6-6 2 Simultaneous Equations Bias But what would happen if we had estimated equations( 4)and (5),i.e the structural form equations, separately using Ols? Both equations depend on P. One of the ClrM assumptions was that E(Xu)=0, where X is a matrix containing all the variables on the rhs of the equation. It is clear from(8)that P is related to the errors in(4)and (5) i.e. it is stochastic What would be the consequences for the ols estimator, B, if we ignore the simultaneity? C Chris brooks2002,陈磊204
© Chris Brooks 2002, 陈磊 2004 6-6 • But what would happen if we had estimated equations (4) and (5), i.e. the structural form equations, separately using OLS? • Both equations depend on P. One of the CLRM assumptions was that E(Xu) = 0, where X is a matrix containing all the variables on the RHS of the equation. • It is clear from (8) that P is related to the errors in (4) and (5) - i.e. it is stochastic. • What would be the consequences for the OLS estimator, , if we ignore the simultaneity? 2 Simultaneous Equations Bias
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(Xu) = 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 − − − − + + + = = =
6-8 Simultaneous Equations bias But. if the equation is part of a system, then E(Xu)#0, in general Conclusion: Application of ols to structural equations which are part of a simultaneous system will lead to biased coefficient estimates Is the ols estimator still consistent even though it is biased? No-In fact the estimator is inconsistent as well Hence it would not be possible to estimate equations (4) and (5)validly using ols. C Chris brooks2002,陈磊204
© Chris Brooks 2002, 陈磊 2004 6-8 • But .... if the equation is part of a system, then E(Xu) 0, in general. • Conclusion: Application of OLS to structural equations which are part of a simultaneous system will lead to biased coefficient estimates. • Is the OLS estimator still consistent, even though it is biased? • No - In fact the estimator is inconsistent as well. • Hence it would not be possible to estimate equations (4) and (5) validly using OLS. Simultaneous Equations Bias
6-9 3 Avoiding Simultaneous Equations Bias So What can we do Taking equations( 8 )and(9), we can rewrite them as x25+6(10 Q=20+x21T+z25+b2(l We can estimate equations(10)&(Il) using Ols since all the rhs variables are exogenous But . we probably dont care what the values of the T coefficients are: what we wanted were the original parameters in the structural equations-a, B,r, n, u,K C Chris brooks2002,陈磊204
© Chris Brooks 2002, 陈磊 2004 6-9 So What Can We Do? • Taking equations (8) and (9), we can rewrite them as (10) (11) • We CAN estimate equations (10) & (11) using OLS since all the RHS variables are exogenous. • But ... we probably don’t care what the values of the coefficients are; what we wanted were the original parameters in the structural equations - , , , , , . 3 Avoiding Simultaneous Equations Bias P = + T + S + 10 11 12 1 Q = + T + S + 20 21 22 2
6-10 4 Identification of Simultaneous equations Can We retrieve the original coefficients from the zs? Short answer: sometimes we sometimes encounter another problem: identification. x Consider the following demand and supply equations Supply equation =a+ BP(12) Demand equation Q=n+up(13) We cannot tell which is which! Both equations are UNidEntiFied or UNDERIDENTIFIED. The problem is that we do not have enough information from the equations to estimate 4 parameters. Notice that we would not have had this problem with equations (4 ) and(5) since they have different exogenous variables C Chris brooks2002,陈磊204
© Chris Brooks 2002, 陈磊 2004 6-10 Can We Retrieve the Original Coefficientsfrom the ’s? Short answer: sometimes. • we sometimes encounter another problem: identification.* • Consider the following demand and supply equations Supply equation (12) Demand equation (13) We cannot tell which is which! • Both equations are UNIDENTIFIED or UNDERIDENTIFIED. • The problem is that we do not have enough information from the equations to estimate 4 parameters. Notice that we would not have had this problem with equations (4) and (5) since they have different exogenous variables. 4 Identification of Simultaneous Equations Q = + P Q = + P