hws02 - Economics 140A Answers to Exercise 2 1. n a) We are...

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Economics 140A Answers to Exercise 2 1. a) We are given a sequence f Y i g n i =1 of independent and identically distributed random variables with mean and variance ± 2 . We wish to learn the value of and to that end construct the estimators Y = 1 n n X i =1 Y i and ~ Y = 1 P n i =1 i n X i =1 iY i : To verify that the two estimators of are linear functions of Y i , that is, that the two estimators are linear estimators, we must verify that each estimator can be written as P n i =1 w i Y i where the weights f w i g n i =1 do not depend on any element of the sequence f Y i g n i =1 . Observe that Y = n X i =1 1 n Y i = n X i =1 w i Y i with w i = 1 n ; and ~ Y = n X i =1 w i Y i with w i = i P n i =1 i = 2 i n ( n + 1) ; where the last equality follows from the fact that P n i =1 i = n ( n +1) 2 . Because w i does not depend on any element of the sequence f Y i g n i =1 , both Y and ~ Y are linear estimators. b) To verify that the estimators are unbiased, we calculate the expected value of each estimator. The expected value of Y is E Y = 1 n n X i =1 EY i = so Y is an unbiased estimator of . The expected value of ~ Y is E ~ Y = 1 P n i =1 i n X i =1 iEY i = 1 P n i =1 i n X i =1 i = Because the expected value of both estimators equals the true value of the parameter, the estimators are unbiased. c) As both estimators are unbiased, we can directly compare the variances of
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hws02 - Economics 140A Answers to Exercise 2 1. n a) We are...

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