Lecture26 IVs

8/11/2019 Lecture26 IVs

1/21

Instrumental Variables & 2SLS

y = 0+ 1x1+ 2x2+ . . . kxk+ u

x1= 0+ 1z + 2x2+ . . . kxk+ v


2/21

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


3/21

mitted Variable ias (re%a)

( ) *2****$

22**$

++

then+++estimate

but we

asgi!enismodeltruetheSuose

+=

++=

+++=

E

uxy

uxxy


4/21

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


5/21

/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


6/21

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) # $


7/21

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


8/21

IV ;stimation in the Simle


9/21

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

**

**


10/21

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

=


11/21

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


12/21

IV ;stimation in the 3ultile


13/21

3ultile


14/21

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


15/21

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


16/21


17/21

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


18/21

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


19/21

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


20/21

Testing !eridenti'ying


21/21

The !erI, Test

;stimate the stru%tural model using IV and

obtain the residuals

Lecture26 IVs

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