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chapter 9 [article] - Contents 1 Systems of equations 1.1...

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Contents 1 Systems of equations 1 1.1 Structural and reduced form . . . . . . . . . . . . . . . . . . . . . 1 1.2 Simultaneous equations bias . . . . . . . . . . . . . . . . . . . . . 4 2 Instrumental variables 8 2.1 IV estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2 Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3 IV in practice 14 3.1 Recap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.2 Examples of IV . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1 Systems of equations Contents 1.1 Structural and reduced form Contents Another reason for B.7 to fail We saw in chapter 8 how measurement error (practical and/or conceptual) can lead the regressors to be correlated with the error term Assumption B.7 (independence of X i and u i ) fails, and OLS is inconsistent Y = β 1 + β 2 X + u plim b OLS 2 = β 2 + σ Xu σ 2 X 6 = β 2 if σ Xu 6 = 0 Another reason for B.7 to fail and σ Xu 6 = 0: simultaneous equations Example: price and wage inflation Let p be price inflation, w be wage inflation and U be the unemployment rate Suppose on the one hand that, as wages rise, bosses raise goods prices to cover the increased production costs: p = β 1 + β 2 w + u p price equation (9.1)
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Suppose on the other hand that, as goods prices rise, workers bargain for higher wages to maintain their real wages w = α 1 + α 2 p + α 3 U + u w wage equation (9.2) p = β 1 + β 2 w + u p price equation (9.1) w = α 1 + α 2 p + α 3 U + u w wage equation (9.2) So, p and w are the endogenous variables here their values are determined by their interactions in the model p affects w , which in turn affects p - circular Unemployment U is exogenous here its value is determined outside the model value of U determines equilibrium values of p and w p = β 1 + β 2 w + u p price equation (9.1) w = α 1 + α 2 p + α 3 U + u w wage equation (9.2) These price and wage equations are in structural form each has an endogenous variable on the right-hand-side We can rewrite this system in reduced form write p and w in terms of exogenous variable and disturbance terms only p = β 1 + β 2 w + u p price equation (9.1) w = α 1 + α 2 p + α 3 U + u w wage equation (9.2) Substituting wage into price : p = β 1 + β 2 ( α 1 + α 2 p + α 3 U + u w ) + u p (9.3) = β 1 + α 1 β 2 + α 3 β 2 U + u p + β 2 u w 1 - α 2 β 2 (9.5) Equation (9.5) is the reduced form price equation 2
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p = β 1 + β 2 w + u p price equation (9.1) w = α 1 + α 2 p + α 3 U + u w wage equation (9.2) Substituting price into wage : w = α 1 + α 2 ( β 1 + β 2 w + u p ) + α 3 U + u w (9.6) = α 1 + α 2 β 1 + α 3 U + u w + α 2 u p 1 - α 2 β 2 (9.8) Equation (9.8) is the reduced form wage equation Structural form: p = β 1 + β 2 w + u p (9.1) w = α 1 + α 2 p + α 3 U + u w (9.2) Reduced form: p = β 1 + α 1 β 2 + α 3 β 2 U + u p + β 2 u w 1 - α 2 β 2 (9.5) w = α 1 + α 2 β 1 + α 3 U + u w + α 2 u p 1 - α 2 β 2 (9.8) Note from (9.8) that a +ve price shock u p > 0 raises w , thanks to the model simultaneity This is why in (9.5) Δ p = 1 1 - α 2 β 2 u p > u p Model stability p = β 1 + β 2 w + u p structural price equation (9.1) w = α 1 + α 2 p + α 3 U + u w structural wage equation (9.2) Consider a positive price shock u p > 0 initial Δ p = u p Δ w = α 2 Δ p = α 2 u p from wage equation Δ p = β 2 Δ w = α 2 β 2 u p from price equation and so on. . .
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