Instrumental Variables in Continuous-Time Stochastic Processes

Abstract

Causal questions often arise in systems that evolve in continuous time. While instrumental variable (IV) methods provide a principled approach to identify causal effects in static or discrete-time settings, even in the presence of unobserved confounding, they do not yet extend to continuous-time dynamics. In this work, we develop an IV framework for stationary continuous-time stochastic processes parameterized by stochastic differential equations (SDE), where the instrument itself takes the form of a stochastic process. We encode the structural sparsity and causal effects directly at the infinitesimal level of the drift, and contrast these assumptions with the IV assumptions on the discretized version of the same process. Based on these structural assumptions, we derive moment restrictions using path-valued instruments. We show how these restrictions simplify for Ornstein-Uhlenbeck processes and present an identification condition together with a corresponding consistent estimation method. Finally, we extend our estimation framework to nonlinear causal effects by expressing the path-based moment restrictions leveraging the signature kernel. Empirically, our method recovers causal effects in confounded continuous-time systems where maximum-likelihood-based estimation and discrete-time IV methods are systematically biased.