Instrumental variables 2Sls y-Bo+ Bx+B2x2+.. Bkrk+u ◆x1=兀n+x1z+x2+..xx+v Economics 20- Prof anderson
Economics 20 - Prof. Anderson 1 Instrumental Variables & 2SLS y = b0 + b1 x1 + b2 x2 + . . . bk xk + u x1 = p0 + p1 z + p2 x2 + . . . pk xk + v
Why use instrumental variables? o Instrumental Variables(Iv) estimation is used when your model has endogenous xs ◆ That is, whenever cov(x,u)≠0 Thus. Iv can be used to address the problem of omitted variable bias o Additionally, iv can be used to solve the classic errors-in-variables problem Economics 20- Prof anderson
Economics 20 - Prof. Anderson 2 Why Use Instrumental Variables? Instrumental Variables (IV) estimation is used when your model has endogenous x’s That is, whenever Cov(x,u) ≠ 0 Thus, IV can be used to address the problem of omitted variable bias Additionally, IV can be used to solve the classic errors-in-variables problem
What is an instrumental variable? In order for a variablez to serve as a valid instrument for x, the following must be true The instrument must be exogenous ◆ That is,Cov(z,u)=0 The instrument must be correlated with the endogenous variable x ◆ That is,Cov(=,x)≠0 Economics 20- Prof anderson
Economics 20 - Prof. Anderson 3 What Is an Instrumental Variable? In order for a variable, z, to serve as a valid instrument for x, the following must be true The instrument must be exogenous That is, Cov(z,u) = 0 The instrument must be correlated with the endogenous variable x That is, Cov(z,x) ≠ 0
More on valid instruments We have to use common sense and economic theory to decide if it makes sense to assume Cov(z, u)=0 We can test if Cov(E, x)#0 e Just testing Ho: T=0 inx=7o+2+v e Sometimes refer to this regression as the first-stage regression Economics 20- Prof anderson 4
Economics 20 - Prof. Anderson 4 More on Valid Instruments We have to use common sense and economic theory to decide if it makes sense to assume Cov(z,u) = 0 We can test if Cov(z,x) ≠ 0 Just testing H0 : p1 = 0 in x = p0 + p1 z + v Sometimes refer to this regression as the first-stage regression
IV Estimation in the simple Regression Case For y-Bo+Bx+u, and given our assumptions e CoV(E, y)=B, Cov(z, x)+Cov(z, u), so ◆B1=C0V(y)/CoV(x) Then the Iv estimator for B, is ∑(-Xv-y Economics 20- Prof anderson 5
Economics 20 - Prof. Anderson 5 IV Estimation in the Simple Regression Case For y = b0 + b1 x + u, and given our assumptions Cov(z,y) = b1Cov(z,x) + Cov(z,u), so b1 = Cov(z,y) / Cov(z,x) Then the IV estimator for b1 is ( )( ) ( )( ) − − − − = z z x x z z y y i i i i 1 b ˆ
Inference with v estimation o The homoskedasticity assumption in this case is E(l21-)=a2=Var() e As in the ols case, given the asymptotic variance we can estimate the standard error 2 a)=- C nop 2 selB SSTR Economics 20- Prof anderson 6
Economics 20 - Prof. Anderson 6 Inference with IV Estimation The homoskedasticity assumption in this case is E(u 2 |z) = s2 = Var(u) As in the OLS case, given the asymptotic variance, we can estimate the standard error ( ) ( ) 2 , 2 1 2 , 2 2 1 ˆ ˆ ˆ x x z x x z SST R se n Var s b s s b = =
IV versus ols estimation Standard error in iv case differs from ols only in the R2 from regressing x on z o Since R2< 1, Iv standard errors are larger However. iv is consistent. while ols is inconsistent, when Cov(x,u)#0 o The stronger the correlation between z and x. the smaller the iv standard errors Economics 20- Prof anderson 7
Economics 20 - Prof. Anderson 7 IV versus OLS estimation Standard error in IV case differs from OLS only in the R2 from regressing x on z Since R2 < 1, IV standard errors are larger However, IV is consistent, while OLS is inconsistent, when Cov(x,u) ≠ 0 The stronger the correlation between z and x, the smaller the IV standard errors
The effect of poor instruments o What if our assumption that Cov(E,u)=0 is false? o The iv estimator will be inconsistent. too s Can compare asymptotic bias in Ols and Iv ◆ Prefer Iv if cori(二,u)Cor(二,x)<Cor(x, IV: plim B,=B orr(z,u) orr(z,x)0 OLS: plim B=B+Corr(x,u.u Economics 20- Prof anderson 8
Economics 20 - Prof. Anderson 8 The Effect of Poor Instruments What if our assumption that Cov(z,u) = 0 is false? The IV estimator will be inconsistent, too Can compare asymptotic bias in OLS and IV Prefer IV if Corr(z,u)/Corr(z,x) < Corr(x,u) x u x u Corr x u Corr z x Corr z u s s b b s s b b = + • = + • ( , ) ~ OLS: plim ( , ) ( , ) ˆ IV : plim 1 1 1 1
IV Estimation in the multiple Regression Case EStimation can be extended to the multiple regression case o Call the model we are interested in estimating the structural model Our problem is that one or more of the variables are endogenous We need an instrument for each endogenous variable Economics 20- Prof anderson 9
Economics 20 - Prof. Anderson 9 IV Estimation in the Multiple Regression Case IV estimation can be extended to the multiple regression case Call the model we are interested in estimating the structural model Our problem is that one or more of the variables are endogenous We need an instrument for each endogenous variable
Multiple regression Iv(cont) e Write the structural model as y-Bo+ By +B221+ui, where y2 is endogenous and z Is eXogenous o Let z, be the instrument, so Cov(z,,u)=0 ane d y2=To+=1+1222+v2, where 2+0 This reduced form equation regresses the endogenous variable on all exogenous ones Economics 20- Prof anderson 10
Economics 20 - Prof. Anderson 10 Multiple Regression IV (cont) Write the structural model as y1 = b0 + b1 y2 + b2 z1 + u1 , where y2 is endogenous and z1 is exogenous Let z2 be the instrument, so Cov(z2 ,u1 ) = 0 and y2 = p0 + p1 z1 + p2 z2 + v2 , where p2 ≠ 0 This reduced form equation regresses the endogenous variable on all exogenous ones
