Impact of International Trade on Central Bank Efficiency: An Application of DEA and Tobit Regression Analysis

The purpose of this study is to introduces a novel methodology to measure the central bank efficiency. The data envelopment analysis (DEA) applies in the combination of three input and two output variables characterizing the economic balance in international trade. Super-efficiency DEA model is applied for ranking & comparing the efficiency of different central banks. In contrast, the Malmquist productivity index (MPI) is used to measure the productivity change over the period of time. Further, the study is extended to quantify the impact of international trade dimension on the efficiency of the central bank by using Tobit regression analysis. Finally, based on our data analysis, we reported that the efficiency changes over the period of time and the total productivity changes significantly due to the technology shift as compared to efficiency change. Additionally, it is also observed that the central bank efficiency is impacted dramatically by the export level of the country as compared to import level, average exchange rate and GDP. It implies that the export level of the country significantly influences the performances of the central bank.


Introduction
International trade is the exchange of good, service and capital across the international territories (borders). The sound international trade indicates a healthy economy and development of the country. The central bank or central financial institution of a country can be considered as the engine of the economy, like an engine it has the power to control and regulate the economic and development functions, any malfunction in the engine creates a risk for the financial system of the country. The central bank plays a vital role in the economic system of the country by doing different functions, like currency regulation, securing the stability of exchange rate, supervisor of commercial and other financial institutions, controller of credit/money supply and balance on international trade. It is an only legal and autonomous financial institution that is allowed to print money as a legal tender [1], by printing money the central bank has opportunities to control the money supply, the total amount of funds available in the economy. The central bank of the country functions is dynamic because these are profoundly * Correspondence to: Young Hyo Ahn (Email: yhahn@inu.ac.kr), Qaiser Farooq Dar (qaiserdea@gmail.com). Division of International Trade, Institute of Digital Economy, Incheon National University, (Songdo-dong)119 Academy-ro,Yeonsu-gu, Incheon-22012, South Korea 225

Literature review
The concept of efficiency evaluation was first proposed by [17]. He considered ratio in the form of input and outputs to measure the efficiency, which in turn have mathematical limitations for handling in the context of multiple inputs and multiple outputs. Almost after twenty years, [18] propose a CCR model based on Farrell's idea to assess the relative efficiency of DMU in case of multi-input and multi-output, titled it as DEA. The basic idea behind the DEA model is to formulate the optimization problem to identify the best-practised DMU, which makes an efficient frontier. Furthermore, it finds the efficiency of non-frontier DMUs and identifies benchmarks against which such inefficient DMUs can be compared. As this model was based on constant returns to scale (CRS) assumption, it was further extended by [19] in case of variable returns to scale (VRS) assumption by introducing convexity constraint in CCR-DEA model. Since the advent of DEA in 1978, there is an impressive growth in both theoretical and applied aspects. In theoretical aspects, various models were proposed to estimate different efficiency measures wherein the latter case. The proposed models were used as a performance assessment tools in a variety of organizations like banking, education, health-care, agriculture, production companies, airports and many more profit as well as non-profit organizations as reported in a survey and analysis of the first 40 years of scholarly literature in DEA [20].
DEA is a linear programming based technique for measuring the relative efficiency of DMUs, and therefore hypothesis testing is considered to be a difficult task. Sensitivity analysis will be used to verify and to estimate the robustness of efficiency scores obtained by DEA [21]. The CCR and BBC model of DEA are categorized as radial measure efficiency because both are dealing directly with the inputs and outputs of a DUM [22]. These models can solve by using either input orientation at a fixed level of output or output orientation at a fixed level of input or mixed-orientation varying both input and outputs at an optimal level [23]. A non-oriented and non-radial measure of efficiency was proposed by [24], which is not dealing with the inputs and outputs of DMU directly but dealing with input excesses and output shortfall called slack based measure (SBM) of efficiency. SBM also introduces the concept of weak and strong theory of efficiency.
DEA classifies the DMUs into two diverse, efficient and inefficient groups. Unlike the inefficient DMUs, the efficient ones cannot be ranked based on their efficiencies score because of having the same efficiency score of unity. It is not, however, reasonable to claim that efficient DMUs have the same performance in actual practice [25]. To overcome this drawback from the DEA methodology concept super-efficiency was introduced by [25,26,27], which is the most powerful approach for rank the DMUs. The same methodology was applied in case of the SBM approach given by [28].
Productivity and efficiency of an organization are interconnected. However, the effectiveness of DMU is static as it does not consider the time for production, whereas productivity is based on time. DEA window analysis is a suitable approach to estimate the efficiency change over time given by [29,30]. However, it neglects the effect of technological change over time and assigns any technological change as of technical efficiency change. Several methods could be used to evaluate productivity change, which includes Fisher index, Tornqvist index and the Malmquist Index. Among the three, Malmquist Total Factor Productivity (TFP) index is used most often to evaluate productive change given by [31] and use of MPI is based on distance function in DEA proposed by [32]. Evaluation and analyse of the efficiency and productivity of 3 public, 6 private and 6 foreign deposit banks operating in the Turkish banking sector given in [33], with the help of DEA and MPI. Performance of non-banking finance companies (NBFC) in the Indian context using two stage DEA was given by [34], and Multi-period performance evaluation of Indian commercial banks by using DEA and MPI given by [35]. Bias-corrected network DEA is used to measure the efficiency of total National Innovation Systems (NIS) and the efficiency of the other sub-processes within the system given by [36]. Whereas, in case of input and output variables are not known with absolute precision in DEA we can use Fuzzy DEA as shown in [37] with different approaches. 226 CENTRAL BANK EFFICIENCY: AN APPLICATION OF DEA AND TOBIT REGRESSION ANALYSIS Tobit model also called a censored regression model to estimate linear relationships between variables and first time introduced in DEA by [38]. The Tobit regression analysis was also used to identify the main influencing factors in banking efficiency [39]. It was shown that the capital adequacy ratio has a statistically significant adverse impact on the performance of banks [40] and which may reflect a riskreturn trade-off in the banking sector. The DEA can be used in every aspect of research where the input and outputs are to be considered for the productivity and efficiency of DMU. There is a continuous and exponential growth of publication related to the application of DEA [20] from the last four decades .
In this study, we are using a non-parametric deterministic frontier DEA approach for estimating the efficiency of different central banks in case of 17 Asian countries. The study is further extended to quantify the cause and impact on the relationship between the efficiency of the central bank and international trade dimension by using the Tobit regression model.The existing literature of efficiency technique DEA are used with different application. The impact of international trade on central bank efficiency was derive from existing literature of DEA and Tobit regression. The paper structure includes, section first is on the overview of international trade, central bank and approaches of efficiency evaluation techniques. The second section is based on the brief literature review about DEA, MPI, Tobit regression analysis and its application in banking. DEA, MPI and Tobit regression analysis methodology are discussed in the third section. The fourth section is based on empirical analysis, including data description and selection of input & output variables. Significant development of studies where discussed with empirical data, results, and conclusion is in the final section.

Material and methods
This study based on the evaluation and estimation of central bank efficiency by using a non-parametric approach called DEA. It is a linear programming based technique for estimating the relative efficiency of organizational units together with multiple inputs and outputs. The mathematical formulations of DEA are based on two orientations (input and output). In this study, we are using input orientation in case of both constant and variable returns to scale. The non-oriented and non-radial measure of efficiency called SBM approach and super efficiency SBM approach is used for the ranking of DMUs. Apart from this, Tobit analysis is used to identify the main influencing factors of international trade on central bank efficiency.

Radial measures of DEA in case of CRS and VRS
The mathematical formulation of DEA with the assumption of CRS was given by [18]. Let x ij and y rj denote the i th ; (i = 1, 2, 3, ..., m) input and r th ; (r = 1, 2, 3, ..., s) output of j th DMU (j = 1, 2, 3, ..., n) respectively. The efficiency of k th DMU is denoted by and standard envelopment form of inputorientation CCR-DEA model as; ..n. To get valid returns on the scale, Banker, Charnes and Cooper in 1984 have extended DEA to the case of VRS by adding one more constraint known as convexity constraint in CRR envelopment model given by [19]. The purpose of this envelopment form is to point out the most efficient scale size for each DMU and at the same time to identify its technical efficiency. Following is the envelopment form of BCC-DEA model with input-orientation.
Where x ik and y rk are the inputs and outputs of k th DMU (which is under evaluation). The input excesses and output shortfalls are denoted by s − i ; (i = 1, 2, 3, ..., m) and s + r ; (r = 1, 2, 3, ..., s) are also known as input and output slacks. The DMU which is under evaluation is said to be efficient if and only if θ * k = 1 and all the slacks must be zero i.e., s + r = 0 and s − i = 0 . If θ * k = 1, but one the slack is non zero. Then DMU under evaluation is said to be week efficient otherwise if θ * k < 1, then the DMU which is under evaluation is said to be inefficient.
CCR efficiency is the combination of purely technical and scale efficiencies and whereas BCC efficiency is only pure technical efficiency excluding scale efficiency. Thus, the radial difference between the CCR frontier and BCC frontier is called as scale inefficiency. Thus, we have an equation for scale efficiency of k th DMU as; Thus any DMU with both CCR and BCC efficiency can be scale efficient and are called as most productive scale size (MPSS).

SBM Efficiency Model in DEA
A non-oriented and non-radial measure of efficiency, which deals with input excesses and output shortfall of DMU is called the slack-based measure of efficiency was given by [24]. The mathematical formulation is defined as: Let x ij and y rj denote the i th ; (i = 1, 2, 3, ..., m) input and r th ; (r = 1, 2, 3, ..., s) output of j th DMU (j = 1, 2, 3, ..., n) respectively. It is assumed that the data set is known and strictly positive. The production possibility set P of DMU k is defined as: P is closed and convex set with boundary points as the efficient production frontier. The relative reduction rate of i th input and j th output for the k th DMU is defined as: Where s − i and s + r is the input and output slacks of DMU k respectively.
Let ρ k be the inefficiency rate of DMU k assessing the m-inputs and s-outputs is defined as: The interpretation of non-oriented and non-radial DEA technique SBM is minimizing the above inefficiency rate directly on the base of slacks, subject to production possibility set P and standard mathematical form is given below: Where x ik and y rk are the inputs and outputs of the DMU k under evaluation. s − i ≥ 0; (i = 1 , 2 , 3 , . . . m) and s + r ≥ 0; (r = 1 , 2 , 3 , . . . s) are the input excess and output shortfalls, also referred to as slacks. The converted linear form of SMB-model by Charnes-Cooper transformation is as given below; Let an optimal solution of the mathematical model (3.2.2) be (τ * k , t * , Λ * , S − * , S + * ). Then we have an optimal solution of SBM-model is defined as: On the base of an optimal solution given in equation (III), we decide whether DMU k which is under evaluation is efficient or inefficient. Where,in the case of SBM modal of VRS, can be expressed by adding the convexity constraint into the linear programming problem (3.2.2).

SBM Super-Efficiency DEA Model
Generally speaking, multiple DMUs can have the "full efficient status" with a DEA score of one in conventional DEA models. Thus, it is difficult to rank the DMUs as many of them are of having the same efficiency of value one. In order to discriminate these DMUs, another model will be used through the super efficiency of a specific DMU [24,27,41] which discriminates between these efficient DMUs and ranks them by assigning the efficiency score greater than1. Super efficiency refers to an amended DEA score in which the DMU can obtain a score of technical efficiency more excellent than one because each DMU is not permitted to use itself as a peer. In this section, we are discussing the super efficiency which is an extension of DEA efficiency in which efficient DMUs can obtain the efficiency scores greater than unity and each DMU is not allowed to use itself as a peer. Let us assume k th DMU is SBM-efficient, i.e. ρ * k = 1 are using x i ; (i = 1, 2, 3, ..., m) and y r ; (r = 1, 2, 3, ..., s) respectively. Then the mathematical formulation of slack-based super-efficiency of (x k , y k ) as the optimal objective function value ρ * k of the following model: The super-efficiency in powerful approach for rank DMUs efficiency obtained SMB DEA model. We can rank efficient DMUs by solving the model (3.3.1) for each efficient DMU. For efficient DMUs have super-efficiency score greater than or equal to unity, while inefficient DMUs have super-efficiency score less than unity.

Malmquist Productivity Index (MPI) in DEA
In this section, we are discussing the MPI to measure the productivity change over the period of time. MPI is based on the two distance function with respect time periods, and the main advantage is to decompose the total productivity change into two mutually exclusive and exhaustive components by [32]. These two components can be identified as catching up and innovation, respectively. MPI is defined as the ratio of two distance function of k th DMU (k = 1, 2, 3, ..., n) with reference to the period t and (t+1) is as follows: This index represents two types of efficiencies. One which is outside the brackets measures the change in relative efficiency (i.e., the radial difference between the observed production and expected production) between years' t and (t + 1), while the other within the brackets represents a shift in technology between the years' t and (t + 1). That is; The value of M k lies greater than unity, i.e. M k ≥ 1 implies that the total factor productivity has increased or decreased over the time period t and (t + 1). M k = 1 main that there is no change in productivity where M k ≥ 1 implies the percentage change of productivity over the time period t and t + 1. Improvement in productivity yield MI greater than unity, and any impairment in performance yields MI less than unity. At times it may happen that efficiency change and technical change are moving in opposite directions.
To sum up, this [32] define productivity growth as a product of efficiency change and technical change. They interpreted the components of productivity growth as improvements in efficiency change are considered to be catching up, while improvements in technical change are considered to be evidence of innovation. This decomposition thus provides a way for testing the source of change in productivity. To know the source, it is necessary to measure the MPI, and various approaches are known to calculate this index. Herein we apply linear programming approach as given by [22]. Suppose we have x t ij and y t rj that are inputs and outputs of j th DMU (j = 1, 2, 3, ..., n) at point time t. The reference technology for the time period t can be obtained from the data set as follows: Thus to calculate Malmquist productivity of a DMU between time period t and t + 1, we need to solve four linear programming problems with four different maximize objective functions D t 0 (x t , y t ), D t+1 0 (x t , y t ), D t 0 (x t+1 , y t+ ) and D t+1 0 (x t+1 , y t+ ) under the same set of constraints as shown in the model (3.4.3) on same time period t and (t + 1).

Tobit Regression Analysis in DEA
In this section, we are discussing Tobit regression analysis, to estimate the linear relationship between efficiency (dependent variable) and set of independent variables. Tobit analysis is also called a censored regression and the efficiency score estimated by (0 and 1). The standard form of the Tobit model for i th DMU is given as follows: Where x i and β are the vectors of explanatory variables and unknown parameters and y * i is an efficiency score of the DEA model treated as the latent independent variable. The error term ε j fellows' normal distribution with mean 0 and variance σ 2 i.e. ε j ∼ N(0, σ 2 )∀j = 1, 2, 3, ..., n.

Data Collection and Statistical Analysis
In this section, DEA methodology is applied to the 17 central banks of top Asian exporter countries for the three financial years 2016-18. The data structure has been collected from Bloomberg, central banks annual reports and other national and international economic magazines where data was compiled in Excel, DEA Frontier, PIM-DEA and Stata software.
In this study, we have used three inputs operating expenses (million $), total investment (million $), and total deposits (million $) where total assets (million $) and net income (million $) are considered as two outputs for the efficiency evaluation by using DEA methodology. Where for the Tobit analysis super-efficiency score is used as dependent variable (which is censored) and total exports (million $), total imports (million $), GDP and exchange rate (AER) are used as independent variables. Summary statistics of all inputs and output variables are given in table 1. China is the largest exporter, not only in the Asian economy but also in the world economy too. China exported $2.41T and imported $1.54T, resulting in a positive trade balance of $873B, which is about 36% of total Asian export. The export of Japan is about $738.18B, and South Korea is about $605.17B pursue on 2nd and 3rd place in the ranking, which is about 10.7% and 8.7% of total Asia export. The total export of Kazakhstan ($ 60.95B), Bangladesh ($43.53B), Pakistan ($23.63B), and Sri Lanka ($11.97B) pursue 14th, 15th, 16th and 17th position in the Asian export economy see [42]. In this study, we have taken the central bank (as DMUs) data of the top 17 Asian exporter countries, as listed in the below table 2.  Sri Lanka Central Bank of Sri Lanka DMU17

Results and Discussion
This section analyses the central bank efficiency and results obtained by using the proposed DEA methodology. We use both radial and non-radial measures of efficiency to obtain pure technical, scale and super efficiency of top 17 Asia central banks listed in table 2 for 2016-18. The slack-based super-     1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 9.  Table 6 presents the results of the MPI of 17 top Asian central banks. As mentioned in the model (3.4.2) MPI can be decomposed total productivity change into two components: technical change and efficiency change. We have estimated the total productivity change separately and analyzed the results based on these separate components. Figure 1 shows the average shift of productivity, efficiency and technology over the period of time 2016-18. Overall, the productivity remains insignificant during 2016-18 where efficiency has decline trade, and the technological shift has significant positive trade over the time period.  In contrast, the second and third part showed the efficiency change and technological shift throughout studied time. It is revealed from table 6 that total productivity change is in influenced much heavily by technical change than the efficiency change.
It should be noted that index value higher than one means increase of total productivity, less than one means decrease and equal to one means no change in total productivity between the respective periods. The results show that DMU 2,4,5,6,8,10,13,14,and 15 have increasing trend during the year 2016-17, whereas DMU 4,5,6,8,9,10,11,13,14,16, and 17 has increasing trend over the period 2017-18. The remaining DMUs has decreased trend over the period 2016-17 and 2017-18. The productivity change, efficiency change and technical shift are also in figure 2, 3, and 4 in the Appendix-A. Table 7 describes the results of Tobit regression analysis since Tobit analysis is designed to estimate linear relationships between variables when there is either left-or right-censoring in the dependent variable. As we noted, the efficiency score lies between 0 and 1, which implies the efficiency score is censored variable from the left as well from right between 0 and 1. Thus in this part, we are showing a linear relationship between the efficiency score of the central bank with total export level, total import level, average exchange rate and gross domestic product of the country.
It figures out that by taking all the four predictors (total export level, total import level, average exchange rate and gross domestic product of the country) together with the super-efficiency score as dependent variable (censor variable) in the model     Table 8 describes the results of correlation results between the central bank efficiency with total export level, total import level, GDP and average exchange rate. It was observed from the results of Pearson correlation that the total export level has found a statistically positive significant correlation with the central efficiency. Whereas, the total import, average exchange rate and GDP have a negative correlation with the central bank efficiency.

Summary, Conclusion and Recommendations
Performance evaluation of central banks is beneficial for the international trade, development of banking system and other financial institutions of the nation. This study endeavours to evaluate the extent of technical, pure technical and scale efficiencies of top 17 Asian exporter country central banks using time series data during 2016-18 by using DEA and its extensions. DEA proved to be a fantastic technique of performance assessment inefficiency. DEA provides a measure of relative efficiency where the performance of DMUs is evaluated concerning others and helps to identify the strength and weakness of the DMUs. It also provides the possible direction of improvement and benchmarks for comparison purposes. Besides this, an attempt has been made to explain the ranking as per the performance central banks using the concept of super-efficiency. The productivity change, efficiency change and technological shift during 2016-18 were also attempted in this study. Further, the study was extended to find out the impact of total exports, total imports, GDP and exchange rate on the efficiency of the country central banks.
From the practical point of view, the conclusions can be drawn in three different ways as per the objectives of the study. First one is about the efficiency, it was concluded that the efficiencies of DMUs changes over a period of time except for DMU4 (Hong Kong Monetary Authority) remain superefficient and benchmark of all the DMUs throughout the study time. The central bank of China, Japan, South Korea, Hong Kong, Singapore, Taiwan, Malaysia and Turkey are operating the inputs and outputs efficiently as compared to the central bank of India, Vietnam, Thailand, Indonesia, Philippines, Kazakhstan, Bangladesh, Pakistan and Sri Lanka. However, the Hong Kong Monetary Authority is superior (super-efficient) in terms of efficiency, whereas Pakistan and Sri Lanka are inferior (lower rank) in terms of efficiency. It is observed the sources of overall technical inefficiency have been noticed due to reduced input and output utilization (i.e., managerial inefficiency) and failure to operate at most productive scale size (i.e., scale inefficiency). However, the overall inefficiency of most inefficient central banks is mainly attributed by pure technical inefficiency rather than scale inefficiency. The second one is about the change of productivity, efficiency and technology shift, and overall, the productivity remains insignificant during 2016-18 where efficiency has decline trade. The technical shift has significant positive trade over a period of time. Thus it was concluded that the overall productivity changes are because of a significant shift of technology rather than the efficiency change. It was also observed that there a significant positive correlation between the export level and the central bank efficiency. Thus 238 CENTRAL BANK EFFICIENCY: AN APPLICATION OF DEA AND TOBIT REGRESSION ANALYSIS finally, it was concluded that there is a positive impact of the export level of country and efficiency of the central bank as compared to import level, exchange rate and GDP.
The inefficient central bank can improve the efficiency by either reducing the level of the input without altering the output level or extended the output level without altering the input level and selecting appropriate scale size of the central bank. The efficiency of the central bank may improve by exploring the export level rather than import level in international trade. The future work could extend our research in various directions not considered in this study. First, we could examine the variations in the technical, purely technical, scale and super efficiency by using longitudinal data. Second, explore the same concept in the top exporter countries of the world.