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1、第11章联立方程模型,Introduction to simultaneous-equation model,Content,Introduction to simultaneous-equation systemsThe indentification problemIndirect least-squares estimation(ILS)Two-stage least squares(2SLS),Introduction to simultaneous-equation systems,So far,we concerned ourselves primarily with single
2、-equation models.In this chapter we turn our attention to models consisting of several equations,in which the behavior of the variables is jointly determined.Considering a supply-demand models,in which the price of a products is simultaneously determined by the interaction of producers and consumers
3、 in a market.,Simultaneous-Equation Model,SEM consists of a series of equations with each equation serving to explain one variable which is determined in the model.Consider a three-equation supply-demand model described as follows:Supply:QtS=a1+a2Pt+a3Pt-1+etDemand:QtD=b1+b2Pt+b3Yt+utEquilibrium:QtS
4、=QtD,Simultaneous-Equation Model,The supply equation,demand equation,and equilibrium condition determine the market price and the quantity supplied(and demanded)when the market is in equilibrium.For this reason,the variables QtD,QtS,and Pt are often called endogenous variables;they are determined wi
5、thin the system of equations.The model also contains two variables whose values are not determined directly within the system,which is often called predetermined variables.Pt-1 and Yt are both predetermined variables in the model.The Pt-1 is determined within the system-by past values of the variabl
6、es,thus,lagged endogenous variables are predetermined variables.The variable Yt is determined completely outside the model and is called an exogenous variable.,Simultaneous-Equation Model,We can see the endogeneity of the Pt and Qt variables graphically in the figure.,SEM:structural model,Considerin
7、g following supply-demand systemSupply:QtS=a1+a2Pt+etDemand:QtD=b1+b2Pt+b3Yt+utEquilibrium:QtS=QtD The model is often called a structural model because its form is given by the underlying theory.A structural model contains endogenous variables on the left-hand side(LHS)and contains endogeneous as we
8、ll as predetermined variables on the RHS.,SEM:reduced model,If we solve the structural model for each of the endogenous variables as a function solely of the predetermined variables in the model.Then,the transformed model is called reduced form model.,Simultaneous-Equation Model,Because Pt and Qt ar
9、e endogenous,applying ordinary least squares to the estimation of the supply(or the demand)equation will generate biased and inconsistent estimators.In the SEM,where(endogenous)variables in one equation feed back into variables in another equation,the error terms are correlated with the endogenous v
10、ariables and least squares is both biased and inconsistent.,Simultaneous-Equation Model,Suppose we estimate the supply equation in the SEM model by using OLS.The slope parameter estimate will beRearrange the equation,we find that,Simultaneous-Equation Model,Now see a simple model of national income
11、determination.The reduced form of the model,Simultaneous-Equation Model,Using OLS,we have,The Identification Problem,Suppose we know the reduced form of a system of equations.Is this sufficient to allow us to discern the value of the parameters in the original set of structural equation?The problem
12、of determining the structural equations,given knowledge of the reduced form,is called the identification problem.,The Identification Problem,An equation is unidentified,if there is no way of estimating all the structural parameters from the reduced form.An equation is identified,if it is possible to
13、 obtain values of the parameters from the reduced form equation system.An equation is exactly identified,if a unique parameter value exists and is overidentified if more than one value is obtainable for some parameters.In a structural model,some equations may be identified while others may not.In a
14、single equation,it is possible that some parameters may be identified while others may remain unidentified.,The Identification Problem,Considering following supply-demand systemSupply:Qt=a1+a2Pt+etDemand:Qt=b1+b2Pt+utThe reduced form,The Identification Problem,Considering following supply-demand sys
15、temSupply:Qt=a1+a2Pt+etDemand:Qt=b1+b2Pt+b3Yt+utThe reduced form,The Identification Problem,Considering following supply-demand systemSupply:Qt=a1+a2Pt+a3Tt+etDemand:Qt=b1+b2Pt+b3Yt+utThe reduced form,The Identification Problem,Order condition(necessary condition)k,the number of all variables exclud
16、ed from the equation be greater than or equal to m,the number of endogenous variables in the model system minus 1.that is,km-1.g,the number of predetermined variables excluded from the equation must be greater than or equal to m,the number of included endogenous variables minus 1.that is,gm-1.,Indir
17、ect least-squares estimation(ILS),Consider a simple model of national income determinationCt=b0+b1Yt+utYt=Ct+ItThe reduced formExample:(ex154.txt),Two-stage least squares(2SLS),Two-stage squares(2SLS)provides a very useful estimation procedure for obtaining the values of structural parameters in ove
18、ridentified equations.Consider following model:Income function:Yt=a1+a2Mt+a3It+a4Gt+etMoney supply function:Mt=b1+b2Yt+utWhere,I and G are exogenous.Its easy to see the money supply function is overidentified,because k=2m-1=1 and income function is unidentified.,Two-stage least squares(2SLS),First,r
19、egress Y on all predetermined variables,in this example,reg Y on I and G.Yt=p1+p2It+p3Gt+wt,wt is error term.Estimate it,we get=p1+p2It+p3Gt.Second,money supply function is estimated by replacing the variable Yt with the first-stage fitted variable.That is,reg Mt on,Mt=b1+b2(+wt)+ut=b1+b2+(b2wt+ut),
20、Two-stage least squares(2SLS),reg m y(i g)ivreg m(y=i g)ivregress 2sls m(y=I g),small stata 11reg3(m y),2sls exog(i g)endog(m y),Two-stage least squares(2SLS),Another exampleSTATA commandreg exp aid inc pop(inc pop ps)reg aid exp ps(inc pop ps)Ivreg exp(aid=inc pop ps)inc popIvreg aid(exp=inc pop ps)psreg3(exp aid inc pop)(aid exp ps),2sls,summary,Endogenous variable,exogenous variable,predetermined variableThe indentification problemIndirect least-squares estimation(ILS)Two-stage least squares(2SLS),
