consider the regression model yi a bxi ui

Question

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] = s².  

(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_i² = 10,  Σ_i=1,n X_i = 9, and  n = 9?