An Analysis on Danish Micro Data - School of Economics and ...

The pooled probit does not make use of the panel data structure of the data, since the fact that
individuals reoccur several times is not exploited. Panel data methods exist that are able to take this
reoccurrence into account. A random effects probit estimator is one of these methods. When
working with panel data methods it is assumed that observations over time belonging to the same
individual are more alike than observations coming from independent individuals. 65 In other words
it is assumed that each individual has an individual specific component, C i , that does not change
over time and is unobserved.
The most important assumption in the unobserved effects probit model is the following:
( Y = 1 X , C ) = P(
Y = 1 X , C ) = Φ(
X β C )
P +
it
i
i
it
From this assumption it appears that Xit is strictly exogenous conditional on Ci, the unobserved
individual specific component. Furthermore, Ci is added on to the probit index.
Also it is assumed that the dependent variable is independent conditional on the explanatory
variables and the individual specific component,
The density of Yit given Xi and Ci looks as follows
f
it
i
( i.
i.
d )
it
Yi , K , YiT
X i , Ci
~
[5.4]
1
T
( Y , K , Y X , C ; β ) = f ( Y X , C ; β )
1
Yt
1−Yt
where f ( Y X , C;
) = Φ(
X β + C)
[ 1−
Φ(
X β + C)
]
t
t
T
i
i
t
∏
t=
1
t
it
t
i
β [5.5]
Different assumptions about the correlation between the explanatory variables and the individual
specific component can be made. In the case of random effects it is assumed that the individual
specific component is independent of the explanatory variables and has a normal distribution,
2 ( 0,
)
Ci X i ~ Normal σ C
[5.6]
The unobserved effects probit model can be estimated with conditional likelihood methods and is
often called the random effects probit estimator. The unobserved Ci are integrated out of the
distribution of the dependent variables conditional on the explanatory variables,
where ( Y X ,C;
β )
f t
f
∞
⎡
⎤
K =
⎛ ⎞ ⎛ ⎞
⎥⎜
1 ⎟φ⎜
C
⎟dc
[5.7]
⎠
T
( Y1,
, Y X i ; θ ) ∫ ⎢∏
f ( Yt
X it , C;
β )
T
− ∞⎣
=
⎦⎝
σ
t
C ⎠ ⎝ σ
1
C
t is equal to equation [5.5] and θ contains β and
65 Johnston and DiNardo (1997)
2
σ C .
i
49