One-dimensional scaling without apologies∗

Tasos Kalandrakis†

July 23, 2019

Abstract

I propose a fully non-parametric framework for estimation of and inference on the loca-

tion of political parties in a one-dimensional scale. I derive a balance condition sufficient for

identification and consistent estimation of parties’ order from left to right, and a weaker error

rate condition that partially identifies the order. The method relies only on ordinal data on

individual preference rankings over parties and is robust even to systematic or strategic error.

Capitalizing on recent advances in testing moment inequalities, I build a strong and weak spec-

ification test of the adequacy of the one-dimensional model and an inferential framework to

test hypotheses about the relative location of parties or to probe the validity of one-dimensional

scales produced by other methods. Under stronger cardinality assumptions on preferences, I rely

on arguments from revealed preference theory to execute similar tests efficiently, obviating the

need for confidence set construction using an approximating grid of party locations. I illustrate

these techniques using CSES survey data on the German party system.

∗Earlier versions of this paper have benefited from the audiences at the 2015 APSA and the2016 MPSA meetings, a Political Economy conference at Emory University, and a Riker seminarat the University of Rochester. I especially thank Kei Kawai, Keith Poole, and Sergio Montero fortheir comments. I would also like to thank the Center for Integrated Research Computing (CIRC)at the University of Rochester for providing computational resources for the analysis performed inSection 5.†Department of Political Science and Department of Economics, University of Rochester. E-

mail: kalandrakis@rochester.edu

1

It is virtually impossible to discuss politics without reference to a liberal-conservative or

left-right scale. Such scales have strong and broad resonance in political discourse and can be

traced back to the origins of modern representative democracy. In addition to their prominence in

both informal and expert political commentary, one-dimensional representations of the policy space

are widely used in scientific work, empirical and theoretical. Of course, despite its omnipresence,

the one-dimensional scale can rarely be defended literally. Any attempt to pin down or elaborate

the content of “left-right” or “liberal-conservative” runs into a multitude of stances and political

positions on diverse and often tenuously connected issue areas both across countries and across

time. Yet, citizens and pundits evidently find it possible to distill opinions on such diverse issue

areas into a powerful summary statistic. Endorsing the statistical nature of the one-dimensional

scale, the question pursued in this paper is whether we can rigorously determine its adequacy?

And, if a one-dimensional summary cannot be rejected (statistically), can it be recovered from the

data and scrutinized through rigorous inference procedures?

In answering the first question, it is important to ensure that evidence to reject the unidi-

mensonal model bears on its statistical inadequacy alone and not on any discrepancy between data

and extra ancillary or convenience assumptions forced into the model. To guard against such false

negatives, the approach pursued in this paper is fully non-parametric. By that I mean that both

the systematic and statistical error components of the model are specified as infinite dimensional

objects disciplined only by the bearest structure consistent with a one-dimensional interpretation.

As a result the preferences of individuals who consume the scale and inform its reconstruction need

not be order-restricted (in the sense explained in Footnote 7) and for the bulk of the analysis are

only ordinal in nature. Similarly, under the null hypothesis that one-dimensionality holds, any

observed discrepancy from these systematic preferences is assumed to originate from error that

follows general, preference-specific distributions, and can enter the data in arbitrary and, within

some tolerance, even systematic form.

The data for this study are assumed to take the form of individual preference rankings of a

set of objects to be scaled. These objects can be political parties, individual candidates, or other

choice items in elections or markets, and to fix ideas I heretofore refer to them as parties. I start with

the assumption that systematic individual preferences over parties can be represented by a quasi-

concave utility function. This is an ordinal assumption weaker than single-peakedness, arguably the

2

weakest assumption on individual preferences consistent with a one-dimensional interpretation of

the scale. Under this assumption, I derive a condition sufficient for identification of the ordering of

political parties from left to right, which I refer to as a balance condition. It requires that sufficient

(relative to the incidence rate of consequential erroneous responses) fractions of respondents least

prefer the left-most and right-most parties in every triplet of parties. The condition can be used to

construct a family of consistent estimators, and I exhibit a simple score estimator to that effect. The

balance condition is a joint restriction on the amount of error in responses and the distribution of

individual ideological leanings across the spectrum. A weaker error rate condition sheds the latter

restrictions thus relinquishing identification of the order of parties while still imposing testable set-

identification restrictions on that order. It amounts to a restriction that is necessary and sufficient

to prevent the fraction of individuals that rank the middle party in every triplet of parties as least

preferred being the maximum such fraction over the parties in the triplet.

A key contribution of this study is to interpret the balance and error rate conditions as

moment inequalities, making this approach amenable to the inferential framework of the burgeoning

literature on testing moment inequalities.1 These methods can be used to test whether the balance

or error rate condition are satisfied for specific orderings of parties. Inversion of these tests can

be used to build confidence sets of party orderings. As proposed in Romano and Shaikh (2008),

Andrews and Guggenberger (2009), and Andrews and Soares (2010), a specification test can then

be based on these confidence sets, rejecting the one-dimensional model when the corresponding

confidence set is empty. I therefore propose a strong test of one-dimensionality as a specification

test of this type based on the balance moment inequalities. A weak test of one-dimensionality

is a specification test built using the error rate moment inequalities. Both specification tests are

asymptotically valid but may be asymptotically conservative. Given the minimal structure on the

data generating process imposed by the latter test, there is very little room to defend the adequacy

of the one-dimensional model if the null is rejected using the weak specification test.

I also explore the additional purchase of cardinality assumptions, strengthening quasi-

concavity to concavity of individual preferences. Paired with an added (aggregate but otherwise

1It is outside the scope of this paper to fully review this literature. Canay and Shaikh (2017)

offer a recent review and I further discuss works most relevant to this study in Sections 2 and 4.

3

standard) assumption of zero expectation of party-specific response error,2 this assumption further

restricts relative party locations for any given ordering of parties. Therefore, when combined with

either the balance or error rate condition, concavity generally narrows the identified set of party

locations but may not be supported by the data. Unfortunately, confidence sets built through test

inversion and specification tests using these combined moment conditions require an approximation

grid on the space of party locations. To avoid the use of such approximation grids, I rely on argu-

ments from revealed preference theory to convert the concavity moment inequalities (expressed as

a function of party locations) to a set of unimodality of score expectations inequalities (expressed

as a function of party orders from left to right). Specification tests of the combined concavity and

error rate or balance moment inequalities can be performed efficiently using this conversion.

I implement these tests using the practical two-step procedure for testing moment inequal-

ities of Romano, Shaikh and Wolf (2014) and illustrate them using German CSES survey data

in Section 5. The data reject the balance identification condition and fail the strong test of one-

dimensionality across all elections. But, the weak test of one-dimensionality is not rejected in a

subset of these elections, and in these cases the corresponding confidence sets of party orders are

narrow and informative. These confidence sets are broadly consistent with a conventional left-right

scale, but in a subset of elections they may represent a Libertarian–Totalitarian or Establishment–

Antiestablishment dimension. The extra assumption of concavity of individual preferences receives

mixed support and the joint error rate and concavity specification test is rejected in half of the

cases when the weak test of one-dimensionality alone is not rejected.

The present work is related to a number of diverse literatures in Political Science, Economics,

and Psychology, besides the works already cited above. Within economics, its reliance on preference

shape restrictions connects it with the broad literature on non-parametric estimation of models of

choice as in, for example, Manski (1975, 1988); Matzkin (1992); Varian (1982). With a more specific

political economy focus, several works employ non-parametric and/or moment inequality methods:

Merlo and de Paula (2017) recover the distribution of preferences in the electorate; Henry and

2This analysis only requires data to convey expected cardinal utility levels. Though still de-

manding a lot of the data, the analysis of Eggers and Vivyan (2018) provides prima facie evidence

that the sympathy score surveys used in the application of Section 5 do contain individual cardinal

information.

4

Mourifie (2013) test and reject the pure Euclidean spatial model; Kawai and Watanabe (2013)

recover the fraction of strategic voters in the electorate. Since its inception, the scaling literature

in Political Science has relied on data similar to that used in this study,3 for example, Brady (1989,

1990); Cahoon and Hinich (1976); Cahoon, Hinich and Ordeshook (1978); Hinich (2005). The

bulk of scaling work in Political Science, though, uses roll-call votes to recover legislator preference

parameters (e.g., Canen, Kendall and Trebbi (2018); Clinton, Jackman and Rivers (2004); Heckman

and Snyder (1997); Iaryczower and Shum (2012); Martin and Quinn (2002); Poole and Rosenthal

(1985, 1997)). Even the non-parametric version of these roll-call models (Poole (2000); Tahk (2018))

imposes a strong order restriction on individual preferences. Poole (2005) expands on the connection

of this literature with the long tradition of non-parametric scaling or unfolding in Psychology, where

seminal are the non-parametric scales of Mokken (1971). His methods and its variants also rely on

certain stochastic order restrictions variably known in this literature as item monotonicity, double

monotonicity, or monotone homogeneity, that bear a connection to order restrictions on individual

respondent preferences or abilities, and are ably reviewed in, among others, Post (1992); Sijtsma

and Molenaar (2002); Sijtsma and Meijer (2007); van Schuur (2011). In the fully nonparametric

approach pursued here the individual respondents cannot be ordered, though their preferences

maintain enough structure essential for a one-dimensional scaling of the parties.

Other methods to construct one-dimensional scales used in Political Science take the con-

ventionally construed left-right dimension as given and attempt to recover party locations on that

dimension either by analyzing the content of parties’ declarations (e.g., Volkens et al. (2018)) or by

eliciting the locations through surveys of experts (e.g., Castles and Mair (1984)) or the electorate

(e.g., Aldrich and McKelvey (1977)). By design, these methods do not preclude the existence of

additional dimensions with different content nor do they probe the statistical significance of extra

dimensions. On the other hand, the data used in this study allow all possible relevant dimensions

to determine the preference rankings or sympathy scores over parties and for that reason do not a

priori constrain the content or number of relevant dimensions.

3Aldrich and McKelvey (1977) level several objections to the use of such data (page 111, second

paragraph) because of the reliance of extant methods on parametric (Euclidean) restrictions on the

preferences of respondents and the absence of specification tests for these assumptions. Both of

these concern are fully addressed in the present framework.

5

1 Setup

Consider J , J ≥ 3, political parties indexed by j = 1, . . . , J . Each party is characterized

by its location in D-dimensional Euclidean space, D ≥ 1, with the location of party j denoted

by xj ∈ RD. The possibility D > 1 is allowed at this point only as an alternative to the main

working hypothesis that parties are located in a one-dimensional space (D = 1). The restriction

to one dimension is only a necessary component to an interpretable one-dimensional scale, as such

an interpretation relies primarily on a certain alignment of preferences of the population over these

parties. Before fully spelling out the uni-dimensionality assumption, I first introduce the necessary

restriction D = 1 jointly with a standard orientation restriction:

D = 1 and xjL < xjR for some known parties jL, jR.(D)

Assumption (D) fixes both the dimensionality of the space and the relative location of two parties

in that space. The latter restriction is necessary in some such form for identification purposes, as

any one-dimensional scale is possibly identified only up to reflection.

By virtue of assumption (D), the parties are naturally ordered from left to right,4 and

this ordering (besides the parties’ actual location) is of central focus in this analysis. We may

summarize this ordering with a bijection π : {1, . . . , J} → {1, . . . , J} that maps from a possible

order r = 1, . . . , J, to a party j that is located in that order, so that xπ(1) < xπ(2) < . . . < xπ(J). I

have precluded weak inequality above, that is, I assume that party locations are pairwise distinct.

This is a simplifying assumption for the purposes of exposition and it ensures there is a unique

ordering π of the party locations x. Because multiplicity of true orderings due to parties sharing a

location is ruled out by a forthcoming identification restriction, all results stated in the main body

of the paper hold (with only minor adjustments to some definitions) if we allow parties to share a

location at this stage of the analysis (see Appendix D).

In general, both the ordering π and the actual location of the parties x = (x1, . . . , xJ),

is (up to assumption (D)) unknown to the analyst. The information that is assumed available

4I emphasize the term “left-right” is used for orientation purposes only. Should a one-

dimensional scaling prove supported by the data, it is a whole other matter whether the produced

scale is indeed interpretable in the traditional left-right fashion.

6

and may allow inferences on these quantities arises from N survey respondents. Each individual

respondent i = 1, . . . , N, reports a vector of scores yi ∈ Y ⊂ RJ where yi = (yi1, . . . , yiJ) are

sympathy/preference scores for the J parties with higher scores indicating preferred (better) parties

to individual i.5 Underlying this notion of preference is an unobserved preference/utility/sympathy

function ui : R→ R corresponding to each individual i. I assume that the score that respondent i

assigns to party j is related to the location of that party, xj , according to:

yij = g(ui(xj), εij), all i, j.

Here g : R2 → R is a function that takes the level of i’s systematic preference function evaluated

at xj and an error εij as inputs. The vector of error terms εi = (εi1, . . . , εiJ) captures all forces that

might push a respondent to report a sympathy score different than what her systematic preference

function dictates. This error could be additive, multiplicative, or take any other form.

To reduce nomenclature, I assume individual respondent preferences u and error terms ε

are jointly drawn independently and identically across respondents according to a distribution Φ.

As a result, each individual respondent is characterized by a pair (ui, εi) ∈ U ×RJ corresponding to

her systematic sympathy or utility function ui, and a vector of error terms (one coordinate for each

party) εi.6 Note that the conditional distribution over the noise terms εi may vary across respondent

preferences, ui. Furthermore, no restriction whatsoever is placed on the distribution of εi across

its coordinates, allowing for dependent and heteroscedastic errors across parties. Throughout, I

assume Y is a compact set, an assumption that is easily met in the case these data are recorded

in discrete survey scales and spares the need for further integrability conditions on the population

5I follow the convention of using superscripts to index respondents and subscripts to index

parties. I omit the superscript when referring to u or y as a random variable.6The analysis could be equivalently carried out by assuming Φ has support on RJ ×RJ , that is,

reducing ui to a vector in RJ containing the values of ui evaluated at the N party locations. By

standard arguments from revealed preference theory, it suffices to impose the shape restrictions (Q)

and (C) introduced later in this section on the restricted function evaluated at the J party locations,

for if these restrictions are met in the function thus restricted, then there exists an extension of

this function to the entire real line that also satisfies these shape restrictions.

7

distributions.

At this level of generality, the structure imposed on the problem permits little if any political

interpretation of the party locations x, certainly not based on familiar notions of ideological position

on a political scale such as a traditional left-right scale. I entertain a pair of related assumptions

on the functions u that do allow such interpretation (of course, the caveat entered in Footnote 4

still applies). The first part of the analysis only relies on the weakest such assumption, namely,

quasi-concavity:

Quasi-concavity: For all u ∈ U , and for all x, x′, x′′ such that x ≤ x′ ≤ x′′,(Q)

u(x′) ≥ min{u(x), u(x′′)}.

Assumption (Q) is consistent with u being discontinuous or having plateaus, that is, open sets

at which it is constant, including around a non-unique peak. When expressed in terms of the

preferences over the underlying set of parties, it is weaker than the single-peaked property of Black

(1948) and Arrow (1951), or the related domain restriction of Sen (1966), as it only requires that the

middle alternative in any triplet cannot be strictly least preferred over the other two alternatives.

In those terms, (Q) amounts to condition (1) of Dummett and Farquharson (1961) (page 35), a

weak sufficient condition for the existence of a majority core point. Assumption (Q) is ordinal in

nature and unburdens us of any need to impute cardinal interpretations on the respondents’ scores,

y. Such relief always comes at a cost. In particular, assumption (Q) will allow us to identify the

ordering of political parties π from the survey responses under certain conditions, but does not

otherwise restrict the location of the parties, x.

We may further restrict party locations by introducing cardinality assumptions on prefer-

ences u. For that purpose, and in keeping with the overall goal of imposing minimal assumptions,

I will employ the following concavity restriction in Section 3:

Concavity: For all u ∈ U , and for all x, x′, x′′ such that x ≤ x′ ≤ x′′,(C)

(x′′ − x′)(u(x′)− u(x)) ≥ (x′ − x)(u(x′′)− u(x′)).

While concavity is a fairly weak condition compared to standard parametric assumptions, it is

certainly stronger than quasi-concavity and as a result may not be supported by the data. Every

8

function u that satisfies (C) also satisfies (Q) but not vice versa. Neither assumption implies in

any way, shape, or form that population preferences are order restricted7 according to the relative

location of the respondents’ bliss point from left to right: Any two respondents may have arbitrary

differently shaped functions within each class, whether they share or not an ideal point (or plateau).

2 Identification & estimation of ordering π

In this section I consider identification and estimation of the ordering of political parties,

π, under the core assumptions (D) and (Q). Some notation is necessary before we proceed. Let all

the possible orderings of party locations be given by set Π and let p denote a generic element of

Π. Shape restriction (Q) is characterized by a property that holds on triplets of alternatives and,

anticipating the use of sample analogues of this condition, define the set of all possible triplets of

parties T := {τ ⊆ {1, . . . , J} | #τ = 3}. For all τ ∈ T and all p ∈ Π, define a certain restriction of

ordering p as a function denoted pτ : {1, 2, 3} → τ that takes the form:

pτ (r) := jr, where τ = {j1, j2, j3} and p−1(j1) < p−1(j2) < p−1(j3).

The function pτ takes a party’s relative (according to ordering p) location in the triplet as an input

(1, 2, or 3), and returns the party that holds this position in triplet τ . Now, for every triplet τ and

all r = 1, 2, 3, define the population quantity

Pr(τ) := P[{πτ (r)} = arg minj∈τ

u(xj)],(1)

that is, for the trio of parties in τ , Pr(τ) is the fraction of respondents in the population that

prefer the r-th ordered party from left to right the least. Because our assumptions do not rule out

indifferences, it is the case that

(2) P1(τ) + P2(τ) + P3(τ) ≤ 1, for all τ.

7 Therefore, both (Q) and (C) allow an individual with ideal x (i.e., such that u(x) ≥ u(x) for

all x) to prefer x over x′, x < x′, while an individual with ideal x′ < x may prefer x′ over x.

9

Figure 1: Identification of order

x1 x2 x3

i=3

i=2 i=1u1(x3)>u1(x2)>u1(x1)u2(x2)>u2(x3)>u2(x1)u3(x1)>u3(x2)>u3(x3)

J = 3, τ = {1, 2, 3}, and jL = 1, jR = 3 according to (D). Without response error, if the populationcomprises only individuals such as i = 1, 2 (that is, P3(τ) = 0) it is impossible to discriminatebetween x1 < x2 < x3 and x1 < x3 < x2. If the population includes respondent i = 3 and at leastone of i = 1, 2, the true ordering x1 < x2 < x3 is identified.

With this notation in place and in order to better understand the nature of the identification

problem at hand, I first discuss the relatively straightforward case when respondent sympathy

scores reflect true preferences without error.

No error in responses: In this case, we can take full advantage of the shape restriction (Q)

and its key manifestation in the data

(3) P2(τ) = 0, for all τ,

that is, the middle party in any triplet τ cannot be uniquely least preferred. Whenever we observe

some party in a triplet that is strictly least preferred by some respondent, we can safely conclude

that this party is not the middle party in that triplet. This observation, when applied to all

triplets, along with the orientation restriction of (D), would appear sufficient for identification.

Unfortunately that is not the case.

To illustrate, consider Figure 1 where for the purposes of this discussion it is assumed

that there are only three parties (J = 3) and at most three possible preference functions u in the

population as depicted for individuals i = 1, 2, and 3. Imposing assumption (D) with jL = 1, jR = 3,

the three parties are located at x1 < x2 < x3 and our estimation problem amounts to locating party

10

2 to either the left of party 1; or to the right of party 3; or in between these two parties as is actually

the case in this example. If the population only contains respondents with the preferences of i = 1, 2,

then even a sample of size one bars party 1 from being in between parties 2 and 3, by (3). Because,

by restriction (D), party 1 cannot be the right-most party of the three, we conclude it must be the

left-most. But, with the population only comprising individuals with the preferences of i = 1, 2,

we cannot further discriminate between orderings x1 < x2 < x3 or x1 < x3 < x2. But, continuing

with the example of Figure 1, if the population comprised both individuals with the preferences of

i = 3 and at least one of i = 1, 2 with positive probability, then we can fully recover the correct

ordering x1 < x2 < x3. The balance condition (B0) of Theorem 1 exactly requires that both the

left-most and the right-most parties in each triplet are least preferred by some positive fraction of

respondents in the population.

Theorem 1. Assume (D) and (Q), and that yij > yij′ ⇔ ui(xj) > ui(xj′) for all i, j, j′. If

(B0) Pr(τ) > 0, for all τ, r = 1, 3,

then the party ordering π is identified.

The sufficiency of condition (B0) follows from more general results established later in this

section, but the preceding discussion makes the argument fairly clear: (B0) coupled with (Q)

allows us to identify the true ordering of parties in each triplet up to orientation using (3), and

then restriction (D) inductively nails the orientation of all triplets. Note that the condition is not

necessary when J > 3. This is illustrated in Figure 2, where there are four parties (J = 4) and only

two preference types in the population. Condition (B0) is not satisfied for the triplet τ = {1, 2, 3}

since party 3 is never ranked below both parties 1 and 2 in the population. Nevertheless, we can

infer that party 1 is between parties 2 and 4 from triplet τ = {1, 2, 4} and that party 3 is between

parties 1 and 4 from triplet τ = {1, 3, 4}. With this information, the true ordering is recovered

using (D) imposed for any pair of parties jL, jR = 1, . . . , 4.

Allowing for response error: Once we admit response error, we cannot infer the exact ordering

with probability one in any finite sample. Nevertheless, a strengthening of the balance condition

(B0) to adjust for response error will suffice to ensure identification. To get to the specifics, I define

11

Figure 2: Balance sufficient but not necessary for identification (J > 3)

x2 x1 x4

i=2i=1

u1(x4)>u1(x3)>u1(x1)>u1(x2)u2(x1)>u2(x3)>u2(x2)>u2(x4)

x3

J = 4, and population comprises only preferences of i = 1, 2, both with positive probability.Without response error, condition (B0) is not met for triplet τ = {1, 2, 3}, yet the ordering π isidentified because the ordering for triplets {1, 2, 4} and {1, 3, 4} is.

certain rates of correct and erroneous survey responses. First, for all τ , define

qc(τ) := minr=1,3

P[{πτ (r)} = arg min

j∈τyj

∣∣∣∣πτ (r) ∈ arg minj∈τ

u(xj)

].

qc(τ) is a highest lower bound (across r = 1, 3) on the probability of responses that ‘correctly’8

report the r-th party in τ as being strictly least preferred, conditional on that party being least

preferred. Thus, conditional on being a respondent who least prefers either of the two extreme

parties in a triplet, there is probability of at least qc(τ) of uniquely assigning the lowest sympathy

score to that actually least preferred party in triplet τ .

On the other end, for all τ we define a consequential error response rate

qe(τ) := P[{πτ (2)} = arg minj∈τ

yj ].

Accordingly, there is error probability qe(τ) that the middle party in the triplet receives a strictly

8‘Correctly’ can be used literally here if the fraction of respondents that are indifferent between

any pair of parties is null in the population. Ansolabehere and Brady (1989) suggest reasons the

incidence of indifferences may not be negligible. If so, qc(τ) encompasses responses that report only

one party (that is extreme in τ) as strictly least preferred in τ even though two or all three parties

in τ are tied at that ranking.

12

minimum sympathy score. The type of error captured by qe(τ) is termed “consequential” as it only

captures errors that violate condition (Q) and its key implication in equation (3). This error rate

is generally much smaller than the probability that respondents report their preferences with some

error. Not only does this rate pertain exclusively to errors with respondents reporting the middle

party as their least preferred party, but also this party must receive a strictly minimum sympathy

score. Thus, error responses in which two or more parties including the middle party receive

the same lowest sympathy score do not count into the calculation of qe(τ). This is particularly

relevant because typically surveys record sympathy scores on a discrete scale (e.g., from 1 to 10)

and the probability of ties in sympathy scores is therefore non-negligible (even if that probability

is null when it comes to actual preferences). Furthermore, one of the most common response error

attributions is to presume that respondents accurately record their sympathy towards a group of

most preferred parties and then lump a large number of their least preferred parties at the bottom

of their sympathy scale.

We obtain the following generalization of Theorem 1 to the case response error is possible:

Theorem 2. Assume (D) and (Q). If

(B) Pr(τ) >qe(τ)

qc(τ), for all τ, r = 1, 3,

then the party ordering π is identified.

Just like condition (B0), condition (B) is a balance condition requiring that there exists a

sufficient fraction of respondents in the population that least prefer the left-most extreme party in

each triplet and a sufficient fraction of respondents in the population9 that least prefer the right-

most extreme party in each triplet. The required fraction depends on the rate of consequential error

response. If there is zero such error (qe(τ) = 0) then as long as qc(τ) > 0, condition (B) reduces

to condition (B0) and any positive fraction is sufficient. The balance requirement becomes tighter

the larger the rate qe(τ) is and the lower the bound on correct bottom responses, qc(τ), is. To give

an example, assuming no indifferences in true preferences, if respondents correctly10 report their

least preferred party at least 60% of the time independent of the actual least preferred party (that

9In the absence of indifferences, these two fractions sum to one by equations (2) and (3).10Responses may still contain error in the ranking of the first and second preferred parties.

13

is, with probability qc(τ) > 0.6) and consequentially err by reporting the middle party as uniquely

least preferred with probability qe(τ) = 0.1, then (B) amounts to P1(τ),P3(τ) > 16 . Condition (B)

is a sufficient condition. It is certainly not necessary because, as already shown for condition (B0),

it need not hold for all triplets τ . It is conservative because it places little other structure on the

nature of response errors, but is by no means guaranteed to hold. It is less likely to hold (even

if one-dimensonality holds) when extreme parties in the scale are part of the triplet, as they are

likely least preferred by a large fraction of respondents, leaving a small complementary fraction of

least sympathizers for the other extreme party in that triplet.

To further parse condition (B) and Theorem 2 observe that (by (2)) a necessary condition

for (B) to hold is

(R) qc(τ) > 2qe(τ) for all τ.

The error rate bound (R) requires that respondents are at least twice as likely to report a least

preferred extreme party as strictly least preferred than to incorrectly report the middle party as

their strictly least preferred party in each triplet τ . This appears to be a minimal requirement on

the data generating process in order for data to provide intelligible information on the ordering of

parties. To substantiate this claim, define a population response analogue of Pr(τ),

Qr(τ) := P[{πτ (r)} = arg minj∈τ

yj ].

Qr(τ) is defined on the basis of reported preferences, while Pr(τ) in terms of true preferences. While

the counterpart of inequality (2) still applies for these population response quantities, equation (3)

need not as it is now possible that Q2(τ) > 0 for any τ . In fact, we have Q2(τ) = qe(τ). Clearly,

responses y provide poor guidance as to the relative location of parties in τ if the response analogue

of unimodality condition (Q) is violated so often so that

(¬Q) Q2(τ) > maxr=1,3{Qr(τ)},

that is, if the middle party is reported as strictly least preferred with higher probability than either

of the two extreme parties in triplet τ . The following Theorem establishes that condition (B)

14

preserves the response analogue of inequality P2(τ) < Pr(τ), r = 1, 3, that ensures identification of

ordering π when there is no response error, while condition (R) implies a weaker inequality and is

necessary and sufficient to rule out (¬Q).

Theorem 3. Assume (D) and (Q).

1. (B) implies (R) and

(MB) Q2(τ) < Qr(τ), for all τ, r = 1, 3.

2. (R) implies

(MR) 2Q2(τ) < Q1(τ) + Q3(τ), for all τ.

3. (¬Q) may hold if qc(τ) < 2qe(τ).

Theorem 3 establishes a hierarchy of decreasingly informative restrictions on expected frac-

tions of strict bottom responses. In the first part, inequalities (MB) underpin the identification

result of Theorem 2 under the balance condition (B). Inequalities (MR) in the second part

of the Theorem are weaker and imply that at least one (but possibly not both) of inequalities

Qr(τ) > Q2(τ), r = 1, 3, holds. Thus, when error rates are bounded according to (R), (MR) at

least ensures that responses y are not woefully misleading as a source of information on party

ordering, though that information may not be sufficient to identify the true ordering π. This is no

longer true if (R) fails, as established in part 3. Very little can be learned from response data that

violate (R).

2.1 Estimation

Under balance condition (B) the ordering π can be recovered consistently by a number of

analogy principle estimators. For example, a score estimator searches over possible rankings p and

recovers that which minimizes violations of unimodality in responses. This estimator takes the

form

π := arg minp∈Π

{QN (p, y) | p−1(jL) < p−1(jR)

},(4)

15

where the criterion function QN (p, y) is defined as the fraction of triplets of parties across respon-

dents that violate the sample analogue of (Q)

QN (p, y) :=∑τ∈T

1#T

N∑i=1

1N I(yipτ (2) < min{yipτ (1), y

ipτ (3)}

),(5)

and parties jL and jR are those specified in assumption (D). Function I that appears in (5) is

a truth function that evaluates to one (1) when its argument is true and evaluates to zero (0)

otherwise. Under the maintained assumptions (D) and (Q), if (B) holds then

limN→+ inf

QN (π, y) =∑τ

Q2(τ) < limN→+ inf

Q(p, yN )

for all p 6= π by (MB). It follows that:

Theorem 4. Assume (D), (Q), and (B). Then π is a consistent estimator of π.

The proof of Theorem 4 also serves as a constructive proof of identification (Theorem 2).

The score estimator π is certainly not unique and has no claim to efficiency in the class of consistent

estimators. In fact, Theorem 3 and the moment inequality conditions it establishes can be directly

employed to define a family of moment inequality estimators. In the terminology of that literature,

either of the inequalities (MB) or (MR) induces an identified set

Π∗M := {p ∈ Π | (MM ) hold, and p−1(jL) < p−1(jR)},M = B,R.

By the arguments above, Π∗B = {π} if (B) holds, while π ∈ Π∗R if (R) holds. Consistency conditions

for estimators ΠM ,M = B,R, of the identified set using moment inequalities have been discussed

among others by Imbens and Manski (2004) and, in greater generality, by Chernozhukov, Hong

and Tamer (2007) and Rosen (2008).11 Because the parameter space in this application is discrete,

some of the more technical of the general consistency conditions they establish are immediately

11Brady (1985) provides an early line of attack for estimators of sets, though not cast in a moment

inequalities framework.

16

verified,12 so that consistent estimation of the respective identified sets using such estimators follows

even in the case the ordering of parties is not point identified, that is, using moment inequalities

(MR). In the case of moment inequalities (MB), there is a natural connection between such moment

estimators and estimator π, particularly when the weighting matrix used in the moment estimator

criterion function is the identity matrix. Because the question of estimation of the identified set is

naturally linked to questions of inference and the construction of confidence sets for the identified

set which I heavily rely on in the ensuing sections, I do not develop additional notation for moment

inequality based estimation of the identified sets Π∗B and Π∗R in this paper.

3 Concavity

The analysis so far has relied only on ordinal assumptions on preferences that are con-

sistent with a one-dimensional party scale. In this section I explore the additional purchase of

concavity restriction (C). Concavity is often imposed through functional families prevalent in the

parametric framework (negative quadratic, piece-wise linear). To start, I introduce for the first

time a-(n otherwise typical) restriction on the distribution of the error terms entering sympathy

scores, namely:

For all j,

∫g(u(xj), εj)Φ(d(u, ε)) =

∫u(xj)Φ(d(u, ε)).(E)

Assumption (E) is easily met in the classic case the error is additive (yj = g(u(xj), εj) = u(xj)+εj)

and the error term has zero expectation. Even in this special case, the restriction allows dependence

across parties and heteroscedasticity across parties and respondents. Under this added restriction

and the stronger concavity assumption, inequalities (C) are directly inherited by expectations of

the sympathy scores:

Theorem 5. Assume (D), (C), and (E). Then

(xπ(r) − xπ(r−1))(E[yπ(r+1)]− E[yπ(r)]

)− (xπ(r+1) − xπ(r))

(E[yπ(r)]− E[yπ(r−1)]

)≤ 0,(MC)

12Specifically condition C.1 of Chernozhukov, Hong and Tamer (2007), or the continuity condi-

tions in Proposition 2 of Rosen (2008).

17

for all r = 2, . . . , J − 1.

Theorem 5 and the ensuing Theorem 6 are straightforward with proofs relegated to Ap-

pendix C. Inequalities (MC) are the first restrictions we have derived that directly involve party

locations, x, and for that reason appear more promising in further identifying these locations, be-

sides the ordering of parties. To further explore the reach of these added restrictions, define the

set of party locations that are consistent with ordering p under assumption (D) as:

Xp := {x ∈ RJ | xp(r) ≤ xp(r+1),r=1,...,J−1, and xj < xj′ for some j, j′}.

The following Theorem helps clarify the point that concavity of score expectations alone imparts

meager restrictions on the ordering of parties (though it does limit which party locations are

consistent with any one such ordering). Specifically, we can show:

Theorem 6. For all p, there exists x ∈ Xp that satisfies (MC) iff there exists r such that

E[yp(r−1)] ≤ E[yp(r)] for all r 1 < r ≤ r,(MU )

E[yp(r−1)] ≥ E[yp(r)] for all r r > r.

The Theorem states that in order for there to exist party locations x consistent with ordering

p and satisfying the concavity of expectations restriction it is necessary and sufficient that that

ordering renders expectations of party sympathy scores unimodal, in the sense that expectation of

party scores are increasing from the left and decreasing on the right of a party receiving the modal

expectation value. This is so under the restriction that at least two party locations inXp are distinct.

Counting the possible orders p that are consistent with (MU ) for any fixed expectations E[yj ], we

conclude there are∑J−1r=0 (J−1

r )2 = 2J−2 party orderings that are consistent with the concavity of

expectations restriction (MC), a large set.

This level of indeterminacy not-withstanding, the impact of joint assumptions (C) and (E)

is non-trivial for two reasons. First, relying on moment inequalities (MC) we can perform a point

test of the hypothesis that party locations x = x0 where x0 may correspond to some existing

scale. These moment conditions alone are sufficient to reject a number of existing scales of the

German party system (Appendix E). Second, because concavity (C) is stronger than the quasi-

18

concavity assumption (Q), either of the assumptions (R) or (B) allow us to combine the moment

restrictions (MC) with (MB) or (MR). These joint moment restrictions bring the full strength

of these assumptions to bear on the problem of estimation and/or inference on party locations:

moment conditions (MB) or (MR) bring information on the ordering of parties from left to right,

while moment conditions (MC) further restrict possible party locations for any given ordering. As

I discuss more extensively in the next section, Theorem 6 allows us to build efficient confidence sets

and specification tests of these joint assumptions by exploiting the uni-modality of expectations

inequalities, (MU ).

4 Inference and Specification Tests

Both the assumption of one-dimensionality embodied in (D) and (Q) or (C), and the balance

and error rate conditions (B) and (R) may or may not be supported by the data and need to be

subjected to the rigors of statistical inference. At the heart of the approach pursued in this study

is a test of the form

E[mM (θ, yi)] ≤ 0,(HMθ )

corresponding to a suitably defined class of moment inequalities represented by moment function

mM , and parameter(s) θ, as developed in Theorems 3, 5, and 6. There is a long and growing

literature on test procedures of that form, and the most recent lines of attack recognize that test

performance is improved when the test procedure incorporates information on the subset of the

moment inequalities that likely bind. Within the class of such “generalized moment selection”

(GMS) test procedures, two recent papers provide practical avenues for the execution of the tests

in finite samples: The recommended GMS procedure of Andrews and Barwick (2012) and the

practical two-step procedure of Romano, Shaikh and Wolf (2014). Because the former relies on

certain data-tuning parameters that are practically available only when the number of inequalities

is relatively small, all tests performed in the remainder of this study use the procedure of Romano,

19

Shaikh and Wolf (2014), where the reader can find the necessary background details.13

In our case, the moment inequalities are some combination of (the weak form of) (MB),

(MR), (MC), and/or (MU ) and the parameter is either the ordering p, possibly paired with the

location of the modal expectation of sympathy scores r, or party locations x. The (not jointly

combined) moment functions mM ,M = B,R,C,U, are reported in Table 6 of Appendix B. Several

tests are possible building on these inequalities. Of primary focus is a test of the balance condition

(B) that guarantees identification of party ordering π. By virtue of the first part of Theorem 3,

(B) can be tested using inequalities (MB), that is by testing

E[mB(p, yi)] ≤ 0.(HBp )

Test (HBp ) is a test of the hypothesis that ordering π = p and (B) holds, along with the maintained

assumptions (D) and (Q). Similarly, invoking the second part of Theorem 3, the error rate condition

(R) can be tested by

E[mR(p, yi)] ≤ 0.(HRp )

Turning to the cardinality assumption (C) along with (E), we can perform a point test

that the location of the parties is at x0 by testing moment inequalities (MC) evaluated at x = x0,

13 This test satisfies a uniform consistency condition

lim supN→+∞

sup(θ,Φ):(MM ) hold

PΦ[(HMθ ) is rejected] ≤ α,

under mild assumptions that are easily met in the present framework (Theorem 2.1 of Romano,

Shaikh and Wolf (2014)). The test procedure of Andrews and Barwick (2012) has the added benefit

of attaining the property with equality, though by remark S.2 in the supplemental Appendix of

Romano, Shaikh and Wolf (2014), the gap between the left-hand-side of this inequality and α is

no larger than α/10 in the recommended implementation of their test as used in the case study of

Section 5.

20

Table 1: Summary of Point Tests

Test Joint Null Hypothesis Hypothesis on x Moment Inequalities†(HB

p ) π = p, (D), (Q), (B) x ∈ Xp (MB)

(HRp ) π = p, (D), (Q), (R) x ∈ Xp (MR)

(HCp,x0) x = x0, (D), (C), (E) x = x0 (MC)

(HBCp,x0) x = x0, (D), (C), (E), (B) x = x0 (MB) & (MC)

(HRCp,x0) x = x0, (D), (C), (E), (R) x = x0 (MR) & (MC)

(HBUp,r0) π = p, r = r0, (D), (C), (E), (B) x ∈ Xp (MB) & (MU )

(HRUp,r0) π = p, r = r0, (D), (C), (E), (R) x ∈ Xp (MR) & (MU )

† All tests are based on the weak version of the indicated moment inequalities. The correspondingmoment functions, mM , are reported in Table 6 of Appendix B.

x0 ∈ Xp, that is by testing

E[mC(p, x0, yi)] ≤ 0.(HC

p,x0)

By Theorem 5, test (HCp,x0) is a joint test of the hypotheses that party locations x = x0 (x0 ∈ Xp)

and (D), (C), and (E) hold. While tests (HBp ) are nested in that inequalities (MB) imply (MR),

(HCp,x0) is a test of distinct hypotheses. But the two sets of assumptions can be combined to perform

a joint test, by virtue of the fact that concavity (C) implies quasi-concavity (Q). Specifically, for

any p and x0 ∈ Xp we can jointly test

E

mM (p, yi)

mC(p, x0, yi)

≤ 0,M = B,R.(HMCp,x0 )

Test (HMCp,x0 ) performs a joint test that party locations are at x = x0 ∈ Xp with expectations of

party scores that respect concavity at these locations and balance condition (B) (when M = B)

or error bound restriction (R) (when M = R) hold. As discussed at the end of Section 3, the

combination of these moment conditions is quite potent, restricting both party orders and possible

party locations for each order.

For ease of reference, Table 1 summarizes the tests introduced in this subsection and the

(implicit or explicit) joint null hypotheses for each. The last row in that Table includes a test which,

while not interesting in and of itself, is instrumental in efficiently constructing certain confidence

21

sets and specification tests in the next subsection. Test (HMUp,r0 )

E

mM (p, yi)

mU (p, r0, yi)

≤ 0,M = B,R,(HMUp,r0 )

is a joint test of the hypothesis that π = p, the party with the maximum score expectation is at

order r = r0, and assumptions (D), (C), (E), and (B) or (R) hold, depending on the value of

M = B,R.

4.1 Confidence sets & Specification tests

The natural next step capitalizing on this inferential framework is to construct confidence

sets of party locations.14 These sets are built by test-inversion, that is, for any parameters θ and

moments M , and any level of significance α, they take the form:

CMθ (α) := {θ | (HMθ ) is not rejected at level α}.

In addition to providing succinct information on possible party locations consistent with the respec-

tive joint hypotheses, a number of authors (Andrews and Soares (2010); Andrews and Guggenberger

(2009); Romano and Shaikh (2008)) have advocated the use of these confidence sets to perform

model specification tests. The idea is to reject the hypothesis that the model is correctly specified

when there is no parameter value that passes the test, that is, when CMθ (α) = ∅. Under the null

14Such confidence sets can either aim to cover the identified set with probability 1−α, or to cover

each point in the identified set with probability 1−α. In the latter vein, we build confidence sets for

points in the identified set following the procedure of Romano, Shaikh and Wolf (2014). By virtue

of the uniform consistency condition of the corresponding test discussed in Footnote 13, confidence

sets CN built through this procedure satisfy a uniform consistency condition (see Theorem 3.1 of

Romano, Shaikh and Wolf (2014))

lim infN→+∞

infΦ∈Φ

infθ∈Θ∗M (Φ)

PΦ[θ ∈ CN ] ≥ 1− α.

22

hypothesis that the model is well-specified, the actual size of these test is likely less than α.15 That

is, these specification tests are (by virtue of the uniform consistency of the test (HMθ )) uniformly

asymptotically valid, but are possibly asymptotically conservative.16

Construction of these confidence sets requires attention to an additional practical issue,

especially if these sets are used in specification tests. If parameters θ take values in a continuum,

the set CMθ (α) must necessarily be approximated by executing test (HMθ ) on a finite number of

parameter values θ. Use of such grids becomes more costly as the dimensionality of the parameter

space increases. In addition, construction of confidence sets using grids invites doubt to their use

in specification tests, as (without substantial additional arguments and costly increases in grid

precision) we cannot take the fact that the set proved empty on an approximating grid to imply

that the non-approximated confidence set is indeed empty.

This last problem can be circumvented entirely for the purposes of model specification by

virtue of the the inferential framework built through Theorems 3, 5, and 6. With the exception of

tests relying directly on moment inequalities (MC) and are parameterized by the party locations

x, all other confidence sets in our context can be built exactly by inverting the corresponding

hypothesis test on the entire (finite) domain of the parameters in question. Even when we wish to

test a joint hypothesis that includes the concavity assumption (C), we can build a confidence set

in the space of pairs p, r instead of party locations x by using test (HMUp,r0 ). This approach obviates

the need for an approximating grid and adds considerable credence to the use of these confidence

sets as specification tests.

15See the explicit discussion in the above references, especially Footnote 9 of Andrews and Soares

(2010), page 138.16More recently, specification tests for moment inequality models have been explored by Bugni,

Canay and Shi (2015) (see also Bugni, Canay and Shi (2017) for a discussion in the context of the

more general problem of sub-vector inference). They term the above test procedure “test BP” for

by-product, and propose two alternative tests which they show are asymptotically more powerful. I

do not entertain these alternative tests in this study mainly because of the difference in asymptotic

framework in Bugni, Canay and Shi (2015) with either that of Andrews and Barwick (2012) or

Romano, Shaikh and Wolf (2014), and because of the leeway the former allows in specifying a

critical tuning parameter.

23

Though not exhaustive, three types of specification tests seem particularly relevant in our

context. I conclude this section by reviewing each in turn, while Table 2 provides a summary of

the resulting specification test procedures.

Table 2: Specification Tests

Joint Null Hypothesis Moment Inequalities† Rejection Procedure

(D), (Q), (B) (MB) CBp (α) = ∅(D), (Q), (R) (MR) CRp (α) = ∅(D), (C), (E), (B) (MB) & (MU ) CBUp,r (α) = ∅(D), (C), (E), (R) (MR) & (MU ) CRUp,r (α) = ∅

† All tests are based on the weak version of the indicated moment inequalities. The correspondingmoment functions, mM , are reported in Table 6 of Appendix B.

Strong test of one-dimensionality: This test relies on a confidence set

CBp (α) := {p | (HBp ) is not rejected at level α}.

When CBp (α) = ∅, we can reject the joint hypothesis that (D), (Q), and (B) hold. This is a

strong test of uni-dimensionality, in the following sense: We may still have data generated from a

one-dimensional scale satisfying (D) and (Q), yet the hypothesis is rejected because the balance

condition (B) is violated for some triplet of parties.

Weak test of one-dimensionality: An alternative line of attack to testing the one-dimensionality

assumption can be pursued analogously to the strong test of one-dimensionality, by constructing a

confidence set

CRp (α) := {p | (HRp ) is not rejected at level α}.

When CRp (α) = ∅ we can reject the joint hypothesis that (D), (Q), and error bound (R) hold. This

is a weak test of the existence of a one dimensional scale, in that we reject the null either if (D)

and (Q) fail to hold or if the data is so noisy that (R) fails to hold. While the test is weak in the

sense that it is easier for data to ‘pass’ the test, rejection of this test provides strong evidence that

the uni-dimensionality assumption does not hold.

24

Joint tests of one-dimensionality & concavity: Finally, to perform a joint test of the null

hypothesis that (D), (Q), (B) or (R), and (C) and (E) hold, we build a confidence set

CMUp,r (α) := {(p, r) | (HMU

p,r0 ) is not rejected at level α},M = B,R.

We reject the null that the model is well-specified under these assumptions when CMUp,r (α) = ∅.

As already discussed, this is an efficient way to test the concavity assumption without using an

approximating grid on RJ , since there is only a finite number of pairs (p, r) on which we need

to invert the test. Paired with either balance condition (B) or error rate bound (R), the test

amounts to a strong and weak test of one-dimensionality and concavity of respondents’ systematic

preferences.

5 Implementation & a case study

In order to illustrate these estimation and inference techniques, I provide a detailed analysis

of CSES survey data on the German party system in this section. The relevant questions in these

surveys ask respondents to evaluate parties on an eleven point integer scale (0 − 10), with higher

values indicating the respondent likes the party more. The surveys included in the analysis were

timed around the federal elections in the period 1998-2013. I analyze each sub-wave of the 2002

data separately.17 All but the 2013 survey included questions for six political parties, namely, the

Left Partie (LP),18 Alliance 90/Greens (A90G), the Social Democratic Party (SPD), the Liberal

Democratic Party (FDP), and the two branches of the Christian Democrats the CDU and CSU. In

2013, the Pirates and AfD parties were added to the survey.

17Besides differences in timing and sampling scheme, the telephone version of the 2002 survey

over-sampled East Germany.18PDS prior to the 2005 elections.

25

Table 3: Manifesto and Chapel Hill left-right scales

LP A90/G SPD FDP CDU CSU Pirates AfDManifesto ’98 -26.816 -16.256 -1.603 14.704 24.868 – – –Manifesto ’02 -21.327 -13.434 4.099 4.591 20.76 – – –Manifesto ’05 -31.239 -10.445 -2.154 16.913 25.611 – – –Manifesto ’09 -24.519 -13.571 -18.297 4.272 8.724 – – –Manifesto ’13 -34.547 -19.595 -23.568 14.036 2.564 – -12.653 -2.74

Chapel Hill ’02 1.64 3.360 4 6.070 5.930 7.360 – –Chapel Hill ’06 1.27 3.090 3.55 6.640 6.360 7.640 – –Chapel Hill ’10 1.313 3.625 3.625 6.6 6.125 7.067 – –Chapel Hill ’14 1.231 3.615 3.769 6.538 5.923 7.231 3.25 8.923

Rows marked Manifesto report RiLe scores (Volkens et al. (2018)). The two branches of theChristian Democrats (CDU, CSU) are not separately scaled. Rows marked Chapel Hill reportleft-right scales constructed from expert surveys (Bakker et al. (2015)).

To provide necessary background information and as a basis of comparison for the ensuing

analysis, Table 3 reports existing conventionally construed left-right scales of these parties in the

period of study by two different but prominent methods, the RiLe scores form the Manifesto project

and the left-right scale produced by expert surveys conducted by the Chapel Hill project. There

are three chief threats to the adequacy of a one-dimensional scale in the period of study: First,

while most green parties typically have a clear left leaning, the German green party (A90/G) either

by virtue of strategic aspiration or coincidental make-up has often defied such placement. Second,

the Liberal Democrats (FDP) endorse conservative economic ideas but take more liberal stances on

social issues. Finally, the two parties that emerged more recently in the German political scene (the

Pirates and the AfD) endorse populist, anti-systemic, and/or nativist ideas. The data of Table 3

reflect this possible tension: the Manifesto data place A90/G to the right of the SPD in the last

two elections (but A90/G is to the left of SPD in all other cases); Chapel Hill expert data place

the FDP to the right of the CDU (but to the left of the CSU) while Manifesto does so only for

the last election; while the Pirates and the AfD seem to confound the Manifesto data that places

them at the center of the policy space (while the experts more conventionally place the AfD to the

far-right).

As a first step, I apply estimator (4) of Section 2, with the results reported in Table 4. Two

orderings emerge as point estimates across all periods. First, the FDP is placed at the far-right

of the spectrum at the beginning and end of the sample period, while the conventional ordering

26

Table 4: Estimated party order

1998 2002a 2002b 2005 2009 2013c 2013π p1 p2 p2 p2 p2 p1 pd1

Observed violationsf 4,452 4,315 1,406 4,421 4,560 2,576 6,443

N e 1,799 1,912 845 1,940 1,903 1,731 1,276Observed triplets 35,980 38,240 16,900 38,800 38,060 34,620 71,456

J 6 6 6 6 6 6 8Possible orders 360 360 360 360 360 360 20,160

Assumption (D) is imposed by setting jL =SPD and jR =CSU. The party orders p1, p2 are givenby ordering parties (from left to right) asp1 : LP A90/G SPD CDU CSU FDP,p2 : LP A90/G SPD FDP CDU CSU.π is estimator (4) of Section 2.a Telephone survey. b Mail-back survey. c Estimation performed excluding two parties added inthat year’s survey (Afd and Pirates). d Party Pirates is left-most, followed by AfD, which is orderedto the left of LP. e Sample only includes respondents that scored all parties. f Number of violationsof uni-modality across sample triplets when parties are ordered according to π.Data Sources: The Comparative Study of Electoral Systems (2015, 2018), Modules 1-4.

placing the FDP in the middle of the spectrum in-between the SPD and CDU emerges in the

intermediate elections. Second, in the 2013 elections with the two added parties (last column of

Table 4) the Pirates and AfD appear at the far left of the spectrum. In all cases the Greens are

placed to the left of the SPD. The fraction of observed violations of uni-modality in the data is

north of 10% of the observed triplets, with some periods performing better than others. This level

provides a rough gauge of the error rate qe(τ) on average, but does not resolve whether either of

the conditions (B) or (R) hold in the data.

For that purpose we turn to hypothesis tests (HBp ) and (HR

p ) which are executed for all

possible party orders to construct confidence sets. In Table 5 I report these confidence sets built

using all three implementations of the two-step procedure of Romano, Shaikh and Wolf (2014).19

The strong test of one-dimensionality is rejected across all elections. There is no support for the

hypothesis that balance condition (B) holds in these data. Since the consistency of the estimator

(4) is premised on condition (B), these results cast doubt on the point estimates of Table 4. Turning

to the error rate condition (R), the picture is more nuanced. First, we can reject the weak test

19For admissible test statistics for these tests see Romano, Shaikh and Wolf (2014), Andrews and

Barwick (2012), and Andrews and Soares (2010).

27

Table 5: Confidence Sets and Specification Tests

1998 2002a 2002b 2005 2009 2013c 2013

StrongTest

CBp (α)–MMM ∅ ∅ ∅ ∅ ∅ ∅ ∅CBp (α)–QLR ∅ ∅ ∅ ∅ ∅ ∅ ∅CBp (α)–MAX ∅ ∅ ∅ ∅ ∅ ∅ ∅

WeakTest

CRp (α)–MMM ∅ ∅ {p2, p3, p5, p6} {p2, p3} {p1, p2, p4} {p1} {pd1}CRp (α)–QLR ∅ ∅ {p3, p5} {p2} {p1} {p1} ∅CRp (α)–MAX ∅ ∅ {p3, p5, p6} {p2} {p1} {p1} ∅

Comb.WeakTeste

CURp,r (α)–MMM ∅ ∅ ∅ ∅ {p1} {p1} {pd1}CURp,r (α)–QLR ∅ ∅ ∅ ∅ {p1} {p1} {pd1}CURp,r (α)–MAX ∅ ∅ ∅ ∅ {p1} {p1} ∅

Nf 1,799 1,912 845 1,940 1,903 1,731 1,276J 6 6 6 6 6 6 8

Possible orders 360 360 360 360 360 360 20,160

All tests carried out using the two-step procedure of Romano, Shaikh and Wolf (2014) with α = 0.05,β = 0.005, B = 10, 000. Assumption (D) is imposed by setting the SPD to the left of CSU. Partyorders p1 − p6 arep1 : LP A90/G SPD CDU CSU FDP,p2 : LP A90/G SPD FDP CDU CSU,p3 : LP SPD A90/G FDP CDU CSU,p4 : LP SPD A90/G CDU CSU FDP,p5 : SPD A90/G FDP CDU CSU LP,p6 : A90/G SPD FDP CDU CSU LP.a Telephone survey. b Mail-back survey. c Analysis performed excluding two parties added in thatyear’s survey (AfD and Pirates). d Party Pirates is left-most, followed by AfD, which is orderedto the left of LP. e Only the ordering coordinate p is reported. f Sample only includes respondentsthat scored all parties.Data Sources: The Comparative Study of Electoral Systems (2015, 2018), Modules 1-4.

28

of one-dimensionality in the 1998 elections. There is very little room to defend a one-dimensional

representation of the German party system in those elections. The same may be true in 2002

and possibly in 2013 when the test is performed using all eight parties of that survey. The 2002

mail-back survey seems to possibly admit a one-dimensional representation but this may be due to

smaller sample size or the different population of respondents in the two surveys. We cannot reject

the weak one-dimensionality test in the elections of 2005, 2009, and 2013 (without the added parties

Pirates and AfD). It is worth noting that the resulting confidence sets are small (even though (R)

does not necessarily point-identify the true ordering). Furthermore, with the exception of the set

built using the MMM statistic, the order that survives in the 2009 confidence sets is different than

the one that minimizes violations of uni-modality, and places the FDP at the far-right. Finally,

turning to tests that incorporate concavity (C), in two elections (2009, 2013) we cannot reject the

joint assumption of concavity combined with the weak one-dimensionality test.20 On the more

mundane but practically important side of things, the MAX test statistic proves a coherent and

inexpensive implementation of the tests proposed by Romano, Shaikh and Wolf (2014) compared

to the QLR and MMM alternatives.

In the subset of elections when these results do not outright reject the one-dimensional

model, the task remains of interpreting the resulting confidence sets in order to understand what

the content of that dimension might be. Of course, the standard statistical caveat still applies

that failure to reject the one-dimensionality test does not confirm that one dimension is indeed

adequate. Still, the non-empty confidence sets in Table 5 may be consistent with a conventional

economic left-right scale, but then the particular ordering is different than that produced by either

the Manifesto or Chapel Hill methods. Notably, the left-right interpretation requires the FDP at

the right-most position in the second half of the period and the Greens (except for the 2002 or

MMM implementations) to the left of the SPD.

Failure of the balance condition in these data warns us that the most extreme parties in

the scale may be harder to place: A larger fraction of the electorate strictly least prefer these

parties, so that when such parties are contained in a triplet the fraction of respondents that least

20Notably, for the 2013 election with all eight parties included, the combined weak test of one-

dimensionality and concavity is now not rejected for both the MMM and the QLR implementation

of the test.

29

prefer the extreme party on the other side of the spectrum in that triplet is more likely to fail (B).

Therefore, one-dimensional data may pass the weak test but the identified set includes party orders

placing the most extreme parties on both ends of the spectrum. This may account for the dual

placement of party LP in the 2002 mail-back survey results. It might appear tempting to attribute

the placement of the AfD and Pirates on the left-most position of the scale to the same effect, but

there is no corresponding placement of these parties to the right-most position.21 Indeed, we ought

to consider this datum with additional attention: There are 21,160 candidate orderings of the eight

parties (after orientation restriction (D) is imposed) and this is the only one that passed test (HRp )

or (HRUp,r0)! One possible interpretation is that if 2013 is consistent with a one-dimensional scale for

all eight parties, then that scale is a libertarian/totalitarian scale starting with the FDP as most

libertarian party, with the AfD and Pirates being most totalitarian parties. Another possibility

is that the world economic crisis of 2008 forced increased emphasis on narrow/populist economic

well-being (that subsumes nativist perspectives through that narrow lense), thus placing both the

AfD and Pirates at the extreme left and the economically conservative FDP at the right-most

position.

6 Conclusions

One dimensional summaries such as liberal, moderate, conservative, etc., are tremendously

efficient in conveying information about political parties, individual candidates, Supreme Court

nominees, and most other political arenas. It therefore behooves us to use one-dimensional scales

when they are adequate but it is also equally important to understand when is it that we may miss

something significant about the politics in question if we commit to a one dimensional summary?

It might be tempting to surmise that one dimension almost never suffices. But when it comes

to statistical sufficiency, such questions are addressed using rigorous test procedures. Because

statistical test procedures always test joint hypotheses, we also need to ensure we are not rejecting

the one dimensional model solely because individual preferences are not quadratic, or symmetric

around an ideal point, or similarly order restricted.

21Recall that the AfD is placed in the right-most position by the Chapel Hlll survey and in a

central position by the Manifesto scores.

30

The methods developed in this paper aim to give maximal leeway to a one-dimensional scale

and provide guidance on its reconstruction if it, and to the degree possible only it, is not rejected

by the data. Identification of the ordering on the scale is established under the balance condition.

The weaker error rate condition still produces narrow confidence sets without fully identifying the

order of parties. Both conditions are testable in the data. Implementation of this framework using

German CSES survey data yielded strong evidence against the balance condition. These data also

outright reject a one-dimensional scale in some but not all federal elections. Furthermore, cases

when a one-dimensional scale is not rejected by the data often require unconventional placement

of the parties and careful interpretation of the empirical content of the scale.

The analysis in this paper has focused exclusively on one-dimensional scales. Given the

results in Kalandrakis (2010), particularly his embedding Theorem 5, it is unlikely that this ap-

proach can be extended to more than one dimensions without added restrictions on preferences. I

leave such questions for future research. Within the one-dimensional framework, it is possible that

the austere setup in this paper leaves room for the development of procedures for oversampling

certain sub-populations in order to restore balance in the sample when the error rate condition is

met. The challenge with such schemes is to properly guard against biasing the results in favor of

one-dimensionality and I also leave this question for future research. In work that is currently in

progress, I use this framework to study more systematically the adequacy of one-dimensional party

scales in a comparative perspective and in Presidential elections in the US.

31

Appendix A Proofs of Theorems 1-4

The logical chain runs in reverse order, that is, Theorem 1 follows from Theorem 2, which

in turn is proved using Theorems 3 and 4.

Proof of Theorem 1. Since yij > yij′ ⇔ ui(xj) > ui(xj′) for all i, j, j′, we have qe(τ) = 0 for all

τ . (B0) further implies that qc(τ) is well-defined and qc(τ) > 0 for all τ . Consequently (B) holds

and the result follows from Theorem 2.

Proof of Theorem 2. Follows from Theorem 4.

Proof of Theorem 3. We proceed to show the three parts in the same order:

1. Assume (B). To show (MB) consider any τ and r = 1, 3. We have

Qr(τ) ≥ qc(τ) · P[πτ (r) ∈ arg maxj∈τ

u(xj)] (by definition of qc(τ))

≥ qc(τ) · Pr(τ) (because P[πτ (r) ∈ arg maxj∈τ

u(xj)] ≥ P[{πτ (r)} = arg maxj∈τ

u(xj)])

> qc(τ) · qe(τ)

qc(τ)(by (B))

≥ qe(τ) = Q2(τ) (by definition of qe(τ)).

Next, to show (R), we have for all τ

qc(τ) ≥ qc(τ)(P1(τ) + P3(τ)) (by (2))

> qc(τ) · 2qe(τ)

qc(τ)= 2qe(τ) (by (B)).

2. Assume (R) and consider any τ . We have

Q1(τ) + Q3(τ) ≥ qc(τ) ·∑r=1,3

P[πτ (r) ∈ arg minj∈τ

u(xj)] (by definition of qc(τ))

≥ qc(τ) (by (3))

> 2qe(τ) (by (R))

= 2Q2(τ) (by definition).

32

3. We will show there exists a distribution Φ satisfying qc(τ) < 2qe(τ) such that (¬Q) holds for

some τ . So fix τ and assume Φ is such that the mass of respondents that are indifferent among

any two parties in τ is null so that∑

r=1,3 Pr(τ) = 1 (by (3)), and that every respondent either

reports her true least preferred party in τ as strictly least preferred (with probability qc(τ))

or reports the middle party as strictly least preferred (with probability qe(τ) = 1 − qc(τ)).

Then, we have

Q1(τ) + Q3(τ) = qc(τ)(P1(τ) + P3(τ)) (by assumption)

= qc(τ) (by assumption)

< 2qe(τ) (because qc(τ) < 2qe(τ))

= 2Q2(τ) (by definition).

Now, if P1(τ) ≈ P3(τ) we have Qr(τ) < Q2(τ) for both r = 1, 3.

Proof of Theorem 4. It suffices to show that

limN→+∞

QN (π, y) < limN→+∞

QN (p, y),

for all p ∈ Π, p 6= π and p−1(jL) < p−1(jR). Then, since Π is finite, π is a consistent estimator

which also provides a constructive proof of identification of π. Observe that for all such p 6= π the

set

Tp := {τ | pτ (2) 6= πτ (2)},

is non-empty. Therefore, for such p 6= π we have

limN→+∞

(QN (p, y)−QN (π, y)) =∑τ∈Tp

(Qπ−1

τ (pτ (2))(τ)−Q2(τ)).

Because (B) implies (MB) (by Theorem 3) and π−1τ (pτ (2)) ∈ {1, 3} for all τ ∈ Tp, we conclude that

limN→+∞ (QN (p, y)−QN (π, y)) > 0 as desired.

33

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38

Appendix B – Supplemental: Moment functions

Table 6: Moment functions

θ Θ mM (θ, yi) (MM )

p Π

I(yipτ (2) < min

k=1,3{yipτ (k)}

)−I(yipτ (r) < min

k 6=r{yipτ (k)}

)

τ∈Tr=1,3

(MB)

p Π

2I(yipτ (2) < min

r=1,3{yipτ (r)}

)−∑r=1,3

I(yipτ (r) < min

k 6=r{yipτ (k)}

)τ∈T

(MR)

(p, x) ({p} ×Xp)p∈Π

(xp(r) − xp(r−1))(yip(r+1) − y

ip(r))

−(xp(r+1) − xp(r))(yip(r) − yip(r−1))

J−1

r=2

(MC)

(p, r) Π× {1, . . . , J}(

(I(r′ ≤ r)− I(r′ > r))(yip(r′−1) − yip(r′))

)Jr′=2

(MU )

Function mM (θ, yi) represents the contribution of sample response yi to each moment inequality onthe right column. The parameter input for moment inequalities (MC) involves some redundancyas it specifies both a party order and the actual location of the parties for ease of comparison withthe remaining moment inequalities.

Appendix C – Supplemental: Proof of Theorems 5 & 6

Proof of Theorem 5. For all r = 2, . . . , J − 1 we have

0 ≥ E[(xπ(r) − xπ(r−1))(u(xπ(r+1))− u(xπ(r))

)− (xπ(r+1) − xπ(r))

(u((xπ(r))− u(xπ(r−1))

)]

= (xπ(r) − xπ(r−1))(E[u(xπ(r+1))]− E[u(xπ(r))]

)− (xπ(r+1) − xπ(r))

(E[u(xπ(r))]− E[u(xπ(r−1))]

)]

= (xπ(r) − xπ(r−1))(E[yπ(r+1)]− E[yπ(r)]

)− (xπ(r+1) − xπ(r))

(E[yπ(r)]− E[yπ(r−1)]

).

The first line follows from (C), the second from linearity of expectation, the third from (E).

39

Both directions of Theorem 6 are, barring small details, well-understood. The proof pro-

vided here is rudimentary, but does cover and rely on the fact that at least two party locations are

distinct. If no pair of parties have distinct locations in Xp, then the second direction holds trivially

while the first direction does not hold, in general. The proof of the second direction is performed

by solving a reverse Afriat inequality problem of sorts, in that we are given ‘utility’ levels and seek

alternatives and supergradients that jointly satisfy concavity restrictions.

Proof of Theorem 6. ⇒ Suppose there exists x ∈ Xp that satisfies (MC) but there does not

exist r such that (MU ) holds. Let ¯r be the smallest candidate value such that E[yp(r−1)] ≥ E[yp(r)]

for all r > ¯r. Since (MU ) does not hold, ¯r > 1 and E[yp(¯r−1)] < E[yp(¯r)] (for otherwise ¯r is not the

smallest), and there exists r, 1 < r < ¯r such that E[yp(r−1)] > E[yp(r)] (it cannot be that r = ¯r

because E[yp(¯r−1)] < E[yp(¯r)]). Let this r be the largest such. We thus have

E[yp(r−1)] > E[yp(r)] ≤ E[yp(¯r−1)] < E[y(¯r)].

Then for τ = {p(r − 1), p(r), p(¯r)} we have xp(r−1) ≤ xp(r) ≤ xp(¯r) and (by E[mC(p, x, y)] ≤ 0)

(xp(r) − xp(r−1))(E[yp(¯r)]− E[yp(r)])− (x¯r − xp(r))(E[yp(r)]− E[yp(r−1)]) ≤ 0,

which is possible only if xp(r−1) = xp(r) = xp(¯r) = x. Then, since xj 6= xj′ for some j, j′, x 6= xk for

some k ∈ {j, j′}. If xk < x then for τ = {k, p(r), p(¯r)}

(xp(r)−xk)(E[yp(¯r)]−E[yp(r)])−(xp(¯r)−xp(r))(E[yp(r)]−E[yk]) = (xp(r)−xk)(E[yp(¯r)]−E[yp(r)]) > 0,

contradicting the assumption that (MC) hold. If, on the other hand, xk > x, then we obtain a

similar contradiction for triplet τ = {k, p(r), p(r − 1)}, that is,

(xp(r)−xp(r−1))(E[yk]−E[yp(r)])−(xk−xp(r))(E[yp(r)]−E[yp(r−1)]) = −(xk−xp(r))(E[yp(r)]−E[yp(r−1)]) > 0.

⇐ Fix p and suppose there exists r such that (MU ) holds. We need to show that there exists

x ∈ Xp such that (MC) holds. We must have xj 6= xj′ for some j, j′, so set xp(1) = 0 and xp(J) = 1

40

(hence, xj 6= xj′ for j = p(1), j′ = p(J)). Define

D :=

∑J

r=2 | E[yp(r)]− E[yp(r−1)] | if∑J

r=2 | E[yp(r)]− E[yp(r−1)] |> 0,

1 if∑J

r=2 | E[yp(r)]− E[yp(r−1)] |= 0.

For all r, define a pair of scalars xp(r), dr so that

dr =

D if r < r and E[yp(r)] < E[yp(r)]

0 if E[yp(r)] = E[yp(r)]

−D if r > r and E[yp(r)] < E[yp(r)],

and xp(1) = 0, xp(J) = 1 as above and for r = 2, . . . , J − 1 inductively set

xp(r) = xp(r−1) +| E[yp(r)]− E[yp(r−1)] |

D.

Observe, first that x = (x1, . . . , xJ) ∈ Xp. Furthermore, it is easy to verify22 that the Afriat

inequalities Afriat (1967)

E[yp(r)]− E[yp(r′)]− dr(xp(r) − xp(r′)) ≥ 0, for all r, r′

are satisfied. As a result, we can construct a function E : R→ R that takes the form

E(z) := minr=1,...,J

{E[yp(r)] + dr · (z − xp(r))}.

22By construction when dr = 0, with equality for r, r′ with sign(dr) = sign(dr′) when dr 6= 0,

and by the triangle inequality in the remaining cases.

41

E is concave as the minimum of concave functions and satisfies E(xp(r)) = E[yp(r)] for all r. We

now have:

E[mC(p, x, y)] =((xpτ (r) − xpτ (r−1))(E[ypτ (r+1)]− E[ypτ (r)])− (xpτ (r+1) − xpτ (r))(E[ypτ (r)]− E[ypτ (r−1)])

)J−1

r=2

=(

(xpτ (r) − xpτ (r−1))(E(xpτ (r+1))− E(xpτ (r)))− (xpτ (r+1) − xpτ (r))(E(xpτ (r))− E(xpτ (r−1))))J−1

r=2

≤ 0,

with the last inequality following by the concavity of E.

Appendix D – Supplemental: Ties in party locations

For expositional purposes, the analysis in the main text has been carried out under the

assumption that party locations are pairwise distinct. If we allow parties to share locations23 then

there may be multiple orderings, π, such that

xπ(1) ≤ xπ(2) ≤ . . . ≤ xπ(J).

Denote the set of such true orderings by Π∗ ⊆ Π. The population quantities Pr(τ) are not affected

by this possibility: If two parties j, j′ share a location xj = xj′ , we have u(xj) = u(xj′) for all u,

therefore for all τ = {j, j′, j′′}, we have Pr(τ) = 0 for all π ∈ Π∗ and all r such that πτ (r) = j or

πτ (r) = j′. Some other quantities defined in the main text may take different values depending on

the choice of π ∈ Π∗. To avoid this possible ambiguity, generalize qc(τ) to

qc(τ) := minr=1,3π∈Π∗

P[{πτ (r)} = arg min

j∈τyj | πτ (r) ∈ arg min

j∈τu(xj)

],

and qe(τ) to

qe(τ) := maxπ∈Π∗

P[{πτ (2)} = arg minj∈τ

yj ].

Using these generalized definitions, it easily follows that the weaker of the two condition (B) and

(R) precludes ties between pairs of parties:

23Except of course for parties jL, jR, of (D).

42

Lemma 1. If (R) holds, then Π∗ = {π} is singleton.

Proof. Suppose qc(τ) > 2qe(τ) but there exist parties j, j′ such that xj = xj′ . Consider any j′′

and the triplet τ = {j, j′, j′′}. Assume, without loss of generality, that P[{j} = arg minj∗∈τ yj∗ ] ≥

P[{j′} = arg minj∗∈τ yj∗ ]. Using the generalized definitions of qe, qc we conclude

qe(τ) ≥ P[{j} = arg minj∗∈τ

y∗ ] ≥ P[{j′} = arg minj∗∈τ

y∗ ] ≥ qc(τ),

which is impossible because (R) holds for all τ .

Since (B) implies (R), it follows that when either of the two holds, Qr(τ) as given in the

main text is well-defined.24 Therefore, Theorems 1, 2, 3, and 4 hold as stated using the above

generalized definitions. Theorems 5 and 6 are not affected either way.

Appendix E – Supplemental: Tests of existing scales using (MC)

24Although it is not necessary in any of the numbered results, we could also generalize the

definition of Qr(τ) to Qr(τ, π) := P[{πτ (r)} = arg minj∈τ yj ] to allow for multiple π ∈ Π∗.

43

Table 7: Testing Alternative Scales, Germany 2009

1998 2002a 2002b 2005 2009 2013

Manifesto

MMM 707.91 405.07 114.78 127.34 130.51 990.90critical level 4.58 2.92 2.92 4.49 4.19 5.36

QLR 706.10 400.59 113.48 120.79 153.10 1355.06critical level 4.38 2.86 2.82 66.07† 4.60 6.06

MAX 26.57 20.13 10.71 10.51 10.93 25.19critical level 1.97 1.72 1.71 1.99 1.97 2.18

ChapelHill

MMM – 262.40 38.87 72.05 60.17 1728.61critical level – 4.50 4.49 4.32 4.54 5.53

QLR – 262.31 42.91 74.67 54.70 2086.13critical level – 4.35 4.28 4.28 4.36 5.73

MAX – 14.49 6.23 7.45 5.99 30.46critical level – 2.01 2.01 1.99 2.02 2.19

N c 1,799 1,912 845 1,940 1,903 1,276

Tests (HCp,x0) carried out with x0 set at the scale values of Table 3 for the corresponding (nearest)

election CSES survey data. Tests executed using the two-step procedure of Romano, Shaikh andWolf (2014) on the square sample with α = 0.05, β = 0.005, B = 10, 000. The null hypothesis isrejected in all cases.a Telephone survey. b Mail-back survey. c Sample only includes respondents that scored all parties.† Numerically unstable value.Data Sources: The Comparative Study of Electoral Systems (2015, 2018), Modules 1-4.

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