Weighted Least Squares (WLS)
Weighted Least Squares Regression · Also known as: WLS, weighted regression, heteroscedasticity-corrected OLS, variance-weighted least squares
Weighted Least Squares is a generalization of Ordinary Least Squares (OLS) regression that assigns each observation a weight inversely proportional to its error variance, thereby down-weighting high-variance data points and up-weighting precise ones. Introduced in its general matrix form by Alexander Craig Aitken in 1935, WLS is the canonical remedy when heteroscedasticity is present and the error variance structure is known or can be reliably estimated.
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When to use it
Use WLS when you have a linear regression setting where the error variance is not constant across observations (heteroscedasticity) and you have a principled basis for specifying the weights. Common triggers include: residual plots that fan out with fitted values or a predictor; data that are group means or proportions (natural weights are group sizes); measurements taken by instruments of differing precision; and ecological or aggregate-level data where units represent different population sizes. The key assumption beyond the standard linear model is that the weight matrix W is correctly specified — if weights are misspecified, WLS can be less efficient than OLS and may distort inference. When the error variance structure is unknown, Feasible GLS (FGLS) estimates weights from the data first.
Strengths & limitations
- Restores the BLUE property (minimum variance among linear unbiased estimators) when the true heteroscedastic error structure is known, yielding more efficient estimates than OLS.
- Produces valid standard errors and confidence intervals under heteroscedasticity without requiring large-sample sandwich corrections.
- Conceptually transparent: weights express prior knowledge about observation reliability, making the modelling assumptions explicit.
- Computationally equivalent to re-scaled OLS, so it is available in every major statistical package with no additional algorithmic complexity.
- Requires knowledge of, or a reliable estimate of, the relative error variances; misspecified weights can produce estimates less efficient than OLS.
- Does not handle correlated errors (off-diagonal covariance); for autocorrelated or clustered errors, Generalized Least Squares or clustered standard errors are needed.
- Estimated weights (Feasible WLS) introduce additional uncertainty not captured by the nominal standard errors unless the sample is large.
- Sensitive to outliers in the weight specification: a single observation assigned a very large weight can unduly dominate the fit.
Frequently asked
How do I choose the weights?
The ideal weight for observation i is 1/σ_i², the reciprocal of its error variance. In practice: if you have replicates, estimate σ_i² from them; if variance scales with a predictor x_i (common in economics), use w_i = 1/x_i or w_i = 1/x_i²; if data are group means, use w_i = n_i (group size). When weights must be estimated from the data, the procedure becomes Feasible WLS and introduces additional sampling error.
Is WLS the same as GLS?
WLS is a special case of GLS. GLS allows a fully general error covariance matrix Ω; WLS restricts Ω to be diagonal, meaning errors are uncorrelated but have unequal variances. If errors are also correlated (e.g. time series, clustered data), you need full GLS or a clustered-error correction.
Why not just use OLS with heteroscedasticity-robust (sandwich) standard errors?
Robust standard errors correct OLS inference without changing the coefficient estimates. WLS changes the estimates themselves, making them more efficient when the weight structure is well specified. If the variance function is well understood, WLS yields narrower confidence intervals than robust-OLS. If it is uncertain, robust-OLS is safer because it does not rely on a possibly wrong weight model.
How do I verify that WLS has improved the fit?
Plot the weighted residuals (raw residual × √w_i) against fitted values and predictors. The weighted residuals should show roughly constant spread (homoscedasticity) if the weights are correctly specified. Also compare the residual standard error and information criteria (AIC/BIC) between the WLS and OLS models.
Sources
- Aitken, A. C. (1935). IV.—On least squares and linear combination of observations. Proceedings of the Royal Society of Edinburgh, 55, 42–48. DOI: 10.1017/S0370164600014346 ↗
- Greene, W. H. (2012). Econometric Analysis (7th ed.). Pearson Education. ISBN: 978-0131395381
- Montgomery, D. C., Peck, E. A., & Vining, G. G. (2012). Introduction to Linear Regression Analysis (5th ed.). Wiley. ISBN: 978-0470542811
How to cite this page
ScholarGate. (2026, June 3). Weighted Least Squares Regression. ScholarGate. https://scholargate.app/en/statistics/weighted-least-squares
Which method?
Set this method beside its closest kin and read them side by side — the library lays the books on the table; the choice is yours.
- Generalized Least SquaresStatistics↔ compare
- Ordinary Least SquaresStatistics↔ compare
- Robust RegressionStatistics↔ compare