Questions / Quant research, Regression & linear algebra
What an intercept guarantees
Ordinary least squares is fitted with an intercept on its training sample. Ignoring numerical error, what is guaranteed about the training residuals?
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Answer: C. Their sum is zero on the fitting sample.
- The derivative of the squared-error objective with respect to the intercept is −2 times the sum of residuals.
- At an OLS optimum that derivative is zero. This forces the training residual sum to vanish, without making the residuals identical or independent.
Interview shortcut. With an intercept, OLS residuals sum to zero.
Takeaway. An intercept imposes a sample mean-residual condition.
Why the other choices fail
- Their sum may be zero, but only for large samples. An intercept forces a zero sum at any sample size.
- Their variance is zero on the fitting sample. Residuals can vary around zero; an intercept fits their average, not every observation individually.
- Their correlation with each other is zero in the population. OLS normal equations concern training-sample orthogonality to design columns, not independence or population correlation among errors.
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