Instrumental Variables in Continuous-Time Stochastic Processes

Lars Pennig; Søren Wengel Mogensen; Niki Kilbertus

cs.LGNov 5, 2026

Instrumental Variables in Continuous-Time Stochastic Processes

Authors: Lars Pennig, Søren Wengel Mogensen, Niki Kilbertus

Organizations: Technical University of Munich; Helmholtz Munich; Munich Center for Machine Learning · Center for Statistics, Department of Finance; Copenhagen Business School

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.

Figures & tables

Panel of Figure 1 — Figure 1: Observed trajectories of processes $(Y,X,Z)$ , confounded by the hidden process $U$ , together with the assumed sparse drift graph, are used to identify the objective function $f_{y}(Y_{t},X_{t})$ .

Panel of Figure 2 — Figure 2: RMSE of $\beta_{yy}$ and $\beta_{yx}$ for a confounded OU process (one instrumental and one outcome process throughout, except in (b)). Left: Estimation error under varying confounding strength. Center: Recovery of multidimensional causal effects from a single instrument process. Right: Estimation error as a function of instrument–covariate coupling strength $\beta_{xz}$ . Error bars indicate 95% confidence intervals, boxplots show medians and interquartile ranges. Each evaluation is based on 20 systems (left) and 10 systems (middle/right).

Method	RMSE
IV-SDE	0.0157 $\;{\scriptstyle(0.0035)}$
MLE	0.0998 $\;{\scriptstyle(0.0003)}$

Table 1: Parameter RMSE on the diabetes dataset.

Figure 3: Results for a nonlinear drift parametrization. (left) Relative drift RMSE over the

X_{t},Y_{t}

distribution. (right) Ground truth sigmoidal drift

f_{\mathrm{gt}}

and learned drift

\hat{f}

parameterized with an MLP.

Appendix figures & tables6 assets

Supplementary material from the paper’s appendix.

Appendix

Panel of Figure 4 — Figure 4: Standard instrumental-variable causal graph. The instrument $Z$ affects the outcome $Y$ only through the treatment $X$ , while the latent variable $U$ confounds the relationship between $X$ and $Y$ . The causal effect of interest is the direct dependence of $Y$ on $X$ highlighted in blue.

Configuration	IV-SDE MSE (total)
$x-y,\ k=0.0$	0.0036
$x-y,\ k=0.3$	0.0068
$x-y,\ k=0.5$	0.0151
$x-y,\ k=0.9$	0.0162
$z-x,\ k=0.3$	0.0031
$z-x,\ k=0.5$	0.0009

Appendix

Table 2: Total MSE of the IV-SDE estimator under different diffusion configurations and mixing strengths.

Confounding strength	MLE full	VAR-1	VAR-1 (reg.)	IVSDE-diff	IVSDE-int
0.1	0.0069	0.6590	0.0516	0.0104	0.0123
0.2	0.0379	1.2766	0.0625	0.0148	0.0192
0.3	0.0691	1.5407	0.0576	0.0182	0.0202
0.4	0.1244	1.1334	0.0464	0.0244	0.0176
0.5	0.1146	0.8761	0.0431	0.0097	0.0137
0.6	0.1746	0.8853	0.0613	0.0214	0.0184

Appendix

Table 3: Mean per-system parameter RMSE over 20 systems. (reg.) denotes that regularly sampled timepoints were used.

Confounding strength	MLE full	VAR-1	VAR-1 (reg.)	IVSDE-diff	IVSDE-int	Direct ratio (reg.)
0.1	0.0472	0.8777	0.3202	0.0166	0.0174	0.1005
0.2	0.1711	1.9936	0.1894	0.0504	0.0526	0.0613
0.3	1.2192	2.2831	0.1939	0.0772	0.0649	0.3011
0.4	0.3261	2.1911	0.2962	0.0218	0.0233	0.1542
0.5	0.5827	1.6280	0.2167	0.0338	0.0348	0.1127
0.6	1.1755	1.2828	0.2407	0.0344	0.0442	0.1520

Appendix

Table 4: Ratio RMSE. Ratio RMSE across varying confounding strength. Each value represents the mean over 20 sampled systems.

Parameter	Description	Value
$k$	Number of instruments	$2$
$\mu_{Z}$	Common instrument equilibrium mean	$0$
$(m_{Z,1},m_{Z,2})$	Instrument mean-reversion rates	$(0.4,\,0.8)$
$(a_{Z,1},a_{Z,2})$	Instrument effects on $X$	$(0.35,\,0.35)$
$(\sigma_{Z,1},\sigma_{Z,2})$	Instrument noise amplitudes	$(0.5,\,0.5)$
$c_{ux}$	Effect of $U$ on $X$	$0.60$

Appendix

Table 5: Parameter values for the two-instrument diabetes dataset configuration.

Study	Setting	RMSE
		$a_{y}$	$\gamma_{xy}$	Overall
Lookback	0.2	0.0615	0.0126	0.0444
	0.5	0.0113	0.0117	0.0115
	0.7	0.0371	0.0154	0.0284
	0.9	0.0477	0.0217	0.0370
Horizon	0.1	0.0239	0.0094	0.0182

Appendix

Table 6: Hyperparameter sensitivity in the diabetes model. in terms of RMSE. The standard setting across all runs is a Lookback of

0.7s

, a horizon of

0.2

and a RBF bandwidth of

0.5

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Outstrumental Variable Regression

Felix Schur, Linus Kühne, Julius von Kügelgen, Jonas Peters

Seminar for Statistics, ETH Zurich · ETH AI Center, ETH Zurich

Causal Effect Estimation Causal Inference Causal Inference with Unobserved Confounding Causal Identifiability

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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

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.

Abstract

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