Identifiability and Estimation in Continuous Lyapunov Models

TL;DR

This paper investigates the identifiability and estimation of parameters in continuous Lyapunov models derived from stochastic differential equations, providing theoretical results and a new estimator validated through simulations.

Contribution

It establishes generic identifiability of the drift matrix using higher-order cumulants and introduces a semiparametric estimator with proven asymptotic properties.

Findings

Identifiability holds for any connected graph under non-Gaussianity.

The proposed estimator is asymptotically valid for large samples.

Estimation accuracy improves with larger sample sizes.

Abstract

Cross-sectional observations from a dynamical system can be modeled via steady-state distributions of Markov processes. The major challenge is then to determine whether the process parameters can be identified and estimated from the steady-state distributions. We study this problem for continuous Lyapunov models that arise as steady-state distributions of the solution to a multivariate stochastic differential equation, whose linear drift matrix is parametrized by a directed graph. We derive equations for the cumulant tensors of any order for this distribution, which generalize the well-known covariance Lyapunov equation. Under a non-Gaussianity assumption we prove generic identifiability of the drift matrix for any connected graph using the equations for the higher-order cumulants. Based on the identifiability result, we propose a new semiparametric estimator of the drift matrix, and we derive its asymptotic distribution. A simulation study demonstrates the asymptotic validity of the estimator but shows that it is only accurate for relatively large sample sizes, illustrating the hardness of the unconstrained estimation problem.