I am having a hard time seeing how this equality works in calculating the coefficent in a regression of y on x.

Question

Confusing equality for the regression coefficient

Answer 1

For the numerator, you obtain after expanding the products as recommended by Yves \begin{eqnarray} \sum_n (x_n-\overline{x})(y_n-\overline{y}) &=& \sum_n x_n y_n - \overline{y} \sum_n x_n - \overline{x} \sum_n y_n + N \overline{x} \overline{y} \\ &=& \sum_n x_n y_n - \left(\frac{\sum_n y_n}{N}\right) \sum_n x_n - \left(\frac{\sum_n x_n}{N}\right) \sum_n y_n + N \left(\frac{\sum_n x_n}{N}\right)\left(\frac{\sum_n y_n}{N}\right) \\ &=& \sum_n x_n y_n - \frac{1}{N} \left(\sum_n x_n\right) \left(\sum_n y_n\right) \end{eqnarray} The denominator proceeds identically if you replace $y$ by $x$