Econometric Analysis of Panel Data

• Panel Data Analysis– Random Effects• Assumptions• GLS Estimator• Panel-Robust Variance-Covariance Matrix• ML Estimator

– Hypothesis Testing• Test for Random Effects• Fixed Effects vs. Random Effects

Panel Data Analysis

• Random Effects Model

– ui is random, independent of eit and xit.

– Define it = ui + eit the error components.

' ( 1,2,..., )

( 1,2,..., )i

it it i it i

i i i T i

y u e t T

u i N

x β

y X β i e

Random Effects Model

• Assumptions– Strict Exogeneity

• X includes a constant term, otherwise E(ui|X)=u.

– Homoschedasticity

– Constant Auto-covariance (within panels)

( | ) 0, ( | ) 0 ( | ) 0it i itE e E u E X X X

2 2 '( | )i i ii e T u T TVar ε X I i i

2 2

2 2 2

( | ) , ( | ) , ( , ) 0

( | )

it e i u i it

it e u

Var e Var u Cov u e

Var

X X

X

Random Effects Model

• Assumptions– Cross Section Independence

2 2 '

1

2

( | )

0 0

0 0( | )

0 0

i i ii i e T u T T

N

Var

Var

ε X I i i

ε X Ω

Random Effects Model

• Extensions– Weak Exogeneity

– Heteroscedasticity and Autocorrelation

– Cross Section Correlation

1 2

1 2

( | , ,..., ) ( | ) 0

( | , ,..., ) 0

( | ) 0

iit i i iT it i

it i i it

it it

E E

E

E

x x x X

x x x

x

2( | )itit iVar X

( , | , ) 0it jt i j ijCorr X X

Model Estimation: GLS

• Model Representation

2 2 '

2 22

2

'

,

( | )

( | )

1

i

i i i

i

i i i

i i i i i T i

i i

i i i e T u T T

e i ue i T i

e

i T T Ti

u

E

Var

TQ Q

where QT

y X β ε ε i e

ε X 0

ε X I i i

I

I i i

Model Estimation: GLS

• GLS

11 1 1 1 1

1 1

11 1 1

1

21

2 2 2

21/2

2 2

ˆ ( )

ˆ( ) ( )

1

1

i

i

N N

GLS i i i i i ii i

N

GLS i i ii

ei i T i

e e i u

ei i T i

e e i u

Var

where Q QT

and Q QT

β XΩ X XΩ y X X X y

β XΩ X X X

I

I

Model Estimation: RE-OLS

• Partial Group Mean Deviations' '

'

2

2 2

' '

'

( )

( )

1

( ) [(1 ) ( )]

it it it it i it

i i i i

ei

e i u

it i i it i i i i it i i

it i it

y u e

y u e

T

y y u e e

y

x β x β

x β

x x β

x β

Model Estimation: RE-OLS

• Model Assumptions

• OLS

'

' 2 2 2 2 2

' 2 2 2 2

2

2 2

( | ) 0

( | ) (1 ) (1 2 / / )

( , | ) (1 ) ( 2 / / ) 0,

: 1

it i

it i i u i i i i e e

it is i i u i i i i e

ei

e i u

E

Var T T

Cov T T t s

NoteT

x

x

x

1' 1 ' '

1 1

12 ' 1 2 '

1

2

ˆ ( )

ˆˆ ˆ ˆ( ) ( )

ˆ ˆ ˆ ˆˆ ' / ( ),

N N

OLS i i i ii i

N

OLS e e i ii

e

Var

NT N K

β XX Xy X X X y

β XX X X

ε ε ε y Xβ

Model Estimation: RE-OLS

• Need a consistent estimator of :

– Estimate the fixed effects model to obtain– Estimate the pooled model to obtain– Based on the estimated large sample variances, it

is safe to obtain

2

2 2

ˆˆ 1ˆ ˆ

ei

e i uT

2ˆe2 2ˆ ˆe u

ˆ0 1

Model Estimation: RE-OLS

• 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

Var E

β β β β β

X X X ε ε X X X

x x x x x x

ε y X β

' ˆit ity x β

Model Estimation: ML

• Log-Likelihood Function

' '

2 2 '

2 2 1

( ) ( 1,2,..., )

( 1, 2,..., )

~ ( , ),

1 1( , , | , ) ln 2 ln

2 2 2

i i i

it it i it it it i

i i i

i i i e T u T T

ii e u i i i i i i

y u e t T

i N

iidn

Tll

x β x β

y X β ε

ε 0 I i i

β y X ε ε

Model Estimation: ML

• ML Estimator

2 2 2 2

1

2 2 1

2 22

2

2 2' 2 '

2 2 21 1

ˆ ˆ ˆ( , , ) argmax ( , , | , )

1 1( , , | , ) ln 2 ln

2 2 2

1ln 2 ln

2 2

1( ) ( )

2i i

N

e u ML i e u i ii

ii e u i i i i i i

i e ue

e

T Tuit it it itt t

e e i u

ll

where

Tll

T T

y yT

β β y X

β y X ε ε

x β x β

Hypothesis Testing

• Test for Var(ui) = 0, that is

– If Ti=T for all i, the Lagrange-multiplier test statistic (Breusch-Pagan, 1980) is:

, , ,( ) ( ) ( )it is i it i is it isCov Cov u e u e Cov e e

22

1 1 2

2

1 1

'

ˆ1 ~ (1)

2 1 ˆ

ˆˆ 1

ˆ

N T

iti t

N T

iti t

it it it

Pooled

eNTLM

T e

where e yu

βx

Hypothesis Testing

– For unbalanced panels, the modified Breusch-Pagan LM test for random effects (Baltagi-Li, 1990) is:

– Alternative one-side test:

22 2

1 1 1 2

2

1 11

ˆ1 ~ (1)

ˆ2 ( 1)

i

i

N N T

i iti i t

N TN

iti i i ti

T eLM

eT T

0~ (0,1)

: Pr ( )n

LM N under H

P Value z LM

Hypothesis Testing

• Fixed Effects vs. Random Effects '

0

'1

: ( , ) 0 ( )

: ( , ) 0 ( )

i it

i it

H Cov u random effects

H Cov u fixed effects

x

x

Estimator Random EffectsE(ui|Xi) = 0

Fixed EffectsE(ui|Xi) =/= 0

GLS or RE-OLS(Random Effects)

Consistent and Efficient

Inconsistent

LSDV or FE-OLS(Fixed Effects)

ConsistentInefficient

ConsistentPossibly Efficient

Hypothesis Testing

• Fixed effects estimator is consistent under H0 and H1; Random effects estimator is efficient under H0, but it is inconsistent under H1.

• Hausman Test Statistic ' 1

2

ˆ ˆ ˆ ˆ ˆ ˆ( ) ( )

ˆ ˆ ˆ~ (# ), # # ( )

RE FE RE FE RE FE

FE FE RE

H Var Var

provided no intercept

β β β β β β

β β β

Hypothesis Testing

• Alternative Hausman Test– Estimate the random effects model

– F Test that = 0

' ' ' '( ) ( ) ( )it i it i it i ity y e x x β x x γ

0 0: 0 : ( , ) 0i itH H Cov u γ x

Example: Investment Demand

• Grunfeld and Griliches [1960]

– i = 10 firms: GM, CH, GE, WE, US, AF, DM, GY, UN, IBM; t = 20 years: 1935-1954

– Iit = Gross investment

– Fit = Market value

– Cit = Value of the stock of plant and equipment

it i it it itI F C