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For a simple linear regression model I am able to derive the normal equations and solve these to to obtain the following- $$\beta_0=\bar{Y}-\beta_1\bar{X}$$ $$\beta_1=\frac{\sum(X_i-\bar{X})(Y_i-\bar{Y})}{\sum(X_i-\bar{X})^2}$$

However in my text book and elsewhere online I see that $\beta_1$ is often (not always) simplified to $$\beta_1=\frac{\sum(X_i-\bar{X})Y_i}{\sum(X_i-\bar{X})^2}.$$

How does one arrive at this simplification? It is not at all obvious to me why?

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  • $\begingroup$ The simplification is wrong... $\endgroup$ – TheSimpliFire Jun 21 '18 at 7:01
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    $\begingroup$ It is correct. Why don't you expand the numerator of the first expression? $\endgroup$ – StubbornAtom Jun 21 '18 at 7:05
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\begin{align} \sum (X_i - \bar{X}) ( Y_i - \bar{Y}) &= \sum(X_i - \bar{X})Y_i - \sum(X_i - \bar{X})\bar{Y} \\ &= \sum(X_i - \bar{X})Y_i - \sum X_i \bar{Y} +\sum\bar{X}\bar{Y} \\ &= \sum(X_i - \bar{X})Y_i - n\bar{Y}\bar{X} +n\bar{Y}\bar{X}\\ &= \sum(X_i - \bar{X})Y_i. \end{align}

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