Reconciling quantile autoregressions of firm size and variance–size scaling

Aim of this paper is to understand what economic mechanisms may cause the Law of Proportionate Effect to break down for fast-growing and shrinking firms, as highlighted by recent evidence that the first-order coefficients of quantile autoregression of firm size decline across quantiles. We theoretically show that if firm size is Laplace-distributed, conditional on size one period ahead, the decline in quantile regression coefficients is equivalent to a negative relationship between growth rates variance and size, while if conditional firm size is Asymmetric Laplace distributed, the equivalence result holds for fast-growing firms if skewness is decreasing with size, but not necessarily for shrinking firms. These results hold if the growth process is affected more by the determinants of variance-size scaling, e.g. interdependencies between a firm's units and submarkets, or cross-firm differences in organization and strategy, than by determinants of negative skewness in growth rates, such as diseconomies of growth, market power, and managerial attention issues. Using data on Dutch companies from the Business Register of Enterprises (BR), our estimates of quantile regression models confirm the evidence of declining quantile regression coefficients for small-medium firms (20 to 200 employees), and for the same subsample we find that growth rates variance and skewness are decreasing with size. The theoretical propositions of the paper are thus corroborated.