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PS7Sol - Econ104(1)

# PS7Sol - Econ104(1) - Econ 104 Problem Set 5 Lorenzo...

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Unformatted text preview: Econ 104 - Problem Set 5 Lorenzo Braccini * December 7, 2011 Question 1 a) First note the following: Q s i = Q d i ⇒ β 1 P i = γ- β + u d i- u s i Now consider this fact: Cov( β 1 P i ,u s i ) = Cov( γ- β + u d i- u s i ,u s i ) =- Var( u s i ) = β 1 Cov( P i ,u s i ) Hence for β 1 6 = 0 (in particular we expect β 1 > 0) we have that: Cov( P i ,u s i ) =- Var( u s i ) β 1 Therefore we can conclude that there is (negative) correlation be- tween P i and u s i . b) No, the OLS estimator is not consistent in this case. In fact: b β 1 = β 1 + ( 1 n ∑ n i =1 P i u s i )- ¯ P ¯ u s 1 n ∑ n i =1 ( P i- ¯ P ) 2 * [email protected] 1 Now note that: 1 n n X i =1 P i u s i-→ p E [ P i u s i ] = Cov( P i ,u s i ) ¯ P ¯ u s-→ p E [ P i ] E [ u s i ] = 0 and that: 1 n n X i =1 ( P i- ¯ P ) 2-→ p Var( P i ) Finally, the last three facts imply that: b β 1-→ p β 1 + Cov( P i ,u s i ) Var( P i ) = β 1 + ρ ( P,u s ) σ u s σ P c) Suppose it exists an observable random variable Z i such that: Cov( Z i ,u s i ) = 0 Cov( P i ,Z i ) 6 = 0 i.e. a valid instrument. I would then estimate β and β 1 by 2SLS using Z i as instrument for P i . I would then estimate γ using the sample average of Q d i ....
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PS7Sol - Econ104(1) - Econ 104 Problem Set 5 Lorenzo...

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