Комментарии к статье «Long and short memory in economics: fractional-order difference and differentiation»

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Комментарии к статье «Long and short memory in economics: fractional-order difference and differentiation»

Тарасова В.В.¹, Тарасов В.Е.²

Аннотация: в данных комментариях приводятся исправленные формулы для точных конечных разностей, которые в статье «Long and short memory in economics: fractional-order difference and differentiation» (IRA-International Journal of Management and Social Sciences, 2016. Vol. 5. No. 2. P. 327334.) приведены с опечатками. Точные дробные разности можно рассматривать как точные дискретные аналоги дробных производных Лиувилля целого и нецелого порядка. Эти дробные разности и производные нецелого порядка могут быть использованы для описания экономических процессов с степенной угасающей памятью.

Ключевые слова: долговременная память, кратковременная память, экономические процессы с памятью, модель ARIMA, модель ARFIMA, точные разности, дробные разности, разности ГрюнвальдаЛетникова, дробная производная, точная дискретизация.

In the paper [1, p. 332], equations (21) and (23) of the exact fractional differences are given with misprints. These comments provide corrected formulas for the exact fractional differences.

The exact discretization of the derivatives of integer and non-integer orders [2, 3] and the corresponding exact finite differences [2, 3] were initially proposed in [4, 5, 6, 7] as derivatives on lattices. In economics, they were used in [8, 9]. Derivatives of non-integer order and the corresponding fractional finite differences allow us to describe economic processes with power-law dynamic memory [10].

The exact fractional differences Д^б_T,exact of order б are defined by the equation

, (1)

дробный дискретный производная лиувилль

where K_б(m) is the kernel of the exact fractional difference of the form

. (2)

The kernels

, (3)

, (4)

where we use the generalized hypergeometric function

. (5)

Using equation (5), the kernel (2) can be written in the form

For б<0 expression (2) with the kernel (6) defines the discrete fractional integration [2, 3].

For the arbitrary positive integer order б=n, the kernel K_б(m) of the exact difference can be represented by the equation

(7)

for m ? 0, and for m = 0 the kernel is written by the expression

. (8)

The exact finite difference (б=1) is defined by the equation

, (9)

where the sum implies the Cesaro or Poisson-Abel summation [3, p. 55-56; 13]. An important characteristic property of the exact finite difference (9) is the Leibniz rule (the product rule) in the form

Д¹_T,exact (X(t) · Y(??)) = (Д¹_T,exact X(t)) · Y(t) + X(t) · (Д¹_T,exact Y(t)) (10)

which is satisfied for all X(t), Y(t) from the space of entire functions. Exact finite difference of second and next integer orders can be derived by the recurrence formulas .

In the paper [1, p. 332], equations (21) and (23) should be replaced by equations (4) and (6), respectively.

It should be noted that the exact fractional differences are exact discrete analogues of the Liouville fractional derivatives. Equations of discrete macroeconomic models, which are used the exact finite differences, are exact discrete analogs of differential equations of models with continuous time for a wide class of solutions (for example, see [9] and [10, 11, 12, 14]).

References / Список литературы

1. Tarasov V.E., Tarasova V.V. Long and short memory in economics: fractional-order difference and differentiation // IRA-International Journal of Management and Social Sciences, 2016. Vol. 5. № 2. P. 327334. DOI: 10.21013/jmss.v5.n2.p10.

2. Tarasov V.E. Exact discrete analogs of derivatives of integer orders: Differences as infinite series // Journal of Mathematics, 2015. Vol. 2015. Article ID 134842. 8 p. DOI: 10.1155/2015/134842.

3. Tarasov V.E. Exact discretization by Fourier transforms // Communications in Nonlinear Science and Numerical Simulation, 2016. Vol. 37. P. 31-61. DOI: 10.1016/j.cnsns.2016.01.006.

4. Tarasov V.E. Toward lattice fractional vector calculus // Journal of Physics A, 2014. Vol. 47. No. 35. Artilce ID 355204. DOI: 10.1088/1751-8113/47/35/355204.

5. Tarasov V.E. Lattice fractional calculus // Applied Mathematics and Computation. 2015. Vol. 257. P. 12-33. DOI: 10.1016/j.amc.2014.11.033

6. Tarasov V.E. United lattice fractional integro-differentiation // Fractional Calculus and Applied Analysis, 2016. Vol. 19. № 3. P. 625-664. DOI: 10.1515/fca-2016-0034.

7. Tarasov V.E. Exact discretization of fractional Laplacian // Computers and Mathematics with Applications, 2017. Vol. 73. № 5. P. 855-863. DOI: 10.1016/j.camwa.2017.01.012.

8. Tarasova V.V., Tarasov V.E. Exact discretization of economic accelerator and multiplier with memory // Fractal and Fractional, 2017. Vol. 1. № 1. Article ID: 6. DOI: 10.3390/fractalfract1010006.

9. Tarasova V.V., Tarasov V.E. Accelerators in macroeconomics: Comparison of discrete and continuous approaches // Scientific Journal [Nauchnyj Zhurnal], 2017. № 8 (21). С. 4-14 [in Russian].

10. Tarasova V.V., Tarasov V.E. Concept of dynamic memory in economics // Communications in Nonlinear Science and Numerical Simulation, 2018. Vol. 55. P. 127-145. DOI: 10.1016/j.cnsns.2017.06.032.

11. Tarasova V.V., Tarasov V.E. Fractional dynamics of natural growth and memory effect in economics // European Research, 2016. № 12 (23). P. 30-37. DOI: 10.20861/2410-2873-2016-23-004.

12. Tarasova V.V., Tarasov V.E. Economic growth model with constant pace and dynamic memory // Problems of Modern Science and Education [Problemy Sovremennoj Nauki i Obrazovaniya], 2017. № 2 (84). P. 40-45. DOI: 10.20861/2304-2338-2017-84-001.

13. Fichtenholz G.M. Infinite Series: Ramifications. New York: Routledge, 1970. 139 p. ISBN 0-677-20940-1

14. Tarasova V.V., Tarasov V.E. Economic accelerator with memory: discrete time approach // Problems of Modern Science and Education [Problemy Sovremennoj Nauki i Obrazovaniya], 2016. № 36 (78). P. 37-42. DOI: 10.20861/2304-2338-2016-78-002.

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