Results from the first order conditions for the least squares problem

Summary

In this section \(e_i\) are the OLS residuals and \({\hat{y}}_i\) are the OLS fitted values
Result for \(j=1, \ldots ,k\) :

\[\sum_{i=1}^{n}{ x_{i,j}e_i }=0\]

\[ \sum_{i=1}^{n}{ e_i }=0\]

and \(\bar{e}=0\) where \(\bar{e}\) is the sample average of the OLS residuals.
Result: The sample mean of the regressand data is equal to the sample mean of the OLS fitted values,

\[\bar{y}=\bar{\hat{y}}\]

\[\sum_{i=1}^{n}{ e_i{\hat{y}}_i }=0\]

Note that \(\sum_{i=1}^{n}{ e_i }=0\) and \(\bar{y}=\bar{\hat{y}}\) will fail to hold if we do not include “1” in our regressor data. \(\sum_{i=1}^{n}{ e_i{\hat{y}}_i }=0\) will still hold.