Synthetic Control Using Lasso (SCUL) ==================================== .. currentmodule:: mlsynth When to Use This Estimator -------------------------- The synthetic control (SC) method of Abadie and co-authors [ABADIE2010]_ builds a counterfactual for one treated unit as a weighted average of donor units that reproduces the treated unit's pre-treatment outcome path. The textbook version restricts the weights to be non-negative and to sum to one, so the synthetic control is a convex combination that stays inside the range of the donors. That restriction is comfortable when the donor pool is small and the treated unit sits inside the donors' convex hull, but it breaks down in two common modern situations: the donor pool is high-dimensional -- more candidate donors than pre-treatment periods -- and the donors come from many variable types, not just the same outcome measured elsewhere. SCUL [HollingsworthWing2022]_ handles both. It builds the synthetic control as a lasso regression of the treated unit's pre-treatment outcome on the donor pool. The lasso's :math:`\ell_1` penalty selects a sparse set of donors automatically, so the pool can be larger than the pre-period and can mix variable types (in the California example below, every donor state's cigarette sales and its retail cigarette price are candidates). The weights are unrestricted -- they may be negative, and an intercept is included -- so the synthetic control is allowed to extrapolate beyond the donors' convex hull when that gives a better fit. The penalty is chosen by a rolling-origin cross-validation that respects the time ordering, guarding against overfitting the pre-period noise. Reach for SCUL when the donor pool is large or multi-type, when a convex synthetic control cannot fit the treated unit because it lies outside the donor hull, or when you want model selection over donors to be automatic and reproducible rather than hand-curated. It is the lasso sibling of the penalized synthetic control :doc:`pda` (LASSO) variant: both select donors with an :math:`\ell_1` penalty, but SCUL pairs that with the rolling-origin cross-validation and the unit-free fit diagnostics of the source paper. Notation -------- One treated unit :math:`i = 0` is observed over :math:`T` periods, treated after :math:`T_0`. Write :math:`y_t` for its outcome and :math:`X_t \in \mathbb{R}^P` for the donor pool at period :math:`t` -- a (possibly high-dimensional, multi-type) vector of candidate predictors built from the donor units. The synthetic control is the lasso fit .. math:: (\hat{\beta}_0, \hat{\beta}) \;=\; \arg\min_{\beta_0,\,\beta}\ \frac{1}{2T_0}\sum_{t=1}^{T_0} \bigl(y_t - \beta_0 - X_t' \beta\bigr)^2 \;+\; \lambda \lVert \beta \rVert_1 , with the donor columns standardised to unit (population) variance before the penalty is applied. The lasso is solved *exactly* by the Langlois & Darbon [LangloisDarbon2025]_ differential-inclusion homotopy: there is no convergence tolerance, and one warm-started sweep returns the whole regularisation path, so the cross-validation is both exact and cheap. The synthetic series is :math:`\hat{y}_t = \hat{\beta}_0 + X_t' \hat{\beta}` over all :math:`t`, and the treatment effect in a post period is :math:`\hat{\tau}_t = y_t - \hat{y}_t`. The reported ATT is the post-period mean :math:`\frac{1}{T-T_0}\sum_{t>T_0}\hat{\tau}_t`. The penalty :math:`\lambda` is selected by a rolling-origin expanding-window cross-validation: for each expanding training window of pre-periods, a lasso is fit over a grid of penalties and scored on the next ``training_post_length`` pre-periods (an out-of-sample window mimicking the post-treatment forecast); the selected penalty is the median across windows of each window's error-minimising value (the reference's ``lambda.median``). Assumptions ----------- Assumption 1 (single treated unit, balanced panel). Exactly one unit is treated, after a known period :math:`T_0`, on a strongly balanced panel. Remark. SCUL estimates one counterfactual series; staggered or multi-unit designs are out of scope for this estimator. Assumption 2 (predictive pre-period relationship). The treated unit's untreated outcome is well approximated, in the pre-period, by a sparse linear combination of the donor pool. SCUL does not require the weights to be convex; it requires only that such a (possibly extrapolating) combination exists and is stable across the pre/post boundary. Remark. Unrestricted weights buy flexibility at the cost of a higher overfitting risk than convex SC. The rolling-origin cross-validation is the guardrail: a penalty that only fits in-sample noise scores poorly on the held-out windows and is not selected. Assumption 3 (continuous, high-dimensional donors are admissible). The pool may contain more donors than pre-periods and donors of several variable types. For continuously distributed donors the lasso solution is unique [TIBSHIRANI2013]_, so the selected synthetic control is well defined. Remark. This is the assumption that makes SCUL different from convex SC: it embraces, rather than avoids, the high-dimensional donor pool that the convexity restriction cannot accommodate. Inference and Diagnostics ------------------------- Pre-treatment fit is summarised by a unit-free Cohen's D -- the mean absolute pre-period gap divided by the pre-period standard deviation of the outcome -- reported in ``fit_diagnostics``. Small values indicate a synthetic control that tracks the treated unit closely before treatment. Inference is a placebo permutation test. Each donor unit is treated, in turn, as a fake-treated target and fit with SCUL on the remaining pool; the test statistic is the unit-free ratio of the post- to pre-treatment root mean squared gap [ABADIE2010]_. The p-value is the rank of the real treated unit's ratio in the placebo distribution. Placebo units whose pre-fit exceeds ``cohensd_threshold`` are trimmed, per the paper's quality-control recommendation. Set ``inference=False`` to skip the (re-fitting) placebo loop. Example ------- .. code-block:: python import pandas as pd from mlsynth import SCUL from mlsynth.config_models import SCULConfig # California (Proposition 99): cigarette sales + retail price, 1970-2000. df = pd.read_csv("basedata/california_panel.csv") df["treat"] = ((df["state"] == "California") & (df["year"] >= 1989)).astype(int) config = SCULConfig( df=df, outcome="cigsale", treat="treat", unitid="state", time="year", donor_variables=["retprice"], # widen the pool with each state's price display_graphs=True, ) results = SCUL(config).fit() print(results.effects.att) # post-period ATT print(results.method_details.parameters_used) # selected lambda, Cohen's D, pool size Verification ------------ SCUL is cross-validated value-for-value against the authors' reference R package (`hollina/scul `_) on the California (Proposition 99) cigarette panel: the rolling-origin cross-validation penalty matches ``glmnet`` to ten digits, and -- since the lasso solution is unique for continuous donors and mlsynth solves it exactly -- the weights and synthetic series agree up to ``glmnet``'s own convergence threshold. See the durable benchmark `benchmarks/cases/scul_prop99.py `_ and the :doc:`replications/scul` page. Core API -------- .. autoclass:: SCUL :members: fit .. autoclass:: mlsynth.config_models.SCULConfig :members: