1. ## Unbiased Estimator help

SO, you collect data points x(i),y(i) and the estimation for the slope parameter, beta, is beta hat. If you just connect the first point x1,y1 to xn,yn and use the slope of this line beta star (B*) is B* and unbiased estimator for beta?????

I am assuming that is should be biased, but I can't prove it. Any help would be great.

2. Originally Posted by Jar23
SO, you collect data points x(i),y(i) and the estimation for the slope parameter, beta, is beta hat. If you just connect the first point x1,y1 to xn,yn and use the slope of this line beta star (B*) is B* and unbiased estimator for beta?????

I am assuming that is should be biased, but I can't prove it. Any help would be great.
$y_1=a x_1+b+\varepsilon_1$

$
y_n=a x_n+b+\varepsilon_n
$

$
\frac{y_n-y_1}{x_n-x_1}=\frac{a(x_n-x_1)+\varepsilon_n-\varepsilon_1} {x_n-x_1}=a+\frac{\varepsilon_n-\varepsilon_1}{x_n-x_1}
$

But under the usual assumptions about the error model in linear regression:

$E(\varepsilon_n-\varepsilon_1)=0$

and so our estimator is unbiased.

RonL