DiD vs Lagged Outcomes: Why Estimates Bracket the Truth

📌 What This Paper Compares

Difference-in-differences (DiD) and regression adjustment for a lagged dependent variable offer two common ways to draw causal conclusions from panel data. Each method rests on a distinct identifying assumption: DiD requires parallel trends (an assumption that is scale-dependent), while lagged-outcome adjustment requires ignorability conditional on past outcomes.

📌 The Central Result

Building on Angrist and Pischke (2009), which established a bracketing relationship in linear models, the paper shows that the same directional bracketing holds more broadly:

• For a true positive effect, if ignorability is correct, then mistakenly imposing parallel trends (DiD) will overestimate the effect.
• Conversely, if parallel trends is correct, then mistakenly imposing ignorability (lagged-outcome regression) will underestimate the effect.

This bracketing relationship is proven in general nonparametric (model-free) settings and is extended to semiparametric estimation based on inverse-probability-weighting.

📊 How the Claim Is Established

• The argument generalizes the linear-model intuition from Angrist and Pischke (2009) to nonparametric environments, removing reliance on specific functional-form assumptions.
• Semiparametric estimators that use inverse-probability weighting are shown to inherit the same bracketing property under their respective identifying conditions.

🔍 Illustrations and Materials

• Three worked examples illustrate the theoretical results in practice.
• Replication files are available in Ding and Li (2019).

⚖️ Why This Matters

The results give applied researchers a principled way to interpret differences between DiD and lagged-outcome estimates: when either identifying assumption is more plausible than the other, the two methods can provide informative upper and lower bounds on treatment effects. The findings therefore offer a robust diagnostic and bounding tool for causal inference with observational panel data.