$epsilon =sum (y_i-beta x_i)$

$beta = frac {sum x_iy_i}{sum x_i^2}$

$sum y_i – frac {sum x_iy_i}{sum x_i^2}sum x_i=0$?

Is the sum equal zero or not? I only know that $sum y_i =nbar y$ and $sum x_i =nbar x$ but what to do with the fraction term?

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#### Best Answer

The residuals are perpendicular to the regressor vectors $x_{j,i}$. That means for each $j$ we have: $$sum_{i=1}^{i=n} x_{j,i} epsilon_i = 0$$

**So if one of the regressors vectors, say $x_{1,i}$, is an intercept term** (that means $x_{1,i} = 1$, or a constant instead of 1, for all $i$) then we *always* have: $$sum_{i=1}^{i=n} epsilon_i = 0$$ which means that **then the residuals sum to zero**.

If this is **not** the case (as is your case), then the residuals do not *necessarily* sum to zero.

You will get:

$$sum epsilon_i = sum y_{i} – beta_1 sum x_{1,i} – beta_2 sum x_{2,i} -… – beta_{k-1} sum x_{k-1,i} – beta_k sum x_{k,i}$$

In your case when there is only a single regressor

$$sum epsilon_i = sum y_{i} – beta_1 sum x_{1,i}$$

which is only zero iff

$$frac{sum y_{i}}{sum x_{i}}= frac{sum x_{i} y_i }{sum x_{i}^2}$$

(where we applied a division, possibly by zero, so also the case $sum x_{i} = sum y_{i} = 0$, counts)

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