1. Introduction

2. Literature

3. Methodology

3.1. Modeling

The model used is inspired by previous works such as Detragiache et al. (2006), Hermes and Lensink (2004), and Claessens et al. (2001). These studies explain bank profitability and efficiency based on bank-specific characteristics and the macroeconomic environment in which banks operate. Our model assesses the impact of foreign bank entry on the efficiency of domestic banks in developing countries using sustainability proxies related to efficiency, bank characteristics, and macroeconomic factors.

Our model is specified as follows:

Y_btp = α + ξ Presence_tp + φβ B_btp + δ M_tp + γ Inter + ε_btp.

(1)

where b, t, and p, respectively, reflect the bank b in year t and the country p;

b = 1, .. ..., n, t = 1, ...., T and p = 1, .. ..., P.

Ybtp: Efficiency of domestic banks; serving as the proxy for financial sustainability.

Presence_tp represents the "presence of foreign banks" (measured by the proportion of foreign-controlled assets), which serves as a proxy for their influence on financial stability and sustainability.

B_tp: includes bank characteristics, such as size (total assets), capital adequacy ratio, and loan portfolio composition (e.g., proportion of green or SME loans), which reflect financial, social, and environmental sustainability.

M_tp includes macroeconomic variables like GDP growth, inflation rate, and interest rate, which affect banks' overall financial sustainability.

Inter: includes interaction terms (Presence * Performance Gap between domestic and foreign banks) to measure the moderating effect of foreign bank presence on domestic bank characteristics, allowing us to test how competition influences sustainability.

εb_tp: The stochastic component of the error is assumed to be uncorrelated with the explanatory variables and follows the normal distribution N (0, σε). It is independent and identically distributed (iid).

This study uses two different methods to measure each bank's efficiency level. The first is to calculate the intermediation margin ratio, which has been widely used in the literature to assess the effect of the entry of foreign banks on the efficiency of local banks in developing countries. The second method is based on the estimated efficiency level by the "Meta frontier approach." This approach calculates the technical efficiency of banks operating different production technologies, and they are subject to different environmental conditions (economic, political, social ... etc.). It provides a measure of the efficiency level of banks operating in different countries while considering the existing technological gap between these countries.

Suppose we have "P" in different countries, and each country 'p' contains Np banks that face the same input prices and pursue the goal of cost minimization given the output level. The stochastic cost frontier model for each bank "b" of the country "p" at the time "t" can be formulated as follows:

$$C{T}_{bt(p)}=f(X_{bt(p)};\varphi {}_{(p)})e^{v_{bt(p)}+u_{bt(p)}}$$

(2)

$$b=1,\hspace{0.33em}2,\hspace{0.33em}\mathrm{3..........}{N}_p\hspace{1em};t=1,\hspace{0.17em}2,\mathrm{...............}T.\hspace{1em}\hspace{1em}\hspace{1em}p=1,\hspace{0.17em}\hspace{0.17em}2,\mathrm{..........}P.$$

where CTbt(p) is the total cost, Xbt (p) is the vector of output and input prices. φ (p) is the vector of unknown parameters to be estimated. Vbt (p) is the usual error term that follows the normal distribution N (0, σ2); it is independent of Ubt(p). It is a white noise used to control measurement errors and determinants of costs beyond the control of managers. The term also captures the luck (bad) of the bank having experienced a favorable (unfavorable) exogenous shock that (increases) decreases the total cost of the bank. Ubt(p) is the deviation of the cost of the firm b compared to the efficiency frontier. It serves as an approximation of technical and allocative inefficiency. It is positively defined with an independent distribution Vbt(p).

To facilitate the equation (2-29), it can be formulated as follows:

$$C{T}_{bt(p)}=e^{X_{bt(p)}{\varphi }_{(p)}+}{v}_{bt(p)}+u_{bt(p)}$$

(3)

Following Battese et al. (2004), this model assumes that for each country, there is only one data generation process for banks that use a given technology.

Therefore, this model is applied to each country to assess its stochastic frontier cost. Meta-frontier takes the same functional form as an individual stochastic frontier for each country. Thus, the function of the Meta-cost frontier that envelops the frontiers of specific costs for each country can be formulated as follows:

$$C{T}_{bt}^{\ast }=f(X_{bt};{\varphi }^{\ast })\equiv e^{X_{bt}{\varphi }^{\ast }}$$

(4)

$C{T}_{bt}^{\ast }$ Is the optimal expense for the bank "b" at the time "t" required to produce the given output amount under exogenous input price conditions?

${\varphi }^{\ast }$ Is the vector of parameters associated with the Meta-frontier cost function that satisfies the condition (C1) below:

$$X_{bt}{\varphi }^{\ast }\le X_{bt}{\varphi }_{(p)}$$

(C1)

Meta-frontier is the deterministic function parameter that envelops the deterministic part of the individual cost frontiers. Its values must be less or equal to the deterministic components of the estimated stochastic cost frontier for each country. The cost Meta-frontier function can reflect the minimum production cost to produce a given output level. This is the minimum cost associated with the technique of more efficient production. The inequality constraint given by equation (C1) must be satisfied for all countries throughout the period. Meta-frontier is considered the curve enveloping the individual frontier cost countries.

Figure 1 illustrates how the Meta-frontier cost envelops the stochastic frontiers function of different countries in the case of a single output.

In this figure, the frontier 1, 2, and 3 represent four stochastic frontiers for 3 countries. The cost of the Meta-frontier function that encompasses the three stochastic frontiers indicates the possibility of producing at a total cost below the deterministic costs correlated with the stochastic cost frontiers of the four countries. Frontiers 1 and 2 are arbitrarily chosen to be tangent to the Meta-frontier, and frontier 3 is not. Thus, as the stochastic cost frontiers 1 and 2 are closest to the Meta-frontier, we can say that banks in countries 1 and 2 adopt more production technology developed than those in the third country.

3.2. The Efficiency Score and the Technology Gap Ratio "TGR"

A company's cost efficiency is evaluated by the minimum cost (taken from the frontier) to the observed current cost for the same given production and conditions price. It is a measure of cost deviation versus best performance.

The Meta-frontier cost efficiency."$C{T}_{bt(p)}^{\ast }$" of a bank "b" in a year "t" in the country "p" can be estimated using the Meta-frontier model, as specified in equation (5) :

$$C{E}_{bt(p)}^{\ast }={\displaystyle \frac{e^{X_{bt}{\varphi }^{\ast } \hspace{0.17em}+\hspace{0.17em}{v}_{bt(p)}}}{C{T}_{bt(p)}}}$$

(5)

Integrate the equation (3) into equation (5):

$$C{E}_{bt(p)}^{\ast }={\displaystyle \frac{e^{X_{bt}{\varphi }^{\ast } \hspace{0.17em}+\hspace{0.17em}{v}_{bt(p)}}}{e^{X_{bt}{\varphi }_{(p)}+v_{bt(p)}+u_{bt(p)}}}}=e^{-u_{bt(p)}}\ast {\displaystyle \frac{e^{X_{bt}{\varphi }^{\ast }}}{e^{X_{bt}{\varphi }_{(p)}}}}$$

(6)

where the first term on the right reflects the level of technical efficiency (EC) on the stochastic cost frontier of the country "p."1

$$C{E}_{bt(p)}={\displaystyle \frac{e^{X_{bt}{\varphi }_{(p)} \hspace{0.17em}+\hspace{0.17em}{v}_{bt(p)}}}{e^{X_{bt}{\varphi }_{(p)}+v_{bt(p)}+u_{bt(p)}}}}=e^{-u_{bt(p)}}$$

(7)

It must be between 0 and 1 because $u_{bt(p)}$It is considered a non-negative random variable.

For the second term, it is the technology gap ratio "TGR."

$$TG{R}_{bt(p)}={\displaystyle \frac{e^{X_{bt}{\varphi }^{\ast }}}{e^{X_{bt}{\varphi }_{(p)}}}}$$

(8)

The ratio "$TGR$" measures the technology gap for country "p" where production technology adopted by its firms is less developed than that available for all countries. It is obtained from the Meta-frontier cost function.

According to the constraint (C1), this measurement is between 0 and 1. A country's high value of the ratio "TGR" reflects its adoption of advanced production technology.

Finally, the cost efficiency relative to the Meta-frontier as measured by the equation (5) can then be formulated as follows:

$$C{E}_{bt(p)}^{\ast }=C{E}_{bt(p)}\ast TG{R}_{bt(p)}$$

(9)

The vector ${\stackrel{⌢}{\varphi }}^{\ast }$It is obtained after solving the following optimization problem:

$$\begin{array}{l}Min\hspace{0.33em}L={\displaystyle \sum _{t=1}^T}{\displaystyle \sum _{b=1}^N\left|Ln\hspace{0.17em}f\left(X_{bt}{\stackrel{⌢}{\varphi }}_{(p)}\right)-Ln\hspace{0.17em}f\left(X_{bt}{\varphi }^{\ast }\right)\right|}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}(10-37)\\ \\ s/c:Ln\hspace{0.17em}f\left(X_{bt}{\varphi }^{\ast }\right)\le Ln\hspace{0.17em}f\left(X_{bt}{\stackrel{⌢}{\varphi }}_{(p)}\right)\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.17em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}(C2)\end{array}$$

As all deviations are positive according to the constraint (C2), all the absolute differences are equal to the differences. Using equations (3) and (4), we can convert the optimization problem above (10, C2) to a linear programming problem "LP" :

$$\begin{array}{l}Min\hspace{0.33em}L={\displaystyle \sum _{t=1}^T}{\displaystyle \sum _{b=1}^N\left(X_{bt}{\stackrel{⌢}{\varphi }}_{(p)}-X_{bt}{\varphi }^{\ast }\right)}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.33em}\hspace{0.17em}(11)\\ \\ \hspace{1em}s/c:X_{bt}{\varphi }^{\ast }\le X_{bt}{\stackrel{⌢}{\varphi }}_{(p)}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}(C3)\end{array}$$

The functional form of the cost function adopted for calculating the level of efficiency is the type of translog cost function; it is as follows:

$${(\mathrm{Ln} \mathrm{CT})}_{\mathrm{b},\mathrm{t}}= \mathsf{\alpha } + ({\displaystyle \sum _{i:1}^3}{\mathsf{\phi }}_{\mathrm{i}} \mathrm{Ln} {\mathrm{w}}_{\mathrm{i}}{)}_{\mathrm{b},\mathrm{t}}+({\displaystyle \frac{1}{\begin{array}{l}2\\ \end{array}}}{\displaystyle \sum _{i:1}^3}{\displaystyle \sum _{j:1}^3}{\mathsf{\phi }}_{\mathrm{ij}} \mathrm{Ln} {\mathrm{w}}_{\mathrm{i}} \mathrm{Ln} {\mathrm{w}}_{\mathrm{j}} {)}_{\mathrm{b},\mathrm{t}}+({\displaystyle \sum _{k:1}^3}{\mathsf{\gamma }}_{\mathrm{k}} \mathrm{L}\mathrm{n} {\mathrm{y}}_{\mathrm{k}}{)}_{\mathrm{b},\mathrm{t}}$$

$$+({\displaystyle \frac{1}{\begin{array}{l}2\\ \end{array}}}{\displaystyle \sum _{k:1}^3}{\displaystyle \sum _{m:1}^3}{\mathsf{\gamma }}_{\mathrm{km}} \mathrm{Ln} {\mathrm{y}}_{\mathrm{k}} \mathrm{Ln} {\mathrm{y}}_{\mathrm{m}}{)}_{\mathrm{b},\mathrm{t}}+({\displaystyle \sum _{i:1}^3}{\displaystyle \sum _{k:1}^3}{\mathsf{\lambda }}_{\mathrm{ik}} \mathrm{L}\mathrm{n} {\mathrm{w}}_{\mathrm{i}} \mathrm{L}\mathrm{n} {\mathrm{y}}_{\mathrm{k}}{)}_{\mathrm{b},\mathrm{t}}$$

$$+({\mathsf{\tau }}_{\mathrm{f}} \mathrm{Ln} \mathrm{F} {)}_{\mathrm{b},\mathrm{t}}+{\displaystyle \frac{1}{\begin{array}{l}2\\ \end{array}}}({\mathsf{\tau }}_{\mathrm{ff}} \mathrm{LnF} \mathrm{LnF} {)}_{\mathrm{b},\mathrm{t}}+({\displaystyle \sum _{k:1}^3}{\mathsf{\omega }}_{\mathrm{fi}} \mathrm{L}\mathrm{n} \mathrm{F} \mathrm{L}\mathrm{n} {\mathrm{y}}_{\mathrm{k}}{)}_{\mathrm{b},\mathrm{t}}$$

$$+({\displaystyle \sum _{i:1}^3}{\mathsf{\psi }}_{\mathrm{fi}} \mathrm{L}\mathrm{n}\mathrm{F} \mathrm{L}\mathrm{n} {\mathrm{w}}_{\mathrm{i}}{)}_{\mathrm{b},\mathrm{t}}+{\mathrm{U}}_{\mathrm{b}\mathrm{t}}+{\mathrm{V}}_{\mathrm{b}\mathrm{t}}$$

(12)

where (Ln CT) is the natural logarithm of the total cost of the b^th bank in the period t, (Ln w_{i bt}) is the natural logarithm of its i^th input prices, (Ln y_k) represents the natural logarithm of its k^th output, (Ln F) denotes its capitalization ratio used to control observable heterogeneity among banks.

Technical inefficiency, due to the excessive use of inputs to produce a given volume of output resulting from the choice of the wrong combination of inputs given their relative prices in the market, would be captured by the inefficiency term U_bt. The term V_bt is used to control measurement errors and determinants of costs beyond the control of managers. This term also captures the (bad) luck of the bank having experienced an exogenous shock (unfavorable) favorable that (increases) decreases the total cost of the bank.

To be consistent with the economic theory, the cost function should be: (i) non-negative, (ii) twice continuously differentiable in its domain, (iii) symmetric, (iv) linear homogeneous in input prices, (v) monotonically increasing in input prices and outputs, (vi) concave in input prices. The parameters need to satisfy the following restrictions :

φ_ij = φ_ji, γ_km = γ_mk ⇒ pour que la fonction de coût soit symétrique

${\displaystyle \sum _{i:1}^3}$ φ_i = 1, ${\displaystyle \sum _{i:1}^3}$φ_ij = 0, ${\displaystyle \sum _{k:1}^3}$λ_ik = 0, ${\displaystyle \sum _{i:1}^3}$_ψfi = 0 ∀ i

In this work, we refer to the intermediation approach to define the inputs and outputs bank.

The independent variables included in the model reflect the characteristics of the banks and the environment in which they operate. We select variables that can be measured for all banks in different countries.

4. Results

5. Conclusions

References

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