Synthetic Historical Control (SHC) ================================== .. currentmodule:: mlsynth Overview -------- The Synthetic Historical Control (SHC) method estimates the time-varying intervention effect on a **single treated unit using only its own time series** — no cross-sectional control units are required. It is the answer to the setting where *every* unit is treated (a nationwide policy, a global shock such as COVID-19) so the synthetic control method has no donor pool to draw on. SHC is due to Chen, Yang & Yang (2024). It builds on a semi-parametric time-series regression in which a smooth latent trend :math:`\ell_t` is a time-varying confounder. Rather than extrapolate an *assumed* parametric trend (as the interrupted-time-series, Prophet, and CausalImpact approaches do), SHC carries the synthetic-control idea over to a single series: it replaces the treated *unit* with a treated *block* and the cross-sectional donors with overlapping *historical blocks* from the same series, then matches the treated block's pre-intervention segment with a simplex combination of its historical counterparts. The matching quality is therefore *detectable* from the pre-period fit, unlike parametric trend extrapolation whose misspecification is invisible after the intervention. When to use this estimator -------------------------- Reach for SHC when **there is one treated unit and no credible untreated controls**, but a reasonably long pre-intervention series with **recurring local structure** (cycles, seasonal-like swings — not strict periodicity): * **Nationwide / global interventions.** A country-level minimum-wage hike, a national pension reform, or the macroeconomic impact of a pandemic, where no other unit is plausibly untreated. The paper's applications are Brexit's effect on UK GDP growth and COVID-19's effect on US GDP growth. * **Cross-sectional controls exist but fail the SC matching condition.** Even when donors are available, SHC can match the treated pre-period better than SC if the donors track the treated series poorly (the paper's Brexit case: SHC pre-period MSE 0.029 vs SC 0.256). If you have a clean panel with valid untreated donors, the cross-sectional estimators (:class:`mlsynth.CLUSTERSC`, :class:`mlsynth.PDA`, :class:`mlsynth.SBC`) are the appropriate tools. Mathematical formulation ------------------------ Setup ^^^^^ For a single treated unit with outcome :math:`\{y_t\}` and intervention indicator :math:`d_t` (0 for :math:`t \le T_o`, 1 afterwards), the semi-parametric model (Eq. 2; the implementation uses the simplified :math:`x_t`-free form) is .. math:: y_t = \ell_t + \delta_t d_t + \varepsilon_t, where :math:`\ell_t = \ell(t)` is a non-stochastic, smooth latent trend, :math:`\delta_t` is the time-varying intervention effect, and :math:`\varepsilon_t` is a zero-mean error. Because both :math:`\ell_t` and :math:`\delta_t` are unobserved post-intervention, naive pre/post or semi-parametric (Robinson 1988) methods cannot separate the two; SHC identifies :math:`\delta_t` by reconstructing the post-intervention :math:`\ell_t`. Treated and historical blocks ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Fix a pre-intervention block length :math:`m` and a post horizon :math:`n`. The **treated block** spans :math:`[T_o - (m-1),\, T_o + n]`, with pre-segment :math:`\boldsymbol\ell_{pre}` and post-segment :math:`\boldsymbol\ell_{post}`. The pre-period is sliced into :math:`N = T_o - n - (m-1)` overlapping **historical blocks**, each with the same pre/post split (Eq. 7). The SHC weights solve a simplex-matching problem on the latent pre-segments, .. math:: \widehat{\boldsymbol w} = \arg\min_{\boldsymbol w \in \mathbb{W}} \bigl\| \widehat{\boldsymbol\ell}_{pre} - \widehat{\boldsymbol L}_{pre} \boldsymbol w \bigr\|^2, \qquad \mathbb{W} = \{\boldsymbol w \ge 0 : \mathbf{1}^\top \boldsymbol w = 1\}, and the post-intervention counterfactual is the same combination applied to the historical forward segments, :math:`\widehat\ell_t(\widehat{\boldsymbol w}) = \sum_i \widehat w_i \, \widehat\ell_{t(i)}`. Identifying assumptions ^^^^^^^^^^^^^^^^^^^^^^^^ *Assumption 1 (regularity).* :math:`\{\varepsilon_t\}` is i.i.d. with zero mean and finite variance. *Remark.* This is what identifies the (semi-parametric) nuisance components in the pre-period. *Assumption 2(a) (smoothness).* :math:`\ell(\cdot)` has a bounded :math:`(H+1)`-th derivative, with :math:`m - 1 \ge H \ge 3`. *Remark.* The degree of smoothness :math:`H` controls the bias bound :math:`b_\epsilon(H, k) = 2\epsilon |k|^{H+1}/(H+1)!` (Proposition 1): the estimator is *approximately unbiased*, with bias vanishing as the latent component gets smoother or the post horizon :math:`k` shrinks. This is why SHC favors a **small post horizon** and why larger-horizon estimates should be read cautiously. *Assumption 2(b) (matching).* The treated pre-segment is reproducible as a convex combination of its historical counterparts, :math:`\boldsymbol\ell_{pre} = \boldsymbol\ell_{pre}(\boldsymbol w_o)` for some :math:`\boldsymbol w_o \in \mathbb{W}`. *Remark.* This is the distributional analogue of the SC matching condition, transplanted from cross-sectional donors to historical blocks. It is **checkable** from the pre-period fit. It also precludes a pure growth trend (which cannot be reproduced by its own history), so differencing/detrending the series first is recommended. Algorithm (implementation) ^^^^^^^^^^^^^^^^^^^^^^^^^^^ The two-stage estimator (Section 2.3) is orchestrated by :func:`mlsynth.utils.shc_helpers.orchestration.solve_shc`: 1. **Latent trend.** Estimate :math:`\widehat\ell_t` over the pre-period by local-linear kernel regression, with the bandwidth chosen by leave-one-out cross-validation (``bandwidth_grid``). 2. **Blocks.** Build the treated block and the :math:`N` historical blocks. 3. **Matching.** Select and weight the historical blocks by the stepwise-matching procedure (Eqs. 32-34), yielding sparse simplex weights. 4. **Augmentation (optional).** ``use_augmented=True`` adds an ASHC ridge refinement on top of the simplex weights. 5. **Counterfactual.** Apply the weights to the historical forward segments to obtain :math:`\widehat\ell_t(\widehat{\boldsymbol w})` over the post horizon; the gap :math:`y_t - \widehat\ell_t(\widehat{\boldsymbol w})` estimates :math:`\delta_t`. Inference ^^^^^^^^^ ``SHC`` reports the conformal permutation test of Chen, Yang & Yang (2024, footnote 21) — their application of Chernozhukov, Wüthrich & Zhu (2021) — for the sharp null :math:`H_0: \delta_t = 0` over the post period. The test statistic is .. math:: S = n^{-1/2} \sum_{t=T_o+1}^{T_o+n} \bigl| \hat\varepsilon_t^0 \bigr|, \qquad \hat\varepsilon_t^0 = y_t - \widehat\ell_t, and the null distribution is built by sampling :math:`n` residuals **with replacement** from the :math:`T_o` pre-intervention residuals, 1,000 times. ``results.inference`` exposes ``p_value``, ``test_statistic``, the 1/5/10% ``critical_values`` and ``reject`` decisions, the resampled ``null_distribution``, and Andrews-Genton conformal bands for the plot. .. note:: The test is designed for the empirical setting where a genuine effect is present (in the paper's Brexit application it rejects at the 1% level: :math:`S = 2.492 > 2.190`). Because the reference residuals are the in-sample kernel-smoother residuals, which are mildly under-dispersed relative to the true noise, the test can over-reject under an exact null; the paper does not run it in the (effect-free) simulation. Core API -------- .. autoclass:: mlsynth.SHC :members: :undoc-members: :show-inheritance: :special-members: __init__ Configuration ------------- .. autoclass:: mlsynth.config_models.SHCConfig :members: :undoc-members: Helper Modules -------------- .. automodule:: mlsynth.utils.shc_helpers.setup :members: :undoc-members: .. automodule:: mlsynth.utils.shc_helpers.orchestration :members: :undoc-members: .. automodule:: mlsynth.utils.shc_helpers.inference :members: :undoc-members: .. automodule:: mlsynth.utils.shc_helpers.structures :members: :undoc-members: .. automodule:: mlsynth.utils.shc_helpers.plotter :members: :undoc-members: Example ------- A self-contained one-draw example using the paper's own data-generating process (a smooth, recurring latent trend plus noise, with no intervention effect, so the counterfactual should track the latent component). Paste it into a fresh interpreter: .. code-block:: python import numpy as np from mlsynth import SHC from mlsynth.utils.shc_helpers import simulate_shc_panel # One panel from the Chen-Yang-Yang (2024) DGP (Section 3.1): # T_o = m(4h+1) = 90 pre-periods, n = 8 post-periods, delta_t = 0. df, info = simulate_shc_panel( m=10, h=2, n=8, P=10, sigma=0.1, w_f=(1, 0), regular=True, seed=0, ) res = SHC({ "df": df, "outcome": "y", "treat": "treated", "unitid": "unit", "time": "time", "m": 10, "display_graphs": False, }).fit() # SHC reconstructs the latent confounder over the post window. cf_post = res.counterfactual[10:] mse_post = np.mean((cf_post - info["latent_post"]) ** 2) print(f"ATT = {res.att:+.4f} (true effect = 0)") print(f"pre-period R^2 = {res.fit_diagnostics['r_squared_pre']:.3f}") print(f"MSE_post vs latent = {mse_post:.5f}") print(f"historical blocks = {res.inputs.N}, " f"selected = {len(res.weights_by_block)}") print(f"conformal p-value = {res.inference.p_value:.3f}") Monte Carlo Validation ====================== The estimator is validated against the paper's simulation design (Section 3.1), re-implemented in :mod:`mlsynth.utils.shc_helpers.simulation`. The latent confounder :math:`\ell_t` is a globally :math:`C^1` curve of alternating cosine "local trends" and cubic-Hermite connectors; the treated block's shape is a convex combination of :math:`h` historical shapes (Assumption 2(b)). The construction reproduces the paper's exact dimensions: with :math:`h = 4`, :math:`T_o = m(4h+1)` (425 for :math:`m=25`, 850 for :math:`m=50`) and :math:`N = T_o - n - (m-1)` historical blocks (376 and 776). With :math:`\delta_t = 0`, the exercise measures how well SHC recovers :math:`\ell_t`, via the mean squared matching error (``MSE_pre``, Eq. 31) and the mean squared prediction error against the *true* latent (``MSE_post(k)``, Eq. 38). .. code-block:: python from mlsynth.utils.shc_helpers import monte_carlo_shc out = monte_carlo_shc( n_reps=8, m=25, h=4, n=25, P=10, sigma=0.1, w_f=(1, 0, 0, 0), regular=True, k_grid=(1, 5, 10, 15, 25), ) print(out["mse_pre"], out["mse_post"]) Representative output (Regular-:math:`\ell`, :math:`\sigma = 0.1`, :math:`m = 25`):: MSE_pre = 0.0011 MSE_post(1) = 0.0010 MSE_post(5) = 0.0017 MSE_post(10) = 0.0016 MSE_post(15) = 0.0017 MSE_post(25) = 0.0016 Both measures are near zero, and ``MSE_post(k)`` rises from :math:`k = 1` before plateauing — consistent with the paper's finding that the bias bound (Proposition 1) grows with the horizon but stays mild for a smooth, regularly recurring latent component at low noise. .. automodule:: mlsynth.utils.shc_helpers.simulation :members: :undoc-members: .. automodule:: mlsynth.utils.shc_helpers.monte_carlo :members: :undoc-members: Developer API ============= Internal optimization and tuning routines for SHC and ASHC. When ``use_augmented`` is true, the original SHC weights chosen by forward selection are passed to the ASHC ridge refinement for bias correction. .. autofunction:: mlsynth.utils.inferutils.shc_conformal_test .. autofunction:: mlsynth.utils.estutils._solve_SHC_QP .. autofunction:: mlsynth.utils.estutils.tune_lambda_ashc References ---------- Chen, Yi-Ting, Jui-Chung Yang, and Tzu-Ting Yang (2024). "Synthetic Historical Control for Policy Evaluation." SSRN 4995085. Chernozhukov, V., Wüthrich, K., & Zhu, Y. (2021). "An Exact and Robust Conformal Inference Method for Counterfactual and Synthetic Controls." *Journal of the American Statistical Association* 116(536):1849-1864. Hamilton, J. D. (2018). "Why You Should Never Use the Hodrick-Prescott Filter." *Review of Economics and Statistics* 100(5):831-843. Robinson, P. M. (1988). "Root-N-Consistent Semiparametric Regression." *Econometrica* 56(4):931-954.