Double-Triple-Difference (DTD) Model

• The double-triple-difference model is an extension of triple-difference methods that identifies both direct treatment effects (ATT) and spillover effects (ASU) in panel data.
• It employs a fully saturated regression specification with interaction terms, enabling unbiased estimation even when standard parallel trends and SUTVA assumptions are violated.
• Simulation studies and applications, such as the Italian SEZ analysis, demonstrate that DTD robustly recovers treatment effects while mitigating biases from spillover contamination.

The double-triple-difference (DTD) model is an advanced extension of triple-difference (DDD) frameworks designed for causal inference in settings where interference or spillover effects contaminate one or more control groups. Unlike standard DDD, which assumes strict SUTVA (Stable Unit Treatment Value Assumption) and parallel-trend structures, the DTD model explicitly accommodates and identifies both direct average treatment effects on the treated (ATT) and average spillover effects (ASU) under clearly formalized assumptions. The DTD design is particularly relevant in applied contexts—such as policy interventions with sectoral and geographic targeting—where treatment status, group eligibility, and exposure to spillovers can be separately delineated. DTD methods provide rigorous, unbiased estimators for ATT and spillover effects, as demonstrated in simulations and empirical applications, particularly where standard DDD estimators fail due to violation of no-interference or parallel trends (Nicolò et al., 22 Jan 2026).

1. Formal Definition and Regression Specification

The DTD model is defined for panels with two periods (pre, post) and a structure involving two strata and three distinguishable groups:

• Strata ($S_i\in\{0,1\}$): typically geographic or administrative regions (e.g., treated vs. control municipalities).
• Target group ($G_i\in\{0,1\}$): denotes units (e.g., firms or individuals) eligible for direct treatment (e.g., policy-targeted sectors).
• Interference group ($I_i\in\{0,1\}$): units in the treated stratum but not in the target group, who are potentially exposed to spillovers.

The fully saturated DTD regression is

$$\begin{aligned} Y_{it} =\; & \beta_0 + \beta_1 S_i + \beta_2 G_i + \beta_3 I_i + \beta_4 T_t \ & + \beta_5 (S_i G_i) + \beta_6 (S_i I_i) + \beta_7 (G_i T_t) + \beta_8 (I_i T_t) + \beta_9 (S_i T_t) \ & +\; \delta (S_i G_i T_t) + \psi (S_i I_i T_t) + \varepsilon_{it} \end{aligned}$$

where $T_t=1\{t=1\}$ is the post-period indicator. Crucially,

• $\delta$ identifies the ATT for "pure" treated units ($S_i=1,\,G_i=1,\,I_i=0$).
• $\psi$ identifies the ASU (average spillover effect) for interfered controls ($S_i=1,\,G_i=0,\,I_i=1$).

This saturated interaction structure ensures that contrasts for $\delta$ and $\psi$ are not contaminated by spillover exposure in control groups (Nicolò et al., 22 Jan 2026).

2. Identification Assumptions

Identification of ATT ($\delta$) and ASU ($\psi$) in the DTD model requires several structural assumptions:

• Assumption 1 (Stratum SUTVA): Potential outcomes depend on the treatment assignment only within the stratum; there are no cross-stratum spillovers.
• Assumption 2 (No anticipation): Pre-treatment outcomes are invariant to future treatment assignment.
• Assumption 3 (No spillover on the treated): Directly treated units ($G_i=1$) are not affected by exposure of others in their stratum; i.e., spillovers do not feed back onto those receiving the primary treatment.
• Assumption 4 (Observation of pure controls): There exist and are observed units with $(S_i=1, G_i=0, I_i=0)$ and $(S_i=0, G_i=0, I_i=0)$—untreated, unexposed in both strata.
• Assumption 5 (Parallel trend-in-trends for ATT): Among units with $I_i=0$, the "trend-in-trends" bias is zero:

$$\begin{aligned} & \big[ \mathbb{E}(Y_{i1}(0,0) - Y_{i0}(0,0)|S=1,G=1,I=0) - \mathbb{E}(Y_{i1}(0,0) - Y_{i0}(0,0)|S=1,G=0,I=0) \big]\ & - \big[ \mathbb{E}(Y_{i1}(0,0) - Y_{i0}(0,0)|S=0,G=1,I=0) - \mathbb{E}(Y_{i1}(0,0) - Y_{i0}(0,0)|S=0,G=0,I=0) \big] = 0 \end{aligned}$$

• Assumption 6 (Parallel trend-in-trends for ASU): Analogous requirement, comparing $I=1$ versus $I=0$ in $G=0$ units, for identification of the spillover effect.

These restrictions ensure that $\delta$ and $\psi$ correspond respectively to the ATT of primary treated units free from spillovers and the ASU for those exposed to interference, even in the presence of endogenous or heterogeneous group assignments (Nicolò et al., 22 Jan 2026).

3. Formal Identification and Estimand Structure

Under the identification assumptions, the triple-interaction terms in the DTD regression admit structural double-triple-difference forms:

• For the ATT:

$$\delta = \Big[\mathbb{E}(Y_{i1} - Y_{i0}|S=1,G=1,I=0) - \mathbb{E}(Y_{i1} - Y_{i0}|S=1,G=0,I=0)\Big] \ - \Big[\mathbb{E}(Y_{i1} - Y_{i0}|S=0,G=1,I=0) - \mathbb{E}(Y_{i1} - Y_{i0}|S=0,G=0,I=0)\Big]$$

By Assumption 5, this collapses to the ATT within $(S=1, G=1, I=0)$ units.

• For the ASU:

$$\psi = \Big[\mathbb{E}(Y_{i1}-Y_{i0}|S=1,G=0,I=1) - \mathbb{E}(Y_{i1}-Y_{i0}|S=1,G=0,I=0)\Big] \ - \Big[\mathbb{E}(Y_{i1}-Y_{i0}|S=0,G=0,I=1) - \mathbb{E}(Y_{i1}-Y_{i0}|S=0,G=0,I=0)\Big]$$

which, by Assumption 6, corresponds to the spillover effect in $(S=1, G=0, I=1)$ (Nicolò et al., 22 Jan 2026).

The estimand structure ensures that policy-relevant quantities are identified and separated from spillover contamination—a property not satisfied by standard DDD if control groups are contaminated by indirect exposure.

4. Simulation Evidence and Bias Properties

Extensive Monte Carlo simulations calibrate DTD versus standard triple-difference (TD) estimators:

• Scenario 1 (SUTVA holds, no spillovers): Both TD and DTD are unbiased.
• Scenario 2 (Treated-stratum spillovers only): TD estimator exhibits severe negative bias proportional to the magnitude and fraction of spillover-exposed controls; mean squared error escalates, and coverage probability collapses. DTD remains unbiased and has nominal coverage.
• Scenario 3 (Spillovers in both strata): Both TD and DTD are biased unless spillover effects cancel in the difference.

In two-period, covariate-adjusted settings, DTD consistently outperforms TD and remains unbiased even with complex treatment assignment mechanisms, provided the trend-in-trends assumptions hold. The DTD estimator preserves proper coverage and robustly estimates both ATT and ASU, whereas the TD estimator is grossly biased under spillover contamination (Nicolò et al., 22 Jan 2026).

5. Applied Example: Special Economic Zones in Italy

The DTD methodology is empirically implemented using Italian firm-level panel data (AIDA, 2013–2020), contrasting Campania SEZ (treated stratum) and Sicilian SEZ municipalities (placebo). Key features:

• Groups:
  • G=1: Export-oriented manufacturing eligible for SEZ tax credits (target group).
  • I=1: Noneligible sectors (potentially exposed to demand and shipping-chain spillovers).
  • G=I=0: Pure controls.
• Results: Pre-treatment tests confirm unconditional parallel trend-in-trends (assumptions 5–6) but reject simple parallel trends.
• Estimates:
  • Standard TD (no spillovers): $\hat\delta_\text{TD}\approx 5.5$ pp (SE ≈ 1.3 pp).
  • DTD (allowing for spillovers):
  • $\hat\delta_\text{DTD}\approx 6.8$ pp (SE ≈ 1.3 pp) for ATT.
  • $\hat\psi\approx 3.8$ pp (SE ≈ 0.8 pp) for ASU.

Notably, the conventional TD estimator understates the direct sector-targeted effect because indirect controls experience positive spillovers, confirming the necessity of DTD decomposition in such designs (Nicolò et al., 22 Jan 2026).

6. Methodological Implications, Extension, and Limitations

The DTD approach marks an advance in policy evaluation where standard difference-in-differences and triple difference methods are invalidated by the presence of treated-connected spillover exposure. The DTD model’s double-triple-difference structure ensures unbiased identification of both direct treatment and spillover effects under the parallel trend-in-trends conditions.

However, when spillovers exist simultaneously in both the treated and control strata, the DTD estimator also becomes misspecified unless spillover contrasts are balanced in expectation. Therefore, explicit checks of pre-intervention trends and precise design of groupings are essential. In typical cases with clear definition of geographic/sectoral interference, DTD structures provide reliable inference; elsewhere, further assumption refinement or alternative approaches may be required (Nicolò et al., 22 Jan 2026).