# Bias of a different estimator for linear regression slope

We know he fitted estimator for $\beta_1$ is $$\hat{\beta_1} = \frac{\sum_{i=1}^{n}{(x_i-\bar{x})(y_i-\bar{y})}}{\sum_{i=1}^{n}{(x_i-\bar{x})^2}}$$

Now, given the following estimator: $$\beta_1' = \frac{1}{n}\sum_{i=1}^{n}\frac{y_i-\bar{y}}{x_i-\bar{x}}$$

How can I find out its bias? How can I calculate its $MSE$? I'm not sure how to treat that random variable, what are its properties, etc.

• As it stands there is no way this question can be answered, as a bias needs to be assessed relative to some process having generated the data, which you do not state. Is it $y_i=\alpha+\beta x_i+u_i$ where something is assumed about the relationship between $u_i$ and $x_i$? Commented Feb 27, 2018 at 13:54
• Exactly, I'm assuming simple linear regression such that $\hat{y_i}=\hat{\beta_0}+\hat{\beta_1}x_i + e_i$. where $e_i$ is normally distributed. Commented Feb 27, 2018 at 14:37
• The more interesting point than the distribution for $e$ is what you assume about (in)dependence between $x$ and $e$. Commented Feb 27, 2018 at 15:42

• @AdamO But why the $x$'s are fixed..? And are you relying on the fact that $E[y_i] = \beta_1x_i + \beta_0$? Commented Feb 28, 2018 at 10:02