Mogens K Justesen

The Social Choice of EU Treaties Discrepancies between Voter Preferences and Referendum Outcomes in Denmark

The article applies Social Choice theory to analyse new and so far undiscovered aspects of the Danish referendums on the Maastricht Treaty in 1992 and 1993. The article queries whether the amended Maastricht Treaty adopted in 1993 was, in fact, the most preferred alternative for a majority of Danish voters. A reconstruction of voter preferences regarding the political alternatives in the European Union -the Maastricht Treaty, the amended Maastricht Treaty and the Status Quo -reveals that the amended Maastricht Treaty, despite the fact that it was the Condorcet winner and won the 1993 referendum, may not have been preferred by a majority but was probably the most preferred alternative only for a minority of the electorate.

Introduction

In 1992 and 1993 Danish voters faced a fundamental choice concerning the future of Danish integration in the European Community/Union (EC/EU). Voting on the Maastricht Treaty in 1992, a small majority of 50.7% of the voters rejected the treaty, whereas a year later a majority of 56.7% of the voters accepted an amended version of the Maastricht Treaty. A common perception of the results of these referendums is that they somehow reflect the 'will of the people', i.e. that the amended version of the Maastricht Treaty represents the Danish electorate's collectively most preferred outcome. Indeed, the very raison d'être of referendums supposedly is that they somehow express some form of popular will. This justification for referendums is associated with what William [START_REF] Riker | Liberalism against Populism: A Confrontation between the Theory of Democracy and the Theory of Social Choice[END_REF] calls the 'populist' conception of democracy. On this view, referendums are perceived to be a fairer and more democratic way of making political decisions, since the collective choice is made directly by the voters (i.e. 'the people') rather than by indirect means, e.g. through legislation in parliament (see Siune and Svensson, 1993: 100). One important virtue of referendums is, no doubt, that they allow voters to express their preferences directly regarding one or more policy issues. However, this does not mean that the outcome of referendums necessarily reflects a common electoral will or a unique social welfare function in any meaningful sense. In fact, from a social choice theory perspective, such a conception of referendums is misconceived mainly owing to a lack of awareness of the way referendums are designed and the importance of the institutions and collective choice mechanisms that amalgamate individual preferences into a social choice (see [START_REF] Nurmi | Referendum Design: An Exercise in Applied Social Choice Theory[END_REF][START_REF] Brams | Voting on Referenda: The Separability Problem and Possible Solutions[END_REF][START_REF] Brams | The Paradox of Multiple Elections[END_REF][START_REF] Saari | Decisions and Elections: Explaining the Unexpected[END_REF].

However, whereas social choice theory notoriously tends to emphasize problems that appear to be nothing but theoretical constructs and that are difficult to corroborate empirically, mainstream approaches to the study of referendums within political science have witnessed a surge in empirical analyses of EU referendums in recent years (Siune and Svensson, 1993;[START_REF] Siune | The European Union: The Danes Said "No" in 1992 and "Yes" in 1993: How and Why[END_REF][START_REF] Franklin | Attitudes towards Europe and Referendum Votes: A Response to Siune and Svensson[END_REF][START_REF] Franklin | Referendum Outcomes and Trust in Government: Public Support for Europe in the Wake of Maastricht[END_REF][START_REF] Hug | Referendums on European Integration: Do Institutions Matter in the Voter's Decision?[END_REF][START_REF] Christin | Referendums and Citizens Support for European Integration[END_REF]Hug, 2002: Svensson, 2002;[START_REF] Franklin | Learning from the Danish Case: A Comment on Palle Svensson's Critique of the Franklin Thesis[END_REF][START_REF] Hobolt | How Parties Affect Vote Choice in European Integration Referendums[END_REF][START_REF] Hobolt | Taking Cues on Europe? Voter Competence and Party Endorsement in Referendums on European Integration[END_REF]. Many resources have been devoted to examining factors influencing voter decisions in referendums on European integration, for instance the role of information, campaigning, party alignment, and party endorsement of the issue in question (e.g. Siune and Svensson, 1993;[START_REF] Hobolt | How Parties Affect Vote Choice in European Integration Referendums[END_REF][START_REF] Hobolt | Taking Cues on Europe? Voter Competence and Party Endorsement in Referendums on European Integration[END_REF]. In this regard, a particularly contested issue is whether voters in EU referendums primarily vote according to policy preferences or according to their (lack of) support for the incumbent government (see [START_REF] Svensson | Five Danish Referendums on the European Community and European Union: A Critical Assessment of the Franklin Thesis[END_REF][START_REF] Franklin | Learning from the Danish Case: A Comment on Palle Svensson's Critique of the Franklin Thesis[END_REF][START_REF] Garry | Second-Order" versus "Issue-Voting" Effects in EU Referendums: Evidence from the Irish Nice Treaty Referendums[END_REF]. Indeed, the relative importance of policy issues versus concerns for rewarding or punishing incumbent governments leaves voters in EU referendums in an uncomfortable dilemma between favouring a proposal for its political qualities and thereby rewarding an unsuccessful government or rejecting the proposal, with the implication that a popular government is punished [START_REF] Schneider | The Punishment Trap: Integration Referendums as Popularity Contests[END_REF]. Moreover, [START_REF] Hug | Referendums on European Integration: Do Institutions Matter in the Voter's Decision?[END_REF] have shown that domestic institutions relating to whether referendums are binding or non-binding and required or non-required affect voter decisions on whether or not to support a proposal. Similarly, [START_REF] Christin | Referendums and Citizens Support for European Integration[END_REF] provide a theory and some evidence suggesting that institutionally induced opportunities to subject EU-related policy proposals to a referendum may increase voter support for European integration.

The present article departs from this literature by, firstly, adopting an explicit social choice theory perspective on referendums. Although game theory models have previously been used to analyse referendums on EU integration [START_REF] Schneider | The Punishment Trap: Integration Referendums as Popularity Contests[END_REF][START_REF] Christin | Referendums and Citizens Support for European Integration[END_REF], little attention has been given to the topic from the angle of social choice theory. As such, the purpose of the paper is to analyse problems that potentially arise in relation to the use of referendums as a method of preference aggregation. Secondly, the paper does not primarily deal with the more traditional political science issues, such as why voters vote 'Yes' or 'No' in EU referendums, or whether EU referendums possess a 'second-order election' or 'issue-voting' character. Rather, the article contributes to the empirical literature on EU referendums by analysing possible discrepancies between the policy preferences of voters and social choice that emerge in situations of sequential referendums and/or three or more distinguishable alternatives. This will be illustrated by highlighting novel features of the Danish referendums on the Maastricht Treaty in 1992 and 1993 that have so far been completely overlooked.

The following section considers possible relations between voter preferences and the social choices that emerged from the referendums in 1992 and 1993. I attempt to disentangle the preference orders of those voters who voted Yes or No in both the 1992 and the 1993 referendums. The third section asks whether there is a Condorcet winner among the set of EU political alternatives, and the concluding section gives the key findings.

Voter preferences, collective rationality and social choice

Almost immediately after the voters' rejection of the Maastricht Treaty in 1992, negotiations between the Danish government and key parliamentary opposition parties began on the future of Danish EU membership (see [START_REF] Siune | The European Union: The Danes Said "No" in 1992 and "Yes" in 1993: How and Why[END_REF]. The negotiations resulted in agreement on the so-called 'national compromise', which was supported by the government, the Social Liberal Party, the Social Democratic Party and, significantly, the initially anti-Maastricht Socialist People's Party.

Substantially, the national compromise entailed that Denmark should opt out of EU cooperation on four key dimensions that were all part of the original Maastricht Treaty. These were (1) justice and home affairs, (2) a common currency, (3) common citizenship, and (4) a common defence policy. Subsequently, the governments of the other EU member countries agreed to grant Denmark special status on the basis of the national compromise, allowing Denmark to adopt an amended version of the Maastricht Treaty (the so-called 'Edinburgh Agreement'), which included opt-out clauses on the four policy issues.

In 1993 a new referendum was held on the amended Maastricht Treaty. The result of the referendum was that the amended Maastricht Treaty received a majority of 56.7% of the votes, thereby beating the Status Quo. 1 Table 1 shows the change in Yes and No votes in the two referendums based on survey data collected by Siune et al. (1993) and [START_REF] Andersen | Gallup interviews og enquete efter unionsafstemningen[END_REF] after the 1993 referendum.

It is clear from Table 1 that most voters chose to stick to their original vote: an overwhelming majority (96-97%) of those voting Yes in 1992 also voted Yes in 1993. Not surprisingly, more voters (18-23%) changed their vote from No to Yes than the reverse (3-4%). Thus, introducing the amended Maastricht Treaty as a third alternative implied that the No vote from the 1992 referendum was split to a degree that resulted in a relatively clear majority in favour of the amended Maastricht Treaty. ------------------------Maastricht Yes (%) No (%) - --------------------- Sources: Siune et al. (1993) and [START_REF] Andersen | Gallup interviews og enquete efter unionsafstemningen[END_REF]. Note: The real net change in votes between the Maastricht and Edinburgh referendums was 56.7 -49.3 = 7.4 percentage points.

Possible voter preferences

On the basis of the distribution of votes in Table 1, the question is whether it is possible to make inferences with regard to voter preferences over the set of EU political alternatives, X = {Maastricht (M), Edinburgh (E) and Status Quo (SQ)}. 2 It is fairly obvious that the voter population consisted of (at least) four groups, N = {1 yy , 2 yn , 3 ny , 4 nn }, where the subscript indicates each group's voting behaviour at the referendum in 1992 and 1993, respectively. That is, the voter groups voted as follows: Group 1 yy : 'Yes-Yes'; Group 2 yn : 'Yes-No'; Group 3 ny : 'No-Yes'; Group 4 nn : 'No-No'.

Given the distribution of votes in Table 1, it is further possible to estimate the size of the four voter groups, as shown in Figure 1. 3 As Figure 1 shows, the largest voter groups (1 yy and 4 nn ) were either in favour of both the Maastricht Treaty and the Edinburgh Agreement or against them. However, for the purpose of this study, the interesting thing to notice is that none of the voter groups had an absolute majority. The largest group (over 47%) consisted of those who voted Yes in both 1992 and 1993, whereas the smallest group (up to 2%) voted Yes to the Maastricht Treaty but No to the Edinburgh amendments.

The interesting question, then, is whether it is possible on the basis of the information provided in Table 1 and Figure 1 to reconstruct the voter groups' ordinal preference orderings with regard to the three relevant alternatives. Notes: Percentages not in brackets are based on Siune et al. (1993); percentages in brackets are based on [START_REF] Andersen | Gallup interviews og enquete efter unionsafstemningen[END_REF].

The numerically largest voter group (group 1 yy ) voted Yes in both the 1992 Maastricht referendum and the subsequent 1993 referendum, and was as such the most pro-EU of the four groups. The implication of the voting behaviour of group 1 yy could lead one to reason that its most preferred alternative might be the original 1992 Maastricht Treaty. The fact that group 1 yy voted Yes in 1993 as well further indicates that the Edinburgh Agreement was a more preferred alternative than the Status Quo. Consequently, it is straightforward to infer that the Status Quo was the least preferred alternative. On this reasoning, an ordinal preference ordering would place the Edinburgh Agreement between the Maastricht Treaty and the Status Quo. Thus, one might tentatively infer that the collective preference order of group 1 yy can be written as:

M ՝ E ՝ SQ.
The problem is, of course, that it is difficult to establish whether all Yes-Yes voters preferred the Maastricht Treaty to the Edinburgh Agreement; i.e. whether M ՝ E or E ՝ M. We will return to this problem below.

Group 2 yn voted Yes to the 1992 Maastricht Treaty but No in the Edinburgh referendum. Inferring from the voting behaviour of group 2 yn , the Maastricht Treaty was its most preferred alternative and the group's rejection of the 1993 Edinburgh Agreement indicates that the Status Quo was more preferred than the Edinburgh Agreement, making the latter group 2 yn 's least preferred alternative. That is, since M ՝ SQ and SQ ՝ E, then by transitivity M ՝ E. This places the Status Quo in between the 1992 Maastricht Treaty and the 1993 Edinburgh Agreement. Consequently, the collective preference order of group 2 must be as follows: M ՝ SQ ՝ E.

Thirdly, group 3 ny voted No in 1992 but Yes in the Edinburgh referendum in 1993. The fact that group 3 voted Yes to the Edinburgh Agreement strongly indicates that this alternative was more preferred than both the Status Quo and the original Maastricht Treaty, thus making the Edinburgh Agreement the most preferred alternative for group 3 ny . The same logic entails that the Status Quo for group 3 ny was more preferred than the Maastricht Treaty, which must then be considered the least preferred of the three alternatives. In short, since E ՝ SQ and SQ ՝ M, then by transitivity E ՝ M. Placing the Status Quo alternative between the Maastricht Treaty and the Edinburgh Agreement results in the following ordinal preference ordering for group 3 ny : E ՝ SQ ՝ M.

Finally, the No votes of group 4 nn in both referendums indicate that it strongly disapproved of both the original Maastricht Treaty and the Edinburgh amendments. On the basis of the voting behaviour of group 4 nn , it is therefore relatively uncontroversial to infer that the Status Quo was the group's most preferred alternative. Furthermore, one might presume that the 1992 Maastricht Treaty was the least preferred alternative, the reason being that the Edinburgh Agreement did after all reduce the level of Danish integration into the EU, and so was probably perceived as being a lesser evil than the Maastricht Treaty. This would place the Edinburgh Agreement between the Status Quo and the Maastricht Treaty in an ordinal preference ordering. The resulting collective preference order of group 4 nn would then be: SQ ՝ E ՝ M. However, as with the Yes-Yes voters, the difficulty with this reasoning is that it is not straightforward to establish whether M ՝ E or E ՝ M for all No-No voters. 5 Given the analysis above, the provisional ordinal preference orderings of the four voter groups regarding the three EU political alternatives are summarized in Table 2. The most striking observation from Table 2 is that the 1993 Edinburgh Agreement may have been the most preferred alternative for a very small proportion of voters, possibly as few as 9.1%. Despite the fact that the Edinburgh Agreement was the winning alternative and became the social choice following the two referendums, this indicates that the Edinburgh Agreement may have ranked highest in the preference order of only a small minority of Danish voters.

The difficulty with the analysis above, however, is that, although the preference orders for groups 2 and 3 seem reasonable given their voting behaviour, it is -as mentioned -less reasonable to infer the preference orders M ՝ E ՝ SQ for group 1 yy and SQ ՝ E ՝ M for group 4 nn as a whole, because at least substantial parts of these groups may, in fact, have had preference orders in which the E and M alternatives were reversed.

Reconsidering the preference order of the Yes-Yes and No-No voters

To resolve the problem of the ranking of the M and E alternatives for the Yes-Yes and No-No voters, I have empirically tried to estimate the relative size of the subgroups of voters in groups 1 yy and 4 nn who (possibly) preferred E ՝ M rather than M ՝ E. To this end, I use data from the same surveys on which Tables 1 and2 are based. The surveys in Siune et al. (1993) and [START_REF] Andersen | Gallup interviews og enquete efter unionsafstemningen[END_REF] provide data on a number of questions pertaining to the attitudes of the respondents to the Maastricht Treaty and the Edinburgh Agreement, and specifically the issues relating to the Edinburgh amendments. Furthermore, these issues have been shown to be important for how the Danish voters voted in 1992 and 1993 (Siune et al., 1993[START_REF] Siune | The European Union: The Danes Said "No" in 1992 and "Yes" in 1993: How and Why[END_REF][START_REF] Svensson | Five Danish Referendums on the European Community and European Union: A Critical Assessment of the Franklin Thesis[END_REF]. However, the analyses below implicitly rest on the assumption that 'policy matters' in voter decisions on whether to vote Yes or No in the referendums. In light of the debate on whether EU referendums are really on EU political issues or are for or against the incumbent government ( Siune and Svensson, 1993;[START_REF] Siune | The European Union: The Danes Said "No" in 1992 and "Yes" in 1993: How and Why[END_REF][START_REF] Franklin | Attitudes towards Europe and Referendum Votes: A Response to Siune and Svensson[END_REF][START_REF] Franklin | Referendum Outcomes and Trust in Government: Public Support for Europe in the Wake of Maastricht[END_REF][START_REF] Franklin | Learning from the Danish Case: A Comment on Palle Svensson's Critique of the Franklin Thesis[END_REF][START_REF] Svensson | Five Danish Referendums on the European Community and European Union: A Critical Assessment of the Franklin Thesis[END_REF][START_REF] Garry | Second-Order" versus "Issue-Voting" Effects in EU Referendums: Evidence from the Irish Nice Treaty Referendums[END_REF], I have -in order to substantiate the assumption -performed a number of logistical regressions so as to investigate whether voter attitudes towards the (amended) Maastricht Treaty were instrumental in determining the propensity of voters to vote either Yes or No. The results are presented in the appendix. 6 The regressions clearly demonstrate that voter attitudes towards the (amended) Maastricht Treaty were important for voters' decisions on whether to vote Yes or No. 7 Thus, the underlying assumption of the analyses below seems quite reasonable. 8 The question is, then, how we can estimate the size of the subgroups of Yes-Yes and No-No voters who might have preferred E ՝ M? Obviously, this is an inherently difficult task. As a starting point, I note that for the Yes-Yes voters it is, as shown above, reasonable to preclude any preference order in which SQ is ranked above M or E. Similarly for the No-No voters, we can exclude any preference order in which M or E is ranked above SQ. Secondly, on the basis of the surveys by Siune et al. (1993) and [START_REF] Andersen | Gallup interviews og enquete efter unionsafstemningen[END_REF] I have constructed a series of indicator variables of voter attitudes to the (amended) Maastricht Treaty, which can help establish the preference orders of the Yes-Yes and No-No voters. Because the surveys contain different questions relating to the same policy issues, I have constructed six different indicators based on different questions. However, all the questions do solely concern voter attitudes to the opt-out clauses of the Edinburgh Agreement (see appendix for details).

I then constructed two types of indicators. In the first, I isolated the group of voters who were in favour of EU political integration in all the policy areas where Denmark was granted opt-out clauses in 1993 and on which the surveys contain information. This group was then compared with the group of voters who were against one or more of the same policy areas. Because the first group comprises voters who were in favour of more EU political integration than the Edinburgh Agreement offered, it seems reasonable to assume that that they preferred the full Maastricht Treaty to the Edinburgh Agreement (M ՝ E). On the other hand, the voters in the second group were against integration in at least one of the policy areas, and therefore in favour of at least one of the opt-out clauses, and were consequently less positive about full integration. The preference order of these voters is thus assumed to be given by E ՝ M.

However, this approach is likely to overestimate support for the Edinburgh Agreement. For instance, if a voter was against only one of the four policy areas and in favour of integration on the other three issues, then s/he is assumed to prefer E ՝ M, even though s/he was in favour of integration on a majority of issues, and may therefore have preferred M ՝ E. To rectify this problem I constructed a second type of indicator variable, which is a replication of the first index, except that the voters who were in favour of all but one issue are assumed to prefer M ՝ E, whereas the remainder of the voters are assumed to prefer E ՝ M. This will probably provide a somewhat more realistic account of the size of the voter groups who preferred M ՝ E or E ՝ M, respectively. Regardless of which aggregation method one prefers, we get -if nothing else -a measure of the interval of support for the alternatives among the voters.

On the basis of the six indicator variables, the Yes-Yes and No-No voters from Table 2 were divided into two subgroups with the preference orders M ՝ E and E ՝ M, respectively. Table 3 shows the support for the Maastricht Treaty and the Edinburgh Agreement within each subgroup of Yes-Yes and No-No voters.

Table 3 reveals some variation in the estimates of the size of the positive Yes-Yes voters (top row) and the less positive Yes-Yes voters (second row). ------------------------------ On the most optimistic estimate (index A), the Edinburgh Agreement seems to have been more preferred than the Maastricht Treaty for around 80% of the Yes-Yes voters. However, as mentioned this estimate is probably biased upwards, i.e. in the direction of exaggerating support for the Edinburgh Agreement. On the other hand, according to the least optimistic estimate (index D), the Yes-Yes voters were (roughly) equally divided in their support for Maastricht and Edinburgh. Therefore, it seems safe to conclude that at least 50% and at most 80% of the Yes-Yes voters preferred E ՝ M, whereas somewhere between 20% and 50% preferred M ՝ E.

- A B C D E F (4/4) (3/4) (4/4) (3/4) (3/3) (3/3)
As might be expected, an overwhelming majority of No-No voters were consistently against further EU political integration in the policy areas contained in the opt-out clauses of the Edinburgh Agreement. In fact, only a very small minority -at most 4% -of the No-No voters were positive towards increasing EU integration despite voting No. We can therefore divide the No-No group into two subgroups, in which a small minority had a preference order where SQ ՝ M ՝ E, whereas the preference order of the large majority is given by SQ ՝ E ՝ M.

Overall, these results suggest that it is indeed reasonable to divide the Yes-Yes and No-No voters into subgroups with different priorities as to the choice between the Maastricht Treaty and the Edinburgh Agreement. However, doing so is particularly important for the Yes-Yes voters. This supports the view that not all Yes-Yes voters preferred M ՝ E, and that a substantial proportion of the voters who voted Yes to the Maastricht Treaty in 1992 did so despite being only moderately positive in their attitudes towards the Maastricht Treaty, and despite having some (at the time unspecified) intermediate alternative as their most preferred alternative. Since the size of the remaining two voter groups is unaffected by the re-estimation of the size of the new groups of Yes-Yes and No-No voters, Table 4 summarizes information on the ordinal preference orderings and size of the now six voter groups.

Table 4 provides a somewhat more realistic account of the relative size of the voter groups. This obviously means that the Edinburgh Agreement was the most preferred alternative for a proportion of the voters (groups 2 yy and 4 ny ) that was significantly greater than the 11.7% (or 9.1%) estimate in Table 2. That is, apart from the voters in group 4 ny , somewhere between 24.6% and 38.7% of the voters (group 2 yy ) preferred the Edinburgh Agreement to the Maastricht Treaty.

Given this information, we can derive an estimate of the proportion of voters supporting each of the three EU political alternatives (M, E and SQ). This is done in Table 5. The estimates show that the two alternatives with the greatest support in the electorate were the Edinburgh Agreement and the Status Quo. Generally, the Maastricht Treaty was not particularly popular: at best it was the most preferred alternative for around 25% of voters, and at worst for as few as 10%. According to the most optimistic estimates (column 1), the Edinburgh Agreement may, in fact, have been the most preferred alternative for a bare majority of 50.4% of Danish voters. However, it would probably be premature to conclude that the Edinburgh Agreement was preferred by a majority based on this result. Firstly, the estimate of 50.4% is almost certainly exaggerated (see above). Secondly, whereas the other estimates suggest that support for Edinburgh was substantial, more realistic estimates (e.g. index B in column 2) show that support for the Edinburgh amendments was probably well below 50%, somewhere in the neighbourhood of 40%, and perhaps even as low as 36% (index D). In fact, of the six estimates in Table 5, five indicate that the Edinburgh Agreement was not preferred by a majority, but rather was preferred by a plurality of Danish voters. Thus, whereas the Edinburgh Agreement was certainly the most preferred alternative for a large proportion of voters, it is probably reasonable to conclude that a majority of Danish voters preferred a different alternative. That is, the polarization of voters as to the most preferred level of EU political integration meant that a majority probably preferred either more or less integration than was offered by the Edinburgh Agreement.

Why, then, did the Edinburgh Agreement become the social choice? The obvious answer is that it did so, not necessarily by being the most highly valued alternative for the majority of voters, but because it was the alternative that -given the choice between the Edinburgh Agreement and the Status Voter groups and preference order - ---------------------------------- Quo -was closest to the bliss-point of the median voter. Compared with the 1992 referendum, the combination of the votes of groups 1 yy , 2 yy and 4 ny was sufficient to form the majority necessary to beat the Status Quo.

The Condorcet winner

On the basis of the structure of voter preferences displayed in Table 4, it is now possible to examine whether the set of alternatives X = (M, E, SQ) contains a Condorcet winner; i.e. an alternative that can beat all other alternatives in a pairwise comparison regardless of the voting sequence and using simple majority voting (Riker, 1982: 100;Gehrlein, 1983: 162;McLean, 1988: 66;[START_REF] Kurrild-Klitgaard | An Empirical Example of the Condorcet Paradox in a Large Electorate[END_REF]. In so far as the Edinburgh Agreement was preferred by a majority, it was obviously the Condorcet winner too. However, since it is more likely that the Edinburgh Agreement was preferred by a plurality rather than by a majority, and because plurality winners are not always Condorcet winners [START_REF] Riker | Plurality and Runoff and the Number of Candidates[END_REF], it is reasonable to query the status of the Edinburgh Agreement as a Condorcet winner.

From the two referendums we know that the Edinburgh Agreement beat the Status Quo with 56.7% of the votes. 9 Secondly, we know that Status Quo beat the Maastricht Treaty by a majority of 50.7% in the 1992 referendum. From Table 4 we can furthermore infer that the Edinburgh Agreement was majority-preferred to the Maastricht Treaty by between 73.7% and 89.4% of the voters (groups 2 yy , 4 ny and 6 nn ). Using the Condorcet method we accordingly obtain the following collective preference order: E ՝ SQ ՝ M.

This result is interesting for (at least) three reasons. First of all, it reveals that the Edinburgh Agreement -the winner of the 1993 referendum -was the Condorcet winner and thus the collectively most preferred alternative, despite the fact that it was probably not the most preferred alternative for a majority of voters. Secondly, the fact that a Condorcet winner exists rules out the possibility of a Condorcet paradox (i.e. a collectively intransitive preference order where A ՝ B ՝ C ՝ A . . .). Thus, in so far as the Condorcet method is considered a kind of democratic fairness criterion (McLean, 1988: 66), the 1993 referendum did, in fact, produce the 'right' social choice. Thirdly, since the collectively most preferred alternative was probably the most preferred alternative only for a plurality of voters, it would be possible -and quite legitimate -for a majority of between 52.5% and 63.7% of the voters to be dissatisfied with the final result of the two referendums in 1992 and 1993. Stated differently, for a majority of the voters, the winning alternative may not have been the most preferred alternative, but instead a lower-ranking and less preferred alternative.

Although the Edinburgh Agreement was probably the Condorcet winner and the collectively most preferred alternative, given the alternatives for consideration, this makes an unambiguous interpretation of the results of the 1992 and 1993 referendums much more difficult. We cannot say that the Edinburgh Agreement was preferred by a majority of the Danish electorate simply because 56.7% of the voters voted Yes in 1993. To understand the EU political preferences of the Danish voters, we need to disentangle voter preferences instead, and take into account that a substantial proportion of voters -most likely a majority -preferred a level of integration different from the one offered by the Edinburgh Agreement. This fundamentally alters the way we perceive the result of the Danish EU referendums in 1992 and 1993 and the four amendments that were added to the Maastricht Treaty in 1993 and that are still in effect today.

Conclusion

The purpose of this article has not been to assess whether the outcome of the Danish referendums in 1992 and 1993 was right or wrong. The purpose has solely been to analyse the relationship between voter preferences and the outcome that emerged as the social choice following the referendums in 1992 and 1993. The most significant finding in this article is that it is doubtful whether the Edinburgh Agreement -despite the fact that it won the 1993 referendum and despite its probable status as a Condorcet winner -constitutes an expression of some general EU political will of the Danish people in any meaningful sense. Nevertheless, on the positive side, there are strong arguments that the Edinburgh Agreement was the 'right' outcome of the 1992 and 1993 referendums. Regardless of how voter preferences are reconstructed, the Edinburgh Agreement comes out as the Condorcet winner and the median voter favourite. In addition, it may also have been the most preferred alternative for the largest proportion of voters. On the other hand, taking into account that a substantial proportion of Danish voters preferred the Maastricht Treaty or the Status Quo to the Edinburgh Agreement, the latter was most likely not preferred by a majority of voters. This renders the interpretation of the outcome of the referendums in 1992 and 1993 more ambiguous than has been recognized until now, and questions the presence of a unique social choice that reflects the Danish voters' most preferred level of integration into the EU. As such, it is quite simply questionable whether referendums can be conceived as a procedure by which information on individual preferences and values in a simple and uncomplicated manner can be translated into an unambiguous social choice that somehow reflects the will of the people on the issue in question. 1

Notes

The Danish referendums in 1992 and 1993 have some similarity to the Irish referendums on the Nice Treaty in 2001 and 2002, where the Irish voters first rejected the Treaty but then voted in favour of it a year later (see [START_REF] Garry | Second-Order" versus "Issue-Voting" Effects in EU Referendums: Evidence from the Irish Nice Treaty Referendums[END_REF]. However, in the Irish case, the 2001 and 2002 referendums were on the same Treaty. That is, unlike in the Danish case of the Maastricht Treaty, no opt-out clauses were added to the Nice Treaty following the 2001 rejection in Ireland. The only difference between the 2001 and 2002 referendums on the Nice Treaty was that the Irish government and the European Council issued two declarations at the Seville summit in 2002 stating that the Nice Treaty would not affect the policy of Irish military neutrality. However, these declarations did not substantially alter the Nice Treaty (see Garry et al., 2005: 207-8;Department of Taoiseach, 2002). 2

Note that the exact content of the alternative identified as Status Quo is somewhat unclear. In addition, it is possible that voter perceptions of the Status Quo -i.e. perceptions of what would happen if the outcome of 1993 referendum was another No -changed between 1992 and 1993 (see Svensson, 2002: 743). In any case, the Status Quo alternative might be interpreted as containing 'anything but' the Maastricht Treaty (or the Edinburgh Agreement), involving, presumably, less integration into the EU. This includes a range of policy positions, from complete withdrawal from the EU to upholding the pre-1992 membership status. 3

The numbers in Figure 1 are calculated as the proportion of voters whose voting behaviour remained the same or changed between 1992 and 1993. For example, 97% of the 49.3% who voted Yes in 1992 also voted Yes in 1993; hence the net size of the 'Yes-Yes' voters is 47.8%. 4

Voter preferences are assumed to be both (strongly) transitive and complete. Preferences are transitive if X ՝ Y and Y ՝ Z → X ՝ Z. Preferences are complete if, given a choice between X and Y, a person prefers either X ՝ Y, Y ՝ X or X ~Y, the latter expression indicating indifference. 5

A potential problem with constructing these voter groups is that it was not the exact same electorate that voted at the two referendums; i.e. changes occurred owing to natural causes and possible abstentions in one of the referendums. However, since the referendums were close -separated by approximately one year -the problem with changes in the electorate is not very severe. Turnout was 83.1% in 1992 and 86.5% in 1993. 6 The appendix is available at http://www.uni-konstanz.de/eup/issues.htm. 7

Ideally, the empirical analyses would focus on the dynamics of voters changing their votes from 1992 to 1993. However, this would require the use of a panel data set, which unfortunately -to the best of the my knowledge -does not exist for the 1992 and 1993 referendums. 8

Another objection to the analyses is that they presume that the policy space is unidimensional, whereas in reality it might be multidimensional. To investigate this, I have conducted a series of principal component factor analyses (PCA) in order to derive the number of underlying dimensions of EU policy space in the Danish electorate. The PCAs were based on the items listed in appendix A, Table A1. In all cases the PCAs extract only one factor. This validates the assumption that the policy space may be considered unidimensional. Detailed results are available upon request. 9

Using the data from indexes A-D in the first four rows of Table 4, Edinburgh beat SQ by 59.5%. The small difference is owing to sampling error in the survey data.

I

  am grateful to Hannu Nurmi, Steven Brams, Peter Kurrild-Klitgaard, Robert Klemmensen, Gerald Schneider and the anonymous referees at EUP for constructive comments, and to Jens Wagner at the Danish Data Archive for kindly providing data and documentation on the Maastricht and Edinburgh referendums in Denmark. An earlier version of the paper was presented at the 2006 meeting of the European Public Choice Society in Turku, Finland, where it won the Wicksell Prize.

Table 1 Voting changes in the Maastricht and Edinburgh referendums

	Edinburgh (amended Maastricht)

  4

		Yes 47.8% [47.3%] (1yy)
	E
	Yes
	49.3%	No 1.5% [2.0%] (2yn)
	M
	No	Yes 11.7% [9.1%] (3 ny)
	50.7%
	E
		No 39.0% [41.6%] (4nn)

Figure 1

Voter behaviour in the Maastricht and Edinburgh referendums.

Table 2

 2 Possible voter preferences regarding EU political alternatives

	Group 1 yy	Group 2 yn	Group 3 ny	Group 4 nn
	47.8/47.3%	1.5/2.0%	11.7/9.1%	39.0/41.6%
	M	M	E	SQ
	E	SQ	SQ	E
	SQ	E	M	M

Note:

The first percentage figure is based on

Siune et al. (1993)

; the second figure is based on

[START_REF] Andersen | Gallup interviews og enquete efter unionsafstemningen[END_REF]

.

Table 3

 3 Voter support for Maastricht and Edinburgh, Yes-Yes and No-No voters (%)

	Index

Table 4

 4 Ordinal preference structure regarding EU political alternatives given six voter groups

  -Group 1 yy Group 2 yy Group 3 yn Group 4 ny Group 5 nn Group 6 nn Index M ՝E ՝SQ E ՝M ՝SQ M ՝SQ ՝E E՝SQ ՝M SQ՝M ՝E SQ՝E ՝M

	A (4/4)	9.1	38.7	1.5	11.7	0.0	39.0
	B (3/4) 16.3	31.5	1.5	11.7	0.6	38.4
	C (4/4) 12.0	35.8	1.5	11.7	0.2	38.8
	D (3/4) 23.2	24.6	1.5	11.7	1.6	37.4
	E (3/3)	13.6	33.7	2.0	9.1	0.8	40.8
	F (3/3)	14.9	32.4	2.0	9.1	0.6	41.0
	Notes: Numbers are percentages. Horizontal summation within rows equals 100.

Table 5

 5 Total support for the three EU political alternatives as the most preferred alternative

		A (4/4) B (3/4) C (4/4) D (3/4) E (3/3)	F (3/3)
	Total support for E.
	(groups 2 yy + 4 ny )	50.4	42.9	47.5	36.3	42.8	41.5
	Total support for M.
	(groups 1 yy + 3 yn )	10.6	18.1	13.5	24.7	15.6	16.9
	Total support for SQ.
	(groups 5 nn + 6 nn )	39.0	39.0	39.0	39.0	41.6	41.6
	Notes: Numbers are percentages. Vertical summation within columns equals 100.

European Union Politics 8(4)

About the author

Mogens K. Justesen is a PhD candidate in the Department of Political Science, University of Southern Denmark, Campusvej 55, DK-5230 Odense M., Denmark. Fax: +45 6550 2280 E-mail: mju@sam.sdu.dk


Jonathan Golub

Bernard Steunenberg

How Time Affects EU Decision-Making

. Although correct, the interpretation of the results of such a model is less straightforward than he recognized. In this paper we explain how to make sense of the estimates from survival models that contain timeinteraction terms, and then investigate how a more precise interpretation affects Golub's findings. Overall, we draw three main conclusions: the effect that formal qualified majority voting (QMV) rules have on speeding up decision-making is even larger and more consistent than originally claimed; the trade-off between efficiency and legitimacy is more complicated than first thought; and the effects of some key variables do not just wear off but actually reverse direction once proposals survive long periods of time.

Survival models with time-interaction terms

The assumption that the effect of covariates remains constant over time is often unjustified, and violations of the assumption can render a model's estimates and inferences meaningless. Violations can arise naturally in all covariates when their relationship with the hazard is affected by time, but by definition they arise when we have time-varying covariates (TVCs), that is, covariates where the initial coding assigned to a case changes during its lifetime [START_REF] Box-Steffensmeier | Event History Modelling: A Guide for Social Scientists[END_REF][START_REF] Golub | Survival Analysis and European Union Decision-making[END_REF]. The solution is to fit a model with time-interactive terms that accommodate these nonproportional effects [START_REF] Box-Steffensmeier | Duration Models and Proportional Hazards in Political Science[END_REF]Box-Steffensmeier and Jones, 2004), as [START_REF] Golub | Survival Analysis and European Union Decision-making[END_REF] did in his paper on decision-making speed. However, the interpretation of such a model is not straightforward, since the impact of a variable on the hazard rate is now a combination of its timeindependent and time-dependent coefficients. This can be shown as follows.

Imagine that we fit a Cox model that included one TVC and that its effect changed over time. The hazard rate would then be (1) with x(t) 1 as the TVC. Note that, to deal with a non-proportional effect of this covariate, time can be transformed in a way that fits better to the underlying structure in the data (in this instance, by taking the natural logarithm, as is done in Golub's study). We would then use the parameter estimates from this equation to calculate the effect of a unit change in x 1 on the hazard ratio for a case that did not undergo state changes. To get the effect of a unit change in a covariate we construct a hazard ratio where the denominator is the baseline hazard rate (i.e. hypothetical case j with all covariates set to zero). So we have (2) where b ˆ1 is a time-independent coefficient and b ˆ2 ln(t) is a time-dependent coefficient. The numerator at the right-hand side of this equation can be rewritten as:

(3) In other words, the impact of x 1 on the hazard rate is now the result of b ˆ1 and b ˆ2 ln(t). The combined term [b ˆ1 + b ˆ2 ln(t)] is the actual coefficient of the covariate, which depends not only on the values of b ˆ1 and b ˆ2 but also on t.

Golub recognized this much, and he calculated the magnitude of the effects of certain variables at different values of t, the point after which the effects of each variable 'wore off', and the point at which the effects of one

exp( ˆˆln( ) ) exp([ ˆˆln( )] ) b x b t x b b t x 1 1 2 1 1 2 1 + = + . . ˆ( ) ˆ( ) exp( ˆˆln( ) ) ˆ( ) e h t h t b x b t x h t i j = + + 1 1 2 1 0 0 x xp( ) ˆ( ) exp( ˆˆln( ) ) 0 1 0 1 1 2 1 h t b x b t x = + h t b x t b t x t b x b x i n n ( ) exp( ( ) ln( ) ( ) ... = + + + 1 1 2 1 3 2 --1 0 ) ( ) h t
variable outweighed those of another. He did this by examining the exact value of the combined term and the point at which it equaled exactly zero. But this approach ignores the standard error of the combined term, and herein lies the problem [START_REF] Steunenberg | When Time Passes: The Transposition of Maritime Directives[END_REF]. This is a common mistake of those who try to interpret interactive terms and it yields misleading inferences.

The standard error of the new, combined coefficient is actually a function of the standard errors of the original coefficients, the covariance between both terms, and time. We know from the more familiar setting of interactive terms in linear regression, logit and probit models [START_REF] Friedrich | In Defense of Multiplicative Regression Equations[END_REF][START_REF] Brambor | Understanding Interaction Models: Improving Empirical Analyses[END_REF] that the standard error of the sum (b ˆ1 + b ˆ2 x) is calculated as follows:

.

(4)

In the context of a non-proportional hazards model, our 'x' is some function of t, in our case ln(t). So the standard error of the sum (b 1 + b 2 ln(t)) is .

(5)

The value of this term, as indicated by t, also depends on time. Consequently, the size, sign and significance of the combined coefficient may vary with time.

The crucial point is that the effect of a covariate is zero not when the value of the combined term [b ˆ1 + b ˆ2 ln(t)] is exactly 0, but when it is indistinguishable from 0, and not when the hazard ratio in equation ( 2) is exactly 1, but when it is indistinguishable from 1. There are two equivalent ways in which the standard error of the combined term allows us to determine the values of t for which these situations obtain, and thus to draw more precise inferences about the effects of covariates. First, we can compute the Wald statistic, which is defined as follows:

(6) for all values of t. This statistic follows a chi-square distribution with one degree of freedom and thus provides a formal test of whether the combined term is distinguishable from 0. Alternatively, for all values of t we can construct a confidence interval around the point prediction for the hazard ratio in equation ( 2). This hazard ratio is indistinguishable from 1 whenever the confidence interval contains the value 1. 1 Moreover, we can estimate the

W b b t b b b t = + ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ = + ( ˆˆln( )) ( ( l n () ) 1 2 2 1 2 s.e. 1 1 2 1 2 2 2 + + + ˆln( )) var( ˆ) (ln( )) var( ˆ) ln( b t b t b t) )cov( ˆ, ˆ) b b 1 2 2 ⎛ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ s.e. ( l n () ) var( ˆ) (ln( )) var( ˆ) b b t b t b 1 2 1 2 2 + = + +2 2 1 2 ln( )cov( ˆ, ˆ) t b b s.e. ( ) var( ˆ) var( ˆ) cov( ˆ, b b x b x b x b 1 2 1 2 2 1 2 + = + + ˆ) b 2
hazard ratio and construct confidence intervals for any linear combination of covariates, which allows us to compare the effects of one covariate with another at any point in time. Their respective effects are indistinguishable whenever their respective confidence intervals overlap.

Time interaction and EU decision-making

Having shown how the combined coefficients and standard errors of covariates in non-proportional hazards models vary with time, in this section we re-analyze Golub's data and provide a more careful interpretation of his results. Table 1 reports the original estimates from Golub's Cox model with TVCs and non-proportional effects for six covariates. We begin by assessing the impact of these six covariates at different values of survival time for legislative proposals. Table 2 presents the results. We chose values located up to nearly one standard deviation below and oneand-a-half standard deviations above the mean survival time of 1100 days, because decision-making time is highly positively skewed, with a median of 646 days. This range, from 80 to 2900, covers 87% of the observations in Golub's data set. For each covariate we also present the combined coefficient, its standard error and the significance level based on the Wald statistic.

The table shows how the impact of our covariates changes over time. While some covariates may have a positive impact on the hazard rate at some moment in time, leading to the speeding-up of decision-making, this impact may change into a negative one later on. In addition, the combined coefficient for each covariate is insignificant for some period in time, which indicates that during this period it temporarily has no effect on the hazard. Decisionmaking speed is, in that period, affected only by other covariates.

We now compare our new findings with Golub's original claims for each covariate.

QMV before the Single European Act (SEA)

Original claims: the hazard rate for proposals formally subject to QMV, as opposed to unanimity, was 146% higher after six months and 82% higher after one year, and the effect of QMV wore off after four years.

Our results confirm the first two claims, since the combined coefficient for QMV is positive and highly significant for t = 180 and t = 360 and neither of the respective confidence intervals around the hazard ratios contains one. 2 The combined coefficient is negative and statistically significant until t = 882, so for the 1968-87 period the positive effects of QMV on speed wore off after 29 months (when t = 882), not four years. Moreover, the impact of QMV reverses sign and once again becomes significant for very long decisionmaking processes (more than 10 years). Apart from only five exceptional cases that survived well over two standard deviations beyond the mean, this goes beyond the period for which we have observations and thus could easily be an artifact of our analysis. For this reason, we will disregard this effect.

QMV after the Single European Act

Original claims: the effect of QMV in the period after the SEA was the same as in the pre-SEA period, and for the first five months the pre-SEA effect was actually larger than the post-SEA effect.

Our results confirm the first but not the second claim. Judging only from the combined coefficient, as Golub (2007: 168) did, the coefficients presented in Survival time is first presented in terms of standard deviations away from the mean, which is about 1100 days (the median is 646 days) with a standard deviation of about 1200 days; the second row presents the corresponding time in terms of number of days.

Table 2 seem to indicate that the speeding-up effect of QMV before the SEA was greater than the effect of QMV after 1987 for at least the first 80 days. Soon after this short period the relative impact of these coefficients seems to reverse. But taking the respective standard errors into account reveals that the effects of QMV pre-and post-SEA are indistinguishable for all values of time. Figure 1 plots the respective hazard ratios and confidence intervals for QMV in the two periods, and shows that they always overlap.

Cooperation 1987-93

Original claim: for the period 1987-93 the negative effect of the cooperation procedure on decision-making speed outweighed the positive effect of QMV after SEA during the first year of a proposal's survival time.

Our results paint a more complicated, and in one respect slightly less severe, picture of the trade-off between speed and Parliamentary involvement. The hazard ratio for QMV after the SEA and the cooperation procedure is significantly less than 1 until t = 249, then indistinguishable from 1 until t = 497, then significantly greater than 1 thereafter. So in the six years following the SEA, the drag from cooperation overwhelms QMV for first 8 months (not 12), then neutralizes it for the next 8 months, but the QMV effect outweighs the cooperation effect after 16 months. We can also report two new findings: the drag from cooperation wears off only after 688 days, and, more surprisingly, after 1218 days the effect of cooperation is reversed and significantly speeds decision-making. This might be a statistical artifact, though, since, although there are 67 cases in the data set that were proposed under cooperation and survived more than 1218 days, all but three of them shifted to codecision at some point. The relevant issue, then, is the effect of codecision on EU decisionmaking speed, which we discuss below.

QMV after the Treaty on European Union (TEU)

Original claims: ever since the Maastricht Treaty, QMV exerted a still significant but somewhat reduced effect, there was no evidence of a unanimity norm, and for the first 18 months the effect of QMV was larger during 1968-87 than during 1994-9.

Our results confirm a significant but temporarily reduced effect of QMV compared with previous periods, show that the unanimity norm is even weaker than originally claimed, and provide a more optimistic prediction about the impact of prospective reforms to extend the scope of QMV. The effect of QMV after the Maastricht treaty (post-TEU) is not time dependent, so does not appear in Table 2. For all values of t, the hazard ratio is 1.51, with a confidence interval from 1.09 to 2.09. Thus for the entire 1994-9 period QMV was consistently faster than unanimity, so there is no sign of a unanimity norm. To compare the effects of QMV in the post-Maastricht period with its effects in earlier periods we then examined the three respective confidence intervals. These comparisons reveal that, for the first 4 months, the effect of QMV was greater during the period before Maastricht than after Maastricht, and that for only the first 5 months (not 18) the effect of QMV was greater during the period 1968-87 than during the period 1994-9. After 5 months all three confidence intervals overlap. In other words, the full effects of QMV return much more quickly than originally reported. So the unanimity norm is even weaker than Golub thought, and prospective QMV reforms would likely ease inertia more than he originally suggested.

Codecision and cooperation after Maastricht

Original claim: the drag exerted by each of these procedures outweighs the effects of QMV for 18 months.

Again our results confirm a slightly more complicated trade-off between speed and democratic inclusiveness. The hazard ratio for proposals subject to QMV and codecision is significantly less than 1 until t = 343, then indistinguishable from 1 for the period between 344 and 866 days, then significantly more than 1 for t > 866. Thus, during the period 1994-9, the drag from codecision outweighed the effects of QMV for 11 months (not 18). The two effects balance out for the next 17 months, and QMV more than offsets the delays from codecision only after 28 months.

We can also report two important new findings. First, inspection of their respective confidence intervals shows that the effects of cooperation and codecision are indistinguishable for all values of t, so the European Parliament's growth from agenda-setter to veto player has not slowed EU decision-making. Second, the effect of codecision wears off after 671 days, then after t = 1674 it is reversed and significantly speeds up decisions. Overall, our results suggest that broadening participation in the legislative process by empowering the European Parliament adds nearly two extra years to a decision, but helps resolve disagreement on the especially contentious proposals that experience more than four-and-a-half years of negotiations. This result is hardly what proponents of deliberative democracy and informal norms have in mind when they claim that adding more voices to the discussion expedites the EU legislative process.

Thatcher effect

Original claims: when Thatcher was prime minister the hazard was 22% lower after 6 months, and the 'Thatcher effect' wore off after 14 months. 2 show that the negative 'Thatcher effect' was slightly less than this, and, more interestingly, that it eventually reverses direction and significantly speeds the adoption of legislation. The combined coefficient for Thatcher is negative and significant until t = 246, indistinguishable from 0 for t = 247 through 793, then positive and significant for t > 794. Thus the hazard rate when Thatcher was prime minister was 22% lower after 6 months, as Golub originally claimed (with a confidence interval from 7% to 35%), but it wore off after 8 months (not 14), and its effect reversed after 26 months.

Results in Table

This curious finding deserves further study, but one possible explanation is that the reversal reflects a landmark change in Thatcher's attitude towards the EU. Thatcher became prime minister in 1979 and was highly antagonistic for nearly five years until, following an agreement at the June 1984 European Council summit at Fontainebleau, 'the British budgetary dispute was over' (Dinan, 1999: 92) and she 'got her money back'. It is certainly possible then that pre-Fontainebleau proposals were bogged down by Thatcher for much longer than 8 months, which drags down the hazard ratio, and were the ones more likely to survive 26 months and experience Thatcher's new views. So we would expect no Thatcher effect on proposals that were made after June 1984, and for those made earlier the effect should vanish if they survived past this date. This would account for the figures in Table 2, and a four-year period of intransigence would also accord well with Golub's other findings that do not derive from survival analysis -that Eurosclerosis appeared only upon Thatcher's arrival with a decline in Council output; that proposals made in 1979 and 1980 survived an unusually long time compared with those in the years immediately preceding and following; and that the volume of Council adoptions rose sharply in 1984 [START_REF] Golub | In the Shadow of the Vote? Decisionmaking in the European Community[END_REF].

In future work, the most effective way to investigate this matter would be to construct a TVC that codes state changes for proposals under consideration before or after the Fontainebleau summit, or, better still, that reflects shifts in Thatcher's attitude. The latter could be folded into the broader TVC Golub's article suggests to capture shifts in Council heterogeneity.

Backlog

Original claim: mounting backlog expedited Council decision-making on new proposals.

A proper interpretation of time-interaction effects yields more precise and interesting results. Table 2 shows that the combined coefficient for legislative backlog (with backlog set to its average value of 169) is positive and significant until t = 295, then indistinguishable from zero for t = 296 through 1421, then significant and negative for t > 1421. So, an average-sized backlog expedites decisions, in line with Golub's original claim, but only for the first 10 months. It has no discernible effect for the next three years, and then slows decision-making for proposals that have survived over 46 months. This reversal of sign shows that a large backlog spurs the Council to dispose of new proposals but does not expedite passage of the most controversial pieces of legislation.

Conclusions

Survival analysis often requires the use of models that contain timeinteraction terms to deal with the non-proportional effects of certain covariates, but it is easy to misinterpret the results of such models. We have shown that the key to correct interpretation is calculating the magnitude, standard error and statistical significance of the term formed by combining a covariate's time-dependent and time-independent coefficients. In his recent study of decision-making speed in the European Union published in this journal, [START_REF] Golub | Survival Analysis and European Union Decision-making[END_REF] overlooked this issue, and by addressing it we obtained more precise findings.

From our re-analysis of his data we draw three main conclusions. First, the effect that formal QMV rules have on speeding up decision-making is even larger and more consistent than originally claimed. Not only did QMV have a substantial effect as far back as the 1960s, this effect did not increase following the Single European Act. Following the Maastricht Treaty, speed under QMV was consistently faster than under unanimity, and the full effects of QMV returned much more quickly than Golub first thought. There is little sign, as often purported, that the EU legislative process operates by a unanimous consensus norm that slows down decision-making. Second, the trade-off between decision-making efficiency and legitimacy is more complicated than Golub recognized. The growth of the European Parliament's powers beyond mere consultation did significantly slow decision-making, but its evolution from an agenda-setter to a veto player added no discernible extra delay. Third, the effects of some key variables do not just wear off but actually reverse direction once proposals survive long periods of time. We suspect that some of these reversals are statistical artifacts, or, as in the case of the 'Thatcher effect', will vanish once we develop a more sophisticated TVC to capture shifts in Council heterogeneity. We attribute others, such as those for legislative backlog and codecision, with more substantive meaning.

Notes 1

The only example we know of where analysts have done this is Box-Steffensmeier and Zorn (2001: 984), and we are grateful to them for sharing their elegant Stata routine, which makes the calculations of standard errors and confidence intervals less cumbersome. 2 At t = 180 the hazard ratio is 2.46 with a 95% confidence interval from 2.00 to 3.03. At t = 360 the hazard ratio is 1.83 with a 95% confidence interval from 1.53 to 2.17.

  time-independent coefficient for a covariate and b ˆ2 as its estimated time-dependent coefficient, and ln(t) as the natural logarithm of time; s.e. is the standard error. The significance of b ˆt, based on the Wald statistic, which follows the chi-square distribution with 1df

Figure 1

 1 Figure1Effect of QMV before and after the Single European Act.Note: The thick lines plot the relative hazard for QMV before and after the Single European Act. The thin lines are 95% confidence intervals.

Table 1

 1 Estimates for Golub's Cox model of EU decision-making speed

	Variable	b	s.e.
	QMV	3.122****	0.478
	QMV after SEA	2.110****	0.513
	QMV after Maastricht	0.413**	0.166
	Cooperation procedure	-6.041****	0.614
	Codecision procedure	-5.001****	0.876
	EU9	0.496**	0.198
	EU10	0.457*	0.243
	EU12	0.659**	0.257
	EU15	0.571**	0.263
	Thatcher (as prime minister)	-1.716****	0.379
	Expanded legislative agenda	0.177	0.191
	Legislative backlog	0.026****	0.007
	QMV ϫ ln(t)	-0.428****	0.079
	QMV after SEA ϫ ln(t)	-0.224***	0.085
	Cooperation ϫ ln(t)	0.890****	0.099
	Codecision ϫ ln(t)	0.725****	0.134
	Thatcher ϫ ln(t)	0.282****	0.061
	Legislative backlog ϫ ln(t)	-0.004***	0.0009
	Likelihood ratio	311.27***
	Number of cases	1669

Notes: See

[START_REF] Golub | Survival Analysis and European Union Decision-making[END_REF]

for a further explanation of the variables. Data are right-censored on 17 December 1999. * p < .10; ** p < .05; *** p < .01; **** p < .001

Table 2

 2 The impact of time on covariate effects

Golub and Steunenberg How Time Affects EU Decision-Making

European Union Politics 8(4)

About the authors