Econometric Analysis of Panel Data
• Instrumental Variables in Panel Data
– Assumptions of Instrumental Variables
– Fixed Effects Model
– Random Effects Model
– First Difference Model
Instrumental Variables in Panel Data
• The Model with Random Regressors
yit  xit' β   it  x1it' β1  x2it' β2   it
E ( it | x1it' )  0, E ( it | x2it' )  0  E ( it | xit' )  0
• Instrumental Variables
Cov(xit' , z it' )  0, in particular Cov(x2it' , z it' )  0
 E (Zi' εi )  0
– Instrumental variables can be obtained through use of
exogenous regressors X1 in periods other than the
current period, using the exogeneity assumption.
Assumptions of Instrumental Variables
• Summation Assumption
E ( it | z )  0, t  E (Z ε )  0  E
'
it
'
i i


'
z
 t 1 it it  0
T
 z i' 1 
  it 
 ' 
 
z
Zi   i 2  , εi   i 2  , L  # Z  # X  K
 
 
 ' 
 
 z iT T  L
 iT 
– Exogenous variables X1 are included in Z.
– Z may include excluded exogenous variables (other
than X1), although they are difficult to find.
Assumptions of Instrumental Variables
• Contemporary Exogeneity Assumption
E ( it | z it' )  0  Cov( it , z it' )  0, t
 z i' 1

0

Zi 


 0
0
z i' 2
0
0

0

' 
z iT T TL
– Any time-invariant exogenous variables in X1
can be used only once as an instrument.
Assumptions of Instrumental Variables
• Weak Exogeneity Assumption
(Predetermined Instruments Assumption)
E ( it | z i' 1 , z i' 2 ,
, z it' )  0
 Cov( it , z is' )  0, s  1, 2,
 z i' 1

0
Zi  

0

,t
0
 z i' 1
0
z i' 2 
0
0
 z i' 1
z i' 2





z iT'   T (T 1)
T
L
2
Assumptions of Instrumental Variables
• Strong Exogeneity Assumption
E ( it | z i' 1 , z i' 2 ,
, z iT' )  0
 Cov( it , z is' )  0, s  1, 2,
Zi  IT   z i' 1 z i' 2
,T
z iT' 
– For dynamic models, at most weak
exogeneity of instruments can be assumed.
– Time invariant instruments can be used only
once.
IV for Fixed Effects Model
• Fixed Effects Model
yit  xit' β  ui  eit
Cov(ui , xit' )  0 and / or E (eit | xit )  0
yit  yi  (xit'  xi' )β  (eit  ei )
yit  xit' β  eit , E (eit | xit' )  0
IV : E ( Zi' ei )  E ( Z i' (ei  ei ))  0
 Cov(z it , eis )  0, s  1, 2,
,T
– Strong exogeneity of instrumental variables must be
assumed for the fixed effects model so that the within
estimates are consistent.
IV for Random Effects Model
• Random Effects Model
yit  xit' β   it where  it  ui  eit , E ( it | xit )  0
yit  i yi  (xit'  i xi' )β  ( it  i i )
yit  xit' β   it , E ( it | xit )  0 where  it  (1  i )ui  (eit  i ei )
IV : E (Zi' ε i )  E[Zi' (1  i )ui  Zi' (ei  i ei )]  0
 Cov(z it , ui )  0 and Cov( z it , eis )  0, s  1, 2,
,T
– Strong exogeneity of instrumental variables must be
assumed for the random effects model so that the
GLS parameter estimates are consistent.
IV for First Difference Model
• First Difference Model
yit  xit' β   it where  it  ui  eit , E ( it | xit )  0
yit  yit 1  (xit'  xit' 1 )β  (eit  eit 1 )
yit  xit' β  eit , E (eit | xit' )  0
E (eit | xit'  2 )  0, E (eit | xit' 3 )  0,
 E (eit | xit'  2 )  0,
IV : E (Zi' ei )  E (Zi'(t 2,...T ) (ei(t 2,...T )  ei(t 1,...T 1) ))  0
 Cov(z it , eit  eit 1 )  0, or
Cov(z it , eis )  0, s  t ( s  1, 2,
, t  1; t  2,3,
,T )
IV for First Difference Model
• First Difference Model
– No time-invariant variables.
– To consistently estimate the first-difference model, we
need only the Weak Endogeneity assumption for the
instrumental variables.
E (Zi' ei )  E (Zi'(t  2,...T ) (ei( t 2,...T )  ei( t 1,...T 1) ))  0
 z i' 2

0
Zi  

0

0
 z i' 2
0
z i' 3 
0
0
 z i' 2
z i' 3





z iT'  
Example: Returns to Schooling
• Cornwell and Rupert Model (1988)
yit  x1it' β1  x2i' β2  ui  eit
• Data (575 individuals over 7 years)
– Dependent Variable yit:
• LWAGE = log of wage
– Explanatory Variables xit:
• Time-Variant Variables x1it:
– EXP = work experience
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
UNION = 1 if wage set by union contract
• Time-Invariant Variables x2i:
– ED = years of education  endogenous
FEM = 1 if female
BLK = 1 if individual is black
Example: Returns to Schooling
• Labor Market Equilibrium Model
– Labor Demand Equation
lwage = exp exp2 wks occ ind south
smsa ms union [blk fem ed]
– Labor Supply Equation
wks = lwage union [fem ed]
– Endogenous or predetermined variable:
[ed]