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:`e_{it} = \sum_{j} w_{ij}\, d_{jt}`. The implied population ATE follows from Serenini & Masek's eq. 14, .. math:: \widehat{\mathrm{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:`\mathbf{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:`\mathbf{W} = \mathbf{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:`e_{it} = \sum_{j} w_{ij}\, d_{jt}` to the doubly-weighted SDID regression, so the estimator returns *two* numbers: the direct ATT :math:`\widehat{\tau}` (same form as SDID) and the per-exposure indirect coefficient :math:`\widehat{\tau}_s`. The population ATE then follows from :math:`\widehat{\mathrm{ATE}} = \widehat{\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:`\mathbf{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:`\mathbf{W} = \mathbf{0}` or no treated neighbours, SpSyDiD reduces numerically to plain :doc:`sdid` with :math:`\widehat{\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:`\widehat{\tau}_s` runs through :math:`\mathbf{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:`\mathbf{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 ^^^^^ Let :math:`\mathcal{N} \coloneqq \{1, \dots, N\}` index the units and :math:`t \in \mathcal{T} \coloneqq \{1, \dots, T\}` the periods (1-indexed); the intervention takes effect after period :math:`T_0`, so the pre-period is :math:`\mathcal{T}_1 \coloneqq \{t \in \mathcal{T} : t \le T_0\}` and the post-period is :math:`\mathcal{T}_2 \coloneqq \{t \in \mathcal{T} : t > T_0\}`, with :math:`T_{\mathrm{post}} \coloneqq T - T_0`. Let :math:`y_{it}` be the outcome, :math:`d_{it} \in \{0, 1\}` the direct-treatment indicator, and :math:`\mathbf{W} \in \mathbb{R}_{\ge 0}^{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:: e_{it} \coloneqq (\mathbf{W}\mathbf{d}_t)_i = \sum_{j \in \mathcal{N}} w_{ij}\, d_{jt} \in [0, 1], \qquad \mathbf{d}_t \coloneqq (d_{1t}, \dots, d_{Nt})^\top . The estimator auto-partitions :math:`\mathcal{N}` into * :math:`\mathcal{I}_{\mathrm{tr}}` -- directly treated units (:math:`d_{it} = 1` for some :math:`t`), with :math:`N_{\mathrm{tr}} \coloneqq |\mathcal{I}_{\mathrm{tr}}|`; * :math:`\mathcal{I}_{\mathrm{sp}}` -- indirectly treated units (:math:`d_{it} = 0` for all :math:`t` but :math:`e_{it} > 0` for some :math:`t`), with :math:`N_{\mathrm{sp}} \coloneqq |\mathcal{I}_{\mathrm{sp}}|`; * :math:`\mathcal{C}` -- pure controls (:math:`d_{it} = 0` and :math:`e_{it} = 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{\boldsymbol{\omega}} \in \Delta^{|\mathcal{C}|}` and time weights :math:`\widehat{\boldsymbol{\lambda}} \in \Delta^{T_0}` (each on the unit simplex :math:`\Delta^{m} \coloneqq \{\mathbf{x} \in \mathbb{R}_{\ge 0}^{m} : \|\mathbf{x}\|_1 = 1\}`) on :math:`\mathcal{C}` only. The regularisation parameter is :math:`\zeta \coloneqq T_{\mathrm{post}}^{1/4} \cdot \mathrm{sd}(\Delta \mathbf{y})`, 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{\boldsymbol{\alpha}}, \widehat{\boldsymbol{\beta}}) = \arg\min_{\tau, \tau_s, \mu, \boldsymbol{\alpha}, \boldsymbol{\beta}} \sum_{i \in \mathcal{N}} \sum_{t \in \mathcal{T}} \bigl[ y_{it} - \mu - \alpha_i - \beta_t - \tau\, d_{it} - \tau_s\, e_{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. With :math:`\overline{WD}` the average exposure over :math:`\mathcal{I}_{\mathrm{tr}} \cup \mathcal{I}_{\mathrm{sp}}` in the post-period, the indirect and total effects are .. math:: \widehat{\mathrm{AITE}} = \widehat{\tau}_s\, \overline{WD}, \qquad \widehat{\mathrm{ATE}} = \widehat{\tau}\,(1 + \overline{WD}). Identification assumptions ^^^^^^^^^^^^^^^^^^^^^^^^^^ A1. No anticipation -- units do not adjust outcomes in advance of the treatment. *Remark.* The pre-period outcomes :math:`y_{it}` for :math:`t \in \mathcal{T}_1` must reflect the no-treatment world, since they are what the SDID weights are fit on. If units respond before :math:`T_0` -- forward-looking firms or households re-optimising ahead of a known policy -- the pre-period fit is contaminated and both :math:`\widehat{\tau}` and :math:`\widehat{\tau}_s` absorb part of the anticipation, as in standard SDID. 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. *Remark.* This is the SDID identifying restriction carried over to the three-way partition: net of the unit effect :math:`\alpha_i` and the time effect :math:`\beta_t`, the pure controls :math:`\mathcal{C}` trace out the common trajectory that the directly- and indirectly-treated units would have followed absent the policy. The doubly-weighted design relaxes the raw parallel-trends requirement to hold only for the SDID-reweighted controls, which is why only :math:`\mathcal{C}` enters the weight fit. 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:`e_{it}`. *Remark.* This is what lets a single coefficient :math:`\tau_s` summarise the indirect effect: the exposure :math:`e_{it} = \sum_j w_{ij} d_{jt}` enters the regression as one extra additive column, scaling linearly with how much treated mass a unit's neighbours carry. If the spillover were non-linear in exposure (e.g. saturating once a few neighbours are treated), :math:`\widehat{\tau}_s` would recover only a local linear approximation. A4. Limited interference -- spillovers operate exclusively through the structure defined by the exogenous :math:`\mathbf{W}`. No other local or global interference mechanisms are assumed. *Remark.* The whole identification of :math:`\widehat{\tau}_s` runs through :math:`\mathbf{W}`, so the weight matrix must encode the true spillover channel and be exogenous to the treatment. Interference that flows through an unmodelled network, or general-equilibrium effects that hit every unit equally, are not captured by :math:`e_{it}` -- the latter are absorbed into the time effects :math:`\beta_t` and cannot be separated from the direct effect. 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). *Remark.* Because the unit weights are fit on :math:`\mathcal{C}` alone (the indirectly-treated units in :math:`\mathcal{I}_{\mathrm{sp}}` are held out, then re-entered with uniform weight :math:`1/N_{\mathrm{sp}}`), the same synthetic must stand in for the counterfactual of units it was not fit to. Serenini & Masek (2024) argue this transfer is credible in geographic settings, where a treated unit's exposed neighbours tend to resemble the treated unit itself; where spillover-affected units look unlike the treated, this is the assumption most at risk. Connection to existing methods ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ * When :math:`\mathbf{W} = \mathbf{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 / |\mathcal{C}|` 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:`\mathbf{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:`\mathbf{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/serenini/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 state-level finding is institutionalised as a per-replication cross-validation benchmark -- ``benchmarks/cases/spsydid_state_mc.py`` runs both ``SpSyDiD`` and the authors' own algorithm on each panel and asserts per-rep agreement; see the dedicated page :doc:`replications/spsydid`. The reference panels and adjacency matrices ship with mlsynth in ``basedata/``: * ``state_unemployment.parquet`` -- 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.