We first assume sigmai2 sigma2 for all i obtain the ols

Question

Problem - We consider a simple regression model given y_i = βx_i + u_i for i = 1, 2, ···, n, where we assume x_i is non-random. Note that we do not have the constant term α. Though the error term u_i is independent over i, we assume that E(u_i) = 0 and E(u_i²) = σ_i² for each i.

(a) We first assume σ_i² = σ² for all i. Obtain the OLS estimator of β, which we denote β^{^}, and find E(β^{^}) and var(β^{^}).

(b) Using your answer in part (a), obtain an unbiased estimator of var(β^{^}).

(c) Now we assume heteroskedasticity, i.e., σ_i² can vary over i. In this case, is the OLS estimator β^{^} in part (a) unbiased? How about your estimator of var(β^{^}) in part (b)?

(d) Under the heteroskedasticity in part (c), what is ver(β^{^}) now?

(e) In order to handle heteroskedasticity, we estimate β using the WLS (weighted least squares). Assuming that we observe σ_i² for all i, obtain the WLS estimator of β, which we denote β^~. Find E(β^~) and var(β^~).

(f) We now assume that σ_i² = cx_i² for some constant c > 0. In this case, compare var(β^{^}) in part (d) with var(β^~) in part (e). Which one is larger? Does your answer justify the use of WLS in this case?