(4) (5) Consider the bivariate linear regression model given by Yi - Bo + B1x;+u, i = 1,2,...,n, where E(|x1,x2,...,XN)
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(4) (5) Consider the bivariate linear regression model given by Yi - Bo + B1x;+u, i = 1,2,...,n, where E(|x1,x2,...,XN) - 0. The OLS estimators of Bo and B1 are given respectively by Bo = J BE and B, (x1 -7)(y; -7) Σ"., (αι - i)? where (6) (7) y - ΗΣνι, Σ - ΗΣ. Σ X. + i=1
(a) Show that B, may be written as B. Σ" (x, - ): (x1 - x)2 (8) (1 mark) (b) Starting with (8), show that B, may be written as B = B. + (xi – )ui Σ., (α - x)? (9) (5 marks) (2 marks) (1 mark) (10) (c) Use (9) above to show that ECB x1,x2,...,xn) - B.. (d) Use the Law of Iterated Expectations (LIE) to prove that ECB) = B. (e) i) Use (4) above to show that = Bo + B17 +ū, where Π = ΗΣ ii) Show that B, may be written as Bo = Bo + (B1-B, JI+ū. iii) Show that ECB x1,x2,...,xn) = Bo iv) Use the LIE to show that ECB) = Bo- 1. (2 marks) i=1 (2 marks) (2 marks) (1 mark)