B 10 derive the formula for the variance of 0 ˆ y

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b) (10) Derive the formula for the variance of 0 ˆ Y . Show at least two steps in this derivation. a. Hint 1: You are looking for 0 0 1 0 . This is the variance of a sum of two random variables. What is the general formula for such a sum? (Go back to the beginning of week 2, if you need a reminder.) Use that formula now. ˆ ˆ ˆ ( ) ( ) Var Y Var X
3) (20 points, 4 points for each piece.) Compare the following two regressions: i. 0 1 ˆ ˆ i i i Y X e ii. 0 1 ˆ ˆ (3 ) i i i Y X e
Since SST is obviously unchanged (Y hasn’t been touched), *THEREFORE, THE VALUE OF R 2 WILL BE UNCHANGED. Similarly, since SSR will also be unchanged between the two regressions, 2 ˆ S will also be unchanged. 1 1 1 2 2 ˆ ˆ 2 2* 2 2 2* 2 ˆ ˆ ˆ ˆ ˆ 2* 2 2 2 ( 1) 1 1 ( 1) ( 1)3 9 ( 1) 9 X X X X S S n S S S S S S n S n S n S *In the last line, recall if you multiply a variable by the factor b, the variance increases by the factor b 2 . So Sx 2* changes to 3 2 Sx 2 . FACTORING THIS OUT, WE SEE THAT THE VARIANCE OF THE SLOPE WILL SHRINK BY A FACTOR OF 9. (Standard deviation falls by factor of 3)

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