Hampel's Three-Part Estimator and Tuning Constants

Ordinary least squares weights every residual equally regardless of size, which is exactly why a single extreme outlier can dominate the fit — a residual of 100 pulls the estimate ten thousand times harder than a residual of 1, since squared error grows quadratically. M-estimators replace the squared-error loss with an alternative function chosen to grow more slowly for large residuals, down-weighting outliers instead of letting them dominate. Hampel's three-part (redescending) estimator is one specific, carefully shaped choice: for residuals within a first cutoff $a$, it behaves like ordinary least squares (full weight); between $a$ and a second cutoff $b$, the weight is held constant rather than growing further; between $b$ and a third cutoff $c$, the weight tapers linearly down to zero; and beyond $c$, the weight is exactly zero — an extreme enough outlier is entirely ignored rather than merely down-weighted.

The three tuning constants $a, b, c$ set where each regime kicks in, and choosing them is a genuine trade-off: smaller values discard more data as "outliers" and sacrifice statistical efficiency on clean data, while larger values retain more influence from extreme points and sacrifice robustness — there's no value that is simultaneously most efficient and most robust, only choices suited to how contaminated the data is expected to be.

Fitting a factor model's exposures on daily stock returns with a Hampel estimator means the handful of days with genuine earnings-surprise moves get zero weight entirely (beyond cutoff $c$), rather than the ordinary-least-squares fit being visibly bent toward those few extreme days at the expense of everything else.