Some OLS results

Summary

Given an OLS trendline with intercept

$$y=b_1+b_2x$$

where $b_1,b_2$ are determined by the OLS formula and $e_i, {\hat{y}}_i$ are the OLS residuals and the OLS fitted values,

• $\sum_{i=1}^{n}{ e_i }=0$
• $\bar{e}=0$
• $\sum_{i=1}^{n}{ x_ie_i }=0$
• The sample correlation between the explanatory variable and the residuals is zero
• $\sum_{i=1}^{n}{ {\hat{y}}_i }=\sum_{i=1}^{n}{ y_i }$
• $\bar{\hat{y}}=\bar{y}$
• $\sum_{i=1}^{n}{ {\hat{y}}_ie_i }=0$
• The sample correlation between the fitted values and the residuals is zero

Note: These results do not hold if the trendline has no intercept.