A directional semi-oriented radial DEA measure: an application on financial stability and the efficiency of banks

Abstract

Data envelopment analysis (DEA) is a widely used non-parametric technique for measuring the relative efficiencies of decision-making units with multiple inputs and multiple outputs. The main caveat of traditional DEA models is that they are applicable to positive inputs and outputs, while negative data are commonly present in most real applications. To accommodate variables that can take both negative and positive values, Emrouznejad et al. (Eur J Oper Res 200(1):297–304, 2010a) introduced the Semi-Oriented Radial Measure (SORM) model, which was later modified by Kazemi Matin et al. (Measurement 54:152–158, 2014). The present study proposes a new version of the modified SORM model, using directional distance function and choosing a relevant direction to efficiently deal with variables with both positive and negative values. Our Directional SORM (DSORM) model is superior to its predecessors from both computational and target settings perspectives while it allows for the dual formulation of linear programming. To illustrate our proposed model, we employ two widely used selections of inputs and outputs to estimate the efficiency scores for a sample of banks operating in Persian Gulf Council Countries (GCC) over the period of 2002–2011. The estimated efficiency scores are then used to study the impact of financial system stability on technical efficiency of individual banks.

Appendix A

In order to show the sensitivity of results to the choice of direction, we estimate the efficiency score by our proposed model under three different directions. Following formulas shows the three Direction1, Direction2 and Direction3 respectively.

Direction 1: Improving all outputs

$$\begin{aligned}&\left( \mathbf{g}_x ,\mathbf{g}_{y^{p}} ,\mathbf{g}_{y^{1}} ,\mathbf{g}_{y^{2}} \right) =\left( \mathbf{0},\mathbf{y}^{p},\mathbf{y}^{1},\mathbf{y}^{2}\right) \\&{\textit{Max}}\,\, h \\&{\textit{s. t.}}\,\,\, \left( {\mathbf{x},(1+h) \mathbf{y}_k^p ,(1+h)\mathbf{y}_k^1 ,(1-h)\mathbf{y}_k^1 } \right) \in T_{\textit{SORM}} \end{aligned}$$

Direction 2: Improving all inputs and outputs

$$\begin{aligned}&\left( \mathbf{g}_x ,\mathbf{g}_{y^{p}} ,\mathbf{g}_{y^{1}} ,\mathbf{g}_{y^{2}} \right) =\left( \mathbf{x},\mathbf{y}^{p},\mathbf{y}^{1},\mathbf{y}^{2}\right) \\&{\textit{Max}} \,\,h \\&{\textit{s. t.}}\,\,\,\left( {(1-h) \mathbf{x},(1+h) \mathbf{y}_k^p ,(1+h)\mathbf{y}_k^1 ,(1-h)\mathbf{y}_k^2 } \right) \in T_{\textit{SORM}} \end{aligned}$$

Direction 3: Improving Inputs and negative output in DSORM

$$\begin{aligned}&\left( \mathbf{g}_x ,\mathbf{g}_{y^{p}} ,\mathbf{g}_{y^{1}} ,\mathbf{g}_{y^{2}} \right) =\left( \mathbf{x},\mathbf{0},\mathbf{y}^{1},\mathbf{y}^{2}\right) \\&{\textit{Max}}\,\, h \\&{\textit{s. t.}}\,\,\, \left( {(1-h) \mathbf{x}, \mathbf{y}_k^p ,(1+h)\mathbf{y}_k^1 ,(1-h)\mathbf{y}_k^2 } \right) \in T_{\textit{SORM}} \\ \end{aligned}$$

The yearly average of efficiency scores for directions 2 and 3, under each scenario for two different banking operational styles is reported in the following table. The results of Direction 1 which we used in our empirical example is being reported in the manuscript.

Scenario 1

	Meaan eff S1D1	Meaan eff S1D2	Meaan eff S1D3
Conventional
2002	0.51	0.33	0.83
2003	0.49	0.61	0.82
2004	0.51	0.39	0.83
2005	0.63	0.62	0.88
2006	0.70	0.49	0.90
2007	0.74	0.70	0.94
2008	0.71	0.75	0.95
2009	0.63	0.67	0.87
2010	0.64	0.69	0.91
2011	0.63	0.79	0.90
Islamic
2002	0.40	0.27	0.86
2003	0.39	0.32	0.82
2004	0.45	0.36	0.84
2005	0.39	0.48	0.85
2006	0.45	0.75	0.89
2007	0.50	0.55	0.85
2008	0.52	0.65	0.81
2009	0.48	0.57	0.76
2010	0.48	0.34	0.83
2011	0.49	0.41	0.81

1. S1D1 Scenario 1 with direction 1, S1D2 Scenario 1 with direction 2

Scenario 2

	Meaan eff S2D1	Meaan eff S2D2	Meaan eff S2D3
Conventional
2002	0.69	0.21	0.52
2003	0.70	0.23	0.52
2004	0.70	0.25	0.52
2005	0.75	0.36	0.62
2006	0.77	0.40	0.61
2007	0.79	0.44	0.62
2008	0.79	0.37	0.59
2009	0.69	0.29	0.41
2010	0.73	0.31	0.45
2011	0.72	0.30	0.44
Islamic
2002	0.65	0.17	0.41
2003	0.56	0.17	0.34
2004	0.48	0.18	0.27
2005	0.56	0.29	0.42
2006	0.65	0.31	0.49
2007	0.61	0.32	0.46
2008	0.55	0.27	0.34
2009	0.53	0.28	0.22
2010	0.55	0.22	0.17
2011	0.52	0.24	0.27

1. S2D1 Scenario 2 with direction 1, S2D2 Scenario 2 with direction 2