Reviewing my econometrics books and was just confused by the following derivation:
We use the fact that $\sum_{i=1}^{n}\left(x_{i}-\bar{x}\right)\left(y_{i}-\bar{y}\right)=\sum_{i=1}^{n}\left(x_{i}-\bar{x}\right) y_{i}$ to write the OLS slope estimator equation as $$ \hat{\beta}_{1}=\frac{\sum_{i=1}^{n}\left(x_{i}-\bar{x}\right) y_{i}}{\sum_{i=1}^{n}\left(x_{i}-\bar{x}\right)^{2}} $$
I don't believe I've ever seen the above formula written that way. I don't understand how that fact above is the case, or how the formula is the same as the general $\hat\beta$ below? $$ \hat{\beta}_{1}=\frac{\sum_{i=1}^{n}\left(x_{i}-\bar{x}\right)\left(y_{i}-\bar{y}\right)}{\sum_{i=1}^{n}\left(x_{i}-\bar{x}\right)^{2}} $$
$$ \sum_{i=1}^n (x_i - \bar{x})( y_i - \bar{y} ) = \sum_{i=1}^n (x_i - \bar{x} ) y_i - \sum_{i=1}^n (x_i - \bar{x} ) \bar{y} = \sum_{i=1}^n (x_i - \bar{x} ) y_i - \bar{y} ( n\bar{x} - n \bar{x}) = \sum_{i=1}^n (x_i - \bar{x} ) y_i $$