Enhancing Two-Stage Estimation in Differential Equation Models: A Bias-Correction Method via Stochastic Approximation

Differential equation models have become increasingly popular for investigating dynamic processes. However, commonly used two-stage estimation methods, such as the generalized local linear approximation (GLLA), often produce biased parameter estimates. This study proposes a bias-correction method for GLLA estimates in second-order differential equation models. The method solves a bias-correction equation (derived from the relation between true parameter values and their biased estimates) via stochastic approximation, producing asymptotically unbiased estimates even with large initial bias. We first demonstrate the application of the bias-correction method by correcting the bias of a single parameter in a simple second-order differential equation model (i.e., the linear oscillator model with time-independent measurement error). We then extend the method to a more commonly used second-order differential equation model (i.e., the damped linear oscillator model), examining its performance in simultaneously addressing multiple parameters and incorporating time-dependent dynamic error. A simulation study shows that the bias-correction method substantially reduces bias in GLLA estimates, yielding highly accurate and precise parameter estimates. An empirical illustration further compares the results of GLLA and the bias-correction method. Our findings highlight the effectiveness of the proposed method in improving parameter estimation for differential equation models, offering an enhanced approach for analyzing dynamic processes.