Continuous-Treatment Synthetic Control (CTSC) ============================================== .. currentmodule:: mlsynth Overview -------- CTSC (Powell, D. (2022). *"Synthetic Control Estimation Beyond Comparative Case Studies: Does the Minimum Wage Reduce Employment?,"* Journal of Business & Economic Statistics 40(3):1302-1314) generalises the synthetic control method to settings the original was never built for: continuous and/or multi-valued treatments in panels where there is no clean treated / never-treated split. The canonical synthetic control estimates the effect of one unit adopting a single binary policy, using never-treated units as donors. But many policy variables are continuous and adopted (and changed) by every unit -- every U.S. state has a minimum wage that moves over time, so there is no "untreated" state to serve as a clean control. CTSC (the paper's "GSC") handles exactly this case. .. note:: The paper names the estimator GSC (generalized synthetic control). mlsynth calls it CTSC to avoid collision with Xu (2017)'s differently constructed Generalized Synthetic Control (``gsynth``), which is an interactive-fixed-effects estimator. What CTSC does ^^^^^^^^^^^^^^ * Builds a synthetic control for every unit out of the other units' *untreated* outcomes. * Jointly estimates a unit-specific treatment-slope vector :math:`\boldsymbol{\alpha}_i` (allowing fully heterogeneous marginal effects) and the synthetic-control weights. * Reports the population-weighted average marginal effect :math:`\alpha^{AE} \coloneqq \sum_{i \in \mathcal{N}} \pi_i\, \boldsymbol{\alpha}_i`. * Permits an interactive-fixed-effects (factor) outcome structure, so it is consistent when the treatment is correlated with unobserved factors and trends -- precisely the case where two-way fixed-effects regression is badly biased. When to use CTSC ^^^^^^^^^^^^^^^^ * The policy variable is continuous or multi-valued (minimum wage, tax rate, dosage) rather than a 0/1 indicator. * Every unit is "treated" with a time-varying intensity, so there is no never-treated donor pool and comparative-case-study SC cannot be applied. * You suspect the treatment is correlated with unobserved trends/factors, making two-way fixed effects inconsistent. * You want unit-specific marginal effects (heterogeneity), not just an average. Notation ^^^^^^^^ CTSC departs from the single-treated-unit setting, so there is no distinguished :math:`j = 1`: every unit carries its own treatment intensity. Index the units by :math:`i \in \mathcal{N} \coloneqq \{1, \dots, N\}` and time by :math:`t \in \mathcal{T} \coloneqq \{1, \dots, T\}` (1-indexed). The observed scalar outcome of unit :math:`i` at time :math:`t` is :math:`y_{it}`, with its no-treatment potential outcome (Abadie superscript) written :math:`y_{it}^N`. The treatment of unit :math:`i` at time :math:`t` is a :math:`K`-vector :math:`\mathbf{D}_{it}` of continuous or multi-valued intensities, and :math:`\boldsymbol{\alpha}_i \in \mathbb{R}^{K}` is its unit-specific marginal-effect (slope) vector -- the estimand. The factor term :math:`\boldsymbol{\lambda}_t^\top \boldsymbol{\mu}_i` carries an interactive fixed effect, with time factors :math:`\boldsymbol{\lambda}_t` and loadings :math:`\boldsymbol{\mu}_i`, and :math:`\epsilon_{it}` is mean-zero idiosyncratic noise. When unit :math:`i` is synthesized from the others, its donor weights :math:`\boldsymbol{\phi}^i` lie on the simplex :math:`\Delta^{N-1} \coloneqq \{\boldsymbol{\phi} \in \mathbb{R}_{\ge 0}^{N-1} : \|\boldsymbol{\phi}\|_1 = 1\}`, and :math:`\Omega_i` is its per-unit fit weight. The population-weighted average marginal effect is :math:`\alpha^{AE} \coloneqq \sum_{i \in \mathcal{N}} \pi_i\, \boldsymbol{\alpha}_i`, with population shares :math:`\pi_i`. Assumptions (and how to spot violations) ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ CTSC is a *much* more permissive estimator than vanilla SC -- it does not require a clean treated/control split, it does not assume homogeneous effects, and it does not require the analyst to specify the number of factors. But it does still rely on a small set of identifying assumptions. Each is given here together with the symptom you would see if it failed. 1. Interactive fixed-effects DGP for the untreated outcome. The paper assumes :math:`y_{it}^N = \boldsymbol{\lambda}_t^\top \boldsymbol{\mu}_i + \epsilon_{it}`, i.e. the counterfactual (no-treatment) outcome is a low-rank factor model plus mean-zero idiosyncratic noise. *Remark.* This is plausibly violated when the outcome has strong unit-specific nonlinear trends, structural breaks affecting only some units, or unit-specific deterministic time polynomials that cannot be written as :math:`\boldsymbol{\lambda}_t^\top \boldsymbol{\mu}_i`. To spot it, fit an interactive-FE model (e.g. ``gsynth``) to the pre-/non-treatment data and inspect the residuals -- if there is visible unit-specific curvature left, the factor structure is misspecified. 2. Convex-hull condition on donor loadings. For each treated-period observation of unit :math:`i`, its factor loading :math:`\boldsymbol{\mu}_i` must lie (approximately) in the convex hull of the other units' loadings, so that a simplex-weighted combination of donors can reproduce its untreated trajectory. *Remark.* This is plausibly violated when the treated unit is an outlier on level, seasonality, or factor exposure -- a coastal mega-state with no interior analog, a country with idiosyncratic policy that no donor shares. To spot it, look at the per-unit fit weights :math:`\Omega_i` returned by CTSC; units with very poor untreated fit get heavily down-weighted, and if most units fall in that bucket the hull condition is failing population-wide. 3. Large :math:`T` regime (consistency, not factor count). Powell shows the estimator is consistent as :math:`T \to \infty` *without* the user having to specify the rank of :math:`\boldsymbol{\lambda}_t^\top \boldsymbol{\mu}_i`. The trade-off is that very short panels leave the simplex weights and slopes weakly identified. *Remark.* This is plausibly violated when :math:`T` is on the order of :math:`N` or smaller (so the per-unit least squares with :math:`N-1` simplex weights becomes ill-posed). To spot it, re-run on a longer pre-period or aggregate to a coarser frequency and check whether the average effect and weights are stable; large swings suggest :math:`T` is too short. 4. Linearity of the outcome in the treatment vector. The paper writes :math:`y_{it} = y_{it}^N + \mathbf{D}_{it}^\top \boldsymbol{\alpha}_i`, i.e. the treatment effect is linear in :math:`\mathbf{D}_{it}` (with unit-specific slopes :math:`\boldsymbol{\alpha}_i`). Multi-valued :math:`\mathbf{D}_{it}` is fine, and interactions/polynomial terms can be entered as extra columns of :math:`\mathbf{D}_{it}`, but CTSC does not estimate a fully nonparametric dose-response curve. *Remark.* This is plausibly violated when the dose-response is strongly nonlinear inside the support and you have not encoded the nonlinearity (e.g. sharp regime-switching, kinked schedules). To spot it, add a quadratic or spline term to ``treatment_vars`` and test whether the higher-order coefficient is jointly significant under the sign-flip distribution. 5. Slope heterogeneity :math:`\boldsymbol{\alpha}_i` is unit-specific but time-invariant. Each unit gets its own marginal effect, but it does not drift over time within a unit. *Remark.* This is plausibly violated when the marginal effect itself moves -- a minimum-wage elasticity that changes after a labour-market reform, a dose-response that shifts when patient populations change. To spot it, split the post-treatment window in half, refit, and compare the recovered :math:`\widehat{\boldsymbol{\alpha}}_i`; large within-unit drift indicates the time-invariance assumption is binding. 6. No simultaneity in :math:`\mathbf{D}_{it}`. CTSC allows :math:`\mathbf{D}_{it}` to be correlated with the factors :math:`\boldsymbol{\lambda}_t^\top \boldsymbol{\mu}_i` (this is the whole point), but it still assumes :math:`\mathbf{D}_{it}` is mean-independent of the idiosyncratic shock :math:`\epsilon_{it}`. Reverse causality from contemporaneous :math:`\epsilon_{it}` to :math:`\mathbf{D}_{it}` breaks identification. *Remark.* This is plausibly violated when the policy responds within the same period to the outcome -- e.g. the minimum wage being raised *because* employment surprised on the upside this quarter. To spot it, regress :math:`\Delta \mathbf{D}_{it}` on lagged residuals from a no-treatment factor fit; significant feedback is a red flag. When not to use CTSC ^^^^^^^^^^^^^^^^^^^^^^^^ * Single treated unit with a clean binary policy. A canonical comparative-case-study set-up (one state passes one law on one date, others never do) is exactly what vanilla SC was built for; CTSC's per-unit weight system is unnecessary overhead and its sign-flip inference is weaker than the placebo / conformal inference available in the binary-treatment world. Use :doc:`tssc` or :doc:`fdid`. * Short panels. Powell's consistency story is in :math:`T \to \infty`. With :math:`T \lesssim N` the per-unit least squares with :math:`N-1` simplex weights is ill-posed and the average effect can swing dramatically with the seed. Prefer :doc:`fdid` (which is designed for short panels by stepwise donor selection) or a factor estimator that explicitly regularises the rank. * Treated trajectory outside the donor convex hull. If the treated unit's untreated trend cannot be expressed as a simplex combination of the donors' untreated trends -- coastal vs. interior states, an outlier sector, a country with idiosyncratic seasonality -- CTSC has no fix; the per-unit :math:`\Omega_i` will collapse and the average effect is dominated by a handful of well-fit units. Use :doc:`fma` or :doc:`scmo` (auxiliary outcomes) to widen the donor information set before forcing a hull fit. * Treatment effect that drifts over time within a unit. CTSC fixes :math:`\boldsymbol{\alpha}_i` across time. If the dose-response is genuinely time-varying (a minimum-wage elasticity that changes after a recession, a drug whose effect attenuates with tolerance), CTSC will return an average across the post-window that masks the dynamics. Use a time-varying-effects estimator (:doc:`tasc` for state-space dynamics, :doc:`dscar` for autoregressive treated processes) instead. * Strongly nonlinear or kinked dose-response that you cannot encode. CTSC is linear in :math:`\mathbf{D}_{it}`. If the policy effect has a sharp kink or saturation that no parsimonious basis expansion captures, fall back to a doubly-robust panel estimator or a changes-in-changes design. * Contemporaneous reverse causality from outcome to treatment. CTSC permits the treatment to be correlated with unobserved factors, but not with the *same-period* idiosyncratic shock. If the policy is set in response to within-period outcome surprises, you need an instrument or a timing-based identification strategy; CTSC alone is not enough. Mathematical Formulation ------------------------ Model (paper eq. 4) ^^^^^^^^^^^^^^^^^^^^ .. math:: y_{it} = \boldsymbol{\lambda}_t^\top \boldsymbol{\mu}_i + \mathbf{D}_{it}^\top \boldsymbol{\alpha}_i + \epsilon_{it}, where :math:`\mathbf{D}_{it}` is a :math:`K`-vector of (continuous/discrete) treatment variables, :math:`\boldsymbol{\lambda}_t^\top \boldsymbol{\mu}_i` is an interactive fixed-effects (factor) term, and :math:`\boldsymbol{\alpha}_i` is the unit-specific slope. Joint estimation (paper eq. 5-6) ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ CTSC minimises, over slopes :math:`\mathbf{b}` and per-unit simplex weights :math:`\boldsymbol{\phi}`, .. math:: \frac{1}{2NT}\sum_{i \in \mathcal{N}} \Omega_i^{-1} \sum_{t \in \mathcal{T}} \Bigl[ y_{it} - \mathbf{D}_{it}^\top \mathbf{b}_i - \sum_{j \ne i} \phi_j^i (y_{jt} - \mathbf{D}_{jt}^\top \mathbf{b}_j) \Bigr]^2, \quad \phi_j^i \ge 0,\ \sum_{j \ne i} \phi_j^i = 1, where :math:`y_{it} - \mathbf{D}_{it}^\top \mathbf{b}_i` is unit :math:`i`'s untreated outcome, its synthetic control is a convex combination of the *other* units' untreated outcomes, and :math:`\Omega_i` is a per-unit fit weight (a two-step measure, eq. 6) that down-weights units lacking a good synthetic control. The average effect (paper eq. 7) is :math:`\alpha^{AE} \coloneqq \sum_{i \in \mathcal{N}} \pi_i\, \boldsymbol{\alpha}_i`. Inference (paper Section 4) ^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Because CTSC mechanically correlates units (each control is built from the others), inference uses unit-level moment scores at the null-restricted estimate and calibrates a Wald statistic with a Rademacher (sign-flip) randomization distribution, valid under arbitrary within- and cross-unit dependence. Implementation note ------------------- The paper minimises the objective with Nelder-Mead over all :math:`NK + N(N-1)` parameters. mlsynth exploits the biconvex structure -- weighted linear least squares in the slopes for fixed weights (a single closed-form linear solve) and :math:`N` independent simplex-constrained least squares in the weights for fixed slopes -- and solves it by block coordinate descent. This optimises the same objective with far better stability and speed. The null-restricted fit used for inference imposes :math:`\sum_i \pi_i \boldsymbol{\alpha}_i = a_0` via a KKT-augmented linear system. Calibration to the paper's simulation -------------------------------------- :mod:`mlsynth.utils.ctsc_helpers.simulation` reproduces the paper's Section 5 / Table 1 Monte Carlo (Models 1-4). The data-generating process is .. math:: y_{it} = \beta_i d_{it} + 5 \sum_{k=1}^{2} \lambda_t^{(k)} \mu_i^{(k)} + \epsilon_{it}, with piecewise factor paths :math:`\lambda_t^{(k)}`, :math:`\mu_i^{(k)} \sim U(0,1)`, :math:`\epsilon_{it} \sim N(0, \tfrac14)`, and :math:`\beta_i = \sum_k \mu_i^{(k)} - \tfrac1N \sum_i \sum_k \mu_i^{(k)}` (so the true average effect is exactly zero). The continuous treatment :math:`d_{it}` is a function of the same factors, so it correlates with the interactive fixed effects. Reproduced calibration (mlsynth, fewer Monte-Carlo draws than the paper's 1000): ============ ============================ ========================= Model CTSC mean bias (paper) Two-way FE bias (paper) ============ ============================ ========================= 1 (n=10) ~0.00 (0.011) ~0.80 (0.850) 2 (n=30) ~0.00 (0.005) ~0.82 (0.846) 3 (within) ~0.00 (-0.001) ~0.49 (0.423) 4 (T=20) ~0.01 (-0.002) ~0.53 (0.263) ============ ============================ ========================= CTSC is essentially unbiased across all models while two-way fixed effects is badly biased; the inference rejects the (true) zero null at roughly the nominal 5% rate (paper Panel B ~0.044). .. code-block:: python from mlsynth.utils.ctsc_helpers.simulation import run_simulation for model in (1, 2, 3, 4): s = run_simulation(model, n_sims=100, seed=0) print(f"Model {s.model}: CTSC bias={s.ctsc_mean_bias:+.3f} " f"MAD={s.ctsc_mad:.3f} | FE bias={s.fe_mean_bias:+.3f}") Core API -------- .. automodule:: mlsynth.estimators.ctsc :members: :undoc-members: :show-inheritance: Configuration ------------- .. autoclass:: mlsynth.config_models.CTSCConfig :members: :undoc-members: Helper Modules -------------- .. automodule:: mlsynth.utils.ctsc_helpers.setup :members: :undoc-members: .. automodule:: mlsynth.utils.ctsc_helpers.estimate :members: :undoc-members: .. automodule:: mlsynth.utils.ctsc_helpers.inference :members: :undoc-members: .. automodule:: mlsynth.utils.ctsc_helpers.simulation :members: :undoc-members: .. automodule:: mlsynth.utils.ctsc_helpers.pipeline :members: :undoc-members: .. automodule:: mlsynth.utils.ctsc_helpers.structures :members: :undoc-members: Example ------- .. code-block:: python import numpy as np import pandas as pd from mlsynth import CTSC from mlsynth.utils.ctsc_helpers.simulation import generate_model # A continuous-treatment panel (here from the paper's Model 1 DGP). rng = np.random.default_rng(3) Y, D, true_ae = generate_model(1, rng) n, T = Y.shape rows = [{"state": f"s{i}", "qtr": t, "emp": Y[i, t], "minwage": D[i, t, 0]} for i in range(n) for t in range(T)] df = pd.DataFrame(rows) res = CTSC({ "df": df, "outcome": "emp", "treat": "minwage", # placeholder for the base config "treatment_vars": ["minwage"], "unitid": "state", "time": "qtr", "inference": True, }).fit() print(f"average marginal effect = {res.average_effect[0]:+.4f} " f"(true = {true_ae})") print(f"sign-flip Wald p-value = {res.inference.p_value[0]:.3f}") # Unit-specific slopes (heterogeneity): print("unit effects:", np.round(res.unit_effects[:, 0], 3)) References ---------- Abadie, A., Diamond, A., & Hainmueller, J. (2010). "Synthetic Control Methods for Comparative Case Studies." *Journal of the American Statistical Association* 105(490):493-505. Arkhangelsky, D., Athey, S., Hirshberg, D. A., Imbens, G. W., & Wager, S. (2021). "Synthetic Difference-in-Differences." *American Economic Review* 111(12):4088-4118. Canay, I. A., Romano, J. P., & Shaikh, A. M. (2017). "Randomization Tests Under an Approximate Symmetry Assumption." *Econometrica* 85(3):1013-1030. Dube, A., & Zipperer, B. (2015). "Pooling Multiple Case Studies Using Synthetic Controls: An Application to Minimum Wage Policies." IZA DP. Powell, D. (2022). "Synthetic Control Estimation Beyond Comparative Case Studies: Does the Minimum Wage Reduce Employment?" *Journal of Business & Economic Statistics* 40(3):1302-1314. Xu, Y. (2017). "Generalized Synthetic Control Method: Causal Inference with Interactive Fixed Effects Models." *Political Analysis* 25(1):57-76.