Nonlinear Weighted Least Squares (NWLS)

NWLS is a special case of nonlinear GLS in which the error covariance matrix is diagonal — errors are uncorrelated but have unequal variances captured by the weights. Full nonlinear GLS also models off-diagonal covariance (e.g. autocorrelated errors in time series), which requires additional structure on the covariance matrix.

A common practice is a two-step feasible approach: fit an unweighted NLS first, regress the squared residuals on candidate variance predictors to estimate the variance function, then re-fit using the reciprocal of the fitted variances as weights. The estimated weights introduce additional uncertainty, but the procedure is consistent under mild conditions.

Good starting values are critical. Use domain knowledge (e.g. rough estimates of the plateau and half-saturation constant for a Michaelis-Menten model), grid search over plausible parameter ranges, or a linearising transformation to get OLS-based initial guesses before passing them to the NWLS routine.

Yes, provided the nonlinear mean function applies within each cross-sectional unit and you can specify unit-level or observation-level weights capturing the error variance. For panels with both heteroscedasticity and serial correlation, a nonlinear feasible GLS approach accounting for both features is more appropriate.

Not exactly. IRLS is an algorithmic strategy used to fit various models (including GLMs) by repeatedly updating weights and refitting. NWLS can be implemented via IRLS, but the term specifically describes a regression model where fixed or estimated variance-stabilising weights correct for heteroscedasticity around a nonlinear mean function.