Week 1: endogeneity & instrumental variables
1 1
.
endogeneity
if the regressors ,
i ,
are exogenous ,
then there's no correlation between the
regressors& the disturbances
When :* E : are mutually correlated (i .
e .
Mi is endogenous) ,
the OLS estimator
is inconsistent
*
causes of endogeneity :
·
unobserved factors :
omitted explanatory variables (omitted variable bias)
non-observable heterogeneity
·
measurement errors
Simultaneitonx
·
depends on
y ,
think : demand & supply
·
selection mechanism
the sample isn't random
·
misspecified dynamics
·
consequences of endogeneity
·
ECYilTli] + cliB ,
so no linear regression
·
OLS I inconsistent
estimator is biased
·
there's correlation between X& E :
PlimiX's = 0
,
EIX * E(XE(IX)] + 0
OLS is biased :
E(BIX] =
B Ex((X(X) "X E(s(X](X)
+
B + (XX) X E(z(X]
+
=
+ B
as is inconsistent :
by LLN : EkLisi] +0 =>
pliM UX's + O
by Slutsky :
plim B =
plim (XX)- Xy =
plim (XX) X'(XB 2)
=
+
=
B +
plim (xX) "X's
=
B + Qxx Plim n X'E
+ B
,1 2
.
instrumental variables
to find an estimator that is consistent can identify an instrumental variable
,
you
assume that y XB
=
ECel2] 0 where the instrumental variable
+ 3 & =
,
,
I
,
is
an (nxm) matrix (with m > k) & exogenous if :
11) Z is uncorrelated with 2 plim (n2'e) 0
=
:
correlated
(2) 24 X are
sufficiently :
plim (n2'x1 =
Qzx ; rank (Q2x) =
K
13) Z is stable :
plim /n z'z)
=
Q22 ; rank (Q2z) =
m
*
finding the IV estimator ,
Bi
we have that :
u =
XB +
E
z's
multiply by I'
Zy Xp 2 +
=
:
then ,
n Z'y (nZ'XIB =
+ I'E
taking plim ,
19st term > 0 .
thus : Div =
(2'X1 "Zy =
B (x1 "I'd
+
this estimator has 2 convenient properties :
1 by is consistent
proof :
plim(birl =
plim (B + (2'x) "I's)
B + plim(n2'x): plim (nI'd)
=
-
1
=
B Q2X O
+
.
=
B q
.
e .
d .
2 bir is asymptotically normally distributed
proof : n (bi -
B) =
(n2'X)" (in I's)
(n2'X)"
1
(i) Qx by Slutsky
-
>
(ii) Since [i-iid 10 ,
0 % & plim (n2'z) =
Q22
"
& in 2's N(0 03Q22)
by CLT CMT : <
,
(iii) Cramer's theorem :
Anon-Ar-NCAM ASIA)
if m v - NCM [i)
3 then
>
, . .
Plim An : A
..
using i ,
ii d iii :
"
n (bir-B) N10 .
02QzQ2z(Qix)")
N(B 102QQ22(Qex")
thus .
Dir -
,
A o 'needs to be estimated :
i [F, e,
"
o =
E[E P ] , by LIN : > o ,
but since ci is unobserved use e
: 82 =
n[ ei =
nein ei
this estimator is consistent
given y -xbir y X(2'x) * Zy (1 X(2(X) z)(XB 2)
+
eir = +
-
=
=
-
XB (I X12'X) I'le
-
=
XB +
- -
=
(I -
X(2'x)"2')E
, 1 or is consistent
proof : 8 neiver
= =
ne(l-X (2'X)"2)' (l-X (2'X) I'ld
=
ne's -
ne'(X(2'x)" z(c + ne'(z(X'2)"XX(2'x1'2') 3
* note : (1) plim (nss) 02 LLN =
:
(2) plim (n2'x) Qzx full rank & instruments relevant
=
:
(3) plim (n2's) 0 instruments valid = :
by Slutsky :
22 IX'21 "XX(Z'X1 "Z'E
plim(oir) plim
( E's-2eX
(2'x)"Ze
(
= +
M M U nnnn n
02 2 +0 Q O 0 Q Qxx Qu O
=
-
.
.
. t . . ·
02
now , given the estimator of ,
the asymptotic covariance matrix is :
varibirl Fir
in (* " (22)"(x2)
=
Gi (2'X1" (2'21"(x2)
=
* What ifm instruments ? (M > k .
i e. overidentified)
if you havem instruments could decide to only use K e g could select the , ,
. .
firstK columns in 2 z (2 . [c] : =
However the formuld var(biv) shows that the bigger XZ the smaller the
, ,
variance since X'z > Cov( 2),
: pick columns that most correlated
:
,
with I
rather than taking K columns from 2 could take linear combo 25 .
,
example : 3 122) "X
X = 2B + 0
.
=
= X =
zB =
P2X(kxk)
i. e .
X as instruments
use
these instruments are relevant as they're correlated with X & they're
valid since :
plim Xe QQO
plim
XL (22) "I =
= =
* two stage least squares
the IV estimator can be computed by means of 2 successive regressions
1st stage
:
perform regression for each column in X on 2 ,
with fitted values
X =
PzX ,
save the predictions
2nd stage perform regression y on X y XB 2 of
*
: : =
+
the resulting estimate is given by :
Biss (xX)" * y (XP2X)"XPzy ( ** ) Yy
"
=
= =
Biscs N (B 0 XP2X1")
*
-
,
1 1
.
endogeneity
if the regressors ,
i ,
are exogenous ,
then there's no correlation between the
regressors& the disturbances
When :* E : are mutually correlated (i .
e .
Mi is endogenous) ,
the OLS estimator
is inconsistent
*
causes of endogeneity :
·
unobserved factors :
omitted explanatory variables (omitted variable bias)
non-observable heterogeneity
·
measurement errors
Simultaneitonx
·
depends on
y ,
think : demand & supply
·
selection mechanism
the sample isn't random
·
misspecified dynamics
·
consequences of endogeneity
·
ECYilTli] + cliB ,
so no linear regression
·
OLS I inconsistent
estimator is biased
·
there's correlation between X& E :
PlimiX's = 0
,
EIX * E(XE(IX)] + 0
OLS is biased :
E(BIX] =
B Ex((X(X) "X E(s(X](X)
+
B + (XX) X E(z(X]
+
=
+ B
as is inconsistent :
by LLN : EkLisi] +0 =>
pliM UX's + O
by Slutsky :
plim B =
plim (XX)- Xy =
plim (XX) X'(XB 2)
=
+
=
B +
plim (xX) "X's
=
B + Qxx Plim n X'E
+ B
,1 2
.
instrumental variables
to find an estimator that is consistent can identify an instrumental variable
,
you
assume that y XB
=
ECel2] 0 where the instrumental variable
+ 3 & =
,
,
I
,
is
an (nxm) matrix (with m > k) & exogenous if :
11) Z is uncorrelated with 2 plim (n2'e) 0
=
:
correlated
(2) 24 X are
sufficiently :
plim (n2'x1 =
Qzx ; rank (Q2x) =
K
13) Z is stable :
plim /n z'z)
=
Q22 ; rank (Q2z) =
m
*
finding the IV estimator ,
Bi
we have that :
u =
XB +
E
z's
multiply by I'
Zy Xp 2 +
=
:
then ,
n Z'y (nZ'XIB =
+ I'E
taking plim ,
19st term > 0 .
thus : Div =
(2'X1 "Zy =
B (x1 "I'd
+
this estimator has 2 convenient properties :
1 by is consistent
proof :
plim(birl =
plim (B + (2'x) "I's)
B + plim(n2'x): plim (nI'd)
=
-
1
=
B Q2X O
+
.
=
B q
.
e .
d .
2 bir is asymptotically normally distributed
proof : n (bi -
B) =
(n2'X)" (in I's)
(n2'X)"
1
(i) Qx by Slutsky
-
>
(ii) Since [i-iid 10 ,
0 % & plim (n2'z) =
Q22
"
& in 2's N(0 03Q22)
by CLT CMT : <
,
(iii) Cramer's theorem :
Anon-Ar-NCAM ASIA)
if m v - NCM [i)
3 then
>
, . .
Plim An : A
..
using i ,
ii d iii :
"
n (bir-B) N10 .
02QzQ2z(Qix)")
N(B 102QQ22(Qex")
thus .
Dir -
,
A o 'needs to be estimated :
i [F, e,
"
o =
E[E P ] , by LIN : > o ,
but since ci is unobserved use e
: 82 =
n[ ei =
nein ei
this estimator is consistent
given y -xbir y X(2'x) * Zy (1 X(2(X) z)(XB 2)
+
eir = +
-
=
=
-
XB (I X12'X) I'le
-
=
XB +
- -
=
(I -
X(2'x)"2')E
, 1 or is consistent
proof : 8 neiver
= =
ne(l-X (2'X)"2)' (l-X (2'X) I'ld
=
ne's -
ne'(X(2'x)" z(c + ne'(z(X'2)"XX(2'x1'2') 3
* note : (1) plim (nss) 02 LLN =
:
(2) plim (n2'x) Qzx full rank & instruments relevant
=
:
(3) plim (n2's) 0 instruments valid = :
by Slutsky :
22 IX'21 "XX(Z'X1 "Z'E
plim(oir) plim
( E's-2eX
(2'x)"Ze
(
= +
M M U nnnn n
02 2 +0 Q O 0 Q Qxx Qu O
=
-
.
.
. t . . ·
02
now , given the estimator of ,
the asymptotic covariance matrix is :
varibirl Fir
in (* " (22)"(x2)
=
Gi (2'X1" (2'21"(x2)
=
* What ifm instruments ? (M > k .
i e. overidentified)
if you havem instruments could decide to only use K e g could select the , ,
. .
firstK columns in 2 z (2 . [c] : =
However the formuld var(biv) shows that the bigger XZ the smaller the
, ,
variance since X'z > Cov( 2),
: pick columns that most correlated
:
,
with I
rather than taking K columns from 2 could take linear combo 25 .
,
example : 3 122) "X
X = 2B + 0
.
=
= X =
zB =
P2X(kxk)
i. e .
X as instruments
use
these instruments are relevant as they're correlated with X & they're
valid since :
plim Xe QQO
plim
XL (22) "I =
= =
* two stage least squares
the IV estimator can be computed by means of 2 successive regressions
1st stage
:
perform regression for each column in X on 2 ,
with fitted values
X =
PzX ,
save the predictions
2nd stage perform regression y on X y XB 2 of
*
: : =
+
the resulting estimate is given by :
Biss (xX)" * y (XP2X)"XPzy ( ** ) Yy
"
=
= =
Biscs N (B 0 XP2X1")
*
-
,
