Linear IV and Simultaneous Equations
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1 Linear IV and Daniel Schmierer Econ 312 April 6, 2007
2 Setup Linear regression model Y = X β + ε (1) Endogeneity of X means that X and ε are correlated, ie. E(X ε) 0. Suppose we observe another variable Z (the instrument) which is uncorrelated with ε. We postulate that the relation between Z and X is of the form Equation (2) is known as the first stage. X = Zγ + u (2)
3 Setup Note that we can combine equations (1) and (2) to form the reduced form: Y = Zπ + η (3) where π = γβ and η = uβ + ε. Our goal: consistently estimate β. In all IV settings there will be two required conditions: Rank: In this case: Z is correlated with X. (testable) Independence: In this case: Z is uncorrelated with ε. (not testable) Suppose throughout that dim(x ) = n 1 so there is only one endogenous regressor (this is just for simplicity).
4 IV Suppose that we have only one instrument, so dim(z) = n 1. Then consider the estimator ˆβ IV = (Z X ) 1 Z Y Consistency: p lim ˆβ IV = p lim(z X ) 1 Z (X β + ε) = p lim β + (Z X ) 1 Z ε = β + [E(Z X )] 1 E(Z ε) = β
5 IV Asymptotic normality: = n( ˆβ IV β) = ˆβ IV = β + (Z X ) 1 Z ε where 1 n zi ε i d N(0, V ). d 1 P zi n ε i 1 P zi x n i N(0, [E(Z X )] 1 V [E(Z X )] 1 ) The weaker the correlation between Z and X, the larger this asymptotic variance will be. An instrument with a small E(Z X ) is known as a weak instrument.
6 Two-stage Least Squares Now suppose we have k instruments (so dim(z) = n k). Consider first regressing X on Z (that s why we called it the first stage) and then regressing Y on the fitted values ˆX from that regression. This would result in the estimator ˆβ 2SLS = ( ˆX ˆX ) 1 ˆX Y = [ (Z(Z Z) 1 Z X ) (Z(Z Z) 1 Z X ) ] 1 (Z(Z Z) 1 Z X ) Y Consistency: ˆβ 2SLS = [ X Z(Z Z) 1 Z Z(Z Z) 1 Z X ] 1 X Z(Z Z) 1 Z Y = [ X Z(Z Z) 1 Z X ] 1 X Z(Z Z) 1 Z (X β + ε) = β + [ X Z(Z Z) 1 Z X ] 1 X Z(Z Z) 1 Z ε So plim ˆβ 2SLS = β.
7 Two-stage Least Squares Showing asymptotic normality is the same as in the IV case, but now the asymptotic variance depends on how well the linear combination of Z variables explain X. Equivalence between ˆβ 2SLS and ˆβ IV when Z is scalar: ˆβ 2SLS = [ X Z(Z Z) 1 Z X ] 1 X Z(Z Z) 1 Z Y = (Z X ) 1 [ X Z(Z Z) 1] 1 X Z(Z Z) 1 Z Y = (Z X ) 1 Z Y
8 Some Interpretation Remember what the IV estimator is in terms of the reduced form and the first stage: ˆβ IV = ˆπˆγ where π is the coefficient on Z in the reduced form and γ is the coefficient on Z in the first stage. This is another way of seeing that a weak relationship between X and Z (a small γ) will lead to an imprecisely estimated γ. If X is binary (say, our treatment, so call it D) and Z is binary (say, a randomized assignment) then the IV estimator is just ˆβ IV = E(Y Z = 1) E(Y Z = 0) E(D Z = 1) E(D Z = 0)
9 Model from class: Y 1 + γ 12 Y 2 = α 1 + β 11 X 1 + β 12 X 2 + U 1 γ 21 Y 1 + Y 2 = α 2 + β 21 X 1 + β 22 X 2 + U 2 Discussion of causal effects is a black hole. Counterfactuals are ambiguous only insofar as we fail to specify an intervention that would bring them about. When moving along the demand curve, does changing price cause quantity demanded to change, or does a change in quantity demanded cause price to change?
10 Supply/Demand Framework To fix ideas, rename the variables so that these two equations specify the supply curve and the demand curve respectively: Q s + γ 12 P s = α 1 + β 11 X 1 + β 12 X 2 + U 1 γ 21 Q d + P d = α 2 + β 21 X 1 + β 22 X 2 + U 2 Because in equilibrium P s = P d we can plug the bottom equation into the top one and solve for Q = Q s = Q d. The algebra is easier in matrix notation: ( ) ( ) ( Q α1 X1 Γ = + B P α 2 X 2 ) ( U1 + U 2 )
11 Supply/Demand The reduced form is: ( ) ( Q = Γ 1 α1 P α 2 where ) ( + Γ 1 X1 B X 2 Γ 1 B = 1 1 γ 12 γ 21 ( 1 γ12 γ 21 1 ) + Γ 1 ( U1 U 2 ) ( β11 β 12 β 21 β 22 = 1 1 γ 12 γ 21 ( β11 γ 12 β 21 β 12 γ 12 β 22 γ 21 β 11 + β 21 γ 21 β 12 + β 22 ) ) )
12 Exclusions Say we know that X 2 doesn t enter the supply curve (so β 12 = 0). That is, X 2 shifts the demand curve, but not the supply curve. Intuitively, this means we should be able to trace out points on the supply curve. We can in fact do this because if we regress Q on both X 1 and X 2 the coefficient on X 2 will be β 12 γ 12 β 22 1 γ 12 γ 21 while regressing P on X 1 and X 2 will give a coefficient on X 2 of γ 21β 12 +β 22 1 γ 12 γ 21. Take the ratio of these coefficients (remember, β 12 = 0) and you get γ 12. This is precisely the slope of the supply curve.
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