1 A Confounder You Will Never See

Every applied researcher has met the same ghost. You want the causal effect of a treatment A on an outcome Y, you have collected a rich set of covariates X, and yet you cannot shake the suspicion that some unmeasured variable U-health-seeking behaviour, managerial talent, latent demand drives both A and Y. The standard response is an act of faith: assume no unmeasured confounding, Y(a) ⊥ A | X and proceed. When the faith is misplaced, every adjustment method regression, matching, inverse-probability weighting, double machine learning returns a biased answer, and no amount of data fixes it [Imbens and Rubin, 2015].

For decades the alternatives were limited. Find an instrument, find a discontinuity, find a natural experiment or give up and report bounds. Proximal causal inference (PCI) offers a genuinely different route. Its premise is disarmingly simple: even if we cannot measure U, we can often find imperfect proxies of it, and two carefully chosen proxies, used together, can purge the bias that a single one cannot. The framework, developed by Miao et al. [2018], Tchetgen Tchetgen et al. [2024], and Cui et al. [2024], has grown from an epidemiological curiosity into one of the most active frontiers in causal econometrics, with a wave of 2025-2026 papers extending it to survival data, modified treatment policies, and panel settings [Ying et al., 2025].  

This feature explains what negative controls are, why two of them suffice where one fails, and what the framework demands in return.  

2 Negative Controls: From Detection to Correction

Epidemiologists have long used negative controls as a falsification device [Lipsitch et al., 2010]. A negative control outcome is a variable that the treatment cannot plausibly affect but that shares the same confounding structure if you "find" an effect on it, you have detected residual confounding. A classic example: if a study claims that influenza vaccination reduces winter mortality, but you also find it "reduces" the risk of being hit by a car, the association is confounded by frailty, not causal.  

PCI's insight is that negative controls can do more than detect bias they can remove it. The framework distinguishes two kinds of proxy:  

• A negative control treatment (NCT), Z: a variable associated with the unmeasured confounder but with no causal effect on the outcome, and not itself caused by the treatment.  
• A negative control outcome (NCO), W: a variable associated with the unmeasured confounder but not causally affected by the treatment.  

Both are "proxies" of U in the sense that they are driven by it. The causal structure PCI assumes is shown in Figure 1.  

Figure 1: The canonical proximal structure. U (dashed) is unmeasured. The negative control treatment Z proxies U but affects Y only through A; the negative control outcome W proxies U but is not caused by A (the bent A → W arrow is forbidden). Covariates X are suppressed for clarity.  

The defining exclusion restrictions are  

In words: Z affects the outcome only through the treatment (it is a valid "treatment-side" proxy), and W is unaffected by treatment and by the other proxy once we condition on the true confounder. The genius of using two proxies is that the NCT supplies exogenous variation that lets us learn how the NCO maps onto the hidden confounding, and the NCO then lets us subtract that confounding from the outcome equation.  

3 The Confounding Bridge Function

The technical heart of PCI is the outcome confounding bridge function h(w,a,x), defined as a solution to the integral equation  

This is a Fredholm integral equation of the first kind: h is the function that, when averaged over the distribution of the negative control outcome W given (Z, A, X), reproduces the observed conditional mean of Y. Crucially, equation (2) involves only observed quantities- no U appears. Under a completeness condition (described below), a solution exists, and  Miao et al. [2018] show that the average treatment effect is recovered by plugging the bridge function back in and averaging over W and X:  

The intuition is that h "relabels" the negative control outcome into the scale of the real outcome's confounding component, so that fixing the treatment level at a and integrating over W marginalises out the hidden U.  

There is a dual, symmetric representation through a treatment confounding bridge q(z,a,x) resembling an inverse-probability weight built from the negative control treatment. Cui et al. [2024] combine the two into a doubly robust estimator whose moment condition is Neyman-orthogonal: the estimate of the ATE is consistent if either the outcome bridge h or the treatment bridge q is correctly specified, and it is √n consistent and asymptotically normal even when both are estimated nonparametrically with machine learning, provided the product of their estimation errors vanishes quickly enough exactly the cross-fitting logic of double machine learning [Chernozhukov et al., 2018].  

4 What the Framework Asks in Return

PCI does not manufacture identification from nothing. It trades the untestable assumption of no unmeasured confounding for a different set of conditions, some testable and some not.  

Relevance (completeness). The proxies must be "relevant" to the confounder in a precise sense. The key requirement is a completeness condition: roughly, the family of conditional distributions of Z given (U,X) must be rich enough to detect any function of U. Completeness is the proximal analogue of instrument relevance, and like a weak instrument, a "weak proxy" inflates variance and can make the integral equation (2) ill-posed. Tchetgen Tchetgen et al. [2024] stress that completeness is fundamentally a statement about how strongly the proxies track U.  

Valid exclusion. The restrictions in (1) are the crux. The NCT must not affect Y directly; the NCO must not respond to treatment. These are causal assumptions, defended by subject-matter knowledge just as an instrument's exclusion restriction is. The framework relocates the burden of judgement rather than eliminating it.  

Solvability. A bridge function must exist. This is guaranteed under completeness plus regularity, but in finite samples the ill-posed inverse problem requires regularisation, and inference must account for it [Deaner, 2023].  

A natural worry is circularity: have we simply renamed "find an instrument" as "find two proxies"? Not quite. An instrument must be independent of the confounder; a proxy must be dependent on it. That sign flip is what makes proxies available in settings where instruments are not: lagged outcomes, alternative survey items, pre-period measurements, and biomarkers are often driven by the same latent factor we fear, which disqualifies them as instruments but qualifies them as negative controls.  

5 Where It Is Going

The 2025-2026 literature has pushed PCI well beyond its cross-sectional origins. Ying et al. [2025] develop regression-based proximal methods that practitioners can implement with familiar two-stage least-squares-style software, dramatically lowering the barrier to entry, and extend the machinery to right-censored time-to-event outcomes. Others have connected PCI to the synthetic control literature: Shi et al. [2023] reinterpret synthetic controls as a proximal procedure in which donor units serve as negative control outcomes, supplying a formal identification theory for a method long justified mainly by good pre-treatment fit. There is now active work on proximal inference for modified treatment policies and on doubly robust proximal estimators that fuse the framework with targeted learning.  

Two cautions temper the enthusiasm. First, the completeness conditions are inherently untestable and the inverse problem is genuinely hard; weak proxies can be worse than honest sensitivity analysis. Second, the framework demands two distinct, well-argued proxies-a real empirical cost. As with instrumental variables a generation ago, the danger is that the elegance of the theory invites mechanical application to proxies that do not deserve the name.  

Still, the conceptual shift is significant. For thirty years the credibility revolution taught us to hunt for exogenous variation. Proximal causal inference teaches us to hunt instead for informative proxies and to recognise that two flawed measurements of a hidden cause can, together, reveal what neither could alone. It is a reminder that identification is not a single trick but a design problem, and that the space of credible designs is still expanding.  

References

1. Chernozhukov, V., Chetverikov, D., Demirer, M., Duflo, E., Hansen, C., Newey, W., and Robins, J. (2018). Double/debiased machine learning for treatment and structural parameters. The Econometrics Journal, 21(1), C1-C68.  
2. Cui, Y., Pu, H., Shi, X., Miao, W., and Tchetgen Tchetgen, E. (2024). Semiparametric proximal causal inference. Journal of the American Statistical Association, 119(546), 1348-1359.  
3. Deaner, B. (2023). Proxy controls and panel data. Working paper, University College London.  
4. Imbens, G. W., and Rubin, D. B. (2015). Causal Inference for Statistics, Social, and Biomedical Sciences. Cambridge University Press.  
5. Lipsitch, M., Tchetgen Tchetgen, E., and Cohen, T. (2010). Negative controls: a tool for detecting confounding and bias in observational studies. Epidemiology, 21(3), 383-388.  
6. Miao, W., Geng, Z., and Tchetgen Tchetgen, E. (2018). Identifying causal effects with proxy variables of an unmeasured confounder. Biometrika, 105(4), 987-993.  
7. Shi, X., Miao, W., Hu, M., and Tchetgen Tchetgen, E. (2023). Theory for identification and inference with synthetic controls: a proximal causal inference framework. Working paper. 
8. Tchetgen Tchetgen, E. J., Ying, A., Cui, Y., Shi, X., and Miao, W. (2024). An introduction to proximal causal inference. Statistical Science, 39(3), 375-390.  
9. Ying, A., Miao, W., Shi, X., and Tchetgen Tchetgen, E. J. (2025). Regression-based proximal causal inference. American Journal of Epidemiology, 194(7), 2030-2041.