# NT - Samba

Urban and Civil

Nov 16, 2013 (5 years and 5 months ago)

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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

.
.
.
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 ;
-
)