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 E.Z. estimator of beta1 in the simple regression model. Algebraically, if the two points are (x1, y1) and (x2, y2 ) , the E.Z. estimation rule is beta1=(y2-y1)/(x2-x1)
I need to show:
(1) Show that beta1
EZ is an unbiased estimator.
To show this the lecturer said we need to calculate the mean: E(( beta0+beta1*xN+ epsilonN-(beta0 +beta1*x1+epsilon1))/(xN-x1), and this should be equal to beta1. How to do it?
(2) Find the probability distribution of beta1
To show this we need to show that a*epsilonN + b*epsilon1~N(..,..) Normal distribution and calculate the mean value and the dispersion. But i don't know how.
Can someone help me?