Bias of a different estimator for linear regression slope

Question

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.

Answer 1

That estimator is not biased.

\begin{eqnarray} E[\beta_1^\prime] &=& \frac{1}{n} \sum_{i=1}^n E \left[ \frac{y_i - \bar{y}}{x_i - \bar{x}}\right]\\ &=& \frac{1}{n} \sum_{i=1}^n \frac{E[y_i] - E[\bar{y}]}{x_i - \bar{x}} \\ &=& \frac{1}{n} \sum_{i=1}^n \frac{\beta_1 x_i - \beta_1 \bar{x}}{x_i - \bar{x}}\\ &=& \beta_1 \end{eqnarray}