In Equation (26), the meanings of each parameter are the same as the preceding equation. The results show that labor, capital, total factor productivity, and CO2 emissions all affect the potential economic growth rate. Therefore, scenario analysis is adopted to predict the numerical changes of the potential economic growth rate of Hebei Province under the above factors and carbon emission control according to Equation (26).

3.3.1. Prediction Analysis on Changes of the Growth Rate of Various Factors

In Equation (27),

ln
C
,
ln
?

—Logarithmic sequence of CO2 emission and carbon intensity. When the residual sequence of the regression equation is modified, the variable ARMA (1, 1) is introduced to eliminate the autocorrelation of the residual, and the final fitting degree of the model is as high as 0.99.

Thus, in the future, the elasticity of

ln
?

relative to

ln
C

is 1.022. Then, the annual decline rate of CO2 emission must reach 4.2% to meet the requirements, while the annual decline rate of carbon intensity is 4.1%.

In Equation (28), Tr—Constant price of new investment in Hebei Province every year; Ta—Constant price of fixed investment in 2020;
?
—Actual year-on-year growth rate. Therefore, as long as the actual year-on-year growth rate of Hebei Province is predicted, the growth rate of capital stock can be estimated.

Given the current economic situation in Hebei Province, the role of capital in promoting economic development is becoming even smaller, and the capital formation rate will continue to decline. Therefore, three scenarios, optimism, benchmark, and pessimism, are defined, which are evaluated based on 8.5% (growth rate of capital stock) in 2020. In the optimistic scenario, the actual growth rate of capital stock will remain unchanged. In the benchmark scenario, the growth rate of capital stock will continue to decline at the rate of 0.2% per year. In the pessimistic scenario the real growth rate of capital stock will decline at an annual rate of 0.5%.

Third, the growth rate of labor input is predicted. Here, a polynomial regression with time series is employed to predict the potential employment of Hebei Province and calculate the growth rate of labor input.

Fourth, the growth rate of trend total factor productivity is predicted. Through calculation, the average total factor productivity in Hebei Province from 1999 to 2020 is 1.7%. Although it decreased in 2008, the overall trend is upward. Under the new economic form of Hebei Province, the main economic driving force of this province, namely capital investment, will be replaced by innovation. The constant exploration of reform and innovation plays a major role in developing total factor productivity in Hebei Province. The trend TFP growth rate scenarios are also defined as optimism, benchmark, and pessimism. Similarly, according to the 2.4% (TFP growth rate) in 2020, the growth rate of TFP will continue to rise at the rate of 0.5% in an optimistic scenario. In the benchmark scenario, it will continue to rise at a rate of 0.2%. In a pessimistic scenario, the trend TFP growth rate will be consistent with that in 2020.

Fifth, the change in output elasticity of each factor is predicted.

I
B

,

I
A

,

I

C

O
2

—output elasticity of labor, capital, and CO2.

3.3.2. Prediction

To sum up, from the optimistic situation, the overall trend of potential growth rate in Hebei Province is not large. The average growth rate of the potential economy is 6.414%, while the potential growth rate in 2023 is the smallest, only 6.093%. In the benchmark scenario, Hebei’s potential growth rate is on a downward trend, falling to 5.739% by 2024. Under the pessimistic scenario, Hebei’s four-year average potential growth rate is 5.312%. This conclusion indicates that although there is some uncertainty in the future economic growth prospects of Hebei Province, it shows a gradual slowing trend on the whole. This may be related to a variety of factors, such as changes in domestic and foreign economic environment, industrial structure adjustment, and population aging. At the same time, the difference of potential growth rate in different situations also reflects the impact of different policies and market environments on economic growth. These results provide an important reference for economic planning and policy making in Hebei Province. Policymakers can formulate corresponding economic policies and development strategies according to the changing trend of potential growth rate under different circumstances, to promote the long-term and stable development of Hebei’s economy. Furthermore, these results also offer beneficial scope for academic research in related fields, and provide theoretical support for in-depth study of China’s economic growth. Compared with the study of Wang et al. (2022), this work provides a more comprehensive research scope and results, and adopts more advanced research methods to afford more advanced technologies for future development.