a. Show that the new model satisfies the assumptions of the standard statistical model. b. Find the least squares estimates of 0 and

  

  

    a. Show that the new model satisfies the assumptions of the standard statistical model.

  

    b. Find the least squares estimates of β₀ and β₁.

  

    c. Show that performing a least squares analysis on the new model, as was done in part (b), is equivalent to minimizing

  

      

  

This is a weighted least squares criterion; the observations with large variances are weighted less.

  

    d. Find the variances of the estimates of part (b).
 

(Weighted Least Squares) Suppose that in the model y = Bo + Bixi + e, the errors have mean zero and are independent, but Var(e) = po, where the pi are known constants, so the errors do not have equal variance. This situation arises when the y, are averages of several observations at x;; in this case, if yi is an average of n, independent observations, p = 1/n, (why?). Because the variances are not equal, the theory developed in this chapter does not apply; intuitively, it seems that the observations with large variability should influence the estimates of Bo and B less than the observations with small variability. The problem may be transformed as follows: Piy = Pi Bo + Pi Bixi + pi'ei or where -1 U = P Z = uio + v B + di V = Px di = pi e