Conditional least squares estimation in nonstationary nonlinear stochastic regression models

Abstract

Let $\{Z_n\}$ be a real nonstationary stochastic process such that $E(Z_n|{\mathcaligr F}_{n-1})\stackrel{\mathrm{a.s.}}{<}\infty$ and $E(Z^2_n|{\mathcaligr F}_{n-1})\stackrel{\mathrm{a.s.}}{<}\infty$, where $\{{\mathcaligr F}_n\}$ is an increasing sequence of $σ$-algebras. Assuming that $E(Z_n|{\mathcaligr F}_{n-1})=g_n(θ_0,ν_0)=g^{(1)}_n(θ_0)+g^{(2)}_n(θ_0,ν_0)$, $θ_0\in{\mathbb{R}}^p$, $p<\infty$, $ν_0\in{\mathbb{R}}^q$ and $q\leq\infty$, we study the asymptotic properties of $\hatθ_n:=\arg\min_θ\sum_{k=1}^n(Z_k-g_k({θ,\hatν}))^2λ_k^{-1}$, where $λ_k$ is ${\mathcaligr F}_{k-1}$-measurable, $\hatν=\{\hatν_k\}$ is a sequence of estimations of $ν_0$, $g_n(θ,\hatν)$ is Lipschitz in $θ$ and $g^{(2)}_n(θ_0,\hatν)-g^{(2)}_n(θ,\hatν)$ is asymptotically negligible relative to $g^{(1)}_n(θ_0)-g^{(1)}_n(θ)$. We first generalize to this nonlinear stochastic model the necessary and sufficient condition obtained for the strong consistency of $\{\hatθ_n\}$ in the linear model. For that, we prove a strong law of large numbers for a class of submartingales. Again using this strong law, we derive the general conditions leading to the asymptotic distribution of $\hatθ_n$. We illustrate the theoretical results with examples of branching processes, and extension to quasi-likelihood estimators is also considered.