lecture26 ivs
TRANSCRIPT
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Instrumental Variables & 2SLS
y = 0+ 1x1+ 2x2+ . . . kxk+ u
x1= 0+ 1z + 2x2+ . . . kxk+ v
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Why Use Instrumental Variables?
Instrumental Variables (IV) estimation is
used when your model has endogenousxs
That is whene!er "o!(x,u) # $
Thus IV %an be used to address the
roblem o' omitted !ariable bias
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mitted Variable ias (re%a)
( ) *2****$
22**$
++
then+++estimate
but we
asgi!enismodeltruetheSuose
+=
++=
+++=
E
uxy
uxxy
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Summary o' ,ire%tion o' ias
(re%a)"orr(x1, x2) - $ "orr(x1, x2) . $
2- $ /ositi!e bias 0egati!e bias
2. $ 0egati!e bias /ositi!e bias
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/ro1y Variables
What i' model is misse%i'ied be%ause nodata is a!ailable on an imortantx!ariable?
It may be ossible to a!oid omitted!ariable bias by using a ro1y !ariable
ro1y !ariable must be related to the
unobser!able !ariableWhat i' a suitable ro1y !ariable is also nota!ailable? 3ay use IV or 2SLS roa%h4
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What Is an Instrumental Variable?
In order 'or a !ariablez to ser!e as a !alid
instrument 'orx the 'ollowing must be true
The instrument must be e1ogenous
That is "o!(z,u) 5 $
The instrument must be %orrelated with the
endogenous !ariablex
That is "o!(z,x) # $
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3ore on Valid Instruments
We ha!e to use %ommon sense and
e%onomi% theory to de%ide i' it ma6es sense
to assume "o!(z,u) 5 $We %an test i' "o!(z,x) # $
7ust testing 8$9 15 $ inx = 0+ 1z + v
Sometimes re'er to this regression as the
'irst:stage regression
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IV ;stimation in the Simle
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The ;''e%t o' /oor Instruments
What i' our assumtion that "o!(z,u) 5 $ is 'alse?
The IV estimator will be in%onsistent too
"an %omare asymtoti% bias in LS and IV/re'er IV i' "orr(z,u)"orr(z,x) . "orr(x,u)
x
u
x
u
uxCorr
xzCorr
uzCorr
+=
+=
)(+&lim9LS
)(
)(@&lim9IV
**
**
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In'eren%e with IV ;stimation
The homos6edasti%ity assumtion in this %ase is
;(u2|z) 5 25 Var(u)
s in the LS %ase gi!en the asymtoti%!arian%e we %an estimate the standard error
( )2*
2
2
* )
@
(@zxxRSST
se
=
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IV !ersus LS estimation
Standard error in IV %ase di''ers 'rom LS
only in theR2'rom regressingxonz
Sin%eR2. * IV standard errors are larger
8owe!er IV is %onsistent while LS is
in%onsistent when "o!(x,u) # $
The stronger the %orrelation betweenzand
x the smaller the IV standard errors
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IV ;stimation in the 3ultile
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3ultile
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Two Stage Least SAuares (2SLS)
Its ossible to ha!e multile instruments
"onsider our original stru%tural model and
lety2= 0+ 1z1+ 2z2+ 3z3+ v2
8ere were assuming that bothz2andz3are
!alid instruments B they do not aear in
the stru%tural model and are un%orrelatedwith the stru%tural error term u1
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est Instrument
"ould use eitherz2orz3as an instrument
The best instrument is a linear %ombination
o' all o' the e1ogenous !ariablesy2* = 0+1z1+ 2z2+ 3z3
We %an estimatey2* by regressingy2onz1,
z2andz3B %an %all this the 'irst stageI' then substitute2'ory2in the stru%tural
model get same %oe''i%ient as IV
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Testing 'or ;ndogeneity
Sin%e LS is re'erred to IV i' we do not
ha!e an endogeneity roblem then wed
li6e to be able to test 'or endogeneityI' we do not ha!e endogeneity both LS
and IV are %onsistent
Idea o' 8ausman test is to see i' theestimates 'rom LS and IV are di''erent
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Testing 'or ;ndogeneity (%ont)
While its a good idea to see i' IV and LS
ha!e di''erent imli%ations its easier to use
a regression test 'or endogeneityI'y2is endogenous then v2('rom the
redu%ed 'orm eAuation) and u1'rom the
stru%tural model will be %orrelatedThe test is based on this obser!ation
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Testing 'or ;ndogeneity (%ont)
Sa!e the residuals 'rom the 'irst stage
In%lude the residual in the stru%tural
eAuation (whi%h o' %ourse hasy2in it)I' the %oe''i%ient on the residual isstatisti%ally di''erent 'rom Cero reDe%t the
null o' e1ogeneityI' multile endogenous !ariables Dointlytest the residuals 'rom ea%h 'irst stage
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Testing !eridenti'ying
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The !erI, Test
;stimate the stru%tural model using IV and
obtain the residuals