Evidence of Convergence Clubs Using Mixture Models

Abstract

Cross-country economic convergence has been increasingly investigated by finite mixture models. Multiple components in a mixture reflect groups of countries that converge locally. Testing for the number of components is crucial for detecting “convergence clubs.” To assess the number of components of the mixture, we propose a sequential procedure that compares the shape of the hypothesized mixture distribution with the true unknown density, consistently estimated through a kernel estimator. The novelty of our approach is its capability to select the number of components along with a satisfactory fitting of the model. Simulation studies and an empirical application to per capita income distribution across countries testify for the good performance of our approach. A three-clubs convergence seems to emerge.

Keywords:

• Convergence clubs
• Kernel-density based test
• Mixture models

JEL Classification:

• C14
• C15
• O11

ACKNOWLEDGMENTS

We would like to thank the Associate Editor and two anonymous reviewers for their constructive comments. We send special thanks to one reviewer for having also helped in improving the presentation of our manuscript. We are also grateful to Aman Ullah, Alessio Farcomeni, Paul Johnson, and Marco Alfò for valuable discussions and suggestions.

Notes

¹See, e.g., Whindam and Cutler (1992), Fraley and Raftery (1998), Biernacki et al. (2000), Andrews and Currim (2003), Pittau (2005), Flachaire and Nunez (2007), Ray and Lindasy (2008).

²Several distance measures have been used to compare the two distributions such as the penalized minimum distance (Chen and Kalbfleisch, 1996), the Kullback–Leibler distance (James et al., 2001), and the Hellinger distance (Woo and Sriram, 2006).

³In the area of multiple comparison, our procedure has some similarities with the gatekeeping procedure (e.g., Dmitrienko et al., 2006; Kim et al., 2011). In the gatekeeping framework, the objectives (i.e., hypotheses) are hierarchically ordered based on their importance and grouped into families F₁, F₂,…, F_m. The families are examined sequentially beginning with F₁, which represents the most important group of objectives, with more than one hypothesis within each family possibly true. Each of the first m − 1 families serves as a gatekeeper for the families placed later in the sequence. To maintain the overall type I error rate at the desired α-level gatekeeping procedures work by sequentially adjusting each family of hypotheses reached and stop when any hypothesis within a family is not rejected. Three important features make our approach different from the gatekeeping procedure. The total number m of families is unknown, each family consists of only one hypothesis and only one hypothesis is true while all the others are false. These differences complicate the adjustment procedure. First of all, correction for multiplicity (like Holm–Bonferroni or Bonferroni–Hochberg methods) involves m and, even though we could consider possible solutions (Farcomeni and Finos, 2013), calibrating the adjustment is not straightforward. More importantly, the level of α should be inflated as we test for an increasing number of components g. This translates in testing the first hypothesis g = 1 at the usual α level but inflating α when we test for more than one component. For example, we test g = 2 only after having rejected g = 1, and consequently, the level of α of this second step should also incorporate the probability of having rejected g = 1 when it was false (i.e., the power of the first step), resulting in a bigger value of the α threshold. This situation becomes more complicated as g increases and further theoretical developments are required in order to find a proper adjustment. For these reasons, we prefer to leave the significance threshold at the same level at each step. We thank the Associated Editor for having raised this point.

⁴We can do this without losing generality because of the location invariance of the test statistics.

⁵The simulation strategy has been carried out in R (R Core Team, 2012). The codes are available on request from the first author.

⁶The average or the maximum levels of pairwise overlap serve as a proxy of the degree of overall interaction among mixture components (Melnykov and Maitra, 2010).

⁷See Durlauf et al. (2005) for an exhaustive review on “conditional” and “unconditional” convergence.

⁸On the impact of the PWT revisions on growth estimates and how they affect the cross-country growth literature see Deaton and Heston (2010) and Johnson et al. (2013).