Conditional least squares estimation in nonstationary nonlinear stochastic regression models

TL;DR

This paper develops asymptotic theory for conditional least squares estimators in nonstationary nonlinear stochastic regression models, extending linear model results and applying to branching processes.

Contribution

It generalizes strong consistency and asymptotic distribution results to nonlinear models with nonstationary data, including quasi-likelihood estimators.

Findings

Established strong law of large numbers for submartingales.

Derived conditions for estimator consistency.

Illustrated results with branching process examples.

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.