📈 economics

Compositional Synthetic Controls

This paper introduces a synthetic control estimator for compositional outcomes based on a random utility model and Aitchison geometry, which is applied to Pennsylvania's electricity mix to reveal a significant, persistent shift toward natural gas and away from renewables following the Alternative Energy Portfolio Standard.

Original authors: Onil Boussim

Published 2026-07-21

📖 4 min read☕ Coffee break read

CC BY 4.0

Original authors: Onil Boussim

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: "What would have happened if a specific event hadn't occurred?" This is the heart of a field called causal inference, where scientists try to figure out cause and effect. Usually, they look at a single city, state, or country that did something new (like passing a new law) and compare it to a "twin" made up of a mix of other places that didn't do that thing. This method, known as the "Synthetic Control," works like a recipe: you take a little bit of State A, a dash of State B, and a pinch of State C to create a perfect "Synthetic State" that looks exactly like the treated one before the event.

But there's a catch. Many real-world outcomes aren't just single numbers like "temperature" or "GDP." They are recipes themselves—vectors of shares that must always add up to 100%. Think of a pie chart: if the slice for "Apples" gets bigger, the slices for "Oranges" and "Bananas" must get smaller. This is called compositional data. The old way of comparing these pies used a ruler that measured straight-line distance, which is terrible at understanding how the ratios between slices change. It's like trying to judge the flavor of a soup by measuring the weight of the bowl instead of tasting the broth. This paper steps in to fix that ruler, creating a new way to compare these shifting pies that respects the math of how parts relate to the whole.

The paper, titled "Compositional Synthetic Controls" by Onil Boussim, introduces a clever new tool to solve this puzzle. The author argues that when we look at things like employment types, energy sources, or vote shares, we shouldn't just look at the raw numbers. Instead, we should look at the "log-odds," which is a fancy way of comparing how much more likely one choice is than another. By using a special kind of geometry (called Aitchison geometry) that understands these ratios, the new method builds a "Synthetic Twin" that is mathematically perfect. It ensures that if the twin's "Gas" slice goes up, the "Coal" and "Renewables" slices automatically adjust to keep the total at 100%, just like a real pie.

The paper finds that this new method is much better at spotting the truth than the old "straight-line" ruler. The author demonstrates this with a real-world case study: Pennsylvania's electricity grid after a 2004 law called the Alternative Energy Portfolio Standard. The old methods might have missed the full story, but this new tool reveals a massive, persistent shift. It shows that by 2022, Pennsylvania's natural gas usage was nearly 60 percentage points higher than it would have been without the law, while renewable energy, despite growing in absolute terms, actually lost ground relative to gas. The paper suggests that without this new way of looking at the data, we would have missed the fact that the policy's biggest impact was a huge swap from coal to gas, rather than a simple boost for renewables. The author is quite confident in these results, having tested the method against fake scenarios (placebos) to ensure the findings aren't just a fluke, though they note that the small number of states available for comparison limits how statistically "sure" we can be about the exact probability.

In short, this paper gives us a better pair of glasses for watching how the pieces of a whole change over time. It proves that when you are dealing with shares and percentages, you can't just measure the distance between numbers; you have to measure the distance between the relationships those numbers have with each other. By doing so, it uncovers a hidden story in Pennsylvania's energy history that the old methods simply couldn't see.

Technical Summary: Compositional Synthetic Controls

Problem Statement
The paper addresses the challenge of applying the synthetic control method (SCM) to outcomes that are compositions—vectors of shares summing to one, generated by an underlying categorical process (e.g., employment states, energy generation sources, or vote shares). Extending standard SCM to these settings faces three fundamental challenges:

Multiplicity: Running separate SCMs for each category yields distinct weight vectors, resulting in fitted shares that do not sum to one, thereby failing to produce a single coherent counterfactual unit.
Geometric Mismatch: Shares reside on a simplex where economic structure is defined by ratios and relative proportions, not arithmetic differences. Standard SCM relies on Euclidean distance, which is blind to this structure and treats relative shifts in small-share categories as negligible compared to absolute shifts in dominant categories.
Scale Dependence: The Euclidean objective penalizes absolute deviations, causing the estimator to prioritize fitting large-share categories while ignoring the relative evolution of small-share categories, even when the latter are of primary substantive interest.

Methodology
The author proposes Compositional Synthetic Controls (CSC), an estimator derived from a structural random utility model of discrete choice. The methodology proceeds in four steps:

Micro-foundation: Under a multinomial logit model, aggregate shares are determined by relative systematic utilities. The log-odds of each category against a baseline are shown to be exactly equal to these relative systematic utilities. Thus, the observable log-odds map serves as the structural preference parameter.
Factor Structure: An interactive fixed effects model is imposed on the relative systematic utilities (the log-odds), rather than the shares directly. This assumes that the unobserved factors driving choice preferences interact with unit-specific loadings.
Identification: Under a convex hull condition on the factor loadings, the treated unit's counterfactual log-odds are identified as a convex combination of donor log-odds. Geometrically, this implies the counterfactual composition is the unique Fréchet barycenter of the donor compositions under the Aitchison metric (the canonical geometry of the simplex).
Estimation: The weights are estimated by minimizing the pre-treatment Aitchison distance between the treated unit and the synthetic combination. By stacking the $p-1$ log-odds equations across all pre-treatment periods, the estimation reduces to a single convex quadratic program. This "stacking" principle, adapted from Sun et al. (2025), treats the $p-1$ log-odds coordinates as $L$ outcomes sharing common factor loadings, effectively increasing the sample size to $(p-1)T_0$ .

Key Contributions

Structural Derivation of Geometry: Unlike previous manifold-based approaches that impose geometry abstractly, CSC derives the Aitchison metric directly from the discrete-choice model. This provides a behavioral interpretation where donor weights measure the similarity of relative preferences (structural parameters) rather than just marginal shares.
Coherent Counterfactuals: The estimator produces a single set of weights that generates a valid counterfactual composition (positive shares summing to one) by construction.
Resolution of Scale Dependence: The Aitchison metric treats pairwise log-ratios symmetrically. A doubling of a 2% share contributes equally to the objective function as a doubling of a 30% share, ensuring that small-share categories receive appropriate weight.
Theoretical Properties: The paper establishes consistency of the estimator as the product of the number of categories and pre-treatment periods grows, and provides a placebo inference procedure based on the Aitchison distance.

Empirical Results
The method is applied to Pennsylvania's electricity generation mix following the implementation of the Alternative Energy Portfolio Standard (AEPS) in 2004.

Findings: The analysis reveals a large and persistent compositional shift. By 2022, the natural gas share exceeds its counterfactual by nearly 60 percentage points. While renewables grew in absolute terms, they lost relative ground to natural gas.
Interplay: A component-wise analysis would miss the trade-off between gas and renewables; the compositional counterfactual captures this redistribution coherently.
Inference: A placebo test yields a p-value of 0.111, indicating that while the shift is substantial, it is not statistically distinguishable from the shifts observed in a few other states (Michigan, Louisiana, Florida) that underwent similar gas-driven transformations, though Pennsylvania's shift began earlier.

Significance and Claims
The paper claims that CSC resolves the geometric and structural limitations of applying standard SCM to compositional data. By grounding the method in a structural choice model, it ensures that the counterfactual respects the simplex geometry and that the weights reflect genuine preference similarity. The author argues that this approach is essential for policy evaluation where the outcome is inherently a distribution of shares, as it prevents the distortion of small but policy-relevant categories and provides a unified, behaviorally interpretable framework for causal inference in compositional settings. The paper does not claim to solve all inference challenges (noting the power limitations of permutation tests) but positions CSC as a principled alternative to Euclidean or separate-category approaches.

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