Robust Regression — OLS + HC3 SE (heteroscedasticity-consistent)

Coming soon

Advanced & specialized

This test is implemented and is currently going through StatMinds’ production verification: every statistic is independently checked against a trusted reference (scipy / R), locked with regression tests, and the screen is exercised across assumption-met/violated and significant/non-significant scenarios before it opens up.

See what’s live now

OLS point estimates with HETEROSCEDASTICITY-CONSISTENT (HC3) standard errors as the primary inferential basis — protects the inferential conclusions when residual variance depends on the predictors or

on the fitted value. Reports coefficients (β, HC3 SE, t, p, 95% CI, standardised β), model fit (R², adjusted R², F, AIC, BIC), Type-III ANOVA, and a full residual / influence diagnostic panel. Classical OLS standard errors are reported alongside as a sensitivity comparison so the analyst can see how much heteroscedasticity is changing the inference.

Worked example

What is the study-time effect once a few extreme outliers are down-weighted?

The same exam-score model, but two influential outliers distort ordinary least squares. Robust (MM-estimator) regression down-weights them instead of deleting anything.

Result

The robust (MM) fit gave b = 2.48 (p < .001), down-weighting two outliers that had shifted the OLS slope to 2.75 — a more trustworthy estimate.

How you'd report it (APA)

A robust (MM-estimator) regression, resistant to two influential outliers, showed study time predicted exam score, b = 2.48, p < .001 (OLS b = 2.75).

Try it yourself: Load this ready-made sample and follow the run above.

When to use it

  • Continuous outcome + k predictors with heteroscedastic residuals
    Wage (y) ~ education + experience + region (n = 5000).
  • Earnings / income / inequality research
    Mincer earnings equation: log(wage) ~ education + experience + experience² (n = 10,000).

When NOT to — use instead

Hypotheses

H₀: β_j = 0 for all predictor coefficients (and overall H₀_F: R² = 0 for the model-fit F test).
Hₐ: at least one β_j ≠ 0.

Parameter tested: vector of population coefficients β

Assumptions (and what to do if they fail)

Residuals are approximately normally distributed. HC3 produces consistent SEs without this assumption asymptotically; for small n, normality still aids small-sample CI accuracy. Under the Central Limit Theorem this concern fades as n grows.low

Check: Shapiro-Wilk p-value on residuals.

If violated: Non-normal residuals affect small-n CI for HC3.

Constant variance of residuals (informational — HC3 is robust to violations).low

Check: Breusch-Pagan p-value (informational only).

If violated: Heteroscedasticity is the very condition HC3 is designed to handle — a Breusch-Pagan flag is INFORMATIONAL here, not a fault.

Residuals are independent (no autocorrelation). HC3 does NOT correct for dependence — only for variance.high

Check: Durbin-Watson statistic in [1.5, 2.5] is OK.

If violated: Autocorrelated residuals bias HC3 SEs because HC3 assumes independent (but possibly heteroscedastic) errors.

No single case dominates the fit (Cook's D thresholds, leverage, DFFITS). HC3 down-weights heteroscedasticity but not high-leverage outliers.high

Check: Max Cook's D reported.

If violated: An influential case can dominate slope estimates regardless of SE choice.

VIF for each predictor is moderate (< 5 preferred; < 10 tolerable).medium

Check: Max VIF across predictors.

If violated: High VIF inflates coefficient SEs (HC3 or otherwise) — some βs look non-significant when the underlying relationship is real, just shared with another predictor.

Ready to run a Robust Regression — OLS + HC3 SE (heteroscedasticity-consistent) on your own data?

Guided setup, automatic assumption checks, effect sizes, figures and an APA write-up.

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