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Econometrics 2 Summary
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  • Grado
  • 6012B0378Y
  • Institución
  • Universiteit Van Amsterdam (UvA)
  • Book
  • 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...

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  • No
  • Chapters 4, 5 & 6
  • 1 de junio de 2024
  • 1 de junio de 2024
  • 26
  • 2023/2024
  • Resumen

Temas

  • econometrics

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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


*
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")
*
-

,

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