第6章 Multivariate models.ppt

Chapter 6 Multivariate models 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 Simultaneous Equations Models: The Structural Form 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. Obtaining the Reduced Form Solving for Q, (6) ? Solving for P, (7) ? Rearranging (6), ? ? ? (8) ? Obtaining the Reduced Form Multiplying (7) through by ??, ? ? ? ? (9) ? ? (8) and (9) are the reduced form equations for P and Q. 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(X?u) = 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? ? Simultaneous Equations Bias 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 si