Least squares intercept estimator variance

Hello everybody,

I am tackling the following problem relating to the regression intercept estimator:

"(a) Show that the least squares estimator $\displaystyle \hat{\alpha} = \sum_{i=1}^{n}\Big[\frac{1}{n}-\frac{(x_i-\bar{x})}{S_{xx}}\Big]y_i$

(b) Using the result of part (a), or otherwise, derive the variance of $\displaystyle \hat{\alpha}$".

My progress is very limited. On part (a), I cannot see how this is true. Given $\displaystyle \hat{\alpha}=\bar{y}-\hat{\beta}\bar{x}$, I cannot equate the formula given to what it should be. It seems the formula given amounts to $\displaystyle \hat{\alpha}=\bar{y}-\hat{\beta}$ and I am short of an $\displaystyle \bar{x}$. I am willing to accept I may have made a mistake but I can't see it and this is a major concern.

Re: Least squares intercept estimator variance

Hey Naranja.

Can you show us the complete expanded form by Beta_hat? (i.e. show it in terms of x_i's and y_i's).