Econometric Analysis of Panel Data
• Panel Data Analysis
– Linear Model
• One-Way Effects
• Two-Way Effects
– Pooled Regression
• Classical Model
• Extensions
Panel Data Analysis
• Linear Model Representation
yit  xit' β   it
 it  ui  vt  eit
t  1, 2,..., T (Ti ); i  1, 2,..., N ( N t )

(1) y i  Xi β  ui iTi  v i  ei
or
(2) y t  Xt β  u t  vt i Nt  et
 v1 
 u1 
v 
u 
2
2 


vi 
, ut 
 
 
 
 
vTi 
u Nt 
 xi' 1   x1,i1
 yi1 
 '  
y 
xi 2   x1,i 2
i2 


yi 
,X 

  i   
 '  
 
 yiTi 
 xiTi   x1,iTi
 x1' t   x1,1t
 y1t 
 '  
y 
x 2t   x1,2t
2t 


yt 
, Xt 






 '  


 yNt t 
 x Nt t   x1, Nt t
x2,i1
x2,i 2
x2,iTi
x2,1t
x2,2t
x2, Nt t
xK ,i1 
 ei1 
 1 
e 
 
xK ,i 2 
i2 
2 


,β
, ei 

 
 

 
 
xK ,iTi 
eiTi 
K 
xK ,1t 
 e1t 
e 
xK ,2 t 
2t
, et   

 

 
xK , Nt t 
eNt t 
Linear Panel Data Model (1)
• One-Way (Individual) Effects
yit  xit' β  ui  eit  y i  Xi β  ui iTi  ei
(t  1, 2,..., Ti ; i  1, 2,..., N )

y  Xβ  u  e
 u1iT1 
 y1 
 X1 
 e1 
 1 


y 
X 
e 
 
u
i
 2 T2 
 2
y   2 , X   2 , β   2 , u  
,
e


 
 
 
 


 
 
 
 
u N iTN 
K 
y N 
XN 
e N 
Linear Panel Data Model (1)
• One-Way (Time) Effects
yit  xit' β  vt  eit  y i  Xi β  v i  ei
(t  1, 2,..., Ti ; i  1, 2,..., N )

y  Xβ  v  e
 y1 
 X1 
 v1 
 e1 
 1 
y 
X 
v 
e 
 
y   2 , X   2 , β   2 , v   2 , e   2 
 
 
 
 
 
 
 
 
 
 
y
X
v

 K
 N
 N
 N
e N 
Linear Panel Data Model (1)
• Two-Way Effects
yit  xit' β  ui  vt  eit  y i  Xi β  ui iTi  v i  ei
(t  1, 2,..., Ti ; i  1, 2,..., N )

y  Xβ  u  v  e
Linear Panel Data Model (2)
• One-Way (Individual) Effects
yit  xit' β  ui  eit  y t  Xt β  u t  et
(i  1, 2,..., N t ; t  1, 2,..., T )

y  Xβ  u  e
 y1 
 X1 
 1 
 u1 
 e1 
y 
X 
 
u 
e 
y   2 , X   2 , β   2 , u   2 , e   2 
 
 
 
 
 
 
 
 
 
 
yT 
 XT 
K 
uT 
eT 
Linear Panel Data Model (2)
• One-Way (Time) Effects
yit  xit' β  vt  eit  y t  Xt β  vt i Nt  et
(i  1, 2,..., N t ; t  1, 2,..., T )

y  Xβ  v  e
 v1i N1 
 y1 
 X1 
 1 
 e1 


y 
X 
 
e 
v
i
 2 N2 
 2
y   2 , X   2 , β   2 , v  
,
e


 
 
 
 


 
 
 
 
y
X

vT i NT 
 T
 T
 K
eT 
Linear Panel Data Model (2)
• Two-Way Effects
yit  xit' β  ui  vt  eit  y t  Xt β  ut  vt i Nt  et
(i  1, 2,..., N t ; t  1, 2,..., T )

y  Xβ  u  v  e
Panel Data Analysis
• Between Estimator
yit  xit' β  ui  eit  yi  xi' β  ui  ei
1
yi 
Ti
1
 t 1 yit , x  T
i
Ti
1
ui  u 
N
'
i
1
 t 1 x , ei  T
i
Ti
'
it

Ti
e
t 1 it
• If
 i 1 ui ,
then the pooled or population-averaged
model is more efficient.
N
Panel Data Analysis
• Linear Pooled (Constant Effects) Model
yit  xit' β  ui  eit  yit  xit' β  u  eit  yit  w it δ  eit
w it   x
'
it
β 
1 , δ   
u 
(t  1, 2,..., Ti ; i  1, 2,..., N ; NT   i 1 Ti )
N

y  Wδ  e
Pooled Regression Model
• Classical Assumptions
– Strict Exogeneity
E (eit | W)  0; Cov(wit , eit )  0
– Homoschedasticity
Var (eit | W)   e2
– No cross section and time series correlation
Var (e | W)   e2 I NT
Pooled Regression Model
• Extensions
– Weak Exogeneity
E (eit | w i1 , w i 2 ,..., w iTi )  E (eit | Wi )  0
E (eit | w i1 , w i 2 ,..., w it )  0
E (eit | w it )  0
– Heteroschedasticity
Var (eit | Wi )   it2
Var (eit | Wi )   t2
Var (eit | Wi )   i2
Pooled Regression Model
• Extensions
– Time Series Correlation (with cross section
independence for short panels)
Cov(eit , eis | w it , w is )   ts , t  s
Cov(eit , e js | w it , w js )  0, i  j
Var (eit | w it )   tt   t2  Var (ei | Wi )  i  Var (e | W)  Ω
  11  12

  21  22
i  

 Ti 1  Ti 2
 1T 
i

 2T 
i


 TiTi 
1 0
0 
2
Ω


0
0
0 
0 


N 
Pooled Regression Model
• Extensions
– Cross Section Correlation (with time series
independence for long panels)
Cov(eit , e jt | w it , w jt )   ij , i  j
Cov(eit , e js | w it , w js )  0, t  s
Var (eit | w it )   i2
  12 I  12 I

 21I  22 I

Var (e | W )  Ω    IT 


 N 1I  N 2 I
 1N I 

 2N I

2 
 N I 
Pooled Regression Model
• Extensions
– Cross Section and Time Series Correlation
Var (eit | w it )   i ,tt   i2
Cov(eit , eis | w it , w is )   i ,ts   i ts , t  s
Cov(eit , e jt | w it , w jt )   ij , i  j
Cov(eit , e js | w it , w js )   ij ts , t  s
  12 R  12 R

2

R

R
21
2
Var (e | W )  Ω    R  


 N 1 R  N 2 R
 1

R   21


 T 1
 1N R 

 2N R


 N2 R 
12
1
T 2
1T 
 2T 


1 
Alternative Pool Regression Models
• Between (Group Means) Estimator
yit  xit' β  u  eit  yi  xi' β  u  ei
• First-Difference Estimator
yit  yit 1  (xit'  xit' 1 )β  (eit  eit 1 )  yit  xit' β  eit
• Within (Group Mean Deviations) Estimator
yit  yi  (xit'  xi' )β  (eit  ei )
Pooled Regression: OLS
• Classical Model Estimation (OLS)
N
δˆ OLS  ( W ' W ) 1 W ' y    i 1 Wi' Wi 


1

N
'
W
yi
i
i 1
ˆ (δˆ )  ˆ 2 ( W ' W) 1  ˆ 2   N W ' W 
Var
OLS
e
e
 i 1 i i 
ˆ e2  eˆ ' eˆ / ( NT  K )
eˆ  y  Wδˆ
1
ˆ (δˆ ) is inconsistent
• Variance estimator Var
because of heteroscedasticity and
autocorrelation.
OLS
Pooled Regression: OLS
• Panel-Robust Variance-Covariance Matrix
– Adjusting general heteroscedasticity and serial correlation
within panel
Var (δˆ )  E[(δˆ  δ)(δˆ  δ) ']  ( W' W) 1 W' E (ee ') W( W' W) 1
1
  i 1 W Wi   i 1 W E (e e )Wi    i 1 W Wi 

 


N
'
i
N
'
i
1
'
i i
N
'
i
1
ˆ (δˆ )    N W ' W    N W 'eˆ eˆ ' W    N W ' W 
Var
 i 1 i i   i 1 i i i i   i 1 i i 
1
1
N
T
N
T
T
N
T
   i 1  t i 1 w it w it'    i 1  t i 1  si1 w it w is' eˆit eˆis    i 1  t i 1 w it w it' 

 


eˆ i  y i  Wi δˆ , eˆit  yit  w it δˆ
1
Pooled Regression: GLS
• The Model
y  Wδ  e
E (e | W)  0
Var (e | W)  Ω    R
  12  12

 21  22




 N 1  N 2
 1N 

 2N 


 N2 
 1

R   21


 T 1
12
1
T 2
1T 
 2T 


1 
• Generalized Least Squares (GLS)
1
ˆ 1W  W'Ω
ˆ 1y, Var
ˆ 1W 
ˆ (δˆ )   W'Ω
δˆ GLS   W'Ω
GLS



1
– If cross sections are independent (short panels)
N
ˆ 1W 
δˆ GLS  i 1 Wi'
i
i


– where
ˆ is

i
1
1
ˆ )   N W' 
ˆ
ˆ
ˆ 1W 
W

y
,
Var
(
δ
i1 i i

GLS
i
i
i
 i 1

N
'
i
the consistent estimator of i
1
Pooled Regression: GLS
• Heteroscedasticity
1 T 2
ˆ   t 1 eˆit
T
2
i
• Cross Section Correlation
1 T
ˆ ij   t 1 eˆit eˆ jt
T
• Time Series Correlation
ˆts
Pooled Regression: GLS
• Examples of Time Series Correlation
– Equal-Correlation
– AR(1)
1 if t  s
ts  
  if t  s
if t  s
1
ts   |t  s|
if t  s

– Stationary(1)
if t  s
1

ts    if | t  s | 1
 0 otherwise

– Nonstationary(1)
if t  s
1

ts   ts if | t  s | 1, ts   st
0
otherwise

Model Extensions
•
•
•
•
Time-invariant regressors
Random regressors
Lagged dependent variables
Dynamic models
Example: Investment Demand
• Grunfeld and Griliches [1960]
Iit  i   Fit   Cit   it
– 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
Example: Investment Demand
• Pooled Model (Population-Averaged Model)
Iit     Fit   Cit   it
• Classical OLS
• Panel-Robust OLS
• Feasible GLS
– Heteroscedastcity
– Autocorrelation