Nonlinear Synthetic Control (NSC) ================================= .. currentmodule:: mlsynth Overview -------- NSC implements Tian (2023), *"The Synthetic Control Method with Nonlinear Outcomes: Estimating the Impact of the 2019 Anti-Extradition Law Amendments Bill Protests on Hong Kong's Economy"* (arXiv:2306.01967). NSC generalises the canonical Abadie-Diamond-Hainmueller (2010) synthetic-control method to panel-data settings where the untreated potential outcome is a nonlinear function of the underlying predictors. Three structural changes versus canonical SC: 1. **Drops the non-negativity restriction** on donor weights — only the adding-up constraint :math:`\sum_j w_j = 1` remains. The resulting *affine-weight* SC widens the set of treated units the method can handle, since it no longer requires the treated unit to sit inside the convex hull of the donors. 2. **Adds an elastic-net penalty** on the weights, with the L1 term weighted by the **pairwise pretreatment matching discrepancies** between the treated unit and each donor. This biases the estimator towards near-neighbour matching when the outcome is highly nonlinear and towards spread-out weights when it is more linear. 3. **Eigenvalue-scales the tuning parameters** so the dimensionless :math:`(a^*, b^*) \in [0, 1]` admit a coarse cross-validation grid. The paper recommends grid size 0.1 with coordinate-descent convergence. Inference defaults to the **Doudchenko-Imbens (2017)** variance estimator: for every period the variance of the gap is approximated by the MSE of predicting each donor's outcome from the other donors under the same :math:`(a^*, b^*)` regime. Per-period and ATT normal-based confidence intervals are returned. Mathematical Formulation ------------------------ Setup ^^^^^ For each unit :math:`i` we observe an outcome :math:`Y_{it}` and a ``(K \times 1)`` vector of pretreatment matching variables :math:`Z_i = [X_i; Y_{i,1}, \dots, Y_{i,T_0}]'`. Unit 1 receives treatment at :math:`T_0 + 1` and remains treated thereafter. The untreated potential outcome follows the interactive fixed-effects model .. math:: Y_{it}^0 = F(X_i' \beta_t + \mu_i' \lambda_t + \varepsilon_{it}), with :math:`F(\cdot)` a strictly monotonic link function and a smooth conditional-expectation function :math:`G(\cdot) = E_\varepsilon[F(\cdot)]`. NSC weight problem ^^^^^^^^^^^^^^^^^^ The weights solve (Tian 2023, eq. 7) .. math:: \min_{\{w_j\}} \quad \biggl\| Z_1 - \sum_j w_j Z_j \biggr\|^2 + a \sum_j |w_j| \, \| Z_1 - Z_j \| + b \sum_j w_j^{\,2} \quad \text{s.t.} \quad \sum_j w_j = 1. * **Pretreatment fit term** (first norm) -- minimised by ordinary affine SC weights. * **L1 penalty** :math:`a \sum_j |w_j| \, \|Z_1 - Z_j\|` -- pushes weight onto donors that are *close* to the treated unit in the matching variables. As :math:`a \to \infty` it collapses to the one-nearest-neighbour estimator. * **L2 penalty** :math:`b \sum_j w_j^2` -- spreads the weights. As :math:`b \to \infty` the weights become uniform and the estimator collapses to a difference-in-differences. Eigenvalue scaling ^^^^^^^^^^^^^^^^^^ The paper scales the raw multipliers so the dimensionless tuning parameters live on :math:`[0, 1]`. With :math:`n = \min(J, K)` and :math:`\lambda_1 \le \dots \le \lambda_n` the sorted non-zero eigenvalues of :math:`Z_0 Z_0'`, .. math:: b = b^* \, \lambda_{\lceil n b^* \rceil}, \qquad a = a^* \, \tilde \lambda_{\lceil n a^* \rceil}, where :math:`\tilde \lambda_*` come from the eigenvalues of :math:`Z_0 Z_0' + \mathrm{diag}(b)`. At :math:`a^* = 1` only the nearest neighbour receives weight; at :math:`b^* = 1` the weights are roughly uniform. Cross-validation ^^^^^^^^^^^^^^^^ R-faithful (NSC.R, ``fn_cv``): for each donor :math:`j`, fit NSC weights on its **pre-period** matching vector using the other :math:`J - 1` donors plus one randomly drawn extra donor (the pool size stays at :math:`J` so the eigenvalue scaling of :math:`(a^*, b^*)` is comparable to the final treated-unit fit). The score is the **post-period** MSPE of donor :math:`j`'s held-out outcomes against that weighted combination. The (a, b) selected minimises the average held-out MSPE across donors. Coordinate descent: 1. Initialise :math:`b^* = 0`. 2. Sweep :math:`a^*` over the grid; pick the minimiser. 3. Hold :math:`a^*`; sweep :math:`b^*`; pick the minimiser. 4. Iterate until :math:`(a^*, b^*)` stops moving. The extra-donor draw uses ``seed`` (default ``123`` to match the reference R script's ``set.seed(123)``). A previous in-sample "controls" target (predicting the donor's *pre*-period from the others, i.e. evaluating on the same data the fit used) was removed in favour of this proper held-out score; the legacy ``cv_target = "treated"`` option no longer validates. Inference ^^^^^^^^^ Doudchenko-Imbens (2017): for each period :math:`t` the variance of the gap :math:`\hat\tau_t = Y_{1, t} - Y_{1, t}^{\text{SC}}` is estimated by averaging the squared leave-one-control prediction residuals at that period. Normal-based CIs follow: .. math:: \hat\tau_t \pm z_{1 - \alpha/2} \cdot \hat \sigma_t, \quad \widehat{\text{ATT}} \pm z_{1 - \alpha/2} \cdot \frac{\sqrt{\overline{\hat\sigma_t^{\,2}}}}{\sqrt{n_{\text{post}}}}. A two-sided z-test of :math:`H_0: \text{ATT} = 0` reports the p-value. Assumptions (Tian 2023) ----------------------- NSC inherits the interactive-fixed-effects (IFE) model of Abadie-Diamond-Hainmueller (2010) and adds the structural conditions needed for the synthetic-control bias to vanish under a **nonlinear** outcome. The paper's formal assumptions: **A1 (IFE for the untreated latent).** Each unit's untreated latent outcome is :math:`Y_{it}^{0*} = X_i' \beta_t + \mu_i' \lambda_t + \varepsilon_{it}`, where :math:`X_i` are observed predictors, :math:`\mu_i` unobserved loadings, and :math:`\varepsilon_{it}` an idiosyncratic shock. **A2 (transitory shocks).** Shocks :math:`\varepsilon_{it}` are independent across :math:`i, t`, have zero conditional mean given :math:`(X_j, \mu_j, D_{js})` for all :math:`j, s`, and bounded :math:`p`-th moments for some even :math:`p \ge 2`. **A4 (factor non-degeneracy).** The smallest eigenvalue of :math:`T_0^{-1} \sum_t \lambda_t \lambda_t'` is bounded away from zero. Translation: the unobserved factors carry persistent identifying variation across the pre-period. **A5 (overlap / continuous support).** The composite predictor :math:`H = [X', \mu']` has a density bounded away from zero on a compact convex support, units are iid draws from that distribution, and the treatment-assignment probability is non-degenerate. Practical reading: for almost every value of the treated unit's predictors there is a control with near-identical predictors *in the population* -- with enough donors, some of those near-twins land in the sample. **A6 (smooth nonlinear link).** The observed outcome is :math:`Y_{it}^0 = F(X_i' \beta_t + \mu_i' \lambda_t + \varepsilon_{it})` with :math:`F` strictly monotonic and :math:`G(\cdot) = \mathbb{E}_\varepsilon[F(\cdot)]` smooth. The paper does **not** require the analyst to know :math:`F` or :math:`G` -- only that they exist with these regularity properties. **A7 (nearest-:math:`M` weighting).** The synthetic control uses only the :math:`M > k + T_0` nearest neighbours (in observed predictors and pre-period outcomes) of the treated unit; donors outside that neighbourhood get weight zero. Under A2, A4, A5, A6, A7 the NSC estimator is asymptotically unbiased as :math:`T_0 \to \infty` provided the donor pool grows **super-polynomially** in :math:`T_0` (Theorem 2: :math:`J = O(T_0^{b(T_0)})` with :math:`b'(\cdot) > 0`). The fast growth is what makes Assumption 5's near-twins *available in finite samples*, so the local-neighbourhood matching in A7 is non-empty. When the assumptions bind: practical diagnostics ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ (a) **IFE / smooth-link DGP for the untreated outcome (A1, A6).** The untreated potential outcome is generated by a smooth monotone function of a low-rank factor model. *Plausibly violated when* the outcome is genuinely discrete (binary, ordinal, low-count) -- ``F`` is then non-monotonic *between intervals*, A6 is violated, and matching on observed pre-period outcomes no longer implies matching on the latent predictors (paper's Remark 8). *Diagnostic*: histogram the outcome; if it heaps on a small number of values, switch to a discrete-choice or DSC-style framework (:doc:`dsc` for the distributional question). (b) **Convex-hull / near-twin availability in the sample (A5).** NSC dropped the non-negativity restriction *precisely* because affine weights tolerate the treated unit lying *outside* the donor convex hull -- nearest neighbours can be reached with negative weights on far donors. But identification still requires the **population** density to put mass near the treated unit's predictors. *Plausibly violated when* the treated unit is qualitatively different from every potential donor -- a coastal mega-state against only interior small states, a tech-led economy with only commodity-led donors. *Diagnostic*: read ``res.pre_rmse``; a large pre-period RMSE despite the eigenvalue-scaled L1/L2 freedom signals no near-twin in the sample. Increasing the donor pool is the only fix -- NSC does not invent population mass that is not there. (c) **Donor pool size grows with** :math:`T_0` **(Theorem 2).** Bias vanishes asymptotically only if :math:`J` grows super-polynomially in :math:`T_0`. In finite samples, the actionable reading is: **more donors are unambiguously better in the nonlinear case** because they make the nearest-:math:`M` matching tighter (eq. 5). *Plausibly violated when* :math:`J` is small relative to :math:`T_0` -- a country-level study with a dozen donors and decades of pre-period. *Diagnostic*: re-run NSC after dropping the bottom-half of donors by closeness; if ATT shifts substantially, the small donor pool was binding. Cross-check against the Monte Carlo table below, which shows bias falling almost in half when :math:`J` doubles from 25 to 50. (d) **Pre-period factor non-degeneracy (A4).** The time factors :math:`\lambda_t` must move enough across the pre-period for the unobserved loadings to be identified. *Plausibly violated when* the pre-period is flat or co-trending across all units (e.g. all donors follow the same secular trend with no idiosyncratic shocks). *Diagnostic*: an SVD on the donor pre-matrix should show a clear spectrum, not a single dominant component drowning out the rest. (e) **Outcome is monotone in the latent index (A6).** The link function :math:`F` is strictly increasing in its argument. Many transformations satisfy this (logistic, square, log) but capped or saturating outcomes do not (e.g. an outcome with a true ceiling that some donors but not the treated unit hit, making the *same* latent value produce different observed outcomes). *Plausibly violated when* the outcome has a hard floor or ceiling that some donors are pressed against. *Diagnostic*: inspect the empirical distribution of donor outcomes against its theoretical bounds; if a non-trivial fraction sit at the boundary, A6 fails for those donors and they should be dropped or re-coded. (f) **Doudchenko-Imbens variance reflects a normal limit.** The closed-form CI assumes the gap is approximately normal with the leave-one-control variance estimate. With very small :math:`J` or heavy-tailed outcomes the normal approximation breaks. *Plausibly violated when* :math:`J < 10` or the donor outcomes are heavy-tailed. *Diagnostic*: bootstrap the post-period gap by resampling donors; if the bootstrap distribution is visibly skewed or shows heavy tails, treat the closed-form CI as a guide and report the bootstrap CI alongside. When to use NSC -- and when not to ---------------------------------- **Reach for NSC when:** * The outcome is a **smooth nonlinear** function of latent predictors -- bounded growth metrics, log-GDP, count outcomes modelled continuously, share variables on :math:`(0, 1)`. * The treated unit sits at or near the **boundary** of the donor predictor distribution -- canonical SC's non-negativity restriction would force interpolation bias from far donors, and affine weights with the L1 closeness penalty can reach the treated unit by **extrapolating** through the closest neighbours. * You have a **large donor pool** relative to the pre-period (:math:`J \gg T_0`) so that nearest-:math:`M` matching is sharp. * You're willing to **trade interpretability** for bias: NSC returns weights that can be negative, which means "subtract some donors", a step beyond the canonical SC story of "a convex combination of similar units". **Do not use NSC when:** * **Outcome is binary, ordinal, or low-count.** A6 requires a strictly monotonic link :math:`F`; discrete outcomes need discrete-choice or distributional methods (:doc:`dsc`) instead. * **You need genuinely sparse, non-negative weights.** NSC's whole premise is dropping non-negativity. If the policy story is "California is a convex combination of these four states", use *canonical SCM* or :doc:`tssc` -- NSC's negative weights are an identification gain but a rhetorical loss. * **Very small donor pool** (:math:`J \le 10` or so). Theorem 2 fails, the leave-one-control inference becomes unstable, and the L1-anchored nearest-neighbour story degenerates. * **Outcome with hard floors or ceilings binding for some donors.** Bounded growth where multiple donors are pinned at the boundary violates A6 for those donors. Drop or re-code them, or move to a censored-regression-aware estimator. * **Pre-period too short to identify the factor structure.** With :math:`T_0 \lesssim r` (factor count), A4 is at the edge and the cross-validation grid is choosing essentially noise. Use :doc:`fdid` (which is designed for short panels) or :doc:`fma` (which jointly estimates a low-rank factor model). * **You suspect interference / spillovers across units.** NSC inherits SUTVA from canonical SC; a treated unit influencing donors breaks A2's mean-independence. Switch to :doc:`spillsynth` or :doc:`spsydid`. * **Continuous treatment.** NSC encodes a single binary intervention. Continuous dose belongs in :doc:`ctsc`. Monte Carlo: NSC vs OSC bias under nonlinearity ----------------------------------------------- The paper's Section 4 simulation: a rank-2 IFE latent process rescaled to :math:`[0, 1]` and raised to the power :math:`r` (:math:`r = 1` linear; :math:`r = 2` nonlinear), with a small post-period treatment effect grid. The table below reproduces the qualitative finding -- NSC beats OSC (the non-negativity-restricted original) on bias in the nonlinear regime, and matches it in the linear one. .. code-block:: python """Tian (2023) Section 4 DGP: NSC vs OSC bias as nonlinearity rises. This runs a handful of replications (per (J, T0, r) cell). The full paper uses 5000 reps; this snippet keeps it tractable while still reproducing the qualitative ordering.""" import numpy as np import pandas as pd from mlsynth import NSC def gen_panel(J, T0, T_post, r, seed): rng = np.random.default_rng(seed) T = T0 + T_post X = rng.uniform(0.0, 2 * np.sqrt(3.0), size=(J + 1, 2)) mu = rng.uniform(0.0, 2 * np.sqrt(3.0), size=(J + 1, 4)) beta_t = rng.normal(10.0, 1.0, size=(T, 2)) lam_t = rng.normal(10.0, 1.0, size=(T, 4)) eps = rng.normal(0.0, 1.0, size=(T, J + 1)) Y_star = (X @ beta_t.T).T + (mu @ lam_t.T).T + eps Yn = (Y_star - Y_star.min()) / (Y_star.max() - Y_star.min()) return Yn ** r # (T, J+1) untreated outcomes def att_bias(Y0, T0, J, seed): T_post = Y0.shape[0] - T0 tau = np.linspace(0.02, 0.2, T_post) Y = Y0.copy(); Y[T0:, 0] += tau rows = [{"unit": j, "time": t, "y": float(Y[t, j]), "D": int(j == 0 and t >= T0)} for j in range(J + 1) for t in range(Y.shape[0])] df = pd.DataFrame(rows) res = NSC({"df": df, "outcome": "y", "treat": "D", "unitid": "unit", "time": "time", "cv_grid_size": 0.1, "run_inference": False, "display_graphs": False, "seed": seed}).fit() # Per-period |bias| averaged across post-period, ×100 (paper scale). return float(np.mean(np.abs(res.gap[T0:] - tau)) * 100.0) for (J, T0) in [(25, 15), (50, 15), (50, 30)]: for r in (1, 2): biases = [] for rep in range(20): Y0 = gen_panel(J=J, T0=T0, T_post=10, r=r, seed=10 * rep + 1) biases.append(att_bias(Y0, T0=T0, J=J, seed=10 * rep + 2)) print(f" J={J:>3d}, T0={T0:>2d}, r={r}: " f"|bias|x100 mean = {np.mean(biases):.2f}, " f"SD = {np.std(biases, ddof=1):.2f}") Representative output at one cell (mlsynth, 20 reps; the paper uses 5000): .. code-block:: text J=50, T0=15, r=1: |bias|x100 mean = 0.82, SD = 0.28 (paper NSC: 0.77) J=50, T0=15, r=2: |bias|x100 mean = 0.87, SD = 0.51 (paper NSC: 0.74) The qualitative paper findings to look for as you fill in the remaining cells: * **Bias falls as** :math:`J` **grows** -- direct evidence of the Theorem 2 mechanism: more donors makes the near-twin pool denser and the nearest-:math:`M` matching tighter. Paper Table 1: NSC bias drops from 0.99 (J=25) to 0.77 (J=50) in the linear case and from 0.92 to 0.74 in the nonlinear case at T0=15. * **NSC matches OSC in the linear case** and **wins in the nonlinear case**. Paper Table 1: at (J=25, T0=30), OSC bias is 1.79 (r=1) and 1.40 (r=2); NSC stays at 0.91 and 0.87. The non-negativity restriction in OSC pays an extrapolation penalty in the nonlinear regime that NSC's affine + L1-anchored weighting avoids. * **Coverage of the closed-form CI sits at 0.93-0.95** across the paper's :math:`(J, T_0, r)` grid -- the Doudchenko-Imbens variance estimator is well-calibrated for this DGP. mlsynth's CI reproduces that coverage band. Empirical: Proposition 99 (California, 1989-2000) ------------------------------------------------- Revisits the Abadie-Diamond-Hainmueller (2010) tobacco-control case study. mlsynth's NSC, R-faithful to the reference implementation, recovers the paper's published :math:`(a^*, b^*) = (0.3, 0.7)` and produces per-year gaps within a pack or two of the paper's reported numbers. .. code-block:: python import numpy as np import pandas as pd from mlsynth import NSC df = pd.read_stata("synth_smoking.dta") # Abadie-Diamond-Hainmueller (2010) df = df.sort_values(["state", "year"]).reset_index(drop=True) df["treatment"] = ((df["state"] == "California") & (df["year"] >= 1989)).astype(int) res = NSC({ "df": df, "outcome": "cigsale", "treat": "treatment", "unitid": "state", "time": "year", "cv_grid_size": 0.1, "run_inference": True, "display_graphs": False, "seed": 42, }).fit() print(f"selected (a*, b*) = ({res.design.a_star}, {res.design.b_star})") print(f"pre-RMSE = {res.pre_rmse:.4f}") print(f"ATT (1989-2000) = {res.att:+.4f} packs/capita") print(f"ATT 95% CI = ({res.inference.att_lower:+.2f}, " f"{res.inference.att_upper:+.2f})") gaps = pd.DataFrame({ "year": res.inputs.time_labels.astype(int), "gap": res.gap, "ci_low": res.inference.gap_lower, "ci_high": res.inference.gap_upper, }) for y in (1990, 1995, 2000): row = gaps.loc[gaps.year == y].iloc[0] print(f" {y}: gap = {row.gap:+6.2f} CI=({row.ci_low:+.2f}, {row.ci_high:+.2f})") Output:: selected (a*, b*) = (0.3, 0.7) # matches the paper exactly pre-RMSE = 1.2450 ATT (1989-2000) = -19.1313 packs/capita ATT 95% CI = (-25.51, -12.75) 1990: gap = -9.05 CI=(-26.38, +8.27) # paper: -9.5 1995: gap = -22.62 CI=(-46.03, +0.78) # paper: -24.5 2000: gap = -27.01 CI=(-54.31, +0.29) # paper: -28.7 Two things are worth flagging about this replication. * **The selected** :math:`(a^*, b^*) = (0.3, 0.7)` **is grid-stable across seeds 42, 789, 1000, 2024** and one grid step away from the paper for other seeds (e.g. seed 123 lands on :math:`(0.2, 0.8)`). This jitter is the same RNG noise the reference R script exhibits across ``set.seed()`` values -- unavoidable when the CV burns ``J`` random draws per grid point and Python's RNG differs from R's. * **Sparsity comparison vs OSC**: the original SC concentrates weight on Utah (0.385), Montana (0.271), Nevada (0.186). NSC spreads weight across roughly 20 states -- the L1-anchored affine fit is recruiting more near-neighbour Western states with smaller weights (some negative -- Alabama -0.015, Tennessee -0.071 in the paper's Table 2). The pre-period fit is similar (NSC pre-RMSE 1.25 vs OSC ~2.0) but the spread weights reduce the variance of the post-period counterfactual. The take-away mirrors the paper's: NSC's affine + L1-anchored formulation finds a tighter pre-period fit by pulling in close-neighbour Western states (with some negative weights to correct for the few far donors it admits), which then translates into a smoother and slightly larger counterfactual decline. The ATT of ~-19 packs/capita over 1989-2000 is in the same ballpark as the canonical Abadie et al. result while using a strictly larger weight space. Core API -------- .. automodule:: mlsynth.estimators.nsc :members: :undoc-members: :show-inheritance: Configuration ------------- .. autoclass:: mlsynth.config_models.NSCConfig :members: :undoc-members: Helper Modules -------------- .. automodule:: mlsynth.utils.nsc_helpers.setup :members: :undoc-members: .. automodule:: mlsynth.utils.nsc_helpers.optimization :members: :undoc-members: .. automodule:: mlsynth.utils.nsc_helpers.crossval :members: :undoc-members: .. automodule:: mlsynth.utils.nsc_helpers.inference :members: :undoc-members: .. automodule:: mlsynth.utils.nsc_helpers.plotter :members: :undoc-members: .. automodule:: mlsynth.utils.nsc_helpers.structures :members: :undoc-members: Example ------- A self-contained one-draw Monte Carlo at the paper's nonlinear DGP (Tian 2023, Section 4): latent linear factor model, rescale to :math:`[0, 1]`, square (so :math:`r = 2`), apply a small treatment effect to the treated unit's post-period. .. code-block:: python """One draw of the Tian (2023) nonlinear-outcome simulation.""" import numpy as np import pandas as pd from mlsynth import NSC # --------------------------------------------------------------------- # 1. Simulate one nonlinear panel # --------------------------------------------------------------------- rng = np.random.default_rng(0) J = 12 # donors T_pre = 12 # pre-treatment periods T_post = 6 # post-treatment periods T = T_pre + T_post tau_true = 0.10 # treatment effect (on the unit interval after rescaling) # Latent linear factor model: Y* = X' beta_t + mu' lambda_t + epsilon X = rng.uniform(0.0, np.sqrt(12.0), size=(J + 1, 2)) mu = rng.uniform(0.0, np.sqrt(12.0), size=(J + 1, 4)) beta_t = rng.normal(10.0, 1.0, size=(T, 2)) lam_t = rng.normal(10.0, 1.0, size=(T, 4)) eps = rng.normal(0.0, 1.0, size=(T, J + 1)) Y_star = (X @ beta_t.T).T + (mu @ lam_t.T).T + eps # Rescale to [0, 1] then apply nonlinear transformation (r = 2 in the paper). Yn = (Y_star - Y_star.min()) / (Y_star.max() - Y_star.min()) Y_control = Yn ** 2 # Apply the additive treatment effect to unit 0 in the post-period. Y = Y_control.copy() Y[T_pre:, 0] += tau_true rows = [ { "unit": j, "time": t, "y": float(Y[t, j]), "D": int(j == 0 and t >= T_pre), } for j in range(J + 1) for t in range(T) ] df = pd.DataFrame(rows) # --------------------------------------------------------------------- # 2. Fit NSC with CV-selected (a*, b*) # --------------------------------------------------------------------- results = NSC({ "df": df, "outcome": "y", "treat": "D", "unitid": "unit", "time": "time", "cv_target": "controls", # paper default "cv_grid_size": 0.1, "cv_max_iterations": 3, "alpha": 0.05, "run_inference": True, "display_graphs": False, }).fit() # --------------------------------------------------------------------- # 3. Inspect the output # --------------------------------------------------------------------- print(f"truth tau = {tau_true:+.3f}") print(f"ATT_hat = {results.att:+.3f}") print(f"a* = {results.design.a_star:.2f}, b* = {results.design.b_star:.2f}") print(f"pre-RMSE = {results.pre_rmse:.4f}") print(f"95% CI ATT = [{results.inference.att_lower:+.3f}, " f"{results.inference.att_upper:+.3f}]") print(f"p-value = {results.inference.p_value:.3f}") # Per-period diagnostics: import pandas as pd print(pd.DataFrame({ "t": np.arange(T), "gap": results.gap, "ci_low": results.inference.gap_lower, "ci_high": results.inference.gap_upper, }).round(3).to_string(index=False)) References ---------- Abadie, A., Diamond, A., & Hainmueller, J. (2010). "Synthetic Control Methods for Comparative Case Studies: Estimating the Effect of California's Tobacco Control Program." *Journal of the American Statistical Association* 105(490):493-505. Doudchenko, N., & Imbens, G. W. (2017). "Balancing, Regression, Difference-In-Differences and Synthetic Control Methods: A Synthesis." NBER Working Paper 22791. Tian, W. (2023). "The Synthetic Control Method with Nonlinear Outcomes: Estimating the Impact of the 2019 Anti-Extradition Law Amendments Bill Protests on Hong Kong's Economy." arXiv:2306.01967.