In ordinary least squares, the sum of residuals equals which value?

In ordinary least squares, the sum of residuals equals which value?

• A. 0
• B. n
• C. Sum of squared residuals
• D. Mean of residuals

Answer: (A)

The residuals balance to zero in a least-squares fit that includes an intercept. If e_i = y_i − ŷ_i and ŷ_i = α + β x_i, the method minimizes the sum of squared residuals. Taking the derivative with respect to the intercept α and setting it to zero gives ∑ e_i = 0. That means the positive and negative deviations cancel out, so the total residuals sum to zero (and their average is zero since there are n observations). This is a defining property of OLS with an intercept.

The residuals balance to zero in a least-squares fit that includes an intercept. If e_i = y_i − ŷ_i and ŷ_i = α + β x_i, the method minimizes the sum of squared residuals. Taking the derivative with respect to the intercept α and setting it to zero gives ∑ e_i = 0. That means the positive and negative deviations cancel out, so the total residuals sum to zero (and their average is zero since there are n observations). This is a defining property of OLS with an intercept.