Professor E.Z. Stuff has decided that the least squares estimator is too much trouble. Noting that two points determine a line, Dr. Stuff chooses two points from a sample of size \(N\) and draws a line between them, calling the slope of this line the EZ estimator of \(\beta_{2}\) in the simple regression model. Algebraically, if the two points are \(\left(x_{1}, y_{1}\right)\) and \(\left(x_{2}, y_{2}\right)\), the EZ estimation rule is

Assuming that all the assumptions of the simple regression model hold:

a. Show that \(b_{E Z}\) is a "linear" estimator.

b. Show that \(b_{E Z}\) is an unbiased estimator.

c. Find the conditional variance of \(b_{E Z}\).

d. Find the conditional probability distribution of \(b_{E Z}\).

e. Convince Professor Stuff that the EZ estimator is not as good as the least squares estimator. No proof is required here.

Transcribed Image Text:

DEZ EZ Y2 x2x1