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An approximation problem in computing electoral volatility
Applied Mathematics and Computation 192 (2007) 299–310 www.elsevier.com/locate/amc
An approximation problem in computing electoral volatility
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Francisco A. Ocan˜a Department of Statistics and Operations Research, University of Granada, Faculty of Economics, Campus de Cartuja, 18071 Granada, Spain
Abstract This paper is focused on an approximation problem which often arises when volatility is computed from electoral databases. Volatility is a key dimension in studies on electoral change or stability in party systems [M.N. Pedersen, The dynamics of European party systems: changing patterns of electoral volatility, European Journal of Political Research 7 (1979) 1–26]. Though the volatility formula offers no mathematical complexity, its application to real data often presents several problems. The present paper is inspired by the guidelines proposed by Bartolini and Mair [S. Bartolini, P. Mair, Identity, Competition and Electoral Availability: The Stabilization of European Electorates 1885–1985, Cambridge University Press, Cambridge, 1990] for the approximation problem of volatility. Once a general formulation for this problem is provided, Bartolini and Mair’s (BM) approach is justified through the analysis of its approximation error. This leads to suggest a class of approximations of volatility whose errors are also analyzed. One of the goals of this paper is to prove that these new approximations are an improvement of the BM approximation. Finally, the performance analysis of the considered approximations is illustrated through examples with data. 2007 Elsevier Inc. All rights reserved. Keywords: Election; Party system; Volatility; L1 norm; Data analysis
1. Introduction The electoral change or stability in party systems is a topic of interest in Political Science. In quantitative studies on this topic, the volatility measure proposed by Pedersen [7] is a basic tool. Though the Pedersen volatility (PV) measure offers no mathematical complexity, some computational problems appear in practice, for example, when the PV measure is computed from electoral databases. These computational problems, which were first pointed out by Bartolini and Mair [1, Appendix 1], are essentially explained by some intrinsic properties of real data which make the straight application of the PV formula impossible. The guidelines
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This research has been supported by Project MTM2004-5992 from Direccio´n General de Investigacio´n, Ministerio de Ciencia y Tecnologı´a, Spain. E-mail address: [email protected] URL: http://www.ugr.es/local/focana 0096-3003/$ - see front matter 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2007.03.032
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proposed by Bartolini and Mair [1, Appendix 1] to face such computational problems have inspired the present study. Bartolini and Mair [1] carried out a study on the stabilization of European electorates between 1885 and 1985. In this outstanding research, some computational problems arose when the PV measure (total volatility) was calculated from data. Such computational problems will be outlined here in Section 2.1, but more details are found in Bartolini and Mair [1, Appendix 1]. These problems were empirically solved by Bartolini and Mair (BM) for the data obtained from European elections over the period 1885–1985. In the present paper, one of these computational problems on volatility will be solved in a general framework: the approximation of volatility measure. The present paper is sketched out as follows. The main theoretic results are contained in Section 2. The computational problems on volatility treated in Bartolini and Mair [1] are outlined in Section 2.1. Among such problems, the approximation problem of volatility is treated in Section 2.2, where theoretic results are derived in a general framework. Among these results, a mathematical justification in favor of the BM approximation of the volatility is provided by means of the study of its error. This study leads to propose a class of approximations of volatility, which is analysed in Section 2.2.1. It is also proved that these new approximations are an improvement of the BM approximation by studying its approximation errors. The performance analysis for such alternative approximations is illustrated through some examples in Section 3. Finally, some conclusions are derived in Section 4. 2. Approximation of volatility In this section, we will study the problem of volatility approximation which often arises when it is computed from electoral databases in practice. 2.1. Problems computing volatility in practice Consider two elections whose patterns of change are going to be quantified. Assume that each competing unit (party, coalition, etc.) in both elections is denoted by an integer in I ¼ f1; . . . ; N g N. Thus a data set of pairs given by fðpi ; qi Þ : i 2 Ig can be obtained, where pi and qi stand for the electoral strengths (votes, seats, etc.) of the (party) unit i 2 I in both elections, respectively. Indeed, ðpi Þi2I and ðqi Þi2I are the distributions of electoral strengths associated to the set of the N party units in both considered elections, which are also supposed that N X
pi ¼
i¼1
N X
qi ¼ 1:
ð1Þ
i¼1
To measure the change between two elections, the PV measure could be computed from its definition in Pedersen [7] by the formula V ¼
N 1X jp qi j: 2 i¼1 i
ð2Þ
Though the PV formula is very simple, the following computational problems can arise when V is calculated from real data in practice. • The first problem appears when the competing parties in both considered elections are not the same. This problem was solved in Bartolini and Mair [1, Appendix 1, pp. 311–312], where they proposed a set of rules describing what to do in different situations given by the merging or splitting of parties, etc. between two consecutive elections. Once these rules are applied to our data, the equality of the sets of parties in both elections can be assumed in the so transformed data [3,6]. • The second computational problem, which is the target of this paper, appears when part of electoral data is unknown. This problem can appear in several situations in practice. For example, a part of data is unknown when the party category Others appears in the electoral data and no information is given for par-
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ties included in this category, such as is the case in Bartolini and Mair [1, Appendix 1]. However, this problematic situation is even usual in volatility studies carried out from electoral databases. For example, details on parties with non significant strength may be omitted to maintain memory capabilities under minimum requirements. Moreover, another example appears when the researcher is only interested in relevant parties [8, p. 122] and, thus, those non–relevant parties are not considered for the BM rules [5]. From a theoretic point of view, Eq. (2) presents no mathematical complexity. However, if a part of electoral data is not available, then V cannot be evaluated. In our framework, this problem can be formulated by assuming that some of the pairs in fðpi ; qi Þ : i 2 Ig are unknown. In summary, the present work will analyze the approximation of V in such a framework. The problem of volatility approximation was partially solved in Bartolini and Mair [1, Appendix 1, p. 312] from an empirical point of view. These authors considered two approximating indexes of V: TV (the index of total volatility with Others) and TVWO (the index of total volatility without Others). They studied the differences between both volatility indexes from the 1885–1985 European election data. Though they concluded that non significant difference between TV and TVWO was encountered, they eventually decided to use TV, i.e., V was approximated by using Eq. (2) with the category Others in the available data. This way, these authors provided a partial and inspiring solution to the approximation problem of volatility. In the present paper, the approximation problem of V will be analyzed in a general framework. Among other results, we will prove that the BM solution (approximation) becomes even interesting in more general data settings. Moreover, an improvement of the BM solution is going to be derived. 2.2. Approximations of volatility and errors Throughout this paper, assume that only some M–first pairs of strengths are the available data, for any M 2 I, which are given by fðpk ; qk Þ : k ¼ 1; . . . ; Mg fðpi ; qi Þ : i 2 Ig: This means that the subset given by fðpj ; qj Þ : j ¼ M þ 1; . . . ; N g is assumed unknown. Notice that to consider the first M pairs is not a lack of generality, because no order is assumed in the electoral data fðpi ; qi Þ : i 2 Ig. Under the above introduced assumption, V cannot be thus calculated, because some of the terms in Eq. (2) are unknown. Therefore, the value of V is assumed unknown in what follows and, also, must then be approximated from the available information fðpk ; qk Þ : k ¼ 1; . . . ; Mg. As a first approach to our problem, a naive approximation of V is given by M 1X V ½M ¼ jp qk j; ð3Þ 2 k¼1 k where Eq. (2) has been applied on the available pairs of strengths. It can be established from Eqs. (2) and (3) that N 1 X V ¼ V ½M þ jp qj j; 8M 2 I: ð4Þ 2 j¼Mþ1 j On the other hand, Eq. (1) could be rewritten by M M X X pk þ P ½M ¼ qk þ Q½M ¼ 1; k¼1
k¼1
where P ½M and Q½M are defined by P ½M ¼
N X
pj ¼ 1
j¼Mþ1
Q½M ¼
N X j¼Mþ1
M X
pk
and
k¼1
qj ¼ 1
M X k¼1
qk ;
8M 2 I:
ð5Þ
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Notice that P ½M and Q½M are interpreted as the overall proportions of electoral strength for those party units not included in the available data in both elections, respectively. Further, though electoral strengths in fðpj ; qj Þ : j ¼ M þ 1; . . . ; N g are unknown, P ½M and Q½M are obtained from the available data by using Eq. (5). Proposition 1. V ½ is a non decreasing function on I and the following inequalities hold: 0 6 V ½M 6 VL½M 6 V 6 VU½M;
8M 2 I;
where VL½ and VU½ are the functions on I defined by 1 VL½M ¼ V ½M þ jP ½M Q½Mj and 2 1 VU½M ¼ V ½M þ ðP ½M þ Q½MÞ; 8M 2 I: 2
ð6Þ ð7Þ
Furthermore, as M tends to N, V ½M, VL½M and VU½M get closer and closer to V, with V ½N ¼ VL½N ¼ V ¼ VU½N . Proof. By using Eqs. (3) and (4), it can be assured that V ½ is a non decreasing function on I such that 0 6 V ½M 6 V , 8M 2 I and V ½N ¼ V . Two hypothetical situations are now going to be considered for a given M 2 I. In each of such situations, the behavior of the PV measure for aggregated data is going to be applied (see Appendix A). • Suppose that data for parties in fM þ 1; . . . ; N g were aggregated to obtain an unique bloc in both considered elections. By using Appendix A, it can be established that M 1X 1 1 V P jpk qk j þ jP ½M Q½Mj ¼ V ½M þ jP ½M Q½Mj: 2 k¼1 2 2 Notice that the volatility from this hypothesized data (in the right hand side of the above equation) can be interpreted as a bloc volatility. • Consider the lowest level of aggregation for those parties whose information is not available. Such a hypothetical situation is given when their corresponding pairs are fðpj ; 0Þ; ð0; qj Þ : j ¼ M þ 1; . . . ; N g. Notice that the theoretic data set fðpi ; qi Þ : i 2 Ig can be obtained by aggregating these hypothesized data. It can be established from Appendix A that V 6
M N 1X 1 X 1 jpk qk j þ jpj 0j þ j0 qj j ¼ V ½M þ ðP ½M þ Q½MÞ: 2 k¼1 2 j¼Mþ1 2
ð8Þ
The volatility from the hypothesized data (in the right hand side of the above equation) can be viewed as total volatility and thus V can be viewed as bloc volatility. Taking into account Eqs. (4) and (5), as M tends to N, the values of P ½M and Q½M get closer and closer to zero and V ½M gets closer and closer to V. It follows that VL½M and VU½M get closer and closer to V, as M tends to N. The proof is completed by taking into account that P ½N ¼ Q½N ¼ 0 and V ½N ¼ V . h The formula in Eq. (6) is used to approximate V in Bartolini and Mair [1]. In fact, it can be said that these authors used VL½M and V ½M, i.e., TV ¼ VL½M and TVWO ¼ V ½M (see Section 2.1). Bartolini and Mair [1] decided to use TV and discarded TVWO in their volatility analysis by using empirical argumentation. In the present study, Proposition 1 assures that TV ¼ VL½M is always closer to the true value of V than the one obtained from TVWO ¼ V ½M. Therefore, Bartolini and Mair’s choice in favor of TV is mathematically justified for any electoral data. The error of the BM approximation, VL½M ¼ TV, is defined by V VL½M. However, as V is assumed unknown, such an error cannot be calculated. Nevertheless, it is known from Proposition 1 that V is contained in the interval ½VL½M; VU½M. This implies that an upper error bound for VL½M can be defined by EBM½M ¼ VU½M VL½M;
8M 2 I;
ð9Þ
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where 0 6 V VL½M 6 EBM½M, for any M 2 I. Therefore, EBM½M provides a prior quality measure of VL½M, 8M 2 I. Corollary 2. EBM½ is a non increasing function on I with EBM½N ¼ 0 and can be obtained by EBM½M ¼ minfP ½M; Q½Mg;
8M 2 I:
Proof. Taking into account Eqs. (6) and (7), we obtain that 1 1 1 EBM½M ¼ ðP ½M þ Q½MÞ jP ½M Q½Mj ¼ ðP ½M þ Q½M jP ½M Q½MjÞ 2 2 2 ¼ minfP ½M; Q½Mg: The proof is completed by using that P ½ and Q½ are both non increasing functions on I with P ½N ¼ Q½N ¼ 0. h 2.2.1. A new approach The study of the error for the BM approximation suggests an alternative approach to the volatility approximation. A class of approximations of V is proposed as follows: V h ½M ¼ hVU½M þ ð1 hÞVL½M;
8M 2 I and 8h 2 ½0; 1:
ð10Þ
Notice that VL½ and VU½ are in the class fV h ½gh for h = 0 and h = 1, respectively. Corollary 3. For any h 2 ½0; 1, the approximation V h ½M can be obtained by V h ½M ¼ VL½M þ hEBM½M;
8M 2 I
and gets closer and closer to V, as M tends to N, with V h ½N ¼ V . Proof. It is derived by using Eq. (9) and Corollary 2 that V h ½M ¼ VL½M þ hðVU½M VL½MÞ ¼ VL½M þ hEBM½M ¼ VL½M þ h minfP ½M; Q½Mg;
8M 2 I:
The proof is completed by using Proposition 1 and Corollary 2. h On the one hand, it can be established by using Eq. (10) that VL½M 6 V h ½M 6 VU½M;
8M 2 I:
On the other hand, Proposition 1 establishes that V 2 ½VL½M; VU½M, for any M 2 I. This implies that the (unknown) approximation error for V h ½M, which is given by jV V h ½Mj, is bounded by ECh ½M ¼ maxfV h ½M VL½M; VU½M V h ½Mg;
ð11Þ
where jV V h ½Mj 6 ECh ½M, 8M 2 I and 8h 2 ½0; 1. By the way, ECh ½M provides a prior quality measure of V h ½M, 8M 2 I and 8h 2 ½0; 1. Corollary 4. For any h 2 ½0; 1, ECh ½ is a non increasing function on I with ECh ½N ¼ 0 and can be obtained by ECh ½M ¼ cðhÞEBM½M;
8M 2 f1; . . . ; N g;
where c : ½0; 17!½0; 1 is defined by cðhÞ ¼ maxfh; 1 hg, 8h 2 ½0; 1. Proof. From Eq. (10), we obtain that V h ½M VL½M ¼ hðVU½M VL½MÞ
and
VU½M V h ½M ¼ ð1 hÞðVU½M VL½MÞ:
It follows from Eqs. (11) and (9) that ECh ½M ¼ cðhÞðVU½M VL½MÞ ¼ cðhÞEBM½M ¼ cðhÞ minfP ½M; Q½Mg: Finally, the proof is completed by using Corollary 2 in the above expression. h
ð12Þ
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Corollary 4 leads to establish that ECh ½M 6 EBM½M;
8h 2 ½0; 1 and 8M 2 I:
This means that the new approximations V h ½M improve the BM approximation VL½M, when no prior information on V is available (see Section 4). In fact, when EBM½M > 0, we have that ECh ½M < EBM½M;
8h 2 ð0; 1Þ:
To select an optimum value of h, the approximation V h ½M with minimum upper error bound is going to be considered. It follows from Eq. (12) that the minimum of ECh ½M, with h 2 ½0; 1, is derived from the minimum of cðhÞ, which is achieved when h ¼ 1=2. In what follows, this optimum approximation of V is denoted by VA½M ¼ V 1=2 ½M ¼ ðVU½M þ VL½MÞ=2;
8M 2 I;
being its upper error bound given by 1 1 EA½M ¼ EC1=2 ½M ¼ EBM½M ¼ minfP ½M; Q½Mg: 2 2 Therefore, it can be established that EA½M 6 ECh ½M 6 EBM½M;
8h 2 ½0; 1 and 8M 2 I:
ð13Þ
ð14Þ
This expression justifies the interest for VA½M and, also, summarizes the performance analysis of the considered approximations in this paper. A more efficient formula for VA½M can be obtained from Eqs. (6) and (7) as follows: 1 1 VA½M ¼ ðVU½M þ VL½MÞ ¼ V ½M þ ðP ½M þ Q½M þ jP ½M Q½MjÞ 2 4 1 ¼ V ½M þ maxfP ½M; Q½Mg; 8M 2 I: ð15Þ 2 Finally, throughout this section some equations have been qualified as (computationally) efficient. These equations are all in terms of V ½M, P ½M and Q½M, which are values first computed from data in practice. The main reason for such a distinction is given by the fact that the implementation of such equations becomes an easy task in practice [4]. 3. A study with data The aim of this section is to illustrate the performance analysis of the approximations of V carried out in Section 2. Indeed these approximations will be applied to data in order to illustrate their behavior in practice. To simplify the resulting comparative study, our attention is focused on the BM approximation, VL½M, and VA½M, which is proposed in this paper. Four artificial examples will be considered throughout this section, being their data sets different variations from a common data subset. In fact, four examples enumerated by 0–3, which are given by their corresponding sets of pairs of electoral strengths (see Section 2.1), are considered with N = 9, 10, 11, 12, respectively. Once Example 0 is presented, Examples 1–3 are obtained by modifying one of the pairs not belonging to 6 the first six pairs. This means that the subset fðpi ; qi Þgi¼1 is common for Examples 0–3 (in Table 1), being their differences located in the rest of pairs (in Table 2). Observe the decreasing level of data aggregation on each example in Table 2. The usual framework for computing volatility in practice (see Section 2) will be assumed. Suppose that the first six pairs of strengths were the available data to approximate V, i.e., M = 6. Table 1 contains the approximations V ½6, VL½6, VA½6 and VU½6, the proportions P ½6 and Q½6 and the upper error bounds for VL½6 and VA½6, denoted by EBM½6 and EA½6, respectively. Notice that all these values are the same for each of Examples 0–3, when M = 6. The rest of pairs of electoral strengths for each example appear in Table 2. They are assumed unknown in the volatility approximation framework. However, the true values of V can be evaluated for Examples 0–3, since all the pairs of strengths are known (Tables 1 and 2). The true value of V and the approximation errors
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Table 1 The first six pairs of electoral strengths, the approximations of volatility with M = 6 and some upper error bounds, for Examples 0–3 i
pi
qi
1 2 3 4 5 6 .. . M=6
0.3 0.25 0.15 0.1 0.1 0.05 .. .
0.4 0.2 0.2 0.1 0.02 0.03 .. .
P ½6 ¼ 0:05
Q½6 ¼ 0:05
V ½6 0.15
VL[6] 0.15
VA[6] 0.175
EBM [6] 0.5
EA [6] 0.25
VU[6] 0.2
The rest of pairs are in Table 2.
Table 2 The rest of pairs of strengths for Examples 0–3 i
pi
qi
P[i]
Q[i]
Example 0 7 8 9
0.02 0.02 0.01
0.02 0.01 0.02
0.03 0.01 0
0.03 0.02 0
Example 1 7 8 9 10
0.02 0 0.02 0.01
0 0.02 0.01 0.02
0.03 0.03 0.01 0
0.05 0.03 0.02 0
Example 2 7 8 9 10 11
0.02 0 0.02 0 0.01
0 0.02 0 0.01 0.02
0.03 0.03 0.01 0.01 0
0.05 0.03 0.03 0.02 0
Example 3 7 8 9 10 11 12
0.02 0 0.02 0 0.01 0
0 0.02 0 0.01 0 0.02
0.03 0.03 0.01 0.01 0 0
0.05 0.03 0.03 0.02 0.02 0
of VL½6 and VA½6 are contained in Table 3, for every example. Notice that Table 3 shows that the upper error bounds, which appear in Table 1, can be achieved by the approximation errors (see Example 3). Eq. (14) leads to establish that, when only M of the pairs of electoral data are known, VA½M would provide an approximation to V with more prior warranties than the one obtained by the BM approach. However, we can not say that the approach presented here is better than the one proposed by BM over all possible data settings. For instance, in Table 3, Example 0 illustrates that the error of the BM approach (V VL½6 ¼ 0:01) could be less than the corresponding error of VA½M (jV VA½6j ¼ 0:015). The opposite situation is observed in Examples 1–3, where VA½6 provides the best approximations to V. Another confirmation of this comment will be derived from Fig. 6.
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Table 3 True volatility measures and approximation errors of VL[6] and VA[6], for Examples 0–3 # Examp.
N
V
V VL[6]
jV VA[6]j
0 1 2 3
9 10 11 12
0.16 0.18 0.19 0.20
0.01 0.03 0.04 0.05
0.015 0.005 0.015 0.025
A dynamic point of view in our study is introduced, when M varies on f1; . . . ; N g and the approximations of V are obtained for each M. Due to the way Examples 0–3 are generated, the sequences of approximations versus M are plotted as follows in order to ease interpretation. Fig. 1 contains the (first) common values of V ½M, VL½M, VA½M and VU½M, with M 2 f1; . . . ; 6g, and the true values of V (see Table 3) for Examples 0–3. The rest of approximated values, for M P 6, are plotted in Figs. 2–5 for Examples 0–3, respectively. In these plots, the rate of convergence of the considered approximations to V can be observed, when M is increasing. In fact, the approximations given by VA½M exhibit a better approximating performance than those by the BM approach (VL½M) in Examples 1–3, but not in Example 0. Finally, a new approach to compare VL½M and VA½M is provided in Fig. 6, where the true approximation errors of VL½M and VA½M are plotted against the values of minfP ½M; Q½Mg which are below 0:1, in Examples 0–3 together. In this figure, we compare the performance of such approximations with the minimum pro-
Fig. 1. Common approximations V ½M, VL½M, VA½M and VU½M versus M 2 f1; . . . ; 6g and the values of volatility V (dotted constant lines), for Examples 0–3. The true volatility values appear in Table 3.
Fig. 2. Approximations V ½M, VL½M, VA½M and VU½M versus M 2 f6; . . . ; 9g and the value of volatility V (constant), for Example 0.
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Fig. 3. Approximations V ½M, VL½M, VA½M and VU½M versus M 2 f6; . . . ; 10g and the value of volatility V (constant), for Example 1.
Fig. 4. Approximations V ½M, VL½M, VA½M and VU½M versus M 2 f6; . . . ; 11g and the value of volatility V (constant), for Example 2.
Fig. 5. Approximations V ½M, VL½M, VA½M and VU½M versus M 2 f6; . . . ; 12g and the value of volatility V (constant), for Example 3.
portion of the overall strength not considered in our available data. Notice that, for each value of minfP ½M; Q½Mg, the errors for VL½M and VA½M are not clearly separated. However, the errors for VA½M are always located below the threshold given by 0:5 minfP ½M; Q½Mg, such as is established in Eq. (13).
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Fig. 6. Approximation errors of VL½M and VA½M versus minfP ½M; Q½Mg, for Examples 0–3.
4. Comments and conclusions Modern electoral databases provide to the researcher a complete information source but their use reveals some new difficulties. In this paper we study an approximation problem which usually arises when volatility is computed in practice. The approximation proposed by Bartolini and Mair [1, Appendix 1] has been mathematically studied by analyzing its approximation error. This study has inspired a new approach which represents an improvement of the BM approach. The main theoretic results on the approximations of V considered in this paper are found in Corollaries 2 and 4 and in Eqs. (13) and (14). Among these approximations, this section focuses on two approximations of V: the BM approximation, VL½M, and VA½M, which is proposed in this paper. The study on the approximation errors for VL½M and VA½M leads to establish that VA½M exhibits the best approximation performance, when no prior knowledge is given for V (Eq. (14)). This does not mean that the approximation VA½M is better than VL½M over all data settings, however, a better approximation performance could be expected for VA½M. In what follows, I am going to explain these ideas. The first idea is that VA½M exhibits the best prior approximation performance, when no prior knowledge is given for V. The best way of illustrating this idea is to show its applicability for real problems. Let us suppose that V (assumed unknown) is going to be approximated from real data in an electoral database. Indeed the rules proposed in Bartolini and Mair [1, pp. 311–312] to define some of the pairs of electoral strengths (Section 2.1) could have been applied. Therefore, there exists a part of electoral data not taken into account on these pairs. Problem 1. Let fðpk ; qk Þ : k ¼ 1; . . . ; Mg be the available electoral data set. In this case, the only information on V is given by jV VL½Mj 6 } and jV VA½Mj 6 0:5} (Corollaries 2 and 4), where } ¼ minfP ½M; Q½Mg is computed from Eq. (5). This means that it is expected that VA½M is more accurate than VL½M to approximate V. Problem 2. Now, assume that we want to approximate V with a prior approximation error given by (M is not given). The question is how P ½M and Q½M have to be in order to achieve such an admissible error in volatility approximation. For example, if volatility scores (expressed in percentages) are going to be rounded up from the first decimal point, then we could consider ¼ 5 104 ¼ 0:05%. • For the BM approximation, our objective is given by EBM½M 6 . By using Corollary 2, we can thus establish that the set of parties which can be dropped out for the BM approximation must satisfy that minfP ½M; Q½Mg ¼ EBM½M 6 :
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• For the approximation VA½M, we impose that EA½M 6 . By using Eq. (13), we can thus establish that the set of parties to be dropped out must satisfy that minfP ½M; Q½Mg ¼ 2EA½M 6 2: For example, if ¼ 5 104 , then it could be previously established that ( minfP ½M; Q½Mg 6
5 104 ¼ 0:05%; for VL½M; 103 ¼ 0:1%;
for VA½M:
This means that the BM approximation needs more data information than the approximation VA½M, for a given prior admissible error . I will now try to justify that the performance of such approximations depends on the data setting. Firstly, under the hypothesized settings considered to derive Eq. (8), it is obvious that the approximation given by VU½M will provide the best approximation to V, i.e. V VU½M. This implies that, under conditions closed to the lowest level of data aggregation considered in Eq. (8), the approximation given by VA½M will be closer to V than the one obtained from VL½M. This behaviour is illustrated by the performance of such approximations in Example 3 in Section 3. Secondly, let us now try to identify a data setting where the BM approach (VL½M) provides the best approximation, i.e., VL½M ¼ V . Assume that the strengths not belonging to the first M pairs are equally distributed, i.e. pj qj ¼ ; 8j ¼ M þ 1; . . . ; N : ð16Þ P ½M Q½M PN Let us denote aj ¼ pj =P ½M ¼ qj =Q½M. It is obvious that aj P 0, 8j ¼ M þ 1; . . . ; N , and j¼Mþ1 aj ¼ 1. It is obtained from this assumption and Eq. (4) that V ¼ V ½M þ
N N 1 X 1 X 1 jpj qj j ¼ V ½M þ aj jP ½M Q½Mj ¼ V ½M þ jP ½M Q½Mj ¼ VL½M: 2 j¼Mþ1 2 j¼Mþ1 2
This means that, when values fpj : j ¼ M þ 1; . . . ; N g and fqj : j ¼ M þ 1; . . . ; N g are equally distributed, then the BM approximation provides the unknown value of V and thus exhibits the best approximation performance. Appendix A. Volatility and data aggregation In this appendix, details are provided about the behaviour of the PV measure for aggregated data, which are needed in the proof of Proposition 1 (Section 2.2). The behaviour of the PV measure, when initial data are aggregated to constitute an arbitrary number of blocs of parties, is analyzed in Ocan˜a [2, Propositions 4 and 8], where TV (total volatility) and BV (bloc volatility) stands for the PV measure from the initial data and from the aggregated data, respectively. In fact, [2] establishes that the relation of order, for such volatility indexes, is always given by 1 P TV P BV P 0. References [1] S. Bartolini, P. Mair, Identity Competition and Electoral Availability: The Stabilization of European Electorates 1885–1985, Cambridge University Press, Cambridge, 1990. [2] F.A. Ocan˜a, An approach to electoral volatility: a method for exploring blocs of parties, Social Science Research, submitted for publication. [3] F.A. Ocan˜a, P. On˜ate, Las elecciones autono´micas de 1999 y las espan˜as electorales, Revista Espan˜ola de Investigaciones Sociolo´gicas 90 (2000) 183–228. [4] F.A. Ocan˜a, P. On˜ate, IndElec: a software for analyzing party systems and electoral systems, Journal of Statistical Software, submitted for publication.
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[5] P. On˜ate, F.A. Ocan˜a, Ana´lisis de datos electorales, Cuadernos Metodolo´gicos, vol. 27, Centro de Investigaciones Sociolo´gicas, Madrid, 1999. [6] P. On˜ate, F.A. Ocan˜a, Elecciones de marzo de 2000: ? cua´nto cambio electoral? Revista de Estudios Polı´ticos 110 (2000) 297–336. [7] M.N. Pedersen, The dynamics of European party systems: changing patterns of electoral volatility, European Journal of Political Research 7 (1979) 1–26. [8] G. Sartori, Parties and Party Systems: A Framework of Analysis, Oxford University Press, Oxford, 1976.
An approximation problem in computing electoral volatility
q
Francisco A. Ocan˜a Department of Statistics and Operations Research, University of Granada, Faculty of Economics, Campus de Cartuja, 18071 Granada, Spain
Abstract This paper is focused on an approximation problem which often arises when volatility is computed from electoral databases. Volatility is a key dimension in studies on electoral change or stability in party systems [M.N. Pedersen, The dynamics of European party systems: changing patterns of electoral volatility, European Journal of Political Research 7 (1979) 1–26]. Though the volatility formula offers no mathematical complexity, its application to real data often presents several problems. The present paper is inspired by the guidelines proposed by Bartolini and Mair [S. Bartolini, P. Mair, Identity, Competition and Electoral Availability: The Stabilization of European Electorates 1885–1985, Cambridge University Press, Cambridge, 1990] for the approximation problem of volatility. Once a general formulation for this problem is provided, Bartolini and Mair’s (BM) approach is justified through the analysis of its approximation error. This leads to suggest a class of approximations of volatility whose errors are also analyzed. One of the goals of this paper is to prove that these new approximations are an improvement of the BM approximation. Finally, the performance analysis of the considered approximations is illustrated through examples with data. 2007 Elsevier Inc. All rights reserved. Keywords: Election; Party system; Volatility; L1 norm; Data analysis
1. Introduction The electoral change or stability in party systems is a topic of interest in Political Science. In quantitative studies on this topic, the volatility measure proposed by Pedersen [7] is a basic tool. Though the Pedersen volatility (PV) measure offers no mathematical complexity, some computational problems appear in practice, for example, when the PV measure is computed from electoral databases. These computational problems, which were first pointed out by Bartolini and Mair [1, Appendix 1], are essentially explained by some intrinsic properties of real data which make the straight application of the PV formula impossible. The guidelines
q
This research has been supported by Project MTM2004-5992 from Direccio´n General de Investigacio´n, Ministerio de Ciencia y Tecnologı´a, Spain. E-mail address: [email protected] URL: http://www.ugr.es/local/focana 0096-3003/$ - see front matter 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2007.03.032
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proposed by Bartolini and Mair [1, Appendix 1] to face such computational problems have inspired the present study. Bartolini and Mair [1] carried out a study on the stabilization of European electorates between 1885 and 1985. In this outstanding research, some computational problems arose when the PV measure (total volatility) was calculated from data. Such computational problems will be outlined here in Section 2.1, but more details are found in Bartolini and Mair [1, Appendix 1]. These problems were empirically solved by Bartolini and Mair (BM) for the data obtained from European elections over the period 1885–1985. In the present paper, one of these computational problems on volatility will be solved in a general framework: the approximation of volatility measure. The present paper is sketched out as follows. The main theoretic results are contained in Section 2. The computational problems on volatility treated in Bartolini and Mair [1] are outlined in Section 2.1. Among such problems, the approximation problem of volatility is treated in Section 2.2, where theoretic results are derived in a general framework. Among these results, a mathematical justification in favor of the BM approximation of the volatility is provided by means of the study of its error. This study leads to propose a class of approximations of volatility, which is analysed in Section 2.2.1. It is also proved that these new approximations are an improvement of the BM approximation by studying its approximation errors. The performance analysis for such alternative approximations is illustrated through some examples in Section 3. Finally, some conclusions are derived in Section 4. 2. Approximation of volatility In this section, we will study the problem of volatility approximation which often arises when it is computed from electoral databases in practice. 2.1. Problems computing volatility in practice Consider two elections whose patterns of change are going to be quantified. Assume that each competing unit (party, coalition, etc.) in both elections is denoted by an integer in I ¼ f1; . . . ; N g N. Thus a data set of pairs given by fðpi ; qi Þ : i 2 Ig can be obtained, where pi and qi stand for the electoral strengths (votes, seats, etc.) of the (party) unit i 2 I in both elections, respectively. Indeed, ðpi Þi2I and ðqi Þi2I are the distributions of electoral strengths associated to the set of the N party units in both considered elections, which are also supposed that N X
pi ¼
i¼1
N X
qi ¼ 1:
ð1Þ
i¼1
To measure the change between two elections, the PV measure could be computed from its definition in Pedersen [7] by the formula V ¼
N 1X jp qi j: 2 i¼1 i
ð2Þ
Though the PV formula is very simple, the following computational problems can arise when V is calculated from real data in practice. • The first problem appears when the competing parties in both considered elections are not the same. This problem was solved in Bartolini and Mair [1, Appendix 1, pp. 311–312], where they proposed a set of rules describing what to do in different situations given by the merging or splitting of parties, etc. between two consecutive elections. Once these rules are applied to our data, the equality of the sets of parties in both elections can be assumed in the so transformed data [3,6]. • The second computational problem, which is the target of this paper, appears when part of electoral data is unknown. This problem can appear in several situations in practice. For example, a part of data is unknown when the party category Others appears in the electoral data and no information is given for par-
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ties included in this category, such as is the case in Bartolini and Mair [1, Appendix 1]. However, this problematic situation is even usual in volatility studies carried out from electoral databases. For example, details on parties with non significant strength may be omitted to maintain memory capabilities under minimum requirements. Moreover, another example appears when the researcher is only interested in relevant parties [8, p. 122] and, thus, those non–relevant parties are not considered for the BM rules [5]. From a theoretic point of view, Eq. (2) presents no mathematical complexity. However, if a part of electoral data is not available, then V cannot be evaluated. In our framework, this problem can be formulated by assuming that some of the pairs in fðpi ; qi Þ : i 2 Ig are unknown. In summary, the present work will analyze the approximation of V in such a framework. The problem of volatility approximation was partially solved in Bartolini and Mair [1, Appendix 1, p. 312] from an empirical point of view. These authors considered two approximating indexes of V: TV (the index of total volatility with Others) and TVWO (the index of total volatility without Others). They studied the differences between both volatility indexes from the 1885–1985 European election data. Though they concluded that non significant difference between TV and TVWO was encountered, they eventually decided to use TV, i.e., V was approximated by using Eq. (2) with the category Others in the available data. This way, these authors provided a partial and inspiring solution to the approximation problem of volatility. In the present paper, the approximation problem of V will be analyzed in a general framework. Among other results, we will prove that the BM solution (approximation) becomes even interesting in more general data settings. Moreover, an improvement of the BM solution is going to be derived. 2.2. Approximations of volatility and errors Throughout this paper, assume that only some M–first pairs of strengths are the available data, for any M 2 I, which are given by fðpk ; qk Þ : k ¼ 1; . . . ; Mg fðpi ; qi Þ : i 2 Ig: This means that the subset given by fðpj ; qj Þ : j ¼ M þ 1; . . . ; N g is assumed unknown. Notice that to consider the first M pairs is not a lack of generality, because no order is assumed in the electoral data fðpi ; qi Þ : i 2 Ig. Under the above introduced assumption, V cannot be thus calculated, because some of the terms in Eq. (2) are unknown. Therefore, the value of V is assumed unknown in what follows and, also, must then be approximated from the available information fðpk ; qk Þ : k ¼ 1; . . . ; Mg. As a first approach to our problem, a naive approximation of V is given by M 1X V ½M ¼ jp qk j; ð3Þ 2 k¼1 k where Eq. (2) has been applied on the available pairs of strengths. It can be established from Eqs. (2) and (3) that N 1 X V ¼ V ½M þ jp qj j; 8M 2 I: ð4Þ 2 j¼Mþ1 j On the other hand, Eq. (1) could be rewritten by M M X X pk þ P ½M ¼ qk þ Q½M ¼ 1; k¼1
k¼1
where P ½M and Q½M are defined by P ½M ¼
N X
pj ¼ 1
j¼Mþ1
Q½M ¼
N X j¼Mþ1
M X
pk
and
k¼1
qj ¼ 1
M X k¼1
qk ;
8M 2 I:
ð5Þ
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Notice that P ½M and Q½M are interpreted as the overall proportions of electoral strength for those party units not included in the available data in both elections, respectively. Further, though electoral strengths in fðpj ; qj Þ : j ¼ M þ 1; . . . ; N g are unknown, P ½M and Q½M are obtained from the available data by using Eq. (5). Proposition 1. V ½ is a non decreasing function on I and the following inequalities hold: 0 6 V ½M 6 VL½M 6 V 6 VU½M;
8M 2 I;
where VL½ and VU½ are the functions on I defined by 1 VL½M ¼ V ½M þ jP ½M Q½Mj and 2 1 VU½M ¼ V ½M þ ðP ½M þ Q½MÞ; 8M 2 I: 2
ð6Þ ð7Þ
Furthermore, as M tends to N, V ½M, VL½M and VU½M get closer and closer to V, with V ½N ¼ VL½N ¼ V ¼ VU½N . Proof. By using Eqs. (3) and (4), it can be assured that V ½ is a non decreasing function on I such that 0 6 V ½M 6 V , 8M 2 I and V ½N ¼ V . Two hypothetical situations are now going to be considered for a given M 2 I. In each of such situations, the behavior of the PV measure for aggregated data is going to be applied (see Appendix A). • Suppose that data for parties in fM þ 1; . . . ; N g were aggregated to obtain an unique bloc in both considered elections. By using Appendix A, it can be established that M 1X 1 1 V P jpk qk j þ jP ½M Q½Mj ¼ V ½M þ jP ½M Q½Mj: 2 k¼1 2 2 Notice that the volatility from this hypothesized data (in the right hand side of the above equation) can be interpreted as a bloc volatility. • Consider the lowest level of aggregation for those parties whose information is not available. Such a hypothetical situation is given when their corresponding pairs are fðpj ; 0Þ; ð0; qj Þ : j ¼ M þ 1; . . . ; N g. Notice that the theoretic data set fðpi ; qi Þ : i 2 Ig can be obtained by aggregating these hypothesized data. It can be established from Appendix A that V 6
M N 1X 1 X 1 jpk qk j þ jpj 0j þ j0 qj j ¼ V ½M þ ðP ½M þ Q½MÞ: 2 k¼1 2 j¼Mþ1 2
ð8Þ
The volatility from the hypothesized data (in the right hand side of the above equation) can be viewed as total volatility and thus V can be viewed as bloc volatility. Taking into account Eqs. (4) and (5), as M tends to N, the values of P ½M and Q½M get closer and closer to zero and V ½M gets closer and closer to V. It follows that VL½M and VU½M get closer and closer to V, as M tends to N. The proof is completed by taking into account that P ½N ¼ Q½N ¼ 0 and V ½N ¼ V . h The formula in Eq. (6) is used to approximate V in Bartolini and Mair [1]. In fact, it can be said that these authors used VL½M and V ½M, i.e., TV ¼ VL½M and TVWO ¼ V ½M (see Section 2.1). Bartolini and Mair [1] decided to use TV and discarded TVWO in their volatility analysis by using empirical argumentation. In the present study, Proposition 1 assures that TV ¼ VL½M is always closer to the true value of V than the one obtained from TVWO ¼ V ½M. Therefore, Bartolini and Mair’s choice in favor of TV is mathematically justified for any electoral data. The error of the BM approximation, VL½M ¼ TV, is defined by V VL½M. However, as V is assumed unknown, such an error cannot be calculated. Nevertheless, it is known from Proposition 1 that V is contained in the interval ½VL½M; VU½M. This implies that an upper error bound for VL½M can be defined by EBM½M ¼ VU½M VL½M;
8M 2 I;
ð9Þ
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303
where 0 6 V VL½M 6 EBM½M, for any M 2 I. Therefore, EBM½M provides a prior quality measure of VL½M, 8M 2 I. Corollary 2. EBM½ is a non increasing function on I with EBM½N ¼ 0 and can be obtained by EBM½M ¼ minfP ½M; Q½Mg;
8M 2 I:
Proof. Taking into account Eqs. (6) and (7), we obtain that 1 1 1 EBM½M ¼ ðP ½M þ Q½MÞ jP ½M Q½Mj ¼ ðP ½M þ Q½M jP ½M Q½MjÞ 2 2 2 ¼ minfP ½M; Q½Mg: The proof is completed by using that P ½ and Q½ are both non increasing functions on I with P ½N ¼ Q½N ¼ 0. h 2.2.1. A new approach The study of the error for the BM approximation suggests an alternative approach to the volatility approximation. A class of approximations of V is proposed as follows: V h ½M ¼ hVU½M þ ð1 hÞVL½M;
8M 2 I and 8h 2 ½0; 1:
ð10Þ
Notice that VL½ and VU½ are in the class fV h ½gh for h = 0 and h = 1, respectively. Corollary 3. For any h 2 ½0; 1, the approximation V h ½M can be obtained by V h ½M ¼ VL½M þ hEBM½M;
8M 2 I
and gets closer and closer to V, as M tends to N, with V h ½N ¼ V . Proof. It is derived by using Eq. (9) and Corollary 2 that V h ½M ¼ VL½M þ hðVU½M VL½MÞ ¼ VL½M þ hEBM½M ¼ VL½M þ h minfP ½M; Q½Mg;
8M 2 I:
The proof is completed by using Proposition 1 and Corollary 2. h On the one hand, it can be established by using Eq. (10) that VL½M 6 V h ½M 6 VU½M;
8M 2 I:
On the other hand, Proposition 1 establishes that V 2 ½VL½M; VU½M, for any M 2 I. This implies that the (unknown) approximation error for V h ½M, which is given by jV V h ½Mj, is bounded by ECh ½M ¼ maxfV h ½M VL½M; VU½M V h ½Mg;
ð11Þ
where jV V h ½Mj 6 ECh ½M, 8M 2 I and 8h 2 ½0; 1. By the way, ECh ½M provides a prior quality measure of V h ½M, 8M 2 I and 8h 2 ½0; 1. Corollary 4. For any h 2 ½0; 1, ECh ½ is a non increasing function on I with ECh ½N ¼ 0 and can be obtained by ECh ½M ¼ cðhÞEBM½M;
8M 2 f1; . . . ; N g;
where c : ½0; 17!½0; 1 is defined by cðhÞ ¼ maxfh; 1 hg, 8h 2 ½0; 1. Proof. From Eq. (10), we obtain that V h ½M VL½M ¼ hðVU½M VL½MÞ
and
VU½M V h ½M ¼ ð1 hÞðVU½M VL½MÞ:
It follows from Eqs. (11) and (9) that ECh ½M ¼ cðhÞðVU½M VL½MÞ ¼ cðhÞEBM½M ¼ cðhÞ minfP ½M; Q½Mg: Finally, the proof is completed by using Corollary 2 in the above expression. h
ð12Þ
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Corollary 4 leads to establish that ECh ½M 6 EBM½M;
8h 2 ½0; 1 and 8M 2 I:
This means that the new approximations V h ½M improve the BM approximation VL½M, when no prior information on V is available (see Section 4). In fact, when EBM½M > 0, we have that ECh ½M < EBM½M;
8h 2 ð0; 1Þ:
To select an optimum value of h, the approximation V h ½M with minimum upper error bound is going to be considered. It follows from Eq. (12) that the minimum of ECh ½M, with h 2 ½0; 1, is derived from the minimum of cðhÞ, which is achieved when h ¼ 1=2. In what follows, this optimum approximation of V is denoted by VA½M ¼ V 1=2 ½M ¼ ðVU½M þ VL½MÞ=2;
8M 2 I;
being its upper error bound given by 1 1 EA½M ¼ EC1=2 ½M ¼ EBM½M ¼ minfP ½M; Q½Mg: 2 2 Therefore, it can be established that EA½M 6 ECh ½M 6 EBM½M;
8h 2 ½0; 1 and 8M 2 I:
ð13Þ
ð14Þ
This expression justifies the interest for VA½M and, also, summarizes the performance analysis of the considered approximations in this paper. A more efficient formula for VA½M can be obtained from Eqs. (6) and (7) as follows: 1 1 VA½M ¼ ðVU½M þ VL½MÞ ¼ V ½M þ ðP ½M þ Q½M þ jP ½M Q½MjÞ 2 4 1 ¼ V ½M þ maxfP ½M; Q½Mg; 8M 2 I: ð15Þ 2 Finally, throughout this section some equations have been qualified as (computationally) efficient. These equations are all in terms of V ½M, P ½M and Q½M, which are values first computed from data in practice. The main reason for such a distinction is given by the fact that the implementation of such equations becomes an easy task in practice [4]. 3. A study with data The aim of this section is to illustrate the performance analysis of the approximations of V carried out in Section 2. Indeed these approximations will be applied to data in order to illustrate their behavior in practice. To simplify the resulting comparative study, our attention is focused on the BM approximation, VL½M, and VA½M, which is proposed in this paper. Four artificial examples will be considered throughout this section, being their data sets different variations from a common data subset. In fact, four examples enumerated by 0–3, which are given by their corresponding sets of pairs of electoral strengths (see Section 2.1), are considered with N = 9, 10, 11, 12, respectively. Once Example 0 is presented, Examples 1–3 are obtained by modifying one of the pairs not belonging to 6 the first six pairs. This means that the subset fðpi ; qi Þgi¼1 is common for Examples 0–3 (in Table 1), being their differences located in the rest of pairs (in Table 2). Observe the decreasing level of data aggregation on each example in Table 2. The usual framework for computing volatility in practice (see Section 2) will be assumed. Suppose that the first six pairs of strengths were the available data to approximate V, i.e., M = 6. Table 1 contains the approximations V ½6, VL½6, VA½6 and VU½6, the proportions P ½6 and Q½6 and the upper error bounds for VL½6 and VA½6, denoted by EBM½6 and EA½6, respectively. Notice that all these values are the same for each of Examples 0–3, when M = 6. The rest of pairs of electoral strengths for each example appear in Table 2. They are assumed unknown in the volatility approximation framework. However, the true values of V can be evaluated for Examples 0–3, since all the pairs of strengths are known (Tables 1 and 2). The true value of V and the approximation errors
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305
Table 1 The first six pairs of electoral strengths, the approximations of volatility with M = 6 and some upper error bounds, for Examples 0–3 i
pi
qi
1 2 3 4 5 6 .. . M=6
0.3 0.25 0.15 0.1 0.1 0.05 .. .
0.4 0.2 0.2 0.1 0.02 0.03 .. .
P ½6 ¼ 0:05
Q½6 ¼ 0:05
V ½6 0.15
VL[6] 0.15
VA[6] 0.175
EBM [6] 0.5
EA [6] 0.25
VU[6] 0.2
The rest of pairs are in Table 2.
Table 2 The rest of pairs of strengths for Examples 0–3 i
pi
qi
P[i]
Q[i]
Example 0 7 8 9
0.02 0.02 0.01
0.02 0.01 0.02
0.03 0.01 0
0.03 0.02 0
Example 1 7 8 9 10
0.02 0 0.02 0.01
0 0.02 0.01 0.02
0.03 0.03 0.01 0
0.05 0.03 0.02 0
Example 2 7 8 9 10 11
0.02 0 0.02 0 0.01
0 0.02 0 0.01 0.02
0.03 0.03 0.01 0.01 0
0.05 0.03 0.03 0.02 0
Example 3 7 8 9 10 11 12
0.02 0 0.02 0 0.01 0
0 0.02 0 0.01 0 0.02
0.03 0.03 0.01 0.01 0 0
0.05 0.03 0.03 0.02 0.02 0
of VL½6 and VA½6 are contained in Table 3, for every example. Notice that Table 3 shows that the upper error bounds, which appear in Table 1, can be achieved by the approximation errors (see Example 3). Eq. (14) leads to establish that, when only M of the pairs of electoral data are known, VA½M would provide an approximation to V with more prior warranties than the one obtained by the BM approach. However, we can not say that the approach presented here is better than the one proposed by BM over all possible data settings. For instance, in Table 3, Example 0 illustrates that the error of the BM approach (V VL½6 ¼ 0:01) could be less than the corresponding error of VA½M (jV VA½6j ¼ 0:015). The opposite situation is observed in Examples 1–3, where VA½6 provides the best approximations to V. Another confirmation of this comment will be derived from Fig. 6.
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Table 3 True volatility measures and approximation errors of VL[6] and VA[6], for Examples 0–3 # Examp.
N
V
V VL[6]
jV VA[6]j
0 1 2 3
9 10 11 12
0.16 0.18 0.19 0.20
0.01 0.03 0.04 0.05
0.015 0.005 0.015 0.025
A dynamic point of view in our study is introduced, when M varies on f1; . . . ; N g and the approximations of V are obtained for each M. Due to the way Examples 0–3 are generated, the sequences of approximations versus M are plotted as follows in order to ease interpretation. Fig. 1 contains the (first) common values of V ½M, VL½M, VA½M and VU½M, with M 2 f1; . . . ; 6g, and the true values of V (see Table 3) for Examples 0–3. The rest of approximated values, for M P 6, are plotted in Figs. 2–5 for Examples 0–3, respectively. In these plots, the rate of convergence of the considered approximations to V can be observed, when M is increasing. In fact, the approximations given by VA½M exhibit a better approximating performance than those by the BM approach (VL½M) in Examples 1–3, but not in Example 0. Finally, a new approach to compare VL½M and VA½M is provided in Fig. 6, where the true approximation errors of VL½M and VA½M are plotted against the values of minfP ½M; Q½Mg which are below 0:1, in Examples 0–3 together. In this figure, we compare the performance of such approximations with the minimum pro-
Fig. 1. Common approximations V ½M, VL½M, VA½M and VU½M versus M 2 f1; . . . ; 6g and the values of volatility V (dotted constant lines), for Examples 0–3. The true volatility values appear in Table 3.
Fig. 2. Approximations V ½M, VL½M, VA½M and VU½M versus M 2 f6; . . . ; 9g and the value of volatility V (constant), for Example 0.
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Fig. 3. Approximations V ½M, VL½M, VA½M and VU½M versus M 2 f6; . . . ; 10g and the value of volatility V (constant), for Example 1.
Fig. 4. Approximations V ½M, VL½M, VA½M and VU½M versus M 2 f6; . . . ; 11g and the value of volatility V (constant), for Example 2.
Fig. 5. Approximations V ½M, VL½M, VA½M and VU½M versus M 2 f6; . . . ; 12g and the value of volatility V (constant), for Example 3.
portion of the overall strength not considered in our available data. Notice that, for each value of minfP ½M; Q½Mg, the errors for VL½M and VA½M are not clearly separated. However, the errors for VA½M are always located below the threshold given by 0:5 minfP ½M; Q½Mg, such as is established in Eq. (13).
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Fig. 6. Approximation errors of VL½M and VA½M versus minfP ½M; Q½Mg, for Examples 0–3.
4. Comments and conclusions Modern electoral databases provide to the researcher a complete information source but their use reveals some new difficulties. In this paper we study an approximation problem which usually arises when volatility is computed in practice. The approximation proposed by Bartolini and Mair [1, Appendix 1] has been mathematically studied by analyzing its approximation error. This study has inspired a new approach which represents an improvement of the BM approach. The main theoretic results on the approximations of V considered in this paper are found in Corollaries 2 and 4 and in Eqs. (13) and (14). Among these approximations, this section focuses on two approximations of V: the BM approximation, VL½M, and VA½M, which is proposed in this paper. The study on the approximation errors for VL½M and VA½M leads to establish that VA½M exhibits the best approximation performance, when no prior knowledge is given for V (Eq. (14)). This does not mean that the approximation VA½M is better than VL½M over all data settings, however, a better approximation performance could be expected for VA½M. In what follows, I am going to explain these ideas. The first idea is that VA½M exhibits the best prior approximation performance, when no prior knowledge is given for V. The best way of illustrating this idea is to show its applicability for real problems. Let us suppose that V (assumed unknown) is going to be approximated from real data in an electoral database. Indeed the rules proposed in Bartolini and Mair [1, pp. 311–312] to define some of the pairs of electoral strengths (Section 2.1) could have been applied. Therefore, there exists a part of electoral data not taken into account on these pairs. Problem 1. Let fðpk ; qk Þ : k ¼ 1; . . . ; Mg be the available electoral data set. In this case, the only information on V is given by jV VL½Mj 6 } and jV VA½Mj 6 0:5} (Corollaries 2 and 4), where } ¼ minfP ½M; Q½Mg is computed from Eq. (5). This means that it is expected that VA½M is more accurate than VL½M to approximate V. Problem 2. Now, assume that we want to approximate V with a prior approximation error given by (M is not given). The question is how P ½M and Q½M have to be in order to achieve such an admissible error in volatility approximation. For example, if volatility scores (expressed in percentages) are going to be rounded up from the first decimal point, then we could consider ¼ 5 104 ¼ 0:05%. • For the BM approximation, our objective is given by EBM½M 6 . By using Corollary 2, we can thus establish that the set of parties which can be dropped out for the BM approximation must satisfy that minfP ½M; Q½Mg ¼ EBM½M 6 :
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• For the approximation VA½M, we impose that EA½M 6 . By using Eq. (13), we can thus establish that the set of parties to be dropped out must satisfy that minfP ½M; Q½Mg ¼ 2EA½M 6 2: For example, if ¼ 5 104 , then it could be previously established that ( minfP ½M; Q½Mg 6
5 104 ¼ 0:05%; for VL½M; 103 ¼ 0:1%;
for VA½M:
This means that the BM approximation needs more data information than the approximation VA½M, for a given prior admissible error . I will now try to justify that the performance of such approximations depends on the data setting. Firstly, under the hypothesized settings considered to derive Eq. (8), it is obvious that the approximation given by VU½M will provide the best approximation to V, i.e. V VU½M. This implies that, under conditions closed to the lowest level of data aggregation considered in Eq. (8), the approximation given by VA½M will be closer to V than the one obtained from VL½M. This behaviour is illustrated by the performance of such approximations in Example 3 in Section 3. Secondly, let us now try to identify a data setting where the BM approach (VL½M) provides the best approximation, i.e., VL½M ¼ V . Assume that the strengths not belonging to the first M pairs are equally distributed, i.e. pj qj ¼ ; 8j ¼ M þ 1; . . . ; N : ð16Þ P ½M Q½M PN Let us denote aj ¼ pj =P ½M ¼ qj =Q½M. It is obvious that aj P 0, 8j ¼ M þ 1; . . . ; N , and j¼Mþ1 aj ¼ 1. It is obtained from this assumption and Eq. (4) that V ¼ V ½M þ
N N 1 X 1 X 1 jpj qj j ¼ V ½M þ aj jP ½M Q½Mj ¼ V ½M þ jP ½M Q½Mj ¼ VL½M: 2 j¼Mþ1 2 j¼Mþ1 2
This means that, when values fpj : j ¼ M þ 1; . . . ; N g and fqj : j ¼ M þ 1; . . . ; N g are equally distributed, then the BM approximation provides the unknown value of V and thus exhibits the best approximation performance. Appendix A. Volatility and data aggregation In this appendix, details are provided about the behaviour of the PV measure for aggregated data, which are needed in the proof of Proposition 1 (Section 2.2). The behaviour of the PV measure, when initial data are aggregated to constitute an arbitrary number of blocs of parties, is analyzed in Ocan˜a [2, Propositions 4 and 8], where TV (total volatility) and BV (bloc volatility) stands for the PV measure from the initial data and from the aggregated data, respectively. In fact, [2] establishes that the relation of order, for such volatility indexes, is always given by 1 P TV P BV P 0. References [1] S. Bartolini, P. Mair, Identity Competition and Electoral Availability: The Stabilization of European Electorates 1885–1985, Cambridge University Press, Cambridge, 1990. [2] F.A. Ocan˜a, An approach to electoral volatility: a method for exploring blocs of parties, Social Science Research, submitted for publication. [3] F.A. Ocan˜a, P. On˜ate, Las elecciones autono´micas de 1999 y las espan˜as electorales, Revista Espan˜ola de Investigaciones Sociolo´gicas 90 (2000) 183–228. [4] F.A. Ocan˜a, P. On˜ate, IndElec: a software for analyzing party systems and electoral systems, Journal of Statistical Software, submitted for publication.
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[5] P. On˜ate, F.A. Ocan˜a, Ana´lisis de datos electorales, Cuadernos Metodolo´gicos, vol. 27, Centro de Investigaciones Sociolo´gicas, Madrid, 1999. [6] P. On˜ate, F.A. Ocan˜a, Elecciones de marzo de 2000: ? cua´nto cambio electoral? Revista de Estudios Polı´ticos 110 (2000) 297–336. [7] M.N. Pedersen, The dynamics of European party systems: changing patterns of electoral volatility, European Journal of Political Research 7 (1979) 1–26. [8] G. Sartori, Parties and Party Systems: A Framework of Analysis, Oxford University Press, Oxford, 1976.