In least squares regression, what characteristic should ideally be true for the residuals?

In least squares regression, a fundamental assumption is that the mean of the residuals should be zero. This is crucial because residuals are the differences between the observed values and the values predicted by the regression model. If the mean of the residuals is not zero, it indicates that the regression model has systematic bias, meaning it consistently underestimates or overestimates the dependent variable.

An ideal set of residuals should balance symmetrically around zero, which affirms that the regression line is appropriately fitted to the data. When this condition is met, it suggests that the model is capturing the underlying relationship between the independent and dependent variables effectively.

In contrast, if the mean of the residuals were not zero, it would signal that there are elements in the data not being accounted for in the model, which could lead to inefficiencies or erroneous predictions. This is why having a mean of zero for the residuals is not merely a desirable characteristic but a foundational trait that validates the effectiveness of the regression analysis in least squares regression.