计量经济学工具变量IV-(2SLS).ppt
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1、Week14InstrumentVariableRegressionModelsSimultaneous Equation Using 2SLS(Chapter 16),IV Estimation in Multiple Regression models(15.1-3)计量经济学(研究生)计量经济学(研究生)ANewApproachtotheOmittedVariableProblemnWe have talked about the problem of omitted variable bias(in Ch.3),and have shown that it will lead to i
2、nconsistency,for nIf we have a suitable proxy,we can minimize the bias,to some degree.(see Chapter 9)nFurthermore,if the omitted variable is time invariant,then we can use a panel data model without much hesitation.nWithout a suitable proxy,no panel data,or if the omitted variable does change with t
3、ime we need a new approachInstrumentalVariablesRegressionnThree important threats to internal validity are:1.omitted variable bias from a variable that is correlated with X but is unobserved,so cannot be included in the regression;(遗留变量偏差)2.simultaneous causality bias(X causes Y,Y causes X);(联立因果)3.
4、errors-in-variables bias(X is measured with error)(变量误差)nInstrumental variables regression can eliminate bias from these three sources.Terminology:endogeneityandexogeneitynAn endogenous variable is one that is correlated with u.nAn exogenous variable is one that is uncorrelated with u.nHistorical no
5、te:“Endogenous”literally means“determined within the system,”that is,a variable that is jointly determined with y.nIn other words,it is a variable subject to simultaneous causality.nHowever,this definition is narrow and IV regression can be used to address OV bias and errors-in-variable bias,not jus
6、t to simultaneous causality bias.What is Simultaneous CausalitynSuppose we have two endogenous variables Y1,Y2 and two exogenous variables X1,X2 such that Y1i=0+1X1i+2Y2i+u1i(1)Y2i=0+1Y1i+2X2i+u2i(2)nLets see why Y2(or Y1)is endogenousnSuppose u1i 0 and u2i=0,then we have Y1i E(Y1i)from(1)nBut in(2)
7、,if 20,this will cause a change in Y2i,so Y2i is correlated with u1i through(2)nThe same is true for Y1i and u2i in(2)through(1)Simultaneous BiasCan we estimate these two equations consistently?y1=a1y2+1z1+u1 y2=a2y1+2z2+u2For consistency,we need cov(y2,u1)=0,and cov(y1,u2)=0However,a large u2 means
8、 a larger y2,which implies a larger y1(if a10),so cov(y1,u2)0The same is true for cov(y2,u1)due to the circular effect of u1TheIVEstimatorwithaSingleRegressorandaSingleInstrumentyi=0+1xi+uinLoosely,IV regression breaks x into two parts:a part that might be correlated with u,and a part that is not.nB
9、y isolating the part that is not correlated with u,it is possible to estimate 1.nThis is done using an instrumental variable,zi,which is uncorrelated with ui.nThe instrumental variable detects movements in xi that are uncorrelated with ui,and use these to estimate 1.Twoconditionsforavalidinstrumenty
10、i=0+1xi+uinFor an instrumental variable(an“instrument”)z to be valid,it must satisfy two conditions:1.Instrument relevance:cov(zi,xi)02.Instrument exogeneity:cov(zi,ui)=0nIn other words,IV variable zi must be an exogenous variable that is correlated with x nOr,zi s effect on y is only through xnWhic
11、h condition can we test?A)1 B)2 C)BothD)Neither E)Dont knownWe can test the 1st but have to assume the 2ndExample:Labor EconomicsSuppose log(wage)=0+1educ+u,u=2abil+vnWhen abil is unobserved,how can we estimate 1 consistently if cov(educ,abil)0?nIf we have a proxy for abil,such as IQ and substitute
12、it into our model,then we are finenOtherwise,we need something that is correlated with educ but not with abilnParents education,or number of siblings might be an instrument for educ nSuppose we have:yi=0+1xi+uicov(x,ui)0nOur estimate of 1 will be inconsistent nEither we find the omitted variable in
13、ui and add it into our model to overcome the inconsistencynOr we find an instrument zi for the included variablenSuppose for now that you have such a zi(well discuss how to find instrumental variables later)nHow can you use zi to estimate 1?nWe will explain this in two waysInstrument Variable Regres
14、sionTheIVEstimator,onexandonezExplanation#1:Two Stage Least Squares(TSLS)nAs it sounds,TSLS has two stages two regressions:(1)First isolates the part of x that is uncorrelated with u:regress x on z using OLSxi=0+1zi+vi(1)nBecause zi is uncorrelated with ui,0+1zi is uncorrelated with ui.nWe dont know
15、 0 or 1 but we have estimated them,sonCompute the predicted values of xi,xi,where xi=0+1 zi,i=1,n.(2)Replace xi by xi in the regression of interest:regress y on xi using OLS:yi=0+1 xi+ui(2)nBecause xi is uncorrelated with ui in large samples,so the first least squares assumption holdsnThus 1 can be
16、estimated by OLS using regression(2)nThis argument relies on large samples(so 0 and 1 are well estimated using regression(1)nThis the resulting estimator is called the“Two Stage Least Squares”(TSLS)estimator,.TheIVEstimator,onexandonez,ctd.nExplanation#2:(only)a little algebrayi=0+1xi+uiButxi=0+1zi+
17、vinThus,cov(yi,zi)=cov(0+1xi+ui,zi)=cov(0,zi)+cov(1xi,zi)+cov(ui,zi)=0 +cov(1xi,zi)+0=1cov(xi,zi)nwhere cov(ui,zi)=0(instrument exogeneity);thus1=in large samplesnThe instrument relevance condition,cov(x,z)0,ensures that you dont divide by zero.Supply and Demand Examplen Start with an equation youd
18、like to estimate,say a supply function in a market.qs=a1p+b1z+u1,where p is the price and z is a supply shifter.n Call this a structural equation its derived from economic theory and has a causal interpretation where p directly affects qs.Example(cont)nProblem that cant just regress observed quantit
19、y on price,since observed quantity are determined by the equilibrium of supply and demandnConsider a second structural equation,in this case the demand function qd=a2p+u2nSo quantity are determined by a SEMExample(cont)nBoth q and p are endogenous because they are both determined by the equilibrium
20、of supply and demandnz is exogenous,and its the availability of this exogenous supply shifter that allows us to identify the structural demand equationnWith no observed demand shifters,supply is not identified and cannot be estimatedIdentification of Demand EquationpqDS(z=z1)S(z=z2)S(z=z3)Using IV t
21、o Estimate DemandnGiven qs=a1p+b1z+u1,qd=a2p+u2nSo,we can estimate the structural demand equation,using z as an instrument for pn First stage equation is p=0+1z+v2n Second stage equation is q=a2p+u2n Thus,2SLS provides a consistent estimator of a2,the slope of the demand curven We cannot estimate a1
22、,the slope of the supply curveThe General SEMnSuppose our structural equations are:y1=a1y2+1z1+u1y2=a2y1+2z2+u2nThus,y2=a2(a1y2+1z1+u1)+2z2+u2nSo,(1 a2a1)y2=a2 1z1+2z2+a2 u1+u2,which can be rewritten(if a2a1 1)as y2=1z1+2z2+v2 v2=(a2u1+u2)/(1a2a1)nThis is the so called“reduced”formnHowever,in the re
23、duced form,we dont know what is the value of a1 or a2Example#1:SupplyanddemandforbutternIV regression was originally developed to estimate demand elasticities for agricultural goods,for example butter:nlog(Qbutter)=0+1 log(Pbutter)+uin1=price elasticity of butter=percent change in quantity for a 1%c
24、hange in price(recall log-log specification discussion)nData:observations on price and quantity of butter for different yearsnThe OLS regression of log(Qbutter)on log(Pbutter)suffers from simultaneous causality bias(why?)Simultaneous causality bias in the OLS regression of log(Qbutter)on log(Pbutter
25、)arises because price and quantity are determined by the interaction of demand and supplyAsidenote:Whatistherelationshipbetween,sayMarxianconceptoflabor theory of valueandtheMicroeconomicstheoryofprice formation?Whatisthelong-termsupplycurveanditsdetermination?A Quick Note on Marxian EconomicsnAtQ1,
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