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

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.