Nonlinear Generalized Least Squares (NGLS)
Nonlinear Generalized Least Squares · Also known as: NGLS, nonlinear generalized least squares, feasible nonlinear GLS, FNGLS
Nonlinear Generalized Least Squares extends the classical GLS framework to regression models where the mean function is nonlinear in the parameters. It accounts for non-spherical errors — heteroscedasticity or autocorrelation — by pre-weighting the nonlinear objective with an estimated error covariance matrix, yielding consistent and asymptotically efficient estimates.
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When to use it
Use Nonlinear GLS when the relationship between the outcome and predictors is genuinely nonlinear in the parameters AND the errors are non-spherical (heteroscedastic or autocorrelated). It is especially suited to time-series models (e.g., exponential growth, logistics curves) with AR or ARCH errors, or to cross-sectional models where error variance varies systematically with covariates. Do NOT use it when the mean function is actually linear — plain GLS is simpler and equivalent. Avoid NGLS when the sample is small (estimation of Ω is imprecise), when the error model is misspecified (resulting in an inconsistent Ω̂ that distorts inference), or when the nonlinear functional form has not been justified by theory, as misspecification in f(x, β) can dominate the efficiency gain.
Strengths & limitations
- Asymptotically efficient relative to NLS when errors are non-spherical, under correct specification of Ω.
- Accommodates a wide range of error structures: heteroscedasticity, AR/MA serial correlation, ARCH effects.
- Produces valid standard errors and confidence intervals even when errors are correlated or heteroscedastic.
- Subsumes NLS as a special case when Ω = I, providing a unified framework.
- Compatible with standard iterative nonlinear solvers (Gauss-Newton, Levenberg-Marquardt).
- Requires correct specification of both the mean function f(x, β) and the error covariance structure Ω; misspecification in either damages consistency.
- Estimation of Ω from a first-stage NLS residuals introduces additional estimation uncertainty, complicating finite-sample inference.
- Iterative numerical optimisation may fail to converge or converge to a local minimum if the objective is non-convex or starting values are poor.
- Computationally more demanding than NLS or linear GLS, especially for large datasets with complex Ω structures.
- Asymptotic theory may be a poor guide in small samples where Ω̂ is imprecise.
Frequently asked
How does NGLS differ from NLS?
NLS minimises the unweighted sum of squared residuals and is consistent but not efficient when errors are non-spherical. NGLS additionally weights the objective by Ω⁻¹, exploiting the error covariance structure to achieve asymptotic efficiency — the same gain that GLS has over OLS in the linear case.
What happens if I mis-specify the covariance structure Ω?
A misspecified Ω can make NGLS inconsistent or less efficient than NLS. If you are uncertain about the form of Ω, consider using robust (sandwich) standard errors after NLS, or GMM with a flexible weighting matrix, rather than committing to a potentially wrong parametric Ω.
How do I choose starting values for the nonlinear optimiser?
Grid search, profile likelihood plots, or theoretical guidance from economic or physical models are common strategies. Running NLS first with several starting points and using the best NLS estimates as the NGLS starting point is a practical and widely recommended approach.
Is NGLS the same as Feasible GLS for nonlinear models?
Yes — Feasible NGLS (FNGLS) is the operational version where Ω is estimated in a first stage. True NGLS with known Ω is a theoretical benchmark; in practice one always uses FNGLS.
When should I use GMM instead of NGLS?
When regressors are endogenous, error moments beyond the second are informative, or the covariance structure is difficult to parametrise, GMM is more general. NGLS is preferable when the model is fully specified (exogenous regressors, known error family) and computational simplicity matters.
Sources
- Gallant, A. R. (1987). Nonlinear Statistical Models. Wiley. ISBN: 978-0471802600
- Davidson, R., & MacKinnon, J. G. (2004). Econometric Theory and Methods. Oxford University Press. link ↗
How to cite this page
ScholarGate. (2026, June 3). Nonlinear Generalized Least Squares. ScholarGate. https://scholargate.app/en/econometrics/nonlinear-gls
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