Stochastic frontier analysis by means of maximum likelihood and the method of moments Andreas Behr (University of Münster) Sebastian Tente (University of Münster) Discussion Paper Series 2: Banking and Financial Studies No 19/2008 Discussion Papers represent the authors’ personal opinions and do not necessarily reflect the views of the Deutsche Bundesbank or its staff.
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Stochastic frontier analysis bymeans of maximum likelihoodand the method of moments
Andreas Behr(University of Münster)
Sebastian Tente(University of Münster)
Discussion PaperSeries 2: Banking and Financial StudiesNo 19/2008Discussion Papers represent the authors’ personal opinions and do not necessarily reflect the views of theDeutsche Bundesbank or its staff.
Table 2: Mean Average Error normal-exponential approach
3.2.3 Misspecification scenarios
As it is an unfeasible task to determine any real inefficiency distribution across an industry,
we can basically assume an underlying misspecification in every applied Stochastic Frontier
Analysis. To exemplify the impacts of misspecification against the background of MOM vs.
ML-estimation, we interchanged the data generating processes of the normal-exponential and
normal-halfnormal case. So table 9 given in the appendix shows the mae of an exponential
model estimated as halfnormal (misspecification scenario M1), and table 10 in the appendix
obtains the results of a halfnormal model estimated as exponential (misspecification M2).
The findings are straightforward: The exponential model M1 shows the predominant advantage
of ML-estimation, even more clearly than in the correctly specified case. Obviously, the mae
of MOM-estimation improves with increasing n or λ, but does not decrease in jointly larger
n, λ-combinations. The indications in the halfnormal model M2 suggest slight advantages of
MOM-estimation facing an overall lower error.
1510
3.2.4 Rules of thumb
To summarize the particular advantages of ML- or MOM-estimation, we estimated multiple
linear regressions based on tables 1 and 2 for the normal and exponential cases, respectively:
yk = β0 + β1k · λ + β2k · n + μk
with
y =
⎧⎪⎨⎪⎩0 if MOM has smaller error
1 if MLE has smaller error
and μk as normal error in all k possible combinations of sample size n and λ. Predicted values
yk > 0.5 imply an advantage of ML- over MOM-estimation. Figure 1 illustrates the separating
line between both approaches in the half normal and exponential case. The corresponding
parameter estimates are shown in table 3.
0 200 400 600 800 1000
0.0
0.5
1.0
1.5
2.0
2.5
sample size
λλ
− MOM −
− MLE −
normal−half normalnormal−exponential
Figure 1: Rules of thumb
1611
Estimate Std. Error t value Pr(>|t|)normal-half normal
Intercept 0.0602 0.0900 0.67 0.5062
λ 0.1908 0.0292 6.53 0.0000
n 0.0006 0.0001 4.52 0.0000
normal-exponential
Intercept 0.3338 0.0874 3.82 0.0003
λ 0.1710 0.0284 6.02 0.0000
n 0.0002 0.0001 1.33 0.1897
Table 3: Parameter estimates rules of thumb
4 An application to German banks
4.1 Methodological issues
4.1.1 Inputs and Outputs
We review the most relevant literature covering bank efficiency estimation by means of stochas-
tic frontier analysis. One of the most cited articles in recent literature is Mester (1996). She laid
out the main features of current efficiency analyses via frontier cost functions. Resorting to cost
functions instead of production functions (technologies) has several advantages.
Beside the methodical problems discussed above, we find another key question of empirical
bank efficiency estimation in modelling inputs and outputs of the production process. As Girar-
done et al. (2004) state: While the multiproduct nature of the banking firm is widely recognized,
there is still no agreement as to the explicit definition and measurement of banks’ inputs and
outputs. So it is common practice to operationalise bank production according to the funda-
mental idea of the Intermediation Approach proposed by Sealey and Lindley (1977). They had
in mind a multistage production process of the financial firm, using capital, labour and material
to acquire customer deposits in a first step. Lending these funds in a (virtual) second step to de-
ficit spending units and issuing securities and other earning assets involve in general an interest
profit. So financial production for intermediation purposes is about adding value to deposits.
Obviously, the use of multiple outputs does not apply to the single-output production func-
tions described above. But, referring to Duality Theory3, one can prove under certain regularity
conditions4 the equivalence of indirect cost functions tc = tc(y, c) and the underlying techno-
3Cp. Beatti and Taylor (1985), chapter 6.4In particular, linear homogeneity and weak concavity in input prices if the implicit production technology is
1712
logy F (y,x) = 0 with tc total operating costs, y′ a vector of outputs, and c′ a vector of prices of
the inputs x′. Estimating restricted stochastic cost frontiers is virtually the same as production
frontiers, as the lower stochastic frontier of the ’data cloud’ is simply defined by turning the
c1 Cost of fixed assets (% depreciation) 0.160 0.089 0.141
c2 Cost of labour (TEUR/employee) 80.671 64.654 65.621
c3 Price of funds (% interest expenses) 0.049 0.058 0.031
Table 4: Descriptive statistics of inputs, outputs, prices and bank size
4.1.2 Shape of the cost function
As for the formal issues, one can state that most authors apply a ’regular’ translog cost frontier.
In most cases the translog form offers an appropriate balance between flexibility (in price and
output elasticities), parameters to estimate and global fit. The exceptions among the reviewed
literature are Altunbas et al. (2000), Altunbas and Chakravarty (2001), Girardone et al. (2004),
and Weill (2004), using a Fourier Flexible form with additional trigonometric terms. Otherwise,
Fitzpatrick and McQuinn (2005) had to restrict themselves to a simple Cobb-Douglas form due
to an insufficient number of observations.
As we, too, work on a small-sized dataset (n = 56), we encountered severe multicollinearity in
the flexible translog form. So we fall back to a simple log linear Cobb-Douglas cost function
(Kumbhakar and Tsionas, 2008). To ensure linear homogeneity in input prices tc(y, k · c) =
1914
k1 · tc(y, c) with k > 0, we normalize total costs and input prices by the price of labour c2 (Lang
and Welzel, 1996).
log tc(y, c) = β0 +3∑
i=1
βi log yi +3∑
j=1
γj log cj + v + u s.t.
3∑i=j
γj = 1
The homogeneity-constrained cost frontier results in:
logtc(y, c)
c2
= β0 +3∑
i=1
βi log yi +∑j=1,3
γj logcj
c2
+ v + u
4.1.3 Inefficiency distribution
In the course of the simulation part, we applied the half normal and exponential assumption
of inefficiency distribution. A closer look at the relevant literature reveals another ’truncated’
approach (Bos and Kool, 2006, Battese et al., 2000, Fitzpatrick and McQuinn, 2005): ui ∼N+(μ, σu) with μ ≥ 0. Referring to Greene (1990), Weill (2004) is the only one using a gamma-
distributed inefficiency term. But the selection of an adequate distribution of ui does not have
to be overvalued, as Greene (1990) reportet extremely high rank correlations in the efficieny
estimates between half normal, truncated, gamma and exponential model. So obviously there is
no need to make use of two-parameter distributions.
4.2 Empirical evidence
To our knowledge there is not a single bank efficiency study applying the method of moments
estimator we discussed. Conventionally, authors prefer Maximum Likelihood Estimation with
the Jondrow et al. (1982) exp[−E(u|ε)] estimator mentioned above.
As our sample is rather small-sized, we expect the method of moments approach to deliver
highly reliable efficiency scores. Table 5 shows the results of all scenarios in discussion. Our
’rules of thumb’-indicator gives rather ambivalent recommendations: Values greater than 0.5
point at MLE application. But especially in the normal-exponential case we are facing values
≈ 0.5. So we should reckon with equivalent results in both MOM and ML-estimation. In fact,
the correlation table 6 confirms a ρ(CEMOM , CEMLE) ≈ 0.99. Mean efficiencies 1n
∑i CEi
differ only between varying inefficiency distribution assumptions. By the way, just like Greene
(1990), we can still report extemely high correlations ρ > 0.95 between the half normal and
exponential approach to inefficiency.
We observe slight differences in the estimated output and price elasticities (note that γ2 =
1 − γ1 − γ3). The overall scale economies are calculated as εc =(∑
i∂ log tc(y,c)
∂ log βi
)−1
. As εc > 1
in all cases, the results hint at increasing returns to scale in the German banking industry.
2015
normal-half normal normal-exponential
MLE MOM MLE MOM
Intercept 0.760 1.526 2.096 1.819
y1 0.107 0.157 0.117 0.157
y2 0.543 0.505 0.552 0.505
y3 0.055 0.060 0.050 0.060
c1 0.208 0.233 0.254 0.233
c3 0.289 0.388 0.418 0.388
λ 2.696 2.804 1.158 0.933
σv 0.326 0.325 0.416 0.467
σu 0.880 0.913 0.481 0.436
mean CE 0.537 0.531 0.655 0.673
rule of thumb 0.608 0.629 0.541 0.503
Table 5: Estimation results, all cases
nhn-mle nhn-mom exp-mle exp-mom
nhn-mle 1.000
nhn-mom 0.987 1.000
exp-mle 0.957 0.952 1.000
exp-mom 0.951 0.952 0.988 1.000
Table 6: Correlation table, cost efficiencies
5 Conclusions
We put forward the MOM-approach to stochastic frontiers in bank efficiency analysis. An ex-
tensive simulation study confirmed the findings of Coelli (1995) and Olson et al. (1980): Rules
of thumb suggest that the MOM-estimation of parametrical frontiers assuming a half normal
inefficiency distribution can be favourable in terms of mse(TE, TE) and mae(TE, TE) if the
sample size is medium scale (≤ 700 observations) and inefficiency does not strongly dominate
noise (λ < 2), i.e. the bias of βOLS0 is small.
So we propose that method of moment estimation should be considered an alternative to maxi-
mum likelihood estimation. We do so especially in cases focusing on a small number of ho-
mogeneous banks (Fitzpatrick and McQuinn, 2005). Applying MOM-estimation additionally
to the ML-procedure even in larger samples could shed new light on the significance of the
findings.
2116
Another practical advantage of MOM-estimation is obvious: Whenever Newton-like numerical
optimization is unavailable or fails due to awkward data structure, MOM provides a robust and
easy to implement loophole. A simple two-step procedure (OLS-fitting and bias correction
based on estimated residuals) is available within every statistical environment.
References
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6 Appendix
6.1 Derivatives of the log-likelihood: half-normal
ln L(y|β, λ, σ2) = n ln
(√2√π
)+n ln
(σ−1
)+
n∑i=1
ln[1 − Φ
([yi − x′
iβ] λσ−1)]− 1
2σ2
n∑i=1
(yi−x′iβ)2
The derivatives are given by
∂ ln L
∂β= − n
σ2
n∑i=1
(yi − x′iβ)xi +
λ
σ
n∑i=1
φ∗i
(1 − F ∗i )
xi
∂ ln L
∂σ2= − n
2σ2+
1
2σ4
n∑i=1
(yi − x′iβ)2
+1
2σ3
n∑i=1
φ∗i
(1 − Φ∗i )
(yi − x′iβ)
∂ ln L
∂λ= − 1
σ+
1
2σ4
n∑i=1
φ∗i
(1 − Φ∗i )
(yi − x′iβ)
where
φ∗i = φ([ln yi − x′
iβ] λσ−1)
Φ∗i = Φ([ln yi − x′
iβ] λσ−1)
6.2 Derivatives of the log-likelihood: exponential
labour, funds, physical capital total loans, securities, off-balance sheet items
(contingent liabilities, acceptances,
guarantees
Intermediation
(varied)
Altunbas and Chakravarty (2001)
labour, total funds, physical capital all types of loans, total aggregate securities,
off-balance sheet activities
Intermediation
(varied)
Battese et al. (2000)
public loans, guarantees, deposits,
number of branches, value of
inventories
costs of labour use Input-requirement
model
Bos and Kool (2006)
public relations, labour, housing,
physical capital
retail loans, wholesale loans, mortgages,
provisions
Intermediation
(varied)
Ferrier and Lovell (1990)
employees, occupancy costs,
materials
demand deposit accounts, time deposit
accounts, real estate loans, real estate loans,
installment loans, commercial loans
Production
Fitzpatrick and McQuinn (2005)
labour, physical capital, financial
capital
consumer/commercial/other loans,
non-interest revenue
Intermediation
(varied)
Girardone et al. (2004)
employees, total customer deposits,
total fixed assets
total customer loans, other earning assets Intermediation
Lang and Welzel (1996)
employees, fixed assets, deposits short-term and long-term loans to non-banks,
loans to banks, bonds/cash/real estate
investments, fees and commissions, revenue
from sales of commodities
Intermediation
(varied)
Mester (1996)
labour, physical capital, deposits real estate loans,
commercial/industrial/government/... loans,
loans to individuals
Intermediation
Perera et al. (2007)
funds, labour, capital net total loans, other earning assets Intermediation
Weill (2004)
labour, physical capital, borrowed
funds
loans, investment assets Intermediation
Table 11: Input and output-modelling in selected bank efficiency studies
2924
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