# Econometric Analysis of Panel Data_2_ by malj

VIEWS: 5 PAGES: 22

• pg 1
```									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

```
To top