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 = − − − − + − + − −