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Econometric Analysis of Panel Data_2_

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					Econometric Analysis of Panel Data
• Random Regressors
  – Pooled (Constant Effects) Model
  – Instrumental Variables
  – Fixed Effects Model
  – Random Effects Model
  – Hausman-Taylor Estimator
               Random Regressors
• Pooled (Constant Effects) Model
  yit  x1it β1  x2it β2  u  eit
          '         '


  E (eit | x1it )  0, but E (eit | x2it )  0

  – Other classical assumptions remained.
  – OLS is biased; Instrumental variables
    estimation should be used.
  – IV estimator is consistent.
     E (eit | Zi )  0  Cov(eit , zit )  0
    Cov(xit , zit )  0, in particular Cov(x2it , zit )  0
            Constant Effects Model
• Instrumental Variables Estimation
 y i  Wi δ  ei
 ei | Zi ~ iid (0,  e2 I )
                                             β1 
  where Wi   X1i X2i 1 , δ  β2         
                                            u
                                             
  E (eit | x1it )  0, but E (eit | x2it )  0
  E (eit | Zi )  0  Cov(eit , z it )  0
 Cov(xit , z it )  0
         Constant Effects Model
• Instrumental Variables Estimation
  – Instrumental Variables: Zi
  – Included Instruments: X1 i
  – # Zi ≥ # W i
    Cov(eit , zit )  0
    Cov(wit , z it )  0
              Constant Effects Model
• Instrumental Variables Estimation
  ˆ        ˆ          ˆ
  δ IV  ( W ' W) 1 W ' y
        ˆ            ˆ
  Var (δ )   2 ( W ' W) 1
         IV      e

        ˆ
  where W  Z(Z ' Z) 1 Z ' W
  
  ˆ
  δ IV  [ W ' Z(Z ' Z) 1 Z ' W]1 W ' Z(Z ' Z) 1 Z ' y
        ˆ
  Var (δ )   2 [ W ' Z(Z ' Z) 1 Z ' W]1
         IV      e
              Constant Effects Model
• Instrumental Variables Estimation
  ˆ
  δ IV  [ W ' Z(Z ' Z)1 Z ' W]1 W ' Z(Z ' Z) 1 Z ' y
        ˆ
  Var (δ )   2 [ W ' Z(Z ' Z) 1 Z ' W]1
         IV      e

   ˆ ˆ
  Var (δ IV )   e2 [ W ' Z(Z ' Z) 1 Z ' W]1
                  ˆ
                                        ˆ
   2  e ' e / ( N  K ), e  y  Wδ , K  # W
   ˆ    ˆ ˆ                 ˆ
    e                                  IV



• HAC Variance-Covariance Matrix
   ˆ ˆ                              ˆ
  Var (δ IV )  W ' Z(Z ' Z) 1 Z ' ΩZ(Z ' Z) 1 Z ' W
   ˆ
  Ω  consistent estimator of Ω  E (ee ')
        Constant Effects Model
• Hypothesis Testing of Instrumental
  Variables
  – Test for Endogeneity
  – Test for Overidentification
  – Test for Weak Instruments
                Random Regressors
• Fixed Effects Model
  yit  x1it β1  x2it β2  ui  eit
          '         '


  E (eit | ui , X1i )  0, but E (eit | ui , X2i )  0

  – Other classical assumptions remained.
  – Can not estimate the parameters of time-
    invariant regressors, even if they are
    correlated with model error.
  – The random regressors x2 has to be time-
    varying.
                    Fixed Effects Model
• The Model
 yit  yi  (x1it  x1i' )β1  (x2it  x 2i' )β2  (eit  ei )
               '                  '


  yit  x1it β1  x2it β2  eit
           '         '


 E (eit | x1it )  0, but E (eit | x2it )  0
 Cov(eit , xit )  0 in particular Cov(eit , x2it )  0

• Instrumental Variables
  – #Zi ≥ #Xi (Zi must be time variant)
  E (eit | Zi' )  0, let z it  z it  zi'
                            '      '


   E (eit | z it )  0, or Cov(eit , z it )  0
               '                        '


  Cov(xit , z it )  0
       '      '
                    Fixed Effects Model
• Within Estimator
  y i  Xi β  ei
  E (ei | Xi )  0, but E (ei | Zi )  0, and Cov(z it , xit )  0
  ˆ
  β FE  IV  [ X ' Z(Z ' Z) 1 Z ' X]1 X ' Z(Z ' Z) 1 Z ' y
   ˆ ˆ
  Var (β         )   2 [ X ' Z(Z ' Z) 1 Z ' X]1
                      ˆ
          FE  IV     e

                                     ˆ
   e2  e ' e / ( N  K ), e  y  Xβ FE  IV , K  # X
   ˆ     ˆ ˆ                ˆ
  – Panel-Robust Variance-Covariance Matrix
   ˆ ˆ                                   ˆ
  Var (β FE  IV )  X ' Z(Z ' Z) 1 Z ' ΩZ(Z ' Z) 1 Z ' X
   ˆ
  Ω is consistent estimator of Ω  E (ee ')
 Example: Returns to Schooling
• Cornwell and Rupert Model (1988)
   yit  x1it β1  x2i' β 2  ui  eit  yit  x1it β1  ui  eit ( fixed effects )
           '                                     '



• Data (575 individuals over 7 ears)
   – Dependent Variable yit:
         • LWAGE = log of wage
   – Explanatory Variables xit:
         • Time-Variant Variables x1 it:
              – EXP = work experience (+EXP2)  exogenous
                WKS = weeks worked  endogenous
                OCC = occupation, 1 if blue collar  IV
                IND = 1 if manufacturing industry  IV
                SOUTH = 1 if resides in south  IV
                SMSA = 1 if resides in a city (SMSA)  IV
                MS = 1 if married  IV
                UNION = 1 if wage set by union contract  IV
         • Time-Invariant Variables x2 i:
              – ED = years of education  endogenous
                FEM = 1 if female
                BLK = 1 if individual is black
                  Random Regressors
• Random Effects Model
  yit  x1it β1  x2it β2   it
          '         '


  where  it  ui  eit , Cov(ui , eit )  0
  E ( it | x1it )  0, but E ( it | x2it )  0. That is,
  E (eit | x1it )  0, E (ui | x1it )  0; E (eit | x2it )  0 or E (ui | x2it )  0
  – Other classical assumptions remained.
  – Mundlak approach may be used when
      E (ui | x2it )  0

  – Instrumental variables must be used if
      E (eit | x 2it )  0
              Random Effects Model
• The Model
  yit  i yi  (x1it  i x1i' )β1  (x2it  i x 2i' )β2  [(1  i )ui  (eit  i ei )]
                   '                     '



                   e2
 i  1 
               e2  Ti u2
  yit  x1it β1  x2it β 2   it
           '         '


  E ( it | x1it )  0, but E ( it | x2it )  0
 Cov( it , xit )  0 in particular Cov( it , x2it )  0
              Random Effects Model
• (Partial) Within Estimator
  y i  Xi β  ε i
  E (εi | Xi )  0, but E (εi | Zi )  0, and Cov(z it , xit )  0
  ˆ
  β RE  IV  [ X ' Z(Z ' Z) 1 Z ' X]1 X ' Z(Z ' Z) 1 Z ' y
   ˆ ˆ
  Var (β         )   2 [ X ' Z(Z ' Z) 1 Z ' X]1
                      ˆ
          RE  IV     e

                                     ˆ
   e2  e ' e / ( N  K ), e  y  Xβ RE  IV , K  # X
   ˆ     ˆ ˆ                ˆ
  – Panel-Robust Variance-Covariance Matrix
   ˆ ˆ                                   ˆ
  Var (β RE  IV )  X ' Z(Z ' Z) 1 Z ' ΩZ(Z ' Z) 1 Z ' X
   ˆ
  Ω is consistent estimator of Ω  E (εε ')
 Example: Returns to Schooling
• Cornwell and Rupert Model (1988)
   yit  x1it β1  x 2i' β 2  ui  eit (random effects)
           '


• Data (575 individuals over 7 years)
    – Dependent Variable yit:
         • LWAGE = log of wage

    – Explanatory Variables xit:
         • Time-Variant Variables x1 it:
               – EXP = work experience (+EXP2)  exogenous
                 WKS = weeks worked  endogenous
                 OCC = occupation, 1 if blue collar  IV
                 IND = 1 if manufacturing industry  IV
                 SOUTH = 1 if resides in south  IV
                 SMSA = 1 if resides in a city (SMSA)  IV
                 MS = 1 if married  IV
                 UNION = 1 if wage set by union contract  IV

         • Time-Invariant Variables x2 i:
               – ED = years of education  endogenous
                 FEM = 1 if female  IV
                 BLK = 1 if individual is black  IV
         Hausman-Taylor Estimator
• The Model
 yit  x1it β1  x2it β2  x3i' β3  x4i' β 4  ui  eit
         '         '


 E (eit | x1it , x2it , x3i' , x4i' )  0, E (ui | x1it , x3i' )  0
            '      '                                 '



   – Time-variant Variables: x1it, x2it
       Cov (ui , x1i' )  0, but Cov (ui | x2i' )  0

   – Time-invariant Variables:x3i, x4i
       Cov(ui , x3i' )  0, but Cov(ui | x4i' )  0
   – Fixed effects model can not estimate b3 and
     b4; Random effects model has random
     regressors: x2 and x4 correlated with u.
         Hausman-Taylor Estimator
• Fixed Effects Model
   yit  x1it β1  x2it β2  x3i' β3  x4i' β4  ui  eit
           '         '


   yi  x1i' β1  x 2i' β2  x3i' β3  x4i' β4  ui  ei
   yit  yi  (x1it  x1i' )β1  (x2it  x 2i' )β2  (eit  ei )
                 '                  '


   y  x1' β1  x2' β2  e  β1 , β2ˆ      ˆ
    it     it         it       it       FE      FE

  ˆ             ' ˆ         ' ˆ
  eit  yit  x1it β1FE  x2it β 2 FE
         Hausman-Taylor Estimator
• Fixed Effects Model
  – Within Residuals
             ˆ
  Let  it  eit
  OLS with IV : x1it , x3i' (# x1it  # x4i' , x3i' included )
                    '            '


                                 ˆ
    x3' β3  x 4' β 4  v  β3 , β 4  ˆ
    it     i            i          i   IV     IV
                        ' ˆ          ' ˆ              ˆ            ˆ
  Define  it  yit  x1it β1FE  x 2it β 2 FE  x3i' β3IV  x 4i' β 4 IV
         ˆ
  Then  e2  ε ' ε / NT
        ˆ     ˆ ˆ
                        e2
   i  1 
                    e2  Ti u2
        Hausman-Taylor Estimator
• Random Effects Model
  yit  x1it β1  x2it β2  x3i' β3  x4i' β4  ui  eit
          '         '


  yit  i yi  (x1it  i x1i' )β1  (x2it  i x 2i' )β2
                   '                     '


              (1  i )x3i' β3  (1  i ) x4i' β4  [(1  i )ui  (eit  i ei )]
  yit  x1it β1  x2it β2  x3i' β3  x4i' β4   it
          '         '
       Hausman-Taylor Estimator
• Instrumental Variables
  – Hausman-Taylor (1981)
   z it  [(x1it  x1i' ), (x2it  x 2i' ), x1i' , x3i' ]
     '        '               '


   Note : x2it  (x2it  x 2i' ) not correlated with ui
            '       '


   Note : x4i'  x1i'
  – Amemiya-Macurdy (1986)
    Assuming E (ui | Xi )  E (ui | x1i' 1 , x1i' 2 ,           , x1iT )  0
                                                                    '


   z it  [(x1it  x1i' ), (x2it  x 2i' ), x3i' ]  [ x1i' 1 , x1i' 2 ,
     '        '               '                                                '
                                                                           , x1iT ]
   (balanced panels only )
         Hausman-Taylor Estimator
• Instrumental Variable Estimation
  w it  [x1it , x 2it x3i' , x4i' ], δ  [β1' , β2' , β3' , β4' ]'
    '       '       '


  z it  [(x1it  x1i' ), (x 2it  x 2i' ), x1i' , x3i' ]
    '        '                '


   y  Wδ  ε, # Z  # W
  ˆ
  δ IV  [ W ' Z(Z ' Z)1 Z ' W]1 W ' Z(Z ' Z) 1 Z ' y
        ˆ
  Var (δ )   2 [ W ' Z(Z ' Z) 1 Z ' W]1
          IV      e

   ˆ ˆ
  Var (δ IV )   e2 [ W ' Z(Z ' Z) 1 Z ' W]1
                  ˆ
                                        ˆ
   2  ε ' ε / ( N  K ), ε  y  Wδ , K  # W
   ˆ    ˆ ˆ                 ˆ
    e                                   IV
 Example: Returns to Schooling
• Cornwell and Rupert Model (1988)
   yit  x1it β1  x 2i' β 2  ui  eit
           '



• Data (575 individuals over 7 ears)
   – Dependent Variable yit:
         • LWAGE = log of wage
   – Explanatory Variables xit:
         • Time-Variant Variables x1 it:
               – EXP = work experience  endogenous (+EXP2)
                 WKS = weeks worked  endogenous
                 OCC = occupation, 1 if blue collar,
                 IND = 1 if manufacturing industry
                 SOUTH = 1 if resides in south
                 SMSA = 1 if resides in a city (SMSA)
                 MS = 1 if married  endogenous
                 UNION = 1 if wage set by union contract  endogenous
         • Time-Invariant Variables x2 i:
               – ED = years of education  endogenous
                 FEM = 1 if female
                 BLK = 1 if individual is black

				
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posted:10/16/2011
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