The exact contents of citations.db main_text.text for this paper — one flattened LaTeX string, title through conclusion, appendix excluded, unmodified except for removing email addresses. This is what our citation measures are computed over.
47,587 characters
Compositional Synthetic Controls
\ifblind
\bigskip\bigskip\bigskip
\begin{center}
{\LARGE\bf Compositional Synthetic Controls}\\[1.5cm]
\textbf{Abstract}
\end{center}
\else
\title{\bf Compositional Synthetic Controls}
\author{Onil Boussim\\[0.2cm]
Department of Economics, Pennsylvania State University,\\
University Park, PA 16802\\
\texttt{[email removed]}}
\date{}
\maketitle
\begin{center}\textbf{Abstract}\end{center}
\fi
\medskip
\noindent
This paper develops a synthetic control estimator for compositional outcomes, vectors of shares generated by an underlying categorical process. Derived from a random utility model with interactive fixed effects on relative systematic utilities, the estimator maps compositions to log-odds, where the standard convex hull condition identifies the counterfactual as a convex combination of donor log-odds. Equivalently, it recovers the Fr\'{e}chet barycenter under the Aitchison metric, the canonical geometry of the simplex
(the non-linear space of shares) using a single set of weights across all categories. I also developed a placebo inference procedure based on the Aitchison distance. An application to Pennsylvania's electricity generation mix following the Alternative Energy Portfolio Standard uncovers a large and persistent compositional shift: natural gas exceeds its counterfactual by nearly 60 percentage points by 2022, while renewables lose relative ground.
\medskip
\noindent\textbf{Keywords:} Synthetic control, categorical outcomes, compositional data, Aitchison geometry, discrete choice.
\noindent\textbf{JEL:} C21, C25, C43.
\newpage
\setcounter{page}{1}
\section{Introduction}
\label{sec:intro}
Many policy interventions operate on outcomes that are categorical at the individual level. A job-training program moves workers across employment states, permanent contract, temporary contract, and unemployed. An education reform redirects students among fields of study, sciences, humanities, and social sciences. A transportation subsidy shifts commuters between car, public transit, and bicycle. When such a policy is applied to a single aggregate unit, a state, a region, or a country, the researcher observes the resulting vector of category shares, a composition on the simplex. Alternatively, compositional outcomes may be observed directly at the aggregate level, GDP shares by sector, vote shares by party, electricity generation shares by sources, without access to underlying individual data.
The synthetic control method \citep{abadie2003economic, abadie2010synthetic} has become a leading approach for causal inference in settings with a single treated unit, a pool of untreated donors, and a well-defined intervention date. However, extending it to outcomes that are compositions, vectors of shares generated by an underlying categorical process, poses three fundamental challenges.
\emph{First, multiplicity.} Running a separate synthetic control for each category yields a different weight vector per category and fitted shares that need not sum to one. No single counterfactual unit exists. Because the categories arise from a common choice process, individuals select among the same alternatives, a single weight vector that exploits the shared latent structure is the principled response \citep{tian2023synthetic, sun2025using}.
\emph{Second, geometric mismatch.} Shares live on the simplex, where ratios and relative proportions, not arithmetic differences, characterize economic structure. The Euclidean distance underlying standard synthetic controls is blind to this. Consider three hypothetical regions allocating GDP across services, industry, and agriculture. Region~A allocates $(82.5\%, 15.0\%, 2.5\%)$, Region~B $(72.5\%, 25.0\%, 2.5\%)$, and Region~C $(92.5\%, 5.0\%, 2.5\%)$. The Euclidean distance from A to B and from A to C is identical, approximately $0.14$, suggesting equally good matches. But the services-to-industry ratio is 5.5 in A, 2.9 in B, and 18.5 in C. In terms of economic structure, B is far closer to A. Euclidean distance entirely misses this.
\emph{Third, scale dependence.} The Euclidean objective penalizes absolute deviations, not proportional ones. As a result, large relative errors in small-share categories can have little influence on the optimization, whereas modest relative errors in dominant categories may drive the fit. Standard synthetic control methods therefore, tend to overlook small-share categories, even when their relative evolution is of primary substantive interest, such as emerging renewable energy technologies, minority employment groups, or rare modes of transportation.
This paper develops \emph{Compositional Synthetic Controls} (CSC), an estimator that resolves all three problems with a single set of weights derived from the structural model that generates the categorical data. The argument has four steps.
\emph{Step~1: Micro-foundation.} Under a random utility model of individual choice, aggregate shares are determined by relative systematic utilities, and the log-odds of each category against a baseline equal these utilities exactly. The log-odds of the observed shares \emph{are} the structural preference parameters.
\emph{Step~2: Factor structure.} I impose an interactive fixed effects model on the relative systematic utilities, the same structure that motivates the standard synthetic control estimator, but applied to the primitive objects governing choice rather than to the aggregate outcome directly.
\emph{Step~3: 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 (Proposition~1). The corresponding counterfactual composition is the unique Fr\'{e}chet barycenter of the donor compositions under the Aitchison metric.
\emph{Step~4: Estimation.} The weights are estimated by minimizing the pre-treatment Aitchison distance. By stacking the log-odds equations across all pre-treatment periods, the estimation reduces to a convex quadratic program, following the multi-outcome stacking principle of \citet{sun2025using}. The resulting counterfactual is automatically a valid composition, positive shares summing to one.
The structural derivation from discrete choice yields two advantages that are absent from alternatives. First, the Aitchison metric treats all pairwise log-ratios symmetrically: a doubling of a 2\% share contributes as much to the objective as a doubling of a 30\% share. I formalize this in Proposition~\ref{prop:bias}, which shows that the Euclidean SCM objective is a share-weighted distortion of the Aitchison objective. Second, the donor weights carry a direct behavioral interpretation as measures of \emph{preference similarity}: a donor earns weight to the extent that its population's relative preferences track those of the treated unit before the intervention. The treatment effect can therefore be interpreted as the change in the relative attractiveness of each category induced by the policy.
I illustrate the method with an application to Pennsylvania's electricity generation mix following the Alternative Energy Portfolio Standard (AEPS, Act~213 of 2004). The estimated effects reveal a sustained compositional shift: the natural-gas share exceeds its counterfactual by nearly 60 percentage points by 2022, while renewables, despite growing in absolute terms, lose ground to gas on the relative scale. A componentwise analysis would miss this interplay; the compositional counterfactual captures it coherently.
\subsection*{Related literature}
This paper connects three literatures. In the \emph{synthetic control} literature, the foundational contributions are \citet{abadie2003economic} and \citet{abadie2010synthetic}, with extensions to regularization \citep{doudchenko2016balancing, abadie2017penalized}, matrix completion \citep{athey2018matrix}, and design-based inference \citep{chernozhukov2018exact}. The most closely related methodological work is \citet{gunsilius2023distributional}, who extends synthetic controls to distributional outcomes using optimal transport, and \citet{gunsilius2024tangential}, who proposes tangential interpolation for manifold-valued outcomes. CSC differs in three respects: the geometry is not imposed abstractly but derived from a structural model of choice; the Aitchison metric has an explicit behavioral interpretation as log-odds matching; and the estimator reduces to a standard convex quadratic program rather than requiring manifold-specific optimization. \citet{tian2023synthetic} and \citet{sun2025using} argue for a common weight vector when components share latent factors, which is the structural rationale this paper formalizes. The present paper adopts the stacking principle of \citet{sun2025using}: because the $p - 1$ log-odds equations share the same unit-level factor loadings, they can be stacked into a single regression, expanding the effective pre-treatment sample from $T_0$ to $(p-1)T_0$ and enabling consistency through the number of categories as well as the number of pre-treatment periods.
In \emph{compositional data analysis}, the foundational framework is due to \citet{aitchison1982statistical}, with the Hilbert space structure developed by \citet{egozcue2003isometric}. CSC embeds the Aitchison geometry within a treatment-effects framework grounded in a structural model, providing both identification conditions and a behavioral interpretation that the statistical CoDA literature does not offer.
In the \emph{geodesic and manifold} literature, \citet{zhu2023geodesic} and \citet{kurisu2024geodesic} build synthetic controls for general unique geodesic spaces. This paper trades generality for structure: specializing to the simplex equipped with the Aitchison metric allows the geometry to carry economic content, the distance between compositions equals the distance between the preference structures that generated them.
\subsection*{Outline}
Section~\ref{sec:setup} defines the data structure and estimands. Section~\ref{sec:model} develops the model, states the main identification result (Theorem~1), and presents the estimator and implementation algorithm. Section~\ref{sec:comparison} compares CSC to alternative estimators. Section~\ref{sec:inference} discusses inference. Section~\ref{sec:application} reports the Pennsylvania application. Section~\ref{sec:conclusion} concludes. Proofs and secondary results are in the Appendix.
\section{Setup and Estimands}
\label{sec:setup}
There are $J$ units indexed by $j = 1,\dots,J$, observed over $T$ periods $t = 1,\dots,T$. Unit $j=1$ is treated beginning at a known date $T_0$, with $1 < T_0 < T$. Units $j = 2,\dots,J$ form the donor pool and are unaffected throughout. For each unit and period we observe the aggregate shares across $p$ mutually exclusive and exhaustive categories,
\begin{equation}
\pi_{j,t} = (\pi_{1,j,t},\dots,\pi_{p,j,t}) \in \mathcal{S}^{p-1},
\end{equation}
where $\mathcal{S}^{p-1} = \{\pi \in \mathbb{R}_{+}^{p} : \sum_{k=1}^{p} \pi_{k} = 1\}$ is the probability simplex. These shares may be computed from individual-level categorical data, as when a census records each person's employment sector and the researcher aggregates to state-level shares, or observed directly as compositional measures at the aggregate level. In the potential-outcomes notation, $\pi_{j,t}^{0}$ and $\pi_{j,t}^{1}$ denote the share vectors absent and under treatment, respectively. For donor units, $\pi_{j,t}^{0} = \pi_{j,t}^{1}$ for all $t$. For the treated unit, $\pi_{1,t}^{0} = \pi_{1,t}^{1}$ for $t \leq T_0$. The object of interest is the counterfactual $\pi_{1,t}^{0}$ for $t > T_0$.
\begin{assumption}[Positivity]\label{ass:pos}
$\pi_{k,j,t}^{0} > 0$ for all $k = 1,\dots,p$, $j = 1,\dots,J$, and $t = 1,\dots,T$.
\end{assumption}
Assumption~\ref{ass:pos} ensures that log-ratios are well-defined. It can be relaxed in practice by adding a small positive constant to zero shares, a standard device in compositional data analysis \citep{aitchison1982statistical}. With a counterfactual $\hat{\pi}_{1,t}^{0}$ in hand, two families of estimands are available.
\paragraph{Share-level ATT.} The level change in each category's share,
\begin{equation}
\label{eq:att}
\mathrm{ATT}_{k,t} = \pi_{k,1,t}^{1} - \hat{\pi}_{k,1,t}^{0}, \qquad k = 1,\dots,p.
\end{equation}
Since both vectors sum to one, $\sum_{k} \mathrm{ATT}_{k,t} = 0$: what one category gains, the others jointly lose. This is the compositional analogue of the standard ATT, expressed in percentage points.
\paragraph{Log-ratio treatment effect (LRTE).} The change in relative odds of category $k$ versus category $\ell$,
\begin{equation}
\label{eq:lrte}
\mathrm{LRTE}_{k\ell,t} = \log\frac{\pi_{k,1,t}^{1}}{\pi_{\ell,1,t}^{1}} - \log\frac{\hat{\pi}_{k,1,t}^{0}}{\hat{\pi}_{\ell,1,t}^{0}}.
\end{equation}
A value $\mathrm{LRTE}_{k\ell,t} > 0$ means treatment raised category $k$ relative to category $\ell$. The LRTE isolates the treatment-induced redistribution of mass between any pair of categories and is the natural estimand under the simplex geometry. A scalar summary of total compositional displacement is the Aitchison distance between observed and counterfactual compositions,
\begin{equation}
\label{eq:aitchison-dist}
\delta_A(\pi_{1,t}^{1}, \hat{\pi}_{1,t}^{0}) = \left[\frac{1}{2p}\sum_{i=1}^{p}\sum_{j=1}^{p} \left(\log\frac{\pi_{i,1,t}^{1}}{\pi_{j,1,t}^{1}} - \log\frac{\hat{\pi}_{i,1,t}^{0}}{\hat{\pi}_{j,1,t}^{0}}\right)^{2}\right]^{1/2}.
\end{equation}
\section{Model, Identification, and Estimation}
\label{sec:model}
This section develops the method from its behavioral foundations. I begin with the model of individual choice that generates aggregate shares, introduce a factor structure on the primitive parameters of that model, state the main identification result, and present the estimator with a detailed implementation algorithm.
\subsection{From categorical choices to aggregate shares}
Following \cite{mcfadden1972conditional}, I consider the following model. The aggregate shares $\pi_{j,t}$ summarize the choices of individuals in unit $j$ at time $t$. Suppose that in each unit and period there is a large population of decision-makers, each selecting one of the $p$ mutually exclusive alternatives. Individual $i$ in unit $j$ at time $t$ faces the random utility
\begin{equation}
U_{ik,j,t}^{0} = V_{k,j,t}^{0} + \varepsilon_{ik,j,t}, \qquad k = 1,\dots,p,
\end{equation}
where $V_{k,j,t}^{0}$ is the systematic utility of alternative $k$, common to all individuals in the unit, and $\varepsilon_{ik,j,t}$ is an idiosyncratic taste shock. When the shocks are independently and identically distributed across individuals and alternatives following the type-I extreme value distribution, the probability that a randomly drawn individual selects alternative $k$ is given by the multinomial logit formula,
\begin{equation}
\pi_{k,j,t}^{0} = \frac{\exp(V_{k,j,t}^{0})}{\sum_{\ell=1}^{p} \exp(V_{\ell,j,t}^{0})}, \qquad k = 1,\dots,p.
\end{equation}
The systematic utilities are identified only up to a common additive constant. Choosing category $p$ as the baseline, define the \emph{relative systematic utility} of alternative $k$,
\begin{equation}
\tilde{V}_{k,j,t}^{0} \coloneqq V_{k,j,t}^{0} - V_{p,j,t}^{0}, \qquad k = 1,\dots,p-1.
\end{equation}
From the logit formula, the log-odds of category $k$ against the baseline exactly equal the relative systematic utility:
\begin{equation}
\label{eq:logodds}
\log \frac{\pi_{k,j,t}^{0}}{\pi_{p,j,t}^{0}} = \tilde{V}_{k,j,t}^{0}, \qquad k = 1,\dots,p-1.
\end{equation}
This is the key structural equation: the observable log-odds \emph{are} the preference parameters that govern choice. Define the log-odds map $\ell\colon \mathrm{int}(\mathcal{S}^{p-1}) \to \mathbb{R}^{p-1}$ by
\begin{equation}
\ell(\pi) = \left(\log\frac{\pi_1}{\pi_p}, \dots, \log\frac{\pi_{p-1}}{\pi_p}\right),
\end{equation}
with inverse $\ell^{-1}(y)_k = e^{y_k}/(1 + \sum_{m=1}^{p-1} e^{y_m})$ for $k < p$ and $\ell^{-1}(y)_p = 1/(1 + \sum_{m=1}^{p-1} e^{y_m})$. Equation~\eqref{eq:logodds} states that $\ell(\pi_{j,t}^{0}) = \tilde{V}_{j,t}^{0} = (\tilde{V}_{1,j,t}^{0}, \dots, \tilde{V}_{p-1,j,t}^{0})$.
\begin{remark}[Scope of the logit foundation]\label{rem:scope}
The logit specification provides a clean bijection between shares and relative utilities. The identification argument extends to any additive random utility model whose choice probabilities satisfy a log-odds representation of the form $\log(\pi_k/\pi_p) = h(\tilde{V}_k)$ for a known, invertible link $h$; the GEV and nested logit families admit such representations after appropriate normalization. The key requirement is that the log-ratio transformation maps shares to objects on which a linear factor model is a plausible specification. The framework also applies directly when the researcher observes only aggregate compositions, vote shares, generation shares, expenditure shares, without access to individual data, provided the multinomial logit is taken as a structural maintained hypothesis at the aggregate level.
\end{remark}
\subsection{Factor model for relative utilities}
The standard synthetic control method is motivated by a factor model for the outcome variable. The idea, formalized by \citet{abadie2010synthetic}, is that the outcome is generated by an interactive fixed effects structure in which unobserved unit-specific factor loadings interact with time-varying factors, and the condition for identification is that the treated unit's loadings lie in the convex hull of the donor loadings. I adopt exactly this structure, but applied to the primitive objects that generate compositional data: the relative systematic utilities.
\begin{assumption}[Factor model]\label{ass:factor}
For all $k = 1,\dots,p-1$, $j = 1,\dots,J$, and $t = 1,\dots,T$, the counterfactual relative utility satisfies
\begin{equation}
\label{eq:factor}
\tilde{V}_{k,j,t}^{0} = \delta_{k,t} + \theta_{k,t}' Z_j + \lambda_{k,t}' \mu_j + \varepsilon_{k,j,t},
\end{equation}
where:
\begin{itemize}[nosep]
\item $\delta_{k,t}$ is a category-time fixed effect capturing common shocks to the attractiveness of alternative $k$ relative to the baseline;
\item $Z_j \in \mathbb{R}^{r}$ is a vector of observed, time-invariant unit characteristics with category-time-varying coefficients $\theta_{k,t} \in \mathbb{R}^{r}$;
\item $\mu_j \in \mathbb{R}^{F}$ is a vector of unobserved factor loadings with category-time-varying factors $\lambda_{k,t} \in \mathbb{R}^{F}$;
\item $\varepsilon_{k,j,t}$ is a mean-zero transitory shock satisfying $\operatorname{\mathbb{E}}[\varepsilon_{k,j,t} \mid Z_j, \mu_j] = 0$.
\end{itemize}
\end{assumption}
Two features of this specification deserve emphasis. First, the factor loadings $\mu_j$ are \emph{common across all $p-1$ relative utility equations within a unit}. They represent persistent features of the unit's preference environment, demographics, industry composition, regulatory history, that shape the relative attractiveness of all alternatives simultaneously. A state with a high loading on a ``green preferences'' factor will display higher relative utilities for renewables over fossil fuels, for public transit over driving, and for other environmentally favorable alternatives, consistently across categories and over time. The factors $\lambda_{k,t}$ are common across units but vary by category and time, allowing the same latent trait to have different implications for different alternatives in different periods.
Second, the factor model is imposed on \emph{relative utilities}, not on shares. Because the log-odds map is nonlinear, a factor model on $\tilde{V}$ does not imply a factor model on $\pi$, and conversely. The factor structure on utilities is the economically primitive specification: it models the latent determinants of choice, not their nonlinear aggregation into probabilities.
Stacking the $p-1$ equations, let $\bm{\delta}_t = (\delta_{1,t},\dots,\delta_{p-1,t})' \in \mathbb{R}^{p-1}$, let $\Theta_t$ be the $(p-1) \times r$ matrix with rows $\theta_{k,t}'$, let $\Lambda_t$ be the $(p-1) \times F$ matrix with rows $\lambda_{k,t}'$, and let $\bm{\varepsilon}_{j,t} = (\varepsilon_{1,j,t},\dots,\varepsilon_{p-1,j,t})'$. Then
\begin{equation}
\label{eq:log-odds-factor}
\ell(\pi_{j,t}^{0}) = \bm{\delta}_t + \Theta_t Z_j + \Lambda_t \mu_j + \bm{\varepsilon}_{j,t}, \qquad j = 1,\dots,J, \quad t = 1,\dots,T.
\end{equation}
\subsection{Identification}
The factor model provides the primitive justification for the synthetic control approach. The following condition is the compositional analogue of the standard convex hull requirement.
\begin{assumption}[Convex hull]\label{ass:convex}
There exist weights $w^{\ast} = (w_2^{\ast},\dots,w_J^{\ast})$ with $w_j^{\ast} \geq 0$ and $\sum_{j=2}^{J} w_j^{\ast} = 1$ such that
\begin{equation}
Z_1 = \sum_{j=2}^{J} w_j^{\ast} Z_j \qquad \text{and} \qquad \mu_1 = \sum_{j=2}^{J} w_j^{\ast} \mu_j.
\end{equation}
\end{assumption}
Assumption~\ref{ass:convex} requires that the treated unit's factor loadings lie in the convex hull of the donor units' loadings. It states that the treated unit's persistent preference structure, the unobserved features shaping the relative attractiveness of all alternatives, can be expressed as a mixture of the donors' preference structures. This is the same condition that motivates the standard synthetic control estimator, transposed to the space of preference parameters.
The main result of the paper is the following.
\begin{proposition}[Identification of the counterfactual composition]\label{thm:main}
Under Assumptions~\ref{ass:pos}--\ref{ass:convex}, the counterfactual log-odds of the treated unit satisfy
\begin{equation}
\label{eq:rep}
\ell(\pi_{1,t}^{0}) = \sum_{j=2}^{J} w_j^{\ast}\, \ell(\pi_{j,t}^{0}) + \eta_{1,t}, \qquad t = 1,\dots,T,
\end{equation}
where $\eta_{1,t} = \bm{\varepsilon}_{1,t} - \sum_{j=2}^{J} w_j^{\ast} \bm{\varepsilon}_{j,t}$ is a mean-zero composite error with $\operatorname{\mathbb{E}}[\eta_{1,t} \mid Z, \mu] = 0$. Consequently, the counterfactual composition satisfies
\begin{equation}
\label{eq:cf-identified}
\pi_{k,1,t}^{0} = \ell_k^{-1}\!\left(\sum_{j=2}^{J} w_j^{\ast}\, \ell(\pi_{j,t}^{0})\right) + O_p(\|\eta_{1,t}\|), \qquad k = 1,\dots,p,
\end{equation}
and the counterfactual is a valid composition in $\mathrm{int}(\mathcal{S}^{p-1})$: each $\hat{\pi}_{k,1,t}^{0} > 0$ and $\sum_k \hat{\pi}_{k,1,t}^{0} = 1$.
\end{proposition}
\begin{proof}
For any weights $w \in W_{J-1} = \{w \in \mathbb{R}^{J-1} : w_j \geq 0,\, \sum_{j \geq 2} w_j = 1\}$, the difference between the treated unit's log-odds and the synthetic combination is
\begin{align}
\ell(\pi_{1,t}^{0}) - \sum_{j=2}^{J} w_j\, \ell(\pi_{j,t}^{0}) &= \Theta_t\!\left(Z_1 - \sum_{j=2}^{J} w_j Z_j\right) + \Lambda_t\!\left(\mu_1 - \sum_{j=2}^{J} w_j \mu_j\right) \notag \\
&\quad + \left(\bm{\varepsilon}_{1,t} - \sum_{j=2}^{J} w_j \bm{\varepsilon}_{j,t}\right). \label{eq:decomp}
\end{align}
Setting $w = w^{\ast}$ zeroes the first two terms by Assumption~\ref{ass:convex}, yielding~\eqref{eq:rep} with $\eta_{1,t} = \bm{\varepsilon}_{1,t} - \sum_j w_j^{\ast} \bm{\varepsilon}_{j,t}$. The mean-zero property follows from $\operatorname{\mathbb{E}}[\bm{\varepsilon}_{j,t} \mid Z, \mu] = 0$ (Assumption~\ref{ass:factor}). Equation~\eqref{eq:cf-identified} follows by applying the continuously differentiable inverse $\ell^{-1}$ and a first-order expansion. Validity of the counterfactual as a composition follows from the definition of $\ell^{-1}$: for any $y \in \mathbb{R}^{p-1}$, the components $\exp(y_k)/(1 + \sum_m \exp(y_m))$ are strictly positive and sum to one.
\end{proof}
\begin{remark}[Behavioral content of the weights]\label{rem:behavioral}
Under the logit model, $\ell(\pi_{j,t}^{0})$ is exactly the vector of relative systematic utilities governing individual choice in unit $j$. The representation~\eqref{eq:rep} states that the treated unit's preference structure evolves as a convex combination of the donors', up to transitory shocks. The weights $w_j^{\ast}$ measure \emph{preference similarity}: a donor earns weight to the extent that its population's relative preferences, not merely its marginal shares, track those of the treated unit. The treatment effect $\widehat{\mathrm{LRTE}}_{kp,t} = \ell_k(\pi_{1,t}^{1}) - \sum_j \hat{w}_j \ell_k(\pi_{j,t})$ is the divergence of the treated unit's relative preferences from the common trajectory defined by the synthetic control.
\end{remark}
\begin{remark}[Geometric interpretation]\label{rem:geom}
The identification result admits an equivalent geometric reading. The simplex carries a Hilbert space structure under the Aitchison geometry \citep{aitchison1982statistical, egozcue2003isometric}, with perturbation $\pi \oplus \pi' = \mathcal{C}(\pi_1\pi_1',\dots,\pi_p\pi_p')$ and powering $\alpha \odot \pi = \mathcal{C}(\pi_1^{\alpha},\dots,\pi_p^{\alpha})$, where $\mathcal{C}$ is the closure (renormalization). The Aitchison distance $\delta_A(\pi,\pi') = \|\ell(\pi) - \ell(\pi')\|_2$ makes $\ell$ an isometry. The counterfactual composition identified by Theorem~\ref{thm:main} is the Fr\'{e}chet barycenter, the minimizer of weighted squared distances, of the donor compositions under the Aitchison metric:
\begin{equation}
\ell^{-1}\!\left(\sum_{j=2}^{J} w_j^{\ast}\, \ell(\pi_{j,t}^{0})\right) = \operatorname*{argmin}_{m \in \mathcal{S}^{p-1}} \sum_{j=2}^{J} w_j^{\ast}\, \delta_A^{2}(\pi_{j,t}^{0}, m) = \mathcal{C}\!\left(\prod_{j=2}^{J} \pi_{k,j,t}^{0\, w_j^{\ast}}\right)_{k=1}^{p}.
\end{equation}
That is, the counterfactual is the renormalized weighted geometric mean of the donor compositions. Standard SCM computes the arithmetic mean (Fr\'{e}chet barycenter under Euclidean distance); CSC computes the geometric mean (Fr\'{e}chet barycenter under Aitchison distance). The geometry is not an ad hoc choice: it is derived from the discrete-choice model.
\end{remark}
\subsection{Estimation}
The weights $w^{\ast}$ are estimated by minimizing the pre-treatment discrepancy between the treated unit's log-odds and the synthetic combination. The CSC weights solve
\begin{equation}
\label{eq:est}
\hat{w} = \operatorname*{argmin}_{w \in W_{J-1}} \sum_{t=1}^{T_0} \left\|\ell(\pi_{1,t}) - \sum_{j=2}^{J} w_j\, \ell(\pi_{j,t})\right\|_2^{2}.
\end{equation}
This is a convex quadratic program in $J-1$ unknowns with a simplex constraint, solvable with standard routines. By the isometry between $(\mathcal{S}^{p-1}, \delta_A)$ and $(\mathbb{R}^{p-1}, \|\cdot\|_2)$, the objective is equivalently $\sum_{t=1}^{T_0} \delta_A^{2}(\pi_{1,t},\, \bigoplus_{j \geq 2} w_j \odot \pi_{j,t})$.
\paragraph{Stacked representation.} Because $\ell(\pi_{j,t}) \in \mathbb{R}^{p-1}$, the squared norm in~\eqref{eq:est} decomposes across the $p - 1$ log-odds coordinates:
\begin{equation}
\label{eq:stacked}
\hat{w} = \operatorname*{argmin}_{w \in W_{J-1}} \sum_{t=1}^{T_0} \sum_{k=1}^{p-1} \left( \ell_k(\pi_{1,t}) - \sum_{j=2}^{J} w_j\, \ell_k(\pi_{j,t}) \right)^{2}.
\end{equation}
This is a single least-squares problem with $N \coloneqq (p-1) T_0$ scalar observations, each of the form
\begin{equation}
\label{eq:stacked-obs}
y_{k,t} = \ell_k(\pi_{1,t}), \qquad x_{k,t,j} = \ell_k(\pi_{j,t}), \qquad k = 1,\dots,p-1,\quad t = 1,\dots,T_0.
\end{equation}
As emphasized by \citet{sun2025using} in the context of multi-outcome synthetic controls, the common factor loadings $\mu_j$ shared across all $p - 1$ equations within a unit mean that each log-odds coordinate provides independent information about the same unit-level parameters. The stacking aggregates this information: rather than estimating $w^{\ast}$ from $T_0$ vector-valued observations, one estimates it from $(p-1)T_0$ scalar observations drawn from the same underlying factor model.
Given the estimated weights, the counterfactual shares for any post-treatment period $t > T_0$ are
\begin{equation}
\label{eq:cf}
\hat{\pi}_{k,1,t}^{0} = \frac{\exp\!\left(\sum_{j=2}^{J} \hat{w}_j \log\frac{\pi_{k,j,t}}{\pi_{p,j,t}}\right)}{1 + \sum_{m=1}^{p-1}\exp\!\left(\sum_{j=2}^{J} \hat{w}_j \log\frac{\pi_{m,j,t}}{\pi_{p,j,t}}\right)}, \qquad k = 1,\dots,p-1,
\end{equation}
with $\hat{\pi}_{p,1,t}^{0} = 1 - \sum_{k=1}^{p-1} \hat{\pi}_{k,1,t}^{0}$.
\begin{proposition}[Consistency]\label{prop:consist}
Suppose Assumptions~\ref{ass:pos}--\ref{ass:convex} hold, the transitory shocks $\{\varepsilon_{k,j,t}\}$ are independent across $(k,t)$ with uniformly bounded fourth moments, and the weights $w^{\ast}$ satisfying Assumption~\ref{ass:convex} are unique in $W_{J-1}$. Then $\hat{w} \xrightarrow{p} w^{\ast}$ as $(p-1)T_0 \to \infty$ with $J$ fixed.
\end{proposition}
The proof, given in Appendix~\ref{app:proofs}, applies standard extremum estimator arguments to the quadratic objective on the compact weight simplex. The key observation is that the effective sample size for the stacked regression~\eqref{eq:stacked} is $N = (p-1)T_0$, not $T_0$ alone. Consistency can therefore be achieved in two regimes: the standard long-panel regime where $T_0 \to \infty$ with $p$ fixed, and the \emph{many-categories regime} where $p \to \infty$ with $T_0$ fixed. In the application below, $p = 3$ and $T_0 = 14$, so $N = 28$. The uniqueness condition rules out collinear donor configurations and is standard in the synthetic control literature.
\begin{remark}[Comparison with the multi-outcome estimator of \citeauthor{sun2025using}]
\label{rem:stacking}
\citet{sun2025using} develop a stacked synthetic control estimator for $L$ scalar outcomes that share a common factor structure, yielding an effective sample of $LT_0$. The CSC estimator is a structural specialization of their approach to the compositional setting. The $p - 1$ log-odds coordinates play the role of the $L$ outcomes, with the multinomial logit model providing both the transformation (log-odds rather than levels) and the structural justification for the shared loadings (they arise from a common latent preference environment). The advantage of this specialization is that the geometry of the problem, the Aitchison metric on the simplex, is inherited from the model rather than imposed, and the resulting counterfactual is automatically a valid composition.
\end{remark}
\subsection{Implementation algorithm}
Algorithm~\ref{alg:csc} provides a step-by-step implementation of CSC. The algorithm requires only the share data and a standard quadratic programming solver; the R function \texttt{solve.QP} from the \texttt{quadprog} package or the Python function \texttt{scipy.optimize.minimize} with the SLSQP method are both suitable. A replication package implementing Algorithm~\ref{alg:csc} in R with the Pennsylvania data is available online.
\begin{algorithm}[t]
\caption{Compositional Synthetic Controls (CSC)}
\label{alg:csc}
\begin{algorithmic}[1]
\Require Share matrix $\Pi = [\pi_{j,t}]$ for $j = 1,\dots,J$, $t = 1,\dots,T$, with $p$ categories; treatment date $T_0$; baseline category $p$.
\Ensure Estimated weights $\hat{w}$; counterfactual shares $\hat{\pi}_{1,t}^{0}$ for $t > T_0$; treatment effects.
\Statex
\State \textbf{Step 1: Log-odds transformation.} For each $j = 1,\dots,J$ and $t = 1,\dots,T$, compute
\[
\ell_k(\pi_{j,t}) = \log\frac{\pi_{k,j,t}}{\pi_{p,j,t}}, \qquad k = 1,\dots,p-1.
\]
If $\pi_{k,j,t} = 0$ for some $(k,j,t)$, replace with $\pi_{k,j,t} + c$ for a small $c > 0$ (e.g., $c = 10^{-6}$) and renormalize.
\Statex
\State \textbf{Step 2: Construct stacked QP inputs.} Stack the $(p-1)$ log-odds equations across all $T_0$ pre-treatment periods into a single regression with $N = (p-1)T_0$ observations. Form the $N$-vector $\bm{y}$ and $N \times (J-1)$ matrix $\bm{X}$:
\[
y_{(k-1)T_0 + t} = \ell_k(\pi_{1,t}), \qquad X_{(k-1)T_0 + t,\, j} = \ell_k(\pi_{j+1,t}),
\]
for $k = 1,\dots,p-1$ and $t = 1,\dots,T_0$. Define $H = \bm{X}'\bm{X}$ and $f = \bm{X}'\bm{y}$.
\Statex
\State \textbf{Step 3: Solve QP.} Obtain $\hat{w} = \operatorname*{argmin}_{w \in W_{J-1}} (w'Hw - 2f'w)$, subject to $w_j \geq 0$ and $\sum_j w_j = 1$.
\Statex
\State \textbf{Step 4: Counterfactual shares.} For each $t > T_0$ and $k = 1,\dots,p-1$:
\[
\hat{y}_{k,t} = \sum_{j=2}^{J} \hat{w}_j \, \ell_k(\pi_{j,t}), \qquad \hat{\pi}_{k,1,t}^{0} = \frac{e^{\hat{y}_{k,t}}}{1 + \sum_{m=1}^{p-1} e^{\hat{y}_{m,t}}}, \qquad \hat{\pi}_{p,1,t}^{0} = 1 - \sum_{k=1}^{p-1}\hat{\pi}_{k,1,t}^{0}.
\]
\Statex
\State \textbf{Step 5: Treatment effects.} Compute:
\[
\mathrm{ATT}_{k,t} = \pi_{k,1,t} - \hat{\pi}_{k,1,t}^{0}, \quad \mathrm{LRTE}_{k\ell,t} = \log\frac{\pi_{k,1,t}}{\pi_{\ell,1,t}} - \log\frac{\hat{\pi}_{k,1,t}^{0}}{\hat{\pi}_{\ell,1,t}^{0}}, \quad \delta_A(t) = \|\ell(\pi_{1,t}) - \hat{\ell}_{1,t}^{0}\|_2.
\]
\Statex
\State \textbf{Step 6: Placebo inference.} For each donor $j = 2,\dots,J$: repeat Steps 1--5 treating unit $j$ as if treated, using remaining donors. Compute $R^{(j)} = \mathrm{RMSPE}_{\mathrm{post}}^{(j)} / \mathrm{RMSPE}_{\mathrm{pre}}^{(j)}$. Retain donors with $\mathrm{RMSPE}_{\mathrm{pre}}^{(j)} \leq m \cdot \mathrm{RMSPE}_{\mathrm{pre}}^{(1)}$. The $p$-value is $\#\{j : R^{(j)} \geq R^{(1)}\} / M$.
\end{algorithmic}
\end{algorithm}
\section{Comparison with Alternative Estimators}
\label{sec:comparison}
To clarify what CSC achieves, I contrast it with two natural alternatives.
\paragraph{Separate SCM.} Running a standard synthetic control independently for each category,
\begin{equation}
w^{(k)} = \operatorname*{argmin}_{w \in W_{J-1}} \sum_{t=1}^{T_0} \left( \pi_{k,1,t} - \sum_{j=2}^{J} w_j \pi_{k,j,t} \right)^{2}, \qquad k = 1,\dots,p,
\end{equation}
yields $p$ distinct weight vectors. The fitted shares $\sum_{j} w_j^{(k)} \pi_{k,j,t}$ generally do not sum to one, so no single counterfactual exists. Moreover, the method ignores the joint structure: the same latent factors that determine relative utilities for category $k$ also determine those for category $\ell$, and separate estimation discards this information.
\paragraph{Euclidean SCM.} A single weight vector can be obtained by minimizing Euclidean distance on raw shares:
\begin{equation}
w^{\mathrm{E}} = \operatorname*{argmin}_{w \in W_{J-1}} \sum_{t=1}^{T_0} \left\| \pi_{1,t} - \sum_{j=2}^{J} w_j \pi_{j,t} \right\|_2^{2}.
\end{equation}
The fitted shares sum to one because the simplex is convex under arithmetic means. However, the Euclidean metric introduces a systematic scale dependence that I now formalize.
\begin{proposition}[Scale dependence of Euclidean SCM]\label{prop:bias}
For compositions near a reference $\pi^{0} \in \mathrm{int}(\mathcal{S}^{p-1})$, the squared Euclidean distance satisfies
\begin{equation}
\|\pi - \pi'\|_2^{2} \approx \sum_{k=1}^{p} (\pi_k^{0})^{2} \left(\log\frac{\pi_k}{\pi_k'}\right)^{2}.
\end{equation}
The Euclidean objective weights the log-ratio discrepancy of category $k$ by $(\pi_k^{0})^{2}$: a category holding 2\% of the total contributes at most $4 \times 10^{-4}$ relative to a category at 50\%. The Aitchison objective treats centered log-ratios symmetrically regardless of share magnitude.
\end{proposition}
The proof is in Appendix~\ref{app:proofs}. This matters whenever the policy question involves small-share categories. The Euclidean estimator matches the dominant categories at the expense of precisely those alternatives the policy targets.
\begin{example}[Illustration]\label{ex:three}
Consider three regions allocating GDP across services, industry, and agriculture. Region~A: $(82.5, 15.0, 2.5)\%$; Region~B: $(72.5, 25.0, 2.5)\%$; Region~C: $(92.5, 5.0, 2.5)\%$. The services-to-industry ratio is 5.5 in A, 2.9 in B, and 18.5 in C. The Euclidean distance from A to B and from A to C is identical ($\approx 0.14$). The Aitchison distance from A to B is 0.61; from A to C it is 1.21, correctly reflecting the far larger structural divergence. A synthetic control based on Euclidean distance would treat B and C as equally good matches for A, while CSC would correctly down-weight C.
\end{example}
\section{Inference}
\label{sec:inference}
I propose a placebo permutation test following \citet{abadie2010synthetic}, adapted to the Aitchison metric (Step~6 of Algorithm~\ref{alg:csc}). Each donor unit $j = 2,\dots,J$ is treated in turn as if it had received the intervention at $T_0$. For each placebo, the CSC weights are re-estimated using the remaining donors, and the post-to-pre root mean squared prediction error ratio is computed:
\begin{equation}
R^{(j)} = \frac{\mathrm{RMSPE}_{\mathrm{post}}^{(j)}}{\mathrm{RMSPE}_{\mathrm{pre}}^{(j)}},
\end{equation}
where
\begin{equation}
\mathrm{RMSPE}_{\mathrm{pre}}^{(j)} = \left[\frac{1}{T_0} \sum_{t=1}^{T_0} \delta_A^{2}\!\left(\pi_{j,t}, \hat{\pi}_{j,t}^{0}\right)\right]^{1/2},
\end{equation}
and $\mathrm{RMSPE}_{\mathrm{post}}^{(j)}$ is defined analogously over $t = T_0+1,\dots,T$. Using the Aitchison distance ensures consistency with the estimation objective.
To avoid comparing the treated unit to donors with poor pre-treatment fit, I retain only those units whose pre-treatment RMSPE does not exceed a multiple $m$ (typically $m = 5$) of the treated unit's pre-treatment RMSPE. Let $M$ be the number of retained units, including the treated unit. Under the sharp null of no treatment effect for any unit, the test statistic $R^{(1)}$ is exchangeable with the placebo statistics, so the exact $p$-value is
\begin{equation}
p = \frac{\sum_{j=1}^{M} \mathbf{1}\{R^{(j)} \geq R^{(1)}\}}{M}.
\end{equation}
\begin{remark}[Power]\label{rem:power}
The power of the permutation test is bounded above by $1/J$, a limitation shared by all synthetic control inference procedures based on permutation. With $J = 36$ retained units, the smallest attainable $p$-value is $1/36 \approx 0.028$. Conformal inference approaches \citep{chernozhukov2018exact} adapted to the Aitchison metric could yield pointwise confidence sets with exact coverage, strengthening inference beyond the permutation approach employed here.
\end{remark}
\section{Application: Pennsylvania's Alternative Energy Portfolio Standard}
\label{sec:application}
\subsection{Data and donor pool}
I use annual state-level net generation from the U.S.\ Energy Information
Administration for 1990--2023, aggregated into three categories: natural gas;
coal and oil; and renewables (conventional hydro, wind, solar, geothermal,
and other).
The treated unit is Pennsylvania and the treatment date is $T_0=2003$, the
year before the AEPS (Act~213 of 2004), giving fourteen pre-treatment and
twenty post-treatment periods. The donor pool of 42 states excludes units with
a structural zero in any category over the sample, states that adopted
comparable alternative energy portfolio standards during the sample (Ohio in
2008, West Virginia in 2009), and Vermont, whose gas and fossil shares are
each below $0.1\%$ throughout and would produce extreme log-odds.
\subsection{Pre-treatment fit}
\Cref{tab:fit} reports the CSC weights and balance. Nine donors
receive positive weight, led by coal-intensive states, Illinois ($0.233$) and
Wyoming ($0.208$), with Iowa ($0.168$), Nebraska ($0.149$), Massachusetts
($0.102$), and smaller contributions from Connecticut, New Jersey, Indiana,
and Montana. The synthetic control reproduces the pre-treatment mix almost
exactly. The Aitchison pre-treatment RMSPE is $0.187$ and the Euclidean share
RMSPE is $1.34$ percentage points. \Cref{fig:shares} shows the graphical
evolution of the shares.
\begin{table}[t]
\centering
\caption{Pennsylvania: donor weights and pre-treatment balance (CSC)}
\label{tab:fit}
\begin{threeparttable}
\small
\begin{tabular}{@{}lc c lcc@{}}
\toprule
Donor & Weight & \quad & \multicolumn{3}{c}{Pre-treatment mean share (\%)} \\
\cmidrule(l){4-6}
& & & & Pennsylvania & Synthetic \\
\midrule
Illinois & 0.233 & & Natural gas & 3.6 & 3.6 \\
Wyoming & 0.208 & & Coal and oil & 93.2 & 93.2 \\
Iowa & 0.168 & & Renewables & 3.2 & 3.2 \\
Nebraska & 0.149 & & & & \\
Massachusetts & 0.102 & & Aitchison RMSPE & \multicolumn{2}{c}{0.187} \\
Connecticut & 0.048 & & Share RMSPE (pp) & \multicolumn{2}{c}{1.34} \\
New Jersey & 0.043 & & & & \\
Indiana & 0.032 & & & & \\
Montana & 0.016 & & & & \\
\bottomrule
\end{tabular}
\begin{tablenotes}\footnotesize
\item Weights sum to one.
\end{tablenotes}
\end{threeparttable}
\end{table}
\begin{figure}[t]
\centering
\includegraphics[width=\textwidth]{fig2_shares.png}
\caption{Generation shares for Pennsylvania and its synthetic control. The
synthetic shares are the Fr\'echet barycenter of the donor compositions
under the Aitchison metric.}
\label{fig:shares}
\end{figure}
\subsection{Treatment effects}
\Cref{tab:effects} reports the estimated effects. The
natural-gas share rises steadily above the counterfactual, reaching a peak
gap of $60$ percentage points in 2022, while coal and oil fall by a
corresponding amount; the renewables gap is mildly negative in later years,
so Pennsylvania's renewable share grew more slowly than the synthetic
control's even as its gas share surged. The gas-versus-fossil log-ratio
effect reaches $2.37$ by 2022, the observed gas-to-fossil ratio is about
$\exp(2.37)\approx10.7$ times its counterfactual, and the Aitchison distance
between observed and counterfactual compositions grows throughout, reaching
$2.53$ in 2022.
\begin{table}[t]
\centering
\caption{Estimated compositional treatment effects for Pennsylvania}
\label{tab:effects}
\begin{threeparttable}
\small
\begin{tabular}{@{}lrrrrrr@{}}
\toprule
Year & $ATT_{\text{gas}}$ & $ATT_{\text{fos}}$ & $ATT_{\text{ren}}$
& $LRTE_{\text{gas,fos}}$ & $LRTE_{\text{gas,ren}}$ & $\delta_A$ \\
& (pp) & (pp) & (pp) & & & \\
\midrule
2004 & +4.6 & $-4.8$ & +0.2 & +0.98 & +0.87 & 0.76 \\
2006 & +2.7 & $-1.9$ & $-0.7$ & +0.34 & +0.49 & 0.36 \\
2008 & +7.1 & $-5.7$ & $-1.4$ & +0.82 & +1.04 & 0.78 \\
2010 & +16.5 & $-10.5$ & $-6.0$ & +1.46 & +2.18 & 1.57 \\
2012 & +25.3 & $-13.7$ & $-11.6$ & +1.41 & +2.43 & 1.73 \\
2014 & +30.4 & $-18.8$ & $-11.6$ & +1.94 & +2.69 & 1.96 \\
2016 & +37.3 & $-20.8$ & $-16.5$ & +1.68 & +2.54 & 1.82 \\
2018 & +39.9 & $-22.4$ & $-17.5$ & +1.66 & +2.33 & 1.70 \\
2020 & +52.1 & $-15.8$ & $-36.3$ & +1.79 & +3.08 & 2.19 \\
2022 & +60.4 & $-24.5$ & $-35.9$ & +2.37 & +3.50 & 2.53 \\
2023 & +59.1 & $-18.9$ & $-40.2$ & +2.35 & +3.35 & 2.43 \\
\bottomrule
\end{tabular}
\begin{tablenotes}\footnotesize
\item Even years shown; full series in \Cref{fig:shares}. Share effects (pp) sum
to zero across categories each year. $LRTE$ is in natural-log units; $\delta_A$
is the Aitchison distance between observed and counterfactual compositions.
\end{tablenotes}
\end{threeparttable}
\end{table}
\subsection{Inference}
\Cref{fig:placebo} reports the placebo distribution and \Cref{fig:gaps} the
placebo gaps. Among the $35$ donors retained under the pre-fit screen, three
attain a post-to-pre Aitchison RMSPE ratio at least as large as
Pennsylvania's ($R^{(1)}=9.18$): Michigan ($10.46$), Louisiana ($10.22$),
and Florida ($10.07$), giving an exact $p$-value of $4/36=0.111$. The nearest
placebos are states that underwent their own gas-driven transformations in
the 2010s; Pennsylvania still stands apart partly because its shift began in
2004, several years before the shale boom reached most non-adopting states,
and went further.
\begin{figure}[t]
\centering
\includegraphics[width=0.85\textwidth]{fig4_placebo.png}
\caption{Placebo distribution of post-to-pre Aitchison RMSPE ratios.
Pennsylvania (dark) exceeds all but three of the $35$ retained donors,
giving a $p$-value of $0.111$.}
\label{fig:placebo}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[width=0.85\textwidth]{fig7_gaps.png}
\caption{Placebo gaps: Aitchison distance between actual and synthetic shares
for Pennsylvania (dark) and the donor pool (grey).}
\label{fig:gaps}
\end{figure}
\subsection{Discussion}
The estimates describe a compositional transition that began at enactment and
widened for nearly two decades, concentrated in the substitution of natural
gas for coal and oil. Because the synthetic control is built overwhelmingly
from coal-intensive states that experienced the same national gas-price
decline, the estimated effect is the \emph{additional} compositional shift in
Pennsylvania beyond that common movement. The mildly negative renewables
effect is itself informative: the policy's headline renewables target was
modest next to the gas-driven transformation of the fossil base, so on the
relative scale the simplex makes visible, renewables lost ground to gas even
as they grew in absolute terms. A single-share analysis would miss this
interplay; the compositional counterfactual captures it coherently.
\section{Conclusion}
\label{sec:conclusion}
This paper developed Compositional Synthetic Controls, a method for constructing counterfactual share vectors when the outcome is categorical or compositional and a single aggregate unit receives an intervention. The method rests on a structural model of discrete choice. Under the logit specification, the log-odds of shares equal relative systematic utilities; an interactive fixed effects model on these utilities delivers identification via the standard convex hull condition in log-ratio space (Theorem~\ref{thm:main}). The counterfactual is the Fr\'{e}chet barycenter of the donor compositions under the Aitchison metric: a valid composition by construction, with weights that measure preference similarity. Adopting the multi-outcome stacking principle of \citet{sun2025using}, the $p - 1$ log-odds equations are pooled into a single regression with effective sample $(p-1)T_0$, ensuring that consistency requires only the product of the number of categories and pre-treatment periods to grow.
CSC has three central advantages over standard Euclidean approaches. It produces a single synthetic unit with a coherent counterfactual composition; it respects the log-ratio structure of the simplex, ensuring that relative movements and small-share categories receive appropriate weight; and the donor weights carry an interpretable behavioral reading as measures of structural similarity in the latent preference parameters. The Pennsylvania application demonstrates these properties in a policy setting that directly targets composition.
Three extensions are natural. First, covariates can be incorporated by augmenting the log-odds objective with predictor balance terms, analogous to the covariate-adjusted SCM of \citet{abadie2010synthetic}. Second, penalization, ridge or LASSO on the weights, can improve finite-sample performance when the donor pool is large relative to the effective sample $(p-1)T_0$. Third, conformal inference methods \citep{chernozhukov2018exact} adapted to the Aitchison metric could yield pointwise confidence sets with exact coverage, strengthening inference beyond the permutation approach.
\subsection*{Declaration of interest}
The author reports no conflicts of interest.
\subsection*{Data availability}
The electricity generation data used in Section~\ref{sec:application} are publicly available from the U.S.\ Energy Information Administration. A complete replication package containing all code and data files is available.
\section*{Use of generative AI}
The author used generative AI (Claude Opus 4.6) for language editing, LaTeX formatting, and replication code generation. The author reviewed and is responsible for all final content.
\newpage
\bibliographystyle{abbrvnat}
\bibliography{ref}
\newpage