Online appendix to the article “Why do you want to lash yourself to the mast? The case of
the Danish ‘Budget Law’”
This appendix provides an extended review of the method, procedure and problems associated with
the factor analysis and the Cronbach’s Alpha measures, which are used to strengthen the validity of
hypothesis 4 (the median voter model).
Factor analysis seeks to identify underlying dimensions in a body of data material and estimate the
correlation between these and the observed variables. The analysis uses the so-called ‘Principal
Component Analysis’ (PCA). Mathematically, the factor model can be described as follows in a case
with n variables and k factors:
Y1 = α11F1 + α12F2 + ….. + α1kFk + ε1
Y2 = α21F1 + α22F2 + ….. + α2kFk + ε2
Yn = αn1F1 + αn2F2 + ….. + αnkFk + εn
The model comprises a set of regression equations in which the variables in the PCA analysis are
dependent, while the factors are independent. The model thus estimates the alphas, i.e. the
correlations (standardised beta coefficients) between the factors and the variables in the PCA
analysis. These alphas are called factor loadings.
The mathematical presentation above shows that the PCA analysis comprises a number of linear
regression models. Therefore, the formal requirements are the same for the factor analysis as for
OLS regression. It is particularly relevant that the variables should in principle be at interval scale
level, which is problematic, as the variables/expense items in the analysis only have three categories.
This requirement is often loosened, and it is generally accepted that factor analysis can be applied on
ordinally scaled variables.
Basically, all variation equals the number of variables in the PCA analysis. The eigenvalue is given by
how much of the variation a factor can explain uniquely (e.g. an eigenvalue of 2.5 means that the
variable can explain variation corresponding to 2.5 variables). Therefore, only factors with an
eigenvalue of more than 1 are applied. Furthermore, it is relevant to consider the factor loading of
each variable. This is the measure of the correlation of each variable with the factor.
Reliability analysis is performed before the creation of indexes with Cronbach’s Alpha test. This is a
measure of how precisely the factor is measured. This is doneby analysing the proportion of the total
variance that can be explained by the latent dimension, while the remainder of the variance is
ascribable to error variation. This should preferably be more than 0.7, but depends on the number of
variables in the factor. The analysis also provide an overview of how much Cronbach’s Alpha will
increase if certain items are removed. However, removal of variables should always be substantiated
theoretically.
Two PCA analyses were performed with all variables/expense items for 2007 and 2011, respectively.
Both analyses showed that there was probably an underlying value-based dimension as there were
high factor loadings for Environmental problems, Culture, Foreign aid as well as Refugees and
immigrants. Moreover, both eigenvalues were above 1, which means that the factors could explain
more than the variance of just a single variable. The model for the two years with Cronbach’s Alpha is
outlined below:
Table 1: Factor analysis
Expense
Factor loadings valuebased policy 2007
Factor loadings valuebased policy 2011
Defence
0.089
-0.169
Healthcare
0.004
0.005
Education
0.454
-0.230
-0.322
0.290
Environment
0.567
-0.534
Culture
0.628
-0.639
Kindergartens and nurseries
0.206
-0.044
Unemployment benefits
0.132
-0.108
Welfare benefits
0.289
-0.323
Foreign aid
0.796
-0.792
Refugees and immigrants
0.753
-0.770
Home care
-0.074
0.124
Motorways and bridges
-0.035
0.039
Police
-0.133
0.170
Old age pensions
N
Eigenvalue
% of variation explained
3595
770
3.73
2.09
26.66
14.94
Table 1 confirms the idea that some of the variation in the expense items is attributable to a valuebased dimension. This is confirmed by eigenvalues, which show that the value-based dimension can
explain what corresponds to the variation for 3.73 variables in 2007 and 2.09 variables in 2011
(corresponding to 26.66 per cent and 14.94 per cent, respectively, of the variation). It can therefore
be argued that the variables Environmental problems, Culture, Foreign aid as well as Refugees and
immigrants should be removed from the aggregate expense policy preference index (EPPI) because
they could reflect value-based and not expense preferences. Also, the reliability test in Table 2
(measured by Cronbach’s Alpha1) reveals that the reliability of the EPPI index is increased if the
variables Defence and Motorways and bridges are also removed.
1
The Cronbach’s Alpha value shows the proportion of the total variance caused by an underlying latent
dimension, and the measure provides an expression of the ‘true’ variance as a proportion of the total variance,
while the remainder is probably ascribable to error variation.
Table 2: Cronbach’s Alpha
Cronbach’s Alpha for indices where the
following value-based variables have been
removed:
EPPI 2007
EPPI 2011
Environmental problems
Culture
Foreign aid
Refugees and immigrants
0.564
0.571
0.645
0.626
0.614
0.618
Environmental problems
Culture
Foreign aid
Refugees and immigrants
Defence
Environmental problems
Culture
Foreign aid
Refugees and immigrants
Motorways and bridges
However, the Motorways and bridges variable is included in the index below, as there is no
theoretical explanation to justify its removal.
Index showing the shift in preference
Based on the analysis outlined above, we can create three indexes reflecting the voters’ expense
preferences, cf. Table 3. Transfers in connection with unemployment have been included in the
calculations above though they are not comprised by the spending ceilings in the Budget Law. The
reason is that the Budget Law may also be considered to apply to these, as it requires the minister of
finance to present proposals that meet spending overruns of a permanent nature regarding the
transfer payments (Ministry of Finance 2012b: 8). The first index comprises all variables. The second
index excludes the following variables: Environmental problems, Culture, Foreign aid as well as
Refugees and immigrants. The third index excludes these four value-based variables as well as
Defence. Thus, it is tested whether the voters prefer significantly lower public expenditure in 2011
than in 2007 via three reflexive aggregated expense policy preference indexes (EPPI). These have
been coded on a 0-100 scale, with a value of 50 corresponding to a total PFI value of 0, i.e. the same
number of respondents find that the expense is too high and too low.
Table 3: Difference between the expense preference indices
Expense preference index
EPPI 2007
EPPI 2011
Sig.
EPPI diff.
N 2007
N 2011
Expense preference index, all
expense items
63.52
(.220)
58.83
(.427)
(.000)***
4.66
3595
770
Expense preference index, without
value-based expense items
67.41
(.204)
63.82
(.453)
(.000)***
3.59
3642
815
Expense preference index 2007,
without value-based expenses incl.
defence
71.37
(.227)
67.98
(.499)
(.000)***
3.39
3671
818
Note: Standard errors in brackets, *significant at 0.1 level, **significant at 0.05 level, ***significant at 0.01
level.
As the table shows, all three indexes have declined significantly (a decrease between 4.66 and 3.39
on a 0-100 scale). Moreover, it appears that the shift in voters’ preferences – as expected – will be
slightly less pronounced in step with the number of value-based questions removed from the index.
However it can be argued that removing the variables is a highly conservative test of the hypothesis.
First, value political changes still make spending cuts easier on the expense items that are least
preferred by voters. Second, research on the psychological effect of economic recessions has shown
that people tend to distinguish themselves from what they view as ‘foreign’. White persons who view
pictures of bi-racial people are more likely to categorize the faces as black when given cues that
signal economic recession. They are more likely to label the faces as white when given cues that
signal economic prosperity (Rodeheffer et al. 2012). Therefore the above effects are likely to be a
result of the economic crisis as well and not only a representation of shifts in value political voter
preferences.
All in all, the adoption of the Budget Law cannot be viewed as an initiative intended to appeal to
voters, as the parties according to the median voter model should raise spending to win votes.
However, the adoption of the Budget Law has most likely been made much easier by the voters
preferring/accepting lower public expenditure after the crisis. This dynamics is often confirmed
empirically because the voters’ expectations to public consumption follow the economic cycle,
implying that it is much easier to pursue a tight expenditure policy in periods of economic downturn
because the voters’ attitude changed due to the crisis. Thus, the crisis opens a window of
opportunity for taking the Budget Law through Parliament.
Overall, the above results strengthen the analysis of hypothesis 3. The significant shifts in voter
preferences are found even when value political variables are removed from the EPPI index. The
results are robust.