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Generated outcomes as generated regressors: Equivalences in recursive causal estimation

This paper demonstrates that recursive plug-in, balancing weight, and doubly robust estimators for time-varying treatment effects are numerically equivalent when fitted via ordinary least squares, and it characterizes the behavior of these estimators under ridge penalization, showing that the bias correction towards OLS decays geometrically with the number of time periods.

Original authors: Wisse Rutgers, Rahul Singh

Published 2026-06-30

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Wisse Rutgers, Rahul Singh

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

The Big Picture: The "Pass-the-Parcel" Problem

Imagine you are trying to figure out the true cause of something complex, like how a specific medicine affects health over two years, or how a training program changes a person’s career path.

In statistics, we often can’t just look at the final result and say, "This caused that." We have to build a chain of predictions.

First, we predict what happens in Year 1.
Then, we take that prediction and use it to predict what happens in Year 2.
Finally, we use that second prediction to estimate the total effect.

This is called recursive estimation. It’s like a game of "Pass-the-Parcel" or a relay race. The baton (the prediction) is passed from one runner (statistical model) to the next. If the first runner drops the baton, the second runner is running with the wrong object, and the final result will be wrong.

The Three Competitors

Statisticians have three main ways to handle this relay race. The authors of this paper compare them:

The Predictor (Plug-in Estimator): This method focuses on predicting the outcome at every step. It says, "Let me guess what happens in Year 1, then use that guess to predict Year 2."
The Balancer (Balancing Weight Estimator): This method focuses on the people (the data). It says, "Let me adjust the weight of each person in my study so that the group looks balanced and fair, then calculate the average."
The Hybrid (Doubly Robust Estimator): This method tries to do both. It uses the predictions and the balancing weights. The idea is that if one part is slightly wrong, the other part might save the day. This is usually considered the "safest" or most robust method.

The Surprise: They Are Actually the Same (Mostly)

In simple, one-time studies (cross-sectional), statisticians already knew that if you use basic math (Ordinary Least Squares, or OLS), the Predictor, the Balancer, and the Hybrid all give the exact same answer.

The big question this paper asks is: Does this still happen when we have a multi-step relay race (longitudinal data)?

The Answer: Yes.
If you use basic, unpenalized math (OLS) at every step of the relay, all three methods are numerically identical. It doesn’t matter if your models are perfect or imperfect; the math works out so that the "Hybrid" method doesn’t actually add anything extra—it just arrives at the same destination as the other two.

The Twist: What Happens When We Add "Safety Nets"?

In real life, data is messy. To prevent the models from getting too crazy (overfitting), statisticians add "regularization" or "penalties." Think of this like adding training wheels or a safety net to keep the predictions stable.

The paper looks at what happens when we add these safety nets (specifically, something called Ridge Regression).

1. The "Shrinkage" Myth

In simple, one-step studies, there is a common belief: Adding the Hybrid safety net is like "undersmoothing" or pulling the answer back toward the basic OLS answer. It’s like saying, "The Hybrid method is just a smarter version of the basic method."

The Paper’s Finding: This intuition breaks down in multi-step studies.
As you add more steps to the relay race (more time periods), the Hybrid method stops looking like the basic OLS method. The "pull" toward the basic answer gets weaker and weaker. In fact, it decays geometrically. If you have a long timeline (many years), the Hybrid method behaves very differently from the basic Predictor method.

2. The "Anchor" Effect

When you use safety nets (Ridge penalties), the Hybrid method can be thought of as a mix of two things:

The "messy" predicted values.
The "clean" basic OLS values.

The paper shows that the weight given to the "clean" OLS values shrinks rapidly as you add more time steps. If you have 1 year, the Hybrid method is heavily anchored to the basic OLS. If you have 10 years, that anchor is barely holding on.

Creative Analogy: The Echo Chamber

Imagine you are shouting a message down a long hallway with mirrors on the walls (the recursive steps).

The Predictor is you shouting the message directly.
The Balancer is adjusting the volume of your voice so it echoes clearly.
The Hybrid is you shouting and adjusting the volume.

Finding 1 (No Safety Nets): If the hallway is perfectly quiet and the mirrors are perfect, shouting directly and adjusting the volume give the exact same echo at the end. They are equivalent.

Finding 2 (With Safety Nets): Now, imagine the hallway has sound-absorbing foam (regularization/penalties) on the walls to stop the echo from getting too distorted.

In a short hallway (1 step), the Hybrid method is very similar to just shouting.
In a very long hallway (many steps), the foam absorbs so much of the "basic" shout that the Hybrid method ends up sounding completely different. The "safety" of the Hybrid method doesn't just "clean up" the shout; it changes the nature of the echo entirely as the hallway gets longer.

Why Does This Matter?

Simplicity: If you are doing a multi-step analysis and using basic math (OLS), you don’t need to worry about choosing between the Predictor, Balancer, or Hybrid methods. They are the same. You can pick the one that is easiest to compute.
Caution with Complexity: If you are using advanced machine learning techniques (which use penalties/regularization) for long-term studies, you cannot assume that the "Doubly Robust" method is just a slightly better version of the basic method. It behaves differently, and the "safety" it provides works in a more complex way than statisticians previously thought.

Summary in One Sentence

In multi-step causal studies, the three main estimation methods are mathematically identical if you use basic statistics, but if you use advanced "safety-net" statistics, the most robust method stops behaving like a simple correction and becomes increasingly distinct as the number of time steps grows.

Technical Summary: Generated Outcomes as Generated Regressors: Equivalences in Recursive Causal Estimation

1. Problem and Context
The paper addresses the algebraic relationships between three classes of estimators used in longitudinal causal inference: recursive plug-in estimators (based on outcome regressions), recursive balancing weight estimators (based on inverse probability weighting or Riesz representers), and doubly robust estimators. These estimators are applied to "recursive functionals," a class of parameters identified by iterating conditional expectations backward through time. Canonical examples include time-varying treatment effects, surrogate-identified treatment effects, and mediation effects.

In cross-sectional causal inference ( $T=1$ ), it is well-established that under linear specifications, regression-based, balancing weight, and doubly robust estimators are numerically equivalent. Recent work (Bruns-Smith et al., 2025) further characterized the doubly robust estimator with ridge regularization as a shrinkage toward ordinary least squares (OLS). This paper investigates whether these equivalences and interpretations extend to the longitudinal setting ( $T \ge 1$ ), where estimators involve "generated outcomes" (predictions from one stage used as responses in the next) and "generated regressors" (weights from one stage used as covariates in the next).

2. Methodology and Framework
The authors define a general framework for recursive functionals involving $T$ stages. At each stage $t$ , there are conditioning variables $Z_t$ , an expectation operator $E_t$ , and a linear functional $m_t$ . The target parameter $\theta_0$ is identified via a backward recursion of outcome regressions (g-computation) or a forward recursion of Riesz representers (balancing weights).

The analysis focuses on linear estimation frameworks where nuisance functions (outcome regressions $f_t$ and Riesz representers $\alpha_t$ ) are estimated using basis functions $\phi_t$ . The paper compares three estimators:

Recursive Plug-in ( $\hat{\theta}^P$ ): Estimates outcome regressions backward from $t=T$ to $t=1$ and plugs them into the identification formula.
Recursive Balancing Weight ( $\hat{\theta}^Q$ ): Estimates Riesz representers forward from $t=1$ to $t=T$ and uses them to weight the final outcome.
Doubly Robust ( $\hat{\theta}^{DR}$ ): Combines both approaches using a Neyman-orthogonal moment to reduce bias.

The paper analyzes these estimators under three regularization regimes: no regularization (OLS), ridge regularization, and general convex penalties.

3. Key Contributions and Results

A. Exact Equivalence under OLS
The primary contribution is the proof that, in the absence of regularization, the three estimators are numerically identical in any finite sample, provided the Gram matrices are invertible.

Theorem 1 (OLS Equivalence): When all nuisance functions are estimated via OLS, $\hat{\theta}^P = \hat{\theta}^Q = \hat{\theta}^{DR}$ . This holds regardless of whether the linear models are correctly specified. The debiasing corrections in the doubly robust estimator vanish exactly because the residuals are orthogonal to the balancing weights (which lie in the span of the regressors).

B. Ridge Regularization and Geometric Decay
When ridge penalties are applied to the Riesz regressions (balancing weights), the equivalence between $\hat{\theta}^P$ and $\hat{\theta}^Q$ holds if the penalty matrices are identical across stages. However, the doubly robust estimator diverges from the pure OLS plug-in.

Theorem 2 (Recursive Ridge Shrinkage): The doubly robust estimator can be represented as a recursive outcome regression with "augmented" coefficients. At each stage, the augmented coefficient is a blend of the penalized regression coefficient and an OLS coefficient regressed on the augmented generated outcome from the previous stage.
Corollary 1 (Geometric Decay): The weight placed on the pure OLS plug-in estimator decays geometrically with the number of time periods $T$ . Specifically, the coefficient on the recursive OLS anchor is the product of shrinkage factors $\prod_{t=1}^T a_t$ , where $a_t \in (0,1)$ .
Implication: The intuition from the cross-sectional case—that debiasing effectively shrinks the estimator toward OLS (undersmoothing)—weakens significantly as $T$ increases. The doubly robust estimator is pulled toward OLS, but the strength of this pull diminishes exponentially with the number of periods.

C. General Convex Penalties
For general convex penalties, closed-form solutions are not available, but the authors derive telescoping identities.

Lemma 3 & Theorem 3: The balancing weight estimator $\hat{\theta}^Q$ can be expressed as the OLS estimator corrected by a sum of inner products between Riesz residuals and OLS coefficients. The doubly robust estimator $\hat{\theta}^{DR}$ is further decomposed into the balancing weight estimator plus a correction term involving the gap between the penalized outcome coefficients and the OLS coefficients. This confirms that each stage contributes a specific bias correction term, but the aggregate effect does not simplify to a single undersmoothing operation as in the $T=1$ case.

4. Significance and Claims
The paper claims two main takeaways:

Extension of Equivalence: The numerical equivalence between regression, balancing weight, and doubly robust estimators documented in cross-sectional causal inference extends exactly to longitudinal causal inference when using OLS. This unifies the algebraic understanding of these estimators in complex recursive settings.
Limitation of Undersmoothing Intuition: The interpretation of debiasing as "undersmoothing" (shrinking toward OLS) is specific to the single-period case. In longitudinal settings, the doubly robust estimator’s relationship to OLS is more complex, with the OLS component decaying geometrically. Thus, the number of time periods is central to understanding the behavior of regularized recursive estimators.

The paper does not propose new balancing criteria or asymptotic convergence results but rather provides deterministic algebraic identities that clarify the finite-sample relationships between existing estimators.