6-7 Simultaneous Equations Bias Recall that B=(rx)y'y and y=xb+u · So that B=(XXX(XB+u) (XXX XB+(X'XX'u B+(XX)Xu aking expectations, e(B)=E(B)+E((XX)Xu B+(X(Xu If the x's are non-stochastic, E(Xu)=0, which would be the case in a single equation system, so that e(B)=B, which is the condition for unbiasedness C Chris brooks2002,陈磊204© Chris Brooks 2002, 陈磊 2004 6-7 • Recall that and • So that • Taking expectations, • If the X’s are non-stochastic, E(Xu) = 0, which would be the case in a single equation system, so that , which is the condition for unbiasedness. Simultaneous Equations Bias   = ( ' ) ' − X X X y 1 y = X + u E E E X X X u X X E X u (  ) ( ) (( ' ) ' ) ( ' ) ( ' )    = + = + − − 1 1 E(  ) =  X X X u X X X X X X X u X X X X u ( ' ) ' ( ' ) ' ( ' ) ' ( ' ) '( ) ˆ 1 1 1 1 − − − − + + + = = =    