Cremaschi et al. (2025) study a 2010 Italian reform that required municipalities below 5,000 inhabitants to jointly manage public services with neighboring communes. Their identification strategy compares municipalities below these population cutoffs (treated) to those above (controls), before and after the reform, using difference-in-differences. They conclude that the reform increased far-right vote share by roughly 1.5 percentage points in the 2013 and 2018 general elections, relative to 2001–2008.
Because treatment is assigned by a population cutoff, treated municipalities are systematically smaller than controls. Far-right support has grown faster in small municipalities since the early 2000’s — a differential trend that a standard DID specification, even with municipality and year fixed effects, cannot absorb.
Here I show how their diff-in-diff draws its implicit comparisons across the municipal-size gradient. Each chart shows 100 weighted-percentile bins along log(population), with the red horizontal line at the treated average and the blue line at the control average; their gap equals the estimator's point estimate exactly.
The test below pits two readings of the same dots against each other. The discontinuity hypothesis says the reform caused a step at 5,000: municipalities just below should jump above those just above. The size-gradient hypothesis says there is no step — far-right growth simply declines smoothly with size, and the diff-in-diff “effect” is that smooth curve read off between far-apart groups. Use View to switch between the two fits, and drag the threshold to see that no cutoff produces a real jump while the global gap stays large everywhere.