Bootstrapping profit change: An application to Spanish banks

Computers & Operations Research 39 (2012) 1857–1871

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Bootstrapping profit change: An application to Spanish banks Emili Tortosa-Ausina a,, Carmen Armero b, David Conesa b, Emili Grifell-Tatje´ c a b c

´ de la Plana, Spain Departament d’Economia, Universitat Jaume I and Ivie, Campus del Riu Sec, 12071 Castello Universitat de Vale ncia, Spain  Universitat Autonoma de Barcelona, Spain

a r t i c l e in fo

abstract

Available online 6 May 2010

The aim of this study is to provide a tool which enables us to conduct statistical analysis in the context of changes in productivity and profit. We build on previous initiatives to decompose profit change into mutually exclusive and exhaustive sources. To do this we use distance functions, which are calculated empirically using linear programming techniques. However, we may not learn a great deal by solving these linear programs unless methods of statistical analysis are used to examine the properties of the relevant estimators. Our purpose is to provide a methodology based on bootstrap that allows us to conduct statistical inference for the profit change decomposition. Thus, it will be possible to answer questions such as whether variations in the profit change components, or the differences across firms, are statistically significant. We provide an application to Spanish commercial banks for the 2003/2004 period. Results suggest that profit change differentials between them are not always significant. Therefore, the validity of the conclusions which do not factor in the bootstrap may be jeopardized to varying degrees. & 2010 Elsevier Ltd. All rights reserved.

Keywords: Banking Bootstrap Productivity Profits

1. Introduction In recent years there has been a growing interest in analyzing the productivity of firms that operate in different industries worldwide. Studies have dealt with this issue from a variety of angles and most of them have implicitly focused on the concept of total factor productivity (TFP), given the multiproduct nature of many firms in different industries—both on the input and output sides. According to the traditional definition of growth accounting, TFP changes are constituted by the differences between output growth rates and input growth rates [1]. This definition implicitly assumes that no inefficiency exists, requiring all production units to lie on the frontier. However, should inefficiency exist, TFP growth could be composed of both technical change (shifts in the frontier) and catching-up (changes in efficiency). Some approaches ignore inefficiency, implicitly assuming that the observed output is Farrell [2] technically efficient. They may be labeled as ‘‘nonfrontier’’ (see Diewert [3,4], Morrison Paul and Diewert [5]), because of presupposing that all production units are on it. On the other hand, frontier models assume inefficiency may actually exist. Although the focus of some proposals has been parametric (see Førsund and Hjalmarsson [6], Nishimizu and Page [7]), most of them are nonparametric. The latter approach builds

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E-mail address: [email protected] (E. Tortosa-Ausina). 0305-0548/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.cor.2010.04.017

on the pioneer study by Caves et al. [8], who devised individual productivity indices and named them after Sten Malmquist. Later, ¨ the indices were refined by Fare et al. [9,10]. Malmquist productivity indices are calculated from distance functions which, according to Grosskopf [11], constitute a natural way to model the production frontier by taking into account both efficiency change and technical change. In the specific case of financial institutions, on which our application is focused, some studies such as Berger and Mester [12] or Bauer et al. [13] estimate translog cost functions to construct indices of productivity change. Although their findings are interesting, they share the disadvantage of a priori specifying functional forms which, as suggested by some authors, may be problematic.1 In contrast, other studies which examine productivity change in banking have used data envelopment analysis (DEA) and the Malmquist index. See, for instance, Wheelock and Wilson [17,18], among many others. Despite the fact that the analysis of firms’ productivity has mostly been carried out by economists, other disciplines such as operations research and management science, engineering or psychology have also dealt with the issue [19]. For instance,

1 McAllister and McManus [14], Mitchell and Onvural [15], and Wheelock and Wilson [16] test and reject the translog specification of bank cost functions, and suggest semi-nonparametric and nonparametric methods for estimating bank costs [17].

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business literature [20,21] has proposed partial productivity measures which, given their incomplete nature, can vary in opposing directions. The advantage of using distance functions is that they can be linked unambiguously to profit [22]. Accordingly, some authors such as Banker et al. [23] or Banker [24] propose variants of a three-way decomposition in which profit change is decomposed into a price effect (including changes in resource prices paid and product prices received), a productivity effect (usually attributed exclusively to an improvement in technology) and an activity effect, capturing the effect of changes in the size and partly the scope of the business. By merging the economics literature on productivity change and the business literature on profit change, Grifell-Tatje´ and Lovell [22] (G-T&L hereafter) provide a methodology aimed at disentangling the links between firms’ profits and their productivity. Specifically, they attempt to embed a productivity change decomposition similar to that developed in economics literature within the profit/productivity linkage developed in business literature [23–26]. In order to do this, they decompose the different sources of profit change with activity analysis techniques, highlighting the role of productivity change and its components, while considering other determinants of profit change simultaneously. Specifically, their analysis sheds light on four aspects of the link. First, profit change is decomposed into a price effect and a quantity effect. The quantity effect is further decomposed into a productivity effect and an activity effect, following some of the ideas in business literature. The ensuing stage decomposes the productivity effect into a technical change effect and an operating efficiency effect—whose relevance was stressed earlier—whereas the activity effect follows a threefold decomposition, namely, a product mix, a resource mix, and a scale effect. G-T&L methodology builds on solving different linear programming problems that modify and extend the DEA technique,2 which is intensively used by economics and business literature to analyze business performance as well as ¨ macroeconomic performance (see, for instance, Fare et al. [10], Lozano-Vivas and Pastor [28]).3 However, since their methodology is based on DEA, it is also subject to its disadvantages. For example, sensitivity to outliers, curse of dimensionality (see, in a different setting, Xu [31], Warfield [32], Staley and Warfield [33]) and the fact that noisy data are not allowed. But above all, statistical inference can be difficult. This latter disadvantage requires a detailed comment given that there is always the tendency to think of an estimation as the final stage of inference. As stated by Simar and Wilson [34,35], if we do not give importance to the underlying statistical model (that is, the process which has generated the data and the sampling scheme used to draw them), we could erroneously convolute the underlying true distance functions and their estimates. Some contributions, nevertheless, attempt to minimize all these disadvantages. In particular, Simar and Wilson [34–37] propose a bootstrap methodology in order to conduct statistical inference in the context of nonparametric frontier models which entails all the features mentioned above. Our purpose in this paper is to extend the profit change decomposition suggested by G-T&L by giving a statistical

2 Although it could easily be extended to its nonconvex variant, the so-called free disposable hull (FDH) (see Tulkens [27]). 3 However, in the case of economics, many applications also consider econometric techniques. For an interesting review of both types of techniques applied to banking, see Berger and Humphrey [29] or, more recently, Weill [30]. There are also some new proposals being developed. See the special issue of the Journal of Econometrics, volume 126, number 2, year 2005.

interpretation to its different components, via a similar bootstrap procedure to that proposed by Simar and Wilson [34]. Bootstrapping [38] is based on the idea of resampling from an original sample of data so as to create replicated data sets from which we can make inferences on the required quantities of interest. We will therefore be able to determine whether the discrepancies found in the firms in our sample, and for the different components of profit change, are statistically significant or not. Accordingly, results will have a variety of angles since they will be subject to both ‘‘vertical’’ and ‘‘horizontal’’ examination, that is to say, not only across firms but also across the different components of profit change. Although our methods hinge on DEA, they can easily be extended to its nonconvex (FDH) counterpart. We have applied our methodology to the context of the Spanish banking system,4 which has witnessed remarkable changes over the last two decades due to deregulation (i.e. regulatory harmonization with banks of other European Union countries) and technological change. The industry is made up of three types of firms: namely, private commercial banks, savings banks and credit cooperatives, yet their importance in terms of the share in total industry assets is unequal. As of 2005, their share of total industry assets were roughly 50%, 45% and 5%, respectively.5 Given that they all now face the same regulatory environment, operational differences have virtually faded away, and the only remaining differences relate to firms’ ownership type (see Crespı´ et al. [40], Kumbhakar et al. [41]). These differences might be important for our specific setting since they determine how profits are allocated and therefore might influence the intensity with which firms pursue profits. Whereas private commercial banks allocate most of their profits to their shareholders, savings banks cannot do so and must either retain their earnings or invest them in social and cultural programs, which account for roughly 25% of their net annual profits. Indeed, since they have no formal owners there is no market for the corporate control of savings banks, and some authors even label them as ‘‘not-for-profit’’ organizations [40], or ‘‘commercial nonprofit organizations’’ [42]. Although we do not entirely share these views, we do consider that savings banks might have an incentive to seek profits less intensively than private commercial banks do, due to their a priori weaker corporate control mechanisms. Based on the above rationale, our analysis will be confined to those firms pursuing profits with more intensity, i.e. private commercial banks. Although some recent deregulatory initiatives—such as the removal of restrictions on the geographic expansion of savings banks—triggered an unprecedented growth in this type of firms to the detriment of commercial banks, their quota of total industry assets is still quite remarkable. Thus our study attempts to decompose the profit change between two particular years (2003 and 2004) for each bank in our sample into several sources, incorporating a statistically based vision of the magnitudes being estimated. This statistical approach is the main contribution of our paper. The rest of the article is organized as follows. Section 2 summarizes the proposals of G-T&L, in order to provide a seamless link to our methodology, which is presented in Section 3. Section 4 provides full details of the empirical application, both on data (Section 4.1 and 4.2) and results (Section 4.3). Section 5 presents the concluding remarks.

4

For applications with a more global perspective see, for instance, Xu [39]. Since the share of the latter group of firms is comparatively minor, the discussion will be based on the other two types of firms. 5

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2. Decomposing the profit change We consider that in period t a firm (which in our particular setting are banks) uses N resources or inputs, xt ¼ ðxt1 , . . . , xtN Þ A RNþ , to produce M products or outputs, yt ¼ ðyt1 , . . . , t ytM Þ A RM þ . We define the profit p of a firm in period t as the difference between the total revenue and the total cost during that period:

pt ¼ yt  pt xt  wt , t

¼ ðwt1 ,

ð1Þ

. . . ,wtN Þ A RNþ

where w represents the prices the firm must pay for the inputs in period t, pt ¼ ðpt1 , . . . ,ptM Þ A RM þ stands for the prices charged by the firm for the outputs in period t and the symbol ‘‘’’ represents the scalar product between the two vectors involved. Because our final interest is to identify and evaluate the elements which are more directly responsible for the change in profit between two periods of time, namely t1,t2, that is, Dpðt1 ,t2 Þ ¼ pt2 pt1 , we first provide a characterization of the production technology. Taking into account the above inputs and outputs, the production set St in period t A ft1 ,t2 g is given by St ¼ fðxt ,yt Þ,yt can be produced from xt g,

ð2Þ

and represents the set of output and input quantity vectors that is feasible with the technology in that period. In the case of an output-oriented approach where we consider that firms attempt to maximize their output quantities by holding input quantities fixed, the production set in period t may also be described in terms of its corresponding output set, Pt ðxt Þ ¼ fyt ,ðxt ,yt Þ A St g,

ð3Þ

which is the set of all attainable outputs that, under the technology available in t, can be produced from the vector of input quantities xt. Both the production and the output sets are assumed to be closed and convex and to satisfy strong disposability of outputs. The outer boundary of the output set in period t is its output isoquant, Isoq Pt ðxt Þ ¼ fyt ,yt A Pt ðxt Þ, lyt2 = Pt ðxt Þ,8l 41g:

ð4Þ

In order to identify the determinants of the changes between two periods t1 and t2, we may also consider Pt2 ðxt1 Þ as the set of hypothetical output quantity vectors which, under the technology available in period t2, might be produced from input vector xt1 . Isoq Pt2 ðxt1 Þ is the outer boundary of this mixed-period output set, and can be interpreted similarly. Shephard’s [43] output distance function for a firm producing yt outputs with xt inputs in period t measures the efficiency of that firm via the distance of its output vector to the frontier as defined by the best-practice firm, Dot ðxt ,yt Þ ¼ minfc, c A R þ ,yt =c A Pt ðxt Þg, t

ð5Þ

6

where Do ðx ,y Þ r 1. Adjacent-period output distance function Dot2 ðxt1 ,yt1 Þ may also be defined by replacing Pt(xt) with Pt2 ðxt1 Þ, and can be interpreted in a similar manner. It is worth noting that the values of this distance function may either be higher, equal, or lower than unity, since quantity data from one period may not be feasible with technology prevailing in another period. Moreover, we can also consider the distance function Dot2 ðxt2 ,yt1 Þ, which involves technology and inputs from period t2 and outputs from period t

t

6 Therefore, if we have a firm with Dot ðxt ,yt Þ ¼ a, we consider that its output represents only a% of the maximum attainable output with that vector of inputs. For instance, if a¼ 0.5(0.25), then the firm would produce 50% (25%) of the maximum attainable output, with the available inputs.

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t1. From now on, and for the sake of simplicity, we will denote Dot2 ðxt1 ,yt1 Þ, Dot2 ðxt2 ,yt1 Þ, Dot1 ðxt1 ,yt1 Þ and Dot2 ðxt2 ,yt2 Þ by Dot2 ðt1 ,t1 Þ, Dot2 ðt2 ,t1 Þ, Dot1 ðt1 ,t1 Þ and Dot2 ðt2 ,t2 Þ, respectively. In addition, when referring to a specific banking firm, a subindex k will be attached. On the assumption of an input-oriented approach, the firms will attempt to minimize their input quantities by holding outputs fixed. In this case, analogous elements of the outputoriented approach need to be defined in order to provide this approach with a similar structure to the output-oriented one. They are:

 the input set Lt(yt) in period t defined as Lt ðyt Þ ¼ fxt ,ðxt ,yt Þ A St g,  

which is a closed and convex set of all attainable inputs that, under the technology available in t, can result of yt outputs, the input isoquant Isoq Lt(yt) in period t, Isoq Lt ðyt Þ ¼ fxt ,xt A Lt ðyt Þ, lxt2 = Lt ðyt Þ, l o 1g, and the Shephard [44] input distance, which measures the efficiency for a firm producing yt outputs with xt inputs in period t through the distance Dit ðyt ,xt Þ ¼ maxfc,xt =c A Lt ðyt Þg, where Dit ðyt ,xt Þ Z1.

It is clear that all the extensions, comments and notations introduced with regard to the output-oriented approach also apply to all the elements involved in the input-oriented case. Following G-T&L, the profit change of a firm between period t1 and t2 can be decomposed, using a three stage strategy, into different components or effects. The chart in Fig. 1 illustrates this hierarchical decomposition. In the first stage, the profit change between period t1 and period t2 can be decomposed as the sum of two known magnitudes, namely the quantity effect Q(t1,t2), and the price effect Pc(t1,t2), between both periods: Dpðt1 ,t2 Þ ¼ ½ðyt2 yt1 Þ  pt1 ðxt2 xt1 Þ  wt1  þ½yt2  ðpt2 pt1 Þxt2  ðwt2 wt1 Þ : |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} Quantity effect Q ðt1 ,t2 Þ

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} Price effect Pcðt1 ,t2 Þ

ð6Þ The quantity effect holds prices constant, whereas the price effect keeps the quantities fixed. The first one shows the impact of an expansion or contraction on profit, holding prices fixed, defining a Laspeyres indicator. The second one shows the impact of changes in the price structure of the business on profit, keeping quantities constant, its structure corresponding to a Paasche indicator. Secondly, the quantity effect is decomposed into two additional effects: namely, the productivity effect, Pd(t1,t2), and the activity effect, A(t1,t2), whose evaluation requires the use of the previously introduced distance functions:      yt1 yt2 yt1  pt1  yt2  pt1 Q ðt1 ,t2 Þ ¼ t2 t2 Do ðt1 ,t1 Þ Do ðt2 ,t2 Þ |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl} 

Productivity effect Pdðt1 ,t2 Þ

  y t2 yt1 t1 t2 t1 t1   p þ ðx x Þ  w : Dot2 ðt2 ,t2 Þ Dot2 ðt1 ,t1 Þ |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

ð7Þ

Activity effect Aðt1 ,t2 Þ

In the final stage, the productivity effect is decomposed into two further effects: namely, the technical change effect, Tc(t1,t2), and the operating efficiency effect, O(t1,t2), as follows:    yt1 yt1   pt1 Pdðt1 ,t2 Þ ¼ t2 t1 Do ðt1 ,t1 Þ Do ðt1 ,t1 Þ |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} Technical change effect Tcðt1 ,t2 Þ

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Profit change Price effect

Quantity effect

Productivity effect

Activity effect

Technical change effect

Operating efficiency effect

Resource mix Effect

Scale effect

Product mix effect

Fig. 1. Profit change decomposition.

 þ

    yt1 yt2 t1 t1 t2 t1 y y  ,  p  p Dot1 ðt1 ,t1 Þ Dot2 ðt2 ,t2 Þ |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl}

ð8Þ

w ¼ fðxtk1 ,ytk1 ,xtk2 ,ytk2 Þ,k ¼ 1, . . . ,Kg,

Operating efficiency effect Oðt1 ,t2 Þ

while the activity effect is also decomposed into three further effects: namely the scale effect S(t1,t2), which measures the contribution of changes in the level or volume of operations and is sensitive to the kind of returns to scale of the industry.7 The other two components are the resource mix effect, R(t1,t2), and the product mix effect, Pm(t1,t2): 2

1

0

3

 C 7 6 B yt1 yt1 xt1 C 7 6 B   xt1 C  wt1 7   pt1 B Aðt1 ,t2 Þ ¼ 6 t1 A 5 4 Dot2 ðt2 ,t1 Þ Dot2 ðt1 ,t1 Þ @ t2 y t 1 ,x Di Dot2 ðt2 ,t1 Þ |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} Scale effect Sðt1 ,t2 Þ

20

1

6B 6B 6Bxt2  4@

s.t.



  yt2 y t1   pt1 : t2 Do ðt2 ,t2 Þ Do ðt2 ,t1 Þ |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} t2

k, in period t, t A ft1 ,t2 g, respectively. Although there are different methods of estimation (e.g. the free disposal hull (FDH) suggested by Deprins et al. [45]), we use data envelopment analysis (DEA). This approach (based on [2] study) measures the distance relative to the boundary of the convex hull of w. The procedure for dealing with the estimation of the input function measure is similar in all cases. In particular, for firm k, c t1 ðt1 ,t1 Þ of the distance Dot1 ðt1 ,t1 Þ can k¼1,y,K, the estimation Do k k be obtained via the linear programming model t1

1 yk ytkm r

Resource mix effect Rðt1 ,t2 Þ

 þ

ð10Þ

where xtk ¼ ðxtk1 , . . . ,xtkn , . . . ,xtkN Þ A RNþ and ytk ¼ ðytk1 , . . . ,ytkm , . . . , ytkM Þ A RM þ are the input and output vector corresponding to firm

c ðt1 ,t1 Þ1 ¼ maxfyk , yk 4 0g ½Do k

3

7 C xt1 7 C C  wt1 7 5 A yt1 t Di ,x 1 t2 Do ðt2 ,t1 Þ |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} t2

and t2:

K X

1 ltj 1 ytjm , m ¼ 1 . . . ,M,

j¼1

ð9Þ

Product mix effect Pmðt1 ,t2 Þ

K X

ltj 1 xtjn1 rxtkn1 , n ¼ 1, . . . ,N,

j¼1

When trying to evaluate the determinants of profit change between two periods of time, we have to take into account that the production sets, and consequently the input and output sets, are typically unobserved. Thus, the values of the output and input distance functions of the different elements in the decomposition by G-T&L are unknown quantities which need to be estimated. If we focus on the definition of R(t1,t2) and the second element in S(t1,t2) in (9), we will observe a nested element which jointly combines the input and output measure functions. This element is in fact a new function defined by the particular composition of both. From a statistical point of view (when evaluating it in a particular period with determined input and output values) it would be a new parameter which, as is usual for well-behaved functions, could be estimated in terms of the appropriate composition of the original estimator. With the aim of estimating these parameters, we consider a sample with the inputs and outputs on K firms in periods t1

ltk1 ¼ 1,

ð11Þ t1

t ¼ ðl11 ,

t . . . , lK1 Þ

is a vector of weights that forms a convex where k combination of observed firms relative to which the subject firm’s t efficiency is evaluated. The constraint lk1 ¼ 1 indicates that we are assuming Variable Returns to Scale (VRS), which is usual in c t2 ðt2 ,t2 Þ can be done similarly by banking literature.8 Obtaining Do k solving the same linear programming model as (11), where all t2 are substituted for t1. The mixed-period distance function Dotk2 ðt1 ,t1 Þ for firm k can be estimated from the linear programming model: t2

c ðt1 ,t1 Þ1 ¼ maxfyk , yk 4 0g ½Do k s.t. 1 yk ytkm r

K X

2 ltj 2 ytjm , m ¼ 1 . . . ,M,

j¼1 K X

ltj 2 xtjn2 rxtkn1 , n ¼ 1, . . . ,N,

j¼1 7 Suppose the simplest possible scenario with just one output and one input without inefficiency and technical change in a context of constant returns of scale. In this scenario: xt2 ¼ lxt1 and yt2 ¼ lyt1 , l 40 and the expression of the scale effect S(t1,t2) in (9) can be rewritten as Sðt1 ,t2 Þ ¼ pt1 ðlyt1 yt1 Þot1 ðlxt1 xt1 Þ ¼ ðpt1 yt1 ot1 xt1 Þðl1Þ ¼ pt1 ðl1Þ. Thus, e.g. in the case that the firm expands its activities, l 4 1, and has positive profits, pt1 40, the scale effect takes a positive value Sðt1 ,t2 Þ4 0. Furthermore, in periods of expansion we expect an important contribution of the scale effect. For additional details see G-T&L.

ltk1 ¼ 1,

ð12Þ

8 In our data set we have tested and rejected the assumption of constant returns to scale using the methodology proposed by Simar and Wilson [46]. Regarding the utility of return to scale in DEA programming in some specific contexts, see Butler and Li [47].

E. Tortosa-Ausina et al. / Computers & Operations Research 39 (2012) 1857–1871

while Doti 2 ðt2 ,t1 Þ can be approached by means of

productivity effect: c ðt ,t Þ b ðt1 ,t2 Þ ¼ Q ðt1 ,t2 ÞPd A k k k 1 2

t2

c ðt2 ,t1 Þ1 ¼ maxfyk , yk 40g ½Do k

b t2 ðt ,t Þ yt2 y b t2 ðt ,t Þyt1   pt1 ðxt2 xt1 Þ  wt1 : ¼ ½y k 2 2 k 1 1 k k k k k k

s.t. 1 yk ytkm r

K X

2 ltj 2 ytjm , m ¼ 1 . . . ,M,

b t2 ðt ,t Þy b t1 ðt ,t Þyt1  pt1 , ck ðt1 ,t2 Þ ¼ ½y Tc k 1 2 k 1 1 k k

ltj 2 xtjn2 r xtkn2 , n ¼ 1, . . . ,N,

c ðt ,t ÞTc b ðt1 ,t2 Þ ¼ Pd ck ðt1 ,t2 Þ O i 1 2 k

j¼1

t1

ltk1 ¼ 1:

ð13Þ

With regard to the estimation of the input distance function Ditk2 ðyt1 =Dot2 ðt2 ,t1 Þ,xt1 Þ in (9) for each firm k, we will proceed in two stages:

t2

b ðt ,t Þ1yt1  pt1 ½y b ðt ,t Þ1yt2  pt1 , ¼ ½y k 1 1 k 2 2 k k k b t2 ðt ,t Þy b t2 ðt ,t Þyt1  pt1 ½f b t2 1xt1  wt1 , b S k ðt1 ,t2 Þ ¼ ½y k 2 1 k 1 1 k k k k k b t2 xt1 Þ  wt1 , b k ðt1 ,t2 Þ ¼ ðxt2 f R k k k k t

b k ¼ ðy bk1 , . . . , y bkm , . . . , y bkM Þ defined as 1. Compute the vector y

c t2 ðt2 ,t1 Þ, where Do c t2 ðt2 ,t1 Þ is the estimated distance b k ¼ ytk1 =Do y k k in (13). 2. Approach the target quantity from the linear programming model: t2

c ðy b k ,t1 Þ1 ¼ minffk , fk 40g ½Di k s.t. bkm r y

K X

2 ltj 2 ytjm , m ¼ 1 . . . ,M,

j¼1 K X

ltj 2 xtjn2 r fk xtkn1 , n ¼ 1, . . . ,N,

j¼1

ltk2 ¼ 1:

ð14Þ

After solving all the different linear programming problems involved, we reparameterize and obtain the estimated Farrell [2] output and input-oriented measure of technical efficiency for firm k, k¼1,y,K as follows: t1

ð17Þ

The estimation of the rest of the effects for each firm k can be obtained in a similar manner. Thus,

j¼1 K X

1861

t

b 2 ðt ,t Þ yt2 y b 2 ðt ,t Þ yt1   pt1 , b ðt1 ,t2 Þ½Sc c ðt1 ,t2 Þ þ R b k ðt1 ,t2 Þ ¼ ½y dk ðt1 ,t2 Þ ¼ A Pm k k k 2 2 k 2 1 k k k

ð18Þ are, respectively, estimations of the following: technical change effect, Tck(t1,t2), operating efficiency effect Ok(t1,t2), scale effect Sck(t1,t2), resource mix effect Rk(t1,t2), and product mix effect Pmk(t1,t2) of firm k between periods t1 and t2. Thus far, we have provided a descriptive estimation, without any statistical validity, of the magnitude of the different effects which make up business profit change. In other words, we do not have information about their variability or any knowledge of their general probabilistic behavior. Recall that, from a statistical point of view, an estimator for a given parameter is a random variable whose stochastic behavior is determined by its sampling distribution, which is the basic tool for deriving its most relevant features (unbiasedness, standard error, etc.). The probabilistic nature of the estimators is derived from the fact that they are functions of random samples (independent and identical distributed random variables) from a common probabilistic population. Within this probabilistic framework, the following section examines the statistical underpinnings of the bootstrap methodology in order to assess the behavior of the estimation of the different effects outlined above.

t1

c ðt1 ,t1 Þ, ybk ðt1 ,t1 Þ ¼ 1=Do k t2

t2

t2

t2

t2

t2

3. Bootstrapping the profit decomposition

c ðt1 ,t1 Þ, ybk ðt1 ,t1 Þ ¼ 1=Do k c ðt2 ,t2 Þ, ybk ðt2 ,t2 Þ ¼ 1=Do k c ðt2 ,t1 Þ, ybk ðt2 ,t1 Þ ¼ 1=Do k 0

b t2 ¼ 1=Di c t2 @ f k k

y t1 c t2 ðt2 ,t1 Þ Do k

1 t1 A

,x

ð15Þ

from which we can obtain an estimation of the different components of the quantity effect of each firm by simply substituting these values in expressions (7)–(9). In particular, from (7) c ðt ,t Þ ¼ ½y b t2 ðt ,t Þ1 yt1  pt1 ½y b t2 ðt ,t Þ1 yt2  pt1 Pd k 1 2 k 1 1 k 2 2 k k k k

ð16Þ

is the estimation of the productivity effect Pdk(t1,t2) for firm k, k¼1,y,K, between periods t1 and t2. Taking into account that Qk(t1,t2) is a known quantity, an estimation of the activity effect Ak(t1,t2) for firm k between periods t1 and t2 can easily be obtained from the estimation of the

Bootstrap is a statistical computer-intensive approach that can provide measures of uncertainty (confidence intervals, standard errors, etc.) for a wide range of problems (see Efron and Tibshirani [48], Davison and Hinkley [49], for a detailed discussion). This methodology, introduced by Efron [38], is based on the 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 data sets from which we can make inferences on the quantities of interest. In our context, the basic idea considers that the sample data set in (10) has been generated by some data generating process (DGP hereafter) U. This DGP is completely characterized by a set of assumptions [35] about the production set and the probability density function from which the data are obtained. The bootstrap methodology therefore allows us to retrieve a consistent estimator, b , of the DGP from the sampling information. Consequently, the U b can be used to generate similar samples to the original estimator U data set,9 from where we can construct an approximate (bootstrap) 9 This is a key point in this context, as failing to do it properly can produce inconsistent bootstrap estimators. See Lovell [50], Ferrier and Hirschberg [51,52]

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sampling distribution of the above-mentioned estimators of the profit decomposition. More explicitly, if we focus only on the magnitude describing the productivity effect, Pdk(t1,t2), of firm k and c ðt ,t Þ (the same comment applies to the other its estimator Pd k 1 2 terms in the decomposition), according to the basic bootstrap idea, the sampling distribution c ðt ,t ÞPd ðt ,t ÞÞjU, ðPd k 1 2 k 1 2 would be approximately equal to the distribution of the bootstrap c  ðt ,t Þ of Pd (t ,t ); that is, estimator Pd k

1

2

k

1 2

2. Calculate the estimated Farrell output-oriented measures of b t1 ðt ,t Þ and y b t2 ðt ,t Þ from (15). technical efficiency y k 1 1 k 2 2 3. Compute a pseudo-data set

wðbÞ ¼ fxtk1 ðbÞ ,ykt1 ðbÞ ,xkt2 ðbÞ ,ytk2 ðbÞ g, to form the reference bootstrap technology using bivariate kernel density estimation and the adaption of the reflection method proposed by Simar and Wilson [34]. 4. Carry out the DEAs for the pseudo-sample and compute the corresponding bootstrap estimate of the Farrell output-

c  ðt ,t ÞPd c ðt ,t ÞÞjU b: ðPd k 1 2 k 1 2

b t1 ðbÞ ðt ,t Þ, y b t2 ðbÞ ðt ,t Þ, y b t2 ðbÞ ðt ,t Þ and oriented distances, y 1 1 2 2 1 1 k k k

This means that in the so-called ‘‘bootstrap world’’, and depending on the observed data, the distribution of the bootstrap c  ðt ,t Þ is known since U b is a consistent estimator of the estimator Pd

ybk

k

1

2

DGP U. However, in practice, it is impossible to compute it, and therefore Monte Carlo simulations are required to approach it from ðbÞ

c ðt ,t Þ,b ¼ 1, . . . ,Bg, i.e. the set of B bootstrap estimates of fPd 1 2 k Pdk(t1,t2). This technique is not new in the context of nonparametric frontier models. The first use of bootstrap in frontier models dates back to Simar [55], although its development for nonparametric envelopment estimators was introduced by Simar and Wilson [36]. Since then, it has been extensively developed in Simar and Wilson [34,35,37,56]. For applications in banking see, for instance, Ferrier and Hirschberg [51], Casu and Molyneux [57], Gilbert and Wilson [58], MurilloMelchor et al. [59], or Tortosa-Ausina et al. [60]. In their work, Simar and Wilson [36] highlight that the biggest problem when bootstrapping in frontier models is that of consistently mimicking the DGP. In our case, in order to find consistent bootstrap estimators of the magnitude of the effects which make up profit change, we have to take into account that these effects are a function of distance estimators and that procedures mentioned in the previous section for the estimation of those distances give values which are close to unity. Consequently, resampling directly from the original data (naive bootstrap) would provide a poor estimate of the DGP. The way to solve this problem is by using smoothed bootstrap in which the above-mentioned distance functions are estimated using kernel smoothing methods and, in particular, using the reflection method proposed by Silverman [61]. But given that we work with panel data, the time-dependence structure of the data must also be taken into account. Simar and Wilson [34] deal with this problem by extending the smoothed bootstrap mentioned above by estimating, and drawing from, the joint density of inputs and outputs over the two time periods under consideration. Considering the distance t1

t2

c ðt1 ,t1 Þ, Do c ðt2 ,t2 ÞÞ ,k ¼ 1, . . . ,Kg, we obtain bootstrap vector fðDo k k samples—producing consistent estimators—using a generalization of the reflection method for the bivariate case. Thus, we overcome the problem caused by the fact that distances are close to unity. The result of repeatedly resampling from the efficiency scores using the smoothed bootstrap is an imitation of the sampling distribution of the original scores from which inference can be made. Basically, for each firm k, k¼1,y,K, this process can be summarized as follows:

c t1 ðt1 ,t1 Þ and 1. Compute the estimated efficiency scores Do k t2 c ðt2 ,t2 Þ by solving the relevant linear programming model Do k in the previous section. (footnote continued) and Simar and Wilson [53,54] for an illustration on this topic and Simar and Wilson [37] for a detailed discussion based on Monte Carlo evidence.

t2 ðbÞ

5.

b t2 ðbÞ . ðt2 ,t1 Þ and of the input-oriented distance f k

c ðbÞ ðt ,t Þ, A b ðbÞ ðt1 ,t2 Þ, Compute the bootstrap estimates Pd 1 2 k k

ðbÞ b ðbÞ ðt1 ,t2 Þ, b cðbÞ ðt1 ,t2 Þ, O b ðbÞ ðt1 ,t2 Þ and Pm dðbÞ ðt1 ,t2 Þ Tc S k ðt1 ,t2 Þ, R k k k k by substituting the bootstrap estimates in Eqs. (16)–(18) for the DEA estimates of the output and input distances above. 6. Repeat steps 3–5 B number of times in order to provide the sets ðbÞ

c ðt ,t Þ,b ¼ 1, . . . ,Bg, fPd 1 2 k ðbÞ

b ðt1 ,t2 Þ,b ¼ 1, . . . ,Bg, fA k ðbÞ

c ðt1 ,t2 Þ,b ¼ 1, . . . ,Bg, fTc k ðbÞ

b ðt1 ,t2 Þ,b ¼ 1, . . . ,Bg, fO k ðbÞ

fb S k ðt1 ,t2 Þ,b ¼ 1, . . . ,Bg, ðbÞ

b ðt1 ,t2 Þ,b ¼ 1, . . . ,Bg, fR k ðbÞ

d ðt1 ,t2 Þ,b ¼ 1, . . . ,Bg, fPm k of bootstrap estimates of Pdk(t1,t2), Ak(t1,t2), Tck(t1,t2), Ok(t1,t2), Sk(t1,t2), Rk(t1,t2) and Pmk(t1,t2), respectively. It is worth noting that the empirical distribution of these bootstrap values is precisely the Monte Carlo approximation of c  ðt ,t Þ given U b. the respective sampling distribution of the Pd k

1

2

Although it is clear that the quality of the approach relies in part on the value of B, the practical choice of this value is limited by computer speed. However, in order to ensure adequate coverage of the confidence intervals, Simar and Wilson [37] recommend a value of B ¼2000. After computing the bootstrap estimates, what is important is that we are now able to make inferences on the different effect scores. Constructing confidence intervals at desired levels of significance is not difficult. Although there are different possibilities,10 we work with that proposed by Simar and Wilson [34] which, in their words, ‘‘automatically corrects for bias’’. In order to construct an approximate 100  ð1aÞ% confidence interval for the productivity effect Pdk(t1,t2), and based on the computed bootstrap estimates, we have to sort the values c ðt ,t Þ,b ¼ 1, . . . ,Bg, c ðbÞ ðt ,t ÞPd fPd 1 2 k 1 2 k

ð19Þ

in increasing order and then delete ða=2  100Þ% of the elements at either end of the sorted set. If we select ak, a=2 and ak,1a=2 equal 10 All the proposed methods for obtaining bootstrap confidence intervals are implemented in FEAR [62]. For further details, see Efron [48] or Davison and Hinkley [49].

E. Tortosa-Ausina et al. / Computers & Operations Research 39 (2012) 1857–1871

to the endpoints of the sorted array, the approximated confidence interval for the productivity effect of firm k, Pdk(t1,t2), k¼1,y,K, is  c ðt ,t Þa c Pd k 1 2 i,1ða=2Þ r Pdk ðt1 ,t2 Þ r Pd k ðt1 ,t2 Þai, a=2 :

It is worth noting that the values

ak, a=2

and

employees, or zero bank accounts. Data are taken from the ˜ola.11 Anuario Estadı´stico de la Banca Espan

ð20Þ ak,1a=2

are the

empirical (100  a=2)-th and (100  ð1a=2Þ)-th percentiles of c ðt ,t Þ in (19). c ðbÞ ðt ,t ÞPd the empirical distribution of Pd 1 2 k 1 2 k With the information this interval provides it is possible to ascertain whether growth (or decline) of the productivity effect is significant, i.e. if it is greater than (or less than) zero at the desired significance level. Clearly, the same holds for the rest of the sources of profit change, as it is also possible to assess their significance. Moreover, given that the use of smoothed bootstrap guarantees that the bootstrap distribution will imitate the original sampling distribution of the estimation of the effects, obtaining a bootstrap bias corrected estimation of the effects is also c ðt ,t Þ can be straightforward. The bias of each estimation Pd k 1 2 estimated using the bootstrap sample as 

c ðt ,t ÞPd c ðt ,t ÞÞ ¼ Pd c ðt ,t Þ, d Pd Biasð k 1 2 k 1 2 k 1 2 

c ðt ,t Þ is the sample mean of the bootstrap estimates of where Pd k 1 2 P c ðbÞ ðt ,t Þ. the productivity effect of firm k, that is, ð1=BÞ Bb ¼ 1 Pd 1 2 k Then, from this bootstrap estimation, the bias-corrected estimator for Pdk(t1,t2) can be obtained as 

c ðt ,t Þ: f ðt ,t ÞÞ ¼ Pd c ðt ,t ÞBiasð c ðt ,t ÞÞ ¼ 2Pd c ðt ,t ÞPd d Pd Pd k 1 2 k 1 2 k 1 2 k 1 2 k 1 2 ð21Þ Nevertheless, following the comments by Simar and Wilson [37] and Efron [48], this correction should not be used unless c ðt ,t ÞÞ2 , d Pd b 2 o 13½Biasð s k 1 2

1863

ð22Þ

2

b represents the sample variance of the bootstrap values. where s Again, it is worth noting that this bias-corrected estimator can also be obtained for the rest of the sources of profit change, giving us relevant additional information. These new estimators could either confirm what the original decomposition revealed or express different behavior.

4. An application to Spanish banking 4.1. Data We apply the techniques presented above to a sample of Spanish banks. The Spanish banking system is made up of three types of firms, namely private commercial banks, savings banks and credit unions. Our paper focuses on private commercial banks for various reasons, among which we should emphasize that private commercial banks are the only type of firms whose shares can be traded. Therefore, these institutions are profit-oriented given that the value of their shares depends critically on each year’s profit and loss accounts. Our sample is made up of 41 private commercial banks. Although the number of banks seems low compared with the total banks in the industry (80 banks in 2003 and 75 banks in 2004), our sample accounts for virtually 95% of total assets. Some banks had to be dropped because of not being in continuous existence during the sample years, 2003 and 2004. These changes were mainly caused by mergers and acquisitions as well as the failure, end of operation, or creation of some small banks. In addition to this, we had to drop some firms (usually very small) due to the presence of unavailable or unreliable data—such as zero

4.2. Modeling banking activities We consider that banks use three inputs to produce three outputs. Our choice of inputs and outputs is crucially determined by the available information and, unfortunately, we have had to make some assumptions which can be judged as crude and make it difficult to model bank production with precision. One of the main drivers of our choice is the requirement of elucidating what the cost generated by each input category is, as well as the income raised by each output category. Because the information comes from different sources (balance sheets, profit and loss accounts, as well as ‘‘other information’’ such as the number of employees), building bridges between prices and quantities is not easy. Our model of bank production closely follows the intermediation approach (see Sealey and Lindley [63]), which has gained broad acceptance in the literature.12 Our annual profit figure consists of operating profit, which comes from intermediation activities, as well as providing payment instruments and money holding systems. It is defined as gross profit less gains and losses from trading in stocks and public debt instruments, and less the extraordinary G-T&L profit. We include all sources of bank income contributing to the operating profit. We consider three output categories to raise income: namely, loans, securities, and other sources of bank income. The first two categories relate to each bank’s intermediation activities, whereas the latter is more related to nontraditional sources of income such as fee income. The first output category, loans (y1), consists of all types of loans issued that raise a given amount of interest income; the ratio of the amount raised, less provision for bad debt, to the output quantity yields the corresponding price component (p1). We perform the same operation for securities, both fixed-income securities and other types of securities (y2), obtaining price component p2. As suggested by G-T&L, it would be desirable to further decompose loans and investment income both by the type of instrument and by term, as a way of accounting for risk differences. Unfortunately, this type of information is publicly unavailable as of today. Our last output category (y3) is net commission income, which corresponds to the difference between commission income generated and commission expenses incurred, other operating products and profits from financial operations. There are studies which consider some of these magnitudes in order to measure the nontraditional activities that banks perform (see Rogers [64], Tortosa-Ausina [65]), using fee income as a proxy for these activities. Unfortunately, although we are interested in calculating both quantity and price components associated to this output category, we cannot calculate the price for this output category so the price takes a value equal to one.13 We specify three input categories to model the cost side. The first one is labor (x1), whose quantity component is the number of 11

The information is available through the URL: http://www.aebanca.es. In order to alleviate the curse of dimensionality [35], we decided to restrict our final choice to three outputs only. Although this is problematic, because it may thwart the precise modeling of bank activities, our final choice stands due to relatively recent contributions arguing for the inclusion of nontraditional activities [64,65]. We thank a referee for this suggestion. 13 In the previous version of the article, following G-T&L, and assuming that net commission income hinges on the number of deposit accounts, we proxied the quantity component of net commission income by the number of deposit accounts (y4), whereas its price component (p4) is the ratio of net commission income to y4. Both approaches are not entirely valid given that in one case we cannot use prices, whereas in the other it must be assumed that all fee income is generated by bank accounts. 12

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Table 1 Summary statistics for Spanish banks, 2003 and 2004. Magnitude

2003

2004

Mean Outputs All forms of loans, y1a Securities, y2 Other sources of income, y3 Output pricesb p1 p2 p3

Inputs Number of employees, x1 Fixed assets, x2a Loanable funds, x3a

Median

Std. dev.

Mean

Median

Std. dev.

8,810,607.85 2,943,672.82 67,168.13

1,924,813.86 72,358.91 15,919.56

20,954,430.12 10,368,194.77 196,517.86

9,597,586.64 3,004,794.96 115,636.97

2,050,947.29 52,850.35 14,897.95

22,231,417.06 10,215,362.10 306,201.36

3.42 4.80 –

3.47 3.41 –

0.94 3.74 –

3.06 4.72 –

3.03 3.77 –

0.77 4.52 –

2506.76 113,811.71 10,940,512.53

375 17,040.36 1,707,235.35

5980.19 301,256.87 28,734,560.59

2524.88 109,078.73 11,463,783.92

363 17,545.38 1,835,845.06

5875.24 284,279.33 28,731,480.52

42.47 188.91 1.92 12,603,094.23

39.05 75.10 1.76 1,989,821.27

10.80 268.48 0.50 33,751,042.63

42.06 207.03 1.68 13,488,999.52

39.27 79.94 1.58 2,135,902.80

10.27 313.09 0.38 34,940,630.41

Input prices

o1 a o2 b o3 b Assetsa a b

Thousands of h, converted from constant 1995 pesetas. In percentage.

employees, while the corresponding price component (o1 ) is calculated as a ratio of labor expense to the total number of employees. The second input category, fixed assets (x2), corresponds to each bank’s physical capital, whose price (o2 ) is obtained as a ratio of its associated costs (proxied by nonlabor expense, which is made up of nonlabor operating expense, direct expenditure on buildings, and amortization expense). Finally, we consider those categories responsible for generating financial costs, which account for two-thirds of banks’ total costs. The quantity component (x3) includes all deposits, plus other liabilities which generate interest and other financial expenses. Therefore, the corresponding price (o3 ) is calculated as a ratio of the financial expense generated to x3.14 Summary statistics on inputs and outputs, both for their price and quantity components, are reported in Table 1. Since each category contains different sorts of information, the magnitudes are expressed in different units of measurement, as indicated in the table’s footnotes. Those quantities corresponding to money values are expressed in thousands of constant euros, which have been converted from 1995 pesetas. Table 1 contains information on mean, median and standard deviation. These statistics are included to show the great variability and the extremely positive skewness of the distribution of the output and input data. This property is a consequence of the large values of the mean in relation to the ones corresponding to the median, which

14 We tried to alleviate the dimensionality problem [35, p. 441] by enlarging the sample (a previous draft of the paper contained 13 observations, those corresponding to the banks whose shares are traded in the stock exchange). A drastic solution would be to reduce the number of inputs and outputs when possible. However, this is problematic because modeling bank activities is complex, and we could run the risk of excluding too much relevant information by dropping some inputs or outputs, or aggregating them too much. Therefore, a balance must be reached between the need to avoid the curse of dimensionality and modeling bank activities reasonably. The optimal solution might be to consider some robust methods to measure efficiency, such as the relatively recent order-m estimators [66]. However, this has some problems such as the inclusion of prices. These objectives belong to our immediate research agenda.

corroborates our comments in the Introduction about the presence of some large firms in our data set. The evolution of the magnitudes mirrors some of the tendencies observed in both the financial sector and the Spanish economy. For instance, the 2003–2004 period (and also the immediately previous and following years) witnessed an unprecedented era of both economic and banking expansion, which is a reflection of both the increase of (median) loans (which have nearly doubled in six years in real terms) and the increase of (median) loanable funds, which augmented by almost 50%. The evolution of securities is more modest, because of disparate tendencies for its components. In the case of other securities (which includes the shares that banks’ trade in the stock market), the years analyzed were turbulent ones for stock markets. In the case of fixed-income securities, their volume has diminished sharply. This is because banks have preferred to allocate their funding by issuing larger volumes of loans in order to meet the peak of demand. However, we must take into account that our sample does not include the fastest-growing group of firms, that is, savings banks. Given that we provide data for only two years it is difficult to make any inference about the dynamics of the banking industry. Nevertheless, we can observe some of the tendencies which have prevailed in the Spanish banking system since the mid nineties, such as the decline in bank margins. From 2003 to 2004 the price of loans (p1) decreased from 3.42% to 3.06% on average, whereas the price of loanable funds ðo3 Þ decreased only from 1.92% to 1.68% on average, and for the banks in our sample. Most were able to offset this general trend in the industry by innovating financially and releasing some new products and services, many of which constituted an additional source of fee income. Many banks also attempted to offset the decline in mark-ups by curbing personnel expenses (o1 declined from 42,470h to 42,060h per employee, on average, and for the banks in the sample). But in spite of these general tendencies in the industry, most firms were able to increase profits from 2003 to 2004. The following section is devoted to an in-depth analysis of the sources of that profit change.

E. Tortosa-Ausina et al. / Computers & Operations Research 39 (2012) 1857–1871

Table 2 Decomposition of profit change into quantity effect and price effect (nonbootstrapped), 2003/2004. Firm #

Profit change

Quantity effect

Price effect

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

4375.67  16,008.05 2467.90 45,019.10 144.94  1829.69 5541.18 448.01 672,652.14 6249.59  746.96  4108.80 8294.65 41,543.73  376.29 46,561.11  1168.62 1708.66  9703.71  2333.24  201.98  4024.62 1031.58 6661.47 32,794.81  6519.72  962.48  1426.31  403.43  6179.05 148,125.40  266.20  239.74 98.12 88.89 74.33  1330.27 1482.22 4137.59 4073.37  736.74

44,245.42 27,178.40 2866.22 135,533.16  1150.19  20,126.21 9904.58 1205.49 2,060,274.96 5268.53 12,850.08 65,310.17 43,142.58 158,073.87 276.49  25,320.85 12,342.13 5031.56 18,584.77  2215.79 5586.61 2967.07 1391.98  4346.71 99,515.07  1501.46 818.45  307.22  1607.06  6130.72 318,981.65  1306.33  628.77 102.70  99.48 302.81  1136.92 2960.35 3019.74 5140.84 4054.24

 39,869.74  43,186.44  398.33  90,514.06 1295.13 18,296.52  4363.40  757.48  1,387,622.82 981.06  13,597.04  69,418.97  34,847.93  116,530.14  652.78 71,881.96  13,510.76  3322.90  28,288.48  117.45  5788.59  6991.69  360.41 11,008.18  66,720.26  5018.26  1780.93  1119.09 1203.63  48.32  170,856.25 1040.13 389.02  4.58 188.37  228.48  193.35  1478.13 1117.86  1067.47  4790.98

Mean Median Std. dev.

23,780.70 88.89 107,114.50

72,708.59 2960.35 323,800.15

 48,927.89  1119.09 217,899.99

1865

Table 3 Decomposition of quantity effect into productivity effect and activity effect, statistical significance, 2003/2004. Firm #

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

4375.67  16,008.05 2467.90 45,019.10 144.94  1829.69 5541.18 448.01 672,652.14 6249.59  746.96  4108.80 8294.65 41,543.73  376.29 46,561.11  1168.62 1708.66  9703.71  2333.24  201.98  4024.62 1031.58 6661.47 32,794.81  6519.72  962.48  1426.31  403.43  6179.05 148,125.40  266.20  239.74 98.12 88.89 74.33  1330.27 1482.22 4137.59 4073.37  736.74

Mean Median Std. dev. n

Profit change

23,780.70 88.89 107,114.50

Quantity effect

Price effect

Productivity effect

Activity effect

 2376.20  609.86 1900.99 112,538.17*  61.35  103,578.70*  9716.11  1216.95  468,386.17  655.04  3036.05 20,194.37 24,797.50*  33,303.31 232.33 23,608.55  587.31 1423.76 10,225.94 2854.93* 4629.29 331.84  412.63  4006.37  25,203.29 2251.86 3326.33*  6075.20* 341.80  7984.61* 218,301.86  237.61 2229.08*  405.73  364.38 NA NA 452.58 NA  1735.88  36,416.61

46,621.62 27,788.26 965.23 22,995.00  1088.84* 83,452.49* 19,620.69 2422.44 2,528,661.13* 5923.56* 15,886.13 45,115.80* 18,345.09 191,377.18* 44.17  48,929.40 12,929.44* 3607.80 8358.83  5070.72* 957.33 2635.23 1804.61*  340.34 124,718.36*  3753.32  2507.87 5767.97*  1948.86* 1853.89 100,679.79  1068.72  2857.85* 508.43 264.90 NA NA 2507.77* NA 6876.72* 40,470.84*

 7282.32  301.00 89,058.68

85,673.55 3121.51 409,200.25

 39,869.74  43,186.44  398.33  90,514.06 1295.13 18,296.52  4363.40  757.48  1,387,622.82 981.06  13,597.04  69,418.97  34,847.93  116,530.14  652.78 71,881.96  13,510.76  3322.90  28,288.48  117.45  5788.59  6991.69  360.41 11,008.18  66,720.26  5018.26  1780.93  1119.09 1203.63  48.32  170,856.25 1040.13 389.02  4.58 188.37  228.48  193.35  1478.13 1117.86  1067.47  4790.98

 48,927.89  1119.09 217,899.99

Indicates significant differences from zero at a 5% significance level.

4.3. Results Results are shown in Tables 2–7. Table 2 provides results on profit change ðDpÞ and its decomposition into the quantity effect (Q) and price effect (Pc) for 2003/2004; Tables 3–5 provide the statistical significance for each of the decompositions proposed, while Tables 6 and 7 complement this information by providing confidence intervals. The results in Table 2 have been obtained by computing the profit change between 2003 and 2004 for all firms in the sample directly from expression (6). These are nonbootstrapped results showing that on average profit change was positive between 2003 and 2004. This was mostly because of the positive contribution of the quantity effect, and despite the negative contribution of the price effect. The average contribution of the quantity effect (72,708,590h) was enough to offset the negative price effect (  48,927,890h), yielding positive profit change (23,780,700h). An evaluation with regard to medians produces the same general comments, and again the great variability in the results is remarkable both for quantity and price effects. The causes for these opposite contributions are

multiple and possibly bank-specific. However, they might be related to some trends in the banking industry, which was witnessing an all-time high in the volumes of issued loans and, simultaneously, an all-time low in terms of interest rates. As reported by the Annual Supervision Report on Banking ´n Bancaria, 2005, published by the (Memoria de la Supervisio Bank of Spain and available through www.bde.es), the Euribor 1-year interest rate (Mibor before 1998) decreased from more than 10% to less than 2.5% from 1996 to 2004, whereas the volume of mortgage loans increased by more than 20% per annum from 1997—and by more than 28% from 2003 to 2004 for the entire banking sector. In these particular circumstances, the results reported in Table 2 were partly easy to predict. Despite the decline in bank margins induced by the decrease in official interest rates and, in the particular case of the Spanish banking industry, the enhanced competition triggered by the deregulatory and liberalizing initiatives which had taken place since the eighties, financial

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Table 4 Decomposition of productivity and activity effects into technical change, operating efficiency, scale, resource mix, and product mix effects, statistical significance, 2003/ 2004. Firm #

Profit change

Quantity effect

Price effect

Productivity effect

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

4375.67  16,008.05 2467.90 45,019.10 144.94  1829.69 5541.18 448.01 672,652.14 6249.59  746.96  4108.80 8294.65 41,543.73  376.29 46,561.11  1168.62 1708.66  9703.71  2333.24  201.98  4024.62 1031.58 6661.47 32,794.81  6519.72  962.48  1426.31  403.43  6179.05 148,125.40  266.20  239.74 98.12 88.89 74.33  1330.27 1482.22 4137.59 4073.37  736.74

Mean Median Std. dev. n

23,780.70 88.89 107,114.50

Activity effect

Technical change effect

Operating efficiency effect

Scale effect

Resource mix effect

Product mix effect

 5991.44  2707.98  217.83 151,171.43* 43.50  103,578.70* 25,935.13* 1449.83*  468,386.17  655.04 419.53 32,773.14* 5576.72  42,397.52 80.10 23,763.45  1865.85  22.85 7165.11  979.89* 416.37  744.11  412.63 3063.77*  27,968.13  29.71  733.85 193.43* 36.89  8037.81* 208,990.04  425.19 765.40  405.73  570.23 NA NA 30.94 NA  669.07*  36,416.61

3615.24 2098.12 2118.82  38,633.26  104.85 0.00  35,651.24*  2666.78 0.00 0.00  3455.58  12,578.77 19,220.78 9094.21 152.22  154.91 1278.54 1446.61 3060.83 3834.82* 4212.92 1075.95 0.00  7070.15 2764.84 2281.57 4060.18*  6268.63* 304.91 53.20 9311.82 187.58 1463.68* 0.00 205.85 NA NA 421.64 NA  1066.81 0.00

47,673.70*  57,935.40* 10,060.66* 233,643.51* 2997.21* 106,112.34* 5373.38 10,494.37* NA  3863.51* 39,385.98*  360,078.86*  9414.11  286,687.06*  491.82 451,826.76*  25,792.94* 20,354.47* 43,771.90* 2965.64 14,956.88*  19,791.24*  17,005.32*  3452.52  10,669.64  21,685.99 17,293.18*  3586.78* 7945.07*  28,220.25* NA  135.31 2806.82*  5884.33*  57,394.04* NA NA  7080.70* NA  6050.27  84,245.77*

10,081.27 83,841.67*  5080.60  157,310.21  3708.04*  101,833.61*  6835.09  6001.49 NA 8628.32*  15,166.05 382,819.38* 62,481.46 386,459.96* 834.84  466,168.27* 42,135.57*  18,630.06*  25,498.04  1477.87  8952.02* 25,678.00* 16,102.25*  5184.00 1348.54 19,247.03  14,181.46* 4424.70*  4463.88* 30,998.93* NA  1592.70  1439.86 5914.66* 56,905.05* NA NA 9539.80* NA 10,502.95* 86,588.88*

 11,133.35 1881.99  4014.82*  53,338.30  378.00 79,173.76* 21,082.40  2070.45 2,025,832.02* 1158.75  8333.79 22,375.27  34,722.27* 91,604.27  298.85  34,587.89  3413.20 1883.40  9915.03  6558.49*  5047.54  3251.53 2707.68* 8296.18 134,039.46*  1314.37  5619.60* 4930.05*  5430.06*  924.79  42,430.41 659.29  4224.81* 478.09 753.88 NA NA 48.67 NA 2424.05* 38,127.73*

227.67  1972.17 120,186.95

11,139.17 1091.69 127,803.19

57,906.56  338.43 329,588.11

 6351.09  123.77 90,275.61

 931.23 196.71 9915.45

 39,869.74  43,186.44  398.33  90,514.06 1295.13 18,296.52  4363.40  757.48  1,387,622.82 981.06  13,597.04  69,418.97  34,847.93  116,530.14  652.78 71,881.96  13,510.76  3322.90  28,288.48  117.45  5788.59  6991.69  360.41 11,008.18  66,720.26  5018.26  1780.93  1119.09 1203.63  48.32  170,856.25 1040.13 389.02  4.58 188.37  228.48  193.35  1478.13 1117.86  1067.47  4790.98

 48,927.89  1119.09 217,899.99

Indicates significant differences from zero at a 5% significance level.

institutions managed to reach record profits because of the increase in the amount of intermediated funds (see Oroz and Salas [67]). Therefore, the cost of the gradual deregulation of the Spanish banking system, and the consequent increase in competition in terms of the price structure and the profitability of commercial banks was less severe than what a priori could have been expected. These tendencies are also apparent from Table 1, which shows that the average loan rate for the banks in the sample decreased by more than 10% (from 3.42% to 3.06%) from 2003 to 2004. Because the results for profit change decomposition into quantity effect and price effect are nonbootstrapped, we cannot ascertain whether differences across firms are statistically significant or not. However, following the procedures presented in Section 3, it is possible to perform such an analysis for the different components of the quantity effect. Both Tables 3 and 4 contain information on the statistical significance of our bootstrap

results. We use single asterisks (n) to indicate firms whose effects are significantly different from zero at the 0.05 level (since we constructed 95% confidence intervals). These results are provided for all firms in the sample and for each decomposition of the quantity effect. Therefore, we can ascertain whether a positive or negative value for each quantity effect component (productivity, activity, technical change, operating efficiency, scale, and product mix effect) differs significantly from zero or not.15 As shown in Table 3, on average productivity contributed negatively to the quantity effect, whereas the activity effect

15 There were some observations for which solutions of the linear programming problems were unfeasible. This also occurred in Grifeell-Tatje´ and Lovell [22], where the mixed period linear programs (which in our case is linear program (13) were not guaranteed to have solutions for the smallest banks. As indicated by those authors, the explanation for this is that an output radial expansion may not intersect period t +1 technology for the smallest banks.

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Table 5 Decomposition of quantity effect into productivity and activity effect (bootstrapped), 2003/2004. Firm #

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

Productivity effect

Activity effect

ed P

Lower C.I.

Upper C.I.

e A

Lower C.I.

Upper C.I.

 2376.20  609.86 1900.99 112,538.17  61.35  103,578.70  9716.11  1216.95  468,386.17  655.04  3036.05 20,194.37 24,797.50  33,303.31 232.33 23,608.55  587.31 1423.76 10,225.94 2854.93 4629.29 331.84  412.63  4006.37  25,203.29 2251.86 3326.33  6075.20 341.80  7984.61 218,301.86  237.61 2229.08  405.73  364.38 NA NA 452.58 NA  1735.88  36,416.61

 16,755.75  21,132.69  1562.14 2966.19  524.43  110,702.83  23,616.46  5259.38  724,677.40  1481.60  11,763.11  3384.88 3248.83  103,913.49  604.73  34,850.59  7147.00  4144.95  3085.77 839.24  204.74  6236.54  822.18  11,499.77  64,873.51  1732.38 46.79  6573.57  1332.06  10,705.22  96,385.09  1321.51 580.74  572.84  6119.87 NA NA  511.49 NA  2144.17  48,995.11

74,334.24 65,507.71 19,362.84 291,503.72 1090.76  79,870.56 21,503.61 17,437.58 227,897.01 542.90 14,727.82 57,193.62 145,686.06 97,487.07 1619.11 103,730.48 10,143.73 10,074.34 28,608.72 11,406.69 30,943.18 34,017.74 93.44 8689.77 51,616.39 9693.94 16,340.64  3872.28 3509.97  4862.19 691,902.63 3236.33 8314.63 244.61 17,668.59 NA NA 2121.20 NA 431.12 7149.79

46,621.62 27,788.26 965.23 22,995.00  1088.84 83,452.49 19,620.69 2422.44 2,528,661.13 5923.56 15,886.13 45,115.80 18,345.09 191,377.18 44.17  48,929.40 12,929.44 3607.80 8358.83  5070.72 957.33 2635.23 1804.61  340.34 124,718.36  3753.32  2507.87 5767.97  1948.86 1853.89 100,679.79  1068.72  2857.85 508.43 264.90 NA NA 2507.77 NA 6876.72 40,470.84

 29,452.72  32,782.77  16,319.45  149,983.66  2178.70 59,858.30  10,127.69  16,095.77 2,100,578.14 4739.74  635.03 12,248.86  101,185.82 62,776.91  1282.45  127,216.93 2443.69  4819.91  9895.52  13,064.52  24,476.84  30,105.84 1305.02  12,936.81 48,510.54  10,605.85  14,527.67 3614.86  4862.22  1142.73  372,486.74  4428.62  8705.95  139.66  10,951.07 NA NA 877.03 NA 4729.17 17,435.75

61,070.68 48,328.02 4,429.10 132,640.86  624.39 90,585.87 33,715.06 6481.97 2,785,371.85 6762.63 24,665.12 69,107.98 39,955.06 263,454.99 899.89 10,247.32 19,534.82 9191.59 21,672.03  3012.70 5822.07 9214.56 2223.94 7404.27 164,950.51 325.17 812.88 6268.41  261.08 4602.23 418,334.23 24.49  1199.88 676.55 6107.94 NA NA 3477.84 NA 7285.54 53,711.29

contribution was positive (  7,282,320h vs. 85,673,550h). The values of the medians have the same signs, although their values are much lower, corroborating the high disparities in the banking sector. It is also interesting to note the high variability of the activity effect compared with the productivity effect and price effect. Simultaneously, the productivity effect is further decomposed into a technical change effect and an operating efficiency effect (Table 4). Negative technical change (technical regress) is attributable to a deterioration in the productivity of best practice banks, although the results in Table 4 with positive and negative values suggest a nonneutral shift in technology. Operating efficiency also deteriorated on average, although the median increases. This efficiency reflects the success or failure of the remaining banks to keep pace with the improved performance of best practice banks, and in our case the effect is difficult to evaluate because of the remarkable disparities across banks. Table 4 also shows the decomposition of the activity effect into scale of operations, resource mix effect, and product mix effect, and is conceived to reflect changes in the scale and scope of the operations of the organization. More generally, the table reflects the change in output generated by the change in resource usage. The second (resource mix effect) and third component (product mix effect) reflect the ability of banks to substitute products whose prices are falling and resources whose prices are

increasing for products whose prices are falling and resources whose prices are increasing. On average, the contribution of the activity effect on the quantity effect, in comparison to the contribution of the productivity effect, has been major as a result of a positive average increase in the level of operations that we have commented in Section 4.2. Resource mix and product mix effects (although signs differ when considering the medians) corroborate how disparate tendencies are for many banks in the sample. All three components contribute to yielding a positive activity effect. Therefore, it seems that as a response to a deteriorated price structure, banks managed to substitute products and resources adequately. Nevertheless, summary statistics mask a variety of individual behavior, and we observe that disparities among firms are remarkable. Such results might partly be expected as a reflection of the fact that they also differ in other relevant respects such as size or specialization. Indeed, two of the banks in our sample account for more than 50% of total industry assets—namely, Banco Bilbao Vizcaya Argentaria (BBVA) and Banco Santander. Although profit change for most firms was positive, some of them actually had losses. However, these magnitudes do not represent actual profits (or losses) but operating profits. That is, they do not include extraordinary profits or losses, or provision for loan losses and consequently most firms exhibiting negative values were

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Table 6 Decomposition of productivity effect into technical change and operating efficiency effect (bootstrapped), 2003/2004. Firm #

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

Technical change effect

Operating efficiency effect

Te

Lower C.I.

Upper C.I.

e O

Lower C.I.

Upper C.I.

 5991.44  2707.98  217.83 151,171.43 43.50  103,578.70 25,935.13 1449.83  468,386.17  655.04 419.53 32,773.14 5576.72  42,397.52 80.10 23,763.45  1865.85  22.85 7165.11  979.89 416.37  744.11  412.63 3063.77  27,968.13  29.71  733.85 193.43 36.89  8037.81 208,990.04  425.19 765.40  405.73  570.23 NA NA 30.94 NA  669.07  36,416.61

 14,649.27  21,132.69  1387.22 64,695.90  92.22  110,702.83 17,412.18 269.79  724,677.40  1481.60  2762.59 23,044.37  1272.00  106,271.04  224.61  8505.82  5055.45  4756.02  301.36  2001.42  885.97  3606.54  822.18 359.93  64,873.51  1157.26  2246.71 8.19  394.88  10,705.22  96,385.09  1466.75  263.07  572.84  6119.87 NA NA  378.26 NA  1063.52  48,995.11

2430.03 16,085.33 1035.91 241,435.72 204.32  79,870.56 49,715.70 2681.71 227,897.01 542.90 3887.89 46,985.20 13,059.10 24,062.75 376.17 59,612.91 1048.10 9689.20 15,497.13  141.23 1892.84 1856.09 93.44 6155.13 6103.92 1336.39 586.42 429.26 512.83  4862.19 691,902.63 647.17 2515.26 244.61 10,851.58 NA NA 550.87 NA  347.73 7149.79

3615.24 2098.12 2118.82  38,633.26  104.85 0.00  35,651.24  2666.78 0.00 0.00  3455.58  12,578.77 19,220.78 9094.21 152.22  154.91 1278.54 1446.61 3060.83 3834.82 4212.92 1075.95 0.00  7070.15 2764.84 2281.57 4060.18  6268.63 304.91 53.20 9311.82 187.58 1463.68 0.00 205.85 NA NA 421.64 NA  1066.81 0.00

 5114.20 0.00  1281.58  127,588.88  556.08 0.00  44,545.08  6588.28 0.00 0.00  10,581.11  33,681.49  1039.20 0.00  578.34  62,569.37  4428.72  6806.95  8598.55 2049.95  665.00  5352.66 0.00  13,846.02 0.00  1693.64 727.61  6727.94  1262.03 0.00 0.00 0.00 251.26 0.00 0.00 NA NA  584.89 NA  1281.70 0.00

84,382.97 50,936.97 19,666.12 207,947.75 1047.74 0.00  18,659.87 16,204.05 0.00 0.00 15,159.00 20,190.14 136,937.44 103,547.03 1571.89 79,705.46 11,763.86 3374.24 22,774.56 12,525.30 30,333.37 33,756.74 0.00 6832.77 2154.57 9997.54 16,906.31  4106.92 3580.83 694.88 0.00 3613.24 7927.08 0.00 0.00 NA NA 2260.86 NA 1233.39 0.00

actually profitable. In addition, most of these banking firms are quite small and therefore their losses barely affect the aggregate profits of the industry as a whole. As shown in Table 3, the average negative productivity effect is actually positive in most instances. In addition, for some firms exhibiting a high decline in productivity (for instance, firm #9) this result is not significant at the specified significance level. This pattern can be corroborated through Table 5, which shows bootstrap confidence intervals for the effects. Because the bootstrap confidence intervals for this firm, for instance, include zero, we conclude that productivity change is not significantly different from zero. On the positive side, firm #31, which is another large bank, goes through productivity growth but it is not significantly different from zero either. This can be corroborated through Table 5, which also shows that the confidence intervals for this firm include zero. Therefore, as with any estimator, it is not enough to know whether productivity change is positive or negative, but whether the changes indicated are significant in a statistical sense. In other words, it is important to know whether the result indicates a real change in productivity, or is an artifact of sampling noise [34]. The components of the productivity effect (technical change effect and operating efficiency effect) can also be examined from the

same angle. Although on average there was technical regress, firm-specific results vary a great deal, with many of them showing technical progress, as indicated in Table 4. Despite its dominating effect, technical regress is statistically significant in only four instances (firms #6, 20, 30, 40) given that their confidence intervals (Table 6) excludes zero, whereas the same does not hold for many other banks going through technical regress. In contrast, despite not being the dominating effect, the results for positive technical change (technical progress) are statistically significant in more instances (firms #4, 7, 8, 12, 24, 28). But although technical progress is insignificant and the contribution of productivity change to profit change is modest, its activity effect is large and significant enough to offset the price effect and yield positive profit change. In contrast, the operating efficiency effect, similar to what G-T&L found, is relatively smaller. Our bootstrap analysis provides extra information, indicating that in most instances when its estimation for this effect was different from zero, it was not statistically significant since the confidence intervals included zero (Table 6). Our resampling methods therefore turn out to be especially relevant for these firms, for which it is not possible to ascertain whether their operating efficiency has increased or declined. That is to say, we cannot reject the hypothesis that their efficiency has

E. Tortosa-Ausina et al. / Computers & Operations Research 39 (2012) 1857–1871

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Table 7 Decomposition of activity effect into scale, resource mix, and product mix effect (bootstrapped), 2003/2004. Firm #

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

Scale effect

Resource mix effect

Product mix effect

Se

Lower C.I.

Upper C.I.

em R

Lower C.I.

Upper C.I.

em P

Lower C.I.

Upper C.I.

47,673.70  57,935.40 10,060.66 233,643.51 2997.21 106,112.34 5373.38 10,494.37 NA  3863.51 39,385.98  360,078.86  9414.11  286,687.06  491.82 451,826.76  25,792.94 20,354.47 43,771.90 2965.64 14,956.88  19,791.24  17,005.32  3452.52  10,669.64  21,685.99 17,293.18  3586.78 7945.07  28,220.25 NA  135.31 2806.82  5884.33  57,394.04 NA NA  7080.70 NA  6050.27  84,245.77

30,164.91  98,933.47 4657.82 70,588.18 2466.64 89,178.07  14,589.04 3312.44 NA  6422.69 20,738.30  443,136.87  51,989.58  458,263.44  1562.98 393,832.62  34,959.70 9605.74 15,531.09  827.74 7277.09  29,822.15  20,478.02  17,722.23  107,638.44  30,912.03 10,571.28  7282.33 4608.89  37,100.65 NA  2202.49 208.68  7831.91  78,195.26 NA NA  9197.11 NA  10,877.92  90,598.88

63,138.51  15,278.54 13,791.48 389,490.95 3313.66 113,951.99 20,528.48 14,756.61 NA  1779.98 52,976.99  205,469.19 66,794.95  12,832.25 494.45 588,302.94  3405.15 27,327.23 59,586.31 5800.27 20,215.54  10,898.07  14,267.12 10,039.55 83,043.31 2079.43 21,208.37  1409.96 10,406.74  16,681.44 NA 1779.34 5499.08  5170.38  46,479.68 NA NA  5247.70 NA 1216.78  75,473.92

10,081.27 83,841.67  5080.60  157,310.21  3708.04  101,833.61  6835.09  6001.49 NA 8628.32  15,166.05 382,819.38 62,481.46 386,459.96 834.84  466,168.27 42,135.57  18,630.06  25,498.04  1477.87  8952.02 25,678.00 16,102.25  5184.00 1348.54 19,247.03  14,181.46 4424.70  4463.88 30,998.93 NA  1592.70  1439.86 5914.66 56,905.05 NA NA 9539.80 NA 10,502.95 86,588.88

 4544.84 44,428.19  9004.28  291,238.14  4038.59  111,043.37  21,618.19  10,692.74 NA 6484.82  29,537.87 234,351.29  11,501.61 139,169.61  134.98  589,223.27 27,418.62  25,676.23  39,912.88  4400.20  14,294.23 16,218.56 13,331.43  17,491.56  82,117.22  4517.28  18,199.06 2235.99  7099.97 20,797.34 NA  3038.11  4004.12 5208.29 45,556.42 NA NA 7585.59 NA 3318.71 79,696.06

35,054.77 131,200.69 298.24 12,716.62  3135.90  82,940.89 12,932.46 1185.10 NA 11,506.27 4196.17 467,781.90 106,997.70 552,356.95 1911.66  394,118.46 51,930.58  7846.10 2483.73 2382.71  1100.33 36,271.12 19,789.28 9298.36 100,704.10 29,414.48  7424.53 8309.66  1089.69 39,927.50 NA 66.36 1167.46 7868.12 83,927.93 NA NA 11,821.90 NA 15,321.60 92,713.13

 11,133.35 1881.99  4014.82  53,338.30  378.00 79,173.76 21,082.40  2070.45 2,025,832.02 1158.75  8333.79 22,375.27  34,722.27 91,604.27  298.85  34,587.89  3413.20 1883.40  9915.03  6558.49  5047.54  3251.53 2707.68 8296.18 134,039.46  1314.37  5619.60 4930.05  5430.06  924.79  42,430.41 659.29  4224.81 478.09 753.88 NA NA 48.67 NA 2424.05 38,127.73

 86,197.05  84,022.54  21,135.54  238,537.00  1495.02 40,257.42  7563.71  20,360.18 1,526,291.88  109.61  26,328.23  10,592.18  151,563.67  60,776.38  1648.04  136,232.97  14,562.76  7028.10  29,716.10  14,667.66  30,275.95  36,063.61 2418.85  4596.44 3311.35  8105.43  17,764.10 2744.58  8405.88  3272.30  556,669.89  3059.36  10,146.93  170.00  9919.36 NA NA  1652.17 NA 277.75 12,458.36

6356.00 22,449.28  330.87 58,521.78 132.61 90,227.09 34,814.87 2422.01 2,295,265.85 2079.06 1945.78 46,159.11  11,019.66 170,494.77 577.87 37,335.63 3754.24 7759.09 4026.84  4440.23 219.34 3776.50 3156.57 14,819.02 173,481.69 2465.66  2197.72 5495.89  3589.11 1072.52 295,119.98 1430.05  2622.56 646.22 6522.35 NA NA 1196.33 NA 2908.22 52,089.70

remained stagnant. Thus, in the case of firm #1 (for instance), although the operating efficiency effect is positive (O~ 1 ¼ 3,615,240), it is not possible to conclude whether this apparently positive contribution to the productivity effect, and ultimately to the profit change effect, is statistically significant. The same conclusion can also be extended to many other firms in the sample. On average, the activity effect contribution to profit change is much higher, but this result is partly caused by strong disparity in individual behavior (see Table 3), especially for some particular banks (firm #9). The sign of the effect varies and is significant in many instances, particularly those in which the activity effect is large (firms #6, 9, 10, 12, 14, 17, 23, 25, 28, 38, 41), i.e. we can reject the null hypothesis of zero activity effect. This effect is made up of the scale of operations, the resource mix effect, and the product mix effect. On average, their contributions to the activity effect are positive, but the negative medians for the scale and product mix effect indicate that variability is quite high. In addition, as shown in Table 7, the results are not always statistically significant. Reject the null hypotheses of zero scale effect To sum up, the operating profit change among the Spanish commercial banks in our sample from 2003 to 2004 was the consequence of a number of factors, whose average effect was

positive in four instances and negative in two of them. The negative contributions came from a deterioration in the performance of some of the best practice banks that creates a nonneutral shift in technology (the technical change effect), and a failure of the remaining banks to maintain their level of efficiency (the operating efficiency effect). A further negative contribution was the deterioration in the banks’ price structure brought on by lower official interest rates and increased competition (the price effect). The positive effects were attributable to the scale of operations (the scale effect), and an emphasis on loans and other financial investments having high and relatively stable rates of return (the resource mix and product mix effects). However, these are average effects masking a variety of firmspecific behavior, which do not necessarily coincide with general tendencies. The bootstrap analysis allows us to ascertain whether each effect is statistically significant or not for each firm in the sample or, in other words, whether we can reject the null hypothesis of zero effect (depending on the effect being analyzed).

5. Conclusions This article has provided a bootstrap methodology to perform statistical inference when decomposing changes in business

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profit. The profit change decomposition used here is based on Grifell-Tatje´ and Lovell [22], in which a multi-stage decomposition is considered. Those authors therefore provide us with a view of the determinants of profit change, yet with no explicit intention of carrying out statistical inference. Consequently, it is not possible to ascertain whether the relative contributions to profit change are statistically significant or not. The reasons for this can be found in the a priori inability of data envelopment analysis—the underlying methodology of G-T&L’s proposals—to disentangle the statistical significance of efficiency differentials. Simar and Wilson [35] propose a bootstrap methodology which silences this ‘‘tired refrain’’ (Lovell [50], dixit), and therefore it is now possible to conduct statistical inference in the context of either DEA or FDH. A statistical model describing the process that yields the data in the sample (a data generating process) needs to be defined first, because without this model it will not be possible to construct confidence intervals for the values of interest or to test hypotheses about the production process. The main contribution of our article is to combine a procedure to decompose the sources of profit change with the bootstrap methodology in such a way that it is now possible to ascertain what the main determinants of changes in business profit. Although we might a priori judge this finding as modest, we must consider that a methodology incapable of disentangling the statistical validity of the results is equivalent to using the method of least squares only in a descriptive way, without any projection or conclusion beyond the data that have conducted the fit. Furthermore, since the G-T&L contribution is based on indicators of quantities, this paper presents a methodology which bootstraps the economic decomposition of an indicator of quantities. To our knowledge, this is the first time a bootstrap methodology has been applied in this context. We provide an application to Spanish banking, one of the European banking industries which has undergone the most profound changes over the last 20 years. We restrict the sample to commercial banks, excluding both savings banks and credit unions. The reasons for this are not only to keep the application simple, but also because private commercial banks’ pursuit of profit may be more explicit than other banking firms due to pressure from both actual and potential shareholders. It is also worth noting that Bootstrap could be very helpful in determining possible differences between groups of firms by classifying, comparing or identifying patterns of behavior between them, or for providing an evaluation of different effects on profit change. In other words, we could think of Bootstrap as a tool for performing any statistical inference in the context of profit change and productivity change. On the other hand, there are also some facets in which the methodology presented here could be improved. For instance, in order to make our methods robust both to the curse of dimensionality and the presence of outliers (which are frequent when using bank data), both GrifellTatje´ and Lovell’s [22] and our paper could be extended by using the relatively new order-m techniques [66] to measure efficiency. These techniques are especially appealing, because since the paper by Cazals et al. [66], new relevant contributions have flourished (see, for instance Daraio and Simar [68], Daouia and Simar [69], Daouia et al. [70], Martins-Filho and Yao [71], among others), constituting a promising area of research.

Acknowledgements David Conesa and Carmen Armero would like to thank the financial support for the Ministerio de Educacio´n y Ciencia (jointly financed with European Regional Development Fund) via the

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