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.