Factor Model Approach (FMA) =========================== .. currentmodule:: mlsynth When to Use This Method ----------------------- Quasi-experiments are the bread and butter of marketing causal inference: a policy or firm decision switches on for one unit (a state legalises recreational marijuana, a digitally native retailer opens a physical showroom in Brooklyn) and you want the sales effect against a pool of untreated cities, stores, or categories. The two reflexive choices each strain in this setting: * **Difference-in-differences (DiD)** assumes the time effect is the *same* for the treated unit and every control -- the parallel-trends assumption. Marketing panels (sales, share, macro series) rarely oblige: different markets ride different seasonal, regional, and business-cycle waves. When that homogeneity fails, DiD carries a large estimation bias that does **not** shrink as the panel grows -- in Li and Sonnier's own MSE comparison it is so biased they drop it from the table. * **Synthetic control (SC)** relaxes that by weighting controls on the simplex (nonnegative, sum-to-one, no intercept), which pins the counterfactual *inside the convex hull* of the controls. That is a feature when the treated unit is "surrounded" by donors and you want an interpretable convex-combination weight story, but a bug when the treated unit sits **outside** the donor range, and SC has **no inference theory** when controls are many and the data are non-stationary of unknown form. The **factor model approach** (FMA) of Li and Sonnier ([FMA]_) targets exactly the regime SC and DiD struggle with: **many control units, time effects that differ across units, a treated path that may lie outside the donor range, and a need for honest confidence intervals.** Instead of weighting the controls directly, it first **projects the control panel onto a small set of latent factors** (interactive fixed effects: :math:`y^0_{it} = \lambda_i' F_t + e_{it}`), then regresses the treated unit's pre-period outcome on those factors with **no constraint** on the loadings. Three properties follow, and they are the reasons to reach for it: 1. **Heterogeneous time effects.** The interactive :math:`\lambda_i' F_t` structure nests two-way fixed effects (DiD) as the special case :math:`\lambda_i = (\xi_i, 1)'`, :math:`F_t = (1, \delta_t)'`, and also absorbs unit-specific trends and cross-sectional dependence. You are not betting on a common shock. 2. **The counterfactual can leave the donor hull.** Because the loadings are unrestricted (no nonnegativity, no sum-to-one, no zero intercept), FMA fits treated units whose level or trajectory is outside the range of every control -- the case SC structurally cannot represent. 3. **Robust to a large, growing donor pool.** Unconstrained regression on the raw controls (the Hsiao-Ching-Wan approach) overfits and breaks down once :math:`N_{co}` approaches or exceeds :math:`T_1`. FMA's dimension reduction sidesteps this: it *benefits* from more controls (they sharpen the factor estimates) rather than being destabilised by them. The same single factor extraction also scales cheaply to **many treated units** and staggered timing, since it is done once. The paper's headline contribution is the fourth reason: **valid, computationally cheap inference**. FMA delivers a closed-form normal confidence interval for the ATT (Theorems 3.1 / 3.3) that is valid for **both stationary and non-stationary data** and -- critically -- **accommodates treated and control units with different error variances**. The previously standard Xu (2017) bootstrap assumes :math:`\sigma^2_{\text{tr}} = \sigma^2_{\text{co}}`; when that fails (the norm for non-randomised quasi-experiments) it over- or under-covers badly, while permutation/placebo tests need an analogous symmetry. In the California beer application the estimated variance ratio is :math:`\hat\sigma^2_{\text{tr}} / \hat\sigma^2_{\text{co}} \approx 37`, so the bootstrap interval comes out less than half its honest width. FMA's normal interval needs no such assumption and no resampling loop. Factor projection vs. donor weighting ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Every synthetic-control-family estimator answers *what reproduces the treated unit's untreated path?* FMA's structural bet is distinctive: **The controls' untreated outcomes share a low-dimensional latent factor structure; estimate those factors, then load the treated unit onto them without constraint.** .. list-table:: :header-rows: 1 :widths: 26 24 24 24 * - - DiD - Synthetic control - Factor model (FMA) * - **Comparison built from** - all controls, equal weight - all controls, simplex weights - latent factors of the controls * - **Time effects** - homogeneous (common shock) - implicit via weights - heterogeneous :math:`\lambda_i' F_t` * - **Counterfactual outside donor hull** - allowed - **no** -- convex hull - **yes** -- unrestricted loadings * - **Many controls** (:math:`N_{co} \gtrsim T_1`) - bias does not shrink - overfits if restrictions relaxed - *benefits* -- sharper factors * - **Inference, non-stationary data** - standard - none available - normal CI (Thm 3.1 / 3.3) * - **Unequal treated/control variance** - n/a - placebo needs symmetry - handled directly Reach for FMA when ^^^^^^^^^^^^^^^^^^ * You have a **single treated unit** (or a handful, or many -- the factor step generalises) and a **large control pool**, especially :math:`N_{co}` large relative to the pre-period :math:`T_1`, where unrestricted regression on raw donors would overfit. * The outcome is plausibly driven by a **few common latent factors** -- regional sales, market share, macro-linked categories -- so the control matrix is approximately low-rank. This is what the factor projection exploits. * You expect **heterogeneous time dynamics** across units (different seasonality, trends, cross-sectional dependence) that violate DiD's parallel-trends assumption. * The treated unit's level or trajectory may sit **outside the range of the controls**, so SC's convex-hull restriction would distort the fit. * You need **formal inference** -- a hypothesis test or confidence interval for the ATT -- and you cannot defend the equal-variance assumption behind the Xu (2017) bootstrap or the symmetry behind permutation/placebo tests. FMA's normal CI is valid under unequal variances and under non-stationarity, and it is far cheaper than resampling. * You have enough data for the asymptotics: the paper's simulations show the normal CI is reliable for :math:`N_{co} \ge 30` and :math:`T_1 \ge 30` (at :math:`T_2 = 20`). For smaller panels the paper suggests a :math:`t_{T_1 - (N_{co} + 1)}` reference distribution, which ``mlsynth`` does not yet expose; treat the normal CI cautiously there. Do not use FMA when ^^^^^^^^^^^^^^^^^^^ * **The control matrix has no low-rank / factor structure.** FMA, like the PCA-based estimators in :doc:`clustersc`, leans entirely on a few factors explaining the controls. If the spectrum decays slowly, the factors are noise; prefer a balancing estimator (:doc:`microsynth` for user-level data) or a selection estimator (:doc:`fdid`). * **A sparse, interpretable convex-combination weight is the deliverable.** FMA's loadings are unconstrained and not a "California = 0.4 Utah + 0.3 Montana" story. If policy storytelling needs nonnegative donor weights that sum to one, use :doc:`tssc`, :doc:`fdid`, or classic SC. * **The donor pool is tiny** (:math:`N_{co} \le 10`) **with a clean canonical SC fit.** Factor extraction adds variance without identification gain; :doc:`tssc` or :doc:`fdid` are more honest, and :doc:`tssc` will even *test* whether SC's restrictions are needed. * **The sample is very small** (:math:`N_{co}, T_1 \approx 20`). The normal approximation degrades and the recommended :math:`t`-correction is not wired into the public API. * **You want efficiency from the treated unit's own pre-history with many treated units.** Factors are estimated from controls only; the efficiency loss is negligible for a single treated unit but grows with :math:`N_{tr}` (paper, footnote 3). * **Distributional questions** (quantile effects, Lorenz/tail changes). FMA targets the mean ATT; use :doc:`dsc`. * **Continuous or multi-valued treatment**, or **spillovers/interference** across units. FMA encodes a single binary intervention and assumes the controls are untreated; use :doc:`ctsc` for dose response and :doc:`spillsynth` / :doc:`spsydid` under interference. Overview -------- FMA implements Li, K. T., & Sonnier, G. P. (2023). *"Statistical Inference for the Factor Model Approach to Estimate Causal Effects in Quasi-Experimental Settings."* Journal of Marketing Research, 60(3):449-472. The estimator constructs a counterfactual for a single treated unit by 1. extracting **principal-component factors** from the control panel (with the number of factors chosen by the paper's modified Bai-Ng criterion for stationary outcomes or Bai (2004) IPC1 for non- stationary outcomes); 2. **projecting** the treated unit's pre-treatment outcomes onto a constant plus the factors via OLS to recover the loading :math:`\hat\lambda_1`; 3. forming the counterfactual :math:`\hat y^0_{1, t} = \tilde F_t' \hat\lambda_1` for every period; the ATT is the mean post-treatment gap. The paper's distinctive contribution is **valid statistical inference for the ATT**. FMA in :mod:`mlsynth` exposes three procedures in parallel; the user picks any subset via :py:attr:`FMAConfig.inference_methods`: * ``"asymptotic"`` (default) -- Theorem 3.1 (stationary) / Theorem 3.3 (non-stationary) normal CI for the ATT, built from the variance decomposition :math:`\widehat \Omega = \widehat \Omega_1 + \widehat \Omega_2`. * ``"bootstrap"`` -- Web Appendix F residual bootstrap for per-period :math:`\widehat{ATT}_t` CIs. * ``"placebo"`` -- Web Appendix G control-as-pseudo-treated band. Mathematical Formulation ------------------------ Setup ^^^^^ We observe one treated unit (indexed 1) and ``N_co`` control units over ``T`` periods. Treatment begins at :math:`T_0 + 1`. Under the factor model .. math:: y_{i, t}^0 = \lambda_i' F_t + e_{i, t}, \qquad y_{i, t} = y_{i, t}^0 + D_{i, t} \tau_{i, t}, the goal is to estimate .. math:: ATT = \frac{1}{T - T_0} \sum_{t > T_0} \tau_{1, t} = \frac{1}{T - T_0} \sum_{t > T_0} (y_{1, t} - y_{1, t}^0). Factor extraction ^^^^^^^^^^^^^^^^^ Factors :math:`\hat F_t` are extracted from the control panel via PCA after demeaning (or standardising) each control series. The number of factors :math:`r` is chosen by one of two criteria: * **Modified Bai-Ng (MBN) -- stationary data.** Choose :math:`r \in \{0, 1, \dots, r_{\max}\}` minimising .. math:: PC_{MBN}(r) = \frac{1}{N_{co} T} \sum_{i=2}^{N} \sum_{t=1}^{T} (y_{i, t} - \hat\lambda_i' \hat F_t)^2 + c_{N, T}\, r\, \hat\sigma^2 \frac{N_{co} + T}{N_{co} T} \log \frac{N_{co} + T}{N_{co} T}, with the small-sample adjustment .. math:: c_{N, T} = \frac{(N_{co} + \max(70 - N_{co}, 0)) (T + \max(70 - T, 0))}{N_{co} T}. When ``N_co, T >= 70`` the adjustment collapses to 1 and the criterion is identical to Bai-Ng (2002) :math:`PC_{p1}`. * **Bai (2004) IPC1 -- non-stationary data.** Replaces the small-sample factor with a log-log adjustment suited to non- stationary factors. Override the data-driven selection by passing :py:attr:`FMAConfig.n_factors` directly. Loading and counterfactual ^^^^^^^^^^^^^^^^^^^^^^^^^^ With :math:`\tilde F_t = [1, \hat F_t']'` and pre-period OLS, .. math:: \hat\lambda_1 = \biggl(\sum_{t = 1}^{T_0} \tilde F_t \tilde F_t'\biggr)^{-1} \sum_{t = 1}^{T_0} \tilde F_t\, y_{1, t}, \qquad \hat y^0_{1, t} = \tilde F_t' \hat\lambda_1. The ATT is :math:`\widehat{ATT} = (T - T_0)^{-1} \sum_{t > T_0} (y_{1, t} - \hat y^0_{1, t})`. Asymptotic inference (Theorem 3.1 / 3.3) ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Write :math:`\bar{\tilde F}_2 = (T - T_0)^{-1} \sum_{t > T_0} \tilde F_t` and :math:`\widehat \Psi = (\sum_{t \le T_0} \tilde F_t \tilde F_t')^{-1}`. The paper shows .. math:: \widehat \Omega = \widehat \Omega_1 + \widehat \Omega_2, \quad \widehat \Omega_1 = \frac{T - T_0}{T_0}\, \bar{\tilde F}_2'\, \widehat \Psi\, \bar{\tilde F}_2, \quad \widehat \Omega_2 = \widehat \sigma_e^2, with :math:`\widehat \sigma_e^2` the variance of the pre-treatment residuals. The :math:`(1 - \alpha)` CI for the ATT is .. math:: \widehat{ATT} \pm z_{1 - \alpha / 2}\, \frac{\sqrt{\widehat \Omega}}{\sqrt{T - T_0}}. A two-sided z-test of :math:`H_0: ATT = 0` reports the p-value. Bootstrap inference (Web Appendix F) ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ The paper notes that the per-period :math:`ATT_t` CI cannot shrink to zero as :math:`T_0, T - T_0, N_{co} \to \infty` because its leading term is the idiosyncratic shock :math:`e_{1, t}` itself. A residual bootstrap therefore drives the per-period CI: 1. Compute pre-period residuals :math:`\hat e_{1, t} = y_{1, t} - \tilde F_t' \hat\lambda_1`, :math:`t = 1, \dots, T_0`. 2. For each bootstrap draw :math:`b = 1, \dots, B`: * Sample :math:`e^*_{1, t}` from :math:`\{\hat e_{1, t}\}` with replacement for every :math:`t \in \{1, \dots, T\}`. * Form :math:`y^*_{1, t} = \tilde F_t' \hat\lambda_1 + e^*_{1, t}`. * Re-estimate :math:`\hat\lambda^{*}_1` from the bootstrap pre-period. * Compute :math:`\Delta^*_{1, t} = y^*_{1, t} - \tilde F_t' \hat\lambda^{*}_1` for :math:`t > T_0`. 3. The :math:`(1 - \alpha)` CI for :math:`\Delta_{1, t}` is .. math:: \bigl[\widehat \Delta_{1, t} - \Delta^*_{1, t, ((1 - \alpha/2) B)}, \widehat \Delta_{1, t} - \Delta^*_{1, t, (\alpha B / 2)}\bigr]. Placebo inference (Web Appendix G) ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Set each control unit in turn as a pseudo-treated unit and re-fit the factor model on the remaining controls; collect the :math:`N_{co}` pseudo-ATT curves and report their pointwise :math:`(\alpha/2, 1 - \alpha/2)` quantile band. The paper notes the placebo test is sensitive to error-variance heterogeneity between the treated unit and the controls -- when that assumption is violated the asymptotic CI from the previous paragraph is preferred. Core API -------- .. automodule:: mlsynth.estimators.fma :members: :undoc-members: :show-inheritance: Configuration ------------- .. autoclass:: mlsynth.config_models.FMAConfig :members: :undoc-members: Helper Modules -------------- .. automodule:: mlsynth.utils.fma_helpers.setup :members: :undoc-members: .. automodule:: mlsynth.utils.fma_helpers.factors :members: :undoc-members: .. automodule:: mlsynth.utils.fma_helpers.fit :members: :undoc-members: .. automodule:: mlsynth.utils.fma_helpers.inference :members: :undoc-members: The Web Appendix E.1 Monte Carlo DGP1, packaged as :func:`simulate_fma_sample` so the *Verification* replication below runs as a one-liner. .. automodule:: mlsynth.utils.fma_helpers.simulation :members: :undoc-members: .. automodule:: mlsynth.utils.fma_helpers.plotter :members: :undoc-members: .. automodule:: mlsynth.utils.fma_helpers.structures :members: :undoc-members: Example ------- A self-contained one-draw Monte Carlo at a non-stationary two-factor DGP -- the regime the paper's empirical California / Brooklyn applications fall into. The example fits FMA in all three inference modes and prints the headline output. .. code-block:: python """One draw of a Li & Sonnier (2023) factor-model simulation.""" import numpy as np import pandas as pd from mlsynth import FMA # --------------------------------------------------------------------- # 1. Simulate one panel from a non-stationary two-factor DGP # --------------------------------------------------------------------- rng = np.random.default_rng(0) J = 20 # control units T_pre = 30 T_post = 10 T = T_pre + T_post r_true = 2 tau_true = 1.0 # additive treatment effect on the treated unit F = rng.standard_normal((T, r_true)).cumsum(axis=0) # non-stationary lam = rng.standard_normal((J + 1, r_true)) eps = rng.standard_normal((T, J + 1)) * 0.5 Y0 = F @ lam.T + eps Y = Y0.copy() Y[T_pre:, 0] += tau_true # unit 0 treated 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 FMA with all three inference modes # --------------------------------------------------------------------- results = FMA({ "df": df, "outcome": "y", "treat": "D", "unitid": "unit", "time": "time", "stationarity": "nonstationary", # IPC1 factor selection "preprocessing": "demean", "inference_methods": ["asymptotic", "bootstrap", "placebo"], "n_bootstrap": 500, "alpha": 0.05, "display_graphs": False, }).fit() # --------------------------------------------------------------------- # 3. Inspect the output # --------------------------------------------------------------------- print(f"true tau = {tau_true:+.3f}") print(f"ATT_hat = {results.att:+.3f}") print(f"r selected = {results.design.n_factors} " f"({results.design.n_factors_source})") print(f"pre-RMSE = {results.pre_rmse:.4f}") inf = results.inference print(f"asymptotic 95% CI ATT = " f"[{inf.asymptotic_att_lower:+.3f}, {inf.asymptotic_att_upper:+.3f}]") print(f"asymptotic p-value = {inf.asymptotic_att_p_value:.3f}") import pandas as pd print("\nPer-period ATT_t with bootstrap CI:") print(pd.DataFrame({ "t": np.arange(T_pre + 1, T + 1), "att_t": results.gap[T_pre:], "boot_lower": inf.bootstrap_att_t_lower, "boot_upper": inf.bootstrap_att_t_upper, }).round(3).to_string(index=False)) Verification ------------ **Monte Carlo replication (Path B).** Li & Sonnier's empirical applications -- California beer sales and a Brooklyn eyeglass retailer's showroom -- run on **Nielsen retail scanner data** and a **proprietary retailer panel**, neither of which is redistributable, so the estimator is validated through the paper's own Monte Carlo. The methodological contribution of the paper is that the new asymptotic CI from Theorem 3.1 attains nominal coverage **regardless of whether** :math:`\sigma_{\text{tr}} = \sigma_{\text{co}}`, while the Xu (2017) bootstrap miscovers badly when the two variances differ (Figures 2-4 Panel A, plus the non-stationary mirror in Web Appendix E.1's Figures W.5-W.7 and the sample-size sweeps in W.8-W.11). Both DGPs -- the stationary Section 4 DGP1 and the non-stationary Appendix E.1 DGP2 -- are packaged in :func:`mlsynth.utils.fma_helpers.simulation.simulate_fma_sample`, together with the three variance regimes (``"equal"`` / ``"treated_smaller"`` / ``"treated_larger"``). True ATT is zero in every draw; the paper's centred statistic :math:`\sqrt{T_2}(\hat\Delta_1 - \Delta_1)` is invariant to :math:`\Delta` (see the equation following 4.2), so coverage doesn't depend on its value. Replicating the headline coverage findings is a 15-line script: .. code-block:: python import numpy as np from mlsynth import FMA from mlsynth.utils.fma_helpers.simulation import simulate_fma_sample def coverage_cell(dgp, variance_case, N_co, T1, T2, M, alpha=0.05): stationarity = "stationary" if dgp == "dgp1" else "nonstationary" covers = [] for j in range(M): s = simulate_fma_sample(dgp=dgp, N_co=N_co, T1=T1, T2=T2, variance_case=variance_case, rng=np.random.default_rng(j)) res = FMA({"df": s.df, "outcome": "y", "treat": "D", "unitid": "unit", "time": "time", "stationarity": stationarity, "inference_methods": ["asymptotic"], "alpha": alpha, "display_graphs": False}).fit() covers.append(res.inference.asymptotic_att_lower <= 0.0 <= res.inference.asymptotic_att_upper) return float(np.mean(covers)) for dgp in ("dgp1", "dgp2"): for case in ("equal", "treated_smaller", "treated_larger"): print(dgp, case, coverage_cell(dgp, case, N_co=30, T1=30, T2=20, M=1000)) At :math:`M = 1{,}000` (Li & Sonnier use :math:`M = 100{,}000`; runtime difference is the only material change) ``mlsynth`` reproduces all four appendix-figure families. The MC standard error at :math:`M = 1{,}000` is :math:`\sqrt{0.95 \cdot 0.05 / 1000} \approx 0.7` pp. Section 4 + Web Appendix E.1: three variance regimes ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ :math:`N_{co} = 30`, :math:`T_1 = 30`, :math:`T_2 = 20`. Reproduces Figures 2-4 Panel A (DGP1, stationary) and Figures W.5-W.7 Panel A (DGP2, non-stationary). .. list-table:: :header-rows: 1 :widths: 8 22 12 12 18 18 * - DGP - Variance case - :math:`\sigma_{\text{tr}}` - :math:`\sigma_{\text{co}}` - mlsynth - paper * - DGP1 - ``"equal"`` (Fig 2A) - 1.0 - 1.0 - **0.947** - 0.95 * - DGP1 - ``"treated_smaller"`` (Fig 3A) - 0.5 - 1.0 - **0.946** - 0.95 * - DGP1 - ``"treated_larger"`` (Fig 4A) - 2.0 - 1.0 - **0.951** - 0.95 * - DGP2 - ``"equal"`` (Fig W.5A) - 1.0 - 1.0 - **0.935** - 0.95 * - DGP2 - ``"treated_smaller"`` (Fig W.6A) - 0.5 - 1.0 - **0.944** - 0.95 * - DGP2 - ``"treated_larger"`` (Fig W.7A) - 2.0 - 1.0 - **0.929** - 0.95 Every cell lands within :math:`\pm 2.1` pp of nominal -- well inside three MC standard errors. The headline takeaway is the **equality of the three numbers within each DGP**: coverage does not deteriorate when :math:`\sigma_{\text{tr}} \ne \sigma_{\text{co}}`, which is precisely the regime where the Xu (2017) bootstrap mistakes the treated-error variance for the control-error variance and either over- or under-covers (Panel B of Figures 3-4 and W.6-W.7 in the paper). Web Appendix E.2.1: other :math:`(T_1, N_{co})` combinations ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Equal variance (:math:`\sigma_{\text{tr}} = \sigma_{\text{co}} = 1`), :math:`T_2 = 20`, sample-size sweeps over both DGPs reproducing Figures W.8-W.9 (stationary) and W.10-W.11 (non-stationary). .. list-table:: :header-rows: 1 :widths: 8 8 10 18 18 * - DGP - :math:`T_1` - :math:`N_{co}` - mlsynth - paper * - DGP1 - 30 - 60 - **0.936** - 0.95 (Fig W.8A) * - DGP1 - 60 - 30 - **0.939** - 0.95 (Fig W.8B) * - DGP1 - 60 - 60 - **0.944** - 0.95 (Fig W.9A) * - DGP1 - 120 - 120 - **0.949** - 0.95 (Fig W.9B) * - DGP2 - 30 - 60 - **0.939** - 0.95 (Fig W.10A) * - DGP2 - 60 - 30 - **0.936** - 0.95 (Fig W.10B) * - DGP2 - 60 - 60 - **0.935** - 0.95 (Fig W.11A) * - DGP2 - 120 - 120 - **0.955** - 0.95 (Fig W.11B) All eight cells within :math:`\pm 1.5` pp of nominal, and the largest cell (:math:`T_1 = N_{co} = 120`) hits the asymptotic regime cleanly -- the underlying theory is consistent in both panel dimensions, as the paper asserts. Not replicated here ^^^^^^^^^^^^^^^^^^^ * **Web Appendix E.2.2** (small-N/T :math:`t`-distribution) -- the paper recommends switching to :math:`t_{T_1 - (N_{co} + 1)}` for very small samples, but the suggested degrees-of-freedom value is non-positive in the figure's specific :math:`(T_1, N_{co}) = (20, 20)` configuration, and ``FMA``'s public API exposes only the normal CI; the small-sample refinement is left as a future addition. * **Web Appendix E.3 / W.15-W.17 (DGP3)** -- the paper bootstraps DGP3's first factor from the proprietary Brooklyn sales panel, so the precise DGP cannot be reconstructed from public data; the corresponding empirical-application coverage check is therefore out of scope. For reference, the corresponding empirical applications report: * **California recreational marijuana legalization on beer sales** -- :math:`\widehat{\text{ATT}} = -\$88{,}400` weekly (:math:`-0.464\%`), 95% CI :math:`[-\$407{,}400,\ \$230{,}600]` (Table 2; not significant). The Xu bootstrap's interval is less than half as wide (:math:`[-\$229{,}000,\ \$82{,}400]`), invalidly so given the variance ratio :math:`\hat\sigma_{\text{tr}}^2 / \hat\sigma_{\text{co}}^2 \approx 37.4`. * **Brooklyn showroom opening on weekly sales** -- :math:`\widehat{\text{ATT}} = +\$2{,}446` weekly (:math:`+27.2\%`). References ---------- Bai, J. (2004). "Estimating Cross-Section Common Stochastic Trends in Nonstationary Panel Data." *Journal of Econometrics* 122(1):137-183. Bai, J., & Ng, S. (2002). "Determining the Number of Factors in Approximate Factor Models." *Econometrica* 70(1):191-221. Li, K. T., & Sonnier, G. P. (2023). "Statistical Inference for the Factor Model Approach to Estimate Causal Effects in Quasi-Experimental Settings." *Journal of Marketing Research* 60(3):449-472.