The convergence level and influencing factors of China’s digital economy and real economy based on grey model and PLS- SEM

Abstract

The paper constructs a research model mainly based on the Deng’s correlation analysis model on the convergence level measurement, the GM (1,1) coordinated development prediction model and PLS-Structural Equation Model (PLS-SEM) analysis model on the influencing factors. The data about China’s digital economy and real economy from 2005 to 2019 (totaled 2,250) is adopted to conduct an empirical analysis of the convergence level from 2005 to 2019 and predict the development trend from 2020 to 2029. The paper could further analyze the influencing factors of convergence, in an attempt to put forward relevant development suggestions. We hope the study could provide an objective reference and theoretical basis for improving the convergence level in China in some extent.

Keywords

Grey model PLS-structural equation model convergence of digital economy and real economy influencing factors

1 Introduction

Since the 21st century, the digital technology has entered a new round of accelerated development. The 5 G, Big Data, Artificial Intelligence, Internet of Things, Cloud Computing, Blockchain and other technologies have reconstructed the material foundation of the Information Age as well as could be found widely into various industries. The world has entered a new era characterized by emerging smart, data-driven and learning economy [1]. In general, it is considered that the digital economy is a series of economic activities with digital knowledge and information as the key production factors, modern information network as a crucial carrier and the effective use of digital technology as an important driving force for efficiency improvement and economic structure optimization [2 –5]. For the real economy, the main advantages are relatively mature technical capabilities, product systems and industrial foundations while the disadvantages comparing with the digital economy mainly include the lack of innovation capabilities and new production factors(e.g. data), which need further optimization and improvement on the transaction costs and operational efficiency. Meanwhile, the main advantage of the digital economy is that it could extend the industrial boundaries through network effects, platform effects, economies of scope and scale effects as well as the data in the digital economy could continuously drive the innovation of technology. However the disadvantage is mainly that it has strong asset-light attributes and has the risks of bubblization. Generally speaking, The digital economy has subverted the original logic of cost, price and quantity, creating an associated profit source beyond the products, which brings the economy of scope to the extreme as well as new development opportunities for the real economy [6].

Although academia has done more research on issues related to inter-industry convergence, there are relatively few studies on the convergence of the digital economy and real economy. Most scholars have mainly carried out research on the convergence of specific industries in the context of the digital economy [7 –12]. For example, Cao studied the convergence process of different industries within the manufacturing industry as well as the influences of the convergence on the industrial structure and performance [13]; Yang and Liu analyzed the basis and process of the integration between manufacturing industry and productive service industry [14]; Wen and Chen explored the path of the digital economy to promote the convergence of rural industries [15]. Furthermore, some scholars have focused on the research on the relationship between the digital economy and the real economy in recent years. For example, Zhang et al. put forward some policy suggestions to tackle the problems in the course of China’s digital economy and real economy deeply integrated, which was based on the analysis of Zhejiang province’s digital economy development experiences combined with a large quantity of interviews including enterprises and governments [16]. Prokopenko et al. discussed the possibility of digital technology to improve the efficiency of business processes as well as the advantages of the digital transformation of business from the perspective of enterprises [17]. Ma and Ning applied the mediating effect model to investigate the effect of digital economy on the quality upgrading of manufacturing industry and found that digital economy could significantly promote the quality upgrading of manufacturing industry [18]. Jiang and Sun utilized the weighted least squares,quantile regression and cross-secction threshold regression to demonstrate the overall effect, conditionality and stage characteristics of the digital economy’s impact on the real economy, which was found that it presented the characteristics of first “promotion effect” and then “crowding out effect” [19]. Xu et al. used the adjusted Logistic model to conduct an empirical study on the internal mechanism of the convergence between China’s digital economy and real economy from 2005 to 2019 as well as predict the overall trend of the co-evoluation under seven evolution modes [20].

It could identified that the mechanism of the co-evolution of China’s digital economy and the real economy has been already explored in some extent, however, there is a lack of objective and reliable empirical research on the convergence level and influencing factors of the two economies. Therefore, we attempt to conduct research in the following aspects: observing the the convergence level from 2005 to 2019 to find the convergence law; predicting the co-evolution trend of the two economies in the next 10 years; and studying the influencing factors that affect the level of co-evolution. Consequently, it could possibly make up for the insufficiency of existing researches through a quantitative analysis method to explore the convergence level and the influencing factors. Moreover, it may provide some decision-making references for government departments at all levels when they formulating digital economy policies to effectively promote the coordinated development of the digital economy and real economy. Besides, the research process is shown in the following Fig. 1.

Fig. 1

Flow chart of the research design.

2 Literature review

2.1 Connotation of industrial convergence

Technical perspective: Industrial convergence first began with technology convergence. Rosenberg found that there are similar metal processing procedures between the machinery and metal-using sectors of an industrial economy. Although the finished products of these industries were different, they were closely related from the perspective of technical foundation as well as there were a lot of shared production process and problems, which was defined as technological convergence [21]. Moreover, technology convergence is also the process of the widespread application and diffusion of the technologies, leading to the occurrence of innovation [22]. When the technological convergence of different industries significantly affects or changes the nature of the product, competition and value creation process in another industry, it means that technological convergence occurs [23]. And some scholars define the convergence as a substitution process in which new technologies continuously substitute existing technologies [24].

Enterprise perspective: The industry convergence occurs when the firms in two previously distinct industries become direct competitors [25]. However, if the only standard to measure whether an industry convergence occurs is the changes in firms competition or cooperative relations, the complementary effects of firms in the industries are often ignored. For instance, sharing economy platforms, such as Uber and Airbnb, provide renting or servicing opportunities for scattered or idle vehicles, housing and other assets. They compete with traditional taxi companies and hotels. Meanwhile, the differentiated characteristics of the services of sharing economy platforms are complementary to traditional enterprises (for example, carpooling and ride-hailing provided by Uber effectively alleviate the shortage of taxis during the rush hours).

Market perspective: Some believed that convergence is a confluence and merging of separated markets, removing entry barriers across the market and industry boundaries [26]. In addition, the meaning of industrial convergence from a larger scale has been explored, such as: convergence analysis on industry boundary [27], industry category [28] and modular theory [29].

2.2 Types of industrial convergence

Technical perspective: The principal causes for technological convergence can be distinguished as technological substitutes and technological complements [30]. For instance, Rosenberg mentioned the application of mechanical skills like drilling, grinding, polishing, etc., which replaced the firearms, sewing machines and bicycles, promoting the emergence of machine tool industry [21].

Product perspective: Industrial convergence from the perspective of product could be divided into three types: alternative convergence, complementary convergence and combined convergence [31]. When the user believes that two products can be substituted for each other, the products undergo alternative convergence.The two products that are used together are more effective than them when used separately, then they undergo complementary convergence. When the functions achieve mutual penetration to converge into a new product, the two products undergo combined convergence.

Market perspective: The industrial convergence can be divided into functional convergence from the demand side, institutional convergence from the supply side, high-functional and high-institutional convergence(pure convergence), high-functional and low-institutional convergence (demand-driven convergence), low-functional and high-institutional convergence (supply-driven convergence) [25].

2.3 Summary of theoretical research

2.3.1 Grey relational analysis

Grey relational analysis (GRA) is an important part of Grey system theory, which is suitable for solving problems with complicated interrelationships between multiple factors and variables [32]. Compared with regression analysis, variance analysis, principal component analysis and other methods, GRA is suitable for small sample and irregular sample analysis. It has been widely used in industry, economy, management and other fields [33 –36]. Furthermore, the major advantage of GRA is that it can handle both incomplete information and unclear problems very precisely [37]. In recent years, Deng’s correlation model has been widely used in the disease control, decision making of asset allocation and analysis of influential factors of housing prices [38 –40].This paper focuses on Deng’s correlation model of GRA:

Hypothesis 1: The reference sequence of the system is: $X_{0} k = (x_{0} 1, x_{0} 2, \dots, x_{0} n)$

Hypothesis 2: The compared sequence of the system is: $X_{i} k = (x_{i} 1, x_{i} 2, \dots, x_{i} n), i = 1, 2, \dots m$

In view of the above assumptions, Deng’s correlation analysis model is as follows:

Step 1: Selecting a method to perform non-dimensional processing on X₀k and X_ik. In general, the most frequently used methods are initialization, mean generating and intervalization. This paper will focus on sorting out the methods, specifically:

Initialization: ${\begin{matrix} X_{0}^{'} k = (x_{0}^{'} 1, x_{0}^{'} 2, \dots, x_{0}^{'} n) \\ X_{i}^{'} k = (x_{i}^{'} 1, x_{i}^{'} 2, \dots, x_{i}^{'} n) \end{matrix}$ (1) in which $x_{0}^{'} k = \frac{x_{0} k}{x_{0} 1}, x_{i}^{'} k = \frac{x_{i} k}{x_{i} 1}; x_{i} 1 \neg = 0,$ x₀1 ¬ =0, k = 1, 2, …, n

Mean generating: ${\begin{matrix} X_{0}^{'} k = (x_{0}^{'} 1, x_{0}^{'} 2, \dots, x_{0}^{'} n) \\ X_{i}^{'} k = (x_{i}^{'} 1, x_{i}^{'} 2, \dots, x_{i}^{'} n) \end{matrix}$ (2) in which $x_{0}^{'} k = \frac{x_{0} k}{\bar{X_{0}}}, x_{i}^{'} k = \frac{x_{i} k}{\bar{X_{i}}}; \bar{X_{0}} = \frac{1}{n} \sum_{j = 1}^{n} x_{0} j, \bar{X_{i}} = \frac{1}{n} \sum_{j = 1}^{n} x_{i} j$ ; k = 1, 2, …, n

Intervalization: ${\begin{matrix} X_{0}^{'} k = (x_{0}^{'} 1, x_{0}^{'} 2, \dots, x_{0}^{'} n) \\ X_{i}^{'} k = (x_{i}^{'} 1, x_{i}^{'} 2, \dots, x_{i}^{'} n) \end{matrix}$ (3) in which $x_{0}^{'} k = \frac{x_{0} k - min_{k} x_{0} k}{max_{k} x_{0} k - min_{k} x_{0} k}, x_{i}^{'} k = \frac{x_{i} k - min_{k} x_{i} k}{max_{k} x_{i} k - min_{k} x_{i} k}; k = 1, 2, \dots, n$

Step 2: calculating the correlation coefficient according to the non-dimensional sequences of $X_{0}^{'} k$ and $X_{i}^{'} k$ obtained in the step 1 [35]:

$\begin{matrix} ξ_{0 i} (k) = \\ \frac{min_{i} min_{k} | X_{0}^{'} k - X_{i}^{'} k | + ρ max_{i} max_{k} | X_{0}^{'} k - X_{i}^{'} k |}{| X_{0}^{'} k - X_{i}^{'} k | - ρ max_{i} max_{k} | X_{0}^{'} k - X_{i}^{'} k |} \end{matrix}$ (4)ρ is the resolution coefficient, which is usually taken as ρ = 0.5.

Step 3: calculating the overall correlation between the sequences: $γ (X_{0}, X_{i}) = \frac{1}{n} \sum_{j = 1}^{n} ξ_{0 i} (j)$ (5)

Deng’s correlation model focuses on holistic analysis. When the system has singular values, the resolution coefficient could be smaller to weaken the influence of singular values.

2.3.2 GM (1,1) grey forecasting model

Grey Model (GM) is mainly used for predictive analysis of Grey systems that contain known and unknown information, in which the general expression is GM (X, N). GM (X, N) means to use X-order differential equations to model and analyze N variables. Furthermore, the most widely used Grey scale analysis model is GM (1,1) on basis of the first-order linear differential equation. After initial processing is performed by accumulated generating operation (AGO), inverse accumulated generating operation (IAGO), mean generating and class ratio generating of discrete data scattered on the time axis, which could weaken the influence of unknown factors in the Grey system and strengthen the influence of known factors. It is suitable for studying uncertain problem with less data and poor information [41 –43]. This paper will focus on summarizing the GM (1, 1) modeling to provide a theoretical basis for the subsequent analysis.

Assuming x⁽⁰⁾ = (x⁽⁰⁾ (1) , x⁽⁰⁾ (2) , …, x⁽⁰⁾ (n)) is the original sequence to be predicted, the GM(1,1) modeling process is as follows:

1-AGO:

Supposing x⁽¹⁾ is an accumulated generation sequence of x⁽⁰⁾, and z⁽¹⁾ is the generated mean sequence of x⁽¹⁾, while z⁽¹⁾ (k) is the mean value of adjacent data, namely: $x^{(1)} (k) = \sum_{j = 1}^{k} x^{(0)} (j), (k = 1, 2, \dots, n)$ (6)

$\begin{matrix} z^{(1)} (k) = \\ 0.5 x^{(1)} (k) + 0.5 x^{(1)} (k - 1), (k = 2, 3, \dots, n) \end{matrix}$ (7)

in which z⁽¹⁾ = (z⁽¹⁾ (2) , z⁽¹⁾ (3) , …, z⁽¹⁾ (n)).

Establishing GM(1,1) first-order linear differential equation: $x^{(0)} (k) + {az}^{(1)} (k) = b_$ (8)

Constructing data matrix B and Y: $B = [\begin{matrix} - z^{(1)} (2) & 1 \\ \begin{matrix} - z^{(1)} (3) \\ \begin{matrix} ⋮ \\ - z^{(1)} (n) \end{matrix} \end{matrix} & \begin{matrix} 1 \\ \begin{matrix} ⋮ \\ 1 \end{matrix} \end{matrix} \end{matrix}], Y = [\begin{matrix} \begin{matrix} x^{(0)} (2) \\ x^{(0)} (3) \end{matrix} \\ ⋮ \\ x^{(0)} (n) \end{matrix}]$ (9)

Finding sequence of parameters μ (μ = (a, b) ^T)

The GM (1, 1) equation can be written as Y = Bμ and the estimated value of μ can be solved using the least square method: $\hat{μ} = {(B^{T} B)}^{- 1} B^{T} Y$ (10)

Evaluating the whitening response of GM (1, 1)

First, the whitening equation of GM (1,1) is $\frac{{dx}^{(1)}}{dt} + {ax}^{(1)} = b$ (11)

Then, substituting $\hat{μ}$ into Equation (11), then ${\hat{x}}^{(1)} (k + 1) = (x^{(1)} (1) - \frac{b}{a}) e^{- ak} + \frac{b}{a}$ (12)

We have ${\hat{x}}^{(0)} (k + 1) = {\hat{x}}^{(1)} (k + 1) - {\hat{x}}^{(1)} (k)$ (13)

Checking model accuracy

In order to determine whether we need a residual error model to modify the GM(1,1) model or use other model, we adopt two most frequently used methods to test accuracy of the prediction model, i.e., residual error checking and after-test residual checking (i.e., Variance ratio test and Error probability test), as described below [44, 45]:

A. Residual error checking ( $\bar{φ}$

Supposing $e (k) = | x^{(0)} (k) - {\hat{x}}^{(0)} (k) |, k = 1, 2, \dots n$ ; then the residual sequence between the predicted value and the actual value is: $φ (k) = \frac{e (k)}{x^{(0)} (k)}, k = 1, 2, \dots, n$ (14)

Then, the average relative residual value is: $\bar{φ} = \frac{1}{n} \sum_{k = 1}^{n} φ (k)$ (15)

B. After-test residual checking

After-test residual checking is a test on statistical character in accordance with residual’s distribution. We firstly get the variance ration from the mean and mean square error and then calculate the residual probability accordingly:

Variance Ratio Test (C)

Based on the above assumptions, the variance values of the original sequence and the residual sequence are: ${\begin{matrix} S_{x^{(0)}} = \frac{1}{n} \sum_{k = 1}^{n} {(x^{(0)} (k) - \bar{x^{(0)}})}^{2} \\ S_{e} = \frac{1}{n} \sum_{k = 1}^{n} {(e (k) - \bar{e})}^{2} \end{matrix}$ (16)

Then the variance ratio is: $C = \frac{S_{e}}{S_{x^{(0)}}}$ (17)

Error probability test (p)

Finally, we could calculate the error probability: $p = (| (e (k) - \bar{e}) | < 0.6745 {* S}_{x^{(0)}})$ (18)

Furthermore, it would indicate the different model accuracy varying with value ranges of $\bar{φ}$ , C, p. The specific classifications standard of accuracy level are illustrated in Table 1.

Table 1

GM (1, 1) model accuracy test

	$\bar{φ}$	C	p
Level 1	$\bar{φ} ⩽ 0.05$	C ⩽ 0.35	0.95 ⩽ p
Level 2	$0.05 < \bar{φ} ⩽ 0.1$	0.35 < C ⩽ 0.5	0.8 ⩽ p < 0.95
Level 3	$0.1 < \bar{φ} ⩽ 0.2$	0.5 < C ⩽ 0.65	0.7 ⩽ p < 0.8
Level 4	$0.2 < \bar{φ}$	0.65 < C	p < 0.7

If the accuracy of the GM(1,1) model cannot reach the test standard, the residual sequence model can be established to modify the original one to improve the accuracy.

2.3.3 Multiple index weighting methods

Currently, academia mainly adopts three methods to assign weights to raw data: subjective weighting, objective weighting and combination weighting [46, 47]. This paper focuses on the entropy weight method of objective weighting:

Step 1: Data standardization processing:

Supposing k-type index (X₁, X₂, …, X_k), where X_i ={ x₁, x₂, …, x_n } standardize as Y₁, Y₂, …, Y_k and $Y_{ij} = \frac{X_{ij} - \min (X_{i})}{max X_{i} - \min (X_{i})}$ .

Step 2: Finding the information entropy value of each indicator.

The entropy of the jth index is: $E_{j} = - \frac{1}{\ln (n)} \sum_{i = 1}^{n} p_{ij} {lnp}_{ij}$ (19) $p_{ij} = \frac{Y_{ij}}{\sum_{i = 1}^{n} Y_{ij}}$ (20)

Step 3: Determining the weight of each index:

Calculating the weight of each indicator $(W_{i} = \frac{1 - E_{i}}{k - \sum E_{i}})$ according to the entropy value (E₁, E₁, …, E_k) obtained in the step 2.

2.3.4 PLS-SEM

Structural Equation Modeling (SEM) is a multivariate statistical model proposed by Swedish statisticians such as Joreskog in the 1970 s on the basis of statistical theory [48]. The model is a comprehensive application and improvement of statistical methods, for example, exploratory factor analysis, confirmatory factor analysis, path analysis, multiple regression analysis and analysis of variance. Comparative speaking, SEM allows consideration of simultaneous equations with many endogenous variables, measurement error in the exogenous and endogenous variables as well as multiple indicators of latent constructs and estimation of reliability and validity, while it permits more general measurement models than traditional factor analytic structures to specify structural relationships among the latent variables, etc. [49].

In general, SEM variables can be divided into two categories: one is the observable variables (also called “indexes”). The other is the latent variables that need to be inferred from indexes. Moreover, latent variables are further categorized into exogenous latent variables (also called “latent independent variables”) and endogenous latent variables (also called “latent dependent variables”) [50]. Under the premise of the above variable classification, the structural equation analysis process can be roughly divided into five steps: (1) model specification, (2) identification, (3) estimation, (4) testing fit, (5) respecification [49]. Furthermore, the estimation generally has two methods: LISREL (LInear Structural RE-Square) and PLS (Partial Least Square). The comparison of the two methods is as shown in Table 2 [51]:

Table 2
Comparison of LISREL and PLS modeling methods

Comparison LISREL PLS

Objective of Overall Analysis Showing that the null hypothesis of the model is plausible and rejecting path-specific null hypotheses of no effect. Rejecting a set of path-specific null hypotheses of no effect

Objective of Variance Analysis Overall model fit Variance explanation

Required Theory Base Requireing sound theory base and supporting confirmatory research. Does not necessarily require sound theory base, but supporting both exploratory and confirmatory research

Minimal Sample Size At least 100–150 cases At least 10 times the number of items in the most complex construct

Comparison	LISREL	PLS
Objective of Overall Analysis	Showing that the null hypothesis of the model is plausible and rejecting path-specific null hypotheses of no effect.	Rejecting a set of path-specific null hypotheses of no effect
Objective of Variance Analysis	Overall model fit	Variance explanation
Required Theory Base	Requireing sound theory base and supporting confirmatory research.	Does not necessarily require sound theory base, but supporting both exploratory and confirmatory research
Minimal Sample Size	At least 100–150 cases	At least 10 times the number of items in the most complex construct

It can be identified that PLS is more focused on explaining variances. There are no strict assumptions on the data distribution and no specific requirements on the sample size. Additionally, PLS has been widely used in knowledge management, hospitality research, Internet research, enterprise performance and other fields in recent years [52 –55]. Consequently, this paper will use the PLS-SEM model to analyze the factors affecting the convergence level of the digital economy and real economy [56]. In the process of model construction, it is necessary to check the predictive ability and the validity of model parameter estimation. The predictive ability test method mainly includes the Average Variance Extracted (AVE), the Coefficient of Determination Method (R²) and the Composite Reliability method (Composite reliability), while the main method of checking the validity is Bootstrapping.

After sorting out the possible adopted methods in the paper, it could be found that:

In terms of correlation analysis, the GRA is suitable for the analysis of small or irregular samples while the Deng’s correlation model in the GRA focuses on the holistic analysis. Considering the characteristics of convergence analysis between the digital economy and real economy are lack of large samples, irregular data samples and focusing on overall analysis, therefore, this paper uses Deng’s correlation analysis method to measure the co-evolution level of the digital economy and real economy.

In terms of data prediction, as it could only possibly obtain very limited information in most cases, we have to adopt the Grey Forecasting Model model to carry out prediction research [41]. Moreover, due to the high computational efficiency of GM(1,1), most of the previous researchers have focused their attention on GM(1,1) models in their predictions [43]. Thus, this paper use GM (1, 1) model to predict the development of the digital economy and real economy in the next 10 years, and then obtain the convergence level between the two economics through the Deng’s correlation analysis method.

In terms of influencing factors, PLS-SEM has advantages on focusing on explaining variances, no strict assumptions on the data distribution and no specific requirements on the sample size, comparing with LISREL [50]. Consequently, the PLS-SEM model is more suitable for the research needs of the paper due to the limited amount of data on the digital economy and real economy.

3 Model construction

Hypothesis 1: Digital economy and real economy are two coordinated development subsystems, and the digital economy subsystem is marked as d while the real economy subsystem is marked as s.

Hypothesis 2: Each subsystem has m dimensions of development indexes in t years.

Hypothesis 3: From hypotheses 1 and 2, the original data of the development indexes of the digital economy subsystem can be expressed as $x_{d}^{(0)} = {(x_{d 1}^{(0)}, x_{d 2}^{(0)}, \dots, x_{dm}^{(0)})}^{T}, x_{di}^{(0)} = (x_{di}^{(0)} (1), x_{di}^{(0)}$ $(2), \dots x_{di}^{(0)} (t)), i = 1, 2, \dots m$ , where the ith dimension index of the kth year is $x_{di}^{(0)} (k)$ .

Hypothesis 4: From hypotheses 1 and 2, the original data of the development indexes of the real economy subsystem can be expressed as $x_{s}^{(0)} = {(x_{s 1}^{(0)}, x_{s 2}^{(0)}, \dots, x_{sm}^{(0)})}^{T}, x_{sj}^{(0)} = (x_{sj}^{(0)} (1), x_{sj}^{(0)}$ $(2), \dots x_{sj}^{(0)} (t)), j = 1, 2, \dots m$ ; while the jth dimension index of the kth year is $x_{sj}^{(0)} (k)$ .

Hypothesis 5: The correlation coefficient between any two indexes in the digital economy subsystem is defined as r_lh (t) ; l = 1, 2, … m ; h = 1, 2, … m ; l ¬ = h; The comprehensive convergence sequence relative to its own index is defined as B_d (t) in the digital economy subsystem. The correlation coefficient between any two indexes in the real economy subsystem is defined as r_uv (t) ; u = 1, 2, … m ; v = 1, 2, … m ; u ¬ = v; The comprehensive convergence sequence relative to its own index in the real economy subsystem is defined as B_s (t).

Hypothesis 6: The correlation coefficient between the two subsystems is defined as R_ds (t), while the comprehensive convergence level between the two subsystems is B_ds (t).

3.1 A measurement model of convergence level of digital economy and real economy based on the Deng’s correlation analysis model

On account of hypotheses 1 to 6, considering the need for an overall analysis of the convergence level, the paper uses Deng’s correlation analysis method, entropy weight method and other methods to construct a measurement model of the convergence level of the digital economy and real economy. The specific modeling process is as follows:

(1) Using the Deng’s correlation analysis model to calculate the correlation coefficients (r_lh (t) and r_uv (t)) between the indexes in the digital economy subsystem (r_lh (t)) and real economy subsystem (r_uv (t)).

First, generating the mean of the original index sequences: ${\begin{matrix} x_{d}^{' (0)} = {(x_{d 1}^{' (0)}, x_{d 2}^{' (0)}, \dots, x_{dm}^{' (0)})}^{T} \\ x_{di}^{'} k = (x_{di}^{' (0)} 1, x_{di}^{' (0)} 2, \dots, x_{di}^{' (0)} t) \end{matrix}$ (21) ${\begin{matrix} x_{s}^{' (0)} = {(x_{s 1}^{' (0)}, x_{s 2}^{' (0)}, \dots, x_{sm}^{' (0)})}^{T} \\ x_{sj}^{'} k = (x_{sj}^{' (0)} 1, x_{sj}^{' (0)} 2, \dots, x_{sj}^{' (0)} t) \end{matrix}$ (22) while $x_{di}^{' (0)} k = \frac{x_{di}^{(0)} (k)}{\bar{x_{di}^{(0)}}}, x_{di}^{(0)} (1) \neg = 0, k = 1, 2, \dots, t$ . $x_{sj}^{' (0)} k = \frac{x_{sj}^{(0)} (k)}{\bar{x_{sj}^{(0)}}}, x_{sj}^{(0)} (1) \neg = 0, k = 1, 2, \dots, t$ .

Then, after the meaning generating process, the calculation of the correlation coefficient between the indexes in the two subsystems is:

${\begin{matrix} r_{lh} (t) = \frac{min_{m} min_{k} | x_{dl}^{' (0)} (k) - x_{dh}^{' (0)} (k) | + ρ max_{m} max_{k} | x_{dl}^{' (0)} (k) - x_{dh}^{' (0)} (k) |}{| x_{dl}^{' (0)} (t) - x_{dh}^{'} (t) | - ρ max_{i} max_{k} | x_{dl}^{' (0)} (k) - x_{dh}^{' (0)} (k) |} \\ r_{uv} (t) = \frac{min_{m} min_{k} | x_{su}^{' (0)} (k) - x_{sv}^{' (0)} (k) | + ρ max_{m} max_{k} | x_{su}^{' (0)} (k) - x_{sv}^{' (0)} (k) |}{| x_{su}^{' (0)} (t) - x_{sv}^{' (0)} (t) | - ρ max_{i} max_{k} | x_{su}^{' (0)} (k) - x_{sv}^{' (0)} (k) |} \end{matrix}$ (23) in which $min_{m} min_{k} | x_{dl}^{' (0)} k - x_{dh}^{' (0)} k |$ is the smallest value of $| x_{dl}^{' (0)} t - x_{dh}^{' (0)} t |$ , while $max_{m} max_{k} | x_{dl}^{' (0)} k - x_{dh}^{' (0)} k |$ is the maximum value.When ρ is determined, the value of r_lh (t) depends on $min_{m} min_{k} | x_{dl}^{' (0)} k - x_{dh}^{' (0)} k |$ , $max_{m} max_{k} | x_{dl}^{' (0)} k - x_{dh}^{' (0)} k | | x_{dl}^{' (0)} t - x_{dh}^{' (0)} t |$ and $| x_{su}^{' (0)} t - x_{sv}^{' (0)} t |$ . Moreover, $min_{m} min_{k} | x_{su}^{' (0)} k - x_{sv}^{' (0)} k |$ is the smallest value of $| x_{su}^{' (0)} t - x_{sv}^{' (0)} t |$ and $max_{m} max_{k} | x_{su}^{' (0)} k - x_{sv}^{' (0)} k |$ is the maximum value.When ρ is determined, the value of r_uv (t) is dependent on $min_{m} min_{k} | x_{su}^{' (0)} k - x_{sv}^{' (0)} k |$ , $max_{m} max_{k} | x_{su}^{' (0)} k - x_{sv}^{' (0)} k |$ and $| x_{su}^{' (0)} t - x_{sv}^{' (0)} t |$ . In addition, ρrefers to the general processing, that is, the value is 0.5.

(2) Calculating the convergence sequence relative to its own index in each subsystem respectively (i.e. digital economy subsystem –B_d (t); real economy subsystem –B_s (t)) ${\begin{matrix} B_{d} (t) = \sum_{l, h = 1}^{t} r_{lh} (t) / \sum_{Z = 1}^{m - 1} Z, (l \neg = h) \\ B_{s} (t) = \sum_{u, v = 1}^{t} r_{uv} (t) / \sum_{Z = 1}^{m - 1} Z, (u \neg = v) \end{matrix}$ (24)

(3) Using the entropy weight method to calculate the internal index weights of each subsystem respectively, then obtaining the comprehensive development value sequence of the digital economy subsystem $(X_{d}^{' (0)} (t))$ and the real economy subsystem $(X_{s}^{' (0)} (t))$ . ${\begin{matrix} X_{d}^{' (0)} (t) = \sum_{i = 1}^{m} w_{di} \times x_{di}^{' (0)} t \\ X_{s}^{' (0)} (t) = \sum_{j = 1}^{m} w_{sj} \times x_{sj}^{' (0)} t \end{matrix}$ (25)w_di and w_si are the weight values of the digital economy subsystem and real economy subsystem calculated by the entropy weight method.

(4) Calculating the correlation coefficient (R_ds (t)) between the two subsystems. $R_{ds} (t) = \frac{ρ max_{t} | X_{d}^{' (0)} t - X_{s}^{' (0)} t |}{| X_{d}^{' (0)} t - X_{s}^{' (0)} t | - ρ max_{t} | X_{d}^{' (0)} t - X_{s}^{' (0)} t |}$ (26)

(5) Calculating the convergence degree (B_ds (t)) between the two subsystems. $B_{ds} (t) = R_{ds} (t)$ (27)

Since the coordination coefficient is a 1 × t matrix, the convergence degree between the two subsystems is the same as the correlation coefficient.

(6) Considering the impacts of indexes, constructing a D (t) model on account of B_d (t), B_s (t) and B_ds (t) to measure the level of coordinated development of the digital economy and real economy. $D (t) = α_{1} (w_{d} \times B_{d} (t) + w_{s} \times B_{s} (t)) + α_{2} B_{ds} (t)$ (28)

in which D (t) is the comprehensive convergence degree between the digital economy subsystem and real economy subsystem; w_d and w_s are based on $X_{d}^{' (0)} (t)$ and $X_{s}^{' (0)} (t)$ sequence that are the weight of the comprehensive development sequence of the two systems calculated through the entropy weight method. α₁ is the weight coefficient of the sum of own convergence degrees of the digital economy and the real economy; α₂ is the weight coefficient of the convergence degree between the digital economy and real economy, where α₁ > 0, α₂ > 0, α₁ + α₂ = 1. In advocating the simultaneous development of the digital economy and real economy, then α₁ = α₂ = 0.5.

(7) Using the methods from steps (1) to (6) to calculate the convergence levels between digital economy and primary, secondary, and tertiary industries respectively.

3.2 Prediction model of convergence of digital economy and real economy based on GM (1,1)

In accordance with the above hypotheses 1 to 6, after averaging the raw data of various indicators, we establish a GM (1,1) model for the digital economy subsystem and real economy subsystem respectively to obtain the prediction model of each index in the subsystem relative to others, and then verify the effectiveness of each prediction model. The numerical values of the various indexes in the next 10 years are simulated under the premise that the prediction model is reliable. Further, the convergence level measurement model constructed in this paper is used to predict the convergence level of the digital economy and real economy in the next 10 years. The process is as follows:

Using Equations (21) and (22) to generate the mean of the original sequence of the indexes to obtain the dimensionless sequences of the digital economy subsystem $(x_{d}^{' (0)})$ and the real economy subsystem $(x_{s}^{' (0)})$ .

Referring to Equation (6) to generate an accumulation of $x_{d}^{' (0)}$ and $x_{s}^{' (0)}$ (1-AGO) ${\begin{matrix} x_{d}^{' (1)} = {(x_{d 1}^{' (1)}, x_{d 2}^{' (1)}, \dots, x_{dm}^{' (1)})}^{T} \\ x_{di}^{' (1)} (k) = \sum_{j = 1}^{k} x_{di}^{' (0)} (j), (k = 1, 2, \dots, t) \end{matrix}$ (29) ${\begin{matrix} x_{s}^{' (1)} = {(x_{s 1}^{' (1)}, x_{s 2}^{' (1)}, \dots, x_{sm}^{' (1)})}^{T} \\ x_{si}^{' (1)} (k) = \sum_{j = 1}^{k} x_{si}^{' (0)} (j), (k = 1, 2, \dots, t) \end{matrix}$ (30)

Referring to Equations (8) to (13) to establish a GM (1,1) prediction model between the internal indexes of the two subsystems.

Taking $x_{d 1}^{' (1)}$ and $x_{s 1}^{' (1)}$ as examples, the Grey models of the first index in the digital economy subsystem and the first index in the real economy subsystem are established respectively:

Constructing the adjacent mean generation sequences of $x_{d 1}^{' (1)}$ and $x_{s 1}^{' (1)}$ respectively (i.e. $z_{d 1}^{' (1)}$ and $z_{s 1}^{' (1)}$ ): ${\begin{matrix} z_{d 1}^{' (1)} (k) = 0.5 x_{di}^{' (1)} (k) + 0.5 x_{di}^{' (1)} (k - 1), (k = 2, 3, \dots, t) \\ z_{s 1}^{' (1)} (k) = 0.5 x_{si}^{' (1)} (k) + 0.5 x_{si}^{' (1)} (k - 1), (k = 2, 3, \dots, t) \end{matrix}$ (31)

Then, constructing the GM(1,1) first-order linear equations of $x_{d 1}^{' (1)}$ and $x_{s 1}^{' (1)}$ respectively: ${\begin{matrix} x_{d 1}^{' (0)} (k) + a_{d 1} z_{d 1}^{' (1)} (k) = b_{d 1} \\ x_{s 1}^{' (0)} (k) + a_{s 1} z_{s 1}^{' (1)} (k) = b_{s 1} \end{matrix}$ (32)

And constructing the data matrices of $x_{d 1}^{' (1)} (B_{d 1}$ and Y_d1) and the data matrices of $x_{s 1}^{' (1)} (B_{s 1}$ and Y_s1), and respectively solving the identification parameters of the two subsystems (μ_d1 and μ_s1), then getting the prediction model ( $x_{d 1}^{' (1)}$ and $x_{s 1}^{' (1)}$ ): ${\hat{x_{d 1}}}^{' (1)} (k + 1) = (x_{d 1}^{' (1)} (1) - \frac{b_{d 1}}{a_{d 1}}) e^{- a_{d 1} k} + \frac{b_{d 1}}{a_{d 1}}$ (33) ${\hat{x_{s 1}}}^{' (1)} (k + 1) = (x_{s 1}^{' (1)} (1) - \frac{b_{s 1}}{a_{s 1}}) e^{- a_{s 1} k} + \frac{b_{s 1}}{a_{s 1}}$ (34)

After IAGO of ${\hat{x_{d 1}}}^{' (1)}$ , ${\hat{x_{s 1}}}^{' (1)}$ , we could get the predicted value of the first index (mean generating) of the digital economy subsystem $({\hat{x_{d 1}}}^{' (0)} (k + 1))$ and the real economy subsystem $({\hat{x_{s 1}}}^{' (0)} (k + 1))$ : ${\begin{matrix} {\hat{x_{d 1}}}^{' (0)} (k + 1) = {\hat{x_{d 1}}}^{' (1)} (k + 1) - {\hat{x_{d 1}}}^{' (1)} (k) \\ {\hat{x_{s 1}}}^{' (0)} (k + 1) = {\hat{x_{d 1}}}^{' (1)} (k + 1) - {\hat{x_{s 1}}}^{' (1)} (k) \end{matrix}$ (35)

Using Equations (14) to (18) and Table 1 to analyze the accuracy of the prediction model to determine whether the model has reached a qualified state. For models that do not meet the requirements, we need to construct a residual error modified GM(1,1) model and then perform the accuracy verification again.

Moreover, the prediction model of other indexes is constructed to obtain the prediction sequence of all indexes in the next 10 years.

Re-using Equations (21) to (28) to predict the convergence level of the digital economy and real economy in the next 10 years.

Repeating the above (1)–(4) to predict respectively the convergence levels of the digital economy and primary, secondary, and tertiary industries in the next 10 years.

3.3 Analysis model of factors affecting convergence of digital economy and real economy based on PLS-SEM

The key to construct the analysis model is to make scientific and reasonable hypotheses, and then verify whether the hypotheses meets the requirements of the predictive ability and the validity through empirical analysis. Therefore, this paper proposes the following hypotheses in terms of the impactingfactors:

3.3.1 Hypotheses of internal factors

Innovation is the core driving force to encourage high-quality social and economic development and the process of innovation is the integration of academic knowledge and empirical knowledge [57]. The widespread use of digital technologies such as the World Wide Web provides a mechanism for sharing large volumes of information and enriches the sources of knowledge, which effectively enables the knowledge transfer between various sections of society [58]. Meanwhile, open innovation with knowledge diffusion as the main feature can achieve a substantial shortening of the innovation cycle and time [59]. Thus the widespread application of digital technology has made digitization a key element in the economic development environment, exerting a positive impact on the innovation efficiency of the real economy. To sum up, in terms of endogenous factors, the following hypotheses are proposed:

H1: The innovation capability of the digital economy has a positive and direct effect on that of the real economy.

H2: The innovation capability of the digital economy has a positive and direct effect on the development level of the digital economy.

H3: The innovation capability of the digital economy has a positive and direct effect on the development environment of the digital economy and real economy.

H4: The development efficiency of the digital economy has a positive and direct effect on the development level of the digital economy.

3.3.2 Hypotheses of direct factors

The main goal of the transformation and upgrading of manufacturing industry is to get out of the trap at the low end of the value chain as well as shift the industry to the R&D and sales links at both ends of the “smile curve” [60]. In the process of expanding into the manufacturing sector, the digital economy have solved many pain points in the transformation and upgrading of China’s manufacturing industry,for example: cracking the bottleneck of innovation chain, improving the quality of manufacturing chain, optimizing the efficiency of supply chain and expanding the service space, etc. [61]. In other words, as the demand side of the digital economy, the real economy’s innovative capabilities will affect the development efficiency of the digital economy; while the digital economy improves the convergence level between the two economies by improving the development quality of the real economy. In conclusion, the direct factors hypotheses are proposed:

H5: The innovation capability of the real economy has a positive and direct effect on the development efficiency of the digital economy.

H6: The development level of the digital economy has a positive and direct impact on that of the real economy.

H7: The development level of the real economy has a positive and direct effect on the convergence level of the digital economy and real economy.

3.3.3 Hypotheses of environmental factors

It was reported that more than 75% of OECD countries have developed and published an explicit regulatory policy by the end of 2015, while for the countries without an explicit regulatory have also quickly started to develop the strategies or policies in certain specific areas to effectively promote their digital development [62]. Moreover, the basic environment such as the level of Internet penetration and the accumulation of data are the key basic conditions for facilitating the high-quality development of the digital economy while the development level is the core driving force that directly enhances the high-quality development of the real economy. Therefore, the hypotheses about environmental factors are as follows:

H8: The macro development environment plays a positive and direct role in the development level of the digital economy.

In short, the relationship between the above 8 hypotheses is illustrated in Fig. 2.

Fig. 2

The analysis model of factors affecting convergence of the digital economy and real economy.

The paper will consider the corresponding observable variables as well as analyze whether the hypotheses are significantly effective by using actual data in following empirical analysis.

4 Empirical research

4.1 Measurement of convergence level of digital economy and real economy

4.1.1 Index selection

At the regional level, considering the timeliness and availability of data, this paper limits the scope to Mainland China. At the data dimension level, current academics often use data on economic benefits, social benefits, growth potential and innovation vitality within an economy or industry as indexes of industrial development characteristics [63 –66]. Hence the paper constructs the original data sequence of development indexes of the digital economy and real economy (refined to the primary, secondary and tertiary industries) on the basis of the above four dimensions in accordance with the principles of scientificity, systematicness and availability. Then the secondary data indexes in each dimension are as shown in Table 3.

Table 3
The system of the digital economy and real economy development indexes

# Type Evaluation Index Name Index Definition Unit

D1 Economic benefits Added value Added value of various industries in the digital economy Million Yuan

D2 Investment in fixed assets Total investment in fixed asset of the digital economy Million Yuan

D3 Social benefits Number of employees Total number of employees in the digital economy Thousands person

D4 Number of market entities Number of enterprises or legal persons in the digital economy

D5 Digital Economy Growth rate of value added (value added this year in the digital economy /value added last year in the digital economy – 1)×100% %

D6 Growth potentials The proportion of the fixed asset investment in fixed asset investment of the whole society The fixed asset investment of the digital economy this year / the national fixed asset investment of this year×100% %

D7 Innovative vitality Number of R&D projects Number of R&D projects in the digital economy

D8 R&D investment expenditure R&D investment expenditure in the digital economy Thousand Yuan

S1 Economic benefits Added value Added value of various industries in the real economy Million Yuan

S2 Investment in fixed assets Total investment in fixed assets of the real economy Million Yuan

S3 Social benefits Number of employees Total number of employees in the real economy Thousands person

S4 Number of market entities Number of enterprises or legal persons in the real economy

S5 Real economy Growth rate of value added (value added this year in the real economy/value added last year in the real economy – 1)×100% %

S6 Growth potentials The proportion of the fixed asset investment in fixed asset investment of the whole society The amount of investment in fixed assets in the real economy this year / the amount of investment in fixed assets in the country this year×100% %

S7 Innovative vitality Number of R&D projects Number of R&D projects in the real economy

S8 R&D investment expenditure R&D investment expenditure in the real economy Thousand Yuan

SF1 Added value Added value of various industries in the primary industry Million Yuan

SF2 Economic benefits Investment in fixed assets Total investment in fixed asset of the primary industry Million Yuan

SF3 Number of employees Total number of employees in the primary industry Thousands person

SF4 Social benefits Number of market entities Number of enterprises or legal persons in the primary industry

SF5 Primary industry Growth rate of value added (value added this year in the primary industry /value added last year in the primary industry – 1)×100% %

SF6 Growth potentials The proportion of the fixed asset investment in fixed asset investment of the whole society The fixed asset investment of the primary industry this year / the national fixed asset investment of this year×100% %

SF7 Innovative vitality Number of R&D projects Number of R&D projects in primary industry

SF8 R&D investment expenditure R&D investment expenditure in primary industry Thousand Yuan

SS1 Added value Added value of various industries in the secondary industry Million Yuan

SS2 Economic benefits Investment in fixed assets Total investment in fixed assets of the secondary industry Million Yuan

SS3 Number of employees Total number of employees in the secondary industry Thousands person

SS4 Social benefits Number of market entities Number of enterprises or legal persons in the secondary industry

SS5 Secondary industry Growth rate of value added (value added this year in the secondary industry /value added last year in the secondary industry – 1)×100% %

SS6 Growth potentials The proportion of the fixed asset investment in fixed asset investment of the whole society The amount of investment in fixed assets in the secondary industry this year / the amount of investment in fixed assets in the country this year×100% %

SS7 Innovative vitality Number of R&D projects Number of R&D projects in the secondary industry

SS8 R&D investment expenditure R&D investment expenditure in the secondary industry Thousand Yuan

ST1 Added value Added value of various industries in the tertiary industry Million Yuan

ST2 Economic benefits Investment in fixed assets Total investment in fixed assets of the tertiary industry Million Yuan

ST3 Number of employees Total number of employees in the tertiary industry Thousands person

ST4 Social benefits Number of market entities Number of enterprises or legal persons in the tertiary industry

ST5 Tertiary industry Growth rate of value added (value added this year in tertiary industry /value added last year in the tertiary industry – 1)×100% %

ST6 Growth potentials The proportion of the fixed asset investment in fixed asset investment of the whole society The amount of investment in fixed assets in the tertiary industry this year / the amount of investment in fixed assets in the country this year×100% %

ST7 Innovative vitality Number of R&D projects Number of R&D projects in the tertiary industry

ST8 R&D investment expenditure R&D investment expenditure in the tertiary industry Thousand Yuan

#	Type	Evaluation	Index Name	Index Definition	Unit
D1		Economic benefits	Added value	Added value of various industries in the digital economy	Million Yuan
D2			Investment in fixed assets	Total investment in fixed asset of the digital economy	Million Yuan
D3		Social benefits	Number of employees	Total number of employees in the digital economy	Thousands person
D4			Number of market entities	Number of enterprises or legal persons in the digital economy
D5	Digital Economy		Growth rate of value added	(value added this year in the digital economy /value added last year in the digital economy – 1)×100%	%
D6		Growth potentials	The proportion of the fixed asset investment in fixed asset investment of the whole society	The fixed asset investment of the digital economy this year / the national fixed asset investment of this year×100%	%
D7		Innovative vitality	Number of R&D projects	Number of R&D projects in the digital economy
D8			R&D investment expenditure	R&D investment expenditure in the digital economy	Thousand Yuan
S1		Economic benefits	Added value	Added value of various industries in the real economy	Million Yuan
S2			Investment in fixed assets	Total investment in fixed assets of the real economy	Million Yuan
S3		Social benefits	Number of employees	Total number of employees in the real economy	Thousands person
S4			Number of market entities	Number of enterprises or legal persons in the real economy
S5	Real economy		Growth rate of value added	(value added this year in the real economy/value added last year in the real economy – 1)×100%	%
S6		Growth potentials	The proportion of the fixed asset investment in fixed asset investment of the whole society	The amount of investment in fixed assets in the real economy this year / the amount of investment in fixed assets in the country this year×100%	%
S7		Innovative vitality	Number of R&D projects	Number of R&D projects in the real economy
S8			R&D investment expenditure	R&D investment expenditure in the real economy	Thousand Yuan
SF1			Added value	Added value of various industries in the primary industry	Million Yuan
SF2		Economic benefits	Investment in fixed assets	Total investment in fixed asset of the primary industry	Million Yuan
SF3			Number of employees	Total number of employees in the primary industry	Thousands person
SF4		Social benefits	Number of market entities	Number of enterprises or legal persons in the primary industry
SF5	Primary industry		Growth rate of value added	(value added this year in the primary industry /value added last year in the primary industry – 1)×100%	%
SF6		Growth potentials	The proportion of the fixed asset investment in fixed asset investment of the whole society	The fixed asset investment of the primary industry this year / the national fixed asset investment of this year×100%	%
SF7		Innovative vitality	Number of R&D projects	Number of R&D projects in primary industry
SF8			R&D investment expenditure	R&D investment expenditure in primary industry	Thousand Yuan
SS1			Added value	Added value of various industries in the secondary industry	Million Yuan
SS2		Economic benefits	Investment in fixed assets	Total investment in fixed assets of the secondary industry	Million Yuan
SS3			Number of employees	Total number of employees in the secondary industry	Thousands person
SS4		Social benefits	Number of market entities	Number of enterprises or legal persons in the secondary industry
SS5	Secondary industry		Growth rate of value added	(value added this year in the secondary industry /value added last year in the secondary industry – 1)×100%	%
SS6		Growth potentials	The proportion of the fixed asset investment in fixed asset investment of the whole society	The amount of investment in fixed assets in the secondary industry this year / the amount of investment in fixed assets in the country this year×100%	%
SS7		Innovative vitality	Number of R&D projects	Number of R&D projects in the secondary industry
SS8			R&D investment expenditure	R&D investment expenditure in the secondary industry	Thousand Yuan
ST1			Added value	Added value of various industries in the tertiary industry	Million Yuan
ST2		Economic benefits	Investment in fixed assets	Total investment in fixed assets of the tertiary industry	Million Yuan
ST3			Number of employees	Total number of employees in the tertiary industry	Thousands person
ST4		Social benefits	Number of market entities	Number of enterprises or legal persons in the tertiary industry
ST5	Tertiary industry		Growth rate of value added	(value added this year in tertiary industry /value added last year in the tertiary industry – 1)×100%	%
ST6		Growth potentials	The proportion of the fixed asset investment in fixed asset investment of the whole society	The amount of investment in fixed assets in the tertiary industry this year / the amount of investment in fixed assets in the country this year×100%	%
ST7		Innovative vitality	Number of R&D projects	Number of R&D projects in the tertiary industry
ST8			R&D investment expenditure	R&D investment expenditure in the tertiary industry	Thousand Yuan

On the selected indexes, this paper comprehensively sorts out the concepts and analysis in the “G20 Digital Economy Development and Cooperation Initiative” issued by the G20 Hangzhou Summit in 2016, the “Defining and Measuring the Digital Economy” issued by the US Department of Commerce (BEA) in 2018, the definition of the digital economy by the British Economic and Social Research Institute, the “Digital Economy Plan” announced by Russia in 2017 and the definition of digital economy by Chinese scholars, combined with China’s latest “National Economic Industry Classification” (GB/T4754-2017). The paper defines the boundaries of the digital economy and the three industries as follows [67 –70]:

The statistics of the digital economy include C39 (computer, communication and other electronic equipment manufacturing) and I63 I65 (information transmission, software and information technology service industry).

The statistics of the real economy statistics include:

For the primary industry, it includes agriculture, forestry, animal husbandry and fishery (A01 A05).

For the secondary industry, it includes mining (B06 B10, B12), manufacturing (C13 C38, C40 C42), electricity, heat, popularity and water production and supply (D44 D46), and construction (E47 E50).

For the tertiary industry, it includes agricultural and sideline product processing (A06), mining professional and auxiliary activities (B11), metal products, machinery and equipment repair industry (C43), wholesale and retail industry (F51, F52), transportation, storage and postal industry (G53 G60), accommodation and catering industry (H61, H62), financial industry (J66 J69), real estate industry (K70), leasing and business service industry (L71, L72), scientific research and technical service industry (M73 M75), water conservancy, environment and public facilities management industry (N76 N79), resident service, repair and other service industry (O80 O82), culture, sports and entertainment industry (R86 R90), public management, social security and social organizations(S91 S96).

In pursuance of the above industry classifications, the data in the paper is basically from the 2005–2019 China Statistical Yearbook, 2005–2019 China National Economic and Social Development Statistical Bulletin, the 1st to 4th National Economic Census Communiqu $\overset{´}{e}$ as well as the “Annual Statistical Report on Software and Information Technology Services Industry” of the 2018 and 2019 [71 –73]. Taking into account the availability, timeliness and reliability of data, the following principles should be followed in the processing: First, the data of various industries from 2005 to 2019 would be adopted to ensure the timeliness of data.Second, in order to ensure availability, the added value of the three industries uses the data of enterprises above designated size. The number of market entities in the three industries is calculated by the number of legal persons, while the number of entities in the digital economy market is calculated by the number of enterprises.Third, since the National Bureau of Statistics had no longer published the total output value of each segment of the industry from 2012, this paper uses operating income as the added value of the computer, communications and other electronic equipment manufacturing industries to ensure data consistency.

4.1.2 Classification of the convergence level between industries

When analyzing the convergence level, we focus on measuring the degree of the convergences of their own and the convergences between economies. Assuming D is the convergence degree between the digital economy and real economy, this index mainly reflects the degree and type of convergence. Moreover, supposing that the convergence degree of the digital economy and the real economy of their own are B_d and B_s respectively, the ratio of B_d/B_s will further reflect the convergence relationship. The specific criteria are as shown in Table 4 [74, 75].

Table 4
Judgment criteria for the convergence of the digital economy and real economy

Interval D value Basic type B_s and B_d ratio Specific type

0.9 < D≤1 High-quality Convergence B_d/B_s>1.05 High-quality collaborative digital economy-led type

0.95≤B_d/B_s≤1.05 High-quality collaborative synchronous development type

B_d/B_s<0.95 High-quality collaborative real economy-led type

0.8 < D≤0.9 Good Convergence B_d/B_s>1.05 Good collaborative digital economy -led type

0.95≤B_d/B_s≤1.05 Good collaborative synchronous development type

B_d/B_s<0.95 Good collaborative real economy -led type

Acceptable interval 0.7 < D≤0.8 Intermediate convergence B_d/B_s>1.05 Intermediate collaborative digital economy-led type

0.95≤B_d/B_s≤1.05 Intermediate collaborative synchronous development type

B_d/B_s<0.95 Intermediate collaborative real economy-led type

0.6 < D≤0.7 Primary convergence B_d/B_s>1.05 Primary collaborative digital economy-led type

0.95≤B_d/B_s≤1.05 Primary collaborative synchronous development type

B_d/B_s<0.95 Primary collaborative real economy-led type

0.5 < D≤0.6 Barely Coordinated B_d/B_s>1.05 Barely collaborative digital economy-led type

0.95≤B_d/B_s≤1.05 Barely collaborative synchronous development type

B_d/B_s<0.95 Barely collaborative real economy-led type

Transition Interval 0.4 < D≤0.5 On the verge of disorder B_d/B_s>1.05 On the verge of disorder digital economy-led type

0.95≤B_d/B_s≤1.05 On the verge of disorder synchronous development type

B_d/B_s<0.95 On the verge of disorder real economy-led type

0.3 < D≤0.4 Mild disorder B_d/B_s>1.05 Mild disorder digital economy-led type

0.95≤B_d/B_s≤1.05 Mild disorder synchronous development type

B_d/B_s<0.95 Mild disorder real economy-led type

0.2 < D≤0.3 Moderate Disorder B_d/B_s>1.05 Moderate disorder digital economy-led type

0.95≤B_d/B_s≤1.05 Moderate disorder synchronous development type

B_d/B_s<0.95 Moderate disorder real economy-led type

Unacceptable interval 0.1 < D≤0.2 Severe Disorder B_d/B_s>1.05 Severe disorder digital economy-led type

0.95≤B_d/B_s≤1.05 Severe disorder synchronous development type

B_d/B_s<0.95 Severe disorder real economy-led type

0 < D≤0.1 Extreme Imbalance B_d/B_s>1.05 Extreme imbalance real economy-led type

0.95≤B_d/B_s≤1.05 Extreme imbalance synchronous development type

B_d/B_s<0.95 Extreme imbalance real economy-led type

Interval	D value	Basic type	B_s and B_d ratio	Specific type
	0.9 < D≤1	High-quality Convergence	B_d/B_s>1.05	High-quality collaborative digital economy-led type
			0.95≤B_d/B_s≤1.05	High-quality collaborative synchronous development type
			B_d/B_s<0.95	High-quality collaborative real economy-led type
	0.8 < D≤0.9	Good Convergence	B_d/B_s>1.05	Good collaborative digital economy -led type
			0.95≤B_d/B_s≤1.05	Good collaborative synchronous development type
			B_d/B_s<0.95	Good collaborative real economy -led type
Acceptable interval	0.7 < D≤0.8	Intermediate convergence	B_d/B_s>1.05	Intermediate collaborative digital economy-led type
			0.95≤B_d/B_s≤1.05	Intermediate collaborative synchronous development type
			B_d/B_s<0.95	Intermediate collaborative real economy-led type
	0.6 < D≤0.7	Primary convergence	B_d/B_s>1.05	Primary collaborative digital economy-led type
			0.95≤B_d/B_s≤1.05	Primary collaborative synchronous development type
			B_d/B_s<0.95	Primary collaborative real economy-led type
	0.5 < D≤0.6	Barely Coordinated	B_d/B_s>1.05	Barely collaborative digital economy-led type
			0.95≤B_d/B_s≤1.05	Barely collaborative synchronous development type
			B_d/B_s<0.95	Barely collaborative real economy-led type
Transition Interval	0.4 < D≤0.5	On the verge of disorder	B_d/B_s>1.05	On the verge of disorder digital economy-led type
			0.95≤B_d/B_s≤1.05	On the verge of disorder synchronous development type
			B_d/B_s<0.95	On the verge of disorder real economy-led type
	0.3 < D≤0.4	Mild disorder	B_d/B_s>1.05	Mild disorder digital economy-led type
			0.95≤B_d/B_s≤1.05	Mild disorder synchronous development type
			B_d/B_s<0.95	Mild disorder real economy-led type
	0.2 < D≤0.3	Moderate Disorder	B_d/B_s>1.05	Moderate disorder digital economy-led type
			0.95≤B_d/B_s≤1.05	Moderate disorder synchronous development type
			B_d/B_s<0.95	Moderate disorder real economy-led type
Unacceptable interval	0.1 < D≤0.2	Severe Disorder	B_d/B_s>1.05	Severe disorder digital economy-led type
			0.95≤B_d/B_s≤1.05	Severe disorder synchronous development type
			B_d/B_s<0.95	Severe disorder real economy-led type
	0 < D≤0.1	Extreme Imbalance	B_d/B_s>1.05	Extreme imbalance real economy-led type
			0.95≤B_d/B_s≤1.05	Extreme imbalance synchronous development type
			B_d/B_s<0.95	Extreme imbalance real economy-led type

4.1.3 Analysis of the overall convergence level of the digital economy and real economy

According to the Measurement Model of Convergence Level of Digital Economy and Real Economy Based on the Deng’s Correlation Analysis Model constructed in this paper, the digital economy and the real economy’s own convergence sequences as well as the comprehensive convergence sequences between the economies from 2005 to 2019 were obtained after calculating the original data sequences through using matlab software.

(1) Analysis of the own convergence level of the digital economy and real economy

It could found from Table 5 and Fig. 3, there was obvious similarity on the convergences of China’s digital economy, real economy, primary industry, secondary industry and tertiary industry from 2005 to 2019: Firstly, the co-evolution level of above five economies had remained relatively low from 2005 to 2007. Secondly, it had risen year by year and reached a peak from 2008 to 2012, indicating that the main driving force in this stage stemmed from the own convergence effect within each economy. Thirdly, although the convergence of some economies began to rebound from 2018 to 2019, the own convergence level generally showed a downward trend year by year from 2012 to 2019, which showed the economic growth dividend that brought from the own convergence effects had tended to be fully released and the new growth momentum of various economies was gradually migrating.

Table 5
The digital economy and the real economy’s own convergence sequence from 2005 to 2019

Category 2005 2006 2007 2008 2009 2010 2011 2012

Real economy (B_s) 0.6040 0.6731 0.6057 0.6548 0.7769 0.7487 0.7800 0.8253

Primary industry (B_s1) 0.7020 0.6905 0.6064 0.6172 0.7557 0.7011 0.7154 0.8397

Secondary industry (B_s2) 0.6345 0.7421 0.6641 0.6846 0.8328 0.8146 0.8257 0.8433

Tertiary industry (B_s3) 0.6374 0.6286 0.6086 0.6786 0.7604 0.7553 0.8012 0.8551

Digital economy (B_d) 0.8054 0.6902 0.7691 0.8302 0.8539 0.7762 0.8528 0.9147

Category 2013 2014 2015 2016 2017 2018 2019 Convergence degree

Real economy (B_s) 0.7600 0.7180 0.7023 0.6632 0.6493 0.6295 0.5838 0.6916

Primary industry (B_s1) 0.8357 0.8026 0.7139 0.6417 0.5703 0.5639 0.5926 0.6899

Secondary industry (B_d2) 0.7752 0.7582 0.7165 0.6945 0.7597 0.7388 0.6521 0.7424

Tertiary industry (B_d3) 0.7949 0.7371 0.7387 0.6823 0.6216 0.6071 0.5761 0.6989

Digital economy (B_d) 0.8791 0.8782 0.8374 0.8365 0.7521 0.6097 0.5787 0.7909

Category	2005	2006	2007	2008	2009	2010	2011	2012
Real economy (B_s)	0.6040	0.6731	0.6057	0.6548	0.7769	0.7487	0.7800	0.8253
Primary industry (B_s1)	0.7020	0.6905	0.6064	0.6172	0.7557	0.7011	0.7154	0.8397
Secondary industry (B_s2)	0.6345	0.7421	0.6641	0.6846	0.8328	0.8146	0.8257	0.8433
Tertiary industry (B_s3)	0.6374	0.6286	0.6086	0.6786	0.7604	0.7553	0.8012	0.8551
Digital economy (B_d)	0.8054	0.6902	0.7691	0.8302	0.8539	0.7762	0.8528	0.9147
Category	2013	2014	2015	2016	2017	2018	2019	Convergence degree
Real economy (B_s)	0.7600	0.7180	0.7023	0.6632	0.6493	0.6295	0.5838	0.6916
Primary industry (B_s1)	0.8357	0.8026	0.7139	0.6417	0.5703	0.5639	0.5926	0.6899
Secondary industry (B_d2)	0.7752	0.7582	0.7165	0.6945	0.7597	0.7388	0.6521	0.7424
Tertiary industry (B_d3)	0.7949	0.7371	0.7387	0.6823	0.6216	0.6071	0.5761	0.6989
Digital economy (B_d)	0.8791	0.8782	0.8374	0.8365	0.7521	0.6097	0.5787	0.7909

Fig. 3

The change trend of the digital economy and real economy’s own convergence sequence from 2005 to 2019.

(2) Analysis of the convergence level between the digital economy and real economy (as shown in Tables 6 and 7).

Table 6

The sequence of comprehensive convergence between the digital economy and real economy from 2005 to 2019

Category	2005	2006	2007	2008	2009	2010	2011	2012
Real economy	0.6867	0.8075	0.6748	0.6885	0.7749	0.7997	0.7619	0.9365
Primary industry	0.8613	0.5300	0.5579	0.5551	0.9030	0.6881	0.6119	0.8018
Secondary industry	0.6530	0.6305	0.7327	0.7088	0.7315	0.6988	0.6825	0.7736
Tertiary industry	0.6818	0.7777	0.6713	0.6976	0.6839	0.7765	0.7780	0.8692
Category	2013	2014	2015	2016	2017	2018	2019	Convergence degree
Real economy	0.7642	0.7273	0.7435	0.8108	0.7753	0.8038	0.7058	0.7641
Primary industry	0.8446	0.6581	0.5630	0.5373	0.7058	0.6020	0.5137	0.6622
Secondary industry	0.7047	0.5854	0.5668	0.5957	0.8250	0.8327	0.6369	0.6906
Tertiary industry	0.7001	0.6625	0.6533	0.6995	0.6896	0.7146	0.7887	0.7230

Table 7

The ratio series of the convergence between digital economy and real economy from 2005 to 2019

Category	2005	2006	2007	2008	2009	2010	2011	2012
B_d/B_s	1.3334	1.0254	1.2698	1.2679	1.0991	1.0367	1.0933	1.1083
B_d/B_s1	1.1473	0.9996	1.2683	1.3451	1.1299	1.1071	1.1921	1.0893
B_d/B_s2	1.2693	0.9301	1.1581	1.2127	1.0253	0.9529	1.0328	1.0847
B_d/B_s3	1.2636	1.0980	1.2637	1.2234	1.1230	1.0277	1.0644	1.0697
Category	2013	2014	2015	2016	2017	2018	2019	Mean
B_d/B_s	1.1567	1.2231	1.1924	1.2613	1.1583	0.9685	0.9913	1.1457
B_d/B_s1	1.0519	1.0942	1.1730	1.3036	1.3188	1.0812	0.9765	1.1519
B_d/B_s2	1.1340	1.1583	1.1687	1.2045	0.9900	0.8253	0.8874	1.0689
B_d/B_s3	1.1059	1.1914	1.1336	1.2260	1.2099	1.0043	1.0045	1.1339

It could been observed from Fig. 4 that the convergence level fluctuated between 0.7 and 0.8 and the B_d/B_s value was basically greater than 1.05 (except for 2006, 2010, 2018 and 2019). The overall level was in the intermediate convergence stage led by the digital economy from 2005 to 2019, which would fluctuate periodically every five years. Taking account of China’s macro policy environment of implementing the “five-year plan”, it could be reflected that the potential correlation between the convergence level and macro policy environment. In addition to the peak of co-evolution in 2012, the average level from the first to the third fluctuation cycles had gradually increased and the convergence degree had continued to strengthen. Moreover, in the context of the gradual decline in the level of the own convergence during the period from 2005 to 2019, it showed that the development momentum of the digital economy and real economy had gradually shifted from the convergence effect of their own within each economy to the convergence effect between the economies.

Fig. 4

The trend of convergence between the digital economy and real economy from 2005 to 2019.

Figure 5 shows that from 2005 to 2019, the convergence level between the digital economy and primary industry fluctuated between 0.5 and 0.8 and the B_d/B_s1 value was basically greater than 1.05 (except for 2006 and 2009). Furthermore, the convergence type had evolved from a good collaborative digital economy-led type to a primary collaborative digital economy-led type, which had a periodic fluctuation in about 5 years, indicating that there was a potential correlation between the co-evolution level and macro policy environment. Comparing the three fluctuation cycles, it can be concluded that the overall convergence level between the digital economy and primary industry showed a gradual decrease. According to the above-mentioned “Analysis of the own convergence level of the digital economy and real economy”, the convergence level of the primary industry was also a gradual decline from 2005 to 2019. Therefore, a preliminary conclusion can be drawn considering the annual slowdown in the primary industry’s growth rate and fixed investment growth rate from 2005 to 2019 that under the premise that the convergences of the digital economy and the primary industry respectively as well as the convergence between the primary industry and digital economy are gradually decreasing, the development momentum of the primary industry continued to weaken, which had an adverse impact on the development of the primary industry.

Fig. 5

The convergence trend of the digital economy and primary industry from 2005 to 2019.

It could be found from Fig. 6 that the convergence level between digital economy and secondary industry fluctuated roughly between 0.5 and 0.8. The B_d/B_s2 value was less than 0.95 in 2006, 2018, and 2019 that were primary and intermediate convergence types led by the secondary industry. In 2009, 2010, and 2011, the B_d/B_s2 was between 0.95 and 1.05 that were primary and intermediate collaborative synchronous development types. The B_d/B_s2 of the remaining years were greater than 1.05 that were primary and intermediate collaborative digital economy-led types. Furthermore, the convergence between the digital economy and secondary industry had undergone a change process of “steady growth-rapid decline-strong rebound”. With the rise of the “consumer Internet”, a good convergence effect was formed between the digital economy and secondary industry from 2005 to 2012, which promoted the transformation and upgrading of the secondary industry with its own convergence effect. However,during the period from 2013 to 2015, with the gradual release of the “consumer Internet” technology dividend, the process of the digital economy’s deep penetration into the industry had found many disruptive innovation formats that tried to “substitute” or “subvert” the traditional secondary industry, while the convergence level of the secondary industry was still high. Therefore, they led to a rapid decline in the convergence level between the digital economy and secondary industry in the short term. Moreover, the penetration path of the digital economy into secondary industry had evolved from “substitution” and “subversion” to “empowerment” and “transformation” from 2015 to 2019. Additionally, the secondary industry’s own convergence level continued to decrease during this period and the convergence level between the digital economy and secondary industry had entered a stage of rapid recovery. To conclude, it may bring new development momentum of the secondary industry that there is the good convergence between the digital economy and secondary industry on the premise that their own convergences continues to decrease.

Fig. 6

The convergence trend of the digital economy and secondary industry from 2005 to 2019.

As illustrated in Fig. 7 the convergence level between the digital economy and tertiary industry fluctuated between 0.6 and 0.8 and the B_d/B_s3 value was basically greater than 1.05 from 2005 to 2019 (except in 2018 and 2019), which was mainly embodied in the primary and intermediate convergence types led by the digital economy. The convergence level would fluctuate cyclically approximately every 7 years, which had been potentially relevant with the implementation of the China’s “five-year plan”. Specifically, in the first 7-year cycle, the convergence level was at a relatively low level from 2006 to 2009. With the rapid rise of the “consumer Internet”, the convergence degree continued to increase rapidly in 2009–2012 and reached a peak in 2012. In the second 7-year cycle, the “consumer Internet” technology dividend was gradually released from 2013 to 2015 and the convergence level fell rapidly to the lowest point. Furthermore, the convergence level had rebounded again and continued to enhance from 2015 to 2019, while the growth rates of added value and fixed investment of the tertiary industry in the second 7-year cycle remained at a high level. A preliminary conclusion could be drawn: under the premise that the own convergences of the digital economy and tertiary industry respectively continued to decrease, it could bring new development momentum to the tertiary industry in case of the good convergence between the economies.

Fig. 7

The change trend of comprehensive convergence sequences of the digital economy and tertiary industry from 2005 to 2019.

4.2 Prediction of convergence of digital economy and real economy

4.2.1 Indexes selection

In order to ensure the consistency of the prediction results with the previous empirical analysis, the selection of indexes is consistent with the “Measurement of Convergence Level of Digital Economy and Real Economy”.

4.2.2 Indexes prediction

Prediction indexes in the digital economy system and real economy subsystem

Using “Forecasting Model of Convergence of Digital Economy and Real Economy Based on GM (1,1)” to predict the indexes (mean values) in the digital economy, real economy, primary industry, secondary industry and tertiary industry subsystems. The proportion of the fixed investment in each subsystem and the growth rate of added value can be indirectly calculated by the predicted values of other indexes; thus the prediction data of the each subsystem is as shown in Table 8.

Prediction of the convergence level of the digital economy and real economy in 2020–2029

Prediction of the own convergence level of the digital economy and the real economy It can be observed from Table 9 and Fig. 8 that the own convergence levels of China’s digital economy, real economy, primary industry, secondary industry and tertiary industry would be similar with each other from 2020 to 2029. Specifically, the convergence levels of the five types of economies would continue to rise from a relatively low level to a peak from 2020 to 2025. However, the convergence level will show a downward trend year by year from 2026 to 2029, indicating that the economic growth dividends brought about by the own convergences effects will once again tend to be fully released and then the growth momentum within the each economy will gradually slow down.

Prediction of the convergence level between digital economy and real economy

Table 8
Predicted values of development indexes in each subsystem of the real economy and digital economy (mean generating)

Indexes 2020 2021 2022 2023 2024 2025 2026

D1 2.0968 2.3222 2.5719 2.8484 3.1547 3.4939 3.8695

D2 2.4142 2.7105 3.0432 3.4167 3.8361 4.3069 4.8355

D3 1.0174 1.0196 1.0218 1.0240 1.0262 1.0284 1.0306

D4 2.2441 2.5287 2.8494 3.2107 3.6179 4.0767 4.5938

D5 0.9997 0.9999 1.0001 1.0000 1.0002 1.0001 0.9999

D6 0.9758 0.9725 0.9692 0.9658 0.9626 0.9593 0.9560

D7 1.4643 1.5414 1.6226 1.7080 1.7979 1.8926 1.9923

D8 1.5845 1.6815 1.7844 1.8936 2.0094 2.1324 2.2629

Indexes 2027 2028 2029 $\bar{φ}$ C p Validity

D1 4.2855 4.7463 5.2566 0.0575 0.1087 1.0000 valid

D2 5.4290 6.0953 6.8434 0.1215 0.2763 1.0000 valid

D3 1.0328 1.0350 1.0372 0.0018 0.2433 1.0000 valid

D4 5.1764 5.8328 6.5726 0.0444 0.1125 1.0000 valid

D5 1.0000 1.0001 1.0000 / / / valid

D6 0.9528 0.9496 0.9463 / / / valid

D7 2.0972 2.2076 2.3238 0.0613 0.3103 1.0000 valid

D8 2.4013 2.5483 2.7042 0.1379 0.5107 0.9333 valid

Indexes 2020 2021 2022 2023 2024 2025 2026

S1 2.0156 2.2173 2.4393 2.6834 2.9520 3.2475 3.5725

S2 2.8113 3.2819 3.8313 4.4726 5.2213 6.0954 7.1157

S3 1.4524 1.5231 1.5973 1.6751 1.7567 1.8423 1.9320

S4 2.6188 3.2125 3.9408 4.8343 5.9302 7.2746 8.9238

S5 1.0001 0.9998 1.0003 0.9998 1.0001 1.0001 0.9999

S6 1.1032 1.1432 1.1847 1.2275 1.2721 1.3182 1.3659

S7 2.7187 3.2200 3.8137 4.5169 5.3498 6.3362 7.5045

S8 4.0464 5.2200 6.7341 8.6877 11.2083 14.4606 18.6571

Indexes 2027 2028 2029 $\bar{φ}$ C p Validity

S1 3.9301 4.3235 4.7562 0.0557 0.1316 1.0000 valid

S2 8.3068 9.6973 11.3206 0.0257 0.0455 1.0000 valid

S3 2.0261 2.1248 2.2283 0.0598 0.2925 1.0000 valid

S4 10.9469 13.4287 16.4731 0.0731 0.2906 1.0000 valid

S5 1.0001 1.0001 0.9999 / / / valid

S6 1.4155 1.4667 1.5199 / / / valid

S7 8.8882 10.5270 12.4680 0.1434 0.1806 1.0000 valid

S8 24.0720 31.0593 40.0758 0.1172 0.2597 1.0000 valid

Indexes 2020 2021 2022 2023 2024 2025 2026

SF1 1.7038 1.8282 1.9616 2.1049 2.2585 2.4234 2.6004

SF2 2.2701 2.284 2.2313 2.3562 2.5133 2.8912 3.2365

SF3 0.7170 0.6891 0.6622 0.6364 0.6116 0.5878 0.5649

SF4 3.1243 3.6574 4.2814 5.0118 5.8669 6.8679 8.0396

SF5 1.0006 1.0000 0.9994 1.0006 0.9995 1.0000 1.0003

SF6 1.0781 0.9629 0.8350 0.7827 0.7411 0.7567 0.7519

SF7 1.2955 1.3398 1.3856 1.4330 1.4820 1.5327 1.5852

SF8 1.9874 2.1928 2.4195 2.6696 2.9455 3.2500 3.5859

Indexes 2027 2028 2029 $\bar{φ}$ C p Validity

SF1 2.7902 2.9939 3.2125 0.0692 0.2119 1.0000 valid

SF2 3.4264 3.5129 3.6113 0.0301 0.1741 1.0000 valid

SF3 0.5429 0.5217 0.5014 0.0116 0.0750 1.0000 valid

SF4 9.4113 11.0170 12.8967 0.1913 0.1935 1.0000 valid

SF5 0.9997 0.9999 1.0000 / / / valid

SF6 0.7066 0.6431 0.5868 / / / valid

SF7 1.6394 1.6955 1.7534 0.0203 0.1702 1.0000 valid

SF8 3.9566 4.3655 4.8168 0.0675 0.1795 1.0000 valid

Indexes 2020 2021 2022 2023 2024 2025 2026

SS1 1.8454 2.0038 2.1757 2.3624 2.5651 2.7851 3.0241

SS2 2.1466 2.3956 2.6714 2.9768 3.3151 3.6898 4.1047

SS3 1.0213 1.0386 1.0425 1.0466 1.0511 1.0617 1.0722

SS4 1.7711 1.9174 2.0758 2.2473 2.4329 2.6339 2.8515

SS5 0.9998 1.0004 0.9998 1.0001 1.0000 0.9996 1.0002

SS6 0.8338 0.8260 0.8176 0.8087 0.7995 0.7898 0.7800

SS7 1.8588 2.0285 2.2137 2.4158 2.6364 2.8771 3.1397

SS8 2.1910 2.4294 2.6937 2.9867 3.3116 3.6719 4.0713

Indexes 2027 2028 2029 $\bar{φ}$ C p Validity

SS1 3.2836 3.5653 3.8712 0.0805 0.2087 1.0000 valid

SS2 4.5643 5.0732 5.6369 0.1585 0.3336 1.0000 valid

SS3 1.0802 1.0925 1.1233 0.0456 0.1970 1.0000 valid

SS4 3.0871 3.3421 3.6183 0.0364 0.1583 1.0000 valid

SS5 1.0001 0.9999 1.0000 / / / valid

SS6 0.7699 0.7595 0.7491 / / / valid

SS7 3.4264 3.7392 4.0806 0.0643 0.2635 1.0000 valid

SS8 4.5142 5.0053 5.5498 0.0478 0.1255 1.0000 valid

Indexes 2020 2021 2022 2023 2024 2025 2026

ST1 2.3607 2.6639 3.0061 3.3922 3.8279 4.3195 4.8743

ST2 2.5056 2.8285 3.1929 3.6044 4.0688 4.5932 5.1850

ST3 1.3082 1.3553 1.4040 1.4545 1.5069 1.5611 1.6172

ST4 2.3043 2.6097 2.9555 3.3472 3.7907 4.2931 4.8620

ST5 1.0000 1.0000 1.0001 1.0000 1.0000 0.9999 1.0000

ST6 1.0107 1.0128 1.0148 1.0169 1.0189 1.0210 1.0231

ST7 1.4866 1.5678 1.6535 1.7439 1.8391 1.9396 2.0456

ST8 1.5354 1.6221 1.7136 1.8103 1.9124 2.0203 2.1343

Indexes 2027 2028 2029 $\bar{φ}$ C p Validity

ST1 5.5004 6.2069 7.0041 0.0533 0.0795 1.0000 valid

ST2 5.8532 6.6074 7.4589; 0.0188 0.2671 1.0000 valid

ST3 1.6754 1.7357 1.7982 0.0142 0.1165 1.0000 valid

ST4 5.5063 6.2360 7.0624 0.0420 0.0881 1.0000 valid

ST5 1.0001 1.0000 1.0000 / / / valid

ST6 1.0252 1.0273 1.0294 / / / valid

ST7 2.1574 2.2753 2.3996 0.0524 0.2573 1.0000 valid

ST8 2.2547 2.3819 2.5162 0.1459 0.5640 0.9333 valid

Indexes	2020	2021	2022	2023	2024	2025	2026
D1	2.0968	2.3222	2.5719	2.8484	3.1547	3.4939	3.8695
D2	2.4142	2.7105	3.0432	3.4167	3.8361	4.3069	4.8355
D3	1.0174	1.0196	1.0218	1.0240	1.0262	1.0284	1.0306
D4	2.2441	2.5287	2.8494	3.2107	3.6179	4.0767	4.5938
D5	0.9997	0.9999	1.0001	1.0000	1.0002	1.0001	0.9999
D6	0.9758	0.9725	0.9692	0.9658	0.9626	0.9593	0.9560
D7	1.4643	1.5414	1.6226	1.7080	1.7979	1.8926	1.9923
D8	1.5845	1.6815	1.7844	1.8936	2.0094	2.1324	2.2629
Indexes	2027	2028	2029	$\bar{φ}$	C	p	Validity
D1	4.2855	4.7463	5.2566	0.0575	0.1087	1.0000	valid
D2	5.4290	6.0953	6.8434	0.1215	0.2763	1.0000	valid
D3	1.0328	1.0350	1.0372	0.0018	0.2433	1.0000	valid
D4	5.1764	5.8328	6.5726	0.0444	0.1125	1.0000	valid
D5	1.0000	1.0001	1.0000	/	/	/	valid
D6	0.9528	0.9496	0.9463	/	/	/	valid
D7	2.0972	2.2076	2.3238	0.0613	0.3103	1.0000	valid
D8	2.4013	2.5483	2.7042	0.1379	0.5107	0.9333	valid
Indexes	2020	2021	2022	2023	2024	2025	2026
S1	2.0156	2.2173	2.4393	2.6834	2.9520	3.2475	3.5725
S2	2.8113	3.2819	3.8313	4.4726	5.2213	6.0954	7.1157
S3	1.4524	1.5231	1.5973	1.6751	1.7567	1.8423	1.9320
S4	2.6188	3.2125	3.9408	4.8343	5.9302	7.2746	8.9238
S5	1.0001	0.9998	1.0003	0.9998	1.0001	1.0001	0.9999
S6	1.1032	1.1432	1.1847	1.2275	1.2721	1.3182	1.3659
S7	2.7187	3.2200	3.8137	4.5169	5.3498	6.3362	7.5045
S8	4.0464	5.2200	6.7341	8.6877	11.2083	14.4606	18.6571
Indexes	2027	2028	2029	$\bar{φ}$	C	p	Validity
S1	3.9301	4.3235	4.7562	0.0557	0.1316	1.0000	valid
S2	8.3068	9.6973	11.3206	0.0257	0.0455	1.0000	valid
S3	2.0261	2.1248	2.2283	0.0598	0.2925	1.0000	valid
S4	10.9469	13.4287	16.4731	0.0731	0.2906	1.0000	valid
S5	1.0001	1.0001	0.9999	/	/	/	valid
S6	1.4155	1.4667	1.5199	/	/	/	valid
S7	8.8882	10.5270	12.4680	0.1434	0.1806	1.0000	valid
S8	24.0720	31.0593	40.0758	0.1172	0.2597	1.0000	valid
Indexes	2020	2021	2022	2023	2024	2025	2026
SF1	1.7038	1.8282	1.9616	2.1049	2.2585	2.4234	2.6004
SF2	2.2701	2.284	2.2313	2.3562	2.5133	2.8912	3.2365
SF3	0.7170	0.6891	0.6622	0.6364	0.6116	0.5878	0.5649
SF4	3.1243	3.6574	4.2814	5.0118	5.8669	6.8679	8.0396
SF5	1.0006	1.0000	0.9994	1.0006	0.9995	1.0000	1.0003
SF6	1.0781	0.9629	0.8350	0.7827	0.7411	0.7567	0.7519
SF7	1.2955	1.3398	1.3856	1.4330	1.4820	1.5327	1.5852
SF8	1.9874	2.1928	2.4195	2.6696	2.9455	3.2500	3.5859
Indexes	2027	2028	2029	$\bar{φ}$	C	p	Validity
SF1	2.7902	2.9939	3.2125	0.0692	0.2119	1.0000	valid
SF2	3.4264	3.5129	3.6113	0.0301	0.1741	1.0000	valid
SF3	0.5429	0.5217	0.5014	0.0116	0.0750	1.0000	valid
SF4	9.4113	11.0170	12.8967	0.1913	0.1935	1.0000	valid
SF5	0.9997	0.9999	1.0000	/	/	/	valid
SF6	0.7066	0.6431	0.5868	/	/	/	valid
SF7	1.6394	1.6955	1.7534	0.0203	0.1702	1.0000	valid
SF8	3.9566	4.3655	4.8168	0.0675	0.1795	1.0000	valid
Indexes	2020	2021	2022	2023	2024	2025	2026
SS1	1.8454	2.0038	2.1757	2.3624	2.5651	2.7851	3.0241
SS2	2.1466	2.3956	2.6714	2.9768	3.3151	3.6898	4.1047
SS3	1.0213	1.0386	1.0425	1.0466	1.0511	1.0617	1.0722
SS4	1.7711	1.9174	2.0758	2.2473	2.4329	2.6339	2.8515
SS5	0.9998	1.0004	0.9998	1.0001	1.0000	0.9996	1.0002
SS6	0.8338	0.8260	0.8176	0.8087	0.7995	0.7898	0.7800
SS7	1.8588	2.0285	2.2137	2.4158	2.6364	2.8771	3.1397
SS8	2.1910	2.4294	2.6937	2.9867	3.3116	3.6719	4.0713
Indexes	2027	2028	2029	$\bar{φ}$	C	p	Validity
SS1	3.2836	3.5653	3.8712	0.0805	0.2087	1.0000	valid
SS2	4.5643	5.0732	5.6369	0.1585	0.3336	1.0000	valid
SS3	1.0802	1.0925	1.1233	0.0456	0.1970	1.0000	valid
SS4	3.0871	3.3421	3.6183	0.0364	0.1583	1.0000	valid
SS5	1.0001	0.9999	1.0000	/	/	/	valid
SS6	0.7699	0.7595	0.7491	/	/	/	valid
SS7	3.4264	3.7392	4.0806	0.0643	0.2635	1.0000	valid
SS8	4.5142	5.0053	5.5498	0.0478	0.1255	1.0000	valid
Indexes	2020	2021	2022	2023	2024	2025	2026
ST1	2.3607	2.6639	3.0061	3.3922	3.8279	4.3195	4.8743
ST2	2.5056	2.8285	3.1929	3.6044	4.0688	4.5932	5.1850
ST3	1.3082	1.3553	1.4040	1.4545	1.5069	1.5611	1.6172
ST4	2.3043	2.6097	2.9555	3.3472	3.7907	4.2931	4.8620
ST5	1.0000	1.0000	1.0001	1.0000	1.0000	0.9999	1.0000
ST6	1.0107	1.0128	1.0148	1.0169	1.0189	1.0210	1.0231
ST7	1.4866	1.5678	1.6535	1.7439	1.8391	1.9396	2.0456
ST8	1.5354	1.6221	1.7136	1.8103	1.9124	2.0203	2.1343
Indexes	2027	2028	2029	$\bar{φ}$	C	p	Validity
ST1	5.5004	6.2069	7.0041	0.0533	0.0795	1.0000	valid
ST2	5.8532	6.6074	7.4589;	0.0188	0.2671	1.0000	valid
ST3	1.6754	1.7357	1.7982	0.0142	0.1165	1.0000	valid
ST4	5.5063	6.2360	7.0624	0.0420	0.0881	1.0000	valid
ST5	1.0001	1.0000	1.0000	/	/	/	valid
ST6	1.0252	1.0273	1.0294	/	/	/	valid
ST7	2.1574	2.2753	2.3996	0.0524	0.2573	1.0000	valid
ST8	2.2547	2.3819	2.5162	0.1459	0.5640	0.9333	valid

Table 9

The own convergence sequence of the digital economy and real economy in 2020–2029

Category	2020	2021	2022	2023	2024	2025
Real economy (B_se)	0.5856	0.6201	0.6670	0.7339	0.8364	0.9740
Primary industry (B_s1e)	0.5743	0.6273	0.6981	0.7730	0.8626	0.9483
Secondary industry (B_s2e)	0.6164	0.6501	0.6952	0.7591	0.8578	0.9721
Tertiary industry (B_s3e)	0.5998	0.6328	0.6774	0.7403	0.8356	0.9717
Digital economy (B_de)	0.6698	0.6887	0.7164	0.7583	0.8229	0.9265
Category	2026	2027	2028	2029	Own convergence sequence
Real economy (B_se)	0.8112	0.6855	0.5938	0.5231	0.7031
Primary industry (B_s1e)	0.8206	0.7081	0.6143	0.5364	0.7163
Secondary industry (B_s2e)	0.8210	0.7105	0.6272	0.5597	0.7269
Tertiary industry (B_s3e)	0.8229	0.6970	0.6054	0.5355	0.7118
Digital economy (B_de)	0.9089	0.7587	0.6332	0.5305	0.7414

Fig. 8

Changes in the own convergence sequence of the digital economy and real economy in 2020–2029.

It is astonishing to identify that from Tables 10, 11 and Figs. 9–12 that the comprehensive convergence coefficient between the digital economy and real economy (also the primary, secondary and tertiary industries) from 2020 to 2029 shows a trend of first increasing and then decreasing, but the overall, there are in the intermediate convergence stage. The ratio of the digital economy own convergence coefficient (B_de) to the real economy (also the primary, secondary and tertiary industries) own convergence coefficients (i.e., B_se, B_s1e, B_s2e and B_s3e respectively) is greater than 0.95, showing that the two economies are in a digital economy-led type or a synchronized development type. Furthermore, the comprehensive convergence coefficient of the digital economy and real economy would continue to rise between 0.6 and 0.7, and the overall convergence is in the primary convergence stage led by digital economy from 2020 to 2023. The comprehensive convergence coefficient will grow rapidly between 0.8 and 1 and reach a peak from 2024 to 2025, which is overall in good convergence type led by the digital economy. Correspondingly, the comprehensive convergence coefficient of the digital economy and primary industry (also the secondary and tertiary industries) from 2020 to 2025 continues to rise from the primary convergence to the high-quality convergence, and will reach a peak in 2025.Moreover, the overall convergence coefficient of the digital economy and real economy (also the primary, secondary and tertiary industries) would begin to fall and gradually to the barely coordinated level from 2026 to 2029.

Table 10

Comprehensive convergence sequence between the digital economy and real economy in 2020–2029

Category	2020	2021	2022	2023	2024	2025
Real economy	0.6079	0.6329	0.6695	0.7254	0.8151	0.9652
Primary industry	0.5602	0.6047	0.6717	0.7435	0.8471	0.9693
Secondary industry	0.6156	0.6386	0.6745	0.7290	0.8173	0.9604
Tertiary industry	0.6118	0.6339	0.6675	0.7190	0.8025	0.9444
Category	2026	2027	2028	2029	Comprehensive convergence degree
Real economy	0.9322	0.7209	0.5695	0.4579	0.7097
Primary industry	0.9109	0.7110	0.5622	0.4544	0.7035
Secondary industry	0.9347	0.7255	0.5755	0.4652	0.7136
Tertiary industry	0.9348	0.7212	0.5682	0.4561	0.7059

Table 11

The ratio series of the convergence between the digital economy and real economy in 2020–2029

Category	2020	2021	2022	2023	2024
B_de/B_se	1.1438	1.1106	1.0741	1.0332	0.9839
B_de/B_s1e	1.1663	1.0979	1.0262	0.9810	0.9540
B_de/B_s2e	1.0866	1.0594	1.0305	0.9989	0.9593
B_de/B_s3e	1.1167	1.0883	1.0576	1.0243	0.9848
Category	2025	2026	2027	2028	2029
B_de/B_se	0.9512	1.1204	1.1068	1.0664	1.0141
B_de/B_s1e	0.9770	1.1076	1.0715	1.0308	0.9890
B_de/B_s2e	0.9531	1.1071	1.0678	1.0096	0.9478
B_de/B_s3e	0.9535	1.1045	1.0885	1.0459	0.9907

Fig. 9

The convergence trend between the digital economy and real economy in 2020–2029.

Fig. 10

The convergence trend between the digital economy and primary industry from 2020 to 2029.

Fig. 11

The convergence trend between the digital economy and secondary industry in 2020–2029.

Fig. 12

The convergence trend between the digital economy and tertiary industry from 2020 to 2029.

4.3 Analysis of factors affecting convergence of digital economy and real economy

4.3.1 Index selection

There are 7 latent variables in accordance with the “The analysis model of factors affecting convergence of the digital economy and real economy”. Each latent variable needs to correspond to the the index to carry out empirical analysis. On the basis of referring to relevant literature, an observable variable is selected for each latent variable in the paper. The specific indexes are shown in Table 12 [69 , 76–78].

Table 12
Indexes of factors affecting the convergence of the digital economy and real economy

Latent variable Variable Number Observable Variable Index Number Unit Effects

Innovation capability of the real economy CQ1 Full-time equivalent of R&D personnel CQ11 Person-year Positive

Number of R&D topics CQ12 Positive

Number of valid invention patents CQ13 Positive

Development level of the real economy CQ2 Added value CQ21 Million Yuan Positive

Fixed asset investment CQ22 Million Yuan Positive

Number of employees CQ23 Thousands persons Positive

Number of enterprises CQ24 Positive

Innovation capability of the digital economy CQ3 Full-time equivalent of R&D personnel CQ31 Person-year Positive

Number of R&D topics CQ31 Positive

Number of valid invention patents CQ33 Positive

Development efficiency of the real economy CQ4 Per capita output CQ41 Thousand Yuan Positive

Value added as a proportion of GDP CQ42 % Positive

Increased value growth rate CQ43 % Positive

Development efficiency of the digital economy CQ5 Added value CQ51 Million Yuan Positive

Fixed asset investment CQ52 Million Yuan Positive

Number of employees CQ53 Thousands persons Positive

Number of enterprises CQ54 Positive

Macro development environment CQ6 Policy releases of digital economy CQ61 Positive

Internet penetration rate CQ62 % Positive

Data accumulation CQ63 ZB Positive

The convergence level of the digital economy and real economy C The convergence degree between the digital economy and real economy C / /

Latent variable	Variable Number	Observable Variable	Index Number	Unit	Effects
Innovation capability of the real economy	CQ1	Full-time equivalent of R&D personnel	CQ11	Person-year	Positive
		Number of R&D topics	CQ12		Positive
		Number of valid invention patents	CQ13		Positive
Development level of the real economy	CQ2	Added value	CQ21	Million Yuan	Positive
		Fixed asset investment	CQ22	Million Yuan	Positive
		Number of employees	CQ23	Thousands persons	Positive
		Number of enterprises	CQ24		Positive
Innovation capability of the digital economy	CQ3	Full-time equivalent of R&D personnel	CQ31	Person-year	Positive
		Number of R&D topics	CQ31		Positive
		Number of valid invention patents	CQ33		Positive
Development efficiency of the real economy	CQ4	Per capita output	CQ41	Thousand Yuan	Positive
		Value added as a proportion of GDP	CQ42	%	Positive
		Increased value growth rate	CQ43	%	Positive
Development efficiency of the digital economy	CQ5	Added value	CQ51	Million Yuan	Positive
		Fixed asset investment	CQ52	Million Yuan	Positive
		Number of employees	CQ53	Thousands persons	Positive
		Number of enterprises	CQ54		Positive
Macro development environment	CQ6	Policy releases of digital economy	CQ61		Positive
		Internet penetration rate	CQ62	%	Positive
		Data accumulation	CQ63	ZB	Positive
The convergence level of the digital economy and real economy	C	The convergence degree between the digital economy and real economy	C	/	/

4.3.2 Data sources and descriptive statistics

The paper mainly uses the 2005–2019 China Statistical Yearbook, the 2005–2019 China National Economic and Social Development Statistical Bulletin, the first to fourth National Economic Census Communiqu $\overset{´}{e}$ , the 2005–2019 China Statistics Yearbook of Science and Technology, 2018–2019 “Software and Information Technology Service Industry Statistical Annual Report”, the 10th to 46th “Statistical Report on China’s Internet Development Status” as well as the data disclosed on the official websites of the State Council of China and the governments of various provinces (autonomous regions) as the basis for analysis [71–73 , 80]. Before parameter estimation, we use SPSS software to descriptively analyze the original data from five dimensions: full range, maximum value, minimum value, mean value and standard deviation, to understand the concentrated and discrete state of the data sample distribution (as shown in Table 13). From the statistical results, all observable variables have no extreme outliers, while the sample distribution is concentrated and the discrete state is good. Therefore, it can be used for the parameter estimation of the hypotheses.

Table 13
Descriptive statistical results of observable variables

Index number Sample size Full range Maximum value Minimum value Mean value Standard deviation

C 15 0.2617 0.9365 0.6748 0.7655 0.0661

CQ11 15 230962 464365 233403 360729 78641

CQ12 15 58929 59024 117953 83076 19344

CQ13 15 6713830 7070922 357092 2630549 2161015

CQ21 15 6867669 8463025 1595356 4615810 2175552

CQ22 15 5410025 6133011 722986 3524720 2018756

CQ23 15 22672 763486 740814 754654 7786

CQ24 15 18659826 24214077 5554251 12399669 6347662

CQ31 15 61413 62183 770 22323 26030

CQ32 15 4742 5525 783 2174 1508

CQ33 15 825746 888470 62724 353986 262420

CQ41 15 588 1079 491 778 186

CQ42 15 0.1675 0.3158 0.1483 0.2071 0.0552

CQ43 15 0.1489 0.2300 0.0811 0.1545 0.0433

CQ51 15 1167793 1445626 277833 848998 382446

CQ52 15 226930 254904 27974 109335 73299

CQ53 15 7732 13388 5656 10377 2594

CQ54 15 972562 1066134 93572 371862 307437

CQ61 15 137 138 1 38 50

CQ62 15 0.5440 0.6290 0.0850 0.3845 0.1773

CQ63 15 37950 38000 50 9082 12485

Index number	Sample size	Full range	Maximum value	Minimum value	Mean value	Standard deviation
C	15	0.2617	0.9365	0.6748	0.7655	0.0661
CQ11	15	230962	464365	233403	360729	78641
CQ12	15	58929	59024	117953	83076	19344
CQ13	15	6713830	7070922	357092	2630549	2161015
CQ21	15	6867669	8463025	1595356	4615810	2175552
CQ22	15	5410025	6133011	722986	3524720	2018756
CQ23	15	22672	763486	740814	754654	7786
CQ24	15	18659826	24214077	5554251	12399669	6347662
CQ31	15	61413	62183	770	22323	26030
CQ32	15	4742	5525	783	2174	1508
CQ33	15	825746	888470	62724	353986	262420
CQ41	15	588	1079	491	778	186
CQ42	15	0.1675	0.3158	0.1483	0.2071	0.0552
CQ43	15	0.1489	0.2300	0.0811	0.1545	0.0433
CQ51	15	1167793	1445626	277833	848998	382446
CQ52	15	226930	254904	27974	109335	73299
CQ53	15	7732	13388	5656	10377	2594
CQ54	15	972562	1066134	93572	371862	307437
CQ61	15	137	138	1	38	50
CQ62	15	0.5440	0.6290	0.0850	0.3845	0.1773
CQ63	15	37950	38000	50	9082	12485

4.3.3 Parameter estimation

Using SmartPLS software to estimate the parameters of the hypotheses, we could obtain the explanatory ability of latent variables for observable variables and test the hypotheses in “3.3 Analysis Model of Factors Affecting Convergence of Digital economy and Real economy based on PLS-SEM”. The specific parameter estimation results are as described in Table 14 andFig. 13.

Table 14
Estimation of the path coefficients of latent variables in the convergence between the digital economy and real economy

C CQ1 CQ2 CQ3 CQ4 CQ5 CQ6

C

CQ1 0.987

CQ2 0.205

CQ3 0.978 0.054 0.982

CQ4 0.499

CQ5 0.988

CQ6 0.451

Fig. 13

Model path coefficients of factors affecting the convergence between the digital economy and real economy.

When the outer loading is greater than 0.7,latent variables have the ability to explain the internal meaning of observable variables. It can be seen from Fig. 12 that the outer loadings of the 21 observable variables are all greater than 0.7, which can effectively reflect the explanatory ability of latent variables for observable variables and have a good measurement effect.

4.3.4 Model checking

Predictive ability test

We use the AVE, the R² value and the Composite reliability to test the predictive ability of the model. When the AVE value is greater than 0.5, the R² value is greater than 0.65 and the composite reliability value is greater than 0.7, the model could be regarded as reliable. The verification of the model is as illustrated in Table 15.

It can be seen from Table 15 that the AVE value of each latent variable in the model is greater than 0.5, the R2 value is greater than 0.65 and the composite reliability value is greater than 0.7, indicating that the observable variables are highly effective in prediction.

Bootstrapping validity verification

The Bootstrapping in the SmartPLS software is adopted to verify the validity of outer loadings and path coefficients in the model. The specific process is: setting the sampling frequency to 1000 times and verifying the path coefficients and outer loadings at significance levels of 1%, 5%, and 10%. A two-tailed T-test would be used to verify the significance of the coefficients, which at least achieve at the level of 10%. The specific verification results are as described in Tables 16 and 17.

From the above tables, we can see obviously that the path coefficients have high T statistics and all passed the T-test at least at significance level of 5%, indicating that the overall validity of the hypotheses is in the high level.That is, the above 8 hypotheses in “3.3 Analysis Model of Factors Affecting Convergence of Digital economy and Real economy based on PLS-SEM” are all valid.Furthermore, the outer loadings also have high T statistic and even pass T-test at the level of 1%, which shows that it has a high accuracy and is in significantly good condition. Accordingly, the model is reliable through the above-mentioned predictive ability test and validity test.

Table 15
Predictive ability test of the analysis model of factors affecting the convergence of the digital economy and real economy

Latent variable R2 AVE Composite reliability

C 0.926 1 1

CQ1 0.957 0.954 0.984

CQ2 0.976 0.955 0.988

CQ3 / 0.952 0.984

CQ4 0.974 0.924 0.973

CQ5 0.986 0.930 0.981

CQ6 0.964 0.892 0.961

Latent variable	R2	AVE	Composite reliability
C	0.926	1	1
CQ1	0.957	0.954	0.984
CQ2	0.976	0.955	0.988
CQ3	/	0.952	0.984
CQ4	0.974	0.924	0.973
CQ5	0.986	0.930	0.981
CQ6	0.964	0.892	0.961

Table 16

Bootstrapping verification of model path coefficients

	Initial sample (O)	Sample mean (M)	Standard deviation (STDEV)	T statistics (\|O/STDEV\|)	P value
CQ1 ->CQ4	0.987	0.988	0.006	175.583^★★★	0
CQ2 ->C	0.205	0.204	0.008	126.826^★★	0
CQ3 ->CQ1	0.978	0.979	0.01	98.842^★★★	0
CQ3 ->CQ5	0.054	0.065	0.129	21.163^★★	0.003
CQ3 ->CQ6	0.982	0.981	0.01	94.483^★★★	0
CQ4 ->CQ5	0.499	0.481	0.179	2.782^★★★	0.006
CQ5 ->CQ2	0.988	0.989	0.005	197.63^★★	0

Note: ^★★★, ^★★, ^★ indicate significant statistical levels of 1%, 5%, and 10% respectively.

Table 17

Bootstrapping verification of outer loadings

	Initial sample (O)	Sample mean (M)	Standard deviation (STDEV)	T statistics (\|O/STDEV\|)	P value
CQ11< - CQ1	0.964	0.967	0.008	121.067^★★★	0
CQ12< - CQ1	0.979	0.978	0.014	68.559^★★★	0
CQ13< - CQ1	0.987	0.989	0.003	303.659^★★★	0
CQ21< - CQ2	0.99	0.99	0.006	157.346^★★★	0
CQ22< - CQ2	0.986	0.987	0.008	119.938^★★★	0
CQ23< - CQ2	0.97	0.972	0.01	97.515^★★★	0
CQ24< - CQ2	0.962	0.964	0.011	84.513^★★★	0
CQ31< - CQ3	0.958	0.959	0.02	48.904^★★★	0
CQ32< - CQ3	0.98	0.981	0.008	124.075^★★★	0
CQ33< - CQ3	0.99	0.989	0.006	154.02^★★★	0
CQ41< - CQ4	0.992	0.992	0.003	316.122^★★★	0
CQ42< - CQ4	0.961	0.964	0.008	114.225^★★★	0
CQ43< - CQ4	0.93	0.935	0.025	37.911^★★★	0
CQ51< - CQ5	0.994	0.995	0.002	576.562^★★★	0
CQ52< - CQ5	0.991	0.991	0.002	412.908^★★★	0
CQ53< - CQ5	0.931	0.935	0.018	52.931^★★★	0
CQ54< - CQ5	0.939	0.943	0.017	54.337^★★★	0
CQ61< - CQ6	0.964	0.964	0.015	62.723^★★★	0
CQ62< - CQ6	0.901	0.905	0.028	32.233^★★★	0
CQ63< - CQ6	0.966	0.97	0.008	126.681^★★★	0
C< - C	1	1	0

Note: ^★★★, ^★★, ^★ indicate significant statistical levels of 1%, 5%, and 10% respectively.

4.3.5 Result analysis

In accordance with the model’s predictive ability test and validity test, the total effects of each latent variable and the outer loadings are further calculated as demonstrated in Tables 18 and 19.

Table 18
The total effect of latent variables

C CQ1 CQ2 CQ3 CQ4 CQ5 CQ6

C

CQ1 0.1 0.487 0.987 0.493

CQ2 0.205

CQ3 0.198 0.978 0.966 0.966 0.978 0.982

CQ4 0.101 0.493 0.499

CQ5 0.202 0.988

CQ6 0.091 0.445 0.451

	C	CQ1	CQ2	CQ4	CQ5	CQ6
C
CQ1	0.1		0.487	0.987	0.493
CQ2	0.205
CQ3	0.198	0.978	0.966	0.966	0.978	0.982
CQ4	0.101		0.493		0.499
CQ5	0.202		0.988
CQ6	0.091		0.445		0.451

Table 19

The outer loadings of the model

Models	Load	Models	Load
CQ11< -CQ1	0.964	CQ41< -CQ4	0.992
CQ12< -CQ1	0.979	CQ42< -CQ4	0.961
CQ13< -CQ1	0.987	CQ43< -CQ4	0.93
CQ21< -CQ2	0.99	CQ51< -CQ5	0.994
CQ22< -CQ2	0.986	CQ52< -CQ5	0.991
CQ23< -CQ2	0.97	CQ53< -CQ5	0.931
CQ24< -CQ2	0.962	CQ54< -CQ5	0.939
CQ31< -CQ3	0.958	CQ61< -CQ6	0.964
CQ32< -CQ3	0.98	CQ62< -CQ6	0.901
CQ33< -CQ3	0.99	CQ63< -CQ6	0.966
/	/	C	1

Considering the Tables 14, 18, 19 and the model verification results, the following conclusions could be drawn on the hypotheses of the factors affecting the convergence between the digital economy and real economy:

The total effects of the innovation capability of the digital economy on that of the real economy, the macro development environment and the development level of the digital economy are 0.978, 0.982 and 0.978 respectively. Assuming H1, H2, and H3 are verified, it explains that the innovation capability of the digital economy exerts positive influence on the convergence of the digital economy and real economy. In other words, the key to enhance the high-quality developments of the digital economy and real economy is promoting the continuous improvement of innovation capabilities in the digital economy. Additionally, it can be found from Table 19 that the outer loading of innovation capability of the digital economy on the full-time equivalent of R&D personnel, the number of R&D topics and the number of effective invention patents are 0.958, 0.98, and 0.99 respectively, explaining that the technical innovation is the main focus of facilitating the innovation capability of the digital economy.

The total effect of the innovation capability of the real economy on the development efficiency of the digital economy is 0.987, the effect of the development efficiency of the digital economy on the development level of the digital economy is 0.499, while the effect of the macro development environment on the development level of the digital economy is 0.451. Assuming H4 and H5 and H8 are verified, it could be concluded that the innovation capability of the real economy has a greater impact on the development efficiency of the digital economy that could further enhance the development level of the digital economy. The real economy is essentially an important “supply-side” factor to improve the efficiency of the digital economy while the macro environment has a relatively smaller influence on the development level of the digital economy.

The total effect of the development level of the digital economy on that of the real economy is 0.988 and the effect of the development level of the real economy on the convergence of the digital economy and real economy is 0.205, indicating that the development level of the digital economy has a greater impact on that of the real economy, and then affecting the convergence level between the two economies through the real economy. Meanwhile, although the development level of the real economy has a small absolute value on the convergence of the two economics, it could be identified from Table 18 that the total effects of other latent variables on the convergence level are less than that of the real economy. In other words, the effect on the convergence level are mainly affected by the development level of the real economy (0.205), followed by the development level of the digital economy (0.202) and the innovation capability of the digital economy (0.198), which reflects that the digital economy also has a positive influence on the convergence between the two economies.

In addition to the H1-H8 hypotheses, other crucial factors (i.e., the total effects are greater than 0.9) are also identified in the Table 18: The total effect of the innovation capability of the digital economy on the development efficiency of the digital economy is 0.966, reflecting that the innovation capability has a great positive impact on the development efficiency of the digital economy. The effect of the innovation capability of the digital economy on the development level of the real economy is also 0.966, indicating that the innovation capability is the key factor in boosting the development level of the real economy.

5 Research conclusion

This paper aims to explore the convergence level of the digital economy and real economy as well as its influencing factors. The following three main conclusions could be drawn from the empirical research:

First of all, the convergence between the digital economy and real economy (including the three industries) showed cyclical fluctuations in the unit of about 5 years. It could be found that the convergences of digital economy and the real economy had continued to decrease, indicating each economy could no longer rely solely on the convergence within its own to achieve sustained and healthy economic developments. Accordingly, the each economy would showed a better development trend if there was good convergence between economies; however, the economy may lag behind in development if the convergence effect was not strong. In a short, the convergence of the digital economy and real economy has become the main driving force for the sustained and healthy development.

Secondly, it could predict that the convergence level between the real economy and digital economy, showing that an upward trend firstly and then a downward trend from 2020 and 2029. To sum up, it demonstrates the convergence of the digital economy and real economy may face the problem of insufficient motivation and would continue to weaken from 2025 considering the current internal and external environments, which would exert negative impacts on the healthy development of two economies.

Finally, the paper constructs an analysis model of factors affecting the convergence based on PLS-SEM. It could be identified that the convergence level is impacted positively to varying degrees by the innovation capabilities of the two economies, the development efficiency of the digital economy, the macro development environment as well as the development level of the two economies. Therefore, the paper puts forward suggestion on boosting the convergence level, i.e., effectively enhancing the knowledge diffusion of innovative activities to achieve innovation convergence, creating a good macro development environment, regarding the real economy as a key supply factor and foothold for the convergence as well as improving the efficiency of innovation convergence, for instance: organizational convergence innovation, technology convergence innovation and business model convergence innovation.

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The convergence level and influencing factors of China’s digital economy and real economy based on grey model and PLS- SEM

Abstract

Keywords

1 Introduction

2.1 Connotation of industrial convergence

2.2 Types of industrial convergence

2.3 Summary of theoretical research

2.3.1 Grey relational analysis

3.1 A measurement model of convergence level of digital economy and real economy based on the Deng’s correlation analysis model

3.3.1 Hypotheses of internal factors

3.3.2 Hypotheses of direct factors

3.3.3 Hypotheses of environmental factors

4.1 Measurement of convergence level of digital economy and real economy

4.1.1 Index selection

4.2.1 Indexes selection

4.2.2 Indexes prediction

4.3.1 Index selection

Table 14 Estimation of the path coefficients of latent variables in the convergence between the digital economy and real economy C CQ1 CQ2 CQ3 CQ4 CQ5 CQ6 C CQ1 0.987 CQ2 0.205 CQ3 0.978 0.054 0.982 CQ4 0.499 CQ5 0.988 CQ6 0.451

Table 18 The total effect of latent variables C CQ1 CQ2 CQ3 CQ4 CQ5 CQ6 C CQ1 0.1 0.487 0.987 0.493 CQ2 0.205 CQ3 0.198 0.978 0.966 0.966 0.978 0.982 CQ4 0.101 0.493 0.499 CQ5 0.202 0.988 CQ6 0.091 0.445 0.451

References

Table 14
Estimation of the path coefficients of latent variables in the convergence between the digital economy and real economy

C CQ1 CQ2 CQ3 CQ4 CQ5 CQ6

C

CQ1 0.987

CQ2 0.205

CQ3 0.978 0.054 0.982

CQ4 0.499

CQ5 0.988

CQ6 0.451

Table 18
The total effect of latent variables

C CQ1 CQ2 CQ3 CQ4 CQ5 CQ6

C

CQ1 0.1 0.487 0.987 0.493

CQ2 0.205

CQ3 0.198 0.978 0.966 0.966 0.978 0.982

CQ4 0.101 0.493 0.499

CQ5 0.202 0.988

CQ6 0.091 0.445 0.451