Voting in central banks: Theory versus stylized facts Roman Horv´ath†
ˇ ıdkov´a∗ Kateˇrina Sm´
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Jan Z´apal∗∗ ‡
June 3, 2016
Abstract The paper examines the ability of several alternative group decision-making models to generate proposing, voting and decision patterns matching those observed in the Bank of England’s Monetary Policy Committee and the US Federal Reserve’s Federal Open Market Committee. A decision-making procedure, common to all the models, is to vote between adoption of the chairman’s proposal and retention of the status-quo policy, with heterogeneous votes generated by private information of the models’ monetary policy committee members. The members can additionally express reservations regarding the final committee decision. The three alternative models differ in the degree of informational influence between the chairman and the remaining members. We find that a ‘supermajoritarian’ model, in which the chairman proposes a policy she knows would be accepted by a supermajority of the committee members, combined with allowance for reservations, closely replicates real-world decision-making patterns. The model predicts no rejections of chairman’s proposals, low but non-trivial dissent, even during meetings where the chairman proposes no change in policy, and predictive power of the voting record of the whole committee regarding future monetary policy changes.
JEL Classification: C78, D78, E52, E58 Keywords: monetary policy, voting record, collective decision-making
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This paper is based on the 3rd chapter of the doctoral dissertation of Jan Z´ apal at the London School of Economics. We thank two anonymous referees, Marianna Blix-Grimaldi, Jakob de Haan, Michael Ehrmann, Petra Gerlach-Kristen, Etienne Farvaque, Jan Fil´ aˇcek, Tom´ aˇs Holub, Jarek Hurn´ık, Jakub Matˇej˚ u, Ronny Razin, Marek Rozkrut, Marek Rusn´ ak, Andrey Sirchenko and seminar participants at the CESifo conference on central bank communication, the European Public Choice Society annual conference, the Czech Economic Society biennial conference, the Czech National Bank and the Eurasia Business and Economics Society conference for helpful discussions. We appreciate the support from the Grant Agency of the Czech Republic, no. P402/12/G097. † Institute of Economic Studies, Charles University, Opletalova 26, 11000, Prague, Czech Republic
[email protected] ∗ We dedicate this paper to our co-author Kateˇrina, who passed away on April 29, 2014. We will miss more than a co-author. ∗∗ CERGE-EI, a joint workplace of Charles University in Prague and the Economics Institute of the Czech Academy of Sciences, Politickych veznu 7, 111 21 Prague, Czech Republic
[email protected] ‡ IAE-CSIC and Barcelona GSE, Campus UAB, Bellaterra, Barcelona, 08193, Spain
1
Introduction
Monetary policy transparency has increased substantially over the last two decades, and monetary policy committees have become increasingly transparent regarding the details of their decision-making procedures. Among the major changes is increasingly common publication of voting records from monetary policy meetings. Increasing availability of voting records has occasioned growth in both the theoretical and empirical literature seeking to better understand monetary policy determination. We contribute to this literature by examining the ability of several alternative group decision-making models to generate proposing, voting and decision patterns that match those observed in the data. To this end, we first establish several stylized facts about proposing, voting and decision behaviour in the Bank of England’s Monetary Policy Committee and the US Federal Reserve’s Federal Open Market Committee. Our data span more than a decade of monetary policy determination at the former institution and more than two decades at the latter. We use the Bank of England and the US Federal Reserve, as these constitute prime examples of central bank committees that use voting procedures to reach decisions, publish voting records from their meetings and have the governor or chairman propose alternatives that are put to a vote (see Maier, 2010, for details). Voting between adoption of a chairman’s proposal and its rejection, where the latter entails retention of the status-quo policy, is a defining feature of all the theoretical models presented below. Endogeneity of the status-quo policy is the key property our models share with Riboni and Ruge-Murcia (2010, 2014). Where our models differ is in assuming time-varying heterogeneity among committee members. Without heterogeneity, any decision would be reached by unanimous vote. In our models, heterogeneity arises from the fact that each committee member has a time-varying private signal regarding the optimal course of monetary policy. Time-varying heterogeneity is the key property our models share with Gerlach-Kristen (2008, 2009) and Weber (2010). Private information of the committee members implies that their actions, in terms of proposing or voting, necessarily carry information other
1
committee members can extract.1 What distinguishes our three alternative models is the degree of informational influence between the chairman and the rest of the committee. In the autarkical model, the committee is informationally independent - its members do not extract information from the chairman’s proposal when voting, and the chairman does not know whether her proposal would be accepted when putting it forward. In the consensual model, the chairman has informational influence - the committee extracts information from her proposal before voting. And in the supermajoritarian model, the committee has informational influence - the chairman learns about the committee members’ preferences and proposes a policy she knows would gain support of a supermajority of the committee members. We simulate random paths of decisions for each of the models and contrast the results with the patterns observed in actual data. Observing that none of the models can generate dissenting behaviour when the chairman proposes a policy identical to the status-quo policy - more of a regularity than an exception in actual monetary policy committee deliberations - we additionally allow the committee members to express reservations against final committee decisions. Thus, each of our three models are presented in two variants, one with allowance for reservations, the other without allowance for reservations. We ask our three models to replicate the main stylized facts in the data. Our analysis of the real-world decision-making data shows, among other things, that the chairman’s proposal is very rarely rejected, that dissent during monetary policy meetings is low but not exceptional, that dissent occurs even during meetings at which the chairman proposes no change in policy and that the voting record of the whole committee has predictive power regarding future monetary policy changes. Furthermore, by analysing preferred policies within pairs of monetary policy makers we show that their behaviour cannot be captured by the often-used dove vs. hawk distinction. In their entirety, these facts are non-trivial to replicate in a theoretical model. Any model in general, and our models in particular, must strike a delicate balance between generating dissent that results in rejections of chairman’s proposals on the one hand and generating acceptance of chairman’s 1
As put by Maier (2010): ‘We can think of MPC as a group of people sharing information and taking a decision together, on the basis of the information reviewed and revealed’ and ‘Committee members can possess different information sets ... in addition ... Individuals differ in terms of their ability to process information.’
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proposals stemming from no dissent on the other. Allowing for reservations is thus particularly helpful in bringing theory closer to the data. Reservations enable dissent without causing the rejections of chairman’s proposals. Reservations additionally produce dissent at meetings at which the chairman proposes no change in policy and improve the predictive power of committee members’ voting records. Among our three models, it is the supermajoritarian one, with reservations, that comes closest to replicating the decision-making patterns observed in our dataset. By construction, chairman’s proposals are always accepted and, with the help of reservations, the model generates the remaining stylized facts, as well. The remainder of this paper is organized as follows. In the next section, we establish several stylized facts about central bank decision-making. Section 3 introduces the basic structure of the theoretical model, a structure common to all the models of committee behaviour discussed in section 4, which also discusses the related literature. Section 5 describes our simulation exercise, with the results of the exercise described in section 6. Section 7 offers several concluding remarks. All technical theoretical details are in Appendix A1, and further simulation results are in Appendix A2.
2
Stylized facts
Two major monetary policy committees decide by voting in between policy proposed by their chairman and the status-quo policy, as in our model, the Bank of England’s Monetary Policy Committee and the US Federal Reserve’s Federal Open Market Committee.2 For each monetary policy committee meeting, we have recorded the status-quo policy, the chairman’s proposed policy and the voting record 2 We have decided not to analyse decision-making data from the Swedish Riksbank’s Executive Board because its governor often proposes more than one policy alternative. Preliminary analysis of the decision-making process at the Riksbank’s Executive Board leads us to believe that its governor puts to vote all policy alternatives seriously discussed during monetary policy discussion, with the median alternative prevailing. As a result, the decision-making procedure used, for example, by Gerlach-Kristen (2009) or Weber (2010), might be a more appropriate model for the Riksbank. Additional insights on the Riksbank, as well as the Norges Bank, are provided by Apel, Claussen, Gerlach-Kristen, Lennartsdotter, and Roisland (2013), who survey the behaviour of board members based on questionnaires. We have decided not to include other countries because either their voting data are not fully available or, according to Maier (2010), their voting procedures deviate from typical voting practice.
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consisting of any policy receiving a strictly positive number of votes and number of votes for that policy.3 Analysis of the data in Table 1 yields several observations. The first points to note are the well-documented low level of dissent in monetary policy committees and the relatively higher level of dissent in the Bank of England compared to that of the US Federal Reserve (see the line labelled ‘average dissent’ in Table 1).4 Second and equally well-known is a high degree of policy inertia, based on the fraction of meetings with no policy change. Third, the chairman’s proposal, in our data, is always accepted. As a result, the fraction of meetings during which the chairman proposes a policy equivalent to the status-quo and the fraction of meetings with no policy change are exactly equal. Fourth, the voting records are informative about future monetary policy. Following Gerlach-Kristen (2004) and Horvath, Smidkova, and Zapal (2012), we have defined the variable skewt as the difference between the mean and median policy voted for at a given meeting and estimated an ordered probit model with the change in policy in a subsequent meeting as a dependent variable. The significant estimates for skewt , for the Bank of England and US Federal Reserve under Greenspan’s chairmanship, imply the predictive power of skewt with respect to future monetary policy changes. The next set of results uses the classification of monetary policy meetings 3
We have combined several sources. For the Bank of England, the main sources were Minutes of Monetary Policy Committee Meetings, complemented by Historical MPC voting spreadsheet, available from the Bank’s website. For the Federal Reserve, we have used Records of Policy Actions (until the end of 1992), Minutes of the Federal Open Market Committee Meetings (1993-2002) and Federal Reserve Press Releases (from 2003 onward), complemented by Federal Open Market Committee Meeting Transcripts. Statements akin to ‘Governor invited members to vote on the proposition that the repo rate should be . . .’ in the Bank of England’s minutes, allow us to determine the proposed policy. For the Federal Reserve, we know from Meade (2005) that chairman Greenspan’s interest rate proposals were always adopted, so we can determine his proposals from the adopted policies. Meade’s data do not cover the whole period of Greenspan’s chairmanship, but analysis of the transcripts confirms that the same pattern prevailed. For Bernanke’s chairmanship, we could have used the transcripts only until the end of 2006. Reportedly, Bernanke tends to speak last during the policy go-around (Blinder, 2009), and reading the available transcripts shows that he makes a policy proposal at that point. Until the end of 2006, his proposals were always adopted, and we assume that the same pattern prevails for the rest of the sample. 4 Expressed as the number of dissenting votes rather than as a fraction, the dissent data would read 1.2, 0.5 and 0.6 (average number of dissenting votes per meeting) for the Bank of England and for the Federal Reserve under Greenspan and Bernanke, respectively.
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Table 1: Central bank decision-making stylized facts
skewt (a1 ) ∆pt (a2 )
Bank of England
Fed: Greenspan
Fed: Bernanke
14.2 [2.42]*** 3.44 [0.55]***
10.1 [5.47]*** 3.06 [0.46]***
7.90 [8.46]*** 3.29 [0.76]***
fraction of meeting types 1.0 1.1 1.2 2.0 2.1 2.2
0.30 0.40 0.05 0.07 0.17 0.01
0.40 0.22 0.04 0.23 0.10 0.00
0.35 0.39 0.00 0.14 0.12 0.00
fraction of meetings with no policy change proposal status-quo proposal accepted average dissent period N
0.76 0.76 1.00
0.66 0.66 1.00
0.73 0.73 1.00
0.14 1:98-12:11 169
0.05 8:87-1:06 149
0.06 3:06-12:11 49
Note: The first two rows are ordered probit estimates of ∆pt+1 = a0 + a1 skewt + a2 ∆pt + ut+1 with [standard errors]. ***/**/* significant at 1%/5%/10% level. Meeting types are coded as x.y, where x is the number of alternatives put to vote and y is the number of alternatives with a positive number of votes (excluding the Chairman’s proposal). Dissent is defined as the ratio of dissenting to present committee members.
into types coded as x.y, where x is the number of alternatives put to vote and y is the number of alternatives that receive a strictly positive number of votes, excluding the chairman’s proposal. For example, meeting type 1.1 indicates a meeting during which the chairman proposed to keep the policy unchanged (x = 1) and where, in the subsequent vote, one alternative policy received a non-zero number of votes (y = 1). In contrast, meeting type 2.0 indicates a meeting during which the chairman proposed to change the policy (x = 2) and his proposal passed unanimously (y = 0). Note that any theoretical model with voting over the policy proposed by the chairman and the status-quo policy can produce only 1.0, 2.0 and 2.1 meeting types. When the chairman proposes to keep the policy unchanged, there is effectively a single alternative to vote for. Our fifth observation, based on Table 1, is that meeting types incompatible with such a model 5
(1.1, 1.2 and 2.2) are rather common. Especially 1.1 meetings constitute a sizeable fraction of monetary policy meetings. Sixth, meetings with three or more alternatives receiving positive number of votes are rare. In Table 1, these are 1.2 and 2.2 meetings, the three alternatives being the chairman’s proposal and two other policies. Less than 6% of meetings fall into these two categories. Finally, unanimous agreements, meeting types 1.0 and 2.0, are common but not universal. The last two observations, taken together, imply that most monetary policy meetings fall into one of two categories. Either the chairman’s proposal is accepted unanimously, or there is one other alternative policy that receives a positive number of votes. We now present an alternative analysis of the monetary policy committee decision-making data focusing on the differences in policies voted on by pairs of monetary policy committee members. We conduct this analysis to see whether the members can be labelled as doves and hawks, i.e., whether some members consistently vote for higher or lower rates, respectively. This would suggest that members exhibit time-invariant heterogeneity to an alternative of time-varying (stochastic) heterogeneity. Therefore, for each member-pair (i, j) with strictly positive number of meetings attended by both i and j, we have calculated (ml(i,j) , me(i,j) , mh(i,j) ). Each of the variables denotes number of meetings: ml(i,j) number of meetings in which i voted for a strictly lower rate than j, me(i,j) number of meetings in which i and j voted for equal rates and mh(i,j) number of meetings in which i voted for a strictly higher rate than j. We order members within the pairs such that ml(i,j) ≥ mh(i,j) , i.e., within each pair, member i is the more ‘dovish’ one. Table 2 shows average of mt(i,j) , mt , for t ∈ {l, e, h} across different member-pair types. Member-pairs of the first type always vote for the same policy. Member-pairs of the second type comprise of members of which one always votes for a lower policy. Member-pairs of the third type comprise of members of which one votes for a lower policy in some meetings and for a higher policy in other meetings. Note that any theoretical model with time-invariant heterogeneity among the committee members predicts that one of the members in any memberpair is (weakly) more dovish and hence cannot generate member-pairs of the third type. However, these member-pairs are common in our data; they comprise 27% =
46 122+46 ,
18% and 5% of member-pairs that display
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Table 2: MPC member-pairs: Can members be labeled doves/hawks? Bank of England
Fed: Greenspan
Fed: Bernanke
pair type
1
2
3
1
2
3
1
2
3
ml me mh N
0.00 15.97 0.00 32
7.88 20.88 0.00 122
5.70 34.02 2.26 46
0.00 12.16 0.00 301
2.79 14.13 0.00 209
3.04 22.07 1.31 45
0.00 9.28 0.00 158
3.04 6.35 0.00 83
1.75 4.75 1.75 4
Note: mt(i,j) is number of meeting in which monetary policy member i votes for lower (t = l), equal (t = e) and higher (t = h) rate than j. mt is average, over pairs, number of meetings conditional on a pair type. N is number of pairs of each type. Pair types: 1 . . . pairs with ml(i,j) = 0 (and hence mh(i,j) = 0); 2 . . . pairs with ml(i,j) > 0 and mh(i,j) = 0; 3 . . . pairs with mh(i,j) > 0 (and hence ml(i,j) > 0).
disagreement (types 2 and 3) respectively for the Bank of England and US Federal Reserve under Greenspan’s and Bernanke’s chairmanship.5 Table 2 also shows that the most common voting within a memberpair is voting for the same policy. Given the discreteness of the policy rates considered by most monetary policy committees, voting for the same policy makes identification of dovish and hawkish members a daunting task. However, existence of a significant fraction of member-pairs of the third type suggest that the dovish-hawkish classification might be misguided. Within the member-pairs of the third type, conditional on disagreement, the more dovish member votes for a higher policy in
2.26 2.26+5.70
= 28% of meetings in
the Bank of England. Similar numbers for the US Federal Reserve under Greenspan’s and Bernanke’s chairmanship are 30% and 50%.6 5
Incidence of member-pairs of the second type in any model with time-varying heterogeneity among committee members, including ours, has to be a decreasing (to zero) function of the number of meetings pairs attend jointly. The committee members serve indefinitely in our model, but arbitrary member replacement would lead to identical results. Hence both the time-invariant as well as the time-varying heterogeneity should be regarded as simplifying assumption. 6 The limitation of the dove-hawk classification our data shows is in line with the existing anecdotal evidence. For example, Sarah Bloom Raskin remarked on her experience as an FOMC member saying that ‘Indeed, some commentators assign a label of ‘hawk’ or ‘dove’ to the various FOMC participants [. . . ] In my view, such labels are ill conceived and misleading [. . . ] When my colleagues and I are doing our job correctly, we are neither hawks nor doves [. . . ]’ (Raskin, 2011).
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3
Model setup
The model is set in an infinite horizon, with discrete periods denoted by t = 0, 1, . . ., in each of which the monetary policy committee makes a decision about the policy instrument, with a policy adopted at t denoted by pt . Although we call pt the interest rate, it can stand for any standard monetary policy instrument. There are N (even) ‘normal’ board members P (each referred to as ‘he’) and one proposer or chairman C (referred to as ‘she’). Therefore, the committee size is odd. In each period t, decision-making occurs through a standard majority rule, with two alternatives pitted against each other. The first alternative is the policy proposed by the chairman, which we denote by yt . The second alternative is the current status-quo policy, xt , which is the policy adopted at t − 1, i.e., xt = pt−1 . In other words, the status-quo in our model is endogenous, as in Riboni and Ruge-Murcia (2010, 2014). The alternative that gains a majority of votes then becomes the new policy, pt . For mathematical convenience, we assume that a C, who cannot propose anything better than xt , in fact proposes xt (instead of proposing a policy that would be rejected for certain). The committee tries to set policy pt so as to match the uncertain ‘state of the world’ denoted by i∗t , where for inflation-targeting central banks, i∗t can be interpreted as the interest rate that is compatible with achieving the bank’s inflation target over time. We assume that the per-period utility function of all committee members is quadratic around i∗t and is given by −(pt − i∗t )2 . Note that even though the board members share an equal goal embedded in a common utility function, their behaviour can (and will) depend on their private information, which is not necessarily homogeneous. We assume that the unobserved state of the world follows an AR(1)7 process given by i∗t = ρ i∗t−1 + ut , where ρ ∈ (0, 1), with ut being an i.i.d. shock with distribution N (0, σu2 ). That is, the optimal monetary policy changes over time, with the current optimal interest rate being influenced by the previous-period optimal interest rate and eventually converging to some long-run value compatible with a stable state of the economy.8 7
In Appendix A2, we provide a robustness check to show that the AR(1) assumption can be changed into AR(2) without altering our conclusions. 8 With our interpretation of i∗t as the optimal interest rate, it might seem unrealistic to assume that it can attain negative values. However, the whole model and all the results
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To generate non-homogeneous votes among the committee members, we assume that each member j has an imperfect signal ijt about i∗t , given by ijt = i∗t + vtj , where the noise vtj is i.i.d., with distribution N (0, σj2 ). The assumption of non-homogeneous views of the individual committee members about the state of the economy is perfectly in line with the observed practice. Individual committee members often rely both on a staff forecast and on additional privately-collected information about the state of the economy, as well as on their privately-formed views about what degree of confidence to attach to the staff forecast (Budd, 1998). It is assumed that for all P s, σj = σP , and that C has σj = σC . We assume that the chairman has the same or a higher capacity to collect private information than the other committee members, and hence the same or a higher capacity to reduce the noise of her signal, that is, σC ≤ σP .9 The proposal power of the chairman, along with heterogeneous preferences among the committee members generated by their differing signals, implies that our model applies best to the final stage of a typical monetary policy meeting. A common practice in many central banks is to start with a free format discussion of economic developments, after which, typically, the chairman proposes a policy that is then approved or rejected in a formal vote.10 Our next assumption makes the model more tractable. We assume that the whole committee learns the previous state of the world at the beginning of each period, before making its next decision, i.e., i∗t−1 is known by the are invariant to adding a constant to the optimal interest rate. 9 We could have generated heterogeneous preferences among the committee members by assuming fixed innate differences in their preferences. However, with a fixed pattern of heterogeneity, there is no reason why the voting record should predict future decisions. On the other hand, our assumption of private signals generating heterogeneous preferences can be alternatively viewed as an assumption of different innate preferences among the committee members, but one following a stochastic pattern. The assumption that the chairman has the same or a higher capacity to collect private information is in line with Claussen, Matsen, Roisland, and Torvik (2012). They suggest that the chairman might be better informed than other members of the committee because of better access to the central bank staff. Additional possible reasons why the chairman may have better information than board members may follow from the fact that she typically represents the bank at high-level international meetings and in some countries may regularly participate in government meetings or even have access to selected classified materials. 10 Hence, the committee members’ signals, independent conditional on (unknown) ¯i∗ but correlated conditional on (known) i∗ , can be interpreted as capturing preference heterogeneity remaining after the committee deliberation. Preference heterogeneity among the committee members may arise despite their communication due to, for example, differences in data interpretation.
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time the t-period decision is being made. The alternative would be to not reveal i∗t−1 and have the board members use Kalman filtering (as in GerlachKristen, 2008) to update their beliefs about the optimal interest rate. While this extension is possible, we think it would add no substantive insight while greatly complicating the analysis. The timing of events in period t is as follows: i) the last-period state of the world i∗t−1 is revealed; ii) nature determines all the random variables in the model, hence setting i∗t and all the signals of the committee members; iii) the signals about the current state of the world, ijt , are revealed to all the members, and remain their private information; iv) C makes proposal yt ; v) voting takes place between yt and the status-quo (i.e., the last-period policy) xt = pt−1 , and the winning alternative becomes the new policy pt ; and finally, vi) the players collect their utilities and the decision-making process moves to t + 1. We focus on a Stationary Markov Perfect equilibria in which strategies are measurable only with respect to payoff-relevant variables (histories) and do not depend on time (Maskin and Tirole, 2001). This allows us to drop the time subscripts, so that the notation becomes x for the status-quo, y for the proposal, i∗ for the previous-period optimal interest rate, and ij for signals about the current optimal interest rate. The current optimal interest rate will be denoted by ¯i∗ , with the bar denoting variables that will become known in the next period (the same applies to the other variables, i.e., ¯ij is the signal received by player j at the beginning of the next period about the next-period optimal interest rate). With this notation, the AR(1) process for the optimal interest rate becomes ¯i∗ = ρ i∗ + u ¯, and the signals are determined according to ij = ¯i∗ + v¯j . The information set of each player j is thus Ij = {i∗ , ij }. C’s strategy in this game is to offer the proposal that, given the statusquo and her information set, maximizes her expected utility. It will be a solution to max EM −(p(x, y) − ¯i∗ )2 |IC y∈Y
(1)
where p(x, y) denotes the policy adopted, depending on the status-quo x and proposal y. The set Y is assumed to be a set of discrete values to which the interest rate can be set, i.e., Y is a set of integer multiples of some value s¯. The notation for the expectation operator EM [ · ] captures the idea that C
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will calculate her expectations differently based on a model of the committee members’ behaviour, which we specify below.11 The strategy of each P member j is a simple binary decision to vote for or reject C’s proposal, given the status-quo x and all the remaining variables in information set Ij . We restrict our attention to stage-undominated strategies (Baron and Kalai, 1993) by which player j simply votes for the alternative that provides higher expected utility. This avoids equilibria in which players vote for an alternative they do not prefer simply because their vote cannot change the final decision. This implies that j, given the status-quo x, C’s proposal y and j’s information set Ij = {i∗ , ij }, votes for y if and only if EM −(y − ¯i∗ )2 |Ij ≥ EM −(x − ¯i∗ )2 |Ij .
(2)
Notice that the voting rule specifies that an indifferent j votes for C’s proposal. Hence, when C’s proposal y equals the current status-quo x, proforma voting takes place within the committee and C’s proposal is unanimously approved.
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Committee members’ behaviour
One way to proceed would be to assume full rationality, in the standard sense, on the part of all the committee members, solve for the model equilibrium (which would involve complicated expectation updating and signal extraction problems) and then simulate the path of decisions for a random draw of the model’s stochastic variables. However, the presence of information asymmetry among the committee members makes derivation of a full solution unfeasible. 11 The specification of C’s objective function (1) implicitly assumes that she ignores effect of her current actions on future decisions via the status-quo. We will make the same assumption for the other committee members. This assumption is common. (And not without loss of generality; the committee members have common values conditional on knowing the state but expect to learn different information conditional on having different signals.) In Gerlach-Kristen (2008, 2009) and Riboni and Ruge-Murcia (2008b), a vote between the status-quo and the chairman’s proposal does not occur, so there can be no effect of current actions on future decisions. In models more similar to ours (Riboni and Ruge-Murcia, 2010, 2014), with a vote between an endogenous status-quo and the chairman’s proposal, current actions can in principle affect future decisions. The common assumption is that the committee members ignore this effect. Notable exceptions are Riboni and Ruge-Murcia (2008a) and Riboni (2010). Neither paper studies voting and decision-making patterns, focusing on the effect of the status-quo endogeneity instead.
11
Apart from its technical complexity, such a model does not capture the different modes or codes of conduct found in real-world monetary policy committees (see Blinder, 2004; Chappell, McGregor, and Vermilyea, 2005, for a discussion) and the possible degrees of informational influence among their members. A purely rational model also implicitly assumes a lack of other motives on the parts of committee members, such as acknowledgement of the chairman’s authority and greater expertise or career concerns manifested by a willingness to adopt the chairman’s opinion. Lahner (2015)’s empirical study supports existence of these other motives. He examines verbatim transcripts of FOMC meetings and voting records of the FOMC members in 1989-2008. From the transcripts, he determines a voiced (policy) preference of each member, revealed during each meeting’s discussion round, and compares it to a voted preference from the voting records. He finds systematic differences in FOMC members’ preference revisions; FOMC members were more reluctant to change their preferences after 1993, i.e., after the announcement that verbatim meeting transcripts would be released to the public (with a five-year lag), and the more senior members are less willing to change their preferences. Since we view the fully rational model as an unrealistic description of reality, we specify three different models of committee behaviour for which we solve for equilibrium and then proceed to the simulations. We label the models autarkical, consensual and supermajoritarian, and note that differences among them can be understood in terms of different degrees of informational influence among the chairman C and the remaining P members.12 In the autarkical model, there is little or no influence, as C is not influenced by the information that the P members have and the P members are not influenced by C’s proposal. In the consensual model, C is informationally independent, while the P members are influenced by her proposal. Finally, in the supermajoritarian model, it is C who is influenced by the remaining P members in that she bases her proposal on their preferences and disregards her own preference, to a certain extent.13 12
We postpone discussion of related models in already-published work to the end of this section. Prior explanation of our modeling strategy will greatly facilitate explanation of the similarities and differences of alternative approaches. 13 The degree of informational influence is not related to how intense communication is prior to voting. The committee members may in principle communicate a lot with one another prior to a policy meeting, but with a low degree of informational influence and vice versa. Nor is informational influence, or lack thereof, related to competence. The
12
Autarkical model In this model of committee behaviour, the chairman C plays the role of a leader whose only special power is to make proposals. A proposal is based solely on C’s own information set. The other committee members are free to express their own views by voting for or against her proposal, and C’s behaviour has no influence on their votes. In the language of our model, each P member j is assumed to vote based on the voting rule (2), using information set Ij = {i∗ , ij } and extracting no information from C’s proposal. Given this behaviour, C solves her optimization problem (1), using information IC = {i∗ , iC } and forming her expectations in a standard rational manner, i.e., EM [ · ] = E[ · ], where E[ · ] is a standard expectation operator. Notice that this does not mean C proposes the expected optimal policy rate E[ ¯i∗ |i∗ , iC ], given her information set; she offers proposal y, taking into account the fact that its eventual acceptance (as opposed to acceptance of the status-quo x) reveals information about the unobserved ¯i∗ .
Consensual model In this model, chairman C is assumed to have a position of dominance that goes beyond her proposal-making power. Her dominant position makes the other P members too keen to adopt her point of view, as they assume that the information available to the chairman is superior to their own. In the language of our model, C’s proposal is a solution to (1), given information IC = {i∗ , iC }, but with the expectation operator EM [ · ] not taking into account the fact that possible rejection or acceptance of y contains information about unknown ¯i∗ . In other words, C’s proposal is the policy in Y closest to C’s expectation of ¯i∗ , i.e., closest to E[ ¯i∗ |i∗ , iC ]. While not fully rational, this specification of the way in which C forms her expectations captures the notion that, because she knows that the other committee members’ voting behaviour is strongly influenced by her own proposal, she disregards the possible information content of that behaviour and proposes her optimal policy.14 committee members can be competent in the sense of appreciating the information others possess, but can for various reasons decide not to take that information into account. 14 A purely rational model would be a mixture of the autarkical model (C extracting information from her proposal’s eventual acceptance) with the consensual model (P s
13
To capture the notion that the P members adopt C’s point of view, we assume that each P member j votes based on voting rule (2), but, when calculating the expected value of ¯i∗ , extracts information from C’s proposal. It C is easy to see that the expectation can be written as E ¯i∗ |i∗ , ij , iC ∈ [iC l , iu ] , C where iC l and iu are, respectively, the lower and upper bounds on C’s signal,
as revealed by her proposal. We have decided to label this model consensual, as the extraction of information from C’s proposal considerably reduces the level of heterogeneity of opinions within the committee.
Supermajoritarian model In this model, we assume that C consults the other P members before the actual committee meeting. Once at the meeting, C knows the mostpreferred policies of the other members and proposes the policy she knows will be adopted by a supermajority,
N 2
+2, of them.15 In terms of our model,
we assume that C knows the most-preferred policy of each member j, which is the policy in Y closest to E[ ¯i∗ |i∗ , ij ]. Order those policies such that y1 ≤ . . . ≤ ym ≤ . . . ≤ yN +1 , where ym is the policy preferred by the median committee member. Then, proposing a policy adopted by a supermajority of
N 2
+ 2 entails, for the x ≤ ym case, proposing ym−1 if x ≤ ym−1 and x
if x ≥ ym−1 . The x ≥ ym case is analogous. An implicit assumption about each P member j is that his voting behaviour is determined by voting rule (2), with the expectation computed using information set Ij = {i∗ , ij } and ignoring the information content of C’s proposal.16 extracting information from C’s proposal). The more P s extract information from C’s proposal, the less information C extracts from P s’ acceptance. What makes the model hard to solve is the construction of the fixed point (equilibrium) with respect to information extraction. We have investigated a purely rational version of a simplified model and it is closer to the (simplified) autarkical model than to the (simplified) consensual model. In all the simplified models, C is strongly biased (likely to propose) towards the previous-period optimal interest rate i∗ when it differs from the status-quo. What the autarkical model and the purely rational model share, but the consensual model lacks, is C’s tendency to propose a policy different from i∗ , when i∗ is equal to the status-quo. Intuitively, C relies on the information P s have. 15 In the Appendix A2 (Tables A17 through A20), we provide a robustness check for a simple majority ( N2 + 1) case to illustrate that this assumption is not crucial to our results. The results are largely similar to the supermajoritarian model. Notice that the simple majority version of the supermajoritarian model, which we call the majoritarian model, is often used in the literature as the accepted policy is equal to the policy most preferred by the median committee member. 16 This model is inspired by Riboni and Ruge-Murcia (2010), who, in their empirical investigation of several descriptive monetary policy committee decision-making models,
14
Reservations For each of the models of committee behaviour, we now additionally specify two versions, one where committee members can express reservations regarding the adopted policy, and one where they cannot. Without reservations, the models work as described above. With reservations, each of the P members has the option to express his reservation regarding the adopted policy at the end of the committee meeting, without changing the policy itself.17 We allow reservations to be expressed as preferences for a lower or for a higher policy value, relative to the one adopted. Denote by ij,e the expected value of ¯i∗ held by P member j in a given model,18 and recall that the policy is set in integer multiples of s¯. In the given meeting, with statusquo x and chairman’s proposal y, we assume j expresses his reservation towards lower policy if ij,e ≤ min {x, y} − s¯ and towards higher policy if ij,e ≥ max {x, y} + s¯. In meetings where x = y, this enables j to express preferences he could not have expressed by voting, and in meetings where x 6= y, this enables j to express his preferences if those preferences are even stronger than those expressed by his vote.
Relation to existing models We now discuss the relation of our model to existing models. We restrict our attention to papers that focus on decision-making in monetary policy committee meetings with explicit voting, namely Gerlach-Kristen (2008, 2009), Riboni and Ruge-Murcia (2010, 2014)19 and Weber (2010). These papers show that their ‘consensual’ model fits the real-world data best. In their model, the adopted policy is equal to the most-preferred policy of the next-to-median member (where the side depends on the position of the status-quo), when this policy is sufficiently far from the status-quo. When this policy is close to the status-quo, the adopted policy is, in fact, the status-quo. This is what our supermajoritarian model does, except that it captures the idea that the chairman’s objective is to offer a policy that would never be rejected, an objective she achieves by using her authority to consult individual committee members, or, in an alternative interpretation, speak last during the committee discussion, after the remaining members have expressed their preferred policies. 17 Reservations in our model play the same role as ‘dissent’ in Riboni and Ruge-Murcia (2014). Both allow committee members to express their preferences after the committee decision has been made. 18 That is, ij,e = E[ ¯i∗ |i∗ , ij ] in the models, whereas autarkical and theC supermajoritarian for the consensual model ij,e = E ¯i∗ |i∗ , ij , iC ∈ [iC l , iu ] . 19 Riboni and Ruge-Murcia (2008a) and Riboni (2010) share many similarities with Riboni and Ruge-Murcia (2010, 2014). We discuss the latter papers as the most related.
15
can be compared along two dimensions. The first is whether the voting is over a binary agenda, with the status-quo policy being one of the alternatives considered, or whether the voting has no explicit status-quo policy. Our model and that of Riboni and Ruge-Murcia (2010, 2014) fall into the former category, while the models of Gerlach-Kristen (2008, 2009) and Weber (2010) fall into the latter category. We have decided to use a model with explicit status-quo as voting with no explicit status-quo is usually modeled by assuming that the committee members vote for an alternative they most prefer. The alternative with the largest number of votes is then the winning one, but does not necessarily receive a majority of the votes.20 The second dimension along which the existing models differ is in whether they assume time-invariant heterogeneity among the committee members or time-varying (and stochastic) one. Riboni and Ruge-Murcia (2010, 2014) belong to the former category, while Gerlach-Kristen (2008, 2009) and Weber (2010) belong to the latter one.21 In our view, the latter modeling approach is more natural, as it avoids the conclusion that it is always the same committee members who dissent for a given direction of change of the interest rate, conclusion which is refuted in empirical data. We stress that we view our model as complementary to the other models just discussed. Different models will be more appropriate for modeling the decision-making of different central banks. From Riboni and Ruge-Murcia (2010, 2014), we adopt the binary voting agenda, with the important role of the status-quo policy, but not time-invariant heterogeneity. From GerlachKristen (2008, 2009) and Weber (2010), we adopt time-varying heterogeneity, but not the voting without explicit status-quo policy. Our three models also relate to the other models under discussion in terms of decision-making procedures. Our autarkical model is similar to the ‘agenda-setting’ model of Riboni and Ruge-Murcia (2010), in that the chairman proposes the policy that maximizes her expected utility among the policies she knows would be accepted. The key difference in our autar20
Gerlach-Kristen (2009) overcomes this problem in an elegant way by endowing the committee members with information sufficiently precise such that the majority vote for the same alternative. 21 The optimal interest rate, for a given committee member, in Riboni and Ruge-Murcia (2010, 2014), is stochastic as well. However, there is a time-invariant ordering of the committee members such that members earlier in the order always prefer lower interest rate than members later in the order.
16
kical model is that acceptance is only probabilistic, as C does not know the signals of the other committee members. Our consensual model is similar to the ‘autocratically collegial’ model in Gerlach-Kristen (2008), in that the chairman proposes her most-preferred policy and her authority compels the other committee members to vote for her proposal.22 Finally, as explained in footnote 16, our supermajoritarian model is similar to the ‘consensual’ model of Riboni and Ruge-Murcia (2010).23
5
Model simulations
For each version of the model of committee behaviour and for each parameter scenario described below, we generate 101 different random 100-period-long paths. These are chosen so as to gain insights into the results and avoid inference based on simulated paths either too short or too few in number, while keeping the simulations manageable.24 With the simulation of one path in the autarkical model taking approximately one hour for N = 4 on a standard desktop computer (and twice as long for N = 6), we see the choice of the number and length of paths as an appropriate trade-off between validity and manageability.25 Along each path, for every period, we record the status-quo xt , the proposal yt , the level of interest rate expressed in reservations towards lower policy rt = min {xt , yt } − s¯, and towards higher policy rt = max {xt , yt } + s¯. For each of these variables, we also record the number of committee members 22
In the autocratically collegial model, this is modeled as the other committee members having a ‘tolerance interval’ around their preferred policy. In our model, it is modeled as the other members considering the chairman’s point of view by extracting information from her proposal. 23 The majoritarian model described in footnote 15 is similar to the ‘frictionless’ model in Riboni and Ruge-Murcia (2010), to the ‘individualistic’ model in Gerlach-Kristen (2008) and to the model in Weber (2010). In all these models, the adopted policy is equal to the policy preferred by the median committee member. 24 Many of the results we present have one committee meeting as the unit of observation rather than one path. For these results, we effectively have a sample of 100 · 101 = 10100 observations. The only thing in the model that depends on the level of the optimal policy, i.e., on the starting position of i∗ , is its speed of convergence to its long-run average. Given the rather persistent AR(1) process we use for i∗ , the speed changes little in the level of i∗ . 25 We also tested the stability of our results across sub-samples of the 101 paths. The results using one committee meeting as the unit of observation rarely change, even in the second decimal place. The remaining results consist of ordered probit estimates, which rarely (never) lose significance if the original estimate is significant at the 10% (5%) level. The full results are available upon request.
17
# # # supporting it, denoted by x# t , yt , r t and r t , respectively. We also know the
final policy, pt , and the optimal interest rate, i∗t . This allows us to calculate decision-making statistics similar to those presented in Table 1. In order to estimate the predictive power of the voting record for future policy changes, we follow Gerlach-Kristen (2004) and Horvath et al. (2012) and summarize the whole voting record by the variable skewt . Denote by lt = min {xt , yt } and ht = max {xt , yt } the lower and higher of the two alternatives put to vote and the number of votes for these alternatives, lt# and h# t , respectively, (ht = yt when xt = yt ). Then skewt is defined as
# # # # rt r# t + ht (ht − r t − r t ) + r t r t − pt for xt = yt N +1 skewt = (3) # # # # # # rt rt + lt (lt − rt ) + ht (ht − rt ) + rt rt − pt for xt 6= yt N +1
or, in words, as the difference between the mean alternative voted for and the resulting policy, adjusting for reservations.26 Positive skewt indicates votes favouring a higher policy that were insufficient for its adoption. The values of the parameters in the models are as follows. From Mahadeva and Sterne (2000), we know that the most common monetary policy committee size is 5 or 7 members, so we use N = 4 and N = 6. For the AR(1) process governing i∗ , we use ρ = 0.95 making its deviations from its long-run value persistent. We assume that the interest rate is set in steps of a quarter of a percentage point; thus, s¯ = 0.25. For the distributions of random shocks, we must specify values for σu , σC and σP .27 The choice of σu is driven by an effort to match the standard deviation of changes in the monetary policy rate in our empirical data. As pt in our model eventually follows a process similar to that of i∗t , ∆pt = pt − pt−1 will follow a process similar to that of ∆i∗t = i∗t − i∗t−1 . With the standard deviation of ∆i∗t p equal to 2/(1 + ρ)σu and the empirically observed standard deviation of changes in the monetary policy rate close to 0.25, we set σu = 41 . For the standard deviation of the committee members’ signals, we assume σP = 26
1 4
For models without reservations, skewt is calculated using the same expression, with r# t = 0. In order to make the results comparable among the different models, we keep the values of the random variables fixed across the simulations of those models. That is, when simulating, say, the first path in the autarkical model, the random values in the model are the same as those used in simulating the first path in the consensual or the supermajoritarian model. r# t = 27
18
and σC = 14 , implying that approximately 70% of the committee members’ signals are within 25 basis points of the optimal interest rate. The values above, most importantly σu = σC = σP = 14 , form our benchmark scenario. To examine the comparative static properties of our model, we generate a high volatility scenario by doubling σu , a bad information scenario by doubling σC and σP and a P bad information scenario by doubling only σP .
6
Simulation results
Tables 3 through 6 show the results of our simulation exercise. Each table includes results for the three models and the four scenarios described above. First two tables show the simulation results for models without reservations, and the next two tables show the results for models with reservations. Within each pair, the first table shows the results for N = 4, and the second table shows the results for N = 6. Tables 3 through 6 also include a mechanical model not previously discussed. This model serves to answer two questions. The first is whether it is possible to generate the predictive power of skewt purely mechanically (with negative answer). The second is how much mismatch between the adopted and the optimal policy is due purely to the fact that the former is set to discrete values (for the answer, see the tables). In the mechanical model, the chairman acts as if she knows the optimal policy ¯i∗ and her proposal y equals the value in Y closest to ¯i∗ . At the voting stage, we assume that the number of dissenters, i.e., the number of votes against y and thus for x, is a draw from uniform distribution on {0, . . . , N/2} so that at least
N 2
+1
members vote for y, which is thus always accepted. Finally, the number of reservations towards lower policy is a draw from uniform distribution on {0, . . . , r¯}, where r¯ = N/2, for meetings in which x = y, and r¯ equals the number of votes cast in favour of min {x, y}, for meetings with x 6= y. Reservations towards higher policy are determined analogously, with r¯ = N/2 and max {x, y} instead. Each cell of Tables 3 through 6 shows the estimated predictive power of skewt , the relative proportions of the different meeting types and a series of decision-making statistics similar to those in Table 1 (the average dissent, the fraction of meetings with no policy change, the fraction of meetings where
19
the chairman’s proposal is identical to the status-quo and the fraction of meetings at which the chairman’s proposal is adopted). The last row of each cell shows mse, defined as the mean squared difference between the adopted and the optimal monetary policy. We first discuss results for the models without reservations, focusing initially on different models under the baseline parameter scenario, and then on differences generated by the alternative scenarios and by larger committee size.
20
Table 3: Predictive power of voting record and decision-making statistics Models without reservations, N = 4 Model
Autarkical
Consensual
Supermajoritarian
Mechanical
Baseline scenario (σu = σC = σP = 14 ) skewt (a1 ) ∆pt (a2 )
4.10 [1.63]*** 0.75 [0.53]***
5.93 [3.29]*** 0.07 [0.45]***
5.25 [6.40]*** 1.63 [0.74]***
0.01 [2.47]*** -0.30 [0.63]***
Meeting types Stats
0.00/0.20/0.80 0.42/0.43 0.00/0.57 0.027
0.38/0.39/0.23 0.07/0.41 0.38/0.97 0.032
0.59/0.23/0.17 0.03/0.59 0.59/1.00 0.033
0.36/0.22/0.42 0.13/0.36 0.36/1.00 0.005
mse
High volatility scenario (σu doubled) skewt (a1 ) ∆pt (a2 )
2.19 [1.04]*** 0.24 [0.24]***
2.59 [2.43]*** 0.01 [0.21]***
0.69 [3.65]*** 0.60 [0.29]***
-0.02 [1.26]*** -0.07 [0.32]***
Meeting types Stats
0.00/0.33/0.67 0.31/0.28 0.00/0.72 0.043
0.19/0.58/0.22 0.08/0.24 0.19/0.96 0.049
0.37/0.41/0.22 0.04/0.37 0.37/1.00 0.044
0.19/0.27/0.54 0.16/0.19 0.19/1.00 0.005
mse
Bad information scenario (σC , σP doubled) skewt (a1 ) ∆pt (a2 )
3.43 [1.50]*** 0.29 [0.48]***
6.84 [3.59]*** 0.08 [0.46]***
6.05 [6.47]*** 1.04 [0.64]***
-
Meeting types Stats
0.00/0.25/0.75 0.40/0.43 0.00/0.57 0.048
0.39/0.40/0.21 0.06/0.41 0.39/0.98 0.052
0.56/0.30/0.15 0.03/0.56 0.56/1.00 0.052
-
mse
P bad information scenario (σP doubled) skewt (a1 ) ∆pt (a2 )
4.97 [1.79]*** 0.82 [0.54]***
9.98 [6.24]*** -0.19 [0.41]***
6.53 [6.43]*** 1.25 [0.67]***
-
Meeting types Stats
0.00/0.21/0.79 0.45/0.50 0.00/0.50 0.041
0.37/0.53/0.10 0.02/0.38 0.37/1.00 0.036
0.57/0.27/0.16 0.03/0.57 0.57/1.00 0.049
-
mse
Note: All entries averaged over 101 random paths. The first two rows of each panel are ordered probit estimates of ∆pt+1 = a0 + a1 skewt + a2 ∆pt + ut+1 , with [standard errors]. ***/**/* indicates significance at the 1%/5%/10% level. Meeting types show the fraction of meetings of given types (1.0/2.0/2.1), coded as x.y, where x is the number of alternatives put to vote and y is the number of alternatives with a positive number of votes (excluding the proposal). Stats show average dissent, the fraction of meetings with no policy change (first row), the fraction of meetings with the proposal equal to the status-quo and the fraction of meetings with the chairman’s proposal accepted (second row). mse is the mean squared difference between the adopted and the optimal policy.
21
Table 4: Predictive power of voting record and decision-making statistics Models without reservations, N = 6 Model
Autarkical
Consensual
Supermajoritarian
Mechanical
Baseline scenario (σu = σC = σP = 14 ) skewt (a1 ) ∆pt (a2 )
5.15 [1.66]*** 1.08 [0.56]***
6.57 [3.30]*** 0.14 [0.46]***
7.41 [5.30]*** 2.05 [0.81]***
0.30 [2.50]*** -0.25 [0.66]***
Meeting types Stats
0.00/0.13/0.87 0.44/0.45 0.00/0.55 0.026
0.38/0.34/0.28 0.08/0.42 0.38/0.97 0.032
0.57/0.17/0.26 0.05/0.57 0.57/1.00 0.030
0.36/0.16/0.47 0.14/0.36 0.36/1.00 0.005
mse
High volatility scenario (σu doubled) skewt (a1 ) ∆pt (a2 )
2.37 [1.02]*** 0.28 [0.24]***
3.01 [2.43]*** 0.02 [0.21]***
1.69 [2.99]*** 0.63 [0.30]***
0.00 [1.28]*** -0.06 [0.34]***
Meeting types Stats
0.00/0.26/0.74 0.33/0.31 0.00/0.69 0.040
0.19/0.54/0.27 0.08/0.24 0.19/0.95 0.049
0.33/0.33/0.34 0.07/0.33 0.33/1.00 0.036
0.19/0.21/0.60 0.17/0.19 0.19/1.00 0.005
mse
Bad information scenario (σC , σP doubled) skewt (a1 ) ∆pt (a2 )
3.57 [1.47]*** 0.40 [0.49]***
7.60 [3.67]*** 0.15 [0.47]***
7.44 [5.09]*** 1.16 [0.65]***
-
Meeting types Stats
0.00/0.19/0.81 0.42/0.44 0.00/0.56 0.047
0.39/0.34/0.27 0.07/0.41 0.39/0.98 0.052
0.52/0.25/0.23 0.05/0.52 0.52/1.00 0.051
-
mse
P bad information scenario (σP doubled) skewt (a1 ) ∆pt (a2 )
5.31 [1.71]*** 1.00 [0.55]***
12.4 [6.61]*** -0.13 [0.42]***
7.86 [5.08]*** 1.36 [0.68]***
-
Meeting types Stats
0.00/0.16/0.84 0.48/0.51 0.00/0.49 0.041
0.37/0.49/0.14 0.03/0.37 0.37/1.00 0.036
0.54/0.23/0.23 0.05/0.54 0.54/1.00 0.048
-
mse
Note: All entries averaged over 101 random paths. The first two rows of each panel are ordered probit estimates of ∆pt+1 = a0 + a1 skewt + a2 ∆pt + ut+1 , with [standard errors]. ***/**/* indicates significance at the 1%/5%/10% level. Meeting types show the fraction of meetings of given types (1.0/2.0/2.1), coded as x.y, where x is the number of alternatives put to vote and y is the number of alternatives with a positive number of votes (excluding the proposal). Stats show average dissent, the fraction of meetings with no policy change (first row), the fraction of meetings with the proposal equal to the status-quo and the fraction of meetings with the chairman’s proposal accepted (second row). mse is the mean squared difference between the adopted and the optimal policy.
22
Differences among the models can be largely understood in terms of the chairman’s proposing behaviour. The first cell in Table 3 shows that in the autarkical model, C never proposes a policy identical to the status-quo. She knows that if she proposes a policy different from the status-quo, the committee will adopt the correct alternative with a high probability, as the votes of the remaining committee members effectively aggregate information contained in their signals. On the other hand, proposing a policy identical to the status-quo implies that she acts as a dictator, and the policy outcome is then based on her signal alone. Put differently, proposing a policy equal to the status-quo means foregoing an opportunity to have the better-informed party decide. Most of the values of the decision-making statistics in the autarkical model follow from this basic observation. The model generates large dissenting frequencies, a rather small fraction of meetings with no policy change and, most importantly, a high frequency of rejections of the chairman’s proposal. The model also generates high predictive power of the voting record, as summarized by the skewt variable. The intuition for this result is as follows. Assume that the optimal policy has been constant at i∗1 and then unexpectedly rises to a new constant level of i∗2 > i∗1 . Further, assume that the committee has been setting its policy optimally at p1 = i∗1 , before the change. If, in the fist meeting under i∗2 , C’s signal causes her to propose a policy equal to the status-quo, skewt will be equal to zero. If, on the other hand, she proposes p2 = i∗2 > p1 , and her proposal is rejected, skewt will be positive, and, most importantly, the policy value is likely to be raised at the subsequent meeting, as it is now below the optimal level. If the chairman’s proposal is accepted, skewt will be negative, but the policy value will be unlikely to change in the future, as it is now at the optimal level. The combination of a likely future increase of the policy under positive skewt and of an unlikely change under negative skewt generates the predictive power of this variable. The consensual model generates decision patterns markedly different from those of the autarkical model. Again, C’s proposing behaviour is key. C proposes a policy she personally sees as the most appropriate, and the information contained in her proposal homogenizes the beliefs of the other
23
committee members. This generates a low degree of dissent and a large fraction of meetings in which C’s proposal is accepted. The low degree of dissent also implies that most of the meetings with no policy change result from C proposing that the policy be unchanged. The resulting fraction of meetings with no policy change is generally the smallest among the three models. The remaining members of the committee place few constraints on C, and changes in the policy largely reflect noise in C’s information set. Finally, under the consensual model, skewt is not a good predictor of future policy changes. The intuition behind this result, in fact one that is more general, relates to the low degree of dissent. With no dissenting votes, skewt equals zero and thus is a weak predictor in general. The supermajoritarian model is, in many respects, similar to the consensual model. It generates a low degree of dissent and a high frequency of acceptance of C’s proposals (by construction). However, the mechanism behind these results is different. It is not homogenization of beliefs among the committee members that generates low dissent and universal acceptance, but the fact that C proposes policies she knows will be accepted by a supermajority of the committee. This is also manifest in the large fraction of meetings at which C’s proposal is identical to the status-quo, and, due to universal acceptance, the large fraction of meetings at which there is no policy change - indeed, it is the largest such proportion among the three models. Low dissent also implies no predictive power of skewt . Using mse to measure the success of the different decision-making protocols in matching the adopted policy to the optimal one shows that all the models produce errors considerably larger than those produced by the mechanical model, which errs only because of the discrete nature of policy choices. Again, the mechanisms generating these errors vary among the models. In the autarkical model, the decision is in principle based on all the signals the committee possesses, but because C never proposes a policy identical to the status-quo, opportunities for error are plentiful. In the consensual model, the decision is mostly based on C’s signal, and the decision-making protocol leads to wasteful disregard of information of the other committee members. In the supermajoritarian model, the reason for mistakes is inertia. It takes time to gather enough support for a change in policy, which then lags behind the optimal one. Finally, the fractions of different meeting types reveal little informa-
24
tion beyond that provided by the decision-making statistics discussed above. None of the models generates, by construction, meetings of types 1.1, 1.2 and 2.2 we observe in the real data. Turning our attention to the different scenarios, their impact can be understood in terms of the impact of σu and σj on player j’s expectation of the optimal policy, ¯i∗ . Well-known signal extraction result gives E[ ¯i∗ |i∗ , ij ] =
σj2 ρ i∗ + σu2 ij σj2 + σu2
(4)
so that an increase in σu , in the high volatility scenario, causes all the players to focus more intently on their own signals relative to the previous-period optimal policy, i∗ . An increase in σj for j ∈ {C, P }, in the bad information scenarios, has the opposite effect: players focus on i∗ rather than on their signals.28 Players’ intent focus on their private information, under the high volatility scenario, leads to increased dissent in the consensual and supermajoritarian models, but decreased dissent in the autarkical model. The intuition behind this is that, with larger σu , deviations from the optimal policy become more pronounced and hence easier to detect.29 In addition to the effect on dissent, the high volatility scenario naturally generates a higher frequency of policy changes because the optimal policy has a larger variance. The final change is skewt losing its predictive power in the autarkical model. The major change introduced by the two bad information scenarios is a decrease in the dissenting frequencies in the consensual and supermajoritarian models. This is driven by the homogenization of the players’ beliefs, occasioned by their focus on the variable common to their information sets, i∗ . This effect is especially pronounced in the consensual model, under the P bad information scenario, which effectively adds another common variable to the information set of the committee members, C’s signal iC . Comparing the relative success of our model committees in matching the 28
Along with the fact that the signals of the committee members are independent, conditional on ¯i∗ , while i∗ is common to their information sets, this implies more dispersed (heterogeneous) beliefs among committee members under larger σu and less dispersed (homogeneous) beliefs under larger σj . 29 By the same argument, dissent should decrease in the other two models. This effect does not show due to the already very low levels of dissent in these models.
25
adopted policy to the optimal policy across different scenarios, mse is generally larger under the bad information scenarios relative to the high volatility scenario. Possession of good information by the monetary policy committee members seems to be more important than volatility of the economic environment. The detrimental effect of bad information, then, is mitigated by C having precise information in the P bad information scenario, but the extent of mitigation depends crucially on the role C plays in the committee. In fact, mse improves little in the supermajoritarian model, improves somewhat in the autarkical model and improves considerably in the consensual model. Note that this ordering of the models corresponds to the relative importance of C in the different models. Finally, there are small differences to be seen in moving from Table 3, for N = 4, to Table 4, for N = 6. The two main differences are an across-the-board decrease in mse and skewt gaining predictive power in the autarkical model under all scenarios. Both effects seem to be driven by the fact that the two additional members bring more information into the model committees.30 More information improves decision-making in all the models but generates increased predictive power of skewt only in the model that provides sufficient opportunity for dissent.
30 The Condorcet Jury Theorem literature (see, for example, Gerling, Gruner, Kiel, and Schulte, 2005, for a survey treatment) has stressed the benefits of larger group size for decision-making situations with incomplete information. The reason this benefit is rather modest in our model is that even in large committee, the proposal is made by a single individual.
26
Table 5: Predictive power of voting record and decision-making statistics Models with reservations, N = 4 Model
Autarkical
Consensual
Supermajoritarian
Baseline scenario (σu = σC = σP =
Mechanical
1 ) 4
skewt (a1 ) ∆pt (a2 )
5.22 [1.63]*** 0.89 [0.53]***
7.92 [3.01]*** 0.13 [0.44]***
10.7 [3.09]*** 1.80 [0.63]***
-0.06 [1.09]*** -0.31 [0.40]***
Meeting types
0.00/0.00/0.00 0.17/0.75/0.08 0.43/0.43 0.00/0.57 0.027
0.34/0.04/0.00 0.37/0.24/0.00 0.09/0.41 0.38/0.97 0.032
0.38/0.21/0.00 0.18/0.21/0.01 0.10/0.59 0.59/1.00 0.033
0.04/0.16/0.16 0.04/0.22/0.20 0.46/0.36 0.36/1.00 0.005
Stats mse
High volatility scenario (σu doubled) skewt (a1 ) ∆pt (a2 )
2.74 [0.96]*** 0.22 [0.22]***
4.64 [1.93]*** -0.02 [0.20]***
3.88 [1.82]*** 0.49 [0.27]***
-0.03 [0.78]*** -0.07 [0.22]***
Meeting types
0.00/0.00/0.00 0.18/0.63/0.18 0.37/0.28 0.00/0.72 0.043
0.13/0.06/0.00 0.47/0.33/0.01 0.13/0.24 0.19/0.96 0.049
0.10/0.24/0.03 0.17/0.36/0.09 0.21/0.37 0.37/1.00 0.044
0.02/0.09/0.08 0.05/0.28/0.25 0.48/0.19 0.19/1.00 0.005
Stats mse
Bad information scenario (σC , σP doubled) skewt (a1 ) ∆pt (a2 )
3.75 [1.48]*** 0.30 [0.48]***
8.20 [3.44]*** 0.13 [0.45]***
9.58 [3.72]*** 1.10 [0.57]***
-
Meeting types
0.00/0.00/0.00 0.23/0.74/0.03 0.41/0.43 0.00/0.57 0.048
0.37/0.02/0.00 0.39/0.22/0.00 0.07/0.41 0.39/0.98 0.052
0.43/0.13/0.00 0.26/0.18/0.00 0.07/0.56 0.56/1.00 0.052
-
Stats mse
P bad information scenario (σP doubled) skewt (a1 ) ∆pt (a2 )
5.65 [1.78]*** 0.88 [0.54]***
10.2 [6.21]*** -0.19 [0.41]***
8.46 [3.70]*** 1.26 [0.59]***
-
Meeting types
0.00/0.00/0.00 0.19/0.72/0.09 0.46/0.50 0.00/0.50 0.041
0.37/0.00/0.00 0.53/0.10/0.00 0.02/0.38 0.37/1.00 0.036
0.44/0.13/0.00 0.25/0.18/0.00 0.07/0.57 0.57/1.00 0.049
-
Stats mse
Note: All entries averaged over 101 random paths. The first two rows of each panel are ordered probit estimates of ∆pt+1 = a0 + a1 skewt + a2 ∆pt + ut+1 , with [standard errors]. ***/**/* indicates significance at the 1%/5%/10% level. Meeting types show the fraction of meetings of given types, 1.0/1.1/1.2 (first row), 2.0/2.1/2.2 (second row), with the residual 2.3 not shown, coded as x.y, where x is the number of alternatives put to vote and y is the number of alternatives with a positive number of votes (excluding the proposal). Stats show average dissent, the fraction of meetings with no policy change (first row), the fraction of meetings with the proposal equal to the status-quo and the fraction of meetings with the chairman’s proposal accepted (second row). mse is the mean squared difference between the adopted and the optimal policy.
27
Table 6: Predictive power of voting record and decision-making statistics Models with reservations, N = 6 Model
Autarkical
Consensual
Supermajoritarian
Baseline scenario (σu = σC = σP =
Mechanical
1 ) 4
skewt (a1 ) ∆pt (a2 )
6.26 [1.66]*** 1.22 [0.55]***
8.97 [3.07]*** 0.22 [0.45]***
11.7 [3.20]*** 2.31 [0.67]***
0.02 [1.11]*** -0.31 [0.40]***
Meeting types
0.00/0.00/0.00 0.10/0.79/0.11 0.45/0.45 0.00/0.55 0.026
0.32/0.06/0.00 0.32/0.30/0.00 0.09/0.42 0.38/0.97 0.032
0.31/0.26/0.00 0.12/0.29/0.02 0.12/0.57 0.57/1.00 0.030
0.02/0.14/0.20 0.02/0.17/0.20 0.50/0.36 0.36/1.00 0.005
Stats mse
High volatility scenario (σu doubled) skewt (a1 ) ∆pt (a2 )
2.89 [0.94]*** 0.26 [0.23]***
5.07 [1.93]*** -0.02 [0.20]***
3.79 [1.86]*** 0.52 [0.26]***
0.01 [0.79]*** -0.06 [0.23]***
Meeting types
0.00/0.00/0.00 0.11/0.63/0.26 0.40/0.31 0.00/0.69 0.040
0.11/0.08/0.00 0.39/0.40/0.02 0.14/0.24 0.19/0.95 0.049
0.05/0.23/0.05 0.10/0.38/0.18 0.23/0.33 0.33/1.00 0.036
0.01/0.07/0.11 0.03/0.22/0.25 0.51/0.19 0.19/1.00 0.005
Stats mse
Bad information scenario (σC , σP doubled) skewt (a1 ) ∆pt (a2 )
3.81 [1.45]*** 0.41 [0.49]***
9.26 [3.54]*** 0.21 [0.46]***
10.3 [3.64]*** 1.26 [0.58]***
-
Meeting types
0.00/0.00/0.00 0.17/0.82/0.02 0.43/0.44 0.00/0.56 0.047
0.36/0.03/0.00 0.33/0.28/0.00 0.07/0.41 0.39/0.98 0.052
0.37/0.15/0.00 0.20/0.27/0.01 0.08/0.52 0.52/1.00 0.051
-
Stats mse
P bad information scenario (σP doubled) skewt (a1 ) ∆pt (a2 )
5.48 [1.68]*** 1.00 [0.55]***
12.6 [6.59]*** -0.13 [0.42]***
9.57 [3.64]*** 1.40 [0.60]***
-
Meeting types
0.00/0.00/0.00 0.14/0.78/0.08 0.48/0.51 0.00/0.49 0.041
0.37/0.00/0.00 0.49/0.14/0.00 0.03/0.37 0.37/1.00 0.036
0.38/0.15/0.00 0.19/0.27/0.01 0.08/0.54 0.54/1.00 0.048
-
Stats mse
Note: All entries averaged over 101 random paths. The first two rows of each panel are ordered probit estimates of ∆pt+1 = a0 + a1 skewt + a2 ∆pt + ut+1 , with [standard errors]. ***/**/* indicates significance at the 1%/5%/10% level. Meeting types show the fraction of meetings of given types, 1.0/1.1/1.2 (first row), 2.0/2.1/2.2 (second row), with the residual 2.3 not shown, coded as x.y, where x is the number of alternatives put to vote and y is the number of alternatives with a positive number of votes (excluding the proposal). Stats show average dissent, the fraction of meetings with no policy change (first row), the fraction of meetings with the proposal equal to the status-quo and the fraction of meetings with the chairman’s proposal accepted (second row). mse is the mean squared difference between the adopted and the optimal policy.
28
We now turn our attention to the models with reservations, in Tables 5 and 6. First, note that incorporating reservations into the models does not alter the outcomes of the monetary policy meetings (and therefore, mse does not change). Hence the results independent of the voting records remain the same. However, what inclusion of reservations does change is the frequency of dissent, the fraction of different meeting types and the predictive power of skewt . In all the models, average dissent increases, but the effect is most notable in the supermajoritarian model. Intuitively, in the autarkical model, there are plentiful opportunities to express preferences through voting, even without reservations, as C’s proposal always differs from the status-quo. On the other hand, in the consensual model, the committee members need rarely express their preferences via reservations, or voting for that matter, due to the homogenization effect of C’s proposal.31 The effect of allowing for reservations on the fraction of different meeting types is in general to change x.y into x.y 0 meetings, where y < y 0 . In the autarkical model, this mainly shifts meetings from the 2.0 into the 2.1 and 2.2 categories, but the size of the effect is rather small. Similar applies to the consensual model, with the main change seen in the 2.0 and 2.1 categories. On the other hand, reservations have a large impact in the supermajoritarian model. The main change is a large increase in the fraction of 1.1 meetings at the expense of 1.0 meetings. A second, if smaller, change is a shift of some 2.0 meetings into the 2.1 category. Additionally, the effect of reservations on meeting types seems to be modulated by the different parameter scenarios. The effect is more pronounced in the high volatility scenario than in the bad information scenarios. The reason, again, is the increased focus of the committee members on their signals in the high volatility scenario and the disregard of their signals in the bad information scenarios. Finally, allowing for reservations in general increases the predictive power of skewt . This is not surprising, as reservations allow more information to enter the voting records. Inclusion of reservations makes skewt significant in the autarkical model, for all the scenarios in Tables 5 and 6, and for many scenarios in the consensual and supermajoritarian models. 31 An average number of reservations per meeting is almost never larger than one and typically smaller than one-half. This explains the small effect on the dissenting frequencies.
29
The effect of the different parameter scenarios on skewt , however, is less clear-cut. The general pattern is that scenarios with low dissent generate insignificant skewt . In fact, with dissent at zero, skewt will be a constant equal to zero and hence have no predictive power. However, the effect of higher dissent is more ambiguous. On the one hand, higher dissent means that more information is summarized in skewt . On the other hand, higher dissent is typically associated with a less predictable environment.32 The significance of the skewt variable, associated with the higher dissent, is then a question of which of these two effects dominates. To compare our model results to real-world data, we present Table 7. We observe that the supermajoritarian model is closest to the properties of the UK and US voting records, followed by the consensual model. On the other hand, the autarkical model does not fit the data well, even though our simulations show that this model is typically able to deliver a lower level of mse. Table 7: Model results vs. reality – summary Fraction of meetings with no policy change
proposal status-quo
proposal accepted
average dissent
Real data: Bank of England Fed: Greenspan Fed: Bernanke
0.76 0.66 0.73
0.76 0.66 0.73
1.00 1.00 1.00
0.14 0.05 0.06
Theory: Autarkical Consensual Supermajoritarian
0.45 0.42 0.57
0.00 0.38 0.57
0.55 0.97 1.00
0.45 0.09 0.12
Note: Reproduced from Table 1 for the real data and from Table 6 for the theory (baseline scenario with reservations and N = 6).
Before concluding, let us note on a series of additional simulation results found in Appendix A2. First, we were interested in the performance of a model with full informational independence of committee members. This model is mostly based on the autarkical model, with C additionally 32
A good illustration arises from comparing the baseline and high volatility scenarios in the consensual and supermajoritarian models, in Table 6. In both models, higher volatility increases dissent, but skewt retains its significance only in the consensual model.
30
disregarding information that acceptance of her proposal reveals about the optimal policy. In terms of results, this full independence model generates policy inertia comparable to that of the supermajoritarian model and predictive power of skewt comparable to that of the autarkical model, avoiding the main unrealistic feature of the latter, namely, that C never proposes the statusquo. However, it still generates an unrealistic fraction of meetings in which the chairman’s proposal is rejected. For this reason, we have decided not to include the full independence model in the main body of the paper. Second, as mentioned in footnote 15, we also ran simulations of the simple majority version of our supermajoritarian model, a model we call majoritarian. Relative to the supermajoritarian model, this model generates larger dissent frequencies, lower policy inertia and somewhat less robust predictive power of skewt . Given the similarity of these two models and the arguably better performance of the supermajoritarian one, we include details of the majoritarian model only in the appendix. Third, we have investigated whether the relatively good performance of the supermajoritarian model is due solely to the fact that it features a higher degree of information sharing relative to the autarkical and consensual models. For the two latter models we have specified and simulated their versions with information sharing assuming that the chairman learns the committee members’ signals prior to making a proposal. For the autarkical model with information sharing this implies that the chairman knows which policies the committee accepts and proposes an acceptable policy closest to her expected optimal policy rate. In the spirit of the autarkical model, the committee members’ voting is based solely on their signals. In the consensual model with information sharing the committee members always approve the chairman’s proposal; they extract information from the proposal and know that the chairman knows all their signals. Given this, the chairman proposes a policy closest to her expected optimal policy rate. The consensual model with information sharing, due to the universal acceptance of the chairman’s proposal, generates no dissent and hence pushes the consensual model away from the stylized facts. On the other hand, the information sharing improves the fit of the autarkical model with the stylized facts; it increases the fraction of meetings with no policy change, with proposal equal to the status-quo and with chairman’s proposal acceptance,
31
while lowering the dissenting frequencies. What the autarkical model with information sharing does not predict is the predictive power of skewt . The intuition is that since the chairman’s proposal is based on the entire committee’s information and is always accepted, the resulting policy is often close to optimal and hence needs not catch up with the optimal policy rate, which is the mechanism generating the predictive power of skewt . Overall, the autarkical model with information sharing fits the stylized facts better than the consensual model but worse than the supermajoritarian model, as it still predicts somewhat frequent policy changes and insignificant skewt .33 For each of the three models in the main text, and each of the three additional models just mentioned, Appendix A2 includes four tables ordered similarly to those above, namely, by whether reservations are allowed and by committee size.34 The first three columns of each table are estimates with an AR(1) process for i∗ and ρ values of 0.90, 0.95 (benchmark) and 0.99, respectively. The last column changes the process for i∗ to AR(2), with ρ1 = 1.95 and ρ2 = −0.98 (following Gerlach-Kristen, 2008). Changing the value of ρ has little impact. For the AR(2) process, we 1 ∗ 20 , to keep the standard deviation of the first difference of i unchanged.35 This makes the optimal policy highly predictable and the beliefs
set σu =
of the committee members homogeneous, which in turn implies significantly lower dissent frequencies for all models and, other than in the autarkical model, skewt losing its predictive power.36 Apart from this effect, there is little difference between the AR(1)-based and the AR(2)-based results. 33
Different degree of information sharing in the different models makes comparison of the bad information scenario for the supermajoritarian model with the baseline scenario for the autarkical and consensual models meaningful. In order to keep the degree of remaining uncertainty at the point of proposal-making and voting constant across the models, and given that the supermajoritarian model by construction eliminates some uncertainty, the appropriate starting position for the supermajoritarian model is a situation with less reliable information held by the committee. Since our results show low sensitivity to the different scenarios, this perspective leads to similar conclusions regarding the performance of the models. 34 Tables A1-A4 for the autarkical model, A5-A8 for the consensual model, A9-A12 for the supermajoritarian model, A13-A16 for the full independence model, A17-A20 for the majoritarian model and A21-A24 for the autarkical with information sharing model. 35 q The standard deviation of the first difference of an AR(2) process is equal to 2 σ , (1+ρ2 )(1+ρ1 −ρ2 ) u 36
which we want to be approximately equal to 0.25.
Increasing the precision of the committee members’ signals easily restores the predictive power of skewt . We have decided to present the AR(2) results without changing the values of σC and σP , for consistency.
32
7
Conclusions
We have investigated the ability of different group decision-making models to replicate proposing, voting and decision patterns that we observe in realworld monetary policy committees. In all the models, the final decision is chosen in the voting stage over a binary agenda consisting of the chairman’s proposal and the status-quo policy. The winning alternative from the voting stage represents the final decision and becomes the status-quo alternative in the voting stage at the subsequent meeting. One of the main conclusions we highlight is the fine balance any model of central bank decision-making must strike. Real-world committees frequently display non-negligible dissenting frequencies, but rarely, if ever, reject a chairman’s proposal. Any theoretical model with non-trivial dissent then runs a risk of rejecting the chairman’s proposals too often. Ensuring that the chairman’s proposals are accepted on the other hand constrains the ability of any model to generate dissent. Generating dissent relies, of course, on the assumption of heterogeneous committee members. In our model, this is achieved by assuming that each committee member holds a private signal regarding the unobserved optimal policy rate. We stress the intricate nature of the interaction between the decision-making protocol, the volatility of the economic environment, the precision of the members’ signals and other information they hold. Less precise signals make our model committees more homogeneous, shifting their attention to an alternative source of information to which they have common access. A volatile economic environment, on the other hand, makes our model committees more heterogeneous, causing them to shift their attention to their private information. However, increased dissent does not follow automatically. Larger economic shocks, for example, might be easier to detect, making the committee more likely to make unanimous decisions. Which of the two effects dominates depends on the particularities of the decision-making protocol in use and the existing level of dissent. An alternative way of generating dissent, one that does not lead to rejections of the chairman’s proposals, is to allow the committee members to express reservations regarding the final committee decision. To our knowledge, we are among the first (along with Riboni and Ruge-Murcia, 2014) to systematically investigate this possibility.
33
Our results indicate that allowing for reservations improves the predictive power of voting records with respect to future policy changes and helps the theory produce decision patterns observed in the data. However, their impact will be largest for models that otherwise provide limited opportunities to express preferences via voting. These will be the models characterized by frequent chairman’s proposals for no change in policy. Allowing for reservations thus has the double benefit of generating dissent without rejections of the chairman’s proposals and of supporting models with large policy inertia. We have also shown that reservations help to improve the predictive power of voting records, summarized by skewt . However, the significance of skewt is, in our view, the most elusive of our results. It is naturally modeldependent, but also parameter-dependent. To a first approximation, the significance of skewt is closely linked to the degree of dissent. Without dissenting votes, skewt is constant and loses its predictive power. However, the opposite need not hold, and large dissenting frequencies can be associated with insignificant skewt . Additionally, the predictive power of skewt seems to improve with committee size, worsen with less precise information of committee members and worsen in a more volatile economic environment. Note, however, that the conclusion that with precise enough information and a predictable economic environment skewt would be a good predictor of future policy is not warranted. With perfect information or a perfectly predictable economic environment, our model committees would be perfectly homogeneous and would reach their decisions unanimously. The relationship between the quality of information, predictability of the economic environment and the significance of skewt thus seems to be a non-monotone one. A key mechanism that generates non-unanimous decisions in our model are the private signals of committee members. Without the private signals, the committee would share common preferences. An alternative way to generate non-unanimous decisions would be to assume time-invariant differences in innate preferences of the committee members, in the spirit of Riboni and Ruge-Murcia (2010). Because, in our view, disagreement among real-world policy makers is driven both by their innate preferences and by the information they posses, we view both modeling approaches as extreme. An open question, therefore, is how much of the disagreement between policy makers
34
is due to preferences rather than information (see Hansen, McMahon, and Velasco Rivera, 2014, for an interesting investigation of this question). Admittedly, we have made no rigourous attempt to fit the parameters of any of our models to real data. Instead, our investigation has been driven by an attempt to understand the mechanisms through which the models generate different decision-making patterns. For this reason, we have decided to use a more traditional comparative statics approach, contrasting different models and changing parameters in a structured way. Given a large variety of approaches to modeling central bank decision-making in the existing literature, and given the lack of a clearly-established workhorse model, we see this as a more meaningful endeavour. From a positive perspective, judging the models by their ability to replicate real-world data, we lean toward the supermajoritarian model, combined with the ability of the committee members to express reservations regarding final committee decisions. This model generates large policy inertia, no rejections of the chairman’s proposals, non-trivial dissent and high predictive power of skewt . In fact, we see the supermajoritarian model as fitting even the Bank of England, despite the fact that its Monetary Policy Committee is often touted as highly individualistic. However, even at the Bank of England, rejections of the governor’s proposals are non-existent (Mervyn King dissented twice during the ten years he served as the Governor, in both cases against his own proposal, in all other meetings he voted his own proposal and was part of a winning majority), and our supermajoritarian model with reservations has no problem generating dissent frequencies resembling those in the data. Additionally, we see individualism at the voting stage as distinct from individualism when expressing opinions. Indeed, this observation is at the heart of the definition of an individualistic committee: ‘Members of an individualistic committee not only express their own opinions verbally, but probably also act on them by voting.’ (Blinder, 2007, page 113, emphasis added).
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Riboni, A. and F. J. Ruge-Murcia (2008b). Preference heterogeneity in monetary policy committees. International Journal of Central Banking 4 (1), 213–233. Riboni, A. and F. J. Ruge-Murcia (2010). Monetary policy by committee: Consensus, chairman dominance, or simple majority? Quarterly Journal of Economics 125 (1), 363–416. Riboni, A. and F. J. Ruge-Murcia (2014). Dissent in monetary policy decisions. Journal of Monetary Economics 66, 137–154. Tallis, G. M. (1961). The moment generating function of the truncated Multi-Normal distribution. Journal of the Royal Statistical Society. Series B (Methodological) 23 (1), 223–229. Weber, A. (2010). Communication, decision making and the optimal degree of transparency of monetary policy committees. International Journal of Central Banking 6 (3), 1–49. Wilhelm, S. and B. G. Manjunath (2012). tmvtnorm: Truncated multivariate Normal and Student t distribution. R package version 1.4-6.
A1
Model technical details
Appendix A1 includes more technical details for the models from the main body of the paper so that it becomes apparent how to generate C’s proposals and P s’ voting behaviour. We further explain finer details of our simulation exercise and the methods used. First note that for all the models equilibrium exists by simple (withinperiod) backward induction argument and it is unique. The Markovian restriction (Maskin and Tirole, 2001) then means all the committee members condition their actions only on the state given by the status-quo policy and their signals, not on payoff irrelevant histories. Throughout the explanation we will often work with a vector of random variables. All those variables form a random vector r = {¯i∗ , iP 1 , . . . , iP N , iC }0 that has a multivariate normal distribution with, conditional on i∗ , mean ρ i∗ and variance-covariance matrix equal to a matrix with the vector of 2 }0 on the main diagonal and all the off{σu2 , σu2 + σP2 , . . . , σu2 + σP2 , σu2 + σC diagonal elements equal to σu2 .37 Often we will need to compute the conditional expectation of r given some specific value of one or more of its 37
When changing the process governing the optimal policy rate to AR(2) the only necessary adjustment is change of ρ i∗ into ρ1 i∗t−1 + ρ2 i∗t−2 in all the expressions.
38
elements. For this we use the well known result for the multivariate normal distribution that states that for a vector of (possibly more than two) random variables {x1 , x2 }0 distributed according to N (µ, Σ) with µ = {µ1 , µ2 }0 and σ12 Σ = ( σσ11 21 σ22 ) where the partitioning of µ and Σ conforms to the partition of {x1 , x2 }0 , the conditional distribution of x1 given a specific value of x2 is 0 ), where µ0 = µ + σ σ −1 (x − µ ) and σ 0 = σ − σ σ −1 σ . N (µ01 , σ11 1 12 22 2 2 11 12 22 21 1 11 In the autarkical model, at the beginning of each period we have statusquo x, last-period optimal interest rate i∗ and r = {¯i∗ , iP 1 , . . . , iP N , iC }0 . First we need to derive C’s proposal y. This will be a solution to max E −(p(x, y) − ¯i∗ )2 |i∗ , iC (A1) y∈Y
where p(x, y) is the policy adopted given proposal y and status-quo x. The optimization problem can be rewritten as max pa E[−(y − ¯i∗ )2 |i∗ , iC , a] + (1 − pa )E[−(x − ¯i∗ )2 |i∗ , iC , a ˆ] y∈Y
(A2)
where a is the event of y being accepted, a ˆ is the event of y being rejected and pz is the probability of event z. We need to calculate the probability of y being accepted against statusquo x, pa . Chairman C knows, and we show below, that the remaining players will vote for y if and only if their signal is above (or below, but this case is symmetric) a certain cut-off that we denote here by k. The other relevant information C has is iC and i∗ . Hence we need to calculate the probability of at least N2 P members voting for y given iC and i∗ . The probability of the first N 0 committee members voting for y is equal to P(#|iP ≥ k| = N 0 , #|iP < k| = N − N 0 |i∗ , iC ) and is straightforward to calculate. We know the distribution of {iP 1 , . . . , iP N }0 and can transform the probability into P(#|iP ≤ k| = N |i∗ , iC ) by multiplying the whole problem (that is the mean and variance-covariance matrix) by {−1, . . . , −1, 1, . . . , 1}0 , where there are N 0 negative ones and N − N 0 positive ones. Denoting the 0 members accepting by P 0 , the probability of y probability of the first NP N N being accepted becomes N P . i i=N/2 i The key computational problem in simulating the autarkical model is computing the expected value of ¯i∗ given iC , i∗ and the event of y being accepted. Acceptance means that the signals iP of N2 or more P members must have been above (or below) a certain threshold k, which carries information about the unknown ¯i∗ . We use two simple results to simplify the computation. For random variable X and two mutually exclusive and exhaustive events A and B we have E[X] = E[X|A]P(A) + E[X|B]P(B)
39
(A3)
and the similar result for variance states that var(X) = var(X|A)P(A) + var(X|B)P(B) +(E[X|A] − E[X])2 P(A) + (E[X|B] − E[X])2 P(B).
(A4)
However, the key problem remains. We need to calculate an expectation of the form E[ ¯i∗ |i∗ , iC , #|iP ≥ k| = N 0 , #|iP < k| = N − N 0 ]. The first step is simple and amounts to calculating the distribution of {¯i∗ , iP 1 , . . . , iP N }0 given i∗ and iC . It is N (µ, Σ), with each element of µ being equal to 2 +iC σ 2 0 ρ i∗ σC u and Σ being a matrix with the vector σ 0 , σ 0 + σP2 , . . . , σ 0 + σP2 σ 2 +σ 2 u
C
2 σ2 σu C 2 +σ 2 off the main diagonal. We then convert σu C finding E[ ¯i∗ |i∗ , iC , #|iP ≥ k| = N ] using the same
on the main diagonal and σ 0 =
the problem into one of multiplication by a vector of positive and negative ones as when calculating pa . This leaves us with a multivariate truncated normal random vector with known mean and variance. To calculate the expectation we used the results in Tallis (1961) and Lee (1979) and wrote our own MATLAB function which calculates the expectation. We checked its correctness using the ‘tmvtnorm’ R-software package (see Wilhelm and Manjunath, 2012). With these results, we can expand the maximand in (A2) and use the rules for conditional expectations and variance given above to determine the value of C’s objective function for each y ∈ Y . This gives us the solution to C’s optimization problem and hence her proposal. With C’s proposal y, we can determine the voting behaviour of the remaining P committee members. For each member j we use the voting rule (2) from the text adapted to the autarkical model E −(y − ¯i∗ )2 |i∗ , ij ≥ E −(x − ¯i∗ )2 |i∗ , ij (A5) x2 − y 2 ≥ 2(y − x)E[ ¯i∗ |i∗ , ij ] with E[ ¯i∗ |i∗ , ij ] =
2 ρ i∗ σj2 +ij σu . 2 2 σu +σj
This also proves that each P member votes
for y if and only if his signal is above (or below, depending on the position of the status-quo) a certain cut-off. In the consensual model, again given x, i∗ and r = {¯i∗ , iP 1 , . . . , iP N , iC }0 , C’s proposal y will be the policy in Y closest to C’s expectation of ¯i∗ , ρ i∗ σ 2 +iC σ 2 E[ ¯i∗ |i∗ , iC ] = σ2C+σ2 u . u C Because the P committee members extract information from C’s proposal in the consensual model, each P member j will vote based on the voting rule (2) given i∗ , ij and the information embedded in C’s proposal y. Adapting (2) to the consensual model gives x2 − y 2 ≥ 2(y − x)E[ ¯i∗ |i∗ , ij , y].
40
(A6)
It is easy to confirm that the information embedded in C’s proposal is C equal to an event of iC ∈ [iC l , iu ], with the bounds given by i 1 h s¯ 2 2 2 ∗ y − (σ + σ ) − σ ρ i u C C σu2 2 (A7) h i 1 s¯ 2 C 2 2 ∗ iu = 2 y + (σu + σC ) − σC ρ i . σu 2 C Using the law of iterated expectations, E ¯i∗ |i∗ , ij , iC ∈ [iC l , iu ] rewrites as C E E[ ¯i∗ |i∗ , ij , iC ]|i∗ , ij , iC ∈ [iC l , iu ] . The inner expectations are equal to iC l =
2 +ij σ 2 σ 2 +iC σ 2 σ 2 ρ i∗ σj2 σC u C u j 2 2 σ 2 +σ 2 σ 2 σu σj2 +σu C C j
and we know that the distribution of iC given i∗ and
ij is normal with mean
2 ρ i∗ σj2 +ij σu 2 2 σu +σj
and variance
2 σ2 σu j 2 σu +σj2
2. + σC
Last thing we need is to be able to calculate mean of doubly truncated normal variable. For N (µ, σ 2 ) distributed x1 conditional expectation of x1 given x1 ∈ [al , au ] is E[x1 |x1 ∈ [al , au ]] = µ + σ
al −µ )−φ( auσ−µ ) σ a −µ Φ( auσ−µ )−Φ( lσ )
φ(
, where φ(·)
and Φ(·) are, respectively, the probability density and cumulative distribution functions of the univariate standard normal distribution. In the supermajoritarian model C knows the most preferred policies of all the committee members. For each player j this policy, given x, i∗ and r = {¯i∗ , iP 1 , . . . , iP N , iC }0 , will be the policy in Y closest to j’s expectation of ¯i∗ , 2 ρ i∗ σj2 +ij σu i.e. closest to E[ ¯i∗ |i∗ , ij ] = . Denote vector of the most preferred 2 2 σu +σj
policies by {p∗1 , . . . , p∗N +1 }, ordered such that p∗j ≤ p∗j+1 for j ∈ {1, . . . , N }. Policy most preferred by the median member is denoted by p∗m = p∗N/2+1 . C’s proposal will be the policy that receives a supermajority of at least N + 2 members. Naturally this policy depends on the status-quo and it is 2 easy to show that y as a function of x satisfies p∗m−1 for x ≤ p∗m−1 x for x ∈ [p∗m−1 , p∗m+1 ] y(x) = (A8) ∗ p∗ m+1 for x ≥ pm+1 . For voting we again use (2) along with the assumption that all the committee members extract no information from proposal y. This makes the voting stage in the supermajoritarian model equivalent to the voting stage in the autarkical model. However, by construction, C’s proposal is always accepted and receives at least N2 + 2 votes. To simulate each of the models, we start in the first period of a given path, with the previous optimal interest rate and monetary policy rate being zero. In all the simulations we restrict the policy space to be in the [−10, 10] interval so that with our choice of s¯ the policy space is equal to Y = {−10, −9.75, . . . , 9.75, 10}0 . We do not need to look at a larger policy space, as the optimal interest rate and players’ signals stay well away from 41
its border. As explained in the text, it is inconsequential that we allow the optimal interest rate and the monetary policy rate to attain negative values, as all the results and estimates are invariant to adding a constant to the optimal interest rate. The values of the random variables used in the simulations are kept constant across the different models. That is, when we simulate, say, the first path for the baseline scenario of the autarkical model, the random variables used are the same as when simulating the first path of any other scenario for the same model or of any other model for the same scenario. This holds even across the N = 4 and N = 6 simulations, where we naturally have to add two more random variables for the two extra players, but the remaining random variables are kept the same.
A2
Further simulation results
42
Table A1: Predictive power of voting record and decision-making statistics Autarkical model without reservations, N = 4 ρ = 0.90
ρ = 0.95
Baseline scenario (σu = σC = σP =
1 4
ρ1 = 1.95, ρ2 = −0.98
ρ = 0.99
for AR(1) and σu =
1 20
for AR(2))
skewt (a1 ) ∆pt (a2 )
4.16 [1.64]*** 0.65 [0.54]***
4.10 [1.63]*** 0.75 [0.53]***
4.30 [1.65]*** 0.78 [0.53]***
18.2 [4.53]*** 3.56 [0.64]***
Meeting types Stats
0.00/0.20/0.80 0.42/0.43 0.00/0.57 0.027
0.00/0.20/0.80 0.42/0.43 0.00/0.57 0.027
0.00/0.20/0.80 0.41/0.42 0.00/0.58 0.027
0.27/0.51/0.22 0.15/0.46 0.27/0.82 0.008
mse
High volatility scenario (σu doubled) skewt (a1 ) ∆pt (a2 )
2.20 [1.06]*** 0.17 [0.24]***
2.19 [1.04]*** 0.24 [0.24]***
2.16 [1.05]*** 0.29 [0.24]***
6.98 [2.74]*** 3.11 [0.37]***
Meeting types Stats
0.00/0.34/0.66 0.30/0.28 0.00/0.72 0.041
0.00/0.33/0.67 0.31/0.28 0.00/0.72 0.043
0.00/0.33/0.67 0.31/0.28 0.00/0.72 0.041
0.00/0.66/0.34 0.22/0.26 0.00/0.74 0.013
mse
Bad information scenario (σC , σP doubled) skewt (a1 ) ∆pt (a2 )
3.43 [1.52]*** 0.19 [0.49]***
3.43 [1.50]*** 0.29 [0.48]***
3.58 [1.53]*** 0.38 [0.47]***
19.3 [6.40]*** 3.26 [0.61]***
Meeting types Stats
0.00/0.24/0.76 0.41/0.43 0.00/0.57 0.048
0.00/0.25/0.75 0.40/0.43 0.00/0.57 0.048
0.00/0.27/0.73 0.39/0.41 0.00/0.59 0.049
0.37/0.53/0.11 0.07/0.45 0.37/0.91 0.008
mse
P bad information scenario (σP doubled) skewt (a1 ) ∆pt (a2 )
4.93 [1.78]*** 0.76 [0.56]***
4.97 [1.79]*** 0.82 [0.54]***
4.83 [1.76]*** 0.87 [0.53]***
19.7 [6.85]*** 3.28 [0.61]***
Meeting types Stats
0.00/0.20/0.80 0.46/0.51 0.00/0.49 0.041
0.00/0.21/0.79 0.45/0.50 0.00/0.50 0.041
0.00/0.22/0.78 0.44/0.49 0.00/0.51 0.041
0.38/0.53/0.10 0.07/0.46 0.38/0.92 0.008
mse
Note: All entries averaged over 101 random paths. The first two rows of each panel are ordered probit estimates of ∆pt+1 = a0 + a1 skewt + a2 ∆pt + ut+1 , with [standard errors]. ***/**/* indicates significance at the 1%/5%/10% level. Meeting types show the fraction of meetings of given types (1.0/2.0/2.1), coded as x.y, where x is the number of alternatives put to vote and y is the number of alternatives with a positive number of votes (excluding the proposal). Stats show average dissent, the fraction of meetings with no policy change (first row), the fraction of meetings with the proposal equal to the status-quo and the fraction of meetings with the chairman’s proposal accepted (second row). mse is the mean squared difference between the adopted and the optimal policy.
43
Table A2: Predictive power of voting record and decision-making statistics Autarkical model without reservations, N = 6 ρ = 0.90
ρ = 0.95
Baseline scenario (σu = σC = σP =
1 4
ρ1 = 1.95, ρ2 = −0.98
ρ = 0.99
for AR(1) and σu =
1 20
for AR(2))
skewt (a1 ) ∆pt (a2 )
5.07 [1.65]*** 1.00 [0.56]***
5.15 [1.66]*** 1.08 [0.56]***
4.94 [1.66]*** 1.12 [0.55]***
19.6 [5.32]*** 3.53 [0.63]***
Meeting types Stats
0.00/0.14/0.86 0.45/0.47 0.00/0.53 0.026
0.00/0.13/0.87 0.44/0.45 0.00/0.55 0.026
0.00/0.14/0.86 0.44/0.45 0.00/0.55 0.026
0.31/0.50/0.19 0.13/0.46 0.31/0.85 0.008
mse
High volatility scenario (σu doubled) skewt (a1 ) ∆pt (a2 )
2.38 [1.03]*** 0.21 [0.24]***
2.37 [1.02]*** 0.28 [0.24]***
2.35 [1.02]*** 0.33 [0.25]***
8.62 [3.13]*** 3.16 [0.38]***
Meeting types Stats
0.00/0.26/0.74 0.33/0.30 0.00/0.70 0.039
0.00/0.26/0.74 0.33/0.31 0.00/0.69 0.040
0.00/0.25/0.75 0.34/0.31 0.00/0.69 0.040
0.00/0.65/0.35 0.24/0.26 0.00/0.74 0.013
mse
Bad information scenario (σC , σP doubled) skewt (a1 ) ∆pt (a2 )
3.62 [1.49]*** 0.33 [0.51]***
3.57 [1.47]*** 0.40 [0.49]***
3.59 [1.48]*** 0.46 [0.48]***
21.0 [7.62]*** 3.24 [0.61]***
Meeting types Stats
0.00/0.18/0.82 0.44/0.46 0.00/0.54 0.047
0.00/0.19/0.81 0.42/0.44 0.00/0.56 0.047
0.00/0.20/0.80 0.42/0.44 0.00/0.56 0.047
0.38/0.52/0.09 0.06/0.46 0.38/0.93 0.008
mse
P bad information scenario (σP doubled) skewt (a1 ) ∆pt (a2 )
5.56 [1.72]*** 0.94 [0.57]***
5.31 [1.71]*** 1.00 [0.55]***
5.23 [1.70]*** 1.04 [0.54]***
21.5 [8.47]*** 3.25 [0.61]***
Meeting types Stats
0.00/0.15/0.85 0.49/0.52 0.00/0.48 0.041
0.00/0.16/0.84 0.48/0.51 0.00/0.49 0.041
0.00/0.17/0.83 0.47/0.50 0.00/0.50 0.041
0.39/0.52/0.09 0.06/0.46 0.39/0.93 0.008
mse
Note: All entries averaged over 101 random paths. The first two rows of each panel are ordered probit estimates of ∆pt+1 = a0 + a1 skewt + a2 ∆pt + ut+1 , with [standard errors]. ***/**/* indicates significance at the 1%/5%/10% level. Meeting types show the fraction of meetings of given types (1.0/2.0/2.1), coded as x.y, where x is the number of alternatives put to vote and y is the number of alternatives with a positive number of votes (excluding the proposal). Stats show average dissent, the fraction of meetings with no policy change (first row), the fraction of meetings with the proposal equal to the status-quo and the fraction of meetings with the chairman’s proposal accepted (second row). mse is the mean squared difference between the adopted and the optimal policy.
44
Table A3: Predictive power of voting record and decision-making statistics Autarkical model with reservations, N = 4 ρ = 0.90
ρ = 0.95
Baseline scenario (σu = σC = σP =
1 4
ρ1 = 1.95, ρ2 = −0.98
ρ = 0.99
for AR(1) and σu =
1 20
for AR(2))
skewt (a1 ) ∆pt (a2 )
5.27 [1.63]*** 0.78 [0.53]***
5.22 [1.63]*** 0.89 [0.53]***
5.39 [1.64]*** 0.92 [0.52]***
18.2 [4.53]*** 3.56 [0.64]***
Meeting types
0.00/0.00/0.00 0.16/0.76/0.08 0.43/0.43 0.00/0.57 0.027
0.00/0.00/0.00 0.17/0.75/0.08 0.43/0.43 0.00/0.57 0.027
0.00/0.00/0.00 0.17/0.75/0.08 0.42/0.42 0.00/0.58 0.027
0.27/0.00/0.00 0.51/0.22/0.00 0.15/0.46 0.27/0.82 0.008
Stats mse
High volatility scenario (σu doubled) skewt (a1 ) ∆pt (a2 )
2.77 [0.97]*** 0.15 [0.22]***
2.74 [0.96]*** 0.22 [0.22]***
2.75 [0.97]*** 0.27 [0.23]***
6.99 [2.74]*** 3.11 [0.37]***
Meeting types
0.00/0.00/0.00 0.18/0.63/0.18 0.37/0.28 0.00/0.72 0.041
0.00/0.00/0.00 0.18/0.63/0.18 0.37/0.28 0.00/0.72 0.043
0.00/0.00/0.00 0.18/0.63/0.19 0.37/0.28 0.00/0.72 0.041
0.00/0.00/0.00 0.66/0.34/0.00 0.22/0.26 0.00/0.74 0.013
Stats mse
Bad information scenario (σC , σP doubled) skewt (a1 ) ∆pt (a2 )
3.77 [1.51]*** 0.21 [0.49]***
3.75 [1.48]*** 0.30 [0.48]***
3.96 [1.51]*** 0.40 [0.46]***
19.3 [6.40]*** 3.26 [0.61]***
Meeting types
0.00/0.00/0.00 0.22/0.75/0.03 0.41/0.43 0.00/0.57 0.048
0.00/0.00/0.00 0.23/0.74/0.03 0.41/0.43 0.00/0.57 0.048
0.00/0.00/0.00 0.24/0.72/0.03 0.40/0.41 0.00/0.59 0.049
0.37/0.00/0.00 0.53/0.11/0.00 0.07/0.45 0.37/0.91 0.008
Stats mse
P bad information scenario (σP doubled) skewt (a1 ) ∆pt (a2 )
5.72 [1.78]*** 0.85 [0.55]***
5.65 [1.78]*** 0.88 [0.54]***
5.54 [1.74]*** 0.94 [0.52]***
19.7 [6.85]*** 3.28 [0.61]***
Meeting types
0.00/0.00/0.00 0.19/0.73/0.08 0.46/0.51 0.00/0.49 0.041
0.00/0.00/0.00 0.19/0.72/0.09 0.46/0.50 0.00/0.50 0.041
0.00/0.00/0.00 0.20/0.71/0.09 0.45/0.49 0.00/0.51 0.041
0.38/0.00/0.00 0.53/0.10/0.00 0.07/0.46 0.38/0.92 0.008
Stats mse
Note: All entries averaged over 101 random paths. The first two rows of each panel are ordered probit estimates of ∆pt+1 = a0 + a1 skewt + a2 ∆pt + ut+1 , with [standard errors]. ***/**/* indicates significance at the 1%/5%/10% level. Meeting types show the fraction of meetings of given types, 1.0/1.1/1.2 (first row), 2.0/2.1/2.2 (second row), with the residual 2.3 not shown, coded as x.y, where x is the number of alternatives put to vote and y is the number of alternatives with a positive number of votes (excluding the proposal). Stats show average dissent, the fraction of meetings with no policy change (first row), the fraction of meetings with the proposal equal to the status-quo and the fraction of meetings with the chairman’s proposal accepted (second row). mse is the mean squared difference between the adopted and the optimal policy.
45
Table A4: Predictive power of voting record and decision-making statistics Autarkical model with reservations, N = 6 ρ = 0.90
ρ = 0.95
Baseline scenario (σu = σC = σP =
1 4
ρ1 = 1.95, ρ2 = −0.98
ρ = 0.99
for AR(1) and σu =
1 20
for AR(2))
skewt (a1 ) ∆pt (a2 )
6.08 [1.65]*** 1.12 [0.55]***
6.26 [1.66]*** 1.22 [0.55]***
6.09 [1.66]*** 1.27 [0.55]***
19.6 [5.32]*** 3.53 [0.63]***
Meeting types
0.00/0.00/0.00 0.10/0.79/0.11 0.46/0.47 0.00/0.53 0.026
0.00/0.00/0.00 0.10/0.79/0.11 0.45/0.45 0.00/0.55 0.026
0.00/0.00/0.00 0.11/0.78/0.11 0.45/0.45 0.00/0.55 0.026
0.31/0.00/0.00 0.50/0.19/0.00 0.13/0.46 0.31/0.85 0.008
Stats mse
High volatility scenario (σu doubled) skewt (a1 ) ∆pt (a2 )
2.91 [0.95]*** 0.18 [0.23]***
2.89 [0.94]*** 0.26 [0.23]***
2.92 [0.95]*** 0.32 [0.23]***
8.64 [3.13]*** 3.16 [0.38]***
Meeting types
0.00/0.00/0.00 0.11/0.62/0.26 0.39/0.30 0.00/0.70 0.039
0.00/0.00/0.00 0.11/0.63/0.26 0.40/0.31 0.00/0.69 0.040
0.00/0.00/0.00 0.11/0.62/0.26 0.40/0.31 0.00/0.69 0.040
0.00/0.00/0.00 0.65/0.35/0.00 0.24/0.26 0.00/0.74 0.013
Stats mse
Bad information scenario (σC , σP doubled) skewt (a1 ) ∆pt (a2 )
3.82 [1.47]*** 0.33 [0.50]***
3.81 [1.45]*** 0.41 [0.49]***
3.81 [1.45]*** 0.46 [0.47]***
21.0 [7.62]*** 3.24 [0.61]***
Meeting types
0.00/0.00/0.00 0.15/0.83/0.02 0.44/0.46 0.00/0.54 0.047
0.00/0.00/0.00 0.17/0.82/0.02 0.43/0.44 0.00/0.56 0.047
0.00/0.00/0.00 0.17/0.81/0.02 0.43/0.44 0.00/0.56 0.047
0.38/0.00/0.00 0.52/0.09/0.00 0.06/0.46 0.38/0.93 0.008
Stats mse
P bad information scenario (σP doubled) skewt (a1 ) ∆pt (a2 )
5.72 [1.68]*** 0.94 [0.56]***
5.48 [1.68]*** 1.00 [0.55]***
5.39 [1.65]*** 1.04 [0.53]***
21.5 [8.47]*** 3.25 [0.61]***
Meeting types
0.00/0.00/0.00 0.13/0.79/0.08 0.49/0.52 0.00/0.48 0.041
0.00/0.00/0.00 0.14/0.78/0.08 0.48/0.51 0.00/0.49 0.041
0.00/0.00/0.00 0.14/0.77/0.08 0.48/0.50 0.00/0.50 0.041
0.39/0.00/0.00 0.52/0.09/0.00 0.06/0.46 0.39/0.93 0.008
Stats mse
Note: All entries averaged over 101 random paths. The first two rows of each panel are ordered probit estimates of ∆pt+1 = a0 + a1 skewt + a2 ∆pt + ut+1 , with [standard errors]. ***/**/* indicates significance at the 1%/5%/10% level. Meeting types show the fraction of meetings of given types, 1.0/1.1/1.2 (first row), 2.0/2.1/2.2 (second row), with the residual 2.3 not shown, coded as x.y, where x is the number of alternatives put to vote and y is the number of alternatives with a positive number of votes (excluding the proposal). Stats show average dissent, the fraction of meetings with no policy change (first row), the fraction of meetings with the proposal equal to the status-quo and the fraction of meetings with the chairman’s proposal accepted (second row). mse is the mean squared difference between the adopted and the optimal policy.
46
Table A5: Predictive power of voting record and decision-making statistics Consensual model without reservations, N = 4 ρ = 0.90
ρ = 0.95
Baseline scenario (σu = σC = σP =
1 4
ρ1 = 1.95, ρ2 = −0.98
ρ = 0.99
for AR(1) and σu =
1 20
for AR(2))
skewt (a1 ) ∆pt (a2 )
5.90 [3.24]*** -0.02 [0.46]***
5.93 [3.29]*** 0.07 [0.45]***
6.33 [3.25]*** 0.15 [0.45]***
15.0 [15.6]*** 2.64 [0.53]***
Meeting types Stats
0.38/0.39/0.23 0.07/0.41 0.38/0.97 0.033
0.38/0.39/0.23 0.07/0.41 0.38/0.97 0.032
0.37/0.40/0.23 0.07/0.41 0.37/0.97 0.033
0.45/0.52/0.02 0.01/0.45 0.45/1.00 0.008
mse
High volatility scenario (σu doubled) skewt (a1 ) ∆pt (a2 )
3.05 [2.50]*** -0.04 [0.21]***
2.59 [2.43]*** 0.01 [0.21]***
2.68 [2.48]*** 0.04 [0.21]***
12.5 [7.47]*** 2.96 [0.37]***
Meeting types Stats
0.20/0.58/0.22 0.08/0.24 0.20/0.96 0.050
0.19/0.58/0.22 0.08/0.24 0.19/0.96 0.049
0.19/0.58/0.22 0.08/0.24 0.19/0.95 0.049
0.26/0.69/0.05 0.01/0.26 0.26/1.00 0.014
mse
Bad information scenario (σC , σP doubled) skewt (a1 ) ∆pt (a2 )
6.79 [3.58]*** -0.03 [0.47]***
6.84 [3.59]*** 0.08 [0.46]***
7.01 [3.67]*** 0.11 [0.45]***
11.2 [10.9]*** 2.12 [0.42]***
Meeting types Stats
0.40/0.39/0.22 0.06/0.42 0.40/0.98 0.052
0.39/0.40/0.21 0.06/0.41 0.39/0.98 0.052
0.38/0.41/0.21 0.06/0.39 0.38/0.98 0.052
0.45/0.53/0.01 0.00/0.46 0.45/1.00 0.008
mse
P bad information scenario (σP doubled) skewt (a1 ) ∆pt (a2 )
8.86 [6.25]*** -0.28 [0.42]***
9.98 [6.24]*** -0.19 [0.41]***
8.77 [6.08]*** -0.14 [0.41]***
7.17 [74.3]*** 1.08 [0.24]***
Meeting types Stats
0.37/0.53/0.10 0.02/0.37 0.37/1.00 0.036
0.37/0.53/0.10 0.02/0.38 0.37/1.00 0.036
0.36/0.53/0.10 0.02/0.37 0.36/1.00 0.036
0.45/0.54/0.00 0.00/0.45 0.45/1.00 0.008
mse
Note: All entries averaged over 101 random paths. The first two rows of each panel are ordered probit estimates of ∆pt+1 = a0 + a1 skewt + a2 ∆pt + ut+1 , with [standard errors]. ***/**/* indicates significance at the 1%/5%/10% level. Meeting types show the fraction of meetings of given types (1.0/2.0/2.1), coded as x.y, where x is the number of alternatives put to vote and y is the number of alternatives with a positive number of votes (excluding the proposal). Stats show average dissent, the fraction of meetings with no policy change (first row), the fraction of meetings with the proposal equal to the status-quo and the fraction of meetings with the chairman’s proposal accepted (second row). mse is the mean squared difference between the adopted and the optimal policy.
47
Table A6: Predictive power of voting record and decision-making statistics Consensual model without reservations, N = 6 ρ = 0.90
ρ = 0.95
Baseline scenario (σu = σC = σP =
1 4
ρ1 = 1.95, ρ2 = −0.98
ρ = 0.99
for AR(1) and σu =
1 20
for AR(2))
skewt (a1 ) ∆pt (a2 )
6.57 [3.25]*** 0.06 [0.46]***
6.57 [3.30]*** 0.14 [0.46]***
6.70 [3.24]*** 0.22 [0.45]***
19.5 [17.4]*** 2.99 [0.58]***
Meeting types Stats
0.38/0.35/0.28 0.08/0.41 0.38/0.97 0.033
0.38/0.34/0.28 0.08/0.42 0.38/0.97 0.032
0.37/0.35/0.28 0.08/0.41 0.37/0.97 0.033
0.45/0.52/0.03 0.01/0.45 0.45/1.00 0.008
mse
High volatility scenario (σu doubled) skewt (a1 ) ∆pt (a2 )
3.25 [2.45]*** -0.03 [0.21]***
3.01 [2.43]*** 0.02 [0.21]***
2.91 [2.48]*** 0.05 [0.21]***
14.2 [7.79]*** 2.99 [0.37]***
Meeting types Stats
0.20/0.54/0.26 0.08/0.24 0.20/0.95 0.049
0.19/0.54/0.27 0.08/0.24 0.19/0.95 0.049
0.19/0.54/0.27 0.08/0.24 0.19/0.95 0.049
0.26/0.68/0.07 0.02/0.26 0.26/1.00 0.014
mse
Bad information scenario (σC , σP doubled) skewt (a1 ) ∆pt (a2 )
7.53 [3.63]*** 0.05 [0.48]***
7.60 [3.67]*** 0.15 [0.47]***
8.24 [3.66]*** 0.21 [0.45]***
15.7 [14.4]*** 2.41 [0.48]***
Meeting types Stats
0.40/0.33/0.27 0.07/0.42 0.40/0.98 0.052
0.39/0.34/0.27 0.07/0.41 0.39/0.98 0.052
0.38/0.36/0.26 0.06/0.39 0.38/0.98 0.052
0.45/0.53/0.02 0.00/0.46 0.45/1.00 0.008
mse
P bad information scenario (σP doubled) skewt (a1 ) ∆pt (a2 )
11.2 [6.57]*** -0.22 [0.43]***
12.4 [6.61]*** -0.13 [0.42]***
11.6 [6.45]*** -0.07 [0.42]***
13.6 [108.]*** 1.66 [0.33]***
Meeting types Stats
0.37/0.49/0.14 0.03/0.37 0.37/1.00 0.036
0.37/0.49/0.14 0.03/0.37 0.37/1.00 0.036
0.36/0.50/0.14 0.03/0.36 0.36/1.00 0.036
0.45/0.54/0.01 0.00/0.45 0.45/1.00 0.008
mse
Note: All entries averaged over 101 random paths. The first two rows of each panel are ordered probit estimates of ∆pt+1 = a0 + a1 skewt + a2 ∆pt + ut+1 , with [standard errors]. ***/**/* indicates significance at the 1%/5%/10% level. Meeting types show the fraction of meetings of given types (1.0/2.0/2.1), coded as x.y, where x is the number of alternatives put to vote and y is the number of alternatives with a positive number of votes (excluding the proposal). Stats show average dissent, the fraction of meetings with no policy change (first row), the fraction of meetings with the proposal equal to the status-quo and the fraction of meetings with the chairman’s proposal accepted (second row). mse is the mean squared difference between the adopted and the optimal policy.
48
Table A7: Predictive power of voting record and decision-making statistics Consensual model with reservations, N = 4 ρ = 0.90
ρ = 0.95
Baseline scenario (σu = σC = σP =
1 4
ρ1 = 1.95, ρ2 = −0.98
ρ = 0.99
for AR(1) and σu =
1 20
for AR(2))
skewt (a1 ) ∆pt (a2 )
7.92 [2.98]*** 0.05 [0.45]***
7.92 [3.01]*** 0.13 [0.44]***
8.41 [3.00]*** 0.21 [0.44]***
15.0 [15.6]*** 2.64 [0.53]***
Meeting types
0.34/0.04/0.00 0.37/0.25/0.00 0.09/0.41 0.38/0.97 0.033
0.34/0.04/0.00 0.37/0.24/0.00 0.09/0.41 0.38/0.97 0.032
0.33/0.04/0.00 0.38/0.25/0.00 0.09/0.41 0.37/0.97 0.033
0.45/0.00/0.00 0.52/0.02/0.00 0.01/0.45 0.45/1.00 0.008
Stats mse
High volatility scenario (σu doubled) skewt (a1 ) ∆pt (a2 )
4.67 [1.95]*** -0.08 [0.20]***
4.64 [1.93]*** -0.02 [0.20]***
4.68 [1.94]*** 0.01 [0.21]***
12.5 [7.47]*** 2.96 [0.37]***
Meeting types
0.13/0.07/0.00 0.46/0.33/0.01 0.13/0.24 0.20/0.96 0.050
0.13/0.06/0.00 0.47/0.33/0.01 0.13/0.24 0.19/0.96 0.049
0.12/0.07/0.00 0.47/0.33/0.01 0.13/0.24 0.19/0.95 0.049
0.26/0.00/0.00 0.69/0.05/0.00 0.01/0.26 0.26/1.00 0.014
Stats mse
Bad information scenario (σC , σP doubled) skewt (a1 ) ∆pt (a2 )
8.05 [3.44]*** 0.02 [0.47]***
8.20 [3.44]*** 0.13 [0.45]***
8.05 [3.51]*** 0.14 [0.44]***
11.2 [10.9]*** 2.12 [0.42]***
Meeting types
0.38/0.02/0.00 0.38/0.22/0.00 0.07/0.42 0.40/0.98 0.052
0.37/0.02/0.00 0.39/0.22/0.00 0.07/0.41 0.39/0.98 0.052
0.36/0.02/0.00 0.40/0.22/0.00 0.07/0.39 0.38/0.98 0.052
0.45/0.00/0.00 0.53/0.01/0.00 0.00/0.46 0.45/1.00 0.008
Stats mse
P bad information scenario (σP doubled) skewt (a1 ) ∆pt (a2 )
8.98 [6.21]*** -0.28 [0.42]***
10.2 [6.21]*** -0.19 [0.41]***
8.87 [6.06]*** -0.14 [0.41]***
7.17 [74.3]*** 1.08 [0.24]***
Meeting types
0.37/0.00/0.00 0.53/0.10/0.00 0.02/0.37 0.37/1.00 0.036
0.37/0.00/0.00 0.53/0.10/0.00 0.02/0.38 0.37/1.00 0.036
0.36/0.00/0.00 0.53/0.10/0.00 0.02/0.37 0.36/1.00 0.036
0.45/0.00/0.00 0.54/0.00/0.00 0.00/0.45 0.45/1.00 0.008
Stats mse
Note: All entries averaged over 101 random paths. The first two rows of each panel are ordered probit estimates of ∆pt+1 = a0 + a1 skewt + a2 ∆pt + ut+1 , with [standard errors]. ***/**/* indicates significance at the 1%/5%/10% level. Meeting types show the fraction of meetings of given types, 1.0/1.1/1.2 (first row), 2.0/2.1/2.2 (second row), with the residual 2.3 not shown, coded as x.y, where x is the number of alternatives put to vote and y is the number of alternatives with a positive number of votes (excluding the proposal). Stats show average dissent, the fraction of meetings with no policy change (first row), the fraction of meetings with the proposal equal to the status-quo and the fraction of meetings with the chairman’s proposal accepted (second row). mse is the mean squared difference between the adopted and the optimal policy.
49
Table A8: Predictive power of voting record and decision-making statistics Consensual model with reservations, N = 6 ρ = 0.90
ρ = 0.95
Baseline scenario (σu = σC = σP =
1 4
ρ1 = 1.95, ρ2 = −0.98
ρ = 0.99
for AR(1) and σu =
1 20
for AR(2))
skewt (a1 ) ∆pt (a2 )
8.92 [3.03]*** 0.14 [0.45]***
8.97 [3.07]*** 0.22 [0.45]***
9.10 [3.02]*** 0.31 [0.44]***
19.5 [17.4]*** 2.99 [0.58]***
Meeting types
0.32/0.06/0.00 0.32/0.30/0.00 0.09/0.41 0.38/0.97 0.033
0.32/0.06/0.00 0.32/0.30/0.00 0.09/0.42 0.38/0.97 0.032
0.32/0.06/0.00 0.32/0.30/0.00 0.09/0.41 0.37/0.97 0.033
0.45/0.00/0.00 0.52/0.03/0.00 0.01/0.45 0.45/1.00 0.008
Stats mse
High volatility scenario (σu doubled) skewt (a1 ) ∆pt (a2 )
4.94 [1.93]*** -0.08 [0.20]***
5.07 [1.93]*** -0.02 [0.20]***
5.08 [1.96]*** 0.01 [0.21]***
14.2 [7.79]*** 2.99 [0.37]***
Meeting types
0.11/0.08/0.00 0.39/0.40/0.02 0.14/0.24 0.20/0.95 0.049
0.11/0.08/0.00 0.39/0.40/0.02 0.14/0.24 0.19/0.95 0.049
0.11/0.08/0.00 0.39/0.40/0.02 0.14/0.24 0.19/0.95 0.049
0.26/0.00/0.00 0.68/0.07/0.00 0.02/0.26 0.26/1.00 0.014
Stats mse
Bad information scenario (σC , σP doubled) skewt (a1 ) ∆pt (a2 )
9.17 [3.51]*** 0.13 [0.47]***
9.26 [3.54]*** 0.21 [0.46]***
9.63 [3.54]*** 0.26 [0.45]***
15.7 [14.4]*** 2.41 [0.48]***
Meeting types
0.36/0.03/0.00 0.32/0.28/0.00 0.07/0.42 0.40/0.98 0.052
0.36/0.03/0.00 0.33/0.28/0.00 0.07/0.41 0.39/0.98 0.052
0.34/0.03/0.00 0.35/0.27/0.00 0.07/0.39 0.38/0.98 0.052
0.45/0.00/0.00 0.53/0.02/0.00 0.00/0.46 0.45/1.00 0.008
Stats mse
P bad information scenario (σP doubled) skewt (a1 ) ∆pt (a2 )
11.3 [6.55]*** -0.21 [0.43]***
12.6 [6.59]*** -0.13 [0.42]***
11.7 [6.43]*** -0.07 [0.42]***
13.6 [108.]*** 1.66 [0.33]***
Meeting types
0.37/0.00/0.00 0.49/0.14/0.00 0.03/0.37 0.37/1.00 0.036
0.37/0.00/0.00 0.49/0.14/0.00 0.03/0.37 0.37/1.00 0.036
0.36/0.00/0.00 0.50/0.14/0.00 0.03/0.36 0.36/1.00 0.036
0.45/0.00/0.00 0.54/0.01/0.00 0.00/0.45 0.45/1.00 0.008
Stats mse
Note: All entries averaged over 101 random paths. The first two rows of each panel are ordered probit estimates of ∆pt+1 = a0 + a1 skewt + a2 ∆pt + ut+1 , with [standard errors]. ***/**/* indicates significance at the 1%/5%/10% level. Meeting types show the fraction of meetings of given types, 1.0/1.1/1.2 (first row), 2.0/2.1/2.2 (second row), with the residual 2.3 not shown, coded as x.y, where x is the number of alternatives put to vote and y is the number of alternatives with a positive number of votes (excluding the proposal). Stats show average dissent, the fraction of meetings with no policy change (first row), the fraction of meetings with the proposal equal to the status-quo and the fraction of meetings with the chairman’s proposal accepted (second row). mse is the mean squared difference between the adopted and the optimal policy.
50
Table A9: Predictive power of voting record and decision-making statistics Supermajoritarian model without reservations, N = 4 ρ = 0.90
ρ = 0.95
Baseline scenario (σu = σC = σP =
1 4
ρ1 = 1.95, ρ2 = −0.98
ρ = 0.99
for AR(1) and σu =
1 20
for AR(2))
skewt (a1 ) ∆pt (a2 )
6.22 [6.56]*** 1.62 [0.77]***
5.25 [6.40]*** 1.63 [0.74]***
7.06 [6.31]*** 1.86 [0.73]***
17.3 [45.6]*** 2.72 [0.52]***
Meeting types Stats
0.59/0.23/0.17 0.04/0.59 0.59/1.00 0.034
0.59/0.23/0.17 0.03/0.59 0.59/1.00 0.033
0.58/0.25/0.17 0.03/0.58 0.58/1.00 0.033
0.46/0.52/0.02 0.00/0.46 0.46/1.00 0.008
mse
High volatility scenario (σu doubled) skewt (a1 ) ∆pt (a2 )
0.91 [3.65]*** 0.54 [0.29]***
0.69 [3.65]*** 0.60 [0.29]***
0.77 [3.60]*** 0.66 [0.29]***
15.0 [11.7]*** 3.13 [0.37]***
Meeting types Stats
0.37/0.41/0.22 0.04/0.37 0.37/1.00 0.045
0.37/0.41/0.22 0.04/0.37 0.37/1.00 0.044
0.37/0.41/0.22 0.04/0.37 0.37/1.00 0.045
0.28/0.68/0.04 0.01/0.28 0.28/1.00 0.013
mse
Bad information scenario (σC , σP doubled) skewt (a1 ) ∆pt (a2 )
6.50 [6.53]*** 1.00 [0.67]***
6.05 [6.47]*** 1.04 [0.64]***
5.67 [6.19]*** 1.06 [0.61]***
11.4 [14.1]*** 1.64 [0.32]***
Meeting types Stats
0.57/0.28/0.15 0.03/0.57 0.57/1.00 0.052
0.56/0.30/0.15 0.03/0.56 0.56/1.00 0.052
0.53/0.31/0.16 0.03/0.53 0.53/1.00 0.052
0.46/0.53/0.01 0.00/0.46 0.46/1.00 0.008
mse
P bad information scenario (σP doubled) skewt (a1 ) ∆pt (a2 )
7.36 [6.57]*** 1.22 [0.70]***
6.53 [6.43]*** 1.25 [0.67]***
6.46 [6.16]*** 1.33 [0.64]***
13.4 [74.6]*** 2.06 [0.42]***
Meeting types Stats
0.58/0.26/0.16 0.03/0.58 0.58/1.00 0.049
0.57/0.27/0.16 0.03/0.57 0.57/1.00 0.049
0.55/0.28/0.17 0.03/0.55 0.55/1.00 0.049
0.46/0.53/0.01 0.00/0.46 0.46/1.00 0.008
mse
Note: All entries averaged over 101 random paths. The first two rows of each panel are ordered probit estimates of ∆pt+1 = a0 + a1 skewt + a2 ∆pt + ut+1 , with [standard errors]. ***/**/* indicates significance at the 1%/5%/10% level. Meeting types show the fraction of meetings of given types (1.0/2.0/2.1), coded as x.y, where x is the number of alternatives put to vote and y is the number of alternatives with a positive number of votes (excluding the proposal). Stats show average dissent, the fraction of meetings with no policy change (first row), the fraction of meetings with the proposal equal to the status-quo and the fraction of meetings with the chairman’s proposal accepted (second row). mse is the mean squared difference between the adopted and the optimal policy.
51
Table A10: Predictive power of voting record and decision-making statistics Supermajoritarian model without reservations, N = 6 ρ = 0.90
ρ = 0.95
Baseline scenario (σu = σC = σP =
1 4
ρ1 = 1.95, ρ2 = −0.98
ρ = 0.99
for AR(1) and σu =
1 20
for AR(2))
skewt (a1 ) ∆pt (a2 )
7.65 [5.40]*** 1.98 [0.85]***
7.41 [5.30]*** 2.05 [0.81]***
8.16 [5.21]*** 2.19 [0.80]***
16.6 [14.4]*** 2.93 [0.57]***
Meeting types Stats
0.57/0.17/0.26 0.05/0.57 0.57/1.00 0.030
0.57/0.17/0.26 0.05/0.57 0.57/1.00 0.030
0.56/0.18/0.26 0.06/0.56 0.56/1.00 0.029
0.46/0.51/0.03 0.01/0.46 0.46/1.00 0.008
mse
High volatility scenario (σu doubled) skewt (a1 ) ∆pt (a2 )
1.86 [3.04]*** 0.54 [0.30]***
1.69 [2.99]*** 0.63 [0.30]***
1.39 [2.92]*** 0.65 [0.31]***
13.3 [9.25]*** 3.19 [0.38]***
Meeting types Stats
0.34/0.33/0.34 0.07/0.34 0.34/1.00 0.036
0.33/0.33/0.34 0.07/0.33 0.33/1.00 0.036
0.33/0.32/0.35 0.07/0.33 0.33/1.00 0.036
0.27/0.66/0.06 0.01/0.27 0.27/1.00 0.013
mse
Bad information scenario (σC , σP doubled) skewt (a1 ) ∆pt (a2 )
6.70 [5.23]*** 1.11 [0.69]***
7.44 [5.09]*** 1.16 [0.65]***
6.22 [5.05]*** 1.15 [0.63]***
14.0 [109.]*** 2.13 [0.42]***
Meeting types Stats
0.54/0.23/0.23 0.05/0.54 0.54/1.00 0.050
0.52/0.25/0.23 0.05/0.52 0.52/1.00 0.051
0.51/0.26/0.23 0.05/0.51 0.51/1.00 0.050
0.46/0.53/0.01 0.00/0.46 0.46/1.00 0.008
mse
P bad information scenario (σP doubled) skewt (a1 ) ∆pt (a2 )
7.73 [5.26]*** 1.28 [0.72]***
7.86 [5.08]*** 1.36 [0.68]***
6.69 [5.04]*** 1.38 [0.66]***
14.8 [73.0]*** 2.43 [0.49]***
Meeting types Stats
0.55/0.22/0.23 0.05/0.55 0.55/1.00 0.048
0.54/0.23/0.23 0.05/0.54 0.54/1.00 0.048
0.52/0.24/0.24 0.05/0.52 0.52/1.00 0.047
0.46/0.52/0.02 0.00/0.46 0.46/1.00 0.008
mse
Note: All entries averaged over 101 random paths. The first two rows of each panel are ordered probit estimates of ∆pt+1 = a0 + a1 skewt + a2 ∆pt + ut+1 , with [standard errors]. ***/**/* indicates significance at the 1%/5%/10% level. Meeting types show the fraction of meetings of given types (1.0/2.0/2.1), coded as x.y, where x is the number of alternatives put to vote and y is the number of alternatives with a positive number of votes (excluding the proposal). Stats show average dissent, the fraction of meetings with no policy change (first row), the fraction of meetings with the proposal equal to the status-quo and the fraction of meetings with the chairman’s proposal accepted (second row). mse is the mean squared difference between the adopted and the optimal policy.
52
Table A11: Predictive power of voting record and decision-making statistics Supermajoritarian model with reservations, N = 4 ρ = 0.90
ρ = 0.95
Baseline scenario (σu = σC = σP =
1 4
ρ1 = 1.95, ρ2 = −0.98
ρ = 0.99
for AR(1) and σu =
1 20
for AR(2))
skewt (a1 ) ∆pt (a2 )
10.8 [3.12]*** 1.71 [0.65]***
10.7 [3.09]*** 1.80 [0.63]***
11.0 [3.06]*** 1.92 [0.62]***
17.3 [45.6]*** 2.72 [0.52]***
Meeting types
0.37/0.22/0.00 0.18/0.22/0.01 0.10/0.59 0.59/1.00 0.034
0.38/0.21/0.00 0.18/0.21/0.01 0.10/0.59 0.59/1.00 0.033
0.36/0.21/0.00 0.19/0.22/0.01 0.10/0.58 0.58/1.00 0.033
0.46/0.00/0.00 0.52/0.02/0.00 0.00/0.46 0.46/1.00 0.008
Stats mse
High volatility scenario (σu doubled) skewt (a1 ) ∆pt (a2 )
3.96 [1.80]*** 0.41 [0.27]***
3.88 [1.82]*** 0.49 [0.27]***
4.15 [1.81]*** 0.54 [0.27]***
15.3 [11.9]*** 3.16 [0.38]***
Meeting types
0.10/0.24/0.03 0.17/0.36/0.09 0.21/0.37 0.37/1.00 0.045
0.10/0.24/0.03 0.17/0.36/0.09 0.21/0.37 0.37/1.00 0.044
0.10/0.24/0.03 0.18/0.36/0.10 0.22/0.37 0.37/1.00 0.045
0.28/0.00/0.00 0.68/0.04/0.00 0.01/0.28 0.28/1.00 0.013
Stats mse
Bad information scenario (σC , σP doubled) skewt (a1 ) ∆pt (a2 )
9.78 [3.75]*** 1.07 [0.59]***
9.58 [3.72]*** 1.10 [0.57]***
9.05 [3.60]*** 1.12 [0.55]***
11.4 [14.1]*** 1.64 [0.32]***
Meeting types
0.44/0.13/0.00 0.25/0.18/0.00 0.07/0.57 0.57/1.00 0.052
0.43/0.13/0.00 0.26/0.18/0.00 0.07/0.56 0.56/1.00 0.052
0.41/0.13/0.00 0.27/0.19/0.00 0.07/0.53 0.53/1.00 0.052
0.46/0.00/0.00 0.53/0.01/0.00 0.00/0.46 0.46/1.00 0.008
Stats mse
P bad information scenario (σP doubled) skewt (a1 ) ∆pt (a2 )
8.86 [3.74]*** 1.21 [0.62]***
8.46 [3.70]*** 1.26 [0.59]***
8.04 [3.56]*** 1.33 [0.57]***
13.4 [74.6]*** 2.06 [0.42]***
Meeting types
0.45/0.13/0.00 0.23/0.19/0.00 0.07/0.58 0.58/1.00 0.049
0.44/0.13/0.00 0.25/0.18/0.00 0.07/0.57 0.57/1.00 0.049
0.42/0.13/0.00 0.25/0.20/0.00 0.07/0.55 0.55/1.00 0.049
0.46/0.00/0.00 0.53/0.01/0.00 0.00/0.46 0.46/1.00 0.008
Stats mse
Note: All entries averaged over 101 random paths. The first two rows of each panel are ordered probit estimates of ∆pt+1 = a0 + a1 skewt + a2 ∆pt + ut+1 , with [standard errors]. ***/**/* indicates significance at the 1%/5%/10% level. Meeting types show the fraction of meetings of given types, 1.0/1.1/1.2 (first row), 2.0/2.1/2.2 (second row), with the residual 2.3 not shown, coded as x.y, where x is the number of alternatives put to vote and y is the number of alternatives with a positive number of votes (excluding the proposal). Stats show average dissent, the fraction of meetings with no policy change (first row), the fraction of meetings with the proposal equal to the status-quo and the fraction of meetings with the chairman’s proposal accepted (second row). mse is the mean squared difference between the adopted and the optimal policy.
53
Table A12: Predictive power of voting record and decision-making statistics Supermajoritarian model with reservations, N = 6 ρ = 0.90
ρ = 0.95
Baseline scenario (σu = σC = σP =
1 4
ρ1 = 1.95, ρ2 = −0.98
ρ = 0.99
for AR(1) and σu =
1 20
for AR(2))
skewt (a1 ) ∆pt (a2 )
11.4 [3.19]*** 2.14 [0.68]***
11.7 [3.20]*** 2.31 [0.67]***
11.5 [3.16]*** 2.32 [0.65]***
16.6 [14.4]*** 2.93 [0.57]***
Meeting types
0.31/0.26/0.00 0.12/0.28/0.03 0.12/0.57 0.57/1.00 0.030
0.31/0.26/0.00 0.12/0.29/0.02 0.12/0.57 0.57/1.00 0.030
0.30/0.25/0.01 0.12/0.29/0.03 0.12/0.56 0.56/1.00 0.029
0.46/0.00/0.00 0.51/0.03/0.00 0.01/0.46 0.46/1.00 0.008
Stats mse
High volatility scenario (σu doubled) skewt (a1 ) ∆pt (a2 )
3.91 [1.87]*** 0.42 [0.26]***
3.79 [1.86]*** 0.52 [0.26]***
3.74 [1.85]*** 0.56 [0.26]***
13.5 [9.21]*** 3.19 [0.38]***
Meeting types
0.06/0.23/0.05 0.10/0.38/0.18 0.23/0.34 0.34/1.00 0.036
0.05/0.23/0.05 0.10/0.38/0.18 0.23/0.33 0.33/1.00 0.036
0.05/0.22/0.05 0.10/0.38/0.18 0.23/0.33 0.33/1.00 0.036
0.27/0.00/0.00 0.66/0.06/0.00 0.01/0.27 0.27/1.00 0.013
Stats mse
Bad information scenario (σC , σP doubled) skewt (a1 ) ∆pt (a2 )
9.66 [3.70]*** 1.23 [0.61]***
10.3 [3.64]*** 1.26 [0.58]***
9.10 [3.60]*** 1.25 [0.56]***
14.0 [109.]*** 2.13 [0.42]***
Meeting types
0.39/0.15/0.00 0.19/0.26/0.01 0.08/0.54 0.54/1.00 0.050
0.37/0.15/0.00 0.20/0.27/0.01 0.08/0.52 0.52/1.00 0.051
0.37/0.14/0.00 0.21/0.27/0.01 0.08/0.51 0.51/1.00 0.050
0.46/0.00/0.00 0.53/0.01/0.00 0.00/0.46 0.46/1.00 0.008
Stats mse
P bad information scenario (σP doubled) skewt (a1 ) ∆pt (a2 )
9.53 [3.72]*** 1.33 [0.63]***
9.57 [3.64]*** 1.40 [0.60]***
8.40 [3.60]*** 1.42 [0.59]***
14.8 [73.0]*** 2.43 [0.49]***
Meeting types
0.40/0.15/0.00 0.18/0.26/0.01 0.08/0.55 0.55/1.00 0.048
0.38/0.15/0.00 0.19/0.27/0.01 0.08/0.54 0.54/1.00 0.048
0.37/0.15/0.00 0.19/0.28/0.01 0.08/0.52 0.52/1.00 0.047
0.46/0.00/0.00 0.52/0.02/0.00 0.00/0.46 0.46/1.00 0.008
Stats mse
Note: All entries averaged over 101 random paths. The first two rows of each panel are ordered probit estimates of ∆pt+1 = a0 + a1 skewt + a2 ∆pt + ut+1 , with [standard errors]. ***/**/* indicates significance at the 1%/5%/10% level. Meeting types show the fraction of meetings of given types, 1.0/1.1/1.2 (first row), 2.0/2.1/2.2 (second row), with the residual 2.3 not shown, coded as x.y, where x is the number of alternatives put to vote and y is the number of alternatives with a positive number of votes (excluding the proposal). Stats show average dissent, the fraction of meetings with no policy change (first row), the fraction of meetings with the proposal equal to the status-quo and the fraction of meetings with the chairman’s proposal accepted (second row). mse is the mean squared difference between the adopted and the optimal policy.
54
Table A13: Predictive power of voting record and decision-making statistics Full information independence without reservations, N = 4 ρ = 0.90
ρ = 0.95
Baseline scenario (σu = σC = σP =
1 4
ρ1 = 1.95, ρ2 = −0.98
ρ = 0.99
for AR(1) and σu =
1 20
for AR(2))
skewt (a1 ) ∆pt (a2 )
5.46 [2.14]*** 0.94 [0.59]***
5.19 [2.13]*** 0.98 [0.58]***
5.51 [2.10]*** 1.07 [0.56]***
14.9 [92.2]*** 3.02 [0.58]***
Meeting types Stats
0.39/0.19/0.42 0.18/0.54 0.39/0.84 0.032
0.39/0.20/0.42 0.18/0.55 0.39/0.84 0.032
0.38/0.20/0.42 0.18/0.53 0.38/0.84 0.032
0.45/0.51/0.04 0.01/0.46 0.45/0.99 0.008
mse
High volatility scenario (σu doubled) skewt (a1 ) ∆pt (a2 )
2.33 [1.23]*** 0.23 [0.25]***
2.24 [1.23]*** 0.29 [0.25]***
2.38 [1.22]*** 0.34 [0.25]***
9.24 [5.46]*** 3.02 [0.37]***
Meeting types Stats
0.20/0.34/0.46 0.19/0.35 0.20/0.85 0.048
0.19/0.34/0.47 0.19/0.35 0.19/0.85 0.047
0.19/0.34/0.47 0.19/0.35 0.19/0.85 0.047
0.25/0.67/0.08 0.03/0.28 0.25/0.98 0.013
mse
Bad information scenario (σC , σP doubled) skewt (a1 ) ∆pt (a2 )
5.53 [2.35]*** 0.49 [0.54]***
5.31 [2.38]*** 0.58 [0.52]***
4.99 [2.31]*** 0.57 [0.51]***
13.7 [74.0]*** 2.56 [0.50]***
Meeting types Stats
0.40/0.25/0.35 0.15/0.52 0.40/0.88 0.052
0.39/0.27/0.34 0.14/0.51 0.39/0.88 0.052
0.38/0.28/0.34 0.14/0.49 0.38/0.89 0.052
0.45/0.53/0.02 0.01/0.46 0.45/0.99 0.008
mse
P bad information scenario (σP doubled) skewt (a1 ) ∆pt (a2 )
7.01 [1.96]*** 0.93 [0.57]***
6.82 [1.99]*** 0.93 [0.55]***
6.62 [1.91]*** 0.94 [0.53]***
16.4 [177.]*** 2.86 [0.55]***
Meeting types Stats
0.36/0.19/0.45 0.22/0.58 0.36/0.79 0.045
0.36/0.20/0.44 0.22/0.57 0.36/0.79 0.044
0.36/0.20/0.44 0.21/0.56 0.36/0.79 0.045
0.45/0.53/0.02 0.01/0.46 0.45/0.99 0.008
mse
Note: All entries averaged over 101 random paths. The first two rows of each panel are ordered probit estimates of ∆pt+1 = a0 + a1 skewt + a2 ∆pt + ut+1 , with [standard errors]. ***/**/* indicates significance at the 1%/5%/10% level. Meeting types show the fraction of meetings of given types (1.0/2.0/2.1), coded as x.y, where x is the number of alternatives put to vote and y is the number of alternatives with a positive number of votes (excluding the proposal). Stats show average dissent, the fraction of meetings with no policy change (first row), the fraction of meetings with the proposal equal to the status-quo and the fraction of meetings with the chairman’s proposal accepted (second row). mse is the mean squared difference between the adopted and the optimal policy.
55
Table A14: Predictive power of voting record and decision-making statistics Full information independence without reservations, N = 6 ρ = 0.90
ρ = 0.95
Baseline scenario (σu = σC = σP =
1 4
ρ1 = 1.95, ρ2 = −0.98
ρ = 0.99
for AR(1) and σu =
1 20
for AR(2))
skewt (a1 ) ∆pt (a2 )
6.11 [2.14]*** 1.19 [0.61]***
5.90 [2.12]*** 1.25 [0.60]***
5.88 [2.10]*** 1.25 [0.59]***
14.9 [9.68]*** 3.12 [0.59]***
Meeting types Stats
0.38/0.15/0.47 0.20/0.56 0.38/0.83 0.032
0.39/0.14/0.47 0.20/0.56 0.39/0.83 0.032
0.38/0.15/0.47 0.20/0.55 0.38/0.83 0.032
0.45/0.51/0.04 0.01/0.46 0.45/0.99 0.008
mse
High volatility scenario (σu doubled) skewt (a1 ) ∆pt (a2 )
2.47 [1.22]*** 0.26 [0.25]***
2.53 [1.22]*** 0.34 [0.26]***
2.48 [1.22]*** 0.38 [0.26]***
9.94 [5.48]*** 3.03 [0.37]***
Meeting types Stats
0.20/0.28/0.52 0.20/0.36 0.20/0.84 0.046
0.19/0.28/0.53 0.20/0.35 0.19/0.84 0.046
0.19/0.27/0.53 0.21/0.35 0.19/0.84 0.046
0.25/0.65/0.09 0.03/0.28 0.25/0.98 0.013
mse
Bad information scenario (σC , σP doubled) skewt (a1 ) ∆pt (a2 )
5.46 [2.35]*** 0.56 [0.56]***
5.33 [2.36]*** 0.67 [0.54]***
5.11 [2.31]*** 0.66 [0.52]***
16.6 [24.1]*** 2.66 [0.52]***
Meeting types Stats
0.40/0.21/0.39 0.16/0.53 0.40/0.87 0.052
0.39/0.22/0.39 0.15/0.52 0.39/0.87 0.052
0.38/0.23/0.39 0.15/0.50 0.38/0.88 0.052
0.45/0.53/0.02 0.01/0.46 0.45/0.99 0.008
mse
P bad information scenario (σP doubled) skewt (a1 ) ∆pt (a2 )
7.46 [1.98]*** 1.03 [0.58]***
7.25 [2.00]*** 1.10 [0.56]***
7.17 [1.94]*** 1.11 [0.54]***
19.9 [25.5]*** 2.95 [0.56]***
Meeting types Stats
0.36/0.15/0.49 0.23/0.59 0.36/0.77 0.047
0.36/0.16/0.48 0.23/0.58 0.36/0.78 0.045
0.36/0.17/0.48 0.23/0.57 0.36/0.78 0.046
0.45/0.53/0.03 0.01/0.46 0.45/0.99 0.008
mse
Note: All entries averaged over 101 random paths. The first two rows of each panel are ordered probit estimates of ∆pt+1 = a0 + a1 skewt + a2 ∆pt + ut+1 , with [standard errors]. ***/**/* indicates significance at the 1%/5%/10% level. Meeting types show the fraction of meetings of given types (1.0/2.0/2.1), coded as x.y, where x is the number of alternatives put to vote and y is the number of alternatives with a positive number of votes (excluding the proposal). Stats show average dissent, the fraction of meetings with no policy change (first row), the fraction of meetings with the proposal equal to the status-quo and the fraction of meetings with the chairman’s proposal accepted (second row). mse is the mean squared difference between the adopted and the optimal policy.
56
Table A15: Predictive power of voting record and decision-making statistics Full information independence with reservations, N = 4 ρ = 0.90
ρ = 0.95
Baseline scenario (σu = σC = σP =
1 4
ρ1 = 1.95, ρ2 = −0.98
ρ = 0.99
for AR(1) and σu =
1 20
for AR(2))
skewt (a1 ) ∆pt (a2 )
7.41 [1.82]*** 1.12 [0.55]***
7.07 [1.83]*** 1.16 [0.55]***
7.35 [1.79]*** 1.25 [0.53]***
14.9 [92.2]*** 3.02 [0.58]***
Meeting types
0.23/0.16/0.00 0.16/0.44/0.02 0.24/0.54 0.39/0.84 0.032
0.23/0.15/0.00 0.16/0.44/0.02 0.24/0.55 0.39/0.84 0.032
0.22/0.16/0.00 0.16/0.44/0.02 0.24/0.53 0.38/0.84 0.032
0.45/0.00/0.00 0.51/0.04/0.00 0.01/0.46 0.45/0.99 0.008
Stats mse
High volatility scenario (σu doubled) skewt (a1 ) ∆pt (a2 )
3.10 [1.00]*** 0.19 [0.23]***
3.03 [1.00]*** 0.26 [0.23]***
3.21 [1.01]*** 0.32 [0.23]***
9.33 [5.45]*** 3.02 [0.37]***
Meeting types
0.04/0.14/0.01 0.17/0.51/0.12 0.32/0.35 0.20/0.85 0.048
0.04/0.14/0.01 0.17/0.51/0.12 0.32/0.35 0.19/0.85 0.047
0.04/0.14/0.01 0.17/0.52/0.12 0.32/0.35 0.19/0.85 0.047
0.25/0.00/0.00 0.67/0.08/0.00 0.03/0.28 0.25/0.98 0.013
Stats mse
Bad information scenario (σC , σP doubled) skewt (a1 ) ∆pt (a2 )
6.79 [2.06]*** 0.58 [0.52]***
6.60 [2.06]*** 0.65 [0.50]***
6.11 [2.00]*** 0.62 [0.49]***
13.7 [74.0]*** 2.56 [0.50]***
Meeting types
0.30/0.10/0.00 0.23/0.37/0.00 0.18/0.52 0.40/0.88 0.052
0.29/0.10/0.00 0.24/0.37/0.00 0.18/0.51 0.39/0.88 0.052
0.28/0.10/0.00 0.25/0.37/0.00 0.18/0.49 0.38/0.89 0.052
0.45/0.00/0.00 0.53/0.02/0.00 0.01/0.46 0.45/0.99 0.008
Stats mse
P bad information scenario (σP doubled) skewt (a1 ) ∆pt (a2 )
6.91 [1.77]*** 0.87 [0.55]***
6.65 [1.78]*** 0.86 [0.53]***
6.48 [1.72]*** 0.89 [0.51]***
16.4 [177.]*** 2.86 [0.55]***
Meeting types
0.26/0.10/0.00 0.17/0.46/0.01 0.26/0.58 0.36/0.79 0.045
0.25/0.10/0.00 0.18/0.45/0.01 0.25/0.57 0.36/0.79 0.044
0.25/0.11/0.00 0.18/0.45/0.01 0.25/0.56 0.36/0.79 0.045
0.45/0.00/0.00 0.53/0.02/0.00 0.01/0.46 0.45/0.99 0.008
Stats mse
Note: All entries averaged over 101 random paths. The first two rows of each panel are ordered probit estimates of ∆pt+1 = a0 + a1 skewt + a2 ∆pt + ut+1 , with [standard errors]. ***/**/* indicates significance at the 1%/5%/10% level. Meeting types show the fraction of meetings of given types, 1.0/1.1/1.2 (first row), 2.0/2.1/2.2 (second row), with the residual 2.3 not shown, coded as x.y, where x is the number of alternatives put to vote and y is the number of alternatives with a positive number of votes (excluding the proposal). Stats show average dissent, the fraction of meetings with no policy change (first row), the fraction of meetings with the proposal equal to the status-quo and the fraction of meetings with the chairman’s proposal accepted (second row). mse is the mean squared difference between the adopted and the optimal policy.
57
Table A16: Predictive power of voting record and decision-making statistics Full information independence with reservations, N = 6 ρ = 0.90
ρ = 0.95
Baseline scenario (σu = σC = σP =
1 4
ρ1 = 1.95, ρ2 = −0.98
ρ = 0.99
for AR(1) and σu =
1 20
for AR(2))
skewt (a1 ) ∆pt (a2 )
8.19 [1.85]*** 1.40 [0.58]***
8.09 [1.86]*** 1.49 [0.57]***
8.04 [1.82]*** 1.48 [0.55]***
14.9 [9.68]*** 3.12 [0.59]***
Meeting types
0.19/0.19/0.00 0.10/0.48/0.03 0.26/0.56 0.38/0.83 0.032
0.19/0.19/0.00 0.10/0.48/0.03 0.26/0.56 0.39/0.83 0.032
0.18/0.19/0.00 0.11/0.48/0.03 0.26/0.55 0.38/0.83 0.032
0.45/0.00/0.00 0.51/0.04/0.00 0.01/0.46 0.45/0.99 0.008
Stats mse
High volatility scenario (σu doubled) skewt (a1 ) ∆pt (a2 )
3.23 [1.00]*** 0.22 [0.23]***
3.29 [0.99]*** 0.31 [0.23]***
3.34 [1.00]*** 0.35 [0.23]***
10.0 [5.48]*** 3.03 [0.37]***
Meeting types
0.02/0.15/0.02 0.10/0.51/0.18 0.34/0.36 0.20/0.84 0.046
0.02/0.15/0.02 0.10/0.52/0.18 0.34/0.35 0.19/0.84 0.046
0.02/0.15/0.02 0.11/0.51/0.18 0.34/0.35 0.19/0.84 0.046
0.25/0.00/0.00 0.65/0.09/0.00 0.03/0.28 0.25/0.98 0.013
Stats mse
Bad information scenario (σC , σP doubled) skewt (a1 ) ∆pt (a2 )
6.93 [2.07]*** 0.66 [0.54]***
6.97 [2.06]*** 0.79 [0.52]***
6.51 [2.01]*** 0.75 [0.50]***
16.6 [24.1]*** 2.66 [0.52]***
Meeting types
0.27/0.13/0.00 0.17/0.42/0.01 0.20/0.53 0.40/0.87 0.052
0.26/0.13/0.00 0.18/0.42/0.01 0.19/0.52 0.39/0.87 0.052
0.25/0.13/0.00 0.19/0.43/0.01 0.19/0.50 0.38/0.88 0.052
0.45/0.00/0.00 0.53/0.02/0.00 0.01/0.46 0.45/0.99 0.008
Stats mse
P bad information scenario (σP doubled) skewt (a1 ) ∆pt (a2 )
7.31 [1.78]*** 0.96 [0.56]***
7.09 [1.79]*** 1.02 [0.54]***
7.00 [1.74]*** 1.04 [0.52]***
19.9 [25.5]*** 2.95 [0.56]***
Meeting types
0.23/0.13/0.00 0.13/0.50/0.01 0.27/0.59 0.36/0.77 0.047
0.23/0.13/0.00 0.14/0.49/0.01 0.27/0.58 0.36/0.78 0.045
0.22/0.13/0.00 0.14/0.49/0.01 0.27/0.57 0.36/0.78 0.046
0.45/0.00/0.00 0.53/0.03/0.00 0.01/0.46 0.45/0.99 0.008
Stats mse
Note: All entries averaged over 101 random paths. The first two rows of each panel are ordered probit estimates of ∆pt+1 = a0 + a1 skewt + a2 ∆pt + ut+1 , with [standard errors]. ***/**/* indicates significance at the 1%/5%/10% level. Meeting types show the fraction of meetings of given types, 1.0/1.1/1.2 (first row), 2.0/2.1/2.2 (second row), with the residual 2.3 not shown, coded as x.y, where x is the number of alternatives put to vote and y is the number of alternatives with a positive number of votes (excluding the proposal). Stats show average dissent, the fraction of meetings with no policy change (first row), the fraction of meetings with the proposal equal to the status-quo and the fraction of meetings with the chairman’s proposal accepted (second row). mse is the mean squared difference between the adopted and the optimal policy.
58
Table A17: Predictive power of voting record and decision-making statistics Majoritarian model without reservations, N = 4 ρ = 0.90
ρ = 0.95
Baseline scenario (σu = σC = σP =
1 4
ρ1 = 1.95, ρ2 = −0.98
ρ = 0.99
for AR(1) and σu =
1 20
for AR(2))
skewt (a1 ) ∆pt (a2 )
6.53 [3.22]*** 1.45 [0.75]***
6.58 [3.19]*** 1.51 [0.72]***
6.58 [3.14]*** 1.65 [0.72]***
12.0 [9.72]*** 3.04 [0.59]***
Meeting types Stats
0.46/0.19/0.36 0.11/0.46 0.46/1.00 0.025
0.45/0.19/0.36 0.11/0.45 0.45/1.00 0.024
0.44/0.19/0.36 0.11/0.44 0.44/1.00 0.025
0.46/0.51/0.03 0.01/0.46 0.46/1.00 0.007
mse
High volatility scenario (σu doubled) skewt (a1 ) ∆pt (a2 )
1.77 [1.78]*** 0.28 [0.27]***
1.82 [1.78]*** 0.37 [0.27]***
1.76 [1.79]*** 0.40 [0.28]***
9.58 [5.40]*** 3.10 [0.37]***
Meeting types Stats
0.23/0.34/0.44 0.12/0.23 0.23/1.00 0.026
0.22/0.34/0.44 0.12/0.22 0.22/1.00 0.026
0.22/0.33/0.44 0.13/0.22 0.22/1.00 0.026
0.26/0.66/0.08 0.02/0.26 0.26/1.00 0.013
mse
Bad information scenario (σC , σP doubled) skewt (a1 ) ∆pt (a2 )
5.71 [3.22]*** 0.72 [0.62]***
5.11 [3.11]*** 0.70 [0.59]***
5.23 [3.09]*** 0.76 [0.57]***
10.1 [9.86]*** 2.37 [0.48]***
Meeting types Stats
0.45/0.25/0.30 0.09/0.45 0.45/1.00 0.047
0.43/0.27/0.30 0.09/0.43 0.43/1.00 0.048
0.42/0.28/0.30 0.09/0.42 0.42/1.00 0.047
0.46/0.53/0.02 0.01/0.46 0.46/1.00 0.008
mse
P bad information scenario (σP doubled) skewt (a1 ) ∆pt (a2 )
5.89 [3.24]*** 0.95 [0.66]***
5.90 [3.13]*** 0.95 [0.63]***
5.65 [3.07]*** 1.00 [0.60]***
11.1 [11.2]*** 2.74 [0.54]***
Meeting types Stats
0.46/0.23/0.32 0.09/0.46 0.46/1.00 0.043
0.44/0.24/0.32 0.09/0.44 0.44/1.00 0.044
0.43/0.25/0.32 0.09/0.43 0.43/1.00 0.043
0.46/0.52/0.02 0.01/0.46 0.46/1.00 0.008
mse
Note: All entries averaged over 101 random paths. The first two rows of each panel are ordered probit estimates of ∆pt+1 = a0 + a1 skewt + a2 ∆pt + ut+1 , with [standard errors]. ***/**/* indicates significance at the 1%/5%/10% level. Meeting types show the fraction of meetings of given types (1.0/2.0/2.1), coded as x.y, where x is the number of alternatives put to vote and y is the number of alternatives with a positive number of votes (excluding the proposal). Stats show average dissent, the fraction of meetings with no policy change (first row), the fraction of meetings with the proposal equal to the status-quo and the fraction of meetings with the chairman’s proposal accepted (second row). mse is the mean squared difference between the adopted and the optimal policy.
59
Table A18: Predictive power of voting record and decision-making statistics Majoritarian model without reservations, N = 6 ρ = 0.90
ρ = 0.95
Baseline scenario (σu = σC = σP =
1 4
ρ1 = 1.95, ρ2 = −0.98
ρ = 0.99
for AR(1) and σu =
1 20
for AR(2))
skewt (a1 ) ∆pt (a2 )
7.35 [3.37]*** 1.88 [0.81]***
7.66 [3.36]*** 1.99 [0.79]***
7.73 [3.34]*** 2.13 [0.79]***
13.1 [9.52]*** 3.15 [0.61]***
Meeting types Stats
0.47/0.14/0.40 0.11/0.47 0.47/1.00 0.024
0.46/0.13/0.40 0.11/0.46 0.46/1.00 0.023
0.46/0.14/0.40 0.11/0.46 0.46/1.00 0.024
0.46/0.50/0.04 0.01/0.46 0.46/1.00 0.008
mse
High volatility scenario (σu doubled) skewt (a1 ) ∆pt (a2 )
2.03 [1.88]*** 0.38 [0.29]***
2.11 [1.86]*** 0.44 [0.29]***
2.10 [1.88]*** 0.50 [0.29]***
10.4 [5.71]*** 3.19 [0.38]***
Meeting types Stats
0.23/0.27/0.50 0.13/0.23 0.23/1.00 0.023
0.22/0.27/0.51 0.13/0.22 0.22/1.00 0.023
0.22/0.26/0.51 0.13/0.22 0.22/1.00 0.023
0.26/0.65/0.09 0.02/0.26 0.26/1.00 0.013
mse
Bad information scenario (σC , σP doubled) skewt (a1 ) ∆pt (a2 )
6.08 [3.28]*** 0.86 [0.65]***
5.65 [3.16]*** 0.85 [0.62]***
5.58 [3.13]*** 0.92 [0.59]***
13.1 [14.0]*** 2.48 [0.50]***
Meeting types Stats
0.45/0.21/0.34 0.09/0.45 0.45/1.00 0.047
0.43/0.22/0.34 0.09/0.43 0.43/1.00 0.047
0.42/0.23/0.34 0.09/0.42 0.42/1.00 0.047
0.46/0.52/0.02 0.01/0.46 0.46/1.00 0.008
mse
P bad information scenario (σP doubled) skewt (a1 ) ∆pt (a2 )
6.51 [3.30]*** 1.10 [0.69]***
6.14 [3.18]*** 1.04 [0.64]***
6.01 [3.14]*** 1.14 [0.63]***
13.4 [12.3]*** 2.66 [0.54]***
Meeting types Stats
0.46/0.19/0.35 0.10/0.46 0.46/1.00 0.044
0.44/0.20/0.36 0.10/0.44 0.44/1.00 0.044
0.43/0.21/0.36 0.10/0.43 0.43/1.00 0.044
0.46/0.52/0.02 0.01/0.46 0.46/1.00 0.008
mse
Note: All entries averaged over 101 random paths. The first two rows of each panel are ordered probit estimates of ∆pt+1 = a0 + a1 skewt + a2 ∆pt + ut+1 , with [standard errors]. ***/**/* indicates significance at the 1%/5%/10% level. Meeting types show the fraction of meetings of given types (1.0/2.0/2.1), coded as x.y, where x is the number of alternatives put to vote and y is the number of alternatives with a positive number of votes (excluding the proposal). Stats show average dissent, the fraction of meetings with no policy change (first row), the fraction of meetings with the proposal equal to the status-quo and the fraction of meetings with the chairman’s proposal accepted (second row). mse is the mean squared difference between the adopted and the optimal policy.
60
Table A19: Predictive power of voting record and decision-making statistics Majoritarian model with reservations, N = 4 ρ = 0.90
ρ = 0.95
Baseline scenario (σu = σC = σP =
1 4
ρ1 = 1.95, ρ2 = −0.98
ρ = 0.99
for AR(1) and σu =
1 20
for AR(2))
skewt (a1 ) ∆pt (a2 )
7.86 [2.70]*** 1.60 [0.67]***
7.95 [2.72]*** 1.67 [0.66]***
7.74 [2.67]*** 1.77 [0.65]***
12.0 [9.72]*** 3.04 [0.59]***
Meeting types
0.34/0.12/0.00 0.16/0.37/0.01 0.14/0.46 0.46/1.00 0.025
0.34/0.11/0.00 0.16/0.37/0.01 0.14/0.45 0.45/1.00 0.024
0.33/0.11/0.00 0.17/0.38/0.01 0.14/0.44 0.44/1.00 0.025
0.46/0.00/0.00 0.51/0.03/0.00 0.01/0.46 0.46/1.00 0.007
Stats mse
High volatility scenario (σu doubled) skewt (a1 ) ∆pt (a2 )
2.43 [1.52]*** 0.27 [0.24]***
2.39 [1.52]*** 0.35 [0.25]***
2.55 [1.54]*** 0.40 [0.25]***
9.62 [5.40]*** 3.10 [0.37]***
Meeting types
0.09/0.12/0.02 0.21/0.44/0.12 0.21/0.23 0.23/1.00 0.026
0.08/0.12/0.02 0.21/0.44/0.12 0.21/0.22 0.22/1.00 0.026
0.08/0.12/0.02 0.21/0.44/0.12 0.21/0.22 0.22/1.00 0.026
0.26/0.00/0.00 0.66/0.08/0.00 0.02/0.26 0.26/1.00 0.013
Stats mse
Bad information scenario (σC , σP doubled) skewt (a1 ) ∆pt (a2 )
6.62 [2.92]*** 0.80 [0.59]***
6.04 [2.85]*** 0.78 [0.56]***
5.86 [2.80]*** 0.80 [0.54]***
10.1 [9.86]*** 2.37 [0.48]***
Meeting types
0.38/0.06/0.00 0.24/0.31/0.00 0.10/0.45 0.45/1.00 0.047
0.37/0.06/0.00 0.25/0.31/0.00 0.11/0.43 0.43/1.00 0.048
0.36/0.06/0.00 0.26/0.32/0.00 0.10/0.42 0.42/1.00 0.047
0.46/0.00/0.00 0.53/0.02/0.00 0.01/0.46 0.46/1.00 0.008
Stats mse
P bad information scenario (σP doubled) skewt (a1 ) ∆pt (a2 )
6.31 [2.94]*** 0.98 [0.63]***
6.28 [2.87]*** 0.98 [0.60]***
5.78 [2.80]*** 1.00 [0.57]***
11.1 [11.2]*** 2.74 [0.54]***
Meeting types
0.39/0.06/0.00 0.21/0.33/0.00 0.11/0.46 0.46/1.00 0.043
0.38/0.06/0.00 0.23/0.33/0.00 0.11/0.44 0.44/1.00 0.044
0.37/0.06/0.00 0.24/0.33/0.00 0.11/0.43 0.43/1.00 0.043
0.46/0.00/0.00 0.52/0.02/0.00 0.01/0.46 0.46/1.00 0.008
Stats mse
Note: All entries averaged over 101 random paths. The first two rows of each panel are ordered probit estimates of ∆pt+1 = a0 + a1 skewt + a2 ∆pt + ut+1 , with [standard errors]. ***/**/* indicates significance at the 1%/5%/10% level. Meeting types show the fraction of meetings of given types, 1.0/1.1/1.2 (first row), 2.0/2.1/2.2 (second row), with the residual 2.3 not shown, coded as x.y, where x is the number of alternatives put to vote and y is the number of alternatives with a positive number of votes (excluding the proposal). Stats show average dissent, the fraction of meetings with no policy change (first row), the fraction of meetings with the proposal equal to the status-quo and the fraction of meetings with the chairman’s proposal accepted (second row). mse is the mean squared difference between the adopted and the optimal policy.
61
Table A20: Predictive power of voting record and decision-making statistics Majoritarian model with reservations, N = 6 ρ = 0.90
ρ = 0.95
Baseline scenario (σu = σC = σP =
1 4
ρ1 = 1.95, ρ2 = −0.98
ρ = 0.99
for AR(1) and σu =
1 20
for AR(2))
skewt (a1 ) ∆pt (a2 )
9.01 [2.83]*** 2.08 [0.71]***
9.26 [2.83]*** 2.19 [0.71]***
9.02 [2.81]*** 2.26 [0.70]***
13.1 [9.52]*** 3.15 [0.61]***
Meeting types
0.29/0.17/0.00 0.11/0.40/0.03 0.15/0.47 0.47/1.00 0.024
0.29/0.17/0.00 0.10/0.41/0.02 0.15/0.46 0.46/1.00 0.023
0.29/0.16/0.00 0.11/0.41/0.03 0.15/0.46 0.46/1.00 0.024
0.46/0.00/0.00 0.50/0.04/0.00 0.01/0.46 0.46/1.00 0.008
Stats mse
High volatility scenario (σu doubled) skewt (a1 ) ∆pt (a2 )
2.59 [1.60]*** 0.34 [0.25]***
2.74 [1.59]*** 0.42 [0.25]***
2.81 [1.61]*** 0.47 [0.26]***
10.5 [5.70]*** 3.20 [0.38]***
Meeting types
0.05/0.14/0.04 0.13/0.44/0.20 0.23/0.23 0.23/1.00 0.023
0.05/0.14/0.04 0.12/0.45/0.20 0.24/0.22 0.22/1.00 0.023
0.05/0.13/0.04 0.13/0.44/0.20 0.23/0.22 0.22/1.00 0.023
0.26/0.00/0.00 0.65/0.09/0.00 0.02/0.26 0.26/1.00 0.013
Stats mse
Bad information scenario (σC , σP doubled) skewt (a1 ) ∆pt (a2 )
7.12 [2.97]*** 0.95 [0.61]***
6.79 [2.88]*** 0.96 [0.58]***
6.44 [2.86]*** 0.98 [0.57]***
13.1 [14.0]*** 2.48 [0.50]***
Meeting types
0.36/0.09/0.00 0.19/0.36/0.00 0.11/0.45 0.45/1.00 0.047
0.34/0.09/0.00 0.20/0.36/0.00 0.11/0.43 0.43/1.00 0.047
0.34/0.09/0.00 0.21/0.36/0.00 0.11/0.42 0.42/1.00 0.047
0.46/0.00/0.00 0.52/0.02/0.00 0.01/0.46 0.46/1.00 0.008
Stats mse
P bad information scenario (σP doubled) skewt (a1 ) ∆pt (a2 )
7.07 [2.99]*** 1.13 [0.65]***
6.82 [2.90]*** 1.10 [0.61]***
6.49 [2.88]*** 1.17 [0.60]***
13.4 [12.3]*** 2.66 [0.54]***
Meeting types
0.37/0.09/0.00 0.17/0.37/0.01 0.12/0.46 0.46/1.00 0.044
0.35/0.09/0.00 0.18/0.37/0.00 0.12/0.44 0.44/1.00 0.044
0.34/0.09/0.00 0.19/0.38/0.01 0.12/0.43 0.43/1.00 0.044
0.46/0.00/0.00 0.52/0.02/0.00 0.01/0.46 0.46/1.00 0.008
Stats mse
Note: All entries averaged over 101 random paths. The first two rows of each panel are ordered probit estimates of ∆pt+1 = a0 + a1 skewt + a2 ∆pt + ut+1 , with [standard errors]. ***/**/* indicates significance at the 1%/5%/10% level. Meeting types show the fraction of meetings of given types, 1.0/1.1/1.2 (first row), 2.0/2.1/2.2 (second row), with the residual 2.3 not shown, coded as x.y, where x is the number of alternatives put to vote and y is the number of alternatives with a positive number of votes (excluding the proposal). Stats show average dissent, the fraction of meetings with no policy change (first row), the fraction of meetings with the proposal equal to the status-quo and the fraction of meetings with the chairman’s proposal accepted (second row). mse is the mean squared difference between the adopted and the optimal policy.
62
Table A21: Predictive power of voting record and decision-making statistics Autarkical model with information sharing without reservations, N = 4 ρ = 0.90
ρ = 0.95
Baseline scenario (σu = σC = σP =
1 4
ρ1 = 1.95, ρ2 = −0.98
ρ = 0.99
for AR(1) and σu =
1 20
for AR(2))
skewt (a1 ) ∆pt (a2 )
2.51 [2.79]*** 0.15 [0.69]***
2.43 [2.79]*** 0.21 [0.69]***
2.32 [2.81]*** 0.27 [0.71]***
11.8 [11.2]*** 2.92 [0.56]***
Meeting types Stats
0.46/0.14/0.40 0.12/0.46 0.46/1.00 0.018
0.45/0.14/0.41 0.12/0.45 0.45/1.00 0.018
0.45/0.13/0.42 0.12/0.45 0.45/1.00 0.018
0.47/0.51/0.02 0.01/0.47 0.47/1.00 0.007
mse
High volatility scenario (σu doubled) skewt (a1 ) ∆pt (a2 )
0.80 [1.64]*** -0.04 [0.24]***
0.86 [1.66]*** 0.01 [0.24]***
0.55 [1.68]*** 0.00 [0.25]***
7.61 [7.07]*** 3.26 [0.39]***
Meeting types Stats
0.22/0.33/0.46 0.12/0.22 0.22/1.00 0.018
0.22/0.32/0.45 0.12/0.22 0.22/1.00 0.018
0.23/0.32/0.46 0.12/0.23 0.23/1.00 0.018
0.29/0.66/0.06 0.02/0.29 0.29/1.00 0.011
mse
Bad information scenario (σC , σP doubled) skewt (a1 ) ∆pt (a2 )
1.82 [2.87]*** 0.02 [0.65]***
1.95 [2.81]*** 0.16 [0.62]***
1.25 [2.80]*** 0.17 [0.61]***
9.25 [10.2]*** 2.23 [0.45]***
Meeting types Stats
0.51/0.19/0.30 0.09/0.51 0.51/1.00 0.038
0.50/0.19/0.31 0.09/0.50 0.50/1.00 0.038
0.49/0.20/0.30 0.09/0.49 0.49/1.00 0.038
0.46/0.53/0.01 0.00/0.46 0.46/1.00 0.008
mse
P bad information scenario (σP doubled) skewt (a1 ) ∆pt (a2 )
1.28 [2.94]*** 0.09 [0.71]***
1.48 [2.85]*** 0.20 [0.67]***
0.64 [2.82]*** 0.15 [0.67]***
8.19 [9.70]*** 2.23 [0.44]***
Meeting types Stats
0.54/0.16/0.30 0.09/0.54 0.54/1.00 0.035
0.52/0.17/0.31 0.09/0.52 0.52/1.00 0.035
0.53/0.17/0.30 0.09/0.53 0.53/1.00 0.034
0.46/0.53/0.01 0.00/0.46 0.46/1.00 0.008
mse
Note: All entries averaged over 101 random paths. The first two rows of each panel are ordered probit estimates of ∆pt+1 = a0 + a1 skewt + a2 ∆pt + ut+1 , with [standard errors]. ***/**/* indicates significance at the 1%/5%/10% level. Meeting types show the fraction of meetings of given types (1.0/2.0/2.1), coded as x.y, where x is the number of alternatives put to vote and y is the number of alternatives with a positive number of votes (excluding the proposal). Stats show average dissent, the fraction of meetings with no policy change (first row), the fraction of meetings with the proposal equal to the status-quo and the fraction of meetings with the chairman’s proposal accepted (second row). mse is the mean squared difference between the adopted and the optimal policy.
63
Table A22: Predictive power of voting record and decision-making statistics Autarkical model with information sharing without reservations, N = 6 ρ = 0.90
ρ = 0.95
Baseline scenario (σu = σC = σP =
1 4
ρ1 = 1.95, ρ2 = −0.98
ρ = 0.99
for AR(1) and σu =
1 20
for AR(2))
skewt (a1 ) ∆pt (a2 )
3.25 [2.96]*** 0.45 [0.80]***
3.09 [3.00]*** 0.48 [0.81]***
2.63 [3.01]*** 0.49 [0.81]***
11.6 [12.0]*** 2.91 [0.57]***
Meeting types Stats
0.48/0.07/0.45 0.13/0.48 0.48/1.00 0.017
0.48/0.07/0.46 0.13/0.48 0.48/1.00 0.017
0.48/0.07/0.45 0.13/0.48 0.48/1.00 0.016
0.47/0.50/0.03 0.01/0.47 0.47/1.00 0.007
mse
High volatility scenario (σu doubled) skewt (a1 ) ∆pt (a2 )
1.14 [1.67]*** -0.00 [0.26]***
1.04 [1.69]*** 0.03 [0.26]***
0.66 [1.70]*** 0.03 [0.26]***
7.60 [7.63]*** 3.33 [0.39]***
Meeting types Stats
0.22/0.22/0.55 0.14/0.22 0.22/1.00 0.015
0.22/0.21/0.56 0.14/0.22 0.22/1.00 0.014
0.23/0.21/0.56 0.14/0.23 0.23/1.00 0.014
0.29/0.64/0.07 0.02/0.29 0.29/1.00 0.010
mse
Bad information scenario (σC , σP doubled) skewt (a1 ) ∆pt (a2 )
1.88 [3.05]*** 0.14 [0.73]***
1.50 [2.97]*** 0.20 [0.70]***
0.75 [2.95]*** 0.19 [0.69]***
11.0 [18.9]*** 2.38 [0.47]***
Meeting types Stats
0.54/0.13/0.33 0.09/0.54 0.54/1.00 0.037
0.53/0.13/0.34 0.09/0.53 0.53/1.00 0.036
0.53/0.14/0.33 0.09/0.53 0.53/1.00 0.036
0.46/0.52/0.01 0.00/0.46 0.46/1.00 0.008
mse
P bad information scenario (σP doubled) skewt (a1 ) ∆pt (a2 )
1.46 [3.12]*** 0.19 [0.77]***
0.97 [3.07]*** 0.24 [0.75]***
0.52 [3.03]*** 0.26 [0.73]***
9.24 [17.7]*** 2.15 [0.44]***
Meeting types Stats
0.56/0.11/0.33 0.09/0.56 0.56/1.00 0.035
0.56/0.12/0.32 0.09/0.56 0.56/1.00 0.035
0.55/0.12/0.33 0.09/0.55 0.55/1.00 0.034
0.47/0.52/0.01 0.00/0.47 0.47/1.00 0.008
mse
Note: All entries averaged over 101 random paths. The first two rows of each panel are ordered probit estimates of ∆pt+1 = a0 + a1 skewt + a2 ∆pt + ut+1 , with [standard errors]. ***/**/* indicates significance at the 1%/5%/10% level. Meeting types show the fraction of meetings of given types (1.0/2.0/2.1), coded as x.y, where x is the number of alternatives put to vote and y is the number of alternatives with a positive number of votes (excluding the proposal). Stats show average dissent, the fraction of meetings with no policy change (first row), the fraction of meetings with the proposal equal to the status-quo and the fraction of meetings with the chairman’s proposal accepted (second row). mse is the mean squared difference between the adopted and the optimal policy.
64
Table A23: Predictive power of voting record and decision-making statistics Autarkical model with information sharing with reservations, N = 4 ρ = 0.90
ρ = 0.95
Baseline scenario (σu = σC = σP =
1 4
ρ1 = 1.95, ρ2 = −0.98
ρ = 0.99
for AR(1) and σu =
1 20
for AR(2))
skewt (a1 ) ∆pt (a2 )
3.58 [2.53]*** 0.35 [0.65]***
3.38 [2.55]*** 0.39 [0.65]***
3.13 [2.55]*** 0.43 [0.66]***
11.8 [11.2]*** 2.92 [0.56]***
Meeting types
0.36/0.10/0.00 0.13/0.40/0.01 0.14/0.46 0.46/1.00 0.018
0.35/0.10/0.00 0.13/0.41/0.01 0.14/0.45 0.45/1.00 0.018
0.35/0.10/0.00 0.13/0.42/0.01 0.14/0.45 0.45/1.00 0.018
0.47/0.00/0.00 0.51/0.02/0.00 0.01/0.47 0.47/1.00 0.007
Stats mse
High volatility scenario (σu doubled) skewt (a1 ) ∆pt (a2 )
1.40 [1.57]*** -0.01 [0.23]***
1.36 [1.58]*** 0.04 [0.23]***
1.06 [1.60]*** 0.04 [0.23]***
7.80 [6.87]*** 3.26 [0.39]***
Meeting types
0.09/0.11/0.02 0.28/0.40/0.10 0.18/0.22 0.22/1.00 0.018
0.09/0.10/0.02 0.28/0.40/0.10 0.18/0.22 0.22/1.00 0.018
0.09/0.11/0.02 0.28/0.40/0.10 0.18/0.23 0.23/1.00 0.018
0.29/0.00/0.00 0.65/0.06/0.00 0.02/0.29 0.29/1.00 0.011
Stats mse
Bad information scenario (σC , σP doubled) skewt (a1 ) ∆pt (a2 )
2.76 [2.53]*** 0.16 [0.61]***
2.63 [2.49]*** 0.26 [0.59]***
1.92 [2.45]*** 0.26 [0.57]***
9.25 [10.2]*** 2.23 [0.45]***
Meeting types
0.43/0.09/0.00 0.18/0.30/0.00 0.11/0.51 0.51/1.00 0.038
0.41/0.09/0.00 0.19/0.31/0.00 0.11/0.50 0.50/1.00 0.038
0.41/0.09/0.00 0.20/0.31/0.00 0.11/0.49 0.49/1.00 0.038
0.46/0.00/0.00 0.53/0.01/0.00 0.00/0.46 0.46/1.00 0.008
Stats mse
P bad information scenario (σP doubled) skewt (a1 ) ∆pt (a2 )
2.22 [2.51]*** 0.26 [0.65]***
2.05 [2.46]*** 0.29 [0.62]***
1.37 [2.39]*** 0.28 [0.61]***
8.19 [9.70]*** 2.23 [0.44]***
Meeting types
0.44/0.10/0.00 0.16/0.30/0.00 0.11/0.54 0.54/1.00 0.035
0.43/0.10/0.00 0.16/0.31/0.00 0.12/0.52 0.52/1.00 0.035
0.42/0.10/0.00 0.17/0.31/0.00 0.12/0.53 0.53/1.00 0.034
0.46/0.00/0.00 0.53/0.01/0.00 0.00/0.46 0.46/1.00 0.008
Stats mse
Note: All entries averaged over 101 random paths. The first two rows of each panel are ordered probit estimates of ∆pt+1 = a0 + a1 skewt + a2 ∆pt + ut+1 , with [standard errors]. ***/**/* indicates significance at the 1%/5%/10% level. Meeting types show the fraction of meetings of given types, 1.0/1.1/1.2 (first row), 2.0/2.1/2.2 (second row), with the residual 2.3 not shown, coded as x.y, where x is the number of alternatives put to vote and y is the number of alternatives with a positive number of votes (excluding the proposal). Stats show average dissent, the fraction of meetings with no policy change (first row), the fraction of meetings with the proposal equal to the status-quo and the fraction of meetings with the chairman’s proposal accepted (second row). mse is the mean squared difference between the adopted and the optimal policy.
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Table A24: Predictive power of voting record and decision-making statistics Autarkical model with information sharing with reservations, N = 6 ρ = 0.90
ρ = 0.95
Baseline scenario (σu = σC = σP =
1 4
ρ1 = 1.95, ρ2 = −0.98
ρ = 0.99
for AR(1) and σu =
1 20
for AR(2))
skewt (a1 ) ∆pt (a2 )
4.68 [2.67]*** 0.76 [0.74]***
4.73 [2.69]*** 0.84 [0.75]***
4.10 [2.67]*** 0.81 [0.74]***
11.6 [12.0]*** 2.91 [0.57]***
Meeting types
0.31/0.16/0.00 0.07/0.44/0.01 0.16/0.48 0.48/1.00 0.017
0.31/0.16/0.00 0.06/0.45/0.01 0.16/0.48 0.48/1.00 0.017
0.31/0.16/0.01 0.07/0.44/0.01 0.16/0.48 0.48/1.00 0.016
0.47/0.00/0.00 0.50/0.03/0.00 0.01/0.47 0.47/1.00 0.007
Stats mse
High volatility scenario (σu doubled) skewt (a1 ) ∆pt (a2 )
1.69 [1.60]*** 0.03 [0.24]***
1.62 [1.62]*** 0.07 [0.25]***
1.19 [1.64]*** 0.07 [0.25]***
8.11 [7.01]*** 3.34 [0.39]***
Meeting types
0.05/0.13/0.05 0.18/0.44/0.16 0.21/0.22 0.22/1.00 0.015
0.05/0.13/0.05 0.17/0.44/0.16 0.21/0.22 0.22/1.00 0.014
0.05/0.13/0.05 0.18/0.43/0.16 0.21/0.23 0.23/1.00 0.014
0.29/0.00/0.00 0.64/0.07/0.00 0.02/0.29 0.29/1.00 0.010
Stats mse
Bad information scenario (σC , σP doubled) skewt (a1 ) ∆pt (a2 )
2.86 [2.63]*** 0.32 [0.67]***
2.55 [2.52]*** 0.39 [0.64]***
1.89 [2.49]*** 0.38 [0.62]***
11.0 [18.9]*** 2.38 [0.47]***
Meeting types
0.41/0.13/0.00 0.12/0.34/0.00 0.12/0.54 0.54/1.00 0.037
0.40/0.14/0.00 0.12/0.34/0.00 0.12/0.53 0.53/1.00 0.036
0.39/0.14/0.00 0.13/0.34/0.00 0.12/0.53 0.53/1.00 0.036
0.46/0.00/0.00 0.52/0.01/0.00 0.00/0.46 0.46/1.00 0.008
Stats mse
P bad information scenario (σP doubled) skewt (a1 ) ∆pt (a2 )
2.45 [2.60]*** 0.38 [0.69]***
2.00 [2.52]*** 0.43 [0.67]***
1.59 [2.49]*** 0.45 [0.65]***
9.24 [17.7]*** 2.15 [0.44]***
Meeting types
0.42/0.14/0.00 0.10/0.33/0.00 0.12/0.56 0.56/1.00 0.035
0.41/0.15/0.00 0.11/0.33/0.00 0.12/0.56 0.56/1.00 0.035
0.40/0.15/0.00 0.11/0.34/0.00 0.12/0.55 0.55/1.00 0.034
0.47/0.00/0.00 0.52/0.01/0.00 0.00/0.47 0.47/1.00 0.008
Stats mse
Note: All entries averaged over 101 random paths. The first two rows of each panel are ordered probit estimates of ∆pt+1 = a0 + a1 skewt + a2 ∆pt + ut+1 , with [standard errors]. ***/**/* indicates significance at the 1%/5%/10% level. Meeting types show the fraction of meetings of given types, 1.0/1.1/1.2 (first row), 2.0/2.1/2.2 (second row), with the residual 2.3 not shown, coded as x.y, where x is the number of alternatives put to vote and y is the number of alternatives with a positive number of votes (excluding the proposal). Stats show average dissent, the fraction of meetings with no policy change (first row), the fraction of meetings with the proposal equal to the status-quo and the fraction of meetings with the chairman’s proposal accepted (second row). mse is the mean squared difference between the adopted and the optimal policy.
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