Simultaneous Equations
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Simultaneous Equations. y 1 = a 1 y 2 + b 1 z 1 + u 1 y 2 = a 2 y 1 + b 2 z 2 + u 2. Simultaneity. Simultaneity is a specific type of endogeneity problem in which the explanatory variable is jointly determined with the dependent variable

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Simultaneous Equations

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Economics 20 prof anderson

Simultaneous Equations

y1 = a1y2 + b1z1 + u1

y2 = a2y1 + b2z2 + u2

Economics 20 - Prof. Anderson


Simultaneity

Simultaneity

  • Simultaneity is a specific type of endogeneity problem in which the explanatory variable is jointly determined with the dependent variable

  • As with other types of endogeneity, IV estimation can solve the problem

  • Some special issues to consider with simultaneous equations models (SEM)

Economics 20 - Prof. Anderson


Supply and demand example

Supply and Demand Example

  • Start with an equation you’d like to estimate, say a labor supply function

  • hs = a1w + b1z + u1, where

  • w is the wage and z is a supply shifter

  • Call this a structural equation – it’s derived from economic theory and has a causal interpretation where w directly affects hs

Economics 20 - Prof. Anderson


Example cont

Example (cont)

  • Problem that can’t just regress observed hours on wage, since observed hours are determined by the equilibrium of supply and demand

  • Consider a second structural equation, in this case the labor demand function

  • hd = a2w + u2

  • So hours are determined by a SEM

Economics 20 - Prof. Anderson


Example cont1

Example (cont)

  • Both h and w are endogenous because they are both determined by the equilibrium of supply and demand

  • z is exogenous, and it’s the availability of this exogenous supply shifter that allows us to identify the structural demand equation

  • With no observed demand shifters, supply is not identified and cannot be estimated

Economics 20 - Prof. Anderson


Identification of demand equation

Identification of Demand Equation

S (z=z1)

D

w

S (z=z2)

S (z=z3)

h

Economics 20 - Prof. Anderson


Using iv to estimate demand

Using IV to Estimate Demand

  • So, we can estimate the structural demand equation, using z as an instrument for w

  • First stage equation is w = p0 + p1z + v2

  • Second stage equation is h = a2ŵ + u2

  • Thus, 2SLS provides a consistent estimator of a2, the slope of the demand curve

  • We cannot estimate a1, the slope of the supply curve

Economics 20 - Prof. Anderson


The general sem

The General SEM

  • Suppose you want to estimate the structural equation: y1 = a1y2 + b1z1 + u1

  • where, y2 = a2y1 + b2z2 + u2

  • Thus, y2 = a2(a1y2 + b1z1 + u1) + b2z2 + u2

  • So, (1 – a2a1)y2 = a2b1z1 + b2z2 + a2 u1 + u2, which can be rewritten as

  • y2 = p1z1 + p2z2 + v2

Economics 20 - Prof. Anderson


The general sem continued

The General SEM (continued)

  • By substituting this reduced form in for y2, we can see that since v2 is a linear function of u1, y2 is correlated with the error term and a1 is biased – call it simultaneity bias

  • The sign of the bias is complicated, but can use the simple regression as a rule of thumb

  • In the simple regression case, the bias is the same sign as a2/(1 – a2a1)

Economics 20 - Prof. Anderson


Identification of general sem

Identification of General SEM

  • Let z1 be all the exogenous variables in the first equation, and z2 be all the exogenous variables in the second equation

  • It’s okay for there to be overlap in z1 and z2

  • To identify equation 1, there must be some variables in z2 that are not in z1

  • To identify equation 2, there must be some variables in z1 that are not in z2

Economics 20 - Prof. Anderson


Rank and order conditions

Rank and Order Conditions

  • We refer to this as the rank condition

  • Note that the exogenous variable excluded from the first equation must have a non-zero coefficient in the second equation for the rank condition to hold

  • Note that the order condition clearly holds if the rank condition does – there will be an exogenous variable for the endogenous one

Economics 20 - Prof. Anderson


Estimation of the general sem

Estimation of the General SEM

  • Estimation of SEM is straightforward

  • The instruments for 2SLS are the exogenous variables from both equations

  • Can extend the idea to systems with more than 2 equations

  • For a given identified equation, the instruments are all of the exogenous variables in the whole system

Economics 20 - Prof. Anderson


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