Instrumental Variables in Continuous-Time Stochastic Processes

Estimating the causal effect of a treatment $X$ on an outcome $Y$ from observational data in the presence of unobserved confounding requires additional structure or assumptions. Existing identification strategies such as instrumental variable regression exploit exogenous variation in only the treatment. We show that the causal effect can also be identified if the outcome mechanism varies across a finite set of environments, even if the treatment distribution is unchanged. Concretely, we introduce Outstrumental Variable Regression (OVR), which proceeds in three steps: (i) compute the conditional expectation of $Y$ given $X$ in each environment; (ii) form the vector of differences in these conditional expectations between all environments and a reference environment (we call this vector contrast); and (iii) regress $Y$ on $X$ while adjusting for the contrast and environment. We prove that this yields the true causal effect if the following two conditions hold: inclusion restriction, i.e., the contrast determines the conditional confounding bias; and positivity, i.e., the treatment varies sufficiently strongly after adjustment. We prove that a finite-sample version of OVR is consistent under regularity conditions and demonstrate the procedure's effectiveness on synthetic and real-world data.

The instrumental-variables (IV) setting is standard for partial identification of causal effects when unobserved confounding makes point identification impossible. Existing approaches face methodological bottlenecks: closed-form bound estimands are required -- e.g., Balke-Pearl equations in binary IV -- and even when available, designing accurate estimators requires manual effort tailored to each estimand. While direct Bayesian inference of the causal effects, instead of the bounds, circumvents these challenges, it is often computationally intensive and suffers from high prior sensitivity or under-dispersed posteriors. As a remedy, we introduce IV-ICL, an amortized Bayesian in-context learning method that learns the marginal posterior distribution of the causal effects directly and derives bounds as its quantiles. Unlike standard variational inference that optimizes exclusive KL divergence, amortized Bayesian inference minimizes the expected inclusive KL, a mass-covering objective. We empirically observe that optimizing inclusive KL can recover the entire identified set across diverse data-generating processes, while exclusive-KL (e.g. with variational inference) of the same Bayesian formulation collapses onto a single mode and fails to cover the identified set. We evaluate IV-ICL on synthetic and semi-synthetic IV benchmarks and show it produces intervals that are more reliably valid and more informative compared to efficient semi-parametric, Bayesian, and plug-in baselines, at 20-500x lower inference time. Beyond methodology, we propose a procedure to convert randomized controlled trials into IV benchmarks with provably preserved ground-truth causal effects that enables a more realistic evaluation of partial-identification methods.

Continuous-time stochastic differential equations (SDEs) provide a natural framework for modeling interacting dynamical systems. A central causal question is whether causal effects among observed variables can be identified from observational data drawn from the stationary distribution, given the underlying causal graph. For linear SDEs, existing identifiability results are largely restricted to fully observed systems whose causal graphs contain no directed 2-cycles. We first establish new graphical conditions for the generic identifiability of causal effects in systems that may contain directed 2-cycles, thereby allowing for reciprocal interactions. Next, we address the previously unexplored setting of latent variables by developing an instrumental-variable framework for identifying a target direct causal effect under partial observability.