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Exercise 6

Exercise 6 - Economics 241B Exercise 6 1 Asymptotic...

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Economics 241B Exercise 6 1. Asymptotic Behavior of Hypothesis Tests Your °eldwork requires that you test H 0 : ° k = ° ° k vs. H 1 : ° k 6 = ° ° k . a. You know that under H 0 , p n ° B k ° ° ° k ± d ! N (0 ; Avar ( B k )) (1) \ Avar ( B k ) p ! Avar ( B k ) ; where \ Avar ( B k ) = S ° 1 xx ^ SS ° 1 xx . To perform your test, you wish to use a t -statistic and you want to make sure that the limit distribution is standard normal. Show that (1) implies that under H 0 , B k ° ° ° k SE ± ( B k ) d ! N (0 ; 1) ; (2) where SE ± ( B k ) = q 1 n \ Avar ( B k ) . b. When performing the test, under what circumstances would you select the critical values from a normal distribution versus a t distribution? c. Show what SE ± ( B k ) converges in probability to. d. With your answer from part c , intuitively explain the convergence in (2). 2. Consider the model Y t = ° 01 X t 1 + ° 02 X t 2 + U t t = 1 ; : : : ; n (3) where U t iid ± N (0 ; ± 2 ) while X t 1 and X t 2 are scalars such that E ( X t 1 X t 2 ) 6 = 0 but E ( X t 1 U t ) = E ( X t 2 U t ) = 0 . Suppose you fail to observe X 2 , estimating instead the model Y t = ° 01 X t

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Exercise 6 - Economics 241B Exercise 6 1 Asymptotic...

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