Mediation Analysis Synthetic Control (MEDSC) ============================================ .. currentmodule:: mlsynth When to Use This Method ----------------------- You have run a synthetic control and measured the total effect of an intervention on a treated unit, and now you want to know why the effect happened — how much of it ran through one particular channel you can observe, and how much did not. Concretely, you observe a mediator: a second series that the intervention plausibly moved, and that in turn moves the outcome. For California's Proposition 99, the outcome is per-capita cigarette sales, the intervention is the 1989 tobacco-control program, and the natural mediator is the retail price of cigarettes: the program raised the price (through its excise tax), and a higher price lowers sales. Part of the total drop in smoking is this price channel; the rest is everything else the program did — media campaigns, clean-air laws, changed norms. A total effect alone cannot separate the two. MEDSC, following Mellace and Pasquini ([MellacePasquini2022]_), splits the synthetic-control effect into a direct effect and an indirect effect that runs through the mediator, using the counterfactual machinery of synthetic control rather than the linear structural equations of classical mediation analysis. The key device is a cross-world synthetic control: alongside the ordinary synthetic control that reproduces the treated unit's pre-treatment outcome path, MEDSC builds a second control that also matches the treated unit's post-treatment mediator path. This second control answers a counterfactual question — what would the outcome have been under no treatment but with the mediator forced to its treated value — and the gap between the two controls isolates the channel. Reach for MEDSC when ^^^^^^^^^^^^^^^^^^^^^ * you have a single treated unit, a donor pool, and a clean pre/post intervention, exactly the setting of an ordinary synthetic control; * you also observe a mediator — a channel variable the intervention moves and that moves the outcome — over the same panel; * the question is mechanism, not just size: how much of the effect flows through that channel; * you can assemble a donor pool wide enough that some donors reach the treated unit's post-treatment mediator values (see Assumption 4). Do not use MEDSC when ^^^^^^^^^^^^^^^^^^^^^^^ * you only care about the total effect — an ordinary synthetic control (:doc:`vanillasc`) is simpler and makes fewer assumptions; * no donor comes close to the treated unit's post-treatment mediator path, so the cross-world control has to extrapolate far outside the donor range; * the mediator is measured with substantial error or is realised after the outcome, so the ordering "treatment moves mediator moves outcome" is not credible. Notation -------- Let :math:`i = 1` denote the treated unit and :math:`i = 2, \dots, n` the donors, observed for :math:`t = 1, \dots, T` with the intervention at :math:`T_0 + 1`. Write :math:`Y_{it}` for the outcome and :math:`M_{it}` for the mediator. Under the treatment the treated unit realises the outcome :math:`Y_{1t}` and the mediator :math:`M_{1t}(1)`; the estimands are three counterfactual contrasts against its untreated potential outcomes. The total effect compares the observed outcome to the no-treatment counterfactual :math:`\widehat Y^{0,M0}_{1t}`, the outcome the treated unit would have had with no treatment and its own untreated mediator: .. math:: \widehat\tau^{\text{tot}}_t = Y_{1t} - \widehat Y^{0,M0}_{1t}. The direct effect compares the observed outcome to a cross-world counterfactual :math:`\widehat Y^{0,M1}_{1t}`, the outcome under no treatment but holding the mediator at its treated post-treatment value :math:`M_{1t}(1)`: .. math:: \widehat\tau^{\text{dir}}_t = Y_{1t} - \widehat Y^{0,M1}_{1t}. Because the mediator is fixed at its treated path, whatever the direct contrast leaves is the effect not routed through the mediator. The indirect effect is the remainder, .. math:: \widehat\tau^{\text{ind}}_t = \widehat\tau^{\text{tot}}_t - \widehat\tau^{\text{dir}}_t = \widehat Y^{0,M1}_{1t} - \widehat Y^{0,M0}_{1t}, the difference between the two synthetic controls, which is the movement in the counterfactual outcome attributable purely to forcing the mediator to its treated value. The reported ATTs are the post-period means of each path. Estimation ---------- The total effect is an ordinary synthetic control: donor weights :math:`\mathbf{w}^{\text{tot}}` on the simplex reproduce the treated unit's pre-treatment outcome path, and :math:`\widehat Y^{0,M0}_{1t} = \sum_{i} w^{\text{tot}}_i Y_{it}`. The direct effect is a cross-world synthetic control, re-estimated separately for each post period :math:`t' > T_0`. Its donor weights match two blocks at once: the treated unit's pre-treatment outcome path (as for the total effect) and the treated unit's post-treatment mediator path up to :math:`t'`, :math:`\{M_{1s}(1)\}_{T_0 < s \le t'}`. The predictor weighting places ``pre_weight`` (default :math:`3/4`) on the pre-treatment block and the remaining :math:`1/4` equally across the post-treatment mediator constraints, following the paper. The counterfactual :math:`\widehat Y^{0,M1}_{1t'} = \sum_i w^{\text{dir}}_i(t') Y_{it'}` is then read off at :math:`t'`. Re-estimating per period lets the mediator match tighten as the post-treatment mediator path lengthens. With no covariates (the default) the pre-treatment block is the full pre-treatment outcome path and the weights are a simplex least-squares fit; this is the specification under which the paper's Proposition 99 decomposition reproduces. With covariates the predictor weights are cross-validated by the bilevel global search (the ``mscmt`` backend, shared with :doc:`vanillasc`), which chooses how to trade off outcome lags, pre-treatment mediator lags and covariates. Assumptions ----------- 1. Sequential ignorability of the mediator. Conditional on the donor pool spanning the untreated outcome, the mediator is as good as randomly assigned with respect to the outcome — the cross-world outcome :math:`Y^{0,M1}` is identified by matching the mediator path. Remark. This is the mediation-analysis analogue of the no-unmeasured- confounder condition, transposed to the synthetic-control setting: there is no third variable that moves both the mediator and the outcome and is left unmatched. It is the strong assumption of any mediation decomposition and cannot be tested from the data alone. 2. Synthetic-control validity. The treated unit's untreated outcome path lies in (or near) the convex hull of the donors, so both the total and the cross-world controls reproduce it without extrapolating. Remark. This is the standard Abadie condition, applied twice — once for the total control and once for the mediator-matched control. A poorly fitting pre-treatment path invalidates the decomposition just as it would a plain synthetic control. 3. Correct ordering. The treatment moves the mediator, which moves the outcome; the mediator is realised before the outcome in each period. Remark. If the outcome feeds back into the mediator within a period, the "through the mediator" language is not well defined and the split is not interpretable. 4. A mediator-spanning donor pool. Some donors attain mediator values close to the treated unit's post-treatment mediator path, so the cross-world control can match it on the simplex. Remark. This is why MEDSC exposes two donor pools. The treated unit's mediator often jumps at the intervention to a value the ordinary donor pool never reaches (California's post-1989 price exceeded every classic donor state's), so the direct pool adds units — the high-mediator states the total pool excludes — to bracket it. Without such donors the mediator match must extrapolate and the direct effect is unreliable. Inference and diagnostics ------------------------- Inference is Abadie's in-space placebo test on the total effect: each donor is treated as a pseudo-treated unit, its own synthetic control is refit on the remaining donors, and the treated unit's post-to-pre RMSPE ratio is ranked against the placebo ratios. Placebo units whose pre-treatment fit is poor — RMSPE above ``placebo_cutoff`` times the treated unit's (default five) — are dropped, and the p-value is the treated unit's rank among the survivors. The placebo test speaks to the total effect, the identified estimand; the direct and indirect channels are reported as point paths, and their credibility rests on the mediation assumptions above rather than on a sampling distribution. The pre-treatment RMSE of the total control (``pre_rmse_total``) is the primary fit diagnostic — the decomposition is only as trustworthy as the synthetic control it rests on. Example ------- .. code-block:: python import pandas as pd from mlsynth import MEDSC # Proposition 99: does the tobacco-control effect run through price? df = pd.read_csv("basedata/prop99_mediation.csv") program = ["Massachusetts", "Arizona", "Oregon", "Florida", "District of Columbia"] # other tobacco programs tax = ["Alaska", "Hawaii", "Maryland", "Michigan", "New Jersey", "New York", "Washington"] # high-tax states allstates = sorted(df["state"].unique()) direct_pool = [s for s in allstates if s not in ["California"] + program] total_pool = [s for s in direct_pool if s not in tax] df = df[df["state"].isin(["California"] + direct_pool)].copy() df["treated"] = ((df["state"] == "California") & (df["year"] >= 1989)).astype(int) res = MEDSC({ "df": df, "outcome": "cigsale", "mediator": "price", "treat": "treated", "unitid": "state", "time": "year", "total_donors": total_pool, # 38 classic Abadie controls "direct_donors": direct_pool, # + 7 high-price states to span the mediator "display_graphs": False, }).fit() res.att # total effect (packs per capita) res.att_direct # direct channel res.att_indirect # indirect (price) channel res.decomposition.direct # per-period direct path res.inference.p_value # placebo p-value on the total effect The headline ``att`` is the total effect; ``att_direct`` and ``att_indirect`` are the two channels, and ``decomposition`` carries the per-period paths and both counterfactuals. The total-effect donor weights are on ``weights``; the direct fit's final-period weights are in its summary statistics. Verification ------------ MEDSC reproduces the Mellace-Pasquini Proposition 99 decomposition on the CDC / Orzechowski-Walker Tax Burden on Tobacco data (``basedata/prop99_mediation.csv``). The novel cross-world direct effect matches the paper's Table 1 nearly cell for cell — :math:`-16.8` in 1995 (paper :math:`-16.77`) and :math:`-18.0` in 2000 (paper :math:`-17.28`) — and the indirect price channel reproduces its qualitative signature, opening at roughly zero in 1989 and growing negative thereafter (durable case ``benchmarks/cases/medsc_prop99.py``). See :doc:`replications/medsc` for the full comparison, including why the indirect channel's magnitude tracks the outcome-path total rather than the paper's predictor-tuned total. Core API -------- .. autoclass:: MEDSC :members: fit .. autoclass:: mlsynth.config_models.MEDSCConfig :members: References ---------- .. [MellacePasquini2022] Mellace, G., & Pasquini, A. (2022). "Identification and estimation of mediation effects with a synthetic control." Bank of Italy Working Paper (and arXiv:1912.12073).