The Ruling Party and its Voting Power

The Ruling Party and its Voting Power Artyom Jelnov Pavel Jelnov January 11, 2016 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. This concept is different from the stability of a ruling coalition, since a ruling party may form an alternative coalition as long as it is on the winning side in a critical vote. 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 (Lijphart (1984)), the existing literature agrees that any change in the party composition of a coalition is considered to be the end of the coalition. Most of literature also counts a personal change of the prime minister as the end of a cabinet. Ariel University, Israel, artyomj@ariel.ac.il Tel-Aviv University and Leibniz Universität Hannover, jelnov@aoek.uni-hannover.de 1

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 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 political survival of a ruling party, as distinct from the coalition, is novel in the literature. Our concept was influenced by event methodology (see King et al. (1990) and Warwick (1994)). However, our research question is different from theirs, since we focus on stability of the ruling party. The motivating resons are as follows. The prime minister is generally a person who shapes the policy of the whole government, and s/he does it in according to the ideological standings or/and interests of the party s/he represent. Obviously, in a parliamentary democracy a prime minister is constrained by other parties, and we do not ignore an importance of the government composition, but we claim that the prime minister s party has a special impact on the government policy. Let us give some examples. Duverger (1959) introduced a notion of a dominant party: A party is dominant when it is identified with an epoch... A dominant party is that which public opinion believes to be dominant (see also Sartori (1976) for a discussion of this notion). Example of dominant parties are the Swedish Social Democratic party in 1932-1976, whose long-term rule gave...an important advantage in socializing subsequent generations of Swedish voters so that much of its once controversial agenda gradually took on the air of common sense (Pempel (1990, p.18)). Another example is the Italian Christian Democratic Party which has succeeded in building itself into the Italian social structure by means of political patronage: all over the country are people who owe their jobs, their pensions, their new village hall or local highway to Christian Democratic patronage (Irving (1979, p.59)). The German CDU, which ruled in the post-wwii period, created the Federal Republic,... did even more in establishing a 2

new political culture (Burkett (1975, p.23)). Note that in each one of the mentioned countries, during the relevant period, many changes of the cabinet coalition compositions took place. Despite these changes, and the personal changes of prime ministers, an extremely high influence of being the same party ruling is clear. One may argue that the importance of the identity of the ruling party takes place only if it is ruling for a long period. To illustrate why this statement may be wrong, 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 the 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 1996 the Labor party was in power, and the main policy directions (toward peace agreements with the Palestinians, for example) did not change. Moreover, Snyder et al. (2005) show that the formateur (which is typically the ruling party, see Warwick and Druckman (2001)) receives a higher share of the cabinet offices than its electoral weight. This paper shows that the duration of being a party ruling can be predicted by its Rae index (Rae (1969)). The Rae index of a party is the probability that it belongs to a majority coalition, if all coalitions have equal probability to be formed. However, the assumption of equal likelihood of all coalitions is strong. We relax it by introducing a more general probabilistic voting model, which depends on ideological positions of parties. The results of our estimation show that Rae s assumptions have a predictive power, in comparison to a large set of alternative assumptions, which are covered by our general probabilistic model. We make three contributions. First, we analyze a new question dealing with political 3

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 the voting power in a general probabilistic model. 1.1 Overview of the 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 (a common assumption in the literature, which we relax in Section 4). Ideologically, parties are located along the uni-dimensional left-right scale. Following the establishment of a government coalition, a shock, or 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. But even if the crisis takes place, it does not mean an automatic government dissolution. As Lupia and Strøm (1995) wrote:...events...are not inherently critical. Instead, events become critical through their effects on parliamentary bargaining. We restate this argument as events become critical through the ability of the ruling party to survive its consequences. We assume that each crisis is followed by a critical voting in the legislature. Examples of the critical votes are a no-confidence vote, a vote for approval of some important policy, a vote on idelogical issues (for example, the 1995 vote on the law proposal that required a special majority to change the legal status of the Golan Heights in Israel), a vote for the establishment of an alternative government coalition, etc. If a crisis does occur, the survival of the ruling party depends on the result of a critical vote in the parliament. If the majority in the parliament votes like the ruling party, the ruling party overcomes the crisis and continues to be in power, probably with changes in the composition of the government (and in this case we write that the ruling party is on the winning side). Otherwise, the ruling party loses 4

its ruling party status. 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. Coalitional agreements are signed under some political conditions, and one can not assume a-priori that parties will keep its commitment to the prime minister if those conditions have changed. Otherwise, no change in coalitional composition would occur ever, and no government would fall. For example, in Germany in 1982 the Free Democratic Party (FDP) was a member of the coalition led by the Social Democrats, but following an economic downturn it found itself closer to the main opposition party, the CDU. As a result, the FDP decided to cooperate with the CDU, and joined a new government formed by the CDU (see Lupia and Strøm (1995) for detailed description of these events). about 30% of the observations in our data set are of minority governments, which are in power thanks to ad hoc external support, which is a special case of our model. 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, drawn from Beta distribution (a general method to generate a random number between 0 and 1). 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). 1 Observing the actual distribution of voting in favor of a proposed bill by the U.S. House of Representatives (McCrary (2008)) shows that the actual distribution of voting yes is a mix of Beta distributions with different parameters. 5

In order to reflect ideological standings of parties, we assume that there is correlation between the voting of parties which are neighbouring on the right-left ideological scale. Namely, if two parties are adjacent on the left-right scale, the correlation coefficient between their votings is ρ (we discuss and test this assumption in Section 4). To summarize, we estimate a hazard rate model, where the survival of a 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). 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 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. For the formal definition of the model, see Section 2. The probabilty of survival of the ruling party is a multiplication of the probability of a critical event and the probability of survivng such an event. In the empirical analysis, the covariates of the first term, the rate of critical events, include the country fixed effects which absorb the country s political culture and its institutions (for example, whether a country is semi-presidential or not). In regard to time-to-elections effect, 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 one year before the formal end of the term. We accept the conjecture that the frequency of crises may change as the formal end of a term approaches, but we relax the censoring assumption and let the rate of critical events to change continuously. It may happen in real politics that the prime minister initiates the legislature dissolution and shortens her own ruling term. At first glance, it seems like a contradiction to our concept. 6

However, such events are in line with our logic. Generally, a prime minister shortens his/her term in one of two cases. First, if the formal end of the term is close, 2 and then the impact of the dissolution is relatively small. Second, if a prime minister feels that his political standing is not stable, and if s/he would not call for election, other parties will do so. An exception is a decision by Germany Chancellor Schröder in 2005 to dissolve the parliament (he asked his own cabinet members to abstain during a no confidence vote). Actually, Mr. Schröder s decision followed his party s lost in the local elections, and his desire to avoid a total blockage of the German political process as the government camp was now left without any votes to rely upon in the Bundesrat (Poguntke (2006)). Namely, Chancellor Schröder was motivated by a change in the composition of the upper chamber. However, in this study we follow the literature, where, with no exception known to us, formation of government coalition and its survival is explained according to the composition of a lower chamber of parliament. Therefore, taking the upper chamber into account is left out of our scope. It could be claimed that the stability of a political system depends on the existence of a sufficiently large party (see Sartori (1976)). To control for it, we use a dummy variable, which receives one if the ruling party is large (we test different definitions of what is large ). The literature introduced also other indices, such as fractionalization (Sartori (1976)), that could be controlled for. However, as shown below, our results are robust, and we chose to stay with a relatively small number of parameters in the baseline model, in addition to the country fixed effects which eliminate most of the omitted variables bias. Next, we connect our model to the notion of voting power. 3 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 2 For instance, by the Danish election law, the prime minister advises to monarch on the date of election, but it has to take place not later than four years after the previous election. 3 For a comprehensive survey of this see Felsenthal and Machover (1998). For recent research see Holler and Nurmi (2013) and Kurz et al. (2014). 7

probability is also addressed in the voting power literature, 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. 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 and ρ = 0, since Beta(1, 1) is the uniform distribution. 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 The parameters of the model are estimated using a panel of post-wwii coalitions of thirteen European parliamentary democracies. From our empirical findings follows that both α = 1, ρ = 0 and α, ρ = 0 are cases where our model fits the data well. Therefore, the Rae index (and the Banzhaf index) has some predictive power, as well as the distribution of votes that lies in the basis of the Shapley-Shubik index. 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). Importantly, our empricial results are not immediately comparable to the previous research because of a different concept of political stability. Furthermore, the empirical results show 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. Moreover, the likelihood of the model is maximal when ρ is close to 0, in 4 As α, Beta(α, α) converges to the degenerate distribution p = 1 2 with probability 1. 8

line with Rae s assumptions. 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. We perform a set of robustness checks. They are supposed to eliminate concerns regarding (a) selection bias in the sample, (b) the importance of the assumptions for the results, and (c) the even concept of the voting power as an explanatory variable. First, we restrict the sample to only those coalitions that existed on January 1st 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. We observe very similar results to the full sample, with a little lower statistical significance due to a smaller number of observations. Second, we relax the assumptions of the model. The first assumption to be relaxed is the intra-party discipline. Alternatively, we let only a proportion of the parliament members to vote according to the party s decision, while the rest vote independently according to the a-priori probability p, the same as for the parties. The second assumption to be relaxed is the fixed correlation between the ideologically neighbouring parties. Instead, we assume that the correlation exists between the ideologically neighbouring voters, who are uniformely distributed on the left-right scale. The correlation between the neighbouring parties is assumed to be inverse to the ideological distance between their median voters. As a final robustness check, we test the concept of the voting power, by estimating the baseline model with the weight of the ruling party plugged instead of its voting power. We find that it gives a lower statistical likelihood than the baseline model when ρ is not very large in absolute terms. The latter implies that the voting power, although it correlates with the party s weight, is a more informative metric. 9

1.2 Related literature The existing literature widely discusses the formation and the stability of governmental coalitions. Following classical books by Riker (1962), Axelrod (1970) and De Swaan (1973), many authors suggested models of coalition formation. A partial list includes Baron and Ferejohn (1989), who performed a game-theoretic analysis of bargaining over resources (offices, budget, etc.) allocation to parties, during formation of a cabinet, and Snyder et al. (2005), who extended it to weighted majority games (see also Morelli (1999)). Austen-Smith and Banks (1990) and Laver and Shepsle (1990) suggested (independently) models of allocation of ministerial portfolios in coalitional bargaining, where each portfolio is associated with one of the policy or ideology issues. As for the survival of coalitions literature, one of the most influential contributions is Dodd (1976), who shows how duration of a coalition depends on the characteristics of the coalition, such as whether it is a minimal winning coalition. Browne et al. (1986) were the first, as far as we know, to suggest the event approach, where the termination of a coalition is caused by a random shock. King et al. (1990), extended by Warwick (1994), used the unified approach, where probability of a coalition termination event depends on attributes of the coalition and of the parliament. Laver and Shepsle (1998) classified different types of events and studied their impact on cabinets. Lupia and Strøm (1995) argued that a decision of parties to dissolve a coalition, given a potentially critical event, depends on strategic considerations, and suggested a strategic model for the case of three parties. A survey of this literature appears in Laver and Shepsle (1996) and Mueller (2003). It should be emphasized that differently from our model, these studies maintain the traditional concept that any change in the coalitional composition constitutes the end of a coalition. 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 a ruling party, is not studied there. To conclude, Albert (2003) argued that voting power theory is a branch of probability 10

theory and can safely be ignored by political scientists. 5 To the contrary, our analysis provides evidence that some aspects of voting power theory can be applied to the real-world political issues. The rest of the paper is organized as follows. Section 2 presents the model, Section 3 presents the data, the empirical methodology, and the results, Section 4 presents the robustness checks, and Section 5 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 frequency 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 is 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. 5 See Felsenthal and Machover (2003), List (2003) and Albert (2004) for a conceptual discussion that followed Albert s statement. 11

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. 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, w i > Q) + P rob(x r = 0, w i Q) i N,x i =1 i N,x i =1 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 To test our theory, it is natural to use data from countries with a proportional representation system and a relatively long history of democratic elections. The model is less appropriate for the countries where election rules encourage appearance of one party which holds an absolute majority. Therefore, we excluded from our analysis countries with a majoritarian electoral system (UK, France, Canada, India, etc.). For the same reason, we excluded Greece, where the majority bonus system provides, in most cases, an absolute majority to one party. The presence, in a majoritarian system, of independent parliament members, who are not associated with any party, makes them also less appropriate for our analysis. We also exclude the so-called new democracies, since there is no sufficiently long history of elections there. We use the composition of the post-world War II parliaments (lower chambers) and of the 12

government coalitions in thirteen countries: Austria, Belgium, 6 Denmark, Finland, Germany, Ireland, Israel, Italy, Luxembourg, the Netherlands, Norway, Portugal, 7 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 (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)). The data for Israel is from Chua and Felsenthal (2008) (including ideological positions of parties) and from the official Knesset (parliament of Israel) website (www.knesset.gov.il). 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. 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) 6 In Belgium, major parties spit along linguistic lines after 1965. For instance, there are formally two distinct parties: the Flemish Socialist party and the French-speaking Socialist party, which hold similar ideological positions and are generally members in the same coalitions. To avoid confusion, we do not include data from this period in the analysis. 7 We consider coalitions in Portugal only after 1980, since before then governments were appointed by the president, and not necessarily with parliamentary support. 13

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 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 restrict the value of β to be between 0 and 1. 3.3 Calculation of the voting power We calculate the voting power V ij for each combination of α and ρ using a simulation. We draw 100,000 random numbers p from the Beta(α, α) distribution. These numbers correspond to the a-priori 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 neighbouring parties is ρ. Technically, it is sufficient to draw a Bernoulli number with probability p. This is the vote of the leftist party. Then we use Lemma 2.1 to 14

recursively simulate the vote of all other parties, conditional on the leftist one. Note that ρ(1 p) + p may be negative for negative values of ρ. Therefore, the list of the 100,000 simulated p is restricted to the values that provide non-negative ρ(1 p) + p, the probability to vote yes conditional on voting yes by the ideological neighbour. 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. 3.4 Statistical inference 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 H 0 : 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. The maximum likelihood 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. 15

3.5 Results We start by presenting descriptive statistics of the data set and of the simulated voting power. These figures are reported in Table 1. All data is presented by country. The first three columns of the Table present, respectively, the mean duration of the ruling party s stay in power in days, the mean potential duration, and the mean weight of the ruling party in the parliament (all with standard deviations given in parentheses). We observe that in most countries the ruling party has, by average, betwee 40% and 50% of the seats in the parliament. Netherlands and Finland are exceptions, with less than 30% of the seats, by average. The next nine columns present the mean simulated voting power (and its standard deviation in parentheses) for different values of the voting distribution parameters α and ρ. These value are calculated according to the procedure described in Subsection 3.3. We observe several facts. First, the party s voting power is by no means a monotonic transformation of its size. Moreover, the variation between the countries in the ruling party s average voting power is much smaller than the variation in the ruling party s average weight. For example, in Finland, where the ruling parties are the smallest in our sample, their average voting power is the same large as in Luxembourg, where the ruling party holds, by average, 40% of the parliament. Second, the voting power converges to 1 as the correlation between ideologically neighbouring 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). 8 We observe that crises that challenge the ruling party s power are very rare at the beginning of the term 8 The estimates are for α = 0.5 and ρ = 0. 16

and fairly rare at the end.

Austria Belgium Denmark Finland Germany Ireland Israel Italy Luxembourg Netherlands Norway Portugal Sweden 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) 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) 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 (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 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 survival probability, 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 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 (but independently if ρ is zero). Thus, when α decreases, the voting power shifts to the 19

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. Figure 2, presents the p-value of the chi-square statistic that tests the hypothesis H 0 : 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 neighbouring 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 neighbouring 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. 20

Figure 2: The p-value for the hypothesis β = 0, full sample The difference between the case of α approaching zero and the case of ρ approaching 1 is interesting. 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 a small α, but a small and insignificant β in the case of a 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 neighbouring 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 power and its duration in power. 21

Figure 3: The 95% confidence interval for β, full sample 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 neighbouring parties ρ is small in absolute terms, β is confidently high (but not too high - 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. 4 Robustness checks This section presents a set of robustness checks. They are focused on three concerns: selection in the sample, the role of the assumptions for the results, and the concept of the voting power 22

as an explanatory variable. 4.1 Selection First, 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 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 H 0 : 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. 23

Figure 4: The estimated β, filtered sample 24

Figure 5: The p-value for the hypothesis β = 0, filtered sample 4.2 Assumptions There are two assumptions in the baseline model which may be considered as constraining: the fixed correlation between the votes of the neighbouring parties and the strict intraparty discipline. Here we relax these assumptions, and present alternative models, which we estimate. 4.2.1 Correlation In the baseline model, idelogically neighbouring parties vote with correlation ρ. It may be claimed that this assumption ingores political fractionalization. For example, in a twoparty system, the two parties are neighbouring but may vote very different from each other; differently, in a system with twelve parties, the correlation between the neighbours may 25

Figure 6: The 90% confidence interval for β, filtered sample 26

be stronger. Furthermore, in a democracy politicians shall be concerned with their voters opinions. Therefore, an alternative to the baseline model is to assume that the party s vote follows correlation between neighbouring voters. Specifically, assuming that voters are uniformly distributed along the left-right ideological scale, let us define d i as the distance between the median voters of the neighbouring parties i and i + 1: d i = w i + w i+1 2 We model the correlation between the neighbouring parties as ρ min(d i) d ρ Xi X i+1 = i, ρ 0 d ρ i max(d i, ρ < 0 ) The normalization promises that the correlation coefficient is not larger than 1 in absolute terms and monotonically decreases in d i. The estimated β (available from authors by request), for positive ρ, is smaller than in the baseline model - about 0.65. However, its statistical significance is higher than in the baseline model, and, differently from the baseline model, β is statistically significant for all values of α and ρ. 4.2.2 Intra-party discipline The second assumption that we relax is the strict intra-party discipline. Instead, we assume here that proportion ϕ of the parliament members vote according to the party s decision. The remining 1 ϕ vote independently, with the a-priori probability p (the same as for the parties) to vote yes. Thus, the total proportion of voters yes is ϕw i + (1 ϕ)p i N,x i =1 Estimation of this model for ϕ = 0.9 gives results (available from authors by request) 27

extremely similar to the baseline model. Specifically, plotting the estimated β and its p- value for the different values of α and ρ produces pictures very similar to Figures 1 and 2, respectivelly. Moreover, the statistical significance of β is slightly higher than in the baseline model. 4.3 The concept In the final robustness check, we estimate the model with the weight of the ruling party plugged in instead of its voting power. It may be considered as an intuitive, still theoretically unjustified, specification. We find that it gives a smaller maximal likelihood than the baseline model when ρ is small in absolute terms. To illustrate, Figure 7 plots the maximal likelihood of the 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. Note that because the weight of the party is given by the election results, and does not depend on ρ and α, the maximal likelihood of the alternative model with respect to ρ is a horizontal line. 5 Conclusions We present a general probabilistic voting model that predicts how long does a ruling party stay in power. According to the post-war European data, the effect of the ruling party s voting power on its survival is statistically significant when the correlation between parties in the parliament is weak. 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 provide another reason for this assumption: predictive power. As for the distribution of the a-priori probability of voting yes or no, our findings 28

Figure 7: The baseline model versus the model with the size (weight) of the ruling party instead of its voting power. 29

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