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 $\sigma$-algebras. Assuming that $E(Z_n|{\mathcaligr F}_{n-1})=g_n(\theta_0,\nu_0)=g^{(1)}_n(\theta_0)+g^{(2)}_n(\theta_0,\nu_0)$, $\theta_0\in{\mathbb{R}}^p$, $p<\infty$, $\nu_0\in{\mathbb{R}}^q$ and $q\leq\infty$, we study the asymptotic properties of $\hat{\theta}_n:=\arg\min_{\theta}\sum_{k=1}^n(Z_k-g_k({\theta,\hat{\nu}}))^2\lambda_k^{-1}$, where $\lambda_k$ is ${\mathcaligr F}_{k-1}$-measurable, $\hat{\nu}=\{\hat{\nu}_k\}$ is a sequence of estimations of $\nu_0$, $g_n(\theta,\hat{\nu})$ is Lipschitz in $\theta$ and $g^{(2)}_n(\theta_0,\hat{\nu})-g^{(2)}_n(\theta,\hat{\nu})$ is asymptotically negligible relative to $g^{(1)}_n(\theta_0)-g^{(1)}_n(\theta)$. We first generalize to this nonlinear stochastic model the necessary and sufficient condition obtained for the strong consistency of $\{\hat{\theta}_n\}$ 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 $\hat{\theta}_n$. We illustrate the theoretical results with examples of branching processes, and extension to quasi-likelihood estimators is also considered.