Discussion
If \(e_{1}, e_{2}, \ldots, e_{n}\) are the residuals obtained on fitting a simple linear regression \(y = \beta_{0} + \beta_{1}X + \varepsilon\) to \((Y_{i}, X_{i})\) \(i = 1 \ldots n\) under the standard assumption, then :
Answer: C
\(\sum X_{i}e_{i}\) is always zero.
The least squares normal equations are obtained by setting the derivatives of \(\sum(Y_{i} - \beta_{0} - \beta_{1}X_{i})^{2}\) to zero:
\[ \begin{aligned} \sum e_{i} &= 0 \\ \sum X_{i}e_{i} &= 0 \end{aligned} \]
The second equation is exactly the statement that the residuals are orthogonal to the regressor. It holds for every data set fitted by least squares, so the sum is never positive or negative and something definite can be said about it.
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