A Predictive Analysis of China’s CO2 Emissions and OFDI with a Nonlinear Fractional-Order Grey Multivariable Model

: Since the implementation of the Belt and Road Initiative, China’s total amount of outward foreign direct investment (OFDI) has increased each year and this has caused its relationship with carbon emissions (CO 2 e) to receive great attention recently. Forecasting China’s CO 2 e accurately by considering the impact of OFDI is important since the government can use it to formulate an appropriate emissions plan to fulfill its carbon reduction commitments. Because the relationship between OFDI and CO 2 e has nonlinear characteristics, a nonlinear grey multivariable model with fractional-order accumulation (NFGM(1,N)) was proposed in this study. To enhance the prediction accuracy, an optimization process was used to determine the parameters. The outcomes of the variable fractional order showed that fractional-order accumulation can better extract the grey information hidden in the original data, which confirmed the principle of new information priority. The result of the power coefficient indicated a nonlinear relationship between the CO 2 e and OFDI. Based on the prediction performance, the prediction accuracy of the NFGM(1,N) model was proven to be superior to those of the ARMA model, linear regression model, the GM(1,1), GM(1,N), and FGM(1,N) models. The empirical results also revealed that OFDI increased the CO 2 e in China. The relationship between the CO 2 e and OFDI exhibits a U-shaped development based on further predictions for the 2018–2030 period. Finally, some suggestions for long-term CO 2 e reduction plans were provided in this paper.


Introduction
In recent years, excessive carbon dioxide emissions (CO2e) have caused a series of severe environmental problems, such as global warming, the melting of glaciers, and the frequent occurrence of extreme weather, and they also have had a harmful influence on human health. For those reasons, countries around the world have put their efforts into CO2e reduction and have committed to setting emissions reduction goals. China, as the largest developing country with the largest CO2e in the world [1], promised at the UN Climate Change Summit in 2009 (COP15) that the Among the existing studies, on the one hand, most of them concluded that the OFDI had an impact on reducing CO2e. Liu and Gong [25] and Jiang et al. [14] concluded that an increase in OFDI had a positive effect on emissions reduction. Nie and Liu [24] used provincial panel data to analyze the effect of OFDI on CO2e and found that although OFDI was beneficial to reduce the emissions, it was restricted by the threshold effect of urbanization. Long and Zhou [26] mentioned that OFDI had a positive effect on carbon productivity because of the reverse technology spillover effect. More specifically, they found that the carbon productivity increased by 0.37% for each 10% of the increases in the OFDI reverse technology spillover effect, but the effect was different in different regions. On the other hand, several studies hold the opposite view; that is, that OFDI would increase the CO2e in the home country. By using an applied panel threshold model, Yang and Sun [27] found that OFDI aggravated environmental pollution. Liu and Li [13] used provincial panel data to examine China's home country effect on CO2e and found that the per capita CO2e increases by 0.012% for each 1% increase in OFDI, but there were regional differences in this result.
The reasons for the uncertain relationship between OFDI and CO2e can be summarized as follows. First, based on Kojima's theory of marginal industry expansion, one country prefers to transfer its marginal industries with high energy consumption and emissions to the other country, which causes the CO2e to decline in the home country due to the emissions transference effect. Second, as Buckley et al. [28] stated, China's OFDI was motivated by resource seeking; thus, the resource-intensive industries were transferred by OFDI, which reduced the CO2e in the home country [25]. However, as stated in the study of Liu and Li [13], China's OFDI was not dominated by the industries with high emissions, so the emissions transference motivation was not clear. Third, owing to the reverse spillover effects from the resources and technologies backflow, the energy-related technical levels in the home country were enhanced, thereby reducing the CO2e [29,30]. However, the technological spillover effect on China's emission reduction will be varied because of the differences in economic development and absorptive capacity in different regions.
According to the reviews above, the general conclusion can be drawn that OFDI has a significant impact on CO2e, but the effect is uncertain. Furthermore, the positive or negative effect of OFDI on CO2e indicates that there is a nonlinear relationship between these two variables [31,32].

China's CO2e Forecasting Using Grey Model
The widely used prediction models can be roughly categorized into two groups: econometrics methods, including regression models and time series analyses, and intelligence computational technologies. However, knowing the sample distribution and having a large amount of data are the bases of accurate predictions with econometrics methods. Successful intelligence computational technology also needs a large amount of training data [17]. Under these circumstances, the grey model, which works well with an uncertain system, small sample size, poor information, and without the need to make any statistical assumptions, has drawn considerable attention. Grey models have been successfully applied in several disciplines and achieved great performance [20,[33][34][35].
A grey prediction model and the improved versions of it have often been applied to the issues related to China's CO2e forecasting. Overall, the grey models used can be divided into univariable and multivariable models; these are represented by the GM(1,1) and GM(1,N) model, respectively. To examine whether China can achieve its peak emissions commitment by 2030, Li et al. [36] applied the traditional GM(1,1) model to predict China's CO2e. Meng et al. [37] predicted the energy-related CO2 in China using a small-sample hybrid model that combined the GM(1,1) model with a linear model, and they estimated the parameters by using a non-constrained optimization equation. Considering the nonlinear feature of a sequence, Pao et al. [38] forecast China's CO2e by the nonlinear grey Bernoulli NGBM(1,1) model, and they used a combined iterative numerical method as an optimization method to improve the prediction accuracy. Wu et al. [39] applied a conformable fractional non-homogeneous grey model to predict the CO2e in the BRICS (Brazil, Russia, India, China, and South Africa) countries. However, because the GM(1,1) model only reflects the sequence response to time, many studies applied the GM(1,N) model to predict China's CO2e while considering the effects of relevant factors [40]. Wu et al.
[5] predicted the CO2e in the BRICS countries by using GM(1,N) with an opposite-direction accumulated generating operator. Wang and Ye [21], Wang and Li [23] applied a nonlinear grey multivariable model and derived the non-equigap grey Verhulst model to predict the CO2e in combination with optimization algorithms.
A review of the literature shows that the fractional-order accumulative generation operation has been combined with the grey model and achieved great prediction performance. The introduction of fractional order only changes the way the sequence accumulation generates and does not change the linear feature of the GM(1,N) model, so it is still inappropriate to reflect the nonlinear relationship between the system variables and relevant variables. Therefore, this study proposed the NFGM(1,N) model and introduced the power model to the FGM(1,N) model to predict China's CO2e by reflecting the nonlinear effect of OFDI.

Methodology
The

Nonlinear Fractional Grey Multivariable Model
The computational steps to construct an NFGM(1,N) model are demonstrated as follows.
Step 1: Present the data sequences. The system characteristics sequence Step 2: Perform the accumulated generating operation. The accumulative sequence ) can be obtained from X i (0) by the ri-order accumulated generating operation (ri-AGO) as follows: Based on the study of Wu et al. [41], r should be taken in the interval (0,1).
Step 3: Construct the NFGM(1,N) model. Drawing on the concept from the GM(1,1) model, a grey control parameter term u is added to the model. Therefore, the NFGM(1,N) model is given as where a, bi, and u are the coefficients of the modeling, which represent the development term, the driving coefficients, and grey control parameter, respectively. Moreover, ri and γi are the order of the fractional for different variables and the power, respectively. The NFGM(1,N) model reverts to the FGM(1,N) model when γi = 1, and it reverts to the traditional GM(1,N) model when ri = 1 and γi = 1.
Step 4: Estimate the coefficients. The parameters a, bi, and u can be estimated by using the OLS method where (4) where the background values of the system characteristics variable z 1 are calculated by an adjoining mean generated sequence as follows: Then, the time response function can be represented as Step 5: Perform the inverse accumulated generating operation. The r-order inverse fractional accumulation can be defined as Therefore, the predicted values X 1 (0) can be calculated by using the inverse ri-order accumulated generating operation (ri-IAGO) as follows: Step 6: Determine the parameters. From the modeling process, once the parameters ri and γi are determined, the other parameters a, bi, and u are ascertained. The objective of the optimal value of ri and γi should be to make the proposed model have the highest accuracy with the given sample. Therefore, this study establishes an optimization problem where the objective is to minimize the error of the proposed model by changing the values of ri and γi, and the constraints follow the modeling steps of the proposed model. The mean absolute percentage error (MAPE) is applied to assess the prediction performance. In this study, from the perspective of improving the prediction accuracy, taking the minimum MAPE of the modeling as the target, and the relationship between the parameters of the model as a constraint, the non-linear optimization processes are built as follows: The above model can be solved by numerical software programing [42].

An Illustrative Example
To compare the modeling and prediction accuracy, a numerical example from Zeng's [43] work is used to verify the validation of the proposed NFGM(1,N) model. The system characteristics behavior sequence is X1 (0) = (11.4918, 12 . The backward data in the sequence represent the newer information. The first eight data are used for model fitting, and the ninth datum is used for ex-post testing. The econometric model, including the autoregressive moving average (ARMA) model and the linear regression model, GM(1,1), GM(1,N), FGM(1,N), and NFGM(1,N) models are applied to the numerical example. However, because the system characteristics behavior sequence fails to pass unit root test, which means that the data is nonstationary, the sequence cannot be regressed by the ARMA model. The prediction results of the remaining models are shown in Table 1. The MAPEs of the linear regression, GM(1,1), GM(1,N), FGM(1,N), and NFGM(1,N) models in relation to model fitting are 2.91%, 5.51%, 1.51%, 1.10%, and 0.64%, respectively. The MAPEs of the ex-post testing are 8.63%, 15.87%, 3.67%, 9.60%, and 2.24%, respectively. As can be seen from Table 1, the prediction accuracy of the NFGM(1,N) model is much better than that of the other models, whether for model fitting or ex-post testing. Figure 1 shows the superiority of the prediction accuracy of the proposed NFGM(1,N) model over the linear regression model, GM(1,1), GM(1,N), and FGM(1,N) models. Therefore, the proposed NFGM(1,N) model performs well in this numerical example compared with the other models.

Data Description
Because the prediction accuracy of the proposed NFGM(1,N) model for the numerical example has been proved to be better than that of the other models, it is applied to forecast China's CO2e by considering the effect of OFDI. In this study, the NFGM(1,N) model is constructed for the CO2e as a system characteristic behavior sequence and the OFDI as a relevant variable sequence. More specifically, based on the commitments made by at the UN Climate Change Summit, the CO2e per unit of GDP in China is chosen to represent the CO2e, and the total amount of OFDI is selected to represent the OFDI. The raw data, shown in Table 2, were obtained from the International Energy Association (https://www.iea.org) and the Development Indicators from the World Bank (https://data.worldbank.org). The data from 2005 to 2014 are reserved for model fitting, and the data from 2015 to 2017 are used for ex-post testing.

Empirical Results
Following the modeling process mentioned above, the NFGM(1,N) model can be formulated as follows: where the optimal fractional orders for the CO2e and OFDI, and the optimal value of the power are 0.01, 0.3, and 0.91, respectively. The values of the fractional order indicate that giving greater weight to new information improves prediction accuracy, which is also consistent with the principle of new information priority. As explained by Wu et al. [17] and Ding et al. [4], the driving coefficient bi can be used to interpret the relationship between the system characteristic behavior and relevant factors. In this study, the driving coefficient is b2 = 1.04 × 10 −4 , which indicates that there is a positive relationship between the CO2e and OFDI. In other words, the CO2e increase with increasing OFDI.
To examine the prediction accuracy of the proposed NFGM(1,N) model, this model is compared with the ARMA model, linear regression model, GM(1,1), GM(1,N), and FGM(1,N) models. The actual and predicted values of the five competing models are displayed in Table 3.  N) is much better than that of the other competing models for ex-post testing. The prediction precision of different models is illustrated in Figure 2. The above results confirm that the proposed NFGM(1,N) model has good practicability, and it can better predict China's CO2e by considering the effect of OFDI than the other models. Therefore, it has been applied to forecast the CO2e from 2018 to 2030. The predicted amount of OFDI from 2018 to 2030 is generated by the GM(1,1) model. The predicted value of the relevant factor from 2018 to 2030 is then substituted into the NFGM(1,N) model, which thereby enables China's CO2e predictions, after considering the effect of OFDI, to be obtained correspondingly. The predicted OFDI and CO2e are shown in Table 4. It is noteworthy that China's carbon emissions will decrease initially, and then climb. Additionally, the prediction of China's CO2e in 2020 and 2030 are 0.64 and 0.85, respectively, which represent reductions of 72.83% and 63.97%, respectively, compared with that of 2005. The calculation results show that through unremitting efforts, China can fulfill the CO2e reduction commitments.  Figure 3 is an intuitive display of the prediction results of CO2e for the model fitting, ex-post, and out-sample forecasting. From Figure 3 and Table 4, the data sequence of the prediction results indicates that there is a U-shaped relationship between CO2e and OFDIs.

Discussion and Suggestions
To determine whether China can deliver on its carbon reduction commitments, this study proposed an NFGM(1,N) model to forecast China's CO2e by considering the nonlinear relationship with OFDI. This model was then applied to forecast the CO2e in China from 2018 to 2030. Several conclusions can be obtained. First, based on the driving coefficient of the NFGM(1,N) model, the increasing OFDI increases the CO2e in China, which is consistent with the previous studies of Yang and Sun [27] and Liu and Li [13]. As far as this relationship is concerned, the carbon emission transferring effect and technology spillover effect of OFDI are not so clear. The reason for this result may be that, on the one hand, China's OFDI mainly focuses on low energy consumption industries such as the service industry; thus, industries with high emissions remain in the home country. On the other hand, China's current OFDI is mainly motivated by resource seeking, rather than technology seeking, so the influence on technological upgrading in the home country is insignificant. Second, the relationship between the CO2e and OFDI is U-shaped from the perspective view of prediction. According to the forecasts, although China can achieve the emission reduction goal in 2020 and 2030, due to the long-term possibility of emissions increasing, China still faces great pressure to reduce the CO2e.
Based on the forecast results, China's CO2e appear to be on an upward trajectory in the future. In the face of this less-than-optimistic situation, appropriate policies and suggestions are proposed in the study. First, the government should encourage OFDI with technology-seeking motivation. Although rising OFDI could lead to higher carbon emissions in the future, the reverse spillover effect on technological upgrading that has been confirmed by many studies should not be overlooked. Therefore, the government should encourage OFDI with a technology-seeking orientation. In particular, OFDI to the countries with advanced technologies should be encouraged, which can enable domestic enterprises to absorb more advanced knowledge and technology and promote domestic carbon productivity. Second, the government should formulate long-term CO2e reduction plans. It can be seen from the relationship that the conflict between OFDI and CO2e cannot be easily resolved in the short term. As an important component of the "opening-up" strategy, in one respect, access to resources, advanced technology, and markets through OFDI can be important drivers of domestic economic development. In another respect, although it promotes economic development, OFDI will cause an increase in carbon emissions. In the long run, the low-carbon economy should be regarded as the strategic goal of any development in China. Although China's CO2e reduction commitments can be achieved, formulating appropriate long-term plans for OFDI and CO2e reduction are still needed to promote sustainable economic development.

Conclusions
China, as the largest developing country, has been plagued by the problem of excessive CO2e for many years. Whether China can fulfill its commitments to emission reduction has become a global focus. Since the Belt and Road Initiative was put forward, OFDI has been increasing, and this has become an important part of the opening-up strategy. Meanwhile, due to the increasing emphasis on environmental protection, the issues related to the impact of OFDI on CO2e have received general attention. Therefore, this study proposed an NFGM(1,N) model to accurately predict China's CO2e by considering the nonlinear effect of OFDI. The results showed that the prediction accuracy of the proposed NFGM(1,N) model was superior to the other competing prediction models.
The NFGM(1,N) model was applied to the prediction of China's CO2e for the 2018-2030 period because it has the smallest rate of error. According to the forecasts, China's CO2e will decrease by 72.83% and 63.97% in 2020 and 2030, respectively, compared to 2005. Moreover, the prediction of China's CO2e presents a U-shaped trend in the long run, which indicates that China still faces numerous pressures related to CO2e reduction. Therefore, China's government should encourage OFDI motivated by technology seeking, maximize the reverse spillover effects on home country technologies, and develop long-term plans to further reduce CO2e and to promote sustainable economic development.
All the empirical results showed that the proposed NFGM(1,N) model had a higher prediction accuracy than the other models and was an appropriate tool to forecast China's CO2e. However, based on the empirical results, for the year 2008, the difference between actual value and predicted value in NFGM(1,N) model is much larger than other year. The larger error may be due to the impact on the global economy of the sudden global financial crisis in 2008. Furthermore, COVID-19 swept the world at the end of 2019 and the beginning of 2020, which disastrously impacts the global economy, and more or less affects China's OFDI and CO2e in 2020, resulting in a deviation between the predicted value and the real value. The influence of shock disturbances has always been a huge challenge for any prediction model [44]. Therefore, in future studies, the prediction model will be further improved to reduce the influence of shock disturbances, such as global financial crisis and COVID-19, on the predicted results. Additionally, different optimization algorithms, including neural networks, PSO, and others can be introduced into the NFGM(1,N) model to determine the optimal parameters. Furthermore, some approaches, such as the GM(1,1) model, Fourier series, and Markov chain, are available to combine with the NFGM(1,N) as the residual modification model to further improve the prediction accuracy. Finally, the other relevant factors related to CO2e should be further analyzed. (1) (k) + az The adjoining mean generated sequence of is called the background values of the system characteristics variable.