob[ 1| ]N
i i i i i iiL w y y y
x x
( ) ( ) / ( )
= ( )
i i i i i i
i
i
i
w y y P y
True proportion of y s
Sample proportion of y s
a function of y two values
Choice Based Sample
Sample Population Weight
Air 27.62% 14% 0.5068
Ground 72.38% 86% 1.1882
Choice Based Sampling Correction Maximize Weighted Log Likelihood Covariance Matrix Adjustment
V = H-1 G H-1 (all three weighted)
H = Hessian
G = Outer products of gradients
• “Robust” covariance matrix (?) (Above without weights. What is it robust to?)
Effect of Choice Based Sampling
Unweighted+---------+--------------+----------------+--------+---------+|Variable | Coefficient | Standard Error |b/St.Er.|P[|Z|>z] |+---------+--------------+----------------+--------+---------+ Constant 1.784582594 1.2693459 1.406 .1598 GC .2146879786E-01 .68080941E-02 3.153 .0016 TTME -.9846704221E-01 .16518003E-01 -5.961 .0000 HINC .2232338915E-01 .10297671E-01 2.168 .0302+---------------------------------------------+| Weighting variable CBWT || Corrected for Choice Based Sampling |+---------------------------------------------++---------+--------------+----------------+--------+---------+|Variable | Coefficient | Standard Error |b/St.Er.|P[|Z|>z] |+---------+--------------+----------------+--------+---------+ Constant 1.014022236 1.1786164 .860 .3896 GC .2177810754E-01 .63743831E-02 3.417 .0006 TTME -.7434280587E-01 .17721665E-01 -4.195 .0000 HINC .2471679844E-01 .95483369E-02 2.589 .0096
Hypothesis Testing – Neyman/Pearson Comparisons of Likelihood Functions
Likelihood Ratio Tests Lagrange Multiplier Tests
Distance Measures: Wald Statistics
(All to be demonstrated in the lab)
Heteroscedasticity in Binary Choice Models
Random utility: Yi = 1 iff ’xi + i > 0 Resemblance to regression: How to accommodate
heterogeneity in the random unobserved effects across individuals?
Heteroscedasticity – different scaling Parameterize: Var[i] = exp(’zi) Reformulate probabilities
Probit:
Partial effects are now very complicated
'Prob[ 1]
exp( ' )i
ii
Y
x
z
Application: Credit Data Counts of major derogatory reports) “Deadbeat” = 1 if MAJORDRG > 0 Mean depends on AGE, INCOME, OWNRENT,
SELFEMPLOYED Variance depends on AVGEXP, DEPENDT (average
monthly expenditure, number of dependents) Probit model with heteroscedasticity
Probit with Heteroscedasticity+---------------------------------------------+| Binomial Probit Model || Dependent variable DEADBEAT || Number of observations 1319 || Log likelihood function -639.3388 || Restricted log likelihood -653.3217 || Chi-squared 27.96596 || Degrees of freedom 6 || Significance level .9535906E-04 |+---------------------------------------------++---------+--------------+----------------+--------+---------+----------+|Variable | Coefficient | Standard Error |b/St.Er.|P[|Z|>z] | Mean of X|+---------+--------------+----------------+--------+---------+----------+ Index function for probability Constant -1.272312665 .13598690 -9.356 .0000 AGE .1126209389E-01 .40404726E-02 2.787 .0053 33.213103 INCOME .5286782288E-01 .20239074E-01 2.612 .0090 3.3653760 OWNRENT -.2049230056 .88518106E-01 -2.315 .0206 .44048522 SELFEMPL .1143040149 .13825044 .827 .4084 .68991660E-01 Variance function AVGEXP -.4768665802E-03 .12613317E-03 -3.781 .0002 185.05707 DEPNDT .6880605703E-02 .42546206E-01 .162 .8715 .99393480+-------------------------------------------+| Partial derivatives of E[y] = F[*] with || respect to the vector of characteristics. || They are computed at the means of the Xs. || Observations used for means are All Obs. |+-------------------------------------------++---------+--------------+----------------+--------+---------+----------+|Variable | Coefficient | Standard Error |b/St.Er.|P[|Z|>z] | Mean of X|+---------+--------------+----------------+--------+---------+----------+ Index function for probability Constant -.3768739381 .54283831E-01 -6.943 .0000 AGE .3335964337E-02 .12357954E-02 2.699 .0069 33.213103 INCOME .1566006938E-01 .65292318E-02 2.398 .0165 3.3653760 OWNRENT -.6070059841E-01 .24667682E-01 -2.461 .0139 .44048522 SELFEMPL .3385819023E-01 .41052591E-01 .825 .4095 .68991660E-01 Variance function AVGEXP -.1133874143E-03 .31868469E-04 -3.558 .0004 185.05707 DEPNDT .1636042704E-02 .10080807E-01 .162 .8711 .99393480
Part 4Panel Data Models for Binary Choice
Panel Data and Binary Choice Models
Uit = + ’xit + it + Person i specific effect
Fixed effects using “dummy” variables
Uit = i + ’xit + it
Random effects using omitted heterogeneity
Uit = + ’xit + (it + vi)
Same outcome mechanism: Yit = [Uit > 0]
Fixed and Random Effects Models Fixed Effects
Robust to both specifications Inconvenient to compute (many parameters) Incidental parameters problem
Random Effects Inconsistent if correlated with X Small number of parameters Easier to compute
Computation – available estimators
Fixed Effects Dummy variable coefficients
Uit = i + ’xit + it
Can be done by “brute force” for 10,000s of individuals
F(.) = appropriate probability for the observed outcome
Compute and i for i=1,…,N (may be large) See “Estimating Econometric Models with Fixed Effects” at
www.stern.nyu.edu/~wgreene
1 1log log ( ' )iN T
i iti tL F
x
Random Effects Uit = + ’xit + (it + v vi) Logit model (can be generalized) Joint probability for individual i | vi = Unobserved component vi must be eliminated
Maximize wrt , and v
How to do the integration? Analytic integration – quadrature; most familiar software Simulation
1( ' )iT
it v itF v
x
1 1
1log log ( ' )iTN i
it v i ii tv v
vL F v f dv
x
Estimation by Simulation
1 1
1log ( ' )iTN i
it v i ii tv v
vF v f dv
x
is the sum of the logs of E[Pr(y1,y2,…|vi)]. Can be estimated by sampling vi and averaging. (Use random numbers.)
1 1 1
1log ( ' )iTN R
it v iri r tF v
R x
Random Effects is Equivalent to a Random Constant Term Uit = + ’xit + (it + v vi) = ( + vi) + ’xit + it
= i + ’xit + it
i is random with mean and variance
View the simulation as sampling over i
2
1 1 1
1log ( ' )iTN R
ir iti r tF
R x
• Why not make all the coefficients random?
A Sampling Experiment CLOGIT data using GC, TTME, INVT and HINC Standardized data: each Xit* is (Xit – Mean(X))/Sx
Constructed utilities
Uit = 0 + 1GCit* + 1TTMEit* + 1INVTit*
+ (Random numberit + HINCi*) Treat 4 observations in each group as a panel with
T = 4. (We will examine a “live” panel data set in the lab.)
Estimated Fixed Effects Model+---------------------------------------------+| FIXED EFFECTS Logit Model || Maximum Likelihood Estimates || Dependent variable Z || Weighting variable None || Number of observations 840 || Iterations completed 5 || Log likelihood function -342.1919 || Sample is 4 pds and 210 individuals. || Bypassed 51 groups with inestimable a(i). || LOGIT (Logistic) probability model |+---------------------------------------------++---------+--------------+----------------+--------+---------+----------+|Variable | Coefficient | Standard Error |b/St.Er.|P[|Z|>z] | Mean of X|+---------+--------------+----------------+--------+---------+----------+ Index function for probability GC .6708935970E-02 .18621919E-01 .360 .7186 112.29560 TTME .3648053834E-01 .57989428E-02 6.291 .0000 34.779874 INVT .3338438006E-02 .25104319E-02 1.330 .1836 492.25314 INVC .6795479927E-02 .19477804E-01 .349 .7272 48.448113 Partial derivatives of E[y] = F[*] with respect to the characteristics. Computed at the means of the Xs. Estimated E[y|means,mean alphai]= .501 Estimated scale factor for dE/dx= .250 GC .1677222976E-02 .46555287E-02 .360 .7186 112.29560 TTME .9120074679E-02 .14482840E-02 6.297 .0000 34.779874 INVT .8346040194E-03 .62727700E-03 1.331 .1833 492.25314 INVC .1698858823E-02 .48687627E-02 .349 .7271 48.448113
WHY?
Estimated Random Effects Model (1)+---------------------------------------------+| Logit Model for Panel Data || Maximum Likelihood Estimates || Dependent variable Z || Weighting variable None || Number of observations 840 || Iterations completed 15 || Log likelihood function -494.6084 || Hosmer-Lemeshow chi-squared = 15.81181 || P-value= .04515 with deg.fr. = 8 || Random Effects Logit Model for Panel Data |+---------------------------------------------++---------+--------------+----------------+--------+---------+----------+|Variable | Coefficient | Standard Error |b/St.Er.|P[|Z|>z] | Mean of X|+---------+--------------+----------------+--------+---------+----------+ Characteristics in numerator of Prob[Y = 1] Constant -2.074416165 .20930847 -9.911 .0000 GC .9739427161E-02 .53423005E-02 1.823 .0683 TTME .8353847679E-02 .30194645E-02 2.767 .0057 INVT .1252315669E-03 .69864222E-03 .179 .8577 INVC -.1215241461E-02 .55156025E-02 -.220 .8256 RndmEfct .9492940742E-01 .18841088 .504 .6144 -.58755677E-07
Estimated Random Effects Model (2)+---------------------------------------------+| Random Coefficients Logit Model || Maximum Likelihood Estimates || Dependent variable Z || Weighting variable None || Number of observations 840 || Iterations completed 14 || Log likelihood function -494.5136 || Restricted log likelihood -496.1793 || Chi-squared 3.331300 || Degrees of freedom 1 || Significance level .6797315E-01 || Sample is 4 pds and 210 individuals. || LOGIT (Logistic) probability model || Simulation based on 100 random draws |+---------------------------------------------++---------+--------------+----------------+--------+---------+----------+|Variable | Coefficient | Standard Error |b/St.Er.|P[|Z|>z] | Mean of X|+---------+--------------+----------------+--------+---------+----------+ Nonrandom parameters GC .1928882840E-01 .40879229E-02 4.718 .0000 110.87976 TTME .2364065236E-01 .24280249E-02 9.737 .0000 34.589286 INVT .5332059842E-03 .54092102E-03 .986 .3243 486.16548 INVC -.6668386903E-02 .41649216E-02 -1.601 .1094 47.760714 Means for random parameters Constant -2.942970074 .15967241 -18.431 .0000 Scale parameters for dists. of random parameters Constant .5338591567 .56357583E-01 9.473 .0000 Conditional Mean at Sample Point .4886 Scale Factor for Marginal Effects .2499 GC .4819681744E-02 .10205421E-02 4.723 .0000 110.87976 TTME .5907067980E-02 .59571899E-03 9.916 .0000 34.589286 INVT .1332316870E-03 .13504534E-03 .987 .3239 486.16548 INVC -.1666223679E-02 .10411841E-02 -1.600 .1095 47.760714
Commands for Panel Data Models Model: LOGIT ; Lhs = …
; Rhs = …
; Pds = number of periods Common effect
Fixed effects ; FEM $ or ; Fixed $ Random ; Random Effects $ Simulation ; RPM ; Fcn=One(N) $
Use with Probit, Logit (and many others)