Lecture+12+联立方程内生性问题Cross+section+and+panel

1、Research MethodLecture 12 (Ch16)Simultaneous Equations Models (SEMs)1IntroducdtionWe have learned two “sources” of endogeneity. 1. Omitted variables 2. Errors in variablesIn this handout, we will learn another source of endogeneity: Simultaneity.2In econometrics, “endogeneity” usually means that an

2、explanatory variable is correlated with the error term.In simultaneous equation models, endogeneity means that the observed variable is determined by the equilibrium. For example, an observed quantity is determined by the equilibrium between demand and supply. When a variable is endogenous in simult

3、aneous equation sense, it is usually endogenous in econometric sense (i.e., correlated with the error term). We will see this soon.3The nature of simultaneous equation.Consider the following model describing equilibrium quantity of labor (in hours) in agricultural sector in a country.Labor supply :

4、hs=1w+1z1+u1Labor demand: hd=2w+2z2+u2hs is the hours of labor supplied, and hd is the hours of labor demanded. These quantities depends on the wage rate, w, and other factors, z1 and z2.4z1 would be the wage rate of the manufacturing sector. If the manufacturing wage increases, people would move to

5、 manufacturing sector, reducing hours worked in agricultural sector. z1 is called the observed demand shifter. u1 is called the unobserved demand shifter.z2 would be agricultural land area. The more land available, more demand for labor. z2 is the observed supply shifter. u1 is the unobserved supply

6、 shifter.5Demand and supply describes entirely different relationships. The observed labor quantity and wage rate are determined by the equilibirum between these two equations. The equilibrium: hs=hd6Consider you have country level data. Then, for each country, we observe only the equilibirum labor

7、supply and wage rate. Demand: hi=1wi+1zi1+ui1 Supply: hi=2wi+2zi2+ui2where i is the country subscript.These two equations constitute a simultaneous equations model (SEM). These two equations are called the structural equations. 1,1, 2, 2 are called the structural parameters.7In SEM framework, hi and

8、 wi are endogenous variables because they are determined by the equilibrium between the two equations.In the same way, zi1 and zi2 are exogenous variables because they are determined outside of the model. u1 and u2 are called the structural errors.One more important point: Without z1 or z2, there is

9、 no way to distinguish whether one equation is demand or supply.8Simultaneous equation biasConsider the following simultaneous equation model. y1=1y2+1z1+u1.(1) y2=2y1+2z2+u2.(2)In this model, y1 and y2 are endogenous variables since they are determined by the equilibrium between the two equations.

10、z1 z2 are exogenous variables. 9Since z1 and z2 are determined outside of the model, we assume that z1 and z2 are uncorrelated with both of the structural errors. Thus, by definition, the exgoneous variables in SEM are exogenous in econometric sense as well. In addition, the two structural errors, u

11、1 & u2, are assumed to be uncorrelated with each other. 10Now, solve the equations (1) and (2) for y1 and y2, then you get the following reduced form equations. y1=11z1+12z2+v1 y2=21z1+22z2+v2 where 11= 1/(1- 1 2) 112= 1 2/(1- 1 2) v1 =(u1+ 1 u2)/(1- 1 2) 21 =21/(1- 2 1) 22 = 2/(1- 2 1) v2=(2u1+u2)/

12、(1- 2 1)These parameters are called the reduced form parameters.11You can check that v1 and v2 are uncorrelated with z1 and z2. Therefore, you can estimate these reduced form parameters by OLS (Just apply OLS separately for each equation).12However, you cannot estimate the structural equations with

13、OLS. For example, consider the first structural equation. y1=1y2+1z1+u1Notice that Cov(y2, u1) =2/(1-21)E(u12) =2/(1-21)21 0Thus, y2 is correlated with u1 (assuming that 2 0.) In other words, y2 is endogenous in econometric sense.13Thus, endogenous variables in SEM are usually endogenous in economet

14、ric sense as well. Thus, you cannot apply OLS to the structural equations. Cov(y2, u1) =2/(1-21)21 can be used to predict the direction of bias. If this is positive, OLS estimate of 1 will be biased upward. If it is negative, it will be biased downward. The formula above does not carry over to more

15、general models. But we can use this as a guide to check the direction of the bias.14An exampleSuppose that you are interested in estimating the effect of police size on the city murder rate. Notice that the supply of murder would be a function of police size. But the demand for police is a function

16、of murder rates. 15Thus, the observed murder rate and the police size are determined simultaneously by the following model.(Murder)=1(police)+10+1(Income per capita)+u1.(3)(Police)=2(Murder)+ 20+2(other vars)+u2.(4)Allthe variables are the city-level variables. (Murder) is the number of murders per

17、capita. (Police) is the number of police officers per capita.We are interested in estimating the effect of police on the murder rate: equation (3). 16However, since murder rate and police force are determined simultaneously, (police) is endogenous in equation (3). Thus OLS estimate for 1 is biased.

18、Question: What would be the direction of the bias?17Identifying and estimating a structural equation: 2 equations caseWhen we learned OLS, a parameter was said to be identified when the explanatory variable is not correlated with the error. In 2SLS chapter, we learned how to identify (i.e., eliminat

19、e the bias) by apply IV method.In SEM, the term identification is used slightly differently. 18Suppose the following model describing the supply and demand.Supply: q =1p+1z1+u1Demand: q =2p+u1Note that supply curve has an observed supply shifter z1, but demand has no obsedved supply shifter.Given th

20、e data on q, p and z1, which equation can be estimated? That is, which is an identified equation?19DemandSupply: location is different depending on the value of z1.These are the data points. Notice: data points trace the demand curve. Thus, it is the demand equation that can be estimated.20Because t

21、here is observed supply shifter z1 which is not contained in demand equation, we can identify the demand equation.It is the presence of an exogenous variable in the supply equation that allows us to estimate the demand equation.In SEM, identification is used to mean which equation can be estimated.2

22、1Now turn to a more general case.(z11z1k) and (z21 ) may contain the same variables, but may contain different variables as well.When one equation contains exogenous variables not contained in the other equation, this means that we have imposed exclusion restrictions.22The condition for identificati

23、on is the following.The condition for identification: The first equation is identified if and only if the second equation contains at least one exogenous variable (non zero coefficient) that is excluded from the first equation.23The above condition have two components. First, at least one exogenous

24、variable should be excluded from the first equation (order condition). Second, the excluded variable should have non zero coefficients in the second equation (rank condition). The identification condition for the second equation is just a mirror image of the statement.24ExampleLabor supply of marrie

25、d working women.Labor supply equation:Wage offer equation:In the model, hours and lwage are endogenous variables. All other variables are exogenous. (Thus, we are ignoring the endogeneity of educ arising from omitted ability.)25Suppose that you are interested in estimating the first equation.Since e

26、xp and exp2 are excluded from the first equation, the order condition is satisfied for the first equation. The rank condition is that, at least one of exp and exp2 has a non zero coefficient in the second equation. Assuming that the rank condition is satisfied, the first equation is identified.In a

27、similar way, you can see that the second equation is also identified. 26Estimating SEM using 2SLSOnce we have determined that an equation is identified, we can estimate it by two stage least square. 27Consider the labor supply equation example again. You are interested in estimating the first equati

28、on.Suppose that the first equation is identified (both order and rank conditions are satisfied). lwage is correlated with u1. Thus, OLS cannot be used.28However, exp and exp2 can be used as instruments for lwage in the first equation.Why? First, exp and exp2 are uncorrelated with u1 by assumption of

29、 the model (instrument exogeneity satisfied). Second exp and exp2 are correlated with lwage by the rank condition (instrument relevance satisfied). 29In general, you can use the excluded exogenous variables as the instruments. 30ExerciseConsider the following simultaneous equation model.Q1: Which eq

30、uation(s) is/are identified?Q2: Estimate the identified equation(s).31AnswerOLS2SLS3233Note on the terminologyIn the previous slides, the exogenous variables excluded from the equation were called the instruments. In SEM (and in usual IV method too), people often refer to all the exogenous variables

31、 (regardless of whether they are included or excluded) as the instruments. The instruments that are excluded from the equation is called specifically as the excluded instruments. 34Simultaneous equations models with panel data.Consider the following SEM.The notation is a short hand notation for . Th

32、e same for .Due to the fixed effect term and , z-variables are correlated with the composite error terms. Therefore, the excluded exogenous variables cannot be used as instruments unless we do something.35To apply 2SLS, we should first (i) first-difference, or (i) demean the equations. First-differe

33、nced version Time demeaned (fixed effect) version36Then or are not correlated with the error term. Thus we can apply the 2SLS method. Estimation procedure is the same. First, determine which equation is identified. Then, use the excluded exogenous variable as the instruments in the 2SLS method.37An

34、applicationThe effect of prison population on the violent crime rate (Levitte 1996).This paper answers to the following question: To what extent an increase in prison population would decrease the violent crime?38Consider the following model. (Crime): the number of violent crimes per capita. (Prison

35、) prison population per capita. : intercepts (different at each year: just include year dummies.) z1: police per capita, log of income per capita, unemployment rate, proportions of black and those living in metropolitan areas, and age distributions.39First-differece the equation to eliminate the fixed effect ai.Even after eliminating the fixed effect, there still is the simultaneous equation bias, because the prison population is determined by the crime rate as well.40The simultaneity can be expressed in the SEM framework as:(