Parallel Growths Assumption in CoDiD

• Parallel Growths Assumption is a principle asserting that, absent treatment, each category evolves proportionally in both treated and control groups.
• It employs log-linear transformations and Aitchison geometry to construct valid counterfactuals in compositional difference-in-differences analyses.
• Empirical applications in electoral and energy studies demonstrate its ability to separate aggregate effects from compositional shifts while mitigating scale dependency.

The parallel growths assumption posits that, in the absence of treatment or intervention, categories within complex compositional systems—such as groups in an economy, types in a biological population, or alternatives in a discrete choice scenario—evolve such that each category grows or declines at the same proportional rate in both treated and control populations. This principle underlies identification methods for difference-in-differences (DiD) with categorical outcomes, as in the compositional DiD (CoDiD) approach, wherein total quantities and shares are both addressed in a manner consistent with economic and statistical theory (Boussim, 13 Oct 2025).

1. Formal Definition of the Parallel Growths Assumption

In the context of CoDiD, the parallel growths assumption is imposed on the logarithms of category quantities. For any category $c_k$, group $g \in \{0,1\}$ (control, treated), and time $t \in \{0,1\}$ (pre, post), denote $q_{g,t}^{N}(c_k)$ as the untreated counterfactual quantity. The assumption is expressed as: $$\log(q_{1,1}^{N}) - \log(q_{1,0}^{N}) = \log(q_{0,1}^{N}) - \log(q_{0,0}^{N})$$ or equivalently,

$$\frac{q_{1,1}^{N}(c_k)}{q_{1,0}^{N}(c_k)} = \frac{q_{0,1}^{N}(c_k)}{q_{0,0}^{N}(c_k)}, \quad \forall c_k$$

This ensures that, absent treatment, each category’s counterfactual evolution in the treated group matches the proportional (multiplicative) change observed in the control group.

2. Implications in Random Utility and Discrete Choice Models

In multinomial logit or random utility frameworks, category shares are functions of latent utility values. The parallel growths assumption implies that, under the absence of treatment, the log-odds ratios between any pair of categories follow parallel trajectories across treated and control groups. Analytically, this requires: $$\ell(\pi_{1,1}^{N}) - \ell(\pi_{1,0}^{N}) = \ell(\pi_{0,1}^{N}) - \ell(\pi_{0,0}^{N})$$ where $\ell(\pi)$ is the multinomial-logit transformation mapping probability vectors to log-odds space. This invariance guarantees that the relative preference structure between pairs of categories is stable over time and across groups, ensuring that observed compositional effects reflect true redistribution attributable to treatment rather than scale artifacts.

3. Methodological Foundations: Aitchison Geometry and Compositional Data Analysis

Compositional data reside in a probability simplex and must obey relative constraints (e.g., all shares are non-negative and sum to one). CoDiD employs Aitchison geometry, where operations such as perturbation ($\oplus$) and powering ($\odot$) define translations and scalings on the simplex. The parallel growths assumption in this setting states that the trajectory of the untreated probability mass vector of the treated group moves in parallel to that of the control group: $$\pi_{1,1}^{N} = \pi_{1,0}^{N} \ominus \pi_{0,0}^{N} \oplus \pi_{0,1}^{N}$$ with compositional difference $\ominus$ and perturbation $\oplus$ as defined by Aitchison group operations. These transformations guarantee the identified counterfactuals are valid, positive probability distributions.

4. Extensions: Relaxations, Staggered Adoption, and Synthetic Controls

Relaxing Parallel Growths

When multiple pre-treatment periods are available, CoDiD can accommodate deviations from strict parallel growth by bounding counterfactual growth within the convex hull of observed pre-treatment log-differences: $$\log(q_{1,1}^{N}(c_k)) - \log(q_{0,1}^{N}(c_k)) \in \text{Conv}\left\{ \log\left( \frac{q_{1,t}^{N}(c_k)}{q_{0,t}^{N}(c_k)} \right) : t \ \text{pre-treatment} \right\}$$ This provides nonparametric bounds under weaker assumptions.

Staggered Adoption

CoDiD generalizes the parallel growths assumption for staggered treatment designs. Treated units are compared to synthetic or never-treated controls by applying the same log-ratio translation for each cohort: $$\log(q_{\text{treated},\,\text{post}}^{N}(c_k)) = \log(q_{\text{treated},\,\text{pre}}^{N}(c_k)) + \left[ \log(q_{\text{control},\,\text{post}}^{N}(c_k)) - \log(q_{\text{control},\,\text{pre}}^{N}(c_k)) \right]$$ This identifies counterfactual outcomes for each adoption timing.

Synthetic Difference-in-Differences

Synthetic CoDiD constructs counterfactuals by optimally weighting control units’ log-quantity paths to better match treated units’ pre-treatment evolution. This approach increases robustness against violations of strict parallel growth, adapting the parallel log-growth to “best match” observed dynamics.

5. Analytical and Geometric Properties

Formulas Central to CoDiD

• Untreated counterfactual quantity (per category):

$$q_{1,1}^{N}(c_k) = \exp\left( \log(q_{1,0}^{N}(c_k)) + \log(q_{0,1}^{N}(c_k)) - \log(q_{0,0}^{N}(c_k)) \right)$$

• Normalized untreated probability mass (share):

$$\pi_{1,1}^{N}(c_k) = \frac{q_{1,1}^{N}(c_k)}{ \sum_{j=1}^{p} q_{1,1}^{N}(c_j) }$$

• Multinomial-logit counterfactual:

$$\pi_{1,1}^{N} = \ell^{-1}\left( \ell(\pi_{1,0}^{N}) + \ell(\pi_{0,1}^{N}) - \ell(\pi_{0,0}^{N}) \right)$$

• Compositional group operations (Aitchison simplex):

$$\pi_1 \oplus \pi_2, \quad \pi_1 \ominus \pi_2, \quad \alpha \odot \pi_1$$

These ensure validity of counterfactual probability vectors and interpretability in both share and total quantity space.

Equivalence Between Log-Linear and Aitchison Representations

Both the log-linear identification and Aitchison geometry yield the same constructed counterfactuals, thereby translating the parallel growths assumption into both analytical and geometric form in compositional settings.

6. Empirical Applications and Significance

Empirical applications in electoral studies (e.g., early voting reforms) and energy economics (e.g., RGGI’s impact on electricity generation) demonstrate the utility of the parallel growths assumption for recovering both aggregate and compositional treatment effects. For instance, in voter turnout analysis, CoDiD separates the total turnout effect (Growth Treatment Effect on the Treated, GTT) from redistribution effects among electoral categories (Compositional Treatment Effect on the Treated, CTT). Similarly, in energy composition analysis, CoDiD attributes proportional reductions in specific sources (coal, natural gas) while maintaining the non-negativity and interpretability of predicted counterfactual shares.

The methodology overcomes limitations inherent in additive DiD approaches—such as scale dependence and negative predictions—by embedding parallel growths in a multiplicative setting, respecting constraints of compositional data and discrete choice theory.

7. Broader Implications, Limitations, and Generalizability

The parallel growths assumption provides a foundation for robust causal inference and counterfactual identification in settings with compositional outcomes. It generalizes the notion of parallel trends from scalar to vector-valued and probability-valued outcomes, permitting inference in both total quantities and category shares. Relaxations allow for sensitivity analysis regarding pre-treatment dynamics, staggered adoption, and synthetic controls.

However, the validity of the parallel growths assumption relies on contextual knowledge regarding the process generating compositional changes. Violations—such as category-specific shocks, non-proportional redistribution, or treatment interfering with underlying preference evolution—compromise identification. When parallel log-growth is not tenable, CoDiD yields bounds rather than point estimates, thereby preserving interpretability at the cost of additional uncertainty.

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In summary, the parallel growths assumption is a critical identifiability condition for compositional DiD analysis, enabling rigorous, scale-compatible inference for categorical variables in panel and grouped data settings. Its implementation through log-linear transformations and geometric operations on the simplex ensures correct handling of both total and relative effects, providing a unified framework for modern causal analysis of compositional and share-based outcomes (Boussim, 13 Oct 2025).