Consider the regression model Y_{i} = a + bX_{i} + u_{i}, where the X_{i} are non-stochastic and the u_{i} are independently and identically distributed with E[u_{i}] = 0 and var[u_{i}] = s^{2}.
(a) What estimator of b would you use if you did not know a? What is the variance of this estimator?
(b) What estimator of β would you use if you knew (i) a = 0, (ii) a = 1? What is the variance of each of these estimators of b?
(c) How do the magnitudes of the variances of the estimators of β in (a) and (b) compare? In particular, what happens if Σ_{i=1,n} X_{i}^{2} = 10, Σ_{i=1,n} X_{i} = 9, and n = 9?