Bias of LS estimators in nonlinear regression models with constraints. Part I: General case

summary:We derive expressions for the asymptotic approximation of the bias of the least squares estimators in nonlinear regression models with parameters which are subject to nonlinear equality constraints. The approach suggested modifies the normal equations of the estimator, and approximates them up to $o_{p}( N^{-1}) $, where $N$ is the number of observations. The “bias equations” so obtained are solved under different assumptions on constraints and on the model. For functions of the parameters the invariance of the approximate bias with respect to reparametrisations is demonstrated. Singular models are considered as well, in which case the constraints may serve either to identify the parameters, or eventually to restrict the parameter space