Instrumental variables system identification with $L^p$ consistency

TL;DR

This paper introduces an IV estimator that synthesizes data-based instruments, achieving $L^p$ consistency and significantly reducing bias and RMSE in nonlinear dynamical systems.

Contribution

It develops a novel data-driven instrumental variables method with proven finite-sample $L^p$ consistency applicable to nonlinear time series models.

Findings

Reduces parameter bias by 200x in continuous-time Lorenz system

Reduces RMSE by up to tenfold compared to least squares

Achieves nonparametric $ oot n$-convergence rate in both discrete and continuous models

Abstract

Instrumental variables (eliminate the bias that afflicts least-squares identification of dynamical systems through noisy data, yet traditionally relies on external instruments that are seldom available for nonlinear time series data. We propose an IV estimator that synthesizes instruments from the data. We establish finite-sample $L^{p}$ consistency for all $p \ge 1$ in both discrete- and continuous-time models, recovering a nonparametric $\sqrt{n}$-convergence rate. On a forced Lorenz system our estimator reduces parameter bias by 200x (continuous-time) and 500x (discrete-time) relative to least squares and reduces RMSE by up to tenfold. Because the method only assumes that the model is linear in the unknown parameters, it is broadly applicable to modern sparsity-promoting dynamics learning models.