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Fractional calculus in economic growth modelling of the group of seven

  • Inés Tejado EMAIL logo , Emiliano Pérez and Duarte Valério
Published/Copyright: March 19, 2019

Abstract

This paper presents models of economic growth for all the countries of the Group of Seven (G7) in the 1973–2016 period. The models consist of differential equations, of both integer and fractional order, where the gross domestic product (GDP) is a function of the country’s land area, arable land, population, school attendance, gross capital formation (GCF), exports of goods and services, general government final consumption expenditure (GGFCE), and broad money (M3). Results show that fractional models have a better performance, measured by several summary statistics, without increasing the number of parameters, or sacrificing the ability to predict GDP evolution in the short term. A standard validation procedure for economic growth models is presented for the assessment of future models.

Acknowledgements

This work was partially supported by FEDER Funds (Programa Operativo FEDER de Extremadura 2014-2020) through the grant “Ayuda a Grupos de Investigación” (ref. GR15178) of the Junta de Extremadura and by FCT, through IDMEC, under LAETA, project UID/EMS/50022/2019. Inés Tejado would like to thank the Junta de Extremadura its support through the scholarship granted within the mobility program of 2017.

References

[1] B. Baeumer, M.M. Meerschaert, Fractional diffusion with two time scales. Physica A: Stat. Mech. Appl. 373 (2007), 237–251; 10.1016/j.physa.2006.06.014.Search in Google Scholar

[2] J. Blackledge, Application of the fractal market hypothesis for modelling macroeconomic time series. ISAST Trans. Electronics Sig. Proc. 2, No 1 (2008), 89–110; 10.21427/D7091P.Search in Google Scholar

[3] J. Blackledge, Application of the fractional diffusion equation for predicting market behaviour. IAENG Int. J. Appl. Math. 40, No 3 (2010), 130–158; 10.21427/D7HK8R.Search in Google Scholar

[4] D.E. Bloom, D. Canning, J. Sevilla, Technological diffusion, conditional convergence, and economic growth. Working Paper # 8713, National Bureau of Economic Research (2002), 27 pp.; 10.3386/w8713; Available at: https://www.nber.org/papers/w8713.Search in Google Scholar

[5] M. Boleantu, Fractional dynamical systems and applications in economy. Diff. Geom. – Dyn. Syst. 10 (2008), 62–70.Search in Google Scholar

[6] M. Caputo, J.M. Carcione, M.A.B. Botelho, Modeling extreme-event precursors with the fractional diffusion equation. Fract. Calc. Appl. Anal. 18, No 1 (2015), 208–222; 10.1515/fca-2015-0014; https://www.degruyter.com/view/j/fca.2015.18.issue-1/issue-files/fca.2015.18.issue-1.xml.Search in Google Scholar

[7] Á. Cartea, D. del Castillo-Negrete, Fractional diffusion models of option prices in markets with jumps. Physica A: Stat. Mech. Appl. 374, No 2 (2007), 749–763; 10.1016/j.physa.2006.08.071.Search in Google Scholar

[8] W.C. Chen, Nonlinear dynamics and chaos in a fractional order financial system. Chaos Solitons Fract. 36, No 5 (2008), 1305–1314; 10.1016/j.chaos.2006.07.051.Search in Google Scholar

[9] S. Dadras, H.R. Momeni, Control of a fractional-order economical system via sliding mode. Physica A: Stat. Mech. Appl. 389, No 12 (2010), 2434–2442; 10.1016/j.physa.2010.02.025.Search in Google Scholar

[10] E.F. Denison, Why Growth Rates Differ. Brooking Institutions, Washington (1967).Search in Google Scholar

[11] Federal Reserve Bank of St. Louis, Federal Reserve Economic Data (2018). Accessed March 2018; [Online]. Available at: https://fred.stlouisfed.org/.Search in Google Scholar

[12] R. Gorenflo, F. Mainardi, E. Scalas, M. Raberto, Fractional calculus and continuous-time finance III: The diffusion limit. In: Mathematical Finance Trends in Mathematics, Workshop of the Mathematical Finance Research Project, Konstanz (2001), 171–180.10.1007/978-3-0348-8291-0_17Search in Google Scholar

[13] Z. Hu, X. Tu, A new discrete economic model involving generalized fractal derivative. Adv. Differ. Eq. 65 (2015), 1–11; 10.1186/s13662-015-0416-8.Search in Google Scholar

[14] J. Korbel, Y. Luchko, Modeling of financial processes with a space-time fractional diffusion equation of varying order. Fract. Calc. Appl. Anal. 19, No 6 (2016), 1414–1433; 10.1515/fca-2016-0073; https://www.degruyter.com/view/j/fca.2016.19.issue-6/issue-files/fca.2016.19.issue-6.xml.Search in Google Scholar

[15] N. Laskin, Fractional market dynamics. Physica A: Stat. Mech. Appl. 287 (2000), 482–492; 10.1016/S0378-4371(00)00387-3.Search in Google Scholar

[16] J.W. Lee, H. Lee, Lee and Lee Long-run Education Dataset (2016). Accessed March 2018; Available at: http://www.barrolee.com/Lee_Lee_LRdata_dn.htm.Search in Google Scholar

[17] R.E. Lucas, On the mechanics of economic development. J. Monet. Econ. 22 (1988), 3–42; 10.1016/0304-3932(88)90168-7.Search in Google Scholar

[18] A. Maddison, Explaining the economic performance of nations, 1820–1989. In: Convergence of Productivity, W.J. Baumol WJ et al. (Eds), Oxford University Press, Oxford (1994), 20–61.Search in Google Scholar

[19] R.L. Magin, Fractional Calculus in Bioengineering. Begell House (2004).Search in Google Scholar

[20] F. Mainardi, M. Raberto, R. Gorenflo, E. Scalas, Fractional calculus and continuous-time finance II: The waiting-time distribution. Physica A: Stat. Mech. Appl. 287 (2000), 468–481; 10.1016/S0378-4371(00)00386-1.Search in Google Scholar

[21] O. Marom, E. Momoniat, A comparison of numerical solutions of fractional diffusion models in finance. Nonlin. Anal.: Real World Appl. 10 (2009), 3435–3442; 10.1016/j.nonrwa.2008.10.066.Search in Google Scholar

[22] M.M. Meerschaert, E. Scalas, Coupled continuous time random walks in finance. Physica A: Stat. Mech. Appl. 370 (2006), 114–118; 10.1016/j.physa.2006.04.034.Search in Google Scholar

[23] M.M. Meerschaert, A. Sikorskii, Stochastic Models for Fractional Calculus. Walter de Gruyter (2012).10.1515/9783110258165Search in Google Scholar

[24] Ministero degli Affari Esteri e della Cooperazione Internazionale, History of the G7/G8 (2018). Accessed April 2018; Available at: https://www.esteri.it/mae/en/politica_estera/g8/storia-del-g7-g8.html.Search in Google Scholar

[25] OECD, OECDiLibrary (2018). Accessed March 2018; Available at: http://dx.doi.org/10.1787/1036a2cf-en.10.1787/1036a2cf-enSearch in Google Scholar

[26] I. Petrás, I. Podlubny, State space description of national economies: The V4 countries. Comput. Statist. Data Anal. 52, No 2 (2007), 1223–1233; 10.1016/j.csda.2007.05.014.Search in Google Scholar

[27] S. Sassia, M. Goaied, Financial development, ICT diffusion and economic growth: Lessons from MENA region. Telecom. Pol. 37, No 4–5 (2013), 252–261; 10.1016/j.telpol.2012.12.004.Search in Google Scholar

[28] E. Scalas, R. Gorenflo, F. Mainardi. Fractional calculus and continuous-time finance. Physica A: Stat. Mech. Appl. 284, No 1-4 (2000), 376–384; 10.1016/S0378-4371(00)00255-7.Search in Google Scholar

[29] E. Scalas, The application of continuous-time random walks in finance and economics. Physica A: Stat. Mech. Appl. 362 (2006), 225–239; 10.1016/j.physa.2005.11.024.Search in Google Scholar

[30] A. Seck, International technology diffusion and economic growth: Explaining the spillover benefits to developing countries. Struct. Change Econ. Dyn. 23, No 4 (2012), 437–451; 10.1016/j.strueco.2011.01.003.Search in Google Scholar

[31] T. Skovranek, I. Podlubny, I. Petrás, Modeling of the national economies in state-space: A fractional calculus approach. Econ. Model. 29, No 4 (2012), 1322–1327; 10.1016/j.econmod.2012.03.019.Search in Google Scholar

[32] V.E. Tarasov, Long and short memory in economics: Fractional-order difference and differentiation. Int. J. Bus. Manag. Soc. Res. 5, No 2 (2016), 327–334; 10.21013/jmss.v5.n2.p10.Search in Google Scholar

[33] V.V. Tarasova, V.E. Tarasov, Exact discretization of an economic accelerator and multiplier with memory. Fractal Fract. 1, No 6 (2017), 1–14; 10.3390/fractalfract1010006.Search in Google Scholar

[34] V.V. Tarasova, V.E. Tarasov, Economic interpretation of fractional derivatives. Progress Fract. Different. Appl. 1 (2017), 1–6; 10.18576/pfda/030101.Search in Google Scholar

[35] I. Tejado, D. Valério, E. Pérez, N. Valério, Fractional calculus in economic growth modelling: The economies of France and Italy. In: International Conference on Fractional Differentiation and its Applications, Novi Sad (2016).Search in Google Scholar

[36] I. Tejado, D. Valério, E. Pérez, N. Valério, Fractional calculus in economic growth modelling. The Spanish and Portuguese cases. Int. J. Dyn. Control5, No 1 (2017), 208–222; 10.1007/s40435-015-0219-5.Search in Google Scholar

[37] I. Tejado, E. Pérez, D. Valério, Economic growth in the European Union modelled with fractional derivatives: First results. Bull. Pol. Acad. Sci.-Tech. Sci. 66, No 4 (2018), 455–465; 10.24425/124262.Search in Google Scholar

[38] I. Tejado, E. Pérez, D. Valério, Economic Data for the G7 Group (2018). Available at: https://github.com/UExtremadura/Economic/blob/master/G7Data_Tejado_et_al2018.xls.Search in Google Scholar

[39] I. Tejado, E. Pérez, D. Valério, Results for Predictions of the Future Evolution of the GDP for the G7 Group (2018). Available at: https://github.com/UExtremadura/Economic/blob/master/G7Results_Tejado_et_al2018.rar.Search in Google Scholar

[40] J.A. Tenreiro Machado, A.M. Lopes, Analysis of natural and artificial phenomena using signal processing and fractional calculus. Fract. Calc. Appl. Anal. 18, No 2 (2015), 459–478; 10.1515/fca-2015-0029; https://www.degruyter.com/view/j/fca.2015.18.issue-2/issue-files/fca.2015.18.issue-2.xml.Search in Google Scholar

[41] J.A. Tenreiro Machado, M.E. Mata, Pseudo phase plane and fractional calculus modeling of western global economic downturn. Commun. Nonlinear Sci. Numer. Simul. 22, No 1-3 (2015), 396–406; 10.1016/j.cnsns.2014.08.032.Search in Google Scholar

[42] J.A. Tenreiro Machado, M.E. Mata, A.M. Lopes, Fractional state space analysis of economic systems. Entropy17 (2015), 5402–5421; 10.3390/e17085402.Search in Google Scholar

[43] theGlobalEconomy.com, Economic Indicators for Over 200 Countries (2018). Accessed March 2018; Available at: https://www.theglobaleconomy.com/Germany/data_money_supply/, https://www.theglobaleconomy.com/France/data_money_supply/, https://www.theglobaleconomy.com/Italy/data_money_supply/.Search in Google Scholar

[44] The World Bank, World Bank Open Data (2018). Accessed March 2018; Available at: https://data.worldbank.org.Search in Google Scholar

[45] D. Valério, J. Sá da Costa, Introduction to single-input, single-output Fractional Control. IET Control Theory Appl. 5, No 8 (2011), 1033–1057: 10.1049/iet-cta.2010.0332.Search in Google Scholar

[46] Z. Wang, X. Huang, G. Shi, Analysis of nonlinear dynamics and chaos in a fractional order financial system with time delay. Comput. Math. Appl. 62, No 3 (2011), 1531–1539. 10.1016/j.camwa.2011.04.057.Search in Google Scholar

[47] West Germany (2018). Accessed August 2018; Available at: https://en.wikipedia.org/wiki/West_Germany.Search in Google Scholar

[48] Wittgenstein Centre for Demography and Global Human Capital, Wittgenstein Centre Data Explorer Version 1.2 (2015). Accessed March 2018; Available at: http://dataexplorer.wittgensteincentre.org/shiny/wic/.Search in Google Scholar

[49] Y. Xu, Z. He, Synchronization of variable-order fractional financial system via active control method. Cent. Eur. J. Phys. 11, No 6 (2013), 824–835; 10.2478/s11534-013-0237-x.Search in Google Scholar

[50] Y. Yue, L. He, G. Liu, Modeling and application of a new nonlinear fractional financial model. J. Appl. Math. (2013), ID 325050; 10.1155/2013/325050.Search in Google Scholar

Received: 2018-10-01
Published Online: 2019-03-19
Published in Print: 2019-02-25

© 2019 Diogenes Co., Sofia

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