## Pippa Norris and Andrea Abel van Es

Print publication date: 2016

Print ISBN-13: 9780190603601

Published to Oxford Scholarship Online: June 2016

DOI: 10.1093/acprof:oso/9780190603601.001.0001

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# (p.271) Technical Appendix

Source:
Checkbook Elections?
Publisher:
Oxford University Press

This appendix outlines the Bayesian factor analysis model for the Political Finance Regulation Index (PFRI), a latent trait characterized by mixed ordinal and continuous indicators. The appendix includes the simulation specifications, the results from the analysis, and robustness checks.

# A.1. Model Specification

We start from the presumption that the trait we are trying to capture—the degree of state involvement in the regime of political finance—is one that cannot be directly observed. Instead, indicators of this trait are observed: indicators of different types and degrees of regulations around the flow of money into politics. The four indicators we have correspond to de jure levels of regulations with respect to disclosure requirements, contribution limits, spending limits, and public subsidies. Our premise is that each of these types of regulation does not reflect an equal level of state involvement in shaping the finance regime, and thus, a priori, we do not believe that these four indicators should be weighted equally.

Since our latent variable (the PFRI) is characterized by a mix of ordinal and continuous indicators, standard factor analysis, which assumes continuous indicators and hence an underlying normal distribution, is not an appropriate methodology to measure the underlying latent trait. Nor is standard item-response theory adequate since this assumes ordinal indicators, and hence an underlying probit distribution. Instead we turn to Bayesian factor analysis for mixed ordinal and continuous responses (Quinn 2004).

Let $j=1,…,J$ index our indicator variables, and $i=1,…,Naa$ index observations (in this case at the country level). Each observed indicator j can be either ordinal or continuous, and if it is ordinal then it has $Cj>1$ categories. We let X denote the N × J matrix of observed indicators (all indicators for all countries).

(p.272) The values we observe in X are determined by a N × J matrix of latent variables X* (in our case, the degree of state interventionism), and a collection of cut-points for the ordinal indicators, denoted by $γ$. We assume that each ordinal indicator j can have $c=1,…,Cj$ levels. That is, we can write for each observation i and indicator variable j:

$Display mathematics$

That is, if the j-th variable is continuous, then it is assumed that $xij*=xij$ for all i. Of the j-th variable is ordinal, than the probability that it takes the value c in observation i is given by

$Display mathematics$

As usual with ordinal variable models, in order to identify the model, some assumptions for cut-points are required, and Quinn (2004) follows the standard assumption of Johnson and Albert (1999) that $γj0≡−∞$, $γj1≡0$, and $γjCj≡∞$ for all indicators j.

The relationship between the observed variables in X can be modeled through a factor analytic relationship for the latent variables X*, where, for each observation,

$Display mathematics$

$Display mathematics$

Here is the vector of J latent responses to specific observation i, $Λ$ is a matrix of factor loadings, and $ϕi$ is a vector of factor scores specific to each observation i. The first element of $ϕi$ is set equal to 1 for all observations i, which ensures that the elements in the first column of $Λ$ function as negative item difficulty parameters for ordinal response variables (as in item-response theory). For continuous indicator variables, the first column of $Λ$ represented the mean of those variables. Moreover, by standardizing continuous variables to have a mean of zero and standard deviation of one, the elements of $Λ$ corresponding to the continuous variables can be interpreted as factor loadings, in the same way as standard factor analysis.

Because we are using a Bayesian approach to latent variable estimation, we must also specify relevant priors. For model identification purposes, we constrain some elements of $Λ$ to take only positive or negative values, thus eliminating the problem of rotational invariance. In our case we set the elements in $Λ$ corresponding to (p.273) the indicator for public subsidies to be positive, which implies that public subsidies are positively associated with state interventionism. We make use of the Markov-Chain Monte Carlo algorithm (see Jackman 2000) to fit the mixed ordinal/continuous model, which is implemented in the R program for statistical computing through a package known as MCMCpack (Martin and Quinn 2004).

One of the advantages of the Bayesian methodology is that we are able to estimate posterior distributions even for observations where we have some missing data, by “borrowing” information from other observations—naturally the variance associated with observations with missing data tends to be larger than for complete observation. This allows us to estimate latent scores for all countries in the IDEA database, rather than just the subset for which we have complete information across all indicators of regulation (which is, for example, the case if we were to use a pure additive index of levels of regulation).

# A.2. Results

We use data on 169 countries for which the International IDEA database has data in 2012, to estimate the latent PFRI for each of these 169 countries.

Applying the Bayesian estimation strategy to the mixed ordinal and continuous factor model, we use vague and uninformative priors for error variances. The prior mean of each element in $Λ$ is assumed to be zero, and the prior variances of each element in $Λ$ are set to be quite large at 4. The MCMC algorithm had a burn-in of 10,000 scans, and the following summaries are based on a posterior sample size of 1,000, with 100,000 additional scans run thinned by 100. The Metropolis-Hastings acceptance rate was 0.3 on average.

Figure A.1 shows how each of our observed indicators—contribution limits, public subsidies, spending limits, and reporting and disclosure requirements—relates to the underlying latent variable of interest (the PFRI).

The interpretation of figure A.1 draws from both factor analysis and item-response theory in the following sense.

For our continuous indicator—contribution limits—the figure plots the posterior mean of the factor loading for this variable on our latent variable, together with a 95% confidence bound for this mean. We see that there is a mild (in terms of magnitude) and positive association between contribution limits and the degree of state interventionism.

For our ordinal indicators, the figure shows what would be item discrimination parameters for these variables. The item discrimination parameter for ordinal variable j taps the extent to which changes in the “score” of the latent state (the degree of state interventionism) generates different levels of response on ordinal variable j. The figure shows that each ordinal indicator positively discriminates on the latent “state interventionism” scale. Moreover, since none of the 95% confidence intervals overlap zero, we know that each indicator contributes significantly to our understanding of the degree of state interventionism in the political finance regime. (p.274)

The reporting and disclosure indicator has the largest magnitude item discrimination, which indicates that the probability of a higher “score” on this indicator (that is, having higher levels of reporting and disclosure regulations) responds positively and most dramatically to a change in the underlying latent trait: the higher the level of state interventionism as captured by the PFRI, the more likely it is you will have a high score on reporting and disclosure requirements. But this is not to say that reporting and disclosure requirements are most indicative of state interventionism: instead we interpret this to mean that this indicator provides the strongest discrimination between countries’ degree of state interventionism.

The public subsidies indicator, while also positively associated with the degree of state interventionism, is less helpful in discriminating between countries’ levels of state intervention. This is likely due to the fact that public subsidies are becoming increasingly common around the world—indeed every one of the countries in our database had some regulation of either direct or indirect public funding.

Figure A.2 shows the results for the latent variable analysis of the PFRI for all countries in the dataset. The bars represent 95% highest posterior density (HPD) (p.275) (p.276) intervals for the posterior mean point estimate. There is larger measurement error associated with countries with more extreme values for the PFRI, as seen by the (on average) larger error bars. Furthermore, since there is considerable overlap in the HPD intervals in each country, it is somewhat difficult to say unequivocally that, for example, the United Kingdom has a higher level of state intervention in its political finance regime (higher PFRI) than, say, the United States, although probabilistically it would seem that way. At the extreme ends of the distribution, of course, we can make such definitive statements: thus Russia has unambiguously higher levels of state involvement than South Africa (a fact evidenced by the case study analyses).

Table A.1 United Kingdom and United States Compared: Total Regulation and PFRI

United Kingdom

United States

Contribution limits

0.33

0.93

Public subsidies

0.6

0.4

Spending limits

1

0.75

Reporting and disclosure

0.9

0.9

Total political finance regulation (Additive, equal weighting)

0.57

0.81

PFRI

0.77

0.56

Figure A.2 Country rankings for the PFRI.

# A.3. Robustness Checks

Because our PFRI is characterized by indicators of the degree of regulation of certain aspects of political finance—disclosure, contributions, spending, and public subsidies, in general we ought to expect there to be a high level of correspondence between total levels of regulation across these four exhaustive categories and the PFRI itself. If all types of regulation were created equal in the sense of how much intervention they required by the state, then we would expect to see a 100% correspondence between total levels of regulation and levels of state interventionism. However, as we have seen with our latent variable analysis, all types of regulation are not created equal, so there should not be a 100% correspondence.

Looking at the specific case of the United States and the United Kingdom in comparative perspective may illustrate this point. Table A.1 shows the individual scores on the four types of regulation for both countries, as well as the total additive score and the state interventionism score.

Table A.1 shows that while the overall level of regulation based on an index giving equal weighting to each of the categories of regulation is lower in the United (p.277) Kingdom than in the United States, in fact based on the latent analysis model, the United States has a lower level of state interventionism in the political finance regime than the United Kingdom. This is because the spending limits and public subsidies categories of regulation (where the UK has higher absolute levels) load more heavily onto the state interventionism latent trait than the contribution limits category of regulation (where the United States has a higher absolute level).

Figure A.3 shows the ordered rankings of a simple additive index across the four categories of regulation: note that the substantially fewer countries results from observations being dropped where even one category of regulation has missing data. Comparing figures A.2 and A.3, we see that there is a high degree of correspondence between a country’s overall level of regulations equally weighted and summed, and its degree of state interventionism in the political finance regime, but not a total correspondence, as expected. The correlation between these two variables is .79 and highly statistically significant.

## A.3.1. External Checks

What other variables might we expect our PFRI to correlate with? We may expect that a state that is more involved in the regulation of its system of political financing is also more likely to be involved in other policy areas such as the economy overall and social welfare. We look at simple bi-correlations between the PFRI and various measures aiming to capture state involvement in other policy areas.

We use the Economic Freedom of the World Index, compiled by the Fraser Institute, as one potential correlate: this index has several components, including the size of government and taxation, private property and the rule of law, sound money, trade regulation and tariffs, and the regulation of business, labor, and capital markets. The index ranges from 0, being the least free, to 10, being the most economically free. We also use the World Bank’s “Regulatory Quality” Index, which “captures the perceptions of the ability of government to formulate and implement sound policies and regulations that permit and promote private sector development.” This index follows a standard normal distribution with mean zero and standard deviation of one, ranging approximately from −2.5 to 2.5. Higher values correspond to better perceptions of regulatory quality.

We look at more conventional measures of the size of government and the size of the welfare state, using government expenditure as a percentage of GDP to measure the size of the government, and the total social welfare transfers as a percentage of GDP to measure the size of the welfare state.

We also look at a “Freedom from Government” index from the Heritage Foundation, which is composed of two parts: government expenditure as a percentage of GDP and revenues generated by state-owned enterprises and property as a percentage of GDP. This index ranges between 0 and 100, where a 100 represents the maximum freedom from government. (p.278)

Figure A.3 Country rankings for total levels of regulation.

(p.279)

Table A.2 Correlations of the PFRI with Other Indicators of State Regulation

PFRI

Economic freedom

Regulatory quality

Government expenditure

Social welfare expenditure

Freedom from Government

Economic freedom

0.19*

Regulatory quality

0.16*

0.70***

Government expenditure

0.30*

0.31**

0.48***

Social welfare expenditure

−0.10

0.38*

0.74***

0.81***

Freedom from Government

−0.06

−0.20*

−0.27***

−0.74***

−0.74**

Total political finance regulation (additive, equal weighting)

0.79**

0.31*

0.23

0.24

0.16

0.17

(*) p < .10;

(**) p < .05;

(***) p < .01.

(p.280) Finally we look at the correlation between total levels of regulation in political finance (as measured throughout the IDEA database) and the PFRI.

Table A.2 shows the simple bivariate correlations between the PFRI and these other measures of the degree of state involvement in regulating economic and social policy.

Table A.2 shows that there are significant correlations between our state interventionism in political finance regulation variable and three of our five indicators. Specifically, the PFRI is positively and significantly correlated with a measure of economic freedom, regulatory quality, the size of government, and the total amount of regulation in political finance.

The positive correlation between the Fraser Institute’s measure of economic freedom and the PFRI may seem counterintuitive at first. The more a state intervenes in the economy through regulation, the less economically free it will be. We might expect that a state that regulates the economy will also regulate other areas such as political finance; therefore, we would expect a negative correlation between economic freedom and the PFRI. However, upon closer inspection of the Economic Freedom index itself, we see that is not obvious that in this conceptualization more economic freedom is synonymous with less regulation. After all, ensuring private property and the rule of law requires more regulation, yet these things contribute positively to economic freedom. Furthermore, this measure of economic freedom is negatively correlated with the Heritage Foundation’s measure of Freedom from Government.