Asymptotic Properties of Nonlinear Least Squares Estimates in Stochastic Regression Models

Stochastic regression models of the form yᵢ = fᵢ(θ) + εᵢ, where the random disturbances εᵢ form a martingale difference sequence with respect to an increasing sequence of σ-fields ᵢ\ and fᵢ is a random Gi - 1-measurable function of an unknown parameter θ, cover a broad range of nonlinear (and linear) time series and stochastic process models. Herein strong consistency and asymptotic normality of the least squares estimate of θ in these stochastic regression models are established. In the linear case fᵢ(θ) = θTψᵢ, they reduce to known results on the linear least squares estimate (∑ⁿ₁ψᵢψTᵢ)⁻¹∑ⁿ₁ψᵢ yᵢ with stochastic Gi - 1-measurable regressors ψᵢ.