🔧 A stochastic threshold model and MAP-EM estimator

This article introduces a model for the conditional distribution of legislative seats given parties' vote shares that explicitly incorporates a stochastic national threshold of representation and a disproportionality parameter governing how seats are allocated among parties that clear the threshold. Conditions are derived that guarantee the model parameters are identifiable from observed seats/votes data, and a Maximum a Posteriori Expectation-Maximization (MAP-EM) algorithm is developed to estimate those parameters.

📊 Data: 116 electoral systems, 417 lower-house elections across 36 European countries

• Covers 417 elections to lower houses in 36 European countries since WWII.
• Includes 116 distinct national electoral system configurations observed across those elections.

🧪 Model fit and specification tests

• The estimated model is rejected in only 5 of the 116 systems, indicating broad empirical fit.
• A simpler model that omits a national threshold is rejected in favor of the threshold model in 49 systems, showing that thresholds matter empirically in many cases.

🔍 Key patterns in electoral systems

• Two modal configurations emerge:
• Higher effective thresholds combined with seat allocation for parties above the threshold that is not statistically different from perfectly proportional allocation (32.76% of systems).
• Systems where a national threshold cannot be statistically distinguished from absence of a threshold, but where seat allocation among eligible parties shows significant disproportionality (38.79% of systems).

🧾 Procedures for detecting institutional change

• The article also develops statistical procedures to test for significant changes over time in electoral institutions and/or in the distribution of seats, allowing formal detection of institutional shifts and their effects on proportionality.

❗ Why it matters

• Provides a unified, identifiable framework for separating the effects of national thresholds from the degree of proportionality in seat allocation.
• Offers a practical estimator (MAP-EM) and diagnostic tests that enable comparative research on electoral design, interpretation of proportionality measures, and the detection of institutional change using readily available seats/votes data.