The average environmental efficiency technique and its application to Chinese provincial panel data

In this study, we propose average environmental efficiency, a more comprehensive, fair, comparable, and robust environmental efficiency measurement considering all projection directions to the efficient frontier, and then it is used to evaluate the environmental efficiency of Chinese provinces from 2006 to 2017. Furthermore, we investigate the most influential factors of regional environmental efficiency via a feasible generalized least squares regression approach. The empirical results show that only nine Chinese provinces have average environmental efficiency greater than the national average, implying that two-thirds of the provinces still have much room for improvement. Additionally, environmental efficiency disparities exist between provinces and between four larger geographical areas. The east area achieved the best environmental efficiency over the studied period, better than the whole country, followed in order by the west area, central area, and northeast area. Moreover, we find that the energy consumption structure, government intervention, and economic openness significantly and negatively influence regional environmental efficiency. Finally, we provide policy implications in terms of energy consumption structure optimization, government supervision, and foreign investment introduction while considering the local conditions in different provinces.


Introduction
China has achieved considerable economic growth in more than 4 decades since its "Reform and Opening-up" policy started in 1978. According to data provided by the World Bank, China has become the world's second largest economy after the USA, and China's gross domestic product (GDP) increased from 365 billion (CNY) in 1978 to 101,598.6 billion (CNY) in 2020 (National Bureau of Statistics of People's Republic of China, 2021a). The price of such rapid economic growth has been a huge consumption of energy resources, as well as serious environmental pollution problems (Li et al., 2020b). For example, in 2020, China consumed 4.98 billion tons of standard coal equivalent energy and produced 3.18 million tons of SO 2 , 11.81 million tons of nitrogen oxides, and 6.13 million tons of particulates, as well as other pollutants (National Bureau of Statistics of People's Republic of China, 2021b). With this backdrop, how to trade off economic growth, resource conservation, and environmental friendliness is important and necessary for Chinese policymakers in central and local governments. Environmental efficiency is an index that considers the environmental issues in the production efficiency evaluation, measuring the coordinated development of the economy and environment (Mardani et al., 2017;Chen et al., 2020). This index incorporates economic production and environmental factors into a unified framework, a comprehensive assessment of the performance of resource consumption, economic development, and environmental pollution. Improving environmental efficiency is one key way to create economic growth Responsible Editor: Eyup Dogan while achieving energy conservation and emission reduction (Zha et al., 2016;Wang et al., 2018;Stergiou and Kounetas, 2021). Therefore, in this study, we address the problems of (1) how to reasonably evaluate the environmental efficiency of provincial-level regions in China and (2) what the contributing factors are that determine environmental efficiency.
Data envelopment analysis (DEA) is an effective approach to evaluate the environmental efficiency of a set of homogenous decision-making units (DMUs) involving multiple inputs (e.g., labor, energy consumption), multiple desirable outputs (e.g., GDP), and multiple undesirable outputs (e.g., smoke and dust emission) (Zhou et al., 2018;Zhu et al., 2020). The basic idea of DEA methodology is that the efficient frontier is established, and the environmental efficiency of a DMU is evaluated by comparing it with a benchmark on the efficient frontier (Zhu et al., 2022). Note that DEA does not need a predetermined, specific function between input and output. This method has been widely applied in environmental efficiency evaluation at the macro-and micro-levels. For example, Li and Wang (2014) combined a slacks-based efficiency measure and the meta-frontier technique to assess the environmental efficiency of 95 countries. Chen et al. (2017) analyzed the city-level environmental efficiency in the Yangtze River Economic Zone during 2003-2014 using the super-efficiency DEA model. Moutinho et al. (2017) used the CCR and BCC models to measure the eco-efficiency of 26 European countries between 2001 and 2012. Cecchini et al. (2018) applied an SBM-DEA model to assess the environmental efficiency of 10 dairy cattle farms in Umbria (Italy). Halkos and Polemis (2018) measured the environmental efficiency of the power generation sector in the USA by using a window DEA model. Song et al. (2018) evaluated the environmental efficiency of 30 Chinese provinces using a Ray slack-based model. Mavi and Mavi (2019) developed a DEA model with a common set of weights to analyze the energy and environmental efficiency of 34 OECD countries during 2012-2015. Ouyang and Yang (2020) proposed a multiplicative network DEA model to evaluate the energy and environmental efficiency of 27 OECD countries while considering the interaction between departments of production, energy, humanities, and economics. Liu et al. (2020) used the BCC model to measure China's energy efficiency at the provincial level and analyzed its regional differences from 2006 to 2016. Cui and Li (2020) built dynamic benevolent and aggressive DEA cross-efficiency models to evaluate the environmental efficiency of 29 global airlines from 2010 to 2016. Koçak et al. (2021) used a bootstrap DEA approach to evaluate the environmental efficiency of disaggregated energy R&D expenditures in OECD countries. See the literature reviews of Meng et al. (2016), Mardani et al. (2017), Sueyoshi et al. (2017), and Mahmoudi et al. (2020) for more discussions on environmental efficiency evaluation.
One of several DEA-based environmental efficiency models, the nonparametric, directional distance function (DDF) approach, has received significant attention. It has been applied by many scholars in different areas of study because this approach measures the efficiency of DMU along the direction that increases desirable outputs and decreases inputs and undesirable outputs simultaneously (Wang et al., 2019b). For example, Oh (2010) introduced a global Malmquist-Luenberger productivity index based on DDF and applied it to measure the dynamic environmental efficiency of 26 OECD countries from 1990 to 2003. Mandal and Madheswaran (2010) measured the environmental efficiency of the Indian cement industry using both DEA and DDF approaches. Ramli et al. (2013) proposed a scale DDF approach to select an optimal projection direction for each DMU and applied this approach to measure the eco-efficiency of the Malaysian manufacturing sector. Du et al. (2014) employed a nonparametric meta-frontier DDF approach to measure the CO 2 emission efficiency and potential emission reduction of 30 Chinese provinces from 2006 to 2020. Beltrán-Esteve and Picazo-Tadeo (2015) analyzed the dynamic environmental efficiency in the transport sectors of 38 countries using a nonparametric DDF approach based on Luenberger productivity indicators. Kounetas (2015) estimated the environmental efficiency technology gaps in 25 European countries based on a DDF approach. Wang et al. (2016) evaluated the carbon emission efficiency of 30 Chinese provinces during 1996-2012 using a nonradial DDF approach. Sun et al. (2017) developed a nonradial DDF preference model to evaluate the environmental efficiency of 17 Chinese port enterprises. Lee and Choi (2018) measured the environmental efficiency of 16 provinces in Korea by applying a nonradial DDF and examined the pure technical efficiency and scale efficiency. Chen and Xu (2019) used a super-efficiency DDF approach to assess the environmental efficiency and Malmquist-Luenberger productivity index of provincial regions in mainland China from 2000 to 2015. Tovar and Wall (2019) used an output-oriented DDF approach to measure environmental efficiency for a cross-section of 28 Spanish port authorities in 2016. Li et al. (2020a) used a meta-frontier nonradial DDF approach to measure the static and dynamic CO 2 emission performance of 16 port enterprises in China covering the years 2013 to 2018. Li et al. (2020b) empirically analyzed the regional environmental efficiency in China based on an entropy weight method and nonparametric DDF model, and they found environmental efficiency disparities between regions. Li et al. (2021) developed a bound DDF model to evaluate the energy and environmental efficiencies of 30 provinces in mainland China from 2011 to 2015. Singh and Gundimeda (2021) measured the environmental efficiency of the grossly polluting Indian leather industry using DDF and three directional vectors.
Although the DDF approach has been extensively used for measuring the environmental efficiency of countries, regions, and economic sectors, one crucial issue in previous studies that should get attention is that the environmental efficiencies of different DMUs are measured based on different projection directions. The inconsistent projection directions between DMUs will result in unfair and incomparable evaluation results. Specifically, the efficiency value depends considerably on the selection of the projection direction, so one DMU can dominate another along one direction but be dominated along another direction, thus leading to the DMUs having little incentive to accept the evaluation results. Although some scholars propose evaluating DMUs based on a common direction (e.g., Chu et al., 2021;Sharma and Majumdar, 2021), other feasible projection directions include relevant information that could be helpful for decision support, but these are ignored.
The above discussion motivates a new question: is it possible to evaluate the environmental efficiency of a DMU based on all possible projection directions? Taking into account all possible projection directions not only evaluates all DMUs using the same equitable criterion, which increases the acceptability of evaluation results, but also incorporates all information associated with these directions, offering rich evaluation information for decision-makers. Previous studies have addressed similar problems that the efficiency and ranking of a DMU are sensitive to the input and output weights (e.g., Lahdelma and Salminen, 2006;Salo and Punkka, 2011;Li et al., 2020b;Wei et al., 2021a), but the problem of considering all possible projection directions has not been tackled in DDF approach-based environmental efficiency evaluation.
To complement the previous research, in this study, we propose an alternative environmental efficiency measurement, named average environmental efficiency, to evaluate the environmental efficiency values of DMUs. According to the research paradigm of Yang et al. (2018), Lozano and Soltani (2020), and Wei et al. (2021b), within the framework of the nonparametric DDF approach, we can measure all possible environmental efficiency scores of a DMU by considering all possible projection directions to the efficient frontier. Then, the mean value of the environmental efficiency scores of a DMU in all possible projection directions is defined as the average environmental efficiency. Because the average environmental efficiency considers all projection directions to the efficient frontier, it not only ensures that all DMUs are evaluated on the same basis but also eliminates the sensitivity issue of the efficiency value, i.e., the average environmental efficiency has properties of fairness, comparability, comprehensiveness, and robustness. Afterward, the average environmental efficiency is applied to an empirical study of 30 Chinese provincial environmental efficiency using panel data from 2006 to 2017. Moreover, to provide central and local policymakers with more valuable suggestions on how to improve environmental efficiency, we further examine the determinants of environmental efficiency by using the feasible generalized least squares (FGLS) regression technique. Based on the results of all these analyses, targeted policy implications are given.
Overall, this study methodologically and empirically enriches the growing research on regional environmental efficiency evaluation. Specifically, in terms of method innovation, the proposed average environmental efficiency measurement makes up for the shortcomings of DEA-based environmental efficiency evaluation, thus providing methodological support for domestic and foreign scholars from various areas (e.g., DEA scholars, environmental science scholars) to study environmental efficiency. With regard to application, our research not only helps the Chinese central and local policymakers understand the disparities of environmental efficiencies between provinces and between four geographical areas, but it also provides targeted policy recommendations for improving environmental efficiency, thereby opening the door for the study of environmental efficiency at the industry or city level in China and other countries in general.
The rest of the paper unfolds as follows. In the "Methodology" section, we present the preliminaries and develop the average efficiency measurement. The "Empirical analysis" section gives the empirical analysis. The "Conclusions and discussion" section concludes the paper by giving the key findings of this study while providing policy implications and future research directions.

Preliminaries
In our model for environmental efficiency evaluation, each DMU refers to a Chinese provincial administrative region. Suppose that there are n provincial regions to be evaluated, denoted as DMU j (j = 1, 2, ⋯ , n) , and each DMU consumes m input resources to produce s desirable outputs, such as economic benefit, and p undesirable outputs, such as pollutant emissions. The input vector, desirable output vector, and undesirable output vector of DMU j (j = 1, 2, ⋯ , n) are denoted as . The multi-input and multi-output production technology can be characterized as follows.
Equation (1) where T always satisfies the axioms of production theory, such as being closed, bounded, and convex (Färe and Grosskopf, 2006). The strong disposability assumption is imposed on undesirable outputs in this study, and the nonparametric DEA approach is used to construct the strong disposal environmental production technology (Yang and Pollitt, 2010;Sueyoshi and Goto, 2012;Xian et al., 2019;Afzalinejad, 2021;Ma et al., 2021), as follows.
In Eq.
(2), j , ∀j are the nonnegative intensity variables for constructing the environmental production technology using a convex combination. It can be observed that T is constructed under the assumption of variable returns to scale (VRS). VRS commonly occurs in the real world because different DMUs have different production technologies, which means that not every DMU reaches the optimal production scale, i.e., when the inputs increase (decrease), the outputs will increase (decrease) proportionally (Banker et al., 1984).
Figures 1 a and b, respectively, depict the weakly and strongly disposable production possibility sets, indicated by an interior solid line. It can be seen from Fig. 1a that constrained by the weak disposability of undesirable outputs, the blue area (triangle area DHF) is excluded. However, in practice, production in this area is feasible; for example, DMU D can increase the undesirable outputs while keeping its desirable outputs unchanged. Actually, the projections of the undesirable outputs of the evaluated DMU will not be greater than its original value; that is, the right area of the evaluated DMU will not be its projection area, as Fig. 1b illustrates. Therefore, the strong disposability assumption is imposed on undesirable outputs.

Directional distance function with undesirable outputs in data envelopment analysis
The directional distance function (DDF) approach was introduced by Chambers et al. (1996) and further extended by Chung et al. (1997) to measure environmental efficiency. The DDF with undesirable outputs is defined such that it aims to decrease inputs and undesirable outputs and increase desirable outputs simultaneously in the same proportion, as shown below.
In Eq.
(2) and (3), the following DEA-type model is used to calculate the value of for a given DMU, denoted as DMU o .
In model (4), the value of is calculated along the directional vector (g . In previous studies, the observed input and output value of the evaluated DMU o is always set as a directional vector, i.e., g x i , g y r , g b t = x io , y ro , b to , ∀i, r, t , and the environmental efficiency is defined as E o = 1 − o (Wang et al., 2019b), but other directional vectors such as unit value direction are also applied (e.g., Färe et al., 2006;Halkos and Tzeremes, 2013). In other words, the selection of the directional vector in model (4) is flexible, depending on the purpose of the study (Ray, 2008;Zhang and Choi, 2014;Wang et al., 2019a, b, c). In addition, the value of o may be larger than one thus leading to E o = 1 − o < 0 , i.e., yielding negative efficiency (Cheng and Zervopoulos, 2014). Therefore, Cheng and Zervopoulos (2014) proposed a generalized definition of the efficiency score for the DDF and transformed model (4) to the following model (5). Equations (4) and (5) (3) Weakly/strongly disposable production possibility set. a Weakly disposable production set. b Strongly disposable production possibility set (a) Weakly disposable production possibility set (b) Strongly disposable production possibility set where o stands for the environmental efficiency, which lies between 0 and 1. Model (5) has a distinctive feature that the efficiency score o depends on the direction of the directional vector, not the length of the directional vector (Cheng and Zervopoulos, 2014;Yang et al., 2018); that is, the value of o will not change whether (g

Average environmental efficiency
Within the DEA DDF framework, the environmental efficiency of a DMU is commonly measured along a direction determined by either an exogenous (e.g., input/output observation value direction) or endogenous (e.g., direction toward the furthest target) mechanism (Wang et al., 2019b;Wei et al., 2019). This means that each DMU is evaluated based on its own projection direction, chosen according to the study purpose. Moreover, the environmental efficiency may lack robustness since it changes along with the projection direction Lozano and Soltani, 2020;Wei et al., 2021b). As a result, the evaluation results may lack fairness and comparability, thereby reducing the acceptability of the results to DMUs. As illustrated in Fig. 2, for a specific DMU G , its environmental efficiency is evaluated by comparing it with any benchmark on the efficient frontier Ĥ ACI , i.e., in the upper-left projection directions; the entire shadowed area allows DMU G to increase its environmental efficiency by expanding desirable outputs while reducing inputs and undesirable outputs simultaneously. In this study, with the help of model (5), we try to explore all possible projection directions toward the efficient frontier and propose using the average environmental efficiency to compare all the evaluated DMUs.
The efficiency score o obtained from model (5) is independent of the directional vector's length (Cheng and Zervopoulos, 2014). Therefore, as Cheng (2014) suggested, the unit vectors can be regarded as directional vectors and used in model (5). Define the set Θ to include all unit With the help of a directional vector scanning approach developed by Cheng (2014) and later applied by Yang et al. (2018), uniformly distributed unit vectors can be extracted from Θ . The set containing all uniformly distributed unit vectors is denoted as Ψ , and Ψ ⊂ Θ , i.e., Ψ is the so-called directional vector set. In Ψ , each unit vector g can be used as a directional vector in model (5), and the corresponding environmental efficiency score (g) can be obtained. For a certain DMU o , E o is a set that includes all environmental efficiency scores corresponding to all directional vectors in Ψ , that is, In line with Lahdelma and Salminen (2006), considering all environmental efficiency scores in E o , the average environmental efficiency of DMU o , denoted ave o , is defined as the expected value of efficiency scores over the directional vector distributions, as expressed below. Equation (6) The average environmental efficiency defined in formula (6) is a comprehensive evaluation result considering all directions to the frontier, which guarantees that all DMUs are evaluated using the same criterion. Additionally, this definition avoids the risk of the efficiency score changing with the projection direction selection, i.e., it improves the robustness of the environmental efficiency score. Although formula (6) is a continuous integral, it can be calculated with sufficient accuracy by using a sufficiently high value of Q in the following formula. Equation (7) (6) where Q is the number of directional vectors in Ψ. Remark 1. The more the number of directional vectors, Q , the more accurate the average efficiency, ave o , but the greater the required calculation time.
Remark 2. The value of ave o always lies within the interval 0 and 1, and ave o = 1 means the evaluated DMU o is efficient, while ave o < 1 means it is inefficient.

Sample, variables, and data description
Referring to the previous studies on the environmental efficiency evaluation using DEA methods (e.g., Vlontzos et al., 2014;Li et al., 2020a, b;Liu et al., 2020), three input variables, one desirable output, and two undesirable outputs are considered for environmental efficiency evaluation in this study. Labor, fixed asset investments, and energy consumption are the three inputs; GDP (gross domestic product) is the one desirable output; and SO 2 emission and soot emission are the two undesirable outputs. Due to data availability, this study's sample includes 30 provincial-level administrative regions in China's mainland from 2006 to 2017. (Tibet, Hong Kong, Macau, and Taiwan are excluded because of the missing data. All 30 entities are referred to as provinces for convenience in the following analysis.) All data was collected from four sources: the China Statistical Yearbook, DRCNET Statistical Database System, Provincial Statistical Yearbook, and China Labor Statistical Yearbook. A detailed description of inputs and outputs is given below.
(1) Labor input. Labor is a widely accepted input variable of human capital, reflecting the actual utilization of all labor resources within a certain period (Chang et al., 2013). In this study, labor refers to the people in the employed labor force at the end of the year. (2) Fixed asset investment input. This measure represents the capital stock. As previous studies did (e.g., Lin et al., 2018;Kong et al., 2019;Liu et al., 2020), the capital stock used in this work is calculated by the perpetual inventory method proposed by Goldsmith (1951), shown as follows. Equation (8) Here, K j,t represents the capital stock of province j at year t, and K j,t−1 represents the capital stock of province j at year t-1; I j,t is the annual added fixed investment of province j at year t; and is the depreciation rate, which is set at 10.96% in this study. The year 2000 is set as the initial year, and the initial capital stock is calculated by the following equation. Equation (9) where g j is the average annual investment growth rate of province j during 2000-2017.
(3) Energy consumption input. The total energy consumption is the total amount of various types of energy consumed in production and daily life, mainly including energy from coal, petroleum, natural gas, and hydroelectric plants. The energy amounts from various sources are converted into total energy consumption, represented by standard coal (Li et al., 2020a, b;Liu et al., 2020). (4) Desirable output. GDP is an important and widely accepted indicator for measuring economic development. It reflects the economic strength and market size of a country or region. To eliminate the effect of inflation, 2006 is used as the base year to convert the nominal GDP into a real GDP. (5) Undesirable output. Wastewater, waste gas, and solid waste are commonly used as undesirable outputs. Due to the data availability and completeness, the emissions of SO 2 and of smoke and dust are used as undesirable outputs in this study.
1 summarizes the statistical descriptions for all input and output variables.
In addition, according to the National Bureau of Statistics (2020), mainland China is divided into four areas for purposes of analysis: east, central, west, and northeast. Table 2 lists the provinces and the areas to which they belong.

Environmental efficiency analysis
In our empirical study, we generate 58,905 unit vectors via the directional vector scanning approach. These are taken as directional vectors and substituted into model (5) to obtain 58,905 environmental efficiency scores for each of the 30 provinces. Table 10 in the Appendix summarizes the statistical descriptions of environmental efficiencies for 30 provinces from 2006 to 2017. Figure 3 clearly illustrates the variation of the environmental efficiency of the 30 provinces in 2017. It can be observed that the environmental efficiency varies substantially for the inefficient provinces, indicating that different projection directions yield different environmental efficiency scores. In other words, the environmental efficiency is sensitive to the projection direction, which implies that the environmental efficiency measured along a single projection direction (determined by either an exogenous or endogenous mechanism) lacks fairness and comparability. Therefore, the average environmental efficiency that incorporates all possible projection directions is preferable to measure the environmental efficiency of each province.
The calculated results for average environmental efficiency from 2006 to 2017 of 30 provinces are reported in Table 3, and Fig. 4 clearly shows their average environmental efficiency during the studied span. We can learn from combining Table 3 with Fig. 4 that China has a high environmental efficiency score of 0.802. Also, only nine provinces have average environmental efficiency greater than the national average: Shanghai with 1.000, Beijing with 1.000, Tianjin with 0.942, Guangdong with 1.000, Jiangsu with 0.968, Zhejiang with 0.905, Hainan with 1.000, Ningxia with 0.945, and Qinghai with 1.000 (Fig. 5). This implies that even though the country's environmental efficiency is good, twothirds of the provinces still have much room for improvement. Shanghai, Beijing, Tianjin, Guangdong, Jiangsu, and Zhejiang are economically developed regions in China, all of which have advanced environmental management concepts, allowing them to better balance resource consumption, economic growth, and pollution emissions and thereby attain high environmental efficiency. Unlike the findings in previous studies (e.g., Yu et al., 2019;Li et al., 2020a, b;Liu et al., 2020), our results show that Hainan, Ningxia, and Qinghai achieved very high environmental efficiency during the investigated time span. These three provinces, especially Ningxia and Qinghai, are underdeveloped regions in China,  it can be seen that their energy consumption accounts for about 1 ~ 2%, and their pollution emission accounts for less than 1%. Additionally, note that Xinjiang is the only province with an environmental efficiency value less than 0.7. We find that the total energy consumption of Xinjiang ranked around tenth among all provinces, and its pollution emissions ranked around fifth, but its GDP ranked in the bottom five during the investigated period. This indicates that Xinjiang did not allocate resources efficiently and paid little attention to clean production, resulting in low environmental efficiency. Table 3 provides the environmental efficiency of the four areas from 2006 to 2017, and Fig. 6 illustrates their environmental efficiency trends. We can find from Table 3 that the east area achieved the best average environmental efficiency over the studied period with the value of 0.911, followed by the west area (0.765), central area (0.730), and northeast area (0.716). From 2006 to 2017, the east area had the best environmental efficiency each year, followed by the west  The main reason is that the east is an economically developed area; it has advanced production technology, as well as abundant human and material capital, which is good for high environmental efficiency. We also see that the central area performed worse than the west area in terms of environmental efficiency. As part of the "Rise of Central China Strategy" that was initiated in 2004, the provinces in the central area invested more in infrastructure construction to facilitate To alleviate the adverse effects of the financial crisis, the Chinese government implemented many policies and measures to stimulate the economy, such as increased investments in infrastructure construction. Ecological protection was not taken seriously, and as a result, the environmental efficiency of China was hampered. Additionally, Fig. 6 displays that the environmental efficiency of the whole country, as well as the east and central areas showed an upward trend, while the other two areas decreased slightly after 2015. This result agrees with Yu et al. (2019) and Zhao et al. (2019). This achievement may be due to the strict implementation of the "energy conservation and emissions reduction" policy in China's 13th Five-Year Plan (2016. That is, compared with 2015, the energy consumption should be limited to 5 billion tons of coal equivalent, and the total chemical oxygen demand, ammonia nitrogen, sulfur dioxide, and nitrogen oxide emissions should be reduced by 10%, 10%, 15%, and 15%, respectively.

Analysis of influencing factors
To provide more valuable suggestions on the improvement of regional environmental efficiency to policymakers, in this section, we further investigate the factors that may influence the environmental efficiency, which is characterized by the average environmental efficiency (hereafter "AEE") in this study. Referring to the relevant literature (e.g., Kumar and Khanna, 2009;Li and Wang, 2014;Yu et al., 2019;Liu et al., 2020) and considering data availability, the influencing factors this study selected include the energy consumption structure, industrial structure, government intervention, economic openness, and human capital. That is, the AEE is an explained variable, and the above five factors are explanatory variables. Table 4 shows detailed descriptions of these five explanatory variables.
Next, we discuss the relationship between the environmental efficiency and each of five explanatory variables and propose corresponding hypotheses.

Energy consumption structure
The energy consumption structure is directly associated with pollutant emissions, which implies that the energy consumption structure influences the environmental efficiency (Hatzigeorgiou et al., 2008;Ma, 2015). Compared with clean energies, the combustion of fossil fuels, especially coal and oil, yields various pollutants, such as SO 2 , smoke, and dust . Consequently, the higher the proportion of coal and oil in energy consumption, the more the pollution and the greater the environmental damage (Li and Wang, 2014;Li et al., 2018;Yu et al., 2019). Accordingly, we propose the first hypothesis that energy consumption structure negatively affects environmental efficiency. The proportion of the population with primary education and above, multiplied by the number of years corresponding to various education levels, to the population aged at least six Industrial structure Industrial structure is a fundamental influencing factor of an environmental Kuznets curve (EKC) pattern (Kaika and Zervas, 2013), and it is considered to be related to environmental efficiency (Bhattarai and Hammig, 2001). The industrial structure in China has long been in a traditional mode of extensive economic growth, characterized by excessive energy consumption and severe environmental pollution (Wang et al., 2019a). Secondary industry is a major source of energy consumption and pollution emissions in China, and many scholars have found that a higher proportion of secondary industry leads to lower environmental efficiency (e.g., Liu et al., 2019;Wang et al., 2019a). Hence, we propose the second hypothesis that there is a negative relationship between industrial structure and environmental efficiency.

Government intervention
The impact of government intervention on environmental efficiency is not uniformly positive or negative (Wang et al., 2021). In the early stage of economic development, GDP is the priority, so the government prefers to use a large amount of fiscal expenditure for production fields to accelerate economic development while ignoring environmental protection, thus leading to environmental damage (Copeland and Taylor, 2004). At this stage, government intervention negatively influences the environmental efficiency. With more advanced economic development, the government increases its attention to environmental protection, and the central and local governments will increase investment to protect the environment, which is conducive to environmental efficiency improvement . Considering that the studied period in this work includes the financial crisis of 2008, the government invested more in production for economic recovery in the following years, and thus paid relatively little attention to environmental protection. Based on this, we propose the third hypothesis that government intervention negatively influences environmental efficiency.

Economic openness
Openness brings not only advanced technology in a region but also brings high energy consumption and high pollution emission transfer (Liu et al., 2019). Based on the pollution haven hypothesis, companies in developed countries are willing to transfer their production with high energy consumption and high pollution to countries with lower environmental protection requirements (Shahbaz et al., 2015;Sarkodie and Strezov, 2019). Such foreign capital inflow inevitably has a negative impact on the host country's environment (Zafar et al., 2020). Therefore, we propose the fourth hypothesis that economic openness harms environmental efficiency.
Human capital Pablo-Romero and Sánchez-Braza (2015) proposed that human capital investment is an effective way to save energy and reduce emissions. On the one hand, human capital promotes the research and development of clean energy and accelerates the transformation of energysaving technologies, which finally improves energy utilization and reduces pollutant emissions (Li and Lin, 2016;Fang et al., 2017;Salim et al., 2017). On the other hand, human capital helps to foster awareness of energy conservation and environmental protection (Zografakis et al., 2008). Accordingly, we propose the fifth hypothesis that human capital positively influences environmental efficiency.
All explanatory variable data were extracted from the China Statistical Yearbook and Provincial Statistical Yearbook. The statistical descriptions of explanatory variables are reported in Table 5, and Table 6 provides the pairwise correlations of explained and explanatory variables. Table 6 shows that high correlations exist between energy consumption structure and industrial structure, i.e., the correlation coefficient is 0.550. However, the variance inflation factors (VIF) of all variables are lower than 2.0, much smaller than the recommended cut-off value of 10. In other words, there is no multicollinearity between variables, which can be used for the following regression analysis. In addition, the LLC (Levin et al., 2002) and ADF (Dickey and Fuller, 1979) methods are used to test whether the variables are stationary. The results, reported in Table 7, show that no unit root exists, which implies that the panel data is stationary.
With the selected variables, the specific regression model is expressed as follows.
In Eq. (10), AEE jt represents the average environmental efficiency of province j in year t. ECS, IS, GI, EO, and HC arse the energy consumption structure, industrial structure, government intervention, economic openness, and human capital, respectively. The coefficients of independent variables are n (n = 1, 2, ⋯ , 5) , and jt is the error term.  data. To correct for these issues, the feasible generalized least squares (FGLS) approach, an extension of OLS, is applied in this study; the regression results are reported in Table 9. In columns 2 and 3, the explained variables are environmental efficiency measured by model (5) using directional vectors of (−x 1o , −x 2o , −x 3o , y 1o , −b 1o , −b 2o ) and (−1, −1, −1, 1, −1, −1) ; in column 4, the AEE is used as the explained variable. We can learn from columns 2 and 3 in Table 9 that the regression results are different when environmental efficiency is measured using different projection directions; these differences increase the decisionmaking risk and also motivate the adoption of average  environmental efficiency as an alternative measurement. It can be observed from column 4 that energy consumption structure, government intervention, and economic openness negatively influence the environmental efficiency at the 1% or 10% significance level, while industrial structure and human capital have no significant effects on environmental efficiency. Furthermore, we check the robustness of our findings using the sample from 2006 to 2015, which covers the 11th Five-Year Plan period (2006)(2007)(2008)(2009)(2010) and 12th Five-Year Plan (2011, i.e., 300 province-year observations. We also replace the average environmental efficiency by the median environmental efficiency as the explained variable.
The results of the robustness checks are reported in columns 5 and 6 in Table 9, and they are consistent with our findings above. The detailed analysis unfolds as follows.
The energy consumption structure significantly and negatively influences the regional environmental efficiency; that is, the higher the proportion of coal consumption, the greater the negative impact on environmental efficiency, which is similar to the conclusions of Wang et al. (2019c), Yu et al. (2019), and Zhao et al. (2019). A 1% increase in coal consumption decreases the environmental efficiency by 0.057 when other independent variables remain unchanged. During the period of the centrally planned economy, the low price of coal and abundant coal stocks led China to heavily rely on coal consumption (Crompton and Wu, 2005). In recent years, the Chinese government has been committed to the development and utilization of renewable energy, but coal consumption still accounts for more than 55% of China's total energy consumption, thus increasing the difficulty and cost of environmental protection and negatively influencing the environmental efficiency.
Government intervention negatively influences the regional environmental efficiency at the 10% significance level, which is consistent with the result of Yu et al. (2019), but some scholars (e.g., Liu et al., 2019Liu et al., , 2020 did not find a significant relationship between government intervention and environmental efficiency, while Wang et al. (2021) found a nonlinear U-shaped curve relationship. To deal with the adverse effects of the financial crisis of 2008, the government increased investments in infrastructure construction to expand economic development while squeezing out the investment in environmental protection, which finally hampered environmental efficiency (Liu et al., 2020), as reflected in the downward trend during 2008 to 2015 in Fig. 5. As of 2017, the Central Environmental Protection Inspection Team has completed coverage of all provinces. Prior to this, the pursuit of economic growth was still the priority of local governments.
There is a significant and negative relationship between economic openness and environmental efficiency. This finding is similar to the conclusions of Li and Wang (2014), Liu et al. (2019), and Wang et al. (2019a). Keeping other independent variables unchanged, a 1% increase in economic openness decreases environmental efficiency by 0.249. Compared with developed countries, developing countries have relatively loose environmental regulations (Copeland and Taylor, 1994). To pursue economic development and political promotion, the local governments have long sought to attract foreign investment. Some companies with high energy consumption and high pollution emission have been introduced blindly, thus causing environmental deterioration and hampering environmental efficiency.

Conclusions and discussion
This study is motivated by both reality and methodology. China's rapid economic growth has been accompanied by huge energy consumption and severe environmental pollution. With this backdrop, balancing economic growth, energy conservation, and environmental friendliness is an important task for China at present and for a long time in the future. Improving environmental efficiency is one of the most effective ways to achieve sustainable development. However, unfair, incomparable, biased, and sensitive evaluation results have occurred in previous studies on environmental efficiency. Accordingly, we propose a more fair, comparable, comprehensive, and robust environmental efficiency measurement in this study, namely, the average environmental efficiency measure. We then use it to evaluate and analyze the environmental efficiency of Chinese provinces and examine its influencing factors, providing timely information to guide policy implementation for the Chinese central and local policymakers. The following are the key findings, policy implications, and future research directions.

Key findings
Our main findings are as follows: First, the environmental efficiency is sensitive to the projection direction to the frontier, which implies that the environmental efficiency lacks fairness and comparability when measured along a projection direction determined by either an exogenous or endogenous mechanism. The average environmental efficiency is more fair, comparable, comprehensive, and robust because it includes all possible projection directions to measure the environmental efficiency of each province.
Second, only nine provinces have average environmental efficiency greater than the national average, namely, Shanghai with 1.000, Beijing with 1.000, Tianjin with 0.942, Guangdong with 1.000, Jiangsu with 0.968, Zhejiang with 0.905, Hainan with 1.000, Ningxia with 0.945, and Qinghai with 1.000. This implies that two-thirds of the provinces still have much room for improvement.
Third, provinces in the east area achieved the best average environmental efficiency over the studied period, scoring 0.911, followed by the west area (0.765), central area (0.730), and northeast area (0.716). Furthermore, the results show that the environmental efficiency of China and its four areas all exhibited a downward trend from 2008 to 2015. However, the national environmental efficiency, as well as that of the east area and central area, showed an upward trend, while the other two areas decreased slightly after 2015.
Fourth, using a feasible generalized least squares regression approach, we find that the energy consumption structure, government intervention, and economic openness negatively influence the regional environmental efficiency at the 1% or 10% significance level. In contrast, the effects of industrial structure and human capital on environmental efficiency are not significant.

Policy implications
Considering that differences in environmental efficiency exist between provinces as well as four geographical areas, and based on the determinants of environmental efficiency, policy implications should be formulated based on local conditions instead of using a one-size-fits-all approach. We propose the following.
Firstly, the provinces in the east area have better environmental efficiency. Most of them are economically developed provinces with advanced production technology, which can well achieve balanced development between resource consumption, economic growth, and pollution emissions. Therefore, the local governments should focus on the exogenous underlying factors to improve environmental efficiency, such as increasing the development and utilization of renewable energy and clean energy, as well as building a diversified performance appraisal system.
Secondly, the provinces in the central area have lower environmental efficiency than the national average. The central area provinces have been in a stage of rapid economic development since the implementation of the "Rise of Central China Strategy" in 2004. A multi-oriented rather than a GDP-oriented performance appraisal mechanism should be formed, especially increasing the weight of environmental protection in local government performance evaluation, thus urging the local governments to increase fiscal expenditure on environmental protection. Additionally, local governments should strengthen the review and evaluation of foreign companies, avoiding blindly introducing foreign investment.
Thirdly, although the west area has the second highest environmental efficiency, the environmental efficiency varies substantially among western provinces. Considering the rich tourism resources, these western provinces should increase investments in tertiary industry, thereby driving rapid economic growth. In addition, the local governments should make full use of technology spillover effects from the east and central areas to improve their production technology.
Finally, the provinces in the northeast area have the lowest average environmental efficiency. Given the background of "the revitalization of the Northeast" policy started in 2004, the local governments are more willing to pursue rapid economic development while ignoring environmental protection. Therefore, on the one hand, the local governments need to pay attention to the transformation and upgrading of production technology. On the other hand, the central government should gradually increase the weight of environmental performance in local government performance appraisal, thus urging the local governments to invest more in environmental protection rather than blindly expanding the local economy. Additionally, the local governments should raise the threshold for the introduction of foreign investment and reject foreign investment characterized by heavily polluting and low technology.

Future research
Several future research directions should be considered. First, restricted by data availability, we analyze the environmental efficiency and its influencing factors based on a sample of 30 provinces in mainland China. We could extend our study to the city level and provide new implications for scholars and policymakers. Second, in this study, emissions of SO 2 and of smoke and dust are selected as undesirable outputs, but other pollutants, such as CO 2 , wastewater, and solid waste, could be considered. Third, this study examines the effect of each factor on environmental efficiency while assuming that the other factors remain unchanged; the configurational path to environmental efficiency under an interplay of factors presents an intriguing path for further research.