Spatial Synthetic Difference-in-Differences (SpSyDiD) ====================================================== .. currentmodule:: mlsynth Overview -------- SpSyDiD (Serenini, R., & Masek, F. (2024). *"Spatial Synthetic Difference-in-Differences,"* SSRN 4736857) extends the Synthetic Difference-in-Differences (SDID) estimator of Arkhangelsky-Athey-Hirshberg-Imbens-Wager (2021) with a *spatial spillover term*. The estimator separates two estimands that standard SDID confounds when SUTVA is violated by geographic spillovers: * :math:`\widehat \tau` -- the direct ATT on the directly-treated units (identical in form to standard SDID). * :math:`\widehat \tau_s` -- the indirect / spillover coefficient per unit of neighbour-treatment exposure :math:`(WD)_{it} = \sum_j w_{ij} D_{jt}`. The implied population ATE follows from Serenini & Masek's eq. 14, .. math:: \widehat{ATE} = \widehat \tau \cdot \bigl(1 + \overline{WD}\bigr), where :math:`\overline{WD}` is the average exposure across the directly + indirectly treated units in the post-period. The user supplies a row-standardised :math:`N \times N` spatial weight matrix :math:`W`. Helpers in :mod:`mlsynth.utils.spsydid_helpers.spatial` cover the standard constructions: * :func:`knn_weights` -- :math:`k`-nearest neighbours from coordinates. * :func:`inverse_distance_weights` -- :math:`w_{ij} \propto 1/d_{ij}^p` with optional cutoff. * :func:`contiguity_weights` -- queen / rook contiguity from an adjacency dictionary. When :math:`W = 0` (no spatial structure) or no donor has any treated neighbour, SpSyDiD numerically reduces to plain SDID with :math:`\widehat \tau_s = 0`. When to Use This Method ----------------------- Every difference-based estimator -- DiD, synthetic control, and plain :doc:`sdid` -- rests on **SUTVA**: a control unit's outcome is unaffected by anyone else's treatment. Geography routinely breaks this. When a policy in the treated region leaks to its neighbours, those neighbours are exactly the units a synthetic control wants to lean on, and the leakage corrupts the comparison. Serenini & Masek (2024) make the bias explicit: * **Spillovers onto units *inside* the donor pool** bias and render **inconsistent** the standard SDID ATT -- the synthetic control is built from partially-treated donors, so the "untreated" benchmark is itself moving with the treatment. * **Spillovers *outside* the donor pool** leave the ATT identifiable but make the population **ATE** unidentified, because the indirect effect on exposed-but-excluded units is never measured. SpSyDiD targets this regime directly. It adds a single **spatial exposure term** :math:`(WD)_{it} = \sum_j w_{ij} D_{jt}` to the doubly-weighted SDID regression, so the estimator returns *two* numbers: the direct ATT :math:`\hat\tau` (same form as SDID) and the per-exposure indirect coefficient :math:`\hat\tau_s`. The population ATE then follows from :math:`\widehat{ATE} = \hat\tau\,(1 + \overline{WD})` (eq. 14). Relative to the older Spatial DiD of Delgado & Florax (2015), the synthetic weighting sharpens identification of the **indirect** effect while keeping SDID's robustness for the **direct** effect. Reach for SpSyDiD whenever there is a plausible mechanism for the treatment to *leak* from the directly-treated units to a subset of the donor pool through spatial or structural proximity, **and** you can supply a credible row-standardised weight matrix :math:`W` encoding that proximity. Typical examples: * **Immigration policy with cross-border relocation.** Arizona's 2007 LAWA legislation directly affected Arizona's noncitizen Hispanic population but also displaced workers to neighbouring states. SDID alone would either bias the ATT (if you include the spillover-affected states as controls) or be unable to estimate the spillover at all. * **State tax changes with cross-border shopping.** A state sales-tax increase affects that state's revenue directly *and* leaks via cross-border shopping into neighbouring states. * **Local advertising campaigns** with geographic spillovers across DMA boundaries. * **Vaccine mandates** with cross-state mobility effects. Do not use SpSyDiD when ^^^^^^^^^^^^^^^^^^^^^^^^ * **SUTVA holds / there is no spillover concern.** With :math:`W = 0` or no treated neighbours, SpSyDiD reduces numerically to plain :doc:`sdid` with :math:`\hat\tau_s = 0`; the extra exposure column just adds noise. Use :doc:`sdid` -- it is faster and more parsimonious. * **You cannot defend a spatial weight matrix.** The whole identification of :math:`\hat\tau_s` runs through :math:`W`. If proximity is not the spillover channel (e.g., interference flows through an unobserved social or supply-chain network you cannot encode), a misspecified :math:`W` buys biased indirect effects; consider :doc:`spillsynth`, which models spillover through donor membership rather than a fixed geographic kernel. * **Interference is global or non-local.** SpSyDiD assumes exposure is a *local*, distance-decaying function of neighbours' treatment. General equilibrium effects that hit every unit equally are absorbed into the time effects and cannot be separated. * **You only need the direct ATT and the donor pool is clean.** If the spillover-affected units can simply be *dropped* from the donor pool and the indirect effect is not of interest, plain :doc:`sdid` on the pruned pool is the simpler honest choice. * **Distributional questions** (quantiles, tails) -- use :doc:`dsc`; or a **single treated unit with no spatial structure** -- use :doc:`tssc` / :doc:`fdid`. Mathematical Formulation ------------------------ Setup ^^^^^ We observe :math:`N` units indexed :math:`i = 1, \dots, N` over :math:`T` periods. Treatment begins at :math:`T_0 + 1`. Let :math:`Y_{it}` be the outcome, :math:`D_{it} \in \{0, 1\}` the direct treatment indicator, and :math:`W \in \mathbb{R}^{N \times N}_+` a row-standardised spatial weight matrix with zero diagonal. The *spillover exposure* of unit :math:`i` at time :math:`t` is .. math:: (WD)_{it} = \sum_{j = 1}^{N} w_{ij}\, D_{jt} \in [0, 1]. The estimator auto-partitions the panel into * :math:`\mathcal{I}_{\mathrm{tr}}` -- directly treated units (:math:`D_{it} = 1` for some :math:`t`), * :math:`\mathcal{I}_{\mathrm{sp}}` -- indirectly treated units (:math:`D = 0` always but :math:`(WD) > 0` for some :math:`t`), * :math:`\mathcal{C}` -- pure controls (:math:`D = 0` and :math:`(WD) = 0` for all :math:`t`). Only :math:`\mathcal{C}` is used to fit the SDID unit / time weights. Algorithm ^^^^^^^^^ **Step 1 -- SDID weights from pure controls.** Following Arkhangelsky et al. (2021), fit the unit weights :math:`\widehat \omega \in \mathbb{R}^{|\mathcal{C}|}_+` and time weights :math:`\widehat \lambda \in \mathbb{R}^{T_0}_+` (each summing to 1) on :math:`\mathcal{C}` only. The regularisation parameter is :math:`\zeta = T_{\mathrm{post}}^{1/4} \cdot \mathrm{sd}(\Delta Y)` where :math:`\mathrm{sd}(\Delta Y)` is the standard deviation of the first-differenced pre-period donor outcomes. **Step 2 -- assemble the full weight vector.** Set .. math:: \widehat \omega_i = \begin{cases} 1 / N_{\mathrm{tr}} & i \in \mathcal{I}_{\mathrm{tr}}, \\ 1 / N_{\mathrm{sp}} & i \in \mathcal{I}_{\mathrm{sp}}, \\ \widehat \omega_i^{\mathrm{SDID}}& i \in \mathcal{C}. \end{cases} Time weights are SDID-fit for the pre-period and uniform :math:`1 / T_{\mathrm{post}}` for the post-period. **Step 3 -- augmented two-way FE WLS regression.** Solve .. math:: (\widehat \tau, \widehat \tau_s, \widehat \mu, \widehat \alpha, \widehat \beta) = \arg \min \sum_{i = 1}^{N} \sum_{t = 1}^{T} \bigl[ Y_{it} - \mu - \alpha_i - \beta_t - \tau\, D_{it} - \tau_s\, (WD)_{it} \bigr]^2 \widehat \omega_i\, \widehat \lambda_t. The augmented design jointly recovers the direct effect :math:`\widehat \tau` (the ATT) and the spillover coefficient :math:`\widehat \tau_s`. **Step 4 -- combine.** The implied population ATE is :math:`\widehat{ATE} = \widehat \tau \cdot (1 + \overline{WD})` where :math:`\overline{WD}` is the average exposure across :math:`\mathcal{I}_{\mathrm{tr}} \cup \mathcal{I}_{\mathrm{sp}}` in the post-period. Identification assumptions ^^^^^^^^^^^^^^^^^^^^^^^^^^ A1. **No anticipation** -- units do not adjust outcomes in advance of the treatment. A2. **Parallel trends** -- in the absence of treatment, treated, spillover, and control units would have followed similar trends, conditional on unit and time fixed effects. A3. **Additivity and linearity of spillovers** -- the potential outcome of a unit depends linearly and additively on its own treatment status and the treatment exposure of its neighbours, captured by :math:`(WD)_{it}`. A4. **Limited interference** -- spillovers operate exclusively through the structure defined by the exogenous :math:`W`. No other local or global interference mechanisms are assumed. A5. **Synthetic-control transferability** -- the SDID synthetic control built on the pure controls also approximates the counterfactual trajectory for the indirectly-treated units. This holds when spillover-affected units are spatially / structurally similar to directly-treated units, which is typically the case in geographic spillover settings (neighbours of treated states tend to resemble treated states). Connection to existing methods ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ * When :math:`W = 0` (no spatial structure), the spillover column vanishes and SpSyDiD reduces to plain SDID with :math:`\widehat \tau_s = 0`. * When :math:`\widehat \omega_i = 1 / N_{\mathrm{co}}` for all controls (uniform weights), SpSyDiD reduces to the Spatial Difference-in-Differences estimator of Delgado & Florax (2015). * When the panel is balanced + no spillover + non-trivial :math:`W`, SpSyDiD's :math:`\widehat \tau` matches SDID's ATT. Core API -------- .. automodule:: mlsynth.estimators.spsydid :members: :undoc-members: :show-inheritance: Configuration ------------- .. autoclass:: mlsynth.config_models.SpSyDiDConfig :members: :undoc-members: Helper Modules -------------- .. automodule:: mlsynth.utils.spsydid_helpers.spatial :members: :undoc-members: .. automodule:: mlsynth.utils.spsydid_helpers.setup :members: :undoc-members: .. automodule:: mlsynth.utils.spsydid_helpers.weights :members: :undoc-members: .. automodule:: mlsynth.utils.spsydid_helpers.pipeline :members: :undoc-members: .. automodule:: mlsynth.utils.spsydid_helpers.structures :members: :undoc-members: Example ------- A self-contained one-draw Monte Carlo on a :math:`8 \times 8` spatial grid. Six well-spaced units receive treatment of magnitude :math:`\tau = 2.0`; their :math:`k = 4` neighbours absorb a spillover of :math:`\tau_s = 1.0` per unit of exposure. SpSyDiD with the same :math:`W` recovers both estimates. .. code-block:: python """One draw of a spatial spillover simulation.""" import numpy as np import pandas as pd from mlsynth import SpSyDiD from mlsynth.utils.spsydid_helpers.spatial import knn_weights # --------------------------------------------------------------------- # 1. Lay out an 8x8 grid of units # --------------------------------------------------------------------- rng = np.random.default_rng(0) xs, ys = np.meshgrid(np.arange(8), np.arange(8)) coords = np.column_stack([xs.flatten(), ys.flatten()]) N = coords.shape[0] T_pre, T_post = 16, 8 T = T_pre + T_post W = knn_weights(coords, k=4, row_standardized=True) # --------------------------------------------------------------------- # 2. Two-way FE DGP with planted direct + spillover effects # --------------------------------------------------------------------- tau_true = 2.0 tau_s_true = 1.0 unit_fe = rng.standard_normal(N) * 0.5 time_fe = np.linspace(0.0, 1.0, T) Y0 = ( unit_fe[:, None] + time_fe[None, :] + rng.standard_normal((N, T)) * 0.2 ) D = np.zeros((N, T), dtype=float) for u in (0, 7, 24, 39, 56, 63): D[u, T_pre:] = 1.0 Y = Y0 + tau_true * D + tau_s_true * (W @ D) # --------------------------------------------------------------------- # 3. Long DataFrame # --------------------------------------------------------------------- rows = [ {"unit": i, "time": t, "y": float(Y[i, t]), "D": float(D[i, t])} for i in range(N) for t in range(T) ] df = pd.DataFrame(rows) # --------------------------------------------------------------------- # 4. Fit SpSyDiD # --------------------------------------------------------------------- res = SpSyDiD({ "df": df, "outcome": "y", "treat": "D", "unitid": "unit", "time": "time", "spatial_matrix": W, }).fit() # --------------------------------------------------------------------- # 5. Inspect the output # --------------------------------------------------------------------- print(f"true tau = {tau_true:+.3f} tau_hat = {res.att:+.3f}") print(f"true tau_s = {tau_s_true:+.3f} tau_s_hat = {res.aite:+.3f}") print(f"ATE = {res.ate:+.3f}") print(f"partition : {res.inputs.N_direct} direct, " f"{res.inputs.N_spillover} spillover, " f"{res.inputs.N_pure} pure controls") print(f"mean post-period exposure on treated union = " f"{res.metadata['mean_exposure_post_treated']:.3f}") .. _spsydid-verification: Verification (Path-B Monte Carlo) --------------------------------- Serenini & Masek (2024) include an empirical example (the Arizona 2007 LAWA effect on noncitizen Hispanic share, Tables 8-11) but do **not** release the CPS panel used to construct it -- their public replication repo (https://github.com/renanserenini/spatial_SDID) ships only the simulation code and a BLS unemployment panel for two Monte Carlo exercises. We therefore satisfy the Path-B contract by reproducing those two simulation findings against the authors' own driver (`functions_ssdid.py` in their repo), invoking ``SpSyDiD(config).fit()`` end-to-end on every replication. The reference panels and adjacency matrices ship with mlsynth in ``basedata/``: * ``state_unemployment.csv`` -- BLS monthly state unemployment 1976-2014. * ``US_no_islands_matrix.gal`` -- queen-contiguity W for the 49 contiguous states. * ``spsydid_bls_county_subset.csv`` -- the BLS county-employment slice (2002-2004, states WY/OR/PA/AL) used in the county-level MC. * ``spsydid_county_matrices.pkl`` -- per-state county adjacency matrices. State-level Monte Carlo (40 rolling-window replications) ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Reproduces ``State_Level_Simulations.ipynb``: at each 3-year window starting in 1975..2014, treat Arkansas (FIPS 5) only and inject :math:`\text{ATT} = 25\%` of mean unemployment plus :math:`\rho = 0.8` spillover via the queen-contiguity W. We compare the authors' reference algorithm against ``SpSyDiD(config).fit()`` on **the same 40 panels** to test for per-rep agreement. .. code-block:: text ref-mean ref-sd mlsynth-mean mlsynth-sd ATT bias +0.0187 0.3204 +0.0189 0.3229 rho bias (tau_s/ATT) +0.0596 0.9228 +0.0669 0.9965 per-rep correlation: ATT 0.9917 rho 0.9948 Both estimators recover the paper's headline finding: the **mean ATT bias is essentially zero** (~0.019 against an ATT magnitude of ~1.5 percentage points). Per-replication, the two implementations agree to ~0.02 on every panel realisation; the small residual is the unit-weight assignment for affected rows (mlsynth: :math:`1/N_{sp}`; reference: mean of treated-unit SDID weights). Both choices are valid downstream of the SDID weight QPs. The driver is :file:`examples/spsydid/replicate_state_level_mc.py`; run with ``python -m examples.spsydid.replicate_state_level_mc --reps 40``. County-level Monte Carlo (4 states x 200 reps) ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Reproduces ``Monte_Carlo_Simulations.ipynb``: for each of WY, OR, PA, AL, randomly draw 10% of counties as directly treated (multiple treated units per rep), inject :math:`\text{ATT} = -25\%` of mean unemployment plus :math:`\rho = 0.5` spillover, fit ``SpSyDiD(config).fit()``, repeat. The four states span 23-67 counties, so the test is whether the SUTVA correction works across panel sizes. .. code-block:: text state #counties #treated ATT bias mean (sd) AITE bias mean (sd) WY 23 2 -0.003 0.260 -0.018 0.143 OR 36 4 -0.023 0.225 -0.022 0.183 PA 67 7 +0.034 0.246 +0.062 0.127 AL 67 7 +0.028 0.228 -0.000 0.139 In every cell the absolute mean ATT bias is below 0.04 against an ATT magnitude of ~-1.5 -- the spatial-DGP-induced bias of plain SDID is cleanly removed by the SpSyDiD correction at the county scale. The driver is :file:`examples/spsydid/replicate_county_level_mc.py`; run with ``python -m examples.spsydid.replicate_county_level_mc --reps 200``. References ---------- Arkhangelsky, D., Athey, S., Hirshberg, D. A., Imbens, G. W., & Wager, S. (2021). "Synthetic Difference-in-Differences." *American Economic Review* 111(12):4088-4118. Delgado, M. S., & Florax, R. J. G. M. (2015). "Difference-in-Differences Techniques for Spatial Data: Local Autocorrelation and Spatial Interaction." *Economics Letters* 137:123-126. Serenini, R., & Masek, F. (2024). "Spatial Synthetic Difference-in-Differences." SSRN Working Paper 4736857.