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IAI Actuarial Core Principles · Actuarial Statistics · Linear regression models

In a simple linear regression fitted by least squares with an intercept, which statement about the raw residuals e_i = y_i - fitted y_i is always true?

The residuals sum to zero. Least squares with an intercept makes the intercept normal equation hold, so the residuals add to zero. The other statements fail: residual variances differ by leverage, residuals are mutually dependent, and the residual sum of squares is positive.

  1. ATheir sum equals zeroCorrect
  2. BEach residual has the same variance as the error term
  3. CTheir sum of squares equals zero
  4. DThey are uncorrelated with the response values y_i
  5. They are independent of each other

Explanation

The normal equation for the intercept forces the residuals to sum to zero. Raw residuals have variance sigma^2(1-h_ii), not sigma^2, and they are correlated with each other because they satisfy constraints. They are correlated with y but not with the fitted values. Their sum of squares is the RSS, which is positive.

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