Suppose (Y_{i}=beta X_{i}+u_{i}), where ( (left.u_{i}, X_{i} ight)) satisfy the Gauss-Markov conditions given in Equation (5.31). a.

Suppose \(Y_{i}=\beta X_{i}+u_{i}\), where ( \(\left.u_{i}, X_{i}\right)\) satisfy the Gauss-Markov conditions given in Equation (5.31).

a. Derive the least squares estimator of \(\beta\), and show that it is a linear function of \(Y_{1}, \ldots, Y_{n}\).

b. Show that the estimator is conditionally unbiased.

c. Derive the conditional variance of the estimator.

d. Prove that the estimator is BLUE.

Equation (5.31)

Transcribed Image Text:

(i) E(u X., Xn) = 0 (ii) var(u, X., Xn) = , (iii) E(uu; X., X) = 0,i 0 < < j, 8