Testing the Significance of the Difference-in-Differences Coefficient via Doubly Randomised Inference

Stanis{\l}aw Marek Sergiusz Halkiewicz and Andrzej Ka{\l}u\.za

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Abstract: This article develops a significance test for the Difference-in-Differences (DiD) estimator based on doubly randomised inference, in which both the treatment and time indicators are permuted to generate an empirical null distribution of the DiD coefficient. Unlike classical $t$-tests or single-margin permutation procedures, the proposed method exploits a substantially enlarged randomization space. We formally characterise this expansion and show that dual randomization increases the number of admissible relabelings by a factor of $\binom{n}{n_T}$, yielding an exponentially richer permutation universe. This combinatorial gain implies a denser and more stable approximation of the null distribution, a result further justified through an information-theoretic (entropy) interpretation. The validity and finite-sample behaviour of the test are examined using multiple empirical datasets commonly analysed in applied economics, including the Indonesian school construction program (INPRES), brand search data, minimum wage reforms, and municipality-level refugee inflows in Greece. Across all settings, doubly randomised inference performs comparably to standard approaches while offering superior small-sample stability and sharper critical regions due to the enlarged permutation space. The proposed procedure therefore provides a robust, nonparametric alternative for assessing the statistical significance of DiD estimates, particularly in designs with limited group sizes or irregular assignment structures.

Date: 2025-12
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