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.