Chapter 6 Multivariate models C Chris brooks2002,陈磊204
© Chris Brooks 2002, 陈磊 2004 6-1 Chapter 6 Multivariate models
6-2 1 Motivations All the models we have looked at thus far have been single equations models of the form y=XB+u All of the variables contained in the X matrix are assumed to be EXogenous由系统外因素决定的变量 y is an EndoGenous variable.既影响系统同时又被该系统 及其外部因素所影响的变量 An example -the demand and supply of a good: Q=a+m+S1+l1(1) Ost =a+up+xt+v,(2) Oa Os=quantity of the good demanded/ supplied price, price of a substitute good T,= some variable embodying the state of technology C Chris brooks2002,陈磊204
© Chris Brooks 2002, 陈磊 2004 6-2 1 Motivations • All the models we have looked at thus far have been single equations models of the form y = X + u • All of the variables contained in the X matrix are assumed to be EXOGENOUS.由系统外因素决定的变量 • y is an ENDOGENOUS variable. 既影响系统同时又被该系统 及其外部因素所影响的变量. An example - the demand and supply of a good: (1) (2) (3) 、 = quantity of the good demanded / supplied Pt = price， St = price of a substitute good Tt = some variable embodying the state of technology Qdt = + Pt + St + ut Q P T v st = + t + t + t Qdt = Qst Qdt Qst
6-3 Simultaneous equations models The structural form Assuming that the market always clears, and dropping the time subscripts for simplicity Q=a+ BP+ys+u Q=元+HP+kT+ν (5) This is a simultaneous structural form of the model The point is that price and quantity are determined simultaneously (price affects quantity and quantity affects price) P and o are endogenous variables, while s and tare exogenous We can obtain REdUCed FORM equations corresponding to (4)and(5) by solving equations (4) and(5)for P and for Q C Chris brooks2002,陈磊204
© Chris Brooks 2002, 陈磊 2004 6-3 • Assuming that the market always clears, and dropping the time subscripts for simplicity (4) (5) This is a simultaneous STRUCTURAL FORM of the model. • The point is that price and quantity are determined simultaneously (price affects quantity and quantity affects price). • P and Q are endogenous variables, while S and T are exogenous. • We can obtain REDUCED FORM equations corresponding to (4) and (5) by solving equations (4) and (5) for P and for Q. Simultaneous Equations Models: The Structural Form Q = + P +S + u Q = + P +T + v
6-4 Obtaining the reduced form Solving for 2, +BP++=元+HP+T+v (6) Solving for P O元KT BBBB Rearranging(6), (-)P=(4-a)+K1-+(v-l 2-a K P (8) B-4B-4 C Chris brooks2002,陈磊204
© Chris Brooks 2002, 陈磊 2004 6-4 • Solving for Q, (6) • Solving for P, (7) • Rearranging (6), (8) Obtaining the Reduced Form + P +S + u = + P +T + v Q S u Q T v − − − = − − − ( − )P = ( −) +T −S + (v − u) P T S v u = − − + − − − − − −
Obtaining the reduced form Multiplying(7) through by Bu, -p-1yS-1=B-B元-B1-Bh (-B)Q=(10-B4)-Bk+y+(A-Bv) ua Q B/ Br s+ β u-B u 、的 u-B u- (8 )and(9)are the reduced form equations for P and o C Chris brooks2002,陈磊204
© Chris Brooks 2002, 陈磊 2004 6-5 • Multiplying (7) through by , (9) • (8) and (9) are the reduced form equations for P and Q. Obtaining the Reduced Form Q − − S − u = Q − − T − v ( − )Q = ( − ) − T + S + (u − v) Q T S u v = − − − − + − + − −
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