Criterion of Existence of Power-Law Memory for Economic Processes

In this paper, we propose criteria for the existence of memory of power-law type (PLT) memory in economic processes. We give the criterion of existence of power-law long-range dependence in time by using the analogy with the concept of the long-range alpha-interaction. We also suggest the criterion of existence of PLT memory for frequency domain by using the concept of non-integer dimensions. For an economic process, for which it is known that an endogenous variable depends on an exogenous variable, the proposed criteria make it possible to identify the presence of the PLT memory. The suggested criteria are illustrated in various examples. The use of the proposed criteria allows apply the fractional calculus to construct dynamic models of economic processes. These criteria can be also used to identify the linear integro-differential operators that can be considered as fractional derivatives and integrals of non-integer orders.

[1]  Aroop K. Mahanty,et al.  THEORY OF PRODUCTION , 1980 .

[2]  R. Gorenflo,et al.  Fractional calculus and continuous-time finance , 2000, cond-mat/0001120.

[3]  Richard T. Baillie,et al.  Long memory processes and fractional integration in econometrics , 1996 .

[4]  A. K. Jonscher,et al.  Universal relaxation law : a sequel to Dielectric relaxation in solids , 1996 .

[5]  Gilles Teyssière,et al.  Long Memory in Economics , 2006 .

[6]  Aleksander Stanislavsky,et al.  Atypical Case of the Dielectric Relaxation Responses and its Fractional Kinetic Equation , 2016 .

[7]  V. E. Tarasov.  Fractional integro-differential equations for electromagnetic waves in dielectric media , 2009, 1107.5892.

[8]  C. Granger.  Investigating causal relations by econometric models and cross-spectral methods , 1969 .

[9]  H. Srivastava,et al.  Theory and Applications of Fractional Differential Equations , 2006 .

[10]  A. Jonscher.  Dielectric relaxation in solids , 1983 .

[11]  R. Vilela Mendes,et al.  A fractional calculus interpretation of the fractional volatility model , 2009 .

[12]  K. Hoover,et al.  Causality in Economics and Econometrics , 2006 .

[13]  Vasily E. Tarasov,et al.  Fractional Dynamics of Natural Growth and Memory Effect in Economics , 2016 .

[14]  Vasily E. Tarasov,et al.  Exact Discretization of an Economic Accelerator and Multiplier with Memory , 2017 .

[15]  V. E. Tarasov.  Universal electromagnetic waves in dielectric , 2008, 0907.2163.

[16]  José António Tenreiro Machado,et al.  Fractional Dynamics in Financial Indices , 2012, Int. J. Bifurc. Chaos.

[17]  K. Hoover,et al.  The Methodology of Empirical Macroeconomics , 2001 .

[18]  R. Gorenflo,et al.  Fractional calculus and continuous-time finance II: the waiting-time distribution , 2000, cond-mat/0006454.

[19]  M. Sharnoff.  Validity Conditions for the Kramers-Kronig Relations , 1964 .

[20]  K. Hoover.  Causality in Macroeconomics , 2001 .

[21]  Vasily E. Tarasov,et al.  Economic interpretation of fractional derivatives , 2017, 1712.09575.

[22]  Jan Beran,et al.  Statistics for long-memory processes , 1994 .

[23]  B. Hu.  Kramers-Kronig in two lines , 1989 .

[24]  R. Gorenflo,et al.  Mittag-Leffler Functions, Related Topics and Applications , 2014, Springer Monographs in Mathematics.

[25]  Vasily E. Tarasov,et al.  Economic Growth Model with Constant Pace and Dynamic Memory , 2017 .

[26]  V. E. Tarasov.  Map of discrete system into continuous , 2006, 0711.2612.

[27]  Inés Tejado,et al.  Fractional Calculus in Economic Growth Modelling: The Spanish Case , 2015 .

[28]  J. T. Tenreiro Machado,et al.  Relative fractional dynamics of stock markets , 2016 .

[29]  P. Frank,et al.  Boston Studies in the Philosophy of Science , 1968 .

[30]  H. Simon,et al.  Causal Ordering and Identifiability , 1977 .

[31]  Xavier Gabaix,et al.  Power Laws in Economics: An Introduction , 2016 .

[32]  W. Palma.  Long-Memory Time Series: Theory and Methods , 2007 .

[33]  N. Laskin.  Fractional market dynamics , 2000 .

[34]  Michael E. Fisher,et al.  Critical Exponents in 3.99 Dimensions , 1972 .

[35]  Vasily E. Tarasov,et al.  Memory effects in hereditary Harrod-Domar model , 2016 .

[36]  V. E. Tarasov,et al.  Time-dependent fractional dynamics with memory in quantum and economic physics , 2017 .

[37]  X. Gabaix.  Power Laws in Economics and Finance , 2008 .

[38]  S. Havriliak,et al.  A complex plane representation of dielectric and mechanical relaxation processes in some polymers , 1967 .

[39]  J. Toll.  Causality and the Dispersion Relation: Logical Foundations , 1956 .

[40]  Hermann Haken,et al.  Synergetics: An Introduction , 1983 .

[41]  J. Mackie,et al.  The cement of the universe : a study of causation , 1977 .

[42]  Vasily E. Tarasov.  No nonlocality. No fractional derivative , 2018, Commun. Nonlinear Sci. Numer. Simul..

[43]  Jean-Pierre Urbain.  Essays in Econometrics, Collected Papers of Clive W.J. Granger, Volume II: Causality, Integration and Cointegration, and Long Memory , 2005 .

[44]  Eric Ghysels,et al.  Essays in econometrics: Collected Papers of Clive W. J. Granger Volume 1, Spectral Analysis, Seasonality, Nonlinearity, Methodology, and Forecasting , 2001 .

[45]  Wei-Bin Zhang,et al.  Synergetic Economics: Time and Change in Nonlinear Economics , 1991 .

[46]  D. A. Kirzhnits,et al.  METHODOLOGICAL NOTES: Are the Kramers-Kronig relations for the dielectric permittivity of a material always valid? , 1976 .

[47]  Craig F. Bohren,et al.  What did Kramers and Kronig do and how did they do it? , 2010 .

[48]  V. E. Tarasov,et al.  Logistic map with memory from economic model , 2017, 1712.09092.

[49]  Yuri Luchko,et al.  Handbook of fractional calculus with applications , 2019 .

[50]  Vasily E. Tarasov,et al.  Accelerator and Multiplier for Macroeconomic Processes with Memory , 2017 .

[51]  Reiner Koblo,et al.  The Visible Hand: Synergetic Microfoundation of Macroeconomic Dynamics , 1991 .

[52]  Robert B. Gramacy,et al.  A Brief History of Long Memory: Hurst, Mandelbrot and the Road to ARFIMA, 1951-1980 , 2014, Entropy.

[53]  C. Granger.  Testing for causality: a personal viewpoint , 1980 .

[54]  R. Hilfer,et al.  H-function representations for stretched exponential relaxation and non-Debye susceptibilities in glassy systems. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[55]  V. Kiryakova.  Generalized Fractional Calculus and Applications , 1993 .

[56]  H. Nussenzveig.  Causality and dispersion relations , 2012, American Journal of Physics.

[57]  W. J. Granger,et al.  Essays in Econometrics Collected Papers of Clive , 2001 .

[58]  A. K. Jonscher,et al.  A new model of dielectric loss in polymers , 1975 .

[59]  Hashem Dezhbakhsh,et al.  On the typical spectral shape of an economic variable , 2003 .

[60]  V. E. Tarasov.  Continuous limit of discrete systems with long-range interaction , 2006, 0711.0826.

[61]  Kenneth G. Wilson.  Critical Phenomena in 3.99 Dimensions , 2008 .

[62]  Duarte Valerio,et al.  Fractional calculus in economic growth modeling. The Portuguese case , 2014, ICFDA'14 International Conference on Fractional Differentiation and Its Applications 2014.

[63]  S. Havriliak,et al.  Dielectric and Mechanical Relaxation in Materials: Analysis, Interpretation and Application to Polymers , 1996 .

[64]  Vasily E. Tarasov,et al.  Exact discretization by Fourier transforms , 2016, Commun. Nonlinear Sci. Numer. Simul..

[65]  C. Granger,et al.  AN INTRODUCTION TO LONG‐MEMORY TIME SERIES MODELS AND FRACTIONAL DIFFERENCING , 1980 .

[66]  H. Stanley,et al.  Introduction to Phase Transitions and Critical Phenomena , 1972 .

[67]  T. Kaczorek,et al.  Fractional Differential Equations , 2015 .

[68]  Vasily E. Tarasov,et al.  Economic Accelerator with Memory: Discrete Time Approach , 2016 .

[69]  R. Bergman,et al.  General susceptibility functions for relaxations in disordered systems , 2000 .

[70]  E. C. Titchmarsh.  Introduction to the Theory of Fourier Integrals , 1938 .

[71]  V. E. Tarasov,et al.  Comments to the article "long and short memory in economics: fractional-order difference and differentiation" , 2017 .

[72]  D. Braddon-Mitchell.  NATURE'S CAPACITIES AND THEIR MEASUREMENT , 1991 .

[73]  Jan Beran,et al.  Long-Memory Processes: Probabilistic Properties and Statistical Methods , 2013 .

[74]  V. E. Tarasov.  Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media , 2011 .

[75]  Vasily E. Tarasov,et al.  Dynamic intersectoral models with power-law memory , 2017, Commun. Nonlinear Sci. Numer. Simul..

[76]  Emiliano Pérez,et al.  Fractional calculus in economic growth modelling: the Spanish and Portuguese cases , 2015, International Journal of Dynamics and Control.

[77]  Virginia Kiryakova,et al.  A BRIEF STORY ABOUT THE OPERATORS OF THE GENERALIZED FRACTIONAL CALCULUS , 2008 .

[78]  Álvaro Cartea,et al.  Fractional Diffusion Models of Option Prices in Markets With Jumps , 2006 .

[79]  P. Kinsler,et al.  How to be causal: time, spacetime and spectra , 2011, 1106.1792.

[80]  Robert B. Gramacy,et al.  A Brief History of Long Memory: Hurst, Mandelbrot and the Road to ARFIMA , 2017 .

[81]  Vasily E. Tarasov,et al.  Exact Discrete Analogs of Derivatives of Integer Orders: Differences as Infinite Series , 2015 .

[82]  Enrico Scalas,et al.  Fractional Calculus and Continuous-Time Finance III : the Diffusion Limit , 2001 .

[83]  V. E. Tarasov.  Fractional equations of Curie–von Schweidler and Gauss laws , 2008, 0907.1837.

[84]  V. E. Tarasov,et al.  Long and Short Memory in Economics: Fractional-Order Difference and Differentiation , 2016, 1612.07903.

[85]  H. Simon,et al.  Cause and Counterfactual , 1966 .

[86]  Vasily E. Tarasov,et al.  Concept of dynamic memory in economics , 2018, Commun. Nonlinear Sci. Numer. Simul..

[87]  O. Marichev,et al.  Fractional Integrals and Derivatives: Theory and Applications , 1993 .

[88]  R. Kronig.  On the Theory of Dispersion of X-Rays , 1926 .

[89]  A. Offord.  Introduction to the Theory of Fourier Integrals , 1938, Nature.

[90]  Frank H. Stillinger,et al.  Axiomatic basis for spaces with noninteger dimension , 1977 .

[91]  S V Buldyrev,et al.  Self-organized complexity in economics and finance , 2002, Proceedings of the National Academy of Sciences of the United States of America.