Consider the simple linear regression model Yi Bo + Bxi +ui, i=1,2,... n. = coefficient...

 

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Consider the simple linear regression model Yi Bo + B₁xi +ui, i=1,2,... n. " = coefficient ₁ by B₁ x2-x1 Suppose that x, x₁ for i = 2, ....., n, and n is even. One student proposes to estimate the slope 2-1. Another student suggests that we can divide the n observations into n/2 two groups: Group 1: {(x, y)} and Group 2: {(ri, yi)}=n/2+1, and then calculate the sample mean of (x, y) of Group g to obtain (9), y(9)) for g = 1,2. Then he proposes to estimate 3₁ by Let X be the collection of {x}. B₁ (2)-(1) = (2) (1). 1 (a) Is 3₁ a linear estimator of 6₁? Why or why not? Give a geometric interpretation of 3₁. (6) Under Assumptions SLR.1-SLR.4, show that E(B₁|X) = ₁. (c) Without actually deriving the variance of 3₁, argue why ₁ is less efficient than the OLS estimator 3₁ of 3₁ under the Gauss-Markov conditions. (d) Under Assumptions SLR.1-SLR.4, show that E(₁|X) = ₁. (e) Under Assumptions SLR.1-SLR.5, find Var(₁X). How would you divide the n individuals into two groups to ensure Var(1X) to be as small as possible? Consider the simple linear regression model Yi Bo + B₁xi +ui, i=1,2,... n. " = coefficient ₁ by B₁ x2-x1 Suppose that x, x₁ for i = 2, ....., n, and n is even. One student proposes to estimate the slope 2-1. Another student suggests that we can divide the n observations into n/2 two groups: Group 1: {(x, y)} and Group 2: {(ri, yi)}=n/2+1, and then calculate the sample mean of (x, y) of Group g to obtain (9), y(9)) for g = 1,2. Then he proposes to estimate 3₁ by Let X be the collection of {x}. B₁ (2)-(1) = (2) (1). 1 (a) Is 3₁ a linear estimator of 6₁? Why or why not? Give a geometric interpretation of 3₁. (6) Under Assumptions SLR.1-SLR.4, show that E(B₁|X) = ₁. (c) Without actually deriving the variance of 3₁, argue why ₁ is less efficient than the OLS estimator 3₁ of 3₁ under the Gauss-Markov conditions. (d) Under Assumptions SLR.1-SLR.4, show that E(₁|X) = ₁. (e) Under Assumptions SLR.1-SLR.5, find Var(₁X). How would you divide the n individuals into two groups to ensure Var(1X) to be as small as possible?