Income and democracy: a bivariate copula approach

Abstract

We propose a new approach for exploring the relationship between income and democracy by modeling the two most popular discrete democracy indexes, Polity IV and Freedom House, as a joint random variable by means of a copula function. Joint modeling is crucial for eliciting complementarity and/or substitutability amongst these indexes claiming to measure similar things, i.e. a country’s degree of democratization. We find strong evidence supporting both the existence of the relationship and the positive dependence between the two democracy indexes, suggesting that they are complements to each other. Our findings are robust to different samples and model specifications.

KEYWORDS:

• Income
• democracy
• copula

JEL Codes:

• C35
• C52
• D72
• O1

Acknowledgements

I would like to thank the Editor in Chief, an Associate Editor, and two referees for the constructive and helpful comments, which greatly improved this work. Also, I am grateful to the late Professor emeritus Yves Surry for motivating me to write this paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 Acemoglu et al. [1] introduce a new indicator of democracy to tackle the issue of measurement error when democracy is used as an independent variable.

2 The method of Marshall and Olkin [44] is only applicable when the correlation between the two count variables is positive and both marginals are negative binomials whereas the copula approach accommodates any combination of marginal distributions.

3 The basic properties of the copulas are the following: (a) a copula $C\left(u_1,u_2\right)$ is defined on the domain $\left[0,1\right]$ assuming a uniform distribution (b) $C\left(0,0\right)=0$, (c) $C\left(u_1,1\right)=u_1$, (d) $C\left(1,u_2\right)=u_2$ and (e) $C\left(1,1\right)=1$. These can be easily verified for the Frank and the Gaussian copulas although it is more difficult for the latter. See Frees and Valdez [25] for a review of the copula literature.

4 The Poisson, the negative binomial and the binomial distributions for count data are closely - related through various limiting forms discussed in detail in Winkelmann [62], chapter 2. Moreover, the negative binomial and the binomial distributions are built from independent Bernoulli trials with fixed probability of success, p and the Binomial distribution is not only discrete, but also it has a range of 0, 1, … , n, i.e., has an upper bound. However, because of the presence of large overdispersion (the sample variances are roughly five times the sample mean) the choice of the copula-based bivariate model for our study is the one with beta-binomial marginal, a generalisation of the binomial. The beta-binomial distribution approximates the binomial distribution arbitrarily well, for large values of the parameters α and β, and it contains the negative binomial distribution in the limit with large β and n.

5 Their results also hold for alternative models for the marginals.

6 The modeling of the correlation structure is important for the efficiency of the estimator [62].

7 The original democracy indices have been used for the first time in a paper by Paleologou [48].

8 For details refer to the dataset user’s manual, Marshall and Jaggers [45].

9 These indexes of democracy exhibit positive and negative dependencies [4,50]. For example, the indexes of democracy between neighboring countries are likely to be correlated with one another, indicating a positive dependency. While positive dependencies are probably more prevalent in democracy statistics, negative dependencies might be very interesting. A negative dependency between neighboring countries may suggest that these countries are seen as enemies.

10 See, ‘Supplementary material’ Table 7, the values for the mean and standard deviation.

11 The Barro and Lee (2013) dataset is available at http://www.cid.harvard.edu/ciddata/ciddata.html and it is only in 5-year intervals.

12 For the annual sample, the data on education from the World Bank databank is no longer available because it is too sparse and inconsistent.

13 See for example, Genest et al. [27].

14 We also estimate two non-linear count data models. Namely, first a NBII and second, a fixed effects NBII to explicitly account for unobserved country heterogeneity [32]. The results from these estimations show that the evidence is rather mixed and thus their interpretation in relation to Lipset’s ‘modernisation hypothesis’ is inconclusive. In 18 out of the 32 models the modernization hypothesis is not supported, i.e., 56% of the models reject the hypothesis. We also estimate a Marshall-Olkin model to derive the bivariate binomial negative distribution. The results from this method are very similar to the copula approach. All these results are available upon request.