# Econometric Analysis of Panel Data

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```					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
yit  xit β  ui  eit (t  1, 2,..., Ti )
'


y i  Xi β  ui iTi  ei (i  1, 2,..., N )

– ui is fixed, independent of eit, and may be
correlated with xit.
Cov(ui , eit )  0, Cov(ui , xit )  0
Fixed Effects Model
• Classical Assumptions
– Strict Exogeneity
E (eit | u, X)  0
– Homoschedasticity
Var (eit | u, X)   e2
– No cross section and time series correlation
Var (e | u, X)   e2 I NT
Fixed Effects Model
• Extensions
– Weak Exogeneity
E (eit | ui , xi1 , xi 2 ,..., xiTi )  E (eit | Xi )  0
E (eit | ui , xi1 , xi 2 ,..., xit )  0
E (eit | ui , xit )  0
Fixed Effects Model
• Extensions
– Heteroschedasticity
 1,1 0
2
0 
                         
0  1,2
2
0 
Var (eit | ui , Xi )   it , i  
2
                         
                     2 
 0
      0            1,Ti 

1 0                    0 
0                      0 
 Var (e | u, X)  Ω      2                     
                          
                          
0  0                   N 
Fixed Effects Model
• Extensions
– Time Series Correlation (with cross section
independence for short panels)
Cov(eit , eis | ui , xit , xis )   ts , t  s
Cov(eit , e js | ui , xit , u j , x js )  0, i  j
Var (eit | ui , xit )   tt   t2  Var (ei | ui , Xi )  i  Var (e | u, X)  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 
Fixed Effects Model
• Extensions
– Cross Section Correlation (with time series
independence for long panels)
Cov(eit , e jt | ui , xit , x jt )   ij , i  j
Cov(eit , e js | ui , xit , u j , x js )  0, t  s
Var (eit | ui , xit )   i2  Var (e | u, X)    IT  Ω

  12  12                    1N       12I  12I      1N I 
                                                              
 21  22
 2N        21I  2 I
2
 2N I
                                   , Ω
                                                            
                               2                         2 
 N 1  N 2
                              N 
      N 1I  N 2I
                  NI 

Dummy Variable Model
• Dummy Variable Representation
 y1   X1    iT1   0       0 u  e 
                    
1      1
y  X 
 2    2 β 
0    iT2     0  u2
    e2 
                                
                                
y N   X N   0
      0      iTN  u N  e N 

β 
y  Xβ  Du  e   X D    e
u 
 y  Wδ  e

– Note: X does not include constant term, otherwise
one less number of dummy variables should be used.
Dummy Variable Model
• Dummy Variable Estimator (LSDV)
1
δ  ( W W) W y    W W   W y
ˆ                     1
'                  '       N       '             N            '
OLS
                     i 1   i     i        i 1        i    i

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

• Heteroscedasticity and Autocorrelation
ˆ       ˆ      ˆ
Var (δ)  E[(δ  δ)(δ  δ) ']  ( W ' W ) 1 W ' E (ee ') W( W ' W) 1
1                                                           1
   i 1 W Wi    i 1 W E (e e ) Wi    i 1 W Wi 
N          '                 N    '       '                   N              '
              
i                      i
    i i
                         i
Dummy Variable Model
• Panel-Robust Variance-Covariance Matrix
ˆ ˆ         ˆ      ˆ
Var (δ)  E[(δ  δ)(δ  δ) ']
1                                1
  i 1 W Wi    i 1 W e e Wi   i 1 W Wi 
N                   N                 N
'
ˆˆ  '  '               '
          i
                  
i i i

i

1                                                        1
  N  Ti w it w it    N  Ti  Ti w it w is eit eis   N  Ti w it w it 
'                           '
ˆ ˆ                        '
 i 1 t 1           i 1 t 1 s 1                    i 1 t 1         
e  y  Wδ
ˆ          ˆ
i     i      i

ˆ
eit  yit  w it δ
ˆ
Within Model
• Within Model Representation
yi  xi' β  ui  ei
yit  yi  (xit  xi' )β  (eit  ei )
'


yit  xi' β  eit
y i  Xi β  ei     or
Qi y i  Qi Xi β  Qi ei
1 '
where Qi  ITi  iTi iTi , (Qi iTi  0, Qi'Qi  Qi )
Ti
Within Model
• Model Assumptions
E (eit | xit )  0
Var (eit | xit )  (1  1/ Ti ) e2
Cov(eit , eis | xit , xis )  ( 1/ Ti ) e2  0, t  s

1 '
Var (ei | Xi )  i   e2Qi   e2 (ITi         iTi iTi )
Ti
1 0                      0                    1  1/ T1 1/ T1     1/ T1 
0                        0 
 1/ T 1  1/ T      1/ T2 
2                               
Var (e | X)  Ω                                        i   e
2                                                   2        2

                                                                             
                                                                             
0  0                     N                     1/ TN 1/ TN      1  1/ TN 
Within Model
• Within Estimator: FE-OLS
y i  Xi β  ei           y  Xβ  e
1
βOLS  ( X X) X y    i 1 X Xi   i 1 Xi' y i
ˆ                    1
'            '           N N  '
                    i

ˆ ˆ
Var (β )   2 ( X' X) 1 X'QX( X' X) 1
ˆ
OLS           e
1                              1
    i 1 X Xi    i 1 X Qi Xi    i 1 X Xi 
N                N             N
ˆ2            '                 '             '
e
           i
               i
          i

1
    i 1 X Xi 
N
ˆ  2            '
e
i

ˆ ˆ                       ˆ      ˆ
 e2  e ' e / ( NT  N  K ), e  y  Xβ
ˆ
Within Model
• Within Estimator: GLS
1
βGLS  ( X'Q 1X) 1 X'Q 1y    i 1 Xi' Qi1Xi 
ˆ
        Xi' Qi1y i
N                                    N

                                       i 1

1
   i 1 Xi' Xi            
N                           N
Xi' y i
                              i 1

1
Var (βGLS )   e2 ( X'Q 1X) 1   e2   i 1 Xi' Qi1Xi 
ˆ ˆ                                        N
ˆ                    ˆ
                   
1
    i 1 Xi' Xi 
N
ˆ2
e
              

• GLS = FE-OLS
– Note: X Q         '
i   i
1
Xi  Xi' Qi'Qi1Qi Xi  Xi' Qi' Xi  Xi' Qi'Qi Xi  Xi' Xi
Xi' Qi1y i  Xi' Qi'Qi1Qi y i  Xi' Qi' y i  Xi' Qi'Qi y i  Xi' y i
Within Model
• Normality Assumption
yit  xit β  ui  eit
'
(t  1, 2,..., Ti )
y i  Xi β  ui iTi  ei     (i  1, 2,..., N )
ei ~ iidn(0,  e2 ITi )

y i  Xi β  ei with y i  Qi y i , Xi  Qi Xi , ei  Qi ei ,
1 '
Qi  ITi  iTi iTi
Ti
ei  normal (0, i ), where i   e2Qi Qi'   e2Qi
Within Model
• Log-Likelihood Function
Ti         1     1 ' 1
lli (β,  | y i , Xi )   ln  2   ln i  ei i ei
2
e
2          2     2
Ti              Ti e2     1 '
  ln  2                  2 ei ei
2                 2       2 e
• ML Estimator
 i1 i e i i
ˆ ,  2 )  arg max N ll (β,  2 | y , X )
(β ˆ e ML

                   i1
N    '             N
ee               ei' Qi ei
 eML 
ˆ2           i 1 i i
                       ˆ      ˆ
, β ML  β FE OLS
                    
N                   N
T
i 1 i
T
i 1 i
Within Model
• Estimated Fixed Effects
ˆ
ui  yi  xi' β
ˆ
ˆ ˆ                        ˆ ˆ
Var (ui )   i2 / Ti  xi'Var (β)xi
ˆ

– For N   ,
ˆ
β is consistent
ˆ
but u i is inconsistent
unless Ti   .
Within Model
• Panel-Robust Variance-Covariance Matrix
– Consistent statistical inference for general
heteroscedasticity, time series and cross
section correlation.
ˆ ˆ         ˆ      ˆ
Var (β)  E[(β  β)(β  β) ']
1                                            1
   i 1 X Xi    i 1 X e e Xi    i 1 X Xi 
N                        N                     N
'
ˆˆ     '   '                 '
         i
                  
i i i
     i

1                                                      1
   i 1  t 1 xit x   i 1  t 1  s 1 xit x e e   i 1  t 1 xit x 
N        Ti                   N      Ti   Ti                      N   Ti
'
ˆ ˆ     '                   '
                     
it
  is it is
it

ˆ ˆ              ' ˆ
ei  y i  Xi β, eit  yit  xit β
ˆ
First-Difference Model
• First-Difference Representation
yit  yit 1  (xit  xit 1 )β  (eit  eit 1 )  yit  xit β  eit
'     '                                     '

• Model Assumptions
E (eit | xit )  0
Var (eit | xit )  2 e2
 e2 if | t  s | 1
Cov(eit , eis | xit , xis )  
 0 otherwise
 2 1 0 0     0
 1 2 1 0    0
                                       1 0     0 
 0 1 2 1    0                        0       0 
Var (ei | Xi )  i   e                  , Var (e | X)  Ω             
2                                               2

                                                  
0 0                                                
1 2 1                       0  0    N 
                
0 0      0 1 2 
(Toepliz form)
First-Difference Model
• First-Difference Estimator: FD-OLS
y i  Xi β  ei       y  Xβ  e
1
βOLS  (X' X) 1 X' y    i 1 Xi' Xi   i 1 Xi' y i
ˆ                               N                 N

                 
ˆ ˆ
Var (β )   2 (X' X) 1 X'ΩX(X' X) 1
ˆ
OLS      e
1                                           1
    i 1 Xi' Xi    i 1 Xi' i Xi    i 1 Xi' Xi 
N                N                      N
ˆ2
e
                                                    
 e
ˆ2     e ' e / ( NT  N  K ), e  y  Xβ
ˆ ˆ                         ˆ              ˆ

• Consistent statistical inference for general
heteroscedasticity, time series and cross section
correlation should be based on panel-robust variance-
covariance matrix.
First-Difference Model
• First-Difference Estimator: GLS
ˆ
βGLS  (X'Ω 1X) 1 X'Ω 1y
1
   i 1 X  Xi   i 1 Xi' i1y i
1
N        '        N

             i

i

ˆ ˆ
Var (β )   2 (X'Ω 1X) 1
ˆ
GLS       e
1
ˆ   i 1 Xi' i1Xi 
2   N


e

Hypothesis Testing
• To Pool or Not to Pool?
yit  xit β  ui  eit
'

vs. (ui  u , i )
yit  xit β  u  eit
'

– 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
F                             ~ F ( N  1, NT  N  K )
RSS R / ( NT  N  K )
ˆ ˆ                 ˆ ˆ
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

```
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