Instrumental variables move into continuous time: as confounding grows, discrete-time estimators drift while path-based moments hold
Synopsis
Continuous-time stochastic processes had no instrumental-variable framework, and discrete-time methods place the structural sparsity on finite-step transitions rather than on the actual mechanism. The work writes relevance, exclusion and exogeneity directly at the infinitesimal level of a stochastic differential equation's drift and builds moment conditions from the whole instrument path, giving an identification condition and a consistent estimator in the linear case. On confounded Ornstein-Uhlenbeck simulations, maximum likelihood becomes systematically biased as confounding grows while this estimator stays stable, and a single instrument process identifies multi-dimensional effects.
Interpretation
Causal effect estimation can now be built on continuous-time stochastic processes, where the instrument is not a static variable but a stochastic process evolving over time. Earlier instrumental-variable methods covered only static or discrete-time settings, and discrete-time models place structural sparsity on finite-step transitions, so they handle neither irregular sampling nor the local structure of a continuous-time process. On simulated Ornstein-Uhlenbeck systems with confounding strength rising from 0.1 to 1.0, the mean coefficient error of maximum likelihood climbs from 0.0069 to 0.2359, while the continuous-time integral estimator stays between 0.0123 and 0.0290 across those levels.
The approach writes the instrumental-variable assumptions at the infinitesimal level of the drift and then builds moment conditions from the history of the instrument process. The moment conditions are orthogonality between forecasting errors and functionals of the instrument path, so the entire instrument path acts as a valid instrument rather than only its current state. In the linear case the moments reduce to fitting lagged covariance dynamics, identification follows from a controllability condition, and a consistent estimator combines Nadaraya-Watson kernel regression with least squares; in the nonlinear case a signature kernel expresses the path functionals, and on semi-synthetic glucose-insulin data the overall RMSE is 0.0157 against 0.0998 for the maximum-likelihood baseline.
Perspective
The demonstrated scope is jointly stationary continuous-time processes described by stochastic differential equations, with possibly irregular observation times and a confounding process that is never observed; all experiments use simulated Ornstein-Uhlenbeck systems and semi-synthetic glucose-insulin data.
In the nonlinear case the target function is generally unidentifiable, so regularization selects one candidate from the identified set; instrumental-variable estimation is less efficient than maximum likelihood and performs worse when confounding is weak, and estimator variance rises as the dimension of the target causal effect grows.