We propose a new nonlinear economic system with fractional derivative. According to the Jumarie’s definition of fractional derivative, we obtain a discrete fractional nonlinear economic system. Three variables, the gross domestic production, inflation, and unemployment rate, are considered by this nonlinear system. Based on the concrete macroeconomic data of USA, the coefficients of this nonlinear system are estimated by the method of least squares. The application of discrete fractional economic model with linear and nonlinear structure is shown to illustrate the efficiency of modeling the macroeconomic data with discrete fractional dynamical system. The empirical study suggests that the nonlinear discrete fractional dynamical system can describe the actual economic data accurately and predict the future behavior more reasonably than the linear dynamic system. The method proposed in this paper can be applied to investigate other macroeconomic variables of more states.
Economic dynamics has recently become more prominent in mainstream economics. This influence has been quite pervasive and has influenced both microeconomics and macroeconomics. Its influence in macroeconomics, however, has been much greater. In the realworld life, economic evolution behaves like some process with inner random property. The investigation of economic system gains much development in the recent decades mainly since it can exhibit ubiquitous complex dynamics evidenced by largeamplitude and aperiodic fluctuations [
Fractional calculus, the differentiation and integration with arbitrary order, has been a developing branch of mathematics. It is applied to many scientific and engineering fields successfully in the recent 40 years or so, and it is admitted as a powerful tool in modelling some physical processes with memory effect, which cannot be described well by integer order differential equations. In the current paper, we are not going to discuss the detail of fractional calculus. We refer the readers to [
Over the last decade, the dynamics of fractionalorder financial and economic systems have been investigated via several mathematical methods (see [
The most common definitions of fractional derivative are the Caputo fractional derivative (see (
In order to obtain the relation of fractional difference and traditional integer difference, the following generalized Taylor expansion of fractional order is applied:
The expression of fractional derivative (
In model (
Let
In model (
In the rest of this paper, we will mainly study model (
It is easy to find that there are not common parameters for three equations of the model (
In (
We denote
Therefore, we obtain the estimation of
It is easy to find that the estimation of
The estimation for the
In this section, we present an application of discrete fractional order economic model based on the macroeconomic data of USA, which demonstrates the effectiveness of model in simulating the evolution of the macroeconomic variables and predicting the future behavior of the macroeconomic system.
In fractional economic model (
We first estimate the optimal fractional order. In (
The optimal estimation of fractional order
Now we estimate the left parameters. To make a better comparison of numerical simulation, we fit the parameters of fractional economic model (
The estimation results of parameters in model (








L  NL  L  NL  L  NL  

−0.0050  −0.0753 

0.0016  −0.0407 

0.0215  0.0754 

−0.6284  1.0035 

0.1551  1.2353 

−0.2587  −1.4196 

−0.2203  −0.6022 

−0.4282  0.5073 

0.1220  −0.1173 

0.4818  2.1463 

0.1034  0.4666 

−0.2939  −1.3398 

−16.778 

−2.7541 

7.3886  

4.7152 

−18.606 

4.4934  

−16.371 

−9.1467 

10.295  

4.7435 

−4.9194 

2.8206  

1.2345 

3.6713 

−1.0608  

−10.108 

3.6943 

4.0766  
 
SSR 


SSR 


SSR 



0.4955  0.6803 

0.5615  0.7399 

0.4705  0.6918 
Note: SSR is the sum squared residuals. L stands for the linear economic model (
The estimation results of parameters in model (








L  NL  L  NL  L  NL  

−0.0049  −0.0737 

0.0016  −0.0398 

0.0211  0.0739 

−0.6154  0.9826 

0.1519  1.2096 

−0.2533  −1.3900 

−0.2157  −0.5897 

−0.4193  0.4967 

0.1194  −0.1148 

0.4718  2.1016 

0.1013  0.4569 

−0.2878  −1.3118 

−16.428 

−2.6967 

7.2346  

4.6169 

−18.219 

4.3997  

−16.030 

−8.9561 

10.080  

4.6446 

−4.8169 

2.7619  

1.2087 

3.5948 

−1.0387  

−9.8971 

3.6173 

3.9917  
 
SSR 


SSR 


SSR 



0.4955  0.6803 

0.5615  0.7399 

0.4705  0.6918 
The estimation results of parameters in model (








L  NL  L  NL  L  NL  

−0.0048  −0.0714 

0.0016  −0.0386 

0.0204  0.0716 

−0.5962  0.9521 

0.1472  1.1721 

−0.2454  −1.3468 

−0.2090  −0.5714 

−0.4063  0.4813 

0.1157  −0.1113 

0.4572  2.0364 

0.0981  0.4427 

−0.2789  −1.2711 

−15.918 

−2.6130 

7.0099  

4.4736 

−17.653 

4.2631  

−15.532 

−8.6779 

9.7671  

4.5004 

−4.6673 

2.6761  

1.1712 

3.4831 

−1.0065  

−9.5897 

3.5050 

3.8677  
 
SSR 


SSR 


SSR 



0.4955  0.6803 

0.5615  0.7399 

0.4705  0.6918 
In Table
The actual and estimated data of the GDP, inflation, and unemployment by the linear and nonlinear models with
Table
The actual and estimated data of the GDP, inflation, and unemployment by the linear and nonlinear models with
In Table
The actual and estimated data of the GDP, inflation and unemployment by the linear and nonlinear models with
To end this subsection, we would like to make the following remark.
The discrete nonlinear fractional model (
Now we simulate the future behavior of the macroeconomic variables considered in the nonlinear model (
The inofsample prediction can be calculated by the above recursive equations. The actual data of GDP, inflation, and unemployment rate are compared with their corresponding inofsample prediction with step size
The actual data and inofsample prediction of the GDP, inflation, and unemployment with the linear model (
The actual data and inofsample prediction of the GDP, inflation, and unemployment with the nonlinear model (
In Figure
In Figure
Furthermore, careful comparison between the Figures
In this work, we construct a new dynamic nonlinear economic model with fractional derivative defined by Jumarie's sense. The corresponding discrete model is drawn by removing the limit operator in Jumarie's fractional derivative. The step size in discretization and the fractional order are regarded as parameters in the obtained discrete model. All the parameters are estimated by the least square method. Based on the macroeconomic data, we calculate the optimal fractional order and step size. We verify that there are lots of choices of values of step size in discretization and every step size has a unique optimal fractional order. The sum squared residuals and the
Although we only consider the macroeconomic variables, that is, GDP, inflation, and unemployment of USA, many other innerconnected macroeconomic variables can be considered using the same method. Our modelling methodology can also be applied to other nations to investigate their evolution of economic variables.
The authors sincerely thank the reviewers for their constructive comments which significantly improved the quality of their 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). The authors would also like to express sincere appreciation to Dr. Lei He and Dr. Shichang Ma for their unselfish help and benefit discussion on the preparation of the paper.