Assessing differences in legislators' revealed preferences: A case study on the 107th U.S. Senate

2 MB·

by  Chelsea Lofland

Assessing differences in legislators’ revealed
th
preferences: A case study on the 107 U.S. Senate
Chelsea L. Lofland∗, Abel Rodr´ıguez† and Scott Moser‡
6
1
0
2

n Abstract
a
J Roll call data are widely used to assess legislators’ preferences and ideology, as well as

8
2 test theories of legislative behavior. In particular, roll call data is often used to determine

whether the revealed preferences of legislators are affected by outside forces such as party
]
P
pressure, minority status or procedural rules. This paper describes a Bayesian hierarchical
A
. model that extends existing spatial voting models to test sharp hypotheses about differences
t
a
in preferences the using posterior probabilities associated with such hypotheses. We use our
t
s
th
[ model to investigate the effect of the change of party majority status during the 107 U.S.

1 Senateontherevealedpreferencesofsenators. Thisanalysisprovidesevidencethatchangein
v
7 party affiliation might affect the revealed preferences of legislators, but provides no evidence
1
abouttheeffectofmajoritystatusontherevealedpreferencesoflegislators.
6
7
Keywords: Spatial Voting Model; Hypothesis Testing; Spike-and-Slab Prior; Revealed
0
. Preferences;FactorAnalysis.
1
0
6
1
: 1 Introduction
v
i
X
r Spatial voting models (Enelow and Hinich, 1984; Poole and Rosenthal, 1985; Jackman, 2001;
a
Clinton et al., 2004) are widely used to infer preferences of members of legislative and judicial
∗GraduateStudent,DepartmentofAppliedMathematicsandStatistics,UniversityofCalifornia,SantaCruz,1156
HighStreet,MailstopSOE2,SantaCruz,CA,95064;Fax: [phone];Email: [email protected]
†Associate Professor, Department of Applied Mathematics and Statistics, University of California, Santa Cruz,
1156HighStreet,MailstopSOE2,SantaCruz,CA,95064;Ph.:[phone],Fax:[phone];Email:abel@
soe.ucsc.edu.edu
‡AssistantProfessor,DepartmentofGovernment,UniversityofTexasatAustin,1UniversitySta. A1800,Austin,
TX,78712;Ph.: [phone],Fax: [phone];Email: [email protected]
1
bodies from their voting records. Put simply, such models posit that legislators have a most pre-
ferred policy – their ideal point –, which can be represented as a point in some Euclidean space,
and vote for/against motions in accordance with their (latent) preferences. Hence, by estimating
the ideal point of a legislator from their observed voting behavior we can recover a legislator’s
revealedpreference(eg. Varian,2006;Richter,1966). Inturn,revealedpreferencesareoftenused
to construct ideological scales from voting records. Indeed, while ideology has a long a varied
history of usage in scholarship, (see eg, Knight 1985), political ideology usually refers to specific
policy views and preferences held by individuals. Although political ideology may be based on
“an underlying philosophy on which all specific political views are based” (pg 17, Jessee, 2012),
or stem from logical and psycological sets of constraints influencing a person’s beliefs (Converse,
1964), the notion is often operationalized in terms of revealed preferences and ideal points (e.g.,
PooleandRosenthal,1997).
Political scientist often assume that legislators preferences remain stable, at least over short
periods of time, (e.g., see Rosenthal and Voeten, 2004, Shor et al., 2010 and Shor and McCarty,
2011). However, although this assumption has some empirical support, the evidence is equivocal.
For example, although Poole and Rosenthal (1997) find that members of congress tend to express
verystablepolicypositionsacrosstheircareers,Nokken(2000)andNokkenandPoole(2004)find
evidence that legislators who switch party affiliations while serving in Congress exhibit changes
in revealed policy preferences, and Jenkins (2000) and Snyder Jr. and Groseclose (2000) argue
that revealed policy preferences shift across institutional settings and can be influenced by party
pressure. Similarly, Rothenberg and Sanders (2000) find evidence that legislators exhibit changes
inrevealedpolicypreferencesaftertheyhavebeendefeatedinaprimaryorhavedecidedtoretire,
whileCarsonetal.(2004)maketheoppositeargumentusingthesamedatabutaslightlydifferent
statistical model. As a final example, May (1973) and Clausen and Wilcox (1987) argue that the
minority status of a party might affect the revealed preferences of its legislators. The argument is
basedonsimplegame-theoreticmodelsthatshowtheminoritypartymaybeabletosecurepolicies
closertotheiridealpointbystakingoutextremepositions(Merrilletal.,1999).
th
The voting record of the 107 U.S. Senate is a particularly interesting dataset to investigate
th
questions about the stability of legislators’ revealed preferences. This is because during the 107
rd st
U.S. Senate, which met between January 3 , 2001 and December 31 , 2002, control of the U.S.
Senate formally shifted within the course of the two-year congressional term. The 2000 congres-
sional election resulted in a Senate evenly split between Democrats and Republicans. Because
RepublicanDickCheneywaselectedvice-presidentinthe2000presidentialelection,initiallyRe-
2
publicans were considered the majority party, with senator Trent Lott receiving “the right of first
recognition” and committee chairmanships being assigned to Republican senators1. However, on
May 24, 2001, Republican senator James M. Jeffords announced that he would leave his party
to become an Independent and would caucus with the Democratic Party, putting them in control
of the Senate. This had deep organizational consequences including a transfer in agenda-setting
powerstotheDemocratleaderTomDaschleandachangeinallcommitteechairmanships.
th
Clintonetal.(2004)analyzedtheroll-callvotesfromthe107 U.S.Senatebyfittingseparate
one-dimensional spatial voting models for motions before and after Jeffords’ defection and con-
structing posterior credible intervals for difference in ideal points as well as for the difference in
theirrank-order. Thisanalysisidentifiedanumbersenators(includingDemocraticandRepublican
leaders Daschle and Lott, as well party switcher Jeffords and senators Gregg, Torricelli, Thomp-
son, Shelby, McConnell, Ensign and Hutchinson) whose preferences seemed to be affected by the
changeinmajoritystatus. Onafollow-upanalysis,Roberts(2007)investigatetheapparentchange
in preferences of the two party leaders (see Figure 1) in terms of their behavior on cloture mo-
tions. In particular, Roberts (2007) argues that majority leaders might appear less partisan simply
because they sometimes vote strategically against their party during cloture motions so it can be
laterreconsideredifitfailstoreceiveaqualifiedmajority2.
Although the conclusions from both of these analyses are certainly plausible and match pre-
vious theoretical and empirical work, the methodology used to reach them is unsatisfactory. One
reason is that comparing the rank order of legislators before and after the change in leadership ig-
noresthattheranksofdifferentlegislatorsarenotindependent. Indeed,foralegislatortoincrease
her rank, other(s) need to decrease their own. The obvious alternative is to base the comparisons
on the actual ideal points rather than the rank order induced by them. However, comparing the
ideal points directly is difficult because of the invariance of the policy space to affine transforma-
tions, and the fact that we have no data linking both policy spaces. These features imply imply
that estimates obtained by fitting separate models to each collection of motions are not directly
comparable, as they do not necessarily share a common ideological scale. Clinton et al. (2004)
1In spite of Republicans being considered the majority party, Democrats were able to extract some concessions
intheseunusualcircumstances. Forexample,committeeassignments,staffandotherresourcesweredividedequally
among the two parties instead of the more usual arrangement in which the majority party receives a bigger share of
resources.
2SenateRuleXIIIstatesthat“WhenaquestionhasbeendecidedbytheSenate,anysenatorvotingwiththeprevail-
ingside... may,onthesamedayoroneitherofthenexttwodaysofactualsessionthereafter,moveareconsideration
... .”
3
99
97
95 GREGG (R NH)
93
91
89
87
85
83
81
79
77
75 SHELBY (R AL) LOTT (R MS)
73
71
69
67
h 65
c 63
wit 5691 SMITH (R OR)
s' s 5557 l ENSIGN (R NV)
ord 5513 MILLER (D GA)
eff 4479 JEFFORDS (R VT) l ll MCCAIN (R AZ)
efore J 3344479135 TORRICElLLI (Dl NJl)l l BAUCllUlSl (lD MT)
B 35 l l
33 ll
31 l
29 ll
222357 l ll llll l
21 l ll
19 l
17 ll
15 l l
13 l l
11 l l l
9 l DASCHLE (D SD)
7 l l l l
5 l
3 l l
1
13579135791357913579135791357913579135791357913579
111112222233333444445555566666777778888899999
After Jeffords' switch
th
Figure 1: Comparison of the ideal point ranks of legislators in the 107 US Senate obtained
by fitting independent one-dimensional Bayesian spatial voting models (Jackman, 2001) to the
motionsvotedonbeforeandafterthedefectionofsenatorJamesM.JeffordsfromtheRepublican
party. Democratic senators appear as blue circles (◦), Republicans appear as red plus signs (+),
andparty-switcherJeffordsappearsasagreentriangle((cid:52)).
attempt to address this issue by standardizing each set of ideal point estimates to have mean zero
andvariance. However,thistypeofad-hocadjustmentaposterioridoesnotreallyensurethatvotes
on motions are scaled on a common latent dimension. A second reason for potential concern with
4
theapproachescurrentlyusedtoassessthestabilityoflegislator’spreferencesisthelackofadjust-
ment for multiplicities. Identifying changes in preferences in the U.S. Senate implies testing 100
different hypotheses about the relative positions of ideal points, which requires that we adjust our
procedures to avoid a large number of false positives. Although orrections based on Bonferroni
(e.g., see Abdi, 2007) or False Discovery Rate (Benjamini and Hochberg, 1995) procedures are
possible solutions to this type of multiplicity issue, but in our experience they have been rarely
used. The challenge of comparing individuals traits associated with different latent scales also
arises in educational testing, e.g., when attempting to compare the skill level inferred using item
responsetheory(IRT)models(Fox,2010)forparticipantsintwoormoremultiplechoicetests.
Inthispaperweconsiderajointmodelfortwogroupsofmotionsthatallowsustoexplicitlytest
sharphypothesesaboutdifferencesinlegislators’revealedpreferences. Ourmodelisanextension
of the Bayesian spatial voting models discussed in Jackman (2001) and Clinton et al. (2004) that
incorporates hierarchical zero-inflated priors that link the ideal points for the different groups of
measures. Indeed, instead of ad-hoc adjustments to the posterior distribution such as the one
proposed in Clinton et al. (2004), our model induces a common scale across both policy spaces
by assuming that not all legislators change their preferences, so that legislators whose preferences
remain unchanged serve as a bridge between the two policy spaces (Shor et al., 2010; Shor and
McCarty,2011).
Although our model is built in a similar spirit, our approach differs from that in Martin and
Quinn (2002). Indeed, we are interested in testing sharp hypotheses about changes in revealed
preferences rather than model their smooth evolution over longer periods. Our approach is an al-
ternative to frequentist approaches based on the bootstrap (e.g., see Lewis and Poole, 2004 and
Carrolletal.,2009)andsharestheadvantages(anddisadvantages)ofBayesianapproachestovari-
able selection based on Bayes factors/posterior probabilities. In particular, our simulation studies
suggestthatourmodelautomaticallyaddressesissuesofmultiplecomparisons,anobservationthat
isconsistentwithScottandBerger(2006)andScottandBerger(2010).
The remainder of the paper is organized as follows: Section 2 describes the structure of our
statistical model. Section 3 discusses hyperparameter elicitation and some general remarks on
our computational implementation of the model. Section 4 presents our analysis of the data from
th
the 107 U.S. Senate, as well as a number of sensitivity and simulation studies that explore the
propertiesofourmodelinthisapplication. Finally,Section5presentsashortdiscussionandfuture
researchdirections.
5
2 Model description
2.1 Bayesian spatial voting models
Let y ∈ {0,1} encode the vote cast by legislator i = 1,...,I on motion j = 1,...,J, with
i,j
y = 1 if the vote corresponds to a “Yea” and y = 0 if it corresponds to a “Nay”. The spatial
i,j i,j
votingmodelofJackman(2001)andClintonetal.(2004)assumesthatlegislatorsmakedecisions
accordingrandomquadraticutilityfunctions
U (β ,z ) = −(cid:107)β −z (cid:107)2 +(cid:15) , U (β ,z ) = −(cid:107)β −z (cid:107)2 +(cid:15) ,
0 i j,0 i j,0 j,0 1 i j,1 i j,1 j,1
where U and U are, respectively, the utilities associated with a negative and a positive vote,
0 1
β = (β ,...,β )T corresponds to the position of legislator i in a K dimensional Euclidean
i i,0,1 i,0,K
policy space (his or her “ideal point”), z and z are, respectively, the positions associated with
j,0 j,1
a negative and a positive vote on motion j in the same policy space, and (cid:15) and (cid:15) are random
j,0 j,1
shocks. Theseassumptionsleadtoalikelihoodoftheform
(cid:0) (cid:12) (cid:0) (cid:1)(cid:1)
y | µ ,α ,β ∼ Bernoulli y (cid:12)G µ +αTβ , (1)
i,j j j i i,j j j i
where µ = zT z − zT z controls the baseline probability of a positive vote in motion j,
j j,0 j,0 j,1 j,1
α = (α ,...,α )T = 2(z −z ) controls the effect of the ideal points of the legislators on
j j,1 j,K j,1 j,0
the probability of a positive vote in motion j and G is an appropriate link function. For example,
if(cid:15) −(cid:15) followsastandardnormaldistributionthenGcorrespondstotheprobitlink,sothat
j,0 j,1
(cid:0) (cid:1) (cid:0) (cid:1)
G µ +αTβ = Φ µ +αTβ ,
j j i j j i
whereΦisthecumulativedistributionofthestandardnormaldistribution,whileif(cid:15) −(cid:15) follows
j,0 j,1
astandardlogisticdistributionthen
(cid:8) (cid:9)
exp µ +αTβ
G(cid:0)µ +αTβ (cid:1) = j j i .
j j i 1+exp(cid:8)µ +αTβ (cid:9)
j j i
InSection4weconsiderbothtypesoflinksandstudythesensitivityoftheresultstothischoice.
The model is completed by specifying prior distributions for the model parameters. It is com-
mon to assume that the intercepts µ ,...,µ are independent and identically distributed from a
1 J
(cid:0) (cid:1)
normal distribution with unknown mean ρ and variance κ2, i.e., µ | ρ ,κ2 ∼ N µ | ρ ,κ2 .
µ µ j µ µ j µ µ
The hyperparameters ρ and κ2 are in turn assigned independent Gaussian and Inverse-Gamma
µ µ
priors. Similarly,weuseconditionallyindependentzero-inflatedGaussianpriors,
(cid:0) (cid:1)
α | ω ,κ2 ∼ ω δ (α )+(1−ω )N α | 0,κ2 ,
j,k α,k α α,k 0 j,k α,k j,k α
6
whereδ (·)denotesthedegeneratemeasureplacingprobabilityoneatzero(forexample,seeHahn
0
et al., 2012), κ2 is given an Inverse-Gamma hyperprior, and ω is given a beta prior. The use of
α α,k
zero-inflatedpriorsforthecomponentsofα allowsustoexplicitlytestthesignificanceofspecific
j
dimensionsofthepolicyspace. Inparticular,notethatifα = 0forallj thenthek-thcomponent
j,k
ofthepolicyspacehasnoeffectonthevotingprobabilityandtheeffectivedimensionofthepolicy
space is no larger than K − 1. In our specification, ω is the prior probability that the k-th
α,k
dimensionofthepolicyspacedoesnotaffectthevotingbehaviorofthelegislatorsinmotionj;for
example, the prior probability that dimension k is irrelevant is simply ωJ . Finally, it is common
α,k
toassumethattheidealpointsareindependentlydistributedwithβ | η,Σ ∼ N(β | η,Σ).
i i
One challenge associated with the interpretation of spatial voting models is the invariance of
thepolicyspacetotranslations,rotations,reflexionsandrescaling. Acommonapproachtoenforce
identifiabilityinthesemodelsistoimposeconstraintsonthemeanandvarianceoftheidealpoints
sothatη = 0andΣ = I(whichremovesissuesrelatedtolocationandscale)alongwithconstraints
on the vectors α ,...,α so that α > 0 (which takes care of reflections) and α = 0 if k > j
1 K j,j j,k
(which addresses invariance to rotations). In this paper we follow the alternative approach of
fixing the position of K + 1 legislators in policy space (Rivers, 2003; Clinton et al., 2004). We
carefullychoosethelegislatorswhoseidealpointsareconstrainedtofacilitatetheinterpretationof
the model. For example, when modeling the U.S. Senate using a unidimensional policy space, we
ensure identifiability of the parameters by setting the ideal point of two legislators that are clearly
in opposite extremes of the political spectrum to +1 and −1 respectively. Examples include the
leaders of the Republican and Democratic parties, the whips of each party, or the two senators
perceivedtobethemostextreme. Theresultsareinvarianttotheidentityofthelegislators,aslong
astheyarereasonablywellseparatedandonoppositesides. Thischoiceenablesustointerpretthe
policy space in terms of a liberal-conservative ideology, at least to the extent in which parties are
ideologicallyopposites.
2.2 Assessing differences in ideal points
We consider now a joint model for two groups of motions, identified through (known) indicator
variablesγ ,...,γ suchthatγ ∈ {0,1}. As discussed intheintroduction,thegoalofthis model
1 J j
istotestwhethereachlegislator’svotingbehaviorexpressesdifferentrevealedpreferencesineach
ofthesesetsofmotions. Inourillustration,thesetwogroupsofmotionscorrespondtothosepassed
th
during the periods when either the Republicans or Democrats controlled the 107 Senate, so that
7
γ = 0 if the j-th motion was voted upon before senator Jeffords’ defection from the Republican
j
partyandγ = 1otherwise. Inthissettingitisnaturaltoextend(1)sothat
j
(cid:16) (cid:12) (cid:16) (cid:17)(cid:17)
y | γ ,µ ,α ,β ,β ∼ Bernoulli y (cid:12)G µ +αTβ , (2)
i,j j j j i,0 i,1 i,j(cid:12) j j i,γj
where β = (β ,...,β )T and β = (β ,...,β )T correspond to the (potentially
i,0 i,0,1 i,0,K i,1 i,0,1 i,0,K
distinct)idealpointsoflegislatorioneachofthetwogroupsofmotions.
The likelihood (2) allows (in principle) for different ideal points for each of the two groups of
motions. Ifweweretoassignindependentpriorstoβ andβ ,fittingthisjointmodelwouldbe
i,0 i,1
equivalenttofittingseparate(independent)modelstoeachofthetwogroups. Instead,wepropose
amixturepriorthatincorporatesthepossibilitythattheidealpointsarethesame,

(cid:0) (cid:1) (cid:0) (cid:1)
N β | η,Σ δ β ζ = 1
β ,β | ζ ,η,Σ ∼ i,0 βi,0 i,1 i (3)
i,0 i,1 i (cid:0) (cid:1) (cid:0) (cid:1)
N β | η,Σ N β | η,Σ ζ = 0,
i,0 i,1 i
The (unknown) auxiliary variables ζ ,...,ζ indicate whether the legislators express the same
1 I
revealedpreferencesonbothgroupsofmotionsornot. Inparticular,ζ = 1impliesthatβ = β
i i,0 i,1
(sothatthei-thlegislatorisabridge),whereasζ = 0impliesthatβ (cid:54)= β . Hence,ifζ = 1for
i i,0 i,1 i
all i then our approach is equivalent to fitting a single spatial voting model to all motions. On the
other hand, if ζ = 0 for all i then our model fits conditionally independent ideal points for each
i
legislator and group of motions. Hence, these latent indicators are the key parameters of interest
inouranalysis.
Because there is no overlap between the measures being voted upon on each period, identifi-
cation of model requires that 2(K + 1) constraints be introduced, K + 1 associated with the set
of ideal points β ,...,β , and another K + 1 associated with β ,...,β . While previous
1,0 I,0 1,1 I,1
authors have imposed independent sets of constraints on each of these two sets of ideal points, we
proceedbyfixingthepositionofK+1idealpointsandrequiringatleastK+1bridgelegislators,
whose ideal points remain fixed. The idea of using legislators whose preferences do not change
over time and get to vote on different groups of motions has been exploited by Shor et al. (2010)
andShorandMcCarty(2011)tocompareideologicalbiasesbetweenstatelegislaturesandtheU.S.
Congress. However, their approach relies on fixing the identity of bridge legislators beforehand,
whilethekeyfeatureofourmodelisthatweaimatidentifyingthebridgesaspartofouranalysis.
In order to make inferences on the identity of the bridge legislators while simultaneously en-
forcingthepresenceofatleastK+1bridgeswedefineajointpriorontheindicatorsζ ,...,ζ of
1 I
8
theform,
(cid:16) (cid:17) (cid:16) (cid:17)
Γ a+(cid:80)I ζ Γ b+I −(cid:80)I ζ I
i=1 i i=1 i (cid:88)
p(ζ ,...,ζ ) = , ζ = K +1,...,I,
1 I 1−(cid:80)K (cid:0)I(cid:1)Γ(a+s(cid:48))Γ(b+I −s(cid:48)) i
s(cid:48)=0 s(cid:48) i=1
whereaandbarehyperparameters. Tojustifythisprior,notethatitcanberewrittenas
(cid:32) (cid:33) (cid:32) (cid:33)
I I
(cid:88) (cid:88)
p(ζ ,...,ζ ) = p ζ ,...,ζ | ζ = s p ζ = s
1 I 1 I i i
i=1 i=1
(cid:16) (cid:17)
where p ζ ,...,ζ | (cid:80)I ζ = s = (cid:0)I(cid:1)−1 is uniform on all possible subsets of s bridge legisla-
1 I i=1 i s
torsand
(cid:32)(cid:88)I (cid:33) (cid:0)I(cid:1)Γ(a+s)Γ(b+I −s)
p ζ = s = s , s = K +1,...,I,
i 1−(cid:80)K (cid:0)I(cid:1)Γ(a+s(cid:48))Γ(b+I −s(cid:48))
i=1 s(cid:48)=0 s(cid:48)
corresponds to a truncated Beta-Binomial prior on the number of bridges. This last prior on the
numberofbridgelegislatorscanbemotivatedbynotingthatthenon-truncatedversioncorresponds
to the marginal distribution of a hierarchical model where (cid:80)I ζ | λ ∼ Bin(I,λ) and λ ∼
i=1 i
Beta(a,b) (which is the model suggested in Scott and Berger, 2006 and Scott and Berger, 2010 to
addressmultiplicityissues). Weintroducethetruncationtoensuretheminimumnumberofbridges
requiredforthemodeltobeidentifiable.
The hyperparameters a and b control the prior mean and variance on the number of bridge
legislators. Inparticular,settinga = 1andb = 1leadstoauniformdistributiondistributiononthe
numberofbridges,sothat
(cid:18) (cid:19) I
I 1 (cid:88)
p(ζ ,...,ζ | a = 1,b = 1) = , ζ = K +1,...,I.
1 I (cid:80)I ζ I −K i
i=1 i i=1
Largevaluesofa/bleadtoalargeexpectednumberofbridgesapriori,whilelargevaluesofa+b
leadtolowerpriorvariance.
(cid:0) (cid:1)
Finally,asinthepreviousSection,weassumeµ | ρ ,κ2 ∼ N ρ ,κ2 independentlyforallj,
j µ µ µ µ
use conditionally independent zero-inflated Gaussian priors, α | ω ,κ2 ∼ ω δ (α ) +
j,k α,k α α,k 0 j,k
(1 − ω )N(α | 0,κ2), and assign hyperpriors to the unknown parameters ρ , κ2, κ2 and
α,k j,k α µ µ α
ω ,...,ω .
α,1 α,K
9
2.3 Alternative prior specifications
Alternative specifications for the joint prior p(β ,β | ζ ) are possible. For example, one of the
i,0 i,1 i
refereessuggestedsettingβ = β +∆ whereβ | η,Σ ∼ N(β | η,Σ)and
i,1 i,0 i i,0 i,0

δ (∆ ) ζ = 1
0 i i
∆ | ζ ∼
i i
N(∆ | 0,Ω) ζ = 0,
i i
whichisinspiritofClintonetal.(2004). Notethatthisspecificationimplies
(cid:18) (cid:19)
β (cid:16)(cid:0)η(cid:1) (cid:16)Σ Σ (cid:17)(cid:17)
βi,0 | ζi = 0 ∼ N η , Σ Σ+Ω , (4)
i,1
asopposedtothatbyimpliedby(3),
(cid:18) (cid:19)
β (cid:16)(cid:0)η(cid:1) (cid:16)Σ 0(cid:17)(cid:17)
βi,0 | ζi = 0 ∼ N η , 0 Σ . (5)
i,1
The main differences between these two priors are 1) the marginal variances of β and β
0,i 1,i
are different from each other in (4) (and in particular, β is forced to have a higher variance than
1,i
β ), and 2) a priori, there is a non-zero correlation between both ideal points in (4). For us, the
0,i
firstfeatureisparticularlyproblematic. Itisnotcleartouswhyweshouldassumeapriorithatone
set of ideal points has a higher variance than the other, specially if our working assumption is that
bothsetsofidealpointsliveinthesamepolicyspace. Furthermore,althoughintheillustrationwe
discussinSection4thereisanaturalorderingtothetwogroupsofmeasures(itwouldseemnatural
tocentertheidealpointsaftertheswitcharoundtheidealpointsbefore),thatisnotthecaseinother
interesting applications. Hence, our preference for the exchangeable model (5), which treats both
groups of measures identically. About the second feature of the proposed prior, we note that (5)
could be easily extended to include (positive) correlations between both groups of measures when
ζ = 0. However, including correlations can lead to identifiability issues: in the limit (when the
i
value of the correlations approach one) the conditional prior on ζ = 0 becomes identical to the
i
prior conditional on ζ = 1. Hence, working with a model that assumes no correlation not only
i
simplifieselicitation,butitcanbeconsideredasthemostfavorablepriortodifferences(withinthe
classweconsider).
3 Hyperpriors and Computation
We discuss now the specification of priors for the unknown hyperparameters in our model. For
the hyperprior on the intercepts µ ,...,µ we let ρ ∼ N(ρ | 0,1) and κ2 ∼ IGam(κ2 | 2,1),
1 J µ µ µ µ
10