Discrete Choice Modeling

William Greene

Stern School of Business

New York University

Part 4

Panel Data Models

Application: Health Care Panel Data

German Health Care Usage Data, 7,293 Individuals, Varying Numbers of PeriodsVariables in the file areData 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.  This is a large data set.  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).  Note, the variable NUMOBS below tells how many observations there are for each person.  This variable is repeated in each row of the data for the person.  (Downloaded from the JAE Archive) DOCTOR = 1(Number of doctor visits > 0) HOSPITAL = 1(Number of hospital visits > 0) HSAT =  health satisfaction, coded 0 (low) - 10 (high)   DOCVIS =  number of doctor visits in last three months HOSPVIS =  number of hospital visits in last calendar year PUBLIC =  insured in public health insurance = 1; otherwise = 0 ADDON =  insured by add-on insurance = 1; otherswise = 0 HHNINC =  household nominal monthly net income in German marks / 10000. (4 observations with income=0 were dropped) HHKIDS = children under age 16 in the household = 1; otherwise = 0 EDUC =  years of schooling AGE = age in years MARRIED = marital status EDUC = years of education

Unbalanced Panel: Group Sizes

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

Random effects: Two step FGLS. First step is OLS Fixed effects: OLS based on group mean differences

How do we proceed for a binary choice model? 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. = 27326Model size Parameters = 2 Degrees of freedom = 27324Residuals Sum of squares = 796.31864 Standard error of e = .17071Fit R-squared = .06883 Adjusted R-squared = .06879Model 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. = 27326Model size Parameters = 7294 Degrees of freedom = 20032Residuals Sum of squares = 277.15841 Standard error of e = .11763Fit R-squared = .67591 Adjusted R-squared = .55791Model 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 coefficientOn 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)] = .525254Lagrange 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 11.3206Constant| .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

i it

it i it

it it i it

2it it u

Assuming strict exogeneity; Cov( ,u ) 0

y *= u

Prob[y 1| x ] Prob[u - ]

Using the same model format:

Prob[y 1| x ] F / 1+ F( )

This is the "population averaged mo

it

it

it

it it

x

x β

x β

x β x δ

del."

Ignoring Heterogeneity

Ignoring heterogeneity, we estimate not .

Partial effects are f( ) not f( )

is underestimated, but f( ) is overestimated.

Which way does it go? Maybe ignoring u is ok?

Not if we want

it it

it

δ β

δ x δ β x β

β x β

to compute probabilities or do

statistical inference about Estimated standard

errors will be too small.

β.

Pooled vs. A Panel Estimator----------------------------------------------------------------------Binomial Probit ModelDependent 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[*] withrespect to the vector of characteristicsThey are computed at the means of the XsObservations 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

Robustness is not the justification.

c

1

1 1

1 1 1

the number if clusters

n number of observations in cluster c

= inverse of second derivatives matrix

= derivative of log density for observation

1c c

ic

C n n

ic icc i i

C

C

C

H

g

V H g g H

Cluster Correction: Doctor----------------------------------------------------------------------Binomial Probit ModelDependent variable DOCTORLog likelihood function -17457.21899--------+-------------------------------------------------------------Variable| Coefficient Standard Error b/St.Er. P[|Z|>z] Mean of X--------+------------------------------------------------------------- | Conventional Standard ErrorsConstant| -.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 ErrorsConstant| -.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 2: Innovation

A Random Effects Model

i

i

it i

i

1 2 ,1

, u ~ [0, ]

T = observations on individual i

For each period, y 1[ 0] (given u )

Joint probability for T observations is

Prob( , ,...) ( )

For convenience, wr

i

it i u

it

T

i i it it

it

it

u N

U

y y F y u

U

x

x

i u

1 , u1

1

ite u = , ~ [0,1]

log | ,... log ( )

It is not possible to maximize log | ,... because of

the unobserved random effects.

i

i i

TN

N it it ii i t

N

v v N

L v v F y v

L v v

x

A Computable Log Likelihood

1 1

u

log log ( , )

Maximize this functio

The unobserved heterogeneity is a

n with respect to , , .

How to compute the integral?

(1) Analyticall

verage

y? No, no

d

fo

o t

m

u

u

r

iTN

it it u i i ii tL F y v f v dv

x

la exists.

(2) Approximately, using Gauss-Hermite quadrature

(3) Approximately using Monte Carlo simulation

Quadrature – Butler and Moffitt

xiN

ii 1

N

ii

T

it it

1

it 1 i

2

u

Th

F(y , v )

g(

v

1 -vexp

22

logL l

is method is used in most commerical software since

og dv

= log dv

(make a change of variable to w = v/ 2

v

198

)

2

=

N

ii 1

N H

i 1 h

2

h h

hh 1

1 l g( 2w)

g( 2z )

The values

og dw

The integral can be

of w (weights) and z

exp -w

(node

computed using

s) are found i

Hermite quadr

n published

tab

ature.

1

les such as A

log w

bramovitz and Stegun (or on the web). H is by

choice. Higher H produces greater accuracy (but takes longer).

Quadrature Log Likelihood

xiN

h

H

i

T

it it u11 1 hth

After all the substitutions, the function to be maximized:

Not simple, but feasibl

1logL lo F(y , z )g

e

2

.

w

Simulation

xiN

ii 1 i

2i

T

it it u it 1

i

N

ii 1

N

i

i

i1

logL log dv

= log dv

T ]

The expected value of the functio

his equals log E[

n of v can be a

F(y ,

ppro

v

-v1

xima

ex

te

v )

g(v )

d

by dr

g(

aw

p22

v )

ing

xiTN R

S it it u ir

i

i 1

r

ir

r 1 t 1

R random draws v from the population N[0,1] and

averaging the R functions of v . We maxi

1logL log F(y , v

i

)R

mze

Random Effects Model----------------------------------------------------------------------Random Effects Binary Probit ModelDependent variable DOCTORLog likelihood function -16290.72192 Random EffectsRestricted log likelihood -17701.08500 PooledChi squared [ 1 d.f.] 2820.72616Significance level .00000McFadden Pseudo R-squared .0796766Estimation based on N = 27326, K = 5Unbalanced 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 methodConstant| .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 ModelDependent variable DOCTOR (Quadrature Based)

Log likelihood function -16296.68110 (-16290.72192) Restricted log likelihood -17701.08500Chi squared [ 1 d.f.] 2808.80780Significance level .00000McFadden Pseudo R-squared .0793400Estimation based on N = 27326, K = 5Unbalanced panel has 7293 individualsPROBIT (normal) probability modelSimulation 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 parametersConstant| -.11873** .05950 -1.995 .0460 (-.11819) |Scale parameters for dists. of random parametersConstant| .90453*** .01128 80.180 .0000--------+-------------------------------------------------------------

Fixed Effects Models Estimate with 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 FixedEffects.pdf in course materials.

1 1log log ( , )iN T

it i iti tL F y

x

Unconditional Estimation

Maximize the whole log likelihood

Difficult! Many (thousands) of parameters.

Feasible – NLOGIT (2001) (“Brute force”)

Fixed Effects Health ModelGroups 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

i

i

1 1 2 2

1 1

1

T

S1 1

All

Prob( 1| ) .1

Prob , , ,

exp exp

exp exp

i it

i it

i i

i i

i i

t i

i

t i

it it

i i i i iT iT

T T

it it it itt t

T T

it it it i

T

itt

td St t

ey

e

Y y Y y Y y

y

d d

y

y

x

xx

x x

x x

β

β

i

t i

different

iT

it t=1 i

ways that

can equal S

.

Denominator is summed over all the different combinations of T valuesof y that sum to the same sum as the observed . If S is this sum,

there are

it

it

d

y

i

terms. May be a huge number. An algorithm by KrailoS

and Pike makes it simple.

T

Example: Two Period Binary Logit

i it

i it

i

i

i i i

t it i

it it

T

it itTt 1

i1 i1 i2 i2 iT iT it Tt 1

it itd St 1

2

i1 i2 itt 1

i1 i

eProb(y 1| ) .

1 e

exp y

Prob Y y , Y y , , Y y y ,data .

exp d

Prob Y 0, Y 0 y 0,data 1.

Prob Y 1, Y

x β

x β

x

x

x

2

2 itt 12

i1 i2 itt 12

i1 i2 itt 1

exp( )0 y 1,data

exp( ) exp( )exp( )

Prob Y 0, Y 1 y 1,data exp( ) exp( )

Prob Y 1, Y 1 y 2,data 1.

i1

i1 i2

i2

i1 i2

x βx β x β

x βx β x β

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

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].

1

exp( )( ) 0,

1 exp( )iT i it

it it itti it

y P P

x

x

Logit Constant Terms

ii

i

i

ˆT i it

i ˆt 1i i i i

Step 1. Estimate with Chamberlain's conditional estimator

Step 2. Treating as if it were known, estimate from the

first order condition

c1 e e 1y

T T 1 c1 e e

it

it

x β

x β

β

β

i iT T itt 1 t 1

t i i it

i i i i

it

i

i

c1T c

Estimate 1/ exp( ) log

ˆc exp( ) is treated as known data.

Solve one equation in one unknown for each .

Note there is no solution if y = 0 or 1.

Iterating bac

itx β

k 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

What We KNOW - Analytic

Newey and Hahn: MLE converges in probability to a vector of constants. (Variance diminishes with increase in N).

Abrevaya and Hsiao: Logit estimator converges to 2 when T = 2.

Only the case of T=2 for the binary logit model is known with certainty. All other cases are extrapolations of this result or speculative.

What We THINK We Know – Monte Carlo

Heckman: Bias in probit estimator is small if T 8 Bias in probit estimator is toward 0 in

some cases

Katz (et al – numerous others), Greene Bias in probit and logit estimators is large Upward bias persists even as T 20

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: (1) Maximize a penalized log likelihood function, or (2) Directly correct the estimator of β

Advantages 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

*it i

it it

i

y ,i = 1,...,N; t = 1,...,T

y 1 if y > 0, 0 otherwise.

(Projection, not necessarily conditional mean)

w

i it it

i iu

Fixed Effects Model :

x

Mundlak (Wooldridge, Heckman, Chamberlain), ...

x

u 1 2

*it

here u is normally distributed with mean zero and standard

deviation and is uncorrelated with or ( , ,..., )

y ,i = 1,...,N; t = 1,..

i i i iT

i it it iu

x x x x

Reduced form random effects model

x x i

it it

.,T

y 1 if y > 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

x

x

it it i,t 1 it i

it i,t 1 i0 it it i,t 1 i

y 1[ y u > 0]

Two similar 'effects'

Unobserved heterogeneity

State dependence = state 'persistence'

Pr(y 1| y ,...,y ,x ,u] F[ y u]

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

T

i1 i2 iT i0 i i,t 1 i itt 1

i1 i2 iT i0

(1) Conditioned on all effects, joint probability

P(y ,y ,...,y | y , ,u) F( y u ,y )

(2) Unconditional density; integrate out the common effect

P(y ,y ,...,y | y , )

i it

i

x x β

x

i1 i2 iT i0 i i i0 i

2i i0 i0 u i i1 i2 iT

i

P(y ,y ,...,y | y , ,u)h(u | y , )du

(3) (The rabbit in the hat) Density for heterogeneity

h(u | y , ) N[ y , ], = [ , ,..., ], so

u =

i i

i i

x x

x x δ x x x x

i0 i it

i1 i2 iT i0

T

i,t 1 i0 u i it i it 1

y + w (contains every period of )

(4) Reduced form

P(y ,y ,...,y | y , )

F( y y w ,y )h(w )dw

This is a random effects model

i

i

it i

x δ x

x

x β x δ

Simplified Dynamic Model

i

2i i0 i0 u

i i0 i

Projecting u on all observations expands the model enormously.

(3) Projection of heterogeneity only on group means

h(u | y , ) N[ y , ] so

u = y + w

(4) Re

i i

i

x x δ

x δ

i1 i2 iT i0

T

i,t 1 i0 u i it i it 1

duced form

P(y ,y ,...,y | y , )

F( y y w ,y )h(w )dw

Mundlak style correction with the initial value in the equation.

This is (again) a random effects mo

i

it i

x

x β x δ

del

A Dynamic Model for Public Insurance

AgeHousehold IncomeKids in the householdHealth Status

Basic Model

Add initial value, lagged value, group means

Dynamic Common Effects Model

1525 groups with 1 observation were lost because of the lagged dependent variable.