I was reading a book on econometrics and I found this algebraic transformation and I don't know how I can get there.I would appreciate some help.
$\sum_{i=1}^n (X_i-\bar{X})\cdot (Y_i-\bar{Y})=\sum_{i=1}^n (X_i-\bar{X})\cdot Y_i=\sum_{i=1}^n X_i\cdot (Y_i-\bar{Y}) $


$\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} $

The key is that $\bar{Y}$ is a constant, so we can bring it in front of the summation.

$=\sum_{i=1}^n(X_i -\bar{X})(Y_i) -\bar{Y}\sum_{i=1}^n(X_i -\bar{X}) $

Then apply, $\sum_{i=1}^n(X_i -\bar{X})=0$.

It's an analogous argument for the other equality.


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