Identification with Endogeneity

# Identification with Endogeneity - We assumed that the error...

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Consider a model with an endogenous regressor y = & 0 + 1 x + u cor ( x; u ) 6 = 0 We can write this as a triangular system y = & 0 + 1 x + u x = ± 0 + ± 1 z + v cor ( u; v ) 6 = 0 Top equation: second stage (sometimes called structural) form) Most generally: structure is a quantity of interest, which may not correspond to top equation

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Structural Quantity of Interest Average structural function (a function of x ) ( x ) = E u ( y j x ) expectation taken wrt marginal distribution of u For our previous model ( x ) = ± 0 + ± 1 x so our 2SLS regression would recover a quantity of inter- est
Separability

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Unformatted text preview: We assumed that the error is additively separable in the second stage What if we are wrong? Example from Hahn and Ridder y = x 2 u x = z + v u = : 5 v + p 3 2 e each error ( u; v; e ) are iid N (0 ; 1) Average structural function & ( x ) = E u ( y j x ) = 0 yet 2SLS will estimate ^ ± = 0 and ^ ± 1 = 1 Resolve Issue could work with logs (restore separability) Hahn also shows that if we estimate conditional median, no problem arises if u is scalar therefore, if u is symmetric and scalar, then this issue does not arise...
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Identification with Endogeneity - We assumed that the error...

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