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Homework100B5

# Homework100B5 - STAT 100B HW 5 Due Friday 5pm Problem 1 For...

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STAT 100B HW 5 Due Friday 5pm Problem 1: For the simplest regression model y i = βx i + i , i = 1 , ..., n , suppose x i are fixed, and E[ i ] = 0, Var[ i ] = σ 2 , and i are independent. Consider the estimator ˆ β = n i =1 w i y i / n i =1 w i x i , where w i may depends on x i , but not on y i . (1) Find E[ ˆ β ] and Var[ ˆ β ]. (2) Show that Var[ ˆ β ] is minimized when w i x i . Calculate the minimum. Problem 2: For the simple regression y i = β 0 + β 1 x i + i , i = 1 , ..., n , define the sample variance of ( x i , i = 1 , ..., n ) as V x = n i =1 ˜ x 2 i /n , define the sample variance of ( y i , i = 1 , ..., n ) as V y = n i =1 ˜ y 2 i /n , and define the sample covariance between ( x i , y i , i = 1 , ..., n ) as C xy = n i =1 ˜ x i ˜ y i /n , where ˜ x i = x i - ¯ x , and ˜ y i = y i - ¯ y . Define the sample correlation as ρ xy = C xy / ( V x p V y ). (1) Show that ρ xy can be interpreted as cosine of an angle between two vectors. (2) Let ˆ β 1 be the least squares estimate of β 1 . Show that ˆ β 1 = C xy /V x = ρ xy p V y / V x .
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