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):

$$X^{(0)} = \{\text{34.87, 43.66, 110.08, ..., 1260}\}$$

Based on Eq. (5) and using the software MATLAB, the least squares estimation (LSE) of FDI is as follows:

$$
\Phi = \begin{pmatrix} \hat{a} \\ \hat{b} \end{pmatrix} = \begin{pmatrix} -0.0697 \\ 243.795 \end{pmatrix},
$$

Based on Eq. (7), time-response function can be written as x k ^ð Þ¼ þ 1 <sup>3530</sup>:59e<sup>0</sup>:0697<sup>k</sup> � <sup>3495</sup>:72. Residual values can be obtained according to relative error based on the prediction value of GM(1,1) model. To improve the predicting accuracy, the relative error can be divided into five states (E1, E2, E3, E4, E5) between 1990 and 2010. The relative error status can be seen in Tables 2.

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 <sup>x</sup>ð Þ <sup>0</sup> ð Þ<sup>k</sup> , <sup>x</sup>^ð Þ <sup>0</sup> ð Þ<sup>k</sup> , and <sup>ϕ</sup>ð Þ<sup>i</sup> represent the original value, predicting value and

Using Gray-Markov Model and Time Series Model to Predict Foreign Direct Investment Trend…

In the light of Eqs. (9)–(11), the relative residual error of GM(1,1) and GMM is

:0584 , α ¼ 0:1 and 0:0458 , α ¼ 0:05, respectively. Therefore, the GM(1,1) model is barely qualified, and the modified GMM model is qualified. Gray correlation degrees of the two models are 67 and 79.9%, respectively. In summary, the

 34.87 0.0349 0 33.4752 �0.0471 43.66 0.2550 0.8288 127.5082 0.6739 110.08 0.2734 0.5974 136.7182 0.2332 275.15 0.2932 0.0615 281.4594 0.0173 337.67 0.3144 �0.0741 326.9385 �0.0328 375.21 0.3371 �0.1131 379.20437 0.0193 417.26 0.3614 �0.1545 406.5945 �0.0172 452.57 0.3875 �0.1679 435.9630 �0.0289 454.63 0.4155 �0.0941 467.4528 0.0360 403.19 0.4455 0.0950 392.0632 0.0000 407.15 0.4777 0.1477 420.3821 0.0582 468.78 0.5122 0.0848 450.7465 �0.0113 527.43 0.5492 0.0397 527.2409 �0.0056 535.05 0.5889 0.0914 518.2134 �0.0040 606.30 0.6314 0.0398 606.1574 �0.0055 603.25 0.6770 0.1090 595.7787 0.0154 630.21 0.7259 0.1318 638.8121 0.0407 747.68 0.7784 0.0394 747.2223 �0.0059 923.95 0.8346 �0.1071 938.8998 0.0246 900.33 0.8949 �0.0061 930.6540 0.0326 1057.35 0.9595 �0.1020 1079.4327 0.0291 1160.11 1.0288 �0.1276 1157.4006 0.0065 1117.20 1.1031 �0.0128 1241.0002 0.1077 1175.90 1.1828 0.0058 1330.6383 0.1241 1195.60 1.2682 0.0573 1116.0363 �0.0417 1262.70 1.3598 0.0714 1196.648 �0.0260 1260.00 1.4580 0.1358 1283.0826 0.0451

ð Þi represent the modified value and rela-

ð Þ <sup>0</sup> ð Þ<sup>k</sup> is 1 billion dollars. The unit of <sup>x</sup>^ð Þ <sup>0</sup> ð Þ<sup>k</sup> is 10<sup>3</sup> billion dollars. Data source:

ð Þ <sup>0</sup> ð Þ <sup>k</sup> <sup>ϕ</sup><sup>0</sup>

ð Þi

ð Þ <sup>0</sup> ð Þ<sup>k</sup> and <sup>ϕ</sup><sup>0</sup>

Year <sup>x</sup>ð Þ <sup>0</sup> ð Þ <sup>k</sup> <sup>x</sup>^ð Þ <sup>0</sup> ð Þ <sup>k</sup> <sup>ϕ</sup>ð Þ<sup>i</sup> <sup>x</sup>^<sup>0</sup>

relative error of GM(1,1). x^<sup>0</sup>

DOI: http://dx.doi.org/10.5772/intechopen.83801

Annotations: The unit of xð Þ <sup>0</sup> ð Þ<sup>k</sup> and <sup>x</sup>^<sup>0</sup>

Residual checklist of Markov model and GM(1,1).

China Statistical Yearbook.

Table 3.

tive error of GMM.

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 Pð Þ<sup>i</sup> ð Þ i ¼ 1; 2; 3; 4; 5 are shown as follows:


#### Table 1.

Relative error status of FDI level in China.


#### Table 2.

Comparison of GM(1,1) prediction value and original value of FDI of China.

Using Gray-Markov Model and Time Series Model to Predict Foreign Direct Investment Trend… DOI: http://dx.doi.org/10.5772/intechopen.83801

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 <sup>x</sup>ð Þ <sup>0</sup> ð Þ<sup>k</sup> , <sup>x</sup>^ð Þ <sup>0</sup> ð Þ<sup>k</sup> , and <sup>ϕ</sup>ð Þ<sup>i</sup> represent the original value, predicting value and relative error of GM(1,1). x^<sup>0</sup> ð Þ <sup>0</sup> ð Þ<sup>k</sup> and <sup>ϕ</sup><sup>0</sup> ð Þi represent the modified value and relative error of GMM.

In the light of Eqs. (9)–(11), the relative residual error of GM(1,1) and GMM is :0584 , α ¼ 0:1 and 0:0458 , α ¼ 0:05, respectively. Therefore, the GM(1,1) model is barely qualified, and the modified GMM model is qualified. Gray correlation degrees of the two models are 67 and 79.9%, respectively. In summary, the


Annotations: The unit of xð Þ <sup>0</sup> ð Þ<sup>k</sup> and <sup>x</sup>^<sup>0</sup> ð Þ <sup>0</sup> ð Þ<sup>k</sup> is 1 billion dollars. The unit of <sup>x</sup>^ð Þ <sup>0</sup> ð Þ<sup>k</sup> is 10<sup>3</sup> billion dollars. Data source: China Statistical Yearbook.

#### Table 3.

error based on the prediction value of GM(1,1) model. To improve the predicting accuracy, the relative error can be divided into five states (E1, E2, E3, E4, E5) between 1990 and 2010. The relative error status can be seen in Tables 2.

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

0 0

 

CCCCCCCCCCA , Pð Þ<sup>5</sup> ¼

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 —— — —

Data source: China Statistical Yearbook over the period from 2000 to 2006, Ministry of Commerce of the PRC.

Comparison of GM(1,1) prediction value and original value of FDI of China.

 

 

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 <sup>0</sup>

 

 

 

CCCCCCCCCCA

0 0 <sup>3</sup> 

 

   <sup>0</sup> CCCCCCCCCCCA ,

 <sup>0</sup>

<sup>000</sup>

overestimated

CCCCCCCCCCCCA , Pð Þ<sup>3</sup> ¼

0 0

 

 

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E1 E2 E3 E4 E5

 

Underestimated Reasonable Overestimated Extremely

 

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ð Þ i ¼ 1; 2; 3; 4; 5 are shown as follows:

Time Series Analysis - Data, Methods, and Applications

CCCCCCCCCCCA , Pð Þ<sup>2</sup> ¼

0 0

 

 

steps Pð Þ<sup>i</sup>

 

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Residual State

Table 1.

Table 2.

 

0 0

0 0

 

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Pð Þ <sup>4</sup> ¼

Meaning Extremely

Relative error status of FDI level in China.

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> 

BBBBBBBBBB@

underestimated

 0 0

   

Pð Þ<sup>1</sup> ¼

Residual checklist of Markov model and GM(1,1).

prediction accuracy of GMM has been improved, and its fitting effect exceeds the model of GM(1,1).
