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Two-stage least squares with treatment-covariate interactions for treatment effect heterogeneity

This paper clarifies the causal interpretation of interacted two-stage least squares for estimating treatment effect heterogeneity, demonstrating that consistent estimation of covariate-specific effects among compliers requires a restrictive linear IV-covariate interaction condition that is generally unmet unless covariates are categorical or the instrument is randomly assigned.

Original authors: Anqi Zhao, Peng Ding, Fan Li

Published 2026-03-03

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Anqi Zhao, Peng Ding, Fan Li

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 detective trying to figure out if a new medicine (the Treatment) actually cures a disease (the Outcome).

The problem is that people who choose to take the medicine might be different from those who don't. Maybe they are healthier to begin with, or maybe they have better doctors. This makes it hard to tell if the medicine works or if the patients were just lucky.

To solve this, statisticians use a clever trick called an Instrumental Variable (IV). Think of the IV as a "randomizer" or a "nudge."

Example: Imagine a doctor who randomly assigns patients to a waiting list for the medicine based on the last digit of their phone number. The phone number doesn't affect health directly, but it nudges some people to get the medicine and others not to. By comparing the health of those nudged to take it vs. those nudged not to, we can isolate the true effect of the medicine.

The Standard Approach: The "Average" Detective

Usually, statisticians use a method called Two-Stage Least Squares (2SLS).

Stage 1: They predict who would take the medicine based on the nudge (the IV).
Stage 2: They see how the actual health outcomes change based on that prediction.

This gives you a single number: the Average Treatment Effect. It tells you, "On average, the medicine helps by X amount."

The New Problem: One Size Does Not Fit All

But what if the medicine works wonders for young people but does nothing for older people? Or works for men but not women? This is called Treatment Effect Heterogeneity.

Statisticians want to know: Does the medicine work differently for different types of people?

To find out, they try to add "interaction terms" to their math. They try to build a model that says:

"The effect = Base Effect + (Effect for Age) + (Effect for Gender) + (Effect for Age × Gender)..."

They call this the Interacted 2SLS. It's like trying to draw a map where the terrain changes color depending on where you are.

The Big Discovery: The "Magic Condition"

The authors of this paper (Zhao, Ding, and Li) asked a very important question: When does this "Interaction Map" actually tell the truth?

They discovered that this method is extremely fragile. It only works if a very specific, almost impossible condition is met. They call this the "Linear IV-Covariate Interactions Condition."

The Analogy of the "Perfectly Predictable Nudge"

Imagine the "nudge" (the IV) is a machine that decides who gets the medicine.

The Bad News: For the interaction map to be accurate, the machine's decision logic must be incredibly simple. Specifically, the machine's "propensity" (how likely it is to give the medicine) must be a straight line when plotted against the patient's characteristics.
The Catch: The authors prove that for this to happen, the machine's decision can only take on a very limited number of values.

Think of it like this:
If you are trying to paint a picture of a complex landscape (the real world) using only a few distinct colors (the IV's decision), you can only get a perfect picture if the landscape itself is made of only a few distinct blocks.

The paper proves that the "Interacted 2SLS" only works in two very specific, rare scenarios:

The "Categorical" World: The patients are divided into distinct, separate groups (like "Red," "Blue," and "Green" teams) with no in-between. The nudge treats each group exactly the same way.
The "Random" World: The nudge is completely random and ignores the patients' characteristics entirely (like flipping a coin).

The Reality Check:
In the real world, covariates (like age, income, blood pressure) are usually continuous (a smooth scale, not distinct blocks), and the nudge is rarely perfectly random.

Conclusion: If you use this "Interaction Map" method on real-world data where the nudge isn't perfectly random and the data isn't perfectly grouped, the results you get are likely garbage. The numbers you calculate for "how the medicine works for different people" will be biased and misleading.

The Solution: The "Centered" Map

So, is all hope lost? No! The authors offer a brilliant workaround.

Instead of trying to map every single interaction (which breaks the math), they suggest centering the data.

The Metaphor: Imagine you are measuring the height of trees in a forest. Instead of measuring every tree from the ground (which is messy because the ground slopes), you measure every tree relative to the average height of the forest floor.
The Method: They show that if you subtract the "average" characteristics of the people who take the medicine from everyone else (a process called demeaning), and then run the analysis, you can recover the true average effect of the medicine, even in complex situations.

Summary for the Everyday Reader

The Trap: Researchers often try to use complex math to see if a treatment works differently for different people.
The Warning: The authors prove that this specific math trick usually fails unless the data is perfectly simple (like distinct categories) or the experiment is perfectly random. If you use it on messy, real-world data, you will get the wrong answer.
The Fix: If you want to find the true average effect, don't try to map every interaction. Instead, adjust your data by removing the "average" background noise first. This simple step saves the day and gives you a reliable answer.

In short: Don't try to draw a detailed, multi-colored map of a complex world using a simple, rigid ruler. It won't fit. Instead, use a flexible tool (demeaning) that adapts to the shape of the world to find the truth.

1. Problem Statement

Instrumental Variable (IV) analysis is widely used to estimate causal effects when treatments are endogenous. While the standard Two-Stage Least Squares (2SLS) estimator consistently identifies the Local Average Treatment Effect (LATE) under standard assumptions, researchers often suspect treatment effect heterogeneity (i.e., the effect varies across individuals based on observed covariates).

A common empirical strategy to address this is the Interacted 2SLS, where researchers include interaction terms between the treatment and covariates in the second stage, and interactions between the instrument and covariates in the first stage. The goal is to estimate how the treatment effect varies with covariates (systematic treatment effect variation).

The Core Problem:
The paper investigates the causal interpretation of the coefficients on these treatment-covariate interactions in the Interacted 2SLS framework. Specifically, it asks: Under what conditions do the estimated interaction coefficients consistently represent the true heterogeneity of the treatment effect among compliers? The authors find that, contrary to intuition, these coefficients are generally biased and lack a clear causal interpretation unless very restrictive conditions are met.

2. Methodology and Framework

The authors operate within the Potential Outcomes framework (Rubin Causal Model) and the Local Average Treatment Effect (LATE) framework (Imbens and Angrist, 1994).

Setup:
- $D_i$ : Binary treatment.
- $Y_i$ : Outcome.
- $Z_i$ : Binary Instrument (conditionally valid given covariates $X_i$ ).
- $X_i$ : Vector of $K$ covariates (including a constant).
- Compliers ( $U_i=c$ ): Units whose treatment status is changed by the instrument.
The Estimator:
The paper analyzes the Interacted 2SLS defined as:
$2sls(Y_i \sim D_i X_i + X_i \mid Z_i X_i + X_i)$
Here, the outcome is regressed on the interaction of treatment and covariates ( $D_i X_i$ ) and covariates ( $X_i$ ), instrumented by the interaction of the instrument and covariates ( $Z_i X_i$ ) and covariates ( $X_i$ ).
Target Parameter:
The authors define the target causal estimand as $\beta_c$ , the coefficient vector from the linear projection of the individual treatment effect $\tau_i$ onto $X_i$ among compliers:
$\beta_c = \arg\min_b E[(\tau_i - X_i^\top b)^2 \mid U_i = c]$
This represents the "systematic treatment effect variation" explained by covariates.

3. Key Contributions and Theoretical Findings

The paper provides three major theoretical contributions:

A. The Necessary and Sufficient Condition (The "Linear IV-Covariate Interactions" Condition)

The authors establish that the Interacted 2SLS estimator $\hat{\beta}_{2sls}$ consistently estimates the true heterogeneity parameter $\beta_c$ for any outcome-generating process if and only if the Linear IV-Covariate Interactions Condition holds:
$E(Z_i X_i \mid X_i) = \text{proj}(Z_i X_i \mid X_i)$
In simpler terms, the conditional expectation of the product of the instrument and covariates must be linear in the covariates.

Implication: If this condition fails, the estimated interaction coefficients are not causal estimates of treatment effect heterogeneity, even if the outcome model is correctly specified. This highlights a "double robustness" failure: standard 2SLS consistency relies on the instrument being valid, but interaction consistency relies on a specific functional form of the instrument's assignment mechanism.

B. The Stringency of the Condition

The paper proves a striking result regarding the feasibility of the Linear IV-Covariate Interactions Condition. Assuming $X_i$ has dimension $K$ (including a constant):

The condition holds only if the IV propensity score $e(X_i) = P(Z_i=1 \mid X_i)$ takes at most $K$ distinct values.
Consequence: This condition is extremely restrictive. It is satisfied only in two special cases:
1. Saturated Covariates: The covariates are categorical with exactly $K$ levels (allowing for a full set of dummy variables).
2. Random Assignment: The instrument $Z_i$ is unconditionally randomized (independent of $X_i$ ).
Practical Implication: In most observational studies where covariates are continuous or the IV is not randomly assigned, the condition fails. Therefore, the interaction coefficients from standard Interacted 2SLS specifications cannot be interpreted as causal estimates of treatment effect heterogeneity.

C. A Solution: Demeaned Covariates

Recognizing the difficulty of satisfying the condition for heterogeneity, the authors propose a modification to recover the overall LATE ( $\tau_c$ ) rather than the full heterogeneity vector.

Method: Use demeaned covariates in the Interacted 2SLS specification. Specifically, center the non-constant covariates by their complier means (or consistent estimates thereof).
Result: Under the Linear IV-Covariate Interactions Condition, the coefficient on the treatment variable ( $D_i$ ) in the demeaned specification consistently estimates the LATE ( $\tau_c$ ).
Contrast: Standard additive 2SLS (without interactions) or interacted 2SLS without demeaning typically estimates a weighted average of conditional LATEs that differs from the true LATE. The demeaned approach corrects this weighting bias.

4. Simulation Results

The authors validate their theory through simulations:

Bias in Heterogeneity: In a scenario where the Linear IV-Covariate Interactions Condition is violated (continuous covariates, non-random IV), the Interacted 2SLS estimates of the interaction coefficients showed significant bias relative to the true $\beta_c$ .
Consistency of LATE: In a scenario where the condition holds (categorical covariates), the Interacted 2SLS with demeaned covariates successfully recovered the true LATE, whereas standard additive 2SLS and interacted 2SLS without demeaning produced biased estimates.

5. Significance and Implications

Caution for Empirical Practice: The paper serves as a critical warning to applied researchers. Simply adding treatment-covariate interactions to a 2SLS model does not guarantee valid estimates of how treatment effects vary with covariates. Unless the covariates are fully saturated (categorical) or the IV is randomized, these coefficients are likely spurious.
Clarification of LATE Estimation: The paper clarifies that while estimating heterogeneity is difficult without strong assumptions, estimating the average effect (LATE) in the presence of heterogeneity is possible using the demeaned covariate approach.
Extension to Partial Interactions: The theory is extended to cases where interactions are included only for a subset of covariates, providing conditions under which those specific interaction terms can be interpreted causally.
Theoretical Unification: The results unify and extend previous literature (e.g., Angrist & Imbens, 1995; Sloczynski, 2024; Blandhol et al., 2025) regarding the weights assigned by IV estimators, showing that interaction terms fundamentally alter these weights in ways that often break causal interpretation.

Conclusion

Zhao, Ding, and Li demonstrate that the intuitive approach of using Interacted 2SLS to study treatment effect heterogeneity is theoretically fragile. The coefficients on interaction terms are generally inconsistent for causal heterogeneity unless the instrument's propensity score is extremely simple (taking few values). However, they provide a robust alternative for estimating the overall LATE by using demeaned covariates, offering a practical tool for researchers working with conditionally valid instruments.