Bank Efficiency in Transitional Countries: Sensitivity to Stochastic Frontier Design

Abstract

This article provides an empirical insight on the heterogeneity in the estimates of banking efficiency produced by the stochastic frontier approach. Using data from five countries of Central and Eastern Europe, we study the sensitivity of the efficiency score and the efficiency ranking to a change in the design of the frontier. We found that the average scores are significantly smaller when the transcendental logarithmic functional form is used in the profit efficiency measurement and when the scaling effect is neglected in the cost efficiency measurement. The implied bank ranking is robust to changes in the stochastic frontier definition for cost efficiency, but not for profit efficiency.

Appendix

According to Battese and Corra (1997) parametrization, the inefficiency and the noise variances σ ² _u and σ ² _v are replaced by σ² = σ ² _u  + σ ² _u , the variance of composed error ɛ _it . A new variable γ = σ ² _u /(σ ² _v  + σ ² _u ) is defined, so that γ ∈ (0,1) in ML procedure. For the time-varying model, the log-likelihood function has the form of:

$$ \begin{aligned} \hbox{ln} L &= -\frac{1}{2}\left(\hbox{ln} 2\pi + \hbox{ln} \sigma^{2}\right) \sum_{i=0}^{N} T_i - \frac{1}{2} \sum_{i=0}^{N} \left(T_i - 1\right)\hbox{ln} \left(1 - \gamma\right)\\ &\quad-\frac{1}{2}\sum_{i=0}^{N} \hbox{ln} \left\{1 + \left(\sum_{t=1}^{T_i}\eta_{it}^{2} - 1\right)\gamma\right\} - N \hbox{ln} \left\{1-\phi(-\tilde{z})\right\} - \frac{1}{2} N \tilde{z}^2 \nonumber \\ &\quad+ \sum_{i=1}^{N} \hbox{ln} \left\{1 - \phi (- z_i^*) \right\} + \frac{1}{2}\sum_{i=1}^{N}z_i^{*2} - \frac{1}{2}\sum_{i=1}^{N}\sum_{t=1}^{T_i} \frac{\varepsilon_{it}^2}{\left(1 - \gamma \sigma^2\right)} \end{aligned} $$

(7)

where $\eta_{it} = \hbox{exp}\left\{-\eta\left(t-T_i\right)\right\}, \tilde{z} = \mu / \left(\gamma\sigma^2\right)^{1/2},$ and ϕ(·) is the cumulative distribution function of the standard normal distribution, a is the parameter differentiating between the production and the cost functions from (3), and

$$ z_i^{*} = \frac{\mu\left(1 - \gamma\right) - a \gamma \sum_{t=1}^{T_i}\eta_{it}\varepsilon_{it}}{\left[\gamma\left( 1 - \gamma\right) \sigma^2 \left\{1 + \left(\sum_{t=1}^{T_i}\eta_{it}^{2} -1 \right) \gamma \right\}\right]^{1/2}}. $$

The estimates of technical efficiency term from (3) are obtained via:

$$ E\left\{\hbox{exp}(-au_{it})|\varepsilon_{it}\right\} = \left[\frac{1-\phi\left\{a\eta_{it}\tilde{\sigma}_i- \left(\tilde{\mu}_i/\tilde{\sigma}_i\right)\right\}} {1-\phi\left(-\tilde{\mu}_i/\tilde{\sigma}_i\right)}\right] \hbox{exp}\left(-a\eta_{it}\tilde{\mu}_i+\frac{1}{2} \eta_{it}^2\tilde{\sigma}_i^2\right), $$

(8)

where

$$ \tilde{\mu}_i = \frac{\mu\sigma_v^2-a\sum_{t=1}^{T_i}\eta_{it}\varepsilon_{it} \sigma_u^2}{\sigma_v^2+\sum_{t=1}^{T_i}\eta_{it}^2\sigma_u^2}\,\hbox{and}\, \tilde{\sigma}_i^2 = \frac{\sigma_v^2\sigma_u^2}{\sigma_v^2+\sum_{t=1}^{T_i}\eta_{it}^{2} \sigma_u^{2}}. $$

Replacing η _it  = 1 and η = 0 changes the time decay model into the time-invariant model, so that the estimated efficiencies differ only on the cross-sectional level (for banks), not in the time dimension (through years) and u _it  = u _i .

Table 6 Descriptive statistics on variables (in ths. USD)

Fig. 2

(A, B) Development of profit scores for different panels

Fig. 3

(A, B) Kernel profit density 1995–2006 (epanechnikov, bandwidth 0.0148 & 0.0128)

Fig. 4

(A, B) Development of profit scores by countries

Fig. 5

(A, B) Development of cost scores by countries

Fig. 6

(A, B) Development of profit scores by countries

Fig. 7

(A, B) Kernel cost density (epanechnikov, bandw. 0.023)

Fig. 8

(A, B) Kernel profit density (epanechnikov, bandw. 0.023)

Table 7 (A) Profit efficiency scores for 1995–2006

Table 8 (A, B) Stochastic panel cost frontier by years

Table 9 (A) Stochastic panel profit frontier by years

Table 10 (B) Stochastic panel profit frontier by years

Table 11 (A) Efficiency scores for 2003–2006

Table 12 (A) Rank order correlations across models of 2003–2006

Table 13 (B) Cost efficiency by models (2003–2006)

Table 14 (B) Profit efficiency by models (2003–2006)

Table 15 (B) Spearman correlations, the Czech Republic in 2003-06

Table 16 (B) Spearman correlations, Hungary in 2003–2006

Table 17 (B) Spearman correlations, Poland in 2003–2006

Table 18 (B) Spearman correlations, Slovenia in 2003–2006

Table 19 (B) Spearman correlations, Slovakia in 2003–2006

Table 20 (B) Kendall correlations, the Czech Republic in 2003–2006

Table 21 (B) Kendall correlations, Hungary in 2003–2006

Table 22 (B) Kendall correlations, Poland in 2003–2006

Table 23 (B) Kendall correlations, Slovenia in 2003–2006

Table 24 (B) Kendall correlations, Slovakia in 2003–2006