Discrete Choice Modeling

1
Discrete Choice Modeling

• William Greene
• Stern School of Business
• IFS at UCL
• February 11-13, 2004

2
Discrete Choice Modeling

• Econometric Methodology
• Binary Choice Models
• Multinomial Choice
• Model Building
• Specification
• Estimation
• Analysis
• Applications
• NLOGIT Software

3
Our Agenda

1. Methodology
2. Discrete Choice Models
3. Binary Choice Models
4. Panel Data Models for Binary Choice
5. Introduction to NLOGIT
6. Discrete Choice Settings
7. The Multinomial Logit Model
8. Heteroscedasticity in Utility Functions
9. Nested Logit Modeling
10. Latent Class Models
11. Mixed Logit Models and Simulation Based
    Estimation
12. Revealed and Stated Preference Data Sets

4
Part 1

• Methodology

5
Measurement as Observation

Population
Measurement
Theory
Characteristics Behavior Patterns
6
Individual Behavioral Modeling

• Assumptions about behavior
• Common elements across individuals
• Unique elements
• Prediction
• Population aggregates
• Individual behavior

7
Modeling Choice

• Activity as choices
• Preferences
• Behavioral axioms
• Choice as utility maximization

8
Inference

Population
Measurement
Econometrics
Characteristics Behavior Patterns Choices
9
Econometric Frameworks

• Nonparametric
• Parametric
• Classical (Sampling Theory)
• Bayesian

10
Likelihood Based Inference Methods
Behavioral Theory
Likelihood Function
Statistical Theory
Observed Measurement
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)
11
Modeling Discrete Choice

• Theoretical foundations
• Econometric methodology
• Models
• Statistical bases
• Econometric methods
• Estimation with econometric software
• Applications

12
Part 2

• Basics of Discrete Choice Modeling

13
Modeling Consumer ChoiceContinuous Measurement
Example Travel expenditure based on price and
income
Expenditure
Low price

• What do we measure?
• What is revealed by the data?
• What is the underlying model?
• What are the empirical tools?

High price
Income
14
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

15
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

16
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)

17
Behavioral Assumptions

• Preferences are transitive and complete wrt
  choice situations
• Utility is defined over alternatives Uijt
• Utility maximization assumption
• If Ui1t gt Ui2t, consumer chooses
  alternative 1, not alternative 2.
• Revealed preference (duality)
• If the consumer chooses alternative 1 and not
  alternative 2, then Ui1t gt Ui2t.

18
Random Utility Functions
Uitj ?j ?i xitj ?izit ?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
19
Part 3

• Modeling Binary Choice

20
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 gt 0, 0 if not.

21
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

22
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

23
Binary Choice Data
Choose Air Gen.Cost Term Time
Income 1.0000 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
70.000 1.0000 158.00 30.000
50.000 1.0000 136.00 45.000
40.000 1.0000 103.00 30.000
70.000 .00000 77.000 69.000
10.000 1.0000 197.00 45.000
26.000 .00000 129.00 64.000
50.000 .00000 123.00 64.000 70.000
24
An Econometric Model

• Choose to fly iff UFLY gt 0
• Ufly ??1Cost ?2Time ?Income ?
• Ufly gt 0 ?
• ? gt -(??1Cost ?2Time ?Income)
• Probability model For any person observed by
  the analyst, Prob(fly) Prob? gt -(??1Cost
  ?2Time ?Income)
• Note the relationship between the unobserved ?
  and the outcome

25
??1Cost ?2TTime ?Income
26
Econometrics

• How to estimate ?, ?1, ?2, ??
• Its not regression
• The technique of maximum likelihood
• Proby1
• Prob? gt -(??1Cost ?2Time ?Income)
• Proby0 1 - Proby1
• Requires a model for the probability

27
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.

28
(No Transcript)
29
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
30
Effect on predicted probability of an increase in
income
??1Cost ?2Time ?(Income1)
(? is positive)
31
Marginal Effects in Probability Models

• ProbOutcome 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

32
The Delta Method

33
Marginal Effects for Binary Choice

• Logit
• Probit

34
Estimated Marginal Effects
Logit Probit
Extreme Value
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
35
Marginal Effect for a Dummy Variable

• Probyi 1xi,di F(?xi?di)
• conditional mean
• Marginal effect of d
• Probyi 1xi,di1Probyi 1xi,di0
• Logit

36
(Marginal) Effect Dummy Variable

• HighIncm 1(Income gt 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.PZgtz Mean of
X -------------------------------------------
-----------------------
Characteristics in numerator of ProbY 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 P1 - P0. HIGHINCM .8565367181E-01
.99346656E-01 .862 .3886 .18571429
(Autodetected)
37
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!)

38
Elasticities

• Elasticity
• How to compute standard errors?
• Delta method
• Bootstrap
• Bootstrap the individual elasticities? (Will
  neglect variation in parameter estimates.)
• Bootstrap model estimation?

39
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 Elasticity0.5774.
40
Odds Ratio Logit Model Only

• Effect Measure? Effect of a unit change in the
  odds ratio.

41
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

42
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

43
Fit Measures for Binary Choice

• Likelihood Ratio Index
• Bounded by 0 and 1
• Rises when the model is expanded
• Cramer (and others)

44
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 -----------------------------------
-----
45
Predicting the Outcome

• Predicted probabilities
• P F(a b1Cost b2Time cIncome)
• Predicting outcomes
• Predict y1 if P is large
• Use 0.5 for large (more likely than not)
• Count successes and failures

46
Individual Predictions from a Logit Model
Observation Observed Y Predicted Y Residual
x(i)b PrY1 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.
47
Predictions in Binary Choice

• Predict y 1 if P gt P
• Success depends on the assumed
  P

48
ROC Curve

• Plot Y1 correctly predicted vs. y1
  incorrectly predicted
• 450 is no fit. Curvature implies fit.
• Area under the curve compares models

49

50
Aggregate Predictions
Frequencies of actual predicted
outcomes Predicted outcome has maximum
probability. Threshold value for predicting Y1
.5000 Predicted ------ ----------
----- Actual 0 1 Total ------
---------- ----- 0 151 1
152 1 20 38 58 ------
---------- ----- Total 171 39 210
51
Analyzing Predictions
Frequencies of actual predicted
outcomes Predicted outcome has maximum
probability. Threshold value for predicting Y1
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
52
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 100N(a1,p1)N(a0,p0)/N
  100(15138)/21090.0

53
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 100N(a0,p1)N(a1,p0)/N
  100(120)/21010.0

54
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 Y1 .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
55
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 ------
---------- ----- 0 152 0
152 1 29 29 58 ------
---------- ----- Total 181 29 210
Predicted ------ ----------
----- Actual 0 1 Total ------
---------- ----- 0 151 1
152 1 20 38 58 ------
---------- ----- Total 171 39 210

• The model predicts 10 fewer people would fly
• NOTE The same model used for both sets of
  predictions.

56
Scaling

• Uitj ?j ?i xitj ?izit ?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

57
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?

58

• Modeling the Binary Choice
• Ui,suv ?suv ?Psuv ?suvIncome ?i,suv
  Ui,sed ?sed ?Psed ?sedIncome ?i,sed
• Chooses SUV Ui,suv gt Ui,sed
• Ui,suv - Ui,sed gt 0
• (?SUV-?SED) ?(PSUV-PSED) (?SUV-?sed)Income
• ?i,suv - ?i,sed gt 0
• ?i gt -? ?(PSUV-PSED) ?Income

59
Probability Model for Choice Between Two
Alternatives
?i gt -? ?(PSUV-PSED) ?Income
60
Individual vs. Grouped Data

• Proportions and Frequencies
• Likelihood is the same
• Yji may be 1s and 0s, proportions, or frequencies
  for the two outcomes.

61
Weighting and Choice Based Sampling

• Weighted log likelihood for all data types
• Endogenous weights for individual data
• Biased sampling Choice Based

62
Choice Based Sample

Sample Population Weight
Air 27.62 14 0.5068
Ground 72.38 86 1.1882
63
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?)

64
Effect of Choice Based Sampling
Unweighted ------------------------------------
-------------------- Variable Coefficient
Standard Error b/St.Er.PZgtz
--------------------------------------------
------------ 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.PZgtz
--------------------------------------------
------------ 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

65
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)

66
Heteroscedasticity in Binary Choice Models

• Random utility Yi 1 iff ?xi ?i gt 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

67
Application Credit Data

• Counts of major derogatory reports)
• Deadbeat 1 if MAJORDRG gt 0
• Mean depends on AGE, INCOME, OWNRENT,
  SELFEMPLOYED
• Variance depends on AVGEXP, DEPENDT (average
  monthly expenditure, number of dependents)
• Probit model with heteroscedasticity

68
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.PZgtz 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 Ey 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.PZgtz 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
69
Part 4

• Panel Data Models for Binary Choice

70
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 gt 0

71
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

72
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 i1,,N (may be large)
• See Estimating Econometric Models with Fixed
  Effects at www.stern.nyu.edu/wgreene

73
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

74
Estimation by Simulation
is the sum of the logs of EPr(y1,y2,vi). Can
be estimated by sampling vi and averaging. (Use
random numbers.)
75
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

Why not make all the coefficients random?
76
A Sampling Experiment

• CLOGIT data using GC, TTME, INVT and HINC
• Standardized data each Xit is (Xit
  Mean(X))/Sx
• Constructed utilities
• Uit 0 1?GCit 1?TTMEit 1?INVTit
• (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.)

77
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.PZgtz 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 Ey F with respect
to the characteristics. Computed at the means of
the Xs. Estimated Eymeans,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?
78
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.PZgtz Mean of
X -------------------------------------------
-----------------------
Characteristics in numerator of ProbY 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
79
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.PZgtz 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
80
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 FcnOne(N)
• Use with Probit, Logit (and many others)