Simultaneous equations y1=ay2+B1=1+l ◆y2=ny1+B2x2+12 Economics 20- Prof anderson
Economics 20 - Prof. Anderson 1 Simultaneous Equations y1 = a1 y2 + b1 z1 + u1 y2 = a2 y1 + b2 z2 + u2
Simultaneity o Simultaneity is a specific type of endogeneity problem in which the explanatory variable is jointly determined with the dependent variable o As with other types of endogeneity, IV estimation can solve the problem o Some special issues to consider with simultaneous equations modelS (SEM) Economics 20- Prof anderson
Economics 20 - Prof. Anderson 2 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)
Supply and demand example o Start with an equation you' d like to estimate, say a labor supply function ◆h。=anw+B+u, where S 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
Economics 20 - Prof. Anderson 3 Supply and Demand Example Start with an equation you’d like to estimate, say a labor supply function hs = a1w + b1 z + 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
Example(cont) Problem that cant 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 ◆hnt=21+l2 d by So hours are determined a seM Economics 20- Prof anderson 4
Economics 20 - Prof. Anderson 4 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
Example(cont) o Both h and w are endogenous because they are both determined by the equilibrium of supply and demand o z is exogenous, and it's the availability of this exogenous supply shifter that allows us to identify the structural demand equation o With no observed demand shifters, supply is not identified and cannot be estimated Economics 20- Prof anderson 5
Economics 20 - Prof. Anderson 5 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
Identification of Demand equation S(ZZl) S(Z=z2) S(Z=Z3) Economics 20- Prof anderson 6
Economics 20 - Prof. Anderson 6 Identification of Demand Equation w h D S (z=z1) S (z=z2) S (z=z3)
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=7 + Iiz+v Second stage equation ish=a,w+u Thus, 2SLS provides a consistent estimator of a, the slope of the demand curve o We cannot estimate a, the slope of the SU pply curve Economics 20- Prof anderson 7
Economics 20 - Prof. Anderson 7 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 + p1 z + 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
The General sem e Suppose you want to estimate the structural equation: y,=a,y2+=+ur o where, y2=ay+B2z2+u2 o Thus, y2=a,(a,v2+B2,+u)+B2z2+u2 ◆So,(1-a2a1)y2=a2B=1+B2=2+a2l1+ u. which can be rewritten as ◆y2=,x Economics 20- Prof anderson 8
Economics 20 - Prof. Anderson 8 The General SEM Suppose you want to estimate the structural equation: y1 = a1 y2 + b1 z1 + u1 where, y2 = a2 y1 + b2 z2 + u2 Thus, y2 = a2 (a1 y2 + b1 z1 + u1 ) + b2 z2 + u2 So, (1 – a2a1 )y2 = a2 b1 z1 + b2 z2 + a2 u1 + u2 , which can be rewritten as y2 = p1 z1 + p2 z2 + v2
The General SEM(continued) o By substituting this reduced form in for y2, we can see that since v, is a linear function of u,, y2 is correlated with the error term and a, is biased -call it simultaneity bias The sign of the bias is complicated,but can use the simple regression as a rule of thumb o In the simple regression case, the bias is the same sign as a/(I-a2a,) Economics 20- Prof anderson 9
Economics 20 - Prof. Anderson 9 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 )
Identification of general sem o Let z, be all the exogenous variables in the varlables in the co-2 be all the exogenous first equation, and second d equation o It's okay for there to be overlap in z and z? e To identify equation 1, there must be some variables in z, that are not in zI o To identify equation 2, there must be some variables in z, that are not in z2 Economics 20- Prof anderson 10
Economics 20 - Prof. Anderson 10 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
