1/62: Topic 2.3 – Panel Data Binary Choice Models
Microeconometric Modeling
William Greene
Stern School of Business
New York University
New York NY USA
2.3 Panel Data Models
for Binary Choice
2/62: Topic 2.3 – Panel Data Binary Choice Models
Concepts
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Unbalanced Panel
Attrition Bias
Inverse Probability Weight
Heterogeneity
Population Averaged Model
Clustering
Pooled Model
Quadrature
Maximum Simulated Likelihood
Conditional Estimator
Incidental Parameters Problem
Partial Effects
Bias Correction
Mundlak Specification
Variable Addition Test
Models
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Random Effects Progit
Fixed Effects Probit
Fixed Effects Logit
Dynamic Probit
Mundlak Formulation
Correlated Random Effects Model
3/62: Topic 2.3 – Panel Data Binary Choice Models
4/62: Topic 2.3 – Panel Data Binary Choice Models
Application: Health Care Panel
Data
German Health Care Usage Data
Data downloaded from Journal of Applied Econometrics Archive. This is an unbalanced panel with 7,293
individuals. They can be used for regression, count models, binary choice, ordered choice, and bivariate binary
choice. 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).
Variables in the file are
DOCTOR
HOSPITAL
HSAT
DOCVIS
HOSPVIS
PUBLIC
ADDON
HHNINC
=
=
=
=
=
=
=
=
HHKIDS
EDUC
AGE
MARRIED
=
=
=
=
1(Number of doctor visits > 0)
1(Number of hospital visits > 0)
health satisfaction, coded 0 (low) - 10 (high)
number of doctor visits in last three months
number of hospital visits in last calendar year
insured in public health insurance = 1; otherwise = 0
insured by add-on insurance = 1; otherswise = 0
household nominal monthly net income in German marks / 10000.
(4 observations with income=0 were dropped)
children under age 16 in the household = 1; otherwise = 0
years of schooling
age in years
marital status
5/62: Topic 2.3 – Panel Data Binary Choice Models
Unbalanced Panels
Most theoretical results are for balanced panels.
Most real world panels are unbalanced.
Often the gaps are caused by attrition.
The major question is whether the gaps are ‘missing
completely at random.’ If not, the observation
mechanism is endogenous, and at least some
methods will produce questionable results.
Group Sizes
Researchers rarely have any reason to treat the data
as nonrandomly sampled. (This is good news.)
6/62: Topic 2.3 – Panel Data Binary Choice Models
Unbalanced Panels and Attrition ‘Bias’

Test for ‘attrition bias.’ (Verbeek and Nijman, Testing for Selectivity
Bias in Panel Data Models, International Economic Review, 1992,
33, 681-703.



Variable addition test using covariates of presence in the panel
Nonconstructive – what to do next?
Do something about attrition bias. (Wooldridge, Inverse Probability
Weighted M-Estimators for Sample Stratification and Attrition,
Portuguese Economic Journal, 2002, 1: 117-139)


Stringent assumptions about the process
Model based on probability of being present in each wave of the panel
7/62: Topic 2.3 – Panel Data Binary Choice Models
8/62: Topic 2.3 – Panel Data Binary Choice Models
Inverse Probability Weighting
Panel is based on those present at WAVE 1, N1 individuals
Attrition is an absorbing state. No reentry, so N1  N 2  ...  N T .
Sample is restricted at each wave to individuals who were present at
the previous wave.
d it = 1[Individual is present at wave t].
d i1 = 1  i, d it  0  d i ,t 1  0.
xi1  covariates observed for all i at entry that relate to likelihood of
being present at subsequent waves.
(health problems, disability, psychological well being, self employment,
unemployment, maternity leave, student, caring for family member, ...)
Probit model for d it  1[xi1  wit ], t = 2,...,T. ˆ it  fitted probability.
t
Assuming attrition decisions are independent, Pˆit   s 1 ˆ is
ˆ  d it
Inverse probability weight W
it
P̂it
Weighted log likelihood
logLW   i 1  t 1 log Lit (No common effects.)
N
T
9/62: Topic 2.3 – Panel Data Binary Choice Models
10/62: Topic 2.3 – Panel Data Binary Choice Models
Panel Data Binary Choice Models
Random Utility Model for Binary Choice
Uit =  + ’xit
+ it + Person i specific effect
Fixed effects using “dummy” variables
Uit = i + ’xit + it
Random effects using omitted heterogeneity
Uit =  + ’xit + it + ui
Same outcome mechanism: Yit = 1[Uit > 0]
11/62: Topic 2.3 – Panel Data Binary Choice Models
Ignoring Unobserved
Heterogeneity
Assuming strict exogeneity; Cov(x it ,ui  it )  0
y it *=x it β  ui  it
Prob[y it  1 | x it ]  Prob[ui  it  -x it β]
Using the same model format:


Prob[y it  1 | x it ]  F x it β / 1+u2  F( x it δ)
This is the 'population averaged model.'
12/62: Topic 2.3 – Panel Data Binary Choice Models
13/62: Topic 2.3 – Panel Data Binary Choice Models
Ignoring Heterogeneity in the RE Model
Ignoring heterogeneity, we estimate δ not β.
Partial effects are δ f( x it δ) not βf( x itβ)
β is underestimated, but f( x itβ) is overestimated.
Which way does it go? Maybe ignoring u is ok?
Not if we want to compute probabilities or do
statistical inference about β. Estimated standard
errors will be too small.
Scale factor is too large
 is too small
14/62: Topic 2.3 – Panel Data Binary Choice Models
Age  43.527
P
 (.37176  .01625Age).01625  .006128
Age


PRE
 1    x 1   
Age


= 1  .45298  (.53689  .02338Age) 1  .45298 .02338
= .00648
Population average coefficient is off by 43% but the partial effect is off by 6%
15/62: Topic 2.3 – Panel Data Binary Choice Models
Ignoring Heterogeneity (Broadly)




Presence will generally make parameter estimates look
smaller than they would otherwise.
Ignoring heterogeneity will definitely distort standard
errors.
Partial effects based on the parametric model may not
be affected very much.
Is the pooled estimator ‘robust?’ Less so than in the
linear model case.
16/62: Topic 2.3 – Panel Data Binary Choice Models
Pooled vs. RE Panel Estimator
---------------------------------------------------------------------Binomial Probit Model
Dependent variable
DOCTOR
--------+------------------------------------------------------------Variable| Coefficient
Standard Error b/St.Er. P[|Z|>z]
Mean of X
--------+------------------------------------------------------------Constant|
.02159
.05307
.407
.6842
AGE|
.01532***
.00071
21.695
.0000
43.5257
EDUC|
-.02793***
.00348
-8.023
.0000
11.3206
HHNINC|
-.10204**
.04544
-2.246
.0247
.35208
--------+------------------------------------------------------------Unbalanced panel has
7293 individuals
--------+------------------------------------------------------------Constant|
-.11819
.09280
-1.273
.2028
AGE|
.02232***
.00123
18.145
.0000
43.5257
EDUC|
-.03307***
.00627
-5.276
.0000
11.3206
HHNINC|
.00660
.06587
.100
.9202
.35208
Rho|
.44990***
.01020
44.101
.0000
--------+-------------------------------------------------------------
17/62: Topic 2.3 – Panel Data Binary Choice Models
Partial Effects
---------------------------------------------------------------------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] Elasticity
--------+------------------------------------------------------------|Pooled
AGE|
.00578***
.00027
21.720
.0000
.39801
EDUC|
-.01053***
.00131
-8.024
.0000
-.18870
HHNINC|
-.03847**
.01713
-2.246
.0247
-.02144
--------+------------------------------------------------------------|Based on the panel data estimator
AGE|
.00620***
.00034
18.375
.0000
.42181
EDUC|
-.00918***
.00174
-5.282
.0000
-.16256
HHNINC|
.00183
.01829
.100
.9202
.00101
--------+-------------------------------------------------------------
18/62: Topic 2.3 – Panel Data Binary Choice Models
The Effect of Clustering





Yit must be correlated with Yis across periods
Pooled estimator ignores correlation
Broadly, yit = E[yit|xit] + wit,
 E[yit|xit] = Prob(yit = 1|xit)
 wit is correlated across periods
Assuming the marginal probability is the same, the
pooled estimator is consistent. (We just saw that it might
not be.)
Ignoring the correlation across periods generally leads to
underestimating standard errors.
19/62: Topic 2.3 – Panel Data Binary Choice Models
‘Cluster’ Corrected Covariance Matrix
C  the number if clusters
nc  number of observations in cluster c
H1 = negative inverse of second derivatives matrix
gic = derivative of log density for observation
20/62: Topic 2.3 – Panel Data Binary Choice Models
Cluster Correction: Doctor
---------------------------------------------------------------------Binomial Probit Model
Dependent variable
DOCTOR
Log likelihood function
-17457.21899
--------+------------------------------------------------------------Variable| Coefficient
Standard Error b/St.Er. P[|Z|>z]
Mean of X
--------+------------------------------------------------------------| Conventional Standard Errors
Constant|
-.25597***
.05481
-4.670
.0000
AGE|
.01469***
.00071
20.686
.0000
43.5257
EDUC|
-.01523***
.00355
-4.289
.0000
11.3206
HHNINC|
-.10914**
.04569
-2.389
.0169
.35208
FEMALE|
.35209***
.01598
22.027
.0000
.47877
--------+------------------------------------------------------------| Corrected Standard Errors
Constant|
-.25597***
.07744
-3.305
.0009
AGE|
.01469***
.00098
15.065
.0000
43.5257
EDUC|
-.01523***
.00504
-3.023
.0025
11.3206
HHNINC|
-.10914*
.05645
-1.933
.0532
.35208
FEMALE|
.35209***
.02290
15.372
.0000
.47877
--------+-------------------------------------------------------------
21/62: Topic 2.3 – Panel Data Binary Choice Models
Random Effects
Assuming strict exogeneity; Cov(x it ,ui  it )  0
y it *=+x it β  ui  it
Prob[y it  1 | x it ]  Prob[ui  it  - - x it β]


Prob[y it  1 | x it ]  F +x it β / 1+u2  F(0  x it δ)
This is the 'population averaged model.'
Prob[y it  1 | x it ,ui ]  Prob[it  - -x it β - u v i ]
Prob[y it  1 | x it ,ui ]  F    x it β  u v i
This is the structural model

22/62: Topic 2.3 – Panel Data Binary Choice Models
Quadrature – Butler and Moffitt (1982)
This method is used in most commerical software since 1982

T
logL   i1 log  t i 1 F(y it ,   x it  u v i )    v i  dv i
 


 -v 2 
N
1
=  i1 log g( v )
exp 
ui ~
 dv i

2
 2 
N
N[0, 2u ]
=  u vi
(make a change of variable to w = v/ 2

1
N
where vi ~
2
=
l
og
g(
2w)
exp
-w
dw

i

 i1
The integral can be computed using Hermite quadrature.


1

 i1 log h1 whg( 2zh )
N
N[0,1]
H

The values of w h (weights) and zh (nodes) are found in published
tables such as Abramovitz and Stegun (or on the web). H is by
choice. Higher H produces greater accuracy (but takes longer).
23/62: Topic 2.3 – Panel Data Binary Choice Models
Quadrature Log Likelihood
After all the substitutions, the function to be maximized:
Not simple, but feasible.
N
H
Ti
1



logL   i1 log
w
F(y
,



x


2
z
)


h
it
it
u
h
 t 1

 h1
N
H
Ti
1



  i1 log
w
F(y
,



x


z
)


h
it
it
h
 t 1

 h1


24/62: Topic 2.3 – Panel Data Binary Choice Models
Simulation Based Estimator

Ti

logL   i1 log  t 1 F(y it ,   x it  u v i )    v i  dv i
 


 -v i2 
N
1
=  i1 log g(v i )
exp 
 dv i

2
 2 
N
This equals

N
i1
log E[g( v i )]
The expected value of the function of v i can be approximated
by drawing R random draws v ir from the population N[0,1] and
averaging the R functions of v ir . We maximize
logL S   i1 log
N
1 R  Ti
F(y it ,   x it  u v ir ) 

r 1  t 1

R
25/62: Topic 2.3 – Panel Data Binary Choice Models
Random Effects Model: Quadrature
---------------------------------------------------------------------Random Effects Binary Probit Model
Dependent variable
DOCTOR
Log likelihood function
-16290.72192  Random Effects
Restricted log likelihood -17701.08500  Pooled
Chi squared [
1 d.f.]
2820.72616
Estimation based on N = 27326, K =
5
Unbalanced panel has
7293 individuals
--------+------------------------------------------------------------Variable| Coefficient
Standard Error b/St.Er. P[|Z|>z]
Mean of X
--------+------------------------------------------------------------Constant|
-.11819
.09280
-1.273
.2028
AGE|
.02232***
.00123
18.145
.0000
43.5257
EDUC|
-.03307***
.00627
-5.276
.0000
11.3206
HHNINC|
.00660
.06587
.100
.9202
.35208
Rho|
.44990***
.01020
44.101
.0000
--------+------------------------------------------------------------|Pooled Estimates
Constant|
.02159
.05307
.407
.6842
AGE|
.01532***
.00071
21.695
.0000
43.5257
EDUC|
-.02793***
.00348
-8.023
.0000
11.3206
HHNINC|
-.10204**
.04544
-2.246
.0247
.35208
--------+-------------------------------------------------------------
26/62: Topic 2.3 – Panel Data Binary Choice Models
Random Parameter Model
---------------------------------------------------------------------Random Coefficients Probit
Model
Dependent variable
DOCTOR (Quadrature Based)
Log likelihood function
-16296.68110 (-16290.72192)
Restricted log likelihood -17701.08500
Chi squared [
1 d.f.]
2808.80780
Simulation based on 50 Halton draws
--------+------------------------------------------------Variable| Coefficient
Standard Error b/St.Er. P[|Z|>z]
--------+------------------------------------------------|Nonrandom parameters
AGE|
.02226***
.00081
27.365
.0000 ( .02232)
EDUC|
-.03285***
.00391
-8.407
.0000 (-.03307)
HHNINC|
.00673
.05105
.132
.8952 ( .00660)
|Means for random parameters
Constant|
-.11873**
.05950
-1.995
.0460 (-.11819)
|Scale parameters for dists. of random parameters
Constant|
.90453***
.01128
80.180
.0000
--------+-------------------------------------------------------------
Using quadrature, a = -.11819. Implied  from these estimates is
.904542/(1+.904532) = .449998 compared to .44990 using quadrature.
27/62: Topic 2.3 – Panel Data Binary Choice Models
A Dynamic Model
y it  1[x it  y i,t 1  it  ui > 0]
Two similar 'effects'
Unobserved heterogeneity
State dependence = state 'persistence'
Pr(y it  1 | y i,t 1 ,..., y i0 , x it ,u]  F[ x it   y i,t 1  ui ]
How to estimate ,  , marginal effects, F(.), etc?
(1) Deal with the latent common effect
(2) Handle the lagged effects:
This encounters the initial conditions problem.
28/62: Topic 2.3 – Panel Data Binary Choice Models
Dynamic Probit Model: A Standard Approach
(1) Conditioned on all effects, joint probability
P(y i1 , y i2 ,..., y iT | y i0 , x i ,ui )   t 1 F( x it β  y i,t 1  ui , y it )
T
(2) Unconditional density; integrate out the common effect
P(y i1 , y i2 ,..., y iT | y i0 , x i ) 



P(y i1 , y i2 ,..., y iT | y i0 , x i ,ui )h(ui | y i0 , x i )dui
(3) Density for heterogeneity
h(ui | y i0 , x i )  N[  y i0  x iδ, u2 ], x i = [x i1 ,x i2 ,...,x iT ], so
ui =   y i0  x iδ + u w i
(contains every period of x it )
(4) Reduced form
P(y i1 , y i2 ,..., y iT | y i0 , x i ) 

T

t 1
 
F( x it β  y i,t 1    y i0  x iδ  u w i , y it )h(w i )dw i
This is a random effects model
29/62: Topic 2.3 – Panel Data Binary Choice Models
Simplified Dynamic Model
Projecting ui on all observations expands the model enormously.
(3) Projection of heterogeneity only on group means
h(ui | y i0 , x i )  N[  y i0  x iδ, u2 ] so
ui =   y i0  x iδ + w i
(4) Reduced form
P(y i1 , y i2 ,..., y iT | y i0 , x i ) 

T

t 1
 
F(  x it β  y i,t 1  y i0  x iδ  u w i , y it )h(w i )dw i
Mundlak style correction with the initial value in the equation.
This is (again) a random effects model
30/62: Topic 2.3 – Panel Data Binary Choice Models
A Dynamic Model for Public Insurance
Age
Household Income
Kids in the household
Health Status
Add initial value, lagged value, group means
31/62: Topic 2.3 – Panel Data Binary Choice Models
Dynamic Common Effects Model
32/62: Topic 2.3 – Panel Data Binary Choice Models
Application
Stewart, JAE, 2007




British Household Panel Survey (1991-1996)
3060 households retained (balanced) out of 4739 total.
Unemployment indicator (0.1)
Data features
 Panel data – unobservable heterogeneity
 State persistence: “Someone unemployed at t-1 is
more than 20 times as likely to be unemployed at t as
someone employed at t-1.”
33/62: Topic 2.3 – Panel Data Binary Choice Models
Application: Direct Approach
34/62: Topic 2.3 – Panel Data Binary Choice Models
GHK Simulation/Estimation
The presence of the autocorrelation and state dependence in the model
invalidate the simple maximum likelihood procedures we examined earlier.
The appropriate likelihood function is constructed by formulating the
probabilities as
Prob( yi,0, yi,1, . . .) = Prob(yi,0) × Prob(yi,1 | yi,0) ×・ ・ ・×Prob(yi,T | yi,T-1) .
This still involves a T = 7 order normal integration, which is approximated in
the study using a simulator similar to the GHK simulator.
35/62: Topic 2.3 – Panel Data Binary Choice Models
Problems with Dynamic RE Probit




Assumes yi,0 and the effects are uncorrelated
Assumes the initial conditions are exogenous – OK if the process
and the observation begin at the same time, not if different.
Doesn’t allow time invariant variables in the model.
The normality assumption in the projection.
36/62: Topic 2.3 – Panel Data Binary Choice Models
Distributional Problem

Normal distributions assumed throughout




Alternative: Discrete distribution for ui.



Normal distribution for the unique component, εi,t
Normal distribution assumed for the heterogeneity, ui
Sensitive to the distribution?
Heckman and Singer style, latent class model.
Conventional estimation methods.
Why is the model not sensitive to normality for εi,t
but it is sensitive to normality for ui?
37/62: Topic 2.3 – Panel Data Binary Choice Models
Fixed Effects Modeling

Advantages:


Disadvantages:




Allows correlation of covariates and heterogeneity
Complications of computing all those dummy variable coefficients – solved
problem (Greene, 2004)
No time invariant variables – not solvable in the FE context
Incidental Parameters problem – persistent small T bias, does not go away
Strategies





Unconditional estimation
Conditional estimation – Rasch/Chamberlain
Hybrid conditional/unconditional estimation
Bias corrrections
Mundlak estimator
38/62: Topic 2.3 – Panel Data Binary Choice Models
Fixed Effects Models


Estimate with dummy variable coefficients
Uit = i + ’xit + it
Can be done by “brute force” for 10,000s of individuals
log L  i 1
N




Ti
t 1
log F ( yit , i  xit )
F(.) = appropriate probability for the observed outcome
Compute  and i for i=1,…,N (may be large)
See FixedEffects.pdf in course materials.
39/62: Topic 2.3 – Panel Data Binary Choice Models
Unconditional Estimation

Maximize the whole log likelihood

Difficult! Many (thousands) of parameters.

Feasible – NLOGIT (2004) (“Brute force”)
40/62: Topic 2.3 – Panel Data Binary Choice Models
Fixed Effects Health Model
Groups in which yit is always = 0 or always = 1. Cannot compute αi.
41/62: Topic 2.3 – Panel Data Binary Choice Models
Conditional Estimation
Principle: f(yi1,yi2,… | some statistic) is free of
the fixed effects for some models.
 Maximize the conditional log likelihood, given
the statistic.
 Can estimate β without having to estimate αi.
 Only feasible for the logit model. (Poisson
and a few other continuous variable models.
No other discrete choice models.)

42/62: Topic 2.3 – Panel Data Binary Choice Models
Binary Logit Conditional Probabilities
ei  xit 
Prob( yit  1| xit ) 
.
 i  xit 
1 e
Ti


Prob  Yi1  yi1 , Yi 2  yi 2 , , YiTi  yiTi  yit 
t 1


Ti


 Ti

exp   yit xit  
exp   yit xit β 
 t 1

 t 1



.
Ti
Ti




 Ti 


exp
d
x

exp
d
x
β
All   different ways that
 t dit Si  

  it it 
it it 
Si 

 t 1

 t 1

t dit can equal Si
Denominator is summed over all the different combinations of Ti values
of yit that sum to the same sum as the observed  Tt=1i yit . If Si is this sum,
T 
there are   terms. May be a huge number. An algorithm by Krailo
 Si 
and Pike makes it simple.
43/62: Topic 2.3 – Panel Data Binary Choice Models
Example: Two Period Binary Logit

e i  xitβ
Prob(y it  1 | xit ) 
.

1  e i  xitβ

Prob  Yi1  y i1 , Yi2  y i2 ,



Prob  Yi1


Prob  Yi1


Prob  Yi1


Prob  Yi1

, YiTi  y iTi

y

0
,
data


it
t 1

2

 1, Yi2  0  y it  1 , data 
t 1

2

 0, Yi2  1  y it  1 , data 
t 1

2

 1, Yi2  1  y it  2 , data 
t 1

 0, Yi2  0
2
 Ti


exp
y
x

  it it 
Ti

 t 1

y it , data  
.

Ti



t 1


exp
d
x



 tdit Si

it it
 t 1

 1.
exp( x i1β)
exp( x i1β)  exp( x i2β)
exp( x i2β)

exp( x i1β)  exp( x i2β)

 1.
44/62: Topic 2.3 – Panel Data Binary Choice Models
Example: Seven Period Binary Logit
Prob[y = (1,0,0,0,1,1,1)|Xi ]=
exp(i  x1 )
exp( i  x7 )
1

 ... 
1  exp( i  x1 ) 1  exp( i  x 2 )
1  exp( i  x 7 )
There are 35 different sequences of yit (permutations) that sum to 4.
For example, y*it| p1 might be (1,1,1,1,0,0,0). Etc.
Prob[y=(1,0,0,0,1,1,1)|Xi ,t71yit =7] =
exp t71 yit xit 
7
*


exp


y
 p1  t 1 it| p xit 
35
45/62: Topic 2.3 – Panel Data Binary Choice Models
46/62: Topic 2.3 – Panel Data Binary Choice Models
With T = 50, the number of permutations of sequences of
y ranging from sum = 0 to sum = 50 ranges from 1 for 0 and 50,
to 2.3 x 1012 for 15 or 35 up to a maximum of 1.3 x 1014 for sum =25.
These are the numbers of terms that must be summed for a model
with T = 50. In the application below, the sum ranges from 15 to 35.
47/62: Topic 2.3 – Panel Data Binary Choice Models
The sample is 200 individuals each observed 50 times.
48/62: Topic 2.3 – Panel Data Binary Choice Models
The data are generated from a probit process with b1 = b2 = .5. But, it is fit as a
logit model. The coefficients obey the familiar relationship, 1.6*probit.
49/62: Topic 2.3 – Panel Data Binary Choice Models
Estimating Partial Effects
“The fixed effects logit estimator of  immediately gives us
the effect of each element of xi on the log-odds ratio…
Unfortunately, we cannot estimate the partial effects…
unless we plug in a value for αi. Because the distribution of
αi is unrestricted – in particular, E[αi] is not necessarily zero
– it is hard to know what to plug in for αi. In addition, we
cannot estimate average partial effects, as doing so would
require finding E[Λ(xit + αi)], a task that apparently requires
specifying a distribution for αi.”
(Wooldridge, 2010)
50/62: Topic 2.3 – Panel Data Binary Choice Models
MLE for Logit Constant Terms
Step 1. Estimate β with Chamberlain's conditional estimator
Step 2. Treating β as if it were known, estimate i from the
first order condition
ˆ
Ti
1
e i e xit β
1
y i   t 1

ˆ
Ti
1  e i e xit β Ti
ic it
1

 t 1 1   c T
i it
i
Ti
c it
 t 1   c
i
it
Ti
Estimate i  1 / exp(i )  i   log i
 
ˆ is treated as known data.
c it  exp xitβ
Solve one equation in one unknown for each i.
Note there is no solution if y i = 0 or 1.
Iterating back and forth does not maximize logL.
51/62: Topic 2.3 – Panel Data Binary Choice Models
An Approximate Solution for E[i]
Pit   (i  x it )
 has been consistently estimated using b
1 T
1 T
Let yi =  t 1 y it estimate Pi , and xi   t 1 x it
T
T
To an error O(1/T), yi estimates  ( i  xi ).
1
0

Overall, approximate plim

(



x
)


(

 bplimx))

i
it
i,t
nT
 y 
1 n
1
0
use plim  i 1 yi = plim
y
.
Solve
for


log
 bx



i,t it
n
nT
1 y 
52/62: Topic 2.3 – Panel Data Binary Choice Models
An Approximate Solution for i
Pit   ( i  x it )
1 T
 ( i  x it )

t 1
T
(Note, no solution if yi = 0 or 1.)
The FOC for i is yi 
 has been consistently estimated using b
Equate yi to
1
T
Tt 1 (i  bx it ). Use a first order Taylor series around 0
yi  T1 Tt 1 (0  bx it )
ˆ i    1 T
0
0



(


b
x
)[1


(

 bx it )]
it
T t 1
0
(Will tend to give large but finite values when yi = 0 or 1.)
53/62: Topic 2.3 – Panel Data Binary Choice Models
54/62: Topic 2.3 – Panel Data Binary Choice Models
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Systematic overestimation by alphai, or random noise? A second run of the experiment:
57/62: Topic 2.3 – Panel Data Binary Choice Models
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59/62: Topic 2.3 – Panel Data Binary Choice Models
Fixed Effects Logit Health Model:
Conditional vs. Unconditional
60/62: Topic 2.3 – Panel Data Binary Choice Models
Incidental Parameters Problems:
Conventional Wisdom

General: The unconditional MLE is biased in
samples with fixed T except in special cases
such as linear or Poisson regression (even when
the FEM is the right model).
The conditional estimator (that bypasses
estimation of αi) is consistent.

Specific: Upward bias (experience with probit
and logit) in estimators of 
61/62: Topic 2.3 – Panel Data Binary Choice Models
62/62: Topic 2.3 – Panel Data Binary Choice Models
A Monte Carlo Study of the
FE Estimator: Probit vs. Logit
Estimates of Coefficients and Marginal
Effects at the Implied Data Means
Results are scaled so the desired quantity being estimated
(, , marginal effects) all equal 1.0 in the population.
63/62: Topic 2.3 – Panel Data Binary Choice Models
Bias Correction Estimators

Motivation: Undo the incidental parameters bias in the
fixed effects probit model:



Advantages




(1) Maximize a penalized log likelihood function, or
(2) Directly correct the estimator of β
For (1) estimates αi so enables partial effects
Estimator is consistent under some circumstances
(Possibly) corrects in dynamic models
Disadvantage



No time invariant variables in the model
Practical implementation
Extension to other models? (Ordered probit model (maybe) –
see JBES 2009)
64/62: Topic 2.3 – Panel Data Binary Choice Models
A Mundlak Correction for
the FE Model
Fixed Effects Model :
y*it  i  xit  it ,i = 1,...,N; t = 1,...,Ti
yit  1 if yit > 0, 0 otherwise.
Mundlak (Wooldridge, Heckman, Chamberlain),...
i    xi  ui (Projection, not necessarily conditional mean)
where u is normally distributed with mean zero and standard
deviation u and is uncorrelated with xi or (xi1 , xi 2 ,..., xiT )
Reduced form random effects model
y*it    xi  xit  it  ui ,i = 1,...,N; t = 1,...,Ti
yit  1 if yit > 0, 0 otherwise.
65/62: Topic 2.3 – Panel Data Binary Choice Models
Mundlak Correction
66/62: Topic 2.3 – Panel Data Binary Choice Models
A Variable Addition Test for FE vs.
RE
The Wald statistic of 45.27922 and
the likelihood ratio statistic of
40.280 are both far larger than the
critical chi squared with 5 degrees
of freedom, 11.07. This suggests
that for these data, the fixed
effects model is the preferred
framework.
67/62: Topic 2.3 – Panel Data Binary Choice Models
Fixed Effects Models Summary







Incidental parameters problem if T < 10 (roughly)
Inconvenience of computation
Appealing specification
Alternative semiparametric estimators?
 Theory not well developed for T > 2
 Not informative for anything but slopes (e.g.,
predictions and marginal effects)
Ignoring the heterogeneity definitely produces an
inconsistent estimator (even with cluster correction!)
A Hobson’s choice
Mundlak correction is a useful common approach.