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16 Νοε 2013 (πριν από 3 χρόνια και 6 μήνες)

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Panel Data

EM005 Econometrics A


18.12.2006

EM005 Econometrics A

2

Panel Data


pooling time
-
series of cross
-
section data


in the U.S. Panel Study of Income Dynamics
(PSID) or National Longitudinal Survey (NLS)


pooling data gives richer source of variation which
allows for more efficient estimation of parameters


ability to control for individual heterogeneity


better able to identify and estimate effects that are
not detectable in pure cross
-
sections or pure time
-
series data

18.12.2006

EM005 Econometrics A

3

Error Components Procedure

general formulation




i

denotes cross
-
sections and
t

denotes time
-
periods with
i

= 1, 2,…, N
and
t

= 1, 2,…,T,
α

is a scalar,


is
K
×

1 and X
it

is the
it
-
th observation
on
K

explanatory variables



where the µ
i
’s are cross
-
section specific time
-
invariant components
(e.g. individual ability, managerial skill, country specific effect) and ν
it

are remainder effects

it
T
it
it
u
X
y





it
i
it
u




18.12.2006

EM005 Econometrics A

4

Fixed Effects Model

vector form



where
y

is
NT

×

1,
X

is
NT

×

K
,
Z

=
[
ι
NT
,X
],
δ
T

= (
α
T
,

T
), and
ι
NT

is

a vector of ones of dimension
NT



where u
T

= (u
11
,…,u
1T
,u
21
,…,u
2T
,…,u
N1
,…,u
NT
) and Z
µ

=
I
N


ι
T
; I
N

is an identity matrix of dimension N,
ι
T

is

a vector of ones of dimension T; µ
T

= (µ
1
,…µ
N
) and
ν
T

=
(
ν
11
,…
ν
1T
,…
ν
N1
,…,
ν
NT
)

u
Z
u
X
y
NT













Z
u
















Z
Z
Z
X
y
NT
18.12.2006

EM005 Econometrics A

5

Problem 1


premultiply



by Q and verify that the transformed equation reduces to




show that P and Q are symmetric, idempotent, orthogonal
and sum to the identity matrix


show that the new disturbances Qv have zero mean and
variance
-
covariance matrix


show that the GLS estimator is the same as the OLS
estimator on this transformed regression equation

Q
2


















Z
Z
Z
X
y
NT
T
T
Z
Z
Z
Z
I
P
I
Q
where
Q
QX
Qy






1
)
(







18.12.2006

EM005 Econometrics A

6

Fixed Effects Model

Within estimator


least squares dummy variable
estimator (LSDV)




α can be retrieved as



μ
i
’s can also be retrieved from





.
.
.
i
it
i
it
i
it
x
x
y
y








..
..
..
..
..
~
~
x
y
x
y












Q
QX
Qy






..
.
..
.
~
~
x
x
y
y
i
i
i






18.12.2006

EM005 Econometrics A

7

Random Effects Model


µ
i
’s can be assumed as random

variance

variance
-
covariance matrix




t
o invert it, brute force is needed (dimension
NT

×

NT
)


verify that it can be rewritten as

)
(
)
(
)
(
)
(
)
(
2
2
T
N
T
N
T
T
T
T
I
I
J
I
E
Z
E
Z
uu
E

















2
2
)
var(






it
u
Q
P
E
I
J
I
T
T
N
T
N
2
2
1
2
2
)
(
)
(














)
,
0
(
~
2



IID
i
18.12.2006

EM005 Econometrics A

8

Random Effects Model




for




verify that



for



verify that


Q
P
E
I
J
I
T
T
N
T
N
2
2
1
2
2
)
(
)
(














Q
P
2
2
1
1
1
1







NT
I







1
1
Q
P



1
1
1
2
/
1




1
2
/
1
2
/
1







18.12.2006

EM005 Econometrics A

9

Random Effects Model

estimates

of

variance




show

that

it

is

unbiased

estimate

of

σ
2
ν




show

that

it

is

unbiased

estimate

of

σ
2
1










K
T
N
Qy
X
QX
X
QX
y
Qy
y
T
T
T
T





1
/
ˆ
ˆ
1
2








1
/
ˆ
ˆ
1
2
1





K
N
Py
Z
PZ
Z
PZ
y
Py
y
T
T
T
T

18.12.2006

EM005 Econometrics A

10

Random Effects Model


premultiply y by
σ
ν
Ω
-
1/2
where Ω
-
1/2
is defined as




and show that the resulting y* has a typical element



where

Q
P
2
2
1
1
1
1







.
*
i
it
it
y
y
y



2
2
2
1
1
1













T
and
Merry Christmas ;
-
)