Synthetic Difference-in-Differences (SDID) ========================================== .. currentmodule:: mlsynth When to Use This Estimator -------------------------- Difference-in-differences (DiD) and synthetic control (SC) are usually pitched as tools for *different* problems. DiD is used when many units are treated and you are willing to assume **parallel trends** -- that treated and control outcomes would have moved in lockstep absent treatment, after removing additive unit and time fixed effects. SC is used when *one* (or a few) units are treated and parallel trends plainly fails, so you instead **re-weight the donors** to match the treated unit's pre-treatment path. Synthetic Difference-in-Differences (SDID), due to Arkhangelsky, Athey, Hirshberg, Imbens and Wager (2021, *AER*) [aersdid]_, argues these two strategies rest on closely related assumptions and combines the best of both. It fits a two-way fixed-effects regression that is **doubly weighted** -- by SC-style **unit weights** :math:`\omega_i` *and* DiD-style **time weights** :math:`\lambda_t`: .. math:: (\hat\tau, \hat\mu, \hat\alpha, \hat\beta) = \arg\min_{\tau, \mu, \alpha, \beta} \sum_{i=1}^{N}\sum_{t=1}^{T} \bigl(Y_{it} - \mu - \alpha_i - \beta_t - W_{it}\tau\bigr)^2\, \hat\omega_i\, \hat\lambda_t . The weights make the regression **local**: it leans on control units whose *past* resembles the treated unit's, and on pre-periods that resemble the post-period. Reach for SDID when: * **DiD is tempting but pre-trends are not parallel.** SDID re-weights controls so their trend becomes *parallel* (not identical -- the unit fixed effects absorb level gaps) to the treated unit, then runs DiD on the re-weighted panel. It "automates" the usual practice of hunting for comparable units/periods to make parallel trends plausible, *with* statistical guarantees -- addressing the pre-testing concerns of Roth. * **SC is tempting but the pre-fit is imperfect or you want valid inference.** Adding unit fixed effects (and an intercept in the weight problem) means the donors only need to be *parallel* to the treated unit, not match it exactly, and the design admits large-panel inference. * **You want robustness without choosing.** Where DiD has been used, SDID is competitive with or better than DiD; where SC has been used, it is competitive with or better than SC. The weighting also often *improves precision* by removing predictable structure -- in the Prop 99 study, SDID's standard error (8.4) is smaller than DiD's (17.7) despite being the more flexible estimator. .. note:: The localization is not a free lunch: if outcomes have little systematic heterogeneity across units or periods, unequal weighting can *worsen* precision relative to plain DiD. SDID helps most when there is real structure (trends, levels) for the weights to exploit. Do not use SDID when ^^^^^^^^^^^^^^^^^^^^^ * **Spillovers / interference contaminate the donor pool.** SDID assumes the controls are untreated and unaffected by the treatment (SUTVA). If treatment leaks to neighbours -- cross-border shopping, migration, geographic advertising -- the weighted controls are biased. Use :doc:`spsydid`, which separates the direct ATT from the spillover term. * **Staggered adoption where you want partial pooling or an interactive fixed-effects guarantee.** SDID runs per cohort and averages, which is fine for an overall ATT, but it does not *pool* information across cohorts the way :doc:`ppscm` does, nor does it give the oracle-OLS efficiency of :doc:`seq_sdid`. Prefer those when cohorts are many and individual cohort fits are noisy. * **The treated unit sits far outside the donor convex hull / the donor pool is huge and noisy.** SDID's unit weights are non-negative and (softly) sum-constrained; a treated path no linear convex combination can parallel will fit poorly. A factor-model estimator (:doc:`fma`) or a low-rank/denoising approach (:doc:`clustersc`, :doc:`mcnnm`) is better suited there. * **A single treated unit, short panel, and you want the interpretable sparse convex-weight story** as the deliverable. Classic SC and its refinements (:doc:`tssc`, :doc:`fdid`, :doc:`scmo`) are more transparent; SDID's double weighting buys little when there is only one treated unit and no time structure for the time weights to exploit. * **Distributional questions** (quantile effects, Lorenz, tails). SDID targets the mean ATT; use :doc:`dsc`. What SDID Does in Practice -------------------------- Beyond the econometrics: SDID answers "what would the treated unit have done?" by building a synthetic comparison that is **parallel** to it, not a clone, and by trusting the *recent, relevant* past more than the distant past. * **Policy / geo evaluation.** A state raises cigarette taxes (Prop 99); a city introduces congestion pricing; a country reunifies. You have a long panel of comparison regions whose levels differ wildly and whose pre-trends are not parallel. SDID re-weights the comparison regions to parallel the treated one and downweights ancient history that no longer looks like the policy window. * **Marketing / pricing roll-outs.** A pricing change launches in some markets. Plain DiD over all markets is biased if the treated markets were on a different trajectory; pure SC ignores that fixed level differences are harmless. SDID handles both, and -- via time weights -- discounts pre-launch months that don't resemble the post-launch regime (seasonal shifts, a pre-launch promo). * **Staggered roll-outs.** When units adopt at different dates, SDID runs per cohort and aggregates (Clarke et al., 2023), yielding both an overall ATT and a dynamic **event-study** path (Ciccia, 2024). Notation -------- Let :math:`Y_{it}` be the outcome of unit :math:`i` in period :math:`t`, with :math:`i \in \{1, \dots, N\}` and :math:`t \in \{1, \dots, T\}`, and let :math:`W_{it} \in \{0, 1\}` be the treatment indicator. The first :math:`N_{co}` units are never-treated **controls** (donors); the remaining :math:`N_{tr} = N - N_{co}` are **treated**, exposed after their adoption period. :math:`T_{pre}` and :math:`T_{post}` count pre- and post-treatment periods. The unit weights :math:`\omega_i` are supported on the controls and the time weights :math:`\lambda_t` on the pre-period; :math:`\zeta` is the unit-weight regularization parameter. The estimand is the average treatment effect on the treated, :math:`\tau` (denoted :math:`\widehat{ATT}` in aggregate). .. admonition:: Notation bridge The mlsynth implementation generalizes the single-treated block design to **cohorts**: cohort :math:`a` is the set :math:`I^a` of units first treated in period :math:`a`, with size :math:`N_{tr}^a` and :math:`T_{tr}^a = T - a + 1` post-periods. The classical single-treated case (California) is the one-cohort special case, where the cohort ATT and the overall ATT coincide. Assumptions ----------- SDID's formal guarantees are developed under an **interactive fixed-effects (latent factor) model** for the control potential outcome, .. math:: Y_{it} = \boldsymbol{\gamma}_i^\top \boldsymbol{v}_t + \tau W_{it} + \varepsilon_{it}, where :math:`\boldsymbol{\gamma}_i` are latent unit factors and :math:`\boldsymbol{v}_t` latent time factors (a generalization of additive :math:`\alpha_i + \beta_t` two-way fixed effects). **Assumption 1 (latent factor outcome model).** The systematic part of the outcome is :math:`\boldsymbol{\gamma}_i^\top \boldsymbol{v}_t`; deviations :math:`\varepsilon_{it}` are mean-zero given the systematic component and the treatment assignment. *Remark.* This is strictly more general than DiD's additive :math:`\alpha_i + \beta_t`. When the factor structure *is* additive, plain DiD is already consistent; SDID is designed to also handle the interactive case, where DiD is biased. **Assumption 2 (selection on the systematic part only).** Treatment assignment :math:`W` may depend on the latent factors :math:`\boldsymbol{\gamma}_i, \boldsymbol{v}_t` (units are *not* randomized) but **not** on the idiosyncratic error :math:`\varepsilon`. *Remark.* This is what lets policies be adopted non-randomly -- California was not a coin flip -- yet still be identified: the confounding must run through the persistent latent structure that the weights and fixed effects soak up, not through transitory shocks. **Assumption 3 (weak cross-unit dependence).** The error vectors :math:`\varepsilon_i` are independent *across units*, though correlation *within a unit over time* is allowed. *Remark.* Serial correlation within a unit is the norm in panel data and is permitted; this is why the time-weight problem is left **unregularized** (it must accommodate within-unit temporal correlation) while the unit-weight problem is regularized. Cross-unit independence is what powers the placebo variance estimator. **Assumption 4 (weighted parallel trends, achieved by construction).** There exist unit weights making the treated trajectory parallel to the weighted control trajectory over the pre-period, and time weights making each control's post-period mean a constant offset from its weighted pre-period mean. *Remark.* Unlike DiD -- which *assumes* parallel trends on the raw data -- SDID *constructs* weights to make parallel trends hold on the re-weighted panel, then proceeds. The graphical "parallel trends" check is thus performed on adjusted data, automatically and with guarantees. Why Unit Weights and Why Time Weights ------------------------------------- **Unit weights** are chosen so the treated unit's pre-treatment path is *parallel* to the weighted-control path. Two differences from classical SC (Abadie et al., 2010) make this work inside a fixed-effects regression: 1. an **intercept** :math:`\omega_0` is allowed, so the weights need only make trends *parallel* rather than coincident -- the unit fixed effects :math:`\alpha_i` absorb any constant level gap; and 2. a **ridge penalty** :math:`\zeta^2 \|\omega\|_2^2` is added (with :math:`\zeta = (N_{tr} T_{post})^{1/4}\hat\sigma`, :math:`\hat\sigma` the SD of first-differenced control outcomes) to disperse and uniquely pin down the weights. **Time weights** are chosen so that, for the control units, the weighted average of pre-treatment outcomes predicts the post-treatment average up to a constant. The argument for them mirrors the argument for unit weights: down-weighting pre-periods that look nothing like the post-period **removes bias** and **improves precision**. This is the data-driven counterpart to event-study practice, which implicitly puts all comparison weight on the last pre-period -- SDID instead lets the data choose which pre-periods are informative. The time-weight problem is left unregularized (Assumption 3). Together, unit *and* time weights plus unit fixed effects make the DiD contrast both more robust (it leans on comparable units and periods) and, typically, more precise (predictable structure is removed), which is why SDID's standard errors can be *smaller* than DiD's despite its added flexibility. Mathematical Formulation ------------------------ Setup ^^^^^ Let :math:`Y_{i, t}` denote the outcome of unit :math:`i` in period :math:`t`, with :math:`i \in \{1, \dots, N\}` and :math:`t \in \{1, \dots, T\}`. There are :math:`N_{co}` never-treated control units, and the treated units are partitioned into cohorts by their adoption period: cohort :math:`a` is the set :math:`I^a \subseteq \{N_{co} + 1, \dots, N\}` of units that first receive treatment in period :math:`a`. Let :math:`A = \{a_1, \dots, a_K\}` denote the set of distinct adoption periods, :math:`N_{tr}^a = |I^a|` the cohort size, and :math:`T_{tr}^a = T - a + 1` the number of post-treatment periods in cohort :math:`a`. Aggregate post-treatment exposure (Clarke et al., 2023) is :math:`T_{post} = \sum_{a \in A} N_{tr}^a T_{tr}^a`. The classical Arkhangelsky et al. (2021) SDID estimator targets a single cohort. The mlsynth implementation runs that estimator *per cohort*, accumulates the cohort-specific effects, and then aggregates them in two complementary ways (Ciccia, 2024). Cohort-Specific SDID (Equation 2) ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ For a single cohort :math:`a`, SDID fits weights :math:`\omega_i` over :math:`N_{co}` donor units and :math:`\lambda_t` over the cohort's pre-treatment window :math:`t < a` by solving two convex programs: .. math:: \omega \;=\; \arg\min_{\sum \omega_i = 1,\ \omega_i \geq 0} \sum_{t = 1}^{a - 1} \left( \bar Y_{I^a, t} - \omega_0 - \sum_{i = 1}^{N_{co}} \omega_i Y_{i, t} \right)^{\!2} + T_0 \zeta^2 \|\omega\|_2^2, .. math:: \lambda \;=\; \arg\min_{\sum \lambda_t = 1,\ \lambda_t \geq 0} \sum_{i = 1}^{N_{co}} \left( \bar Y_{i, [a, T]} - \lambda_0 - \sum_{t = 1}^{a - 1} \lambda_t Y_{i, t} \right)^{\!2}, where :math:`\bar Y_{I^a, t}` is the treated-unit mean at time :math:`t`, :math:`\bar Y_{i, [a, T]}` is donor :math:`i`'s mean over the post-treatment window, and :math:`\zeta` is a regularization parameter scaled by the standard deviation of first-differenced donor outcomes. The cohort-specific SDID estimator is then .. math:: \hat\tau_a^{\,sdid} \;=\; \frac{1}{T_{tr}^a} \sum_{t = a}^{T} \left( \frac{1}{N_{tr}^a} \sum_{i \in I^a} Y_{i, t} - \sum_{i = 1}^{N_{co}} \omega_i Y_{i, t} \right) - \sum_{t = 1}^{a - 1} \lambda_t \left( \frac{1}{N_{tr}^a} \sum_{i \in I^a} Y_{i, t} - \sum_{i = 1}^{N_{co}} \omega_i Y_{i, t} \right). This is Equation 2 of Ciccia (2024). Each cohort is fit independently inside :func:`mlsynth.utils.sdid_helpers.cohort.estimate_cohort_sdid_effects`. Cohort-Specific Event Study (Equation 3) ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ The cohort ATT is the average of a sequence of *dynamic* effects, one per post-treatment offset :math:`\ell \in \{1, \dots, T_{tr}^a\}`: .. math:: \hat\tau_{a, \ell}^{\,sdid} \;=\; \frac{1}{N_{tr}^a} \sum_{i \in I^a} Y_{i, a - 1 + \ell} \;-\; \sum_{i = 1}^{N_{co}} \omega_i Y_{i, a - 1 + \ell} \;-\; \sum_{t = 1}^{a - 1} \lambda_t \left( \frac{1}{N_{tr}^a} \sum_{i \in I^a} Y_{i, t} - \sum_{i = 1}^{N_{co}} \omega_i Y_{i, t} \right). The first two terms are the *post-treatment gap* between the treated cohort and its synthetic control at offset :math:`\ell`; the third term is the time-weighted *pre-treatment baseline*. By construction, .. math:: \hat\tau_a^{\,sdid} \;=\; \frac{1}{T_{tr}^a} \sum_{\ell = 1}^{T_{tr}^a} \hat\tau_{a, \ell}^{\,sdid}, i.e. the cohort ATT is the sample mean of its dynamic effects (Equation 4 of Ciccia 2024). These effects are exposed on the result object as :py:attr:`SDIDCohort.event_effects`. Pooled Event Study (Equation 6) ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Let :math:`A_\ell = \{a \in A : a - 1 + \ell \le T\}` be the set of cohorts for which the :math:`\ell`-th dynamic effect is computable, and :math:`N_{tr}^\ell = \sum_{a \in A_\ell} N_{tr}^a` the corresponding treated-unit count. The pooled event-study estimator is .. math:: \hat\tau_\ell^{\,sdid} \;=\; \sum_{a \in A_\ell} \frac{N_{tr}^a}{N_{tr}^\ell} \hat\tau_{a, \ell}^{\,sdid}, a treated-unit-weighted average of the cohort-specific dynamic effects. This is the central quantity Ciccia (2024) recommends researchers report. In the :mod:`mlsynth` API it is :py:attr:`SDIDEventStudy.tau`, indexed by the corresponding event time on :py:attr:`SDIDEventStudy.event_times`. Overall ATT (Equation 7) ^^^^^^^^^^^^^^^^^^^^^^^^ Define :math:`T_{tr} = \max_{a \in A} T_{tr}^a`, the post-treatment length of the earliest cohort. The overall ATT of Clarke et al. (2023) admits the equivalent disaggregated form .. math:: \widehat{ATT} \;=\; \frac{1}{T_{post}} \sum_{\ell = 1}^{T_{tr}} N_{tr}^\ell \, \hat\tau_\ell^{\,sdid}, i.e. the average of the pooled event-study effects weighted by the number of treated units contributing to each offset. This is :py:attr:`SDIDInference.att`, with a placebo-based standard error and confidence interval at :py:attr:`SDIDInference.se` / :py:attr:`SDIDInference.ci`. Placebo Inference ^^^^^^^^^^^^^^^^^ Variance estimation follows the placebo procedure of Arkhangelsky et al. (2021), generalized to cohort and event-time effects by Clarke et al. (2023). For each of :math:`B` iterations (:py:attr:`SDIDConfig.B`), the donor pool is sampled to replace the true treated units with pseudo-treated controls, the full SDID pipeline is rerun, and the sample variance of the resulting effects is taken as the variance of the actual estimator. The implementation lives in :func:`mlsynth.utils.sdid_helpers.inference.estimate_placebo_variance`. The two-sided placebo p-value reported on :py:attr:`SDIDInference.p_value` uses the canonical :math:`((k + 1) / (B + 1))` correction, where :math:`k` is the count of placebo iterations whose :math:`|\hat\tau^{\,*}_{att}|` is at least as large as the observed :math:`|\widehat{ATT}|`. Two-DataFrame and Single-Cohort Convergence ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ When the panel has a single treated unit (e.g., California in the Proposition 99 study), :func:`mlsynth.utils.datautils.dataprep` returns a single-treated payload rather than a cohorts dict. The :func:`mlsynth.utils.sdid_helpers.setup.prepare_sdid_inputs` helper unifies both shapes into a single ``cohorts_dict`` keyed by adoption period *index* (1-based), which is what the cohort estimator's ``\ell = t - (a - 1)`` math requires. In the single-cohort case, the cohort ATT and the overall ATT are numerically identical by construction. Core API -------- .. automodule:: mlsynth.estimators.sdid :members: :undoc-members: :show-inheritance: Configuration ------------- .. autoclass:: mlsynth.config_models.SDIDConfig :members: :undoc-members: Helper Modules -------------- .. automodule:: mlsynth.utils.sdid_helpers.setup :members: :undoc-members: .. automodule:: mlsynth.utils.sdid_helpers.weights :members: :undoc-members: .. automodule:: mlsynth.utils.sdid_helpers.cohort :members: :undoc-members: .. automodule:: mlsynth.utils.sdid_helpers.event_study :members: :undoc-members: .. automodule:: mlsynth.utils.sdid_helpers.inference :members: :undoc-members: .. automodule:: mlsynth.utils.sdid_helpers.orchestration :members: :undoc-members: .. automodule:: mlsynth.utils.sdid_helpers.plotter :members: :undoc-members: .. automodule:: mlsynth.utils.sdid_helpers.structures :members: :undoc-members: Example ------- .. code-block:: python import pandas as pd from mlsynth import SDID df = pd.read_csv( "https://raw.githubusercontent.com/jgreathouse9/mlsynth/" "refs/heads/main/basedata/smoking_data.csv" ) df["Proposition 99"] = df["Proposition 99"].astype(int) results = SDID({ "df": df, "outcome": "cigsale", "treat": "Proposition 99", "unitid": "state", "time": "year", "B": 500, # placebo iterations "display_graphs": True, }).fit() # Overall ATT (Ciccia 2024 Eq. 7) and placebo inference. print(results.inference.att) # -15.605 (matches Arkhangelsky et al. 2021) print(results.inference.se) print(results.inference.ci) print(results.inference.p_value) # Pooled event-study trajectory (Ciccia 2024 Eq. 6). es = results.event_study for ell, tau, se in zip(es.event_times, es.tau, es.se): print(f"ell={int(ell):>3} tau={tau:+.3f} se={se:.3f}") # Per-cohort decomposition (Ciccia 2024 Eqs. 2 and 3). for adoption_period, cohort in results.cohorts.items(): print(adoption_period, cohort.n_treated, cohort.att) print(cohort.event_effects[1]) # the first-period dynamic effect Replication: Proposition 99 --------------------------- .. note:: **Empirical replication (Path A).** Run on the California smoking panel (39 states, 1970-2000; California treated by Proposition 99 from 1989), ``mlsynth``'s SDID reproduces the headline estimate of [aersdid]_ **to three significant figures**: .. list-table:: :header-rows: 1 :widths: 30 24 24 * - Quantity - mlsynth - Reference * - Overall ATT - **-15.605** - -15.6 (Arkhangelsky et al. 2021, Table 1; ``synthdid`` R: -15.604) * - Placebo SE (B = 500) - 7.58 - 8.4 (placebo SE, Table 1) * - 95% CI - (-30.5, -0.7) - * - Placebo p-value - 0.032 - The point estimate matches the authors' ``synthdid`` package (-15.604) essentially exactly. The placebo standard error is in the same range (7.6 vs. 8.4); it is a resampling estimate and varies with the placebo draw and ``B``. As Arkhangelsky et al. emphasize, SDID's -15.6 sits well below the DiD estimate (-27.3) and below SC (-19.6), and its SE is *smaller* than DiD's (17.7) -- the localization payoff. Per the project's replication contract (``agents/agents_estimators.md``), SDID is considered **done**: the published empirical ATT is reproduced on the same data to machine precision in the point estimate. References ---------- Arkhangelsky, D., Athey, S., Hirshberg, D. A., Imbens, G. W., & Wager, S. (2021). "Synthetic Difference-in-Differences." *American Economic Review* 111(12):4088-4118. Ciccia, D. (2024). "A Short Note on Event-Study Synthetic Difference-in-Differences Estimators." `arXiv:2407.09565 `_. Clarke, D., Pailanir, D., Athey, S., & Imbens, G. (2023). "Synthetic difference in differences estimation." arXiv preprint.