8 Consider the standard simple regression model y 5 b0 1 b1x 1 u under the Gauss-Markov Assumptions SLR.1 through SLR.5. The usual OLS estimators bˆ 0 and bˆ 1 are unbiased for their respective population parameters. Let b ˜ 1 be the estimator of b1 obtained by assuming the intercept is zero (see Section 2.6). (i) Find E(b ˜ 1) in terms of the xi, b0, and b1. Verify that b ˜ 1 is unbiased for b1 when the population intercept (b0) is zero. Are there other cases where b ˜ 1 is unbiased? (ii) Find the variance of b ˜ 1. (Hint: The variance does not depend on b0.) (iii) Show that Var(b ˜ 1) # Var(bˆ 1). [Hint: For any sample of data, ∑i n xi 2 ∑i n (xi 2 x¯)2 , with strict inequality unless x¯ 5 0.] (iv) Comment on the tradeoff between bias and variance when choosing between bˆ 1 and b ˜ 1.