A Dynamic Taylor's Law

TL;DR

This paper introduces a dynamic version of Taylor's law for dependent time series, establishing a CLT, asymptotic properties, and goodness-of-fit tests, extending the classical static law to dependent data.

Contribution

It develops a dynamic Taylor's law for dependent samples using self-normalized expressions and Bernstein blocks, with theoretical CLT and estimation methods.

Findings

Proves a CLT for the dynamic Taylor's law under dependence.

Provides asymptotic results for goodness-of-fit testing.

Offers a consistent estimator for the Taylor exponent.

Abstract

Taylor's power law (or fluctuation scaling) states that on comparable populations, the variance of each sample is approximately proportional to a power of the mean of the population. It has been shown to hold by empirical observations in a broad class of disciplines including demography, biology, economics, physics and mathematics. In particular, it has been observed in the problems involving population dynamics, market trading, thermodynamics and number theory. For this many authors consider panel data in order to obtain laws of large numbers and the possibility to fit those expressions; essentially we aim at considering ergodic behaviors without independence. Thus we restrict the study to stationary time series and we develop different Taylor exponents in this setting. From a theoretic point of view, there has been a growing interest on the study of the behavior of such a phenomenon. Most of these works focused on the so-called static Taylor related to independent samples. In this paper, we introduce a dynamic Taylor's law for dependent samples using self-normalised expressions involving Bernstein blocks. A central limit theorem (CLT) is proved under either weak dependence or strong mixing assumptions for the marginal process. The limit behavior of such a new index involves the series of covariances unlike the classic framework where the limit behavior involves the marginal variance. We also provide an asymptotic result for for a goodness-of-fit testing suited to check whether the corresponding dynamical Taylor's law holds in empirical studies. Moreover, we also obtain a consistent estimation of the Taylor's exponent.