1 Instrumental Variables Estimation and 2SLS

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1 1 Instrumental Variables Estimation and 2SLS Consider the following wage equation: log(wage)=β 0 +β 1 educ+β 2 ability+e. We would expect that more able people on average earn more (β 2 > 0), however, ability is difficult to measure directly so it may have to be omitted. Estimating the equation without ability puts the latter in the error term, whichwouldbeacompositeofeandability: log(wage)=β 0 +β 1 educ+u. If educ and ability are uncorrelated we can get an unbiased and consistent estimatorofβ 1 fromols,butiftheyarecorrelated(whichislikely)we willnotgetagoodestimatorofβ 1 fromthismodel. Ingeneral,inthesimplelinearmodel y=β 0 +β 1 x+u, anols-estimatorofβ 1 isderivedfromasampleofnobservationas, n i=1 ˆβ 1 = (x i x)(y i ȳ) n i=1 (x i x) 2, orifwedividethenumeratoranddenominatorbyn, ˆβ 1 = Cov(x,y) Var(x). Now, if we substitute the linear equation for each observation of y the covariance term becomes, this can be re-written as, Cov(x,y)=Cov(x,β 0 +β 1 x+u), Cov(x,β 0 +β 1 x+u)=cov(x,β 0 )+Cov(x,β 1 x)+cov(x,u). Sinceβ 0 isaconstantthefirstcovariancetermontherighthandsideis equaltozero,thesecondequalsβ 1 Cov(x,x)=β 1 Var(x),wecantherefore write,anestimatorofβ 1 isdefinedas: ˆβ 1 = Cov(x,y) Var(x) =β 1 + Cov(x,u) Var(x). ifcov(x,u)=0,itfollowsthat ˆβ 1 =β 1,or ˆβ 1 isunbiased. 1

2 IfCov(x,u) 0,OLSwillnotgiveanunbiasedandconsistentestimator ofβ 1. However,assumethatwehaveanothervariable,z,whichisuncorrelatedwithubutcorrelatedwithx(Cov(z,u)=0,Cov(z,x) 0),itis possibletoestimateβ 1 byusingzinsteadofx.ziscalledaninstrumental variable (IV-variable). In terms of population covariances we can write: Cov(z,y)=β 1 Cov(z,x)+Cov(x,u) Now,ifCov(x,u)=0wecansolveforβ 1 β 1 = Cov(z,y) Cov(z,x). Given a random sample (of size n) we get the following IV-estimator of β 1 : n i=1 ˆβ 1 = (z i z)(y i ȳ) n i=1 (z i z) 2, Inthecaseofthewageequation,aninstrumentalvariableforeducmust be (1) uncorrelated with ability and (2) correlated with education. An example could be mother s education (motheduc) or father s education (fatheduc),butthatcouldbeapooriv-variableifitiscorrelatedwith ability. Another example is the number of siblings(sibs), which usually is(negatively) correlated with education, but probably uncorrelated with ability. 1.1 IV Estimation of the Multiple Regression Model Consider the following model with two explanatory variables: y 1 =β 0 +β 1 y 2 +β 2 z 1 +u 1, here we assume that y 1 and y 2 are both endogenous variables, why z 1 is exogenous. This means that Cov(y 1,u) 0, Cov(y 2,u) 0, but Cov(z,u)=0. For example consider the following wage equation: log(wage)=β 0 +β 1 educ+β 2 exper+u 1, where exper is assumed to be exogenous. As before, we need an IVvariable for educ. Note that exper cannot be used as an instrument for educ since it is itself included in the model, we need another exogenous variable(z 2 )whichdoesnotalreadyappearintheequation. The keyassumptionsare thatz 1 andz 2 bothuncorrelatedwithu 1,and thatz 2 iscorrelatedwithy 2. 2

3 An equation which contains one or more endogenous variables as explanatory variables is called a structural equation, while an equation where the endogenous variables have been replaced by exogenous instruments is called a reduced form equation: 1.2 Two Stage Least Squares y 1 =π 0 +π 1 z 1 +π 2 z 2 +ν 1. Let s go back to the structural equation: y 1 =β 0 +β 1 y 2 +β 2 z 1 +u 1, butsupposethattherearetwoexogenousvariablesexcluded,z 2 andz 3. Ifbothfulfilltherequirementsforproperinstrumentsfory 2 wecoulduse either. However,wewouldhavetwodifferentIV-estimatorsofβ 1. Sinceeachofz 1,z 2 andz 3 areuncorrelatedwithu 1,anylinearcombination ofthemisalsouncorrelatedwithu 1 andwouldbeavalidiv. TofindthebestIV,weshouldchoosethelinearcombinationthatismost highlycorrelatedwithy 2.Thisisgivenbythereducedformequationfor y 2 : y 2 =π 0 +π 1 z 1 +π 2 z 2 +π 3 z 3 +ν 2, wheree(ν 2 )=0,Cov(z 1,ν 2 )=0,Cov(z 2,ν 2 )=0andCov(z 3,ν 2 )=0. ThebestIV fory 2 isthen: y 2=π 0 +π 1 z 1 +π 2 z 2 +π 3 z 3. ForthisIV nottobeperfectlycorrelatedwithz 1 weneedatleastoneof π 2 orπ 3 tobedifferentfromzero: π 2 0orπ 3 0. Thisisthekeyidentificationassumption,onceweassumethatthez i :sare all exogenous. Thestructuralequationisnotidentifiedif π 2 =0andπ 3 =0. Thiscan betestedwithanf-test. (H 0 =π 2 =0andπ 3 =0.) y 2 can be estimated by an OLS regression of y 2 on z 1, z 2 and z 3, and obtain the fitted values: ŷ 2 =ˆπ 0 +ˆπ 1 z 1 +ˆπ 2 z 2 +ˆπ 3 z 3. Withmultipleinstruments,theIV-estimatorusingŷ 2 astheivfory 2,is called the two stage least squares(2sls) estimator. 3

4 As the term indicates the estimator is obtained in two stages, or steps. Firstweobtainedthefittedvalues(ŷ 2 )andthenestimatethestructural model: y 1 =β 0 +β 1 ŷ 2 +β 2 z 1 +u 1. Consider the following wage equation: log(wage)=β 0 +β 1 educ+β 2 exper+β 3 exper 2 +u 1, here we have two exogenous variables and one endogenous(educ) as explantory variables. If we assume that both mother s and father s education (fatheduc)areuncorrelatedwithu 1,bothcanbeusedasinstrumentsfor educ.thereducedformequationforeducisnow: educ = π 0 +π 1 exper+π 2 exper 2 + π 3 motheduc+π 4 fatheduc+ν 2. Identificationrequiresthatπ 3 0orπ 4 0,orboth. 2 Simultaenous Equations Models Consider the following labor supply function: h s =α 1 w+β 1 z 1 +u 1, wherez 1 issomeobservedvariableaffectinglaborsupply; allothersuch factors are included in the error term. The above equation is a structural equation which is derivable from economictheory; thecoefficientα 1 measureshowlaborsuppliedchangesas thewageratechanges(economictheorydoesnotsuggestthesignofα 1, however). If we could run controlled experiments there we held z 1 constant and changedw, we couldestimate α 1. But itis notfeasible toconductsuch experiments in practice, but we must collect actual data over wages, z 1 andh s.ifthelabormarket clears theobservedvaluesofwandh s are equilibrium values. Todescribehowequilibriumwagesandhoursaredeterminedweneedto bringinthedemandforlabor: h d =α 2 w+β 2 z 2 +u 2, herez 2 isavariable whichshiftsthedemandcurve(asdoesu 2,butthe error terms includes unobservable variables). The two equations are two structural equations that describes two different phenomena,buttheyarerelatedinthatthewagerateisdeterminedsuch thath is =h id,whereiisdifferentlabormarketregion. 4

5 Recognizing the relationship between the two equations we now write the following simultaneous equations model(sem): h i = α 1 w i +β 1 z i1 +u i1, h i = α 2 w i +β 2 z i2 +u i2. h i andw i are the two endogenous variables whichare tobe determined bythemodelandz i1 andz i2,arethetwoexogenousvariableswhichare determined outside the model and which are necessary to identify the model. Withoutthemwewillnotknowwhichisthedemandandwhich is the supply equation. Thekeystatisticalassumptionconcerningz i1 andz i2 isthattheyareboth uncorrelatedwiththeerrorsu i1 andu i2,respectively. 2.1 Simultaneity Bias in OLS We now consider a general two-equation structural model: y 1 = α 1 y 2 +β 1 z 1 +u 1 y 2 = α 2 y 1 +β 1 z 2 +u 2 (the intercepts are suppressed for simplicty). Estimating the first equation only will not give us unbiased estimators usingols,becausey 2 isgenerallycorrelatedwithu 1. Toseethissubstitutethesecondequationintothefirst: y 1 = α 1 (α 2 y 1 +β 1 z 2 +u 2 )+β 1 z 1 +u 1 (1 α 1 α 2 )y 1 = β 1 z 1 +α 1 β 1 z 2 +u 1 +α 1 u 2. Ifα 1 α 2 1wecansolvey 1,bydividingeachtermoftheright-handside by(1 α 1 α 2 ). We obtain the following reduced form equation: whereπ 11 = y 1 =π 11 z 1 +π 12 z 2 +ν 1, β 1 (1 α 1α 2),π 12= α1β 1 (1 α 1α 2),andν 1= u1+α1u2 (1 α 1α 2). Thereducedformerrortermisuncorrelatedwithz 1 andz 2,andwecan estimateπ 11 andπ 12 byols. Note that ν 1 is a linear function of u 1 and u 2, therefore y 2 is generally correlatedwithν 1,giventhatα 1 0. Wheny 2 iscorrelatedwithu 1,becauseofsimultaneity,wesaythatOLS suffers from simultaneity bias. 5

6 2.2 Identification in a Two-equation System Consider the following simple two-equation SEM: q = α 1 p+β 1 z 1 +u 1 q = α 2 p+u 2. q is the equilibrium quantity supplied and demanded and p is the equilibriumprice. Assumethatthefirstequationisthesupplyequationandthe second the demand equation. Economictheorysuggeststhatα 1 0andα 2 0,howeverwecanonly identifythedemandequation(estimateα 2 ).z 1 isa supplyshifter and as it shifts it traces out the demand equation. But there is no demand shifter so we cannot identify the supply equation. In order to identify both equations there must be exogenous variables in secondequationwhichdoesnotappearinthefirst,andviceversa. InaSEMwithmorethantwostructuralequations,thenumberofexcluded exogenous variables must be greater than the number of endogenous variables included on the right-hand side of an equation. This is called the rank condition. Considerthemodeloflaborsupplyofmarriedwomen: hours = α 1 log(wage)+β 10 +β 11 educ+β 12 age +β 13 kidslt6+β 14 nwifeinc+u 1. log(wage) = α 2 hours+β 20 +β 21 educ+β 22 exper +β 23 exper 2 +u 1. Here we assume that all variables a part from hours and wage are exogenous(even educ). The first equation is the labor supply equation, it is identifiedbecausetwoexogenousvariablesexper andexper 2 areexcluded. If β 22 = 0 and β 23 = 0, there are no exogenous variable in the second equationthatdonotappearinthefirst;therankconditionisnotsatisfied. Byderivingthereducedformequationofthesystem: log(wage) = π 20 +π 21 educ+π 22 age +π 23 kidslt6 +π 24 nwifeinc+π 25 exper+π 26 exper 2 +ν 2, wecantestwhetherπ 25 0andπ 26 0,byanF-test. The second equation(the wage-offer equation) is identified if at least on ofage,kidslt6,ornwifeinchasanon-zerocoefficient(β 12 0,β 13 0, and/orβ 14 0). 6

7 OnceanequationinageneralSEMhasbeenshowntobeidentifieditcan be estimated by 2SLS. The instruments for a particular equation consists of the exogenous variables appearing anywhere in the system. 7