Proof of least Squares estimators

[julion]

Hey guys, long time lurker, first time poster!
Just having some trouble with something..Im probably just looking at it the wrong way, but I was wondering if anyone could help me with this..

Im trying to prove that by choosing b0 and b1 to minimize
http://img24.imageshack.us/img24/7/partas.jpg [Broken]
you obtain the least squares estimators, namely:
http://img15.imageshack.us/img15/3641/partbx.jpg [Broken]

also just wondering how you can prove that OLS minimizes the sum of squares function.
I know it has something to do with second derivatives, but I am a bit stuck.
Thanks!

 

[Enuma_Elish]

It's a standard maximization problem. Set up the sum of squared errors (SSE), differentiate with respect to beta, set to zero, solve for beta. For a maximum, verify that the second derivative at the beta value you found in the first step is negative.

 

[mrwicky]

Hey guys, long time lurker, first time poster!
Just having some trouble with something..Im probably just looking at it the wrong way, but I was wondering if anyone could help me with this..

Im trying to prove that by choosing b0 and b1 to minimize
http://img24.imageshack.us/img24/7/partas.jpg [Broken]
you obtain the least squares estimators, namely:
http://img15.imageshack.us/img15/3641/partbx.jpg [Broken]

also just wondering how you can prove that OLS minimizes the sum of squares function.
I know it has something to do with second derivatives, but I am a bit stuck.
Thanks!

could you expand how to do that with a little bit more help please?

 

[statdad]

Treat

$$S(b_0, b_1) = \sum_{i=1}^n \left(y_i - (b_0 + b_1 x_i)\right)^2$$

as a function of $$b_0$$ and $$b_1$$, and solve this system of equations - the solutions will give the formulas for the estimates of slope and intercept.

$$\begin{align*} \frac{\partial S}{\partial b_0} & = 0\\ \frac{\partial S}{\partial b_1} & = 0 \end{align*}$$

 

[mrwicky]

thanks :)