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Research MethodResearch Method

Lecture 12 (Ch16)Lecture 12 (Ch16)

Simultaneous Simultaneous Equations Models Equations Models

(SEMs)(SEMs)

©

IntroducdtionIntroducdtion

We have learned two “sources” of endogeneity.

1. Omitted variables 2. Errors in variables In this handout, we will learn another

source of endogeneity: Simultaneity.

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In econometrics, “endogeneity” usually means that an 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 ‘simultaneous equation’ sense, it is usually endogenous in econometric sense (i.e., correlated with the error term). We will see this soon.

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The nature of The nature of simultaneous equation.simultaneous equation.

Consider the following model describing equilibrium quantity of labor (in hours) in agricultural sector in a country.

Labor supply : hs=α1w+β1z1+u1

Labor demand: hd=α2w+β2z2+u2

hs 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.

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z1 would be the wage rate of the manufacturing sector. If the manufacturing wage increases, people would move to 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 shifter.

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Demand 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=hd

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Consider you have country level data. Then, for each country, we observe only the equilibirum labor supply and wage rate.

Demand: hi=α1wi+β1zi1+ui1

Supply: hi=α2wi+β2zi2+ui2

where i is the country subscript.

These two equations constitute a simultaneous equations model (SEM). These two equations are called the structural equations. α1,β1, α2, β2 are called the structural parameters. 7

In SEM framework, hi and 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 no way to distinguish whether one equation is demand or supply.

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Simultaneous equation Simultaneous equation biasbias

Consider the following simultaneous equation model.

y1=α1y2+β1z1+u1…………….(1)

y2=α2y1+β2z2+u2…………….(2)

In this model, y1 and y2 are endogenous variables since they are determined by the equilibrium between the two equations. z1 z2 are exogenous variables. 9

Since 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, u1 & u2, are assumed to be uncorrelated with each other.

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Now, solve the equations (1) and (2) for y1 and y2, then you get the following reduced form equations.

y1=п11z1+п12z2+v1

y2=п21z1+п22z2+v2

where п11= β1/(1- α1 α2)

п112= α1 β2/(1- α1 α2)

v1 =(u1+ α1 u2)/(1- α1 α2)

п21 =α2β1/(1- α2 α1)

п22 = β2/(1- α2 α1)

v2=(α2u1+u2)/(1- α2 α1)

These parameters are called the reduced form parameters. 11

You 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).

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However, you cannot estimate the structural equations with OLS. For example, consider the first structural equation.

y1=α1y2+β1z1+u1

Notice that Cov(y2, u1) =[α2/(1-α2α1)]E(u1

2)

=[α2/(1-α2α1)]σ21 ≠0

Thus, y2 is correlated with u1 (assuming that α2 ≠0.) In other words, y2 is endogenous in ‘econometric sense’.

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Thus, endogenous variables in SEM are usually endogenous in ‘econometric sense’ as well.

Thus, you cannot apply OLS to the structural equations.

Cov(y2, u1) =[α2/(1-α2α1)]σ21 can be used to

predict the direction of bias. If this is positive, OLS estimate of α1 will be biased upward. If it is negative, it will be biased downward.

The formula above does not carry over to more general models. But we can use this as a guide to check the direction of the bias.

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An exampleAn example Suppose 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 of murder rates.

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Thus, the observed murder rate and the police size are determined simultaneously by the following model.

(Murder)=α1(police)+β10+β1(Income per capita)+u1..(3)

(Police)=α2(Murder)+ β20+β2(other vars)+u2………..(4)

Allthe variables are the city-level variables. (Murder) is the number of murders per 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).

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However, since murder rate and police force are determined simultaneously, (police) is endogenous in equation (3). Thus OLS estimate for α1 is biased.

Question: What would be the direction of the bias?

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Identifying and estimating a Identifying and estimating a structural equation: structural equation:

2 equations case2 equations case When 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., eliminate the bias) by apply IV method.

In SEM, the term ‘identification’ is used slightly differently.

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Suppose the following model describing the supply and demand.

Supply: q =α1p+β1z1+u1

Demand: q =α2p+u1

Note that supply curve has an observed supply shifter z1, but demand has no obsedved supply shifter.

Given the data on q, p and z1, which equation can be estimated? That is, which is an identified equation?

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Demand

Supply: 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.

Because there 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.

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Now turn to a more general case.

(z11~z1k) 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.

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222212112202

111111121101

uzzyy

uzzyy

ll

kk

lz2

The condition for identification 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.

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The above condition have two components. First, at least one exogenous 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.

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ExampleExample

Labor supply of married 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.)

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114

131211101

)(

6

uomeNonWifeInc

kidsageeduclwagehours

22

232221302 expexp ueduchourslwage

Suppose that you are interested in estimating the first equation.

Since exp 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 similar way, you can see that the second equation is also identified.

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Estimating SEM using Estimating SEM using 2SLS2SLS

Once we have determined that an equation is identified, we can estimate it by two stage least square.

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Consider the labor supply equation example again. You are interested in estimating the first equation.

Suppose that the first equation is identified (both order and rank conditions are satisfied).

lwage is correlated with u1. Thus, OLS cannot be used.

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114

131211101

)(

6

uomeNonWifeInc

kidsageeduclwagehours

22

232221302 expexp ueduchourslwage

However, 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 the model (instrument exogeneity satisfied). Second exp and exp2 are correlated with lwage by the rank condition (instrument relevance satisfied).

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In general, you can use the excluded exogenous variables as the instruments.

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ExerciseExercise Consider the following simultaneous

equation model.

Q1: Which equation(s) is/are identified?

Q2: Estimate the identified equation(s).

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114

131211101

)(

6

uNonWifeInc

kidsageeduclwagehours

22

232221302 expexp ueduchourslwage

AnswerAnswer

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_cons 1523.775 309.4226 4.92 0.000 915.5734 2131.976 nwifeinc -5.918459 3.385146 -1.75 0.081 -12.57231 .7353893 kidslt6 -328.8584 126.681 -2.60 0.010 -577.8629 -79.85399 age .5622541 5.360839 0.10 0.917 -9.975019 11.09953 educ -6.62187 18.43784 -0.36 0.720 -42.86331 29.61957 lwage -2.046796 82.02275 -0.02 0.980 -163.2708 159.1772 hours Coef. Std. Err. t P>|t| [95% Conf. Interval] Robust

Root MSE = 766.63 R-squared = 0.0361 Prob > F = 0.0324 F( 5, 422) = 2.46Linear regression Number of obs = 428

. reg hours lwage educ age kidslt6 nwifeinc, robust

Instruments: educ age kidslt6 nwifeinc exper expersqInstrumented: lwage _cons 2225.662 603.0964 3.69 0.000 1043.615 3407.709 nwifeinc -10.16959 5.287486 -1.92 0.054 -20.53287 .1936911 kidslt6 -198.1543 208.4247 -0.95 0.342 -606.6592 210.3506 age -7.806092 10.48746 -0.74 0.457 -28.36114 12.74896 educ -183.7513 67.78742 -2.71 0.007 -316.6122 -50.89039 lwage 1639.556 593.3108 2.76 0.006 476.6879 2802.423 hours Coef. Std. Err. z P>|z| [95% Conf. Interval] Robust

Root MSE = 1344.7 R-squared = . Prob > chi2 = 0.0274 Wald chi2(5) = 12.60Instrumental variables (2SLS) regression Number of obs = 428

. ivregress 2sls hours educ age kidslt6 nwifeinc (lwage=exper expersq), robust

OLS

2SLS

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Instruments: educ exper expersq age kidslt6 nwifeincInstrumented: hours _cons -.6557254 .4097655 -1.60 0.110 -1.458851 .1474001 expersq -.0007058 .0004265 -1.65 0.098 -.0015418 .0001302 exper .0345824 .0185052 1.87 0.062 -.0016872 .0708519 educ .11033 .0148178 7.45 0.000 .0812877 .1393723 hours .0001259 .0002924 0.43 0.667 -.0004472 .000699 lwage Coef. Std. Err. z P>|z| [95% Conf. Interval] Robust

Root MSE = .67545 R-squared = 0.1257 Prob > chi2 = 0.0000 Wald chi2(4) = 83.56Instrumental variables (2SLS) regression Number of obs = 428

. ivregress 2sls lwage (hours=age kidslt6 nwifeinc) educ exper expersq, robust

_cons -.4619955 .2113449 -2.19 0.029 -.8774124 -.0465786 expersq -.0008585 .0004166 -2.06 0.040 -.0016773 -.0000397 exper .0447035 .0152503 2.93 0.004 .0147277 .0746793 educ .1062139 .0133269 7.97 0.000 .0800187 .1324091 hours -.0000565 .0000654 -0.86 0.388 -.0001852 .0000721 lwage Coef. Std. Err. t P>|t| [95% Conf. Interval] Robust

Root MSE = .6659 R-squared = 0.1601 Prob > F = 0.0000 F( 4, 423) = 20.24Linear regression Number of obs = 428

. reg lwage hours educ exper expersq, robust

Note on the terminologyNote on the terminology In 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 (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’.

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Simultaneous equations Simultaneous equations models with panel data.models with panel data.

Consider the following SEM.

The notation is a short hand notation for . The 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.

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)(

)(

2222122

1111211

itiititit

itiititit

uazyy

uazyy

11itz

22itz

1ia 2ia

tkkt zz 111111

To apply 2SLS, we should first (i) first-difference, or (i) demean the equations.

First-differenced version

Time demeaned (fixed effect) version

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222122

111211

itititit

itititit

uzyy

uzyy

222122

111211

itititit

itititit

uzyy

uzyy

Then 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.

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21, itit zz 11, itit yz

An applicationAn application

The 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?

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Consider the following model.

(Crime): the number of violent crimes per capita. (Prison) 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.

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)4)......(()log()log( 11111 itiittit uazprisoncrime

t

First-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.

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)5......()log()log( 1111 itittit uzprisoncrime

The simultaneity can be expressed in the SEM framework as:

(Exogenous vars) in equation (7) could contain . However, in order to identify the crime equation (6), (exogenous vars) should contain variables that are not included the crime equation. What can be the variable?

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)7.....(........................................Vars)(Exogenous

)log()log(

22it

22

it

tit

u

crimeprison

)6......()log()log( 11111 ititittit uzprisoncrime

1itz

Levitte (1996) used the overcrowding litigation as the excluded instruments.

In the US, prisoner’s right groups have filed law suits to mitigate the overcrowding of the prisons.

When the law suit is successful, the court orders the prisons to mitigate the overcrowding of the prisons. It usually takes the form of population caps.

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Thus, overcrowding litigation, if victories are achieved, will affect the change in the prison population. At the same time, it is reasonable to assume that the overcrowding litigation affect crime rate only through prison population.

Thus, the model is now:

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)8......()log()log( 11111 ititittit uzprisoncrime

)9....(..........) ((Final)

)log()log(

22it

22

it

tit

uFactorsOther

crimeprison

Whether the final decisions about the overcrowiding litigation is reached.

The resultsThe results

44 _cons -.0056706 .0282869 -0.20 0.841 -.0612093 .0498682 y93 .0087141 .0271283 0.32 0.748 -.0445498 .0619781 y92 .0081502 .0245474 0.33 0.740 -.0400464 .0563468 y91 .038884 .0235126 1.65 0.099 -.0072807 .0850488 y90 .0871704 .0226556 3.85 0.000 .0426883 .1316525 y89 .0253571 .0235964 1.07 0.283 -.0209723 .0716865 y88 .0347581 .0221166 1.57 0.117 -.0086658 .0781819 y87 -.0239597 .0237496 -1.01 0.313 -.0705898 .0226704 y86 .0440948 .0272085 1.62 0.106 -.0093267 .0975162 y85 .0094042 .0229413 0.41 0.682 -.0356389 .0544473 y84 -.0136596 .0240967 -0.57 0.571 -.0609713 .033652 y83 -.0421775 .024122 -1.75 0.081 -.0895389 .0051838 y82 -.0407726 .0225609 -1.81 0.071 -.0850688 .0035237 y81 -.0686258 .0203006 -3.38 0.001 -.1084842 -.0287675 cag25_34 2.879946 2.549853 1.13 0.259 -2.126456 7.886347 cag18_24 2.412758 2.937888 0.82 0.412 -3.355514 8.18103 cag15_17 4.98384 4.998959 1.00 0.319 -4.831156 14.79884 cag0_14 .989306 2.38922 0.41 0.679 -3.701708 5.68032 cmetro .5383056 1.417525 0.38 0.704 -2.244874 3.321485 cblack -.0147435 .0336211 -0.44 0.661 -.0807554 .0512685 cunem .41126 .399978 1.03 0.304 -.3740601 1.19658 gincpc .7383676 .2294963 3.22 0.001 .2877727 1.188963 gpolpc .0514239 .0553029 0.93 0.353 -.0571583 .1600061 gpris -.1808974 .0557664 -3.24 0.001 -.2903896 -.0714052 gcriv Coef. Std. Err. t P>|t| [95% Conf. Interval] Robust

Root MSE = .07893 R-squared = 0.2311 Prob > F = 0.0000 F( 23, 690) = 10.66Linear regression Number of obs = 714

> y81 y82 y83 y84 y85 y86 y87 y88 y89 y90 y91 y92 y93, robust. reg gcriv gpris gpolpc gincpc cunem cblack cmetro cag0_14 cag15_17 cag18_24 cag25_34

Simple first differenced model. The coefficient would be biased.

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final1 final2 cag25_34 y81 y82 y83 y84 y85 y86 y87 y88 y89 y90 y91 y92 y93Instruments: gpolpc gincpc cunem cblack cmetro cag0_14 cag15_17 cag18_24Instrumented: gpris _cons .0148377 .0367185 0.40 0.686 -.0571292 .0868047 y93 .0222739 .0340501 0.65 0.513 -.044463 .0890109 y92 .0266574 .0300795 0.89 0.375 -.0322974 .0856122 y91 .0618481 .0303677 2.04 0.042 .0023285 .1213676 y90 .1442652 .036031 4.00 0.000 .0736458 .2148846 y89 .0430862 .0292807 1.47 0.141 -.0143029 .1004754 y88 .0532706 .0284254 1.87 0.061 -.0024422 .1089833 y87 .004771 .0320625 0.15 0.882 -.0580704 .0676124 y86 .0921857 .0385744 2.39 0.017 .0165812 .1677902 y85 .0354026 .0308006 1.15 0.250 -.0249655 .0957707 y84 .0128703 .0320581 0.40 0.688 -.0499624 .0757031 y83 .024703 .0388549 0.64 0.525 -.0514511 .1008572 y82 .0284616 .0390173 0.73 0.466 -.0480108 .104934 y81 -.0560732 .0251341 -2.23 0.026 -.105335 -.0068113 cag25_34 2.319993 2.87316 0.81 0.419 -3.311297 7.951282 cag18_24 3.358348 3.321347 1.01 0.312 -3.151374 9.868069 cag15_17 3.549945 5.910124 0.60 0.548 -8.033685 15.13358 cag0_14 3.379384 2.938685 1.15 0.250 -2.380332 9.1391 cmetro -.591517 1.603944 -0.37 0.712 -3.73519 2.552156 cblack -.0158476 .0403465 -0.39 0.694 -.0949254 .0632302 cunem .5236958 .4809308 1.09 0.276 -.4189114 1.466303 gincpc .9101992 .3257538 2.79 0.005 .2717334 1.548665 gpolpc .035315 .0602729 0.59 0.558 -.0828178 .1534478 gpris -1.031956 .3314008 -3.11 0.002 -1.68149 -.3824227 gcriv Coef. Std. Err. z P>|z| [95% Conf. Interval] Robust

Root MSE = .09385 R-squared = . Prob > chi2 = 0.0000 Wald chi2(23) = 145.87Instrumental variables (2SLS) regression Number of obs = 714

> 4 cag15_17 cag18_24 cag25_34 y81 y82 y83 y84 y85 y86 y87 y88 y89 y90 y91 y92 y93, robust. ivregress 2sls gcriv (gpris= final1 final2) gpolpc gincpc cunem cblack cmetro cag0_1

First-difference plus 2SLS to eliminate the simultaneous equation bias.

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Robust regression F(1,689) = 9.09532 (p = 0.0027) Robust score chi2(1) = 6.14067 (p = 0.0132)

Ho: variables are exogenous Tests of endogeneity

. estat endog

Score chi2(1) = .012929 (p = 0.9095)

Test of overidentifying restrictions:

. estat overid

The results of the overidentifying restriction test and endogeneity test.

47 _cons .0272013 .0224239 1.21 0.226 -.016826 .0712287 final2 -.0529558 .0195862 -2.70 0.007 -.0914117 -.0144999 final1 -.077488 .0162845 -4.76 0.000 -.1094613 -.0455148 y93 .0134109 .0201717 0.66 0.506 -.0261945 .0530164 y92 .0190481 .0175037 1.09 0.277 -.0153188 .053415 y91 .0263719 .0176008 1.50 0.135 -.0081856 .0609295 y90 .0635926 .0175546 3.62 0.000 .0291257 .0980595 y89 .0184651 .0174176 1.06 0.289 -.0157328 .052663 y88 .019245 .0171349 1.12 0.262 -.0143979 .0528878 y87 .0312716 .0178416 1.75 0.080 -.0037587 .066302 y86 .0541489 .0212384 2.55 0.011 .0124491 .0958488 y85 .0279051 .0175523 1.59 0.112 -.0065574 .0623676 y84 .0289763 .0190171 1.52 0.128 -.008362 .0663146 y83 .0767785 .0170559 4.50 0.000 .0432907 .1102663 y82 .0773503 .0174195 4.44 0.000 .0431486 .111552 y81 .0124113 .0156429 0.79 0.428 -.0183022 .0431248 cag25_34 -1.031684 1.813735 -0.57 0.570 -4.592794 2.529427 cag18_24 .9533678 1.630762 0.58 0.559 -2.248492 4.155228 cag15_17 -1.608738 3.739701 -0.43 0.667 -8.951316 5.73384 cag0_14 2.617307 1.665526 1.57 0.117 -.6528087 5.887423 cmetro -1.418389 .8617595 -1.65 0.100 -3.110379 .2736011 cblack -.0044763 .0259464 -0.17 0.863 -.0554198 .0464671 cunem .1616595 .3106113 0.52 0.603 -.4481988 .7715178 gincpc .2095521 .1941286 1.08 0.281 -.1716025 .5907068 gpolpc -.0286921 .033455 -0.86 0.391 -.0943781 .0369938 gpris Coef. Std. Err. t P>|t| [95% Conf. Interval] Robust

Root MSE = 0.0624 Adj R-squared = 0.1226 R-squared = 0.1522 Prob > F = 0.0000 F( 24, 689) = 5.64 Number of obs = 714

First-stage regressions

The first stage regression

Overcrowding litigation reduces the prison population growth.