Consider the standard simple regression model y = ̠₀ + ̠₁x + u under the Gauss-Markov Assumptions SLR.1 through SLR.5. The usual OLS estimators ̢̠₀ and ̢̠₁ are unbiased for their respective population parameters. Let ̠̃₁ be the estimator of ̠₁ obtained by assuming the intercept is zero (see Section 2.6).
(a) Find E(̠̃₁) in terms of the xᄒ, ̠₀, and ̠₁. Verify that ̠̃₁ is unbiased for ̠₁ when the population intercept (̠₀) is zero. Are there other cases where ̠̃₁ is unbiased?
(b) Find the variance of ̠̃₁. (Hint: the variance does not depend on ̠₀.)
(c) Show that Var(̠̃₁) ≤ Var(̢̠₁). [Hint: for any sample of data, ̣ᄂᄒ₁ xᄒ" ≥ ̣ᄂᄒ₁(xᄒ - x̄)", with strict inequality unless x̄ = 0.]