A Weaker Assumption For unbiasedness. we assumed a zero conditional mean -E(ux x, ., xk=0 e For consistency, we can have the weaker assumption of zero mean and zero correlation -(u=0 and Cov(x ) =0, for 2...k o Without this assumption, Ols will be biased and inconsistent Economics 20- Prof anderson 6
Economics 20 - Prof. Anderson 6 A Weaker Assumption For unbiasedness, we assumed a zero conditional mean – E(u|x1 , x2 ,…,xk ) = 0 For consistency, we can have the weaker assumption of zero mean and zero correlation – E(u) = 0 and Cov(xj ,u) = 0, for j = 1, 2, …, k Without this assumption, OLS will be biased and inconsistent!
Deriving the Inconsistency Just as we could derive the omitted variable bias earlier. now we want to think about the inconsistency, or asymptotic bias, in this case True model: y=o+Bx,+B,x,+v You think y=Bo+B,x+u, so that u=B,x,+v and, plim B,=B,+,8 where=Cov(xi,x,) var(x) Economics 20- Prof anderson 7
Economics 20 - Prof. Anderson 7 Deriving the Inconsistency Just as we could derive the omitted variable bias earlier, now we want to think about the inconsistency, or asymptotic bias, in this case ( ) ( ) 1 2 1 2 2 1 1 2 0 1 1 0 1 1 2 2 where , ~ and, plim You think : ,so that True model : Cov x x Var x u x v y x u y x x v = = + = + = + + = + + + b b b b b b b b b