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