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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.