Composite distributions in the social sciences: A comparative empirical study of firms' sales distribution for France, Germany, Italy, Japan, South Korea, and Spain

TL;DR

This study compares 17 statistical distributions to model firms' sales across six countries, finding mixtures of lognormal, loglogistic, and Student's t distributions best describe the data, revealing complex subgroups within firm sizes.

Contribution

It introduces a comprehensive empirical comparison of multiple distributions for firm size data and demonstrates the superiority of mixture models over single distributions.

Findings

Mixtures of LN, LL, and LSt distributions outperform single distributions.

Single lognormal distribution is strongly not suitable.

Multiple subgroups of firms are identified within the size distribution.

Abstract

We study 17 different statistical distributions for sizes obtained {}from the classical and recent literature to describe a relevant variable in the social sciences and Economics, namely the firms' sales distribution in six countries over an ample period. We find that the best results are obtained with mixtures of lognormal (LN), loglogistic (LL), and log Student's $t$ (LSt) distributions. The single lognormal, in turn, is strongly not selected. We then find that the whole firm size distribution is better described by a mixture, and there exist subgroups of firms. Depending on the method of measurement, the best fitting distribution cannot be defined by a single one, but as a mixture of at least three distributions or even four or five. We assess a full sample analysis, an in-sample and out-of-sample analysis, and a doubly truncated sample analysis. We also provide the formulation of the preferred models as solutions of the Fokker--Planck or forward Kolmogorov equation.