- Question 3 20 Marks 8 A Consider The Bivariate Linear Regression Model Y Bo Bix W I 1 2 Where Elu X1 X2 1 (42.94 KiB) Viewed 33 times
Question 3 (20 marks) (8) (a) Consider the bivariate linear regression model y - Bo+Bıx: +w.i=1,2,..., where Elu,x1,x2,...,x.) -0,1 - 1,2,...,, and Varly;lx 1.12....,xn) = 09. i = 1,2,..., Show that the OLS estimator of B. B, - E" (8. – 1)(*:-) (*. -
may be written as B. Σ. (α, - Χ)». "(- )2 (9) (3 marks) (10) (b) The LIE for the mean states that given two random variables 2 and W E(Z) = Ew[E(ZW). There is an analogous result for Var(2) which states that Var(Z) - E[Var(ZW)] + VarE(ZW)). Applying this result to Var(B), and conditioning on (X1X2,---,xn), we obtain VarlB.) = E(Var[By\$1,x2,...,xn)] + Var[ECB,\v1.X2,...,x.)]. Note: When we condition on (x1,x2,...,xn), we also condition on & since R - Σ. 1) Use (4), (9) and the properties of the expectation operator to prove that [E(B, 1x1x2,...,x)] = ? (11) (8 marks) II) Use (8), (9) and the properties of the variance operator to prove that Var(B,x1,x2,...,xn) - ? (4 mark) ill) Use your answers to i) and ii) together with (11) to derive an expression for Var(B.). (2 marks) Note: Your final expression for Var B) should involve o?, n and E(6), where & denotes the sample variance of .