The Ruling Party and its Voting Power

The Ruling Party and its Voting Power Artyom Jelnov 1 Pavel Jelnov 2 September 26, 2015 Abstract We empirically study survival of the ruling party in parliamentary democracies. In our hazard rate model, the variable of interest is the ability of a ruling party to survive a critical vote, when the outcome is not predetermined. The probability of being on the winning side determines the duration of the ruling party s stay in power. We find evidence supporting our model in a panel of post-wwii coalitions of thirteen European parliamentary democracies. 1 Introduction In parliamentary democracies, political stability is usually associated with the duration of the period when a government coalition stays in power. Although the definition of what constitutes the end of a coalitional government differs across studies, the existing literature agrees that any change in the party composition of a coalition is considered to be the end of the coalition. We consider a different question that focuses on the identity of the ruling party, which is the party to which the prime minister belongs. The question we ask is how long does 1 Ariel University, Israel, artyomj@ariel.ac.il 2 Tel-Aviv University and Leibniz University Hanover, jelnov@aoek.uni-hannover.de 1

the ruling party survive in power, which comprises the amount of time since it formed the government until one of two things happens: a) a new election; b) a government with a different ruling party is formed. To the best of our knowledge, the question of the political survival of the ruling party, as distinct from the coalition, is novel in the literature. To illustrate our definition of the survival in power of the ruling party, let us take the situation in Israel between 1992 and 1996. After the 1992 election, the Israeli Labor party formed a coalitional government headed by Y. Rabin. In 1993, the Shas party left the government coalition. In the beginning of 1995 the Yiud fraction joined the coalition. In November 1995 Rabin was assassinated, and S. Peres, also of the Labor party, became prime minister. According to the traditional approach to government duration, one can count four different cabinets in this period (three changes of coalition composition and one change of prime minister). Indeed, there are good reasons for this counting, and we do not contradict the traditional approach. But it misses the fact that for the whole period from 1992 to 1995 the Labor party was in power, and the main policy directions (toward peace agreements with the Palestinians, for example) did not change. We make three contributions. First, we analyze a new question dealing with political stability in a parliamentary democracy. Second, we testify to the fact that the ruling party s stay in power depends on its voting power. Third, we generalize the concept of voting power, such that most existing definitions are special cases of our model. Formally, we estimate a hazard rate model, where the survival of the ruling party in power is determined by two components. The first is the rate of critical vote events, i.e., events that test the prime minister s political strength because parliament members vote ideologically, without obligations to the ruling party (but with party discipline). We assume that the rate of such events depends on the country s political culture (country fixed effects) and the time left until the regularly scheduled election. The second component is the probability of surviving such an event, which translates to being on the winning side of the critical vote. This second component is the ruling party s voting power, which is 2

calculated according to the distribution of seats in the parliament and with respect to some probabilistic assumptions on the voting, which, as we show below, are a generalization of most existing concepts of voting power. The parameters of the model are estimated using a panel of post-wwii coalitions of thirteen European parliamentary democracies. The results show that the voting power of a ruling party is a statistically significant predictor of its survival in power. The estimated coefficient of the effect of voting power on the ruling party s probability of surviving a critical event is about 0.8 (if the model could perfectly explain the data, it would be 1). This result survives controlling for the additional effect of being a relatively large ruling party. Basing on our results, we suggest that voting power, although it correlates with the party s size, is a more informative metric than the size. Moreover, we perform a robustness check, whereby the sample is restricted to only those coalitions that existed on January 1 of every fifth year. This restriction eliminates a possible bias caused by over-representing non-stable coalitions, in which the ruling party s stay in power is of short duration. Our concept was influenced by event methodology suggested by King et al. [1990] (followed by Warwick [1994]). However, our research question is different from theirs, since we focus on the stability not of the coalition, but of the ruling party. We now describe our model and the results. Consider a country where the government is accountable to the parliament. Each party controls a specific proportion of seats in the parliament (the weight of the party). We assume that all representatives of the same party vote the same in the parliament. Ideologically, parties are located along the uni-dimensional left-right scale. Following the establishment of a government coalition, a crisis event may occur on any given day. The occurrence of the crisis is a Poisson event. As long as the crisis does not occur, the ruling party stays in power until the next regular election. If a crisis does occur, the survival of the ruling party depends on the result of a critical vote in the parliament. Examples of the critical votes are a no-confidence vote, a vote for approval of 3

some important policy, a vote for the establishment of an alternative government coalition, etc. If the majority in the parliament votes like the ruling party, the ruling party overcomes the crisis and continues to be in power (and in this case we write that the ruling party is on the winning side). Otherwise, the ruling party loses its ruling party status. Our estimations show that such crises are by no means frequent. Their mean rate at the beginning of the ruling party s power is 0.00075 events per day, and it rises, on average, to 0.0268 on the last day before a regular election. We model critical vote on probabilistic voting. Each party in the parliament votes yes with the same probability p (which may vary from vote to vote) or no with probability 1 p. The parameter p itself is a random variable, distributed with the Beta distribution (a general method to generate a random number between 0 and 1). We assume that the a-priori likelihood of yes or no votes is symmetric; therefore, we assume that both parameters of the Beta distribution are equal (and we denote the value of this parameter by α). Moreover, we assume that there is a correlation in the voting of parties which are neighboring on the right-left ideological scale. Namely, if two parties are adjacent on the left-right scale, the correlation coefficient of their voting is ρ. For the formal definition of the model, see Section 2. To be sure, on a regular day parties vote in accordance with the coalitional discipline, and not probabilistically. But when a crisis is severe enough, in many political situations parties renege on their commitment to uphold coalitional agreements. Moreover, about 30% of our observations are of minority governments, which are in power thanks to ad hoc external support, which is a special case of our model. The rate of critical events includes a country fixed effect. Moreover, it could be claimed that the stability of a political system depends on the existence of a sufficiently large party (see Sartori [1976] 1 ). To control for it, we use a dummy variable, which receives one if the ruling party is larger than some predefined threshold. 1 Sartori also uses the index of fractionalization in his analysis. This is beyond the scope of our study. 4

The duration of a government is bounded by the formally defined end of the parliamentary term. King et al. [1990] observed that in the last year before the formally defined date of election, the frequency of coalition dissolution increases. Therefore, they censored coalitions that lasted up to the year before the formal end of the term. We accept the conjecture that the frequency of crises increases as the formal end of a term approaches, but we relax the censoring assumption and let the rate of critical events to change as the coalition approaches regular elections. Finally, we connect our study to the notion of voting power. 2 The most widely used voting power indices, the Shapley-Shubik index (Shapley and Shubik [1954]) and the Banzhaf index (Banzhaf [1964]), measure the probability that a voter is decisive. In our model we are interested in the probability that a voter (a party) is on the winning side. This probability is also addressed in the voting power literature. 3 It should also be mentioned that Straffin [1977] showed that the Shapley-Shubik index can be interpreted as the probability that a voter is decisive if every voter votes yes independently with the same probability p, and no with probability 1 p, when p is randomly chosen with uniform distribution on [0, 1]. This is a special case of our model where α = 1, since Beta(1, 1) is the uniform distribution, and ρ = 0. The Banzhaf index is the probability that a voter is decisive when every voter votes yes independently with probability 0.5. This is a special case of our model where α and ρ = 0. 4 From our findings follows that both α = 1, ρ = 0 and α, ρ = 0 are cases where our model fits the data well. Thus, the Rae index (and the Banzhaf index, see footnote 3) has some predictive power, as well as the distribution of votes that lies in the basis of the Shapley-Shubik index. Observing the actual distribution of voting in favor of a proposed bill by the U.S. House 2 For a comprehensive survey of this see Felsenthal and Machover [1998]. For recent research see Holler and Nurmi [2013] and Kurz et al. [2014]. 3 See Laruelle and Valenciano [2008]. They use term being successful instead of being on the winning side ). Moreover, the Rae index (Rae [1969]) measures the probability of being on the winning side when all coalitions are equally likely. Dubey and Shapley [1979] showed that the Rae index is a linear transformation of the Banzhaf index. 4 As α, Beta(α, α) converges to the degenerate distribution p = 1 2 with probability 1. 5

of Representatives (McCrary [2008]) shows that the actual distribution of voting yes is a mix of Beta distributions with different parameters. This supports our assumption that the probability of voting yes is connected to Beta distribution. In our empirical results, we observe that the value of α does not have a large impact on the statistical likelihood of the model, suggesting that the true distribution could be a mix of different α s, staying in lines with our model. Furthermore, the likelihood of the model is maximal when ρ is close to 0. This is not surprising. One would expect that the correlation between the voting of ideologically adjacent parties is positive, but not too high. Note that ρ = 1 means that all parties vote the same, and the realistic correlation should be far from that extreme point. Moreover, we also estimate the model while plugging the size of the ruling party instead of its voting power. There is no theoretical justification for this model, but it may be considered as intuitive. We find that it gives a lower statistical likelihood than our model when ρ is not very large in absolute terms. The existing literature widely discusses the political stability of governmental coalitions. Unlike in our model, these studies maintain the traditional concept that any change in the coalition composition constitutes the end of the coalition. One of the most influential contributions is Dodd [1976], who shows how duration of a coalition depends on the characteristics of the coalition. A survey of this literature appears in Mueller [2003]. Our paper is also related to the probabilistic voting literature (see Coughlin [1992]). To the best of our knowledge, the topic of our research, the survival of the ruling party, was not studied there. To conclude, Albert [2003] argued that voting power theory is a branch of probability theory and can safely be ignored by political scientists. To the contrary, we claim that our analysis provides evidence that some aspects of voting power theory can be applied to real-world political issues. The rest of the paper is organized as follows. In Section 2 we describe the model, in 6

Section 3 we present the data, the empirical methodology, and the results, and Section 4 concludes. 2 Model Let N = {1,..., n} be the set of parties, ordered on the left-right ideological scale (i < j means that party i is to the left of party j). Denote the ruling party by r N. Let w = {w 1,..., w n } be the weights of the parties in N, where w i > 0 for each i N. Let n i=1 Q = w i be the simple majority quota. 2 A crisis event is a Poisson event with parameter λ. Once a crisis occurs, the critical vote takes place. Let p be drawn from [0, 1] according to the distribution Beta(α, α) with α > 0. Let X i, i N be a Bernoulli random variable, which receives the value 1 with probability p and 0 with probability 1 p: 1, i votes yes x i = 0, i votes no. Let ρ be the correlation coefficient between X i and X i+1, i N. Lemma 2.1. For each i N, let X i be a Bernoulli trial with parameter p and let ρ Xi,X i+1 = ρ. Then for 1 i < n, P rob(x i+1 = 1 X i = 1) = ρ(1 p) + p and P rob(x i+1 = 1 X i = 0) = p(1 ρ). Proof: See Appendix. Corollary 2.2. ρ has to satisfy: max{ p 1 p, p 1 p } ρ 1 Proof: Follows directly from 0 P rob(x i+1 = 1 X i = 1) 1 and 0 P rob(x i+1 = 1 X i = 0) 1. 7

For 1 r n, let V (N, w, r, p, ρ) be the probability that r is on the winning side, or, in other words, that the majority votes like r: V (N, w, r) = P rob(x r = 1 and i N,x i =1 w i > Q) + P rob(x r = 0 and i N,x i =1 w i Q) Thus, the probability that r stops being the ruling party on any given day is λ[1 V (N, w, r, p, ρ)]. 3 Empirics 3.1 Data We use the composition of post-world War II parliaments (lower chambers) and of government coalitions in thirteen countries: Austria, Belgium, 5 Denmark, Finland, Germany, Ireland, Israel, Italy, Luxembourg, the Netherlands, Norway, Portugal, 6 and Sweden. The source of the data (except for Israel), including the location of parties on the ideological scale, is Müller and Strom [2003]. Whenever the date of government dissolution is absent from this source, we used data from The European Journal of Political Research Political Data Yearbook. 7 The data for Israel is from Chua and Felsenthal [2008] (including ideological positions of parties) and from the official Knesset (parliament of Israel) website. 8 The maximal potential duration of a government term was calculated by the authors in accordance with the Israeli electoral legislation. The full data set consists of 215 observations. 5 In Belgium, major parties spit along linguistic lines after 1965. To avoid confusion, we do not include data from this period in the analysis. 6 We consider coalitions in Portugal only after 1980, since before then governments were appointed by the president, and not necessarily with parliamentary support. 7 Poguntke [2003], Fallend [2000], O Malley and Marsh [2003], Aalberg [2001], Widfeldt [2003], Bille [2002], Sundberg [2000], Lucardie [2003], Hirsch [2000], Ignazi [2002] and Magone [2000] 8 www.knesset.gov.il 8

3.2 Estimation We perform a maximum likelihood estimation procedure, maximizing the following loglikelihood: L = i,j ln(f ij (y ij )) (1) where f ij (y ij ) = [λ ij (y ij )] d ij e y ij 0 λ ij (t)dt (2) and λ ij (t) = (1 βv ij )c j α I ij γ T ij t (3) where we consider observation i of country j. The number of days that elapse between the formation of the government by the ruling party and loss of power to another party or an election occurs is y ij. The number of days between the formation of the government by the ruling party and the regular elections (i.e., the potential stay in power) is T ij. The censoring dummy variable d receives 1 if y ij < T ij. The dummy variable I receives 1 for big ruling parties. Plugging (2) in (1) gives the actual expression that we maximize: L = i,j (d ij ln(λ ij (y ij )) λ ij(y ij ) λ ij (0) ) (4) ln(γ) Equation 4 has 16 parameters. Of these, 14 parameters determine the rate of events: 13 country fixed effects c j and the time-to-elections effect γ. The 15th parameter is β, which is our parameter of interest: the effect of the ruling party s voting power on its probability of surviving the event. The last parameter is the big party effect α. In order to receive positive λ, we restrict the values of the country fixed effects c j, the time-to-elections effect γ, and the big party effect α to be nonnegative by using an exponential function. Additionally, we 9

restrict the value of β to be between 0 and 1. We calculate the voting power V ij for each α and ρ using a simulation. We draw 100,000 random numbers p from a Beta(α, α) distribution. These numbers correspond to the probability of voting yes in 100,000 hypothetical critical votes. Then we simulate the voting results for each of them, such that the correlation between the ideologically neighboring parties is ρ. The simulation is based on Lemma 9 2.1. Finally, we calculate the proportion of votes when the ruling party is on the winning side. This is its voting power V ij, which we incorporate in the empirical model. After estimating the parameters, we use the likelihood-ratio chi-square statistic to test the hypothesis that β is zero, which means that voting power has no impact on the ruling party s survival. Additionally, we calculate the 95% confidence interval for β, which is [β, β], where β is the smallest β such that the hypothesis β = β is not rejected with a significance level of 0.05, and β is the largest β such that the hypothesis H 0 : β = β is not rejected. To calculate the confidence interval, we find the maximal log-likelihood of the model while fixing β on each value between 0 and 1 (stepping by 0.01). β is included in the confidence interval if 2(L 0 L 1 ) χ 2 1,0.95, where L 0 is the maximal log-likelihood under the hypothesis H 0 : β = β and L 1 is the maximal log-likelihood without this restriction. Thus, β is included in the confidence interval if L 1 L 0 + χ2 1,0.95 2. The estimation is performed for values of α between 0.1 and 10 (for values higher than 10 differences are negligible) and for values of ρ between -0.99 and 0.99. Thus, we are agnostic about the true parameters of the voting distribution, and report the resulting coefficient of β for each combination of a and ρ. Finally, we repeat the estimation for different definitions of a big ruling party: namely, a party with more than 10% of the parliamentary seats, a party with more than 30%, and a party with more than 50% of the seats. 9 Technically, it is enough to draw a Bernoulli number with probability p. This is the vote of the leftist party. Then we use the results in Section 2 to recursively simulate the vote of all other parties, conditional on the leftist one. 10

3.3 Results We start by presenting descriptive statistics of the data set and of the simulated voting power. The first three columns of Table 1 present, respectively, the mean duration of the ruling party s stay in power in days, the mean potential duration, and the mean proportion of the ruling party s seats in the parliament (all with the standard deviation given in parentheses). All data is presented by country. The next nine columns present the mean simulated voting power (and standard deviation in parentheses) for different values of the voting distribution parameters α and ρ. We observe several facts. First, the party s voting power is by no means a monotonic transformation of its size. Second, the voting power converges to 1 as the correlation between ideologically neighboring parties ρ rises. The explanation is that when the correlation between parties is positive and high, all parties vote similarly, which places the ruling party almost always on the winning side. Third, the voting power is close to 0.9 in all countries when ρ is equal to zero. The last two columns of Table 1 present, respectively, the estimated average rate of critical events at the beginning of the ruling party s power and on the last day before the regular elections (of course, in many observations the elections come earlier). 10 We observe that crises that challenge the ruling party s power are very rare at the beginning of the term and fairly rare at the end. 10 The estimates are for α = 0.5 and ρ = 0. 11

Data (days) Voting power (simulated) Rate of crises Rate of crises Duration Potential Size of the α = 0.5 α = 1 α = 10 at t = 0 at t = T duration ruling party ρ = 0.75 ρ = 0 ρ = 0.75 ρ = 0.75 ρ = 0 ρ = 0.75 ρ = 0.75 ρ = 0 ρ = 0.75 ρ = 0, α = 0.5 ρ = 0, α = 0.5 1168 1439 0.47 0.87 0.92 0.98 0.83 0.89 0.97 0.77 0.84 0.96 (313) (28) (0.05) (0.14) (0.06) (0.03) (0.19) (0.08) (0.03) (0.29) (0.11) (0.05) 0.0004 0.0292 1124 1490 0.44 0.85 0.90 0.98 0.80 0.87 0.97 0.74 0.81 0.95 (465) (21) (0.07) (0.13) (0.05) (0.02) (0.18) (0.07) (0.03) (0.29) (0.11) (0.04) 0.0003 0.0224 776 1407 0.31 0.87 0.89 0.93 0.85 0.85 0.90 0.82 0.79 0.86 (391) (159) (0.10) (0.08) (0.04) (0.04) (0.11) (0.06) (0.05) (0.18) (0.12) (0.08) Austria Belgium Denmark Finland Germany Ireland 0.0010 0.0539 793 1168 0.25 0.81 0.87 0.95 0.75 0.83 0.94 0.67 0.75 0.91 (477) (352) (0.05) (0.07) (0.02) (0.03) (0.10) (0.03) (0.04) (0.19) (0.05) (0.06) 0.0014 0.0235 1261 1407 0.46 0.92 0.91 0.96 0.91 0.88 0.95 0.92 0.83 0.92 (363) (178) (0.04) (0.04) (0.04) (0.02) (0.04) (0.06) (0.03) (0.05) (0.09) (0.04) Israel Italy 0.0004 0.0192 1139 1744 0.44 0.89 0.92 0.98 0.85 0.90 0.97 0.82 0.86 0.96 (437) (164) (0.09) (0.13) (0.06) (0.02) (0.17) (0.09) (0.03) (0.26) (0.14) (0.05) 0.0004 0.0535 1111 1448 0.34 0.89 0.90 0.93 0.87 0.88 0.91 0.87 0.85 0.87 (316) (236) (0.07) (0.05) (0.03) (0.05) (0.07) (0.05) (0.06) (0.13) (0.08) (0.09) 0.0004 0.0219 975 1518 0.35 0.89 0.90 0.96 0.87 0.88 0.95 0.85 0.84 0.93 (658) (485) (0.14) (0.09) (0.05) (0.03) (0.13) (0.07) (0.03) (0.21) (0.12) (0.05) 0.0017 0.0422 1596 1638 0.40 0.81 0.90 0.99 0.75 0.86 0.98 0.65 0.81 0.97 (364) (348) (0.07) (0.12) (0.03) (0.01) (0.17) (0.04) (0.02) (0.29) (0.07) (0.02) 0.0001 0.0091 1046 1217 0.29 0.86 0.87 0.95 0.83 0.83 0.93 0.80 0.76 0.90 (498) (449) (0.08) (0.07) (0.02) (0.02) (0.10) (0.03) (0.03) (0.18) (0.05) (0.04) 0.0008 0.0074 958 1280 0.36 0.88 0.90 0.95 0.85 0.87 0.93 0.81 0.81 0.90 (490) (326) (0.16) (0.11) (0.08) (0.05) (0.16) (0.11) (0.06) (0.25) (0.18) (0.09) 0.0009 0.0192 1159 1440 0.46 0.93 0.94 0.99 0.92 0.93 0.98 0.94 0.91 0.98 (411) (0) (0.12) (0.05) (0.04) (0.02) (0.06) (0.06) (0.02) (0.06) (0.07) (0.03) 0.0002 0.0146 1091 1156 0.41 0.89 0.92 0.98 0.86 0.90 0.97 0.84 0.87 0.96 Luxembourg Netherlands Norway Portugal Sweden (328) (301) (0.12) (0.12) (0.05) (0.04 ) (0.16) (0.08) (0.05) (0.27) (0.13) (0.07) 0.0004 0.0068 12 Note: The values in the table are the means. The standard deviations are given in parentheses. Table 1: Descriptive statistics

The alternative definitions of a big ruling party give a similar statistical likelihood of the model, with a slight advantage to the definition that a ruling party is big if it occupies more than 30% of the parliament. Thus, we stay with this definition while reporting all further results. Figure 1 presents the estimated β, i.e., the effect of the ruling party s voting power on its probability of losing power, as a function of the voting distribution parameters α and ρ. The estimated β is between 0.8 and 0.9 for values of ρ close to zero. The interpretation of β = 0.9 is that every percentage point increase in the ruling party s voting power increases by 0.9 percentage points the ruling party s probability of surviving a critical vote in the parliament. Figure 1: The estimated β, full sample Note: a ruling party is defined as big if it holds more than 30% seats in the parliament. The estimated β decreases in α and approaches 1 when α 0. The case of a very small α corresponds to a parliament in which parties vote either yes or no almost unanimously 13

(but independently if ρ is zero). Thus, when α decreases, the voting power shifts to the right, but more so for the small values (consider, for example, the maximal possible voting power value, 1, which cannot increase). Thus, the slope between the voting power and the explained variable increases, which means a rising β. The main result appears in Figure 2, which presents the p-value of the chi-square statistic that tests the hypothesis that β = 0. The figure shows that the model obtains a similarly good statistical significance of the voting power effect while assuming different values of α. However, the model is sensitive to ρ, the correlation coefficient between ideologically neighboring parties. The effect of the voting power is statistically significant for negative and low positive values of ρ, but is not statistically significant for a high positive correlation between neighboring parties. The intuition behind this asymmetrical result is that when ρ is positive and high, the parties in the parliament vote similarly. As a result, the ruling party is almost always on the winning side, and, according to the model, the ruling party should only rarely lose power. However, this would contradict the data. Thus, the hypothesis that voting power does not predict stability for the ruling party is not rejected when ρ is positive and high. 14

Figure 2: The p-value for the hypothesis β = 0, full sample Note: a ruling party is defined as big if it holds more than 30% seats in the parliament. The difference between the case of α approaching zero and the case of ρ approaching 1 is worth explaining. In both cases, the probability of the ruling party (and of all the others) being on the winning side converges to 1 because all parties vote similarly. However, we observe a large, statistically significant β in the case of small α, but a small and insignificant β in the case of large ρ. The reason is as follows. As was explained above, when α is small, the voting power of all observations increases and the estimated slope between the voting power and the explained variable rises. But when the correlation between the neighboring parties ρ increases, the voting power of the parties in the center of the political scale rises more than that of the relatively extreme parties. Thus, extreme ruling parties experience a smaller increase in their voting power when ρ rises than ruling parties in the center, and this non-monotonicity destroys the estimated relationship between the ruling party s voting 15

Figure 3: The 95% confidence interval for β, full sample power and its duration in power. The conclusion that our model fits the data better if ρ is small is underlined by the 95% confidence interval of β, presented in Figure 3, as a function of ρ, for different values of α. The confidence interval clearly shows that if the correlation between the neighboring parties ρ is small, β is confidently close to 1 (but not very close - again, a too high effect of the voting power would contradict data). This result is in line with the earlier theoretical literature which assumed (for simplicity) no correlation between the parties in the parliament, while calculating their voting power. 3.4 Robustness check We are concerned with the possibility that the estimated β is biased, because parties that lose power are mechanically overrepresented in the data, since they have, on average, a shorter 16

stay as ruling parties. This selection may inflate or deflate the importance of crisis survival for the ruling party. To rule out this concern, we filter the sample by considering only the coalitions that existed on January 1st of every fifth year, starting with 1.1.1950. Because no coalition exists for more than four and a half years, this filtering eliminates any selection bias. The filtered sample consists of 122 observations. We repeat the estimation procedure using the restricted sample. The estimated β, as a function of the voting distribution parameters α and ρ, is presented in Figure 4 and the p-value corresponding to the hypothesis β = 0 appears in Figure 5. We find that the results are very similar to the full sample results, except that β has a slightly lower statistical significance because of the small sample size. The 90% confidence interval for β is very similar to the 95% confidence interval in the full sample, as Figure 6 shows. Figure 4: The estimated β, filtered sample Note: a ruling party is defined as big if it holds more than 30% seats in the parliament. 17

Figure 5: The p-value for the hypothesis β = 0, filtered sample Note: a ruling party is defined as big if it holds more than 30% seats in the parliament. In another robustness check, we estimate the model with the weight of the ruling party, instead of its voting power. We find that it gives a smaller maximal likelihood when ρ is close to 0. To illustrate, Figure 7 plots the maximal likelihood of our baseline model versus the maximal likelihood of the model with weight of the ruling party plugged in instead of its voting power, for the case α = 0.5. 4 Conclusions We present a general probabilistic voting model that predicts how long does a ruling party stay in power. According to post-war European data, the effect of the ruling party s voting power on political stability is statistically significant when the correlation between parties 18

Figure 6: The 90% confidence interval for β, filtered sample 19

Figure 7: The baseline model versus the model with size of the ruling party instead of its voting power. 20

in the parliament is low. The best results are obtained if the voting of different parties in the parliament is close to being independent. The assumption of independent voting is widely used in the voting literature in general and in the voting power literature in particular, for two reasons: simplicity and naturalness. Indeed, as we are interested in a-priori properties of a political system, the independence assumption is natural. We suggest another reason for this assumption: predictive power. As for the distribution of the a-priori probability of voting yes or no, our finding indicates that the effect of the voting power is only slightly sensitive to the different distributions. However, the effect of the voting power becomes very strong when the a-priori probability to voting yes is zero or one. In this case, the importance of the ruling party s voting power rises. References T. Aalberg. Norway. European Journal of Political Research, 40(3-4), 2001. M. Albert. The voting power approach: Measurement without theory. European Union Politics, 4(3):351 366, 2003. J.F. Banzhaf. Weighted voting doesn t work: A mathematical analysis. Rutgers Law Review, 19:317 343, 1964. L. Bille. Denmark. European Journal of Political Research, 41(7-8), 2002. V. Chua and D. S. Felsenthal. Coalition formation theories revisited: An empirical investigation of Aumann s hypothesis. Power, Freedom, and Voting, M. Braham and F. Steffen (eds.), pages 159 183,Springer, 2008. P. J. Coughlin. Probabilistic Voting Theory. Cambridge University Press, 1992. L. C. Dodd. Coalitions in Parliamentary Government. Princeton University Press, 1976. 21

P. Dubey and L.S. Shapley. Mathematical properties of the Banzhaf power index. Mathematics of Operations Research, pages 99 131, 1979. F. Fallend. Austria. European Journal of Political Research, 38(3-4), 2000. D.S. Felsenthal and M. Machover. The Measurement of Voting Power: Theory and Practice, Problems and Paradoxes. Edward Elgar Cheltenham, UK, 1998. M. Hirsch. Luxembourg. European Journal of Political Research, 38(3-4), 2000. M. J. Holler and H. Nurmi. Power, Voting, and Voting Power: 30 years after. Springer, 2013. P. Ignazi. Italy. European Journal of Political Research, 41(7-8), 2002. G. King, J. E. Alt, N. E. Burns, and M. Laver. A unified model of cabinet dissolution in parliamentary democracies. American Journal of Political Science, 34(3):846 871, 1990. S. Kurz, N. Maaser, S. Napel, and M. Weber. Mostly sunny: A forecast of tomorrow s power index research. Tinbergen Institute Discussion Paper 14-058/I, 2014. A. Laruelle and F. Valenciano. Voting and Collective Decision-making: Bargaining and Power. Cambridge University Press, 2008. P. Lucardie. The Netherlands. European Journal of Political Research, 42(7-8), 2003. J. M. Magone. Portugal. European Journal of Political Research, 38(3-4), 2000. J. McCrary. Manipulation of the running variable in the regression discontinuity design: A density test. Journal of Econometrics, 142(2):698 714, 2008. D. C. Mueller. Public Choice III. Cambridge University Press, 2003. W. C. Müller and K. Strom. Coalition Governments in Western Europe. Oxford University Press, 2003. 22

E. O Malley and M. Marsh. Ireland. European Journal of Political Research, 42(7-8), 2003. T. Poguntke. Germany. European Journal of Political Research, 42(7-8), 2003. D. W. Rae. Decision-rules and individual values in constitutional choice. American Political Science Review, 63(01):40 56, 1969. G. Sartori. Parties and Party Systems: A Framework for Analysis. Cambridge University Press, 1976. L.S. Shapley and M. Shubik. A method for evaluating the distribution of power in a committee system. The American Political Science Review, 48(3):787 792, 1954. P. D. Straffin. Homogeneity, independence, and power indices. Public Choice, 30(1):107 118, 1977. J. Sundberg. Finland. European Journal of Political Research, 38(3-4):374 381, 2000. P. Warwick. Government Survival in Parliamentary Democracies. Cambridge University Press, 1994. A. Widfeldt. Sweden. European Journal of Political Research, 42(7-8), 2003. 23

Appendix Proof of the Lemma 2.1 Denote P rob(x i+1 = 1 X i = 1) = q P rob(x i+1 = 1 X i = 0) = z Let 1 i < n. Since X i, X i+1 are distributed with the Bernoulli distribution, E[X i ] = E[X i+1 ] = p (5) and σ 2 X i = σ 2 X i+1 = p(1 p) (6) By assumption, ρ Xi,X i+1 = Cov[X i, X i+1 ] σ Xi σ Xi+1 = ρ (7) and by (6) and (7) Cov[X i, X i+1 ] = ρp(1 p) (8) On the other hand, Cov[X i, X i+1 ] = E[X i X i+1 ] E[X i ]E[X i+1 ] and by (5) and (8) E[X i X i+1 ] = ρp(1 p) + p 2 (9) Since X i, X i+1 obtains values 0 and 1 only, E[X i X i+1 ] = P rob(x i = 1, X i+1 = 1) = P rob(x i+1 = 1 X i = 1)P rob(x i = 1) = qp (10) 24

and by (9) and (10) q = ρ(1 p) + p (11) as required. We will derive now the formula for z. P rob(x i+1 = 1) = P rob(x i+1 = 1 X i = 1)P rob(x i = 1)+P rob(x i+1 = 1 X i = 0)P rob(x i = 0) Therefore, p = qp + z(1 p) (12) And from (11) and (12) z = p(1 ρ) 25