Estimation of cost efficiency in Non-Parametric frontier Models Текст научной статьи по специальности «Электротехника, электронная техника, информационные технологии»

2019

ВЕСТНИК САНКТ-ПЕТЕРБУРГСКОГО УНИВЕРСИТЕТА ЭКОНОМИКА

Т. 35. Вып. 1

МАКРО- И МИКРОЭКОНОМИЧЕСКИЕ ИССЛЕДОВАНИЯ

JEL: C61, C63

Estimation of Cost Efficiency in Non-parametric Frontier Models

G. Besstremyannaya1, J. Simm2

1 New Economic School and Institute of the Far East Studies of the Russian Academy of Sciences, 45, Skolkovskoe shosse, Moscow, 121353, Russian Federation

2 Stadius Centre for Dynamical Systems, Signal Processing and Data Analytics, Belgium, B-3001, Heverlee

For citation: Besstremyannaya, G., Simm, J. (2019) Estimation of Cost Efficiency in Non-parametric

Frontier Models. St Petersburg University Journal of Economic Studies, vol. 35, issue 1, pp. 3-25.

https://doi.org/10.21638/spbu05.2019.101

The paper proposes a bootstrap methodology for estimating cost efficiency in data envelopment analysis. We consider the conventional concept of Fare, Grosskopf and Lovellcost efficiency, for which our algorithm re-samples "naive" input-oriented efficiency scores, rescales original inputs to bring them to the frontier, and then re-estimates cost efficiency scores for the rescaled inputs. Next, we examine Tone cost efficiency, where input prices vary across producers. Here we show that the direct modification on bootstrap algorithms by Simar and Wilson are applicable. We consider cases both with the absence and presence of environmental variables (i.e. input variables not directly controlled by firms). The bootstrap methodology exploits these assumptions: 1) the sample are i.i.d. random variables with the continuous joint probability density function with support over production set; 2) the frontier is smooth; and 3) the probability of observing firms on the frontier approaches unity with an increase in sample. The results of simulations for a multi-input, multi-output Cobb-Douglas production function with correlated outputs, and correlated technical and cost efficiency, show consistency of our proposed algorithm, even for small samples. Finally, we offer real data estimates for the Japanese banking industry in 2013. Our package "rDEA," developed in the R language, is available from the GitHub and CRAN repository.

Keywords: bootstrap, price efficiency, border analysis, banking.

© Санкт-Петербургский государственный университет, 2019 https://doi.org/10.21638/spbu05.2019.101

Introduction

Data envelopment analysis (DEA) was introduced in [Charnes, Cooper, Rhodes, 1978] as a linear optimization technique, stemming from the seminal work of [Farrell, 1957], who defined a firm's technical and price efficiency and proposed a method for constructing a frontier as a linear convex hull surface to envelop observations. The efficiency scores of each firm are estimated according to the distance from the empirical frontier, which is treated as fully efficient firms. The analysis is based on common premises about production: monotonicity of technology, requirement of inputs for production ("no free lunch" condition), closedness, and strict convexity of the production set [Simar, Wilson, 2000b]. However, the empirical frontier may fail to incorporate unobservable but very efficient firms [Simar, Wilson, 1998]. Thus, the efficiency scores, linked to the empirical frontier, are upwardly biased. Standard approaches for consistent correction of bias in the case of technical efficiency scores are a homogeneous bootstrap based on re-sampling from a smooth consistent estimator of the joint density of input-output pairs, or a semi-parametric bootstrap in the presence of additional inputs, i.e. so-called environmental variables, which are not directly controlled by producers [Simar, Wilson, 2000b; Simar, Wilson, 1998; 2007]1. The bootstrap methodology exploits these assumptions about the data-generating process for observations of firms, which enable a consistent approximation of the unknown distribution of efficiency scores: 1) the sample is i.i.d. random variables with a continuous joint probability density function with support over the production set; 2) the frontier is smooth; and 3) the probability of observing firms on the frontier approaches unity with an increase in sample size (see [Kneip, Simar, Wilson, 2008; Simar, Wilson, 2000b]).

Concerning cost minimization DEA, as formulated in [Fare, Grosskopf, Lovell, 1985, practitioners suggest a direct modification of the [Simar, Wilson, 1998; 2007] bootstrap (e.g. [Borger, Kerstens, Staat, 2008]).

In this paper we show that a direct modification the [Simar, Wilson,1998; 2007] bootstrap is inconsistent for cost minimization DEA, and we propose an alternative algorithm. The proposed algorithm re-samples "naive" input-oriented efficiency scores, rescales original inputs to bring them to the frontier, and then re-estimates cost efficiency scores for the rescaled inputs. The results of the simulations for a multi-input, multi-output Cobb-Douglas production function with correlated outputs, and correlated technical and cost efficiency, show consistency of our proposed algorithm, even for small samples. As for a recently defined "new" cost efficiency [Tone, 2002], which to the best of our knowledge is commonly assessed only in terms of naive scores, we demonstrate that the direct modification of the [Simar, Wilson, 1998; 2007] bootstrap is consistent. Finally, we apply the algorithm to real data from 106 Japanese banks for fiscal year 2013.

The remainder of the paper is structured as follows. Section 1 reviews the theoretical framework for bias correction of technical efficiency scores, using an example of input orientation. Section 2 demonstrates the inconsistency of a direct application of [Simar, Wilson, 1998; 2007] bootstrap and offers an alternative bootstrap algorithm for robust estimation of [Fare, Grosskopf, Lovell, 1985] cost efficiency in absence and in presence

1 In the absence of environmental variables, the smooth bootstrap provides a better inference in a non-simulation context [Kneip, Simar, Wilson, 2008] than an alternative bootstrap based on subsampling [Simar, Wilson, 2011a].

of environmental variables. Section 3conducts simulations of production frontier and technical and cost inefficiencies. Section 4provides real data estimates with a nationwide sample of Japanese banks. The Appendix sets up the microeconomic framework for estimating technical and cost inefficiencies. Our computations are conducted with an R package "rDEA" [Simm, Besstremyannaya, 2016], which is available from GitHub and CRAN repositary.

1. Estimates of input-oriented efficiency

1.1. Naive score

Denote the estimator of existing technology (of the production set), which produces outputsym(m = 1,...,M) using inputs x„(n = 1,...,N) as T = {(x,y)\x > XX,y < YX,X > 0} where efficiency for a given unit (x, y) is measured relative to the boundary of the convex hull (X, Y) under the assumption of constant returns to scale. Input set L(y) [Coelli, Rao, Battese,1994; Shephard,1981] contains inputs that can produce a given amount of output, so that L(y) = {(x) : (x, y) e T}. The technology satisfies common premises about inputs, outputs, and the production set, which were briefly outlined in the Introduction. The important assumptions about the production process are strict convexity of L(y) and strong (free) disposability of inputs and outputs. In particular, strong disposability of inputs implies that if x e L(y), and if x > x, then x e L(y). The input-oriented efficiency dj for a given firm j (j = 1,J) is defined as a solution to the optimization problem below (constant returns to scale, CRS formulation of [Charnes, Cooper, Rhodes, 1978]:

min 0:

0 ,1 j

st.~ymj +ilymi > 0, m = 1,...,M, (1)

i=1

J

0jXnj ~£1Xni > °> n = u.,N, i=1

1 > 0, i = 1,., J.

J

Additional constraints Y.1ixni =1 impose variable returns to scale (VRS).

i=i

It should be noted that system (1) represents a linear maximization program written in concise notation. In fact, for each firm j there is a set of M constraints, where each constraint corresponds to a particular output ymj. Similarly, there are N constraints on each input xnj.

The economic intuition behind the system (1) may be explained as follows. We examine the behavior of economic agents and construct the production possibility set, using inputs and outputs for each agent. The analysis focuses on minimizing the amount of inputs required to produce a given amount of output; therefore, the problem is called "input-oriented DEA". The boundary of the production possibility set becomes the produc-

tion possibility frontier. Thus, the unity values of dj imply that the agent is fully efficient, as these points lie on the non-linear hull of observations (i. e. on the empirical estimate of the production possibility frontier). The weights Xi in system (1) are exploited for the construction of the frontier.

Another optimization problem, called "output-oriented DEA," studies the maximization of the output under a given amount of inputs. The reciprocals of efficiency scores become the multipliers of the output coordinates:

min 0-

ei '

s.t.-ymj /0 - +ZAiymi >0, m = 1,..., M, (1a)

i=1

Xnj -ZMm > 0 " = N,

i=1

Ai >0, i = 1,...,J.

Output-oriented DEA may be used to analyze the behavior of such economic agents as municipal hospitals, which have difficulties increasing the amount of inputs (medical personnel) and hence are analyzed in terms of their ability to maximize outputs (number of treated patients, see [Besstremyannaya, 2013]).

In this paper we focus on an input-oriented DEA when proposing our estimations of bias-corrected cost efficiency. Note, however, that the bottstrap algorithm may be straightforwardly modified for using the output-oriented score at the first stages.

1.2. Bias correction of naive DEA score

The estimates of input-oriented efficiency are upwards biased, since the estimated boundary L 8(y) of the input set is based on the sample of observed units, which may fail to incorporate the most efficient units in the true L(y) [Simar, Wilson 1998; 2000]. The bootstrap methods were proposed in [Simar, Wilson, 1998] and the assumptions about the data-generating process are further explained in [Simar, Wilson, 2000b; Kneip, Simar, Wilson, 2008].The bootstrap corrects for the bias by constructing pseudo-samples that would belong to L(y). The central idea for using the bootstrap is an assumption that the empirical bootstrap distribution of the DEA efficiency score consistently estimates the sampling distribution, and the assumption is ensured with the use of smoothing techniques [Simar, Wilson, 2000a]. Then, according to the re-centering idea of bootstrap, bias 4 = E(4) - 0i. So the estimator of the bias becomes bias Q{ = bias 0* = E(d*) - di. In particular, the homogeneous smoothed bootstrap projects each observation on the frontier and then "pushes" it inside the L(y) [Simar, Wilson, 2008; 1998].

1. Estimate naive scores d1,...,dj, for each i = 1,..., J according to system (1). Assume (01,...0j) are i.i.d. with pdff( ).

2. Loop B times to obtain J sets of bootstrap estimates { d*b .

1. Obtain a smooth estimate /(0) and for each i = 1,...,J draw 0ib from this estimate2.

2. Assume homogeneous distribution of joint density of 0 in input-output space,

A A Q

i.e. /(0 |(xi, yi) = /(0.) and assign x*b = -*- xi.

0ib

3. Calculate 0ib for (x*b, yi).

A 1 B A* A A A A

3. bias0 = — YPib - 0 and bias-corrected score S. = S{ - biasS..

B b=1

Rescaling at step (2.2) guarantees that pseudo-samples {(x*b,yi)}B=1 e L(y). Indeed, input-oriented efficiency evaluates the potential of DMUi for maximal reduction of inputs, holding the amount of outputs constant. The constraints xi > XA imply inputs are larger than possible. Therefore, multiplication of each input by 0, 0 < 0 < 1, projects it to L 5(y), so that the projected observation become an estimate of an efficient input level with coordinates ( 0x., y.). The assumption about homogeneous distribution of joint density of 0 allows drawing each 0*b for pseudo-samples from the same estimate of /( 0), which is obtained for the original sample. Therefore, division of each projected input by 0ib ,0 < 0ib < 1 in step (2.2) "pushes" the projected input inside L (y).

In presence of an r-dimensional vector of environmental variables z , i.e. a special type of inputs that are not directly controlled by producers, [Simar, Wilson, 2007] propose a semi-parametric bootstrap for correcting the bias of the distance function score 8, the reciprocal of 6 3. The algorithm, in the case of input-orientation, is based on the premise about the separability of inputs and environmental variables, i.e. the fact that the support of x does not depend on z [Simar, Wilson, 2011b].

1. Estimate the naive distance function scores Sp...,Sj, for each i = 1,...,J using the

equivalent of system (2) for reciprocals of 0 . Assume, Si = zi a + si > 1 where si

are i. i. d. and independent from zi, e{ ~ N(0,ct2) with left truncation at (1 - zi a).

2. Use observations for which S > 1 to obtain a and &s in the truncated regression

4 = zi a+ei > 1 .

3. Loop B times to obtain J sets of bootstrap estimates {S*b ^ .

1. For each i = 1,..., J draw from N(0,cr2) with left truncation at (1 - zi a).

2. For each i = 1,..., J compute(S* = z.a + si).

3. Assign xb =S xi.

A* Si *

4. Calculate Sib for (x*b, yi).

A 1 B A* A A A _ A

4. bias<5. = — YSib - Si and bias-corrected score S = S{ - biasS..

B b=1

2 Smoothing is necessary to avoid inconsistency in estimating the upper bound of the support of the underlying data-generating process /(•) [Simar, Wilson, 1998].

3 6, which is bounded between 0 and 1, could not be used to estimatethe truncated regression [Simar,

Wilson, 2008].

2. Estimates of cost efficiency

2.1. Naive score with given input prices

Denote wj the vector of input prices. Fare, Grosskopf, and Lovell (1985) define cost efficiency y as

f = wjx j / wjxj, (2)

where x"pt is a solution to the optimization problem (formulated below for constant returns to scale):

min W , X ,

xj j j

s.t.-ymj Jjmt >0, m = 1,..., M, (3)

i=1

J

Xnj -Z2iXni > 0 n = N,

i=1

>0, i = 1,., J.

According to (2) and system (3), 0 < fj <1 by construction. Note that (3) assumes that producers face input prices as given.

2.2. Proposed bootstrap algorithm forcost efficiency

Similarly to input-oriented efficiency scores, [Fare, Grosskopf, Lovell, 1985] cost efficiency scores are linked to L 5(y) and therefore are upwards-biased. Yet, a direct modification of the [Simar, Wilson, 1998; 2007] algorithm to bias correction of cost efficiency score y, which simply replaces 0 by y at steps 2.2 or step 3.3 (e.g. as proposed in [Borger, Kerstens, Staat, 2008], is inconsistent. Indeed, let us look at a given observation i with coordinates xi (point P at Figure 1). By definition of input-oriented efficiency, point P , which is an intersection of the ray from the origin to P and L 5(y), has coordinates Oxi. The hyperplane, set by the cost function wixi and tangent to L 5(y), intersects the ray from the origin to point P at point P'. Since points P and P' are on the same hyperplane, the costs in these points are equal. Therefore, by definition of cost efficiency score, point P' has coordinates fxi. Consequently, point P", obtained through rescaling inputs by f /f*b, belongs to [P',P]. However, it may happen that P"e[P, P], i.e. P"e[P',P"]. So the vector of bootstrapped inputs, obtained at step 2.2 of a direct modification of the Simar and Wilson (1998) algorithm, may be outside L(y). (The same argument applies to step (3.3) for the case with environmental variables, where 9 = 0(i{) and y = y(zi). Note

that the assumptions about strict convexity of L(y) and free disposability of inputs are importantly exploited in our argument.

To correct for the bias of the [Fare, Grosskopf, Lovell, 1985] cost efficiency we propose the following bootstrap, based on the premises that data generating process for the observed firms satisfy the [Kneip, Simar, Wilson, 2008; Simar, Wilson, 2000b] assumptions 4-6, which we mentioned in the Introduction and which allow a consistent approximation of the unknown distribution of efficiency scores. The bootstrap is homogeneous both in terms of (•) and f, (•) and constructs pseudo-samples through re-sampling the input-oriented

technical efficiency score and rescaling original inputs by the ratio fy / dib. In this way, the

bootstrapped inputs are "pushed" inside the L (y). Therefore, ftb, which calculated for the

bootstrapped inputs at step (4) of our algorithm, allow for consistent bias correction.

1. Estimate naive cost efficiency scores ft,-., ft for each i = 1,...,J . Assume (y^..., J are i.i.d. with pdffy(-) .

2. Estimate naive input-oriented efficiency scores #P...,6J . Assume (6v...,6j) are i.i.d. with pdffe(-) .

3. Obtain 0*b through smoothed bootstrap, and under the assumptions of homogeneous distribution of joint density of 0 and joint density of y in input-output space,

assign =4X,b = 1,...,B .

fyb

4. Calculate ftb for (x\b, y).

A A 1 b a* a

5. For each i, bias ft = ~Y,Tib ~ Ti

Bb=1

Next, we proceed to the model with environmental variables. We consider a vector zi of environmental variables for the input-oriented model and the vector zf of environmental variables for the cost minimization model. The corresponding coefficients in the truncated regression for each model are denoted, respectively, as ßY. Note that {zi} c {zf}.

Indeed, as one of the reasons for the bias of cost efficiency scores is the bias of input-oriented scores (owing to the empirical estimate of the frontier), the list of predictors for SY includes the list of predictors for S.

We denote Sf as the reciprocal of the efficiency score in the cost minimization model in order to distinguish it from Si , which is the reciprocal of the efficiency score in the input-oriented model. Under the [Simar, Wilson, 2007] assumption about separability of x and z (i.e. the fact that Ld(y) does not depend on z ), we propose the following algorithm for the reciprocal of the [Fare, Grosskopf, Lovell, 1985] cost efficiency score Sf.

1. Estimate reciprocals of naive cost efficiency scores Sf,..., Sf, for each i = 1,.. ,,J using system (3).Assume Sf = zf af +Wi > 1, where W are i.i.d. and independent from zf, Wi ~ N(0,oW) with left truncation at (1 - zf a7).

2. Estimate naive input-oriented distance function scores S1, ...,Sj, for each i = 1,...J, using the equivalent of system (2) for reciprocals of 0 . Assume, f = zi a + ei > 1,

where ei are i.i.d. and independent from zi, ei ~ N(0,o^) with left truncation at

(i - z, a).

3. Use observations for which S > 1 to obtain a and 6e in the truncated regression

S = z, a + ei > 1.

4. Loop B times to obtain J sets of bootstrap estimates [S*b}, b = 1,...,B. 1. For each i = 1,..., J draw ei from N(0,^) with left truncation at (1 - z,a).

2. For each i = 1,..., J compute (S* = z,a + e{). *

s * Sb

3. Given the semi-parametric dependence of f on z, assign xib x{.

A * * f.

4. Calculate ffb for (xib, y{). '

5. Owing to semi-parametric dependence of f on z, we can compute

biasff = — Ysffb -Sf and bias-corrected score Sf =ff - biasff

B b=1

2.3. Naive cost efficiency score with input prices under producer control

Tone (2002) concentrates on input costs, assuming that producers may choose prices for their inputs. Let x. = (w1 ■x1 j,...,wN-xN-)T , X = (x1,...,Xj)T , where wj is a vector of

prices for each input xj. The Tone (2002) "new" cost efficiency for DMU j is defined as

fj = ex °Jpt / ex j

with Xopt a solution to (constant returns to scale formulation):

min ex:

Xj ,A. j

st.-ymj +hiyml >0, m = M, (5)

i=1

-htxrn > n = U.N,

i=1

¿i >0, i = 1,...,J.

Here e is a unit vector, and by construction in (4) and (5), 0 < ft < 1.

2.4. Proposed bootstrap algorithm /or the new cost efficiency

Denote Tn technology in [Tone, 2002] of "new" technical (and cost) efficiency estimates.

Tn =|(3c, y):x>Xl j<:EU>oJ (6)

Define the "new" input set Ln (y) = {(x):(x, y) eTn}. As is demonstrated in [Tone, 2002] (theorem 4), the set of constraints on each xnj in (5) is equivalent to this aggregate constraint:

ex-eja>0. (7)

Consequently, for a given level of y, the LLn (y) is a hyperplane, parallel to the hyperplane set by a given level of the objective function ex.. Therefore, the tangency of the objective function and Lsn (y) implies that the two hyperplanes are coincident (Figure 2). Accordingly, the ray from origin to the point P e Ln (y) intersects LLn (y) and the hyperplane, set by the objective function, at the same point. So P = P . In other words, as is noted in theorem 6 in Tone (2002), the "new" cost efficiency point is also "new" technically efficient.4 So a consistent bias correction of the Tone (2002) "new" cost efficiency score may be conducted through a direct application of the Simar and Wilson (1998) and

Fig. 2. Bias correction of the newcost efficiency, isoquant in two-input space

4 Therefore, papers that estimate input-oriented efficiency scores using input costs as inputs and interpret the scores as cost efficiency (e. g. [Medin et al., 2011; Linna et al., 2010; Barros, Dieke, 2008] in fact, measure the [Tone, 2002]) "new" cost efficiency.

the Simar and Wilson (2007) algorithm, so that the following rescaling is implemented at

—* 7' — d

step (3.3): xib = —*-x{. Indeed, as Ln(y) is set by the aggregate constraint (7), P"e[P',P]

7ib _

is equivalent to P"e [P, P]. Therefore, rescaling guarantees that each component of xb is larger than the corresponding component of the original vector x , and vector xb lies in the necessary subspace relative to Ln (y) [Besstremyannaya, 2013].

3. Simulations

3.1. Microeconomic framework

The Cobb-Douglas production function, commonly used in the non-parametric efficiency analysis in the banking industry [Kneip, Simar, Wilson; 2008, 2011; Fethi, Pa-siouras, 2010; Thanassoulis, Portela, Despic 2008; Badin, Simar,2003; Simar, Wilson, 2002; 2000b; Kittelsen,1999; Banker, Gadh, Gorr, 1993] is taken in the form of [Kumbhakar, 2011; Resti, 2000]

(8)

M

where xnm is the quantity of n -th input, used to produce m -th output (xn = ^xnm), Am

m=l

and anm are the parameters. Outputs ym and input prices wn are assumed to come from multivariate lognormal distribution, where vector of means and variance-covariance matrix are taken from our real banking data (in particular, are based on the asset approach in defining the input and output pairs)5. The minimal dimension of the input vector, required for differentiating between technical and cost efficiency, is two. Yet, banking is commonly considered to be a multi-output industry, and so we exploit two-output and three-input models:

In (y y

((

w

7.3498 6.2898

A (

1.2686 1.4686 1.4680 1.8260

and ln (w) -

//

-4.9157 -2.0093

-5.5727

W j v

0.0309 0.0368 0.0231 0.0368 0.2079 0.0702 0.0231 0.0702 0.1193

yj

In the absence of environmental variables, we employ the [Resti, 2000] approach of introducing cost inefficiencies to (N - 1) inputs and analytically computing the value of the N-th input, so that the firm remained on the same isoquant (with unchanged level of input-oriented efficiency): xnm = x*nmqnm, where n = 1,...,N -1; 7jnm > 0. Then, cost efficiency y is calculated as follows (Appendix, eq. A. 13):

5 To check for robustness, we conducted a second set of simulations with data from the intermediation approach, and we found similar results.

wx

opt

2n=lSm=l(ym P" / Am ~)O<CCnmTm

y N-1y M ( *llpm / A )C T r + y M (»

^_jn=1 Zj m=1^s m m' nm m Inm m

/ A )C T rrN-1n"a™/C»

' ^m '^Nm1 mL In=1 'nm

(9)

where pm = y a and Tm = n

»T / \anm /Pm

N I W 1 "" m

n=1

n=1

\ Cnm J

Our estimations with Japanese data and results from the empirical literature [Liu, Ondrich, Ruggiero, 2012; Wang, 2003; Banker, Gadh, Gorr, 1993; Giokas, 1991] show that input elasticities do not vary appreciably for banking outputs employed in this paper. Hence, we regard anm = an (and hence pm = p) and ijnm = ijn. Accordingly, we obtain (eq.A.15):

7 =

pO

viN-1 ttN-1 -a,/aN

Zn=1Cnln +CN n n=1^n n

(10)

We use constant returns to scale with (a1> a2, a3) = (0.05, 0.05, 0.9). Inefficiencies are added, so that y = y*OP, 0 <OP < 1 (see [Kneip, Simar, Wilson 2011; Badin, Simar

2003; Simar, Wilson, 2002; 2000b; Resti, 2000; Kittelsen, 1999]). Then, owing to homo-thetic property of cost function, the input-oriented efficiency is 6. Cost inefficiencies are added to x2 and x3 and analytically computed for x1 . Input-oriented efficiency is O = 1/(1+ <0 , where < is drawn from Exp(2) and E(<) = 0.5 . Note that 1 + Exp(Z) has high probability of obtaining a point in the neighborhood of unity. Consequently, the DGP with exponential distribution allows easier estimation of the frontier if compared to DGPs with fewer points in the proximity of unity6. In the presence of environmental variables, we inlroduce inefficiencies as y = y*S~P, o < S p < l, where S can be expressed as zß + s.

We assume a simplified case when the lists of environmental variables, influencing input-oriented efficiency and cost efficiency, coincide. S -N,a2z) with left truncation at unity. Following Simar and Wilson (2007), we set r = 2, ß1 = ß2 = 0.5, z1 = 1, z2 - N(2,4), £-N(0,1), with left-truncation at (1 -z), S = za + £. Then eq. (10) modifies to

S IN-1 N-1 , A

S7(z) =S| yanVn +Cn nriaCn)|.

p \ n=1 n=1 J

As regards cost efficiency, rn = ev-, where vn - N(0, cC). In this case the realized value of may be smaller or larger than unity, and it allows to move x* in different directions along the isoquant. To model different size of cost inefficiencies, we take c2 = {0.05,0.1}. Following [Simar, Wilson, 2011b], we use 1000 trials with B=2000 iterations on each trial. Our samples J = {50,100,200,300,400,600,800,1000} and confidence levels (1 -a) are 0.9, 0.95 and 0.99.

6 If data-generating process results in a small number of points in the proximity of unity, the consistent

estimation of the frontier would require increasing sample size appreciably.

A fixed point to measure cost efficiency on each trial is constructed as follows. We take a vector in the middle of the output and price data and assign it input-oriented efficiency Ed. The coordinates of a point on the frontier are* ^[E0]P i y, i wj, [Ed] i yj,

where x (v) is a an optimal demand function from eq.(A.2). Then, we introduce inefficiencies Ey to (N -1) input coordinates of the point, and analytically compute the values of N-th input coordinate according to eq.(A.5).

3.2. Results

Owing to potential problems of ignoring the zero bound in implementing the [Silverman, 1986] reflection method with the input-oriented efficiency scores 0 [Simar, Wilson, 2000a], the estimations are conducted in terms of the reciprocals S = 1/0 . Accordingly: first, each point S > 1 is reflected by its symmetric image 2 -Si < 1; second, kernel density is estimated from the set of 2J points [Simar , Wilson, 2008]. Our estimates demonstrate that the absolute difference between the true and bias-corrected values of cost efficiency both in absence and presence of environmental variables is close to 0.04 with the smallest sample size (J = 50) and becomes less than 0.01 with J > 600 (Tables 1-2). As the values of efficiency scores belong to the [0, 1] segment, it may be concluded that our bias correction on average leads to the estimates with negligible difference from the true value under moderate sample sizes and reasonable numbers of bootstrap iterations.

Table 1. Absolute difference between the true and estimated cost efficiency for homogeneous smooth bootstrap in the absence of environmental variables, with sample adjusted cross-validation bandwidth

J Ov a = 1 a = 0.05 a = 1.0

50 0.05 0.036 [0.012] 0.038 [0.012] 0.037 [0.012]

100 0.05 0.026 [0.008] 0.026 [0.008] 0.026 [0.008]

200 0.05 0.018 [0.005] 0.018 [0.005] 0.018 [0.005]

300 0.05 0.014 [0.004] 0.014 [0.004] 0.014 [0.004]

400 0.05 0.012 [0.003] 0.012 [0.003] 0.012 [0.003]

600 0.05 0.01 [0.002] 0.01 [0.002] 0.01 [0.002]

800 0.05 0.008 [0.002] 0.009 [0.002] 0.008 [0.002]

1000 0.05 0.007 [0.002] 0.007 [0.002] 0.008 [0.002]

50 0.1 0.037 [0.012] 0.037 [0.012] 0.037 [0.012]

100 0.1 0.025 [0.008] 0.026 [0.008] 0.025 [0.008]

200 0.1 0.018 [0.005] 0.018 [0.005] 0.018 [0.005]

300 0.1 0.014 [0.004] 0.014 [0.004] 0.014 [0.004]

Окончание табл. 1

J Ov а = 1 а = 0.05 а = 1.0

400 0.1 0.012 [0.003] 0.012 [0.003] 0.012 [0.003]

600 0.1 0.01 [0.002] 0.01 [0.002] 0.01 [0.002]

800 0.1 0.008 [0.002] 0.008 [0.002] 0.008 [0.002]

1.000 0.1 0.007 [0.002] 0.008 [0.002] 0.007 [0.002]

Note: standard deviation in brackets.

Table 2. Absolute difference between the true and estimated cost efficiency for semi-parametric bootstrap in the presence of environmental variables

J Ov а = 1 а = 0.05 а = 1.0

50 0.05 0.043 [0.017] 0.043 [0.017] 0.042 [0.016]

100 0.05 0.029 [0.010] 0.03 [0.010] 0.029 [0.010]

200 0.05 0.02 [0.006] 0.02 [0.005] 0.02 [0.005]

300 0.05 0.016 [0.004] 0.016 [0.004] 0.016 [0.004]

400 0.05 0.014 [0.003] 0.014 [0.003] 0.014 [0.003]

600 0.05 0.011 [0.002] 0.011 [0.002] 0.011 [0.002]

800 0.05 0.01 [0.002] 0.01 [0.002] 0.01 [0.002]

1.000 0.05 0.009 [0.002] 0.009 [0.002] 0.009 [0.002]

50 0.1 0.043 [0.018] 0.042 [0.017] 0.042 [0.016]

100 0.1 0.029 [0.010] 0.03 [0.010] 0.03 [0.010]

200 0.1 0.021 [0.006] 0.021 [0.006] 0.021 [0.006]

300 0.1 0.017 [0.004] 0.017 [0.004] 0.017 [0.004]

400 0.1 0.014 [0.004] 0.014 [0.003] 0.014 [0.003]

600 0.1 0.012 [0.003] 0.012 [0.003] 0.012 [0.003]

800 0.1 0.01 [0.002] 0.01 [0.002] 0.01 [0.002]

1.000 0.1 0.009 [0.002] 0.009 [0.002] 0.009 [0.002]

Notes: standard deviation in brackets; dim(x) = dim(y) = 2.

4. Efficiency estimates for Japanese banks

4.1. Data

We use data from the Japanese Bankers Association, which provides financial variables for all Japanese banks from their consolidated financial statements and statements of cash flow, along with the number of employees, bank branches, and bank charters from

interim financial statements. Regional (prefectural) variables come from the Bank of Japan (deposits, vault cash, loans, and bills discounted), Economic and Social Research Institute, Cabinet Office (gross domestic product and gross domestic product deflator), Ministry of Land, Infrastructure and Transport, and Japan Statistical Yearbook (price of commercial land site).

Following common approaches to cost efficiency analyses in banking, we exploit a three input-two output model, where outputs are either performing loans and total securities (asset approach, e. g. [Hori, Yoshida, 1996; Fukuyama, Weber, 2002; Barros, Man-agi, Matousek, 2012) or revenue from loans and revenue from other business activities (intermediation approach, e. g. [Kasuya, 1986; Fukuyama,1993; 1995; Takahashi, 2000; Fukuyama, Weber, 2010; Thanassoulis, Portela, Despic, 2008; Tortosa-Austina, 2002]). In each model the inputs are labor (total employees), capital (premises, real estate, and intangibles) and funds from customers (we follow [Kasuya, 1986; 1989; Fukuyama, 1993; 1995; Hori, Yoshida, 1996; McKillop, Glass, Morikawa, 1996; Glass, McKillop, Morikawa, 1998; Fukuyama, Weber, 2002; Miyakoshi, Tsukuda, 2004; Fukuyama, Weber, 2008; Barros, Managi, Matousek, 2012]). The proxies for input prices are, respectively, personnel expenditure/total employees, capital expenditure/capital, and fund-raising expenditure/ funds from customers (e. g. as in [Kasuya, 1986; 1989; McKillop, Glass, Morikawa, 1996; Fukuyama, Weber, 2002]). The choice of inputs, outputs, and prices follows the methodology of efficiency analysis in Japanese banking7. Bank-level environmental variables include bank size and bank product diversity (see [Aly et al., 1990; Simar, Wilson, 2007]), ratio of loan loss provisions to total loans (following [Altunbas et al., 2000; Drake, Hall, 2003; Drake, Hall, Simper, 2009])8. Prefecture-level environmental variables are share of monetary aggregate in gross regional product, real rate of growth of gross domestic product, and commercial land price [Liu, Tone, 2008]. We include dichotomous variables by bank charter (city bank, regional bank, regional second tier bank, trust bank, long-term credit bank). Bank holdings and financial groups are excluded from the analysis, as they may have zero reported capital (Table 3).

Our sample uses data for the fiscal year which runs from April 2013 to March 2014. The sample represents the entire banking industry in Japan, yet its size is only 106. However, the results of our simulations demonstrate relatively small differences between the true and the estimated cost efficiency, even for such small samples.

Table 3. Descriptive statistics for the sample of 106 Japanese banks in fiscal year 2013

Variable Definition Mean Std. Dev. Min Max

Inputs

x_1 labor = total employees (including board) 2714 4481 312 31461

x_2 capital = premises and real estate + intangibles 26 123 0.044 1125

7 Note that the intermediation approach prevails in international literature [Fethi, Pasiouras, 2010], yet the asset approach is more widespread in analyses of Japanese banking. See review of the literature on measuring the efficiency of Japanese banks in [Besstremyannaya, 2017].

8 Using the non-performing loans in an alternative approach does not change the results of the estimates appreciably, since loan loss provisions and non-performing loans are highly correlated.

Variable Definition Mean Std. Dev. Min Max

x_3 funds from customers = total deposits + negotiable certificates of deposits + call money + bills sold + borrowed money + foreign exchange deposits + other deposits 7366 21589 219 157288

Outputs

Asset approach

y_1 performing loans = total loans — nonper-forming loans 4614 12566 158 89543

y_2 securities and other interest bearing assets 2575 8434 1.4 66543

Intermediation approach

y_3 revenue from loans = interest on loans and discounts + interest on bills bought 67 181 3.6 1326

y_4 revenue from other business activity = total operating income — other operating income — interest and dividends on securities — y_3 63 214 0.706 1428

Input prices

w_1 labor price = (general and administrative ex-penses-depreciation)/total employees 0.017 0.007 0.009 0.065

w_2 capital price = (expenditure on premises and fixed assets)/x_2 1.126 1.399 0.136 10.442

w_3 price of funds = fund raising expenditure/x_3 0.001 0.001 0 0.005

Bank variables

z_1 = ln(branches) 4.56 0.62 3.05 6.72

z_2 Herfindahl index of product diversity 0.72 0.23 0.26 1.44

z_3 nonperforming loan ratio = nonperforming loans /total loans 0.03 0.01 0.004 0.07

z_4 = 1 if city bank 0.05 0.21 0 1

z_5 = 1 if regional bank 0.56 0.5 0 1

z_6 = 1 if regional tier 2 (former Sogo) bank 0.35 0.48 0 1

z_7 = 1 if trust bank 0.03 0.17 0 1

z_8 = 1 if longterm credit bank 0.02 0.14 0 1

Окончание табл. 3

Variable Definition Mean Std. Dev. Min Max

Prefectural

variables

z_9 rate of growth of gross regional product (in 1 0.05 0.93 1.18

2010 real terms)

z_10 share of monetary aggregate (M2 + negotiable 0.7 0.32 0.4 1.52

certificates of deposit) in regional product

z_11 share of loans in gross regional product 0.75 0.46 0.36 1.95

z 12 rate of growth of price of commercial land 0.58 0.14 0.41 0.92

(in 2010 real terms)

Note: financial variables are in billion yen. The sample size is 106.

4.2. Results

Estimations are conducted under variable returns to scale with B=2000. We exploit least squares cross-validation bandwidth and use z1 - z 3, z 9 - z12 and a dummy for city banks (z 4) in the model with the environmental variables. (The remaining dichotomous variables for other bank charters are omitted, owing to multicollinearity).

Table 4. Cost efficiency scores for Japanese banks

Score Asset approach Intermediation approach

f mean 0.7189 0.6679

st.dev. 0.1406 0.166

range [0.4496, 1] [0.3781, 1]

f mean 0.6622 0.5906

st.dev. 0.13 0.137

range [0.4091, 0.9503] [0.3338, 0.9011]

1/ 8f mean 0.7276 0.6649

st.dev. 0.1301 0.1544

range [0.5130, 1] [0.4220, 1]

l/jf mean 0.6737 0.615

st.dev. 0.1187 0.1205

range [0.4832, 0.9732] [0.3739, 0.8928]

Table 4 shows the estimates of "naïve" score y (1/ S7 ) and bias-corrected score y

( 1/ S7 ) for the models, corresponding to the asset approach and the intermediation approach. In each model, the mean bias-corrected score is lower than the mean "naïve" score, while the standard deviation of the "naïve" and bias-corrected scores are close. The bias-corrected score is "to the left" (if compared to the range of the "naïve" score), and there are no exact unity values of bias-corrected cost efficiency. The mean value of cost efficiency is higher in the model, with the asset approach both in presence and in absence of environmental variables. Accounting for environmental variables leads to higher cost efficiency scores, if compared to corresponding models without environmental variables.

Conclusion

The paper shows that a direct modification of the [Simar, Wilson, 1998; 2007] methodology is inconsistent for correcting the bias of the [Fare, Grosskopf, Lovell, 1985] cost efficiency scores and proposes an alternative bootstrap algorithm for estimation. To approximate the bias of the "naïve" cost efficiency score, the proposed algorithm re-samples "naïve" input-oriented efficiency scores, rescales original inputs to bring them to the frontier, and then re-estimates cost efficiency scores for the rescaled inputs.

The results of simulation analyses for a multi-input, multi-output Cobb-Douglas production function with correlated outputs, and correlated technical and cost efficiency, show consistency of the proposed algorithm. An application of the algorithm to real data of 106 Japanese banks for fiscal year 2013 allows quantifying the bias of the naïve cost efficiency scores.

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Received: September 26, 2018 Accepted: December 17, 2018

Author's information:

Galina E. Besstremyannaya — PhD in Economics; gbesstre@nes.ru Jaak Simm — PhD in Computer Science; jaak.simm@kuleuven.be

Оценка ценовой эффективности в непараметрических моделях производственной границы

Г. Е. Бесстремянная1, Я. Симм2

1 Российская экономическая школа и Институт Дальнего Востока РАН, Российская Федерация, 121353, Москва, Сколковское шоссе, 45

2 Stadius Centre for Dynamical Systems, B-3001, Хеверли, Бельгия

Для цитирования: Besstremyannaya G., Simm J. (2019) Estimation of Cost Efficiency in Nonparametric Frontier Models. Вестник Санкт-Петербургского университета. Экономика. Т. 35. Вып. 1. С. 3-25. https://doi.org/10.21638/spbu05.2019.101 (In English)

В работе предлагается методология для несмещенной оценки ценовой эффективности в непараметрических моделях производственный границы (data envelopment analysis).