Linear regression coefficient

Question

I'm studying linear regression with Draper's Applying regression analysis book There is a formula for coefficient b1 which is clear to me:

$$ b_{1}= \frac{n\sum x_{i}y_{i}-\sum x_{i}\sum y_{i}}{n\sum x_{i}^2 - \sum x_{i}\sum x_{i}} $$ What I can't understand is how to get this one: $$ b_{1}= \frac{\sum \left (X_{i}-\overline{X} \right )\left (Y_{i}-\overline{Y} \right )}{\sum \left ( X_{i}-\overline{X} \right )^2} $$

Jens Schwaiger · Accepted Answer · 2018-06-10T03:30:45.193

2

$\sum \left (X_{i}-\overline{X} \right )\left (Y_{i}-\overline{Y} \right )=\sum X_i Y_i-(\sum X_i) \overline{Y}-\overline{X} (\sum Y_i)+n\overline{X}\overline{Y}$. Thus $\sum \left (X_{i}-\overline{X} \right )\left (Y_{i}-\overline{Y} \right)=\sum X_i Y_i-n\overline{X}\overline{Y}$. Using this also for $X=Y$ the second expression reads as $\displaystyle\frac{\sum X_i Y_i-n\overline{X}\overline{Y}}{\sum X_i^2-n\overline{X}^2}$. Multiplying both, nominator and denominator, with $n$ gives the first expression for $b_1$.

edited Jun 10 '18 at 03:30

answered Jun 10 '18 at 03:19

Jens Schwaiger

4,015

How have you eliminated 2nd and 3rd components? – amplifier Jun 10 '18 at 13:38
$\sum X_i=n \overline{X}$ and similarly for $Y$. – Jens Schwaiger Jun 10 '18 at 18:19
@JensSchwaiger When you expand and open the brackets, how does an n come in the third component? – user585380 Sep 26 '18 at 14:54
$\sum \left (X_{i}-\overline{X} \right )\left (Y_{i}-\overline{Y} \right)$ $=\sum \left(X_i Y_i-\overline{X} Y_i-X_i\overline{Y}+\overline{X}\overline{Y}\right)$ – Jens Schwaiger Sep 26 '18 at 16:07

Linear regression coefficient

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