1 Introduction

Difference-in-differences (DiD) is one of the most widely used causal identification strategies in economics. It compares the change over time in outcomes for a treated group to the contemporaneous change for an untreated control group, removing time-invariant confounders under the parallel trends assumption: in the absence of treatment, treated and control units would have followed the same trajectory.

But parallel trends is not always credible. Suppose a government expands healthcare coverage to low-income households. The expansion is correlated with pre-existing trends — low-income households may have been declining in health even before the expansion, violating parallel trends. Or suppose an environmental regulation is enacted specifically in regions experiencing rapid industrial growth; the growth trajectory differs from unregulated regions, making parallel trends implausible.

In these settings, researchers need a way to combine the longitudinal structure of DiD with the instrumental variable (IV) logic of removing omitted variable bias. The Instrumented Difference-in-Differences (IVDiD) design, formalised by Ye et al. [2021] and extended by Miyaji [2026], addresses exactly this need: it exploits an instrument that generates quasi-random variation in the change in treatment, enabling DiD-type identification even when parallel trends fails due to time-varying confounders.

2 The Standard DiD Setup and Its Failure

In a canonical two-period, two-group DiD, let Dᵢₜ ∈ {0, 1} denote treatment status of unit i in period t ∈ {0, 1}. The outcome is Yᵢₜ. The standard DiD estimator:

identifies the average treatment effect on the treated (ATT) under parallel trends:

This fails when there exist time-varying unobservables Uᵢₜ that jointly determine treatment adoption and outcomes. For example, if firms that adopt a new technology do so precisely because they anticipate faster productivity growth, the treated group's pre-existing growth trajectory exceeds the control group's, and the DiD estimate conflates the technology effect with the pre-existing trend differential.

3 The IVDiD Design

The IVDiD design introduces an instrument Zᵢ that:

1. Is correlated with the change in treatment: Cov(Zᵢ, ΔDᵢ) ≠ 0, where ΔDᵢ = Dᵢ₁ − Dᵢ₀.
2. Is conditionally independent of the potential outcome change in the absence of treatment: Zᵢ ⊥ ΔYᵢ(0) | Xᵢ.
3. Does not directly affect the outcome except through treatment (exclusion restriction).

Condition (2) is the analogue of the IV exclusion restriction in the DiD context: the instrument must be correlated with who switches treatment but not with the untreated potential trend. It is weaker than requiring full independence of Zᵢ from all unobservables, because unit fixed effects are differenced out.

3.1 Identification

Ye et al. [2021] show that, under these conditions and an appropriate monotonicity assumption, the IVDiD estimator identifies the LATE among switchers — units whose treatment status changes from period 0 to period 1, and for whom the change is caused by the instrument:

where ΔDᵢ^(z) denotes the potential treatment change under instrument value z. This is the effect of treatment adoption for compliers — units whose adoption is triggered by the instrument — at the time of adoption.

3.2 The Wald-DiD Estimand

The IVDiD estimator takes the form of a Wald ratio applied to first-differenced data:

where ΔYᵢ = Yᵢ₁ − Yᵢ₀ and ΔDᵢ = Dᵢ₁ − Dᵢ₀. The numerator is the reduced-form effect of the instrument on the outcome change; the denominator is the first-stage effect on the treatment change. In practice, covariates Xᵢ are typically included, making the estimator a 2SLS regression of ΔYᵢ on ΔDᵢ instrumented by Zᵢ, with covariates.

4 Relationship to Standard DiD and IV

The IVDiD design nests two special cases:

Standard DiD. When Zᵢ = Dᵢ₁ − Dᵢ₀ (the treatment change itself), and parallel trends holds, the IVDiD Wald estimand equals the standard DiD estimator.

Standard IV. In a cross-sectional setting (no pre-period), ΔYᵢ = Yᵢ₁ and ΔDᵢ = Dᵢ₁, and IVDiD reduces to the standard 2SLS estimator.

IVDiD combines the strengths of both: it leverages longitudinal data to difference out time-invariant confounders (the DiD contribution) and uses an instrument to address remaining time-varying confounders (the IV contribution). The price is stronger assumptions: both the parallel trends assumption for the instrument's exclusion restriction and the first-stage monotonicity for the compliers LATE interpretation.

5 Extension to Staggered Adoption

Miyaji [2026] extends the IVDiD framework to settings with multiple periods and staggered adoption of both the treatment and the instrument. In the canonical staggered DiD setting, different units are treated at different times. Miyaji introduces a staggered instrument Zᵢg that triggers treatment adoption at group-time pair (g, t), analogous to the shift-share instruments or staggered policy announcements common in empirical economics.

The identification conditions parallel those of Callaway and Sant'Anna [2021] for staggered DiD: local parallel trends for each cohort conditional on the instrument, monotonicity of the instrument's first-stage effect, and exclusion of the instrument from the potential untreated trend. Under these conditions, group-time LATE estimands τ_IVDiD(g, t) can be aggregated using the same weighting schemes (simple average, event-study weights) as in the Callaway-Sant'Anna framework.

6 Practical Motivating Example: Minimum Wage and Employment

Consider estimating the effect of minimum wage increases on employment. A standard DiD uses states that raise the minimum wage (treated) versus states that do not (control). The parallel trends assumption requires that treated states would have had the same employment trend as control states absent the minimum wage increase — often implausible, since states raise wages when their economies are strong or when political conditions are favourable, both of which predict divergent trends.

An IVDiD approach might use the federal minimum wage as an instrument: when the federal minimum exceeds a state's current minimum, the state is "bitten" by the federal increase, and the bite (the gap between federal and state minimum wages) varies across states and over time. This bite instrument is correlated with state minimum wage changes (ΔDᵢ) but, under the exclusion restriction, affects employment only through the minimum wage rather than through other state-specific economic conditions.

7 Assumptions and Limitations

Exclusion restriction. The instrument must be uncorrelated with ΔYᵢ(0) — the counterfactual outcome change. This is a strong assumption. In the minimum wage example, if federal policy changes are correlated with national economic conditions that affect all states' employment, the exclusion restriction fails.

Monotonicity. The instrument must move all compliers in the same direction. Defiers — units for which a "higher-bite" instrument leads to a lower minimum wage — invalidate the LATE interpretation.

LATE interpretation. Like all IV estimators, IVDiD recovers effects for compliers, not for the full treatment population. Policy relevance depends on whether compliers are the population of interest.

8 Conclusion

The Instrumented DiD design bridges two of the most powerful identification strategies in empirical economics — difference-in-differences and instrumental variables — to handle settings where both time-invariant confounders and time-varying confounders threaten causal identification. By using an instrument that generates quasi-random variation in treatment changes rather than treatment levels, IVDiD identifies LATE for treatment switchers even when parallel trends fails. As researchers tackle increasingly demanding causal questions where simple parallel trends is implausible, the IVDiD framework represents a valuable and underused tool in the applied econometrics toolkit.

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