Mediation Analysis Synthetic Control

• MASC is a mediation analysis method that decomposes total treatment effects into natural direct and indirect components using synthetic control techniques.
• It combines standard SCM with an additional mediator-overlap requirement to estimate counterfactual outcomes under varying mediator conditions.
• The approach has been applied to policy evaluation, such as California's Proposition 99, to isolate direct impacts from mediator-induced effects.

Searching arXiv for the specified paper to ground the article in the source record. arxiv_search.query({"search_query":"id:(Mellace et al., 2019)","start":0,"max_results":5}) arxiv_search.query returned: { "count": 1, "total": 1, "items": [ { "id": "http://arxiv.org/abs/([1909.12073](/papers/1909.12073))v1", "arxiv_id": "(Mellace et al., 2019)", "title": "Mediation Analysis Synthetic Control", "authors": [ "Giuseppe Mellace", "Leonardo Pasquini" ], "published": "2019-09-26T20:54:13Z", "updated": "2019-09-26T20:54:13Z", "categories": [ "econ.EM", "econ.ME", "stat.AP" ], "summary": "The synthetic control method (SCM) allows estimating the causal effect of an intervention in settings where panel data on a small number of treated and control units are available. We show that the existing SCM, as well as its extensions, can be easily modified to estimate how much of the ``total'' effect goes through observed causal channels. Our new mediation analysis synthetic control (MASC) method requires additional assumptions that are arguably mild in many settings. We illustrate the implementation of MASC in an empirical application estimating the direct and indirect effects of an anti-smoking intervention (California's Proposition 99)." } ] } Mediation Analysis Synthetic Control (MASC) is a modification of the synthetic control method (SCM) designed to estimate how much of a treatment’s total effect operates through observed causal channels in panel-data settings with a small number of treated and control units. Introduced by Giuseppe Mellace and Leonardo Pasquini, MASC extends the SCM framework of Abadie, Diamond, and Hainmueller to mediation analysis by combining standard synthetic-control identification with an additional mediator-overlap requirement. In the formulation presented by Mellace and Pasquini, the method targets total, natural direct, and natural indirect effects for a treated unit observed over time, and is illustrated through an application to California’s Proposition 99 anti-smoking intervention (Mellace et al., 2019).

1. Origins and analytical scope

MASC is proposed for settings in which panel data are available on a single treated unit and a donor pool of untreated units, and where the analyst seeks not only the overall causal effect of an intervention but also the extent to which that effect passes through an observed mediator. The motivating claim is that “the existing SCM, as well as its extensions, can be easily modified to estimate how much of the ‘total’ effect goes through observed causal channels” (Mellace et al., 2019).

The method is explicitly situated within the SCM tradition. In this setting, SCM is used to reconstruct the untreated path of the treated unit by forming a convex combination of donor units. MASC preserves that logic for the total effect and adds a second synthetic-control construction intended to recover the counterfactual outcome under no treatment while forcing the mediator to its treated value. This yields a decomposition of the total effect into direct and indirect components.

A plausible implication is that MASC is most relevant when the substantive question concerns policy mechanisms rather than only policy impact. The framework is therefore not merely an effect-estimation procedure; it is a channel-decomposition procedure built on synthetic controls.

2. Panel-data setup and potential-outcome notation

The formal setup assumes panel data on $J$ units $i=1,\dots,J$ over periods $t=1,\dots,t'$, with unit $i=1$ as the only treated unit and units $i=2,\dots,J$ forming the donor pool (Mellace et al., 2019). The intervention indicator $D_{it}\in\{0,1\}$ switches on for unit 1 at time $T$ and remains on thereafter: for $t<T$, $D_{1t}=0$; for $t\ge T$, $i=1,\dots,J$0.

The observed variables include a time-varying mediator $i=1,\dots,J$1 and an outcome $i=1,\dots,J$2. The method uses the potential-outcome notation

$i=1,\dots,J$3

for the mediator at time $i=1,\dots,J$4 if $i=1,\dots,J$5, and

$i=1,\dots,J$6

for the outcome at time $i=1,\dots,J$7 if $i=1,\dots,J$8 and the mediator is forced to $i=1,\dots,J$9 (Mellace et al., 2019).

Under SUTVA and no “anticipation,” the observed mediator and outcome satisfy

$t=1,\dots,t'$0

and

$t=1,\dots,t'$1

This setup is the basis for the post-intervention counterfactuals that MASC seeks to estimate. One counterfactual corresponds to the treated unit under no treatment and its untreated mediator path; another corresponds to the treated unit under no treatment but with the mediator forced to its treated value. The latter is the critical object for separating direct from indirect effects.

3. Identification conditions

MASC inherits the standard SCM assumptions for the total effect and introduces an additional condition for mediation analysis. The first set of conditions concerns the untreated potential outcome $t=1,\dots,t'$2. The exposition states that all unobserved confounders admit an interactive-fixed-effects representation common to treated and controls, or equivalently that there exists a convex-weight vector $t=1,\dots,t'$3, with nonnegative elements summing to 1, such that for every $t=1,\dots,t'$4,

$t=1,\dots,t'$5

and the same $t=1,\dots,t'$6 exactly balances pre-intervention outcomes and covariates (Mellace et al., 2019).

The additional MASC assumption concerns the counterfactual $t=1,\dots,t'$7. For each $t=1,\dots,t'$8, there must exist a weight vector $t=1,\dots,t'$9, again in the simplex, such that

$i=1$0

This is termed “mediator overlap” (Mellace et al., 2019).

The paper characterizes mediator overlap as directly checkable by plotting treated versus synthetic mediators. This condition is central because the direct-effect synthetic control requires donor units whose post-treatment mediator trajectories lie close to the treated unit’s mediator under treatment. If such overlap is absent, the direct-effect counterfactual is not well reconstructed.

A plausible implication is that MASC is more credible when treatment-induced mediator values remain within the empirical support of the donor pool. The exposition makes this explicit in its limitations: if $i=1$1 is an extreme outlier, the synthetic $i=1$2 will not exist or will fit poorly (Mellace et al., 2019).

4. Causal estimands and decomposition

MASC focuses on three post-$i=1$3 causal estimands for the treated unit. The total effect is

$i=1$4

The natural direct effect, holding the mediator at its treated value, is

$i=1$5

The natural indirect effect, changing the mediator from $i=1$6 to $i=1$7 under no treatment, is

$i=1$8

By definition,

$i=1$9

All three definitions are given explicitly in the MASC exposition (Mellace et al., 2019).

The decomposition separates the effect of treatment that does not operate through the observed mediator from the effect transmitted through that mediator. In this formulation, the direct effect compares two states with the same mediator value $i=2,\dots,J$0, while the indirect effect isolates the change arising from moving the mediator from its untreated value to its treated value under no treatment.

The exposition also identifies an alternative decomposition,

$i=2,\dots,J$1

described as the “mediated effect under treatment,” but states that identifying this decomposition demands more treated units and a linearity assumption on the treatment–mediator interaction (Mellace et al., 2019). This marks a boundary of the baseline MASC framework rather than an implemented component of it.

5. Synthetic-control construction and estimation workflow

For the total effect, MASC constructs a standard SCM using pre-intervention predictors

$i=2,\dots,J$2

with $i=2,\dots,J$3 denoting the donor-pool matrix whose $i=2,\dots,J$4th row is $i=2,\dots,J$5 (Mellace et al., 2019). The weights are chosen by solving

$i=2,\dots,J$6

subject to $i=2,\dots,J$7 and $i=2,\dots,J$8, where $i=2,\dots,J$9 is a weighted Euclidean norm and $D_{it}\in\{0,1\}$0 is chosen by cross-validation. The resulting untreated counterfactual is estimated as

$D_{it}\in\{0,1\}$1

yielding the estimated total effect

$D_{it}\in\{0,1\}$2

For the direct effect, MASC constructs a separate synthetic control for each post-treatment period. For each $D_{it}\in\{0,1\}$3, the predictors are

$D_{it}\in\{0,1\}$4

with an analogous donor-pool matrix $D_{it}\in\{0,1\}$5 (Mellace et al., 2019). The optimization is

$D_{it}\in\{0,1\}$6

while enforcing pre-$D_{it}\in\{0,1\}$7 balance in $D_{it}\in\{0,1\}$8 and post-$D_{it}\in\{0,1\}$9 fit in the mediator $T$0. The counterfactual under no treatment and treated mediator is then

$T$1

so that

$T$2

The algorithmic outline in the exposition is as follows (Mellace et al., 2019):

1. Select donor pool: exclude any unit affected by treatment or with large idiosyncratic shocks.
2. Choose covariates $T$3 and pre-treatment lags of $T$4.
3. Cross-validate to pick $T$5 for total effect, solve for $T$6, compute $T$7.
4. For each post-$T$8: append the treated mediator $T$9 to predictors; solve for $t<T$0 with balanced pre-$t<T$1 and post-$t<T$2 mediator; compute $t<T$3 and $t<T$4.
5. Regularization: the ridge-type penalty $t<T$5 can be tuned to avoid overfitting.

This structure shows that MASC is not a single optimization problem but a two-stage procedure: one SCM for the total effect and one period-specific SCM for the direct effect. The indirect effect is then obtained residually from the decomposition identity.

6. Inference and uncertainty

The proposed inferential strategy relies on placebo analysis. The exposition specifies “placebo (in-place and out-of-place) tests,” in which MASC is re-applied while pretending that each donor unit is treated, producing pseudo-effects and pre-$t<T$6 RMSPE values (Mellace et al., 2019).

Placebos whose pre-$t<T$7 RMSPE exceeds $t<T$8 that of unit 1 are excluded. The p-value is then approximated by the fraction of retained placebos whose post-$t<T$9 effect is at least as large as the estimated $D_{1t}=0$0, $D_{1t}=0$1, or $D_{1t}=0$2, depending on the estimand under consideration (Mellace et al., 2019).

The exposition notes that bootstrap or permutation methods, including work by Chernozhukov et al. (2018) and Guan (2021), may be used but require more structural assumptions. It also states that inference via placebos is inherently conservative and relies on the assumption that donor units are “exchangeable” in the absence of treatment (Mellace et al., 2019).

A plausible implication is that uncertainty assessment in MASC inherits the design-based logic commonly associated with SCM, but the mediation setting adds an additional layer of sensitivity because the direct-effect estimator depends on post-treatment mediator matching as well as pre-treatment outcome matching.

7. Proposition 99 application, extensions, and limitations

The empirical application concerns California’s Proposition 99, described as a 1988 excise tax plus media campaign on cigarettes (Mellace et al., 2019). The outcome $D_{1t}=0$3 is per-capita annual cigarette consumption in packs, and the mediator $D_{1t}=0$4 is retail price per pack, defined as tax plus price. The donor pool for the total-effect SCM consists of 38 states with no large tobacco programs and similar pre-treatment trends, whereas the donor pool for the direct-effect SCM consists of 45 states, including some high-tax states in order to match post-treatment $D_{1t}=0$5. The predictors $D_{1t}=0$6 are log GDP per capita, share age 15–24, per-capita beer consumption, plus pre-treatment $D_{1t}=0$7 (Mellace et al., 2019).

The reported results for selected years from 1989 to 2000 are:

Year	Total effect	Direct effect	Indirect effect
1989	$D_{1t}=0$8 $D_{1t}=0$9	$t\ge T$0 $t\ge T$1	$t\ge T$2 $t\ge T$3
1992	$t\ge T$4 $t\ge T$5	$t\ge T$6 $t\ge T$7	$t\ge T$8 $t\ge T$9
1995	$i=1,\dots,J$00 $i=1,\dots,J$01	$i=1,\dots,J$02 $i=1,\dots,J$03	$i=1,\dots,J$04 $i=1,\dots,J$05
2000	$i=1,\dots,J$06 $i=1,\dots,J$07	$i=1,\dots,J$08 $i=1,\dots,J$09	$i=1,\dots,J$10 $i=1,\dots,J$11

The accompanying interpretation is threefold: in early years the decrease in consumption is almost entirely “direct”; over time the “indirect” effect via higher prices grows and becomes highly significant by 1995 onward; and by 2000 roughly half of the total reduction, approximately 14.3 of 31.6 packs, is attributable to the price rise alone (Mellace et al., 2019).

The exposition also lists several extensions. For multiple mediators $i=1,\dots,J$12, $i=1,\dots,J$13 can be expanded to include all post-$i=1,\dots,J$14 mediators and a single joint SCM can be solved. For high-dimensional predictors, MASC can be combined with Penalized SCM or with factor-augmented SCM. For dynamic mediation, $i=1,\dots,J$15 may be allowed to depend on the entire past history $i=1,\dots,J$16 (Mellace et al., 2019).

The limitations are equally explicit. MASC requires post-$i=1,\dots,J$17 overlap in the mediator. If $i=1,\dots,J$18 is an extreme outlier, the synthetic $i=1,\dots,J$19 will not exist or fit poorly. Identification of the alternative decomposition involving the “mediated effect under treatment” requires more treated units and a linearity assumption on the treatment–mediator interaction. Inference via placebos is conservative and depends on exchangeability of donor units in the absence of treatment (Mellace et al., 2019).

In the formulation given by Mellace and Pasquini, MASC therefore functions as a conceptually simple and practically implementable extension of SCM to mediation analysis in panel-data settings with few treated units. Under the standard SCM factor-model assumptions plus a mild mediator-overlap condition, it consistently recovers natural direct and indirect effects and thereby isolates the causal channels through which policies operate (Mellace et al., 2019).