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Lecture: Simultaneous Equation Model (Wooldridge’s Book Chapter 16)

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Model

• Consider a system of two regressions

y1 = β1y2 +u1 (1)

y2 = β2y1 +u2 (2)

• This is a simultaneous equation model (SEM) since y1 and y2 are determinedsimultaneously.

• Both variables are determined within the model, so are endogenous, and denoted by letter y.

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Example

• The demand-supply model in microeconomics includes demand function and supplyfunction

• y1 is the quantity of good; y2 is the price

• If β1 < 0,β2 > 0, then (1) is the demand function while (2) is the supply function; u1 is thedemand shock and u2 is the supply shock.

• Another example is the Keynesian cross (45 degree line) model in which y1 is the nationalincome and y2 is total consumption.

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Structural Form

• (1) and (2) are structural in the sense that they are directly implied by economics theory.

• We assumeE(u1u2) = 0 (3)

So the structural errors are uncorrelated (orthogonal).

• Our goal is to estimate the structural coefficient β that measures the causal effect of oneendogenous variable on the other endogenous variable

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Simultaneity Bias

• Plugging (1) into the right hand side of (2) leads to

y2 = β2(β1y2 +u1)+u2.

• After collecting terms, we have

y2 =β2u1 +u2

1−β2β1, (4)

which indicates that

E(y2u1) =β2Eu2

11−β2β1

= 0 ⇒ cov(y2,u1) = 0. (5)

• So structural model (1) suffers endogeneity issue (simultaneity bias)—the regressor in (1) iscorrelated with the error term. Consequently, OLS applied to (1) gives inconsistent andbiased estimate. So does (2).

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Example

• Suppose there is a demand shock u1

• u1 affects the quantity y1 through the demand function (1)

• Next quantity y1 affects the price y2 through the supply function (2). Some people call thisreverse causation.

• So u1 affects y2, and the two variables are correlated. In other words, the regressor in (1) isendogenous.

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Endogeneity

• Typically an economic theory implies SEM, so several variables are determinedsimultaneously within the model. Those variables are endogenous from the economicsperspective

• We just show SEM suffers simultaneity bias. Therefore those variables are correlated withthe error, so are endogenous from the econometrics perspective

• In short, economic endogeneity is closely related to econometric (statistical) endogeneity.

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Graph

We can not identify either demand curve or supply curve from the scatter plot of quantity versusprice. However, if there are some exogenous variables, say, input price that can shift the supplycurve, then we can identify the demand curve.

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SEM with Exogenous Regressors

• Now augment the structural model with exogenous regressors

y1 = β1y2 + c1z1 +u1 (6)

y2 = β2y1 + c2z2 +u2 (7)

• For instance, z1 is income; z2 is input price

• z1 and z2 are determined outside the model, so are exogenous (pre-determined)

• Statistically, exogeneity means that

E(z1u1) = 0;E(z1u2) = 0;E(z2u1) = 0;E(z2u2) = 0 (8)

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Reduced Form

The reduced form expresses the endogenous variables in terms of exogenous variables only

y1 =c1z1 +β1c2z2 + e1

1−β2β1(9)

y2 =β2c1z1 + c2z2 + e2

1−β2β1(10)

e1 = u1 +β1u2 (11)

e2 = β2u1 +u2 (12)

(9) and (10) are reduced forms, and only exogenous variables z1 and z2 appear on the right handside (RHS)

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Reduced Form Continued

• Let

π11 =c1

1−β2β1;π12 =

β1c2

1−β2β1;e∗1 =

e1

1−β2β1

π21 =β2c1

1−β2β1;π22 =

c2

1−β2β1;e∗2 =

e2

1−β2β1

• The reduced form can be rewritten as

y1 = π11z1 +π12z2 + e∗1 (13)

y2 = π21z1 +π22z2 + e∗2 (14)

where e∗ is reduced-form error, which is linear function of structural error.

• Note that reduced-form error is correlated cov(e∗1,e∗2) = 0, whereas the structural error is

uncorrelated.

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Reduced Form Continued

• Note that all exogenous variables appear on the right hand side of each reduced form; bycontrast, the structural form has endogenous variable and some exogenous variables on theright hand side. As a result, OLS applied to structural form is inconsistent, whereas OLSapplied to reduced form is consistent

• Reduced form (14) is the first-stage regression if we want to use 2SLS estimator to obtainthe causal effect of y2 on y1. Notice that all exogenous variables are used as regressors inthe first-stage regression.

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Indirect Least Squares Estimator (ILS)

• OLS applied to (13) and (14) separately gives consistent estimate for πs.

• However, we are interested in the coefficients in the structural form.

• The indirect least squares estimator (ILS) estimates the structural-form coefficients β basedon the estimated reduced-form coefficient π

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Indirect Least Squares Estimator Continued

• The ILS estimators for structural-form coefficient β are

βILS1 =

π12

π22(15)

βILS2 =

π21

π11(16)

provided that

c2 = 0 (17)

c1 = 0 (18)

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Identification and Exclusion Restrictions

• β1 cannot be identified if c2 = 0. So identification for β1 requires c2 = 0

• c2 = 0 indicates that there is an exogenous variable z2 which is excluded from the firststructural equation (order condition) but appears in the second structural equation withnon-zero coefficient (rank condition)

• For the demand-and-supply example, the demand function can be identified if input price ispresent in the supply function. Graphically the demand curve can be traced out (identified)when supply curve shifts due to varying input price.

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Remarks

• β1 is over-identified if there are more than one exogenous variables that are excluded fromthe first equation and appear in the second equation with non-zero coefficients

• In that case, the ILS estimator for β1 is not unique (Exercise), a big disadvantage of ILS.

• Another disadvantage of ILS is, βILS is nonlinear function of π, so deriving the varianceentails delta method

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Delta Method

• For simplicity, let π and β = f (π) be scalars. Consider the first order Taylor expansion

β = f (π)≈ f (π)+ f ′(π)(π −π)

which implies that

var(β ) = ( f ′(π))2var(π)

• More generally, for vectors we have

var-covariance(β ) =

(∂ β∂ π

)′

var-covariance(π)

(∂ β∂ π

)

where ∂ β∂ π is called gradient (column) vector.

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2SLS Estimator

• The nice by-product of the structural model is that instrumental variables are readilyavailable.

• The reduced form (9) and (10) clearly show that the exogenous variables z1 and z2 arecorrelated with the endogenous regressors y1 and y2. Moreover, we assume z1 and z2 areuncorrelated with the error u, see (8)

• So z1 and z2 are instrumental variables for y1 and y2, if the exogeneity assumption (8) holds

• Essentially the 2SLS estimator replaces the endogenous regressors with their exogenousparts, and we use instrumental variables to isolate those exogenous parts.

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2SLS Estimator Continued

• Step 1: Estimate the reduced form (13) and (14) (first stage) using OLS and keep the fittedvalues

y1 = π11z1 + π12z2 (19)

y2 = π21z1 + π22z2 (20)

• Step 2: Replace the endogenous regressors with fitted values, and fit the second-stageregressions using OLS

y1 = β1y2 + c1z1 +u1 (21)

y2 = β2y1 + c2z2 +u2 (22)

• y1 is the exogenous part of y1; y2 is the exogenous part of y2; Both are linear combinationsof exogenous z1 and z2. (Where are the endogenous parts?)

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2SLS Estimator Stata

• For example, to get β 2SLS1 in (6) , using

ivreg y1 (y2 = z2) z1

where y1 is the dependent variable, y2 is the endogenous regressor, z2 is the excludedexogenous variable, and z1 is the included exogenous variable (control variable).

• Exercise: what is the stata command to get β 2SLS2 in (7)? You need to think carefully which

variable is which.

• This command will first run regression (19), then (21).

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(Optional) Seemingly Unrelated Regression (SUR)

• Reduced form (13) and (14) are example of seemingly unrelated regressions

• They have different LHS variables, so seem unrelated.

• They are indeed related because the reduce-form errors are correlated across equations, i.e.,

cov(e∗1,e∗2) = 0,

see (11) and (12)

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(Optional) SUR Continued

• Generally the optimal estimator for SUR model is generalized least squares estimator(GLS), due to the correlation between errors across regressions.

• However, if each equation in SUR has the identical RHS variables, GLS becomesequation-by-equation OLS

• The STATA command to estimate SUR model using GLS estimator is

sureg (y1 x1)(y2 x2)

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(Optional) Gauss-Markov Theorem

• If the error is homoskedastic and uncorrelated, then OLS estimator is the best linearunbiased estimator (BLUE) conditional on the regressors.

• See theorem 10.4 in the textbook for details

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(Optional) Generalized Least Squared Estimator I

• There is an estimator better than OLS if error is heteroskedastic.

• Suppose the model is

yi = βxi +ui,E(u2i ) = σ2

i , (Heteroskedasticity) (23)

• Consider the transformed model

y∗i = βx∗i +u∗i (24)

y∗i =yi

σi,x∗i =

xi

σi,u∗i =

ui

σi(25)

where the transformed error is homoskedastic: E(u∗2i ) = 1

• The estimator better than OLS is GLS, which is the OLS applied to the transformedregression (24).

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(Optional) Generalized Least Squared Estimator II

• Consider a model with correlated error

yi = βxi +ui,ui = ρui−1 + vi, (Correlation) (26)

• Consider the transformed model

y∗i = βx∗i +u∗i (27)

y∗i = yi −ρyi−1,x∗i = xi −ρxi−1 (28)

where the transformed error is uncorrelated: E(u∗i u∗i−1) = 0

• GLS is the OLS applied to the transformed regression (27).

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(Optional) Matrix Algebra for GLS

We use GLS when E(UU′|X) = Ω = σ2I. Because Ω is symmetric and positive definite, there isspectral decomposition Ω = AA′. Now consider the transformed model

Y∗ = X∗β +U∗

where Y∗ = A−1Y,X∗ = A−1X,U∗ = A−1U. It follows that GLS estimator is OLS applied to thetransformed regression, which satisfies the conditions of Gauss-Markov Theorem. That is,

E(U∗U∗′ |X∗) = I.

In short,

βGLS =(

X∗′X∗)−1(

X∗′Y∗)=(X′Ω−1X

)−1 (X′Ω−1Y)

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(Optional) STATA and GLS

• For instance, the STATA command

prais y x

reports the GLS estimator assuming the error is an AR(1) process, and so serially correlated.

• You use command sureg to obtain GLS estimator for SUR model.

• Alternatively, you can generate the transformed variables, and fit the transformed regressionusing OLS

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OLS is inconsistent when applied to simultaneous equation model (SEM).2SLS estimator is consistent. ILS estimator is also consistent, but requiresdelta method to obtain the variance and standard error. The benefit ofconsidering SEM is tremendous in that instrumental variables are readilyindicated by the structural model!