Panel Data Analysis Introduction Model Representation N-first or T-first representation Pooled Model Fixed Effects Model Random Effects Model Asymptotic.

Panel Data AnalysisIntroduction

Model Representation N-first or T-first representation

Pooled Model Fixed Effects Model Random Effects Model

Asymptotic Theory N→∞, or T→∞ N→∞, T→∞ Panel-Robust Inference

Panel Data AnalysisIntroduction

The Model

One-Way (Individual) Effects: Unobserved Heterogeneity Cross Section and Time Series Correlation

''it it it

it it i t it

it i t it

yy u v e

u v e

xx

'it it i ity u e x

( , ) 0, ( , ) 0,

( , ) 0,

i j it jt

it i

Cov u u Cov e e i j

Cov e e t

Panel Data Analysis Introduction

N-first Representation

Dummy Variables Representation

T-first Representation'

1,2,..., ; 1, 2,...,

( )

it it i it

i i i T i

N T

y u e

i N t T

u

x β

y X β i e

y Xβ I i u e

'

1,2,..., ; 1, 2,...,

( )

ti ti i ti

t t t

T N

y u e

t T i N

x β

y X β u e

y Xβ i I u e

N T T Nor

y Xβ Du e

D I i D i I

Panel Data Analysis Introduction

Notations

'1, 1 2, 1 , 11 1 11

'1, 2 2, 2 , 22 2 22

'1, 2, ,

1

2

, , ,

,

i i K ii ii

i i K ii iii i i

iT iT K iTiT iT KiT

t t

tt t

tN

x x xy e

x x xy e

x x xy e

y

y

y

x

xy X e β

x

x

y X

'1, 1 2, 1 , 1 1 11

'1, 2 2, 2 , 2 2 22

'1, 2, ,

, ,

t t K t t

t t K t ttt

tN tN K tN tN NtN

x x x e u

x x x e u

x x x e u

xe u

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

Pooled (Constant Effects) Model

'

'

'

2

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

assuming

,

1

( | ) , ( | )

it it i it

i

it it it

it it it

e

y u e i N t T

u u i

y u e or

y eu

E Var

x β

x β

βx y Xβ e

e X 0 e X I

Fixed Effects Model

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

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

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

,

, 1, 2,...,

1, 2,...,i i i T i

t t t

u i i N

t T

y X e

y X u e

Fixed Effects Model

Fixed Effects Model Classical Assumptions

Strict Exogeneity: Homoschedasticity: No cross section and time series correlation:

Extensions: Panel Robust Variance-Covariance Matrix

( | , ) 0itE e u X2( | , )it eVar e u X

2( | , ) e NTVar e u X I

( | , )Var e u X

Random Effects Model

Error Components

ui is random, independent of eit and xit.

Define the error components as it = ui + eit

'

( 1,2,..., ; 1, 2,..., )it it it

it i it

y

u e i N t T

x β

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

( ), 1, 2,...,

( ), 1, 2,...,i i i T i

t t t

u i i N

t T

y X e

y X u e

Random Effects Model

Random Effects Model Classical 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 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

Random Effects Model Classical Assumptions (Continued)

Cross Section Independence

Extensions:Panel Robust Variance-Covariance Matrix

2 2 '( | )

( | )i e T u T T

N

Var

Var

ε X I i i

ε X Ω I

Fixed Effects Model Estimation

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 Estimation

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 Estimation: OLS

Within Estimator: OLS

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 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 Estimation: ML

Log-Likelihood Function

Since Q is singular and |Q|=0, 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( , | , ) ln 2 ln( )

2 2 2i e i i e i ie

T Tll

β y X e e

Fixed Effects Model Estimation: ML

ML Estimator2 2

1

'2 21

2 2

ˆ( , ) argmax ( , | , )

ˆ ˆ 1 ˆ ˆˆ ˆ1 ,

ˆ ˆ'ˆ ˆ

1 ( 1)

N

e ML i e i ii

N

i iie e i i i

e e

ll

NT T

T

T N T

β β y X

e ee y X β

e e

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

Random Effects Model Estimation: GLS

The Model

2 2 '

2 22

2

' '

,

( | )

( | )

1 1,

i i i i i T i

i i

i i e T u T T

e ue T

e

T T T T T T

u

E

Var

TQ Q

where Q QT T

y X β ε ε i e

ε X 0

ε X I i i

I

I i i I i i

Random Effects Model Estimation: GLS

GLS

11 1 1 1 1

1 1

11 1 1

1

2 21 '

2 2 2 2 2 2

1 22

2 2

ˆ ( )

ˆ( ) ( )

1 1

1

N N

GLS i i i ii i

N

GLS i ii

u eT T T T

e e u e e u

eT

e e u

Var

where Q QT T

and Q QT

β XΩ X XΩ y X X X y

β XΩ X X X

I i i I

I

Random Effects Model Estimation: GLS

Feasible GLS Based on estimated residuals of fixed effects

model

1 1 1

1 1

1 2 2 212 2

1

ˆ ˆ ˆ( )

ˆ ˆ( ) ( )

1 1ˆ ˆ ˆ ˆ,ˆ ˆ

GLS

GLS

T e ue

Var

where Q Q T

β XΩ X XΩ y

β XΩ X

I

2

2 2 21 1

ˆ ˆˆ ' / ( 1)

1ˆ ˆ ˆ ˆˆ ˆ ˆ ' / ,

e

T

u e i itt

N T

T T N where e eT

e e

e e

Random Effects Model Estimation: GLS

Feasible GLS Within Model Representation

2 2

2 2 21

' '

'

2

1 1

( ) ( )

(1 ) ( )

( ) 0, ( )

( , ) ( , ) 0

e e

u e

it i it i it i

it it it

it it i i it i

it it e

it i it jt

T

y y

y

u e e

E Var

Cov Cov

x x

x

Random Effects Model Estimation: ML

Log-Likelihood Function

' '

2 2 1

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

( 1, 2,..., )

~ ( , )

1 1( , , | , ) ln 2 ln

2 2 2

it it i it it it

i i i

i

i e u i i i i

y u e t T

i N

normal iid

Tll

x β x β

y X β ε

ε 0

β y X ε ε

Random Effects Model Estimation: ML

where2 2

2 2 '2

2 21 '

2 2 2 2 2 2

2 22 ' 2

2 2

( )

1 1( )

| | ( ) ( ) 1

e ue T u T T T

e

u eT T T T

e u e e u e

T Tu ue T T T e

e e

TQ Q

Q QT T

T

I i i I

I i i I

I i i

Random Effects 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 ln2 2

1( ) ( )

2

N

e u ML i e u i ii

i e u i i i i

e ue

e

T Tuit it it itt t

e e u

ll

where

Tll

TT

y yT

β β y X

β y X ε ε

x β x β

Random Effects ModelHypothesis Testing

Pool or Not Pool Test for Var(ui) = 0, that is

For balanced panel data, 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

Random Effects ModelHypothesis Testing

Pool or Not Pool (Cont.)

2

22

1 1

2

1 1

'

ˆ ˆ'( )1 ~ (1)

ˆ ˆ2 1 '

ˆ1

2 1 ˆ

ˆˆ 1

ˆ

T N

N T

iti t

N T

iti t

it it it

Pooled

NTLM

T

eNT

T e

where e yu

e J I e

e e

βx

Random Effects ModelHypothesis 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 Effects

E(ui|Xi) = 0

Fixed Effects

E(ui|Xi) =/= 0

GLS or RE-OLS

(Random Effects)

Consistent and Efficient

Inconsistent

LSDV or FE-OLS

(Fixed Effects)

Consistent

Inefficient

Consistent

Possibly Efficient

Random Effects ModelHypothesis 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

Random Effects ModelHypothesis Testing

Alternative Hausman Test Estimate the random effects model

F Test that = 0

' ' ( )it it i i ity u e x β x γ

0 0: 0 : ( , ) 0i itH H Cov u γ x

Extensions

Random Coefficients Model

Mixed Effects Model

Two-Way EffectsNested Random Effects

'' '( )it it i it

it it it i it

i i

y ey e

x βx β x u

β β u

' '( )it it it i ity e x β z γ

'it it i t ity u v e x β

'ijt ijt i ij ijty u v e x β

Example: U. S. Productivity

Munnell [1988] Productivity Data48 Continental U.S. States, 17 Years:1970-1986 STATE = State name, ST ABB=State abbreviation, YR =Year, 1970, . . . ,1986, PCAP =Public capital, HWY =Highway capital, WATER =Water utility capital, UTIL =Utility capital, PC =Private capital, GSP =Gross state product, EMP =Employment,

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.