📈 economics

Sensitivity Analysis for Instrumental Variables Under Joint Relaxations of Monotonicity and Independence

This paper proposes a breakdown frontier approach to assess the sensitivity of Local Average Treatment Effects to joint violations of the monotonicity and independence assumptions, deriving identified sets and consistent estimators that reveal the high sensitivity of Angrist & Evans' (1998) family size findings to such violations.

Original authors: Pedro Picchetti

Published 2026-03-27

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CC BY 4.0

Original authors: Pedro Picchetti

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 solve a mystery: Does having more children cause a mother to work less?

In the world of statistics, you can't just ask people "Did you have more kids and then quit your job?" because that's messy. People choose to have kids for many reasons, and they choose jobs for many reasons. To get a clean answer, researchers use a clever trick called an Instrumental Variable (IV).

Think of the IV as a "natural lottery." In a famous study, researchers used the sex of the first two children as the instrument. The idea was: "If a couple has two boys, they might be more likely to have a third child than if they have a boy and a girl." This lottery is random (mostly), so it acts like a nudge. If the nudge changes the number of kids, and the number of kids changes the job status, we can figure out the cause-and-effect.

But here's the problem: What if the "nudge" isn't perfect?

The Monotonicity Problem: What if some parents hate having two boys and decide to have a third child specifically to get a girl? They are "defiers" going against the grain.
The Independence Problem: What if the "nudge" (having two boys) is secretly linked to something else, like the parents' genetics or their economic stability, which also affects their job?

If these assumptions are slightly wrong, the detective's conclusion might be a lie.

The Paper's Solution: The "Breakdown Frontier"

Pedro Picchetti, the author of this paper, invented a new tool called a Breakdown Frontier.

Imagine you are building a bridge. You know the bridge holds if the wind is calm and the materials are perfect. But what if the wind gets gusty? What if the materials are slightly weaker?

Most studies just say, "The bridge holds!" (assuming perfect conditions).
This paper asks: "How much wind can we tolerate before the bridge collapses? How weak can the materials get before we can no longer trust the bridge?"

The Breakdown Frontier is a map that shows you exactly how much "wiggle room" you have. It tells you:

"If you are willing to accept that 1% of people are 'defiers' (going against the grain), then your conclusion holds only if the hidden connection between the instrument and the outcome is less than 0.5%."
"If you are willing to accept a 2% hidden connection, then you must assume zero defiers for your conclusion to hold."

It draws a line in the sand. On one side of the line, your conclusion (e.g., "More kids = Less work") is safe. On the other side, the conclusion falls apart.

The "Goldilocks" Zone

The paper uses two "sensitivity knobs" to test the bridge:

The "Defier" Knob: How many people are going against the expected trend?
The "Hidden Link" Knob: How strong is the secret connection between the instrument and the outcome?

The Breakdown Frontier is the curve that connects these two knobs. It shows the trade-off. You can turn up one knob a little bit, but if you do, you have to turn the other knob down to zero to keep your conclusion valid.

The Real-World Test: The Family Size Study

The author tested this new tool on the famous study about family size and unemployment (Angrist & Evans, 1998).

The Old View: The original study said, "Yes, having more kids makes women work less." They assumed the "two boys" instrument was perfect.
The New View (using the Breakdown Frontier): The author ran the numbers and found the bridge is actually very fragile.
- Even a tiny amount of "defiers" (parents who really wanted a specific mix of boys and girls) or a tiny hidden link (genetics affecting both fertility and jobs) was enough to make the conclusion collapse.
- The "safe zone" was so small that the original conclusion is highly sensitive. It's like saying, "The bridge holds, but only if the wind is perfectly still and the materials are brand new." In the real world, that's rarely true.

Why This Matters

This paper is like a stress test for scientific studies.

Instead of just accepting a result because the math looks good, this method forces researchers to say: "My result is only true if the world is this perfect. If the world is even this slightly imperfect, my result disappears."

It doesn't tell you the answer is wrong; it tells you how much you have to believe in the assumptions for the answer to be right. It turns a "Yes/No" answer into a "How strong is your belief?" answer, making science more honest and transparent.

1. Problem Statement

Instrumental Variable (IV) methods are a cornerstone of causal inference in social sciences, relying on strict identifying assumptions: Independence (the instrument is uncorrelated with unobservables) and Monotonicity (the instrument affects treatment uptake in the same direction for all individuals, implying no "defiers").

In practice, these assumptions are often unverifiable. Recent literature has grown skeptical of IV results when these assumptions are violated. While sensitivity analysis exists for violations of independence (e.g., using $c$ -dependence) or monotonicity (e.g., allowing defiers) separately, there is a lack of frameworks that assess the joint sensitivity of IV estimates when both assumptions are relaxed simultaneously. Furthermore, when these assumptions fail, the Local Average Treatment Effect (LATE) is no longer point-identified, and the Average Treatment Effect (ATE) becomes difficult to bound without additional structure.

This paper addresses the gap by developing a breakdown frontier approach to quantify the robustness of IV conclusions (specifically regarding LATE and ATE) to simultaneous violations of independence and monotonicity.

2. Methodology

2.1 Framework and Assumptions

The author considers a standard IV setting with a binary instrument $Z$ , binary treatment $D$ , and binary outcome $Y$ , conditional on covariates $X$ . The analysis relaxes the standard assumptions as follows:

Relaxed Independence ( $c$ -dependence): The instrument $Z$ is not fully independent of potential outcomes and treatments. Instead, it satisfies $c$ -dependence (Masten & Poirier, 2018). This bounds the difference between the conditional probability of assignment given unobservables and the observed propensity score by a scalar $c \in [0, 1]$ .
$\sup |P(Z=1|Y, D, X) - P(Z=1|X)| \leq c$
Relaxed Monotonicity: The author allows for the existence of defiers (individuals who do the opposite of their assignment). However, a crucial restriction is maintained: the share of defiers ( $\pi_{def}$ ) must be strictly smaller than the share of compliers ( $\pi_{co}$ ). This ensures the first-stage effect remains positive.

2.2 Partial Identification Strategy

The paper proceeds in a stepwise partial identification approach:

Bounds on Potential Quantities: Using $c$ -dependence, the author derives sharp identified sets (bounds) for the joint distribution of potential treatments and outcomes, $P(Y(z), D(z) | X)$ .
Bounds on ITT and LATE:
- The Intention-to-Treat (ITT) effect is bounded as a function of $c$ and $\pi_{def}$ .
- The LATE for compliers is derived by dividing the bounds of the ITT by the bounds of the share of compliers.
- The Average Treatment Effect (ATE) is also bounded, connecting the results to existing literature (Balke & Pearl, 1997; Chen et al., 2017) when $c=0$ and $\pi_{def}=0$ .
Breakdown Frontier: The core contribution is the definition of the breakdown frontier. For a specific conclusion (e.g., "LATE > 0"), the frontier represents the set of pairs $(c, \pi_{def})$ such that the conclusion holds. It delineates the "robust region" (where the conclusion is valid) from the "non-robust region."

2.3 Estimation and Inference

Estimators: The author proposes nonparametric sample analogue estimators for the bounds and the breakdown frontier.
Asymptotics: Because the bounds involve $\min$ and $\max$ operators, the estimators are Hadamard directionally differentiable but not fully differentiable. Standard bootstrap methods fail.
Inference Procedure: The paper utilizes the numerical delta method (Hong & Li, 2018) combined with the bootstrap framework for directionally differentiable functions (Fang & Santos, 2018). This allows for the construction of valid uniform confidence bands for the breakdown frontiers.

3. Key Contributions

Joint Sensitivity Analysis: Unlike previous works that relax independence or monotonicity in isolation, this paper provides a unified framework to analyze the trade-offs between violating both assumptions simultaneously.
Breakdown Frontier for LATE and ATE: It extends the breakdown frontier methodology (Horowitz & Manski, 1995; Masten & Poirier, 2020) to IV settings with non-compliance and binary outcomes, covering both LATE and ATE.
Unified Metric: A significant feature is that both sensitivity parameters ( $c$ and $\pi_{def}$ ) are measured in the same units (probability units), making the interpretation of the trade-off curve intuitive.
Valid Inference for Non-Smooth Functionals: The paper provides a rigorous asymptotic theory and a consistent bootstrap procedure for constructing confidence bands for breakdown frontiers, addressing the non-Gaussian limiting distributions caused by the non-smoothness of the bounds.

4. Results

4.1 Theoretical Results

Sharp Bounds: The paper derives sharp identified sets for the ITT, LATE, and ATE under the joint relaxations. The bounds are shown to be sharp under specific conditions regarding the support of the data and the magnitude of $c$ and $\pi_{def}$ .
Analytical Expressions: Explicit formulas are provided for the breakdown frontiers, showing how the maximum allowable violation of one assumption depends on the assumed level of violation of the other.

4.2 Monte Carlo Simulations

Simulations demonstrate that the proposed estimators have desirable finite-sample properties.
The sampling distribution of the breakdown frontier estimator is found to be symmetric around the true frontier (unlike some previous findings in distributional treatment effects).
The bootstrap confidence bands achieve the nominal coverage rates.

4.3 Empirical Application: Angrist & Evans (1998)

The method is applied to the classic study of the effect of family size on female labor supply using same-sex siblings as an instrument.

Calibration: The author calibrates the sensitivity parameters using external data:
- $c$ is calibrated using selection on observables (Altonji et al., 2008 approach), yielding a baseline violation of $c \approx 0.011$ .
- $\pi_{def}$ is calibrated using survey data on ideal family composition (de Chaisemartin, 2017), yielding a baseline violation of $\pi_{def} \approx 0.018$ .
Findings: The estimated breakdown frontier shows that the conclusion "family size reduces female employment" (LATE < 0) fails even under very small violations of the assumptions.
- If independence holds ( $c=0$ ), the conclusion only holds if the share of defiers is $< 0.008$ (below the calibrated 0.018).
- If monotonicity holds ( $\pi_{def}=0$ ), the conclusion only holds if the unobserved selection bias is $< 0.004$ (below the calibrated 0.011).
Conclusion: The qualitative takeaways of the seminal Angrist & Evans (1998) study are highly sensitive to joint violations of independence and monotonicity.

5. Significance

This paper provides a robust toolkit for applied researchers to move beyond "point estimates" in IV analysis. By visualizing the breakdown frontier, researchers can explicitly state: "My conclusion holds only if the unobserved selection bias is smaller than $X$ AND the share of defiers is smaller than $Y$ ."

The application to Angrist & Evans (1998) is particularly significant as it challenges the robustness of one of the most cited IV studies in economics, suggesting that its conclusions are fragile when standard assumptions are relaxed even slightly. The methodological contribution of providing valid inference for non-smooth functionals in this context sets a new standard for sensitivity analysis in causal inference.