Stochastic Frontier Analysis (SFA)

The parametric SFA approach originated from two innovative papers by Meeusen and Van den Broeck (1977) and Aigner, Lovell, and Schmidt (1977) who sought to captures best practice to gauge inefficiency purely by observation of best practise within the sample of banks tested. This approach however does not necessarily represent a best-possible practice (Berger & Mester, 1997) depending on the sample size or selection bias. This empirical methodolgy was later operationalised by Battese, Rao, and O’Donnell (2004) and O’Donnell, Rao, and Battese (2008). SFA is a form of regression which separates the influence of exogenous factors on the dependent variable, from the measurement error (noise) and firm inefficiency is captured in the error term. The error term in SFA, consists of two components, one is a two-sided random error that represents noise, the other is a one-sided error representing inefficiency. The noise is assumed to be normally distributed with a zero mean and for cost inefficiency the error is assumed to be positively half-distributed. As this is a structural approach the selection of the environment and bank characteristic variables to determine best practice is particularly important (Mester, 2008). To ensure SFA is appropriate the structural form imposed on the analysis also has to reflect the firms’ behaviour. Theoretically within a panel data framework (Feng & Serletis, 2009), the cost frontier model can be written as:

Cit =f(Xit,ρ)τitζit, i= 1, . . . , I, t= 1, . . . T (4.4.1)

16_{On the contrary Olgu and Weyman-Jones (2008) evidence suggested consistency between}

parametric and non-parametric for 10 old EU countries and 12 new EU countries’ banking systems (164 Banks).

This model decomposes the observed total cost (Cit) for firm iat time t, into three

elements. Firstly is the actual frontier f(Xit,ρ), dependent on Xit, which is the

vector of, input prices and output quantities (exogenous variables), andρ, which is a vector of parameters, that represents the minimum possible cost of producing a given level of output for a certain input. Secondly a non-negative termτit ≥117, measures

firm-specific inefficiency. Lastly the random errorζit, captures the statistical noise.

The deterministic kernel of the cost frontier is f(Xit,ρ), and the stochastic cost

frontier is f(Xit,ρ)ζit. As required by microeconomic theory,f(Xit,ρ) is a linearly

homogeneous and concave function in prices and also non-decreasing in both input prices and outputs. Following common practice in this literature it is assumed that

f(Xit,ρ) is a log-linear function form. The stochastic cost function in (4.4.1) can

be rewritten as: cit =α+x 0 itβ+εit (4.4.2) wherecit = lnCitandα+x 0

itβ= lnf(Xit,ρ). The composite error termεit=uit+υit

consists of two parts,νit is a two-sided normal disturbance term with zero mean and

variance σ2

ν and represents the effects of statistical noise; the inefficiency term υit

is assumed to be half-normally distributed. Thus, uit = lnτit ≥ 0 and υit = lnζit.

Further in equation 4.4.2 xit is the counterpart of Xit with the input prices and

output quantities transformed to logarithms,β is a K×1 vector of parameters and

α is the intercept. Following the most commonly used functional form in the bank efficiency literature to identify a frontier, a transcendental logarithmic (translog) form is applied. The empirical cost frontier model is as follows:

lnT Ci,t =α+ X m βmyimt+ X j γjwijt+ 1 2 X m X n βmnyimyin+ 1 2 X j X k βjkyijtyikt +X m X j ψmtlnyimtlnwijt+ϕ1lnEit+ 1 2ϕ2lnE 2 it+ X m λmlnyimtlnEit +X j ξjlnwijtlnEit+θ1T +θ2T2+ X m κmlnyimtT + X j ρjlnwijtT

+ηlnEitT + lnOEA+ lnT LOAN+ lnDeposits+υit+νit (4.4.3)

17_{The cost efficiency is defined asCE}

Where the dependent variable lnT Ci,t is the observed total costs (personnel

expenses, other administrative expenses and other operating expenses) of banki at time t. yi and wi are vectors of output and inputs for the ith bank18. Ei is the

total equity of a bank (which is treated as a quasi-fixed input)19_{; T is the time}

trend used to capture technological changes; and lnOEA, lnT LOAN & lnDeposits is the natural logarithm of Other Earning Assets, Total Loans and Total Deposits respectively. As previously stated, νit is a two-sided normal

disturbance term with zero mean and variance σ2

ν and represents the effects of

statistical noise; the inefficiency term υit is assumed to be half-normally

distributed20_{. α, β, γ, ψ, ϕ, λ, ξ, θ, κ, ρ and η are coefficients to be estimated.}

Furthermore, the standard symmetry restrictions, βnm = βmn and γjk = γkj, are

applied.