厦门大学：《高级经济计量学》讲义 第十一章 联立方程模型

Chapter11联立方程模型 Simultaneous Equation Models

Chapter 11 联立方程模型 Simultaneous Equation Models

o Simultaneous Equation Models Example(供给—需求模型) Demanding equation 2d=6o+bu,P+b2r Supplying equation Q=b tb 21 ptb P,te 20 221-1 2 Simultaneous equation model Od =b 10 +b,+brt Q=b20+b21P+b2P1+E2

⚫ Simultaneous Equation Models Example(供给—需求模型） Demanding equation: Supplying equation: Simultaneous equation model: Q b b P b Y d 10 11 12 1 = + + + Q b b P b P s 20 21 22 1 2  = + + + − 10 11 12 1 20 21 22 1 2 d s d s Q b b P b Y Q b b P b P Q Q   −  = + + +   = + + +   =

P价格,Y收入,Q、供给,Q需求 平衡方程:Q=O Variables的类型 1) Endogenous variable(内生变量,P,Q、Q) (2) Exogenous variable(外生变量,Y) (3) Predetermined variable(前定变量,Y,R) Equations (1)Behavior(structural)equations 2)Relation identities(Given)

P—价格，Y—收入，Qs—供给，Qd—需求 平衡方程： Variables的类型: (1) Endogenous variable(内生变量， P， Qs，Qd ) (2) Exogenous variable(外生变量, Y) (3) Predetermined variable(前定变量，Y，P-1 ) Equations: (1) Behavior (structural) equations (2) Relation identities (Given) Q Q d s =

Simultaneity bias(联立性偏误): Some endogenous variables are both dependent variables and independent variables at the same time and some endogenous variables are relevant to the error terms. So the ols method does not hold good. That is Simultaneity bias 内生变量与误差项相关

Simultaneity Bias(联立性偏误）: Some endogenous variables are both dependent variables and independent variables at the same time and some endogenous variables are relevant to the error terms. So the OLS method does not hold good. That is Simultaneity Bias. 内生变量与误差项相关

● Structural forn: Structural forms are models which describe the structural relations of economic variables The equations of a structural form are called the structural equations and the parameters are called the structural parameters In a structural form if the number of the endo genous variables is the same as the number of the equations the structural form is complete

⚫ Structural form: Structural forms are models which describe the structural relations of economic variables. The equations of a structural form are called the structural equations and the parameters are called the structural parameters. In a structural form, if the number of the endo￾genous variables is the same as the number of the equations, the structural form is complete

Complete and linear structural form ∑bn+∑X ∥-c l,2, In matrix form, BY+X=8 O (B T) X Y--endogenous variable sample matrix X- predetermined variable sample matrix

Complete and linear structural form: In matrix form, Or Y—endogenous variable sample matrix X— predetermined variable sample matrix 1 1 , 1,2, , g k ij jt ij jt it j j b Y r X i g  = =  + = = BY+ = ΓX ε ( )     =   Y B Γ ε X

Reduced form(简化式方程) Y=f(X1,X2,…Xk;1),i=1,2,…,g By supposing B *0, from BY+X=E We have Y=-BTX+Bs=IX+v I=-B.v=B 8 Approach to estimating a structuralforms (1)Estimate the reduced form: 1=xn11+x22+…+丌1+v t=1,…, n and i=1,…,g

⚫ Reduced form(简化式方程）: By supposing , from We have Approach to estimating a structural form: (1) Estimate the reduced form: 1 2 ( , , , ; ), 1,2, , i i k i Y f X X X i g = =  | | 0 B  BY+ = ΓX ε 1 1 1 1 , − − − − = − + = + = − = Y B ΓX B ε ΠX ν Π B Γ ν B ε 1 1 2 2 1, , and 1, , Y X X X it i t i t ik kt t t n i g = + + + +     = =

(2)Use ∏=-B-I to estimate the structural parameters That is Indirect least squares method (ILS) The ilse are inconsistent estimators Unfortunately, sometimes the ils does not work

(2) Use to estimate the structural parameters. That is Indirect least squares method(ILS) The ILSE are inconsistent estimators. Unfortunately, sometimes the ILS does not work. −1 Π = −B Γ

Identification of Simultaneous Equations Models 什么是模型识别 First review the Complete and linear structural form k ∑b+∑Xn=n,1=1 In matrix form, BY+TX=￡ Reduced form:Y=-bTX+B2=iiX+v Parameter relation system I=-BT=B s

Identification of Simultaneous Equations Models 什么是模型识别: First review the Complete and linear structural form: In matrix form, Reduced form: Parameter relation system: 1 1 , 1,2, , g k ij jt ij jt it j j b Y r X i g  = =  + = = BY+ = ΓX ε 1 1 1 1 , − − − − = − + = + = − = Y B ΓX B ε ΠX ν Π B Γ ν B ε

由已知的简化式模型去确定其结构式模型的问 题就称为模型识别问题 若一个随机结构方程的系(参)数可由参数关系体系 (方程组)解出,则称此随机结构方程可识别否则此 随机结构方程不可识别 若一个随机结构方程可识别,且系(参)数的解唯 则称此随机结构方程恰好识别;如解不唯一,则称 此随机结构方程过度识别。 若结构式模型中的每一个随机结构方程都可识别,则 称此结构式模型可识别,否则,称此结构式模型不 可识别

由已知的简化式模型去确定其结构式模型的问 题就称为模型识别问题 若一个随机结构方程的系(参)数可由参数关系体系 (方程组)解出,则称此随机结构方程可识别,否则,称此 随机结构方程不可识别。 若一个随机结构方程可识别，且系(参)数的解唯一， 则称此随机结构方程恰好识别；如解不唯一，则称 此随机结构方程过度识别。 若结构式模型中的每一个随机结构方程都可识别，则 称此结构式模型可识别，否则，称此结构式模型不 可识别