07 Instrumental Variable Estimation

8/19/2019 07 Instrumental Variable Estimation

1/47

Instrumental variable

estimation

Amine Ouazad

Ass. Professor of Economics


2/47

Problemo

•

OLS is plagued by the problem ofomitted variables… – It is not a testable assumption.

remember the e!ercise"#

• An instrumental variable cancircumvent the problem by providingus $ith an %e!ogenous& source of

variation of the covariate. – A variable that provides us $ith

variation almost as good as a natural

e!periment' … $ithout randomization.


3/47

(andomization is nice) but…

• *ostly + time consuming.

• Ethical issues.

•

Individuals,-irms may not $ant toparticipate.

• Only provides us $ith an estimate

valid for our particular dataset.


4/47

Instrumental variables

• Provide %natural e!periment& fromthe comfort of your oce.

• /he e!ogeneity of the variationneeds to be argued) cannot beproven statistically.

• *an solve the endogeneity problemfor samples that have already beencollected.

Observational data vs experimental


5/47

Outline

0. An e!ample1. Instrumental variable estimation

2. Implementation

3. /he 4ausman test forthe e5uality of OLS and I6

7. Instrumental variable estimation

in small samples8. Acemoglu) 9ohnson) (obinson


6/47

0. An E!ample


7/47

1. One covariate• In the regression y :α;β!;e) the covariate ! is

endogenous) i.e. does not satisfy A2) and*ove)!# is nonzero.

• /he variable z is an instrument if< – It predicts !< *ovz)!# is non zero.

– It is e!ogenous< *ovz)e#:=.

• /he I6 estimator is then< – : *ovz)y#,*ovz)!#

$here the covz)y# is the covariance in the sample.

• >otice that if z : ! then the I6 estimator is theOLS estimator.

•


8/47

1 stage least s5uaresinterpretation

• 1@stage least s5uares interpretation< – 0st stage< ! : γ ;δ z ; υ.

– 1nd stage< y : α ; β ! ; ε.

• 1st stage: regress ! on z) and predict !) so thatthe prediction is .

• 2nd stage: regress y on the prediction .

• Each stage is an OLS regression.

• /he coecient of in the second stage is the I6estimator of β.

•


9/47

acB to the e!ample


10/47

(educed form regression

• /he OLS regression of the dependentvariable on the instrument. – y : π ; ϕ z ; u.

• z is e!ogenous.

• >ote that ϕ = βδ . /he reduced formeCect combines the Drst and thesecond stage eCects.

/ ,* l


11/47

/reatment,*ontrolInterpretation

•

Assume that the interest is in looBing at thecausal eCect of a variable !) and a treatment andcontrol group have been set up) but thecompliance of subects is imperfect.

• ! : γ ;δ F ; υ .

• F is a dummy for the treatment group) $hichaCects !.

• /hen the I6 estimator : *ovF)y#,*ovF)!#estimates the eCect of ! on y.

• >otice that <

• /his is called the Wald estimator.

•


12/47

1. Gultiple covariates

• *onsider the regression H : β ; ε.

•

And $e have a vector of instrumentsJ.

• -or the time being) $e assume thateach endogenous variable hase!actly one instrument.

• /he e!ogenous variables in ! areinstrumented by themselves) i.e.they are in the matri! J.


13/47

/$o conditions

• /he instruments predict thecovariates<

plim 0,># JK is nonzero

or EJK# is of full ranB.

•

/he instruments are e!ogenous<plim 0,># JKe is zero

or EJKe# is zero


14/47

*ausal graph

• Another notation for the t$o previous conditions.

H J

ε

Fependent variable

Endogenous covariate

Instrument

nobservables


15/47

I6 Estimator $ith multiplecovariates

•

/hen the I6 estimator is :JK#@0

JKH.

• >otice that it is e5uivalent to the

1SLS regression<0. /he prediction of the Drst stage

regression :JKJ#@0JK.

1. /he regression of H on the Drst stage

regression : K #@0KH.

• E!ercise< Sho$ this is e5ual to the I6

estimator at the top of this slide.

•


16/47

Mhat if the number of instruments L isdiCerent than the number of

covariates N"

• L number of covariates N)model is underidentifed.

• L : number of covariates N)model is exactly identifed.

• L number of covariates N)

model is overidentifed.

• Mhy do $e use these names"


17/47

1 SLS $ith L diC. than N

0. (egress on the vector J.

1. (egress H on the predictions of .

•. >otice this fails $henever LN)because predictions $ill be linearlydependent A1 fails#.

•. ut no problem if L:N.


18/47

2. Implementation

• StataKs ivreg command< – ivreg y ! : z# $

•

! < endogenous variables• $ < e!ogenous variables

• z < instruments – /here should be at least as many variables in z as in

!.

• Allo$s all the clustering,heteroscedasticityoptions as in OLS.

• Standard errors correct.


19/47

/ricBy Questions

Can I predict x using z only?

• 6ariables $ $ill be used in the Drststage'

• /hey are assumed e!ogenous) sothey are used to predict !.

• Strange cases happen.

• ut if $ is good for the second stage)$ is good for the Drst stage. It isecient to use the variables in $.


20/47

/$o stage regression

• regress x1 z w and predict x1p , xb

• …

•

regress xK’ z w and predict xK’p, xb – for each endogenous variable

• And regress y x1p … xK’p w – gives the I6 estimates.

• ut… the standard errors areincorrect.


21/47

3. Standard errors

• Standard errors in Instrumentalvariable regression are typicallylarger than in OLS.

• -ormula<

6ar#:JK#@0JK6arε#JKJ#@0.

• /he Sand$ich formula depends on6arε#.

» Gore interestingly… ne!t page#

•


22/47

Standard error $ith onecovariate

• /he strongest the correlationbet$een J and ) the smaller theconDdence interval.

• MeaBly correlated instruments givelarge s.e.<

• /he OLS standard error is infated by the correlation bet$een theinstrument and the covariates.

:


23/47

Standard errors$ith multiple covariates

• Mith multiple covariates) the instrument is strong ifthe -@statistic of the Drst stage is high. /heinstrument is weak other$ise.

Advanced• A little issue is that the -@stat of the Drst@stage

regression includes the e!ogenous covariates as$ell…

• 4ence it is possible to get a high -@stat but nosigniDcant instrument in the Drst stage regression.

• Solution< use ivreg1 and the Angrist@PischBe -@statdisplayed in the output#.


24/47

7. 4ausman test

• /his test compares the OLS estimator and the I6 estimator.

• /he null hypothesis is that the OLS estimator is e5ual to the I6 estimator.

• 4ausman test statistic<

4:#K6ar##@0#

• And asymptotically) under the null hypothesis) this converges to a chi@s5uare distribution) $ith number of degrees of freedom e5ual to the ranBof the variance@covariance matri!.

• In Stata< – ivreg y ! : z# $

– estimates store ivresults – regress y ! $

– estimates store olsresults

– hausman ivresults olsresults

•


25/47

(ight approach to the 4ausman test

• /he 4ausman test may sho$ thatyour use of the I6 estimator hassigniDcantly aCected the point

estimate of the eCect of yourcovariate.

• If you cannot reect the null) the OLS

$as as good as the I6 strategy.


26/47

Gisconceptionsabout the 4ausman test

• /he 4ausman test is sometimes called a test of%e!ogeneity.& ut this is $rong.

• Indeed) the I6 estimator is valid only if theinstruments are e!ogenous.

• /he OLS estimator is valid if the covariates aree!ogenous.

• If the null is reected) then either i# the instrumentsare endogenous and the covariates are endogenous

or ii# the instruments are e!ogenous and thecovariates are e!ogenous or iii# the instruments areendogenous and the covariates are e!ogenous.


27/47

8. I6 estimation in smallsamples

• /he I6 estimator is biased.

• Indeed<

ERJ)# : β ; EJK#@0

JKEεRJ)##• And EεRJ)# is nonzero ' Other$ise

$ould be e!ogenous…

• So $e have a problem. In Dnitesamples) the bias of I6 can be large '

•


28/47

Staiger and StocB 0T#

• Sho$ using simulations that the ma!imal bias in I6 isno more than 0=U of OLS $e need -0=.

• Ga!imal bias in I6 is no more than 1=U of that ofOLS)

$e need -8.7.

Advanced considerations ! "ated#

• /he distribution of the I6 estimator is Mishart)

assuming the residuals are normally distributed.• /he Dnite sample mean of I6 does not e!ist $ith a

number of instruments e5ual to the number ofcovariates.#


29/47

T. Acemoglu) 9ohnson and (obinson


30/47


31/47

*ausal graph

• Fra$ the causal graph using the abstract.


32/47


33/47


34/47


35/47


36/47

*ausal reasoning


37/47


38/47


39/47

W. Fo $orBers accept lo$er $ages ine!change for health beneDts"

• *raig Olson) 9ournal of Labor Economics) 1==1.

• Compensating wage theory predicts that workers receiving

more generous ringe bene!ts are paid a lower wage thancomparable workers who preer ewer ringe bene!ts" #hisstudy tests this prediction or employer$provided healthinsurance by modeling the wages o married womenemployed ull$time in the labor market" %usband&s unionstatus, husband&s !rm size, and husband&s health coveragethrough his 'ob are used as instruments or his wie&s ownemployer health insurance bene!ts" #he estimates suggestwives with own employer health insurance accept a wageabout ()* lower than what they would have receivedworking in a 'ob without bene!ts"


40/47

*ausal reasoning

• Mrite do$n the causal graph.


41/47

SpeciDcations

• OLS %>aXve& regression<

•

Problem" /he eCect is typically positive) $hich isunliBely to be causal.

• -irst stage regression<

• Alternative Drst stage regression<


42/47

Fataset

• III$ %&e 'ata and t&e (irst)*tage +stimates

• /he data used in this study are from the GarchY9une 0=Y2 *urrentPopulation Surveys *PS#. /he Garch *PSs include 5uestions on employerZprovided health insurance and Drm size. nion status and $age data are asBedeach month of respondents in the outgoing rotations group O(V# subsamples.

/herefore) the data $ere constructed by merging the Garch *PS $ith the O(V

subsamples for April) Gay) and 9une for each of the 3 years. (espondents ineach Garch survey in rotation groups 0) 1) and 2 $ere matched $ith the O(VDles for) respectively) 9une) Gay) and April. Garch respondents in rotationsgroups 3 and W $ere also included because they $ere asBed the unionizationand $age 5uestions in Garch. /hese merged GarchY9une Dles $ere then splitby gender and marital status and merged bacB together by householdidentiDers to produce a single record for each married couple. /he Dles for the

3 years $ere then pooled and the analysis restricted to households $here boththe husband and $ife $ere employed. /he sample $as then restricted tocouples $here the $ife $as employed full time 23 hours a $eeB# and had anhourly $age greater than or e5ual to [1.== an hour. /hese criteria produced asample of 11)221 households.


43/47


44/47

(educed form estimate


45/47

OLS and I6 regression


46/47

C,-C./*I,-


47/47

sing instrumental variables

• Mhenever you believe there is an omittedvariable bias….0. -irst try to assess the direction of the bias in OLS.

1. /hen try to Dnd an appropriate I6 estimator.

2. se ivreg.• FonKt forget clustering) heteroscedasticity.

3. /est $hether the OLS is diCerent from I6) and in $hatdirection. *onsistent $ith your initial interpretation"

7. (eport Drst stage) reduced form) 1SLS.8. Is the instrument $eaB" Is the sample size small"

• MeaB instruments) see for instance ascle 1==W#in +trategic Organization, vol , p(-..

07 Instrumental Variable Estimation

Documents

Transcript of 07 Instrumental Variable Estimation