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ii) Suppose your outcome linear model with endogeneity is yi=xi+ei,E.pdf

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The document describes a linear model with endogenous regressors where yi=xi+ei and xi=zi+ui. It has an instrumental variable zi that satisfies the relevance condition of E[ziei]=0 and the exogeneity condition E[ziui]=0. Using this instrumental variable, the document shows that the two stage least squares estimator (2SLS), which first estimates xi as xi=zi and then regresses yi on the predicted xi, produces estimates that are equivalent to the instrumental variables estimator (IV) which directly regresses yi on zi weighted by xi.

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ii) Suppose your outcome linear model with endogeneity is: yi=xi+ei,E[xiei]=0 You have an instrumental variable zi satisfying E[ziei]=0 and: xi=zi+ui,E[ziui]=0 where =0 (relevance) and E[ziei]=0 (exogeneity condition in a model without intercept). You use the two stage least squares (2SLS) estimator and estimate in a first step via OLS, =i=1nzi2i=1nzixi, and predict xi=zi. In a second step, you construct your 2SLS estimator of : 2SLS=i=1nxi2i=1nxiyi. Show that 2SLS conincides with the IV estimator IV=i=1nzixii=1nziyi.

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ii) Suppose your outcome linear model with endogeneity is yi=xi+ei,E.pdf

  • 1. ii) Suppose your outcome linear model with endogeneity is: yi=xi+ei,E[xiei]=0 You have an instrumental variable zi satisfying E[ziei]=0 and: xi=zi+ui,E[ziui]=0 where =0 (relevance) and E[ziei]=0 (exogeneity condition in a model without intercept). You use the two stage least squares (2SLS) estimator and estimate in a first step via OLS, =i=1nzi2i=1nzixi, and predict xi=zi. In a second step, you construct your 2SLS estimator of : 2SLS=i=1nxi2i=1nxiyi. Show that 2SLS conincides with the IV estimator IV=i=1nzixii=1nziyi.