Panel Data Analysis Using GAUSS

2

Kuan-Pin LinPortland State University

Fixed Effects Model

Within Model Representation'

'

' '

'

( ) ( )

it it i it

i i i i

it i it i it i

it it it

y u e

y u e

y y e e

y e

x β

x β

x x β

x β

'1, ( 0, ' )

i i i

i i i

T T T T

or

Q Q Q

where Q Q Q Q QT

y X β e

y X β e

I i i i

Fixed Effects Model

Model Assumptions

2

2

2 2 '

( | ) 0

( | ) (1 1/ )

( , | , ) ( 1/ ) 0,

1( | ) ( )

( | )

it it

it it e

it is it is e

i i e e T T T

N

E e

Var e T

Cov e e T t s

Var QT

Var

x

x

x x

e X I i i

e X Ω I

Fixed Effects Model Model Estimation

Within Estimator: FE-GLS

1' 1 ' ' '

1 1

' 1 ' ' 1

1 12 ' ' '

1 1 1

12 '

1

2

ˆ ( )

ˆˆ ( ) ( ) ( )

ˆ

ˆ

ˆˆ '

i i i

N N

OLS i i i ii i

OLS

N N N

e i i i i i ii i i

N

e i ii

e

Var

Q

y X β e y Xβ e

β XX Xy X X X y

β XX XΩX XX

X X X X X X

X X

e

ˆ ˆ ˆ/ ( ),NT N K e e y Xβ

Fixed Effects Model Model Estimation: Transformation Approach

Let [FT,T-1,1T/T] be the orthonormal matrix of the eigenvectors of QT = IT-iTi’T/T, where FT,T-1 is the Tx(T-1) eigenvector matrix corresponding to the eigenvalues of 1. Define* * * * * * * 2

( 1)

1*' * 1 *' * *' * *' *

1 1

12 *' * 1 2 *' *

1

2 *' * * * *

, ~ ( , )

ˆ ( )

ˆˆ ˆ ˆ( ) ( )

ˆ ˆ ˆ ˆˆ / ( ),

i i i e N T

N N

OLS i i i ii i

N

OLS e e i ii

e

iid

Var

NT N K

N

y X β e y X β e e 0 I

β X X X y X X X y

β X X X X

e e e y X β

*' * 'ˆ ˆ ˆ ˆ ˆ ˆ:ote with e e ee e y Xβ

* ' * ' * ', 1 , 1 , , 1 , 1 , , 1 , 1 ,, ,i T T T i T i T T T i T i T T T i T y F y x F x e F e

Fixed Effects Model Model Estimation

Panel-Robust Variance-Covariance Matrix Consistent statistical inference for general

heteroscedasticity, time series and cross section correlation.

1 1' ' ' '

1 1 1

1 1' ' '

1 1 1 1 1 1 1

ˆ ˆ ˆˆ ( ) [( )( ) ']

ˆ ˆ

ˆ ˆ

ˆˆ ˆ,

i i i i

N N N

i i i i i i i ii i i

N T N T T N T

it it it is it is it iti t i t s i t

i i i it it

Var E

e e

e y

β β β β β

X X X e e X X X

x x x x x x

e y X β

' ˆitx β

Fixed Effects Model Model Estimation: ML

Normality Assumption'

2

'

2 2

( 1,2,..., )

( 1,2,..., )

~ ( , )

, , ,

1

~ (0, ), '

i

it it i it

i i i T i

i e T

i i i i i i i i i

T T T

i e e

y u e t T

u i N

normal iid

with Q Q Q

QT

normal where QQ Q

x β

y X β i e

e 0 I

y X β e y y X X e e

I i i

e

Fixed Effects Model Model Estimation: ML

Log-Likelihood Function

Since Q is singular and |Q|=0, we use orthonomral transformation of the eigenvectors of Q, we maximize

2 ' 1

2 ' 12

1 1( , | , ) ln 2 ln

2 2 21 1

ln 2 ln( ) ln2 2 2 2

i e i i i i

e i ie

Tll

T TQ Q

β y X e e

e e

* 2 2 *' *2

1 1 1( , | , ) ln 2 ln( )

2 2 2i e i i e i ie

T Tll

β y X e e

Fixed Effects Model Model Estimation: ML

ML Estimator

2 * 2

1

*' *2 * * *1

*' * '

1 1

ˆ( , ) argmax ( , | , )

ˆ ˆˆ ˆˆ ,

( 1)

ˆ ˆ ˆ ˆ ˆ ˆ:

N

e ML i e i ii

N

i iie i i i

N N

i i i i i i ii i

ll

N T

Note with

β β y X

e ee y X β

e e e e e y X β

Fixed Effects ModelHypothesis Testing

Pool or Not Pool F-Test based on dummy

variable model: constant or zero coefficients for D w.r.t F(N-1,NT-N-K)

F-test based on fixed effects (unrestricted) model vs. pooled (restricted) model

'

'

. ( , )it it i it

i

it it it

y u e

vs u u i

y u e

x β

x β

' '

( ) / 1~ ( 1, )

/ ( )

ˆ ˆ ˆ ˆ,

R UR

UR

UR FE FE R PO PO

RSS RSS NF F N NT N K

RSS NT N K

RSS RSS

e e e e

First-Difference Model First-Difference Representation

Model Assumptions

' ' '1 1 1( ) ( ) ( ) , 2,...,it it it it i i it it it it ity y u u e e y e t T x x β x β

2

2

2

( | ) 0, ~ (0, )

( | ) 2

| | 1( , | , )

0

it it it e

it it e

eit is it is

E e given e iid

Var e

if t sCov e e

otherwise

x

x

x x

2 2 21 1

2 1 0 0 0

1 2 1 0 0

0 1 2 1 0( | ) ( | ) ( )

0 0 1 2 1

0 0 0 1 2

( )

i i e T e e N TVar Var I

Toepliz form

e X e X

First-Difference ModelModel Estimation

First-Difference Estimator: OLS

Consistent statistical inference for general heteroscedasticity, time series and cross section correlation should be based on panel-robust variance-covariance matrix.

1' 1 ' ' '

1 1

2 ' 1 ' ' 1

1 12 ' ' '

1 1 1

22 2

ˆ ( )

ˆˆ ˆ( ) ( ) ( )

ˆ

ˆˆ ˆˆ ˆ, ' / (

2

i i i

N N

OLS i i i ii i

OLS e

N N N

e i i i i i i ii i i

ee e

Var

N

y X β e y Xβ e

β X X X y X X X y

β X X XΩ X X X

X X X X X X

e e ˆˆ),T N K e y Xβ

First-Difference ModelModel Estimation

First-Difference Estimator: GLS' 1 1 ' 1

1' 1 ' 1

1 1

2 ' 1 1

12 ' 1

1

22 2

ˆ ( )

ˆˆ ˆ( ) ( )

ˆ

ˆ ˆˆ ˆ ˆˆ ˆ, ' / ( ),2

GLS

N N

i i i i i ii i

GLS e

N

e i i ii

ee e

Var

NT N K

β XΩ X XΩ y

X X X y

β XΩ X

X X

e e e y Xβ

First-Difference ModelModel Estimation: Transformation Approach

The first-difference operator is a (T-1)xT matrix with elements:

Using the transformation matrix (, then we have the Forward Orthogonal Deviation Model:

1

1 1, 1,... 1; 1,...,

0ts

if s t

if s t t T s T

otherwise

' 2

' ' '

' '

1 1 1

, ~ (0, )

( ), ( ), ( ),1

1 1 1, ,

it iitt it it e

F F Fit t it it iitt t it it it t it it t

T T TF F Fit is it is it is

s t s t s t

y e e iid

T ty c y y c e c e e c

T t

y y e eT t T t T t

x

x x x

x x

First-Difference ModelModel Estimation: Transformation Approach

FD-GLS

Consistent statistical inference for general heteroscedasticity, time series and cross section correlation should be based on panel-robust variance-covariance matrix.

1' 1 ' ' '

1 1

12 ' 1 2 '

1

2

ˆ ( )

ˆˆ ˆ ˆ( ) ( )

ˆ ˆ ˆ ˆˆ ' / ( ),

i i i

N N

OLS i i i ii i

N

OLS e e i ii

e

Var

NT N K

y X β e y Xβ e

β XX XX X X X X

β XX X X

e e e y Xβ

References

B. H. Baltagi, Econometric Analysis of Panel Data, 4th ed., John Wiley, New York, 2008.

W. H. Greene, Econometric Analysis, 7th ed., Chapter 11: Models for Panel Data, Prentice Hall, 2011.

C. Hsiao, Analysis of Panel Data, 2nd ed., Cambridge University Press, 2003.

J. M. Wooldridge, Econometric Analysis of Cross Section and Panel Data, The MIT Press, 2002.