Reference no: EM131132011
Problem - We consider a simple regression model given yi = βxi + ui for i = 1, 2, ···, n, where we assume xi is non-random. Note that we do not have the constant term α. Though the error term ui is independent over i, we assume that E(ui) = 0 and E(ui2) = σi2 for each i.
(a) We first assume σi2 = 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., σi2 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 σi2 for all i, obtain the WLS estimator of β, which we denote β~. Find E(β~) and var(β~).
(f) We now assume that σi2 = cxi2 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?
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