Econometric Methods with Applications in Business and Economics
This comprehensive handwritten summary covers key topics and concepts from lectures and the textbook "Econometrics Methods with Applications." The notes include detailed explanations on endogeneity and instrumental variables, generalized method of moments, maximum likelihood estimation, binary resp...
Test Bank for Introductory Econometrics A Modern Approach, 5th Edition by Jeffrey M. Wooldridge
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Universiteit van Amsterdam (UvA)
Econometrics & Data Science
6012B0378Y
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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
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
+
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")
*
-
,
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