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Econometric Analysis of Panel Data

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Econometric Analysis of Panel Data Powered By Docstoc
					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
           ( RSS R  RSSUR ) / N  1
      F                             ~ F ( N  1, NT  N  K )
            RSS R / ( NT  N  K )
      RSSUR  e'FE e FE , RSS R  e'POe PO
              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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