Abstract
There is a long-established practice in the empirical growth and convergence literature of classifying countries into groups or clubs by arbitrarily specifying group boundaries. A problem with this approach is that determining boundaries in a particular fashion also determines the nature of the group in a way that is often prejudicial for analysis ultimately affecting the way transition and class mobility behavior is evaluated. Here a semi-parametric technique for class categorization without resort to arbitrarily specified frontiers is proposed and the convergence of classes and mobility between them is studied in the context of the size distribution of per capita GDP of nations. Category membership is partially determined by the commonality of observed behavior of category members: partial in the sense that only the probability of category membership in each category is determined for each country. Such an approach does not inhibit the size of classes or the nature of transitions between them. A study of the world distribution over the 40 years preceding 2010 reveals substantial changes in class sizes and mobility patterns between them which are very different from those observed in a fixed class size analysis.
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Notes
GNI was chosen since as stated by the World Bank it “\({\ldots }\) was considered to be the best single indicator of economic capacity and progress” though it was recognized that GNI, by itself, did not constitute an adequate measure of welfare or developmental success. The means by which the initial thresholds were established are not entirely clear, however thereafter, the original thresholds have been updated every year to incorporate the effect of inflation using the World Bank’s Atlas conversion factor—essentially a three period exchange rate smoothing operator. The latest 2013 categories published on the World Bank website are as follows. Every country which has a “GNI per Capita” of $1035 or lower belongs to the low-income group. The next income category is the lower middle income group from $1036–$4085. After that the next group starts from $4086–$12,615 and is called the upper middle income group. The highest and last group is the high income group and starts after $12,616 or more.
Theoretically it is possible to pursue this exercise in the context of more than one variable, but the practicalities of so doing are a matter of ongoing research (Anderson et al. 2014).
Absolute convergence, the origin of the convergence debate, requires in fact convergence in structural characteristics across countries. It is no surprise that the hypothesis has been rejected in both cross country regression approaches (e.g. Barro 1991; Barro and Sala-i-Martin 1992) and transitional studies (e.g. Quah 1996a, b).
It is well known that the likelihood function of normal mixtures is unbounded and the global maximizer does not exist (McLachlan and Peel 2000). Therefore, the maximum likelihood estimator of \({\varvec{\Psi }}^{0}\) should be the root of the likelihood equation corresponding to the largest of the local maxima located. The solution usually adopted is to apply a range of starting solutions for the iterations. Here, randomly selected starts, and k-means (crisp and fuzzy) clustering-based starts have been chosen for initialization.
For an evaluation of the performance of these criteria in this context see e.g. Pittau et al. (2014).
This semi-parametric approach reconciles the traditional way of estimating transition probability matrices based on discrete states (typically quantiles) that are known to be sensitive to the arbitrary grouping of observations used to discretize the data (Bulli 2001), and the stochastic kernel, the continuous counterpart of the transition probability matrix, that solves the problem of discretization but, being a kernel, lacks a clear economic interpretation.
If we remove the time-invariant hypothesis the iterative application should be written as: \(T_{t, t-s}=\prod _{j=0}^{s-1} T_{t-j, t-j-1}\).
Since the average length of stay in class k is given by \(1/(1 -p_{k,k})\) (Prais 1955) and the mobility index can be rewritten as \(\sum _{k=1}^{K} (1-p_{k,k})/(K-1)\), it is the reciprocal of the harmonic mean of the mean exit times, normalized by the factor \(k/(k-1)\). This index satisfies the Normalization, Monotonicity, (Strong) Immobility and (Strong) Perfect Immobility properties described in Shorrocks (1978).
Dardanoni (1993) based his theorem upon the class of additively separable Social Welfare Functions which attach greater weight to individuals who start in a lower position in society, hence only monotone transition matrices are considered wherein period \(t+1\) outcome distributions for class k agents in period t, stochastically dominate those of class h agents in period t for all \(h<k\).
For China we included the official version (China1) that probably under-estimates the real value since it over-estimates the PPP (see Brandt and Holz 2006).
All the computing was carried out in R (2015). Functions are available upon request.
Table A.1 in Appendix A1 (Electronic Supplementary Material) reports the estimated parameters of the three-component mixture distributions along with their polarization trapezoids and income cut-offs. Estimated parameters for the \(K=2\) and \(k=4\) component mixtures are available from the authors upon request.
A more refined solution that allows one to use a modelling framework based on the multinormal distribution of the stochastic component relies on the use of “log-ratio” transform, like the additive log-ratio or the centered log-ratio transformation of the probabilities. This approach in economics is primarily associated with the work of Fry et al. (1996) on demand shares equations. However, the use of log-ratio transformations is not appropriate when the data do include many zeros. This is our case since, for example, many “poor” countries have zero probability to belong to the rich class. The problem of how to deal with zero values is discussed in Fry et al. (2000).
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Acknowledgments
We would like to thank the journal’s editor and three anonymous referees for helpful comments. For helpful discussions on the topic of this paper the authors are grateful to Tuomas Malinen and seminar participants at the 6th Meeting of the Society for the Study of Economic Inequality, 2015. Gordon Anderson is grateful to the SSHRC and the University of Toronto for research support. Maria Grazia Pittau and Roberto Zelli are grateful to Sapienza research grants for financial support.
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Anderson, G., Pittau, M.G. & Zelli, R. Assessing the convergence and mobility of nations without artificially specified class boundaries. J Econ Growth 21, 283–304 (2016). https://doi.org/10.1007/s10887-016-9128-5
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DOI: https://doi.org/10.1007/s10887-016-9128-5