Ensemble Synthetic Control (ESC) ================================ .. currentmodule:: mlsynth When to Use This Estimator -------------------------- The synthetic control (SC) method builds a counterfactual for one treated unit as a weighted combination of donor units that reproduces the treated unit's pre-treatment outcome path, then extrapolates that combination into the post-treatment period. The difficulty is rarely the point estimate; it is inference. Estimating the donor weights uses only the pre-treatment window, of length :math:`T_0`, and when :math:`T_0` is short relative to the number of donors the counterfactual must be extrapolated from very little history. In that regime the usual devices strain: placebo p-values have coarse resolution and assume comparable pre-fit across units, and prediction intervals that reuse the pre-treatment residuals as a model of post-treatment error rest on an approximation that is fragile when the series trends. Ensemble Synthetic Control [LianPu2026]_ is built for the short-panel case. It treats SC inference as a small-sample prediction problem and quantifies uncertainty directly, without asking pre-treatment errors to stand in for post-treatment errors. ESC fits a collection of deliberately perturbed, low-complexity base learners and aggregates them. Each learner is a lasso regression of the treated unit's pre-treatment outcome on a random subset of the donors, fit on a block-subsampled slice of the pre-period; the block structure respects serial dependence, and the donor-subspace draw stabilises estimation when donors outnumber periods. The point counterfactual is the ensemble average, and the pointwise prediction interval is read straight off the empirical quantiles of the ensemble's counterfactuals. Reach for ESC when the pre-treatment window is short, when the donor pool is large relative to :math:`T_0`, or when you want interval evidence that does not depend on equating pre- and post-treatment error distributions. It is the ensemble sibling of the single-lasso :doc:`scul`: both build the synthetic control from an :math:`\ell_1`-penalised regression on a high-dimensional donor pool, but ESC bags many perturbed learners so that the spread across them becomes the prediction interval. Notation -------- One treated unit :math:`i = 1` and donor units :math:`i = 2, \dots, J+1` are observed over :math:`t = 1, \dots, T`, with treatment beginning at :math:`T_0 + 1`. Write :math:`y_t` for the treated outcome and :math:`Y_{0t} \in \mathbb{R}^{J}` for the donor outcomes at period :math:`t`. ESC works from the local pre-treatment approximation .. math:: y_t \;=\; Y_{0t}' w^\star + u_t, \qquad t \le T_0 , read as a stable predictive relationship, not a structural model. Each base learner is indexed by a perturbation :math:`\xi = (S, I)`, where :math:`S \subseteq \{1, \dots, J\}` is a random donor subset of size :math:`\lfloor \rho J \rfloor` and :math:`I \subseteq \{1, \dots, T_0\}` is a block subsample of the pre-period. Restricted to :math:`S` and fit on :math:`I`, the learner solves the lasso .. math:: \hat{w}_S(I) \;=\; \arg\min_{w \in \mathbb{R}^{|S|}} \frac{1}{|I|} \bigl\lVert y_{1,I} - Y_{0,I}^{(S)} w \bigr\rVert_2^2 \;+\; \lambda \lVert w \rVert_1 , with :math:`\lambda` chosen by cross-validation on the subsample (the one-standard-error rule by default). Its counterfactual is :math:`\hat{y}_t(\xi) = Y_{0t}' \hat{w}(\xi)`. Over :math:`B` perturbations the ensemble point counterfactual is the average :math:`\hat{y}_t^{\,\mathrm{ESC}} = B^{-1} \sum_b \hat{y}_t(\xi_b)`, and the estimated effect is :math:`\hat{\tau}_t = y_t - \hat{y}_t^{\,\mathrm{ESC}}`. For level :math:`1 - \alpha` the pointwise prediction interval for the counterfactual is the pair of empirical quantiles :math:`\bigl[\hat{Q}_{\alpha/2,t},\, \hat{Q}_{1-\alpha/2,t}\bigr]` of :math:`\{\hat{y}_t(\xi_b)\}_b`, and the interval for the effect is :math:`\bigl[y_t - \hat{Q}_{1-\alpha/2,t},\, y_t - \hat{Q}_{\alpha/2,t}\bigr]`. Assumptions ----------- 1. Sparse predictive approximation. There is a sparse weight vector :math:`w^\star` with :math:`\lVert w^\star \rVert_0 = s \ll T_0` such that the pre-treatment approximation above holds with error :math:`u_t`. Remark. This is a local predictive premise, not a claim of structural identification or of consistency as :math:`T_0 \to \infty`. Sparsity is what lets a lasso learner control its non-asymptotic prediction error at rate :math:`O_p(s \log J / T_0)`, so extrapolation stays stable even with a short pre-window. 2. Weak temporal dependence. The sequence :math:`\{(Y_{0t}, u_t)\}` is weakly dependent (for example mixing), allowing serial correlation and persistent components. Remark. Time is the effective sample dimension, so the perturbations must respect it. ESC subsamples contiguous blocks rather than individual periods; independent resampling would implicitly set the autocovariances to zero and inflate noise-driven instability when :math:`T_0` is small. 3. Restricted eigenvalue. The donor design satisfies a time-series restricted eigenvalue (compatibility) condition on sparse subsets. Remark. This is the standard regularity condition under which the lasso estimator is well behaved; it supports stable prediction and interval construction rather than structural recovery of :math:`w^\star`. Inference and Diagnostics ------------------------- ESC's uncertainty is predictive. The pointwise band on the counterfactual (and its mirror on the effect) comes from the empirical quantiles of the ensemble, so it carries no reliance on an asymptotic variance or on the assumption that the pre-treatment residuals describe post-treatment error. The band is stored on ``time_series`` as ``counterfactual_lower`` / ``counterfactual_upper`` (with ``prediction_interval_level`` and ``prediction_interval_kind``); its presence is reported by ``time_series.has_prediction_interval``. A scalar interval on the ATT is formed from the quantiles of the per-learner average post-treatment gaps and is exposed on ``inference`` (``ci_lower`` / ``ci_upper``, with the ensemble standard error). Pre-treatment fit is the ensemble RMSE on ``fit_diagnostics``. Because each learner regularises rather than interpolating the pre-period, ESC's pre-fit is deliberately looser than an unregularised synthetic control's: the base learners' bias is not a defect but the source of the interval. Example ------- .. code-block:: python import pandas as pd from mlsynth import ESC # California Proposition 99 tobacco panel (treated 1989). df = pd.read_csv("basedata/california_panel.csv") df["treat"] = ((df["state"] == "California") & (df["year"] >= 1989)).astype(int) res = ESC({ "df": df, "outcome": "cigsale", "treat": "treat", "unitid": "state", "time": "year", "n_learners": 300, "block_length": 3, "subspace_ratio": 0.5, "alpha": 0.10, "display_graphs": False, }).fit() print(res.att) # average post-treatment effect print(res.att_ci) # ensemble ATT interval ts = res.time_series print(ts.counterfactual_lower[-1], ts.counterfactual_upper[-1]) # 90% band, last year Verification ------------ ESC is validated on two empirical case studies. On California's Proposition 99, the ensemble counterfactual reproduces the classic synthetic-control path (correlation about 0.997 with the convex synthetic California) and the ATT matches the literature, while the prediction interval places observed California below its lower bound throughout the post-period. On the São Paulo homicide panel (:math:`T_0 = 9`), ESC returns a moderate central effect with a wide, honest interval on lives saved that spans the range the individual synthetic-control variants give as point estimates. See the benchmark cases `esc_prop99.py `_ and `esc_saopaulo.py `_, and the :doc:`replications/esc` page. A caveat on coverage. The band ESC ships is the equation-(7) spread of the ensemble counterfactuals -- an estimation-uncertainty band. In our Monte Carlo checks a faithful port of this band under-covers the realized counterfactual in factor-model designs, because the ensemble spread reflects weight-estimation variability rather than the treated unit's own disturbance :math:`u_t`. The paper's nominal-coverage claim therefore did not reproduce from the equations as written; a calibrated coverage benchmark is deferred pending the authors' reference implementation. Core API -------- .. autoclass:: ESC :members: :undoc-members: :show-inheritance: .. autoclass:: mlsynth.config_models.ESCConfig :members: :undoc-members: :show-inheritance: