Achillefs Papageorgiou
Postdoctoral researcher, Department of Political and Economic Studies, University of Helsinki,
Unioninkatu 37, (P.O. Box 54) 00014 Helsinki.
E-mail Address: Achillefs.Papageorgiou@helsinki.fi; tel: 00358- 9191 24823.
Acknowledgments
A version of this paper was presented in the conference: “New Developments in Modeling Party
Competition”, July 14-15 2012, Social Science Research Center Berlin (WZB). The author is
grateful to: James Adams, Susumu Shikano, Paul Thurner, Samuel Merrill III, Bernard Grofman,
Rob Van Houweling, Woojin Moon, Heiko Giebler, Aiko Wagner, Bernhard Weßels, Thomas
Meyer, Wolfgang Müller, Nicolas Sauger, Timothy Hellwig and Luigi Curini for their insightful
comments. I also thank the participants of the XXXIV Annual Meeting of the Finnish Economic
Association, Vaasa, February 9–10, 2012 and the participants of PCRC (Public Choice Research
Centre) research seminar for their insightful comments. This study is part of the author’s
postdoctoral research and is supported financially by the Finnish Cultural Foundation.
1
Title
Why political parties in Finland have kept stable left and right positions during 2003- 2011? : A
Nash equilibrium approach.
Abstract
This manuscript draws on a previous publication where it was argued that the discrepancy between
Finnish left and right positions and their Nash equilibrium positions owes to the effect of projection.
Here, equilibrium positions are compared with parties’ left and right positions after correcting for
projection. The results reveal that the discrepancy between their left and right positions and their
equilibrium positions does not diminish when projection is taken into account. However, when
equilibrium positions are computed assuming that party choice depends not only upon proximity
but also upon directional characteristics, Finnish parties’ left-right positions resemble their Nash
equilibrium positions.
JEL-codes: C15; C18; C31
Introduction
In a previous publication (Papageorgiou 2010), it was hypothesized that Finnish parties have
reached a state of equilibrium whereby no party has an incentive to change its left or right position
unilaterally. The observation that drove the afore-mentioned hypothesis was the fact that Finnish
parties were perceived by voters to have maintained very similar left and right positions1 during the
course of 2003-2007. The biggest absolute difference on left and right positions was spotted for the
SDP and was equal to just 0.57 increments; the average difference between the parties’ positions
was just 0.35 (s.d.: 0.21). The latest data from the Finnish National Election Study series shows that
parties have also retained very similar left and right positions in 2011 (Table 1). 2 The biggest
By this it is not meant that the differences between voters’ perceptions of parties’ positions are not significant, but that
the positions of the Finnish parties between the years have changed very little in absolute values. The author thanks an
anonymous reviewer for pointing this out.
2
To enhance confidence in the reported mean scores of parties’ positions, the standard errors are also reported for every
party (Table 1, standard errors inside the parentheses). Standard errors are computed using a re-sampling method, such
as the bootstrap (the jackknife method produces almost identical results). The number of bootstrap replications
1
2
difference between 2007 and 2011 left and right positions was spotted for the PS and was equal to
just 0.27 increments. On average, the difference between the 2007 and 2011 perceived positions
was just 0.15 increments (s.d.: 0.08).
[Table 1]
However, results from the equilibrium analysis showed (Papageorgiou 2010) that Finnish parties’
left and right positions in 2007 diverged much more than their Nash equilibrium positions. Finnish
parties’ Nash equilibrium positions converge towards the centre of the left and right dimensions
(mean: 5.66, s.d.: 0.34) whereas parties’ left and right positions are much more spread out, with a
standard deviation more than five times as great as that of the equilibrium positions (mean: 5.45,
s.d.: 1.86).
What explains this significant difference between parties’ equilibrium positions and their perceived
left and right positions?
First, the discrepancy between the Nash equilibrium positions and parties’ left and right positions
might be due to the presence of projection. With projection, voters misjudge party positions by
considering their preferred parties to be closer to their own ideological positions while
overestimating their distance from rival parties (e.g., Gerber and Green 1999; Granberg and
Holmberg 1988; Jensen 2009; etc.).
Second, the equilibrium analysis was based on an algorithm that assumed that the only known
factors that affect voting are party identification and policy issue proximity. The huge party
identification coefficient B  4.036 in comparison to the small policy issue coefficient a  0.0809
(Papageorgiou 2010) indicated that party identifiers are so ‘biased’ towards their parties that the
latter can only compete over independents. Since independents are found on the centre of the left
and right dimensions, parties also have vote maximizing incentives to present centrist positions.
This, coupled with the fact that proximity theory also tends to produce convergence, explains why
performed was 1,000. The small standard errors reported in Table 1 for each of the three data sets increase confidence
that the samples used give unbiased estimates of the population mean (King et al. 2000).
3
Finnish parties’ equilibrium positions converged in contrast to their left and right positions, which
are much more divergent.
The present manuscript takes on board the effect that the above mentioned points have on Finnish
parties’ equilibrium positions by performing two new types of equilibrium analysis.
The first type corrects party positions for the effect of projection. Although the previous article
acknowledged that projection might be responsible for the discrepancy between parties’ equilibrium
positions and their left and right positions, this point was not directly incorporated into the analysis.
The assumption is that the magnitude of the discrepancy between their Nash equilibrium positions
and their left and right positions will diminish if parties’ positions are corrected for the effect of
projection.
The second type considers the effect that another type of spatial theory - namely a directional one will have on the computation of the parties' equilibrium positions. Generally, and contrary to
proximity theory, directional theory produces divergence. The directional component is introduced
in a mixed model with interaction, which is discussed in detail later on. Although it is
acknowledged that a possible high value party identification coefficient will impart a significant
effect on parties’ centripetal positioning, we retain the specification of capturing non-policy issue
characteristics through party identification.3 The main reason for this is to make the results of the
current analysis comparable with the results of the previous manuscript, where party identification
was also used. In the same vein, and in order to make the results comparable with the previous
study, data was also drawn from the Finnish National Election Study 2007.4
3
Instead of, e.g., the recalled voting that was used in Adams, Merrill and Grofman's publications (e.g., Adams et al.
2005; Adams and Merrill 2000)
4
Finnish National Election Study 2007 [computer file]. FSD2269, version 1.0 (2007-08-02). Paloheimo, Heikki
(University of Tampere. Department of Political Science and International Relations) and the members of the Political
Participation and Modes of Democracy: Finland in a Comparative Perspective research group [Authors, 2010s].
Helsinki : Taloustutkimus [data collection], 2007. Tampere: Finnish Social Science Data Archive [distributor], 2007.
4
Equilibrium analysis
The algorithm locating the Nash equilibrium has been incorporated in a series of publications
produced by Adams, Grofman and Merrill (e.g., Merrill and Adams 2001; Adams et al. 2005;
Adams and Merrill 2000; etc.). The algorithm assumes that voters’ party choices depend upon a
quadratic function that measures ideological proximity and a stable function that measures party
identification:
Vi ( )  a( xi  x ) 2  Bt i   i (1)
(Adams et al., 2005; Adams and Merrill, 2000; etc.)
a stands for the policy issue parameter and B for the party identification parameter; xi stands for
the position of a voter i , x  for the mean perceived position of a party  , t i for party
identification and  i for the errors which, by definition, are independent when a conditional logit
(CL) model is assumed. 5 The independence of the errors gives rise to the independence of the
irrelevant alternatives assumption (IIA). The IIA holds that the probability ratio between two
alternatives is not affected by the existence of other, similar alternatives6 (Thurner and Eymann
2000). In electoral studies, this might seem to be a restrictive assumption, since the birth of a party
can provide a similar ideological alternative to two or more already existing parties (Glasgow
2001). To go around the restriction that the IIA entails, other statistical models have been proposed
that partially relax (such as the nested logit) or fully relax (such as the mixed logit) the IIA. The
nested model partially relaxes the IIA in the sense that it assumes that the random factors belonging
to the same nest are correlated, yet those that belong to different nests are independent of one
another. On the other hand, the mixed logit fully relaxes the IIA assumption. In practice, this means
that the coefficients of the explanatory variables are stochastic and not the same for each case
(Christiadi and Cushing 2007). Nonetheless, although the aforementioned models offer a solution
to the restriction imposed by the IIA, they entail a computational burden when more than three
5
Under the conditional logit model, the error term exhibits the extreme value distribution with a probability density
function:
6
f ( i ) e  i e e
  i
.
Therefore, the probability ratio of an individual
i over two party alternatives  and  can be written as:
 a ( x i  s  ) 2  Bt i
P i e

2
P i e a ( xi  s  )  Bti
5
candidates or parties are involved. Thus, most studies on party competition (e.g., Adams et al. 2006;
Alvarez and Nagler 1998; Dow and Endersby 2004), as well the work at hand, incorporate a
multinomial type of statistical model, such as the CL.
Correcting party positions for the effect of projection
As was shown in Papageorgiou (2010) the relationship between voters’ self-placement and their
average perceived distance in relation to parties in Finland is U-shaped. This is because moderate
voters see, on average, the least distance between their own left and right positions and those of the
parties, most of which are placed on non-extreme positions. On the other hand, extreme voters meaning voters who present a position at either end of the left or right dimension - are distant from
most parties. Since the algorithm locating the Nash equilibrium positions is based on the voters’
perceptions regarding the parties’ left and right positions, the latter are corrected for the effect of
projection.
If x  is a party’s position after correcting for projection, then:
x   x    1
(2)
where x  is the position of voter  before correcting for projection and  1 is the coefficient
obtained (Table 2) after fitting the model:
x   0  1Di  .
where Di  ( xi  x ) and x  is the mean value of position x .
With this technique (Merrill and Grofman 1999), the ‘best estimate of the respondent’s projection’
(1999: 180) is deleted from the respondent’s party placement.
6
Replacing (2) in (1):
Vi ( )  ax i  ( x   1) 2  Bt i   i
or:
Vi ( )  a( xi  x  ) 2  Bt i   i
(3)
Assuming a CL model, the values of the coefficients a and B in (3) are estimated. As can be seen
in Table 3, the coefficients are essentially the same as the coefficients reported in Papageorgiou
(2010), where a  0.0809 and B  4.036 . The coefficient of party identification is huge, indicating
that a respondent who identifies with party  and who is equidistant between two rival parties 
and  will almost always prefer  to  . This result is in line with previous studies (e.g., Stokes,
1963; Sanders et al., 2008), which have shown that non-policy characteristics are usually more
salient when compared to spatial characteristics.
Columns 2 and 3 of Table 3 present Finnish parties’ mean left and right positions before and after
correcting for projection, respectively (standard deviations and bootstrapped standard errors are also
reported). Column 4 gives the parties’ Nash equilibrium positions after correcting for projection.
The equilibrium positions are calculated after replacing the values of the coefficients for (3) and
then running the algorithm.7 Entries in italics represent standard errors8 that emanate from a Monte
Carlo simulation, where the empirical estimated coefficients are “re-sampled independently from
their asymptotic Gaussian distribution and [the] Nash equilibrium is re-evaluated each time” (Curini
and Iacus 2012: 18). The number of Monte Carlo simulations was 1,000.
[Table 2]
[Table 3]
7
The algorithm computes the party’s expected vote share, which is given by
derivative of
 P  (s, a) as equal to zero with respect to s 
i
 P  (s, a) . Setting the first partial
i
and solving for s , we get party
 's vote maximizing
position (for the proof of why parties’ vote maximizing positions are unique, see Merrill and Adams, 2001: 351; for the
R scripts or the excel worksheets that run the algorithm, see: http://course.wilkes.edu/merrill/ ).
8
Standard errors for the Nash equilibrium positions have been calculated using the nopp package in R (Curini and
Iacus, 2012)
7
Upon correcting for projection, the Nash equilibrium positions have a mean of 5.66 and a standard
deviation of 0.33, indicating a small diffusion compared to the parties’ left and right positions,
which are spread out with a mean of 5.48 and a standard deviation of 2.0. Thus, the discrepancy
between their equilibrium positions and their left and right positions does not diminish after
correcting parties’ left and right positions for the effect of projection.9 This is not surprising, since
the coefficients presented in Table 2 are close to zero. This minimal effect of projection on parties’
mean perceived positions and their Nash equilibrium positions is also in line with Merrill and
Grofman’s (1999) findings when performing the same analysis in Norway.
The former result drives the analysis in the second part of this article. Since projection is not to be
‘blamed’ for the discrepancy between the parties' equilibrium positions and their left and right
positions, either the Finnish parties have retained stable positions for reasons other than vote
maximizing or else they have indeed reached a state of equilibrium. However, the (equilibrium)
analysis needs to take into account alternative specifications of spatial modelling, as detailed
bellow.
A mixed model specification with interaction
In the analysis just given, it was assumed that directional components have no effect on the
computation of Nash equilibrium positions. In this section, this assumption is relaxed by adding a
directional component to the voting equation. Alongside this directional component, the analysis
also adds an interaction term.
The interaction is between a variable that varies between both observations and alternatives, such as
the mixed proximity directional component and a dummy variable that is constant across choices
such as party identification. Thus, in adding the interaction term, the analysis allows the mixed
directional proximity component to have a different effect on the party identification variable for
each of the two groups: the party identifiers of one party and the party identifiers of another party;
or else between party identifiers and independents. In other words, the analysis allows that the
Finnish parties’ Nash equilibrium positions, without correcting for projection, are: KESK: 5.89; SDP: 5.21;
KOK:6.21; VAS: 5.27; VIHR: 5.49; RKP: 5.93; KD: 5.64; PS: 5.61; with mean: 5.66 and s.d.: 0.34 (Papageorgiou
2010).
9
8
spatial characteristics will have a different effect on party choice for different groups of voters.
Under the mixed model specification with interaction, equation (1) is now re-written as:10


Vi ( )  a 2(1   )( x i 5)( x  5)   ( x i  x ) 2  Bt i  CI   i

(4)

where, I   2(1   )( x i 5)( x  5)   ( x i  x ) 2 *t i .
Coefficient C stands for the interaction between the party identification component and the mixed
directional proximity component.  stands for the mixing parameter. In case the interaction term
C equals zero,  has the following properties: if   1 then (4) reduces to (1); if   0 (4)
reduces to a voting model according to which party choice depends upon the directional theory,
party identification and a random term. According to the directional theory of voting (Rabinowitz
and Macdonald, 1989; Macdonald et al., 1991), a positive product score of ( xi 5)( x  5) indicates
that the voter and the party stand on the same side of the fence. A negative sign indicates that the
voter and the party belong to different sides. The numeral 5 stands for the neutral point on the 0-10
scale of the left-right dimension. The voter votes for the party which yields the biggest product
score11 (Listhaug et al. 1990). If the voters feel intensely about an issue, they will vote for the party
which also places a lot of emphasis on that issue (Merrill and Grofman 1999). If  <0.5, the voter
relies more upon directional components for his party choice. If  >0.5, the voter relies more upon
proximity components. Lastly, if   0.5 , the voter places an equal emphasis upon both proximity
and directional characteristics.
A party’s position x  can either be captured through mean or idiosyncratic (voter-specific) party
placement. Proponents (e.g., Rabinowitz and Macdonald 1989; Macdonald et al. 1991; etc.) of
mean party placement argue that idiosyncratic placement does not produce valid results, as the voter
always projects his own position onto the party. The advantage of mean party placement is that it
lessens voters’ subjective judgements (Rabinowitz and Macdonald 1989). Furthermore, as Enelow
We adopt the convention of referring to equation (4) as the “mixed model with interaction.”
This note only presents a reduced version of the directional theory. A complete version entails the concept of the
“region of acceptability.” The “region of acceptability” is defined as the area outside which the party suffers a penalty
(Listhaug et al. 1994). If a party presents a position outside the range of the “region of acceptability,” then the actual
votes that the party receives are reduced according to a discounting factor.
10
11
9
and Hinich (1984) point out, the spatial theory of voting is consistent with the argument that
‘candidates have fixed, stable locations on a set of underlying predictive dimensions’ (1984: 168),
and therefore the mean party position should be preferred over idiosyncratic party placement. On
the other hand, proponents of idiosyncratic placement (e.g., Krämer and Rattinger 1997; Gilljam
1997; Westholm 1997; Merrill and Grofman 1999) assert that idiosyncratic party placement should
be preferred for the reason that ‘a voter’s evaluations are more closely attuned to the voter’s own
assessment of a candidate’s position than to the national mean placement, which is not known to the
voter’ (Merrill and Grofman 1999: 176). We agree with the latter argument and have decided to
capture a party’s position by employing the idiosyncratic specification. An additional reason for the
use of idiosyncratic placement is that the main objection against its use - namely, the effect of
projection - does not have any real effect on parties’ positions, as was shown in the first part of the
manuscript.
The parameter estimates of algorithm (4) as emanating from a maximum likelihood analysis are
shown in Table 4. As can be seen, the mixing parameter  equals 0.58, thus favouring - slightly the proximity component over the directional component. The sign in front of the interaction
parameter C is negative. This means that when neither the party identification coefficient nor the
mixed component coefficients are zero, the interaction of both of them has a diminishing effect on
voters’ party choice. However, because C equals 0.1 in absolute values, the afore-mentioned
diminishing effect is actually trivial.
[Table 4]
Table 5 shows the parties’ Nash equilibrium positions12 upon replacing the values of the parameters
in algorithm (4).
[Table 5]
12
The Nash equilibrium positions have been computed using an adaptation of the R script posted on Sam Merrill’s
website.
10
An initial observation is that the Nash equilibrium positions are much more widespread than those
reported in Papageorgiou (2010) and those presented in Section 2 of the present manuscript. As
such, although empirical research has shown that is not really possible to distinguish between the
directional component and the proximity component (Lewis and King 1999), the mixed model with
interaction nonetheless produces very different equilibrium positions when compared to those
produced when voting depends only upon proximity and party identification. The result whereby
the equilibrium positions spread out under the mixed model with interaction13 is consistent with the
argument that the directional component produces divergence (Rabinowitz and Macdonald 1989;
Iversen 1994).
A second observation is that parties in equilibrium retain their order on the left and right dimension;
starting from the left, the order of the parties is VAS, SDP, VIHR, PS, KD, KESK, RKP and KOK.
One observation is in order at this point regarding the left and right and the equilibrium position of
the ‘True Finns’, PS. PS cannot easily be classified in terms of left or right - it is a populist party
which has a mixed policy and issues a programme that entails both left and right wing policies.14
For instance, the party advocates policies that are traditionally supported by left wing parties (e.g.,
the maintenance of the welfare state) while at the same time it displays features that one sees in
extreme right wing parties (e.g., the party has often expressed a xenophobic and an anti-immigrant
rhetoric). The fact that PS cannot be easily classified in terms of left and right is evidenced in the
disparity of voters’ perceptions regarding the party’s position. When respondents were asked to
place PS on the left-right dimension, around 48% placed it on a left wing position (that is from 0-5)
and around 52% placed it on a right wing position (that is from 6-10) (Data: FNES 2007). In
general, voters’ perceptions regarding the left and right position of PS were the most dispersed, with
a standard deviation as high as 2.3.15
The last column of Table 5 shows the absolute difference between the parties’ left and right
positions and their Nash equilibrium positions. The largest difference is spotted for VAS, which has
an incentive to relocate from 1.86 to 3.37; in other words, VAS has an incentive to move 1.51
increments from its perceived left and right position towards meeting its equilibrium position. For
13
Actually, the same holds for a mixed model without the interaction term: the equilibrium positions spread out and are
very similar to those reported here.
14
In this respect, PS is somewhat different from the Norwegian Progress Party who is also a populist party but with a
rather clear right wing agenda.
15
On average, the standard deviations for all the other seven parties equalled 1.78.
11
all of the other seven parties, the absolute difference between their left and right positions and their
Nash equilibrium positions is below 0.8 increments, and thus very small. On average, the difference
between the parties’ left and right positions and their Nash equilibrium positions was equal to just
0.69, with a standard deviation as low as 0.36. This small discrepancy between the parties’ left and
right positions and their Nash equilibrium positions is also visualized in Figure 1. As can be seen
here, the parties’ positions on the left-right dimension are similar to their equilibrium positions. The
small but systematic drift to the right in party equilibrium positions when compared to the parties’
left and right positions might be explained by the fact that the percentage distribution of voters’
self-placement is also skewed to the right.16
The former finding supports the argument that what might be behind Finnish parties’ left and right
stability is the presence of equilibrium, whereby no party has an incentive to deviate unilaterally
from its vote-maximizing position.
Nonetheless, the analysis acknowledges that there might be reasons other than vote-maximizing
behaviour behind parties’ stable left and right positions over the course of 2003-2011. For instance,
in a country such as Finland where there has never been a single party government, ‘frozen’ left and
right positions might be explained by parties’ strategies to maintain their coalitional potential.
Another reason behind parties’ perceived stable positions might be parties’ emphasis on policyseeking incentives rather than vote maximization. However, the fact that Finnish parties are located
close to their equilibrium positions is a good basis for an explanation of their ‘unwillingness’ to
alter their left and right positions.
[Figure 1]
16
24.38% of the respondents are placed on positions from 0-4, 23.85 are placed on position 5 and 51.76% are placed
from 6-10.
12
Conclusions
In a previous manuscript (Papageorgiou 2010), Finnish parties’ left and right positions were
compared with their Nash equilibrium positions. Their equilibrium positions were computed by
incorporating an algorithm that assumed that voter choice depends upon two known characteristics:
policy-issue proximity and party identification. The results showed that although the parties’
perceived left and right positions were spread out, their Nash equilibrium positions revolved around
the centre of the left-right dimension.
This manuscript revisited the conclusion that Finnish parties are further away from a state of Nash
equilibrium by performing two alternative types of equilibrium analysis.
In the first type of equilibrium analysis, Finnish parties’ vote-maximizing positions were compared
with their perceived left and right positions, upon correcting the latter for the effect of projection. In
order to find the parties’ Nash equilibrium positions, an algorithm was utilized that assumed that
voting is based on two factors: ideological proximity and party identification. A comparison
between left and right and Nash equilibrium positions was made in light of the following question:
would parties’ equilibrium positions resemble their left and right positions when the latter were
corrected for the effect of projection?
Voters’ misperceptions regarding party positions were corrected for the effect of projection by
subtracting from each party’s position the value of the coefficient obtained after regressing its
position on the difference between the voter’s positions and the party’s position. The results showed
that correcting for projection does not diminish the discrepancy between the equilibrium positions
and the perceived left and right positions. The reason for this was that the coefficients obtained
from the regression analysis were all very close to zero and, therefore, both the parties’ positions
and their equilibrium positions remained, in essence, the same upon correcting for projection.
The former result drove the analysis for the second type of equilibrium analysis. Here, Finnish
parties’ Nash equilibrium positions were calculated upon introducing two important modifications
to the previous equilibrium analysis.
13
First, it was assumed that voters’ choices are made on the grounds of mixed model of voting with
interaction. The mixed model with interaction relaxed the assumption that voting is based only on
proximity characteristics and party identification, and also allowed for mixed proximity-directional
effects (a policy issue component) in the voting equation as well as for the interaction between party
identification and the policy issue component. Second, in order to account for the fact that mean
party placement diminishes the role of directional characteristics in party choice (e.g., Westholm
1997; Gilljam 1997; Krämer and Rattinger 1997; Merrill and Grofman 1999), idiosyncratic party
placement was therefore used.
The results showed that under the mixed model with interaction, parties’ left and right positions
very much resemble their Nash equilibrium positions. On average, the difference between parties’
left and right positions and their Nash equilibrium positions was just 0.69, with a standard deviation
of 0.36. Put differently, Finnish parties’ perceived left and right positions, as reported in the Finnish
National Election Data Series, are very similar to their vote maximizing positions as computed
under the Nash equilibrium analysis. This result can be read as a persuasive reason - albeit not the
only one - as to why Finnish parties have maintained stable left and right positions over the course
of three successive elections.
Tables and Figures
Table 1. Left and right positions of Finnish parties 2003-2007-2011 (Data: Finnish National
Election Study).
Party
Centre
(KESK)
Party
Social Democratic
Party (SDP)
Mean left and right
positions 2003
Mean left and right
positions 2007
Mean left and right
positions 2011
(1)
6.19
s.d.: 1.65
(.0475572)
N=1270
(2)
6.51
s.d.: 1.53
(.0411706)
N=1422
(3)
6.33
s.d. : 1.61
(.0453472
N=1298
4.70
s.d.: 2.04
(.0610885)
4.13
s.d. :1.89
(.0511774)
4.29
s.d.: 1.79
(.0521112)
14
N=1270
N=1422
N=1298
National Coalition
Party (KOK)
7.46
s.d. : 2.11
(.0622608)
N=1270
7.94
s.d. : 1.94
(.0541301)
N=1422
8.14
s.d. : 1.84
(.0525656)
N=1298
Left Alliance (VAS)
2.30
s.d. : 1.83
(.0533356)
N=1270
1.86
s.d. : 1.72
(.0459666)
N=1422
1.89
s.d. : 1.57
(.0433947)
N=1298
Green
(VIHR)
4.72
s.d. : 1.67
(.0493883)
N=1270
4.74
s.d. : 1.65
(.0467291)
N=1422
4.86
s.d. : 1.84
(.0519387)
N=1298
Swedish
People’s
Party (RKP)
6.15
s.d. : 2.17
(.0664)
N=1270
6.67
s.d. : 1.98
(.0532514)
N=1422
6.85
s.d. : 1.97
(.0565691)
N=1298
Christian
Democratic
(KD)
5.88
s.d. : 1.92
(.0585229)
N=1270
5.98
s.d. : 1.80
(.0493579)
N=1422
6.03
s.d. : 1.87
(.0543072)
N=1298
na
5.78
s.d. : 2.30
(.0627505)
N=1422
5.51
s.d. : 2.25
(.0669892)
N=1298
League
Party
True Finns (PS)
Notes: s.d.: standard deviation; parenthesized entries are bootstrapped standard errors; N: number of observations.
15
Table 2. Coefficients from a regression analysis (data: FNES 2007)
Independent variable:
Party placement
x
Coefficient
(Parenthesized entries are
standard errors)
N
.0944576
(.0197816)
.0425093†
(.0253715)
-.0140165†
1013
Dependent
variable:
D i  xi  x 
D iKESK
DiSDP
DiKOK
D iVAS
D iVIHR
D iRKP
D iKD
D iPS
(.0244865)
.0535735*
(.0220108)
.0017983†
(.0225938)
-.0207427†
(.0258508)
0219937
(.024464)
.1245452
(.0324925)
1007
1005
1011
990
984
982
946
Notes: Subscript i stands for the voter; Coefficients are significant at .001 level; *: Significant at .05 level;
significant; Parenthesized entries are standard errors
16
† : Not
Table 3. Maximum likelihood estimates in a CL model and Nash equilibrium positions after
correcting for projection (data: FNES 2007)
Political
Parties
Mean left and
right positions
before
correcting for
projection
Mean left and right
positions after
correcting for the effect
of projection
Nash equilibrium positions after
correcting for the effect of projection
(1)
(2)
(3)
(4)
6.51
s.d.: 1.53
( .0411706)
N=1422
4.13
s.d.: 1.89
(.0511774)
N=1422
7.94
s.d.: 1.94
(.0541301
N=1422
1.86
s.d.: 1.72
(.0459666)
N=1422
4.74
s.d.: 1.65
(.0467291)
N=1422
6.67
s.d.: 1.98
(.0532514)
N=1422
5.98
sd: 1.80
(.0493579)
N=1422
5.78
s.d.: 2.30
(.0627505)
N=1422
5.45
6.49
s.d.: 1.43
(.0437869)
N=1030
3.98
s.d.: 1.81
(.0575188)
N=1030
8.21
s.d.: 1.75
(.0553275)
N=1030
1.71
s.d.: 1.58
(.0484488)
N=1030
4.73
s.d.: 1.60
(.0512773)
N=1030
6.90
s.d.: 1.83
(.05841)
N=1030
6.10
s.d.: 1.73
(.0556102)
N=1030
5.74
s.d.: 2.28
(.0739029)
N=1030
5.48
5.88
0.023
1.86
2.00
0.33
KESK
SDP
KOK
VAS
VIHR
RKP
KD
PS
Mean
s.d.
5.22
0.028
6.19
0.048
5.29
0.031
5.50
0.012
5.92
0.027
5.64
0.006
5.61
0.005
5.66
17
Maximum
likelihood
estimates after correcting
a (Policy salience parameter)
for projection and robust
.0731681
(.0092918)
standard errors estimates
inside the parentheses
B (Party identification parameter)
4.040431
(.1423812)
Log likelihood
-1036.0101
N
8240
Notes: Entries are significant at p  .001 , alpha  .05 ; entries in italics are standard errors from a Monte Carlo
simulation.
Table 4. Parameter estimates from a maximum likelihood analysis for the mixed model (data:
FNES 2007)
a (Policy salience parameter)
0.1064
(0.0109)
B (Party identification parameter)
4.1366
(0.1883)
 (Mixing parameter)
0.5842
(0.0807)
C (Interaction parameter)
-0.09984926
(0.0229)
Maximum likelihood
-892.855
N
920
Notes: Entries are significant at p  .001 , alpha  .05 , robust standard errors inside the parentheses
18
Table 5. Nash equilibrium positions under the mixed model with interaction (data: FNES 2007)
Political Parties
(1)
Left and right
Nash equilibrium
Absolute difference
positions
positions (mixed
between (2) and (3)
model)
(2)
(4)
(3)
KESK
6.51
7.10
0.59
SDP
4.13
4.80
0.67
KOK
7.94
8.20
0.26
VAS
1.86
3.37
1.51
VIHR
4.74
5.49
0.75
RKP
6.67
7.25
0.58
KD
5.98
6.53
0.55
PS
5.78
6.36
0.58
Mean
5.45
6.14
0.69
s.d.
1.86
1.44
0.36
19
Figure 1. Left and right positions and Nash equilibrium positions (data: FNES 2007)
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