Difference-in-differences with as few as two cross-sectional units -- A new perspective to the democracy-growth debate

TL;DR

This paper introduces the T-DiD estimator, enabling unit-specific treatment effect estimation with minimal cross-sectional units, addressing heterogeneity in democracy-growth studies.

Contribution

The paper presents a novel T-DiD estimator that leverages temporal variation to estimate unit-specific effects with as few as two units, improving analysis of heterogeneity.

Findings

T-DiD estimator is asymptotically normal under certain conditions.

A new identification test detects violations of parallel trends.

Empirical analysis estimates democracy's impact on Benin's economy.

Abstract

Pooled panel analyses often mask heterogeneity in unit-specific treatment effects. This challenge, for example, crops up in studies of the impact of democracy on economic growth, where findings vary substantially due to differences in country composition. To address this challenge, this paper introduces the Temporal Difference-in-Differences (T-DiD) estimator that leverages temporal variation in the data to estimate unit-specific average treatment effects on the treated (ATT) with as few as two cross-sectional units. Under asymptotic parallel trends, limited anticipation, and temporal dependence conditions, the proposed DiD estimator is shown to be asymptotically normal. Provided at least two control units are available, the method is further complemented with an identification test that, unlike pre-trends tests, is more powerful and can detect violations of parallel trends in post-treatment periods. Empirical results using the DiD estimator suggest Benin's economy would have been 6.4% smaller on average over the 1993-2018 period had she not democratised.