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

Abstract

<div><p><em>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 </em><em>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 </em><em>the Fourier transform </em><em>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 </em><em>the Fourier transform</em><em> of these discrete operators satisfy the </em><em>power law in the neighborhood of zero only. We prove that the economic processes with t</em><em>he continuous time long and short memory, which is characterized by the power law, should be described by the fractional differential equations.</em></p></div>