Long and Short Memory in Economics: Fractional-Order Difference and Differentiation

TL;DR

This paper establishes that long and short memory in economic processes are best described by fractional differential equations with exact fractional derivatives, clarifying the limitations of discrete fractional differencing.

Contribution

It proves the equivalence of discrete fractional differencing to Grunwald-Letnikov differences and highlights the need for fractional differential equations for continuous-time processes.

Findings

Discrete fractional differencing equals Grunwald-Letnikov differences.

Exact power law behavior is captured by fractional derivatives, not discrete differences.

Continuous-time long and short memory are described by fractional differential equations.

Abstract

Long and short memory in economic processes is usually described by the so-called discrete fractional differencing and fractional integration. We prove that the discrete fractional differencing and integration are the Grunwald-Letnikov fractional differences of non-integer order d. Equations of ARIMA(p,d,q) and ARFIMA(p,d,q) models are the fractional-order difference equations with the Grunwald-Letnikov differences of order d. We prove that the long and short memory with power law should be described by the exact fractional-order differences, for which the Fourier transform demonstrates the power law exactly. The fractional differencing and the Grunwald-Letnikov fractional differences cannot give exact results for the long and short memory with power law, since the Fourier transform of these discrete operators satisfy the power law in the neighborhood of zero only. We prove that the economic processes with the continuous time long and short memory, which is characterized by the power law, should be described by the fractional differential equations.