Conditional least squares estimation in nonstationary nonlinear stochastic regression models

Abstract

Let {Z_n} be a real nonstationary stochastic process such that $E(Z_{n}|{\mathcal{F}}_{n-1})\stackrel{\mathrm{a.s.}}<\infty$ and $E(Z_{n}^{2}|{\mathcal{F}}_{n-1})\stackrel{\mathrm{a.s.}}<\infty$, where $\{{\mathcal{F}}_{n}\}$ is an increasing sequence of σ-algebras. Assuming that $E(Z_{n}|{\mathcal{F}}_{n-1})=g_{n}(\theta_{0},\nu_{0})=g_{n}^{(1)}(\theta_{0})+g_{n}^{(2)}(\theta _{0},\nu_{0})$, θ₀∈ℝ^p, p<∞, ν₀∈ℝ^q and q≤∞, we study the asymptotic properties of $\widehat{\theta}_{n}:=\arg\min_{\theta}\sum_{k=1}^{n}(Z_{k}-g_{k}({\theta,\widehat{\nu}}))^{2}\lambda _{k}^{-1}$, where λ_k is ${\mathcal{F}}_{k-1}$-measurable, ̂ν={̂ν_k} is a sequence of estimations of ν₀, g_n(θ, ̂ν) is Lipschitz in θ and g_n⁽²⁾(θ₀, ̂ν)−g_n⁽²⁾(θ, ̂ν) is asymptotically negligible relative to g_n⁽¹⁾(θ₀)−g_n⁽¹⁾(θ). We first generalize to this nonlinear stochastic model the necessary and sufficient condition obtained for the strong consistency of {̂θ_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 ̂θ_n. We illustrate the theoretical results with examples of branching processes, and extension to quasi-likelihood estimators is also considered.