Discussion
Which of the following statements is correct about a regression model ?
Answer: A
The keyed statement is residual sum of squares reduces with every new term added in the model.
Least squares chooses the coefficients that minimise the residual sum of squares. A model with one more term can always reproduce the smaller model by setting the new coefficient to 0, so its minimum cannot be larger. Adding a term therefore never increases RSS; it falls whenever the new term explains any part of the residual, and stays the same only in the degenerate case where the new term is exactly uncorrelated with the current residuals.
That is why RSS (and \(R^{2}\)) cannot be used alone to choose a model, and adjusted \(R^{2}\) or an F-test is needed. The statements saying RSS increases are wrong in every case, and the condition 'provided the response is dependent on the new term' is unnecessary: even an irrelevant term lowers RSS by chance.
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