2 Model, Identification, and Estimation

In this section, we specify a statistical model of the allocation of seats as a function of vote shares that incorporates a threshold of representation, show that this model is identified, and outline our estimation strategy. Consider $N$ parties that compete across $T$ elections under the same electoral system. Index parties by $i=1,\ldots ,N$ and elections by $t=1,\ldots ,T$ . Let the vote share of party $i$ in election $t$ be denoted by $v_{t,i}\geqslant 0$ , and denote the number of seats allocated to that party in election $t$ by $s_{t,i}\geqslant 0$ . Let the vector of realized seat allocations in election $t$ be denoted by $s_{t}$ and the corresponding vector of vote shares by $v_{t}$ . Let $F$ denote the joint distribution of $(s_{t},v_{t})$ , denote its marginal over vote shares by $F_{v}$ , and the conditional distribution over seats given vote shares by $F_{s\mid v}$ , so that $F=F_{s\mid v}\cdot F_{v}$ . Assume that seats $s$ are drawn conditionally independently across elections according to $F_{s\mid v}$ , conditional on realized vote shares $v$ .Footnote ⁸ Our goal is to estimate the conditional distribution $F_{s\mid v}$ . Since both the vote shares $v$ and seat allocations $s$ are observable, this distribution is in principle nonparametrically identified but such an approach would be impractical for estimation purposes as we do not have the luxury of observing centuries of elections. We will therefore specify a parametric family of distributions within which $F_{s\mid v}$ belongs.

To parameterize $F_{s\mid v}$ and accommodate seat allocation processes that provide for a (for our purposes unobserved to the analyst) threshold of representation, we assume that in each election $t$ , a latent variable $\unicode[STIX]{x1D703}_{t}^{\ast }$ is drawn from a distribution $f(\unicode[STIX]{x1D703}_{t}^{\ast }\mid \unicode[STIX]{x1D703},\unicode[STIX]{x1D70E})$ and a national electoral threshold $\unicode[STIX]{x1D703}_{t}$ is realized as a function of $\unicode[STIX]{x1D703}_{t}^{\ast }$ . Specifically, the threshold $\unicode[STIX]{x1D703}_{t}$ is zero if the latent variable $\unicode[STIX]{x1D703}_{t}^{\ast }$ is negative and is equal to $\unicode[STIX]{x1D703}_{t}^{\ast }$ otherwise, that is, $\unicode[STIX]{x1D703}_{t}=\max \{0,\unicode[STIX]{x1D703}_{t}^{\ast }\}$ . Turning to the distribution of seats in election $t$ given the realized threshold, $\unicode[STIX]{x1D703}_{t}$ , we start with the (provisional) assumption that parties whose vote share falls below the realized threshold receive no seats, that is, $s_{t,i}=0$ when the vote share of party $i$ satisfies $v_{t,i}<\unicode[STIX]{x1D703}_{t}$ . For parties that exceed the threshold in election $t$ , we assume (as in, for example, King (Reference King1990)) that the allocation of seats follows a multinomial distribution

where the expected seat share of party $i$ is given by

Here, $\unicode[STIX]{x1D6FD}$ is a disproportionality parameter, while $\mathbb{I}(\cdot )$ is an indicator function taking the value one if the expression in parentheses is true, and zero, otherwise. The second line of (2) covers the case the realized threshold exceeds the vote share of the plurality party (admittedly a negligible event in our estimation and certainly in the data), specifying that seats are allocated with equal probability among parties tied in the plurality position.Footnote ⁹

We complete the statistical model of seat allocation by specifying a parametric normal distribution for the latent threshold variable, that is, $\unicode[STIX]{x1D703}_{t}^{\ast }\,\sim \,f(\unicode[STIX]{x1D703}_{t}\,\mid \,\unicode[STIX]{x1D703},\unicode[STIX]{x1D70E}):=N(\unicode[STIX]{x1D703},\unicode[STIX]{x1D70E}^{2})$ . This parametric form allows us to compute in closed form the expected national threshold $\bar{\unicode[STIX]{x1D703}}(\unicode[STIX]{x1D703},\unicode[STIX]{x1D70E})$ and its standard deviation $\bar{\unicode[STIX]{x1D70E}}(\unicode[STIX]{x1D703},\unicode[STIX]{x1D70E})$ as functions of the parameters $\unicode[STIX]{x1D703},\unicode[STIX]{x1D70E}$ (see Online Appendix E). Naturally, the expected threshold $\bar{\unicode[STIX]{x1D703}}$ is of primary importance for our purposes, though the nuisance parameter is also of potential relevance—and certainly necessary given the stochastic perspective we take on the data. In turn, parameter $\unicode[STIX]{x1D6FD}$ serves as a natural (dis)proportionality parameter for the seat allocation among parties that exceed the threshold, implying proportional representation (in expectation) when $\unicode[STIX]{x1D6FD}=1$ , disproportional allocations favoring larger parties when $\unicode[STIX]{x1D6FD}>1$ , and disproportional allocations favoring smaller ones when $\unicode[STIX]{x1D6FD}<1$ . Theil (Reference Theil1969) provides justifications for this parameter choice, but it has a long tradition in empirical models of seats–votes relationships, especially in two-party systems (e.g., Kendall and Stuart Reference Kendall and Stuart1950; Taagepera Reference Taagepera1973; Tufte Reference Tufte1973; Schrodt Reference Schrodt1981).

With the model thus specified, and before we move to estimation, we first show that the parameters $\unicode[STIX]{x1D703},\unicode[STIX]{x1D70E},\unicode[STIX]{x1D6FD}$ that determine the conditional distribution of seats given votes $F_{s\mid v}$ according to the above assumptions are identified.

The proof of Theorem 1 appears in Online Appendix A. Note that assumption ( A ₁ ) is standard, so that the crucial identification condition is ( A ₂ ), which simply requires that there exists positive probability that at least three parties receive positive vote share. This is a very mild assumption that is easily met in our sample. By virtue of this assumption, we first establish that parameter $\unicode[STIX]{x1D6FD}_{0}$ is identified from the subset of the data where the smallest of the three parties receives seats. We then further use ( A ₂ ) to show that we can identify (at least) two quantiles of the normal distribution $f(\cdot \mid \unicode[STIX]{x1D703}_{0},\unicode[STIX]{x1D70E}_{0})$ , which in turn pins down the two parameters $\unicode[STIX]{x1D703}_{0}$ and $\unicode[STIX]{x1D70E}_{0}$ .

Before we move to estimation, we comment on a few aspects of the model and its interpretation, and enter some caveats. First, we view the joint distribution over votes and seats, $F$ , as an equilibrium distribution induced by the electoral system. By that we mean that all agents involved (parties, candidates, and voters) behave to the best of their ability and understanding of anticipated electorate behavior and preferences: Parties contest elections, field candidates and lists, form alliances local or national, and respond to observed transient shocks in voter preferences; similarly, voters tailor their behavior to the rules and to the menu of options that arise form the parties’ behavior. It is not our objective to identify these supply and demand forces that shape $F$ ; we merely assume that they combine to determine it. Second, in this perspective, we construe an electoral system broadly as both the legal rules for translating votes to seats as well as the structural features (observable or not) that encourage or discourage participation by certain demographics, the alignment of likely voter preferences across districts given districting choices, but also the structural variability in these preferences (aggregate and individual preference shocks, and other vicissitudes of public opinion toward issues or parties). Third, our assumption that the conditional distribution over seats $F_{s\mid v}$ is nondegenerate and involves a stochastic threshold is consistent with and follows from this perspective. For example, a party may contest the election hoping to achieve a certain vote share and some seats in expectation. Given aggregate uncertainty about voter behavior, the party may hit the target national vote share but may or may not succeed at earning seats because the realized alignment of vote shares across districts on election night proved or did not prove favorable, respectively. Fourth, any time we force a parametric form to a distribution (as we do for $F_{s\mid v}$ ), we take the risk of misspecification. Our goodness-of-fit test in Section 4 provides a way to safeguard against such a risk. Finally, permanent structural changes affecting parties’ or voters’ behavior could result in a different votes–seats distribution $F$ with a different conditional, $F_{s\mid v}$ . In Section 4, we also develop a test of electoral system change that can be used as a way to detect such changes.

We now derive a likelihood for our model in order to motivate our estimator and provide additional insight as to the nature of information present in the data that allows us to recover the threshold parameters. First, we logically infer from the observed data $(v_{t},s_{t})$ that the threshold in period $t$ did not exceed

that is, the minimum vote share among parties that received seats. Letting $p(s_{t}\mid v_{t},\unicode[STIX]{x1D6FD},\unicode[STIX]{x1D703}_{t})$ denote the probability of $s_{t}$ given votes $v_{t}$ , $\unicode[STIX]{x1D6FD}$ , and realized threshold $\unicode[STIX]{x1D703}_{t}$ , we note that if the latter falls in any interval $(\ell ,u ]\subset [0,\bar{v}_{t}]$ and there does not exist a party $i$ with vote share $v_{t,i}$ in $(\ell ,u)$ , then

Here, the exact realization of the threshold does not matter as long as the set of parties with vote share at or above the threshold remains identical. Accordingly, suppose there are $n_{t}$ distinct parties with vote shares $v_{t,i_{1}},\ldots ,v_{t,i_{n_{t}}}$ that are less than or equal to the upper bound of the realized threshold $\bar{v}_{t}$ . We index these by $z_{t}=1,\ldots ,n_{t}$ in increasing order and define $u_{t,z_{t}}:=v_{t,i_{z_{t}}}$ , $z_{t}=1,\ldots ,n_{t}$ , so that

Similarly, we can define for each $z_{t}=1,\ldots ,n_{t}$ a corresponding lower bound

We may now view $z_{t}$ as the realization of a random variable $Z_{t}$ with support $\{1,\ldots ,n_{t}\}$ , with the interpretation that $z_{t}$ denotes the interval $(\ell _{t,z_{t}},u_{t,z_{t}} ]$ within which the latent threshold is realized. Clearly, the probability that $\unicode[STIX]{x1D703}_{t}^{\ast }\in (\ell _{t,z_{t}},u_{t,z_{t}} ]$ is

To illustrate using an example, in Figure 1, we use the seat allocation in the Portuguese elections of 1979, where each row in the table corresponds to the seats and percentage of votes received by a party. Note that in this election the smallest party that gained seats received 2.24% of the votes and so $\overline{v}_{t}=2.24\%$ . Moreover, the number of parties whose vote shares are less than or equal to that upper bound for the threshold is eight $(n_{t}=8)$ . The threshold must then be in one of the intervals defined by the vote shares of these eight parties and the random variable $Z_{t}$ can take value $1,2,3,\ldots ,8$ , each corresponding to one of these intervals. The first of these intervals is $(-\infty ,0.06\%]$ . The second is $(0.06\%,0.22\%]$ , all the way up to $(1.24\%,2.24\%]$ . The probability calculation in (5) is represented by the shaded area in Figure 1 for the case $z_{t}=7$ .

Figure 1.

Threshold location (Portugal 1979).

Using (1), (4), and (5), we can now write a log-likelihood as

It now becomes evident that, even though the a priori probability (5) may suggest otherwise, the estimates of threshold and proportionality parameters $\unicode[STIX]{x1D703},\unicode[STIX]{x1D70E}$ and $\unicode[STIX]{x1D6FD}$ , respectively, interact heavily with each other. Values of $\unicode[STIX]{x1D703},\unicode[STIX]{x1D70E}$ that place high probability on low realizations of threshold interval $Z_{t}$ tend to suggest more disproportionality (higher $\unicode[STIX]{x1D6FD}$ ) because more parties exceeding the threshold receive no seats. Conversely, the value of the proportionality parameter $\unicode[STIX]{x1D6FD}$ provides sharper (compared to the logical bound $\bar{v}_{t}$ ) information about the likely realization of the random variable $Z_{t}$ and the threshold. Returning to the 1979 Portuguese elections example of Figure 1, if we were to also observe a very proportional pattern of seat allocation for parties at or above $\bar{v}_{t}=2.42\%$ (we actually do not), then we would update an appreciably higher probability that $Z_{t}=8$ and the threshold $\unicode[STIX]{x1D703}_{t}\in (1.24,2.42]$ , for we would otherwise expect a party with $1.24\%$ of the vote to also receive seats.

The summations inside the log function make this problem a textbook candidate for the application of an EM-type algorithm (Dempster, Laird, and Rubin Reference Dempster, Laird and Rubin1977), where we “augment” the data with the realization of the unobserved variable $Z_{t}$ that designates the interval within which the threshold in election $t$ was realized. Even though the EM-algorithm is better suited to implement MLE in this problem, it is still plagued by numerical instability and slow convergence problems. There are three main instances where such problems occur:

• ∙ Cases when $\unicode[STIX]{x1D70E}\rightarrow 0$ : These are possible singularities in the likelihood, and motivate the MAP-EM estimator of Fraley and Raftery (Reference Fraley and Raftery2007).

• ∙ Cases when $\bar{\unicode[STIX]{x1D703}}(\unicode[STIX]{x1D703},\unicode[STIX]{x1D70E})\rightarrow 0$ , with $\unicode[STIX]{x1D703}<0$ and relatively small $\unicode[STIX]{x1D70E}>0$ : The likelihood is practically flat with respect to $\unicode[STIX]{x1D703},\unicode[STIX]{x1D70E}$ for machine precision purposes.

• ∙ Electoral systems for which we have very few data, both when it comes to the number of elections and the number of parties.

There is very little that can be done in the last case, except possibly to exclude these systems from the analysis, as any estimator applied to such data should be taken with caution. We choose to include them in the analysis, but expect estimates of uncertainty in the estimated quantities to regulate how much credence to place to the resulting estimates. Our practical solution for the first two problems is to regularize the likelihood by introducing a prior and execute maximum a posteriori estimation (MAP-EM).

We therefore proceed to introducing a prior to regularize the likelihood with respect to the expected threshold parameter $\unicode[STIX]{x1D703}$ and the nuisance parameter $\unicode[STIX]{x1D70E}$ . We use a conjugate inverse-gamma-normal prior, as do Fraley and Raftery (Reference Fraley and Raftery2007). Specifically, we assume that $\unicode[STIX]{x1D70E}^{2}\sim \text{InverseGamma}(\frac{\unicode[STIX]{x1D708}}{2},\frac{s^{2}}{2})$ and that conditional on $\unicode[STIX]{x1D70E}$ , $\unicode[STIX]{x1D703}\sim N(\unicode[STIX]{x1D707},\frac{\unicode[STIX]{x1D70E}^{2}}{\unicode[STIX]{x1D705}})$ . We denote the log of this prior by $p(\unicode[STIX]{x1D703},\unicode[STIX]{x1D70E},\unicode[STIX]{x1D6FD})$ , which is equal to (excluding the normalizing constant)Footnote ¹¹

Here, $\unicode[STIX]{x1D707}$ is the prior mean of the mean, $\unicode[STIX]{x1D703}$ , of the latent threshold. Parameter $\unicode[STIX]{x1D705}$ regulates the dispersion of this prior: The smaller this parameter is, the smaller the impact of the prior on the location of $\unicode[STIX]{x1D703}$ . In Online Appendix K, we further discuss the prior specification,Footnote ¹² and introduce three additional alternatives in order to probe the sensitivity of our estimator to the prior as well as in order to compare our MAP-EM estimates to EM. The broad conclusion of this analysis is that the effect of the prior is present when the amount of data is small (few elections and/or few parties with positive vote shares), but it dissipates quickly with more data, especially for the main quantities of interest, disproportionality $\unicode[STIX]{x1D6FD}$ and expected threshold $\bar{\unicode[STIX]{x1D703}}(\unicode[STIX]{x1D703},\unicode[STIX]{x1D6FD})$ . The comparison of EM with MAP-EM can be summarized in the words of Fraley and Raftery (Reference Fraley and Raftery2007, p. 177) “… the method did eliminate singularities and degeneracies observed in maximum likelihood methods, while having little effect on stable results. When the number of observations is small, the choice of prior can influence the modeling outcome even when no singularities are observed.”

With this regularization in place, we proceed to specifying the EM steps. As is standard, we start by augmenting the data in order to write a (log)likelihood conditional on the data augmented by the (unobserved) component $Z=\{z_{t}\}_{t=1}^{T}$ , and the unobserved latent variables that determine the election thresholds $\unicode[STIX]{x1D6E9}^{\ast }=\{\unicode[STIX]{x1D703}_{t}^{\ast }\}_{t=1}^{T}$ . Given $z_{t}$ , the latent variable $\unicode[STIX]{x1D703}_{t}^{\ast }$ is now distributed according to the normal distribution truncated in $(\ell _{t,z_{t}},u_{t,z_{t}}]$ :

With a bit of algebra, we can now write a log-likelihood for the complete data as

This constitutes a considerable simplification over (6), as we have now avoided taking logs of any summation terms. The MAP-EM estimator amounts to an iterative procedure that starts by setting some initial guess for the parameter values $(\unicode[STIX]{x1D703}_{0},\unicode[STIX]{x1D70E}_{0},\unicode[STIX]{x1D6FD}_{0})$ and at the ( $m+1$ )th iteration computing:

1. (1) Expectation (E-step):

   $$\begin{eqnarray}Q(\unicode[STIX]{x1D703},\unicode[STIX]{x1D70E},\unicode[STIX]{x1D6FD};\unicode[STIX]{x1D703}_{m},\unicode[STIX]{x1D70E}_{m},\unicode[STIX]{x1D6FD}_{m})=\mathbb{E}_{Z,\unicode[STIX]{x1D6E9}^{\ast }}[L(\unicode[STIX]{x1D703},\unicode[STIX]{x1D70E},\unicode[STIX]{x1D6FD}\mid X,Z,\unicode[STIX]{x1D6E9}^{\ast })\mid X,\unicode[STIX]{x1D703}_{m},\unicode[STIX]{x1D70E}_{m},\unicode[STIX]{x1D6FD}_{m}].\end{eqnarray}$$

2. (2) Maximization (M-step):

   $$\begin{eqnarray}(\unicode[STIX]{x1D703}_{m+1},\unicode[STIX]{x1D70E}_{m+1},\unicode[STIX]{x1D6FD}_{m+1})=\underset{(\unicode[STIX]{x1D703},\unicode[STIX]{x1D70E},\unicode[STIX]{x1D6FD})}{\text{arg}\,\text{max}}\{Q(\unicode[STIX]{x1D703},\unicode[STIX]{x1D70E},\unicode[STIX]{x1D6FD};\unicode[STIX]{x1D703}_{m},\unicode[STIX]{x1D70E}_{m},\unicode[STIX]{x1D6FD}_{m})+p(\unicode[STIX]{x1D703},\unicode[STIX]{x1D70E},\unicode[STIX]{x1D6FD})\}.\end{eqnarray}$$

Using the first-order conditions, we execute the Maximization step by setting

It is not possible to solve analytically for $\unicode[STIX]{x1D6FD}_{m+1}$ , but due to the separation of the likelihood into terms involving bias $\unicode[STIX]{x1D6FD}$ and threshold parameters $\unicode[STIX]{x1D703},\unicode[STIX]{x1D70E}$ , we can obtain $\unicode[STIX]{x1D6FD}_{m+1}$ numerically by solving a single nonlinear equation. We efficiently implement these computations, including the derivation of standard errors, at speeds comparable to conventional ML-estimators by taking full advantage of analytical derivations. More details and derivations for this estimator and the computation of standard errors appear in Online Appendices B and C.

3 Data and Results

We implement our estimator on electoral returns data from elections to the lower house in $36$ European democracies for all elections held after the Second World War, or since democratization, until 2010 (inclusive).Footnote ¹³ We have compiled our electoral returns data from various printed sources including Mackie and Rose (Reference Mackie and Rose1991), Caramani (Reference Caramani2000), Nohlen and Stöver (Reference Nohlen and Stöver2010) and online resources such as Alvarez-Rivera (Reference Alvarez-Rivera2011), Carr (Reference Carr2011), the Inter-Parliamentary Union (2011), and the Norwegian Social Science Data Services (2011). We relied on official web pages of national parliaments, election commissions, each country’s Ministry of Interior, and so on to iron out any inconsistencies in these sources and to obtain electoral results for smaller parties, which are usually lumped as “others” in electoral statistics.Footnote ¹⁴ In addition to electoral returns, using these and additional sources (e.g., Carstairs Reference Carstairs1980; Lijphart Reference Lijphart1994; Renwick Reference Renwick2010), we also coded the electoral institutions in effect for each election at a fine level of detail. Specifically, for each election, we recorded the number of seats allocated, the different tiers of allocation, the number of districts in each tier, the allocation formula, the nature of allocation in upper tiers as a function of lower tier allocation, the presence of bonus seat provisions for the (national) plurality or majority party, as well as details for various types of electoral thresholds in effect at each tier of allocation. Details of our institutional coding appear in Online Appendix M.

These institutional data are necessary in order to identify distinct electoral systems in use across time within countries. For the purposes of this analysis, we identify changes in electoral systems within countries if either a change in the allocation formula (across tiers) occurs or when another recorded institutional provision (e.g., number of allocated seats, number of districts) changes by more than 5% of that system’s average. The 5% cutoff is admittedly arbitrary. One of the advantages of our approach is that we can bring standard statistical inference principles to bear on the question of what constitutes a significant change in electoral institutions. We develop a test procedure for that purpose in Section 4.1 and find support (though not universal) for our cutoff rule.

Figure 2.

Threshold and disproportionality estimates. Estimates of thresholds $\bar{\unicode[STIX]{x1D703}}(\hat{\unicode[STIX]{x1D703}},\hat{\unicode[STIX]{x1D70E}})$ and disproportionality $\hat{\unicode[STIX]{x1D6FD}}$ along with 95% confidence intervals. Based on point estimates reported in Online Appendix I.

Figure 2 presents the estimated expected national electoral threshold on the left, proportionality estimates on the right, and their 95% confidence intervals.Footnote ¹⁵ The figure excludes systems with very large 95% confidence intervals to increase visibility, but a full set of estimates is included in Online Appendix I.Footnote ¹⁶ The threshold estimates vary significantly with values ranging from $6.81\%$ for the second Moldavian system (MOL2) to virtually zero in systems like GBR1, ITA4–ITA6, and HUN2, among others. The median threshold estimate is $1.58\%$ , which is close to the third system in Cyprus (CYP3 with $1.59\%$ ) and the third Portuguese system (PRT3 with $1.56\%$ ).

A sign that our threshold estimates perform quite well is the notable success of the estimator at recovering the level of national legal thresholds when these are unequivocally set by the electoral law. Examples include the second Croatian system (HRV2) and the last two Greek systems GRC5 and GRC6, all with national legal thresholds of $3\%$ . In other cases, the electoral system prescribes a national electoral threshold but qualifying provisions allow small parties to gain seats below the nominal threshold. For example, in most of the German systems, a national $5\%$ threshold applies unless parties win a seat in one of the single-member districts of the majoritarian partition. In those cases, the statistical estimates we recover generally indicate an average of the legal threshold attenuated according to the probability that alternative qualifying conditions for representation are met.

Next we turn to the estimates of the electoral proportionality. We see that a majority of systems have estimated proportionality parameters that are greater than one. Overall, there are only 18 systems for which the estimated proportionality parameter is below one, and in many of these cases, the difference is in the third decimal point. Among these 18, it is only for the notoriously aberrant second French system (FRA2, with proportionality $0.78$ ) that we can reject the hypothesis that allocation above the threshold is perfectly proportional (in expectation). Conversely, in 56 out of 116 electoral systems, we find the proportionality parameter to be significantly larger than one, that is, we find statistically significant evidence of disproportional allocations favoring larger parties.

The highest proportionality parameters are found in the third Croatian system (HRV3 with $7.83$ ), the first Maltese system (MLT1 with $2.84$ ), and the first Hungarian system (HUN1 with $2.33$ ). Only HRV3 is a plurality system, while MLT1 is a Single Transferable Vote (STV) system with low district magnitude, and HUN1 is an elaborate fusion of majoritarian and PR provisions. Both HRV3 and MLT1 have a proportionality parameter that is consistent with the “Cube Law,”Footnote ¹⁷ that is, these are the two cases when we fail to reject the null of $\unicode[STIX]{x1D6FD}$ being equal to three (3).Footnote ¹⁸ The median proportionality parameter across all systems is $1.13$ and the next largest point estimates following the above three are all statistically different from 3. Even among majoritarian systems, the Cube Law finds little empirical support.Footnote ¹⁹

To provide a summary of the results, we classify the systems according to whether or not they have high thresholds and disproportionality. When a system has high thresholds, a model that considers thresholds explicitly should fit the systems’ observed seat allocation better than a model that assumes no threshold exists. This reasoning suggests that a test that compares the fit of our model with one that assumes no threshold, which we will call the restricted model, could give us an indication of which systems have high barriers of entry for small parties.Footnote ²⁰ We devise a statistical test to compare the model with and without thresholds in the next subsection. It suffices to say for now that the test’s null hypothesis is that the observed seats data are generated from the estimated restricted model. We find that in 49 out of 116 systems, the comparison test rejects the restricted model.Footnote ²¹ Importantly, the average estimated threshold for those 49 systems is 3.67% while that of those systems for which the restricted model is not rejected is 0.8%.

Table 1.

Thresholds and disproportionality (counts).

Threshold denotes the count of systems for which the comparison test rejects the null that the observed seats data are generated from a model with no threshold and No Threshold denotes all other systems at 5% level of significance. Nonproportional denotes the count of systems for which the disproportionality estimate is larger than one at the 5% level of significance (one-tailed test) and Proportional denotes all other systems. Percentages in brackets.

Table 1 shows that out of the 49 systems for which the comparison tests reject the restricted model, 11 also have disproportionality parameters larger than one. These are the systems that provide the most advantages for larger parties. Among them, the ones with larger disproportionality parameters are the second Polish system, the third Portuguese, and the fifth Greek system (POL2, PRT3, and GRC5). POL2 also has one of the highest thresholds ( $5.12\%$ ) and none use a majoritarian allocation formula in that group. On the other extreme, there are 22 systems that offer the most opportunities to small parties with perfect proportionality and low thresholds. The third Danish and first Italian systems—DNK3 and ITA1, both PR systems with some form of compensatory upper tier—fall into that category. But the most populated categories are in the off-diagonal entries of Table 1, which include systems that provide for thresholds (those for which the restricted model is rejected) but proportional allocation for parties exceeding the threshold (38 out of 116 systems) and systems for which a comparison test does not reject the restricted model but with disproportionality favoring larger parties (45 out of 116 systems).

Figure 3.

MAP-EM estimates versus alternative proportionality and threshold measures. The first plot illustrates overestimation of the proportionality parameter in the model without thresholds, in systems in which this restricted model is rejected. The second plot illustrates the exaggeration, compared to the MAP-EM estimates, of national electoral thresholds by Taagepera’s (Reference Taagepera2002) formula in the most comparable subset of electoral systems without upper tiers or legal thresholds. The 45 degree line is dotted. Based on results reported in Appendix I.

How different are our estimated measures from existing measures in the literature? When it comes to the proportionality parameter, $\unicode[STIX]{x1D6FD}$ , one obvious comparison is with the same parameter estimated with the restricted model (as in, for example, King and Browning (Reference King and Browning1987), King (Reference King1990), but without a partisan bias component). In Figure 3, we present the proportionality estimates of the full and restricted models for those systems for which the above test indicates significant differences in fit across the models.Footnote ²² If a system falls on the dotted line, this indicates that the unrestricted model’s disproportionality estimate is the same as that of the restricted model. As expected, ignoring the threshold brings an overestimation of disproportionality, indicated by the fact that most systems in the figure are above the dotted line. In particular, there is a group of systems that according to our estimates are close to being perfectly proportional for parties above the threshold for which the restricted model would overstate the advantages given to large parties—those near the vertical line at one (e.g., ROM1, SWE3, AUT2, DNK4).

Turning to the estimated thresholds, while there is no estimated alternative in the literature, there are a number of formulas that impute an electoral threshold from provisions of the electoral law (for example, Taagepera and Shugart (Reference Taagepera and Shugart1989), Lijphart (Reference Lijphart1994), Taagepera (Reference Taagepera2002), Ruiz-Rufino (Reference Ruiz-Rufino2007)). We now explore the differences between our estimated expected thresholds and the national thresholds of representation proposed by Taagepera (Reference Taagepera2002), which is a prominent alternative to our method. For this comparison, we compute Taagepera’s nationwide threshold using formula 8 in his paper according to our baseline electoral system definition.Footnote ²³ We compare systems with no upper tiers and those without legal thresholds of any kind, as the formula is more likely to approximate the true threshold with noncomplex allocation rules (Taagepera Reference Taagepera2002, p. 394). The second panel in Figure 3 shows that for the majority of the systems considered, the formula-based threshold overstates the percentage of national vote required in order to win any seats. For a group of majoritarian systems for which Taagepera’s formula gives positive thresholds, our approach suggests thresholds close to zero (e.g., GBR1, HRV1, MKD1, UKR3). This highlights the ability of our estimator to flexibly account for the emergence of successful small seat-winning parties with geographically concentrated support. Even in nonmajoritarian systems the differences can be quite large. For example, while Taagepera’s formula would give an average threshold of 2.7% to the system in place in Luxembourg in 1945 (LUX1), our estimated threshold is 0.25%. Overall, there are only few systems where Taagepera’s approach gives results close to the MAP-EM estimates (IRL2, LUX4, FRA3, FRA5, FRA6, PRT2, PRT3).Footnote ²⁴

4 Inference

We perform all inferences using the conventional approach that relies on model predictions at the point estimates and not the entire posterior, but a proper Bayesian execution of these tests is also possible and the reader can consult Online Appendix H for details. We start by discussing methods to evaluate the fit of the model. First, we devise a test of the hypothesis that the data are generated according to the estimated model. The test is based on a Pearson chi-square type of statistic as a weighted sum of squared deviations between actual and model predicted seat allocations,

where $\overline{s}_{i,t}(\widehat{\unicode[STIX]{x1D703}},\widehat{\unicode[STIX]{x1D70E}},\widehat{\unicode[STIX]{x1D6FD}}):=S_{t}\sum _{z_{t}=1}^{N}q_{i,t}(v_{t},\widehat{\unicode[STIX]{x1D6FD}},u_{t,z_{t}})p(z_{t}\,\mid \,v_{t},\widehat{\unicode[STIX]{x1D703}},\widehat{\unicode[STIX]{x1D70E}})$ denotes party $i$ ’s expected seats according to the estimated model. The intuition of the test is standard: If the observed allocation is close to the model’s prediction, then the value of the statistic will be small, whereas large values of the statistic would cast doubt on the ability of the model to account for the variation in the data. We compute empirical $p$ -values of the test via Monte Carlo simulations.Footnote ²⁵ We find that the test rejects the null of the observed seat allocation being consistent with the model’s prediction for only 5 systems out of 116 (FRA2-4, FRA6, and MKD2) at the 5% significance level.Footnote ²⁶ These results suggest that the model performs well accounting for variation in the data for the bulk of the estimated electoral systems.

We also assess the performance of the model relative to the restricted one (which restricts the national threshold to be zero), a common assumption in the literature heretofore. As an initial comparison, we perform a goodness-of-fit test for the restricted model that is analogous to the one described above. This time, the test rejects the null that the observed seats data are generated from the restricted model in 18 systems, a larger fraction relative to the case of the model allowing thresholds. A more systematic comparison is given by a statistical test of differences of fit between the two estimated models. The test statistic takes difference in sums of squared deviations between predicted and realized seats, and the rejection region is determined under the null that the data are generated from the estimated restricted model. Under that null hypothesis, large values of the statistic provide evidence in favor of the alternative model with thresholds. As mentioned earlier in 49 cases, the empirical $p$ -value of the test is below the 5% significance level. Overall, we see that the full model does a good job accounting for the variation in the data. These test results suggest that accounting for the threshold is warranted both statistically, and certainly politically, for a large fraction of systems.