Time-varying Coefficients Estimation in Differential Equation Models with Noisy Time-varying Covariates

TL;DR

This paper addresses the challenge of estimating time-varying coefficients in differential equation models when covariates are noisy, deriving asymptotic properties for estimators under measurement error conditions.

Contribution

It extends existing estimation theory to handle noisy covariates in differential equations, providing asymptotic bias and variance analysis for two-step estimators.

Findings

Derived asymptotic bias and variance for estimators with noisy covariates

Extended quadratic regression functional theory to differential equations

Improved understanding of measurement error impact on coefficient estimation

Abstract

We study the problem of estimating time-varying coefficients in ordinary differential equations. Current theory only applies to the case when the associated state variables are observed without measurement errors as presented in \cite{chenwu08b,chenwu08}. The difficulty arises from the quadratic functional of observations that one needs to deal with instead of the linear functional that appears when state variables contain no measurement errors. We derive the asymptotic bias and variance for the previously proposed two-step estimators using quadratic regression functional theory.