Relative error status of FDI level in China.
Abstract
Foreign direct investment (FDI) is one of the important factors affecting China’s economic development, the prediction of which is the basis of its development and decisionmaking. Based on elaborating the significant role in China’s economic growth and the status quo of utilizing foreign investment over the period between 2000 and 2016, this chapter attempts to construct GrayMarkov model (GMM) and time series model (TSM) to forecast the trend of China’s utilization of FDI and then compares the precision of two different prediction models to obtain a better one. Results indicate that although it is qualified, traditional Gray model needs to be optimized; GMM is built to help modify the result, improve Grayrelated degrees, and narrow the gap with real value. Comparing the accuracy of GMM with that of TSM, we can conclude that the fitting effect of GMM is better. To increase the credibility of these results, this chapter is based on the data of Beijing and Chongqing from 1990 till 2016, also verifying that the fitting effect of GMM is superior to that of the TSM. Then, we can safely draw a conclusion that the prediction model of GMM is more credible, which has a certain referencing value for the utilization of FDI.
Keywords
 foreign direct investment (FDI)
 GrayMarkov model (GMM)
 time series model (TSM)
1. Introduction
In the light of the definition of the International Monetary Fund (IMF) and the Organization for Economic Cooperation and Development, foreign direct investment (FDI) is an investment in the form of a controlling ownership in a business in one country by an entity based in another country. The primary purpose of the host country in attracting FDI is to promote the country’s economic development and industrial upgrading. This will facilitate domestic enterprises to improve their technology and quality, gradually supporting the development of foreign enterprises to enter the global value chain [1]. Influencing the supply chain system, FDI has significantly promoted the sound and rapid development of the national economy. Therefore, it is necessary to focus on the future tendency of FDI in the supply chain system when we investigate the transformation and innovation of Chinese economy.
Since the late 1970s, FDI attracted by China has been steadily increasing, regardless of the changes and fluctuation of the international economic environment and the total flow of FDI globally. Statistically, over the period from 1979 to 2010, China’s actual use of FDI amounted to $1048.31 billion [2], and FDI keeps a rapid growth. According to the data of Ministry of Commerce of the People’s Republic of China (PRC) (Figure 1), the FDI in China presented a rising trend over the period from 1990 to 2016. The vital roles in the economic development of China are as follows. Firstly, the proportion of basic industries in China declines generally, and the proportion of agricultural output drops by 18% over the period between 1978 and 2011 [3]. Secondly, for a long time, FDI mainly concentrates in secondary and tertiary industries, accelerating the restructuring and upgrading of China’s industries [4]. Finally, FDI provides investment capital and promotes the rapid development of China’s import and export trade, improving China’s status in international trade.
Due to the remarkable role of FDI, a multitude of scholars began to track and study the FDI in developing countries, build analytical framework, and launch a new field of research of FDI in developing countries. The statistics shows that China has become an emerging market for FDI. Dees indicates that FDI has positive effects on the GDP, technological progress, and the improvement of management system [5]. Nourzad considers that FDI promotes economy development through technology transfer [6], while Mah argues that the latter one promotes the former one [7]. Taking the reform policy (implemented in July 2005) as the boundary, Pan and Song explore the impact of the effective exchange rate of RMB on FDI [8]. Research shows that they are in a longterm equilibrium relationship before implementing reform policy. After the policy, the exchange rate of RMB has the Granger causality for FDI, and the appreciation of RMB can promote the flow of FDI. Additionally, De Mello shows that FDI can increase the added value associated with it [9]. Based on the data from 1971 to 2012, Dreher et al. conclude that the membership in international organizations is an essential and decisive factor of FDI liquidity and has a promoting effect on FDI mobility [10]. Badr and Ayed do a quantitative study of the relationship between FDI and economic development in South American countries, and they find that FDI can be determined by some economic factors, having no important effect on economic development [11]. Kathuria et al. apply panel data to examining the effectiveness of public policy in attracting FDI [12]. Lin et al. divide the FDI company into five strategies [13]; Brülhat and Schmidheiny estimate the rivalness of statelevel inward FDI [14].
The trend of FDI in the future is an important reference for China’s economic development. However, much literature focuses on the development of FDI itself and its influencing factors, and there is little research on the future development. This is what we do in this chapter. Currently, the predictive analysis model for economic and trade development can be divided into linear prediction method and nonlinear prediction one. The linear prediction method mainly includes historical average level prediction method, time series prediction method, and Kalman filter prediction method, to name just a few. The nonlinear prediction methods concern Gray theory, Markov chain, support vector machine, and boom prediction method. The historical average prediction algorithm is simple and easy to understand and the parameters can be estimated by using the least squares method. However, it is too simple to accurately reflect the randomness and nonlinearity, and therefore it cannot be applied to unexpected events. The Kalman filter uses the flexible recursive state space model, with the advantages of linear, unbiased, and minimum mean variance. Nevertheless, because the Kalman filter prediction model belongs to the linear model, its performance becomes worse in the nonlinearity and uncertainty [15]. The time series model is simple in modeling, with high prediction accuracy in the case of full historical data. The Gray model can be modeled with less information, handling data easily and having higher accuracy, which can be extensively used in several fields [15, 16, 17, 18]. However, Gray model becomes less attractive for time series with large stochastic fluctuation. Markov stochastic process predicts the development and changes of dynamic system according to the transfer probability of different states, and the transfer probability reflects the influence degrees of various stochastic factors and the internal law of the transition states. Therefore, it is more suitable to predict the problems with large stochastic fluctuation. What cannot be ignored is that Markov model requires data to meet the characteristics of no effect. Consequently, when using a simple model, it is very difficult to obtain a better prediction result, and the combination method becomes a popular method.
Through the vector autoregressive moving average (VARMA), Bhattacharya et al. compare and analyze the consumer price index sequence (CPI) and improve the forecasting accuracy [16]. The Gray model (proposed by a Chinese scholar, Professor Deng) and the Markov model (proposed by a Russian mathematician, Markov) have been combined very early, which is called GrayMarkov model (GMM). Based on the Gray prediction model, GMM is used to solve the inaccurate problems resulting from the large random fluctuation of the data and widely promoted in the fields of financial economy, agricultural economy, and resource and energy [17, 18, 19, 20]. On the basis of GM(1,1), Li et al. propose an improved GM(2,1) model [21]. Based on the model of GM(1,1) and Markov stochastic process and combining Taylor formula approximation method, Li et al. construct a model of TMCRGM(1,1) and verify its validity by the example of thermal power station in Japan [22].
The level of FDI in China is influenced by many factors such as fixed investment, laws and regulations, corporate culture, innovation ability, and financial market stability, among others. To clearly recognize and describe the role of FDI, the foreign investment system is abstracted as a Gray system with no physical prototype and incomplete information, which can be predicted with GM(1,1) model. Meanwhile, the FDI level in the previous year has no direct influence on that in the next year, in line with the noeffect characteristic of Markov stochastic process. On the basis of the previous study of GrayMarkov model, it is used to predict the tendency of FDI in China, addressing the shortcomings of the Gray model for the low precision of the data sample with large fluctuation and compensating for the limitation that the Markov model requires the data to have a smooth process. As a comparison, the time series prediction model is introduced to evaluate FDI. Then, the fitting results are compared to decide the optimal prediction model.
2. GrayMarkov model
GrayMarkov model is a forecasting method integrating the Gray theory with the Markov theory [17, 18, 19, 20, 21, 22, 23, 24, 25]. Firstly, GM(1,1) is constructed to obtain the predicted residual value. Then, the error state can be divided according to the residual values, and the error state can be obtained in light of the Markov prediction model. Then, based on the error state and transition matrix, the predicted sequence from GM(1,1) can be adjusted to obtain more precise predicting internals. The traditional GM(1,1) has its advantage in shortterm prediction, while it has a poor fitting effect in forecasting the longrange and fluctuating data series. And the benefit of Markov stochastic process is the prediction of the large data series with random volatility. GMM has been proposed by He to predict the yield of cocoon and oil tea in Zhejiang Province. Subsequently, this model is widely used in the prediction of transportation, air accidents, and rainfall. Accordingly, we use GMM to predict FDI of China [26, 27, 28].
2.1 Gray model
The Gray system theory, founded and developed by Chinese scholar Deng, extends the viewpoints and methods of general system theory, information theory, and cybernetics to the abstract system of society, economy, and ecology, incorporating the development of mathematical methods to develop the theory and method of Gray system. The modeling process is as follows.
(1) Raw series are
(2) To weaken the randomness of the original data, the accumulated generating series is derived:
(3) Based on the sequence of
(4) Then, whitened differential equation is obtained:
In Eq. (4)
and
By differentiating
(5) The whitened time response is as follows:
Reducing the sequence of
(6) Model testing
Model test is divided into residual test and Grayrelating test. Residual test is to obtain the difference between predicting value and the actual value. Firstly, the absolute residuals and relative residuals about
Then, below is the average value of relative residuals:
Given the value of
As shown in Eq. (12), Gray correlation degree measures the correlating coefficient between the original sequence and the reference sequence:
2.2 Markov model
Markov chain is proposed by Andrey Markov (1856–1922), and it is a discrete time stochastic process with Markov property in mathematics. Given the current knowledge and information, historical information has no impact on the future. To improve prediction accuracy, Markov model is used to handle the data obtained by GM(1,1). It is critical to divide state and build transition matrix.
2.2.1 Dividing states
To divide states, four rules are suggested to follow. Firstly, the partition state must have at least one true value in each state. Secondly, elements in a onestep transition matrix cannot be the same. Thirdly, the actual values must fall into one state. Finally, the state must pass Markov test. The numbers vary according to the original data. In this chapter, the overall level of FDI in China is on the rise while fluctuating in detail. Therefore, the level of FDI is a nonstable stochastic process. Taking the curve of
2.2.2 Transition matrix
Assuming that there are
2.2.3 The forecasting value
The eventual forecast is in the center of the Gray zone, which is denoted as
3. Time series model (TSM)
Burg suggests that recursive algorithm estimated by the AR(
3.1 Preliminary analysis of data and modeling identification
Time series prediction is a statistical method processing dynamic data, which is a random sequence arranged in chronological order or a set of ordered random variables defined in probabilistic space {
3.2 Parameter estimation
In order to fit the TSM, we need to estimate the autoregressive coefficient
3.3 Diagnostic test
The purpose of diagnostic test is to check and test the rationality of the model, including residual test, autocorrelation function of residual error and partial autocorrelation function test, and the significance test of parameters in the model.
3.4 Optimal model selection
Model recognition is only a preliminary selection of TSM. Considering the actual observed errors and statistical errors, several models are taken as candidate models. And the most common methods of selecting optimal models include Ftest method, criterion function method (AIC criterion, BIC criterion, SBC criterion).
4. Comparison of GMM and TSM
4.1 GMM predicting FDI of China
Take the FDI value of China over the period from 1990 to 2016 as the original data (unit, $100 million; data source, Ministry of Commerce of the PRC):
Based on Eq. (5) and using the software MATLAB, the least squares estimation (LSE) of FDI is as follows:
Based on Eq. (7), timeresponse function can be written as
Residual State  E1  E2  E3  E4  E5 

Meaning  Extremely underestimated  Underestimated  Reasonable  Overestimated  Extremely overestimated 
Range  [−0.17, −0.10]  [−0.10, 0.02]  [0.02, 0.07]  [0.07, 0.12]  [0.12, 0.83] 
Year  Original  Relative error of GM  State  Year  Original  Relative error of GM  State 

1990  34.87  0  E3  2001  468.78  0.0848  E4 
1991  43.66  0.8288  E5  2002  527.43  0.0397  E3 
1992  110.08  0.5974  E5  2003  535.05  0.0914  E4 
1993  275.15  0.0615  E3  2004  606.30  0.0398  E3 
1994  337.67  −0.0741  E2  2005  603.25  0.109  E4 
1995  375.21  −0.1131  E1  2006  630.21  0.1318  E4 
1996  417.26  −0.1545  E1  2007  747.68  0.0394  E3 
1997  452.57  −0.1679  E1  2008  923.95  −0.1071  E1 
1998  454.63  −0.0941  E1  2009  900.33  −0.0061  E2 
1999  403.19  0.095  E4  2010  1057.35  −0.102  E1 
2000  407.15  0.1477  E4  —  —  —  — 
According to the original FDI value over a period from 1990 to 2010 and the relative error of prediction value in GM(1,1), the transition matrixes of different steps
Based on the transition matrix, we can obtain the error state over a period from 2011 to 2016 (see Table 3). Taking the middle value of the error state to modify the prediction value of GM(1,1) model, then the modified value can be seen in Table 3. And
Year 






1990  34.87  0.0349  0  33.4752  −0.0471 
1991  43.66  0.2550  0.8288  127.5082  0.6739 
1992  110.08  0.2734  0.5974  136.7182  0.2332 
1993  275.15  0.2932  0.0615  281.4594  0.0173 
1994  337.67  0.3144  −0.0741  326.9385  −0.0328 
1995  375.21  0.3371  −0.1131  379.20437  0.0193 
1996  417.26  0.3614  −0.1545  406.5945  −0.0172 
1997  452.57  0.3875  −0.1679  435.9630  −0.0289 
1998  454.63  0.4155  −0.0941  467.4528  0.0360 
1999  403.19  0.4455  0.0950  392.0632  0.0000 
2000  407.15  0.4777  0.1477  420.3821  0.0582 
2001  468.78  0.5122  0.0848  450.7465  −0.0113 
2002  527.43  0.5492  0.0397  527.2409  −0.0056 
2003  535.05  0.5889  0.0914  518.2134  −0.0040 
2004  606.30  0.6314  0.0398  606.1574  −0.0055 
2005  603.25  0.6770  0.1090  595.7787  0.0154 
2006  630.21  0.7259  0.1318  638.8121  0.0407 
2007  747.68  0.7784  0.0394  747.2223  −0.0059 
2008  923.95  0.8346  −0.1071  938.8998  0.0246 
2009  900.33  0.8949  −0.0061  930.6540  0.0326 
2010  1057.35  0.9595  −0.1020  1079.4327  0.0291 
2011  1160.11  1.0288  −0.1276  1157.4006  0.0065 
2012  1117.20  1.1031  −0.0128  1241.0002  0.1077 
2013  1175.90  1.1828  0.0058  1330.6383  0.1241 
2014  1195.60  1.2682  0.0573  1116.0363  −0.0417 
2015  1262.70  1.3598  0.0714  1196.648  −0.0260 
2016  1260.00  1.4580  0.1358  1283.0826  0.0451 
In the light of Eqs. (9)–(11), the relative residual error of GM(1,1) and GMM is
4.2 TSM predicting FDI of China
Now we will build a TSM based on the FDI value of China over the period from 1990 to 2016, obtain the predicting data, compare the difference between the predicted data and the original date, and evaluate the accuracy of this model.
Figure 1 shows the changing tendency of FDI in China over the period between 1990 and 2016. The raw data series show the seasonal change and overall growth, but the data series are not stable. Through the seasonal difference method to process the data, the seasonal difference order of three was selected. After the differential processing, the data sequence has been stabilized, eliminating the growing trend (Figure 2).
We determine the order of TSM based on sample autocorrelation function and partial autocorrelation function. After the onestep delay, the sample autocorrelation function falls to a standard error of twice times and has the property of truncation. After the twostep delay, the sample partial autocorrelation function falls to a standard error of twice times and has the property of truncation.
In the light of the calculation of SAS software, now we compare the model of ARMA(2,1), AR(2), and MA(1) (see Tables 4 and 5).
Model  Parameter  Estimate  Pvalue  AIC  SBC 

AR(2)  MU  2.0711  <0.0001***  1.0603  4.5944 
AR1,1  1.5357  <0.0001***  
AR1,2  −0.5392  0.0102**  
MA(1)  MU  0.4676  0.0038***  28.7297  31.08558 
MA1,1  −0.7099  0.0001***  
ARMA(2,1)  MU  1.9380  <0.0001***  0.7062  5.4184 
MA1,1  −0.4940  0.1192  
AR1,1  1.2352  0.0015***  
AR1,2  −0.2358  0.5028 
To lag  6  12  18 

Chisquare  3.91  5.44  8.02 
Pr

0.4187  0.8599  0.9481 
Comparing the AIC and SBC values for ARMA(2,1), AR(2), and MA(1) models (see Table 4), we find the model MA(1) to be the most inferior. Considering the AIC and SBC criterion values of ARMA(2,1) and AR(2) and the significance of parameters, it is found that fitting effect of the AR(2) model is the best.
As shown in Table 5, the Pvalue (Pr
where
4.3 Comparison of prediction results of two models
4.3.1 Accuracy assessment
Regarding how to select the appropriate accuracy evaluation criteria, Yokuma and Armstrong [33] have done a survey of expert opinions. They think that accuracy, clear physical meaning, and being easy to implement can be the critical evaluation criteria [33]. Accordingly, three criteria are used to evaluate the accuracy of the prediction model.
4.3.2 Comparing predicted values with actual values
As shown in Table 6, the prediction accuracy of GMM has been improved manifestly compared with that in GM(1,1) model. Therefore, the forecasting value in GMM is closer to the actual level of China’s FDI. Then, from Figures 3 and 4, we can clearly see that GMM model has a better fitting effect than that in TSM.
Criterion  Mean squared error  Mean absolute error  Mean absolute percentage error 

Index 



GM(1,1)  7.3991e+03  66.0812  0.3117 
GMM  2.4731e+03  29.5558  0.1181 
Time series  1.6644e+04  85.6101  0.1448 
5. Empirical analysis of FDI in Chongqing and Beijing
From discussions above, it is found that GMM has higher prediction accuracy and better fitting effects than those of TSM of Chinese FDI level. To further verify the credibility of this result, we construct GMM and TSM based on the FDI level of Beijing (1990–2016) and Chongqing (1990–2015). The divided states involved in the GMM are shown in Table 7, and the transition matrixes of GMM associated with Beijing and Chongqing are denoted as
Area  Error state  E1  E2  E3  E4  E5 

Beijing  Range  [−0.47, −0.2]  [−0.2, −0.1]  [−0.1, 0.1]  [0.1, 0.28]  [0.28, 0.65] 
Chongqing  Range  [0, 0.28]  [0.28, 0.55]  [0.55, 0.75]  [0.75, 0.81]  [0.81,0.97] 
Year  Original value  GM(1,1)  GMM  TSM  To state  GR  MR 

1990  27,696  0.0277  0.0277  0.0277  E3  0  0 
1991  24,482  0.0693  0.0371  0.0245  E5  0.6466  0.3395 
1992  34,984  0.0780  0.0417  0.0364  E5  0.5514  0.1614 
1993  66,693  0.0878  0.0711  0.0311  E4  0.2401  0.0618 
1994  144,460  0.0988  0.1121  0.0863  E1  −0.4625  −0.2885 
1995  140,277  0.1112  0.1262  0.1342  E1  −0.2617  −0.1117 
1996  155,290  0.1251  0.1420  0.1969  E1  −0.2410  −0.0934 
1997  159,286  0.1408  0.1620  0.1514  E2  −0.1310  0.0165 
1998  216,800  0.1585  0.1799  0.2127  E1  −0.3677  −0.2050 
1999  197,525  0.1784  0.2052  0.2126  E2  −0.1071  0.0373 
2000  168,368  0.2008  0.1626  0.2680  E4  0.1615  −0.0352 
2001  176,818  0.2260  0.1831  0.1767  E4  0.2176  0.0341 
2002  172,464  0.2544  0.1361  0.2213  E5  0.3220  −0.2673 
2003  219,126  0.2863  0.2319  0.1890  E4  0.2346  0.0551 
2004  255,974  0.3222  0.2610  0.2560  E4  0.2056  0.0193 
2005  352,834  0.3627  0.3627  0.2878  E3  0.0271  0.0271 
2006  455,191  0.4082  0.4694  0.3979  E2  −0.1151  0.0303 
2007  506,572  0.4594  0.5283  0.5179  E2  −0.1026  0.0412 
2008  608,172  0.5171  0.5947  0.5870  E2  −0.1761  −0.0227 
2009  612,094  0.5820  0.5820  0.6851  E3  −0.0517  −0.0517 
2010  636,358  0.6550  0.6550  0.7277  E3  0.0285  0.0285 
2011  705,447  0.7373  0.8368  0.7195  E5  0.0431  0.1570 
2012  804,160  0.8298  0.6721  0.8222  E4  0.0309  −0.1964 
2013  852,418  0.9339  0.7565  0.9097  E4  0.0873  −0.1268 
2014  904,085  1.0512  1.0512  1.0009  E3  0.1399  0.1399 
2015  1,299,635  1.1831  1.3428  1.0292  E1  −0.0985  0.0322 
2016  1,302,858  1.3316  1.5114  1.4400  E1  0.0216  0.1380 
Year  Original value  GM(1,1)  GMM  TSM  To state  GR  MR 

1990  332  0.0033  0.0029  0.0003  E1  0  −0.1628 
1991  977  0.3159  0.0347  0.0010  E5  0.9691  0.7188 
1992  10,247  0.3647  0.1276  0.0029  E3  0.7190  0.1972 
1993  25,915  0.4210  0.2463  0.0846  E2  0.3844  −0.0523 
1994  44,953  0.4860  0.4180  0.0686  E1  0.0751  −0.0755 
1995  37,926  0.5611  0.3282  0.0842  E2  0.3241  −0.1554 
1996  21,878  0.6478  0.2267  0.0406  E3  0.6623  0.0350 
1997  38,466  0.7478  0.4375  0.0165  E2  0.4856  0.1207 
1998  43,107  0.8633  0.5050  0.0755  E2  0.5007  0.1465 
1999  23,893  0.9967  0.2193  0.0563  E4  0.7603  −0.0897 
2000  24,436  1.1506  0.2531  0.0181  E4  0.7876  0.0347 
2001  25,649  1.3284  0.2922  0.0296  E4  0.8069  0.1223 
2002  28,089  1.5335  0.1687  0.0329  E5  0.8168  −0.6651 
2003  31,112  1.7704  0.1947  0.0360  E5  0.8243  −0.5976 
2004  40,508  2.0439  0.4497  0.0417  E4  0.8018  0.0991 
2005  51,575  2.3596  0.5191  0.0599  E4  0.7814  0.0065 
2006  69,595  2.7241  0.9534  0.0776  E3  0.7445  0.2700 
2007  108,534  3.1448  1.1007  0.1059  E3  0.6549  0.0139 
2008  272,913  3.6306  3.1223  0.1930  E1  0.2483  0.1259 
2009  401,643  4.1914  3.6046  0.6941  E1  0.0417  −0.1143 
2010  304,264  4.8388  2.8307  0.6809  E2  0.3712  −0.0749 
2011  582,575  5.5862  3.2679  0.2878  E2  −0.0429  −0.7827 
2012  352,418  6.4490  3.7727  1.2269  E2  0.4535  0.0659 
2013  414,353  7.4452  4.3554  0.2769  E2  0.4435  0.0487 
2014  423,348  8.5952  1.8909  0.5837  E4  0.5075  −1.2388 
2015  377,183  9.9228  2.1830  0.5124  E4  0.6199  −0.7278 
2016  332  0.0033  0.0029  0.0003  E1  0  −0.1628 
Similar to Section 4.2, TSM of Beijing FDI can be modeled as MA (1):
where
TSM of Chongqing FDI can be modeled as ARMA(1,2,1):
where
Figure 5 (Figure 6) shows the difference between the original value and the predicting value in GrayMarkov model (time series model) of foreign direct investment in Beijing. It is apparent that the fitting effect of GMM is better than that of TSM. The similar conclusion can be drawn from Figures 7 and 8. Tables 9 and 10 show the predicting effect of GMM is better than that of TSM from the point of predicting errors and accuracy. There is no doubt that it is a good thing to predict accurately the foreign direct investment of the forthcoming 5 or 10 years for the domain specialists. Because if the predicting results is lower or higher than they expected, they could pay attention to seeking the critical factors and policy which have impacts on FDI and adjust them in advance.
Area  Index 




Beijing  GM(1,1)  3.3244e+09  4.7473e+04  0.2587 
GMM  4.3599e+09  3.9369e+04  0.1032  
TSM  5.4378e+09  4.6731e+04  0.1335  
Chongqing  GM(1,1)  3.9173e+10  1.3478e+05  2.9584 
GMM  5.8230e+09  3.4080e+04  0.2663  
TSM  4.4524e+10  1.0126e+05  0.6018 
6. Conclusions and future work
Our contributions are threefold. Firstly, comparing the predicting results of the GrayMarkov model and the time series model and the original value, respectively, we can find that the fitting effect of the former (GMM) is better than the latter (TSM) and so does its scientific and practical importance. Secondly, the predicting results of GMM show that the level of foreign investment in China has been increasing by years. Thirdly, in order to further enhance Chinese international status and attract more foreign investment, the government should play a role at a macro level to reduce excessive market administrative intervention, establish a serviceoriented government, and reduce the relevant approval procedures for international investment.
In the future work, the GrayMarkov model and time series model can be combined with other predicting model (e.g., support vector machine and dynamic Bayesian) to improve the accuracy. Also these models have the potential to be applied in the other areas such as finance (e.g., stocks, funds, and security), risk (e.g., financial risk and operational risk), and business (e.g., consumer price index and incomes).
Acknowledgments
This chapter is financially supported by the National Natural Science Foundation of China under grant nos. 71771080, 71172194, 71521061, 71790593, 71642006, 71473155, 71390335, and 71571065.
References
 1.
Choe JI. Do foreign direct investment and gross domestic investment promote economic growth. Review of Development Economics. 2003; 7 (1):4457  2.
Shu T, Liu CX, Chen S, Wang SY, Li JQ. The influence and demonstration of FDI on China’s economic growth in supply chain system. Systems Engineering  Theory & Practice. 2014; 34 (2):282290  3.
Wei KL. Foreign direct investment and economic growth in China. Beijing: Economic Science Press; 2013  4.
Pan FH, Wang JC. From “passive embedding” to supply Chain Park investment: A new model of FDI. China Soft Science. 2010; 25 (3):95102  5.
Dees S. Foreign direct investment in China: Determinants and effects. Economics of Planning. 1998; 31 (2):175194  6.
Nourzad F. Openness and Efficiency of FDI: A panel stochastic production frontier study. International Advances in Economic Research. 2008; 14 (1):2535  7.
Mah JS. Foreign direct investment inflows and economic growth of China. Economic Research Journal. 2002; 48 (3):6975  8.
Pan WR, Song Y. Empirical analysis for the impact of RMB real effective exchange rate on foreign direct investment in China. Journal of Chemical and Pharmaceutical Research. 2014; 6 (5):18301836  9.
De Mello LR Jr. Foreign direct in developing countries and growth: A selective survey. The Journal of Development Studies. 1997; 34 (1):134  10.
Dreher A, Mikosch H, Voigt S. Membership has its privilegesThe effect of membership in international organizations on FDI. World Development. 2015; 66 :346358  11.
Badr OM, Ayed TL. The mediator role of FDI in North Africa: Case of Egypt. 2015; 3 (1):17  12.
Kathuria V, Ray P, Bhangaonkar R. FDI (foreign direct investment) in wind energy sector in India: Testing the effectiveness of state policies using panel data. Energy. 2015; 80 :190202  13.
Lin HL, Hsiao YC, Lin ES. The choice between standard and nonstandard FDI production strategies for Taiwanese multinationals. Research Policy. 2015; 44 (1):283293  14.
Brülhart M, Schmidheiny K. Estimate the rivalness of statelevel inward FDI. Journal of Regional Science. 2015; 55 (1):139148  15.
Wang YB, Papageorgiou M. Realtime freeway traffic state estimation based on extended Kalman filter: A general approach. Transportation Research Part B. 2005; 39 (2):141167  16.
Bhattacharya PS, Thomakos DD. Forecasting industrylevel CPI and PPI inflation: Does exchange rate passthrough matter? International Journal of Forecasting. 2008; 24 (1):134150  17.
Li GD, Yamaguchi D, Nagai M. Application GM(1,1)Markov chain combined model to China automobile industry. International Journal of Industrial and Systems Engineering. 2007; 2 (3):327347  18.
Wang CB, Li LH. The forecast of gold price based on the GM(1,1) and Markov chain. Gray Systems and Intelligent Services. 2007; 18 (1):739743  19.
Lin CT, Lee IF. Artificial intelligence diagnosis algorithm for expanding a precision expert forecasting system. Expert Systems with Application. 2009; 36 (4):83858390  20.
Aries MB, Veitch JA, Newsham GR. Windows, view, and office characteristics predict physical and psychological discomfort. Journal of Environmental Psychology. 2010; 30 (4):533541  21.
Li GD, Masuda S, Yamaguchi D, Nagai M, Wang CH. An improved grey dynamic GM(2,1) model. International Journal of Computer Mathematics. 2010; 87 (7):16171629  22.
Li GD, Masuda S, Nagai M. The prediction model for electrical power system using an improved hybrid optimization model. Electrical Power and Energy Systems. 2013; 44 (1):981987  23.
Lin LC, Wu SY. Analyzing Taiwan IC assembly industry by GrayMarkov fore casting model. Mathematical Problems in Engineering. 2013; 70 (2):717718  24.
Chen YY, Liu CX. An improved GM(1,1)Markov model in supply chain disruption and its application in demand prediction. Information Technology Journal. 2014; 13 (13):22042210  25.
Dong S, Chi K, Zhang QY, Zhang XD. The application of a Gray Markov model to forecasting annual maximum water levels at hydrological stations. Journal of Ocean University of China. 2012; 11 (1):1317  26.
Syamala P, Vishnu B. Gray model for stream flow prediction. Aceh International Journal of Science and Technology. 2012; 1 (1):1419  27.
Sun W, Wang JM, Chang H. Forecasting carbon dioxide emissions in china using optimization grey model. Journal of Computers. 2013; 8 (1):97  28.
He Y, Bao YD. Gray Markov model and its application. Systems Engineering  Theory & Practice. 1992; 12 (4):5963  29.
Burg JP. Maximum Entropy Spectral Analysis. In: Proceedings of the 37th Meeting of the Society of Exploration Geophysicists, Oklahoma City, Oklahoma; 1967  30.
Wang ZX, Chan LW. Learning causal relations in multivariate time series data. ACM Transactions on Intelligent Systems and Technology. 2012; 3 (4):76104  31.
Kadri F, Harrou F, Chaabane S, Tahon C. Time series modelling and forecasting of emergency department overcrowding. Journal of Medical Systems. 2014; 38 (9):120  32.
Rivero CR, Pucheta J, Laboret S. Time series forecasting using bayesian method: Application to cumulative rainfall. IEEE Latin America Transactions. 2013; 11 (1):359364  33.
Yokum JT, Armstrong JS. Beyond accuracy: Comparison of criteria used to select forecasting methods. International Journal of Forecasting. 1995; 11 (4):591597