# Econometric Analysis of Panel Data_5_

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```					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 )
 v1       u1 
                                                     v        u 
(1) y i  Xi β  ui iTi  v i  ei               vi   2 , u   2 
or
  t  
          
(2) y t  Xt β  ut  vt i Nt  et                    vTi 
         u Nt 
 
 yi1         xi' 1   x1,i1      x2,i1        xK ,i1          1        ei1 
y            '  
 i 2  , X   xi 2    x1,i 2    x2,i 2       xK ,i 2                 e 
 , β   2  , e   i2 
yi 
  i                                                           i  
             '                                                         
 yiTi 
             xiTi   x1,iTi
                    x2,iTi       xK ,iTi 
       K        eiTi 
 

 y1t        x1t   x1,1t
'
x2,1t        xK ,1t             e1t 
y           '  
 2t  , X   x 2t    x1,2t        x2,2t        xK ,2t            e 
 , e   2t 
yt 
         t                                                 t  
            '                                                     
 y Nt t 
            x Nt t   x1, Nt t
                       x2, Nt t     xK , N t 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

 y1       X1       1       u1iT1        e1 
y        X                             e 
 u2 iT2 
y   2 , X   2 , β   2 , u           ,e  2
                                           
                                         
y N      XN       K       u N iTN 
             e N 
Linear Panel Data Model (1)
• One-Way (Time) Effects
yit  xit β  vt  eit  y i  Xi β  vi  ei
'

(t  1, 2,..., Ti ; i  1, 2,..., N )

y  Xβ  v  e

 y1      X1      1      v1      e1 
y       X              v       e 
y  2
,X  2 
,β  2
,v  2
,e 2
                                 
                                 
 yN      XN      K      vN     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  ut  et
'

(i  1, 2,..., Nt ; 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,..., Nt ; t  1, 2,..., T )

y  Xβ  v  e

 y1       X1       1       v1i N1        e1 
y        X                              e 
 v2 i N2 
y   2 , X   2 , β   2 , v            ,e  2
                                            
                                          
 yT       XT       K      vT i NT 
              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,..., Nt ; 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                     1                    1
 t 1 yit , x  T     t 1 x , ei  T    
Ti                    Ti                    Ti
yi                    '
i
'
it                    e
t 1 it
Ti                     i                    i

1
• If         i1 ui ,
N
ui  u 
N
then the pooled or population-averaged
model is more efficient.
Panel Data Analysis
• Linear Pooled (Constant Effects) Model
yit  xit β  ui  eit  yit  xit β  u  eit  yit  w it δ  eit
'                        '

β 
w it   x

'
it   1 , δ   

u 
(t  1, 2,..., Ti ; i  1, 2,..., N ; NT   i 1Ti )
N


y  Wδ  e
Pooled Regression Model
• Classical Assumptions
– Strict Exogeneity
E (eit | W)  0; Cov(w it , 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 )   it
2

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              1T         1 0       0 
                               
i
0         0 
  21  22              2T 
i                             i

    Ω    2        
             
                                                 
 Ti 1  Ti 2
                        TiTi 
      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
  12I  12I         1N I 
                           
  21I  2 I         2N I
2
Var (e | W)  Ω    IT 
                         
                      2 
 N 1I  N 2I
                     NI 

Pooled Regression Model
• Extensions
– Cross Section and Time Series Correlation
Var (eit | w it )   i ,tt   i2
 1      12    1T 
Cov(eit , eis | w it , w is )   i ,ts   i ts , t  s                   1      2T 
Cov(eit , e jt | w it , w jt )   ij , i  j                   R   21                 
                  
                  
Cov(eit , e js | w it , w js )   ij ts , t  s                    T 1   T 2    1 

  12 R  12 R                    1N R 
                                        
 21R  2 R
2
 2N R
Var (e | W)  Ω    R  
                                     
                                     
 N 1 R  N 2 R                  NR 
2
                                     
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)
1
δOLS  ( W ' W) 1 W ' y    i 1 Wi' Wi 
ˆ

N                          N
Wi'y i
                             i 1

1
ˆ (δ )   2 ( W ' W) 1   2   N W ' W 
Var ˆ OLS         ˆe         ˆe
 i 1 i i 
 e2  e ' e / ( NT  K )
ˆ     ˆ ˆ
e  y  Wδ
ˆ              ˆ

• Variance estimator Var(δ ) is inconsistent
ˆ ˆ
OLS

because of heteroscedasticity and
autocorrelation.
Pooled Regression: OLS
• Panel-Robust Variance-Covariance Matrix
– Adjusting general heteroscedasticity and serial correlation
within panel
ˆ       ˆ      ˆ
Var (δ)  E[(δ  δ)(δ  δ) ']  ( W ' W) 1 W ' E (ee ') W( W ' W) 1
1                                                   1
   N Wi' Wi    N Wi' E (ei ei' ) Wi    N Wi' Wi 
 i 1         i 1                     i 1       
1                                           1
Var (δ)    i 1 Wi' Wi    i 1 Wi'ei ei' Wi    i 1 Wi' Wi 
ˆ ˆ          N                 N                       N
ˆˆ
                                                     
1                                                                       1
   i 1  t 1 w it w it    i 1  t 1  si1 w it w is eit eis   i 1  t 1 w it w it 
N    Ti                   N      Ti     T                           N      Ti
'                                  '
ˆ ˆ                             '
                                                                                          
ˆ ˆ                ˆ
ei  y i  Wi δ, eit  yit  w it δ
ˆ
Pooled Regression: GLS
• The Model
  12  12      1N         1      12    1T 
y  Wδ  e                                                     
  21  2
2
 2N                  1     2T 
E (e | W)  0                                              R   21                
                                             
                2                            
Var (e | W)  Ω    R              N 1  N 2
               N 
          T 1   T 2    1 

• Generalized Least Squares (GLS)
1                                                    1
ˆ           ˆ          ˆ        ˆ ˆ               ˆ
δGLS   W 'Ω 1W  W 'Ω 1y , Var (δGLS )   W 'Ω 1W 
                                              

– If cross sections are independent (short panels)
1                                                       1
δGLS  i 1 Wi'i1Wi 
ˆ                ˆ               i1 W i1yi , Var (δGLS )  i 1 Wi'i1Wi 
ˆ         ˆ ˆ                    ˆ
N                          N     '                      N

                                i
                
– where         ˆ
i is   the consistent estimator of  i
Pooled Regression: GLS
• Heteroscedasticity
1 T 2
   t 1 eit
ˆi
2
ˆ
T
• Cross Section Correlation
1 T
 ij 
ˆ
T
t 1 eit e jt
ˆ ˆ

• Time Series Correlation
 ts
ˆ
Pooled Regression: GLS
• Examples of Time Series Correlation
– Equal-Correlation                        1 if t  s
ts  
  if t  s

1       if t  s
– AR(1)   ts   |t s|
       if t  s

1       if t  s

– Stationary(1)          ts    if | t  s | 1
 0 otherwise


– Nonstationary(1)                        1                     if t  s

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]
I it   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)
I it     Fit   Cit   it
• Classical OLS
• Panel-Robust OLS
• Feasible GLS
– Heteroscedastcity
– Autocorrelation

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