A new discrete economic model involving generalized fractal derivative
 Zhenhua Hu^{1} and
 Xiaokang Tu^{1}Email author
https://doi.org/10.1186/s1366201504168
© Hu and Tu; licensee Springer. 2015
Received: 8 December 2014
Accepted: 13 February 2015
Published: 27 February 2015
Abstract
The current article is mainly concerned with applying generalized fractal derivatives in a macroeconomic model. We propose a discrete model involving four macroeconomic variables, the gross domestic production, exchange rate, money supply and exports/imports by using the generalized fractal derivative. The fractal derivative can describe the powerlaw phenomenon and memory property of economic variables more accurately. Based on the concrete macroeconomic data of Canada, the coefficients of this nonlinear system are estimated by the method of least squares. The statistical test results show that the four variables we have selected have an apparent causal connection, and the sum of squared residuals of the fitting equations is also acceptable. In simulation, the actual data of Canada from 1990 to 2008 are considered, and the effectiveness of our model is verified. The empirical study shows that in the coming few years, the money supply will grow quickly and hence it may lead to proper inflation.
Keywords
fractional calculus fractal derivative macroeconomic modelMSC
26A33 34A08 00A71 93E241 Introduction
The fractal and fractional derivatives can be regarded as the generalization of usual derivatives [1–3]. They are powerful tools in modeling anomalous physical processes, macro and microscale phenomena, longterm relation description and many discontinuous problems since they involve extra parameters, the order of derivatives. Selecting different fractional and fractal derivatives and their associated order values leads to various fractional and fractal derivative models. In recent forty years, fractional calculus has gained considerable attention in both applied mathematics and engineering fields such as viscoelastic mechanics, quantum physics, ecology, powerlaw phenomenon in fluids, electromagnetic field, economics modeling and financial systems. For a comprehensive understanding of this subject and its applications in economic and financial models, we refer readers to [4–16]. The advantage of applying fractional derivative in modeling economic and financial problems is that the nonlocal property of fractional derivative can depict the memory characteristics in many real economic and financial data [8, 9].
We notice that most of the above mentioned works concentrate on the continuous economic and financial models. There are very few works discussing the modeling of actual macroeconomic data using discrete dynamical system. In mathematics, a discrete dynamical system usually describes a recurrence relation, which is also an equation that recursively defines a sequence or multidimensional array of values once one or more initial terms are given. In difference equation, each further term of the sequence or array is defined as a function of the preceding terms. Discrete dynamical models are widely applied in many mathematical and engineering realms [29–31]. In realworld cases, many macroeconomic variables are observed sequently, i.e., daily, monthly or annually, which motivates us to construct a new model by using difference equation involving a generalized fractal derivative. The coupled relation of macroeconomic variables is expressed via a group of difference equations and the powerlaw property is characterized by the generalized fractal derivative containing two parameters.
The main contribution of the current paper consists of three aspects. Firstly, the fractal derivative is introduced and a generalized form of fractal derivative is applied to a constructed discrete macroeconomic model to describe the actual data of the given nation. Different from the fractional economic models in literature, the fractal derivative model proposed in this paper can depict the powerlaw property in macroeconomic series more accurately. Secondly, we consider the actual data of Canada and the effectiveness of our model is verified in simulation. Moreover, the method can be also applied to study the relation of other macroeconomic variables from other countries without many major modifications. Finally, instead of linear terms in model (1), our new discrete model involves nonlinear relation of macroeconomic variables, which is more accurate in describing the actual data. The rest of this paper is organized as follows. In Section 2, the definitions of fractal derivative and its generalized form are given. In Section 3, the discrete macroeconomic model using generalized fractal derivative is discussed, and the coefficients are estimated via the actual macroeconomic data of the given country. In Section 4, we simulate the dynamics of the new model and some useful observations are given. The effective predictions of macroeconomic variables can be obtained by computing the model corresponding to the future time. Finally, the conclusions are drawn in Section 5, and some additional remarks are listed in Section 6.
2 Fractal derivative and its generalized form
The general relationship between fractional calculus and fractals is explored in [34]. It is verified that the fractal dimension of function is shown to be a linear function of the order of fractional integrodifferentiation. This motivates us to employ fractal derivative to model the previous fractional relationship discovered between macroeconomic variables [8, 9]. Theoretically, fractal modeling has been applied in many random processes, e.g., see [35]. However, to the best of authors’ knowledge, there is no particular research on the application of fractal derivative modeling in macroeconomic models. It will be important to study economic and financial models further by using fractal derivatives.
It is easy to notice that the corresponding difference approximations of the above fractional derivatives involve infinitely many terms as step size approaches zero, which is inconvenient in approximate computation. The fractal derivative defined by (3) contains only one term in its difference form, and hence it is more convenient in approximation.
3 New discrete macroeconomic model
4 Empirical study
In this section, we present the dynamics of the new discrete model. The difference between the actual data and the numerical solution of the model is given, which demonstrates the effectiveness of the model. Then the prediction of the future behavior of the model is shown. By our model, it is reliable to make an estimation of the considered macroeconomic variable in the next few years.
4.1 Data description
In discrete model (8), the GDP (x), exchange rate (y), money supply (z) and exports/imports (u) are governed by a nonlinear dynamical system involving power laws. We preprocess the unit and scale of all data such that they are reasonable in a coupled nonlinear system. The annual data starts from year 1990 and continues to 2008. The unit of variables x and z is thousand billion US dollars and thousand billion Canadian dollars, respectively. The original data of these four macroeconomic variables are downloaded from http://data.worldbank.org.cn/indicator.
4.2 Dynamics of the new discrete macroeconomic model
Parameter sets in new discrete model ( 8 )
\(\boldsymbol {P_{x}}\)  Value  \(\boldsymbol {P_{y}}\)  Value  \(\boldsymbol {P_{z}}\)  Value  \(\boldsymbol {P_{u}}\)  Value 

\(\alpha_{1}\)  8.22  \(\alpha_{2}\)  −1.21E−02  \(\alpha_{3}\)  33.27  \(\alpha_{4}\)  6.85E−05 
\(h_{1}\)  0.15  \(h_{2}\)  5.63  \(h_{3}\)  0.12  \(h_{4}\)  39.47 
\(a_{11}\)  28638.61  \(a_{21}\)  24.82  \(a_{31}\)  1969186.23  \(a_{41}\)  −0.97 
\(a_{12}\)  −58927.57  \(a_{22}\)  −12.20  \(a_{32}\)  11605222.53  \(a_{42}\)  −0.62 
\(a_{13}\)  5641.35  \(a_{23}\)  0.29  \(a_{33}\)  −62813.00  \(a_{43}\)  5.80E−03 
\(a_{14}\)  −1113498.44  \(a_{24}\)  −9.75  \(a_{34}\)  164429009.80  \(a_{44}\)  −5.08 
\(c_{1}\)  1141119.70  \(c_{2}\)  −24.61  \(c_{3}\)  −212825070.60  \(c_{4}\)  66.17 
\(\beta_{1}\)  8.28  \(\beta_{2}\)  1.12  \(\beta_{3}\)  15.00  \(\beta_{4}\)  −0.31 
Remark 1
We take the data from the period 1990 to 2008 mainly because those data are available in the Internet. The data of 2009 and 2010 will be used to make comparison with the empirical data given by the new model, which we can observe demonstrates the effectiveness and robustness of the new discrete fractal macroeconomic model.
Statistical test values on estimation of variable x
Name  Value 

Mean squared error  0.1804 
Sum of squared error  0.5859 
Correlation coefficient (R)  0.9980 
\(R^{2}\)  0.9961 
Determination coefficient  0.9960 
Chisquare  0.0447 
Statistical test values on estimation of variable y
Name  Value 

Mean squared error  0.0883 
Sum of squared error  0.1403 
Correlation coefficient (R)  0.8360 
\(R^{2}\)  0.6989 
Determination coefficient  0.6989 
Chisquare  0.0525 
Statistical test values on estimation of variable z
Name  Value 

Mean squared error  0.8443 
Sum of squared error  12.8308 
Correlation coefficient (R)  0.9924 
\(R^{2}\)  0.9848 
Determination coefficient  0.9823 
Chisquare  1.2705 
Statistical test values on estimation of variable u
Name  Value 

Mean squared error  0.0265 
Sum of squared error  0.0126 
Correlation coefficient (R)  0.7380 
\(R^{2}\)  0.5476 
Determination coefficient  0.9823 
Chisquare  0.0057 
4.3 Prediction of the future evolution
Prediction of macroeconomic variables in 2009 to 2010
Variable  GDP  Exchange rate  Money supply  Exports/Imports 

Prediction in 2009  14.602  1.21  22.424  1.0921 
Actual data in 2009  12.525  –  –  3166/3299 
Error  2.077  –  –  0.1324 
Relative error  14.22%  –  –  12.12 
Prediction in 2010  14.835  1.08  21.073  1.1081 
Actual data in 2010  14.797  –  –  3879/4025 
Error  0.037  –  –  0.1443 
Relative error  2.49%  –  –  13.02% 
5 Conclusion
In this paper, we proposed a novel nonlinear discrete macroeconomic model based on the generalized fractal derivatives. The advantage of employing fractal derivative consists of two aspects. One is that the generalized fractal derivative is suitable for depicting the power law in macroeconomic variables. The other is that the difference form of generalized fractal derivative has finite term, which is different from the fractional derivative in RiemannLiouville, Caputo, and Riesz senses, where their difference expressions involve infinitely many terms as the step size goes to infinity. In our new model, the step size in discretization and the order in fractal derivative are regarded as parameters in the obtained discrete equation. All the parameters are estimated by the leastsquares method. Based on the macroeconomic data, we calculate the optimal parameters in fractal derivatives and step size. The sum squared residuals and the mean squared errors of estimation in simulation are computed, which demonstrate that the nonlinear discrete model is effective in modeling the macroeconomic variables of Canada. It would be expected that the nonlinear discrete model proposed in this paper is better than the linear model proposed in previous references.
6 Additional remarks

As a similar topic, fractional calculus has been applied in many scientific areas and engineering fields. In economic and financial realms, people found that the fractional derivative modeling can describe the memory property in many different financial series. However, as we stated, the powerlaw phenomenon is also one of the obvious characteristics that lots of macroeconomic variables usually exhibit. The fractal modeling, as a strong mathematical tool in studying the powerlaw feature, has not been applied widely. In our research, we show that the fractal derivative modeling can be used to investigate the coupled relation of different macroeconomic variables.

The modeling method discussed in this paper is nonstandard or unique. We may obtain other discrete models which are different from equation (9) by using other approximation scheme in the generalized fractal derivative.

Although we only consider the macroeconomic variables, i.e., GDP, exchange rate, money supply and exports/imports of Canada, many other innerconnected macroeconomic variables can be considered using the same method. Our modeling methodology can also be applied to other countries to investigate their evolution of different macroeconomic variables.
Declarations
Acknowledgements
The authors sincerely thank the reviewers for their constructive comments which significantly improved the quality of the paper. This work is partly supported by the Philosophy and Social Science Fund Project (No. 11YBA097) and the Scientific Research Funding of Hunan Provincial Education Department (No. 11C0437).
Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.
Authors’ Affiliations
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