Consider the following regression-through-the origin model: Yi = X i + u i , for i =

Consider the following regression-through-the origin model:

_Yi = βX_i + u_i, for i = 1, 2

You are told that u1 ∼ N(0, σ²) and u₂ ∼ N(0, 2σ²) and that they are statistically independent. If X₁=+1 and X₂=−1, obtain the weighted least-squares (WLS) estimate of β and its variance. If in this situation you had assumed incorrectly that the two error variances were the same (say, equal to σ²), what would be the OLS estimator of β? And its variance? Compare these estimates with the estimates obtained by the method of WLS. What general conclusion do you draw?*