Sequential Synthetic Difference-in-Differences (Sequential SDiD) ================================================================ .. currentmodule:: mlsynth Overview -------- Sequential Synthetic Difference-in-Differences (Sequential SDiD, `arXiv:2404.00164v2 `_, Arkhangelsky & Samkov, 2025) is an event-study estimator for staggered-adoption designs that remains robust when the parallel-trends assumption fails. It adapts the canonical SDiD of Arkhangelsky et al. (2021) by operating on *cohort-level aggregates* and *sequentially* imputing treated cells with their estimated counterfactuals so that bias from early cohorts does not cascade into later ones. The estimator is asymptotically equivalent to an infeasible oracle OLS regression that knows the unobserved interactive fixed effects (Proposition 3.1 of the paper), giving it the first formal efficiency guarantees for an SC-type method. Five structural differences distinguish Sequential SDiD from the canonical SDID estimator already in :mod:`mlsynth`: * It works on aggregated cohort outcomes :math:`y_{a, t} \coloneqq n_a^{-1} \sum_{i:\,A_i = a} y_{i, t}` rather than unit-level data, with cohort shares :math:`\pi_a \coloneqq n_a / n` carrying the unit-count information. * Weights satisfy only the simplex sum constraint :math:`\mathbf{1}^\top\mathbf{w} = 1` — non-negativity is dropped. * The unit-weight penalty is the population-share-scaled :math:`\eta^{2} \sum_j w_j^2 / \pi_j`; the time-weight penalty is :math:`\eta^{2} \sum_l \lambda_l^2`. * Donors for cohort :math:`a` are restricted to later-adopting cohorts :math:`j > a` (including the never-treated cohort), not the universe of controls. * Cohort-by-horizon effects are estimated in a sequential cascade: each :math:`\widehat{\tau}_{a, k}^{\,\mathrm{SSDiD}}` is computed, then the treated cell :math:`y_{a, a + k}` is overwritten with :math:`y_{a, a + k} - \widehat{\tau}_{a, k}^{\,\mathrm{SSDiD}}` so subsequent ``(a', k')`` steps see an imputed panel free of treatment contamination. When to Use This Estimator -------------------------- Event studies with staggered adoption -- units switching on at different dates -- are the workhorse design of applied micro. The modern estimators built for them (Callaway & Sant'Anna 2020, de Chaisemartin & d'Haultfœuille 2020, Sun & Abraham 2020, Borusyak et al. 2024) fixed the *heterogeneous-effects* bug in two-way fixed effects, but they still rest on parallel trends: absent treatment, treated and comparison cohorts would have moved together. When unobserved interactive fixed effects (latent unit factors loading on latent time factors) drive selection into treatment, parallel trends fails and those estimators are biased -- exactly the regime Sequential SDiD (Arkhangelsky & Samkov 2025) targets. Its bet is to *model* the confounder. Within a linear interactive-fixed- effects model, Sequential SDiD is proven asymptotically equivalent to the infeasible oracle OLS that knows the latent factors (Prop. 3.1). That equivalence buys three things at once: robustness to parallel-trends violations of the IFE type, asymptotic normality with standard inference, and the first formal efficiency guarantee for an SC-type estimator. The sequential imputation -- estimate the earliest cohort, subtract its effect, reuse the cleaned panel for later cohorts -- gives a *cohesive* analysis of the whole event study, in contrast to per-cohort SC methods (:doc:`ppscm`, Cattaneo et al. 2021) that treat each adoption cohort as a separate problem. Reach for Sequential SDiD when ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ * You have a staggered-adoption event study and you suspect parallel trends fails because of unobserved confounders that look like interactive fixed effects (selection on latent trends, not just levels). * Adoption cohorts are reasonably large. The method averages outcomes within each cohort and leans on a law of large numbers to kill idiosyncratic noise; this is the estimator's core requirement and its main limitation. * You want valid, standard inference (asymptotic normality) and a defensible efficiency claim, plus a dynamic event-study path of cohort-by-horizon effects. * You are in the fixed-:math:`T`, large-:math:`N` regime typical of county/firm/individual panels aggregated into a handful of adoption cohorts. Do not use Sequential SDiD when ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ * Cohorts are small (few units per adoption date, or single-unit cohorts). The within-cohort averaging has nothing to average and the oracle equivalence does not engage. Use :doc:`ppscm`, whose partial pooling is designed for noisy per-unit cohort fits, or canonical :doc:`sdid` per cohort. * There is a single treated unit or one common adoption time. The sequential cascade has nothing to cascade; use canonical :doc:`sdid` (many units, one time) or classic SC (:doc:`tssc`, :doc:`fdid`, :doc:`scmo`) for a single unit. * Parallel trends is credible and you want the simplest transparent thing. Standard staggered-DiD (Callaway-Sant'Anna and relatives) is fine; on the Bailey-Goodman-Bacon application where PT holds, Sequential SDiD merely reproduces standard DiD. * SUTVA is violated by spillovers onto the comparison cohorts -- use :doc:`spsydid` or :doc:`spillsynth`. * Distributional questions (quantiles, tails) -- use :doc:`dsc`. Mathematical Formulation ------------------------ Notation ^^^^^^^^ Let :math:`\mathcal{N} \coloneqq \{1, \dots, N\}` index the underlying units, with generic unit :math:`i \in \mathcal{N}`, and let :math:`t \in \mathcal{T} \coloneqq \{1, \dots, T\}` index the (1-indexed) periods. Each unit carries a (possibly infinite) adoption period :math:`A_i`, the period after which it is treated; the never-treated cohort uses :math:`A_i = +\infty`. Units are grouped into cohorts indexed by adoption period :math:`a`, with :math:`n_a` units in cohort :math:`a`, total :math:`n = \sum_a n_a`, and cohort share :math:`\pi_a \coloneqq n_a / n`. The unit-level outcome is :math:`y_{i, t}`; the cohort-level aggregate .. math:: y_{a, t} \;\coloneqq\; \frac{1}{n_a} \sum_{i:\, A_i = a} y_{i, t} plays the role of a series in :math:`\mathbb{R}^{T}`, and the later-adopting cohorts of a treated cohort :math:`a` (those with adoption period :math:`j > a`, including the never-treated cohort) form its donor pool, stacked into the donor matrix :math:`\mathbf{Y}_0`. Unit (donor) weights are the vector :math:`\mathbf{w}` with entries :math:`w_j`, optimiser :math:`\mathbf{w}^\ast`; time weights are the vector :math:`\boldsymbol{\lambda}` with entries :math:`\lambda_l`. The per-(cohort, horizon) effect is :math:`\tau_{a, k}` and the pooled horizon effect is :math:`\tau_k`; the treatment effect is reserved to :math:`\tau` throughout, so the regularization strength is :math:`\eta \ge 0` (never :math:`\tau`) and the significance level is the bare :math:`\alpha` (matching :py:attr:`SequentialSDIDConfig.alpha`), distinct from the subscripted cohort fixed effect :math:`\alpha_a` of the model in *Setup*. Setup ^^^^^ For each cohort the implementation computes the aggregate :math:`y_{a, t}` above with share :math:`\pi_a`. Under Assumption 2.2 of the paper, the aggregate outcomes inherit the interactive-fixed-effects structure .. math:: y_{a, t} \;=\; \alpha_a + \beta_t + \theta_a^\top \psi_t + \mathbf{1}\{a \le t\}\,\tau_{a, t - a} + \epsilon_{a, t}, where :math:`\theta_a^\top \psi_t` captures unobserved confounders that break parallel trends. Aggregation drives :math:`\epsilon_{a, t}` to zero under the paper's "large cohort" asymptotics, leaving the IFE structure identifiable. Identifying assumptions ^^^^^^^^^^^^^^^^^^^^^^^^ 1. Interactive fixed effects (Assumption 2.2). The no-intervention outcomes follow the additive-plus-factor structure above: cohort effect :math:`\alpha_a`, time effect :math:`\beta_t`, and a low-rank interactive term :math:`\theta_a^\top \psi_t` of latent cohort loadings on latent time factors, the component that breaks parallel trends. *Remark.* It is precisely the :math:`\theta_a^\top \psi_t` term that defeats standard staggered-DiD: when adoption correlates with the loading :math:`\theta_a`, comparison cohorts do not trace the treated cohort's counterfactual trend. Sequential SDiD models this term rather than assuming it away, which is what buys robustness when parallel trends fails. 2. Large adoption cohorts. Cohort sizes :math:`n_a` grow with the sample so that aggregation drives the idiosyncratic noise :math:`\epsilon_{a, t}` to zero, leaving the IFE structure identifiable from the cohort aggregates. *Remark.* This is the estimator's core requirement and its main limitation: the within-cohort average must have enough units to average. With single-unit or tiny cohorts the law of large numbers does not engage and the oracle equivalence (Prop. 3.1) does not hold — use :doc:`ppscm` or canonical :doc:`sdid` instead. 3. Donor balance. Each treated cohort :math:`a` is balanced only against the cohorts adopting after it (:math:`j > a`) plus any never-treated cohort, so it needs at least two such donor cohorts to balance even a rank-one interactive fixed effect — matching a one-dimensional loading and the intercept spans :math:`(1, \lambda)`. *Remark.* With a single donor the sum-to-one constraint forces :math:`\mathbf{w} = [1]` and the cohort effect collapses to an unbalanced DiD, biased whenever the loadings differ; that bias also cascades backward through the sequential imputation. The latest cohorts are the most exposed, so :meth:`SequentialSDID.fit` warns when a cohort is donor-starved (see *Limitations*). 4. No interference across cohorts (SUTVA). Treating one cohort does not change the outcomes of the comparison cohorts that serve as its donors, so the donor aggregates carry only no-intervention outcomes. *Remark.* Spillovers onto the comparison cohorts contaminate the counterfactual; when they are present use :doc:`spsydid` or :doc:`spillsynth` rather than Sequential SDiD. Algorithm 1 ^^^^^^^^^^^ For each horizon :math:`k = 0, 1, \dots, K` (outer loop) and each treated cohort :math:`a = a_{\min}, \dots, a_{\max}` (inner loop), Sequential SDiD runs three steps. Step 1: Solve two regularized QPs. Both QPs are equality-constrained convex quadratic programs (no non-negativity). The unit-weight QP is .. math:: \widehat{\mathbf{w}}^{(a, k)}, \widehat{w}_0 \;=\; \operatorname*{argmin}_{\sum_{j > a} w_j = 1} \quad \sum_{l < a + k} \left( w_0 + \sum_{j > a} w_j \, y_{j, l} - y_{a, l} \right)^{\!2} \;+\; \eta^2 \sum_{j > a} \frac{w_j^2}{\pi_j}, and the time-weight QP is .. math:: \widehat{\boldsymbol{\lambda}}^{(a, k)}, \widehat{\lambda}_0 \;=\; \operatorname*{argmin}_{\sum_{l < a + k} \lambda_l = 1} \quad \sum_{j > a} \left( \lambda_0 + \sum_{l < a + k} \lambda_l \, y_{j, l} - y_{j, a + k} \right)^{\!2} \;+\; \eta^2 \sum_{l < a + k} \lambda_l^{\,2}. Both are solved in closed form via their KKT linear systems in :mod:`mlsynth.utils.seq_sdid_helpers.weights`. Step 2: Weighted double-difference. .. math:: \widehat{\tau}_{a, k}^{\,\mathrm{SSDiD}} \;=\; \left( y_{a, a + k} - \sum_{j > a} \widehat{w}_j^{(a, k)} \, y_{j, a + k} \right) \;-\; \sum_{l < a + k} \widehat{\lambda}_l^{(a, k)} \left( y_{a, l} - \sum_{j > a} \widehat{w}_j^{(a, k)} \, y_{j, l} \right). This is the same SDID-style contrast as the canonical estimator, just evaluated on cohort-level aggregates. Step 3: Sequential imputation. .. math:: y_{a, a + k} \;\coloneqq\; y_{a, a + k} \;-\; \widehat{\tau}_{a, k}^{\,\mathrm{SSDiD}}. The treated cell is replaced with its estimated counterfactual *in place* in the panel matrix. When the outer loop advances to a longer horizon or the inner loop advances to a later cohort, subsequent QPs use this imputed panel — which is the mechanism that prevents bias from cascading through the estimator. Pooled Event Study ^^^^^^^^^^^^^^^^^^ Cohort-specific effects are aggregated into a single event-study trajectory via Equation 2.5 of the paper: .. math:: \widehat{\tau}_k^{\,\mathrm{SSDiD}}(\mu) \;=\; \sum_{a \in [a_{\min}, a_{\max}]} \mu_a \, \widehat{\tau}_{a, k}^{\,\mathrm{SSDiD}}, \qquad \mu_a \coloneqq \frac{\pi_a}{\sum_{a' \in [a_{\min}, a_{\max}]} \pi_{a'}}. The default ``mu`` is proportional to cohort shares (i.e. larger cohorts get more weight), recovering the unit-uniform interpretation common in the DiD literature. The result lives on :py:attr:`SeqSDIDEventStudy.tau`. The :math:`\eta \to \infty` Limit ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Remark 2.2 of the paper notes that as :math:`\eta \to \infty`, the unit- weight QP's penalty :math:`\sum w_j^2 / \pi_j` forces :math:`w_j \propto \pi_j` (each unit in the donor pool gets equal weight), and the time-weight QP's penalty :math:`\sum \lambda_l^{\,2}` forces :math:`\lambda_l = 1 / (a + k - 1)` (uniform). The resulting estimator is a sequential DiD imputation estimator closely related to Borusyak, Jaravel, and Spiess (2024). The :mod:`mlsynth` implementation exposes this mode via :attr:`SequentialSDIDConfig.mode`: * ``mode = "ssdid"``: paper's main estimator with finite ``eta`` (default). * ``mode = "sdid_imputation"``: forces the :math:`\eta \to \infty` limit internally and returns the sequential-DiD-style result. Inference (Section 2.3) ^^^^^^^^^^^^^^^^^^^^^^^ Inference uses the Bayesian bootstrap of Rubin (1981) and Chamberlain & Imbens (2003). At each bootstrap iteration: 1. Draw independent weights :math:`\xi_i \sim \mathrm{Exp}(1)` for every underlying unit (not cohort). 2. Reconstruct cohort-level outcomes as weighted means: .. math:: y_{a, t}(\xi) \;=\; \frac{\sum_{i:\, A_i = a} y_{i, t} \, \xi_i}{\sum_{i:\, A_i = a} \xi_i}. 3. Re-run Algorithm 1 on the perturbed panel. 4. Record the pooled event-study vector. Standard errors are sample standard deviations of the bootstrap replicates, and confidence intervals are Wald-type at :py:attr:`SequentialSDIDConfig.alpha`. The full replicate matrix is retained on :py:attr:`SeqSDIDEventStudy.bootstrap_draws` in case quantile-based intervals are preferred downstream. Limitations ^^^^^^^^^^^ The paper's formal guarantees require *large adoption cohorts* — cohort sizes that grow with the sample so the aggregation kills the idiosyncratic noise. The algorithm still runs on single-treated-unit panels (e.g., the Proposition 99 dataset), but with only one treated cohort and one never-treated cohort the time-weight QP becomes effectively underdetermined; the practical recommendation is to use canonical :class:`SDID` for those panels and reserve Sequential SDiD for genuine staggered designs with multiple sizable cohorts. A second requirement is *donor balance*. Each treated cohort :math:`a` is balanced only against the cohorts adopting after it (:math:`j > a`) plus any never-treated cohort, so it needs at least two such donor cohorts to balance even a rank-one interactive fixed effect — matching a one-dimensional loading and the intercept spans :math:`(1, \lambda)`. With a single donor the sum-to-one constraint forces :math:`\mathbf{w} = [1]` and the cohort effect collapses to an unbalanced DiD, biased whenever the loadings differ; that bias also cascades backward through the sequential imputation. On a *noiseless* rank-one factor the estimator recovers the effect exactly for every cohort that is balanced, and the entire residual bias is attributable to the donor-starved late cohorts. Because the latest cohorts are the most exposed (the default ``a_max`` is the last adopting cohort), :meth:`SequentialSDID.fit` emits a :class:`UserWarning` naming any donor-starved cohort and the largest ``a_max`` that keeps every estimated cohort balanced — report, don't silently relax. Verification ------------ Path B — the paper's Section-5.2.2 calibrated-panel Monte Carlo (Table 1) is reconstructed from its description (the authors' CPS log-wage panel is not public). Under an interactive-fixed-effects violation of parallel trends with adoption correlated to the leading loading, standard DiD's 95% CI coverage collapses (:math:`\approx 0.45`; paper :math:`\approx 0.70`) while Sequential SDiD stays near nominal (:math:`0.945`) with roughly five-times-smaller bias and lower RMSE — the paper's "DiD severely biased, Sequential SDiD reliable" result. A noiseless rank-one IFE corollary pins exact machine-precision recovery for every donor-balanced cohort. The durable check is ``benchmarks/cases/seq_sdid_mc.py``; see the dedicated replication page, :doc:`replications/seq_sdid`, for the design, code, and the full table. Core API -------- .. automodule:: mlsynth.estimators.seq_sdid :members: :undoc-members: :show-inheritance: Configuration ------------- .. autoclass:: mlsynth.config_models.SequentialSDIDConfig :members: :undoc-members: Helper Modules -------------- .. automodule:: mlsynth.utils.seq_sdid_helpers.setup :members: :undoc-members: .. automodule:: mlsynth.utils.seq_sdid_helpers.weights :members: :undoc-members: .. automodule:: mlsynth.utils.seq_sdid_helpers.algorithm :members: :undoc-members: .. automodule:: mlsynth.utils.seq_sdid_helpers.inference :members: :undoc-members: .. automodule:: mlsynth.utils.seq_sdid_helpers.plotter :members: :undoc-members: .. note:: ``SequentialSDID.fit()`` returns an :class:`~mlsynth.config_models.EffectResult` on the standardized two-family contract. Sequential SDiD is an event-study estimator, so its standardized ``time_series`` is laid out over *event-time horizons* (``res.gap`` is the pooled horizon effect ``tau_hat_k``) and ``res.att`` is the simple average of those pooled effects. The per-(cohort, horizon) effects, the pooled event study, and the bootstrap config stay on ``res.cohort_effects`` / ``res.event_study`` / ``res.inference_detail``. .. automodule:: mlsynth.utils.seq_sdid_helpers.structures :members: :undoc-members: Example ------- .. code-block:: python import pandas as pd from mlsynth import SequentialSDID df = pd.read_csv("staggered_panel.csv") # state-level panel with treat in {0, 1} results = SequentialSDID({ "df": df, "outcome": "log_wage", "treat": "treated", "unitid": "state", "time": "year", "eta": 1.0, "K": 10, "n_bootstrap": 500, "alpha": 0.05, "display_graphs": True, }).fit() # Pooled event-study trajectory (Eq. 2.5 of the paper). for k, tau, se in zip(results.event_study.horizons, results.event_study.tau, results.event_study.se): print(f"k = {k:>2} tau = {tau:+.3f} se = {se:.3f}") # Cohort-by-horizon decomposition. for (a, k), effect in results.cohort_effects.items(): print(f"cohort a = {a} horizon k = {k} tau = {effect.tau:+.3f}") # Bayesian-bootstrap replicate matrix (B x (K + 1)) — for quantile CIs # or downstream diagnostics. results.event_study.bootstrap_draws.shape References ---------- Arkhangelsky, D., & Samkov, A. (2025). "Sequential Synthetic Difference in Differences." `arXiv:2404.00164v2 `_. Arkhangelsky, D., Athey, S., Hirshberg, D. A., Imbens, G. W., & Wager, S. (2021). "Synthetic Difference-in-Differences." *American Economic Review* 111(12): 4088-4118. Borusyak, K., Jaravel, X., & Spiess, J. (2024). "Revisiting Event-Study Designs: Robust and Efficient Estimation." *Review of Economic Studies.*