Efﬁciency in Vietnamese Banking: A Meta-Regression Analysis Approach

: This study explains the differences and variances in the efﬁciency scores of the Vietnamese banking sector retrieved from 27 studies published in refereed academic journals under the framework of meta-regression analysis. These scores are mainly based on frontier efﬁciency measurements, which essentially are Data Envelopment Analysis (DEA) and Stochastic Frontier Analysis (SFA) for Vietnamese banks over the period of 2007–2019. The meta-regression is estimated by using truncated regression to obtain bias-corrected scores. Our ﬁndings suggest that only the year of publication is positively correlated with efﬁciency, whilst the opposite is true for the data type, and sample size.


Introduction
The banking system fosters economic growth through allocating their savings to competitive firms, entrepreneurs, individuals and governments to enhance capital accumulation and profitability (Bumann et al. 2013;Pagano 1993;Rajan and Zingales 1998). Thus, efficiency measurement in the banking sector becomes one of the most intriguing issues of research in economics and finance because of the substantial impacts an efficient banking system has on the microeconomic as well as the macroeconomic development of the economy. Furthermore, the results of efficiency studies inform policy decisions and influence individual behaviors. From that branch of literature, banking has become an interesting case study for evaluating the policies of a country in attempting to regulate its national banking to increase efficiency (Aiello and Bonanno 2016;Iršová and Havránek 2010).
The ever-growing attractiveness of efficiency studies in the banking industry is also driven by various methodological concerns. Initially, the conventional view of efficient calculation using ratio analysis can be misleading as the cross-sectional variations in input and output combinations and their prices are not appropriately accounted for (Iršová and Havránek 2010). Started from Farrel (1957)'s seminal work on firms' efficiency using the so-called frontier analysis (FA) approach, researchers have developed a number of different methods to examine and evaluate the efficiency and performance of firms, or decision-making units (DMUs), in various industries, including the banking sector. Berger and Humphrey (1997) found 130 FA studies on financial institutions, mostly banks and bank branches. A more recent review of Liu et al. (2013) emphasizes that the number of studies on the banking industry accounts for the highest (about 10.31%) among 3134 empirical research papers that employed Data Envelopment Analysis (DEA) as the main methodology. returns-to-scale assumption, sample size, efficiency estimates and the number of authors of each study on its result, we found that the efficiency scores are associated with the year of publication, data type and sample size.
The remainder of this paper is structured as follows. Section 2 provides an overview of the methodologies used in measuring bank efficiency and introduces the meta-regression analysis as our main method in this study. Section 3 presents the description of the data used. Section 4 discusses empirical findings while Appendices A-C concludes.

DEA and SFA in Banking Efficiency
Since efficiency is evaluated in relation to the best practice (Aiello and Bonanno 2016), one of the main concerns is which method is more appropriate to estimate the frontier. A common criterion of classifying the frontier models distinguishes between parametric and non-parametric approaches. The former form, based on empirical knowledge, takes inefficiency as a distance from the efficient frontier and assumes how random error will be separated from inefficiency. Among the parametric techniques, the three most common are SFA (Clark and Siems 2002), distribution-free approach (DFA) (Berger 1995) and the thick frontier approach (TFA) (Humphrey and Pulley 1997). On the other hand, the nonparametric techniques, based on employing mathematical linear programming, neither assume a particular production function nor assign a distribution to the error term. The two most commonly used formulations are DEA (Thompson et al. 1997) and its alternative variants, and the free disposal hull (De Borger et al. 1998;Cooper et al. 2007)-Appendix A provides the summary of features of these methods. In the scope of this study, we focus on DEA and SFA only, which are the most two commonly used in the technical efficiency literature irrespective of the sector being analyzed as well as in the Vietnamese banking literature.
To begin with, in DEA, the relative efficiency of a group of related decision-making units (DMUs) (i.e., a set of same-sector banks) is measured by using a linear programming technique. DEA calculates efficiency scores for DMUs by combining several inputs and outs at the same time. Next, DEA compares each DMU with those that have the same inputs and outputs using linear programming. An efficiency frontier is then described by DEA as a linear set of the most efficient units. As a result, DMUs that are not on the frontier are ineffective. The DEA determines each DMU's relative efficiency based on its distance from the efficient frontier, so the DMUs will become less efficient if they are farther away. The DEA method was first used by Charnes et al. (1978) under an input-oriented efficiency approach and the assumption of constant returns to scale (CRS), based on the principle of efficiency suggested by Farrel (1957). A variable-returns to scale (VRS) model was later proposed by Banker et al. (1984). These are the two major DEA models, and they have been used extensively in many studies.
Moving on, SFA is another popular method for estimating the efficiency proposed by Aigner et al. (1997), and Battese and Corra (1977). This approach is based on the premise that a variety of factors contribute to DMUs not being on the efficient frontier and not fully controlled by these DMUs. When building the effective frontier, SFA allows the production functions to account for the presence of errors. Random noise and inefficiency are separated from these errors. Following a symmetric normal distribution, the former reflects the factors that influence the dependent variable but not observable. Meanwhile, the latter denotes inefficiency and s often characterized by a truncated normal distribution (Berger and Humphrey 1997). Figure 1 below illustrates the main difference between DEA and SFA. SFA can vary from the best practice DEA frontier due to measurement error and other random factors affecting output. If the error is negative, for example, SFA would be lower than DEA. On the one hand, we calculate inefficiency using DEA as the difference between the estimated output function f(x) and the x generated by x i inputs (measured by the angled line). On the other hand, using SFA, the estimated frontier lies below and the distance from it to x is shorter (measured by the curved line). DEA, therefore, would provide a higher estimate of inefficiency in this situation. Int. J. Financial Stud. 2021, 9, x FOR PEER REVIEW 4 of 17 output function f(x) and the x generated by xi inputs (measured by the angled line). On the other hand, using SFA, the estimated frontier lies below and the distance from it to x is shorter (measured by the curved line). DEA, therefore, would provide a higher estimate of inefficiency in this situation. In the banking sector, DEA and SFA are widely used to measure the efficiency of banks in terms of technical (or production) efficiency, cost efficiency and profit efficiency (Berger and Humphrey 1997;Iršová and Havránek 2010). While the estimation of the former only requires physical data on inputs and outputs, which makes it become the most popular measurement in efficiency analysis (Aiello and Bonanno 2016;Odeck and Bråthen 2012), the latter two need additional information on the prices of the inputs involved (Ngo and Tripe 2016) and thus are less popular.

Meta-Regression Analysis (MRA)
It is commonly difficult to compare the findings of scientific works, even if they are on the same topic since these studies vary in several aspects. One way to deal with it is to treat each study as an independent observation and the collection of those studies as a sample representing the 'true' population which is the topic being examined. Meta-regression analysis (MRA) is a valuable statistical tool that investigates the association between the main findings of various studies (as a dependent variable of the regression model) and the characteristics of those studies, e.g., sample size, methods or studying period (as independent variables of the regression model) (Glass 1976;Glass et al. 1981;Stanley and Jarrell 1989). In this sense, MRA synthesizes different studies into a single model and assesses the impact of particular aspects of the original studies on (the variability of) the results. More details about MRA as well as its pros and cons are available in Haidich (2010); Stanley et al. (2013) and Tatsioni and Ioannidis (2017), and among others.
Even though there has been a widespread use of MRA in economics, education and medical research, only a few deals with efficiency and, when they do, the majority focuses on the agriculture sector (Bravo-Ureta et al. 2007;Thiam et al. 2001), urban transport (Brons et al. 2005), seaports (Odeck and Bråthen 2012) or hospital (Nguyen and Coelli 2009). Even though banking efficiency is among the most popular applications of FA (Berger and Humphrey 1997;Liu et al. 2013), MRA in banking efficiency is still a newly In the banking sector, DEA and SFA are widely used to measure the efficiency of banks in terms of technical (or production) efficiency, cost efficiency and profit efficiency (Berger and Humphrey 1997;Iršová and Havránek 2010). While the estimation of the former only requires physical data on inputs and outputs, which makes it become the most popular measurement in efficiency analysis (Aiello and Bonanno 2016;Odeck and Bråthen 2012), the latter two need additional information on the prices of the inputs involved (Ngo and Tripe 2016) and thus are less popular.

Meta-Regression Analysis (MRA)
It is commonly difficult to compare the findings of scientific works, even if they are on the same topic since these studies vary in several aspects. One way to deal with it is to treat each study as an independent observation and the collection of those studies as a sample representing the 'true' population which is the topic being examined. Meta-regression analysis (MRA) is a valuable statistical tool that investigates the association between the main findings of various studies (as a dependent variable of the regression model) and the characteristics of those studies, e.g., sample size, methods or studying period (as independent variables of the regression model) (Glass 1976;Glass et al. 1981;Stanley and Jarrell 1989). In this sense, MRA synthesizes different studies into a single model and assesses the impact of particular aspects of the original studies on (the variability of) the results. More details about MRA as well as its pros and cons are available in Haidich (2010); Stanley et al. (2013) and Tatsioni and Ioannidis (2017), and among others.
Even though there has been a widespread use of MRA in economics, education and medical research, only a few deals with efficiency and, when they do, the majority focuses on the agriculture sector (Bravo-Ureta et al. 2007;Thiam et al. 2001), urban transport (Brons et al. 2005), seaports (Odeck and Bråthen 2012) or hospital (Nguyen and Coelli 2009). Even though banking efficiency is among the most popular applications of FA (Berger and Humphrey 1997;Liu et al. 2013), MRA in banking efficiency is still a newly emerging issue. Iršová and Havránek (2010) are among the pioneers to use MRA to review the US banking efficiency literature with 53 observations from 32 studies in a 20-year period. Their results suggest that US banks are less efficient when generating profits than controlling costs. In another study focused on five transitional economies in Central and Eastern Europe, Iršová and Havránek (2011) found that the variety of banks' efficiency can be explained by the differences in the choice of variables (e.g., the intermediation approach versus other approaches, see Sealey and Lindley (1997), the choice of estimation method (e.g., parametric versus non-parametric) and so on. A more recent MRA study covered 1661 efficiency scores retrieved from 120 papers published over the 2000-2014 period of Aiello and Bonanno (2016) reconfirms the above findings. Interestingly, although Aiello and Bonanno (2016) have pointed out that FA studies on bank efficiency of Asian countries accounted for 37% of their dataset, most of those studies are on Indian, Japan or Australian banks. To the best of our knowledge, however, there is no MRA study on banking efficiency in Vietnam, and our study, therefore, can contribute to the above picture.
Following Odeck and Bråthen (2012) and Aiello and Bonanno (2016), the empirical MRA model investigating the relationship between study characteristics of the Vietnamese banking efficiency scores can be presented as follows: where EF i is the (average) efficiency scores derived from study i and X ij represents the matrix of covariates consisting of the study i's characteristics, and ε i are the measurement errors. It is noted that the number of observations is equal to the number of data points extracted from the studies, i.e., one study may provide more than one observation. By definition, the efficiency scores EFi are censored between zero and unity. Consequently, the traditional Ordinary Least Squares regression could produce biases when estimating Equation (1) and thus, Tobit regression is more appropriate to deal with such censored dependent variable (Iršová and Havránek 2010;Assaf and Josiassen 2015;Aiello and Bonanno 2016). However, it is suggested that it is practically impossible to have EF i below or equal to zero so that EF i is actually truncated rather than being censored-the truncated regression therefore performs even better than the Tobit regression (Simar and Wilson 2007). Our truncated MRA model is consequently represented as follows.
where YEAR represents the year of publication of observation i; DATA is a dummy variable to represent the data type used in observation i that takes a value of 1 for panel data and 0 for cross-sectional data; METHOD is a dummy variable to account for the frontier approach of observation i that takes a value of 1 for DEA method and 0 for SFA method; RTS is a dummy variable that takes a value of 1 for the assumption of constant returns to scale in observation i and 0 otherwise; PUB is a dummy variable to represent the type of publications of observation i that takes a value of 1 for a published journal and 0 otherwise; SIZE represents the number of banks involved in observation i; PRODUCT and COST are the dummy variables represent the type of efficiency that are estimated in observation i, the reference measurement in this case is profit efficiency; AUTHOR is a dummy variable denoting the number of authors in observation i that takes a value of 1 for a single author observation and 0 otherwise. Due to the small number of our sample, we also perform a conventional bootstrapped Tobit/truncated regression in estimating Equation (1) to improve the statistics of our results. The descriptions of our data as well as our MRA results are then presented in the next section.

The Data: FA Studies on Vietnamese Banking Efficiency
To construct the data sample, we searched for all studies on bank efficiency in Vietnam that use frontier analysis, both DEA and SFA. The collection of relevant literature was primarily identified through searches in several databases (Google Scholar, Science Direct and Web of Science) for a combination of the three keywords of "bank efficiency", "frontier analysis" and "Vietnam". We also used the bibliographic snowballing technique to manually check the reference lists of the retrieved articles for additional relevant studies.
Given that some studies provided more than one data point, our final sample consists of 27 studies which provide us 39 observations for the MRA estimation. Figure 2 shows that FA studies on Vietnamese banks only emerge from 2007. In the first two years of 2007 and 2008, there were only three articles published, providing a total of five data points or observations in our sample. The number of studies has been increased since 2010, particularly after 2015, resulting in 16 articles and 21 observations for the 2015-2019 period. It is noted that the number of data points belong to the published academic journals representing for 69.23% of the sample whereas working papers accounted for 30.77%. Additionally, the number of studies conducted by a group of authors (58.97%) are higher than those performed by a single author (41.03%). Further information regarding our sample is also presented in Appendix B.

The Data: FA Studies on Vietnamese Banking Efficiency
To construct the data sample, we searched for all studies on bank efficiency in Vietnam that use frontier analysis, both DEA and SFA. The collection of relevant literature was primarily identified through searches in several databases (Google Scholar, Science Direct and Web of Science) for a combination of the three keywords of "bank efficiency", "frontier analysis" and "Vietnam". We also used the bibliographic snowballing technique to manually check the reference lists of the retrieved articles for additional relevant studies. Given that some studies provided more than one data point, our final sample consists of 27 studies which provide us 39 observations for the MRA estimation. Figure 2 shows that FA studies on Vietnamese banks only emerge from 2007. In the first two years of 2007 and 2008, there were only three articles published, providing a total of five data points or observations in our sample. The number of studies has been increased since 2010, particularly after 2015, resulting in 16 articles and 21 observations for the 2015-2019 period. It is noted that the number of data points belong to the published academic journals representing for 69.23% of the sample whereas working papers accounted for 30.77%. Additionally, the number of studies conducted by a group of authors (58.97%) are higher than those performed by a single author (41.03%). Further information regarding our sample is also presented in Appendix B. With respect to the type of frontier analysis, i.e., METHOD, the majority of studies (69.2%) were DEA-based while another 30.8% were SFA studies. This is understandable that DEA is often used in the context of the Vietnamese banking system due to a relatively small number of banks and the availability of their data (for example, see the dataset provided by Ngo and Le 2017)-before July 2009, Vietnamese banks were not required to publish their data (Vietnamese Government 2009). Regarding the type of data used, i.e., DATA, most studies (89.7%) used panel data whereas 10.3% used cross-section data (see Table 1). Overall, the average efficiency score of Vietnamese banks across the sampled studies/observations is 0.770, suggesting a moderate level of performance among those banks. With respect to the type of frontier analysis, i.e., METHOD, the majority of studies (69.2%) were DEA-based while another 30.8% were SFA studies. This is understandable that DEA is often used in the context of the Vietnamese banking system due to a relatively small number of banks and the availability of their data (for example, see the dataset provided by Ngo and Le 2017)-before July 2009, Vietnamese banks were not required to publish their data (Vietnamese Government 2009). Regarding the type of data used, i.e., DATA, most studies (89.7%) used panel data whereas 10.3% used cross-section data (see Table 1). Overall, the average efficiency score of Vietnamese banks across the sampled studies/observations is 0.770, suggesting a moderate level of performance among those banks. Notes: EF represents the average efficiency scores derived from observation i; YEAR indicates the year of publication of observation i; DATA is a dummy variable representing the data type used in observation i that takes a value of 1 for panel data and 0 for cross-sectional data; METHOD is a dummy variable accounting for the frontier approach of observation i that takes a value of 1 for DEA method and 0 for SFA method; RTS is a dummy variable that takes a value of 1 for the assumption of constant returns to scale in observation i and 0 otherwise; PUB is a dummy variable representing the type of publications of observation i that takes a value of 1 for a published journal and 0 otherwise; SIZE represents the number of banks involved in observation i; PRODUCT and COST are the dummy variables representing the type of efficiency that are estimated in observation i, the reference measurement in this case is profit efficiency (PROFIT); AUTHOR is a dummy variable denoting the number of authors in observation i that takes a value of 1 for a single author observation and 0 otherwise. Table 2 provides the MRA results for the determinants of banking efficiency in Vietnam. The consistency among the four models (i.e., Tobit regression, bootstrapped Tobit regression, truncated regression and bootstrapped truncated regression) suggests that our results are robust to the estimation method and that the findings are reliable (the final models are presented in Appendix C). The key findings for the causes of the differences in efficiency scores of previous studies on Vietnamese banks are presented as follows. Note:

The MRA Results: Efficiency of Vietnamese Banks
YEAR indicates the year of publication of observation i; DATA is a dummy variable representing the data type used in observation i that takes a value of 1 for panel data and 0 for cross-sectional data; METHOD is a dummy variable accounting for the frontier approach of observation i that takes a value of 1 for DEA method and 0 for SFA method; RTS is a dummy variable that takes a value of 1 for the assumption of constant returns to scale in observation i and 0 otherwise; PUB is a dummy variable representing the type of publications of observation i that takes a value of 1 for a published journal and 0 otherwise; SIZE represents the number of banks involved in observation i; PRODUCT and COST are the dummy variables representing the type of efficiency that are estimated in observation i, the reference measurement in this case is profit efficiency (PROFIT); AUTHOR is a dummy variable denoting the number of authors in observation i that takes a value of 1 for a single author observation and 0 otherwise. The dependent variable is EF which represents the average efficiency scores derived from observation i. The number of bootstraps is 2000. Coef. Stands for coefficient and Std. Err. Stands for standard errors. *, **, *** represent the significance levels at 10, 5 and 1 percent, respectively.
Firstly, the positive and significant coefficients of YEAR across the four models suggest that newer studies tend to find higher efficiency scores in Vietnamese banks. This finding is in line with another MRA study on banking efficiency of Aiello and Bonanno (2016) where it is suggested that market regulations in the financial sector over the past years have helped the banks to improve their efficiency, so do the results of efficiency studies on those banks. Note that this trend in efficiency improvement of Vietnamese banks has also been found in Nguyen et al. (2018Nguyen et al. ( , 2019, among others. Secondly, DATA has a negative and significant impact on the efficiency scores, implying that FA studies utilizing panel data found the Vietnamese banks with lower performance than for studies using cross-sectional data. Similar results have been found in Iršová and Havránek (2010) and Aiello and Bonanno (2016), with a plausible explanation that cross-sectional efficiency analysis does not account for technological progress while FA studies using panel data do-technological improvement can thus contribute to the efficiency improvement of the banks (Krishnasamy et al. 2004;Ngo and Tripe 2017).
Thirdly, it is suggested that the number of banks being examined in each study/observation, i.e., SIZE (slightly) impacts the efficiency estimate: the more banks involved the lower (average) efficiency score. As discussed in Diewert (1993) and Brons et al. (2005), when the number of banks increases, the probability that those additional banks were inefficient are higher than them being efficient. Consequently, the average efficiency which is defined as the sum of all individual bank's efficiency scores divided by the number of banks will likely to decrease (Aiello and Bonanno 2016;Zhang and Bartels 1998).
Due to the limitation of our sample, we could not find a clear (and significant) relationship between other characteristics such as METHOD, RTS, PUB, PRODUCT, COST, PROFIT and AUTHOR and the average efficiency scores of Vietnamese banks. It is noted that this impact is inconclusive, for example, Brons et al. (2005) found that SFA studies tend to find lower efficiency scores than DEA studies whilst Aiello and Bonanno (2016) found the opposite. We suggest that further MRA on the Vietnamese banking sector when more FA studies are available in the coming years may contribute to that debate.

Conclusions
Our study investigated what causes the differences in the efficiency scores of Vietnamese banks retrieved from 27 bank-efficiency studies that were conducted from 2007 to 2019 using meta-regression analysis. Particularly, we examined the impacts of the year of publication, type of data used, type of method used, type of returns-to-scale assumption involved, type of publication, sample size, type of efficiency estimate involved and the number of authors of each study on its result (i.e., efficiency scores). While this study could not find a significant relationship between efficiency scores and the other characteristics (e.g., type of method or of publication involved) due to data limitation, it still provides insightful information on the efficiency of Vietnamese banks as well as its relevant studies. For instance, our findings show that efficiency scores of the Vietnamese banking sector during the 2007-2019 period is moderately high at an average of 0.770. More important, the efficiency scores tend to be higher for newer publications which may reflect the increasing trend of the performance of Vietnamese banks over time; however, it mainly comes from technological improvements rather than from the operational performance of the banks. As such, the inclusion of more (inefficient) banks in the sample tends to decrease the average efficiency score estimation. Those findings are robust to different estimation methods (i.e., Tobit and truncated regressions) and under the bootstrapping approach.
This study could be extended to a larger sample, especially for Vietnam and other emerging markets in the Asia-Pacific regions, so that more empirical evidence regarding the other characteristics (e.g., type of efficiency estimate or number of authors) can be examined. It is also interesting to apply advanced techniques such as the DEA bootstrap (Simar and Wilson 2007) or the hierarchical regression (Woltman et al. 2012) into MRA to acquire more robust and efficient results.

Data Availability Statement:
The data presented in this study are available on request from the corresponding author. The data are not publicly available due to privacy.

Conflicts of Interest:
The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses or interpretation of data; in the writing of the manuscript or in the decision to publish the results.