Econometrics

Week 9

Institute of Economic StudiesFaculty of Social Sciences

Charles University in Prague

Fall 2012

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Recommended Reading

For the todaySimultaneous Equations ModelsChapter 16 (pp. 501 – 523).

In the next weekLimited Dependent Variable ModelsChapter 17 (pp. 529 – 565).

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Today’s Talk

Previous lecture, we have shown how instrumentalvariables can be useful in solving endogeny problems:

omitted variablesmeasurement error

In both cases, we could use OLS if we have better data.Today, we will consider another important form ofendogeneity which is simultaneity.Simultaneity is a specific type of endogeneity problem inwhich the explanatory variable is jointly determined withthe dependent variable.

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Example of Simultaneity

The most important is to remember that each equation inthe system should have its own casual interpretation.

Supply EquationConsider labor supply function:

hs = a1w + b1z1 + u1,

where hs is (annual) labor hours supplied by workers, w averagehourly wage, z1 some observed variable affecting supply (i.e.average manufacturing wage in country)

We call supply equation a structural equation as itcomes from the economic theory and has casualinterpretationα1 measures how labor supply changes with change of wageWhen hs and w are in logarithms, α1 is labor supplyelasticity.

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Example of Simultaneity cont.

Note that z1 and u1 shifts the supply, z1 is observed, whileu1 is not.So how does this equation differ from what we have studiedpreviously?wage w can not be viewed as exogenous as supply equationholds for all positive values of wage.Thus we can not simply regress hs on w as w isendogenous variable.Thus we have to collect some more data.We have to understand that the data are best described bylabor supply and demand interaction.

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Example of Simultaneity cont.

Demand EquationConsider labor demand function:

hd = a2w + b2z2 + u2,

where hs is (annual) labor hours demanded, z2 some observedvariable affecting demand (i.e. land area)

Demand equation is also structural equation.These two equations are linked through intersection ofsupply and demand, which is equilibrium his = hid.As we observe only equilibrium hours for each country i,we denote hi observed hours.

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Example of Simultaneity cont.

Labor Supply and Demand Equation

hi = a1wi + b1zi1 + ui1,

hi = a2wi + b2zi2 + ui2,

while we call this simultaneous equations model(SEM).hi and wi are endogenous variables determined byequations.zi1 and zi2 are exogenous variables determined outsideof the system.Without zi1 and zi2 we can not recognize demand fromsupply ⇒ they identify the equations.Without supply and demand shifters, we are not able toestimate the system.

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Simultaneity Bias in OLS

Consider structural model

y1 = α1y2 + β1z1 + u1

y2 = α2y1 + β2z2 + u2

Variables z1 and z2 are exogenous (Cov(z1, u1) = 0,Cov(z2, u2) = 0)If we put first equation to the second one, we have:y2 = α2(α1y2 + β1z1 + u1) + β2z2 + u2

Reduced form for y2

y2 = π21z1 + π22z2 + ν2,

where π21 = α2β1/(1− α2α1), π22 = β2/(1− α2α1) andν2 = (α2u1 + u2)/(1− α2α1)

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Simultaneity Bias in OLS cont.

Because ν2 is linear function of u1 and u2, u1 and u2 areuncorrelated with z1 and z2.⇒ ν2 is uncorrelated with π21 and π22 and OLSestimatesare consistent.While π̂21 and π̂22 will be consistent, α̂1 and β̂1 from thefirst equation will be inconsistent.This is because y2 and u1 are correlated (it can be seendirectly from the reduced form)When y2 and u1 are correlated because of simultaneity,OLS will suffer from the simultaneity bias.Interactive Example in Mathematica.(NOTE: To make this link work, you need to have Lecture9.cdf at the same directory

and Free CDF player installed on your computer.)

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Simultaneity Bias in OLS cont.

NoteThus we can see that estimating structural equation in asimultaneous equations system by OLS results in biased andinconsistent estimator.

We can solve this problem by using two-stage leastsquare estimator (2SLS)

As we specify the structural equation for each endogenousvariable, we can immediately see if sufficient instrumentalvariables are in the equation.We call this identification problem.

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Identification in Two-Equation System

Example: Supply and Demand of milk consumption

q = α1p+ β1z1 + u1

q = α2p+ u2,

where q is milk consumption, p its price, z1 price of cattle feed(exogenous to supply and demand of milk).

First equation must be supply (↑ z1 ⇒↓ supply, but notdemand).Which equation can be estimated?.The answer is, the one which is identified!Identified is demand (second), as it has z1, but we have noIV for the price in the supply equation.We need more exogenous IV in second equation to identifyalso first one (supply).

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Identification in Two-Equation System cont.

General two-equation model

y1 = β10 + α1y2 + β11z11 + . . .+ β1kz1k + u1

y2 = β20 + α2y1 + β21z21 + . . .+ β2kz2k + u2

where y1 and y2 are endogenous variables and u1, u2 are errorterms and we have k exogenous variables z

Under what assumptions can we estimate the parametersin this model?This is identification issue.

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Rank Condition for Identification

A necessary and sufficient condition for one of theequations to be identified is:

Rank ConditionThe first equation in a two-equation simultaneous equationsmodel is identified if and only if the second equation contains atleast one exogenous variable (with nonzero coefficient) that isexcluded from the first equation.

Order condition is necessary for the rank condition.

Order ConditionThe first equation in a two-equation simultaneous equationsmodel is identified if at least one exogenous variable is excludedfrom the first equation.

Order condition is trivial to check.Rank condition requires at least one nonzero coefficient ofexcluded exogenous variable from the first equation. 13 / 21

Identification in Systems with More Equations

Identification of systems with more equations requirematrix algebra, you have to wait until the advancedeconometrics.But, we can discuss some issues.

Three-equation system

y1 = α12y2 + α13y3 + β11z1 + u1

y2 = α21y1 + β22z2 + β23z3 + u2

y3 = α32y2 + β31z1 + β32z2 + β33z3 + β34z4 + u3,

where yi are endogenous and zj are exogenous.

Which of these equations can be consistently estimated?Without difficult computations we can see that thirdequation contains all IV variables, thus y2 has no IV andthis equation can not be estimated consistently.

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Order Condition for Identification

Order ConditionAn equation in any system of equations model satisfies theorder condition for identification if the number of excludedexogenous variables from the equation is at least as large as thenumber of right-hand side endogenous variables.

In our three-equation system, second equation passes thiscondition ⇒ there is z4 excluded for y1

first equation also passes the order condition ⇒ there arethree excluded variables for y2, z2, z3 and z4.BUT remember: order condition is only necessary, notsufficient condition for identificationSuppose β34 = 0. Then second equation is not identified.We need to extend rank condition (but you have to waituntil advanced econometrics course).

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Order Condition for Identification cont.

If we have more excluded exogenous variables from theequation than included endogenous variables, equation isoveridentified.The first equation from our example is over identified.Second equation is just identified.Third equation is unidentified and can not be estimated.Each identified equation can be estimated by 2SLS.We also know methods which are more efficient, likethree-stage least squares (3SLS).But again, these are little more complicated and you haveto wait until advanced econometrics course.

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Simultanous Equations Models with Time Series

Let’s consider a simple Keynesian model

Ct = β0 + β1(Yt − Tt) + β2rt + ut1

It = γ0 + γ1rt + ut2

Yt ≡ Ct + It +Gt,

where Ct is consumption, Yt is income, Tt is tax receipts, rtinterest rate, It investment and Gt government spending.

We have 3 equations ⇒ 3 endogenous variables.So Tt, Gt and rt should be exogenous.First and third equation can then be estimated by 2SLS,second by OLS.But the problem is that it is hard to assume the exogenity,i.e. taxes clearly depend on income!

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Simultanous Equations Models with Time Series

While it is hard to say that Tt, Gt and rt are exogenous, wecan specify some other exogenous variables.In fact, we can make model dynamic.For example:It = γ0 + γ1rt + γ2Yt−1 + ut2

Lagged exogenous variable can be assumed, if it isuncorrelated with ut2

We call this type of variables predetermined variable.Also, we can put lagged variable into the consumptionequation:Ct = β0 + β1(Yt − Tt) + β2rt + β3Ct−1 + ut1

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Simultanous Equations Models with Time Series

Still, we have to be really careful, as in time series models,we have certain assumptions.Recall weak dependence assumption.Consumption, income, investment violate theseassumptions most of the times.They have trends, they have unit roots...This does not mean that SEM are not useful in time seriesmodels.We just have to be careful about time series assumptions!A simple solution would be to use differenced data (growthrates).We can also consider adding trend as IV.

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Simultanous Equations Models with Panel Data

We are able to use SME also in panel data in somesituations.In this case, we use 2 steps of estimation:

(1) eliminate unobserved effects from the equation usingfixed effects transformation or first differencing(2) find instrumental variable for endogenous variables inthe transformed equation.

Second step is especially difficult, as IV has to vary overtime.

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Thank you

Thank you very much for your attention!

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