Sensitivity analysis of efficiency and Malmquist productivity indices: An application to Spanish savings banks

European Journal of Operational Research 184 (2008) 1062–1084 www.elsevier.com/locate/ejor O.R. Applications Sensitivity analysis of eﬃciency and Ma...

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European Journal of Operational Research 184 (2008) 1062–1084 www.elsevier.com/locate/ejor

O.R. Applications

Sensitivity analysis of eﬃciency and Malmquist productivity indices: An application to Spanish savings banks Emili Tortosa-Ausina

a,*

, Emili Grifell-Tatje´ b, Carmen Armero c, David Conesa

c

a

c

Departament d’Economia, Universitat Jaume I and IVIE, Campus del Riu Sec, 12071 Castello´ de la Plana, Spain b Departament d’Economia de l’Empresa, Universitat Auto`noma de Barcelona, Ediﬁci B, Campus de la UAB, 08193 Bellaterra, Barcelona, Spain Departament d’Estadı´stica i Investigacio´ Operativa, Universitat de Vale`ncia, C/Dr.Moliner 50, 46100 Burjassot, Vale`ncia, Spain Received 28 September 2004; accepted 3 November 2006 Available online 19 December 2006

Abstract Hypothesis testing and statistical precision in the context of non-parametric eﬃciency and productivity measurement have been investigated since the early 1990s. Recent contributions focus on this matter through the use of resampling methods—i.e., bootstrapping techniques. However, empirical evidence is still practically non-existent. This gap is more noticeable in the case of banking eﬃciency studies, where the literature is immense. In this work, we explore productivity growth and productive eﬃciency for Spanish savings banks over the (initial) post-deregulation period 1992–1998 using Data Envelopment Analysis (DEA) and bootstrapping techniques. Results show that productivity growth has occurred, mainly due to improvement in production possibilities, and that mean eﬃciency has remained fairly constant over time. The bootstrap analysis yields further evidence, as for many ﬁrms productivity growth, or decline, is not statistically signiﬁcant. With regard to eﬃciency measurement, the bootstrap reveals that the disparities in the original eﬃciency scores of some ﬁrms are lessened to a great extent. Ó 2006 Elsevier B.V. All rights reserved. Keywords: Banking; Bootstrap; Data envelopment analysis; Eﬃciency; Productivity

1. Introduction Over the last two decades, the Spanish banking industry, like many other Western European banking systems, has been through fascinating changes, such as deregulation, liberalization, and technologi-

cal advances, which have profoundly reshaped the industry.1 They have had a major eﬀect on both types of ﬁrms dominating the industry, namely, private commercial banks and savings banks. The latter has been the most successful in gaining market share. It was also subject to the tightest regulation, 1

*

Corresponding author. Tel.: +34 964387163; fax: +34 964728591. E-mail address: [email protected] (E. Tortosa-Ausina).

A number of contributions have dealt thoroughly with this issue. Some of the earliest were those by Vives (1990, 1991). A more up-to-date contribution is provided by Danthine et al. (1999).

0377-2217/$ - see front matter Ó 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2006.11.035

E. Tortosa-Ausina et al. / European Journal of Operational Research 184 (2008) 1062–1084

and hence, deregulation initiatives have aﬀected this group more notably. For these reasons, in particular, they warrant our attention. Given the importance of the banking sector for the whole economy in general, and for the ﬁnancial system in particular, a number of studies have already set out to analyze how the eﬃciency of banking ﬁrms has evolved since the early 1980s.2 Results are mixed, as these studies measure diﬀerent types of eﬃciencies, using diﬀerent techniques (parametric/non-parametric), diﬀerent measures and definitions of inputs and outputs, and diﬀerent samples (private commercial banks and/or savings banks, for diﬀerent periods). Yet it seems that ineﬃciencies persist, and have quite a dynamic nature, since changes continue to take place. On the other hand, research studies that focus on the productivity of Spanish banking ﬁrms are thinner on the ground. These include studies by Pastor (1995) and GrifellTatje´ and Lovell (1996), which analyze the productivity of savings banks, although the former extends the analysis to commercial banks. Our study complements these two linked stems of research in several ways. Firstly, although research articles analyzing the eﬃciency of banking ﬁrms continue to be published regularly,3 those dealing with productivity not only are more infrequent, but also conﬁne their analysis to the deregulatory period—before 1991.4 In contrast, our work focuses on the 1992–1998 post-deregulation period, in which the wave of mergers and acquisitions (M&As) had almost come to an end. Hence, from a dynamic point of view, it is a fascinating period to study, as nearly the same number of ﬁrms are available in every period, an important prerequisite when measuring productivity change, since panel data are necessary. Furthermore, our output deﬁnition accounts for business lines in which savings banks might have specialized over the deregulation period. However, our main contributions derive from the suggestions put forward by Berger and Humphrey 2

See, for instance, the pioneering works by Dome´nech (1992), Lozano-Vivas (1992) and Grifell-Tatje´ et al. (1994), some later studies such as Maudos (1996, 1998), Lozano-Vivas (1997, 1998), or more recently those by Kumbhakar et al. (2001), LozanoVivas et al. (2002), Maudos and Pastor (2003), Maudos et al. (2002), or Tortosa-Ausina (2002b). 3 See, for instance, the recent survey by Amel et al. (2004). 4 All important deregulation initiatives took place before 1991. The only remaining change was the establishment of the Single European Market in 1993.

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(1997), who surveyed the enormous increase in the number of studies that analyze the eﬃciency of ﬁnancial institutions at an international level. Although the literature is still growing rapidly, their study covered 130 relevant studies that applied frontier eﬃciency analysis to this type of ﬁrm in 21 countries, using both parametric and non-parametric techniques to measure eﬃciency. Their attempts were manifold, and included directions for further research. Among these was suggested the issue of deriving conﬁdence intervals for eﬃciency measures in order to compare them statistically. Speciﬁcally, they suggested that new research in the area of eﬃciency should try to provide conﬁdence intervals for the eﬃciency estimates they generate. This question is closely related to the long-standing debate on the best technique for eﬃciency measurement. Both parametric and non-parametric techniques have been widely used, and the consensus is that neither technique is better than the other5, because of the trade-oﬀ that aﬀects both of them. Speciﬁcally, parametric techniques have the advantage that they allow for random error but the drawback that they impose a particular functional form that presupposes the shape of the frontier. On the other hand, non-parametric techniques tend to envelope data more closely, but they do not allow for random error. Data Envelopment Analysis (DEA) is one of the most widely used among the latter. This inability to allow for random error has induced many authors to label it as deterministic. However, recent literature has silenced ‘‘this tired refrain’’, as Lovell (2000) deﬁnes it. In particular, Simar and Wilson (1998, 1999a, 2000) deﬁne a statistical model which allows for the determination of the statistical properties of the non-parametric estimators in the multi-input and multi-output case. The important practical implication of their ﬁndings is that statistical inference is possible. Hence, non-parametric methods to measure eﬃciency would still have all their advantages, but would somehow allow for random error. This possibility is opened up by bootstrap (Efron, 1979) a computer-intensive technique essentially based on the basic idea of approximating the unknown statistic’s sampling distribution of interest by extensively resampling from an original sample, and then using this simulated sampling distribution to make population inferences. It is worth 5

In fact, of the 130 studies surveyed by Berger and Humphrey (1997), 69 were non-parametric.

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mentioning that stochastic DEA models (Land et al., 1993; Olesen and Petersen, 1995) also allow random disturbances to be incorporated in the input and output data. We should point out that previous attempts to provide DEA eﬃciency scores with statistical precision have been made, and were excellently surveyed by Grosskopf (1996). These included the approach suggested by Banker (1993, 1996), which made some assumptions on the Data Generating Process (DGP). In contrast, the bootstrap does not need to rely on asymptotic theory in order to estimate the statistic’s sampling distribution of interest. Add to this the current ease of implementation6 due to the availability of modern computing power, and it is not diﬃcult to understand why it is becoming a widely accepted econometric procedure. It is therefore also important to judge the relevance of these methods in econometric terms. In every econometric study, a crucial issue is that of the statistical signiﬁcance of the results achieved. When providing the results of a regression equation, it is often the case that we may well choose to stress the interval estimate over the point estimate. Until the appearance of early contributions on hypothesis testing in DEA which relied on asymptotic theory, or the most recent which rely on bootstrapping techniques, this was not possible when reporting the results generated by non-parametric eﬃciency measurement techniques.7 Now we can use these methods to test for statistical signiﬁcance among diﬀerences in ﬁrms’ eﬃciency scores, in much the same way that t test ratios are used in classical regression studies. Yet there is a lack of empirical applications using bootstrap techniques. One exception is the study by Ferrier and Hirschberg (1997), who devised a formal structure to allow for random error in DEA in an application with Italian banking data, which focused exclusively on eﬃciency. Another is the work of Gilbert and Wilson (1998), which examines the productivity angle for Korean banks, and aims to analyze whether certain a ﬁrm’s productivity growth, or decline, was signiﬁcantly diﬀerent from one. However, regarding the Spanish case, the empirical evidence is yet to appear. 6

For instance standard software packages such as RATS, TSP, LIMDEP, SHAZAM, STATA, S-PLUS, or R. 7 Other earlier attempts to use resampling in the general context of non-parametric eﬃciency measurement include Fa¨re and Whittaker (1995), Fa¨re et al. (1989), Grosskopf and Yaisawarng (1990), or Ferrier et al. (1993).

In line with the reasons provided above, we attempt to analyze the eﬃciency and productivity of Spanish savings banks over the 1992–1998 post-deregulation period, and to provide statistical precision in our results. This type of analysis will therefore enable us to assess both whether ﬁrms’ eﬃciencies diﬀer signiﬁcantly, and whether productivity growth/decline for each particular ﬁrm is statistically signiﬁcant. The article proceeds as follows. Following this introduction, Section 2 brieﬂy examines some of the most important features of Spanish savings banks, and compares them with the remaining groups of banking ﬁrms, i.e., private commercial banks and credit co-operatives. The ensuing Section (Section 3) presents the methodology to measure both eﬃciency and productivity indices, while Section 4 is devoted both to motivate and brieﬂy explain bootstrapping techniques in frontier models. Section 5 provides data information, and Section 6 presents and discusses the results. Finally, Section 7 concludes. 2. The evolution of Spanish savings banks in the pre- and post-deregulation period The Spanish banking system is made up of private commercial banks, savings banks, and credit co-operatives. For regulatory reasons, they have traditionally specialized in diﬀerent lines of business. Today, they face exactly the same operational regulation, which allows them to undertake the same activities. The only regulatory diﬀerences they face arise from their ownership type, as commercial banks are privately owned, savings banks can be considered as ‘‘commercial non-proﬁt organizations’’, and credit co-operatives are mutually owned. This diﬀerence is subtle, as savings banks are allowed to acquire commercial banks, but the opposite does not hold, as the former are a mix of privately- and publicly owned companies. In other words, they have no owners8 and are immune to market corporate control, although mergers and takeovers with other savings banks are allowed (see Crespı´ et al., 2004). In contrast, due to this ownership type, savings banks have substantial difﬁculties in gaining equity. In fact, 50% of their profits has to be dedicated to increasing reserves. 8 Both the general assembly and the board of Spanish savings banks are composed by four groups of stakeholders, namely, public authorities, depositors, employees, and founding societies.

E. Tortosa-Ausina et al. / European Journal of Operational Research 184 (2008) 1062–1084

However, the three types of ﬁrms are still inﬂuenced by their historical specializations, although over the last few years ﬁrms’ product mixes have varied greatly. We focus on savings banks for several reasons. In addition to the speciﬁcities outlined above, they are the group of ﬁrms which reacted most intensely in response to deregulation. As a result, the number of savings banks fell dramatically from 77 in 1985 to 50 in 1998 (our last sample year), due to the ﬂow of mergers. Many of them engaged in geographic expansion, similar to that undertaken by US banks after the Riegle-Neal Act, which permits bank branching on almost a nationwide basis. This issue has naturally led to other changes, as some savings banks expanded geographically to diversify, and to oﬀer diﬀerent products and services from those they had in their home region. Some savings banks also embarked on this expansion by acquiring private commercial banks previously established in the host region. The 1992–1998 period on which we focus was not as turbulent as the 1985–1991 period, when the vast majority of mergers took place. However, it has also

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been characterized by marked changes. Apart from the territorial expansion or the acquisition of private commercial banks, there were also some other important trends such as the implementation of the Single Market Programme in 1993, the enormous growth in mutual funds to the detriment of deposits from 1996 to 2000, internet banking, the euro, etc. Table 1 provides some basic information on the three types of ﬁrms: total assets held, loans, noninterbank deposits, oﬀ-balance sheet activity, equity, employees, and branches, for years 1985, 1992 and 1998. Although our initial sample year is 1992, we also provide information for year 1985, as many deregulatory initiatives took place just after. The ﬁgures clearly show that savings banks are undoubtedly the most dynamic ﬁrms, whose importance in the banking system has been growing impressively to the detriment of commercial banks. Credit co-operatives have also grown, although their importance is far lower when compared to the other two type of ﬁrms. Other recent information on Spanish savings banks is provided by Kumbhakar et al. (2001), or Prior (2003).

Table 1 Relevant information on the structure of the Spanish banking industry 1985

1992

1999

Private commercial banks Savings banks Credit co-operatives

231,030.98 103,145.39 10,720.46

281,086.46 161,650.98 13,584.44

361,500.52 220,200.69 21,949.18

Loansa

Private commercial banks Savings banks Credit co-operatives

94,372.02 37,203.26 4616.42

126,003.85 75,703.18 6804.77

157,635.23 119,677.27 13,162.00

Depositsa

Private commercial banks Savings banks Credit co-operatives

97,696.42 78,556.14 8226.86

93,944.14 114,496.22 10,046.84

88,655.05 150,544.09 15,898.33

Oﬀ-balance sheet activitya

Private commercial banks Savings banks Credit co-operatives

18,984.91 1092.36 640.40

54,708.25 15,255.66 1048.17

110,805.43 39,567.46 5476.92

Equity (capital and reserves)a

Private commercial banks Savings banks Credit co-operatives

18,655.72 8454.09 927.82

31,195.42 15,180.39 1586.40

29,607.65 20,499.58 2617.07

Employees

Private commercial banks Savings banks Credit co-operatives

161,621 71,042 10,823

159,281 82,900 11,016

135,164 93,812 13,292

Branches

Private commercial banks Savings banks Credit co-operatives

16,741 10,802 3350

18,254 14,298 3080

17,544 17,594 3607

Total assets

a

a

In millions of euros (converted from 1990 pesetas).

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3. Measuring the eﬃciency and productivity of savings banks Non-parametric frontier techniques are used to measure both the eﬃciency and productivity of Decision Making Units (henceforth DMU), speciﬁcally of savings banks. We ﬁrstly present the details on the measurement of productivity and secondly examine how to measure eﬃciency. We chose to proceed in this way because the measurement of technical eﬃciency through non-parametric techniques is founded on the estimation of distance functions, analogously to productivity indices. Productivity can be measured by means of either parametric and non-parametric methods. Within both approaches may be found those which ignore eﬃciency, and in contrast, others which account for it. We focus on non-parametric methods which account for ineﬃciency or, in other words, nonparametric frontier approaches. These have been widely used since the early eighties, after the inﬂuential works by Caves et al. (1982), which developed the Malmquist productivity index from the notion of ‘‘proportional scaling’’ introduced by Malmquist (1953). Caves et al. (1982) did not account for inefﬁciency, in contrast to Fa¨re et al. (1992), who combine ideas on measurement of eﬃciency from Farrell (1957) and on measurement of productivity from Caves et al. (1982) to develop a Malmquist index of productivity change. To measure productivity growth we consider two periods, t1 and t2. In period t1 a generic ﬁrm produces output yt1 by using input xt1 , whereas in period t2 quantities are yt2 and xt2 , respectively. The production set St which models the transformation of inputs xt 2 RNþ into outputs yt 2 RM þ at time t is St ¼ fðxt ; yt Þ : xt can produce yt g;

ð1Þ

which holds for t1 and t2. Following Shephard (1970), the output distance function for a generic ﬁrm at t1 is Dt1 ðxt1 ; yt1 Þ ¼ inffh 2 R : ðxt1 ; yt1 =hÞ 2 St1 g ¼ ðsupfh 2 R : ðxt1 ; hyt1 Þ 2 St1 gÞ1 ;

ð2Þ

N þM where ðxt1 ; yt1 Þ is the vector in Rþ made up of the generic ﬁrm inputs and outputs, and the quotient (product) in the vector yt1 =h ðhyt1 Þ is in relation to all its components. The same applies to Dt2 ðxt2 ; yt2 Þ. Eq. (2) refers to the output distance function, and is simply the inverse of the Farrell (1957)

output-oriented measure of technical eﬃciency. Note that Dt ðxt ; yt Þ 6 1 if and only if ðxt ; yt Þ 2 St , which holds for both t1 and t2. Computing the Malmquist productivity index requires two additional distance functions to be deﬁned. One measures the maximum proportional change in outputs required to make ðxt2 ; yt2 Þ feasible in relation to the technology at t1, i.e.: Dt1 ðxt2 ; yt2 Þ ¼ inffh 2 R : ðxt2 ; yt2 =hÞ 2 St1 g:

ð3Þ

The second refers to the maximum proportional change in output required to make ðxt1 ; yt1 Þ feasible in relation to the technology at t2: Dt2 ðxt1 ; yt1 Þ ¼ inffh 2 R : ðxt1 ; yt1 =hÞ 2 St2 g:

ð4Þ

The Malmquist productivity index between periods t1 and t2, t1 < t2 , can be deﬁned as t 1 t 2 t2 1=2 D ðx ; y Þ Dt2 ðxt2 ; yt2 Þ Mðxt2 ; yt2 ; xt1 ; yt1 Þ ¼ ; Dt1 ðxt1 ; yt1 Þ Dt2 ðxt1 ; yt1 Þ ð5Þ which is the geometric mean of the output-based Malmquist productivity indices for t1 and t2 deﬁned by Caves et al. (1982). Alternatively, Fa¨re et al. (1992) decompose (5) into the product of an index measuring changes in technical eﬃciency (‘‘catching up’’) and another that captures the shift in the production frontier (‘‘technical change’’) as follows: Mðxt2 ; yt2 ; xt1 ; yt1 Þ 1=2 Dt2 ðxt2 ; yt2 Þ Dt1 ðxt2 ; yt2 Þ Dt1 ðxt1 ; yt1 Þ : ¼ t1 t t D ðx 1 ; y 1 Þ Dt2 ðxt2 ; yt2 Þ Dt2 ðxt1 ; yt1 Þ |ﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄ} |ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ} Efficiency change

ð6Þ

Technical change

The eﬃciency change component is an index of relative technical eﬃciency change, and shows how much closer (or farther away) a ﬁrm gets to the frontier made up of ‘‘best practice’’ ﬁrms. This component is greater than, equal to, or less than unity depending on whether the evaluated ﬁrm improves, stagnates, or declines. The technical change component measures how much the frontier shifts, and indicates whether the best practice relative to which the evaluated ﬁrm is compared is improving, stagnating, or deteriorating. Whatever the case, the index will take a value greater than, equal to, or less than unity—hence technical change is positive, zero, or negative. Data consist of a known vector of inputs and outputs of L ﬁrms in both periods t1 and t2: Z ¼ fðxti1 ; yti1 ; xti2 ; yti2 Þ; i ¼ 1; . . . ; Lg;

ð7Þ

E. Tortosa-Ausina et al. / European Journal of Operational Research 184 (2008) 1062–1084 0

where xti ¼ ðxti1 ; . . . ; xtin ; . . . ; xtiN Þ 2 RNþ and yti ¼ 0 ðy ti1 ; . . . ; y tim ; . . . ; y tiM Þ 2 RM þ are the input and output vectors corresponding to ﬁrm i, i ¼ 1; . . . ; L in period t, respectively. For both periods t1 and t2, the production set, and consequently all distances deﬁned from it are unknown. Following Fa¨re et al. (1992) the four distances which make up Eq. (6) can be estimated via linear programming techniques. For this, we should consider the following linear programming model for ﬁrm i, i ¼ 1; . . . ; L: b ti1 ðxti1 ; yti1 Þ1 ¼ max h ½D L X 1 ktj1 y tmj ; m ¼ 1 . . . ; M; s:t: hy tim1 6 j¼1 L X

ktj1 xtjn1

ð8Þ

6

xtin1 ;

n ¼ 1; . . . ; N ;

j¼1

kti1 P 0;

i ¼ 1; . . . ; L; 0

where kt1 ¼ ðkt11 ; . . . ; ktL1 Þ is a vector of weights that forms a convex combination of observed ﬁrms relative to which the subject ﬁrm’s eﬃciency is evaluated. Linear programming model (8) calculates the b ti1 ðxti1 ; yti1 Þ. Computing D b ti2 ðxti2 ; yti2 Þ is exdistances D actly the same as (8), where t2 is substituted by t1. Two further linear programming models are needed to estimate the mixed-period cases (3) and (4). The ﬁrst of these is computed for each i ﬁrm as b ti1 ðxti2 ; yti2 Þ ½D s:t: hy tim2

1

¼ max h L X 1 6 ktj1 y tmj ; m ¼ 1 . . . ; M; j¼1

L X

ktj1 xtjn1

ð9Þ

6

xtin2 ;

n ¼ 1; . . . ; N ;

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Grifell-Tatje´ and Lovell (1995b) have shown that in the context of non-constant returns to scale, the Malmquist index does not accurately measure productivity change. In addition, we have tested this hypothesis by using non-parametric tests of returns to scale proposed by Simar and Wilson (2002). In the case of Spanish savings banks it is worth remarking that some applied papers have found results of CRS or very close to them (Grifell-Tatje´ and Lovell, 1995a; Grifell-Tatje´ and Lovell, 1997; Raymond and Garcı´a Greciano, 1994). Furthermore, considering CRS allows a more direct comparison between our results and those by Grifell-Tatje´ and Lovell (1996) and Pastor (1995). An extra appealing feature of CRS is that results are coincidental regardless of whether we solve the linear programming models under the input- or output-oriented approaches. No additional formulation is needed for eﬃciency measurement. The distances involved in its computation have been already presented. Speciﬁcally, to estimate the technical eﬃciency of a given DMU, (saving banks in our case) in a generic period t, we would simply have to solve a linear programming model like the one in expression (8), only by replacing t1 with the period considered. The interpretation is straightforward. Clearly, b tjt b tjt D i 6 1, with D i ¼ 1 indicating that the ith ﬁrm lies on the boundary of the production set of period t and is thus technically eﬃcient. The other ﬁrms with scores below unity will be technically ineﬃcient, producing less output at given input levels. For the sake of simplicity, when referring to eﬃciency scores b tjt and eﬃciency measurement, D will be labeled i ^hi , although it must not be forgotten that all the ^hi correspond to the generic period analyzed t.

j¼1

kti1 P 0;

i ¼ 1; . . . ; L:

As Fa¨re et al. (1994) state, observations involved in (9) are from both period t1 and t2. The reference technology relative to which ðxti2 ; yti2 Þ is evaluated is constructed from observations in t1. To compute the b ti2 ðxti1 ; yti1 Þ, second mixed-period distance function, D the t1 and t2 superscripts in (9) must simply be reversed. Hereafter, and for the sake of simplicity, the distances involved in these four linear programb ti1 jt1 , D b ti2 jt2 , D b ti2 jt1 and ming models will be labelled D t jt b i1 2 , respectively. D In this paper we consider the Malmquist index in terms of constant-returns-to-scale (CRS) distance functions. This is a common assumption since

4. Formulation of the bootstrap As commented above, bootstrap is a computerintensive approach that can help us to provide measures of uncertainty (conﬁdence intervals, standard errors, etc.) for a wide range of problems. It is based on the basic idea of resampling from an original sample of data (either directly, naive bootstrap, or via a ﬁtted model, smoothed bootstrap) in order to create replicate datasets from which we can make inferences on the quantities of interest. For a detailed discussion of these techniques, brought to prominence through the pioneering work of Efron (1979); see Efron and Tibshirani (1993); or Davison and Hinkley (1997).

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In this section, we brieﬂy discuss how to bootstrap both productivity indices and eﬃciency scores.9 For the sake of simplicity, we ﬁrstly show how to bootstrap eﬃciency scores in a generic period t and then productivity indices between periods t1 and t2. 4.1. Bootstrapping eﬃciency scores While the ﬁrst use of bootstrap in frontier models dates back to Simar (1992), its development for non-parametric envelopment estimators was introduced by Simar and Wilson (1998). In their work10, they highlight that the biggest problem when bootstrapping in frontier models is that of consistently mimicking the DGP. Some previous applications of this methodology in this context have failed in this attempt, hence providing inconsistent bootstrap estimation.11 The main reason underlying this problem is that procedures mentioned in the previous Section for distance estimation give values close to unity. Consequently, resampling directly from the original data (naive bootstrap) will provide a poor estimate of the DGP. The most usual possibility is to nonparametrically estimate the original densities of technical eﬃciency scores using kernel smoothing methods. Nevertheless, given that eﬃciency scores are bounded from above the unity, no probability mass should exist beyond that point. Silverman (1986) proposed a procedure—the reﬂection method—to account for this issue. Therefore, all these features must be taken into account when mimicking the DGP and the way to do this is via the smoothed bootstrap. The result of repeatedly resampling from the eﬃciency scores using the smoothed bootstrap is a mimic of the sampling distribution of the original scores from which inference can be made. Basically, this process can be summarized as follows: 1. Compute the eﬃciency scores ^ hi for each DMU i ¼ 1; . . . ; L, by solving the linear programming models commented on in the previous section. 9

Our presentation of this methodology closely follows that of Simar and Wilson (1998, 1999a). 10 See Simar and Wilson (2000) for an excellent review up to 2000 and http://www.stat.ucl.ac.be/ISpub/ for later ones. 11 See the debate in Lovell (2000), Ferrier and Hirschberg (1997, 1999) and Simar and Wilson (1999b,c) for an illustration on this topic, and Simar and Wilson (2000) for a detailed discussion based on Monte Carlo evidence.

2. Use kernel density estimation and the reﬂection method to generate a random sample of size L from f^hi ; i ¼ 1; . . . ; Lg, providing fh1b ; . . . ; hLb g.12 3. Compute a pseudo data set fðxib ; yi Þ; i ¼ 1; . . . ; Lg to form the reference bootstrap technology. 4. For this pseudo data, compute the bootstrap estimate of eﬃciency ^hib of ^hi for each i ¼ 1; . . . ; L, by solving the bootstrap counterpart of the linear programming model commented on in the previous Section. 5. Repeat steps 2–4 a large number B of times in order to provide a set of estimates f^hib ; b ¼ 1; . . . ; Bg.13 The important point after computing these bootstrap estimates is that we are now able to make inferences on the eﬃciency scores. In particular, constructing conﬁdence intervals at desired levels of signiﬁcance is not diﬃcult to deal with. Although there are other possibilities,14 we work with that proposed by Simar and Wilson (1999a) which, in their words, ‘‘automatically corrects for bias without explicit use of a noisy bias estimator’’. This procedure is explained in detail in Simar and Wilson (2000) and basically consists of sorting the values ð^hib ^hi Þ for b ¼ 1; . . . ; B in increasing order and deleting ða2 100Þ-percent of the elements at either end of the sorted list. After setting ^ba and ^aa equal to the endpoints of the sorted array, the estimated ð1 aÞ-percent conﬁdence interval for the eﬃciency score of each DMU is then: ^hi þ ^a 6 hi 6 ^hi þ ^b : a a

ð10Þ

With this kind of information it is easy to ascertain not only the performance of each savings bank but whether there is any empirical evidence to conclude in favor of the null hypothesis that establishes that two savings banks are equally eﬃcient. Moreover, as the use of smoothed bootstrap guarantees that the bootstrap distribution will mimic the original sampling distribution of the estimators of the scores, obtaining bootstrap bias corrections for the scores is also easily dealt with. In particular, the bias of each estimation ^hi can be estimated using the bootstrap sample as 12

See Simar and Wilson (1998) for more technical details. In order to ensure adequate coverage of the conﬁdence intervals, Simar and Wilson (2000) recommend a value of B = 2000. 14 See for instance Efron and Tibshirani (1993), or Davison and Hinkley (1997). 13

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d i ð^ bias hi Þ ¼ hi ^ hi ; ð11Þ PB ^ 1 where hi ¼ B b¼1 hib . Then, from this bootstrap estimation the bias-corrected estimator for each eﬃciency score hi can be obtained as ~i ¼ 2h^i h : h ð12Þ i

Nevertheless, following comments by Simar and Wilson (2000), and Efron and Tibshirani (1993), this correction should not be used unless 1 h d ^ i2 ^2 < bias ð13Þ r i ð hi Þ ; 3 ^2 represents the sample variance of the bootwhere r strap values. These bias-corrected estimators are additional to our main objective of explaining the eﬃciency of the savings banks. They could either conﬁrm what the original scores revealed or express a diﬀerent eﬃciency behavior. 4.2. Bootstrapping Malmquist productivity indices The procedure for bootstrapping productivity indices is based on the fact that the Malmquist index is a function of distance estimators. The above presented methodology for eﬃciency scores can be easily adapted to this case, except that now the time-dependence structure of the data must be taken into account. The process can be summarized as follows: 1. Compute the Malmquist productivity index b ðxti2 ; yti2 ; xti1 ; yti1 Þ for each DMU i ¼ 1; . . . ; L, M i by solving the linear programming models (8) and (9), together with their reversals. 2. Compute a pseudo data set fðxit ; yit Þ; i ¼ 1; . . . ; L; t ¼ 1; 2g to form the reference bootstrap technology using bivariate kernel density estimation and the adaption of the reﬂection method proposed by Simar and Wilson (1999a). 3. Compute the bootstrap estimate of the Malmb ðt1 ; t2 Þ by applying quist index for each DMU M ib the original estimators to the pseudo sample obtained in step 2. 4. Repeat steps 2–3 a large number B of times in b ðt1 ; t2 Þ; order to provide a set of estimates f M i1 b . . . ; M iB ðt1 ; t2 Þg. Working in a similar manner as before, these bootstrap estimates can be used to perform statistical inference on the productivity indices. Two complementary ways of doing this are through the

1069

development of an estimate of the bias15 and through the development of conﬁdence intervals. With the information provided in the latter case, it is possible to ascertain whether productivity growth (or decline) measured by the Malmquist productivity index is signiﬁcant, i.e., it is greater than (or less than) unity at the desired signiﬁcance levels.16 The same holds for the sources of productivity, as it is now possible to assess the signiﬁcance of both eﬃciency change and technical change, if they occur. 5. Data 5.1. Production units Data are provided by the Spanish savings banks association (CECA, Confederacio´n Espan˜ola de Cajas de Ahorro). The ﬁrms which were not in continuous existence over the sample period 1992–1998 were dropped, and savings banks were backward merged in order to have the same number of ﬁrms in every year. Although this might a priori involve both losing information and creating artiﬁcial companies, neither of these two cases occurs for two reasons. First, virtually all ﬁrms were in continuous existence over the sample period. Second, the only ﬁrms which were not were those involved in M&As. However, by 1992 the consolidation process was nearly over. Since then, there have been only three consolidations, which aﬀected only small ﬁrms. Hence, in order to include all ﬁrms, and to obtain a complete panel, we judged it a reasonable strategy to backward merge these ﬁrms. Two of these consolidations took place in 1992, and the third occurred in 1994.17 Although we consider virtually all savings banks operating in the Spanish banking industry, we must recall that the industry includes two additional types of banking ﬁrms, namely, private commercial banks, and credit co-operatives. Today, they face exactly the same operational regulation, which allows them to undertake the same activities. Therefore, the common regulatory environment in which they operate allows them to be considered 15

The rationale for this is provided by Mooney and Duval (1993). 16 Obviously, the interpretation would be the opposite if the input-oriented case were considered. In such a case, productivity decline would be indicated by a Malmquist productivity index greater than unity. 17 This strategy, however, is only followed so as to ease presentation and interpretation of results.

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as members of the same competitive environment and, consequently, conclusions could apply for the three types of ﬁrm. In addition to this, and following a theoretical statistical reasoning, our approach to the problem assumes the savings banks are not the sample but the production units. In fact, our sample consists of the observed output and input of the saving banks and is considered as a subset of the theoretical population set that includes all possible values of the inputs and outputs. 5.2. The sample: inputs and outputs We chose to measure the ﬂow of services provided by savings banks following the intermediation approach, which considers ﬁrms as primarily intermediating funds between savers and investors, in contrast to the production approach, which treats ﬁnancial institutions as primarily producing services for account holders. A diﬀerent issue, often confused with the former, refers to the deﬁnition of outputs Tortosa-Ausina, 2002a. The asset, user cost, and value-added methods are three diﬀerent ways of assigning input or output nature to the diﬀerent balance sheet categories. They diﬀer, among other aspects, in the role attached to deposits. Our approach is in line with the asset approach, as it primarily treats earning assets as bank output. However, it is generally accepted that most banks raise a substantial portion of their funds through produced deposits and provide liquidity, payments, and safekeeping services to depositors to obtain these funds. In order to capture these activities, we have included core deposits as an additional output category—hence, our approach will be closer to the value-added approach. In addition, in line with the contributions by Rogers (1998) and Rogers and Sinkey (1999), and with the European Central Bank (2000), we consider an additional output to control for the non-traditional activities performed by banks. All variables are described in Table 2, which also reports descriptive statistics. Our choice thus considers three outputs and three inputs. Our ﬁrst output category is loans (y1), deﬁned as all forms of loans performed by savings banks. This is virtually the only asset category unanimously treated as bank output by the diﬀerent output deﬁnitions. It would be desirable to disaggregate it, but the lack of precise statistical information rules out this possibility. The second output is savings, time, and transactions deposits (y2).

Ideally, this category should include only transaction deposits, given that our purpose is to proxy the liquidity, payments, and safekeeping services provided. Unfortunately, public information only disentangles savings deposits, other deposits, and interbank deposits. We label this category as ‘‘core’’ deposits, following Kumbhakar et al. (2001). Finally, we followed some recent contributions which claim that traditional banking is in ‘‘decline’’ Gorton and Rosen, 1995, and others which, following these ideas, suggest that a proxy should be included to account for non-traditional activities Rogers, 1998; Rogers and Sinkey, 1999. Hence, our third output category (y3) includes mainly non-interest income, following Rogers (1998). The rationale for this also comes from the increase in mutual funds to the detriment of deposits which occurred during the sample years. Additionally, we included the income from securities in this category, as many savings banks took part in the privatization process of certain public companies which occurred in the latter years of the sample, given that fewer regulated-conditioned output mixes could be chosen from.18 In the case of the inputs, the consensus in the literature is far broader. We include two traditional inputs, labor (x1) and capital (x2), measured by total labor expenses and physical capital, respectively. We also include purchased funds (x3), including all deposits categories, since this category generates roughly two thirds of total bank costs (TortosaAusina, 2002c). 5.3. Testing CRS As commented above, an important factor in our study is to check whether the underlaying technology exhibits CRS. For all the years in our sample period 1992–1998, we test for evidence against the null hypothesis that the technology shows CRS using a bootstrap procedure proposed by Simar and Wilson (2002). In particular, we work with a test statistic based on the mean of ratios made up of the estimated distance obtained assuming CRS 18 As pointed out in the text, our choice or inputs and outputs agrees with most previous literature. Although in some occasions this can lead to include partly correlated variables—especially on the outputs’ side—we should follow Dyson et al.’s (2001) recommendation, who points out that omission of variables purely on grounds of correlation should be avoided, provided that under this type of circumstances ‘‘the choice of the variables to retain is crucial, as the results of the eﬃciency assessment can diﬀer signiﬁcantly’’.

E. Tortosa-Ausina et al. / European Journal of Operational Research 184 (2008) 1062–1084

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Table 2 Deﬁnition of the relevant variables (average, 1992–1998) Variable

Variable name

Deﬁnition

Mean

Std. dev.

Outputs y1 y2 y3

Loansa Core depositsa Non-traditional outputa

All forms of loans Savings, time, and transactions deposits Non-interest income, income from securities

2,512,669 2,295,319 26,847

4,191,603 3,493,521 64,104

Inputs x1 x2 x3

Laboura Capitala Purchased fundsa

Total labor expenses Physical capital All deposit categories

54,331 127,042 3,295,698

83,433 227,297 5,637,878

a

In thousands of euros (converted from 1990 pesetas).

in the numerator and the estimated distance obtained assuming variable returns to scale in the denominator. The p-values obtained for each of the years in the study (0.7810, 0.6795, 0.8105, 0.8930, 0.4860, 0.4915 and 0.6635, for years 1992– 1998, respectively) indicate that there is no enough empirical evidence against the CRS hypothesis and consequently, in what follows, we work under this assumption. 6. Assessing eﬃciency and productivity diﬀerences: Cross-section and time disparities 6.1. Productivity growth Changes in eﬃciency, technology, and productivity are reported in Tables 3–5, respectively, for both pairs of consecutive years and the sub-periods 1992– 1995, 1995–1998 and the whole sample period 1992– 1998. The last row in each table reports the mean for each selected pair of years.19 Although the exposition, and computation, was performed in terms of input-oriented indices, the reciprocal (output-oriented) was considered to disclose results for ease of interpretation. Accordingly, values above unity indicate improvement in productivity, eﬃciency, or technical change between periods t1 and t2, and vice versa. Table 5 suggests that by 1998, savings banks were providing, on average, 119%20 as much output per unit of input as in 1992. In no year is the mean of the savings banks’ Malmquist productivity index 19 In the case of the mean, signiﬁcance was determined by appealing to the central limit theorem, which requires independence among the eﬃciency estimates (Wheelock and Wilson, 1999). Given the inherent dependency of DEA eﬃciency scores, the indicated signiﬁcance is at best only. 20 The ﬁgure provided for the geometric mean, quite close to that obtained for the arithmetic mean (121%).

equal to, or below unity. Productivity growth is especially high during the 1995/1998 period, in which the rate of productivity growth was 11.57%, due to increases in periods 1995/1996, 1996/1997 and 1997/1998 of 1.87%, 3.72% and 4.81%, respectively. In the initial post-deregulation years, productivity growth also existed, although the gains occurred at more modest rates—in 1994/1995 the increase was of 0.25%. Suitable explanations for these trends come from the decomposition of the Malmquist productivity index. The last row in Table 4 reveals that to a great extent, productivity growth is attributable to an improvement in ‘‘best practice’’ or, in other words, in production possibilities. An accurate examination of the ﬁgures in the last row of both Tables 4 and 5 discloses that they are quite similar in periods 1996/ 1997, 1997/1998 and 1992/1995. On the other hand, the last row in Table 3 suggests that catching up with ‘‘best practice’’ improved productivity, although at far more modest rates. Speciﬁcally, eﬃciency improvements took place at rates exceeding 1% only in 1993/1994 and 1995/1996. There are even deteriorations in 1992/1993 and 1994/1995, although the overall tendency (1992/1998) was to increase. However, individual scrutiny shows that there are remarkable disparities between the three indices. For instance, ﬁrm #1 underwent substantial productivity decline from 1992 to 1993 due to falling behind ‘‘best practice’’ (or catching up with ‘‘worst practice’’); in fact, there was technical progress of 3.68%. On the other hand, and also in 1992/1993, ﬁrm #50 experienced productivity growth (4.78%) due to an improvement in eﬃciency, despite technical regress (1.85%). Our results coincide with those of Pastor (1995), but diﬀer from those of Grifell-Tatje´ and Lovell (1996). They also coincide with those of Berg et al.

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Table 3 Changes in eﬃciency, consecutive years and sub-periods Firm

1992/1993

1993/1994

1994/1995

1995/1996

1996/1997

1997/1998

1992/1995

1995/1998

1992/1998

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50

0.8626*** 1.0047 1.0000 0.9993 1.1283*** 1.0000 1.0492** 0.9398** 1.0061 1.0129 1.0000 1.0000 1.0851*** 1.0405* 1.0438** 1.0348* 0.9825 0.9988 1.0472 1.0000 1.0050 1.0000 0.9415** 0.9476 0.9655 0.9575* 1.0000* 1.0025 1.0000 1.0360* 1.0003 0.9761 1.0000 0.9976 0.9950 0.9751 0.9545** 0.9630 1.0000 0.8728*** 0.9084*** 1.0300 1.0000 1.0182 0.9951 1.0244 0.9782 1.0000 0.9285*** 1.0675***

1.0663** 0.9651** 1.0000 0.9526** 1.0264 1.0000 1.0000 1.0463* 1.0357 0.9400*** 0.9214*** 1.0000 0.8872*** 1.0537* 1.0073 0.9777 0.9701* 1.2497*** 1.0000 1.0000 1.0600** 1.0000 1.0794** 1.0229 0.9682 1.0080 1.0000 1.0209 0.9955 1.0410* 1.0031 1.0578* 1.0000 0.9746 1.0192 1.0531** 1.0376* 0.9979 1.0000 1.0461* 1.0646** 0.9793 0.9805 0.9433*** 1.0678** 1.0379 0.9961 1.0000 1.0414* 1.0224

0.9578* 0.9961 1.0000 1.0424* 0.9762 1.0000 1.0000 0.9910 0.9806 1.0509** 0.9976 1.0000 1.0440** 0.9267** 1.0467** 1.0195 0.9956 0.9211** 1.0000 0.9998 0.9303** 1.0000 0.8624*** 1.0080 1.1568*** 1.0809*** 0.9728 0.9302** 0.9727 1.0679*** 1.0603*** 0.9970 1.0000 0.9975 0.9239** 1.0341* 1.0410* 1.0135 1.0000 0.9310** 1.0160 1.0073 0.9921 0.9535* 1.1018*** 1.0075 0.9753 0.8955*** 1.0458* 0.9409**

1.0358** 1.0336* 1.0000 0.9941 1.0491 1.0000 1.0000 1.0272 1.0062 1.0751*** 1.0397* 1.0000 0.9469*** 1.0788*** 1.0078 1.0206 0.9835 0.9471*** 1.0000 1.0002 1.0237 1.0000 1.1465*** 1.0608** 0.9752 1.0443* 0.9836 1.0182 1.0327 0.9933 1.0685* 1.0031 1.0000 1.0237 0.9671** 1.0634** 1.0212 1.0157 1.0000 1.0172 0.9469*** 1.0138 0.9918 1.0337 0.9832 1.0493** 0.9323*** 0.9827 1.0000 1.0403*

1.0058 1.0242 1.0000 1.0289* 0.9991 1.0000 1.0000 0.9958 1.0000 0.9547*** 1.0305* 0.9289*** 0.9772 1.0394* 1.0000 1.0263 1.0184 1.1315*** 1.0000 1.0000 1.0040 0.9889 1.0760*** 0.9690** 0.9567*** 1.0000 0.9957 1.0515** 1.0000 0.9317*** 1.0000 1.0000 0.9996** 1.0073 0.9510*** 0.9654* 1.0584*** 1.0429 1.0000 1.0676*** 0.9752 0.9504*** 0.9652* 0.9901 1.0081 0.9876 1.0192 1.0025 1.0000 1.0338*

1.0682*** 1.0103 1.0000 1.0664*** 0.9121*** 1.0000 1.0000 1.0708*** 1.0135 0.9835 0.9651** 0.9569** 0.9750* 0.9609** 1.0000 1.0194 1.0083 1.0077 1.0000 1.0000 1.0424*** 0.9171*** 0.9994 1.0532*** 1.0358** 0.9952 1.0496*** 1.0373 0.9982 1.0221 1.0000 0.9659** 1.0004** 1.0000 1.0122 0.9930 0.9375*** 0.9505** 1.0000 1.0675*** 1.0130 0.9730* 1.0740*** 1.0271* 1.0089 0.9977 1.0350** 1.0115 1.0000 1.0000

0.8809*** 0.9658 1.0000 0.9924 1.1305*** 1.0000 1.0492 0.9745 1.0218 1.0006 0.9191** 1.0000 1.0052 1.0160 1.1005*** 1.0314 0.9490* 1.1497*** 1.0472 0.9998 0.9910 1.0000 0.8764*** 0.9771 1.0813** 1.0432 0.9728 0.9520* 0.9684 1.1517*** 1.0639** 1.0293 1.0000 0.9697 0.9369* 1.0619** 1.0310 0.9740 1.0000 0.8500*** 0.9826 1.0160 0.9727 0.9158*** 1.1708*** 1.0712** 0.9503 0.8955** 1.0113 1.0269

1.1128*** 1.0695** 1.0000 1.0907** 0.9561** 1.0000 1.0000 1.0954** 1.0198 1.0095 1.0341 0.8889*** 0.9022*** 1.0775** 1.0078 1.0678** 1.0099 1.0799* 1.0000 1.0002 1.0713* 0.9069*** 1.2330*** 1.0826** 0.9664 1.0393 1.0280 1.1106*** 1.0308 0.9459** 1.0685* 0.9688 1.0000 1.0312 0.9310*** 1.0194 1.0132 1.0068 1.0000 1.1593*** 0.9354*** 0.9375** 1.0281 1.0512 1.0000 1.0339 0.9835 0.9964 1.0000 1.0755*

0.9803 1.0329 1.0000 1.0824** 1.0809** 1.0000 1.0492 1.0674* 1.0420 1.0101 0.9504* 0.8889*** 0.9068*** 1.0947** 1.1092*** 1.1013*** 0.9583 1.2416*** 1.0472 1.0000 1.0618* 0.9069** 1.0805** 1.0579* 1.0450 1.0843** 1.0000 1.0573 0.9982 1.0894** 1.1367*** 0.9972 1.0000 1.0000 0.8723*** 1.0825** 1.0446 0.9806 1.0000 0.9854 0.9191*** 0.9525* 1.0000 0.9628 1.1707*** 1.1076*** 0.9346** 0.8923*** 1.0113 1.1044***

A. mean G. mean

0.9955 0.9944

1.0111 1.0111

0.9972 0.9959

1.0136** 1.0129

1.0032 1.0025

1.0047 1.0041

1.0035 1.0013

1.0215** 1.0196

1.0236** 1.0209

(*), (**), (***): signiﬁcant diﬀerences from unity at 10%, 5%, and 1%, respectively. ‘‘A. mean’’ and ‘‘G. mean’’ refer to the arithmetic and geometric means, respectively. For the latter, signiﬁcance cannot be provided.

E. Tortosa-Ausina et al. / European Journal of Operational Research 184 (2008) 1062–1084

1073

Table 4 Changes in technology, consecutive years and sub-periods Firm

1992/1993

1993/1994

1994/1995

1995/1996

1996/1997

1997/1998

1992/1995

1995/1998

1992/1998

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50

1.0368* 1.0447*** 1.0951*** 0.9888 0.9987 1.0315 1.0356* 1.0400*** 1.0432** 1.0433*** 1.0363* 1.0481* 1.0479*** 1.0570*** 1.0088 1.0460*** 1.0354 1.0075 1.0343 1.0478 1.0220* 1.0589 1.1334*** 1.0262 0.9984 1.0248** 0.9932 1.0577*** 1.0342* 1.0471*** 1.0446** 1.0064 1.3114*** 1.0557** 1.0531* 1.0166 1.0487*** 1.0494*** 1.0332 1.0571*** 1.0533*** 1.1152*** 0.9875 1.0309* 1.0322* 1.0423*** 1.0292** 0.9619* 1.0289 0.9815*

1.0172 1.0069 1.0510 1.0158 0.9813 0.9842 1.1051*** 1.0042 1.0136 1.0143 1.0124 0.9277*** 1.0205 1.0487** 1.0250 0.9847 1.0141 1.0029 0.9557** 1.0032 0.9736** 0.9811 0.9047*** 1.0171 1.0002 1.0057 0.9748 1.0540** 1.0000 1.0038 0.9640* 1.0348*** 0.9969 1.0092 0.9516** 1.0096 1.0136 0.9970 1.0081 1.0104 0.9542*** 0.9795* 1.0314 1.0143 0.9830** 0.9850* 0.9768*** 1.0046 0.9923 0.9983

1.0390* 1.0081 1.0188 0.9880 0.9802 0.9510 1.0452 1.0024 0.9805 1.0151 1.0152 1.0237 0.9716 1.0446** 1.0285* 1.0000 1.0050 1.0137 0.9780 1.0661* 0.9982 0.9710 1.0346** 1.0236 1.0376** 1.0009 0.9470*** 1.0789*** 0.9946 1.0050 0.9726* 0.9836 1.0467 1.0092 1.0156 1.0138 0.9625*** 1.0055 0.9919 1.0043 0.9985 1.0534** 0.9877 0.9954 1.0086 1.0250 0.9942 1.0078 1.0113 0.9985

0.9699** 0.9810* 1.1203*** 1.0119 1.0251 0.9655 1.0027 1.0135 1.0129 0.9693** 0.9607*** 0.9817 0.9982 0.9675** 0.9810 0.9758* 0.9931 1.0465*** 1.0205* 0.9670 1.0119 1.0108 0.9703** 1.0031 1.0221 0.9936 1.0177 0.9782 0.9815 1.0152 1.0784*** 1.0648*** 1.0806* 1.0012 0.9673** 1.0059 0.9947 0.9591*** 1.0511 0.9773*** 0.9934 0.9876 1.0512** 0.9866 1.0824*** 0.9781 0.9946 1.0103 1.0872*** 1.0019

1.0092 1.0085 1.1321*** 1.0336 1.0890*** 1.0094 1.0583* 1.0190 1.0181 1.0026 0.9963 1.0571*** 1.0136 1.0451*** 1.0805*** 1.0184* 1.0005 1.0391 1.0136 0.9906 1.0170 1.0453 1.0144 1.0702*** 1.0430** 1.0418*** 1.0233** 1.0115 1.0162 1.0241*** 1.0470 1.0193 1.0591* 1.0754*** 1.0028 1.0431*** 1.0533*** 1.0471*** 1.0232 1.0152 1.0142 1.0633*** 1.0565*** 1.0138 1.0755*** 1.0449** 1.0118 1.0270 1.0590*** 1.0566***

1.0379*** 1.0160 1.0585* 1.0776*** 1.0921*** 1.0087 1.1459*** 1.0279* 1.0954*** 1.0080 1.0321*** 1.0444** 1.0325*** 1.0605*** 1.0960*** 1.0275*** 1.0109 1.0879*** 1.0293*** 0.9466* 1.0394*** 1.0908*** 1.0513*** 1.0749*** 1.0674*** 1.0259 1.0212 1.0244 1.0156 1.0346*** 1.0258 1.0408* 1.1294*** 0.9955 1.0188** 1.0491*** 1.0504** 0.9943 0.9932 1.0274*** 1.0478*** 1.0382*** 1.0704*** 1.0203 1.0622*** 1.0481** 1.0481*** 1.0787*** 1.0353 1.0695***

1.1094*** 1.0630*** 1.2005*** 1.0164 0.9671*** 0.9653 1.1606*** 1.0072 1.0750*** 1.0746*** 1.0642*** 0.9946 1.0364** 1.0749*** 1.0472* 1.0301** 1.0592*** 1.0129 0.9520*** 1.1386*** 0.9999 1.0427 1.0423 1.1031*** 1.0838*** 1.0040 0.9122*** 1.1899*** 1.0261 1.0210 0.9765** 1.0205 1.5049*** 1.0888*** 0.9702 1.0390 0.9962 1.0342*** 1.0379 1.0742*** 1.0058 1.1475*** 1.0962 1.0357*** 1.0069 1.0094 1.0004 0.9939 1.0331 0.9967

1.0671*** 1.0097 1.3209*** 1.1475*** 1.2247*** 1.0024 1.2508*** 1.0631*** 1.2028*** 0.9906 1.0114 1.0750*** 1.0876*** 1.0532*** 1.1430*** 1.0073 0.9971 1.2428*** 1.0823*** 0.9510* 1.0744*** 1.1703*** 1.0406** 1.0946*** 1.1775*** 1.0774*** 1.0681*** 0.9968 1.0302 1.0996*** 1.1415*** 1.1403*** 1.2945*** 1.0538** 0.9890 1.1183*** 1.1024*** 1.0278* 1.0656** 1.0638*** 1.0587*** 1.0577** 1.1731*** 1.0478*** 1.1707*** 1.0709*** 1.0536** 1.1527*** 1.1861*** 1.1346***

1.1162*** 1.0410*** 1.5480*** 1.3778*** 1.2960*** 1.0640 1.4312*** 1.0903*** 1.2225*** 1.0347*** 1.0128 1.1247*** 1.0874*** 1.1465*** 1.1853*** 0.9939 1.0682*** 1.3469*** 1.0699*** 1.0400 1.0990*** 1.3416*** 1.0748** 1.3559*** 1.3381*** 1.1229*** 1.0500 1.1863*** 1.0795*** 1.1174*** 1.1268*** 1.2028*** 1.7527*** 1.0900*** 0.9835 1.2233*** 1.1349*** 1.0360** 1.2206*** 1.1178*** 1.0741*** 1.1736*** 1.2807*** 1.0563*** 1.3474*** 1.1061** 1.1121*** 1.2766*** 1.2336*** 1.1872***

A. mean G. mean

1.0412*** 1.0402

1.0004 0.9998

1.0070* 1.0067

1.0064 1.0058

1.0350*** 1.0346

1.0445*** 1.0439

1.0508*** 1.0477

1.0973*** 1.0943

1.1760*** 1.1676

(*), (**), (***): signiﬁcant diﬀerences from unity at 10%, 5%, and 1%, respectively. ‘‘A. mean’’ and ‘‘G. mean’’ refer to the arithmetic and geometric means, respectively. For the latter, signiﬁcance cannot be provided.

1074

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Table 5 Changes in productivity, consecutive years and sub-periods Firm

1992/1993

1993/1994

1994/1995

1995/1996

1996/1997

1997/1998

1992/1995

1995/1998

1992/1998

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50

0.8943*** 1.0495 1.0951** 0.9882 1.1268*** 1.0315 1.0866*** 0.9774* 1.0496* 1.0568* 1.0363 1.0481 1.1371*** 1.0997*** 1.0530** 1.0824** 1.0173 1.0063 1.0830*** 1.0478 1.0271 1.0589* 1.0671 0.9725 0.9639 0.9813 0.9933 1.0604* 1.0342 1.0848*** 1.0449 0.9824 1.3114*** 1.0532 1.0478 0.9913 1.0009 1.0106 1.0332 0.9226*** 0.9568*** 1.1487*** 0.9875 1.0497* 1.0271 1.0677** 1.0067 0.9619 0.9553** 1.0478***

1.0846** 0.9718* 1.0510 0.9677** 1.0072 0.9842 1.1051** 1.0507* 1.0498 0.9534*** 0.9327*** 0.9277* 0.9055*** 1.1049** 1.0324 0.9628* 0.9838 1.2533*** 0.9557 1.0032 1.0320* 0.9811 0.9766 1.0404 0.9684* 1.0137 0.9748 1.0761* 0.9955 1.0449* 0.9670 1.0946** 0.9969 0.9835 0.9699 1.0632** 1.0517* 0.9949 1.0081 1.0570** 1.0158 0.9592* 1.0113 0.9568*** 1.0497** 1.0223 0.9730 1.0046 1.0334 1.0207

0.9951 1.0041 1.0188 1.0299 0.9569 0.9510 1.0452 0.9934 0.9614 1.0668** 1.0128 1.0237 1.0144 0.9681** 1.0765*** 1.0195 1.0006 0.9337** 0.9780 1.0659* 0.9286** 0.9710 0.8923*** 1.0318 1.2003*** 1.0819*** 0.9212* 1.0036 0.9675 1.0732*** 1.0312* 0.9806 1.0468 1.0067 0.9383** 1.0483* 1.0020 1.0191 0.9919 0.9350** 1.0145 1.0611* 0.9798 0.9492* 1.1113*** 1.0327 0.9697 0.9025*** 1.0576** 0.9394**

1.0047 1.0140 1.1203** 1.0059 1.0755** 0.9655 1.0027 1.0411* 1.0192 1.0421** 0.9989 0.9817 0.9452*** 1.0437** 0.9887 0.9959 0.9767 0.9911* 1.0205 0.9671 1.0359 1.0108 1.1125*** 1.0640** 0.9968 1.0376* 1.0010 0.9960 1.0136 1.0084 1.1522*** 1.0680 1.0806* 1.0250 0.9355*** 1.0697** 1.0158 0.9742 1.0511 0.9941 0.9406*** 1.0012 1.0426 1.0198 1.0642 1.0263* 0.9273*** 0.9928 1.0872** 1.0423*

1.0150 1.0329 1.1321*** 1.0635** 1.0881** 1.0094 1.0583* 1.0148 1.0181 0.9572*** 1.0267 0.9820*** 0.9905 1.0863*** 1.0805* 1.0452* 1.0189 1.1758*** 1.0136 0.9906 1.0210 1.0337 1.0915*** 1.0371 0.9979 1.0418 1.0189 1.0636*** 1.0162 0.9542*** 1.0471 1.0193 1.0586* 1.0833* 0.9537*** 1.0070 1.1148*** 1.0920*** 1.0232 1.0838*** 0.9890 1.0106 1.0197 1.0037 1.0842** 1.0319 1.0313 1.0296 1.0590* 1.0923***

1.1087*** 1.0265 1.0585* 1.1492*** 0.9960*** 1.0087 1.1459*** 1.1007*** 1.1102*** 0.9913 0.9961 0.9993* 1.0068 1.0191 1.0960*** 1.0475* 1.0193 1.0963** 1.0293 0.9466* 1.0835*** 1.0003*** 1.0507 1.1321*** 1.1057*** 1.0210 1.0719*** 1.0626*** 1.0137 1.0575** 1.0258 1.0052 1.1299*** 0.9955 1.0312 1.0417 0.9847*** 0.9450*** 0.9932 1.0968*** 1.0614** 1.0102 1.1495*** 1.0480** 1.0716** 1.0457 1.0848*** 1.0911*** 1.0353 1.0695**

0.9773** 1.0267 1.2005*** 1.0087 1.0933*** 0.9653 1.2178*** 0.9814 1.0984* 1.0753 0.9781 0.9946 1.0418 1.0921 1.1524*** 1.0625 1.0052 1.1645*** 0.9969 1.1384*** 0.9909 1.0427 0.9134*** 1.0779 1.1720*** 1.0474 0.8874* 1.1328 0.9936 1.1759*** 1.0388 1.0505 1.5049*** 1.0559 0.9090** 1.1033** 1.0270 1.0073 1.0379 0.9131*** 0.9883 1.1659*** 1.0663 0.9486** 1.1788*** 1.0812** 0.9506 0.8900*** 1.0447 1.0235

1.1875*** 1.0799** 1.3209*** 1.2516*** 1.1709* 1.0024 1.2508*** 1.1645*** 1.2266*** 1.0000 1.0459 0.9555*** 0.9812** 1.1349*** 1.1520** 1.0755** 1.0069 1.3421*** 1.0823 0.9512 1.1511*** 1.0614 1.2830*** 1.1850*** 1.1379* 1.1198** 1.0979* 1.1071*** 1.0619 1.0401 1.2196*** 1.1048 1.2945*** 1.0866 0.9208*** 1.1400** 1.1170* 1.0348 1.0656 1.2333*** 0.9903* 0.9915 1.2060*** 1.1015** 1.1707** 1.1072* 1.0362 1.1486* 1.1861*** 1.2202***

1.0942 1.0753 1.5479*** 1.4914*** 1.4008*** 1.0640 1.5016*** 1.1638*** 1.2738*** 1.0451 0.9627* 0.9997** 0.9861* 1.2551*** 1.3147*** 1.0946** 1.0237 1.6722*** 1.1204** 1.0397 1.1668*** 1.2167** 1.1614*** 1.4344*** 1.3983*** 1.2175*** 1.0500 1.2543*** 1.0775 1.2173*** 1.2808*** 1.1994** 1.7527*** 1.0900 0.8579*** 1.3243*** 1.1856*** 1.0159 1.2206*** 1.1016 0.9872* 1.1179 1.2807*** 1.0170 1.5775*** 1.2250*** 1.0393 1.1391 1.2475*** 1.3111***

A. mean G. mean

1.0364*** 1.0344

1.0125 1.0109

1.0041 1.0025

1.0198*** 1.0187

1.0382*** 1.0372

1.0493*** 1.0481

1.0538*** 1.0490

1.1201*** 1.1157

1.2058*** 1.1920

(*), (**), (***): signiﬁcant diﬀerences from unity at 10%, 5%, and 1%, respectively. ‘‘A. mean’’ and ‘‘G. mean’’ refer to the arithmetic and geometric means, respectively. For the latter, signiﬁcance cannot be provided.

E. Tortosa-Ausina et al. / European Journal of Operational Research 184 (2008) 1062–1084

(1992a), who ﬁnd productivity regress prior to deregulation, but rapid productivity growth after deregulation—although we do not analyze the prederegulation era. However, as stated by GrifellTatje´ and Lovell (1996) it is perhaps surprising, and hopefully temporary, but not unusual that a productivity decline should accompany deregulation. Some evidence was reported by Bauer et al. (1993), or Humphrey (1993). Yet more recent evidence, employing linear programming techniques such as ours, found productivity decline (Wheelock and Wilson, 1999). On the other hand, our results are coincidental with the recent study by Mukherjee et al. (2001) who—also through linear programming techniques—ﬁnd productivity growth. As stated by these authors, a key diﬀerence may lie in the identiﬁcation of outputs. We also think that the evolution of savings banks’ product mixes may have played a non-negligible role in this process. In fact, one of the outputs (y3, non-traditional output) includes items in an attempt to reﬂect the ﬁnancial innovation performed by most savings banks, which in most cases has enabled them to increase their revenues. In addition, application of the bootstrap methodology allows assessment of the ‘‘null hypothesis’’ of no eﬃciency change, no technical change, and no productivity growth/decline, which indicates that the corresponding measures are not statistically different from unity. Resampling was carried out for B = 2000—i.e., we have 2000 pseudo samples. We provide results for, 90%, 95% and 99% conﬁdence intervals. The interpretation is straightforward. In the 95% conﬁdence interval case, if it contains unity, then the corresponding measure is not signiﬁcantly diﬀerent from one at the 5% signiﬁcance level, i.e., it is not possible to conclude that changes occurred in eﬃciency, technology, or productivity. In contrast, when the interval excludes unity, one can conclude with 95% conﬁdence that the corresponding measure is signiﬁcantly diﬀerent from unity. An analogous interpretation is applicable for the 90% and 99% conﬁdence intervals, and conclusions must be drawn at a 10% and 1% signiﬁcance level, respectively. One conclusion from the original estimates is that a number of savings banks had productivity regress in all periods. Speciﬁcally, Table 5 shows that positive productivity change occurred in 36, 27, 28, 33, 42, and 41 cases for periods 1992/1993, 1993/1994, 1994/1995, 1995/1996, 1996/1997 and 1997/1998, respectively, whereas for the remainder, productivity decline occurred. If we consider longer periods,

1075

tendencies are more emphasized, as productivity growth occurs in 34, 43, and 45 cases, for periods 1992/1995, 1995/1998, and 1992/1998, respectively. Put another way, if we consider the entire period 1992/1998, all institutions underwent productivity growth with the exception of ﬁve ﬁrms. However, there are several cases in which the positive changes are not signiﬁcant. For instance, in 1992/1993, out of the 14 ﬁrms which suﬀered productivity decline (i.e., negative values), only ﬁve suﬀered it signiﬁcantly. Speciﬁcally, in three cases (ﬁrms #1, 40 and 41) productivity decline diﬀered from unity at 1% signiﬁcance level, in just one case (ﬁrm #49) it diﬀered from unity at 5% signiﬁcance level, in another (ﬁrm #8) it diﬀered from unity at 10% signiﬁcance level and, in a further nine cases (ﬁrms #4, 24, 25, 26, 27, 32, 36, 43 and 48), it was not statistically signiﬁcant. Similar trends hold for the 36 ﬁrms which underwent productivity growth, out of which productivity growth was statistically signiﬁcant at 1% on nine occasions, at 5% on four occasions, and at 1% in the other ﬁve cases. In the remaining 18 ﬁrms productivity growth estimates are not signiﬁcant and hence cannot be determined. Diﬀerences in conclusions between the original estimates and the bootstrap results are more evident when scrutinizing certain particular cases. For instance, from the original estimates we would conclude that ﬁrm #19 in 1993/1994 had a productivity decline amounting to about 4.4% (see Table 5); in contrast, ﬁrm #15 experienced productivity growth amounting to about 3.2%. However, the conclusion from the bootstrap results is that neither of these two savings banks had a change in productivity signiﬁcantly diﬀerent from unity. Turning to the results for eﬃciency change (Table 3) we note that three ﬁrms were eﬃcient (‘‘best practice’’) in all time periods (ﬁrms #3, 6, 39) as revealed by values of unity between all successive pairs of years and sub-periods. Bootstrap reveals four additional banks (#19, 20, 29 and 34) to be eﬃcient in all years, as the ﬁgures diverging from unity are not signiﬁcant. In addition, if we only consider the non-consecutive years analysis (1992/1995, 1995/1998, 1992/1998), there is an additional ﬁrm with no eﬃciency change (ﬁrm #33), and a further eight ﬁrms whose eﬃciency gains or losses are not signiﬁcant (ﬁrms #7, 9, 10, 27, 32, 37, 38, 43 and 49). We also observe that signiﬁcant changes in eﬃciency occur mainly for the whole period 1992/ 1998. In other periods (for instance, 1992/1995) there are eﬃciency changes for most ﬁrms, although

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on the whole they are not signiﬁcant. In sum, for the consecutive pairs of years, out of the 255 estimates that showed eﬃciency gain or loss, only 47, 39 and 33 diﬀered signiﬁcantly from unity at 1%, 5% and 10% signiﬁcance levels, respectively. When considering longer periods, the percentages of ﬁrms undergoing signiﬁcant eﬃciency change are much higher, particularly in 1992/1998. Finally, for the technical change component (Table 4), a negative shift in technology is obtained for 78 ﬁrms in the original estimates for the consecutive years, out of which the bootstrap method recognized only nine cases to be signiﬁcantly diﬀerent from unity at 1%, a further nine cases to diﬀer at 5%, and nine at 10%. For the remainder, technical regress was not statistically signiﬁcant. The positive change occurs more often, in 222 instances—out of the 300 entries for consecutive years—but only 71 are statistically diﬀerent from unity at 1%, 19 at 5%, and 20 at 10%. Again, for the remainder, technical progress is not signiﬁcant. However, for the sub-period results vary to a certain extent. Considering the entire 1992/1998 period (last column of Table 4), technical change is positive for all ﬁrms except ﬁrms #16 and 35. Yet in no case does technical regress turns out to be signiﬁcant, not even at 10% signiﬁcance level. A further four ﬁrms experience technical progress, although at insigniﬁcant rates (ﬁrms #6, 11, 20 and 27). These changes are decomposable into two sub-periods. In the initial 1992/1995 post-deregulation period, the geometric mean for technical progress is lower than in 1995/1998 (1.0477 vs. 1.0943). The former was the period before which the bulk of mergers had just occurred, hence one might expect technology to improve slowly. In fact, signiﬁcant (at either 1%, 5% or 10%) technical progress is found only for 22 ﬁrms. In 1995/1998 this pattern is present in 40 instances. Therefore, according to the above statements, it may easily be concluded that comparisons of production units based on the original point estimates should be made with caution. In many instances, the productivity, eﬃciency, and technical change estimates deviate from unity at insigniﬁcant levels, which clearly lowers the extent to which our conclusions are convincing. In other words, as suggested by Simar and Wilson (1999a), ‘‘as with any estimator, it is not enough to know whether the Malmquist productivity index indicates increases or decreases in productivity, but whether the indicated changes are signiﬁcant in a statistical sense’’.

Finally, so as to ease both readability and understandability of results, Table 6 provides a summary of our ﬁndings, although the analysis is conﬁned to the sub-periods 1992/1995, 1995/1998 and 1992/ 1998. 6.2. Productive eﬃciency Tables 7–9 show results for the bootstrap of eﬃciency scores for years 1992, 1995, and 1998—for the sake of brevity.21 A summary is presented in Table 10, which shows that eﬃciency at industry level (mean eﬃciency) enhanced slightly over the sample period—although notable ups and downs also occurred. The result coincides with what may a priori be expected in an industry in which deregulation has already taken place. However, the industry is still experiencing important changes. Out of the 50 savings banks in the original sample, an average of 15 banks were found to operate on the best practice frontier ð^hi ¼ 1Þ. As stated by Ferrier and Hirschberg (1997), this result does not supply much information to decision makers as it is not possible to distinguish between the performances of many of the banks. In these circumstances, the bootstrap procedure turns out to be a very useful tool. Columns 2–6 in Tables 7–9 provide the original DEA eﬃciency score ð^hi Þ, the bias-corrected estimate ð~hi ¼ 2^hi hi Þ, the bootstrap bias estimate d i ð^hi ÞÞ, and the lower bound and upper bound ð bias of the 95% conﬁdence interval of the eﬃciency scores, hi . All information provided in Tables 7–9 follows Section 4.1. Moreover, as a check of how important it is to use the smoothed rather than the naive bootstrap, we have estimated density functions of technical eﬃciency scores for each year in our sample using kernel smoothing and Silverman’s (1986) reﬂection method, which provides densities for the eﬃciency scores in the natural region [0, 1], and consequently, upper bounds in the corresponding conﬁdence intervals always less or equal to 1. Fig. 1 represents the resulting estimated densities for each year. The features revealed are manifold but, for our particular purposes, their very peculiar shape must be noted. Speciﬁcally, in all cases a remarkable mode is observed at unity. These results reveal the sensitivity of the eﬃciency measures with respect to sampling variation. 21

Results for all years are available from the authors upon request.

E. Tortosa-Ausina et al. / European Journal of Operational Research 184 (2008) 1062–1084

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Table 6 Summary of bootstrap results for changes in eﬃciency, technology, and productivity 1992/1995 Original estimate

1995/1998

5%

10%

Original estimate

5 4 –

4 2 –

– 3 –

30 13 7

Changes in technology Growth 39 Decline 11 Stagnation 0

19 3 –

2 1 –

1 – –

Changes in productivity Growth 34 Decline 16 Stagnation 0

11 3 –

2 3 –

1 1 –

Changes in eﬃciency Growth 22 Decline 22 Stagnation 6

Signiﬁcance 1%

1992/1998

5%

10%

Original estimate

4 5 –

6 3 –

4 – –

29 17 4

45 5 0

34 – –

5 – –

1 1 –

43 7 0

18 2 –

7 1 –

6 1 –

The bias-corrected estimates ð~ hi Þ in column 4 reveal that diﬀerences in measured eﬃciency are of a diﬀerent magnitude than when original eﬃciency scores ð^ hi Þ are considered.22 Speciﬁcally, we observe that for some ﬁrms eﬃciency enhances, whereas for others it deteriorates. In 1992 (Table 7) there are many instances for which eﬃciency declines. These include some cases in which eﬃciency deteriorates severely, by more than 0.05 (ﬁrms #3, 6, 20, 22, 27, 29, 33, 43, 48). In the case of the originally eﬃcient ﬁrms (with ^ hi ¼ 1:000Þ eﬃciency worsens more dramatically; indeed, all those ﬁrms for which the bootstrap bias d i ð^ estimate ð bias hi ÞÞ is greater than 0.040 are originally eﬃcient ﬁrms. We also have additional statistical information on these ﬁrms, as conﬁdence intervals are constructed. We will examine this below. For years 1995 and 1998 (Tables 8 and 9) we also have results on whether ﬁrms’ eﬃciency improves or deteriorates. However, to more properly interpret the bias-corrected estimators ~ hi in each table we must use information on conﬁdence intervals, which deﬁne the statistical location of the true eﬃciency. Broadly, they show that eﬃciency scores, when adding statistical precision, overlap to a notable extent. For instance, in Table 7, using the intervals in columns 6 and 7, we notice that ﬁrm #2 overlaps with ﬁrms #3, 7, 8 or 13, amongst many others, despite 22 However, as stated by Gonza´lez and Miles (2002), results should be interpreted with care, as values less than zero or greater than one are statistically valid and reﬂect the eﬃciency level of a ﬁrm, but they have no economic meaning—eﬃciency scores are bounded between zero and one.

Signiﬁcance 1%

Signiﬁcance 1%

5%

10%

7 5 –

7 2 –

3 2 –

48 2 0

41 – –

3 – –

– –

45 5 0

26 1 –

4 1 –

– 3 –

their diﬀering bias-corrected eﬃciency estimates. Of special note, ﬁrm #4 overlaps to a very large degree with, for instance, ﬁrm #23. In general, the wider the interval, the higher the chance of overlapping, and vice versa. Consequently, for 90% conﬁdence intervals, overlapping would occur in fewer instances, whereas for 99% conﬁdence intervals it would occur more often. Yet regardless of the interval considered (90%, 95% or 99%), it becomes apparent that in many instances there is not enough empirical evidence to reject the hypothesis that two ﬁrms are equally eﬃcient. Consequently, as stated by Simar and Wilson (1998), we should be particularly cautious when making relative comparisons of the performances among ﬁrms based on the original eﬃciency scores ^hi . In our particular case, results show that disparities are much wider when the bootstrap analysis is performed. However, further information is available. Fig. 2 contains box plots which disclose information about the distributions of each ﬁrm’s pseudo sample for years 1992, 1995 and 1998. The main advantage of this representation is that when simultaneously displaying several box plots, it is easier to see whether diﬀerences exist both between ﬁrms and for the same ﬁrm over time. Examples are the existence, emergence, or vanishing of outliers, dispersion or concentration of data, or the symmetry (or asymmetry) of the distribution. The vertical axis of each sub-ﬁgure in Fig. 2 represents the variable’s scale, in our case the bootstrapped eﬃciencies. The box represents the interquartile range (IQR). The 0.75 and 0.25 quartiles

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E. Tortosa-Ausina et al. / European Journal of Operational Research 184 (2008) 1062–1084

Table 7 Bootstrap of eﬃciency scores, 1992 ~ Firm ^ hi Biasi Lower bound hi 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50

1.0000 0.8934 1.0000 0.8571 0.8431 1.0000 0.9531 0.9162 0.9597 0.8561 1.0000 1.0000 0.9215 0.8778 0.9016 0.7952 0.9770 0.7764 0.9550 1.0000 0.9367 1.0000 0.8708 0.9081 0.9248 0.9179 1.0000 0.9458 1.0000 0.8287 0.8797 0.9685 1.0000 1.0000 0.9179 0.8855 0.8660 0.9693 1.0000 0.9396 0.9469 0.9709 1.0000 0.9093 0.8541 0.8005 0.9483 1.0000 0.9889 0.9055

0.9591 0.8771 0.9359 0.8257 0.8094 0.9174 0.9185 0.8957 0.9260 0.8423 0.9691 0.9521 0.9030 0.8575 0.8712 0.7822 0.9545 0.7482 0.9251 0.9215 0.9196 0.9267 0.8370 0.8731 0.8915 0.8992 0.9293 0.9275 0.9473 0.8172 0.8538 0.9386 0.9175 0.9586 0.8817 0.8588 0.8508 0.9544 0.9574 0.9153 0.9368 0.9391 0.9185 0.8934 0.8288 0.7766 0.9220 0.9324 0.9506 0.8729

0.0409 0.0163 0.0641 0.0314 0.0337 0.0826 0.0346 0.0205 0.0337 0.0138 0.0309 0.0479 0.0185 0.0203 0.0304 0.0130 0.0225 0.0282 0.0299 0.0785 0.0171 0.0733 0.0338 0.0350 0.0333 0.0187 0.0707 0.0183 0.0527 0.0115 0.0259 0.0299 0.0825 0.0414 0.0362 0.0267 0.0152 0.0149 0.0426 0.0243 0.0101 0.0318 0.0815 0.0159 0.0253 0.0239 0.0263 0.0676 0.0383 0.0326

0.9067 0.8571 0.8548 0.7751 0.7565 0.7396 0.8606 0.8709 0.8785 0.8263 0.9301 0.8747 0.8599 0.8216 0.8297 0.7656 0.9210 0.7054 0.8725 0.7847 0.8920 0.8307 0.7717 0.8086 0.8461 0.8637 0.8182 0.9027 0.8725 0.8026 0.8167 0.8940 0.7518 0.9154 0.8055 0.8133 0.8202 0.9357 0.8846 0.8738 0.9253 0.8744 0.7444 0.8741 0.7973 0.7318 0.8766 0.8509 0.8799 0.8308

Upper bound

Table 8 Bootstrap of eﬃciency scores, 1995 ~hi Firm ^hi Biasi Lower bound

Upper bound

0.9963 0.8903 0.9965 0.8540 0.8399 0.9961 0.9497 0.9128 0.9561 0.8530 0.9961 0.9966 0.9184 0.8748 0.8983 0.7922 0.9734 0.7737 0.9518 0.9964 0.9333 0.9962 0.8677 0.9049 0.9214 0.9148 0.9965 0.9423 0.9964 0.8260 0.8763 0.9650 0.9966 0.9956 0.9145 0.8825 0.8633 0.9658 0.9962 0.9361 0.9437 0.9681 0.9960 0.9061 0.8512 0.7978 0.9448 0.9962 0.9853 0.9022

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50

0.8777 0.8598 0.9964 0.8480 0.9499 0.9965 0.9966 0.8896 0.9773 0.8542 0.9165 0.9968 0.9234 0.8886 0.9965 0.8177 0.9248 0.8896 0.9967 0.9969 0.9251 0.9968 0.7608 0.8841 0.9959 0.9544 0.9698 0.8975 0.9652 0.9511 0.9328 0.9941 0.9969 0.9666 0.8574 0.9370 0.8899 0.9416 0.9967 0.7960 0.9276 0.9832 0.9695 0.8302 0.9961 0.8547 0.8981 0.8923 0.9968 0.9268

(n:75 and n:25 ), deﬁne the upper and lower segments respectively. 50% of the distribution of probability mass is inside the box. Thus, its height represents the IQR, a usual dispersion measure. A small IQR results in a short box, revealing that 50% of the den-

0.8809 0.8628 1.0000 0.8506 0.9532 1.0000 1.0000 0.8928 0.9806 0.8566 0.9191 1.0000 0.9262 0.8918 1.0000 0.8202 0.9272 0.8926 1.0000 1.0000 0.9283 1.0000 0.7631 0.8873 1.0000 0.9576 0.9728 0.9004 0.9684 0.9544 0.9359 0.9970 1.0000 0.9697 0.8600 0.9404 0.8929 0.9441 1.0000 0.7987 0.9304 0.9864 0.9727 0.8328 1.0000 0.8575 0.9011 0.8955 1.0000 0.9298

0.8509 0.8358 0.9370 0.8267 0.9284 0.9194 0.9219 0.8742 0.9394 0.8374 0.8921 0.9721 0.9013 0.8665 0.9715 0.8010 0.9074 0.8697 0.9801 0.9245 0.9093 0.9456 0.7417 0.8568 0.9399 0.9344 0.9527 0.8637 0.9422 0.9322 0.9104 0.9752 0.9201 0.9344 0.8355 0.9109 0.8681 0.9212 0.9346 0.7843 0.9139 0.9495 0.9357 0.8147 0.9552 0.8364 0.8829 0.8692 0.9527 0.9040

0.0300 0.0270 0.0630 0.0239 0.0248 0.0806 0.0781 0.0186 0.0412 0.0192 0.0270 0.0279 0.0249 0.0253 0.0285 0.0192 0.0198 0.0229 0.0199 0.0755 0.0190 0.0544 0.0214 0.0305 0.0601 0.0232 0.0201 0.0367 0.0262 0.0222 0.0255 0.0218 0.0799 0.0353 0.0245 0.0295 0.0248 0.0229 0.0654 0.0144 0.0165 0.0369 0.0370 0.0181 0.0448 0.0211 0.0182 0.0263 0.0473 0.0258

0.8064 0.7981 0.8404 0.7870 0.8896 0.7370 0.7841 0.8566 0.8702 0.8016 0.8495 0.9398 0.8603 0.8243 0.9342 0.7695 0.8693 0.8318 0.9552 0.8015 0.8868 0.8765 0.7147 0.8140 0.8571 0.9076 0.9276 0.7810 0.9046 0.9047 0.8800 0.9494 0.7399 0.8674 0.7959 0.8779 0.8264 0.8838 0.8516 0.7643 0.8942 0.8760 0.8675 0.7846 0.9028 0.8083 0.8608 0.8270 0.8973 0.8731

sity is fairly concentrated. The horizontal line inside the box is the median, or 0.50 quartile. Its location relative to the upper or lower limits of the box provides graphical information about the shape of the distribution. If it is not in the center of the box,

E. Tortosa-Ausina et al. / European Journal of Operational Research 184 (2008) 1062–1084 Table 9 Bootstrap of eﬃciency scores, 1998 ~ Firm ^ hi Biasi Lower bound hi

Upper bound

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50

0.9795 0.9220 0.9992 0.9270 0.9104 0.9991 0.9989 0.9772 0.9991 0.8639 0.9495 0.8882 0.8349 0.9601 0.9991 0.8750 0.9356 0.9631 0.9992 0.9991 0.9937 0.9059 0.9400 0.9598 0.9655 0.9944 0.9990 0.9990 0.9972 0.9020 0.9990 0.9652 0.9991 0.9991 0.8000 0.9578 0.9039 0.9496 0.9989 0.9251 0.8695 0.9240 0.9991 0.8747 0.9991 0.8858 0.8855 0.8914 0.9991 0.9991

0.9803 0.9228 1.0000 0.9277 0.9113 1.0000 1.0000 0.9779 1.0000 0.8647 0.9504 0.8889 0.8356 0.9609 1.0000 0.8758 0.9364 0.9639 1.0000 1.0000 0.9945 0.9069 0.9409 0.9606 0.9664 0.9952 1.0000 1.0000 0.9982 0.9028 1.0000 0.9659 1.0000 1.0000 0.8007 0.9586 0.9047 0.9505 1.0000 0.9259 0.8703 0.9247 1.0000 0.8755 1.0000 0.8866 0.8862 0.8923 1.0000 1.0000

0.9682 0.9105 0.9454 0.9092 0.8925 0.9439 0.9451 0.9668 0.9421 0.8562 0.9385 0.8729 0.8258 0.9411 0.9439 0.8667 0.9214 0.9406 0.9757 0.9498 0.9859 0.8879 0.9272 0.9408 0.9470 0.9817 0.9841 0.9609 0.9828 0.8902 0.9554 0.9494 0.9422 0.9451 0.7894 0.9463 0.8857 0.9335 0.9632 0.9151 0.8594 0.9100 0.9841 0.8656 0.9610 0.8655 0.8768 0.8776 0.9822 0.9802

0.0121 0.0123 0.0546 0.0185 0.0188 0.0561 0.0549 0.0111 0.0579 0.0085 0.0119 0.0160 0.0098 0.0198 0.0561 0.0091 0.0150 0.0233 0.0243 0.0502 0.0086 0.0190 0.0137 0.0198 0.0194 0.0135 0.0159 0.0391 0.0154 0.0126 0.0446 0.0165 0.0578 0.0549 0.0113 0.0123 0.0190 0.0170 0.0368 0.0108 0.0109 0.0147 0.0159 0.0099 0.0390 0.0211 0.0094 0.0147 0.0178 0.0198

0.9472 0.8865 0.8081 0.8660 0.8512 0.8054 0.8101 0.9515 0.8153 0.8338 0.9213 0.8349 0.8101 0.8907 0.8434 0.8536 0.8802 0.8746 0.9517 0.8576 0.9730 0.8405 0.9041 0.8954 0.8996 0.9618 0.9640 0.8942 0.9549 0.8679 0.8858 0.9128 0.7994 0.8417 0.7668 0.9218 0.8413 0.9044 0.8965 0.8992 0.8429 0.8840 0.9596 0.8499 0.8836 0.8198 0.8624 0.8493 0.9538 0.9477

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Table 10 Productive eﬃciency in Spanish banks, 1992–1998

the distribution is asymmetric. Two vertical lines are shown in the upper and lower limits of the box. The end of each line is known as the adjacent value (whisker), either upper or lower. In fact, the maximum distance between whiskers is given by the

1992 1993 1994 1995 1996 1997 1998

Mean

Maximum (#)

Minimum

0.9313 0.9264 0.9361 0.9326 0.9442 0.9465 0.9501

1.000 1.000 1.000 1.000 1.000 1.000 1.000

0.7764 0.7754 0.8045 0.7631 0.8124 0.7910 0.8007

(14/50) (16/50) (13/50) (13/50) (18/50) (17/50) (17/50)

½n:25 1:5Rðn:25 Þ; n:75 þ 1:5Rðn:25 Þ interval, where Rðn:25 Þ is the IQR. Therefore, the whiskers deﬁne the natural bounds of the distributions, and the crosses represent outliers lying outside them. Fig. 2 indicates how sensitive a particular ﬁrm’s eﬃciency score is to variations in the eﬃciency of other ﬁrms in the data set. When observing year 1992, we ﬁnd remarkable intra-ﬁrm and inter-ﬁrm variability. Regarding the former, we observe for some ﬁrms that the range is very wide (ﬁrms #6, #20, #33 and #43), whereas for others dispersion of the estimated eﬃciencies is very low (ﬁrms #2, #10, #16, #30, #38, #41 and #44). However, when we compare the diﬀerent ﬁrms altogether we perceive clear dissimilarities: median values range from around 0.75 (ﬁrm #18) to around 0.95 (ﬁrm #11), which can be considered a remarkable diﬀerence. In addition, it seems that we have more disperse values for ﬁrms with higher eﬃciency scores. The general comments for year 1992 also apply to 1995 and 1998. However, it seems that variability within ﬁrms for year 1998 is lower and the eﬃciencies tend to reach higher values.

7. Conclusions This paper has applied recently devised approaches to bootstrapping both Malmquist productivity indices and eﬃciency scores to a database of Spanish savings banks over the initial 1992–1998 post-deregulation period. Although results on the two issues overlap to a great extent, we will summarize them separately. With regard to productivity indices, the picture that emerges is one of productivity growth due to an improvement of production possibilities. Only the ﬁrms lagging behind the eﬃcient frontier lessen such growth. This occurs because of the technical progress that took place at a particularly high rate in 1992/1993, 1996/1997 and 1997/1998. As a result,

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25

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0 0.7 0.75 0.8 0.85 0.9 0.95 1

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0 0.7 0.75 0.8 0.85 0.9 0.95 1

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25

25

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0 0.7 0.75 0.8 0.85 0.9 0.95 1

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0 0.7 0.75 0.8 0.85 0.9 0.95 1

1.05 1.1

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0 0.7 0.75 0.8 0.85 0.9 0.95 1

0 0.7 0.75 0.8 0.85 0.9 0.95 1

1.05 1.1

1.05 1.1

Fig. 1. Non-parametric kernel estimates of technical eﬃciency scores, using Silverman’s (1986) reﬂection method (bandwidths chosen according to the plug-in method provided by Sheather and Jones (1991)).

on average, ﬁrms are able to provide 119% as much service per unit of resource consumed as they were providing just six years earlier. The bootstrap allows for a more careful analysis of what happens at ﬁrm level. Speciﬁcally, the original estimates show that, over the entire period (1992/1998), about 90% of ﬁrms grew in terms of productivity. However, not every ﬁrm had Malmquist productivity indices signiﬁcantly diﬀerent from unity at 5%. In contrast, only 10% of ﬁrms suﬀered productivity decline, and in all cases it was statistically signiﬁcant at 5%. Therefore, the bootstrap backs up the conclusions provided by the original indices. A similar pattern is observed in the technical change component. In this case, only two ﬁrms experienced technological regress which, in addition, was signiﬁcant in just one case (and at the 10% signiﬁcance level). We can thus verify that productivity regress is mainly due to declines in eﬃciency.

Concerning eﬃciency scores, the contribution of the bootstrap is similar. Speciﬁcally, mean technical eﬃciency is quite high, and does not vary greatly over the sample period. However, it seems that there are substantial disparities between ﬁrms but, at the same time, some ﬁrms have more similar eﬃciencies than what was suggested by the ‘‘original’’ eﬃciency scores, and vice versa. These results are coincidental with what might a priori be expected in a context of deregulation. They are similar to those obtained by Mukherjee et al. (2001), Gilbert and Wilson (1998), or Berg et al. (1992b). However, they run counter to Wheelock and Wilson (1999)—although they consider a diﬀerent Malmquist productivity index decomposition. If we consider the Spanish experience, our results are coincidental with those obtained by Pastor (1995), but diﬀer from those obtained by Grifell-Tatje´ and Lovell (1996). The latter, however, is not paradoxical, as their choice of outputs diﬀers from ours, and,

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1081

1 0.95 0.9 0.85 0.8 0.75 0.7 0.65

0

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1 0.95 0.9 0.85 0.8 0.75 0.7 0.65

1 0.95 0.9 0.85 0.8 0.75 0.7 0.65

Fig. 2. Box plots of ﬁrms’ eﬃciencies, years 1992, 1995, 1998.

of particular importance, their sample period is also diﬀerent. The reasons underlying these trends may arise from diﬀerent sources. Productivity growth could partly come from the diﬃculties aﬀecting many savings banks engaged in M&As. New ﬁrms were cre-

ated, and some of them had branches which were concentrated too locally. After closing down the geographically closest branches, eﬃciency and productivity gains might have been substantial. Other ﬁrms engaged in rapid geographic expansion. There has also been a dramatic boom in mutual funds to

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the detriment of deposits; their loss may have not been oﬀset by the increase in fee-income, which is accounted for in our deﬁnition of outputs. Our results are relevant for several reasons, but most of them relate to the increasing importance of Spanish savings banks not only within the Spanish banking system but also when focusing on the whole European scenario—which has turned in an increased interest on this type of ﬁrms for all academics, practitioners and—overall—policymakers. As a consequence, economic and ﬁnancial press (both national and international) have pointed out, among other related issues, Spanish savings banks’ morphing process ‘‘from community supporters to aggressive investors, reﬂecting changing European business patterns’’.23 Given the increasing importance of this type of ﬁrms, it seems necessary to provide conclusions as painstaking as possible regarding their eﬃciency and productivity, especially when considering they are usually involved in major deregulatory initiatives and other industry structure changes aﬀecting the banking system. Our study is an attempt to attain this objectives. The relevance of our results is also related to the peculiar ownership type of Spanish savings banks, as mentioned in Section 2, which provides an abiding debate on whether it should be modiﬁed. As suggested by some authors (see Crespı´ et al., 2004), savings banks ‘‘represent a case of a lack of ownership’’, and can be considered as ‘‘commercial nonproﬁt organizations’’. There often exist conﬂicting interests within their assemblies and boards, emerging from the diﬀerent groups which make them up— namely, public authorities (both local and regional), depositors, employees, and founding entities. In addition to this, the public authorities’ group is made up by diﬀerent political parties, whose composition varies depending on the elections’ results. In these circumstances, in which ﬁrms’ performance could be thwarted due to particular groups’ interests, it is important to attain an accurate picture of their eﬃciency and productivity. This type of debate also exists in other Western European countries. For instance, in Germany, where Savings banks are quite important—accounting for about half of all German savings deposits— there is a wide array of opinions as to whether the ownership of savings banks (what they call Spark-

asse) should be opened to all comers. Therefore, studies as our would probably be welcome.24 In a further study, we will explore these details more deeply, to ﬁnd out not only which ﬁrms experience faster productivity growth, but also which undergo stronger sampling variation, and what might determine this. Another more technically arduous stem of research should attempt to address the problem of bootstrapping cost and, possibly, proﬁt and revenue eﬃciency scores. This is especially relevant in banking where, in contrast to public sector studies, prices are usually available. All computations were performed using MATLAB. For bootstrapping technical eﬃciency scores we faced additional problems, as it requires univariate non-parametric density estimation and, consequently, a bandwidth choice. Plug-in rules, which seem to be reasonably reliable in terms of a fair balance between bias and variance, are provided in Sheather and Jones (1991). Their MATLAB code (bwsjpiSM.m) is available at URL: http:// www.stat.unc.edu/faculty/marron.html.25 However, in the case of Malmquist productivity indices we have panel data which requires bivariate density estimation. This was also performed via kernel smoothing. In this case the state of the art does not provide such satisfactory bandwidths as those for the univariate case. Wand and Jones (1994) provide plug-in methods, although they are somewhat diﬃcult to implement in MATLAB, as they are originally written in S-PLUS. We therefore used those suggested by Simar and Wilson (1999a). We do however recognize that there is still room for improvement on this matter. Acknowledgements We would like to thank George E. Battese, Kevin J. Fox, Quentin Grafton, Joseph G. Hirschberg and other participants at the Economic Measurement Group ’02 Workshop held at the School of Economics of the University of New South Wales for helpful comments. Jose´ Manuel Pastor also provided insightful comments which we wish to recognize. All authors thank ﬁnancial support from the Instituto Valenciano de Investigaciones Econo´micas (IVIE). 24

23

‘‘Spain’s savings banks get—and use—muscles’’, The Wall Street Journal, 8th February, 2005.

‘‘Strife at the Sparkasse’’, The Economist, 4th May, 2006. See also the KernSmooth 2.22 package for R by M.P. Wand and B.D. Ripley. 25

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Sensitivity analysis of efficiency and Malmquist productivity indices: An application to Spanish savings banks

Sensitivity analysis of efficiency and Malmquist productivity indices: An application to Spanish savings banks

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