Changes-in-Changes (CiC)

• Changes-in-Changes (CiC) is a nonparametric causal inference framework that identifies quantile treatment effects using monotone quantile mapping in two-group, two-period designs.
• It recovers missing counterfactual outcome distributions without relying exclusively on parallel trends, offering a robust alternative to difference-in-differences.
• Extensions of CiC address ordered outcomes, sample selection, and high-dimensional confounding, broadening its applicability in policy evaluation.

Searching arXiv for recent and foundational CiC-related papers to ground the article. The Changes-in-Changes (CiC) estimator is a distributional causal-inference framework for two-group, two-period designs. In its classical form, it provides fully nonparametric identification of quantile treatment effects in a two-group, two-period setting under a rank-invariance and support assumption, rather than the usual parallel-trends condition of difference-in-differences (Sasaki et al., 2022). Across subsequent work, CiC has been generalized to ordered discrete outcomes with underreporting, extreme quantiles, mediation, targeted policies via triple differences, multi-category discrete treatments, endogenous sample selection, attrition, and settings with group-level heterogeneity (Gutknecht et al., 2024, Akbari et al., 2024, Boussim, 2024, Viviens, 12 Feb 2025, Ghanem et al., 2022, Chen et al., 2023). The common theme is recovery of a missing counterfactual distribution by monotone quantile mapping rather than by mean-trend extrapolation.

1. Classical formulation and identification logic

In the classical CiC model one supposes a latent “no-treatment” outcome

$Y_N = h(U,T),$

where $U$ is an unobserved scalar with distribution that may differ by group $G\in\{0,1\}$ but satisfies $U\perp T\mid G$, and $h(\cdot,t)$ is strictly increasing (Sasaki et al., 2022). If treatment $I=G\cdot T$, the observed outcome is

$Y = Y_N(1-I) + Y_I \cdot I.$

Under the support condition that the support of $U\mid G=1$ is contained in that of $U\mid G=0$, it can be shown that for all $y$,

$U$0

and hence the $U$1-th quantile treatment effect is identified by a composition of observed empirical distribution functions and inverses (Sasaki et al., 2022).

A closely related formulation writes untreated potential outcomes as

$U$2

with $U$3 strictly increasing and support$U$4support$U$5 (Viviens, 12 Feb 2025). This restates the same identification logic: untreated outcomes evolve through a monotone time transformation of a latent rank whose within-group distribution is stable over time.

The central contrast with difference-in-differences is explicit in the literature. Parallel trends imposes

$U$6

which is only a first-moment condition, whereas CiC imposes a higher-order and distributional restriction through monotone quantile-to-quantile evolution (Akbari et al., 2024). This suggests that CiC is most naturally interpreted as a model for the full untreated distribution rather than as a correction to mean comparisons.

2. Structural assumptions and quantile mapping

The key assumptions recur in multiple variants. One common statement is that there exists a scalar latent $U$7 with common support such that

$U$8

with $U$9 strictly increasing for each $G\in\{0,1\}$0, the distribution of $G\in\{0,1\}$1 the same at $G\in\{0,1\}$2 and $G\in\{0,1\}$3, and support overlap sufficient for inversion and matching (Akbari et al., 2024). In another formulation for binary treatment,

$G\in\{0,1\}$4

with a strictly increasing nonparametric structural model and rank-invariance, also called “copula stability” (Boussim, 2024).

These assumptions imply that untreated evolution can be represented by a quantile–quantile map. In the simplest CiC notation,

$G\in\{0,1\}$5

and, under the classical no-drift-across-groups implication, the same increasing map transports the period-$G\in\{0,1\}$6 untreated distribution into the period-$G\in\{0,1\}$7 untreated distribution across groups (Akbari et al., 2024). In the debiased semiparametric efficient extension, the corresponding “distributional bridge” assumption states that there exists a function $G\in\{0,1\}$8, nondecreasing in $G\in\{0,1\}$9 for each $U\perp T\mid G$0, such that

$U\perp T\mid G$1

or, for continuous outcomes, that the $U\perp T\mid G$2 quantile–quantile map $U\perp T\mid G$3 does not depend on $U\perp T\mid G$4 (Sun et al., 9 Jul 2025).

Under these assumptions, the average treatment effect on the treated can be written as

$U\perp T\mid G$5

and the counterfactual distribution on the treated satisfies

$U\perp T\mid G$6

(Sun et al., 9 Jul 2025). The defining mechanism is therefore composition of an observed pre-treatment distribution for the treated with an observed untreated time map estimated from controls.

3. Ordered outcomes, false zeros, and other nonstandard outcomes

A major recent extension develops a Difference-in-Differences model for discrete, ordered outcomes, building upon elements from a continuous Changes-in-Changes model, with a focus on outcomes derived from self-reported survey data eliciting socially undesirable, illegal, or stigmatized behaviors like tax evasion or substance abuse, where too many “false zeros”, or more broadly, underreporting are likely (Gutknecht et al., 2024). The ordered-outcome framework introduces a latent continuous variable observed only through ordered categories $U\perp T\mid G$7: $U\perp T\mid G$8 For the three-category case $U\perp T\mid G$9, one may normalize $h(\cdot,t)$0, giving categories $h(\cdot,t)$1, $h(\cdot,t)$2, and $h(\cdot,t)$3 according to whether the latent variable lies below $h(\cdot,t)$4, in $h(\cdot,t)$5, or above $h(\cdot,t)$6 (Gutknecht et al., 2024).

Within this threshold-crossing model, the CiC assumption becomes a conditional quantile-mapping restriction on latent untreated outcomes: $h(\cdot,t)$7 Under smoothness and invertibility, this is equivalent to restrictions on the location-scale parameters $h(\cdot,t)$8 and $h(\cdot,t)$9 (Gutknecht et al., 2024).

When reported outcomes are contaminated by underreporting, two identification strategies are described. One is partial identification via nonparametric bounds under one-sided misreporting and an upper bound $I=G\cdot T$0 on underreporting probability. The other is point identification via a semiparametric consumption–reporting model in which

$I=G\cdot T$1

with a latent reporting bound $I=G\cdot T$2 and parametric latent indexes

$I=G\cdot T$3

where $I=G\cdot T$4 are jointly real-analytic copula-distributed, independent of $I=G\cdot T$5, and the CiC condition is imposed on each margin separately (Gutknecht et al., 2024).

Another nonstandard-outcome direction concerns extreme tails. Existing changes-in-changes estimators are tailored to middle quantiles and do not work well for subpopulations with extreme outcomes, such as infants with extremely low birth weights (Sasaki et al., 2022). “Extreme Changes in Changes” proposes a new CIC estimator for extreme quantiles by combining the usual CiC structure with extreme-value tail extrapolation under regular variation. The paper recommends use of the extreme CIC estimator for extreme, such as below $I=G\cdot T$6 and above $I=G\cdot T$7, quantiles, while the conventional CIC estimator should be used for intermediate quantiles (Sasaki et al., 2022).

4. Major extensions of the framework

CiC has been extended well beyond the canonical binary-treatment, two-group design. One line of work generalizes the model to causal mediation with a binary mediator. Under strict monotonicity,

$I=G\cdot T$8

no anticipation, distributional invariance $I=G\cdot T$9, and common support, within-cell quantile–quantile transforms

$Y = Y_N(1-I) + Y_I \cdot I.$0

identify average and quantile direct and indirect effects for various subgroups (Huber et al., 2019). With random assignment and mediator monotonicity, the paper further identifies direct and indirect effects on principal strata such as never-takers, always-takers, and compliers (Huber et al., 2019).

A second line extends CiC to targeted policies through a triple-difference analogue. In the triple-changes estimator, there are two states $Y = Y_N(1-I) + Y_I \cdot I.$1, two eligibility groups $Y = Y_N(1-I) + Y_I \cdot I.$2, and two times $Y = Y_N(1-I) + Y_I \cdot I.$3. Defining

$Y = Y_N(1-I) + Y_I \cdot I.$4

and the within-state drift

$Y = Y_N(1-I) + Y_I \cdot I.$5

the key new assumption is state-independent drift,

$Y = Y_N(1-I) + Y_I \cdot I.$6

Under this condition, the missing counterfactual distribution for the treated eligible subgroup in the treated state is point-identified as

$Y = Y_N(1-I) + Y_I \cdot I.$7

(Akbari et al., 2024).

A third line generalizes CiC to discrete treatments with more than two categories. Let $Y = Y_N(1-I) + Y_I \cdot I.$8 and

$Y = Y_N(1-I) + Y_I \cdot I.$9

The paper assumes a fully nonparametric rank-invariant representation for each treatment arm,

$U\mid G=1$0

and distinguishes weak rank stability, which pertains to untreated potential outcomes within each group, from strong rank stability, which requires invariance of the rank structure of each potential outcome over time even across groups (Boussim, 2024). Under strong rank stability, the counterfactual distribution for arm $U\mid G=1$1 in group $U\mid G=1$2 is

$U\mid G=1$3

from which quantile treatment effects, attended quantile effects, ATEs, and ATTs are recovered (Boussim, 2024).

A fourth line introduces group-heterogeneous CiC. In that setting, untreated potential outcomes are

$U\mid G=1$4

where $U\mid G=1$5 is an individual-level unobservable and $U\mid G=1$6 is a group-level unobservable, with $U\mid G=1$7 strictly increasing in both arguments (Chen et al., 2023). Identification proceeds by matching both levels of latent heterogeneity through a two-stage quantile system over many control groups. There exists a pair $U\mid G=1$8 satisfying

$U\mid G=1$9

and then

$U\mid G=0$0

(Chen et al., 2023). This suggests a distributional matching interpretation in which control subgroups are selected to align both within-group and across-group quantiles with the treated groups’ pre-treatment position.

5. Sample selection, attrition, and semiparametric efficiency

A substantial literature studies cases in which outcomes are not always observed. One contribution shows that sample selection arises endogenously when treatment affects whether certain units are observed, and that the conventional ATT estimand may not be well defined, while the DiD estimand cannot be interpreted causally without additional assumptions (Viviens, 12 Feb 2025). Using principal stratification, it targets treatment effects for the Always-Observed subgroup: $U\mid G=0$1 Combining CiC counterfactual quantile identification with Lee-style trimming yields sharp lower and upper bounds for $U\mid G=0$2 and, by integration, for $U\mid G=0$3 (Viviens, 12 Feb 2025). The paper also develops a CiC selection model to identify the trimming proportions $U\mid G=0$4 under selection monotonicity and a latent-variable structure for selection (Viviens, 12 Feb 2025).

A related paper corrects attrition bias using Changes-in-Changes in two-period panels where baseline outcomes are always observed and follow-up outcomes are observed only if $U\mid G=0$5 (Ghanem et al., 2022). The model is

$U\mid G=0$6

with time-invariance of $U\mid G=0$7 within $U\mid G=0$8 cells and strict monotonicity of the structural functions. Under these assumptions there exists a strictly increasing untreated map

$U\mid G=0$9

such that

$y$0

and similarly a treated map

$y$1

(Ghanem et al., 2022). These transformations identify ATT-R, ATE-R, and, under random assignment, ATE for the entire study population. The paper emphasizes that CiC requires no exclusion or “missing-at-random” restriction on response; instead it imposes structural restrictions on the outcome model, while $y$2 may depend arbitrarily on $y$3 and other unobservables $y$4 provided the joint law of $y$5 is time-invariant (Ghanem et al., 2022).

A further development addresses semiparametric efficiency and inference with high-dimensional covariates and unmeasured confounding. “Debiased Semiparametric Efficient Changes-in-Changes Estimation” introduces a novel extension of CiC that permits high-dimensional unmeasured confounders and non-monotonic relationships between confounders and outcomes, and constructs efficient estimators that are Neyman orthogonal to infinite-dimensional nuisance parameters (Sun et al., 9 Jul 2025). The efficient influence function for the ATT is

$y$6

with $y$7, and it satisfies both $y$8 and the Neyman-orthogonality condition $y$9 (Sun et al., 9 Jul 2025). Estimation uses $U$00-fold cross-fitting and arbitrary machine learning methods for nuisance functions, with $U$01-consistency and asymptotic normality under $U$02 convergence rates $U$03 and standard complexity bounds (Sun et al., 9 Jul 2025).

6. Estimation, inference, applications, and limitations

Most CiC estimators are nonparametric plug-in procedures. In the basic multi-arm formulation, estimation proceeds by empirical CDFs $U$04, empirical quantile functions $U$05, construction of counterfactual distributions by composition, inversion to obtain counterfactual quantiles, and optional rearrangement to enforce monotonicity and prevent crossing quantile curves (Boussim, 2024). In the triple-changes estimator, empirical maps $U$06 are composed exactly as in the identification formula, and nonparametric bootstrap within cells is used for confidence intervals (Akbari et al., 2024). In the extreme-tail setting, Hill estimators and tail extrapolation are plugged into the CiC identifying formula, yielding asymptotic normality and either plug-in or bootstrap confidence intervals (Sasaki et al., 2022).

Applications in the literature span several substantive domains. For ordered discrete outcomes with false zeros, recreational marijuana legalization for adults in several U.S. states is studied using “Monitoring the Future” repeated cross-sections of U.S. 8th-graders, 2015–2018, with past-30-day marijuana occasions coded $U$07 none, $U$08 1–2 times, $U$09 3+ times (Gutknecht et al., 2024). Adding student- and state-level covariates uncovers $U$10, $U$11, and $U$12, all statistically significant at $U$13; accounting for underreporting via the semiparametric CiC model further amplifies the estimated effects by roughly $U$14 at each level, while there is no statistically significant treatment effect on reporting intentions (Gutknecht et al., 2024). In the extreme-quantile application, the 1993 EITC reform is associated with strictly positive and significant effects across low birth-weight quantiles $U$15, including the most extreme quantiles (Sasaki et al., 2022). In the mediation application, the JOBS II programme yields $U$16 with $U$17, $U$18 with $U$19, $U$20 with $U$21, and $U$22 with $U$23 (Huber et al., 2019). In the triple-changes application to Medicaid expansion, classical triple-difference is approximately $U$24 preventive-care visits with $U$25 CI $U$26, while triple-changes is approximately $U$27 with $U$28 CI $U$29 (Akbari et al., 2024). In the selection application to a Colombian job-training program, estimated $U$30 and $U$31, DiD-Lee bounds are $U$32, and CiC-Lee bounds are $U$33; the naïve complete-case CiC point estimate $U$34 has $U$35 CI $U$36, but the selection-corrected bounds include zero (Viviens, 12 Feb 2025). In an early policy application to COVID-19 deaths, CiC estimates are heterogeneous across countries, with Germany and the United States showing negative effects relative to Sweden, while several other countries exhibit positive estimated effects (Kapoor et al., 2020).

A recurrent misconception is that CiC is merely a nonlinear DiD. The papers instead present it as a model based on monotone latent-rank evolution, quantile mapping, and support conditions, with DiD sometimes appearing only as a special or limiting case (Sasaki et al., 2022, Viviens, 12 Feb 2025). Another misconception is that CiC automatically weakens assumptions relative to DiD. Several extensions explicitly require stronger distributional assumptions, support overlap, strict monotonicity, or parametric structure when confronting discrete outcomes, underreporting, or sample selection (Gutknecht et al., 2024, Akbari et al., 2024, Viviens, 12 Feb 2025). The literature therefore treats CiC not as assumption-free, but as a framework that trades mean-trend restrictions for structural restrictions on the evolution of outcome distributions.