Econometric Analysis of Panel Data

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					Econometric Analysis of Panel Data
• Fixed Effects and Random Effects: Extensions
  – Time-invariant Variables
  – Two-way Effects
  – Nested Random Effects
                Time Invariant Variables
• The Model
                                             β1 
   yit  x β  ui  eit  x1
           '
           it             
                                  '
                                  it   x2     ui  eit
                                          '
                                           β2
                                          i
                                             
• Fixed Effects
   yit  x1it β1  x2i' β2  ui  eit 
           '

                                        ( yit  yi )  (x1it  x1i' )β1  (eit  ei )
                                                            '

    yi  x1i β1  x2i β2  ui  ei 
             '       '


   – b2 can not be identified, thus the individual effects ui can
     not be estimated.
               Time Invariant Variables
• Fixed Effects: Two-Step Approach
                                                  ˆ
  (1) yit  x1it β1   i  eit   i  yi  x1i' β1FE
              '
                                  ˆ
  (2)   x2' β2  w  β2
       ˆ                        ˆ
           i       i        i        OLS

                                                                ˆ
  E ( wi )  0, Var ( wi )  Var ( i )   e2 / Ti  x1i' Var (β1) x1i
                                  ˆ
  (heteroscedasticity )
                ˆ        ˆ
   u  y  x1' β1  x2' β2
     ˆ i       i       i   FE    i     OLS
           Time Invariant Variables
• Random Effects
  yit  xit β  ui  eit , requires Cov(ui , xit )  0
         '


  – Mundlak’s Approach
     If Cov(ui , xit )  0 and yit  x1it β1  x2i' β2  ui  eit ,
                                       '


     assume ui  x1i γ1  x2i' γ 2  wi , E ( wi | xit )  0, Var ( wi | xit )   w
                                                                                   2


       • Estimate random effects model including group means:
       yit  x1it β1  x1i γ1  x2i' δ2  wi  eit ( Note : δ2  β2  γ 2)
               '


        ui  x1i γ1  wi
           ˆ        ˆ     ˆ
                        ˆ         ˆ
               y  x1i β1  x2i' δ2
     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 x1it:
             – EXP = work experience
               WKS = weeks worked
               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
               UNION = 1 if wage set by union contract
        • Time-Invariant Variables x2i:
             – ED = years of education
               FEM = 1 if female
               BLK = 1 if individual is black
                         Two Way Effects
• The Model
  yit  xit β  ui  vt  eit (t  1, 2,..., Ti )
         '



• Assumptions
  E (eit | xit )  0
  E (ui | xit )  E (vt | xit )  0 (random effects only )
  Var (eit | xit )   e2 , Cov(eit , e js | xit , x js )  0
  Var (ui | xit )   u2 , Cov(ui , u j | xit , x jt )  Cov(ui , e jt | xit , x jt )  0
  Var (vt | xit )   v2 , Cov(vt , vs | xit , xis )  Cov(vt , eis | xit , xis )  0
  Cov(ui , vt | xit )  0
                    Two-Way Effects
• Dummy Variable Representation
  yit  xit β  ui  vt  eit (t  1, 2,..., Ti )
         '


                     β 
  y i   Xi di     ui iTi  ei  y i  Wi δ  ui iTi  ei
                     v
  (i  1, 2,..., N )
             v1        di1 
            v         d 
  where v   2  , d   i 2  , d   1 if j  1, 2,..., Ti
              i   ij 0                    otherwise
                       
            vTi 
                       diTi 
                         
                    Two-Way Effects
• Using one-way fixed effects or random effects
  model to estimate the dummy variable
  representation of two-way effects model.
  y i  Wi δ  ui iTi  ei (i  1, 2,..., N )
  
  y  Wδ  u  e
                   Two-Way Effects
• Two-Way Fixed Effects Model
  – Between Estimator
    yi  xi' β  ui  v  ei
    yt  xt' β  u  vt  et
    y  x 'β  u  v  e
  – Within Estimator (Group Means Deviations)
    yit  xit β  ui  vt  eit (t  1, 2,..., Ti ; i  1, 2,..., N )
           '


    yit  yi  yt  y  (xit  xi'  xt'  x ' )β  (eit  ei  et  e )
                          '


     yit  xit β  eit
             '
                  Two-Way Effects
• Two-Way Fixed Effects Model
  – OLS
   yit  xit β  eit  y i  Xi β  ei  y  Xβ  e
          '


   ˆ                        ˆ ˆ
   βOLS  ( X' X) 1 X' y, Var (βOLS )   e2 ( X' X) 1
                                          ˆ
         ˆˆ
    2  e'e / ( NT  N  T  1  K )
    ˆe

   ˆ        ˆ
   e  y  Xβ
  – Estimated Individual and Time Effects
                                   ˆ
    ui  ( yi  y )  ( xi'  x ' )β  u
    ˆ
                                   ˆ
    v  ( y  y )  ( x '  x ' )β  v
     t      t            t
                     Two-Way Effects
• Two-Way Random Effects Model
  – Partial Group Means Deviations
   yit  xit β  ui  vt  eit (t  1, 2,..., Ti ; i  1, 2,..., N )
          '


   yit  i yi  t yt  it y
       (xit  i xi'  t xt'  it x ' )β  (eit  i ei  t et  it e )
          '


                                   e2                          e2
   where i  1                             , t  1                    ,
                             Ti
                              2
                              e
                                         2
                                         u                  N t
                                                           2
                                                           e
                                                                      2
                                                                      v

                                 e2
   it  i  t  1 
                        e2  Ti u2  Nt v2
               Two-Way Effects
• Two-Way Random Effects Model
  – Consistent estimates of s are derived from:
     • e2 asym. var. of two-way fixed effects model
     • u2 asym. var. of between (individual) effects model or
       one-way fixed (individual) effects model
     • v2 asym. var. of between (time) effects model or one-
       way fixed (time) effects model
  – For improved efficiency, iterate the consistent
    estimation until convergence.
              Nested Random Effects
• Three-Level Model
  yijt  xijt β   ijt  xijt β  (ui  wij  eijt )
          '                '


  (i  1, 2,..., M ; j  1, 2,..., N i ; t  1, 2,..., Tij )
• Assumptions
  – Each successive component of error term is imbedded or
    nested within the preceding component
   E (eijt | X)  E ( wij | X)  E (ui | X)  0
  Var (eijt | X)   e2 ,Var ( wij | X)   w , Var (ui | X)   u
                                            2                    2


  Cov(eijt , wij | X)  Cov(eijt , ui | X)  Cov( wij , ui | X)  0

• Model Estimation
  – GLS, ML, etc.
       Example: U. S. Productivity
• The Model (Munnell [1988])
  – Two-level model
     ln( gsp jt )  b0  b1 ln(cap jt )  b 2 ln(hwy jt )  b3 ln( waterjt )
                  b 4 ln(util jt )  b5 ln(emp jt )  b6 ln(unemp jt )  u j  e jt

  – Three-level model
     ln( gspijt )  b0  b1 ln(capijt )  b 2 ln(hwyijt )  b3 ln( waterijt )
               b 4 ln(utilijt )  b5 ln(empijt )  b6 ln(unempijt )  ui  wij  eijt
  – See, B.H. Baltagi, S.H. Song, and B.C. Jung, The Unbalanced Nested
    Error Component Regression Model, Journal of Econometrics, 101,
    2001, 357-381.
       Example: U. S. Productivity
• Description
  – i=1,…,9 regions; j=Ni states
      • 6. Gulf: AL, FL, LA, MO
      • Mid West: IL, IN, KY, MI, MN, OH, WI
      • Mid Atlantic: DE, MD, NJ, NY, PA, VA
      • 8. Mountain: CO, ID, MT, ND, SD, WY
      • 1. New England: CD, ME, MA, NH, RI, VT
      • South: GA, NC, SC, TN, WV
      • 7. Southwest: AZ, NV, NM, TX, UT
      • Tornado Alley: AK, IA, KS, MS, NE, OK
      • 9. West: CA, OR, WA
  – t=1970-1986 (17 years)
         Example: U. S. Productivity
• Productivity Data
  – 48 Continental U.S. States, 17 Years:1970-1986
     •   STATE = State name,
     •   ST_ABB = State abbreviation (Region = 1, . . . , 9),
     •   YR = Year (1970, . . . ,1986),
     •   PCAP = Public capital,
     •   HWY = Highway capital,
     •   WATER = Water utility capital,
     •   UTIL = Utility capital,
     •   PC = Private capital,
     •   GSP = Gross state product,
     •   EMP = Employment,
     •   UNEMP = Unemployment

				
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