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1、Week 14Instrument Variable Regression Models,Simultaneous Equation Using 2SLS(Chapter 16),IV Estimation in Multiple Regression models(15.1-3),计量经济学（研究生）,A New Approach to the Omitted Variable Problem,We have talked about the problem of omitted variable bias(in Ch.3),and have shown that it will lead
2、to inconsistency,for If we have a suitable proxy,we can minimize the bias,to some degree.(see Chapter 9)Furthermore,if the omitted variable is time invariant,then we can use a panel data model without much hesitation.Without a suitable proxy,no panel data,or if the omitted variable does change with
3、time we need a new approach,Instrumental Variables Regression,Three important threats to internal validity are: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)simultaneous causality bias(X causes Y,Y causes X);(联立因果)er
4、rors-in-variables bias(X is measured with error)(变量误差)Instrumental variables regression can eliminate bias from these three sources.,Terminology:endogeneity and exogeneity,An endogenous variable is one that is correlated with u.An exogenous variable is one that is uncorrelated with u.Historical note
5、:“Endogenous”literally means“determined within the system,”that is,a variable that is jointly determined with y.In other words,it is a variable subject to simultaneous causality.However,this definition is narrow and IV regression can be used to address OV bias and errors-in-variable bias,not just to
6、 simultaneous causality bias.,What is Simultaneous Causality,Suppose 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)Lets see why Y2(or Y1)is endogenousSuppose u1i 0 and u2i=0,then we have Y1i E(Y1i)from(1)But in(2),if 20
7、,this will cause a change in Y2i,so Y2i is correlated with u1i through(2)The same is true for Y1i and u2i in(2)through(1),Simultaneous Bias,Can we estimate these two equations consistently?y1=a1y2+b1z1+u1 y2=a2y1+b2z2+u2For consistency,we need cov(y2,u1)=0,and cov(y1,u2)=0However,a large u2 means a
8、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 u1,The IV Estimator with a Single Regressor and a Single Instrument,yi=0+1xi+uiLoosely,IV regression breaks x into two parts:a part that might be correlated with u,and a part that i
9、s not.By isolating the part that is not correlated with u,it is possible to estimate 1.This is done using an instrumental variable,zi,which is uncorrelated with ui.The instrumental variable detects movements in xi that are uncorrelated with ui,and use these to estimate 1.,Two conditions for a valid
10、instrument,yi=0+1xi+uiFor an instrumental variable(an“instrument”)z to be valid,it must satisfy two conditions:Instrument relevance:cov(zi,xi)0Instrument exogeneity:cov(zi,ui)=0In other words,IV variable zi must be an exogenous variable that is correlated with x Or,zi s effect on y is only through x
11、Which condition can we test?A)1 B)2 C)BothD)NeitherE)Dont knowWe can test the 1st but have to assume the 2nd,Example:Labor Economics,Suppose log(wage)=0+1educ+u,u=2abil+vWhen abil is unobserved,how can we estimate 1 consistently if cov(educ,abil)0?If we have a proxy for abil,such as IQ and substitut
12、e it into our model,then we are fineOtherwise,we need something that is correlated with educ but not with abilParents education,or number of siblings might be an instrument for educ,Suppose we have:yi=0+1xi+uicov(x,ui)0Our estimate of 1 will be inconsistent Either we find the omitted variable in ui
13、and add it into our model to overcome the inconsistencyOr we find an instrument zi for the included variableSuppose for now that you have such a zi(well discuss how to find instrumental variables later)How can you use zi to estimate 1?We will explain this in two ways,Instrument Variable Regression,T
14、he IV Estimator,one x and one z,Explanation#1:Two Stage Least Squares(TSLS)As 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)Because zi is uncorrelated with ui,0+1zi is uncorrelated with ui.We dont know
15、0 or 1 but we have estimated them,soCompute 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)Because xi is uncorrelated with ui in large samples,so the first least squares assumption holdsThus 1 can be es
16、timated by OLS using regression(2)This argument relies on large samples(so 0 and 1 are well estimated using regression(1)This the resulting estimator is called the“Two Stage Least Squares”(TSLS)estimator,.,The IV Estimator,one x and one z,ctd.,Explanation#2:(only)a little algebrayi=0+1xi+uiButxi=0+1
17、zi+viThus,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)where cov(ui,zi)=0(instrument exogeneity);thus1=in large samplesThe instrument relevance condition,cov(x,z)0,ensures that you dont divide by zero.,Supply and Demand Example,Start with an equation youd l
18、ike to estimate,say a supply function in a market.qs=a1p+b1z+u1,where p is the price and z is a supply shifter.Call this a structural equation its derived from economic theory and has a causal interpretation where p directly affects qs.,Example(cont),Problem that cant just regress observed quantity
19、on price,since observed quantity are determined by the equilibrium of supply and demandConsider a second structural equation,in this case the demand function qd=a2p+u2So quantity are determined by a SEM,Example(cont),Both q and p are endogenous because they are both determined by the equilibrium of
20、supply and demandz is exogenous,and its the availability of this exogenous supply shifter that allows us to identify the structural demand equationWith no observed demand shifters,supply is not identified and cannot be estimated,Identification of Demand Equation,p,q,D,S(z=z1),S(z=z2),S(z=z3),Using I
21、V to Estimate Demand,Given qs=a1p+b1z+u1,qd=a2p+u2So,we can estimate the structural demand equation,using z as an instrument for p First stage equation is p=p0+p1z+v2 Second stage equation is q=a2p+u2 Thus,2SLS provides a consistent estimator of a2,the slope of the demand curve We cannot estimate a1
22、,the slope of the supply curve,The General SEM,Suppose our structural equations are:y1=a1y2+b1z1+u1y2=a2y1+b2z2+u2Thus,y2=a2(a1y2+b1z1+u1)+b2z2+u2So,(1 a2a1)y2=a2 b1z1+b2z2+a2 u1+u2,which can be rewritten(if a2a1 1)as y2=p1z1+p2z2+v2 v2=(a2u1+u2)/(1a2a1)This is the so called“reduced”formHowever,in t
23、he reduced form,we dont know what is the value of a1 or a2,Example#1:Supply and demand for butter,IV regression was originally developed to estimate demand elasticities for agricultural goods,for example butter:log(Qbutter)=0+1 log(Pbutter)+ui1=price elasticity of butter=percent change in quantity f
24、or a 1%change in price(recall log-log specification discussion)Data:observations on price and quantity of butter for different yearsThe 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(
25、Pbutter)arises because price and quantity are determined by the interaction of demand and supply,A side note:What is the relationship between,say Marxian concept of labor theory of value and the Microeconomics theory of price formation?What is the long-term supply curve and its determination?,A Quic
26、k Note on Marxian Economics,At Q1,the production is less then socially necessary,and is causing a shortageThe competition will drive the price above it value,until more producers enters the market or more product is being producedThis leads to an increase in the level of output,all the way to Q*.At
27、Q2,the production is more then socially necessary,and is causing a surplus.The competition will drive the price below it value,until some producers leaves the market or less product is being producedThis leads to a drop in the level of output,all the way to Q*.,SLis the long-term supply curve that i
28、s consistent with the Marxian concept of socially necessary labor timeIs it true that main stream economic has no theory to explain why it is at SL rather then some other level?,Back to our supply and demand for butterThis interaction of demand and supply produces,Would a regression using these data
29、 produce the demand curve?A)Demand B)Supply C)Neither,What would you get if only supply shifted?,TSLS estimates the demand curve by isolating shifts in price and quantity that arise from shifts in supply.Z is a variable that shifts supply but not demand.,TSLS in the supply-demand example:log(Qbutter
30、)=0+1log(Pbutter)+uiLet Z=rainfall in dairy-producing regions.Is Z a valid instrument?Lets check 2 conditions(1)Exogenous?corr(raini,ui)=0?A)Yes B)No C)In sufficient informationPlausibly:whether it rains in dairy-producing regions shouldnt affect demand(2)Relevant?corr(raini,log(Pbutter)0?A)Yes B)No
31、 C)In sufficient informationPlausibly:insufficient rainfall means less grazing means less butter,log(Qbutter)=0+1log(Pbutter)+uiZ=raini=rainfall in dairy-producing regions.Stage 1:regress log(Pbutter)on rain,get log(Pbutter)log(Pbutter)isolates changes in log price that arise from supply(part of sup
32、ply,at least)Stage 2:regress log(Qbutter)on log(Pbutter)The regression counterpart of using shifts in the supply curve to trace out the demand curve.,TSLS in the supply-demand example,ctd.,TSLS(2 stage lest squares)in EViews:,Everything the same as in OLS except:In“Estimation Methods”,select“TSLS Tw
33、o-stage lest squares(TSNLS and ARMA)”.Provide a list of instrument variables,be sure to include all exogenous variables as well.Only the variables on the right hand side not in the list of instruments are considered endogenous.In Options,select“Heteroskedasiticity consistent coefficient covariance”.
34、,Example 15.5 using 2SLS,Dependent Variable:LOG(WAGE)Method:Two-Stage Least SquaresSample:1 753 IF INLFIncluded observations:428Instrument list:EXPER EXPERSQ FATHEDUC MOTHEDUCVariableCoefficientStd.Errort-Statistic Prob.EDUC0.0613970.0314371.953024 0.0515EXPER0.0441700.0134323.288329 0.0011EXPERSQ-0
35、.0008990.000402-2.237993 0.0257C0.0481000.4003280.120152 0.9044R-squared0.135708Mean dependent var1.190173Adjusted R-squared0.129593S.D.dependent var0.723198S.E.of regression0.674712Sum squared resid193.0200F-statistic8.140709Durbin-Watson stat1.945659Prob(F-statistic)0.000028,Note:Red are instrumen
36、tsBlue are exogenousGreen is endogenous,Example:Demand for Cigarettes,How much will a hypothetical cigarette tax reduce cigarette consumption?To answer this,we need the elasticity of demand for cigarettes,that is,1,in the regression,log(Qcigarettes)=0+1log(Pcigarettes)+uiWill the OLS estimator plaus
37、ibly be unbiased?Why or why not?,Example:Cigarette demand,ctd.,log(Qcigarettes)=0+1log(Pcigarettes)+uiPanel data:Annual cigarette consumption and average prices paid(including tax)48 continental US states,1985-1995Proposed instrumental variable:Zi=general sales tax per pack in the state=GSTaxiIs thi
38、s a valid instrument?(1)Relevant?corr(GSTaxi,log(Pcigarettes)0?(2)Exogenous?corr(GSTaxi,ui)=0?,Example:Cigarette demand,two instruments,Dependent Variable:LOG(PACKPC)Method:Two-Stage Least SquaresSample:1 528 IF YEAR=1995Included observations:48White Heteroskedasticity-Consistent Standard Errors&Cov
39、arianceInstrument list:LOG(INCOME/POP)(TAX-TAXS)/CPI TAXS/CPIVariableCoefficientStd.Errort-Statistic Prob.LOG(INCOME/POP)0.2804050.2538901.1044360.2753LOG(AVGPRS/CPI)-1.2774240.249610-5.117680 0.0000C9.7768100.96176310.165510.0000R-squared0.429422 Mean dependent var4.538837Adjusted R-squared0.404063
40、 S.D.dependent var0.243346S.E.of regression0.187856 Sum squared resid1.588044F-statistic13.28079 Durbin-Watson stat1.946351Prob(F-statistic)0.000029,Identification,The general IV regression model,ctd.,Y1=0+1Y2+kYk+1+k+1Z1+k+rZr+uWe need to introduce some new concepts and to extend some old concepts
41、to the general IV regression model:Terminology:identification and overidentificationTSLS with included exogenous variablesone endogenous regressormultiple endogenous regressorsAssumptions that underlie the normal sampling distribution of TSLSInstrument validity(relevance and exogeneity)General IV re
42、gression assumptions,Identification,ctd.,The coefficients 1,k are said to be:exactly identified if m=k.There are just enough instruments to estimate 1,k.overidentified if m k.There are more than enough instruments to estimate 1,k.If so,you can test whether the instruments are valid(a test of the“ove
43、ridentifying restrictions”)well return to this later underidentified if m k.There are too few enough instruments to estimate 1,k.If so,you need to get more instruments!,Identification,In general,a parameter is said to be identified if different values of the parameter would produce different distrib
44、utions of the data.In IV regression,whether the coefficients are identified depends on the relation between the number of instruments(m)and the number of endogenous regressors(k)Intuitively,if there are fewer instruments than endogenous regressors,we cant estimate 1,kFor example,suppose k=1 but m=0(
45、no instruments)!,Identification of General SEM,Once again,our structural equations are:y1=a1y2+b1z1+u1y2=a2y1+b2z2+u2Let z1 be all the exogenous variables in the first equation,and z2 be all the exogenous variables in the second equationIts okay for there to be overlap in z1 and z2How are we able to
46、 identify which equation is which?We need to state the rank condition,Identification of General SEM,Given our two equations:y1=a1y2+b1z1+u1y2=a2y1+b2z2+u2To identify equation 1,there must be some variables(at least 1)in z2 that are not in z1To identify equation 2,there must be some variables(at leas
47、t 1)in z1 that are not in z2 We refer to this as the rank conditionWe are able to identify the two equations if the rank condition is satisfied,Example:Labor Market,Suppose the structural equations for the labor market are:hours=a1log(wage)+b10+b11educ+b12age+b13kidslt6+b14nwifeinc+b15exper+b16exper
48、2+u1log(wage)=a2hours+b20+b21educ+b22age+b23kidslt6+b24nwifeinc+b25exper+b26exper2+u2Can we identify which is the supply/demand equation for labor?No!That is the reason for the rank condition,Example:Labor Market,Suppose the structural equations for the labor market instead are as follows:hours=a1lo
49、g(wage)+b10+b11educ+b12age+b13kidslt6+b14nwifeinc+u1log(wage)=a2hours+b20+b21educ+b22exper+b23exper2+u2Which is the supply/demand equation for labor?1.is supply and 2.is demand equations for labor,for age,kidslt6 and nwifeinc affects supply but not demand for labor,while experience affects demand bu
50、t not supply of labor.,Order Condition,Note that the exogenous variable excluded from the first equation must have a non-zero coefficient in the second equation for the rank condition to holdOrder condition states that there must be at least as many exogenous variables excluded in the first equation
