Outstrumental Variable Regression

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

stat.MLOct 30, 2026

Outstrumental Variable Regression

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

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

Abstract

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.

Figures & tables

Panel of Figure 1 — Figure 1: IV setting.

Panel of Figure 2 — Figure 2: OVR setting.

Figure 3: Outstrumental Variable Regression (OVR). We illustrate the main steps of OVR using three environments. Step 1: Regression. Estimate the conditional expectations

{\color[rgb]{0,0.6211,0.4492}\mu_{1}},{\color[rgb]{0.8359,0.3672,0}\mu_{2}},{\color[rgb]{0.8008,0.4727,0.6563}\mu_{3}}

Y

given

X

in each environment. Step 2: Contrast. Subtract

{\color[rgb]{0,0.6211,0.4492}\mu_{1}}

from

{\color[rgb]{0.8359,0.3672,0}\mu_{2}}

and

{\color[rgb]{0.8008,0.4727,0.6563}\mu_{3}}

to obtain the contrasts

{\color[rgb]{0.4922,0.2461,0.5977}C_{2}}

and

{\color[rgb]{0.4023,0.4922,0.5898}C_{3}}

; the shared causal contribution

{\color[rgb]{0,0.4453,0.6992}\beta_{0}x}

cancels. Step 3: Adjustment. Regress

Y

X

, adjusting for the contrast vector

C(X)\coloneqq({\color[rgb]{0.4922,0.2461,0.5977}C_{2}(X)},{\color[rgb]{0.4023,0.4922,0.5898}C_{3}(X)})

and environment

E

. This recovers

{\color[rgb]{0,0.4453,0.6992}\beta_{0}}

C(X)

determines the nuisance

q_{1}(x)\coloneqq{\color[rgb]{0,0.6211,0.4492}\mu_{1}(x)}-{\color[rgb]{0,0.4453,0.6992}\beta_{0}x}

(dashed black) and enough variation in

X

remains after partialling out

C(X)

and

E

\widehat{Z}_{i}\coloneqq Z_{i}-\frac{1}{M}\sum_{m=1}^{M/2}\sum_{s\in\{-1,1\}}\widehat{f}_{Z,k}\bigl(\overline{C}(X_{i})+s\xi^{\prime}_{im},E_{i}\bigr)\qquad\text{for}\qquad Z\in\{X,Y\}.

Algorithm 1 Smoothed Outstrumental Variable Regression

Panel of Figure 4 — Figure 4: Setting with recoverable latent-state $Z_{C}$ .

Panel of Figure 5 — Figure 5: Coefficient error at $n\in\{0.5,1,2,4,10\}\mathrm{k}$ across six synthetic settings. Lines show means and bands $\pm$ one Monte Carlo s.e. over ten paired replications; panels use separate linear error scales. As $n$ increases, Smoothed OVR has the lowest mean $\ell_{2}$ estimation error in all six settings.

Smoothed OVR	OLS	FE	Anchor ( $\gamma=2$ )	LNN
$\mathbf{.019}\;(\mathbf{.001})$	$.129\;(.001)$	$.104\;(.001)$	$.171\;(.001)$	$.095\;(.011)$

Table 1: Mean

\ell_{2}

estimation error over ten training/reference splits, with Monte Carlo standard errors in parentheses and the lowest means in bold.

Appendix figures & tables5 assets

Supplementary material from the paper’s appendix.

Appendix

\widehat{C}(x)\coloneqq\bigl(\widehat{\mu}_{e}(x)-\widehat{\mu}_{e_{0}}(x)\bigr)_{e\neq e_{0}}.

Appendix

Algorithm 2 Direct Outstrumental Variable Regression

Panel of Figure 6 — Figure 6: Synthetic comparison at $n\in\{0.5,1,2,4,10\}\mathrm{k}$ , including Direct OVR. Means and $\pm$ one Monte Carlo s.e. over ten paired replications.

Added component	$\\|\widehat{\beta}-\beta_{0}\\|_{2}$	Lower error	$\\|\widehat{r}\\|_{2}$	$\dfrac{\\|\widehat{G}_{\lambda}^{-1}\widehat{r}\\|_{2}}{\\|\widehat{r}\\|_{2}}$
Contrast-head regularization	$0.926$	$42/60$	$0.774$	$1.196$
Validation and early stopping	$0.656$	$42/60$	$0.611$	$1.074$
Disjoint split ensemble ( $\tau=0$ )	$1.080$	$25/60$	$0.881$	$1.227$
Uncertainty smoothing ( $\tau=1$ )	$0.792$	$44/60$	$1.147$	$0.691$

Appendix

Table 2: Component ablation at

n=10{,}000

. For each displayed quantity, entries are geometric means of the added-to-previous variant ratios over

60

paired comparisons (six settings, ten replications each); values below one indicate reductions. “Lower error” counts pairs in which the added component reduces

\|\widehat{\beta}-\beta_{0}\|_{2}

Figure 7: Sensitivity of Smoothed OVR to the smoothing scale. Points and bars show mean

\ell_{2}

estimation error and

\pm

one Monte Carlo standard error over ten paired replications. All scales reuse the same fitted contrast ensemble, second-stage folds, and second-stage random seeds. The dotted vertical line marks the default

\tau=1

; panels use separate linear error scales.

Method	Reference distance	Mean $\widehat{\gamma}$	Minimum $\widehat{\gamma}$
DOVR	$.0202\;(.0022)$	$.339$	$.287$
SOVR	$.0188\;(.0013)$	$.397$	$.351$
OLS	$.1289\;(.0010)$	—	—
FE	$.1039\;(.0009)$	—	—
Anchor ( $\gamma=2$ )	$.1708\;(.0010)$	—	—
LNN	$.0954\;(.0111)$	—	—

Appendix

Table 3: KuaiRand-Pure: mean

\ell_{2}

distance to the held-out randomized reference, with Monte Carlo standard errors in parentheses, over ten splits. The last two columns summarize the positivity diagnostic from Section B.1 across splits.

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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.

Abstract

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