Refer to Machine speed Problem 11.7. Demonstrate numerically that the weighted least squares. estimates obtained in part (e) are identical to those obtained when using transformation (11.23) and ordinary least squares.
Problem 11.7
Machine speed. The number of defective items produced by a machine (Y) is known to be linearly related to the speed setting of the machine (X). The data below were collected from recent quality control records.
a. Fit a linear regression function by ordinary least squares, obtain the residuals, and plot the residuals against X. What does the residual plot suggest?
b. Conduct the Breusch-Pagan test for constancy of the error variance, assuming log σ21? = Yo + Yl Xi use α = .10. State the alternatives, decision rule, and conclusion.
c. Plot the squared residuals against X. What does the plot suggest about the relation between the variance of the error term and X?
d. Estimate the variance function by regressing the squared residuals against-X, and then calculate the estimated weight for each case using (11.16b).
e. Using the estimated weights, obtain the weighted least squares estimates of ß0 and ß1. Are the weighted least squares estimates similar to the ones obtained with ordinary least squares in part (a)?
f. Compare the estimated standard deviations of the weighted least squares estimates bwo and bW1 in part (e) with those for the ordinary least squares estimates in part (a). What do you find?
g. Iterate the steps in parts (d) and (e) one more time. Is there a substantial change in the estimated regression coefficients? If so, what should you do?