Compilation of a City-Level & Four-Digit Industry Code MRIO Table Based on Firm-Level Data

Abstract

Scholars have attempted to compile various multi-region input-output (MRIO) tables for different countries. However, due to city-level data scarcity and methodology constraints, almost no MRIO table covers a large number of cities with more disaggregated sectors in countries with large economies, such as China. Based on two large-scale firm-level datasets, the China Annual Survey of Industrial Firms (CASIF) survey and the China Customs Data (CCD) database, from 2000 to 2013, this paper uses China as a case study and presents a new compilation method to construct an MRIO table covering 284 prefecture-level administrative cities and 334 four-digit sectors, which is by far the most comprehensive MRIO table with the largest number of cities and the most segmented industries in China. Unlike existing MRIO tables constructed based on provincial single-region IO (SRIO) tables, we use information along with various linear constraints implied by sector-level and firm-level statistics. This paper expands on the direct decomposition method by developing auxiliary econometric models necessary for estimations and consistency adjustment. In addition, a comparative analysis shows the reliability of our method, which guarantees better coherence and comparability with the MRIO officially published by the National Bureau of Statistics of China (NBS). Therefore, our proposed methodology provides the possibility of producing more disaggregated MRIO tables in other similar contexts.

1. Introduction

Input-output tables with their extension inter-country input-output (ICIO) tables or multi-region input-output (MRIO) tables have been widely used in research on relationships between consuming and producing sectors, such as global value chains (GVC), international production networks and environmental analyses [1,2]. Due to statistical and methodological limitations, including monetary price valuation, sector classification or aggregation, and uncertainties and absences in trade statistics, available input-output (IO) databases are limited. Besides the input-output table at the country level, several well-known and always being updated global MRIO databases are the Global Trade Analysis Project MRIO table (GTAP-MRIOT) [2], the World Input-Output Database (WIOD) [3], EXIOBASE [4], and Eora [5]. The existing literature has applied these databases extensively for various international comparisons. For example, Wang et al. [6] analyzed the spatial differences in energy intensity among G20 countries based on the WIOD and revealed its driving factors by employing multiplicative structural decomposition analysis. In addition, subnational, domestic regional IO databases have been constructed for some countries, such as Australia (e.g., [7]), Spain (e.g., [8]), Germany (e.g., [9]), and China (e.g., [10]). Similarly, they are utilized for a variety of purposes (e.g., [11]). Today’s global intertwined situation is characterized by the multi-polarization of manufacturing centers where GVC is extended mainly in a regional form. That is, not only is the country an important participant in joining the GVC, but also domestic regions as intermediate products or service providers join the GVC directly or indirectly [12].

While many MRIO tables exist in different countries, very few tables include more subdivided regions and sectors. For example, the detailed national benchmark IO tables in the U.S. (United States) contain relationships only for the subdivided 405 industries and do not show input-output relationships between regions. The yearly input-output accounts data published by BEA (Bureau of Economic Analysis) contains 71 industries and 52 regions (50 states, the District of Columbia, and Puerto Rico). Scholars have also compiled more disaggregated tables for various research purposes. However, almost no MRIO table covers a large number of cities with more disaggregated sectors in countries, especially those with advanced urbanization and specialization. Therefore, this paper uses China as a case study to present our approach of compiling an MRIO table with the most disaggregated level of region and sector. China is a good example because processing trade is the main way of embedding the global value chain, and domestic regional trade accounts for a large percentage. As a result, the degree of correlation between China’s economy and the world economy is not only reflected in the scale of trade but also in the changes in the country’s industrial and economic structure, which can be reflected by MRIO. Moreover, in recent years, China’s intermediate product imports have shown a continuous decline, and intermediate products have been transferred to the domestic market. This indicates an inward trend in GVC [13]. The COVID-19 epidemic has exacerbated the trend of reverse globalization and challenged the current domestic value chain [14]. A focus on the general situation at the regional scale in China indicates the contradictory phenomenon of the high level of overall national growth versus unbalanced regional development. A within-nation comparison reflects remarkable regional policies, strategies and industrial structure. Therefore, it is necessary to understand the economic interdependence among different regions in China through MRIO.

Current related studies about China are mostly based on WIOD, or China’s 31 provinces and 42 sector IO models. The National Bureau of Statistics of China (NBS) has compiled national input-output tables every five years since 1987. The most recent is the national IO table for 2017, with a refined sector classification of 149 product sectors. Based on the “Classification of National Economic Industries” (GB/T4754), the national economic production activities in the 2017 table are divided into 149 product sectors. Among them are 5 departments of agriculture, forestry, animal husbandry, and fishery, 95 departments of industry, 4 departments of the construction industry, and 45 departments of service activities. Before 2012, not all provinces in China construct the provincial single region IO (SRIO) tables. SRIO tables are usually compiled and published by official provincial agencies. In 2016, the NBS released the “China Regional Input-Output Table 2012” containing SRIO tables for all 31 provinces.

Almost all available Chinese MRIO tables are constructed based on provincial SRIO tables. The existing Chinese MRIO table involves the input-output relationship of China’s 31 provinces or the IO table of China’s 8 regions. For example, Zhang and Qi [15] compiled MRIO tables for China into eight regions for the years 2002 and 2007 on the basis of the IO tables published by each province in the corresponding year. Liu et al. [16,17] compiled MRIO tables for 30 provinces (except Tibet, because Tibet started to compile input-output tables for the first time in 2012) and 30 economic sectors for 2007 and 2010.

The MRIO table has become one of the most valuable approaches to analyzing various regions’ economic ties and interdependence. Through MRIO, we can identify the economic impacts along the domestic value chain. The MRIO table is also an essential tool for national and subnational economic planning. It can not only directly reflect the trade relations and trade patterns among different industries from each region but also provide a data basis for the analysis of inter-regional interactions.

However, currently available Chinese MRIO tables have a very low resolution, both at the regional and sectoral levels. Databases at the provincial level contradict the reality that China is an enormous country with a large domestic economy where huge variation and industrial diversity exist inside a province. For example, Guangdong province, one of the most developed provinces in China, has 40 of the 41 industrial categories listed in the United Nations Industrial Classification, which basically constitute the entire industrial chain of specific industries. Analysis of the existing database might result in a large estimation bias considering the huge heterogeneity among different regions or cities in China, such as economic level, income disparities, geographical location and the role of local governments. Therefore, compiling an MRIO table with more subdivided sectors and smaller spatial units is beneficial for a reliable and robust economic analysis of China.

The study by Zheng et al. [18] is the only work to date that proposes a method to construct an MRIO table among all prefecture-level cities in China. Although their MRIO tables have been refined to the city level, they are still limited to 42 economic sectors. The aim of this paper is to develop an MRIO table that accounts for prefecture-level cities with more segmented sectors. Unlike the entropy-based method and data they used, we apply an expanded direct decomposition method and develop an optimization model to estimate an MRIO table that reports intersector transactions between different cities in an economy, using information from two large-scale enterprise-level datasets, the China Annual Survey of Industrial Firms (CASIF) survey and the China Customs Data (CCD) database from 2000 to 2013.

This paper contributes to the existing literature in the following ways. First, to the best of our knowledge, compared with previous related studies, our study is the first attempt to examine the IO relationship among the four-digit code industry at the prefecture-city level in China, which is helpful for deepening IO-related research. Second, we employ several specific econometric methods for data processing and balancing purposes, including the non-parametric generalized method of moments (GMM) estimation, generalized maximum entropy (GME) and generalized RAS (GRAS). The application of these methods addresses data challenges and allows the development of a high-resolution MRIO table that builds on enterprise-level data. Third, this paper is based on two micro-level datasets. Previous related studies mostly use statistics from different sources, such as national IO tables, domestic regional IO tables and customs data. We combined the customs data with the CASIF data. CASIF is the original panel data at the enterprise level, with more than 130 variables. The database is mainly for manufacturing enterprises, accounting for about 95% of the country’s total industrial output value. Therefore, the final data obtained by matching the two databases guarantees the data volume we need to compile the MRIO tables and the reliability of the input-output relationships we find. In this regard, our study will expand the database for similar studies in the future. Fourth, China, serving as the subject of our study, poses a particular computational challenge due to its large number of cities, sectoral diversity and the complexity of its economy. Therefore, the method presented here has great applicability in many other empirical settings.

The rest of the paper is structured as follows. Section 2 provides a background description of the development of MRIO tables in China. Section 3 specifies the compilation procedures from data collection to final construction. Section 4 is a comparative analysis in which we compare our MRIO table with the national IO table. Section 5 concludes.

2. Background of MRIO-Compilation in China

Research on IO analysis methods in China began in the late 1950s. In 1974, China issued the first input-output table at the national level, the physical input-output table of 61 products in China, in 1973. Since 1987, China has adjusted its previous practice of using the material product balance system (MPS) to compile the IO table for each year. Based on the System of National Accounts (SNA), the NBS in China has begun compiling the national IO table every 2nd and 7th year. This is similar to many countries whose benchmark IO tables are compiled every five years [19]. The time interval of 5 years might lead to the relative lag in the publication of the IO table and to the difficulty in the real-time analysis. To mitigate such problems, the NBS compiles the IO extension table every five years in the year, ending with 0 or 5. The extension table is based on the national IO table, survey data and other professional statistical data. In 1987, the NBS issued a national value-based input-output table of 117 industry sectors.

In 2005, the State Information Center of China developed an inter-regional input-output table covering 30 departments in 8 regions of China in 1997. First, it adjusts the input-output table of each province or municipality according to a unified standard. Then, according to the similarity of industrial structures, economic development levels, and geographical distributions, all provinces or municipalities are categorized into eight major regions. In addition, in combination with relevant statistical data, a typical survey method is applied to collect basic data about the flow of commodities between provinces. Finally, the inter-regional IO table is compiled based on the Chenery-Moses input-output model [20].

Since 1997, China’s economy has developed rapidly, and inter-regional trade has been further strengthened. As a result, based on the regional input-output table in 1997, Zhang et al. [21] proposed a new estimation model to compile the 2002 and 2007 MRIO tables. Each table involves 31 provinces and municipalities across the country, categorized into eight regions according to their locations. Their work estimates the inter-regional trade coefficient based on the maximum entropy model and the gravity model. It also revises the inter-regional transaction matrix by using the inflows and outflows provincial basic data from the 2007 national IO table. The 2002 and 2007 MRIO tables adopt the sector classification method by the NBS (GB/T 4754-2011) and merge the service sector, thus having a total of 29 sectors. It is basically comparable to the sectoral classification of the above-mentioned 1997 MRIO table with 30 sectors. The regional division of 8 regions is the same as the 1997 MRIO table. It must be pointed out that various MRIO tables can be developed with different regional combinations because the compilation work of the 2002 and 2007 MRIO models is based on the individual provinces.

Furthermore, Liu et al. [16,17] compiled an MRIO table for 30 provinces and 30 economic sectors for 2007 and 2010. Their compilation is based on the province-level input-output tables of China’s 30 provinces or municipalities, except for Tibet. Through an in-depth analysis of the influence from the same industry and spatial dependence theory, this research reconstructs the gravity model of inter-regional economic connections. In addition, it uses the statistical data of inter-regional commodity flows and combines the matrix-balanced RAS method, thus systematically realizing the simulation of inter-regional industrial trade. This table is an improvement over the previous MRIO table in that it describes the input-output relationship between 30 sectors in China’s 30 provinces, rather than being limited to 8 large regions. In 2018, Liu et al. [22] compiled an inter-regional I-O table for 2012, extending to 31 provinces and 42 sectors in China.

Due to the time-consuming and labor-intensive characteristics of input-output tables at the national level, the compilation of the IO tables mentioned above has participated in by the NBS in China. Scholars have also applied different methods to compile MIRO, mainly at the country or subnational levels, such as provinces, states or counties [18]. For example, Wang et al. [23] demonstrated the construction of a 1997 to 2011 time series of nested interregional-international MRIO tables with the case of the Chinese economy, where each of the 30 provinces is distinguished with 135 industry sectors inside it and linked with 185 world countries. Although it features a large-scale, subnational and international IO relationship, it fails to provide high-resolution inter-regional connections inside China. Meng and Yamano [12] used China and Japan’s inter-regional IO tables, the OECD inter-country IO tables, and China and Japan’s regional customs statistics to compile the regionally extended inter-country IO table. However, the table segments the Chinese domestic region into four parts, namely China Coast, center, west and northeast, which is rough of China’s great regional heterogeneity and large economy. In addition, based on the modified gravity model, Mi et al. [24] compiled the Chinese MRIO table for 2012 among 30 economic sectors in 30 regions (30 provinces excluding Hong Kong, Macao, Taiwan and Tibet due to data unavailability). The compilation of the 2012 MRIO table is based on the IO table for 30 Chinese provinces published by the NBS. It is the first MRIO table to reflect China’s economic development pattern after the 2008 global financial crisis [24]. However, it aggregates the 42 economic sectors into 30 sectors because there are 30 sectors in the Chinese MRIO tables for both 2007 and 2010. Therefore, the aggregation of sectors might lead to bias in the IO analysis.

The input-output analysis method based on the IO table is used not only in the research of industrial structure problems but is also widely applied in economic planning, policy simulation, industrial value development, price measurement, etc. In particular, in the field of environmental research, specific input-output tables have also been developed, such as the U.S. environmentally extended input output (EEIO) models [25]. Most studies on the environmental and social impacts associated with consumption require extended MRIO tables for better empirical results [26]. For example, based on an MRIO table for Germen Federal States, the research from Aniello et al. [27] presents a net economic impact assessment of the German Renewable Energy Act on the economy of the state “North-Rhine Westphalia”. Faturay et al. [28] used a new U.S. MRIO model to determine the economic and energy impacts resulting from the installation of wind energy farms in 10 U.S. states. From a practical point of view, input-output data and methods play an important role in China’s current economic structural transformation, as well as in the analysis of regional, resource and environmental economic linkages. Combining network analysis tools, Wang et al. [29] analyzed the regional and sectoral structure of the Chinese economy with the MRIO tables of China in 2007 and 2012 [16,22]. Some researchers have used Chinese MRIO to perform environmental studies, such as carbon emissions, air pollution, water consumption, and other energy use. For example, Liu and Wang [30] compared emission flows in China for 2002 and 2007 by two methods: the bilateral trade (EEBT) and MRIO model. They find that the results from the two approaches differ significantly and the analysis based on the MRIO method is more applicable for consumption-based studies. Mi et al. [31] applied an environmentally extended multiregional input-output approach to measure carbon inequality for households in China’s 30 regions and concluded that economic growth contributes to an overall reduction in carbon inequality in China. Zheng et al. [32] constructed MRIO for 2012 and 2015 and quantitatively evaluated the regional disparity in decarbonization and the driving forces during 2012–2015.

Although many scholars have attempted to compile various MRIO tables about China, there is almost no MRIO table that covers more subdivided sectors (e.g., four-digit level) among all prefecture-level cities in China due to city-level data scarcity and methodology constraints. To the best of our knowledge, a study by Zheng et al. [18] is the only work proposing a method to construct the IO relationship among all prefecture-level city MRIO tables in China. Unlike their entropy-based framework [18], our study applies data from the firm level and uses an expanded direct decomposition method to estimate an MRIO table that reports 334 sector transactions among 284 cities in China. Our work differs from theirs in terms of data sources and compilation methods. Moreover, the finally obtained MRIO table is advanced at both the sector and city levels.

3. Materials and Construction Methodology

Our research relies on two large-scale micro-level datasets: the China Annual Survey of Industrial Firms (CASIF) survey and the China Customs Data (CCD) database. We use the period of the year 2000 to the year 2013. We expand the direct decomposition method to compile the MRIO table. We first estimate the region-sector output factor matrix V (first quadrant). We then obtain the final use (second quadrant) and value-added information (third quadrant) at the region-sector level by counting accounting-related accounts at the enterprise level. We finally obtain the expanded MRIO, as shown in

Table 1. Schematic outline of city-level MRIO table.

		Intermediate Use									Final Output	Total Output
		Region 1			Region 2			Region m
		1	j	n	n + 1	n + j	2n	Mn − n + 1	Mn − n + j	mn
Intermediate inputs	1	x11	x1j	x1n	x1,n+1	x1,n+j	x1,2n	x1,mn−n+1	x1,mn−n+j	x1,mn	Y1	W1
	2	x21	x2j	x2n	x2,n+1	x2,n+j	x2,2n	x2,mn−n+1	x2,mn−n+j	x2,mn	Y2	W2
	…	…	…	…	…	…	…	…	…	…	…	…
	…	…	…	…	…	…	…	…	…	…	…	…
	…	…	…	…	…	…	…	…	…	…	…	…
	n	xn1	xnj	xnn	xn,n+1	xn,n+j	xn,2n	xn,mn−n+1	xn,mn−n+j	xn,mn	Yn	Wn
Initial input		Z1	Zj	Zn	Zn+1	Zn+j	Z2n	Zmn−n+1	Zmn−n+j	Zmn
Total input		X1	Xj	Xn	Xn+1	Xn+j	X2n	Xmn−n+1	Xmn−n+j	Xmn

, which describes the IO relationship at the regional-industry level with 284 regions and 334 sectors. Because this is already the most disaggregated table of city–region relationships we can get, we can also expand other MRIO tables with 3-digit or 2-digit industries and provincial or city-level relationships based on the existing data and method. The key steps of the compilation process are as follows.

3.1. Data Collation & Preparation

This step merges the two datasets by adding trade information from the CCD to the CASIF dataset. The key is identifying a common and recognizable attribute or variable from both datasets. However, the CASIF dataset identifies enterprises according to the business license code, while the CCD uses the enterprise code for a custom declaration. Therefore, we merge the two datasets through other variables common to the two datasets, including the enterprise name (algorithm: keyword matching), zip code, telephone number, legal person/customs declarant, etc. The basic logic is that observations are matched one by one according to the above-listed variables with the premise of the same prefecture-level city. We apply the Python programming language to extract the critical information of each variable string and then match each observation (entry) from the CASIF with those in the CCD dataset. Due to the massive amount of data, successfully matched observations are stored in the database software MongoDB.

The newly formed dataset has the following characteristics. All observations are state-owned enterprises or non-state-owned enterprises above the designated size. Since 2011, the “Industrial Enterprises above Designated Size” (IEADS) are defined by the National Bureau of Statistics as enterprises with an annual income of the main business of and above 20 million yuan. Moreover, all enterprises directly participate in customs import and export declarations. As a result, all enterprises or observations in the new dataset possess financial indicators, import and export indicators, and the proportion of intermediate inputs, value added, labor input, the final use of surplus reserves, and other information at the enterprise level. The final dataset includes 284 prefecture-level administrative cities and municipalities directly under the central government. It covers 334 four-digit sectors. As of 2014, there are altogether 291 prefecture-level administrative cities in China. The 2004 and 2008 national economic census covers 98 two-digit industry sectors, 396 three-digit sectors and 913 four-digit sectors.

3.2. Value Adjustment

Considering that the price information in the dataset is the purchasers’ price, while the transaction costs (including transportation costs and other commercial surcharges) in different countries and regions in international trade vary, we need to adjust the value to the producers’ price to ensure accuracy in the subsequent estimation of input prices. Taking the impact of heterogeneities in countries and sectors, we adjust the price information with the make-up estimation method. The mark-up data are mainly from the OECE yearly reports on each country’s economic and trade situation. We also refer to related empirical papers published in academic journals. We collect data about the mark-up estimation of 107 countries and regions, accounting for 93% of China’s foreign trade data share and covering all years in the dataset. In addition, we need to adjust the industry code for the collected mark-up data because most of them apply the HS code of the world customs organization (International Convention for Harmonized Commodity Description and Coding System).

Finally, we take the Chinese mark-up rate as the benchmark of 100, and the mark-up rate of the sector $j$ in country $r$ is $M_{rj}$, then the adjusted producers’ price of the sector $j$ in country $r$ is $v_{rj}$

$$v_{rj}=\frac{100·v_{rj}^0}{Mrj}$$

(1)

where $v_{rj}^0$ is the original (before the adjustment) price of the sector $j$ in country $r$.

3.3. Correction of Sample Selection Bias

The newly merged dataset shows that the matching ratio between the financial data and product data at the enterprise level is 49%. This means that for nearly 51% of enterprises, we cannot observe their imported intermediate input data directly. Sample selection bias will occur if we do not consider the import data of these enterprises, which results in the final estimation coefficient bias. Therefore, we refer to the two-stage method by Heckman [33] to correct the problem of sample selection bias. This paper applies the Probit model to estimate the probability of the whole sample’s usage of imported intermediate inputs and then calculates the Inverse Mills Ratio ${\lambda }_{ij}$.

We have a model for estimation purposes:

$${\mathrm{y}}_{ij}^k={\beta }_1{X}_{ijk}+{\beta }_2{W}_{ij}+{\beta }_3{Z}_j+{\epsilon }_{1i}$$

(2)

Among them, ${\mathrm{y}}_{ij}^k$ is the output value of product $k$ in enterprise $i$ of sector $j$. $X_{ijk}$ is the total value of intermediate inputs of the product $k$ in enterprise $i$, sector $j$. $W_{ij}$ is the control variables at the enterprise level of the enterprise $i$ in sector $j$. $Z_j$ is the control variable at the macro-level.

To correct the bias resulting from the sample selection problem, we set the following binary discrete selection model:

$$IM{P}_{ij}={\beta }_4{S}_{ij}+{\epsilon }_{2i}$$

(3)

where $IM{P}_{ij}$ is a binary variable representing whether the enterprise uses imported intermediate inputs. $S_{ij}$ is the firm characteristics variables of the enterprise $i$ in sector $j$

If

$E\left({\epsilon }_{1i}\right)=0$ and $E\left({\epsilon }_{2i}\right)=0$, ${\epsilon }_1$ and ${\epsilon }_2$ are positively correlated; we have the following:

$$E\left({\epsilon }_{1i}|IM{P}_{ij}\ge 0\right)=E({\epsilon }_{1i}|{\epsilon }_{2i}\ge -{\beta }_4{S}_{ij})$$

(4)

$$\mathrm{E}\left({\mathrm{y}}_{ij}^k|X_{ijk}, IM{P}_{ij}\ge 0\right)={\beta }_1{X}_{ijk}+E({\epsilon }_{1i}|{\epsilon }_{2i}\ge -{\beta }_4{S}_{ij})$$

(5)

$$\mathrm{E}\left(y_{ij}^k|X_{ijk}, IM{P}_{ij}\ge 0\right)={\beta }_1{X}_{ijk}+{\beta }_5{\lambda }_{ij}$$

(6)

We then obtain the revised model:

$${\mathrm{y}}_{ij}^k={\beta }_1{X}_{ijk}+{\beta }_2{W}_{ij}+{\beta }_3{Z}_j+{\beta }_5{\lambda }_{ij}+\mathsf{\mu }$$

(7)

A correction as described above during the estimation of the consumption coefficient can address the sample selection bias to some extent.

3.4. Estimates of Input Allocations for Variable Factor

As we know, a large number of enterprises in the database are multi-product enterprises, and these products may belong to different sectors. Meanwhile, we cannot know the input allocation of the variable factors. Therefore, we need to estimate the input allocation based on data about single-product enterprises.

Here, we refer to studies from De Loecker et al. [34] and Ackerberg et al. [35] and implement a two-step approach to estimate the output elasticity of the variable factor at the enterprise level.

To obtain an unbiased estimation of the output elasticity for the variable factors, the general production function of product $\mathrm{k}$ in enterprise $\mathrm{i}$ at time $\mathrm{t}$ is:

$${\mathrm{lny}}_{\mathrm{ikdt}}{=\mathrm{f}}_{\mathrm{k}}\left({\mathrm{lnl}}_{\mathrm{ikdt}}{,\mathrm{lnm}}_{\mathrm{ikdt}},\mathsf{\beta }\right){+\mathrm{p}}_{\mathrm{it}}{+\mathsf{\epsilon }}_{\mathrm{ikdt}}$$

(8)

where $\mathrm{f}()$ is the translog production function, ${\mathrm{y}}_{\mathrm{ikdt}}$ is the output at the enterprise level, ${\mathrm{l}}_{\mathrm{ikdt}}$ and ${\mathrm{m}}_{\mathrm{ikdt}}$are labor and capital respectively, $\mathsf{\beta }$ is the parameter vector that needs to be estimated to calculate output elasticity, and ${\mathrm{p}}_{\mathrm{it}}$ is the productivity at the enterprise level, ${\mathsf{\epsilon }}_{\mathrm{ikdt}}$ is a random error.

First, we limit the samples to all single-product companies existing for at least three consecutive years. We obtain an estimate consistent with the expected output from the following regression:

$${\mathrm{lny}}_{\mathrm{ik}}={\mathsf{\varphi }}_{\mathrm{k}}\left({\mathrm{lnl}}_{\mathrm{ik}}{,\mathrm{lnm}}_{\mathrm{ik}}{,\mathrm{K}}_{\mathrm{ik}}{, \mathrm{H}}_{\mathrm{ik}}\right)+{ϵ}_{\mathrm{ik}}$$

(9)

$\mathsf{\varphi }()$ is a third-order polynomial function that approximates the input element. ${\mathrm{K}}_{\mathrm{it}}$ represents variables that affect factor demand, including output’s prices, product’s market share, firm’s import and export, and product and time dummies. ${\mathrm{H}}_{\mathrm{ik}}$ is a quality control variable, including output’s price, product’s market share, capital and variable factors.

Productivity can thus be calculated as the difference between Equations (8) and (9):

$${\widehat{\omega }}_{\mathrm{it}}={\widehat{\mathsf{\varphi }}}_{\mathrm{it}}- \widehat{\mathrm{f}}\left({\mathrm{l}}_{\mathrm{it}},{\mathrm{m}}_{\mathrm{it}};\tilde{\mathsf{\beta }}\right)-{\mathrm{h}}_{\mathrm{it}}\tilde{\gamma }$$

(10)

Then, we refer to the law of motion for productivity [36] to obtain the estimated coefficient vectors β and γ of all production functions,

$${\widehat{\omega }}_{\mathrm{it}}={\mathrm{g}}_{\mathrm{t}-1}\left({\widehat{\omega }}_{\mathrm{it}-1},{\delta }_{\mathrm{it}-1}^{\mathrm{K}},{\delta }_{\mathrm{it}-1}^{\chi },{\widehat{\mathrm{s}}}_{\mathrm{it}-1},{\widehat{\aleph }}_{\mathrm{it}-1}\right)-{\epsilon }_{\mathrm{it}}$$

(11)

${\mathsf{\delta }}_{\mathrm{it}-1}^{\mathsf{\kappa }}$, ${\mathsf{\delta }}_{\mathrm{it}-1}^{\mathsf{\chi }}$ are the lagged export and import dummy, respectively. ${\widehat{\mathrm{s}}}_{\mathrm{it}-1}$ is the probability p-value predicted by the lagged single-product enterprise. ${\widehat{\aleph }}_{\mathrm{it}-1}$ is the lagged predicted probability p-value of enterprise survival. After estimating the law of motion for productivity, productivity innovation $\epsilon$ is represented as a residual term.

Finally, all coefficients of the production function are calculated using the non-parametric GMM estimation method.

$$\mathrm{E} ({\epsilon }_{\mathrm{it}}{ (\mathsf{\beta },\gamma ) \mathrm{B}}_{\mathrm{it}})=0$$

(12)

The output elasticity of variable factor can be expressed as:

$${\widehat{\mathsf{\theta }}}_{\mathrm{mkdt}}^{\mathrm{n}}={\widehat{\mathsf{\beta }}}_{\mathrm{n}}+2{\widehat{\mathsf{\beta }}}_{\mathrm{nn}}{\mathrm{c}}_{\mathrm{mkdt}}^{\mathrm{n}}+{\widehat{\mathsf{\beta }}}_{\ln }{\mathrm{c}}_{\mathrm{mkdt}}^1+{\widehat{\mathsf{\beta }}}_{\mathrm{kn}}{\mathrm{c}}_{\mathrm{mkdt}}^{\mathrm{m}}+{\widehat{\mathsf{\beta }}}_{\mathrm{lpn}}{\mathrm{c}}_{\mathrm{mkdt}}^1{\mathrm{c}}_{\mathrm{mkdt}}^{\mathrm{m}}$$

(13)

where $\mathrm{l}$, $\mathrm{m}$, and $\mathrm{n}$ represent labor, capital, and variable factors, respectively.

3.5. Estimation of Direct Consumption Coefficient

The precondition of the application of the direct decomposition method is that all data are available at the micro-level, namely the enterprise level. However, there are many enterprises’ input compositions that cannot be observed. Therefore, we need to speculate the ratio of the production inputs of a sector in a region by utilizing the available data of the observable sector and enterprise. Benefiting from the huge amount of data in our existing dataset, we can calculate the ratio of the inputs by establishing the following regression model:

$$x_q^t={\displaystyle \sum }_{k=1}^K{a}_{qk}{y}_k^t+u_q^t$$

(14)

where $x_q^t$ is the total input cost of input $q$, $y_k^t$ is the total output of enterprise $k$, $a_{qk}$ is the production coefficient to be estimated, which is defined as the value of $q$ inputs for producing one unit of output, and $u_q^t$ is the disturbance term in the process of enterprises’ input-output production activities.

Unlike the usual estimation method for the production function, we use the GME method (generalized maximum entropy) to estimate the $a_{qk}$ in consideration of the presence of $u_i^t$ and of substituted inputs. With this approach, $a_{qk}$ and $u_q^t$ in Equation (14) can be converted into expected values for estimating the probability distribution [37].

Thus, for input $q$ we have:

$$a_{qk}={\displaystyle \sum }_{s=1}^S{Z}_{qk}^s{P}_{qk}^s=Z_{qk}^{\prime }{A}_{qk}$$

(15)

$$u_q^t={\displaystyle \sum }_{g=1}^G{V}_{qt}^g{w}_{qt}^g=V_{qt}^{\prime }{B}_{qt}$$

(16)

where $Z_{qk}^s$ is the $s$-dimensional support-vector of the production coefficient $a_{qk}$, $V_{qt}^g$ is the $g$-dimensional support-vector of the disturbance term $u_q^t$, $A_{qk}$ and $B_{qt}$ are the probability vectors corresponding to $a_{qk}$ and $u_q^t$ respectively. The elements in the support vector are fluctuations centered on the approximate values of the coefficients and disturbances. The corresponding constraints are as follows:

$${\displaystyle \sum }_{s=1}^S{A}_{qk}=1$$

(17)

$${\displaystyle \sum }_{s=1}^S{B}_{qt}=1$$

(18)

$${{\displaystyle \sum }}_{q=1}^Q{a}_{qk}={{\displaystyle \sum }}_{q=1}^Q{Z}_{qk}^{\prime }{A}_{qk}=1$$

(19)

The constraints guarantee a unified sum and trial balance between total income and total expenses.

Under this balance, we have the following:

$${\displaystyle \sum }_{q=1}^Q{x}_q^t={\displaystyle \sum }_{q=1}^Q\left({\displaystyle \sum }_{k=1}^K{a}_{qk}{y}_k^t+u_q^t\right)$$

(20)

Because of the constraints condition ${{\displaystyle \sum }}_{q=1}^Q{a}_{qk}=1$, we have:

$${\displaystyle \sum }_{q=1}^Q{x}_q^t={\displaystyle \sum }_{k=1}^K\left({\displaystyle \sum }_{q=1}^Q{a}_{qk}\right)y_k^t+{\displaystyle \sum }_{q=1}^Q{u}_q^t={\displaystyle \sum }_{k=1}^K{y}_k^t+{\displaystyle \sum }_{q=1}^Q{u}_q^t$$

(21)

Put ${\displaystyle \sum }_{q=1}^Q{x}_q^t={\displaystyle \sum }_{k=1}^K{y}_k^t$ into (21):

$${\displaystyle \sum }_{q=1}^Q{u}_q^t=0$$

(22)

Therefore, the system is singular. That is, a variance–covariance matrix does not exist. As a result, the least squares (LS) and maximum likelihood estimation (MLE) cannot be used to estimate the coefficients of the equation. Generalized maximum entropy (GME) is a better choice here.

By using GME estimation to obtain $a_{qk}$, we finally have a $n\ast m$ matrix A.

3.6. Balancing the IO Table

Since we use panel data to compile input-output tables, we first estimate the input-output table for the starting year (2000). We then apply the data from other years to make adjustments for the table to ensure the comparability and reliability of our table in different years. This paper uses the RAS method to balance the IO table. The RAS method is a matrix balancing method proposed by Richard Stone and is one of the best-known estimation methods for the I–O table [38].

Here, we have:

$$\underset{\zeta ,\eta }{\min }f\left(x_{\zeta \eta },x_{\zeta \eta }^0\right)$$

(23)

$$s.t\{\begin{array}{l}{\displaystyle {\displaystyle \sum }_{\eta }}{x}_{\zeta \eta }={\phi }_{\zeta }, \mathrm{for} \mathrm{all} \zeta \\ {\displaystyle {\displaystyle \sum }_{\zeta }}{x}_{\zeta \eta }={\omega }_{\eta }, \mathrm{for} \mathrm{all} \eta \end{array}$$

where $x_{ij}$ is the flow data, the RAS idea can be written into an optimization model as follows:

$$RAS=Min{\displaystyle \sum }_{\eta \zeta }{x}_{\zeta \eta }log\left(x_{\zeta \eta }/x_{\zeta \eta }^0\right)$$

(24)

$$s.t\{\begin{array}{l}{\displaystyle {\displaystyle \sum }_{\eta }}{x}_{\zeta \eta }={\phi }_{\zeta }, \mathrm{for} \mathrm{all} \zeta \\ {\displaystyle {\displaystyle \sum }_{\zeta }}{x}_{\zeta \eta }={\omega }_{\eta }, \mathrm{for} \mathrm{all} \eta \\ x_{\zeta \eta }\ge 0, \mathrm{for} \mathrm{all} \zeta \mathrm{and} \eta \end{array}$$

When the $x_{\zeta \eta }$ is negative, the optimization of the RAS method converges to negative infinity and cannot be solved. Sign-preserving or zero-preserving is required when adjusting the IO table. The RAS method is only applicable to nonnegative tables. Therefore, to mitigate this restriction, we refer to generalized RAS (GRAS) as one of the extensions of the RAS method [39].

$$GRAS=Min{{\displaystyle \sum }}^{ }\left|x_{\zeta \eta }^0\right|\cdot a_{\zeta \eta }ln\left(a_{\zeta \eta }\right)$$

(25)

$$Gs.t\{\begin{array}{l}{\displaystyle {\displaystyle \sum }_{\eta }}{a}_{\zeta \eta }{x}_{\zeta \eta }^0={\phi }_{\zeta }, \mathrm{for} \mathrm{all} \zeta \\ {\displaystyle {\displaystyle \sum }_{\zeta }}{a}_{\zeta \eta }{x}_{\zeta \eta }^0={\omega }_{\eta }, \mathrm{for} \mathrm{all} \eta \\ x_{\zeta \eta }=a_{\zeta \eta }{x}_{\zeta \eta }^0 , \mathrm{for} \mathrm{all} \zeta \mathrm{and} \eta \end{array}$$

And $a_{\zeta \eta }>0$, $\frac{x_{\zeta \eta }}{x_{\zeta \eta }^0}=a_{\zeta \eta }>0$, sign-preserving. And when $x_{\zeta \eta }^0=0, x_{\zeta \eta }=a_{\zeta \eta }{x}_{\zeta \eta }^0=0$, zero-preserving. Therefore, the signs remain unchanged. The target function of the GRAS method is a strictly convex function with linear constraints that satisfy the KKT (Karush-Kuhn-Tucker) condition and have a unique optimal solution.

We apply the Lagrangean function. The derivative of the GRAS Lagrangean:

$$L\left(a,\delta ,\theta \right)={\displaystyle \sum }_{\zeta }{\displaystyle \sum }_{\eta }\left(a_{\zeta \eta }-1\right)ln{a}_{\zeta \eta }+{\displaystyle \sum }_{\zeta }{\delta }_{\zeta }\left({\theta }_{\eta }-{\displaystyle \sum }_{\eta }{a}_{\zeta \eta }\cdot x_{\zeta \eta }^0\right)+{\displaystyle \sum }_{\eta }{\theta }_{\eta }\left({\tau }_{\eta }-{\displaystyle \sum }_{\zeta }{a}_{\zeta \eta }\cdot x_{\zeta \eta }^0\right)$$

First-order partial derivative:

$$\partial L/a_{\zeta \eta }=ln{a}_{\zeta \eta }+(a_{\zeta \eta }-1)/a_{\zeta \eta }-{\theta }_{\zeta }\cdot x_{\zeta \eta }^0-{\delta }_{\eta }\cdot x_{\zeta \eta }^0=0$$

$$\frac{\partial L}{{\delta }_{\zeta }}={\theta }_{\zeta }-{\displaystyle \sum }_{\eta }{a}_{\zeta \eta }\cdot x_{\zeta \eta }^0=0$$

$$\frac{\partial L}{{\theta }_{\eta }}={\tau }_{\eta }-{\displaystyle \sum }_{\zeta }{a}_{\zeta \eta }\cdot x_{\zeta \eta }^0=0$$

When $x_{\zeta \eta }^0=0$, $ln{a}_{\zeta \eta }+\frac{a_{\zeta \eta }-1}{a_{\zeta \eta }}=0 \stackrel{}{\Rightarrow }{a}_{\zeta \eta }=1$, which is zero-preserving and has a unique optimal solution.

Imposing the optimality condition:

Construct the transcendental equation $f\left(a_{\zeta \eta }\right)=ln{a}_{\zeta \eta }-\frac{1}{a_{\zeta \eta }}-{\xi }_{\zeta \eta }=0$ to solve, where ${\xi }_{\zeta \eta }=\left({\theta }_{\zeta }+{\theta }_{\zeta }\right)x_{\zeta \eta }^0-1$The iterative solution is as follows:

Step 1:

$$\mathrm{N}=0,f\left(a_{\zeta \eta 0}^0\right)f\left(a_{\zeta \eta 0}^1\right)<0$$

Step 2:

$$a_{\left\{\zeta \eta N+1\right\}}^{\left\{1\right\}}=a_{\left\{\zeta \eta N\right\}}^{\left\{0\right\}}-\frac{f\left(a_{\zeta \eta 0}^0\right)}{F\left(a_{\zeta \eta 0}^0,a_{\zeta \eta 0}^1\right)}$$

$$a_{\zeta \eta N+1}^0=a_{\zeta \eta N+1}^1-\frac{f\left(a_{\zeta \eta N+1}^1\right)}{\omega ·F\left(a_{\zeta \eta N}^0,a_{\zeta \eta N}^1\right)})$$

$$\omega \{\begin{array}{c}\frac{3f\left(a_{\zeta \eta N+1}^1\right)}{2\left(a_{\zeta \eta N+1}^1-a_{\zeta \eta N}^1\right)\cdot F\left(a_{\zeta \eta N}^0,a_{\zeta \eta N}^1\right)}when f(·)\ast F(·)<0\\ \frac{3f\left(a_{\zeta \eta N+1}^1\right)}{2\left(a_{\zeta \eta N+1}^1-a_{\zeta \eta N}^0\right)\cdot F\left(a_{\zeta \eta N}^0,a_{\zeta \eta N}^1\right)}when f(·)\ast F(·)>0\end{array}.$$

Step 3: When $\left|f\left(a_{\zeta \eta N+1}^0\right)\right|$ or $\left|f\left(a_{\zeta \eta N+1}^1\right)\right|$ is infinity close to 0, $a_{\zeta \eta }^{}=a_{\zeta \eta N+1}^1$, otherwise go to step 4.

Step 4: when $f\left(a_{\zeta \eta N+1}^0\right)\ast f\left(a_{\zeta \eta N+1}^1\right)<0$, then N + 1, go to step 2.

Through the iteration, we can correct $a_{\zeta \eta }^{}$.

4. Comparative Analysis

To examine the comparability of the IO direct consumption coefficient matrix in this paper with others, we carry out a detailed empirical investigation. We select the 2007 China regional IO table to gain more insight into the difference between our method and the method by NBS.

Since the input coefficient is a symbol of the production input technology in each region, the input coefficient is more stable and comparable to the intermediate input. Here, the input coefficients and their variation coefficients are used as indicators to describe the characteristics of intermediate inputs in different regions statistically. Because the regional IO tables published by the NBS include 42 industries, it is difficult to list and analyze all 1764 coefficients for one region. Moreover, the IO table by our method includes 284 regions and 334 sectors, thereby producing much more coefficients than those by NBS. Furthermore, the two IO tables have different latitudes and orders of magnitude. They cannot be compared directly. To realize our comparative analysis, we refer to the study by Dietzenbacher and Hoen [19].

The results are summarized in

Table 2. Coefficients of variation (NBS method).

$\mathit{c}{\mathit{v}}_{\mathit{i}\mathit{j}}$	$\mathbf{Coefficient} {\mathit{a}}_{\mathit{i}\mathit{j}}$
	P10	P20	P30	P50	P70	P80	P90	Total
[0, 0.5)	0.00%	0.11%	0.05%	0.11%	0.21%	0.63%	1.59%	2.70%
[0.5, 0.8)	0.16%	2.06%	1.43%	2.33%	2.96%	3.96%	2.38%	15.29%
[0.8, 1.0)	0.37%	2.96%	1.85%	2.80%	1.69%	2.06%	0.79%	12.53%
[1.0, 2.0)	3.65%	8.30%	5.44%	6.55%	3.86%	3.65%	0.79%	32.26%
[2.0,+∞)	11.48%	4.86%	17.93%	1.37%	1.16%	0.42%	0.00%	37.22%
Total	15.66%	18.30%	26.71%	13.17%	9.89%	10.73%	5.55%	100.00%

(the method by NBS) and

Table 3. Coefficients of variation (our method).

$\mathit{c}{\mathit{v}}_{\mathit{i}\mathit{j}}$	$\mathbf{Coefficient} {\mathit{a}}_{\mathit{i}\mathit{j}}$
	P10	P20	P30	P50	P70	P80	P90	Total
[0, 0.5)	0.15%	0.27%	0.14%	0.26%	0.64%	1.54%	3.99%	6.99%
[0.5, 0.8)	0.43%	4.35%	3.02%	4.63%	7.00%	7.99%	6.29%	33.71%
[0.8, 1.0)	0.86%	6.05%	4.13%	5.91%	2.43%	3.39%	1.65%	24.42%
[1.0, 2.0)	1.97%	4.72%	6.21%	7.15%	2.78%	1.04%	0.75%	24.63%
[2.0,+∞)	4.55%	1.43%	2.82%	0.44%	0.93%	0.03%	0.05%	10.25%
Total	7.96%	16.82%	16.32%	18.39%	13.78%	13.99%	12.73%	100.00%

(our method). The overall coefficients ($a_{ij}$) are categorized by the quantile method in the row. The coefficient of variation for element ($i,j$) is then given by $c{v}_{ij}$ and is categorized into five classes in ascending order in the columns of the table. The coefficients of variation in Table 2 are generally high, with only 2.70% of all coefficients of variation less than 0.5 and 69.48% greater than 1.0. In contrast, the coefficients of variation in Table 3, which are less than 0.5, account for 6.99% of all, and those greater than 1.0 account for only 34.88%. Dietzenbacher and Hoen [19] compared the stability of two types of input-output models, the Leontief and Ghosh models, based on a time series of annual IO tables for the Netherlands, covering the period 1948–84 and 13 industries. The results show that 75–80% of the elements in both models have a coefficient of variation smaller than 0.3. This means that the degree of variation in the coefficients of the IO table by the NBS is significantly higher than that of other countries.

Moreover, we categorize the statistics of Table 2 and Table 3 by the median of the coefficients and the coefficients of variation of 1.0, as shown in

Table 4. Comparison of statistics from two tables (links: Table 2; right: Table 3).

[0, 1.0)	14.24%	16.28%	[0, 1.0)	30.20%	34.92%
[1.0,+∞)	59.60%	9.89%	[1.0,+∞)	20.29%	5.58%
	[P0, P50]	(P50, P100]		[P0, P50]	(P50, P100]

. Table 4 shows that coefficients with a small average value (p = 50%) and large coefficients of variation (>1) accounted for 59% in Table 2 compared to 20.29% in Table 3. The part with small coefficients and large coefficients of variation can affect the estimation accuracy of the IO table to a great extent. In this regard, our IO table is more accurate and stable than those by NBS.

5. Conclusions

This study proposes a new construction method for compiling a high-resolution MRIO table. Our approach takes data from China as an example and relies on two large-scale enterprise-level datasets from 2000 to 2013, namely the China Annual Survey of Industrial Firms (CASIF) survey and the China Customs Data (CCD) database. By expanding the direct decomposition method, together with some auxiliary mathematical models necessary for estimations and consistency adjustment, we construct an MRIO table covering 284 prefecture-level administrative cities and 334 four-digit sectors, which is by far the most comprehensive MRIO table covering the largest number of cities and the most segmented industries in China. A comparative analysis shows that the MRIO by our method is stable and reliable.

The proposed method can be implemented not only in China but also in any country with available sectoral output, value-added, and enterprise-level customs data. The presented MRIO table reflects the heterogeneity between cities and sectors. Benefiting from its coverage of the detailed nested among various cities and industries in China, its application can support the investigation of upstream and downstream linkages between sectors and regions. Moreover, when used in conjunction with other datasets, it enables the identification and analysis of key drivers of social and environmental changes. Meanwhile, some packages or systems can be developed to facilitate the retrieval and interpretation of input-output data (e.g., [40,41]). As stated above, researchers have applied MRIO tables to study environmental impact, social ecology and social influences [26]. Furthermore, scholars have proposed an input-output-based analysis to develop a set of indicators for assessing economic performance. A study by Meng et al. [42] applies the 2007 and 2010 Chinese MRIO tables to identify the importance of inter-provincial spillover effects in regional CO₂ emissions growth. More advanced, the MRIO table presented in this paper facilitates the construction of inter-city and intra-city spillover effects in China. Therefore, our approach and the developed MRIO table have significant policy implications for various fields of economics, including international trade and economic growth. Our method can improve the utility of MRIO analysis for regional scientists and decision-makers alike.

This study has several limitations. Due to data availability, some assumptions about the construction process must be made. In addition, service trade data are not collected in our database. Therefore, various weaknesses in the data exist, and there is still space for improvement. Another challenge is the construction of input-output tables for time series. Existing IO tables have been developed for particular benchmark years. The time series input-output tables help achieve dynamic monitoring of long-term economic structures and industrial linkages. The lack of such tables also poses an obstacle to the integration of input-output methods with other quantitative analysis methods, such as econometric models that often require continuous annual time series. Future research could investigate these aspects and yield more relevant implications.