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

By Vasily E. Tarasov, Valentina V. Tarasova. 2016

71 citations recorded by the source corpus.

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 t he continuous time long and short memory, which is characterized by the power law, should be described by the fractional differential equations.

Subfields: Economics and Econometrics, Finance, Modeling and Simulation

View Paper: https://www.semanticscholar.org/paper/6152d24aea6f4955a2a6859a74ac0491611f7140

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Citations in the corpus

Logistic map with memory from economic model (2017, background): “We should note that the fractional calculus and fractional differential equations have a wide application to describe different economic and financial processes with memory and nonlocality [56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 15, 16, 17, 18, 19] in the continuous time approach.”
Generalized Memory: Fractional Calculus Approach (2018, unknown): “We prove [39,40] that the discrete fractional differencing and integration, which are used in economics, are the well-known Grunwald-Letnikov fractional differences of non-integer order.”
On History of Mathematical Economics: Application of Fractional Calculus (2019, background): “As was proved in [39,40], these GJH operators are actually the Grunwald–Letnikov”