Modeling Consumer Decision Making and Discrete

Report
Discrete Choice Modeling
William Greene
Stern School of Business
New York University
Part 5
Panel Data Models
Application: Health Care Panel Data
German Health Care Usage Data, 7,293 Individuals, Varying Numbers of Periods
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
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.)
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.



Do something about attrition bias. (Wooldridge, Inverse Probability
Weighted M-Estimators for Sample Stratification and Attrition,
Portuguese Economic Journal, 2002, 1: 117-139)



Variable addition test using covariates of presence in the panel
Nonconstructive – what to do next?
Stringent assumptions about the process
Model based on probability of being present in each wave of the panel
We return to these in discussion of applications of ordered choice
models
Panel Data Models

Benefits




Modeling heterogeneity
Rich specifications
Modeling dynamic effects in individual behavior
Costs


More complex models and estimation procedures
Statistical issues for identification and estimation
Fixed and Random Effects


Model: Feature of interest yit
Probability distribution or conditional mean






Observable covariates xit, zi
Individual specific heterogeneity, ui
Probability or mean, f(xit,zi,ui)
Random effects: E[ui|xi1,…,xiT,zi] = 0
Fixed effects: E[ui|xi1,…,xiT,zi] = g(Xi,zi).
The difference relates to how ui relates to the
observable covariates.
Household Income
We begin by analyzing Income using linear regression.
Fixed and Random Effects in Regression

yit = ai + b’xit + eit



How do we proceed for a binary choice model?



Random effects: Two step FGLS. First step is OLS
Fixed effects: OLS based on group mean differences
yit* = ai + b’xit + eit
yit = 1 if yit* > 0, 0 otherwise.
Neither ols nor two step FGLS works (even
approximately) if the model is nonlinear.


Models are fit by maximum likelihood, not OLS or GLS
New complications arise that are absent in the linear case.
Pooled Linear Regression - Income
---------------------------------------------------------------------Ordinary
least squares regression ............
LHS=HHNINC
Mean
=
.35208
Standard deviation
=
.17691
Number of observs.
=
27326
Model size
Parameters
=
2
Degrees of freedom
=
27324
Residuals
Sum of squares
=
796.31864
Standard error of e =
.17071
Fit
R-squared
=
.06883
Adjusted R-squared
=
.06879
Model test
F[ 1, 27324] (prob) = 2019.6(.0000)
--------+------------------------------------------------------------Variable| Coefficient
Standard Error b/St.Er. P[|Z|>z]
Mean of X
--------+------------------------------------------------------------Constant|
.12609***
.00513
24.561
.0000
EDUC|
.01996***
.00044
44.940
.0000
11.3206
--------+-------------------------------------------------------------
Fixed Effects
---------------------------------------------------------------------Least Squares with Group Dummy Variables..........
Ordinary
least squares regression ............
LHS=HHNINC
Mean
=
.35208
Standard deviation
=
.17691
Number of observs.
=
27326
Model size
Parameters
=
7294
Degrees of freedom
=
20032
Residuals
Sum of squares
=
277.15841
Standard error of e =
.11763
Fit
R-squared
=
.67591
Adjusted R-squared
=
.55791
Model test
F[***, 20032] (prob) =
5.7(.0000)
--------+------------------------------------------------------------Variable| Coefficient
Standard Error b/St.Er. P[|Z|>z]
Mean of X
--------+------------------------------------------------------------EDUC|
.03664***
.00289
12.688
.0000
11.3206
--------+------------------------------------------------------------For the pooled model, R squared was .06883 and
the estimated coefficient on EDUC was .01996.
Random Effects
---------------------------------------------------------------------Random Effects Model: v(i,t)
= e(i,t) + u(i)
Estimates: Var[e]
=
.013836
Var[u]
=
.015308
Corr[v(i,t),v(i,s)] =
.525254
Lagrange Multiplier Test vs. Model (3) =*******
( 1 degrees of freedom, prob. value = .000000)
(High values of LM favor FEM/REM over CR model)
Baltagi-Li form of LM Statistic =
4534.78
Sum of Squares
796.363710
R-squared
.068775
--------+------------------------------------------------------------Variable| Coefficient
Standard Error b/St.Er. P[|Z|>z]
Mean of X
--------+------------------------------------------------------------EDUC|
.02051***
.00069
29.576
.0000
11.3206
Constant|
.11973***
.00808
14.820
.0000
--------+------------------------------------------------------------Note: ***, **, * = Significance at 1%, 5%, 10% level.
---------------------------------------------------------------------For the pooled model, the estimated coefficient on EDUC was .01996.
Fixed vs. Random Effects

Linear Models
Fixed Effects





Robust to both cases
Use OLS
Convenient



Random Effects




Inconsistent in FE case:
effects correlated with X
Use FGLS: No necessary
distributional assumption
Smaller number of parameters
Inconvenient to compute
Nonlinear Models
Fixed Effects

Usually inconsistent because
of ‘IP’ problem
Fit by full ML
Extremely inconvenient
Random Effects




Inconsistent in FE case :
effects correlated with X
Use full ML: Distributional
assumption
Smaller number of parameters
Always inconvenient to
compute
Binary Choice Model

Model is Prob(yit = 1|xit) (zi is embedded in xit)

In the presence of heterogeneity,
Prob(yit = 1|xit,ui) = F(xit,ui)
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]
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itβ]
Using the same model format:


Prob[y it  1 | x it ]  F xitβ / 1+u2  F( x it δ)
This is the 'population averaged model.'
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.
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.
Pooled vs. A 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
--------+-------------------------------------------------------------
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
--------+-------------------------------------------------------------
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.
‘Cluster’ Corrected Covariance Matrix
C  the number if clusters
n c  number of observations in cluster c
H1 = negative inverse of second derivatives matrix
gic = derivative of log density for observation
 C 
V H 

 C 1
1
 
C
nc
c 1
i 1
gic


1

 i 1 gic H
nc
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
--------+-------------------------------------------------------------
Modeling a Binary Outcome




Did firm i produce a product or process innovation in year t ?
yit : 1=Yes/0=No
Observed N=1270 firms for T=5 years, 1984-1988
Observed covariates: xit = Industry, competitive pressures,
size, productivity, etc.
How to model?



Binary outcome
Correlation across time
Heterogeneity across firms
Application: Innovation
A Random Effects Model
U it    xit  ui , u i ~ N [0, u ]
Ti = observations on individual i
For each period, yit  1[U it  0] (given u i )
Joint probability for Ti observations is
Prob( yi1 , yi 2 ,...)   t 1 F ( yit ,   xit  ui )
Ti
For convenience, write u i = u vi , vi ~ N [0,1]
T
N
log L | v1 ,...vN   i i log  t i 1 F ( yit ,   xit   u vi ) 


It is not possible to maximize log L | v1 ,...vN because of
the unobserved random effects.
A Computable Log Likelihood
The unobserved heterogeneity is averaged out

Ti

log L   i 1 log   t 1 F ( yit ,   xit  u vi )  f  vi  dvi
 

Maximize this function with respect to ,, u .
N
How to compute the integral?
(1) Analytically? No, no formula exists.
(2) Approximately, using Gauss-Hermite quadrature
(3) Approximately using Monte Carlo simulation
Quadrature – Butler and Moffitt
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 
 dv i

2
 2 
N
(make a change of variable to w = v/ 2
=
1



 i1 log g( 2w) exp -w2 dwi
N


The integral can be computed using Hermite quadrature.
1
N
H

log
w hg( 2zh )


i1
h 1

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).
Quadrature Log Likelihood
After all the substitutions, the function to be maximized:
Not simple, but feasible.
Ti
N
1
H



logL   i1 log
w
F(y
,



x


2
z
)


h
it
it
u
h
 t 1

 h1


Simulation
N

N

Ti

logL   i1 log  t 1 F(y it ,   x it  u v i )    v i  dv i
 

=

i1
log g(v i )
This equals


N
i1
 -v i2 
exp 
 dv i
2
 2 
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
Random Effects Model
---------------------------------------------------------------------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
Significance level
.00000
McFadden Pseudo R-squared
.0796766
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 using the Butler and Moffitt method
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
--------+-------------------------------------------------------------
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.
Fixed Effects Models




Uit = i + ’xit + it
For the linear model, i and  (easily) estimated
separately using least squares
For most nonlinear models, it is not possible to
condition out the fixed effects. (Mean deviations does
not work.)
Even when it is possible to estimate  without i, in
order to compute partial effects, predictions, or
anything else interesting, some kind of estimate of i
is still needed.
Fixed Effects Models


Estimate with dummy variable coefficients
Uit = i + ’xit + it
Can be done by “brute force” even 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)
Unconditional Estimation

Maximize the whole log likelihood

Difficult! Many (thousands) of parameters.

Feasible – NLOGIT (2001) (‘Brute force’)
(One approach is just to create the thousands of
dummy variables – SAS.)
Fixed Effects Health Model
Groups in which yit is always = 0 or always = 1. Cannot compute αi.
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.)

Binary Logit Conditional Probabiities
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 d it  S i  

  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.
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.
Comments on Enumeration in the Logit Model
" This can easily be generalized for any T. However
the computations rise geometrically with T and are
excessive for T > 10. See Greene (1993)."
(Baltagi, Panel Data, 1st edition)
"If T is large, getting ... can be cumbersome as
one can guess from 3.5 with T = 3."
(M.J.Lee, "Panel Data Econometrics")
(Both unaware of Krailo and Pike...)
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, 2002)
Binary Logit Estimation


Estimate  by maximizing conditional logL
Estimate i by using the ‘known’  in the FOC for the unconditional
logL
ˆ x )
exp(



i
it
ˆ
(
y

P
)

0,
P

t 1 it it
it
1  exp(i  ˆ xit )
Ti





Solve for the N constants, one at a time treating  as known.
No solution when yit sums to 0 or Ti
“Works” if E[i|Σiyit] = E[i].
Use the average of the estimates of i for E[i]. Works if the cases
of Σiyit = 0 or Σiyit = T occur completely at random.
Use this average to compute predictions and partial effects.
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
1
yi 
Ti

Ti
t 1
ˆ
e i e xitβ
ˆ

1
Ti
ic it
1

 t 1 1   c T
i it
i
Ti
1  e i e xitβ
Estimate i  1 / exp(i )  i   log i
c it
 t 1   c
i
it
Ti
ˆ) is treated as known data.
c it  exp( x it β
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.
Fixed Effects Logit Health Model: Conditional
vs. Unconditional
Advantages and Disadvantages
of the FE Model
 Advantages




Allows correlation of effect and regressors
Fairly straightforward to estimate
Simple to interpret
Disadvantages



Model may not contain time invariant variables
Not necessarily simple to estimate if very large
samples (Stata just creates the thousands of dummy
variables)
The incidental parameters problem: Small T bias
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 . Exactly 100%
when T = 2. Declines as T increases.
Some Familiar Territory – 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.
A Monte Carlo Study of the FE Probit Estimator
Percentage Biases in Estimates of Coefficients, Standard
Errors and Marginal Effects at the Implied Data Means
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)
A Mundlak Correction for the FE Model
“Correlated Random Effects”
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.
Mundlak Correction
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.
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.
Dynamic Models
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.
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) (The rabbit in the hat) Density for heterogeneity
h(ui | y i0 , x i )  N[  y i0  x iδ, u2 ], x i = [x i1 ,x i2 ,...,x iT ], so
ui =   y i0  x iδ + 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 iδ  u w i , y it )h(w i )dw i
This is a random effects model
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 mo del
A Dynamic Model for Public Insurance
Age
Household Income
Kids in the household
Health Status
Basic Model
Add initial value, lagged value, group means
Dynamic Common Effects Model

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