iv - Economics 20 Prof Anderson 1 Instrumental...

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Unformatted text preview: Economics 20 - Prof. Anderson 1 Instrumental Variables & 2SLS y = β + β 1 x 1 + β 2 x 2 + . . . β k x k + u x 1 = π + π 1 z + π 2 x 2 + . . . π k x k + v 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 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 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 H : π 1 = 0 in x = π + π 1 z + v Sometimes refer to this regression as the first-stage regression Economics 20 - Prof. Anderson 5 IV Estimation in the Simple Regression Case For y = β + β 1 x + u , and given our assumptions Cov( z,y ) = β 1 Cov( z,x ) + Cov( z,u ), so β 1 = Cov( z,y ) / Cov( z,x ) Then the IV estimator for β 1 is ( 29 ( 29 ( 29 ( 29 ∑ ∑---- = x x z z y y z z i i i i 1 ˆ β Economics 20 - Prof. Anderson 6 Inference with IV Estimation The homoskedasticity assumption in this case is E( u 2 |z ) = σ 2 = Var( u ) As in the OLS case, given the asymptotic variance, we can estimate the standard error ( 29 ( 29 2 , 2 1 2 , 2 2 1 ˆ ˆ ˆ z x x z x x R SST se n Var σ β ρ σ σ β = = Economics 20 - Prof. AndersonEconomics 20 - Prof....
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iv - Economics 20 Prof Anderson 1 Instrumental...

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