ps2_sol - Introduction to Econometrics Professor Alexei...

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Introduction to Econometrics Professor Alexei Onastki Problem Set 2 – September 30 Problem 1) [ Stock and Watson 3.2 ] Each random draw from the Bernoulli distribution takes a value of either zero or one with probability Pr and Pr The random variable Y has mean i Y p (1 ) i Y == ( 0) 1 . i Y ==− ( ) 0 Pr( p i 0) 1 Pr( 1) i , E YY Y p ×= + = = and variance 2 22 var( ) [( ) ] (0 ) Pr( 0) ) Pr( ) ) ) ii Y YE Y pY pp p p p p μ =− = + + = . (a) The fraction of successes is 1 ) (success) ˆ n i Y #Y # p Y nn n = = = = . (b) 1 11 ˆ () n i Y E pE E Y p p n = ⎛⎞ = ⎜⎟ ⎝⎠ ∑∑ = . (c) 1 ( 1 ˆ var( ) var var( ) ) n i Y ) p p p = === = p . The second equality uses the fact that , , Y n are i.i.d. draws and for 1 Y cov( , ) 0, ij = .
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Problem 2) [ Stock and Watson 3.3 ] Denote each voter’s preference by Y Y . 1 = if the voter prefers the incumbent and Y if the voter prefers the challenger. is a Bernoulli random variable with probability Pr( and Pr From the solution to Exercise 3.2, Y has mean 0 = p == p Y 1 Y ) ( 0) 1 . Y ==− p and variance (1 ). p p (a) 215 400 ˆ 0 5375. p . (b) ± ˆˆ (1 ) 0.5375 (1 0.5375) 4 400 ˆ var( ) 6 2148 10 . pp n p −× = . × The standard error is SE 1 2 ( ) (var( )) 0 0249. . (c) The computed t -statistic is 0 ˆ 05375 05 1 506 ˆ SE( ) 0 0249 p act p t p μ , .− . = . . . Because of the large sample size ( 400), n = we can use Equation (3.14) in the text to get the p -value for the test vs. 0 05 Hp := . 1 :≠ : .
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ps2_sol - Introduction to Econometrics Professor Alexei...

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