# Operationalizing Legislative Bodies:
A Methodological and Empirical Perspective

Carolina Luque, Universidad Ean<EMAIL_ADDRESS>
Juan Sosa, Universidad Nacional<EMAIL_ADDRESS>
###### Abstract

This manuscript extensively reviews applications, extensions, and models
derived from the Bayesian ideal point estimator. We primarily focus our
attention on studies conducted in the United States as well as Latin America.
First, we provide a detailed description of the Bayesian ideal point
estimator. Next, we propose a new taxonomy to synthesize and frame technical
developments and applications associated with the estimator in the context of
North American and Latin American governing bodies. The literature available
in Latin America allows us to conclude that few legislatures in the region
have been analyzed using the methodology under discussion. Also, we highlight
those parliaments of Latin America embedded in democratic presidential systems
as novel scenarios for operationalizing the electoral behavior of legislative
bodies through nominal voting data. Our findings show some alternatives for
future research. Finally, to fix ideas and illustrate the capabilities of the
Bayesian ideal point estimator, we present an application involving the
Colombian House of Representatives 2010–2014.

Keywords: Spatial voting models, Bayesian ideal point estimator, roll call
data, parliamentary electoral behavior, Markov Chain Monte Carlo methods.

##  1 Introduction

The quadratic Bayesian spatial voting model (also known as the Bayesian ideal
point estimator or IDEAL; Jackman, 2001, Clinton et al., 2004) is a valuable
method for identifying latent (unobserved) traits of political actors from
nominal voting data. This model is compatible with the nature of scientific
research in the Social Sciences (Jackman,, 2004, 2009), and also, it is
flexible and powerful enough for recognizing political preferences of voters
(Carlin and Louis,, 2008; Clinton and Jackman,, 2009). Moreover, some authors
point out that it is an essential tool for providing empirical evidence on
different voting phenomena in modern Political Science (e.g., Clinton et al.,,
2004; Yu and Rodríguez,, 2021; Moser et al.,, 2021).

IDEAL and its derivatives have been widely used to analyze the electoral
behavior of the United States Parliament. However, there are a limited number
of applications in other contexts, such as Latin America. Therefore, there is
a legitimate opportunity to characterize electoral phenomena of legislatures
in Latin America (e.g., Tsai,, 2020) by taking IDEAL as a reference framework
due to its extensive usage in the North American Congress. We share this
position with some authors, who see the theoretical and methodological
developments of the legislative voting behavior of the United States Congress
as a basis for advancing the quantitative understanding of parliamentary
electoral conduct in scenarios with limited empirical evidence (Gamm and
Huber,, 2002). Furthermore, the available literature shows that this path
inspires adaptations and applications of IDEAL in different parliamentary
electoral settings (e.g., Zucco,, 2013; McDonnell,, 2017; Tsai,, 2020; Ribeiro
et al.,, 2021).

This manuscript extensively reviews the available literature on models,
extensions, and applications derived from IDEAL in North American and Latin
American legislative settings. Our goal is to provide a comprehensive and up-
to-date overview of statistical foundations, applications, and developments on
the topic under consideration from 2001-2022. We present our findings by
proposing a brand new taxonomy on the matter, considering the parliamentary
voting phenomena underlying to use of the estimator. From a quantitative point
of view, we focus on substantive hypotheses and methodological aspects that
allow us to analyze legislative electoral behavior through roll call data.

To the best of our knowledge, no previous research discusses and synthesizes
methodological aspects and applications of the estimator in the framework of
parliamentary voting in both the United States and Latin America. Our work is
relevant to the scientific community for several reasons. First, it shows the
fusion between political theory and statistical and computational principles
as a fundamental element in implementing a spatial voting model in a
particular legislative reality. Second, it reveals the limitations of the
standard Bayesian ideal point estimator and shows them as an opportunity to
promote methodological and applied research in the context of governing
bodies. Finally, it provides evidence of emerging scientific research niches
in different legislative spaces. Mainly in Latin America, where legislatures
rooted in presidential democratic systems have different dynamics than the
North American parliament (e.g., Pereira and Mueller, 2004a, ; Jones and
Hwang, 2005a, ; Zucco,, 2013; Ribeiro et al.,, 2021).

This document is structured as follows. Section 2 shows spatial voting model
terminology and the relevance of the ideal Bayesian point estimator through a
historical review of studies in legislative committees. Section 3 presents
theoretical references related to the Bayesian ideal point estimator’s
specification, identification, and implementation. Section 4 exhibits
applications, adaptations, and extensions of the ideal point estimator in the
context of North and Latin America. Section 5 illustrates the case of the
Colombian House of Representatives 2010–2014. Finally, Section 6 synthesizes
our main contributions and discusses some alternatives for future research.

##  2 Voting models: Terminology and developments

The collective election is a phenomenon of interest in the political context
because voters’ preferences are not homogeneous (Krehbiel,, 1988; Clinton,,
2012). For example, in the case of deliberative bodies such as congress,
legislators typically show different judgments about public policies under
debate, which unequivocally leads to a collective conflict of perspectives.
The spatial voting models allow us to characterize this phenomenon, assuming
that different subjects (legislators, judges, citizens, among others) adopt
dissimilar preferences regarding choice alternatives (policies, proposals,
motions, among others).

Most of the theoretical work on these models proposes normal or quadratic
utility functions to characterize the preferences of political actors (to
measure characteristics essential for understanding the causes and
consequences of politics, Clinton,, 2012; Carroll et al.,, 2013). Such an
assumption has its roots in the fields of Psychology and Mathematics. On the
one hand, studies in psychology indicate that the response function of
individuals when they judge similarities between stimuli or express their
preferences is approximately normal (Poole,, 2005). On the other hand, studies
in Mathematics show that the normal distribution can be approximated by a
quadratic expression through a second-order Taylor polynomial (Carroll et al.,
2009a, ). For an additional in-depth discussion about other utility structures
in spatial voting models, see Carroll et al., (2013) and Tiemann, (2019).

Assuming a quadratic utility function in a spatial voting model is not only a
convenient mathematical abstraction but also a standard formulation that
describes the response of a subject facing a choice. Thus, each individual has
an utility function centered around an ideal point (optimal alternative, see
Clinton,, 2012) that represents his/hers preferences. Hence, the utility
decreases as the distance (metric defined on the reference space) between
his/hers ideal point and the option of choice increases. In this way, the
subjects vote for the alternatives closest to their ideal point according to a
specific stochastic mechanism.

A spatial voting model fully characterizes the behavior of individuals facing
an electoral process. Additionally, it allows visualizing the political space
(reference space to represent the ideal points of individuals) in specific
contexts such as that of a legislature. The proximity of ideal points in the
political space is evidence of similarities in legislators’ voting records. It
indicates the presence of latent traits (e.g., ideological preferences) that
underlie their voting behavior (Poole,, 2005). Such an abstraction is
advantageous because it allows for studying several issues, including the
evolution of political parties, strategic voting, location of minorities,
electoral interests and incentives, and relations between institutions.
However, for this representation to make sense and be interpretable in a
political framework, besides the statistical foundations of the model, it
requires a deep understanding of the political system of the institutions
under analysis (Poole and Rosenthal,, 1985).

###  2.1 Formalizing electoral conduct of political actors

Spatial voting models are involved in a quite extensive literature. The
foundational work of Black et al., (1958) and Downs, (1957) provide important
advances for developing voting theory. Nevertheless, such work does not
provide a formal mathematical structure to test political theories (Poole and
Rosenthal,, 1985). The first spatial voting models that rigorously
operationalize electoral behavior appear almost a decade later, by identifying
the position of the political space that candidates must adopt to win an
electoral contest in a majority election (Davis and Hinich,, 1965; Davis et
al.,, 1970; Hinich and Ordeshook,, 1970). These investigations formally
introduce the concept of the utility function, pointing out its relevance in
penalizing the discrepancy between the position of a voter and the voting
alternatives as well as electorate preferences. However, these approaches do
not consider a random component in voting procedures (Poole,, 2005).

The interest of researchers in proposing a rigorous mathematical structure
allowing them to design measurement instruments to contrast political theory
favored the development of several studies in subsequent decades. For example,
in the late 1970s, McKelvey et al., (1978) investigated electoral behavior in
committees through cooperative games (Aumann,, 1964), which carries out the
analysis in a new direction, but whose findings are not sufficient to support
general patterns of strategic behavior (Arrow,, 1990). Subsequently, based on
agenda theory, Romer and Rosenthal, (1978) and Shepsle, (1979) propose to
explain the role that structures and procedures play in shaping the results of
legislative voting. These authors provide initial results on the influence of
the order of deliberation of motions on legislative voting behavior. At about
the same time, Cahoon et al., (1978) and Wolters, (1978) use multidimensional
scaling methods to represent the responses of the voting process (dichotomous
or polytomous) as low-dimensional continuous variables in Euclidean space.
Subsequently, other studies investigate fundamental aspects of probabilistic
voting behavior (Manski,, 1977; Coughlin and Nitzan,, 1981) and
predictive/latent dimensions (Enelow and Hinich,, 1984).

The probabilistic theory of electoral behavior considers votes as random
variables whose behavior depends on stochastic utility functions. Such
functions incorporate the distance between the voter’s ideal point and the
voting alternative and consider a random term (McFadden,, 1976; Poole and
Rosenthal,, 1984) reconciling systematic spatial utility differences
(intractable voting patterns) with probabilities to vote in favor of a given
alternative. Typically, random errors are assumed to follow a normal or
logistic distribution, leading to a probit or logit link function,
respectively (Jackman,, 2004; Carroll et al., 2009a, ). These two random
mechanisms lead to similar one-dimensional results with no more than scale
differences (Hahn and Soyer,, 2005; Carroll et al., 2009a, ; Lofland et al.,,
2017; Luque and Sosa,, 2022).

Predictive dimension methods frame voters and election objects in the same
low-dimensional space to explain (dis)similarities between voters as a
function of distances between points in the political space (Weisberg and
Rusk,, 1970; Poole and Rosenthal,, 1984). Actors’ voting patterns interpret
dimensions of political space in terms of issues, policies, or belief systems
(Cahoon et al.,, 1978; Poole and Rosenthal,, 1985; Hinich and Munger,, 1996;
Hinich et al.,, 1997). The number of dimensions of political space is a
technical question. However, some authors have pointed out that it is common
to identify low-dimensional spaces in the study of electoral behavior. Poole,
(2005) argues that only a few dimensions are required to capture the structure
of the North American parliamentary voting behavior because legislators
typically vote based on the underlying policies of the political group they
represent (left-right, Republican-Democrat). Furthermore, perceptual studies
in psychology support the idea of a small number of dimensions underlying
individual choice behavior, as people have a limited ability to perceive
different objects (Miller,, 1956) and a tendency to confront a group against
another group (Tajfel,, 1981).

###  2.2 Roll call data

Spatial voting models based on data reduction techniques such as
multidimensional scaling (Cahoon et al.,, 1978), factor analysis (Brazill and
Grofman,, 2002), and principal component analysis (De Leeuw,, 2006) do not
provide a methodological framework for modeling directly binary responses of
individuals based on the parameters of interest. Then, these methods do not
allow practitioners recovering voters’ ideal points nor modeling the
probability of voting either in favor of or against a given issue (Jackman,,
2001). The interest in explaining the individual behavior of deputies led to
the development of parliamentary voting models in the 1980s (Poole and
Rosenthal,, 1985) focusing on estimating the ideal points of legislative
actors and recovering their latent characteristics (political preferences). In
essence, these are generalized linear models (McCullagh,, 2018) that rely on
iterative methods such as parametric bootstrap and Markov chain Monte Carlo
(e.g., Clinton et al.,, 2004; Martin and Quinn,, 2002; Lewis and Poole,, 2004;
Carroll et al., 2009a, ; Carroll et al., 2009b, ) to estimate and infer
political preferences from roll call data or nominal votes (Poole and
Rosenthal,, 1985, 1987; Poole,, 2005).

By means of nominal votes legislators reveal their positions to other deputies
and the general public (Mayhew,, 1974). This political activity provides data
that allows social scientists to estimate ideal points and retrieve the
political preferences of political actors (Alemán et al.,, 2009). Research
using roll call data to learn about legislative decisions shows that these
data are helpful to calculate correlations between parliamentary members
through factor and cluster analysis methods, which are relevant to identifying
aggregate patterns. However, its use to predict individual voting decisions is
popular as well (e.g., MacRae,, 1952, 1958, 1965; VanDoren,, 1990; Poole,,
2005; Clinton,, 2012; Binding and Stoetzer,, 2022). Most studies on nominal
voting only consider binary data, i.e., data instances consisting of two
alternatives, in favor of or against a motion, known as new alternative and
status quo, respectively (Carroll et al., 2009a, ). Few studies in the
literature consider abstentions (Thurner,, 2000; Poole,, 2005; Rosas et al.,,
2015). On the other hand, recent studies describe the implications and
limitations of using roll call data to test political theory (e.g., Roberts,,
2007; Ainsley et al.,, 2020). Additionally, other research use this type of
data with textual data to analyze legislative bodies from different data
sources (e.g., Lauderdale and Clark,, 2014; Kim et al.,, 2018).

In recent years, the use of roll call data to analyze parliamentary electoral
behavior has increased substantially (e.g., Clinton et al.,, 2004; Hix et
al.,, 2005; Jones and Hwang, 2005a, ; Clinton,, 2012; Zucco,, 2013; Binding
and Stoetzer,, 2022; Seabra and Mesquita,, 2022; Rasmussen,, 2022; Hansford et
al.,, 2022; Grier et al.,, 2022). In particular, the analysis of nominal votes
of the legislative chambers of Latin American countries has become quite
popular (see Table 1). Recent access to these data has made their analysis
novel and relevant to regional legislative phenomena studies. Unfortunately,
roll call data are not widely available in Latin America (Morgenstern,, 2003).
For example, Zucco, (2013) confirms the existence of a reduced number of roll
calls to analyze parliamentary electoral behavior in the legislative chambers
of Uruguay. Unlike the United States, in some countries in the region, roll
call voting is not always the most frequent task (see Jones and Hwang, 2005a,
; Ribeiro et al.,, 2021), and yet, it is becoming a common practice to
facilitate access to this type of data (see McDonnell et al.,, 2019). Even
when roll calls are available, these data are not systematized. Compiling them
is a laborious task involving gathering information from external sources
(local newspapers, websites, among others) (Morgenstern,, 2003; Luque,, 2021).

|  |  | Roll call data | Legislators |
---|---|---|---|---|---
Country | Period | House | Available | Analyzed | Available | Analyzed | Reference
Argentina | 1989–2001 | Lower | 415 |  |  |  | Morgenstern, (2003)
| 1989–2003 | Lower | 473 |  |  |  | Jones and Hwang, 2005a
| 1993–1995 | Lower | 64 |  | 256 |  | Rosas and Shomer, (2008)
| 1989–2007 | Lower | 1193 |  |  |  | Jones et al., (2009)
| 2002–2007 | Congress | 110 | 31 | 618 | 356 | Onuki et al., (2009)
| 2008–2009 | Lower | 251 |  |  |  | Alemán et al., (2018)
| 1993–2017 | Lower |  |  |  |  | Clerici, (2021)
Brazil | 1991–1998 | Lower | 623 |  |  |  | Morgenstern, (2003)
| 1989–2008 | Congress |  |  |  |  | McDonnell, (2017)
| 1989–2010 | Lower | 2332 | 1806 |  |  | Zucco and Lauderdale, (2011)
| 2003–2006 | Lower | 421 |  | 703 |  | Tsai, (2020)
Chile | 2002–2006 | Lower | 157 | 36 | 120 | 118 | Onuki et al., (2009)
| 2004–2006 | Upper | 313 | 118 |  | 49 | Alemán, (2008)
Colombia | 2006–2015 | Congress | 11600 |  | 650 |  | Morales, (2021)
| 2010–2014 | Upper | 417 | 417 | 110 | 91 | Luque and Sosa, (2022)
Paraguay | 2003–2012 | Lower | 147 |  | 167 |  | Ribeiro et al., (2021)
Uruguay | 1985–1994 | Congress | 63 |  |  |  | Morgenstern, (2003)
| 1985–2005 | Upper | 125 |  |  |  | Zucco, (2013)
Table 1: Available literature to analyze roll call data in Latin American
legislatures. Some studies employ selection criteria for selecting either roll
calls or legislators (e.g., Onuki et al.,, 2009; Zucco and Lauderdale,, 2011;
Alemán,, 2008; Luque and Sosa,, 2022). Recently, the nominal votes of
Argentina, Brazil, and Colombia are available at
https://votaciones.hcdn.gob.ar/, https://bancodedadoslegislativos.com.br/, and
https://congresovisible.uniandes.edu.co/, respectively.

Morgenstern, (2003) manifests that some roll call voting records of the
legislative chambers in Brazil, Argentina, Chile, and Uruguay have been
available since the early 90s. For other countries in the region, these data
have been made available later. For example, in the case of Colombia,
parliamentary voting records have been collected since 2006 (Carroll and
Pachón,, 2016), and until very recently, they have been used to implement
spatial voting models to retrieve political preferences of deputies (Luque and
Sosa,, 2022). In Latin America, the analysis of the lower house (House of
Representatives) is more frequent compared to the upper house (Senate) (see
Table 1). Few studies analyze the legislative behavior of the entire congress
without discriminating by chambers. Under this scenario, we argue that Latin
American legislatures are a young field for the empirical analysis of
parliamentary electoral behavior based on roll call data.

###  2.3 Operationalization of individual electoral behavior

Poole and Rosenthal, (1985) provide the building blocks that allow researchers
to estimate ideal points and latent variables considering a political space
from roll call data. Two popular models emerge from such foundational work:
NOMINATE and IDEAL (technical details about the latter are available in
Section 3). In principle, both models are used to analyze the behavior of
deputies in the Senate and House of Representatives, mainly from the United
States. However, their application has been extended to other non-legislative
contexts such as the United Nations (e.g., Voeten,, 2000, 2013; Bailey et
al.,, 2017; Seabra and Mesquita,, 2022) and courtrooms (e.g., Martin and
Quinn,, 2002; Hansford et al.,, 2022).

NOMINATE is a frequentist spatial voting model based on an utility function
with a random component (see Poole and Rosenthal,, 2001; Poole,, 2005; Carroll
et al., 2009b, ; Caughey and Schickler,, 2016, for more details). IDEAL is the
Bayesian counterpart of NOMINATE (Clinton and Jackman,, 2009). The latter also
incorporates random utility functions and extensively uses simulation-based
approaches (e.g., Markov Chain Monte Carlo, MCMC) to estimate the legislators’
ideal points (Jackman,, 2001; Clinton et al.,, 2004). There are comparative
studies that highlight the similarities and differences between these tow
models, as well as their advantages and disadvantages in modeling roll call
data to retrieve political preferences from parliamentarians (see Carroll et
al., 2009a, ; Clinton and Jackman,, 2009). Research in this direction
highlights the flexibility of IDEAL to test substantive hypotheses (Clinton et
al.,, 2004). Furthermore, IDEAL is a more straightforward estimator than
NOMINATE because it only requires the Bayes theorem for estimation and
inference, making it a more generalizable method (Clinton and Jackman,, 2009).
Also, IDEAL is an estimator that has a direct correspondence with the nature
of scientific inquiry in the Social Sciences (e.g., Jackman,, 2004, 2009).

Thus, IDEAL has been extended in various ways. For example, Martin and Quinn,
(2002) postulate a dynamic version that allows ideal points to change
systematically over time. Quinn, (2004) extends the model to handle non-binary
data (including continuous data). Treier and Jackman, (2008) formulate a
variant of the model to measure the level of democracy in countries, while
Rosas et al., (2015) do the same to analyze the behavior of the vote under
informative abstentions. In another way, Yu, (2020) and Yu and Rodríguez,
(2021) develop spatial voting models based on non-Euclidean metrics. Also,
there are applications of the model to study electoral behavior in
legislatures other than the North American one. For example, Zucco, (2013)
analyzes executive-legislative relations in Uruguay between 1985 and 2005 in
the context of presidential coalitions. Furthermore, Tsai, (2020) studies the
Brazilian Chamber of Deputies between 2003 and 2006, highlighting the effect
of political incentives on electoral behavior.

The literature makes it explicit that IDEAL is a dominant methodological and
theoretical instrument in modern Political Science (Yu,, 2020; Moser et al.,,
2021). Moreover, all the works presented above show that the analysis of the
political preferences of legislators from nominal votes is an area of active
research. Therefore, IDEAL constitutes a fundamental model worth
investigating.

##  3 IDEAL: Bayesian ideal point estimator

###  3.1 Modeling

Roll call data arise when $n$ legislators vote on $m$ motions (such as bills
or legislative initiatives). Thus, each legislator $i\in\\{1,\ldots,n\\}$
takes a position in favor of (yea) or against (nay) motion
$j\in\\{1,\ldots,m\\}$. Hence, we let $y_{i,j}\in\\{0,1\\}$ be the vote cast
by legislator $i$ on motion $j$, with $y_{i,j}=1$ if such a vote turns out in
favor of the motion, and $y_{i,j}=0$ otherwise. These votes in favor of or
against motion $j$ are assumed to be points in a $d$-dimensional Euclidean
space, known aspolitical space, and are denoted by $\bm{\psi}_{j}$ and
$\bm{\zeta}_{j}$, respectively.

It is assumed that all the legislators have a political preference, i.e., each
legislator $i$ has a latent (unobserved) factor
$\bm{\beta}_{i}\in\mathbb{R}^{d}$ known as ideal point. Thus, decisions are
made based on a quadratic utility function given by
$U_{i}(\bm{\psi}_{j})=-\parallel\bm{\psi}_{j}-\bm{\beta}_{i}\parallel^{2}+\eta_{i,j}$
and
$U_{i}(\bm{\zeta}_{j})=-\parallel\bm{\zeta}_{j}-\bm{\beta}_{i}\parallel^{2}+\upsilon_{i,j}$,
where $U_{i}(\bm{\psi}_{j})$ and $U_{i}(\bm{\zeta}_{j})$ are the corresponding
profits associated with legislator $i$ for voting in favor of or against
motion $j$, respectively, and $\eta_{i,j}$ and $\upsilon_{i,j}$ are
independent random deviations resulting from the uncertainty involved during
the decision processes, such that $\textsf{E}(\eta_{i,j}-\upsilon_{i,j})=0$
and $\textsf{Var}(\eta_{i,j}-\upsilon_{i,j})=\sigma^{2}_{j}$.

Rational choice theory (e.g., Yu and Rodriguez,, 2019) states that under the
previous setting, legislator $i$ votes in favor of motion $j$ if and only if
$U_{i}(\bm{\psi}_{j})>U_{i}(\bm{\zeta}_{j})$. Hence, assuming that
$\eta_{i,j}-\upsilon_{i,j}$ is normally distributed, i.e.,
$(\eta_{i,j}-\upsilon_{i,j})\mathrel{\overset{\makebox[0.0pt]{\mbox{\tiny
ind}}}{\sim}}\textsf{N}(0,\sigma^{2}_{j})$, it follows that

$\textsf{Pr}(y_{i,j}=1\mid\bm{\zeta}_{j},\bm{\psi}_{j},\sigma_{j},\bm{\beta}_{i})=\textsf{Pr}(\epsilon_{i,j}<\mu_{j}+\bm{\alpha}_{j}^{\textsf{T}}\bm{\beta}_{i})=\Phi(\mu_{j}+\bm{\alpha}_{j}^{\textsf{T}}\bm{\beta}_{i})$

where $\epsilon_{i,j}=(\upsilon_{i,j}-\eta_{i,j})/\sigma_{j}$ is the random
component,
$\mu_{j}=(\bm{\zeta}_{j}^{\textsf{T}}\bm{\zeta}_{j}-\bm{\psi}_{j}^{\textsf{T}}\bm{\psi}_{j})/\sigma_{j}$
is the approval parameter (representing the basal probability of a vote in
favor of motion $j$),
$\bm{\alpha}_{j}=2(\bm{\psi}_{j}-\bm{\zeta}_{j})/\sigma_{j}$ is the
discrimination parameter (representing the effect that ideal points have upon
the probability of a vote in favor of motion $j$), and $\Phi(\cdot)$ is the
cumulative distribution function of the standard normal distribution.

The previous specification fully characterizes the probability of observing a
positive vote since
$y_{i,j}\mid\mu_{j},\bm{\alpha}_{j},\bm{\beta}_{i}\mathrel{\overset{\makebox[0.0pt]{\mbox{\tiny
ind}}}{\sim}}\textsf{Bernoulli}(\Phi(\mu_{j}+\bm{\alpha}^{\textsf{T}}_{j}\bm{\beta}_{i}))$.
The likelihood associated with such latent factor model is given by

$p(\mathbf{Y}\mid\\{\mu_{j}\\},\\{\bm{\alpha}_{j}\\},\\{\bm{\beta}_{i}\\})=\prod_{i=1}^{n}\prod_{j=1}^{m}\Phi(\mu_{j}+\bm{\alpha}_{j}^{\textsf{T}}\bm{\beta}_{i})^{y_{i,j}}\left[1-\Phi(\mu_{j}+\bm{\alpha}_{j}^{\textsf{T}}\bm{\beta}_{i})\right]^{1-y_{i,j}}\,,$

where $\mathbf{Y}=[y_{i,j}]$ is a binary rectangular matrix of size $n\times
m$. Finally, in order to complete the specification of the model and carry out
full Bayesian inference, it is required to specify a joint prior distribution
on the parameter space. A computationally convenient alternative that works
well in practice consists in letting
$(\mu_{j},\bm{\alpha}_{j})\mid\bm{a},\mathbf{A}\mathrel{\overset{\makebox[0.0pt]{\mbox{\tiny
iid}}}{\sim}}\textsf{N}_{d+1}(\bm{a},\mathbf{A})$ and
$\bm{\beta}_{i}\mid\bm{b}_{i},\mathbf{B}_{i}\mathrel{\overset{\makebox[0.0pt]{\mbox{\tiny
ind}}}{\sim}}\textsf{N}_{d}(\bm{b}_{i},\mathbf{B}_{i})$, where
$\bm{a},\mathbf{A},\bm{b}_{i}$, and $\mathbf{B}_{i}$ are the model
hyperparameters (known fixed quantities). Of course, more complex hierarchical
specifications are also possible.

###  3.2 Identifiability

The model parameters in are not identifiable. Specifically, note that
Euclidean distances among ideal points $\bm{\beta}_{i}$ and the voting
alternatives $\bm{\psi}_{j}$ and $\bm{\zeta}_{j}$ remain invariant under any
translation, rotation, or reflection of the political space. Such a geometric
occurrence ensures that both discrimination parameters and ideal points are
not distinguishable for any voting pattern $\mathbf{Y}$. For instance,
consider a rotation of the political space through an $d\times d$ orthogonal
matrix $\mathbf{Q}$, i.e., $\mathbf{Q}^{\textsf{T}}\mathbf{Q}=\mathbf{I}_{d}$.
Then,
$(\mathbf{Q}\bm{\alpha}_{j})^{\textsf{T}}(\mathbf{Q}\bm{\beta}_{i})=\bm{\alpha}_{j}^{\textsf{T}}\bm{\beta}_{i}$,
for all $i$ and all $j$, and therefore,
$p(\mathbf{Y}\mid\\{\mu_{j}\\},\\{\bm{\alpha}_{j}\\},\\{\bm{\beta}_{i}\\})=p(\mathbf{Y}\mid\\{\mu_{j}\\},\\{\mathbf{Q}\bm{\alpha}_{j}\\},\\{\mathbf{Q}\bm{\beta}_{i}\\})$.
Such a phenomenon is also typical of latent space models for social networks
(e.g., Sosa and Betancourt,, 2022).

The lack of identifiability require us to establish parameter constraints. A
popular alternative consists in imposing restrictions on the mean and variance
of the ideal points. Specifically, in the same spirit of Jackman, (2004) and
Lofland et al., (2017), letting $\bm{b}_{i}=\bm{0}_{d}$ and
$\mathbf{B}_{i}=\mathbf{I}_{d}$, for $i=1m,\ldots,n$, is useful to overcome
translation and scale issues. Furthermore, it is convenient fixing the
position of $d+1$ legislator, known as anchor legislators, with known (but
distinctive!) political patterns in the political space, since it allows the
model to differentiate legislative tendencies (Rivers,, 2003; Clinton et al.,,
2004).

###  3.3 Prior elicitation and computation

Along the lines of Clinton et al., (2004), we recommend to set
$\bm{a}=\bm{0}_{(d+1)}$ and $\mathbf{A}=\sigma^{2}\mathbf{I}_{(d+1)}$ with
$\sigma^{2}$ an arbitrarily large constant (e.g., $\sigma^{2}=25$) in order to
assign a zero-centered non-informative prior distribution to $\mu_{j}$ and
$\bm{\alpha}_{j}$, for $j=1,\ldots,m$, aiming to emulate roughly the behavior
of a diffuse state of information. This choice is quite reminiscent of the
hyperparameter elicitation in a standard linear regression model when there is
no need of informative or empirical alternatives, such as the unit information
prior (Kass and Wasserman,, 1996) or the $g$-prior (Zellner,, 1986). Finally,
notice that it is not required to set a large variance a priori for the ideal
points. Actually, all what is required is a prior notion of scale for this set
of parameters.

###  3.4 Posterior Inference

Considering data from $n$ legislators on $m$ motions, we have $dn+m(d+1)$
unknown model parameters to estimate. Thus, the posterior distribution is
framed in a high dimensional space, which makes it analytically intractable.
Even though other deterministic approaches recognized by their efficiency are
available (e.g., variational approximations; Ormerod and Wand, 2010), we
strongly suggest Markov Chain Monte Carlo algorithms (MCMC; e.g., Gamerman and
Lopes,, 2006) to approximate the posterior distribution. In particular, by
means of a Gibbs sampler, we are able to produce a sequence of dependent but
approximately independent draws from the posterior distribution, through
iterative sampling from the full conditional distributions of
$\mu_{j},\bm{\alpha}_{j}$ and $\bm{\beta}_{i}$. Hence, make it possible to
compute point and interval estimates from the corresponding empirical
distributions.

Following Albert and Chib, (1993), we recommend to increase the parameter
space by introducing a set of auxiliary variables $\\{z_{i,j}\\}$ such that
$y_{i,j}\mid z_{i,j}=1$ if $z_{i,j}>0$, and $y_{i,j}\mid z_{i,j}=0$ if
$z_{i,j}\leq 0$, where
$z_{i,j}\mid\mu_{j},\bm{\alpha}_{j},\bm{\beta}_{i}\mathrel{\overset{\makebox[0.0pt]{\mbox{\tiny
ind}}}{\sim}}\textsf{N}(\mu_{j}+\bm{\alpha}_{j}^{\textsf{T}}\bm{\beta}_{i},1)$.
Notice that we obtain exactly the original Bernoulli model specified above
when integrating the $z_{i,j}$ variables out. This model reformulation makes
it easier to draw directly from the full conditional distributions of
$\mu_{j}$, $\bm{\alpha}_{j}$, and $\bm{\beta}_{i}$.

Let
$\mathbf{\Theta}=(\mu_{1},\ldots,\mu_{m},\bm{\alpha}_{1},\ldots,\bm{\alpha}_{m},\bm{\beta}_{1},\ldots,\bm{\beta}_{n})$
be the full set of model parameters. The posterior distribution of
$\mathbf{\Theta}$ is

$\displaystyle p(\mathbf{\Theta}\mid\mathbf{Y})$
$\displaystyle\propto\prod_{i=1}^{n}\prod_{j=1}^{m}p(y_{i,j}\mid
z_{i,j})\times\prod_{i=1}^{n}\prod_{j=1}^{m}\textsf{N}(z_{i,j}\mid\mu_{j}+\bm{\alpha}_{j}^{\textsf{T}}\bm{\beta}_{i},1)$
$\displaystyle\hskip
99.58464pt\times\prod_{j=1}^{m}\textsf{N}_{d+1}(\mu_{j},\bm{\alpha}_{j}\mid\bm{a},\mathbf{A})\times\prod_{i=1}^{n}\textsf{N}_{d}(\bm{\beta}_{i}\mid\bm{b}_{i},\mathbf{B}_{i})\,.$

Moreover, let $\phi^{(b)}$ denote the state of parameter $\phi$ in the $b$-th
iteration of the Gibbs sampling algorithm, for $b=1,\ldots,B$. Then, such an
algorithm in this case is as follows:

  1. 1.

Choose a starting configuration for each model parameter, say $z_{i,j}^{(0)}$,
$\mu_{j}^{(0)}$, $\bm{\alpha}_{j}^{(0)}$, and $\bm{\beta}_{i}^{(0)}$, for
$i=1,\ldots,n$ and $j=1,\ldots,m$.

  2. 2.

Update $z_{i,j}^{(b-1)}$, $\mu_{j}^{(b-1)}$, $\bm{\alpha}_{j}^{(b-1)}$, and
$\bm{\beta}_{i}^{(b-1)}$, for $i=1,\ldots,n$ and $j=1,\ldots,m$, cycling until
convergence:

    1. (a)

Sample $z_{i,j}^{(b)}$ from
$p(z_{i,j}\mid\mu_{j}^{(b-1)},\bm{\alpha}_{j}^{(b-1)},\bm{\beta}_{i}^{(b-1)},y_{i,j})$,
where

$p(z_{i,j}\mid\mu_{j},\bm{\alpha}_{j},\bm{\beta}_{i},y_{i,j})=\left\\{\begin{matrix}\textsf{TN}_{(0,+\infty)}(z_{i,j}\mid\mu_{j}+\bm{\alpha}_{j}^{\textsf{T}}\bm{\beta}_{i},1)&\text{if}&y_{i,j}=1\,,\\\
\\\
\textsf{TN}_{(-\infty,0]}(z_{i,j}\mid\mu_{j}+\bm{\alpha}_{j}^{\textsf{T}}\bm{\beta}_{i},1)&\text{if}&y_{i,j}=0\,.\end{matrix}\right.$

    2. (b)

Sample $(\mu_{j},\bm{\alpha}_{j})^{(b)}$ from
$p(\mu_{j},\bm{\alpha}_{j}\mid\\{\bm{\beta}_{i}^{(b-1)}\\},\\{z_{i,j}^{(b)}\\})$,
where

$p(\mu_{j},\bm{\alpha}_{j}\mid\\{\bm{\beta}_{i}\\},\\{z_{i,j}\\})=\textsf{N}_{d+1}(\mu_{j},\bm{\alpha}_{j}\mid\bm{c}_{j},\mathbf{C})\,,$

with
$\bm{c}_{j}=(\mathbf{A}^{-1}+\mathbf{E}^{\textsf{T}}\mathbf{E})^{-1}\left(\mathbf{A}^{-1}\bm{a}+\mathbf{E}^{\textsf{T}}\bm{z}_{\bullet
j}\right)$ and
$\mathbf{C}=(\mathbf{A}^{-1}+\mathbf{E}^{\textsf{T}}\mathbf{E})^{-1}$, where
$\mathbf{E}$ is a rectangular matrix whose $i$-th row is $(1,\bm{\beta}_{i})$,
and $\bm{z}_{\bullet j}=(z_{1,j},\ldots,z_{n,j})$.

    3. (c)

Sample $\bm{\beta}_{i}^{(b)}$ from
$p(\bm{\beta}_{i}\mid\\{\mu_{j}^{(b)}\\},\\{\bm{\alpha}_{j}^{(b)}\\},\\{z_{i,j}^{(b)}\\})$,
where

$p(\bm{\beta}_{i}\mid\\{\mu_{j}\\},\\{\bm{\alpha}_{j}\\},\\{z_{i,j}\\})=\textsf{N}_{d}(\bm{\beta}_{i}\mid\bm{d}_{i},\mathbf{D}_{i})\,,$

with
$\bm{d}_{i}=(\mathbf{B}_{i}^{-1}+\mathbf{F}^{\textsf{T}}\mathbf{F})^{-1}\left(\mathbf{B}_{i}^{-1}\bm{b}_{i}+\mathbf{F}^{\textsf{T}}(\bm{z}_{i\bullet}-\bm{\mu})\right)$
and
$\mathbf{D}_{i}=(\mathbf{B}_{i}^{-1}+\mathbf{F}^{\textsf{T}}\mathbf{F})^{-1}$,
where $\mathbf{F}=[\bm{\alpha}_{1},\ldots,\bm{\alpha}_{m}]^{\textsf{T}}$,
$\bm{z}_{i\bullet}=(z_{i,1},\ldots,z_{i,m})$, and
$\bm{\mu}=(\mu_{1},\ldots,\mu_{m})$.

##  4 Ideal Bayesian point estimator: North and Latin American case

An extensive review of the literature shows that the ideal Bayesian point
estimator is essential to analyze parliamentary electoral behavior at any
level (whether individual or collective). Its application allows political
scientists to support conjectures about individuals’ political preferences and
the latent characteristics that underlie legislative decisions, taking into
account political theory as well as the singularities of the parliament under
analysis. Furthermore, the flexibility of the estimator makes it adaptable to
provide empirical evidence on the relationship between policymakers,
institutional arrangements, and legislative outcomes.

To present our findings, we propose a brand new taxonomy considering the type
of parliamentary phenomena as well as the scope of the application. The
taxonomy is composed of seven parts: Political space dimension (Section 4.1),
pivot legislators identification (Section 4.2), voting restricted to the
agenda’s nature (Section 4.3), evolution and change in preferences of
political actors (Section 4.4), influence of national political leaders and
groups (Section 4.5), strategic abstentions (Section 4.6), and extremes voting
together (Section 4.7).

###  4.1 Political space dimension

The political space dimension is a popular topic in the Political Science
literature (e.g., Potoski and Talbert,, 2000; Jackman,, 2001; Talbert and
Potoski,, 2002; Aldrich et al.,, 2014; Dougherty et al.,, 2014; Roberts et
al.,, 2016). Research in this direction highlights the discrimination
parameters (Jackman,, 2001), the voting proposal typology Moser et al.,
(2021), and the use of weighted Euclidean distances (Binding and Stoetzer,,
2022) as crucial elements to analyze the dimension of political space. Thus,
social scientists are interested in learning about the number of latent
features to model parliamentarians’ electoral behavior, the nature of the
retrieved dimensions, identifiability in multidimensional models, and
additivity of legislators’ preferences, among others.

The dimension of the political space $d$, from a technical perspective,
corresponds to the number of latent characteristics needed to model the voting
behavior of legislators properly. Its choice translates into a model selection
problem, seeking to achieve a balance between the goodness-of-fit and the
complexity of the model (Moser et al.,, 2021). Several works consider
methodological and epistemological discussions on the political space
dimension as well as specific mechanisms for its choice (Benoit and Laver,,
2012; De Vries and Marks,, 2012; Jackman,, 2001; Lofland et al.,, 2017; Moser
et al.,, 2021).

The dimension of the political space can be studied through the discrimination
parameters $\bm{\alpha}_{1},\ldots,\bm{\alpha}_{m}$. These parameters allow
analysts to assess the political space dimension in conjunction with the
goodness-of-fit of the model, discern substantive content of the recovered
dimensions, and also, contrast substantive hypotheses about the dimensions
that underlie the political space (Jackman,, 2001; Luque and Sosa,, 2022).
Those motions with statistically significant discrimination parameters (that
is, those whose credibility interval does not contain zero) provide evidence
about the number of latent features required to explain voting patterns. For
example, in the one-dimensional case, a considerable number of discrimination
parameters indistinguishable from zero strongly suggests to embed the model in
higher dimensions (Jackman,, 2001). Thus, inspecting the content of those
voting lists associated with not significant discrimination parameters can
reveal the qualitative character of additional dimensions to be considered.
Defining informative prior states about discrimination parameters favors
handling the complexities that arise when considering higher-dimensional
models (Jackman,, 2001).

Some authors point out that interpreting the discrimination parameters as
factor loadings is quite useful to identify the substantive content of the
retrieved dimensions (Jackman,, 2001). However, some others argue that the
dimension of the latent space is not necessarily aligned with the content of
the bills presented for voting (Moser et al.,, 2021). In such a situation,
interpreting the dimensions concerning substantive issues is inconsistent.
These interpretive perspectives lead to discussions regarding a distinction
between the basic space dimension and a thematic space (see Moser et al.,,
2021; Poole,, 2007, for more details).

In a recent study, Moser et al., (2021) indicates that standard spatial voting
models, such as the Bayesian ideal point estimator, assume that the dimension
of the political space is fixed and common. However, this assumption is a
limitation for analyzing political actors’ preferences in different electoral
domains and identifying individual voting patterns. In this sense, this author
proposes a methodology based also on aggregation principles for analyzing
nominal data (see Roberts et al.,, 2016) that allows modelers determining
latent characteristics common to the chamber and each legislator, taking into
account different voting domains. In other words, this approach assumes that
legislators can reveal different preferences depending on the nature of the
vote (economy, security, among others). Furthermore, this framework assumes
the existence of subsets of legislators whose voting patterns for certain
groups of votes are not explainable through the linear combination of the
latent traits recovered for the entire group.

The standard Bayesian ideal point estimator assumes that political actors’
preferences in every dimension are additively separable. Then, the utility of
an actor is given by the weighted sum of the deviations along the dimensions.
In order to test this assumption, Binding and Stoetzer, (2022) introduce a
statistical model allowing non-separability across dimensions. These authors
affirm that political actors’ preferences can be characterized better by
multiple non-separable dimensions rather than a single dimension or multiple
independent dimensions.

Just a handful of studies in the Latin American context focus on providing
empirical evidence to justify the choice of the political space dimension.
Most studies assume one or two dimensions considering the political context of
the legislatures under analysis (e.g., Zucco,, 2013; Zucco and Lauderdale,,
2011). For example, on the one hand, Rosas, (2005) assures that legislative
politics in most of the countries of Latin America is one-dimensional. On the
other hand, Zucco and Lauderdale, (2011) point out that the presence of
religious, linguistic, or ethnic parties justifies the existence of a second
ideological dimension. Thus, the presence of coalitions, electoral districts,
and regional or provincial divisions, among others, indicate possible higher
non-ideological dimensions (Zucco and Lauderdale,, 2011; Zucco,, 2013).

Lastly, Jones and Hwang, 2005a  and Luque and Sosa, (2022), inspired by
Jackman, (2001), examine the dimension of political space through the analysis
of discrimination parameters. The estimates of this parameters provide
empirical evidence to ensure that there is a single dimension underlying the
policy space in the case of the nominal votes of the Argentine House of
Representatives 1989–2003 and Senate of the Republic of Colombia 2010–2014,
respectively. These authors state that a one-dimensional model is enough to
model properly such roll call data.

###  4.2 Pivot legislators identification

The identification of pivot legislators includes studies focused on the
estimation and inference of both ideal points and auxiliary quantities
obtained as a function of model parameters. Such quantities allow researchers
to recognize those members of parliament whose position in the political space
is considered as relevant in order to understand what happens within the
legislature (Clinton et al.,, 2004). In this line of work, the interest relies
on the identity and position of pivot legislators (also known as fundamental
legislators), extremists, minorities, among others.

The notion of pivot legislator is fundamental for some theories of
parliamentary behavior that characterize and predict law formulation processes
based on the position of these legislators in the political space (Clinton et
al.,, 2004). In the North American context, such theories indicate that
pivotal legislators are those whose vote is critical to the success or failure
of the legislative process. In particular, the vote these deputies can
guarantee the closure of a debate and define a majority vote in an
extraordinary legislative situation (see Krehbiel,, 1998; Clinton et al.,,
2004).

In this way, Clinton et al., (2004) exposes a methodology based on the
standard Bayesian one-dimensional ideal point estimator, in order to discern
the identity and spatial location of the deputies who play a fundamental role
within the legislative body. The methodology corresponds to a sequential and
iterative three-step scheme, namely, (i) sample the legislators’ ideal points
from their joint posterior distribution; (ii) order the sampled ideal points
in ascending order, and (iii) observe which legislators occupy a particular
pivot or order statistics of interest. After repeating this scheme an
arbitrarily large number of times, we can identify those legislators who are
more or less likely to occupy a particular ordered position.

In Latin America, studies in this direction are scarce. From a political
perspective, social scientists are still determining whether the fundamental
theory has significant applications in the Latin American context or not. Very
recently, Luque and Sosa, (2022) identify legislators from the Colombian
Senate 2010–2014 who are more likely to be in a conservative position or in
the extremes of the political spectrum. Their findings are limited to
individualizing deputies rather than making inferences about other quantities
that can be derived from ideal point estimates. Nevertheless, more efforts
need to be carried out in order to determine order statistics within
parliament.

###  4.3 Voting restricted to the agenda’s nature

Research in this direction highlights the importance of estimating parameters
associated with voting alternatives (see Clinton and Meirowitz,, 2001), say
$\bm{\psi}_{j}$ and $\bm{\zeta}_{j}$, as a mechanism to investigate issues
related to the behavior of the status quo in a specific period, the
positioning of new policies in particular moments and contexts, and the
dependency relationship between votes under a specific agenda.

Standard political preference estimators, such as the standard Bayesian ideal
point estimator, do not incorporate the sequential nature of the agenda, i.e.,
these models assume that the retrieved parameters are not affected by a
reordering of the voting sequence. This assumption omits helpful information
to locate parameters associated with voting alternatives and determine the
political space dimension, which may lead to unsuitable ideal points estimates
for certain political instances. Consequently, the corresponding findings
could support misleading interpretations regarding the legislative theories
tested with these models (Clinton and Meirowitz,, 2001).

In order to incorporate the agenda’s sequential nature into roll call data
analysis, Clinton and Meirowitz, (2001) extend the standard Bayesian ideal
point estimator to a constrained agenda model, where voting alternatives are
tied to a status quo parameter. In formal terms and under the assumption of
complete knowledge of the order of the agenda, the status quo of the voting
list $j$, say $\bm{\zeta}_{j}$, is assumed to be associated with previous
voting alternatives parameters. Thus, if $\bm{\psi}_{j-1}$ was approved, then
the new status quo is $\bm{\zeta}_{j}=\bm{\psi}_{j-1}$. On the contrary, if
$\bm{\psi}_{j-1}$ was negated, then the new status quo is
$\bm{\zeta}_{j}=\bm{\zeta}_{j-1}$. Therefore, voting in favor of motion $j$
represents a movement in the political space, but voting against it, avoids
such a movement.

In the North American context, the restricted model provides estimates of
ideal points more consistent with deliberate policies than the standard
estimator because it reveals the relationship between the status quo and the
last approved proposal. However, the model needs to provide a framework for
examining endogenous agenda formation and strategic voting within parliament.
Additionally, the analysis does not provide evidence of the consistency about
the restricted and unrestricted estimators Clinton and Meirowitz, (2001).

Although the proposal of these authors allows us to cover a little-studied
legislative phenomenon, the Bayesian ideal point estimator considering the
order of the agenda implies a price to pay in terms of parsimony. Unlike the
standard model, this extension includes more parameters to estimate and more
technical difficulties. To the best of our knowledge, there are no
implementations of the constrained estimator in the Latin American context.

###  4.4 Change in preferences of political actors

In this line of work, research premises lie on the stability of the political
preferences of legislators under particular circumstances of the legislative
process (e.g., change of party). Also, it is of interest to analyze the
discrepancies exhibited by deputies in their voting patterns when faced with
different voting domains. In addition, the differences they reveal in their
electoral behavior when they legislate in different institutional bodies
(e.g., chambers and commissions), among others. All these questions lead to an
essential technical discussion about the importance of establishing common
latent scales that allow contrasting the electoral behavior of political
actors (e.g., Asmussen and Jo,, 2016; Shor et al.,, 2010; Shor and McCarty,,
2011).

The latter task, establishing common latent scales, makes reference to a
methodological aspect that implies analyzing bridges that allow ideal points
to be scaled on a standard scale to ensure that the contrasts between
institutions, periods, or other scenarios are compatible. Thus, Shor et al.,
(2010) asserts that it is not valid to equate ideal point estimates obtained
separately since each set of ideal points produces estimates on a different
latent scale. Therefore, one alternative consists in identifying “bridge
actors” (e.g., legislators with common voting records) to anchor latent spaces
that allow analysts to confront the electoral behavior of subjects not voting
simultaneously.

The change of preferences has been considered from different perspectives. For
example, Martin and Quinn, (2002) postulate a dynamic version of the standard
Bayesian spatial voting model to assess the change and possible dependencies
in political actors’ preferences over time. Although these authors do not
present results in the parliamentary context, their model is a reference for
characterizing the dynamics of legislators’ political preferences, since their
research describes different approaches to measure changes in such preferences
over time. Additionally, they indicate the theoretical structure to
incorporate dynamic linear models (West and Harrison,, 2006) in the estimation
of ideal points and the methodology to support the corresponding Bayesian
inference.

The dynamic model specification is analogous to the base model but taking into
account indexing over time. In this spirit, the Bayesian ideal point estimator
in its dynamic version is a dynamic linear model of the form
$y_{i,j,t}^{*}=\mu_{j}+\bm{\alpha}_{j}^{\textsf{T}}\bm{\beta}_{i,t}+\epsilon_{i,j,t}$,
where $\bm{\beta}_{i,t}$ is the ideal point corresponding to legislator $i$ at
time $t$, and $\epsilon_{i,j,t}$ is the stochastic deviation due to the
uncertainty associated with the voting process over time. The other parameters
have the same connotation as in Section 3. This model differs from the base
model mainly in the prior distributions of the ideal points. Martin and Quinn,
(2002) propose a dynamic prior distribution given by
$\bm{\beta}_{i,t}\mid\bm{\beta}_{i,t-1},\bm{\Delta}_{\beta_{i,t}}\stackrel{{\scriptstyle\text{ind}}}{{\sim}}\textsf{N}(\bm{\beta}_{i,t-1},\bm{\Delta}_{\beta_{i,t}})$,
where $\bm{\Delta}_{\beta_{i,t}}$ is a hyperparameter describing temporal
evolution variation.

In the Latin American context, particularly in Brazil, we are aware of an
approach that applies the one-dimensional standard dynamic Bayesian ideal
point estimator to contrast the electoral behavior of deputies in the Chambers
of Parliament 1989–2008. In particular, McDonnell, (2017) establishes a common
latent scale between chambers, taking as a bridge the joint votes of its
members. In Brazil, representatives and senators occasionally vote
sequentially in Congress. With this strategy, the author equates legislative
behavior to an educational test where subjects act on the same policies almost
simultaneously.

Another perspective to study change in political actors’ preferences is
assuming that the preferences of deputies are stable, at least in short
periods, but they are affected by particular circumstances of the legislative
process (Clinton et al.,, 2004; Moser et al.,, 2021). In this sense, Clinton
et al., (2004) analyze the change of deputies’ preferences in the context of
party change through the standard one–dimensional Bayesian ideal point
estimator (considering a liberal–conservative latent trait). This author
states that legislators who change party affiliation in the exercise of their
office are crucial to identifying the effect of the party on the voting
behavior of members of the legislative body. At the moment of change, other
determinants of the vote remain constant (e.g., the constituency and the
affiliation of the other legislators, among others), and the new affiliation
may reflect a variation in the ideal points of the legislators as a result of
such a change. Naturally, all legislators can present a variation in the
estimates of their ideal points after the change. However, the variation for
deputies who change parties is greater than those who maintain their political
affiliation. In this sense, it is possible to parameterize the ideal points as
$\beta_{i,1}=\beta_{i,0}+\delta_{i}$, where $\beta_{i,0}$ and $\beta_{i,1}$
are the ideal points of legislator $i$ before and after the change,
respectively. Thus, the relative change, $\delta_{i}$, of the $i$-th
legislator will make it possible to analyze the variation of the corresponding
ideal point.

From this angle, there are two main critical issues in the quantitative
investigation of change in preferences of political actors, namely, the
precision and the compatibility of the estimates mentioned above (McCarty et
al.,, 2001; Clinton et al.,, 2004). Regarding precision, Bayesian simulation
allows us to identify the inaccuracy that arises when dividing the data set
into pre and post-change. Furthermore, it provides us with uncertainty
assessments for all the model parameters, showing the drop in accuracy of the
estimates after splitting the dataset around the match change. Now, regarding
producing comparable estimates, Clinton et al., (2004) points out that the
posterior standardization of the ideal points limits the problem. However,
Lofland et al., (2017) question this approach. These authors point out three
methodological inconsistencies. First, comparing the hierarchical order of the
legislators through the estimates made before and after the change of party
ignores that locations of different legislators are not independent. One
legislator increases his rank when another or others decrease theirs. Second,
implementing models before and after the change produces non-comparable ideal
point estimates. Standardized estimates of ideal points do not guarantee that
they share the same latent scale. And third, the party-switching hypothesis
must consider the lack of fit when testing multiple hypotheses.

Then, Lofland et al., (2017) propose a hierarchical model to induce a common
scale without splitting the vote set. Furthermore, they assume that not all
legislators change their preferences. The deputies with constant political
preferences throughout the period under analysis are the bridge between the
political spaces (see Shor et al.,, 2010; Shor and McCarty,, 2011). The model
uses zero-inflated Gaussian priors to identify the bridge legislators that
connect the arbitrary ideological scales and make them comparable.
Additionally, this estimator allows addressing multiple comparison problems
simultaneously (see Scott and Berger,, 2006, 2010), taking into account Bayes
factors and posterior probabilities. Although the proposal is inspired by
Martin and Quinn, (2002), the main goal is not modeling the evolution of
deputies’ preferences over time, but to test hypotheses of party change at a
specific moment in the legislative process.

Inspired by the work of Lofland et al., (2017), Moser et al., (2021) propose
another extension of the ideal Bayesian point estimator. The formulation does
not focus on party switching, but on variation in legislators’ preferences
when they vote in different domains or issues. The estimation of ideal points
of political actors in different voting domains constitutes a version of
change in parliamentarians’ preferences little studied. The extension of the
model provides a methodological framework to carry out contrasts at both
individual and group levels about similarities and discrepancies that
deputies’ revealed preferences present when they make decisions in different
voting domains. These extensions propose estimating the identity of “party
legislators” (bridge legislators), voters whose revealed preference remains
constant for all lists. The approach of these authors does not assume prior
knowledge of the identity regarding partisan voters.

These extensions of the ideal Bayesian point estimator also allow researchers
to investigate the relationship between the political space and the thematic
space through a clustering model (see, Moser et al.,, 2021), under which
legislators may have different preferences for each group of votes. The use of
previous groupings reduces the number of different positions for a legislator
and contributes to less uncertainty in the estimates of the model parameters.
Unlike Lofland et al., (2017), Moser et al., (2021) generalizes the model to
an arbitrary number of groups by introducing prior distributions that divide
the set of votes into $K$ groups determined by the analyst. The generality in
the model formulation opens the door to potential applications mainly in Latin
American legislatures, where there are no studies in this direction to the
best of our knowledge.

###  4.5 Influence of national political leaders and groups

An issue of interest to scholars of the North and Latin American Congress is
the influence of party leaders or national political groups on the electoral
behavior of legislators. Studies in this line focus on the effect of political
groups (parties, coalitions, among others) on electoral behavior and the
influence of incentives on deputies’ voting decisions. In other words, the
research emphasizes voting patterns that are not the product of sincere voting
behavior.

Clinton et al., (2004) provides a methodology to interpret and operationalize
the influence of political groups on parliamentary voting behavior. Not
considering this phenomenon leads to ideal points estimates absorbing an
impact common to the members of a particular group and revealing a greater
polarization (e.g., partisan) in the preferences of the deputies. In this
direction, party influence is a plausible mechanism to generate extra utility
incentives associated with a specific political group. Then, an extension of
the utility function to model directly partisan incentives is

$U_{i}(\psi_{j})=-(\psi_{j}-\beta_{i})^{2}+\delta_{j}^{\textsf{D}}+\eta_{i,j}\hskip
28.45274pt\text{and}\hskip
28.45274ptU_{i}(\zeta_{j})=-(\zeta_{j}-\beta_{i})^{2}+\delta_{j}^{\textsf{R}}+\upsilon_{i,j}\,,$

where $U_{i}(\psi_{j})$ and $U_{i}(\zeta_{j})$ are the corresponding profits
associated with legislator $i$ for voting in favor of or against motion $j$,
respectively, $\delta_{j}^{\textsf{D}}$ and $\delta_{j}^{\textsf{R}}$ are the
incentives that the legislator $i$ receives for voting positively depending on
their affiliation party, where D represents Democrats and R Republicans.
Finally, $\eta_{i,j}$ and $\upsilon_{i,j}$ are random shocks product of the
uncertainty associated with the voting processes, which are assumed
independent and identically with logistic distribution. These utility
functions lead to a linear model of the form

$y_{i,j}^{*}=\mu_{j}+\alpha_{j}\beta_{i}+\delta_{j}D_{i}+\epsilon_{i,j}\,,$

where $\epsilon_{i,j}=(\upsilon_{i,j}-\eta_{i,j})/\sigma_{j}$,
$\mu_{j}=(\zeta_{j}^{2}-\psi_{j}^{2})/\sigma_{j}$ and
$\alpha_{j}=2(\psi_{j}-\zeta_{j})/\sigma_{j}$, $D_{i}$ is a dummy variable
that takes the value of 1 if the legislator $i$ is a Democrat, and 0
otherwise, and $\delta_{j}=\delta_{j}^{\textsf{D}}-\delta_{j}^{\textsf{R}}$ is
the net difference of specific democratic party incentives. If $\delta_{j}>0$,
then the net incentive is for Democrats that vote in favor of motion $j$,
while if $\delta_{j}<0$, then the incentive is for Democrats that vote against
motion $j$.

Clinton et al., (2004) declare that this extension is advantageous compared to
other methodological proposals to study party influence. Unlike other
mechanisms, it does not require a differential analysis of items (e.g.,
Wainer,, 1993), nor the use of two-stage procedures (e.g., Snyder Jr and
Groseclose,, 2000). These methods are criticized due to the bias occurring
during the estimation process (e.g., McCarty et al.,, 2001). In this sense,
the Bayesian approach improves several implementations of party influence
models available in the literature. Furthermore, the model is quite general
since it augments utility functions with party-specific incentives for each
motion.

This extended estimator has the identification restrictions of the standard
model but also introduces restrictions for the parameters $\delta_{j}$ (in the
same direction as Snyder Jr and Groseclose, 2000). Application of the model
leads to the identification of voting lists subject to partisan incentives and
the estimation of statistically significant party effects (the 95% credibility
interval of $\delta_{j}$ does not overlap with zero) for the US 105th Senate.
Clinton et al., (2004) state that although there is evidence consistent with
party influence in only a third of closed votes (roll call votes decided by
margins closer to 35% - 65%), the magnitude of these incentives is large and
politically consistent with the behavior of the deputies. Estimating party-
specific incentives for each roll call makes it possible to determine party
polarization more precisely.

In the same line of thought, Hahn et al., (2012) adapt the standard Bayesian
ideal point estimator using sparse factor models (e.g., Pati et al.,, 2014;
Bernardo et al.,, 2003; Carvalho et al.,, 2008) to reveal partisan patterns in
roll call votes of the US Senate. The proposal incorporates theoretical
referents of the multivariate probit model (e.g., Chib and Greenberg,, 1998),
the Gaussian factor model (e.g., Murray et al.,, 2013), and scattered mass
points at zero (e.g., Castillo et al.,, 2015) to identify cases in which
partisanship is not predictive of roll calls. This formulation makes it
possible to characterize covariance patterns in multivariate binary data to
establish which bills and legislators do not rely on latent factors of
partisan nature. In this way, Hahn et al., (2012) state that the sparse factor
probit model is a crucial exploratory tool for analyzing high-dimensional
correlated categorical data. These authors point out that, like the standard
estimator, their method is adaptive and indicates that an interesting
extension would consist in adding either spatial or temporal autocorrelation
components to scores. The dependency would explain the partisanship of
senators who serve in consecutive congresses or are spatially close (for
example, belonging to the same district).

In the case of Latin America, findings in this regard have shown a broader
literature. The influence of the executive, leaders, and political groups
(leaders of the majority party, coalitions, delegations, among others) in the
legislative vote is remarkable in the region. Studies in this direction have
analyzed the phenomenon in question from different perspectives. The
investigations have shown that legislatures in Latin America have a structure
and dynamics that differ from the US parliament (Pereira and Mueller, 2004b,
). Additionally, the executive–legislative relationship (Pereira and Mueller,
2004b, ; Zucco,, 2013), the consolidation of government–opposition coalitions
(Carey,, 1998; Carroll and Pachón,, 2016; Tsai,, 2020; Zucco and Lauderdale,,
2011), multipartyism (Ames,, 2002; Neto,, 2002; Figueiredo and Limongi,, 2000;
Pereira and Mueller, 2004b, ; Tsai,, 2020; Zucco,, 2013), political actors and
internal subdivisions (committees) or external (constituencies, delegations,
among others) to the legislative chambers (Jones and Hwang, 2005a, ; Cheibub
et al.,, 2009; Pachón and Johnson,, 2016; Tsai,, 2020; Clerici,, 2021), are
determining factors of parliamentary electoral conduct. There are even authors
who argue that, in contrast to the ideological component, these have a greater
impact on the legislative voting process (e.g., Alemán et al.,, 2018; Tsai,,
2020).

Thus, Tsai, (2020) proposes adjusting the Bayesian ideal point estimator to
incorporate non-ideological factors (such as membership in the ruling
coalition) in the analysis of roll call data. This model offers empirical
evidence regarding the influence of political groups on the legislative voting
behavior of the Brazilian Chamber of Deputies from 2003–2006. This author
states that the extension of the model is necessary since the Brazilian
assembly is an example of a legislature that reflects electoral behavior
mediated by political negotiation (strategies to access political resources)
and ideological preferences. The standard ideal point estimates are not
optimal since they need to distinguish between the impact of coalition
dynamics and the effect of individual political preferences that underlie
voting behavior. Other spatial voting models postulated under a similar
argument are those proposed by Zucco, (2009) and Zucco and Lauderdale, (2011).
In this latest study, the authors present a version of the standard ideal
point estimator in two dimensions (left–right and government–opposition).

The modeling strategy assumes that the political problem underlying nominal
voting has a one-dimensional ideological content (left–right) underlying
different voting decisions to be made by legislators with the same political
position. Then, the government–opposition relationship explains the
discrepancy in the election. In this sense, Tsai, (2020) models the
executive’s influence on legislative voting through the party affiliation of
the deputies and the party membership in the ruling coalition. Thus, the
author assumes that legislators whose party belongs to the alliance have
additional incentives, $\delta>0$ and $\lambda>0$, if they vote in favor of
the government proposals and against the opposition proposals, respectively.

Analogous to the formulation of Clinton et al., (2004), the additional
incentives, product of the government–opposition conflict, are incorporated
into the utility functions

$U_{i}(\psi_{j})=-(\psi_{j}-\beta_{i})^{2}+\delta\cdot\textsf{MGov}_{k[i]}\textsf{PGov}_{j}+\eta_{i,j}$

and

$U_{i}(\zeta_{j})=-(\zeta_{j}-\beta_{i})^{2}+\lambda\cdot\textsf{MGov}_{k[i]}\textsf{POpp}_{j}+\upsilon_{i,j}\,,$

where $U_{i}(\psi_{j})$ and $U_{i}(\zeta_{j})$ are the corresponding profits
associated with legislator $i$ for voting in favor of or against motion $j$,
respectively, $\eta_{i,j}$ and $\upsilon_{i,j}$ are random shocks product of
the uncertainty associated with the voting processes, which are assumed
independent and identically distributed. Additionally, $k[i]$ denotes the
party affiliation of the legislator $i$, in a way that $\textsf{MGov}_{k[i]}$
corresponds to a binary indicator that takes the value of 1 when legislator
$i$ belongs to party $k$ of the governing coalition, and otherwise, it takes
the value of 0. Finally, $\textsf{PGov}_{j}$ and $\textsf{POpp}_{j}$ are dummy
variables that take the value of 1 when proposal $j$ comes from the government
and the opposition, respectively, otherwise, they take the value of 0.

The incentives $\delta$ and $\lambda$ directly affect the parameter intercept
$\mu_{j}$ in the regression model

$\mu_{j}=\gamma_{1}+\gamma_{2}\textsf{PGov}_{j}+\nu_{j}\,,$

where $\gamma_{1}$ represents the reference probability of granting a positive
vote when the opposition casts the proposal and
$\nu_{j}\mathrel{\overset{\makebox[0.0pt]{\mbox{\tiny
iid}}}{\sim}}{\textsf{N}}(0,\sigma_{\nu}^{2})$. Thus, $\gamma_{2}>0$ indicates
that the government–opposition conflict increases the probability of voting in
favor when proposal $j$ is governmental, whereas $\gamma_{2}<0$ indicates that
the coalition dynamic does not positively affect the government direction. The
specification includes conventional semiconjugate prior distributions of the
forms $\sigma_{\nu}^{2}\sim\textsf{GI}\left(a_{0}/2,b_{0}/2\right)$,
$\gamma_{1}\sim\textsf{N}(\gamma_{0},\sigma_{\nu}^{2})$, and
$\gamma_{2}\sim\textsf{N}(\gamma_{0},\sigma_{\nu}^{2})$, where $a_{0}$,
$b_{0}$, and $\gamma_{0}$ are the model hyperparameters.

Additionally, the model incorporates the party affiliation through informative
priors on the ideal points. The aim behind this approach is consists in
evaluating the variation between parties and analyzing the effect of the
ruling coalition on the legislative vote through a party base. Thus, in the
same spirit of Zucco and Lauderdale, (2011), the ideal point corresponding
associated with $i$ comes from a distribution specific to his political party,
centered on the party mean $\mu_{\beta_{k[i]}}$ with variance
$\sigma_{\beta}^{2}$, i.e.,
$\beta_{i}\mid\mu_{\beta_{k[i]}},\sigma^{2}_{\beta}{\sim}{\textsf{N}}(\mu_{\beta_{k[i]}},\sigma_{\beta}^{2})$.
Unlike other authors, Tsai, (2020) proposes priors hierarchies on party means.
In this sense, he suggests that
$\mu_{\beta_{1}}\mid\mu_{\beta_{2}}{\sim}\textsf{U}(-3,\mu_{\beta_{2}})$,
$\mu_{\beta_{k}}\mid\mu_{\beta_{k-1}},\mu_{\beta_{k+1}}{\sim}\textsf{U}(\mu_{\beta_{k-1}},\mu_{\beta_{k+1}})$,
and
$\mu_{\beta_{K}}\mid\mu_{\beta_{K-1}}{\sim}\textsf{U}(\mu_{\beta_{K-1}},3)$,
for $k=2,\cdots,K-1$.

In this way, $\mu_{\beta_{k}}$ is subject to a prior ordering restriction,
through which political parties are organized from left to right, taking into
account positions that other authors have identified (e.g., Zucco,, 2009;
Zucco and Lauderdale,, 2011). Additionally, considering the pre-established
ordering, two parties with different ideologies are selected to anchor the
extremes of the scale. Thus, this specification allows the problem of
rotational invariance to be solved directly since it makes the left and right
parties stand on the corresponding side of the underlying ideological scale.
On the other hand, electing deputies from “anchor parties” with different
ideologies but belonging to the same field (government–opposition, selection
based on (Zucco and Lauderdale,, 2011)) ensures that the underlying
ideological dimension and the government–opposition conflict do not overlap.
Lastly, the sign of the discrimination parameters $\alpha_{j}$ reveals the
ideological status of the proposal $j$. The prior distributions for these
parameters are $\alpha_{j}{\mathrel{\overset{\makebox[0.0pt]{\mbox{\tiny
ind}}}{\sim}}}{\textsf{N}}(-2,1)\bm{I}(\alpha_{j}<0)$, for $j=1,\ldots,m_{0}$,
and $\alpha_{j}{\mathrel{\overset{\makebox[0.0pt]{\mbox{\tiny
iid}}}{\sim}}}{\textsf{N}}(0,1)$, for $j=m_{0},\ldots,m$, where
$\bm{I}(\cdot)$ denotes the indicator function. Model parameters are
standardized in each iteration of the MCMC algorithm for solving scale
invariance problems (Tsai,, 2020).

The model results regarding the Chamber of Deputies of Brazil reveal that
legislators with membership in the government coalition are more likely to
support the executive’s proposals regardless of their political ideology.
Empirical evidence confirms that parties influence legislative votes. In
addition, estimating the ideal points associated with the 11 parties in the
ideological scale (left–right) reveals a partisan order similar to previous
studies (e.g., Zucco and Lauderdale,, 2011).

On the other hand, Jones and Hwang, 2005a  implement the standard estimator to
roll call data from the Chamber of Deputies of Argentina 1989–2003. The
estimated ideal points help build an indicator of homogeneity of provincial
legislative behavior to analyze the influence of governors on the legislative
behavior of their regional partners in the lower house. In turn, such a metric
is considered as the response variable of a linear regression model that
evaluates the governor’s influence on parliamentary electoral behavior using
the characteristics of the governor as covariates (e.g., belonging to the
government). The results do not reveal provincial effects on the legislative
behavior of deputies, nor do they reflect a significant relationship between
the characteristics of the governor and the estimated measure of homogeneity.
Zucco, (2013) also uses the estimated ideal points to build an index of
legislative performance. This measure is the response variable of a Tobit
model (Amemiya,, 1984) to explain the influence of political groups or actors
on parliamentary voting behavior through covariates such as the average
ideological distance of the faction from the president as well as the
proportion of nominal passes in which the faction held a ministerial position.
The pattern revealed by the estimated ideal points indicates that the behavior
of legislators from the same fraction is alike. In addition, the ideological
distance influences the legislative behavior indicator. Finally, legislators
whose factions hold ministerial positions behave more pro-government than
their ideology might predict.

Finally, Zucco, (2013) also implements the standard one-dimensional estimator
to roll call data from the Uruguayan Senate 1985–2005. A low number of votes
(see Table) and a highly institutionalized party system lead to a distribution
of ideal points in the political space that does not follow an ideological
pattern. Therefore, the assumed dimension cannot be interpreted ideologically,
but rather in terms of the executive–legislative relationship.

###  4.6 Strategic abstentions

Research in the North American context (see Rodríguez and Moser,, 2015; Rosas
et al.,, 2015) have proposed to extend the standard Bayesian ideal point
estimator to study the phenomenon of strategic abstentions (intentional not
voting, absenteeism from the room, not vote recording, among others; a
theoretical explanation in this regard is given in Cohen and Noll, 1991;
Kromer, 2005; Voeten, 2000). Abstention from voting is a characteristic of the
electoral behavior of deliberative bodies usually ignored without empirical
support (Rodríguez and Moser,, 2015; Rosas et al.,, 2015). The matter of
interest associated with this issue are the identity of legislators who
frequently participate in strategic abstentions, the behavior of the
abstention rates of groups or political leaders over time or under specific
conditions, and the types of bills for which legislators locations in the
political spectrum are key as abstention factors, among others.

The standard Bayesian ideal point estimator typically ignores missing values
and assumes that the random process generating these missing data is
irrelevant for estimating the ideal points (Rosas et al.,, 2015). However,
abstentions reflect legislators’ preferences when faced with the dilemma of
not affecting others (e.g., party leaders or voters) during the competition
(Rodríguez and Moser,, 2015; Rosas and Shomer,, 2008). Consequently, Rosas et
al., (2015) point out that the decision to model the lack of response should
be based on theoretical foundations and not on goodness-of-fit measures since
the strategic nature of abstentions is specific to the context of each
legislature.

The extensions of the estimator that address the phenomenon of abstentions
show two main approaches: first, moving from a binary model to an ordinal
model including the non-response as an intermediate category between the
positive and negative vote (Rodríguez and Moser,, 2015), and second, keep a
binary model but consider the joint probability of both positive vote and vote
registration (Rosas et al.,, 2015). These estimators characterize the
mechanism that drives the presence of missing values in voting lists.
Moreover, its implementation allows modeling the choice of non-response and
voting simultaneously, as well as evaluating the impact of factors (e.g., as
ideology) on the probability of non-response and vice versa (see Rodríguez and
Moser,, 2015; Rosas et al.,, 2015).

Rosas and Shomer, (2008) present an extension of the canonical ideal point
estimator (standard model assuming completely random abstentions, also called
ignorant abstentions model) to analyze the processes of abstention and voting
simultaneously. They illustrate their proposal by analyzing roll call data
from the Federal Congress of Argentina 1993–1995. These data favor the study
due to the high rates of abstention as well as lack of recording (32% for this
period).

The one-dimensional bivariate probit model admits two sources of information:
(i) the voting choice (1 = Yes, 0 = No) and (ii) the vote registration
indicator (1 = No registration and 0 = Registration) of the legislators. Then,
$y_{1,i,j}^{*}$ and $y_{2,i,j}^{*}$ are linear predictors driving the
probability of positive vote and abstention for legislator $i$ in motion $j$,
respectively. These predictors are
$y_{1,i,j}^{*}=\mu_{j}+\alpha_{j}\beta_{i}+\gamma_{j}c_{i}+\epsilon_{1,i,j}$,
with $\epsilon_{1,i,j}\mathrel{\overset{\makebox[0.0pt]{\mbox{\tiny
iid}}}{\sim}}{\textsf{N}}(0,1)$, and
$y_{2,i,j}^{*}=\eta_{j}+\delta_{j}c_{i}+\epsilon_{2,i,j}$, with
$\epsilon_{2,i,j}\mathrel{\overset{\makebox[0.0pt]{\mbox{\tiny
iid}}}{\sim}}\textsf{N}(0,1)$. The former expression is the canonical latent
model with an additional predictor $c_{i}$ that represents the latent
propensity to abstention of legislator $i$ in motion $j$, and $\gamma_{j}$ is
the effect of such propensity on the probability of a positive vote on motion
$j$. These parameters are incorporated into the model in order to allow the
abstention process to be informative of the result of a vote. On the other
hand, the latter expression characterizes the mechanism that generates
abstention, i.e., the probability that legislator $i$ abstains when
considering vote $j$, with $\delta_{j}$ y $\eta_{j}$ are specific parameters
associated with voting list $j$.

Rosas and Shomer, (2008) define informative prior distributions for the
parameters of the model under the assumption of the influence of the
provincial delegations of the parties in the electoral behavior of the
deputies (see Jones and Hwang, 2005b, ). They presume that the propensities to
the abstain are grouped within such delegations as they define a common prior
for the members of the same delegation. Although there are 20 parties with
representation in congress during the period of interest, the results only
provide information on the voting process and abstention of the majority and
opposition parties. The proposed model presents an adequate performance
compared to the model that assumes ignorable abstentions. However, the number
of roll call data for the analysis is scarce, and the information retrieved
through them is insufficient (see Table 1) to generalize the results.
Nevertheless, these findings reveal information about voting lists and bills
that provide the highest abstention rates and discriminate better between
legislators.

###  4.7 Extremes voting together

Very recently, spherical latent factor Bayesian models emerged for binary and
ordinal multivariate data (see Yu and Rodríguez,, 2021; Yu and Rodriguez,,
2019; Yu,, 2020). Although this kind of model is similar to the ideal Bayesian
point estimator, it is not an extension. These models constitute a more
general class, and the standard estimator is a limiting case of the new family
of models (Yu and Rodriguez,, 2019).

Yu and Rodríguez, (2021) mention that most spatial voting models assume ideal
points embedded within a (possibly multidimensional) Euclidean political
space. However, there are situations where Euclidean geometry does not explain
unusual voting behavior. For example, the phenomenon where members from
opposite ends of the ideological spectrum reveal similar preferences by voting
against the rest of the legislature. In this case, neither increasing the
dimensionality of the latent space nor performing linear transformations allow
the classical estimator to characterize the phenomenon of “extremes voting
together” adequately. As a result, the standard ideal point estimator does not
perform well in the mentioned case since it exhibits the extreme deputies as
conservative individuals. Instead, it is a better fit when nominal vote data
comes from political systems where parties are relatively unified (Yu and
Rodríguez,, 2021). So then, this new class of models explores unconventional
voting patterns evaluating the geometry that underlies the political space.

Bayesian spatial voting models based on spherical geometries differ from the
classical approach in their specification and implementation (see Yu,, 2020).
In particular, the model parameters, $\bm{\beta}_{i}$, $\bm{\psi}_{j}$ and
$\bm{\zeta}_{j}$, are defined on a Riemann manifold $\mathcal{D}$ (see Lee,,
2018), and the Euclidean distance is replaced by the geodesic distance $\rho$
in $\mathcal{D}$. The distance between two points is defined on a unit
$(d+1)$-dimensional hypersphere. This distance corresponds to the smallest
angle formed by the projections of the points through the origin. The case
$d=1$ leads to a spatial voting model in a circular space with utility
functions given by

$U_{i}(\psi_{j})=-\rho{(\bm{\psi}_{j},\bm{\beta}_{i})}^{2}+\eta_{i,j}\qquad\text{and}\qquad
U_{i}(\zeta_{j})=-\rho{(\bm{\zeta}_{j},\bm{\beta}_{i})}^{2}+\upsilon_{i,j}\,,$

where $\rho(a,b)=\arccos{(\cos{(a-b)})}\in[0,\pi]$ is the geodetic distance
that establishes the smallest angle between $a$ and $b$. Thus, $\beta_{i}$,
$\psi_{j}$, and $\zeta_{j}$ $\in[-\pi,\pi]$ are interpreted as angular
positions in an unit circle. The specification of the utility function leads
to substantial changes in terms of the choice of the link function, the prior
distributions of the model parameters, the identification, and interpretation
of the model, and its computational implementation (see details in Yu and
Rodriguez,, 2019; Yu,, 2020; Yu and Rodríguez,, 2021).

The circular version of this family of models was applied to 1988–2019 United
States House of Representatives roll call data (see Yu and Rodríguez,, 2021).
The results reveal that the circular voting model explains modern voting
patterns better than traditional Euclidean models. On the other hand, the
circular variance as a measure of sphericity (extreme voting behavior) helps
analyze the temporal evolution of this voting phenomenon.

Spherical Bayesian spatial voting models have yet to be widely applied. To the
best of our knowledge, there is still a need for applications in the Latin
American legislatures’ context. Existing implementations are novel and provide
empirical evidence of a recent phenomenon. Furthermore, these models suggest
uninvestigated methodological paths. For example, work has yet to be reported
in the literature illustrating how to incorporate and interpret strategic
abstentions in this model. The applications discussed so far assume that
missing data result from mere chance. Furthermore, there is an open path to
explore suitable priors in high-dimensional spheres contexts (Yu and
Rodríguez,, 2021).

##  5 Illustration

In order to fix ideas and provide a brand new case study, we focus on the
Colombian House of Representatives 2010–2014. This chamber’s plenary roll call
votes have yet to be previously studied using some of the ideas under review.
Therefore, we take advantage of the novelty of this data set to broaden the
discussion on the nature of the dimension of the political space of the
Colombian parliament, particularly the lower house. We have available 626 roll
calls deliberated by 181 deputies. However, we omitted 31 legislators for
participating in less than $95\%$ of the votes, which does not imply
eliminating any political group. In addition, we eliminated 66 unanimous roll
calls since these are not relevant to recover information about the latent
traits that underlie the political decisions of parliamentarians (Jackman,,
2001; Luque and Sosa,, 2022).

Four political groups make up the House of Representatives 2010–2014, the
coalition of government, independents, minorities, and opposition. Four of the
fourteen political parties in the legislative exercise constitute the
coalition, Conservador Colombiano (CC, $21.33\%$), Liberal Colombiano (LC,
$22.67\%$), Cambio Radical (CR, $10\%$), and Social de Unidad Nacional (PU,
$28.67\%$). The parties openly declared as independent are Alianza Verde (PAV,
$2\%$), Integración Nacional (PIN, $6.67\%$), and MIRA ($0.67\%$). Five of the
six minority parties have a seat in the chamber (each with $0.67\%$
representation). These are Alas Equipo Colombia (ALAS), Alianza Social
Indígena (ASI), Movimiento Integración Regional (MIR), Movimiento Popular
Unido (MPU), and AfroVives (VIVES). The sixth minority is the Movimiento de
Apertura Liberal (MOAL, $1.34\%$) with two seats. The Polo Democrático
Alternativo (PDA, $3.3\%$) is the only opposition party.

We implement the one–dimensional Bayesian ideal point estimator under the
technical details exposed by Luque and Sosa, (2022). The model provides
inferences about the ideal points of 148 parliamentarians (two legislators are
anchors for model identification) and about the discrimination and approval
parameters of the 560 voting lists, i.e., we estimate 1268 parameters. We
eliminate missing data as they have no substantial impact on our findings (see
Luque and Sosa,, 2022). In future methodological research, roll call votes of
this deliberative body could be helpful to illustrate patterns of
parliamentary conduct in the region when abstention is frequent (the data set
has $44\%$ missing data, of which $39\%$ correspond to abstentions and the
remainder to absences).

Figure 1: Ideal point estimates along with their 95% credible bands.

As shown in Figure 1, the ideal point estimates $\beta_{i}$ vary between
$-1.47$ and $2.22$. For all opposition members, the posterior mean of their
ideal point is negative and significantly different from zero (i.e., the
corresponding credibility interval does not contain zero). In the case of the
governing coalition members, the posterior mean of their ideal point is
positive, and $99\%$ of the deputies of this political group have
statistically significant ideal points. On the other hand, the ideal points of
minorities and independent parties are more dispersed throughout the political
space, all being significantly different from zero. The distribution of the
ideal points in the political space of the lower house is similar to the upper
house for the same period (see Luque and Sosa,, 2022).

The distribution of the ideal points indicates that the underlying latent
feature of the nominal voting of the lower house is not ideological
(left–right) because on the same side of the political spectrum are
historically antagonistic parties (LC and CC, see Mazzuca and Robinson,,
2009). Our findings indicate a latent non–ideological trait that divides the
spectrum into opposition–non-opposition since there are political groups that
do not openly declare themselves as part of or against the government (e.g.,
independents and minorities). Then, the recovered dimension reflects power
control between hegemonic parties within the legislative body, with some
minority and independent members taking a moderate (centrist) position.

On the other hand, we identified that 447 of the 560 discrimination
parameters, $\alpha_{j}$, are significantly different from zero. Then,
$79.8\%$ of the motions discriminate between legislators across the political
spectrum. Roll calls segregating among deputies are mostly related to
security, defense, and public forces ($23.04\%$), social security and health
($19.91\%$), and justice ($20.36\%$). So, we argue that power control within
the chamber revolves around these issues. Further, we determined that
$85.46\%$ of motions discriminating this political dimension are from a
government initiative and also that $95.08\%$ are list passes debated during
the first three legislative years.

Furthermore, we note that motions related to the environment, mines and
energy, and recreation and sports do not discriminate for the proposed
dimension. The 113 motions that do not discriminate in the retrieved policy
continuum possibly store information about the second dimension of political
space. Finally, we examine the posterior predictive distribution of some test
statistics. In none of these cases the posterior predictive $p$-value is an
extreme value (less than 0.05 or greater than 0.95). Therefore, the proposed
one-dimensional model reveals a good fit for 2010–2014 Colombian House of
Representatives nominal vote data set.

##  6 Discussion

The taxonomy proposed in this document specifies substantive themes that
inspire research on legislative electoral behavior in different contexts,
particularly in Latin America. Future research aims to identify additional
lines, phenomena, or substantive hypotheses subject to analysis through the
methodologies under review that allows social scientists to extend our
categorization.

We identify that four of the seven lines present empirical evidence of
legislative electoral behavior in Latin America via the Ideal Bayesian point
estimator. Then, in some countries of the region, there are available studies
about the political space dimension, the change in the preferences of
political actors, the influence of national political leaders or groups, and
strategic abstentions. However, to the best of our knowledge, in Latin
America, no research uses the Bayesian ideal point estimator to study issues
related to voting restricted to the nature of the agenda and the case of
extremes voting together. In addition, the available literature only points
out to the parliaments of Argentina (Jones and Hwang, 2005a, ; Rosas and
Shomer,, 2008), Brazil (McDonnell,, 2017; Tsai,, 2020; Zucco and Lauderdale,,
2011), Colombia (Luque and Sosa,, 2022), Paraguay (Ribeiro et al.,, 2021), and
Uruguay (Zucco,, 2013) as case studies. In this sense, Latin American
parliaments within a democratic presidential system are exhibited as novel
scenarios to operationalize the electoral behavior of legislative bodies
through roll call data under a Bayesian approach.

Quantitative studies on parliamentary electoral behavior in Latin America
based on the application, adaptation, and generalization of the Bayesian ideal
point estimator are emerging as promising research fields. In this regard,
several aspects are of interest in the framework of the proposed taxonomy. For
example, although there are studies on the dimension of the political space,
it is still necessary to delve into the dynamic and collective nature of the
dimensions proposed for the different parliaments (e.g., Moser et al.,, 2021).
Furthermore, the nature of higher dimensions in the region’s legislatures is
also an object of study. The latter requires specifying the methodological
conditions to propose multidimensional spaces in the legislative chambers of
Latin America (see Jackman,, 2001). In addition, future research can evaluate
the variation in the dimension of the political space when votes are
segregated by issue, initiative, or other kinds of categorization (e.g.,
Lofland et al.,, 2017).

Regarding the identification of pivotal legislators, future research should
provide empirical evidence on the deputies who are essential for the approval
of new political alternatives or hinder voting in Latin American parliaments
(e.g., Clinton et al.,, 2004)). Furthermore, research framed in voting
restricted to the nature of the agenda indicates an opportunity to investigate
the importance of parameters of voting alternatives when studying legislative
electoral conduct. The latter would allow analysts to know political scenarios
under which there are a greater legislative activity or probability of
changing the status quo and identify possible political strategies to
establish the legislative agenda in the deliberative bodies of the region.
Furthermore, from a technical point of view, more evidence is needed regarding
the consistency of the constrained and unconstrained estimators (see Clinton
and Meirowitz,, 2001).

In the United States, the effect of party switching on legislators’
preferences has been extensively studied (see Clinton et al.,, 2004). However,
this direction has yet to be studied in Latin America. Additionally, given the
characteristics of Latin American parliaments, it is quite important to
investigate the variation in the preferences of members of Congress. For
example, when there is a change of political groups (e.g., entry and exit of
parties to the ruling coalition) or leadership within parliaments (e.g.,
change of president in the legislative chambers). On the other hand, studies
in line with the evolution of legislators’ preferences admit the contrast
between legislative chambers (e.g., plenary and commissions) to discover
changes in the latent features that underlie the vote of deputies. In
addition, interested researchers could evaluate the potential of the estimator
in its dynamic version to provide empirical evidence on the impact of party
fragmentation and the proliferation of political groups on legislative
electoral behavior in the region’s countries.

Concerning the influence of national political leaders or groups, future
research should focus on incorporating multiple incentives (e.g., party and
coalition) into the estimator in the Latin American case. Moreover, we have
yet to review applications of the estimator that incorporate external factors
(e.g., media, opinions of the electorate, foreign political groups, or
electoral processes, among others; see Ribeiro et al.,, 2021; Fouirnaies and
Hall,, 2022) to examine other determinants of the voting decision of
parliamentarians. On the other hand, the line of strategic abstentions traces
an alternative to investigating the nature of the dynamics of non-
participation in the vote of the deputies in the legislative institutions of
Latin America (e.g., Rosas et al.,, 2015).

Finally, the line of extremes voting together is a novel methodological
alternative to evaluate different technical aspects. (e.g., the geometry
underlying political space in Latin American legislatures, see Yu and
Rodriguez,, 2019). Again, to the best of our knowledge, research in Latin
America has yet to explore spherical latent factor voting models for binary
and ordinal data. In this document, we do not describe non-parametric
implementations of the estimator (see Tahk,, 2018; Shiraito et al.,, 2022).
Such approaches will be discussed elsewhere.

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