PS4A - Problem Set 4 Answer Key Econometrics 120A 1(a Since...

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Problem Set 4 Answer Key - Econometrics 120A 1. (a) Since X = 1 if Y a and X = 0 otherwise, we have: E [ X ] = P ( X = 1) × 1 + P ( X = 0) × 0 = P ( X = 1) = P ( Y a ) Similarly, using that E [ X ] = P ( Y a ), and V ar ( X ) = E [ X 2 ] - ( E [ X ]) 2 , the variance of X can be found by: E [ X 2 ] - ( E [ X ]) 2 = P ( X = 1) × 1 2 + P ( X = 0) × 0 2 - P 2 ( Y a ) = P ( Y a )(1 - P ( Y a )) (b) Since E [ X ] = P ( Y a ), a natural estimator for P ( Y a ) is ¯ X . (c) Because ¯ X is a sample mean, we know its variance is given by: V ar ( ¯ X ) = V ar ( X ) n = P ( Y a )(1 - P ( Y a )) n where the last equality follows from our answer to part b). Since ¯ X is a natural estimator for P ( Y a ), a good estimator for the variance of ¯ X is given by replacing ¯ X for P ( Y a ), so we get ¯ X (1 - ¯ X ) /n . Notice that this is problem is actually identical to our discussion for estimating proportions. (d) Our estimator ¯ X is approximately normally distributed since it is a sample mean of identically distributed random variables. (e) Combining b), c) and d) we have: ¯ X approx N P ( Y a ) , P ( Y a )(1 - P ( Y a )) n Therefore, if we let z 1 - α/ 2 solve P ( Z z 1 - α/ 2 ) = α/ 2 and we Z-score ¯ X we get: P ˆ - z 1 - α/ 2 ( ¯ X - P ( Y a )) n p P ( Y a )(1 - P ( Y a )) z 1 - α/ 2 !
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