**2. Review of Literature**

Charnes et al. (1978) invented the term data envelopment analysis and proposed an inputorientated DEA that measured efficiency in terms of radial contraction of input vectors necessary to reach the efficient frontier while assuming constant returns to scale. Banker et al. (1984) extended the DEA analysis to variable returns to scale (see (Emrouznejad and Yang 2017; Liu et al. 2013)).

Efficiency studies mostly model a single-stage decision-making process. However, many organizational operations, including banks, have multiple stages that offer possibilities for separate measurements of efficiency for each stage. The pioneering paper of Charnes et al. (1988) introduced the notion of network DEA structures for the measurement of efficiency in multi-stage operations.

The network DEA was used for measuring the efficiency of US banks across profitability and marketing stages (Seiford and Zhu 1999), Taiwanese banks for deposit mobilization and loan financing stages (Yang and Liu 2012), and Brazilian banks for cost efficiency and productive efficiency stages (Wanke and Barros 2014).

A separate strand of literature attempted to measure bank efficiency in the context of non-performing loans by employing single-stage directional distance functions (Chung et al. 1997), seeking to reduce inputs and undesirable outputs while increasing desirable outputs (Fukuyama and Weber 2008).

As mentioned earlier, credit risk arises in the second (loan financing) stage of banking operations. Given the considerable interest of the researchers in investigating efficiency measurements in the presence of non-performing loans, network DEA—with non-performing loans at the financing stage modeled as an undesirable output—is a natural direction for extending this literature. However, very few studies modeled bank efficiency in the presence of bad loans using the network DEA framework.

Wang et al. (2014) measured the efficiency of Chinese banks through hyperbolic Farrell-type efficiency measures proposed by Färe et al. (1989). The study divided the overall efficiency into two sub-processes, i.e., deposit producing and profit earning. Huang et al. (2014) examined the super efficiency of Chinese banks using a two-stage network model with bad outputs by extending the network slack-based measure model (NSBM) of Tone and Tsutsui (2009). The non-performing loans were modeled as a second-stage undesirable output.

Fukuyama and Weber (2010) proposed two-stage directional distance functions for measuring the efficiency of Japanese banks with non-performing loans as bad output in the second stage. Akther et al. (2013) estimated the efficiency of banks in Bangladesh through two-stage directional distance functions, which modeled bad loans as an undesirable output in the second stage. While these studies employed two-stage DEA and modeled undesirable outputs following Chung et al. (1997), they all focused on single countries. This is a serious limitation as it does not permit cross-country and regional comparisons of bank efficiency measurements, which, as noted above, could shed light on differential impacts on countries of common macroeconomic shocks such as global financial crises.

The selected countries were operated under different banking regulations and supervision. This was an important consideration investigated for cross-country comparison. Barth et al. (2013) conducted a survey to collect data and measures of bank regulatory and supervisory policies for the period of 1999–2011 in 180 countries. The authors collected the data based on several bank related questions. The study concluded that the supervision and regulation of banks differ in many dimensions across selected countries. Moreover, the study found divergence in bank regulatory regimes over the past decade despite the worst global financial crisis since the Great Depression.

Most DEA efficiency studies in the existing literature conducted a follow-on analysis to investigate the determinants of efficiency, using a Tobit model (Tobin 1985) that produced point estimates of contributions of different variables. This approach relies on ad hoc distributional assumptions. An alternative is non-parametric regression based on Kernel density estimation (Rosenblatt 1956), which does not suffer from ad hoc assumptions. For example, Illueca et al. (2009) examined the productivity of Spanish savings banks by employing kernel density estimations and non-parametric regression. However, the study did not consider non-performing loans within a multi-stage network DEA.

The present study makes the following contributions: (i) incorporation of credit risk in efficiency measurements of banks; (ii) use of a sufficiently flexible multi-stage network DEA framework, which better captures stylized banking operations where credit risk arises at the loan financing stage and manifests in the form of non-performing loans; (iii) extension of efficiency measurements to a multi-country setting making possible comparisons across countries; and application of non-parametric regression, avoiding ad hoc assumptions in estimating the impact of bank characteristics and other relevant variables on efficiency.
