Time-Aware Synthetic Control (TASC) =================================== .. currentmodule:: mlsynth Overview -------- Time-Aware Synthetic Control (TASC) `arXiv:2601.03099 `_ is a state-space synthetic-control estimator. Unlike classical SC (:doc:`fdid`, :doc:`tssc`) or robust-SC variants (:doc:`clustersc`, :doc:`proximal`), which treat the ordering of pre-intervention time indices as interchangeable, TASC explicitly models the temporal evolution of the latent factors driving the panel. It embeds the standard SC outcome matrix inside a linear-Gaussian state-space model with a constant trend matrix :math:`\mathbf{A}`, fits the model parameters via the Expectation-Maximization (EM) algorithm with a Kalman-filter + Rauch-Tung-Striebel (RTS) smoother E-step and a closed-form M-step, and produces both a point counterfactual and a posterior-based confidence band in one pass. Two structural properties distinguish TASC from the rest of the ``mlsynth`` toolkit: - Time-awareness. Because :math:`\mathbf{A}` is shared across periods, permuting the pre-intervention time indices changes the fit. Permutation-invariant methods (classical SC, robust SC, nuclear-norm matrix completion) produce identical counterfactuals under the same permutation; TASC does not. Section 5.1 of the paper formalizes this via a data-processing-inequality argument (Proposition A.1). - Approximately low-rank signal under omnidirectional noise. The observation matrix decomposes as :math:`\mathbf{Y} = \mathbf{H}\mathbf{X} + \mathbf{E}` where :math:`\mathbf{H}\mathbf{X}` is exactly rank-:math:`d` and :math:`\mathbf{E}` is full-rank observation noise. TASC therefore tolerates substantial measurement noise — even when PCA-style denoising (used by Robust SC) breaks down — because it does not assume the principal directions are noise-free. When to use TASC instead of something else ------------------------------------------ The Rho-Illick-Narasipura-Abadie-Hsu-Misra (2026) paper runs a 4-cell ablation comparing TASC against vanilla SC, Robust SC, and the Causal Impact Model under independent variation of the observation-noise covariance :math:`\mathbf{R}` and the state-perturbation covariance :math:`\mathbf{Q}` (Section 5.2, Figures 3-4 of the paper). The clean recommendation: * Use TASC when observation noise is high. Across the two large-:math:`\mathbf{R}` cells (small-:math:`\mathbf{Q}` and large-:math:`\mathbf{Q}`) TASC delivers the smallest median RMSE in the paper's simulation. PCA-style denoising (Robust SC) and simplex shrinkage (vanilla SC) break down because they assume the principal directions of the observation matrix are noise-free; TASC's full-rank :math:`\mathbf{r}_t \sim \mathcal{N}(0, \mathbf{R})` assumption is a much better fit when the noise is omnidirectional. * Use TASC when the donor panel has a persistent, smoothly-varying trend. "Persistent" means the trend extends past the intervention point. This is the strong-trend regime (small :math:`\mathbf{Q}`, non-trivial :math:`\mathbf{A}`). The Kalman + RTS smoother extrapolates the trend forward; PCA / nuclear-norm methods don't. * Use TASC when you need a posterior credible band for free. TASC is a generative model. The RTS smoother returns the full posterior covariance at every period, so a ``+/- 1.96 sigma`` band on the counterfactual is part of the fit's output. The other mlsynth estimators that ship credible bands are :doc:`bvss` (Bayesian spike-and-slab) and :doc:`tasc` itself; the rest require an external bootstrap or subsampling pass. When not to reach for TASC: * The pre-intervention trend is weak or absent (the paper writes ":math:`\mathbf{A} \approx 0`" — large :math:`\mathbf{Q}` regime). The smaller the trend, the smaller TASC's edge over classical SC; in the small-:math:`\mathbf{R}`, large-:math:`\mathbf{Q}` cell of the paper's ablation, vanilla SC matches or beats TASC. * Observation noise is small AND structured low-dimensional. Under small :math:`\mathbf{R}`, hard singular-value thresholding (Robust SC) cleans the signal exactly, and TASC's omnidirectional-:math:`\mathbf{R}` prior is paying a price for flexibility it doesn't need. * Long-horizon forecasting in noisy regimes. The paper's Figures 5-6 show that under large :math:`\mathbf{R}` and large :math:`\mathbf{Q}`, TASC's RMSE rises noticeably from horizon 51-60 to horizon 91-100 (small-:math:`\mathbf{Q}` is stable). If you need a 5-year-out counterfactual on a noisy panel, look at :doc:`fma` or :doc:`mcnnm` first. * Time indices are not really ordered (you're modelling a cross-section that happens to be indexed by time, or the periods are interchangeable up to relabelling). Permuting time indices costs TASC 48.5% on mean RMSE and 25.7% on the RMSE standard deviation in the paper's controlled test (Section 5.1, Figure 2). If the time ordering is meaningless, use a permutation-invariant estimator like :doc:`tssc` or :doc:`clustersc`. Notation -------- Let :math:`j = 1` denote the treated unit, with all units :math:`\mathcal{N} \coloneqq \{1, \dots, N\}` and donor pool :math:`\mathcal{N}_0 \coloneqq \mathcal{N} \setminus \{1\}` of cardinality :math:`N_0`. Time runs over :math:`t \in \mathcal{T} \coloneqq \{1, \dots, T\}`, 1-indexed; the intervention takes effect after period :math:`T_0`, splitting :math:`\mathcal{T}` into the pre-period :math:`\mathcal{T}_1 \coloneqq \{t \in \mathcal{T} : t \le T_0\}` (of length :math:`T_0`) and the post-period :math:`\mathcal{T}_2 \coloneqq \{t \in \mathcal{T} : t > T_0\}`. The treated series is :math:`\mathbf{y}_1 = (y_{11}, \dots, y_{1T})^\top \in \mathbb{R}^{T}` with scalar outcomes :math:`y_{1t}`; each donor :math:`j \in \mathcal{N}_0` contributes a series :math:`\mathbf{y}_j`, stacked into the donor matrix :math:`\mathbf{Y}_0 \coloneqq [\mathbf{y}_j]_{j \in \mathcal{N}_0} \in \mathbb{R}^{T \times N_0}` (one column per donor). The full panel stacks the treated series above the donors, so the observation vector at time :math:`t` is :math:`\mathbf{y}_t \in \mathbb{R}^{N}` with :math:`N = N_0 + 1`. TASC carries the panel through a latent state :math:`\mathbf{x}_t \in \mathbb{R}^d` of dimension :math:`d \ll \min(N_0, T)`, with transition matrix :math:`\mathbf{A} \in \mathbb{R}^{d \times d}`, observation (loading) matrix :math:`\mathbf{H} \in \mathbb{R}^{N \times d}` whose row :math:`\mathbf{h}_1^\top` loads the treated unit, state-perturbation covariance :math:`\mathbf{Q}`, and observation-noise covariance :math:`\mathbf{R}`. The synthetic counterfactual for the treated unit is :math:`\widehat{\mathbf{y}}_1` with entries :math:`\widehat{y}_{1t}`, the per-period treatment effect is :math:`\tau_t \coloneqq y_{1t} - \widehat{y}_{1t}`, and the ATT is :math:`\widehat{\tau} \coloneqq |\mathcal{T}_2|^{-1} \sum_{t \in \mathcal{T}_2} \tau_t`. The significance level is :math:`\alpha`. Assumptions (and how to spot violations) ---------------------------------------- TASC inherits the assumptions of a linear-Gaussian state-space model. Section 3 of the paper lays them out; the practitioner-facing restatement is: (a) Linear-Gaussian dynamics. The hidden state evolves as :math:`\mathbf{x}_t = \mathbf{A}\mathbf{x}_{t-1} + \mathbf{q}_t` with :math:`\mathbf{q}_t` zero-mean Gaussian. Equivalently: the trend in the latent factors is well-approximated by a stable linear AR(1) at the level of the state vector, and the perturbations around the trend are homoscedastic and uncorrelated across time. *Remark.* Plausibly violated when the latent factor evolution is strongly nonlinear (regime switches, breakpoints, structural breaks), has fat-tailed shocks, or has volatility clustering. Diagnostic: examine the smoothed state residuals from the pre-period fit; non-Gaussian QQ-plot tails or Ljung-Box-significant autocorrelation suggest misspecification. (b) Constant trend matrix :math:`\mathbf{A}`. The dynamics that hold over the pre-period are assumed to continue unchanged through the post-period. This is the "trend persists past the intervention point" assumption that gives TASC its long-horizon advantage. *Remark.* Plausibly violated when the intervention itself triggers a regime change in the donor units (e.g.\\ a tax change that affects neighbouring states' growth dynamics, not just their levels). TASC is by construction unable to detect a post-period change in :math:`\mathbf{A}` — the post-period target outcomes are treated as missing, so they cannot inform the update. Diagnostic: split the pre-period into two halves and refit on each. If the estimated :math:`\mathbf{A}` differs materially, the constant-:math:`\mathbf{A}` assumption is shaky and the post-period forecast is suspect. (c) Observation model :math:`\mathbf{y}_t = \mathbf{H}\mathbf{x}_t + \mathbf{r}_t`, :math:`\mathbf{R}` full rank, :math:`d \ll \min(N_0, T)`. The signal is low-rank with rank :math:`d` (the latent-state dimension); the noise :math:`\mathbf{r}_t` is Gaussian with a positive-definite covariance. Importantly, :math:`\mathbf{R}` does NOT have to be diagonal: TASC handles correlated cross-donor noise via the full :math:`\mathbf{R}` (set ``diagonal_R = False`` in the config). *Remark.* Plausibly violated when the residual cross-section is rank-deficient (some donors are exact linear combinations of others, e.g.\\ aggregated subseries paired with their components), or when the true signal is full-rank (no shared factors — every donor moves independently). In both cases the EM estimate of :math:`d` ends up wrong and either underfits (low :math:`d`) or overfits (high :math:`d`). (d) No unobserved confounders that affect donors AND treated unit between :math:`T_0` and :math:`T_0 + 1`. This is the standard SC unconfoundedness assumption, not specific to TASC, but worth restating: TASC's counterfactual is informative about the treatment effect only if any post-period shock to the donor pool is also reflected in what the target would have done absent treatment. *Remark.* Plausibly violated when a covariate that drives the treated unit's outcome (but is uncorrelated with the donors) shifts at the intervention time. TASC has no covariate hook, so this kind of confounding can only be diagnosed externally. (e) Hidden-state dimension :math:`d` correctly specified. TASC takes :math:`d` as a user hyperparameter (``hidden_state_dim``). The paper's Section 5.3 shows that underestimating :math:`d` is worse than overestimating — if in doubt, err on the high side. *Remark.* Plausibly violated when the data has more latent factors than you've allowed for. Diagnostic: increase :math:`d` and refit; if the RMSE on a held-out pre-period segment drops materially, you were underfitting. Mathematical Formulation ------------------------ The state-space machinery operates on the panel stacked with units in rows: let :math:`\mathbf{Y} \in \mathbb{R}^{N \times T}` be the outcome matrix whose first row is the treated unit :math:`j = 1` and whose remaining :math:`N_0` rows are the donor pool :math:`\mathcal{N}_0` (the same series collected column-wise in the canonical donor matrix :math:`\mathbf{Y}_0 \in \mathbb{R}^{T \times N_0}` of the Notation block). The column :math:`\mathbf{y}_t \in \mathbb{R}^{N}` is the time-:math:`t` observation vector. Pre-intervention periods are :math:`t \in \mathcal{T}_1`; over the post-intervention window :math:`t \in \mathcal{T}_2` the treated row is unobserved (the very quantity TASC reconstructs). State-Space Model ^^^^^^^^^^^^^^^^^ The TASC generative model (Eqs. (2)-(3) of the paper) is a classical linear-Gaussian state-space model: .. math:: \begin{aligned} \mathbf{x}_t &= \mathbf{A} \, \mathbf{x}_{t-1} + \mathbf{q}_{t-1}, & \mathbf{q}_{t-1} &\sim \mathcal{N}(\mathbf{0}, \mathbf{Q}), \\ \mathbf{y}_t &= \mathbf{H} \, \mathbf{x}_t + \mathbf{r}_t, & \mathbf{r}_t &\sim \mathcal{N}(\mathbf{0}, \mathbf{R}), \end{aligned} with initial state :math:`\mathbf{x}_0 \sim \mathcal{N}(\mathbf{m}_0, \mathbf{P}_0)`. The hidden state :math:`\mathbf{x}_t \in \mathbb{R}^d` has dimension :math:`d \ll \min(N_0, T)`, which is precisely what preserves the low-rank structure of the signal :math:`\mathbf{H}\mathbf{X}`. The complete parameter set is .. math:: \theta \;\coloneqq\; \{\mathbf{A}, \mathbf{H}, \mathbf{Q}, \mathbf{R}, \mathbf{m}_0, \mathbf{P}_0\}, \quad \mathbf{A} \in \mathbb{R}^{d \times d}, \quad \mathbf{H} \in \mathbb{R}^{N \times d}, \quad \mathbf{Q} \in \mathbb{R}^{d \times d}, \quad \mathbf{R} \in \mathbb{R}^{N \times N}. All three covariance matrices :math:`\mathbf{Q}, \mathbf{R}, \mathbf{P}_0` are positive definite. The :class:`TASCConfig` flags ``diagonal_Q`` and ``diagonal_R`` control whether the M-step constrains :math:`\mathbf{Q}` and :math:`\mathbf{R}` to be diagonal (the paper's default — see Algorithm 7) or updates the full symmetric covariance. Relationship to the Linear Factor Model ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ The classical SC linear factor model from Abadie & Gardeazabal (2003), .. math:: y_{jt} \;=\; \delta_t + \boldsymbol{\theta}_t^\top \mathbf{Z}_j + \boldsymbol{\lambda}_t^\top \boldsymbol{\mu}_j + \epsilon_{jt}, can be cast as a state-space model with latent state :math:`\mathbf{x}_t = (\delta_t, \boldsymbol{\theta}_t, \boldsymbol{\lambda}_t)` and observation rows :math:`\mathbf{h}_j = (1, \mathbf{Z}_j, \boldsymbol{\mu}_j)`. The crucial distinction is that linear factor models impose *no* dynamics on :math:`\mathbf{x}_t` (or equivalently :math:`\mathbf{A} = \mathbf{0}`, :math:`\mathbf{x}_t = \mathbf{q}_t`), whereas TASC enforces a stable trend through :math:`\mathbf{A}`. This is what gives TASC its long-horizon forecast accuracy under correct specification, at the cost of greater sensitivity to misspecification when temporal dynamics are complex. The Counterfactual via Infinite-Variance Kalman Filtering ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ In the post-intervention window the target's observed value is unavailable. TASC handles this by formally setting the target's observation-noise variance to :math:`+\infty` (Section 4.2 of the paper). Partition .. math:: \mathbf{y}_t = \begin{pmatrix} y_{1t} \\ \mathbf{y}_{0t} \end{pmatrix}, \quad \mathbf{H} = \begin{pmatrix} \mathbf{h}_1^\top \\ \mathbf{H}_0 \end{pmatrix}, \quad \mathbf{R}' = \begin{pmatrix} \infty & \mathbf{0} \\ \mathbf{0} & \mathbf{R}_0 \end{pmatrix}, where :math:`\mathbf{y}_{0t}, \mathbf{r}_{0t} \in \mathbb{R}^{N_0}`, :math:`\mathbf{H}_0 \in \mathbb{R}^{N_0 \times d}`, and :math:`\mathbf{R}_0 \in \mathbb{R}^{N_0 \times N_0}`. Under :math:`\mathbf{R}'`, the Schur-complement inverse of the innovation covariance .. math:: (\mathbf{S}_k)^{-1} \;=\; \begin{pmatrix} 0 & \mathbf{0} \\ \mathbf{0} & (\mathbf{H}_0 \mathbf{P}_{k|k-1} \mathbf{H}_0^\top + \mathbf{R}_0)^{-1} \end{pmatrix} has a zero in its (1, 1) block. The Kalman gain therefore picks up no contribution from the target row, and the post-intervention filter update depends only on the donor block. This is implemented in :func:`mlsynth.utils.tasc_helpers.filtering.kalman_filter_inf_variance_step` (Algorithm 5), and the full forward pass is composed by :func:`mlsynth.utils.tasc_helpers.filtering.kalman_filter_full` following Algorithm 3. Once the forward pass produces :math:`(\mathbf{m}_k, \mathbf{P}_k)_{k=0}^T`, the backward Rauch-Tung-Striebel smoother (:func:`mlsynth.utils.tasc_helpers.smoothing.rts_smoother`, Algorithm 6) returns the smoothed posterior .. math:: \mathbf{m}_k^s, \; \mathbf{P}_k^s, \; \mathbf{G}_k \quad \text{for } k = T, T-1, \dots, 0, with .. math:: \begin{aligned} \mathbf{m}_{k+1|k} &= \mathbf{A} \, \mathbf{m}_k, \\ \mathbf{P}_{k+1|k} &= \mathbf{A} \, \mathbf{P}_k \, \mathbf{A}^\top + \mathbf{Q}, \\ \mathbf{G}_k &= \mathbf{P}_k \, \mathbf{A}^\top \, \mathbf{P}_{k+1|k}^{-1}, \\ \mathbf{m}_k^s &= \mathbf{m}_k + \mathbf{G}_k \left( \mathbf{m}_{k+1}^s - \mathbf{m}_{k+1|k} \right), \\ \mathbf{P}_k^s &= \mathbf{P}_k + \mathbf{G}_k \left( \mathbf{P}_{k+1}^s - \mathbf{P}_{k+1|k} \right) \mathbf{G}_k^\top. \end{aligned} The counterfactual for the target unit is then read off the smoothed latent state via :math:`\mathbf{h}_1`: .. math:: \widehat{y}_{1t} \;=\; \mathbf{h}_1^\top \, \mathbf{m}_t^s, \qquad t = 1, \dots, T, and the posterior variance of the *observation* (not just the latent target) is .. math:: \operatorname{Var}\!\left(y_{1t} \mid \mathbf{y}_{1, \mathcal{T}_1}, \mathbf{Y}_{0, \mathcal{T}_2}\right) \;=\; \mathbf{h}_1^\top \, \mathbf{P}_t^s \, \mathbf{h}_1 \;+\; \mathbf{R}_{1, 1}. The corresponding :math:`(1 - \alpha)`-confidence band is .. math:: \widehat{y}_{1t} \;\pm\; z_{1 - \alpha / 2} \, \sqrt{\,\mathbf{h}_1^\top \mathbf{P}_t^s \mathbf{h}_1 + \mathbf{R}_{1, 1}\,}. These are populated unconditionally on :attr:`TASCResults.inference`, with :math:`\alpha` controlled by the :class:`TASCConfig.alpha` field. Learning :math:`\theta` from Pre-Intervention Data (EM) ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ The parameter set :math:`\theta` is learned by Expectation-Maximization on the pre-intervention slice :math:`\mathbf{Y}_{\text{pre}} \in \mathbb{R}^{N \times T_0}`. Each outer iteration of :func:`mlsynth.utils.tasc_helpers.em.em_pre` (Algorithm 2) runs: 1. E-step (filtering pass): apply the standard Kalman filter (Algorithm 4) for :math:`k = 1, \dots, T_0` to obtain :math:`(\mathbf{m}_k, \mathbf{P}_k)`. 2. E-step (smoothing pass): apply the RTS smoother backward to obtain :math:`(\mathbf{m}_k^s, \mathbf{P}_k^s, \mathbf{G}_k)` for :math:`k = T_0, \dots, 0`. 3. M-step (closed-form MLE update): Algorithm 7. Define the sufficient statistics .. math:: \begin{aligned} \boldsymbol{\Sigma} &= \frac{1}{T_0} \sum_{k=1}^{T_0} \left( \mathbf{P}_k^s + \mathbf{m}_k^s {\mathbf{m}_k^s}^\top \right), & \boldsymbol{\Phi} &= \frac{1}{T_0} \sum_{k=1}^{T_0} \left( \mathbf{P}_{k-1}^s + \mathbf{m}_{k-1}^s {\mathbf{m}_{k-1}^s}^\top \right), \\ \mathbf{B} &= \frac{1}{T_0} \sum_{k=1}^{T_0} \mathbf{y}_k \, {\mathbf{m}_k^s}^\top, & \mathbf{C} &= \frac{1}{T_0} \sum_{k=1}^{T_0} \left( \mathbf{P}_k^s \mathbf{G}_{k-1}^\top + \mathbf{m}_k^s {\mathbf{m}_{k-1}^s}^\top \right), \\ \mathbf{D} &= \frac{1}{T_0} \sum_{k=1}^{T_0} \mathbf{y}_k \, \mathbf{y}_k^\top. \end{aligned} The update is then .. math:: \begin{aligned} \mathbf{A}' &\leftarrow \mathbf{C} \, \boldsymbol{\Phi}^{-1}, & \mathbf{H}' &\leftarrow \mathbf{B} \, \boldsymbol{\Sigma}^{-1}, \\ \mathbf{Q}' &\leftarrow \operatorname{Diag}\!\left( \boldsymbol{\Sigma} - 2 \mathbf{C} \mathbf{A}'^\top + \mathbf{A}' \boldsymbol{\Phi} \mathbf{A}'^\top \right), & \mathbf{R}' &\leftarrow \operatorname{Diag}\!\left( \mathbf{D} - 2 \mathbf{B} \mathbf{H}'^\top + \mathbf{H}' \boldsymbol{\Sigma} \mathbf{H}'^\top \right), \\ \mathbf{m}_0' &\leftarrow \mathbf{m}_0^s, & \mathbf{P}_0' &\leftarrow \mathbf{P}_0^s + (\mathbf{m}_0^s - \mathbf{m}_0)(\mathbf{m}_0^s - \mathbf{m}_0)^\top, \end{aligned} where :math:`\operatorname{Diag}(\cdot)` zeroes the off-diagonal entries when ``diagonal_Q=True`` / ``diagonal_R=True`` (the paper's default) or returns the symmetric matrix unchanged otherwise. The loop terminates after :class:`TASCConfig.n_em_iter` outer iterations or, if :class:`TASCConfig.em_tol` is set, as soon as the maximum absolute change in :math:`(\mathbf{A}, \mathbf{H})` between successive iterations falls below the threshold. Spectral Initialization ^^^^^^^^^^^^^^^^^^^^^^^ EM is sensitive to initialization (as noted in the paper's Section 7). TASC therefore warm-starts :math:`\theta^{(0)}` from a truncated SVD of the pre-intervention matrix (:func:`mlsynth.utils.tasc_helpers.setup.initialize_parameters`): .. math:: \mathbf{Y}_{\text{pre}} \;=\; \mathbf{U} \, \operatorname{diag}(\mathbf{s}) \, \mathbf{V}^\top, \qquad \mathbf{H}^{(0)} = \mathbf{U}_{:, 1:d} \, \operatorname{diag}(\mathbf{s}_{1:d}), \qquad \mathbf{X}^{(0)} = \mathbf{V}_{:, 1:d}. The transition matrix :math:`\mathbf{A}^{(0)}` is obtained from a ridge- regularized AR(1) least-squares fit on the latent trajectory :math:`\mathbf{X}^{(0)}`; :math:`\mathbf{Q}^{(0)}` and :math:`\mathbf{R}^{(0)}` are seeded from the corresponding residual variances; and :math:`\mathbf{m}_0^{(0)}, \mathbf{P}_0^{(0)}` are taken from the first row of :math:`\mathbf{X}^{(0)}` and :math:`\mathbf{Q}^{(0)}` respectively. Treatment Effect and Pre-Period Fit ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ For post-treatment periods :math:`t \in \mathcal{T}_2`, the per-period treatment effect is the gap between the observed target and its TASC reconstruction, :math:`\tau_t \coloneqq y_{1t} - \widehat{y}_{1t}`, and the average treatment effect on the treated is its post-period mean: .. math:: \widehat{\tau} \;\coloneqq\; |\mathcal{T}_2|^{-1} \sum_{t \in \mathcal{T}_2} \tau_t \;=\; \frac{1}{T - T_0} \sum_{t = T_0 + 1}^{T} \left( y_{1t} - \widehat{y}_{1t} \right), reported as :attr:`TASCResults.att`. The pre-period RMSE between the observed target and the smoother's pre-treatment fit, .. math:: \mathrm{RMSE}_{\text{pre}} \;\coloneqq\; \sqrt{ \frac{1}{T_0} \sum_{t = 1}^{T_0} \left( y_{1t} - \widehat{y}_{1t} \right)^2 }, is reported as :attr:`TASCResults.pre_rmse` and serves as the primary fit diagnostic. Complexity ^^^^^^^^^^ The dominant cost of TASC is :math:`O(N_1 \, T_0 \, N^3)`, where :math:`N_1` is the number of EM iterations and the :math:`N^3` term arises from inverting the innovation covariance during the Kalman filter. The post-EM full-window pass adds :math:`O(T \, N^3)`, which is negligible when :math:`T \ll N_1 \, T_0`. Constraining :math:`\mathbf{R}` to be diagonal in the M-step (the default) does not change the filter's inner-loop complexity but does reduce parameter-count variance and improves numerical stability in moderate-:math:`N` regimes. Algorithm 1 and the Theoretical Appendix ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ The paper's Algorithm 1 is the abstract "SC Family of Methods" frame (target-side regression on donors), which TASC instantiates implicitly through the state-space machinery rather than as a discrete code path. Appendix A's Proposition A.1 (Kalman sufficiency, information loss by permutation invariance, dominance) is the theoretical justification for TASC's edge over permutation-invariant SC variants; it does not correspond to a separate routine in :mod:`mlsynth`. Core API -------- .. automodule:: mlsynth.estimators.tasc :members: :undoc-members: :show-inheritance: Configuration ------------- .. autoclass:: mlsynth.config_models.TASCConfig :members: :undoc-members: Helper Modules -------------- .. automodule:: mlsynth.utils.tasc_helpers.setup :members: :undoc-members: .. automodule:: mlsynth.utils.tasc_helpers.filtering :members: :undoc-members: .. automodule:: mlsynth.utils.tasc_helpers.smoothing :members: :undoc-members: .. automodule:: mlsynth.utils.tasc_helpers.mstep :members: :undoc-members: .. automodule:: mlsynth.utils.tasc_helpers.em :members: :undoc-members: .. automodule:: mlsynth.utils.tasc_helpers.orchestration :members: :undoc-members: .. automodule:: mlsynth.utils.tasc_helpers.inference :members: :undoc-members: .. automodule:: mlsynth.utils.tasc_helpers.plotter :members: :undoc-members: .. automodule:: mlsynth.utils.tasc_helpers.structures :members: :undoc-members: Example ------- .. code-block:: python from mlsynth import TASC # TASC accepts either a TASCConfig instance or a plain dict. config = { "df": df, "outcome": "sales", "unitid": "state", "time": "year", "treat": "treated", # binary 0/1 treatment indicator "d": 2, # hidden state dimension (small) "n_em_iter": 50, # N_1 in Algorithm 2 "em_tol": 1e-4, # optional early-stopping on max |delta(A, H)| "diagonal_Q": True, # paper default; set False for full covariance "diagonal_R": True, "alpha": 0.05, # significance level for posterior CIs "display_graphs": True, } results = TASC(config).fit() # Point estimate and fit diagnostic print(results.att) # mean post-period gap, y_{0,t} - h_1' m_t^s print(results.pre_rmse) # pre-period RMSE of the smoother's target fit # Counterfactual path and posterior confidence band (raw TASC inference). # ``results.counterfactual`` is the same series via the standardized accessor. cf = results.inference_detail.counterfactual # length-T vector lo = results.inference_detail.ci_lower # length-T vector hi = results.inference_detail.ci_upper # length-T vector # Learned model and EM diagnostics theta = results.design.parameters # A, H, Q, R, m0, P0 print(theta.A.shape, theta.H.shape) print(results.design.n_em_iter_used) # number of EM iterations executed print(results.design.em_param_deltas) # per-iteration max |delta(A, H)| # Smoothed latent trajectory (useful for downstream diagnostics) m_s = results.design.smoothed.m_s # shape (T + 1, d) P_s = results.design.smoothed.P_s # shape (T + 1, d, d) # Inputs preserved on the result object for plotting / re-analysis results.inputs.Y_full.shape # (N, T) results.inputs.Y_pre.shape # (N, T0) results.inputs.Y_post_donors.shape # (N - 1, T - T0) (None if no post) results.inputs.treated_unit_name results.inputs.donor_names Verification ------------ Empirical replication against the authors' published numbers (Path A) plus a Section 5 state-space Monte Carlo (Path B). Path A reruns the classical Proposition 99 California-tobacco illustration from Section 6.1 of [TASC]_ using the long-form panel :file:`basedata/prop99_packsales.csv` shipped with ``mlsynth``, and reproduces the post-1988 divergence between observed California cigarette sales and the TASC counterfactual that the paper's Figure 10 displays. Path B replicates the four-cell :math:`(\mathbf{Q}, \mathbf{R})` ablation grid (Figures 3 and 4) by drawing panels directly from TASC's own generative state-space model and comparing ``mlsynth.TASC`` against a simplex-constrained Synthetic Control baseline -- the same baseline the paper benchmarks. Path A: Proposition 99 California (Section 6.1) ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ The paper runs TASC on per-capita cigarette sales alone (no auxiliary predictors) with hidden-state dimension :math:`d = 2`. ``mlsynth.TASC`` on the same long-form panel reproduces the qualitative pattern of the paper's Figure 10 directly: .. code-block:: python import pandas as pd from mlsynth import TASC df = pd.read_csv("basedata/prop99_packsales.csv") df["treat"] = ((df["state"] == "California") & (df["year"] >= 1989)).astype(int) res = TASC({"df": df, "outcome": "cigsale", "unitid": "state", "time": "year", "treat": "treat", "d": 2, "n_em_iter": 50, "em_tol": 1e-4, "alpha": 0.05, "seed": 0, "display_graphs": False}).fit() yhat = res.inference_detail.counterfactual print(f"pre-RMSE = {res.pre_rmse:.3f} ATT = {res.att:.3f}") prints:: pre-RMSE = 0.767 ATT = -16.793 with the year-by-year trajectory .. list-table:: :header-rows: 1 :widths: 10 22 22 18 * - Year - California (observed) - TASC counterfactual - Gap * - 1985 - 102.80 - 102.66 - +0.14 * - 1988 - 90.10 - 91.88 - -1.78 * - 1989 - 82.40 - 88.30 - -5.90 * - 1990 - 77.80 - 84.37 - -6.57 * - 1995 - 56.40 - 76.50 - -20.10 * - 2000 - 41.60 - 65.14 - -23.54 The 1985-1988 fit is essentially tight on California's observed series (pre-RMSE :math:`= 0.77` packs against an outcome scale of roughly 100 packs), the divergence opens at the 1989 intervention, and the gap widens monotonically -- reaching a roughly :math:`-24` pack difference by 2000 against the paper's Figure-10 gap of about :math:`-25` to :math:`-30` packs at the same horizon. The average post-1989 treatment effect is :math:`\widehat{\tau} = -16.8` packs per year, in the same neighbourhood as Abadie, Diamond and Hainmueller's classical estimate. Path B: Section 5 state-space ablation grid ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ The paper's Section 5.2 ablation sweeps a :math:`2 \times 2` grid of state-perturbation and observation-noise covariance scales :math:`(\mathbf{Q}, \mathbf{R})` (Figures 3-4): a "small" covariance has diagonal variance :math:`0.01` (average :math:`|\mathbf{r}_t| \approx 0.084`) and a "big" covariance has diagonal variance :math:`1.0` (average :math:`|\mathbf{r}_t| \approx 0.836`). Panels are drawn from TASC's own generative model (Equations 2-3), so this is a *correctly-specified* Monte Carlo. The DGP is packaged as :func:`mlsynth.utils.tasc_helpers.simulation.simulate_tasc_sample`; the panel below compares the post-period RMSE of ``mlsynth.TASC`` (:math:`d_{\mathrm{fit}} = d_{\mathrm{true}} = 5`) against a simplex-constrained Synthetic Control baseline. .. code-block:: python import numpy as np import scipy.optimize as opt from mlsynth import TASC from mlsynth.utils.tasc_helpers.simulation import simulate_tasc_sample def sc_simplex(Y, T0): y, X = Y[0], Y[1:].T n = X.shape[1] cons = [{"type": "eq", "fun": lambda w: w.sum() - 1.0}] bnds = [(0.0, 1.0)] * n r = opt.minimize(lambda w: ((X[:T0] @ w - y[:T0]) ** 2).sum(), np.full(n, 1.0 / n), method="SLSQP", bounds=bnds, constraints=cons) return float(np.sqrt(np.mean((y[T0:] - X[T0:] @ r.x) ** 2))) def tasc_rmse(sample, d_fit=5): r = TASC({"df": sample.df, "outcome": "y", "treat": "treat", "unitid": "unit", "time": "time", "d": d_fit, "n_em_iter": 30, "em_tol": 1e-4, "alpha": 0.05, "seed": 0, "display_graphs": False}).fit() y0, T0 = sample.Y[0], sample.T0 return float(np.sqrt(np.mean( (y0[T0:] - r.inference.counterfactual[T0:]) ** 2))) M = 30 for q, r in [(0.01, 0.01), (0.01, 1.0), (0.10, 0.01), (0.10, 1.0)]: sc_, tasc_ = [], [] for seed in range(M): s = simulate_tasc_sample(q_scale=q, r_scale=r, rng=np.random.default_rng(seed)) sc_.append(sc_simplex(s.Y, s.T0)) tasc_.append(tasc_rmse(s)) print(f"q={q:.2f} r={r:.2f} TASC={np.median(tasc_):.3f} " f"SC={np.median(sc_):.3f}") prints (at :math:`M = 30`, :math:`N = 38`, :math:`T = 100`, :math:`T_0 = 50`, :math:`d_{\mathrm{true}} = 5`): .. list-table:: :header-rows: 1 :widths: 22 18 18 18 * - Regime - TASC median RMSE - SC median RMSE - Margin (SC / TASC) * - small Q, small R - 0.116 - 0.196 - 1.7x * - small Q, big R - 1.103 - 1.198 - 1.1x * - big Q, small R - 0.117 - 0.524 - 4.5x * - big Q, big R - 1.130 - 1.301 - 1.2x TASC carries the lowest median RMSE in all four regimes, and the margin over SC is largest precisely in the high-:math:`\mathbf{Q}` / low-:math:`\mathbf{R}` cell -- the same regime where the paper's Figure 4 identifies TASC's strongest dominance (a fitted state-space model extracts the persistent low-rank signal that the simplex projection cannot exploit). Under high observation noise (:math:`\mathbf{R} = 1.0`), the SC simplex projection still trails TASC but by a narrower margin, reflecting the noise floor common to both estimators. The paper's Figures 3-4 also include the Robust Synthetic Control of Amjad, Shah and Shen (2018) and the Causal Impact Model of Brodersen et al. (2015) as additional comparators that are not in ``mlsynth``; the ordering above against the canonical simplex-SC baseline is the slice of those comparisons that ``mlsynth`` can reproduce directly. The takeaway carried into the published TASC procedure is the paper's headline finding: when the data-generating process carries a persistent low-rank temporal signal -- as it does in many policy panels with strong trends -- explicitly fitting that temporal structure through a state-space model lowers post-period prediction error relative to permutation-invariant alternatives, and the advantage widens as the latent signal strengthens (large :math:`\mathbf{Q}`). References ---------- Rho, S., Illick, C., Narasipura, S., Abadie, A., Hsu, D., & Misra, V. (2026). *Time-Aware Synthetic Control.* `arXiv:2601.03099 `_. Durbin, J., & Koopman, S. J. (2012). *Time Series Analysis by State Space Methods.* Oxford Statistical Science Series 38, 2nd edition. Oxford University Press.