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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


• Assumptions– Cross Section Independence

2 2 '

1

2

( | )

0 00 0

( | )

0 0

i i ii i e T u T T

N

Var

Var

ε X I i i

ε X Ω


• Extensions– Weak Exogeneity

– Heteroscedasticity

1 2

1 2

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

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

iit i i iT it i

it i i it

it it

E E

EE

x x x X

x x xx

22 2

2( | ) ( | ) it

i i

i

eit i u it i u

e

Var Var e

X X


• Extensions– Serial Correlation

– Spatial Correlation

1'

1

, it itit it i it it

i it it

e vy u e e

e v

x β

' , ,it it it it ij jt it it i itj

y w e e u v x β

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 NGLS i i i i i ii i

NGLS 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 it 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 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 NOLS i i i ii i

NOLS e e i ii

e

Var

NT K

β XX Xy X X X y

β XX X X

ε ε ε y Xβ


• Need a consistent estimator of :

– Estimate the fixed effects model to obtain– Estimate the between model to obtain– Or, 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

2ê2 2ˆ û vT

ˆ0 1

2 2ˆ ê u


• 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 Ni i i i i i i ii i i

N T N T T N Tit 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 ln2 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 Ti N

iidn

Tll

x β x βy X β ε

ε 0 I i i

β y X ε ε

Model Estimation: ML

• ML Estimator

2 2 2 21

2 2 1

2 22

2

2 2' 2 '

2 2 21 1

ˆ ˆ ˆ( , , ) argmax ( , , | , )

1 1( , , | , ) ln 2 ln2 2 2

1ln 2 ln2 2

1 ( ) ( )2

i i

Ne 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

whereT

ll

T T

y yT

β β y X

β y X ε ε

x β x β

Hypothesis TestingTo Pool or Not To Pool, Continued

• 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

222'

1 1 2' 2

1 1

' '

ˆˆ ˆ( )1 1 ~ (1)

ˆ ˆ2 1 2 1 ˆ

ˆˆ 1 ,

ˆ

N Titi tN T

N Titi t

it it it T T T

Pooled

eI JNT NTLMT T e

where e y Ju

e ee e

βx i i


– 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

21 11

ˆ1 ~ (1)

ˆ2 ( 1)

i

i

N N Ti iti i t

N TNiti i i ti

T eLM

eT T

0~ (0,1)

: Pr ( )n

LM N under H

P Value z LM


• References– Baltagi, B. H., and Q. Li, A Langrange Multiplier Test for the Error

Components Model with Incomplete Panels, Econometric Review, 9, 1990, 103-107.

– Breusch, T. and A. Pagan, “The LM Test and Its Applications to Model Specification in Econometrics,” Review of Economic Studies, 47, 1980, 239-254.

Hypothesis TestingFixed 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


• 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


• Alternative (Asym. Eq.) Hausman Test– Estimate any of the random effects models

– F Test that = 0

' ' ' '

' ' '

' ' '

' ' '

( ) ( ) ( )

( , random effects model : ( ) )

( ) ( )

( ) ( )

it i it i it i it

it it it i it

it i it i i it

it i it i it it

y y e

or y e

y y e

y y e

x x β x x γ

x β x x γ

x x β x γ

x x β x γ

0 0: 0 : ( , ) 0i itH H Cov u γ x


• Ahn-Low Test (1996)– Based on the estimated errors (GLS residuals) of

the random effects model, estimate the following regression:

2 2

ˆ ˆ( )

~ (# )it it i i itX X X e

NTR

β γ

γ


• References– Ahn, S.C., and S. Low, A Reformulation of the Hausman Test for

Regression Models with Pooled Cross-Section Time-Series Data, Journal of Econometrics, 71, 1996, 309-319.

– Baltagi, B.H., and L. Liu, Alternative Ways of Obtaining Hausman’s Test Using Artificial Regressions, Statistics and Probability Letters, 77, 2007, 1413-1417.

– Hausman, J.A., Specification Tests in Econometrics, Econometrica, 46, 1978, 1251-1271.

– Hausman, J.A. and W.E. Taylor, Panel Data and Unobservable Individual Effects, Econometrics, 49, 1981, 1377-1398.

– Mundlak, Y., On the Pooling of Time Series and Cross-Section Data, Econometrica, 46, 1978, 69-85.

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

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