Flexible Count Synthetic Control (CSCM) ======================================= .. currentmodule:: mlsynth When to Use This Estimator -------------------------- ``CSCM`` is a synthetic control designed for count data and other non-negative outcomes -- deaths, homicides, disease cases, accident counts, claim counts. It is a port of the flexible synthetic control of Bonander [CSCM]_. Reach for it when you have a single treated unit, a pool of untreated donor units, a panel of pre- and post-intervention periods, and an outcome that cannot go negative and is naturally read on a multiplicative (rate-ratio) scale. The problem it solves is a tension in the ordinary synthetic control method. The textbook estimator constrains the donor weights to be non-negative and to sum to one. Those two constraints keep the counterfactual inside the range of the donors, which for a non-negative outcome has a useful side effect: a weighted average of non-negative series is itself non-negative, so the imputed counterfactual can never turn negative. But the same constraints often produce a poor pre-treatment fit. The usual remedies -- an unconstrained regression, an elastic net, adding an intercept -- restore the fit but let the counterfactual go negative, which is meaningless for a count. CSCM keeps the property you need for counts and relaxes only the one that hurts the fit. It retains non-negativity of the weights (so the counterfactual stays non-negative) but drops the adding-up constraint, penalising how far the weights stray from the classic simplex solution. A single tuning parameter dials the amount of extrapolation: none recovers ordinary synthetic control, more allows the weights to leave the simplex where that demonstrably improves the fit. Reach for CSCM when ^^^^^^^^^^^^^^^^^^^ * The outcome is a count or rate that cannot be negative, and a negative counterfactual would be nonsensical. * Ordinary synthetic control fits the pre-treatment path poorly, but you do not want to abandon the non-negativity guarantee to fix it. * A rate ratio (a multiplicative effect) is the natural way to report the result. Notation -------- There is one treated unit :math:`i=1` and :math:`n_0` donor units, observed over :math:`T_0` pre-treatment and :math:`T_1` post-treatment periods. Let :math:`Y_{it}\in\mathbb{R}^{0+}` be the outcome of unit :math:`i` at time :math:`t`. Stack the treated unit's pre-treatment outcomes (and any auxiliary covariates) into :math:`\mathbf{X}_1` and the donors' into :math:`\mathbf{X}_0`. The synthetic control is a weighted donor average :math:`\sum_{i=2}^{n} Y_{it} w_i`, and :math:`\mathbf{V}` is a diagonal matrix of feature-importance weights. The classic simplex weights solve .. math:: \mathbf{W}^{scm} = \arg\min_{\mathbf{W}} (\mathbf{X}_1 - \mathbf{X}_0 \mathbf{W})' \mathbf{V} (\mathbf{X}_1 - \mathbf{X}_0 \mathbf{W}),\quad w_i \ge 0,\ \textstyle\sum_i w_i = 1 . CSCM keeps :math:`w_i \ge 0` but replaces the sum-to-one constraint with a penalty toward :math:`\mathbf{W}^{scm}`: .. math:: \mathbf{W}^{*} = \arg\min_{\mathbf{W}\ge 0} (\mathbf{X}_1 - \mathbf{X}_0 \mathbf{W})' \mathbf{V} (\mathbf{X}_1 - \mathbf{X}_0 \mathbf{W}) + \lambda \lVert \mathbf{W} - \mathbf{W}^{scm} \rVert_2^2 . As :math:`\lambda\to\infty` the solution returns to :math:`\mathbf{W}^{scm}`; at :math:`\lambda=0` it allows the most extrapolation the non-negativity constraint permits. Because :math:`\mathbf{W}^{scm}` sums to one, penalising the distance from it indirectly penalises departures from the simplex, so :math:`\lambda` controls how far the weights may leave the convex hull of the donors. Assumptions ----------- 1. Non-negative outcome. The outcome is a count or otherwise non-negative quantity. CSCM rejects a panel whose outcome column has negative values. Remark. Non-negativity of the weights guarantees a non-negative counterfactual only because the donor outcomes are themselves non-negative; this is what makes the relaxation safe for counts where an intercept-based relaxation is not. 2. A synthetic control that matches the treated unit on lagged outcomes (and covariates) in the pre-period tracks its untreated potential outcome in the post-period. Remark. This is the usual synthetic-control identifying assumption (a linear factor structure with matched loadings); the relaxation changes only how the weights are estimated, not what they are assumed to recover. 3. No interference. One unit's treatment does not affect another's outcome, so the donors are untreated throughout. Remark. Standard SUTVA/consistency; excludes spillover onto the donor pool. Inference --------- The effect is reported as a rate ratio -- the multiplicative analogue of the ATT, natural for counts -- of the summed post-period observed outcome to the summed post-period counterfactual. Its confidence interval comes from a K-fold cross-fitting procedure (Chernozhukov, Wuthrich and Zhu [CWZ]_) that also removes the synthetic control's finite-sample bias: the pre-period is split into :math:`K` blocks; for each block the weights are refit on the block's complement, the post-period log rate ratio is corrected by the block's placebo log rate ratio, and the corrected estimates are combined into a :math:`t`-interval with :math:`K-1` degrees of freedom. ``K=2`` is preferable on short pre-periods and ``K=3`` on long ones; with small :math:`K` the interval is wide, which the degrees of freedom make explicit. The predictor-importance matrix :math:`\mathbf{V}` is chosen by a leave-one-out Poisson ridge of the donors' post-period outcome on the balance features (``v_method="poisson_ridge"``), or set to equal weights (``v_method="uniform"``), the simple default. On panels where the ridge shrinks heavily the two coincide. Example ------- .. code-block:: python import pandas as pd from mlsynth import CSCM # Vision Zero: Sweden's 1997 road-safety policy; road-death rate. df = pd.read_csv("basedata/viszero.csv") donors = [2, 3, 5, 7, 9, 10, 13, 14, 16] df = df[df["ID"].isin([25] + donors)].copy() df["treated"] = ((df["ID"] == 25) & (df["TIME"] >= 1996)).astype(int) res = CSCM({ "df": df, "outcome": "deathrate_mln", "treat": "treated", "unitid": "ID", "time": "TIME", "K": 2, "v_method": "uniform", "display_graphs": False, }).fit() rr = res.effects.additional_effects["rate_ratio"] # ~1.06 (no detectable effect) lo, hi = res.inference.ci_lower, res.inference.ci_upper # wide, spans 1 res.weights.donor_weights # relaxed weights, sum < 1 Verification ------------ CSCM is cross-validated against the authors' own R implementation on the Vision Zero panel: the classic warm-start concentrates on Finland, the relaxed weights extrapolate below the simplex (their sum falls below one), and the cross-fitted rate ratio matches the reference to about one percent. See the durable benchmark ``benchmarks/cases/cscm_viszero.py`` and the replication page :doc:`replications/cscm`. Core API -------- .. autoclass:: CSCM :members: .. autoclass:: mlsynth.utils.cscm_helpers.config.CSCMConfig :members: .. [CSCM] Bonander, C. (2021). A (Flexible) Synthetic Control Method for Count Data and Other Non-Negative Outcomes. *Epidemiology*, 32(4), e18-e19. .. [CWZ] Chernozhukov, V., Wuthrich, K., & Zhu, Y. (2021). Practical and robust t-test based inference for synthetic control and related methods.