Linear regression throug the origin versus mean? Assume that I have data that can be described by:
$y_i = \beta x_i + \epsilon_i, \epsilon_i \sim (0,\sigma_{\epsilon})$,
then the least squares estimator is given by
$\hat{\beta_1} = \frac{\sum_{i=1}^N x_iy_i}{\sum_{i=1}^N x_i^2}$.
Why is it wrong to use the following estimator?
$\hat{\beta_2} = \frac{1}{N}\sum_{i=1}^N \frac{y_i}{x_i}$.
Do they not estimate the same parameter?

It's not 'wrong' to use that estimator, which is the mean of the ratios given by each point.
But they are not the same; the first one is the unique value of $\beta$ which minimizes
$$ E(\beta) =
\sum_i \left( \beta x_i - y_i \right)^2
$$
and will give a different result in the general case.
In the case where all the $x_i, y_i$ can be exactly fitted with a specific $\beta$ value, both estimates will yield that value.
Qualitatively, $\beta_2$ will tend to give larger fitting errors for points with larger values of $x$, and a better fit for smaller values, compared to $\beta_1$
Also, in the case where all your $x$ fall in a narrow relative range, e.g. $100 \le x \le  104 $; so the ratio $\max (|x_i|) / \min(|x_i|)$ is not much larger than 1: there will be very little difference between the two estimates, even when the points don't fall close to a straight line.  
