There have been series of studies investigating the TFP growth in the Chinese banking sector. However, most of them focused on the overall level of the TFP growth and its components.

Kumbhakar and Wang (2007), using the stochastic frontier analysis, presented that the average TFP growth in the Chinese banking sector was positive. In detail, pure efficiency changes did not affect the TFP growth significantly, and the TFP growth of state-owned commercial banks were mostly benefited in positive scale effect, while that of joint stock commercial banks were combined with both positive scale effect and technological progress.

Sufian (2009) used a Malmquist productivity index, taking off-balance sheet into account, to measure the TFP growth of the Chinese banking sector, and it was found that the whole banking sector had low TFP growth, where state-owned commercial banks and city commercial banks had poor technological change, but benefited in scale efficiency change, while joint stock commercial banks were poor in pure technical efficiency change.

Matthews et al. (2009) used a conventional Malmquist productivity index with a smooth bootstrap approach to study the TFP growth of the Chinese banking sector from 1997–2006, and they specifically accounted for non-performing loans which has been a critical factor affecting performance of the Chinese banking sector. Furthermore, Matthews and Zhang (2010) evaluated the TFP growth of the Chinese banks during the period 1998–2007. They determined 5 types of models with different variables according to production and intermediation approaches, and, while investigating the source of the TFP growth, it was found that efficiency gains were driven by cost reduction, while technological progress was promoted by non-interesting income.

Zhu et al. (2015) applied a DEA-based Luenberger productivity indicator to investigate the TFP growth of 25 Chinese banks over the period 2004–2010. It was found that the overall Chinese banking sector performed well, where the change of return to scale in technology was the main driving force during the period studied, and pure technical efficiency change and pure technological change both were not significant, but the scale efficiency change had a negative effect to TFP.

Zhu et al. (2018) used a non-radial, biennial Luenberger productivity indicator to evaluate the TFP growth of the Chinese banking sector during the period of 2004–2012. The overall Chinese banking sector operated with an average growth rate of 5.4%, where technological progress of the Chinese banking sector dominated during the earlier period, and efficiency gains surpassed technological progress during the later period.

Conventional TFP index, like the most widely applied Malmquist productivity index, usually concerns only the overall level, but ignores the contribution of individual factor. In respect of solving the problem, there have been several appropriate approaches to obtain the variable-specific productivity change, like Russell index (Färe and Lovell, 1978), Zieschang index (Zieschang, 1984), slack-based measures (Tone, 2001), among others. Regarding the applications in the banking TFP field, two representative empirical studies include Chang et al. (2012) and Fujii et al. (2014). Chang et al. (2012) incorporated an input slack-based productivity framework into Luenberger productivity indicator, and investigated the source of the TFP growth by disaggregating to individual input productivity contribution. It was found that the technological change was the driving force to promote the TFP growth. Regarding contribution to input productivity change, capital investment was the major source of the TFP growth in the Chinese banking sector. Fujii et al. (2014) applied a weighted Russell directional distance model to measure the TFP change with non-performing loans. It is found that the TFP growth has not improved significantly over the period 2004–2011, where non-performing loans have a positive effect to the TFP growth during the earlier period, but there is a different pattern in non-performing loans after 2008. Moreover, labour force, loans and fixed assets contribute to positive TFP growth, whereas deposits are negative.

However, Asmild et al. (2016) argued the main drawback of these aforementioned disaggregation approaches is that they require a specification of an objective function which involves an arbitrary aggregation of the variable-specific efficiency score. Therefore, a novel MEA-Malmquist productivity index which deals with the problem of specification of an objective function in conventional disaggregation approaches is utilized in the current paper. The MEA approach was originally proposed by Bogetoft and Hougaard (1999), and further developed by Asmild et al. (2003). It is widely applied in various fields, such as transportation (Holvad et al., 2004), banking (Asmild and Matthews, 2012; Zhu et al., 2019), and many others. MEA is directly designed to obtain specific-variable efficiency scores, and it is in that sense a natural choice for specific-variable TFP growth.

Due to different external factors, like environment, resources, and opportunity, the technology of various firms, indeed, may be heterogeneous. However, conventional studies on performance evaluation assumed that all DMUs have a homogeneous technology, namely there exists only one technological frontier for all DMUs. Therefore, Hayami (1969) initially proposed the conception of metaproduction, while Hayami and Ruttan (1970) then provided a clearer interpretation of metaproduction conception.

With exception of the SFA-based metafrontier approach introduced by Battese and Rao (2002) and Battese et al. (2004), the DEA-based metafrontier approach on the basis of distance function was developed by O’Donnell et al. (2008). Subsequently, plenty of banking efficiency research using DEA-based metafrontier approach became popular (e.g. Kontolaimou and Tsekouras, 2010; Chiu et al., 2016).

Apart from the efficiency aspect, O’Donnell et al. (2008) further extended the concept of metafrontier to the domain measuring the TFP growth. Afterwards, Oh and Lee (2010) developed an alternative global metafrontier Malmquist productivity index. In terms of empirical banking TFP studies based on metafrontier framework, Portela and Thanassoulis (2010) applied a metafrontier Malmquist productivity index to assess 59 branches of a Portuguese bank. Kontolaimou and Tsekouras (2010) measured the TFP growth of 1540 cooperative, 541 commercial and 735 savings European banking firms using the metafrontier approach. Chen and Yang (2011) employed the metafrontier Malmquist productivity index and its decompositions to compare banking TFP growth between China mainland and Taiwan region. Chen (2012) used an input-oriented generalized metafrontier Malmquist productivity index to measure TFP, considering the latent effect of risk-taking behaviour of 12 public and 26 private local banks in Taiwan region during 1999–2007. Casu et al. (2013) used both Malmquist productivity index and parametric Divisia index within metafrontier framework to measure the TFP growth in Indian banks during 1992–2009. Zhu et al. (2015) used a metafrontier Luenberger productivity indicator to compare the TFP growth of three types of banks in the Chinese banking sector. Duygun et al. (2016) used a metafrontier Malmquist productivity index to evaluate the TFP growth between UK-based trademarking and non-trademarking commercial banks among 2005–2013. Lee and Huang (2016) examined the TFP growth, applying a global metafrontier Malmquist productivity index, of 1824 commercial banks from 12 Western European countries covering 1993–2006. Zhu et al. (2018) utilized a metafrontier biennial Luenberger productivity indicator to evaluate the TFP growth of 30 Chinese commercial banks during the period of 2004 to 2012.

The MEA-Malmquist productivity index (MEA-MPI) introduced by Asmild et al. (2016) is developed on the basis of the MEA approach initially by Bogetoft and Hougaard (1999) and Asmild et al. (2003). With respect to the MEA, it gains endogenous project directions of inputs and outputs, respectively, based on local single production objective, and further gains endogenous direction vectors based on global dominant set. In Eq. (1), with exception of the n-th input in the set N, where n∈N, the rest including ( N−1) inputs x removing the n-th, M desirable outputs y, and L undesirable outputs b, are fixed. It is also interpretable according to Eqs. (2)–(3), respectively. (1) xk′nt∗(x,y,b)=minθk′nt,s.t.∑k=1Kλktxknt⩽θk′nt,n=1,…,N,∑k=1Kλktxknt⩽xk′(−n)t,−n=N∖{n},∑k=1Kλktykmt⩾yk′mt,m=1,…,M,∑k=1Kλktbklt=bk′lt,l=1,…,L,λkt⩾0,k=1,…,K, (2) yk′mt∗(x,y,b)=maxψk′mt,s.t.∑k=1Kλktxknt⩽xk′nt,n=1,…,N,∑k=1Kλktykmt⩾ψk′mt,m=1,…,M,∑k=1Kλktykmt⩾yk′(−m)t,−m=M∖{m},∑k=1Kλktbklt=bk′lt,l=1,…,L,λkt⩾0,k=1,…,K, (3) bk′lt∗(x,y,b)=minηk′lt,s.t.∑k=1Kλktxknt⩽xk′nt,n=1,…,N,∑k=1Kλktykmt⩾yk′mt,m=1,…,M,∑k=1Kλktbklt=ηk′lt,l=1,…,L,∑k=1Kλktbklt=bk′(−l)t,−l=L∖{l},λkt⩾0,k=1,…,K.

According to Eqs. (1)–(3), it maximizes the potential inputs contraction and the potential outputs expansion, and it can calculate the endogenous direction vector for individual input and output in Eq. (4). (4) gxMEA=(xk′1t−θk′1t,…,xk′Nt−θk′Nt)gyMEA=(ψk′1t−yk′1t,…,ψk′Mt−yk′Mt)gbMEA=(bk′1t−ηk′1t,…,bk′Lt−ηk′Lt).

Afterwards, the direction vector (gxMEA,gyMEA,gbMEA) is put into the conventional DEA-based directional distance function by Chambers et al. (1996) in Eq. (5). (5) D→(x,y,b)=maxβ,s.t.∑k=1Kλktxknt⩽xk′nt−βgxk′mtMEA,n=1,…,N,∑k=1Kλktykmt⩾yk′mt+βgyk′mtMEA,m=1,…,M,∑k=1Kλktbklt=bk′lt−βgbk′ltMEA,l=1,…,L,λkt⩾0,k=1,…,K.

With regard to the ideal reference point (θk′1t,…,θk′Nt,ψk′1t,…,ψk′Mt,ηk′1t,…,ηk′Lt) and the solution of Eq. (5), β, determining the benchmark selection for each individual variable, one can calculate the variable-specific efficiency score at time period t for input n, output m, and undesriable output l as (6) exk′nt=xk′nt−βk′t(xk′nt−θk′nt)xk′nt, (7) eyk′mt=yk′mtyk′mt+βk′t(φk′t−yk′mt), (8) ebk′lt=bk′lt−βk′t(bk′lt−ηk′lt)bk′lt.

Due to advantages of the non-radial and non-oriented slack-based measure (SBM) by Tone (2001) which considers both desirable and undesirable outputs to an underlying objective of maximizing virtual outcomes (Avkiran, 2011), the overall efficiency score is measured as: (9) ek′t=1−1N(βk′t(xk′nt−θk′nt)xk′nt)1+1M+L(βk′t(φk′mt−yk′mt)yk′mt+βk′t(bk′lt−ηk′lt)bk′lt).

On the basis of the aforementioned efficiency measure, we can further construct the overall MEA-Malmquist productivity index and its decomposition including efficiency change (EC) and technology change (TC) in Eqs. (10)–(12). (10) MPI=ek′t+1,t+1ek′t,t+1ek′t+1,tek′t,t12, (11) EC=ek′t+1,t+1ek′t,t, (12) TC=ek′t+1,tek′t+1,t+1ek′t,tek′t,t+112.

Analogously, the variable-specific productivity measures are opbtained by plugging the efficiency scores from Eqs. (6)–(8) in Eqs. (10)–(12).

In general, the metafrontier MEA-Malmquist productivity index (MMPI) implying potential production frontier can be decomposed into metafrontier efficiency change (MEC) and metafrontier technological change (MTC) as well as conventional MPI decomposition in Eq. (13). (13) MMPI=MEC×MTC.

Analogously, the groupfrontier MEA-Malmquist productivity index (GMPI) implying actual production frontier can be decomposed into group efficiency change (GEC) and group technological change (GTC) in Eq. (14). (14) GMPI=GEC×GTC.

The productivity growth gap (PGG) measures the gap between metafrontier and group frontier technology in Eq. (15), where the efficiency change gap (ECG) implies the gap between MEC and GEC to capture the pure catch-up gap change which is identical to the pure technological catch-up (PTCU) in Chen and Yang (2011), while the technological change gap (TCG) implies the change of frontier-shift between the group-frontier and the metafrontier that captures the innovation gap change which is similar to the frontier catch-up (FCU) component in Chen and Yang (2011). Therefore, ECG > (or <) 1 means a shrinkage (increase) of the technological gap in terms of pure catch-up, whereas TCG > (or <) 1 means the group frontier shift is faster (slower) than the metafrontier shift, indicating the reduction (increase) of innovation gap. (15) PGG=MMPIGMPI=MEC×MTCGEC×GTC=MECGEC×MTCGTC=ECG×TCG. Consequently, the MMPI is decomposed as Eq. (16) by rearranging Eqs. (13)–(15). (16) MMPI=GMPI×MMPIGMPI=GMPI×PGG=GEC×GTC×ECG×TCG.

Table 2 presents the overall TFP growth in the Chinese banking sector during the period 2005–2015. It is found that the mean of MMPI is 0.9606 implying the Chinese banking TFP has a decline by 4% on average each year approximately. Actually, during the earlier period 2005–2008, the MMPI shows a serious slope, while after 2008 the MMPI has a potential upward trend, although the values are still below 1. With regard to the decomposition, the mean of GEC has a positive growth by 1%, but that of GTC has a negative growth by 3.6%, while those of ECG and TCG are not significant. It is obvious that GTC, whose growth rates are below 1 during all the studied period, is the main source to reduce the overall banking TFP growth, particularly the GTC during 2006–2007 drops to the bottom at 88.37%. A potential explanation is that, with regard to the banking sector, there are usually two ways to affect technological change, namely physical technology and policy impact, respectively. As a matter of fact, the former, such as advanced network communication equipment, has been updated so timely in the Chinese banking sector that it should have a positive effect to technological progress. Consequently, the technological regress should be ascribed to policy impact. Indeed, in order to adapt to the economic slowdown before 2008, tight monetary policy and strict capital regulations slowed loan growth rate, and it thus shows a drop in technological change during 2005–2008 (Cai and Guo, 2009). Additionally, there is a remarked drop in GTC from 0.9755 to 0.9261 during 2008–2010 which probably sources from the impact of global financial crisis since 2008. Subsequently, series of appropriate national and local policies were issued to improve technological change of the Chinese banking sector after 2010. The GEC mostly maintains a positive growth, which is largely due to effective financial reform in the Chinese banking sector since 2003. However, the contributions of ECG and TCG are limited which presents that the gaps between metafrontier and groupfrontier on both efficiency change and technological change are small, and it is corresponding with results of some existing research (e.g. Asmild and Matthews, 2012; Wang et al., 2014; Zhu et al., 2015).

Furthermore, comparing the TFP growth in two types of banks, LSCBs and SMCBs, in Fig. 1, it is found that LSCBs outperformed SMCBs in MMPI slightly, which mainly attributes to systematic financial reform in LSCBs, involving stripped of non-performing loans, financial reorganization, and listing in market with assistance of government, since 2003. Indeed, both LSCBs and SMCBs show a rising trend as time passed. The fluctuations of GEC and GTC in both LSCBs and SMCBs are significant, where SMCBs perform better in GEC, and LSCBs outperform SMCBs in GTC. As a matter of fact, SMCBs are more sensitive to impacts of monetary policy and capital regulations than LSCBs, because, unlike LSCBs undertaking more social responsibility (Iannotta et al., 2013), SMCBs prefer to pursue profit maximization within market orientation, rather than policy orientation. Therefore, it causes lower GTC in SMCBs than LSCBs. However, due to more flexible market-oriented mechanism, SMCBs surpass LSCBs in GEC.

The changes of ECG and TCG in SMCBs are nearly flat, while change of ECG in LSCBs has a slight downtrend, and that of TCG in LSCBs has a slight uptrend. It implies that LSCBs has a slower growth rate in efficiency gains than overall sample banks, whereas it has a faster growth rate in technological progress than overall ones, which are corresponding with aforementioned GEC and GTC, respectively.

Figure 2 presents the changes of variable-specific productivity growth and its decomposition during the period 2005–2015. As a whole, IE, NIE, and II in all decomposition fluctuate insignificantly, because they are traditional banking businesses with a stable growth. However, both NPL and NIE have greater fluctuations.

With regards to MMPI, IE, NIE, and II operated similarly, and they outperform NPL and NII markedly before 2008. However, both NPL and NII show an upward trend, and even NPL has been ahead of all other factors since 2012. Considering why NPL and NII fell behind other factors before 2008, the former mainly attributes to partly imperfect financial environment and partly global financial crisis. Afterward, the completion of financial reform, particularly LSCBs’ reform, reduced NPL substantially. The later NII is a renewed banking business because traditional II has occupied most banking business in the Chinese banking sector before. After fully opening to the world since 2007, the Chinese banking sector begun to closely concern and rapidly develop NII businesses competing with domestic and foreign banks.

Investigating changes of GEC and GTC, NPL and NII surpass others in GEC, whereas both are below others in GTC. Indeed, the GECs of both NPL and NII have improved due to effective financial reform, and another potential factor driving the NII growth positively is, recently, the supply side reform which encourages the Chinese banking sector to improve financial services and increase financial innovations, so the NII is among the best in GEC during 2013–2015. However, several factors, like macroeconomic regulation and credit policy, may undermine GTC, and consequently it causes overall MMPI of NPL and NII at the bottom as well. With regards to ECG and TCG, it is obvious that NPL is a critical component to change the gaps of efficiency change and technological changes in the Chinese banking sector during the earlier studied period.

Figure 3 describes individual variable-specific productivity growth involving LSCBs and SMCBs, respectively. It is obvious that, as a whole, LSCBs outperform SMCBs in reducing NIE and increasing II and NII, while SMCBs do better in reducing IE and NPL. Regarding individual variable-specific productivity growth, SMCBs are better than LSCBs in GEC for each variable, but the gaps between them are small, while LSCBs surpass SMCBs in GTC significantly, particularly in NII (6%) and NPL (4%). The ECG and TCG between LSCBs and SMCBs are similar, with the exception of NPL (6.7%) in TCG that the groupfrontier of NPL in SMCBs moves towards metafrontier faster than that in LSCBs. A possible explanation is that, due to stronger risk of the management level in SMCBs which operate within market mechanism, SMCBs are more motivated to reduce NPLs, and pay more attention to improve asset quality.

The technological change implies what happens to the production frontier, but not whether the bank has contributed to a shift in the production frontier (Färe et al., 1994). In order to provide evidence as to which banks are the “innovators”, we aim to check whether that bank actually causes the production frontier to shift (e.g. Färe et al., 1994; Oh, 2010; Fujii et al., 2014). The following Eqs. (17)–(19) give three conditions to identify this issue, where Eq. (17) implies that the production frontier moves towards producing more desirable outputs and less undesirable outputs and using less inputs, Eq. (18) implies that it is not possible to produce outputs at the t+1 period using inputs at the t+1 period with technology at t period, namely the technology progress has been generated, and Eq. (19) implies that the innovator should be on the production frontier. (17) TEt,t+1>1,1,\]]]> (18) Dt(xt+1,yt+1,bt+1)>1,1,\]]]> (19) Dt+1(xt+1,yt+1,bt+1)=1.

Table 3 shows the number of innovator banks in both overall and variable-specific technological changes over the studied period. It is found that, generally, the proportion of LSCBs as innovators is higher during the earlier period (2007–2011), whereas it has a down trend since 2011, particularly in 2015. Particularly, compared with other variable-specific technological changes, there are less innovator banks in NPL in both LSCBs and SMCBs. It implies that there is still a lot of space to improve the reducing of NPL, because there are more opportunities to generate technological innovations in reducing NPL than that in the other factor. Besides, another important factor, NII, has relatively less innovators as well.

Investigating individual banks, it is found that the Bank of China is an outstanding innovator within the overall technological progress in LSCBs, whereas the China Construction Bank plays as the innovator mostly within variable-specific technological progress in LSCBs. Regarding the SMCBs, the China Merchant of China, the Industrial Bank, and the Bank of Beijing have the most opportunities to be the innovators in both overall and variable-specific technological progress.

Furthermore, it is meaningful to make a comparison between current MEA-MPI and conventional MPI. In Table 4, it is obvious that the trend of the TFP growth in both MEA-MPI and MPI is similar, where the overall TFP growth, the MMPI, has a negative effect, and technological regression is the main source slowing the TFP growth in MEA-MPI as well. However, regarding the exact value of the individual component of the TFP growth, it is found that, as the whole, the MPI would overestimate the TFP growth approximately, where the mean of MMPI in MEA-MPI is lower than that in MPI by −2.13% significantly at 1% level, and it is analogous to GTC by −2.41% significantly at 1% level as well. A possible explanation is that the MEA-MPI takes potential productivity into overall evaluation, whereas the MPI only considers past productivity (Asmild et al., 2016).