Functional Synthetic Control (FSC) ================================== .. currentmodule:: mlsynth When to Use This Estimator -------------------------- Almost every synthetic control method assumes that what you observe each period is a number. FSC is for the case where it is a whole object: a fertility rate recorded separately for every age, a distribution of ages at death, a covariance matrix across a country's export categories. You can always turn such an object into a number and run an ordinary synthetic control on that. The question is what you lose. East Germany legalised abortion in 1972, and the effect on fertility was concentrated among women aged roughly 20 to 30 and close to nil at either end of the childbearing range. Collapse the age profile to a total fertility rate and that result becomes a single negative number; the shape of the response, which is the part a demographer would want, is gone. The same argument applies whenever the interesting variation is *within* the object, not in its total: a policy that compresses a wage distribution without moving its mean, a shock that reallocates trade between categories without changing the total. Reach for FSC when the object per unit-period is a curve, a distribution, or a matrix, and when you already have it in that form. If your rows are individual people, not points on a grid, :doc:`dsc` is the right tool for unconditional quantile effects and :doc:`drsc` for conditional ones. If your object is a vector of shares summing to one, use :doc:`compsc`. Notation -------- There are :math:`N` units observed over :math:`T` periods. Unit :math:`i = 1` is treated from period :math:`T_0 + 1` onward; the rest are the donor pool. The outcome :math:`\nu_{it}` is not a number but a point in a metric space :math:`(\mathcal M, d)` -- the set of curves, of distributions, of positive semidefinite matrices. The difficulty is that :math:`\mathcal M` has no linear structure. Averaging two probability distributions by averaging their density values does not generally give a sensible "average distribution", so the weighted-average counterfactual that defines synthetic control is not even well posed. FSC removes the difficulty with an isometric embedding: a map :math:`\Psi : \mathcal M \to \mathcal H` into a Hilbert space that preserves distances exactly, .. math:: d(x, y) \;=\; \lVert \Psi(x) - \Psi(y) \rVert_{\mathcal H} \qquad \text{for all } x, y \in \mathcal M . An isometry is a relabelling that gets the geometry right: distances measured after the embedding are the same distances as before, so nothing about the problem is distorted by moving into :math:`\mathcal H`. Inside :math:`\mathcal H` addition and scalar multiplication are available again, and the ordinary synthetic control construction goes through. Write :math:`Y_{it} = \Psi(\nu_{it})` for the embedded outcomes and :math:`\mathcal Y = \Psi(\mathcal M)` for their image. Which map :math:`\Psi` to use depends on the space, and the ``space`` option picks it: .. list-table:: :header-rows: 1 :widths: 16 32 26 26 * - ``space`` - Objects - :math:`\Psi` - :math:`\mathcal Y` * - ``"function"`` - curves in :math:`L^2` - the identity - all of :math:`\mathcal H` * - ``"distribution"`` - distributions under the 2-Wasserstein metric - the quantile function - increasing functions * - ``"matrix"`` - symmetric positive semidefinite matrices, Frobenius metric - the half-vectorisation - the PSD cone The estimand is the counterfactual object :math:`\nu^N_{1t}` for the treated unit after treatment. What you do with it is up to you: the difference of embedded objects :math:`Y_{1t} - Y^N_{1t}` is a curve, reported here as ``curve.effect``, and its length .. math:: d_t \;=\; d(\nu^I_{1t}, \nu^N_{1t}) \;=\; \lVert Y_{1t} - Y^N_{1t} \rVert_{\mathcal H} is the scalar magnitude of the effect, reported as ``curve.magnitude``. Assumptions ----------- Assumption 1 (Embeddability). :math:`(\mathcal M, d)` admits an isometric embedding into a Hilbert space, and the image :math:`\Psi(\mathcal M)` is closed and convex. Remark. This is a real restriction, not a formality. A metric space embeds isometrically into a Hilbert space exactly when it has 2-negative type (Schoenberg), and many do not -- notably the space of probability measures on :math:`\mathbb{R}^k` under the 2-Wasserstein metric for :math:`k \ge 2`. So "metric space-valued" here does not mean any metric space. The three spaces this estimator supports all satisfy it. Convexity of the image is what makes the plain estimator work at all: a convex combination of donor objects then lands back in :math:`\mathcal Y` automatically, so it corresponds to a real object. Assumption 2 (Common grid). Every unit-period cell is observed at the same argument values. Remark. The estimator checks this at ingestion and raises when it fails. Objects observed on different grids are not comparable coordinate by coordinate, and interpolating them onto a common grid is a modelling choice that belongs to you. Do it yourself first if you need to. Assumption 3 (Data-generating process). The embedded control outcomes follow either a functional autoregression or a latent factor model, with independent mean-zero errors bounded in :math:`\mathcal H`. Remark. These are the same two processes the scalar synthetic control literature assumes, lifted to :math:`\mathcal H`. They are what deliver the error bounds below; the estimator itself does not need them, but the guarantee does. Assumption 4 (Pre-treatment fit). The treated unit's pre-treatment objects are approximately matched by the weighted donors. Remark. This is the assumption that does the work, and it is checkable. The finite-sample bound says the estimation error is controlled by the pre-treatment fit and the norm of the weights and by nothing else, so a poor pre-fit is a direct warning about the estimate. ``pre_treatment_fit`` on the result is that quantity. Estimation ---------- The donor weights match the pre-treatment objects in :math:`\mathcal H`: .. math:: \widehat\gamma^{\mathrm{scm}} \in \operatorname*{arg\,min}_{\gamma \in \Delta^{N-1}} \sum_{t=1}^{T_0} \Bigl\lVert Y_{1t} - \sum_{i=2}^{N} \gamma_i Y_{it} \Bigr\rVert_{\mathcal H}^{2} , over the simplex, and the counterfactual is :math:`\widehat\nu^{N}_{1t} = \Psi^{-1}\bigl(\sum_i \widehat\gamma_i Y_{it}\bigr)`. Once each object is sampled on a common grid this is the ordinary simplex least-squares problem, with the :math:`(\text{period}, \text{grid point})` pairs stacked into one long vector. So the base solve here is mlsynth's own exact quadratic program, the same one :doc:`vanillasc` uses. Nothing about the optimisation is special; the embedding is what makes it applicable. When the pre-treatment fit is imperfect the estimator is biased, and section 3.2 of the paper corrects it the way augmented synthetic control corrects the scalar case: fit a ridge regression of the post-period object on the pre-period ones, and add the imbalance it predicts. Expanding each centered object in a cubic B-spline basis :math:`\{\varphi_k\}_{k=1}^{K}` and writing :math:`r_{i\cdot}` for the stacked coefficients gives a closed form, .. math:: \widehat\gamma^{\mathrm{aug}}_i = \widehat\gamma^{\mathrm{scm}}_i + \bigl(r_{1\cdot} - r_{0\cdot}' \widehat\gamma^{\mathrm{scm}}\bigr)' \bigl(r_{0\cdot}' r_{0\cdot} + \lambda I\bigr)^{-1} r_{i\cdot} , which is exactly the ridge-augmented synthetic control formula with the pre-period axis replaced by the (period, basis coefficient) axis. Two properties follow directly. The correction vanishes when the simplex fit is already perfect, so augmentation never disturbs a good fit. And the augmented weights still sum to one but may go negative, which means the synthetic unit is allowed outside the donor hull -- extrapolation, bought deliberately in exchange for balance. Because negative weights can push the result out of :math:`\mathcal Y`, the augmented estimate is projected back: .. math:: \widetilde Y^{N,\mathrm{aug}}_{1t} = \operatorname*{arg\,min}_{y \in \mathcal Y} \lVert y - \widehat Y^{N,\mathrm{aug}}_{1t} \rVert_{\mathcal H} . Both projections have closed forms. For quantile functions it is the increasing rearrangement -- sort the values -- and for matrices it is the eigenvalue clip: symmetrise, set negative eigenvalues to zero, reassemble. For plain curves :math:`\mathcal Y` is everything and there is nothing to do. The penalty :math:`\lambda` is chosen by leave-one-pre-period-out cross-validation unless you pass ``ridge_lambda``. One caution about reading it: the cross-validation objective is typically very flat near its minimum. On the paper's fertility data it varies by 0.03 percent across :math:`\lambda \in [5,7]` while the pre-treatment fit moves in its fourth decimal. The penalty is weakly identified and the estimate is insensitive to it, so do not read the selected value as an estimated quantity. Inference and diagnostics ------------------------- Two procedures, selected by ``inference``. The conformal band inverts the sharp null :math:`Y^N_{1t}(x) = y_0` pointwise in the argument, comparing the post-treatment residual against the :math:`T_0` pre-treatment ones. The inversion is closed-form: the accepted set is an interval centred on the estimate whose half-width is a quantile of the absolute pre-treatment residuals. It requires :math:`\alpha > 1/(T_0 + 1)`; below that threshold nothing is ever excluded and the band is unbounded, which the estimator reports instead of returning an infinite interval. Two caveats belong with any band you report from this. The weights are held fixed across candidate nulls, not refit, unlike Chernozhukov, Wüthrich and Zhu (2021); that is a deliberate choice which guarantees the band contains the point estimate, at the cost of the exchangeability argument that would justify it. And the paper conjectures asymptotic validity instead of proving it. Treat the band as a descriptive uncertainty measure, not a calibrated confidence set. The placebo test recomputes everything with each donor cast as the treated unit and ranks the real treated unit's effect magnitude among them. It is honest and assumption-light, and it costs :math:`N` times a full fit. Its resolution is set by the donor pool: with :math:`N` units the smallest attainable p-value is :math:`1/N`, so 21 units can never produce a p-value below 0.048. Example ------- .. code-block:: python import pandas as pd from mlsynth import FSC # one row per (country, year, age): the fertility rate at that age df = pd.read_csv("basedata/okano_fsc_fertility.csv") res = FSC({ "df": df, "outcome": "asfr", # the value "argument": "age", # where along the object it sits "treat": "treat", "unitid": "unit", "time": "time", "space": "function", "display_graphs": False, }).fit() res.pre_treatment_fit_fsc # fit before augmentation res.pre_treatment_fit # fit after res.effects.att # grid-averaged post-treatment effect last = res.curves[-1] last.effect # the effect as a curve over age last.magnitude # its length, the paper's d_t For a distribution, pass the quantile function and set ``space="distribution"``; ``value_bounds`` truncates to the support before the rearrangement. For a covariance matrix, pass the row-major lower triangle with ``space="matrix"`` -- the off-diagonal coordinates are rescaled internally so that the half-vectorisation really is a Frobenius isometry. Verification ------------ Reproduced against Okano and Kurisu (2026) on the authors' own data. The fertility application matches the published pre-treatment fits and every donor weight of Table 1; the divergences on the other two applications are measured and explained on the replication page, not smoothed over. See :doc:`replications/fsc`, `benchmarks/cases/fsc_okano.py `_ and `benchmarks/cases/fsc_estimator.py `_. Not to be confused with ----------------------- :doc:`fscm` is Forward-Selected synthetic control (Cerulli), an entirely different method that happens to share three letters. :doc:`dsc` and :doc:`drsc` work on individual-level microdata, not on objects supplied per cell. :doc:`compsc` covers compositions, which are the paper's Example 5. Core API -------- .. autoclass:: FSC :members: fit .. autoclass:: mlsynth.utils.fsc_helpers.config.FSCConfig :members: .. autoclass:: mlsynth.utils.fsc_helpers.structures.FSCResults :members: .. autoclass:: mlsynth.utils.fsc_helpers.structures.FSCCurve :members: