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Doubly Robust Adaptive Conformal Inference for Causal Effects Under Temporal Dependence

This paper proposes Doubly Robust Adaptive Conformal Inference (DR-ACI), a novel method that constructs valid prediction intervals for doubly robust pseudo-outcomes in the presence of temporal dependence.

Original authors: Andreas Koukorinis, Ricardo Silva

Published 2026-06-30

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Andreas Koukorinis, Ricardo Silva

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are a weather forecaster trying to predict how much rain a specific city will get tomorrow. You have a model that gives you a single number (e.g., "5 inches"). But you know your model isn't perfect. You want to give a range, like "between 3 and 7 inches," and you want to be sure that, over time, your range actually captures the real rain 90% of the time.

Now, imagine two extra complications:

The "Hidden" Truth: You aren't just predicting rain; you are trying to measure the difference rain would make if a cloud-seeding machine were turned on versus off. You can never see both outcomes for the same city on the same day (it either rained or it didn't). This "difference" is a ghost—you have to build a clever proxy to estimate it.
The "Chain Reaction": Weather today depends heavily on weather yesterday. The data isn't a clean, random pile of independent events; it's a chain where one event tugs on the next.

This paper introduces a new tool called DR-ACI (Doubly Robust Adaptive Conformal Inference) to solve this exact problem. Here is how it works, broken down into simple concepts.

1. The "Double-Safety" Net (Doubly Robust)

Usually, to estimate the effect of a treatment (like a drug or a policy), you need two models: one to predict who gets the treatment, and one to predict the outcome. If either model is wrong, your answer is garbage.

The authors use a "Doubly Robust" trick. Think of it like a two-engine airplane.

If the first engine (Model A) fails, the second engine (Model B) can keep the plane flying.
If Model B fails, Model A saves the day.
You only crash if both engines fail at the same time.

In this paper, they use this "two-engine" approach to create a "pseudo-outcome"—a fake, constructed data point that acts as a stand-in for the hidden truth (the treatment effect).

2. The "Guard Band" (Handling Time Dependence)

Standard statistical tools often assume that every data point is independent, like rolling a die. But time-series data (like stock prices or weather) is more like a domino effect. If you try to train a model on yesterday's data to predict today, the "training" and "testing" data are still touching, which creates a cheat (look-ahead bias).

To fix this, the authors use Guard Bands.

Imagine you are splitting a long line of people into groups to test a new dance move.
Instead of just cutting the line in half, you throw out a few people in the middle as a "buffer zone" (the guard band).
You train your model on the left group, skip the buffer, and test on the right group.
This ensures the two groups are far enough apart that the "infection" of yesterday's data doesn't contaminate today's test.

3. The "Adaptive Ruler" (Adaptive Conformal Inference)

Even with a good model, you don't know exactly how wide your prediction range should be. Sometimes the world is chaotic (wide range needed); sometimes it's calm (narrow range).

The authors use an Adaptive Ruler.

Imagine a ruler that stretches or shrinks based on how often it misses the target.
If the ruler misses too often, it automatically stretches wider.
If it hits the target too easily, it shrinks to be more precise.
This happens in real-time, learning as it goes, ensuring the coverage stays valid even if the data changes.

4. The "Variance Standardizer" (VS-DR-ACI)

The authors found that sometimes the "ruler" gets too wide because the noise in the data is uneven. Some days are noisy; some are quiet.

They added a Variance Standardizer. Think of this as a noise-canceling headphone for the data.
Before measuring the range, they normalize the data so that the "loud" days and "quiet" days are treated fairly.
The Result: This version (VS-DR-ACI) produced prediction intervals that were 63% narrower than standard methods, while still being just as accurate. It's like getting a much tighter, more useful forecast without losing reliability.

5. Real-World Test: The Nasdaq Experiment

The authors didn't just simulate this; they tested it on real financial data from Nasdaq regarding a new trading rule called "Dynamic M-ELO."

The Problem: They wanted to know how this new rule affected trading quality for each specific stock.
The Result: Their method found that for many stocks, the effect was significant, whereas standard methods missed it because their "rulers" were too wide and fuzzy.
The Stress Test: They also tested what happens when the market suddenly changes (a "drift"). Standard methods failed and gave wrong answers. Their method, especially with the "noise-canceling" feature, kept working and gave stable, reliable answers.

Summary of the "Three-Part" Guarantee

The paper proves mathematically that their method's error comes from three specific sources, like a bill with three line items:

The Mixing Gap: The cost of the data being connected in time (handled by the Guard Bands).
The Nuisance Tax: The cost of imperfect models (handled by the Double-Safety Net).
The Adaptation Rate: The cost of learning on the fly (handled by the Adaptive Ruler).

The Bottom Line:
This paper gives us a way to say, "We are 90% sure the effect of this policy on this specific stock is between X and Y," even when the data is messy, connected in time, and the underlying models aren't perfect. It's a more reliable, tighter, and smarter way to measure uncertainty in a changing world.

Technical Summary: Doubly Robust Adaptive Conformal Inference for Causal Effects Under Temporal Dependence

1. Problem Statement

The paper addresses the challenge of quantifying uncertainty for heterogeneous treatment effects (specifically the Conditional Average Treatment Effect, CATE, $\tau(X)$ ) in settings characterized by temporal dependence.

Standard conformal prediction methods (e.g., split conformal) rely on the assumption of exchangeability (independent and identically distributed data). This assumption fails in time-series contexts common in finance and epidemiology, where adjacent observations are correlated. Consequently, naive application of conformal methods to causal estimators leads to invalid coverage guarantees. Furthermore, standard causal inference intervals (asymptotic, bootstrap, Bayesian) often lose their distribution-free calibration guarantees under dependence or require correct model specification, which machine learning estimators cannot easily guarantee.

The specific goal is to construct finite-sample, distribution-free prediction intervals for individual treatment effect estimates without assuming independent observations, while utilizing doubly robust (DR) pseudo-outcomes to mitigate nuisance estimation errors.

2. Methodology: DR-ACI

The proposed method, Doubly Robust Adaptive Conformal Inference (DR-ACI), integrates three core components:

A. Doubly Robust Pseudo-Outcomes

Instead of calibrating on raw outcomes $Y_t$ or latent CATEs $\tau(X_t)$ , the method constructs doubly robust pseudo-outcomes $\psi^{DR}_t$ :
$\psi^{DR}_t = \hat{\mu}_1(X_t) - \hat{\mu}_0(X_t) + \frac{W_t(Y_t - \hat{\mu}_1(X_t))}{\hat{e}(X_t)} - \frac{(1-W_t)(Y_t - \hat{\mu}_0(X_t))}{1 - \hat{e}(X_t)}$
where $\hat{e}$ is the estimated propensity score and $\hat{\mu}_w$ are estimated conditional outcome models. Under correct specification of at least one nuisance model, $\psi^{DR}_t$ is an unbiased estimate of $\tau(X_t)$ with error bounded by the product of nuisance estimation errors.

B. Temporal Block Cross-Fitting with Guard Bands

To handle temporal dependence ( $\beta$ -mixing) while preserving the orthogonality properties of DR estimation, the authors propose temporal block cross-fitting with guard bands:

Partitioning: The time series is divided into $K$ contiguous blocks.
Guard Bands: When training nuisance models on a set of blocks, a "guard band" of $g$ observations adjacent to the held-out calibration block is excluded from training.
Purpose: This ensures a temporal separation of at least $g$ steps between training and calibration data. Under $\beta$ -mixing, this reduces the residual dependence (coupling error) between the training set and the calibration set to $O(\beta(g)^{\delta/(2+\delta)})$ .

C. Adaptive Conformal Inference (ACI)

The method employs the ACI framework (Gibbs and Candès, 2021) to update the miscoverage level $\alpha_t$ dynamically:
$\alpha_{t+1} = \alpha_t + \gamma(\alpha - \mathbb{1}\{\psi^{DR}_t \notin \hat{C}_t\})$
This provides a deterministic long-run coverage guarantee even when the score sequence is not exchangeable.

D. Variance-Standardised DR-ACI (VS-DR-ACI)

To address the issue that DR pseudo-outcomes include noise variance ( $\xi_t$ ) that inflates interval width, the authors propose variance standardisation:
$s^{VS}_t = \frac{|\psi^{DR}_t - \hat{\tau}(X_t)|}{\hat{\sigma}_\xi(X_t)}$
where $\hat{\sigma}_\xi(X_t)$ estimates the local standard deviation of the DR noise. This adapts the interval width to local noise levels, preventing unnecessary widening in low-noise regions.

3. Key Theoretical Contributions

A. Three-Term Coverage Decomposition

The main theoretical result (Theorem 2) decomposes the coverage gap into three distinct terms:

Mixing Gap: $\min_\tau \{ \tau/T + 2\beta(\tau) \}$ . This term arises from temporal dependence and is controlled by the switch-coefficient bound.
Nuisance-Bias Tax: $\|\hat{e} - e\|_2 \cdot \|\hat{\mu} - \mu\|_2$ . This term represents the product of nuisance estimation errors. The paper proves that temporal block cross-fitting with guard bands preserves the standard DML product-bias rate under $\beta$ -mixing, adding only an explicit coupling remainder $O(\beta(g)^{\delta/(2+\delta)})$ .
Adaptation Rate: $O(T^{-1/2})$ . This term arises from the ACI update mechanism.

B. Switch-Coefficient Inheritance

The paper establishes (Lemma 9) that conformity scores constructed from DR pseudo-outcomes inherit the switch-coefficient bounds of the underlying process. This allows the coverage gap to be bounded by the mixing coefficient of the data, even when nuisance models are estimated.

C. Asymptotic CATE Containment

While the conformal procedure directly targets the pseudo-outcome $\psi^{DR}_t$ , the paper shows (Corollary 6) that under estimator consistency ( $\|\hat{\tau} - \tau\|_{L2} = o_p(1)$ ), the resulting intervals provide asymptotically conservative containment for the latent CATE $\tau(X_t)$ . The interval over-covers $\tau(X_t)$ because it is calibrated for the wider target $\psi^{DR}_t = \tau(X_t) + \xi_t$ .

4. Empirical Results

Simulation Study

Mixing Gap: Under stationary conditions with varying autocorrelation ( $\rho$ ), all conformal methods maintained valid coverage. The theoretical mixing bound was found to be highly conservative (predicting coverage drops to 67% at $\rho=0.99$ , while observed coverage remained near 90%).
Nuisance-Bias: Oracle and estimated DR-ACI showed identical coverage, confirming the nuisance-bias tax is negligible for standard sample sizes and learners.
Drift and Dependence (Regime D): In a scenario combining high dependence ( $\rho=0.95$ $ρ = 0.95$ ) and covariate drift:
- VS-DR-ACI maintained valid coverage (89.9%) with stable interval width.
- Standard DR-ACI maintained coverage but suffered massive interval inflation (width increased by ~170x).
- Split Conformal and DML-Wald intervals lost significant coverage (dropping by ~29pp and ~35pp, respectively).
- Mechanism: Variance standardisation prevented the "spiral" of the adaptive quantile under non-stationarity by keeping scores approximately unit-variance.
Efficiency: VS-DR-ACI produced intervals 63% narrower than split conformal while maintaining valid coverage.

Empirical Application: Nasdaq Dynamic M-ELO

The method was applied to a natural experiment regarding Nasdaq's rollout of Dynamic Midpoint Extended Life Orders (M-ELO).

Context: Staggered adoption of dynamic hold timers across stock cohorts.
Result: DR-ACI identified significant heterogeneity in treatment effects. While baseline asymptotic inference found only 12.3% of stocks with significant effects, DR-ACI identified 33.2% (a 2,000-stock increase in actionable insights).
Precision: DR-ACI intervals were 50% narrower than baseline asymptotic intervals, allowing for more precise stock-level classification of treatment effects.

5. Significance and Claims

The paper claims to occupy a unique intersection in the literature: combining doubly robust scores, valid coverage under $\beta$ -mixing, and online adaptation.

Theoretical Significance: It provides the first coverage guarantee for DR pseudo-outcomes under $\beta$ -mixing, explicitly quantifying the cost of dependence (mixing gap) and the preservation of DML bias rates via guard bands.
Practical Significance: It offers a distribution-free, finite-sample valid method for causal inference in time-series data where standard assumptions fail. The variance-standardised variant (VS-DR-ACI) is shown to be robust to both dependence and drift, outperforming existing methods in stability and interval width.
Limitations: The authors explicitly note that the theoretical guarantees assume stationarity. The robustness to drift observed in simulations and the Nasdaq application is empirical evidence, not covered by the main theorem. The mixing bound is acknowledged as conservative for smooth score functions, and deriving a tighter bound for such cases is identified as future work. The method also assumes the Stable Unit Treatment Value Assumption (SUTVA), which is approximate in financial markets due to potential spillovers.