1、 2013 Cengage Learning.All Rights Reserved.May not be scanned,copied or duplicated,or posted to a publicly accessible website,in whole or in part.Chapter 16Simultaneous Equations ModelsWooldridge:Introductory Econometrics:A Modern Approach,5e第1页，共15页。2013 Cengage Learning.All Rights Reserved.May not
2、 be scanned,copied or duplicated,or posted to a publicly accessible website,in whole or in part.Simultaneity is another important form of endogeneitySimultaneity occurs if at least two variables are jointly determinedA typical case is when oberved outcomes are the result of separate behavioral mecha
3、nisms that are coordinated in an equilibriumThe prototypical case is a system of demand and supply equations:D(p)=how high would demand be if the price was set to p?S(p)=how high would supply be if the price was set to p?Both mechanisms have a ceteris paribus interpretation Observed quantity and pri
4、ce will be determined in equilibriumSimultaneous equations systems can be estimated by 2SLS/IVSimultaneousEquations Systems第2页，共15页。2013 Cengage Learning.All Rights Reserved.May not be scanned,copied or duplicated,or posted to a publicly accessible website,in whole or in part.Example:Labor demand an
5、d supply in agricultureAnnual hours supplied by workers in a given county if the average hourly wage offered to such workers is wAnnual hours demanded by employers in a given county if the average hourly wage paid to workers is wObserved supply shifters e.g.manufacturing wageObserved demand shifters
6、 e.g.agricultural land areaWage elasticity of supplyWage elasticity of demandUnobserved supply shifters e.g.immigration flows Unobserved demand shifters e.g.food market shocksSimultaneousEquations Systems第3页，共15页。2013 Cengage Learning.All Rights Reserved.May not be scanned,copied or duplicated,or po
7、sted to a publicly accessible website,in whole or in part.Example:Labor demand and supply in agriculture(cont.)Competition on the labor market in each county i will lead to a county wage wi so that the total num-ber of hours his supplied by workers in this county equals the total number of hours hid
8、 demanded by agricultural employers in this county:(=observed equilibrium outcomes in each county)Simultaneous equations model(SEM):Endogenous variablesExogenous variablesNote:Without separate exogenous variables in each equation,the two equations could never be distiguished/separately identifiedStr
9、uctural error terms(uncorrelated with the exogenous variables)SimultaneousEquations Systems第4页，共15页。2013 Cengage Learning.All Rights Reserved.May not be scanned,copied or duplicated,or posted to a publicly accessible website,in whole or in part.Example:Murder rates and the size of the police forceMu
10、rders per capitaPolice officers per capitaIncome per capitaBehavioral equation“of murderer populationBehavioral equation“of city governmentIn this equation,polpc will not be exogenous because the number of police officers will dependent on how high the murder rate is(reverse causation“).The interest
11、ing equation for policy purposes is the first one.City governments will want to know by how much the murder rate decreases if the number of police officers is exogenously increased.This will be hard to measure because the number of police officers is not exogenously chosen(it depends on how much cri
12、me there is in the city,see second equation).SimultaneousEquations Systems第5页，共15页。2013 Cengage Learning.All Rights Reserved.May not be scanned,copied or duplicated,or posted to a publicly accessible website,in whole or in part.Simultaneity bias in OLSInsert first equation into second:(=reduced form
13、 equation for y2)Variable y2 is correlated with the error u1 because u1 is indirectly a part of y2.OLS applied to this equation will be therefore be inconsistent.SimultaneousEquations Systems第6页，共15页。2013 Cengage Learning.All Rights Reserved.May not be scanned,copied or duplicated,or posted to a pub
14、licly accessible website,in whole or in part.Identification in simultaneous equations systemsExample:Supply and demand systemWhich of the two equations is identified?The supply function cannot be consistently estimated because one of the regressors is endogenous and we do not have an instrumentThe d
15、emand function can be consistently estimated because we can take z1 as an instrument for the endogenous price variableSupply of milkDemand for milkFor example,price of cattle feed SimultaneousEquations Systems第7页，共15页。2013 Cengage Learning.All Rights Reserved.May not be scanned,copied or duplicated,
16、or posted to a publicly accessible website,in whole or in part.Graphical illustration of identification problemIntuitively,it is clear why the demand equation can be identified:We have an observed variable z1 that shifts the supply equation while not affecting the demand equation.In this way the dem
17、and equation can be traced out.SimultaneousEquations Systems第8页，共15页。2013 Cengage Learning.All Rights Reserved.May not be scanned,copied or duplicated,or posted to a publicly accessible website,in whole or in part.General rules for identification in simultaneous equation systemsOrder conditionA nece
18、ssary condition for the first equation to be identified is that at least one of all exogenous variables is excluded from this equationRank conditionThe first equation is identified if,and only if,the second equation con-tains at least one exogenous variable that is excluded from the first eq.Simulta
19、neousEquations Systems第9页，共15页。2013 Cengage Learning.All Rights Reserved.May not be scanned,copied or duplicated,or posted to a publicly accessible website,in whole or in part.Example:Labor supply of married,working womenSupply equation:Wage offer equation:The supply equation is identified because i
20、t does not contain exper and exper2The wage offer function is identified because it does not contain age,kidslt6 and nwifeincHours and wages are endogenous(market equilibrium)SimultaneousEquations Systems第10页，共15页。2013 Cengage Learning.All Rights Reserved.May not be scanned,copied or duplicated,or p
21、osted to a publicly accessible website,in whole or in part.Example:Labor supply of married,working women(cont.)The rank condition,i.e.the condition that exogenous variables excluded from the equation in ques-tion are included in the other equation can be tested using the reduced form equations.For e
22、xample,reduced form for log(wage):Note that this equation can be consistently estimated by OLS as it contains only exogenous variables.The labor supply function is identified if the hypothesis can be rejected.This is equivalent to rejecting in the system equations.The same argument applies to the id
23、entification of the wage offer function.SimultaneousEquations Systems第11页，共15页。2013 Cengage Learning.All Rights Reserved.May not be scanned,copied or duplicated,or posted to a publicly accessible website,in whole or in part.Estimation of simultaneous equation systems by 2SLSGiven the identification
24、condition holds,the parameters of a simulta-neous equations system can be consistenly estimated by 2SLSFor this,in a first stage,each endogenous variable is regressed on the full list of exogenous variables(=reduced form regressions)In a second stage,the system equations are estimated by OLS but wit
25、h the endog.regressors being replaced by predictions from stage one If not all eq.are identified,one can estimate only the identified onesIf certain additional conditions hold,one can also use more efficient system estimation methods(Three Stage Least Squares,3SLS)SimultaneousEquations Systems第12页，共
26、15页。2013 Cengage Learning.All Rights Reserved.May not be scanned,copied or duplicated,or posted to a publicly accessible website,in whole or in part.Example:Labor supply of married,working women using 2SLSSimultaneousEquations Systems第13页，共15页。2013 Cengage Learning.All Rights Reserved.May not be sca
27、nned,copied or duplicated,or posted to a publicly accessible website,in whole or in part.Systems with more than two equationsA necessary condition for identification of an equation is that there are more excluded exog.var.than endog.regressors(=order condition)There is also a rank condition(but it i
28、s much more complicated)Simultaneous equations models with time seriesAmong the earliest applications of SEMs was the estimation of large systems of simultaneous equations for macroeconomic time seriesFor a number of reasons,such systems are seldom estimated nowThe main problem is that most time ser
29、ies are not weakly dependentAnother problem is the lack of enough exogenous variablesSimultaneousEquations Systems第14页，共15页。2013 Cengage Learning.All Rights Reserved.May not be scanned,copied or duplicated,or posted to a publicly accessible website,in whole or in part.Example:A simple Keynesian model of aggregate demandConsumptionInvestmentIncomeTaxesInterest rateGovernment spendingExogenous:Endogenous:The exogeneity of these variables is very questionableSimultaneousEquations Systems第15页，共15页。
