Econometric Analysis of Panel Data

• Panel Data Analysis– Fixed Effects

• Dummy Variable Estimator• Between and Within Estimator• First-Difference Estimator• Panel-Robust Variance-Covariance Matrix

– Heteroscedasticity and Autocorrelation– Cross Section Correlation

– Hypothesis Testing• To pool or Not to pool

Panel Data Analysis

• Fixed Effects Model

– ui is fixed, independent of eit, and may be correlated with xit.

' ( 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

( , ) 0, ( , ) 0i it i itCov u e Cov u x

Fixed Effects Model

• Classical Assumptions– Strict Exogeneity

– Homoschedasticity

– No cross section and time series correlation

( | ) 0itE e X

2( | ) e NTVar e X I

2( | )it eVar e X

Fixed Effects Model

• Extensions– Weak Exogeneity

1 2

1 2

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

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

iit i i iT it i

it i i it

it it

E e E e

E eE e

x x x X

x x xx

Fixed Effects Model

• Extensions– Heteroschedasticity

1

2

0 00 0

( | )

0 0 N

Var

e X Ω

21,1

21,22

21,

0 00 0

( | ) ,

0 0i

it i it i

T

Var e

X

Fixed Effects Model

• Extensions– Time Series Correlation (with cross section

independence for short panels)

2

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

( | ) ( | ) ( | )

it is it is ts

it js it js

it it tt t i i i

Cov e e t sCov e e i j

Var e Var Var

x xx x

x e X e X Ω

11 12 1

21 22 2

1 2

i

i

i i i i

T

Ti

T T TT

1

2

0 00 0

0 0 N

Ω

Fixed Effects Model

• Extensions– Cross Section Correlation (with time series

independence for long panels)

2

( , | , ) ,

( , | , ) 0,

( | ) ( | )

it jt it jt ij

it js it js

it it i T

Cov e e i j

Cov e e t s

Var e Var

x x

x x

x e X I Ω

2 21 12 1 1 12 1

2 221 2 2 21 2 2

2 21 2 1 2

,

N N

N N

N N N N N N

I I II I I

Ω

I I I

Dummy Variable Model

• Dummy Variable Representation

– Note: X does not include constant term, otherwise one less number of dummy variables should be used.

1

2

1 1 1 1

2 2 2 2

0 0

0 0

0 0N

T

T

N N N NT

uu

u

iy X eiy X e

β

y X ei

βy Xβ Du e X D e

uy Wδ e

Dummy Variable Model

• Dummy Variable Estimator (LSDV)

• Heteroscedasticity and Autocorrelation

1' 1 ' ' '

1 1

12 ' 1 2 '

1

2

ˆ ( )

ˆˆ ˆ ˆ( ) ( )

ˆ ˆˆ ' / ( )ˆˆ

N NOLS i i i ii i

NOLS e e i ii

e

Var

NT N K

δ WW Wy WW Wy

δ WW WW

e e

e y Wδ

' 1 ' ' 1

1 1' ' ' '

1 1 1

ˆ ˆ ˆ( ) [( )( ) '] ( ) ( ') ( )

( )N N Ni i i i i i i ii i i

Var E E

E

δ δ δ δ δ WW W ee W WW

WW W e e W WW

Dummy Variable Model

• Panel-Robust Variance-Covariance Matrix

1 1' ' ' '

1 1 1

1 1' ' '

1 1 1 1 1 1 1

ˆ ˆ ˆˆ ( ) [( )( ) ']

ˆ ˆ

ˆ ˆ

ˆˆˆˆ

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 it it

Var E

e e

e y

δ δ δ δ δ

WW We e W WW

w w w w w w

e y Wδ

w δ

Within Model

• Within Model Representation'

' '

'

( ) ( )i i i i

it i it i it i

it i it

y u e

y y e e

y e

x β

x x β

x β

' '1 , ( 0, )i i i i

i i i

i i i i i i

i T T T i T i i ii

orQ Q Q

where Q Q QQ QT

y X β ey X β e

I i i i

Within Model

• Model Assumptions

2

2

2 2 '

1

2

( | ) 0

( | ) (1 1 / )

( , | , ) ( 1/ ) 0,

1( | ) ( )

0 00 0

( | )

0 0

i i i

it it

it it i e

it is it is i e

i i i e i e T T Ti

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

2

1 1/ 1/ 1/1/ 1 1/ 1/

1/ 1/ 1 1/

i i i

i i ii e

i i i

T T TT T T

T T T

Within Model

• Within Estimator: FE-OLS

1' 1 ' ' '

1 1

2 ' 1 ' ' 1

1 12 ' ' '

1 1 1

12 '

1

ˆ ( )

ˆˆ ˆ( ) ( ) ( )

ˆ

ˆ

ˆ

i i i

N NOLS i i i ii i

OLS e

N N Ne i i i i i i ii i i

Ne i ii

Var

y X β e y Xβ e

β XX Xy X X X y

β XX XQX XX

X X XQ X X X

X X

2 ˆ ˆ ˆ ˆ' / ( ),e NT N K e e e y Xβ

Within Model

• Within Estimator: GLS

• GLS = FE-OLS– Note:

1' 1 1 ' 1 ' 1 ' 1

1 1

1' '

1 1

12 ' 1 1 2 ' 1

1

12 '

1

ˆ ( )

ˆˆ ˆ ˆ( ) ( )

ˆ

N NGLS i i i i i ii i

N Ni i i ii i

NGLS e e i i ii

Ne i ii

Var

β XQ X XQ y XQ X XQ y

X X X y

β XQ X XQ X

X X

' 1 ' ' 1 ' ' ' ' '

' 1 ' ' 1 ' ' ' ' '

i i i i i i i i i i i i i i i i i

i i i i i i i i i i i i i i i i i

Q QQ Q Q QQ

Q QQ Q Q QQ

X X X X X X X X X X

X y X y X y X y X y

Within Model

• Normality Assumption'

2

'

2 ' 2

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

~ ( , )

, , ,1

~ (0, ),

i

i

i i i

it it i it i

i i i T i

i e T

i i i i i i i i i i i i

i T T Ti

i i i e i i e i

y u e t Tu i N

iidn

with Q Q Q

QT

normal where QQ Q

x βy X β i e

e 0 I

y X β e y y X X e e

I i i

e

Within Model

• Log-Likelihood Function

• ML Estimator

2 ' 1

2 '2

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

1ln 2 ln( )2 2 2

ii e i i i i i i

i ie i i

e

Tll

T T

β y X e e

e e

2 21

' '2 1 1

1 1

ˆ ˆ( , ) argmax ( , | , )

ˆ ˆˆ ,

Ne ML i e i ii

N Ni i i i ii i

eML ML FE OLSN Ni ii i

ll

Q

T T

β β y X

e e e eβ β

Within Model

• ML Estimator of e2 is downward biased

even for large N:

• For balanced panel (T=Ti: ), e2 should be

estimated as:

' '2 2 21 1 1

1 1 1

ˆ ˆ ˆ( 1) ( 1)

N N Ni i i i ii i i

eFE eML eMLN N Ni i ii i i

T

T T T

e e e e

'2 21ˆ ˆ

( 1) ( 1)

Ni ii

e eMLT

N T T

e e

Within Model

• Estimated Fixed Effects

– For , is consistent but is inconsistentunless .

' ˆˆi i iu y x β

2 ' ˆˆ ˆˆ ˆ( ) / ( )i i i i iVar u T Var x β x

N β̂

ˆiuiT

Within Model

• 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 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

First-Difference Model

• First-Difference Representation

• Model Assumptions

' ' '1 1 1( ) ( )it it it it it it it it ity y e e y e x x β x β

2

2

( | ) 0

( | ) 2

| | 1( , | , )

0

it it

it it e

eit is it is

E e

Var e

if t sCov e e

otherwise

x

x

x x

1

22 2 2

2 1 0 0 00 01 2 1 0 0

0 00 1 2 1 0( | ) , ( | )

0 00 0 1 2 10 0 0 1 2

( )

i i e i e e

N

Var Var

Toepliz form

e X e X Ω

First-Difference Model• First-Difference Estimator: FD-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

ˆ ˆ ˆˆ ˆ, ' / (2

i i i

N NOLS i i i ii i

OLS e

N N Ne 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 Model

• 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 Ni i i i i ii i

GLS e

Ne 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

Hypothesis Testing

• To Pool or Not to 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 evs u u i

y u e

x β

x β

' '

( ) / 1~ ( 1, )

/ ( )ˆ ˆ ˆ ˆ,

R UR

R

UR FE FE R PO PO

RSS RSS NF F N NT N K

RSS NT N K

RSS RSS

e e e e

Hypothesis Testing

• Heteroscedasticity

• Serial Correlation

• Spatial Correlation

2'

2, ( ) itit it i it it

i

y u e Var e

x β

1'

1

, it itit it i it it

i it it

v ey u v v

v e

x β

' ,it it i it it ij jt itj

y u v v w v e x β

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