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
E (eit | Zi )  0  Cov(eit , z it )  0
Cov(xit , z it )  0, in particular Cov(x2it , z it )  0
– Other classical assumptions remained.
– OLS is biased; Instrumental variables estimation
should be used.
– IV estimator is consistent.
Constant Effects Model
• Instrumental Variables Estimation
y i  Wi δ  ei
ei | Zi ~ iid (0,  e2 I )
 β1 
where Wi   X1i X 2i 1 , δ  β 2 
 u 
E (eit | x1it )  0, but E (eit | x 2it )  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: X1i
– # Zi ≥ # W i
Cov(eit , z it )  0
Cov(w it , z it )  0
Constant Effects Model
• Instrumental Variables Estimation
ˆ ' W ) 1 W
ˆ 'y
δˆ IV  ( W
ˆ ' W ) 1
Var (δˆ )   2 ( W
IV
e
ˆ  Z(Z ' Z) 1 Z ' W
where W

δˆ IV  [ W ' Z(Z ' Z) 1 Z ' W ]1 W ' Z(Z ' Z) 1 Z ' y
Var (δˆ )   2 [ W ' Z( Z ' Z) 1 Z ' W ]1
IV
e
Constant Effects Model
• Instrumental Variables Estimation
δˆ IV  [ W ' Z(Z ' Z) 1 Z ' W ]1 W ' Z(Z ' Z) 1 Z ' y
Var (δˆ )   2 [ W ' Z(Z ' Z) 1 Z ' W]1
IV
e
ˆ (δˆ )  ˆ 2 [ W ' Z(Z ' Z) 1 Z ' W]1
Var
IV
e
ˆ e2  eˆ ' eˆ / ( NT  K ), eˆ  y  Wδˆ IV , K  # W
• HAC Variance-Covariance Matrix
ˆ (Z ' Z) 1 Z ' W
ˆ (δˆ )  W ' Z(Z ' Z) 1 Z ' ΩZ
Var
IV
ˆ  consistent estimator of Ω  E (ee ')
Ω
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 | x1it' )  0, but E (eit | x2it' )  0
Cov(ui , x1it' )  0, Cov(ui , x2it' )  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  (x 2it'  x 2i' )β2  (eit  ei )
 yit  x1it' β1  x 2it' β 2  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  X i β  ei
E (ei | Xi )  0, but E (ei | Zi )  0, and Cov(z it , xit )  0
βˆ
 [ X ' Z(Z ' Z) 1 Z ' X]1 X ' Z(Z ' Z) 1 Z ' y
FE  IV
2
1
1
ˆ (βˆ
ˆ
Var
)


[
X
'
Z
(
Z
'
Z
)
Z
'
X
]
FE  IV
e
ˆ e2  eˆ ' eˆ / ( NT  T  K ), eˆ  y  Xβˆ FE  IV , K  # X
– Panel-Robust Variance-Covariance Matrix
1
ˆ (Z ' Z) 1 Z ' X
ˆ (βˆ
Var
)

X
'
Z
(
Z
'
Z
)
Z
'
ΩZ
FE  IV
ˆ is consistent estimator of Ω  E (ee ')
Ω
Example: Returns to Schooling
• Cornwell and Rupert Model (1988)
yit  x1it' β1  x2i' β2  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 x1it:
– 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 x2i:
– ED = years of education  endogenous
FEM = 1 if female
BLK = 1 if individual is black
Random Regressors
• Random Effects Model
yit  x1it' β1  x 2it' β 2   it
where  it  ui  eit , Cov (ui , eit )  0
E ( it | x1it )  0, but E ( it | x 2it )  0. That is,
E (eit | x1it )  0, E (ui | x1it )  0; E (eit | x 2it )  0 or E (ui | x 2it )  0
– Other classical assumptions remained.
– Mundlak approach may be used when
E (ui | x2it )  0
– Instrumental variables must be used if
E (eit | x2it )  0
Random Effects Model
• The Model
yit  i yi  (x1it'  i x1i' )β1  (x 2it'  i x 2i' )β2  [(1  i )ui  (eit  i ei )]
i  1 
 e2
 e2  Ti u2
 yit  x1it' β1  x 2it' β2   it
E ( it | x1it )  0, but E ( it | x 2it )  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 (ε i | Zi )  0, and Cov(z it , xit )  0
βˆ
 [ X ' Z(Z ' Z) 1 Z ' X]1 X ' Z(Z ' Z) 1 Z ' y
RE  IV
2
1
1
ˆ (βˆ
ˆ
Var
)


[
X
'
Z
(
Z
'
Z
)
Z
'
X
]
RE  IV
e
ˆ e2  eˆ ' eˆ / ( NT  K ), eˆ  y  Xβˆ RE  IV , K  # X
– Panel-Robust Variance-Covariance Matrix
1
ˆ (Z ' Z) 1 Z ' X
ˆ (βˆ
Var
)

X
'
Z
(
Z
'
Z
)
Z
'
ΩZ
RE  IV
ˆ is consistent estimator of Ω  E (εε ')
Ω
Example: Returns to Schooling
• Cornwell and Rupert Model (1988)
yit  x1it' β1  x2i' β2  ui  eit (random effects )
• Data (575 individuals over 7 years)
– Dependent Variable yit:
• LWAGE = log of wage
– Explanatory Variables xit:
• Time-Variant Variables x1it:
– 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 x2i:
– 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' β4  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  x 2it' β 2  x3i' β3  x 4i' β 4  ui  eit
yi  x1i' β1  x 2i' β 2  x3i' β3  x 4i' β 4  ui  ei
yit  yi  (x1it'  x1i' )β1  (x2it'  x 2i' )β2  (eit  ei )
y  x1' β1  x 2' β 2  e  βˆ1
, βˆ 2
it
it
it
FE  IV
it
FE  IV
FE with IV : x1it' , x3i' (# x3i'  # x2i' , x1it' included )
eˆ  y  x1' βˆ1
 x2' βˆ 2
it
it
it
FE  IV
it
FE  IV
Hausman-Taylor Estimator
• Fixed Effects Model
– Within Residuals
Let  it  eˆit
OLS with IV : x1it' , x3i' (# x1it'  # x4i' , x3i' included )
  x3' β3  x4' β 4  v  βˆ 3 , βˆ 4
it
i
i
i
IV
IV
Define ˆit  yit  x1it' βˆ1FE  x2i' t βˆ 2 FE  x3i' βˆ 3IV  x4i' βˆ 4 IV
Then ˆ e2  εˆ ' εˆ / NT
 i  1 
 e2
 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' β 4   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 ,
(balanced panels only )
, x1iT' ]
Hausman-Taylor Estimator
• Instrumental Variable Estimation
w it'  [x1it' , x2it' x3i' , x4i' ], δ  [β1' , β2' , β3' , β4' ]'
z it'  [(x1it'  x1i' ), (x2it'  x 2i' ), x1i' , x3i' ]
 y  Wδ  ε, # Z  # W
δˆ IV  [ W ' Z(Z ' Z) 1 Z ' W ]1 W ' Z(Z ' Z) 1 Z ' y
Var (δˆ )   2 [ W ' Z(Z ' Z) 1 Z ' W]1
IV
e
ˆ (δˆ )  ˆ 2 [ W ' Z(Z ' Z) 1 Z ' W]1
Var
IV
e
ˆ 2  εˆ ' εˆ / ( N  K ), εˆ  y  Wδˆ , K  # W
e
IV
Example: Returns to Schooling
• Cornwell and Rupert Model (1988)
yit  x1it' β1  x2i' β2  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  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 x2i:
– ED = years of education  endogenous
FEM = 1 if female
BLK = 1 if individual is black