How to find the least square estimators for linear regression model.

Question

I have the linear regression model:

$Y_i= \alpha_0 + \alpha_1(X_i - \overline{X})$

Anyway I got through the method for find the least square estimator for $\alpha_0$ and end up with

$\sum_{i=1}^n \frac{\alpha_1(X_i-\overline{X})}{n}$

as part of my answer for $\alpha_0$ and I'm not sure how to compute this.

Answer 1

It's the usual:

$\begin{array}\\ \sum_{i=1}^n \frac{\alpha_1(X_i-\overline{X})}{n} &=\sum_{i=1}^n \frac{\alpha_1(X_i)}{n}-\sum_{i=1}^n \frac{\alpha_1(\overline{X})}{n}\\ &=\alpha_1\sum_{i=1}^n \frac{X_i}{n}-\alpha_1\sum_{i=1}^n \frac{\overline{X}}{n}\\ &=\alpha_1\overline{X}-\alpha_1\overline{X}\\ &=0 \end{array} $