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IAI Actuarial Core Principles · Actuarial Statistics · Linear regression models

A plot of standardised residuals against fitted values from a linear model shows a funnel shape, with the spread increasing as the fitted value increases. Which is the most appropriate conclusion and response?

A funnel shape signals non-constant error variance, called heteroscedasticity. A sensible remedy is a variance-stabilising transformation of the response, such as taking logs, or using weighted least squares. It does not indicate non-linearity, autocorrelation or multicollinearity.

  1. ANon-linearity in the mean; add a quadratic term only
  2. BNon-constant variance; consider a variance-stabilising transformation of the response such as logCorrect
  3. CNon-normal errors; remove the largest fitted values
  4. DAutocorrelated errors; use a Durbin-Watson correction
  5. Multicollinearity; remove one of the covariates

Explanation

A funnel in residuals versus fitted values indicates heteroscedasticity. A log or similar transformation of the response often stabilises the variance. A curved pattern would indicate non-linearity, and autocorrelation would show up in residuals ordered by time.

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