I need to find out if the following estimator for a regression with no intercept (ie. $Y_i = \beta X_i + \epsilon_i$) is unbiased.
$$\hat{\beta} = \frac{\sum x_i^2y_i}{\sum x_i^2}$$
I'm given $\bar{x},\bar{y} $ both = 0. I know I need to find if $\mathbb{E}[\hat{\beta}] = \beta $, but I keep getting stuck. Any help would be really appreciated!
Update: I have changed the summation signs to be more clear, sorry this is my first time using this site and I'm still working on getting the code correct.
\begin{align} \mathbb{E}\big(\hat{\beta}\big)&= \mathbb{E}\left(\frac{\sum_i X_i^2\big(\beta X_i+\epsilon_i\big)}{\sum_iX_i^2}\right)\\ &= \beta\,\mathbb{E}\left(\frac{\sum_i X_i^3}{\sum_iX_i^2}\right)+ \mathbb{E}\left(\sum_i\epsilon_i\left(\frac{X_i^2}{\sum_j X_j^2}\right)\right)\\ &=\beta\,\mathbb{E}\left(\frac{\sum_i X_i^3}{\sum_iX_i^2}\right)+ \sum_i\mathbb{E}\big(\epsilon_i\big) \mathbb{E}\left(\frac{X_i^2}{\sum_j X_j^2}\right)\\ &= \beta\,\mathbb{E}\left(\frac{\sum_i X_i^3}{\sum_iX_i^2}\right) \end{align} Thus $\ \frac{\sum_i x_i^2y_i}{\sum_ix_i^2}\ $ will be an unbiassed estimator of $\ \beta\ $ if and only if $\ \mathbb{E}\left(\frac{\sum_i X_i^3}{\sum_iX_i^2}\right)=1\ $, which will not usually be the case. Note, however, exactly the same technique will show $\ \frac{\sum_i x_i^2y_i}{\sum_ix_i^3}\ $ to be an unbiassed estimator. Are you sure you have the right exponent in the denominator of your putative estimator?