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Econometric Analysis of Panel Data

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					Econometric Analysis of Panel Data
• Hypothesis Testing
  – Specification Tests
     • Fixed Effects vs. Random Effects
  – Heteroscedasticity
  – Autocorrelation
     • Serial Autocorrelation
     • Spatial Autocorrelation
• More on Autocorrelation
                          Hypothesis Testing
• Heteroscedasticity
                                                             e2it
                                                            
  yit  xit β  ui  eit , Var (ui )   u2i , Var (eit )   2
         '

                                                             ei
                                                            

• Serial Correlation
                                 vit 1  eit
  yit  x β  ui  vit , vit  
               '

                                i vit 1  eit
               it


• Spatial Correlation
  yit  xit β  ui  vit , vit    wij v jt  eit
         '

                                                         j
                        Hypothesis Testing
• Heteroscedasticity
  yit  xit β   it ,  it  ui  eit
         '


  Var (ui )   u2i   u2 hu (fi' u ), i  1,..., N (e.g ., fi'  xi' )
  Var (eit )   e2it   e2 he (z it e ), i  1,..., N , t  1,..., T (e.g ., z it  xit )
                                   '                                              '     '


                  u2i   e2   u2 hu (fi' u )   e2
                 2
                
  Var ( it )    u   e2it   u2   e2 he (z it e )
                                                   '

                 2
                 ui   eit   u hu (fi  u )  he (z it e )
                           2        2       '              '
                

  Note : h(.)  0, h(0)  1, h' (0)  0,  u  0,  e2  0
                                           2


  A special case of  e2i is  e2i   e2 he ( zi' e ), i  1,..., N
  then  2   u   e2i   u   e2 he ( zi' e )
                2             2
                           Hypothesis Testing
                        Test for Homoscedasticity
• If u2=0 (constant effects or pooled model), then
   yit  xit β  ui  eit
          '


  Var (eit )   e2it   e2 he (z it e ) or  e2 he ( zi' e )
                                   '


• LM Test (Breusch and Pagan, 1980)                                H0 : e  0
                           1
  LM  2 0 ( e  0)  G ' Z ( Z ' Z ) 1 Z 'G ~  2 (#  e )
      u
                           2
                                                  ˆ ˆ
  G  e 2 /  e2  1,  e2  e'e / NT , e  y  Xβ  u
      ˆ ˆ              ˆ     ˆˆ         ˆ
  Z  [z it , i  1,..., N , t  1,..., T ]
         '


                                          ESS
  (1) Regress [e 2 /  e2  1] on Z , obtain
               ˆ ˆ                             ~  2 (#  e )
                                           2
  (2) Regress e 2 on  0  Z e , obtain NTR 2 ~  2 (#  e )
              ˆ
                       Hypothesis Testing
                     Test for Homoscedasticity
• If u2>0 (random effects), then
   yit  xit β  ui  eit
          '


  Var (ui )   ui   u hu (fi' u )
                2      2


  Var (eit )   e2it   e2 he (z it e ) or  e2 he ( zi' e )
                                   '


• LM Tests (Baltagi, Bresson, and Pirott, 2006)
  H 0 : e  0 | u  0
  H 0 : u  0 | e  0
  H 0 :  u  0,  e  0
                  Hypothesis Testing
                 Test for Homoscedasticity
• Marginal LM Test H 0 :  e  0 |  u  0,  u2  0
           ˆ2
   Regress  it on  0  z it e , obtain NTR 2 ~  2 (#  e )
                           '


         ˆ      ˆ ˆ
   where   u  e  y  x'          ˆ
            it    i   it     it   it

  (e.g., z it  xit , not including constant )
           '     '



   • See, Montes-Rojas and Sosa-Escudero (2011)
                         Hypothesis Testing
                        Test for Homoscedasticity
• Marginal LM Test H 0 :  u  0 |  e  0,  u  0
                                              2


           ˆ
   Regress  i2 on  0  fi' u , obtain NR 2 ~  2 (#  u )
              1 T ˆ ˆ
   where  i    it ,  it  ui  eit  yit  xit 
         ˆ                     ˆ ˆ               ' ˆ

              T t 1
   (e.g., fi'  xi' )

• Joint LM Test           H 0 :  u  0,  e  0 |  u  0
                                                     2


   – Sum of the above two marginal test statistics
     (approximately)
   – See, Montes-Rojas and Sosa-Escudero (2011)
               Hypothesis Testing
            Testing for Homoscedasticity
• References
  – Batagi, B.H., G. Bresson, and A. Priotte, Joint LM Test for
    Homoscedasticity in a One-Way Error Component
    Model, Journal of Econometrics, 134, 2006, 401-417.
  – Breusch, T. and A. Pagan, “A Simple Test of
    Heteroscedasticity and Random Coefficient Variations,”
    Econometrica, 47, 1979, 1287-1294.
  – Montes-Rojas, G. and W. Sosa-Escudero, Robust Tests for
    Heteroscedasticity in the One-Way Error Components
    Model, Journal of Econometrics, 2011, forthcoming.
                      Hypothesis Testing
• Serial Correlation AR(1) in a Random Effects Model
  yit  xit β  ui  vit , ui ~ iid (0,  u )
         '                                2


  vit   vit 1  eit ~ iid (0,  e2 )

• LM Test for Serial Correlation and Random Effects
  H 0 :  u  0,   0
          2


  H 0 :   0 (assuming  u  0 or pooled model)
                          2


  H 0 :   0 (assuming  u  0 or random effects model)
                          2
                     Hypothesis Testing
                 Test for Serial Correlation
• LM Test Statistics: Notations
  Based on OLS residuals of the restricted model (i.e. pooled
  model with no serial correlation)
                                                        2
                                       N
                                                T


      e ( I N  J T )e
      ˆ '
                     ˆ          eit  ˆ
   A                             
                        1  i 1 N t T1   1
             ˆˆ
             e 'e
                                        ˆ2
                                         eit
                                            i 1 t 1
                 N      T

    ˆˆ   '
    e e 1       e e
                   ˆ ˆ       it it 1
  B '         i 1 t  2
     ˆˆ             N T

                  
     ee
                     ˆ2
                     eit
                     i 1 t 1
                  Hypothesis Testing
                 Test for Serial Correlation
• Marginal LM Test Statistic for a Pooled Model
                       NT
   LM  0 (  0) 
             2
                              A2 ~  2 (1)
                     2(T  1)
             u


   – See Breusch and Pagan (1980)


• Marginal LM Test Statistic for Serial Correlation
                         NT 2 2
   LM  2 0 (   0)          B ~  2 (1)
       u
                        (T  1)
   – See Breusch and Godfrey (1981)
                  Hypothesis Testing
                 Test for Serial Correlation
• Robust LM Test Statistic
                          NT
 LM  0 (  0)                        2 B  A ~  2 (1)
     *     2                                      2

                   2(T  1)(1  2 / T )
           u




                            NT 2
 LM  2 0 (   0)                       B  A / T  ~  2 (1)
     *                                                 2
     u
                      (T  1)(1  2 / T )


• See Baltagi and Li (1995)
                   Hypothesis Testing
                 Test for Serial Correlation
• Joint LM Test Statistic for Pooled Model with Serial
  Correlation
  LM (  0,   0) 
         2
         u
                           NT
                      2(T  1)(T  2)
                                       A2  4 AB  2TB 2  ~  2 (2)
  LM ( u2  0,   0)  LM  0 ( u2  0)  LM  2 0 (   0)
                                                    u


   LM  0 ( u2  0)  LM  2 0 (   0)
       *
                              u


   LM  0 ( u  0)  LM  2 0 (   0)
               2           *
                              u



• See Baltagi and Li (1995)
                Hypothesis Testing
              Test for Serial Correlation
• LM Test Statistic for a Fixed Effects Model
                                   ˆˆ
                              NT 2 e'e 1
  LM fixed effects (   0)              ~  2 (1)
                                    ˆˆ
                              T  1 e 'e
  ˆ            ˆ
  e  y  Xβ  residuals of mean deviation regression

   – See Baltagi, Econometric Analysis of Panel Data (2008)
                 Hypothesis Testing
                Test for Serial Correlation
• References
  – Breusch, T. and A. Pagan, “A Simple Test of Heteroscedasticity and
    Random Coefficient Variations,” Econometrica, 47, 1979, 1287-1294.
  – Breusch, T. and A. Pagan, “The LM Test and Its Applications to Model
    Specification in Econometrics,” Review of Economic Studies, 47, 1980,
    239-254.
  – Breusch, T. and L.G. Godfrey, A Review of Recent Work on Testing for
    Autocorrelation in Dynamic Simultaneous Models, in D.A. Currie, R.
    Nobay and D. Peel (eds.), Macroeconomic Analysis, Essays in
    Macroeconomics and Economics (Croom Helm, London), 63-100.
  – Baltagi, B.H. and Q. Li, Testing AR(1) Against MA(1) Disturbances in an
    Error Components Model, Journal of Econometrics, 68, 1995, 133-151.
                       Autocorrelation
• AR(1)
  yit  xit β  ui  eit
         '


 eit   eit 1  it (t  1, 2,..., Ti ; i  1, 2,..., N )

• Assumptions
  E (it | xit )  0
            '


  Var (it | xit )   2  Var (eit | xit )   e2  (1   2 ) 2
              '                         '


  Cov(it , eit 1 )  0, Cov(eit , ui )  0, Cov(eit , xit )  0
                                                         '


  Cov(ui , xit )  0 (random effects only )
            '
                      Autocorrelation
• AR(1) Model Estimation (Paris-Winsten)
  – Begin with =0, estimate the model
                                          ' ˆ
    yit  xit β  ui  eit  eit  yit  xit β  ui
           '
                             ˆ                   ˆ

         ee
             N        Ti
           ˆ ˆ        t  2 it it 1
    
    ˆ        i 1

          e
                 N        Ti
             ˆ   i 1
                                2
                           t 1 it

  – Transform variables according
     zit  zit   zit 1 , t  1
      *
                 ˆ
     zi*1  1   2 zi1
                ˆ
                  Autocorrelation
– Estimate the transformed model
                                        ' ˆ
  yit  x*'β  ui*  eit  eit  yit  xit β  ui
   *
         it
                      *
                           ˆ                   ˆ

       ee
          N        Ti
         ˆ ˆ       t  2 it it 1
  
  ˆ       i 1

        e
              N        Ti
           ˆ  i 1
                             2
                        t 1 it

                         ˆ
                        β 
– Iterate until          
                               converges
                         
                         ˆ
                   Autocorrelation
• Notational Complexity with time lags in unbalanced panel data
  (Unbalanced unequal space panel data)
                              i   t   zit    zit-1   z*it

                              1   1   z11    .       (1-2)1/2 z11
    i    t   zit              1   2   z12    z11     z12 -z11
    1    1   z11              1   3   .      .       .
    1    2   z12              1   4   z14    .       (1-2)1/2 z14
    1    4   z14              1   5   z15    z14     z15 -z14
    1    5   z15              2   1   z21    .       (1-2)1/2 z21
    2    1   z21              2   2   .      .       .
    2    4   z24              2   3   .      .       .
    2    5   z25              2   4   z24    .       (1-2)1/2 z24
    3    3   z33              2   5   z25    z24     z25 -z24

                              3   1   .      .       .

                              3   2   .      .       .

                              3   3   z33    .       (1-2)1/2 z33

                              3   4   .      z33     .

                              3   5   .      .       .
                  Autocorrelation
• Hypothesis Testing
  – Modified Durbin-Watson Test Statistic (Bhargava,
    Franzini, Narendranathan, 1982)

             e               eit 1 
           N       Ti                   2
                ˆ                ˆ
          i 1    t 1   it
    d1
              
                  N       Ti
                  i 1    t 1
                               ˆ2
                               eit

  – LBI Test Statistic (Baltagi-Wu, 1999)
     • For unbalanced unequal spaced panel data
    Example: Investment Demand
• Grunfeld and Griliches [1960]
  I it   i   Fit   Cit   it
   it   it 1  eit ~ iid (0,  e2 )
  – 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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