Generalized Least Squares (GLS)
Generalized Least Squares Estimator · Also known as: GLS, Aitken estimator, EGLS, feasible GLS, FGLS, weighted least squares, WLS
Generalized Least Squares (GLS) is a linear regression estimator that extends ordinary least squares to handle situations where the error terms are correlated or have non-constant variance (heteroscedasticity). Introduced by Alexander Craig Aitken in 1935, GLS achieves the Best Linear Unbiased Estimator (BLUE) under a general error covariance structure by weighting observations according to their precision, providing a theoretical bridge between OLS and modern linear mixed models.
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When to use it
Use GLS when the residuals from an initial OLS fit show evidence of heteroscedasticity (non-constant error variance, detectable with the Breusch–Pagan or White test) or serial correlation (detectable with the Durbin–Watson or Breusch–Godfrey test). GLS is particularly valuable in time-series regression with autocorrelated errors, panel data with group-specific variances, and survey data with known sampling weights. Core assumptions are: (1) the model is correctly specified and linear in parameters; (2) the covariance matrix Ω is known or consistently estimable; (3) X is full column rank; and (4) for finite-sample BLUE status, errors are normally distributed. FGLS requires large samples for reliable inference because estimating Ω introduces additional uncertainty.
Strengths & limitations
- Achieves the lowest variance among all linear unbiased estimators (BLUE) when Ω is correctly specified — surpassing OLS under non-spherical errors.
- Produces valid standard errors and confidence intervals even when errors are heteroscedastic or autocorrelated, unlike naive OLS in those settings.
- Encompasses many important special cases: weighted least squares (diagonal Ω), panel GLS, and AR-corrected time-series regression all follow from the same general formula.
- Provides the theoretical foundation for linear mixed models and random-effects panel estimators, making it a conceptual cornerstone of modern regression analysis.
- Requires knowledge or consistent estimation of Ω; a misspecified covariance structure can make GLS worse than OLS in finite samples.
- Feasible GLS introduces an additional estimation stage, reducing efficiency gains in small samples and complicating inference.
- GLS is linear and parametric: it does not handle nonlinear relationships, binary or count outcomes, or highly non-normal errors without further modification.
- Inverting large Ω matrices can be computationally demanding in high-dimensional settings.
Frequently asked
What is the difference between GLS and OLS?
OLS minimises the unweighted sum of squared residuals and is BLUE only when errors are independent and identically distributed. GLS minimises a weighted sum that accounts for the full error covariance matrix Ω, and is BLUE whenever Ω is correctly specified. When Ω = σ²I, GLS and OLS produce identical estimates; in all other cases GLS is more efficient.
What is Feasible GLS and when should I use it instead of true GLS?
True GLS requires knowing Ω exactly, which is almost never the case in practice. Feasible GLS (FGLS) estimates Ω from OLS residuals — by modelling variance as a function of covariates or by fitting an autocorrelation structure to residuals — then plugs that estimate into the GLS formula. FGLS is consistent and asymptotically efficient, but its finite-sample properties depend on how well Ω is estimated. Use FGLS when sample size is large enough to estimate Ω reliably and you have a defensible model for the error structure.
Is GLS the same as Weighted Least Squares?
Weighted Least Squares (WLS) is a special case of GLS where Ω is diagonal — meaning errors are independent but have different variances. GLS is the more general form that also handles correlated errors through off-diagonal elements of Ω. All WLS estimates are GLS estimates, but not vice versa.
How does GLS relate to linear mixed models?
Linear mixed models (LMMs) generalise GLS by decomposing the error covariance into fixed variance components tied to random effects. The marginal form of an LMM is a GLS model where Ω is structured according to the random-effects design. Estimation via REML in LMMs can be viewed as an iterated FGLS procedure, making GLS the conceptual parent of the mixed-model framework.
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. (2003). Econometric Analysis (5th ed.). Prentice Hall. ISBN: 978-0131108493
- Wooldridge, J. M. (2010). Econometric Analysis of Cross Section and Panel Data (2nd ed.). MIT Press. ISBN: 978-0262232586
How to cite this page
ScholarGate. (2026, June 3). Generalized Least Squares Estimator. ScholarGate. https://scholargate.app/en/statistics/generalized-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.
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