Modeling stochastic frontier based on vine copulas

Accepted Manuscript Modeling stochastic frontier based on vine copulas Michel Constantino, Osvaldo Candido, Benjamin M. Tabak, Reginaldo Brito da Costa

PII: DOI: Reference:

S0378-4371(17)30595-2 http://dx.doi.org/10.1016/j.physa.2017.05.076 PHYSA 18362

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Physica A

Received date : 6 February 2017 Revised date : 27 April 2017 Please cite this article as: M. Constantino, O. Candido, B. Tabak, R.B. da Costa, Modeling stochastic frontier based on vine copulas, Physica A (2017), http://dx.doi.org/10.1016/j.physa.2017.05.076 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Modeling Stochastic Frontier based on Vine Copulas Michel Constantinoa,∗, Osvaldo Candidob , Benjamin M. Tabakb , Reginaldo Brito da Costaa a Dom

Bosco Catholic University, Mato Grosso do Sul, Brazil University of Brasilia, Brasilia, Brazil

b Catholic

Abstract This article models a production function and analyzes the technical efficiency of listed companies in the United States, Germany and England between 2005 and 2012 based on the vine copula approach. Traditional estimates of the stochastic frontier assume that data is multivariate normally distributed and there is no source of asymmetry. The proposed method based on vine copulas allow us to explore different types of asymmetry and multivariate distribution. Using data on product, capital and labor, we measure the relative efficiency of the vine production function and estimate the coefficient used in the stochastic frontier literature for comparison purposes. This production vine copula predicts the value added by firms with given capital and labor in a probabilistic way. It thereby stands in sharp contrast to the production function, where the output of firms is completely deterministic. The results show that, on average, S&P500 companies are more efficient than companies listed in England and Germany, which presented similar average efficiency coefficients. For comparative purposes, the traditional stochastic frontier was estimated and the results showed discrepancies between the coefficients obtained by the application of the two methods, traditional and frontier-vine, opening new paths of non-linear research. Keywords: Vine Copulas, Function Production, Efficiency, Stochastic Frontier, Econophysics. 2010 MSC: 00-01, 99-00

1. Introduction

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The efficiency of companies is a field of extensive empirical exploration in the economic literature. The methods for measuring productivity and efficiency or inefficiency have evolved over time, showing that the heterogeneity of firms must be treated with new forms of estimation capable of addressing the endogenous and exogenous characteristics of inputs and outputs. Studies on efficiency increased beginning in the 1950s, when Farrell [1], influenced by studies from Koopmans [2] and Debreu [3], sparked a generation of studies on measuring productivity and efficiency in various areas and themes, such as Fare & Lovell [4], Charnes, Cooper & Rhodes [5], Forsund & Hjalmarsson [6], Kopp [7], Kumbhakar [8], Fare, Grosskopf & Lovell [9] [10], Kumbhakar & Lovell [11], Greene [12], Staub, Souza & Tabak [13], Shi & Zhang [14], Fare, Rolf and Karagiannis & Giannis [15], Zelenyuk [16], [17], [18], and among others. ∗ Corresponding

author Email addresses: [email protected] (Michel Constantino), [email protected] (Osvaldo Candido), [email protected] (Benjamin M. Tabak), [email protected] (Reginaldo Brito da Costa) Preprint submitted to Physica A

June 2, 2017

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With the increase in competitiveness between companies, especially in large industries that usually have open capital and are listed on the stock exchange, the efficiency analysis by Kumbhakar & Lovell provides tools for benchmarking between competitors, helping investors and decision makers to find problems (inefficiency) and improvements, and finally, assists policy makers in economic problems that may arise from the risk behaviors of large national and international companies. The researchers Aigner, Lovell and Schmidt [19], Meeusen & Broeck [20] and Battese & Corra [21] simultaneously developed the stochastic frontier model (SFA) known in the literature. The authors innovated the efficiency frontier approach by assuming two error components in the model. From this SFA model, we propose to model a stochastic boundary using the vine-copula approach. This paper proposes a new stochastic frontier approach to measure technical efficiency. This modeling of the stochastic production function is different from that in previous studies – it constructs a new function through the vine-copula approach. Efficiency scores are calculated for data from 2005 to 2012 for United States, United Kingdom and German listed companies and compared with the Kumbhakar model efficiency scores. Copula theory has been empirically applied in the past decade to model the dependence of financial assets and insurance companies, as per Cherubini et al [22]; Hurd, Salmon & Schleicher [23]; Jondeau & Rockinger [24]; Rodriguez [25], Okimoto [26] e Silva Filho, Ziegelmann & Dueker [27] [28] e Tofoli [29]. Studies that use the efficiency and production function are recent and limited, for example, Shi & Zhang [14] and Iyetomi et al. [30], respectively. Compared with similar studies, we have contributed several empirical advances: first, the new stochastic frontier modeling using the vine-copula approach; second, the use of several periods of time data for companies in different countries; and third, a comparison between the results of the traditional efficiency approach and its differences with the new vine production. This article is divided into this introduction, section 2, which discusses the concept of efficiency, section 3, which deals with the stochastic frontier theory, section 4, which addresses copula theory, vine-copulas and modeling, section 5, which describes the empirical analysis, and section 6, which presents the conclusions. 2. Efficiency

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In the economic literature, three types of efficiency are described: allocative, technical and economical. Technical efficiency, or Farrel’s efficiency, is defined as the firms capacity or management and decision unit (MDU) to extract the highest level of outputs for a given level of input. The allocative efficiency is the capacity or ability of an MDU to use the best ratio of inputs or outputs to minimize costs. Additionally, the concept of economic efficiency is the sum of technical and allocative efficiency. In this study, the analysis lies in the technical efficiency of the inputs Capital (k) and Labor (l) and the output Product (y). From a mathematical point of view, the concept of efficiency is considered the vector y ∈ Y 0 0 0 is said to be efficient if @ a vector y ∈ Y such that y ≥ y and y 6= y. Farrel [1] used a unit isoquant to measure the economic efficiency and decompose it into technical and allocative efficiency. The isoquant must be estimated from observations of a sample, making it easier to observe the efficiency of a firm through function border production. From the Farrel’s model, nonparametric and parametric approaches have been used by many researchers and allowed new theoretical and practical insertions in the method. While the deterministic approach gained many followers in the context of economic theory. 2

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In contrast, or in evolution, to the deterministic model, the compound error model or stochastic model are more complete and suitable for firm efficiency analysis. The stochastic frontier proposed by Aigner, Lovell and Schmidt [19] and Meeusen & Van den Broeck [20] fixes several problems encountered in the deterministic model. The frontier model uses various types of inputs and outputs to analyze efficiency, ranging from profit, costs of capital, labor and incomes. The greater contribution was the insertion of the inefficiency term in the equation. The following section is dedicated to showing the stochastic frontier model used in the current literature and introduces the functional form of the production function used for empirical application. 3. Stochastic frontier

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The stochastic frontier model is an extension of the deterministic model Yi = f (xi ; β)e−ui , which was represented by Battese [21]. This modeling, assigning the random variable ui to all changes in the efficiency measurement and ignoring exogenous aspects of firm control, proved to be limited in its measurement of performance and advantageous for being computationally simple (Sarafidis)[31]. The main idea of Aigner, Lovell and Schmidt [19] and Meeusen & Broeck [20] was to divide the error term into two components, a symmetrical one, which allows random variations of the boundary between firms and other types of statistical noise, and a unilateral one that captures the effects related to the inefficiency of the firm compared to the efficient frontier. This formulation observes deviations from the production function that can arise from two sources: i) the production inefficiency that would necessarily have to be negative; and ii) specific effects for the company, which could be of any form. In order to incorporate this feature, there is no need to introduce another random variable representing all errors of noise and statistical measurements. The stochastic model includes a compound error term that is a summary error term of two sides measuring all the effects. The effects of the company’s control (exogenous) are a nonnegative error term measuring technical inefficiency. The existing modeling standard currently used in the literature is represented as follows: Yi = f (xi ; β)evi −ui , i = 1, 2, · · · , n

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(1)

where vi is a random error term that has zero mean and exogenous effects on the firm. The term ui represents a non-negative random variable associated with the factors that cause technical inefficiency of the (i-th) firm. In this model, the component that represents the exogenous shock vi is assumed to be independent and identically distributed (iid) with normal distribution. Aigner et al. [19] assumed that the terms of stochastic error vi are independent and identically distributed (iid) and normal random variables with zero mean and constant variance σ2v , which means vit ∼ iidN(0, σ2v ). For other authors, the error term ui that measures the products insufficiency from its maximum value given by the stochastic frontier can follow the medium normal distribution, truncated normal, exponential and gamma (Aigner, Lovell, Schmidt [19]; [32]). Greene [32] states that the frontier can be estimated by MQO, but the most efficient method to do so is the maximum likelihood. Greene [33] and De Negri [34] state that the normal and medium normal distributions have been used frequently in empirical studies and argue that the errors distribution is an assumption, and because of this, the definition of the distribution is carried out somehow randomly. 3

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To estimate the efficient stochastic frontier, the chosen functional form was a conventional production function that relates product (y), capital (k) and labor (l). According to Kopp & Smith [35] and Good et al [36], the functional form is important but has a low impact on the estimated efficiency, meaning that the efficiency level is more affected by the frontier estimation method than by the chosen functional form of the production frontier. As described in Coelli et al. [37], the famous Cobb-Douglas function in the stochastic frontier model takes the logarithmized form: lnyi = β0 + β1 lnxi + vi − ui

(2)

where • β0 + β1 lnxi : deterministic component • vi : noise • −ui : inefficiency 110

The measurement of technical efficiency (TE) assumes values between 0 and 1, ensuring that the level of production from a firm yi is always below or over the stochastic frontier f (xi ; β)evi . T Ei =

f (xi ; β)evi e−ui yi = = e−ui f (xi ; β)evi f (xi ; β)evi

(3)

The innovative implication of this study lies in using vine-copulas to construct the production function. Traditional f (xi ; β)evi will be replaced using the estimated value function, as modeled in the following sections. 115

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4. Copulas The study of copulas and their applications in the economy is a modern and recent phenomenon; until recently, it was known only in the statistical literature. The pioneering of studies with copulas is due to Frechet [38] and Sklar [39]. In the late 1990s, the notion of copulas became more popular. Two books were published and became standard references for the next decade: Joe [40] placed emphasis on multivariable models and Nelsen [41] published the first edition of the introduction to copulas. The interest in copulas in the applied field increased beginning in the year 2000. In addition to the book by Cherubini et al [42], a variety of publications with methodological variations have emerged. Silva Filho et al [27] [28] explore models of conditional copulas with regime change, Shi & Zhang [14] analyze efficiency costs with copulas, and Iyetomi et al [30] use copulas to estimate the dependence of the variables from the production function on Japan’s listed companies in 2006. An interesting application of copulas can also be found in [43]. The increase in publications on the application of copulas in financial market models with multivariate distributions motivates the improvement of the production function modeling. Copulas are functions that connect a multivariate distribution function to its marginal distributions of any dimension. They have all relevant information about the dependence structure between random variables. The characteristics of the copulas allow a greater flexibility in the modeling of multivariate distributions and their marginal ones. Because (i) there is more information about the marginal 4

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ones; (ii) we can derive joint distributions from non-normal margins; (iii) it is possible to separate the effect of dependence between the variables. (Nelsen; Silva Filho) [44] [27]. Copulas emerged as a tool to create multivariate distributions that are more flexible and more realistic, and currently, they appear in a variety of publications. Although there are a large number of bivariate copulas, both parametric and non-parametric, the number of copulas with more than 2 dimensions is quite limited. The construction of trivariate copulas, as is the case in the production function, is recognized as a difficult dimensionality problem. To resolve this dimensionality problem, Joe [40] introduced a method to construct multivariate distributions based on pair-copulas. Bedford and Cooke [45] propose the use of vine diagrams to organize the possible decompositions into pair-copulas. The method consists of decomposing multivariate density in a cascade of bivariate copulas (hence, the origin of the pair-copula or vine-copula term), and their marginal densities. 4.1. Basic concept

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A copula function is a multivariate distribution C, with margins uniformly distributed in [0,1] interval, U(0, 1). One of the main results of copula theory is Sklar’s Theorem. The Sklar’s theorem allows one to create multivariate distribution function from margins with any dimension. 0 That is, for random variables (X1 , · · · , Xd ) with joint distribution F and marginal distributions Fi , i = 1, · · · , d, Sklar’s theorem states that F(x1 , · · · , xd ) = C(F1 (x1 ), · · · , Fd (xd )),

(4)

where C is a d-dimensional copula. Additionally, if F is absolutely continuous and F1 , · · · , Fd are strictly increasing and continuous, d

f (x1 , · · · , xd ) = [ ∏ fk (xk )] × c(F1 (x1 ), · · · , Fd (xd ))

(5)

k=1

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where lower-case letters denote the corresponding density function. In other words, copulas conveniently allow separating the modeling of marginals and dependence in terms of copulas. Extensive references about copulas include the books by Joe [40] and Nelsen [44]. Joe [40] originally proposed multivariate modeling using a bivariate copula specification as dependence models for the distribution of certain pairs of conditional variables to a given set of variables. According to Dissmann et al [46], these sets are called pair-copulas and are independent building blocks used to build multivariate distributions. The choice of multivariate copulas is rather limited, in contrast to the bivariate case, where there is a wide variety of different types of copulas showing flexible and complex dependence standards. The pair-copula constructions (PCC) overcome this problem because each hierarchical level built involves only arbitrary bivariate copula specifications as building blocks and thus allows very flexible models that are able to capture different dependence structures (Brechmann & Czado, 2013 [47]). 4.2. Vine-Copulas

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Vine-copulas are the denomination given to the pair-copula constructions as introduced by Aas et al [48]. Illustrating the PCC in three dimensions, we have the following.

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Let (X1 , X2 , X3 ) be random variables with joint distribution F, margins Fi , i = 1, 2, 3 and all corresponding density functions well-defined. The density function for F can be written as follow f (x1 , x2 , x3 ) = f1 (x1 ) f (x2 |x1 ) f (x3 |x1 , x2 )

(6)

By the Sklar’s theorem one can obtain

f (x2 |x1 ) =

f (x1 , x2 ) c1,2 (F1 (x1 ), F2 (x2 )) f1 (x1 ) f2 (x2 ) = = c1,2 (F1 (x1 ), F2 (x2 )) f2 (x2 ) f1 (x1 ) f1 (x1 )

and f (x2 , x3 |x1 ) c2.3|1 (F1(x2 |x1 ), F(x3 |x1 )) f (x2 |x1 ) f (x3 |x1 ) = f (x2 |x1 ) f (x2 |x1 ) = c2,3|1 (F(x2 |x1 ), F(x3 |x1 )) f (x3 |x1 )

f (x3 |x1 , x2 ) =

= c2,3|1 (F(x2 |x1 ), F(x3 |x1 ))c1,3 (F1 (x1 ), F3 (x3 )) f3 (x3 )

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with F(x|υ) =

∂Cx|υ j |υ− j (F(x|υ− j ,F(υ j |υ− j )) ∂F(v j |υ− j )

(7)

where Cxυ j |υ− j is a bivariate copula and υ− j denotes a vector with j − th component of υ removed. Replacing the results from Sklar’s Theorem into (6) the joint density function, in terms of bivariate copulas, becomes f (x1 , x2 , x3 ) = f1 (x1 ) f2 (x2 ) f3 (x3 )c1,2 (F1 (x1 ), F2 (x2 ))c1,3 F1 (x1 ), F3 (x3 )) · c2,3|1 (F(x2 |x1 ), F(x3 |x1 )) 180

(8)

The bivariate copulas C1,2 , C1,3 and C2,3|1 can be chosen independently of one another so that a wide variety of dependence structures can be modeled using a multivariate vine specification. Vines are graphical representations to specify such PCC. They were introduced by Joe [49] and Bedford & Cooke [45] and described in more detail by Kurowicka & Cooke [50] and Kurowicka & Joe [51]. Vines’ statistical inference was considered by Dissmann et al [46]. 4.3. Vine copula specification

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The construction of a vine specification was described by Dissmann et al [46]. Assembling a specification copula vine for a given set of data requires the following different tasks: 1. Selecting the vine structure and choosing which conditional and unconditional pairs to use. 2. Choosing from a bivariate copula family for each selected pair for the first item. 3. Estimating the parameter(s) correspondent(s) for each copula.

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According to Dissmann et al [46], the three steps proposed are needed for a vine copula specification and indicate that a way to find the best model is to perform steps (2) and (3) for all possible vine constructions. 6

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Using the sequential method 1 to select a vine specification based on Kendall’s tau (τ), it is necessary to decide the pairs of variables for which we want to specify the copula. Proceeding sequentially, beginning with the definition of the first tree to the vine, continuing with the second tree, and so on, captures the stronger dependencies in the first tree, as these are typically the most important dependencies to be modeled explicitly and accurately. For each pair-copula, the corresponding empirical Kendall’s tau (τ) is calculated as dependence measure, given that dependence is measured regardless of distribution and, therefore, is especially useful when combining different (non-Gaussian) families of copulas. The process to obtain the (τ) of each chosen family is shown further. The value of the Kendall’s tau lies in the range between -1 and 1. If the coefficient has a value of 1, there is a perfect positive dependence between two variables; however, if the value is -1, there is a perfect negative dependence, and when it is zero, there is no dependence. The criterion for choosing a particular set of trees (structure) is to maximize the sum of the absolute values of Kendall’s tau for the set of pairs in each tree 2 , i.e. max

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|bτi j |

where bτi j denotes the empirical Kendall’s tau for each pairs (edges, e) on each tree for all trees3 . The next step is the choice of copula families. These functional forms of copulas fit the data and do not need to belong to the same family; therefore, the estimation will define the best family for each pair. These families are chosen by the AIC criterion, and the selection process consists of attributing a value of AIC for each possible family and then selecting the copula with the smaller AIC4 . This above mentioned model selection procedure was introduced by Brechmann & Czado [47] and written in the R software5 with the VineCopula statistical package used in these estimates. The aforementioned procedure allow one to obtain a joint distribution function among random variables. In the present work, the joint distribution among product (y), capital (k) and labor (l) is obtained. There are three vine specifications available, which is represented by its joint density functions below f (y, k, l) = fy (y) · fk (k) · fl (l) · cyk (Fy (y), Fk (k)) · ckl (Fk (k), Fl (l)) · cyl|k (Fy|k (y|k), Fl|k (l|k)) (V1 )

f (y, k, l) = fy (y) · fk (k) · fl (l) · cyl (Fy (y), Fl (l)) · ckl (Fk (k), Fl (l)) · cyk|l (Fy|l (k|l), Fl|k (l|k))

(V2 )

f (y, k, l) = fy (y) · fk (k) · fl (l) · cyk (Fy (y), Fk (k)) · cyl (Fy (y), Fl (l)) · ckl|y (Fk|y (k|y), Fl|y (l|y)) (V3 ) 215

where Fi (·), i = y, k, l are marginal distribution and F(·|·) (·|·) are conditional distributions given by (7). 1 See

Dissmann et al [46] a more extensive discussion: see Brechmann [52] 3 The choice for using Kendall’s tau rank correlation is because it is asymptotically more efficient than other dependence measures including Spearman’s rho. See Croux and Dehon [53] for more details. 4 In this context of Maximum Likelihood Estimation with copula families with the same number of parameters, different information criteria point out to the same model choice. The preference for using AIC is because it is asymptotically optimal with the least mean squared error. See for instance Yang [54] 5 See: Ulf Schepsmeier, Jakob Stoeber, Eike Christian Brechmann, Benedikt Graeler ”[email protected]” 2 For

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It is noticeable that each density above can be separable into three parts: the marginal densities fi , the unconditional copula densities c(·) and the conditional copula density c(·|·) . This is quite flexible for modeling since one can estimate in a first step the margins and in a second step, using the estimated margins parameters, estimate the parameters for all bivariate copulas. This method is known as Inference Function for Margins (IFM). All the parameters are obtained by Maximum Likelihood Estimation (MLE). The log-likelihood function for the copula estimation step for V1 6 is given by I

L (Θ1 |U) = ∑ [ln cyk (Fy (yi ), Fk (ki )|θ1 )+ln ckl (Fk (ki ), Fl (li )|θ2 )+ln cyl|k (Fy|k (yi |ki ), Fl|k (li |ki )|θ3 )] i=1

where Θ1 = {θ1 , θ2 , θ3 } includes all bivariate copula parameters to be estimated from a sample with I points. 4.4. Computing conditional expectation and the vine production 220

An stochastic production function is just the conditional expectation of the product conditional on capital and labor. That is, E[y|k, l] =

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y f (y|k, l)dy

(9)

where f (y|k, l) is the density of y conditional on k and l. According to Aas et al [48] and Kurowicka and Joe [51], the joint distribution of the variables y, k and l can be written as F(y, k, l) =

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Z y

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Ck,l|y (Fk|y (k, z), Fl|y (l, z))dFy (z)

(10)

where Ck,l|y (Fk|y (k, z), Fl|y (l, z)) = F(k, l|y) satisfies Sklar’s theorem. The conditional distribution F(y|k, l) is given by F(y, k, l) (11) F(y|k, l) = F(k, l) where it is well known that limy−→+∞ F(y, k, l)=F(k, l). Considering this and the three possible multivariate distributions for y, k e l, V1 , V2 and V3 , three different specifications for the conditional density function f (y|k, l) can be obtained and represented as follows f (y|k, l) = fy (y) · cy,k (Fy (y), Fk (k)) · cy,l|k (Fy|k (y|k), Fl,k (l|k))

(E1 )

f (y|k, l) = fy (y) · cy,l (Fy (y), Fl (l)) · cy,k|l (Fy|l (y|l), Fk,l (k|l))

(E2 )

f (y|k, l) = fy (y) · cy,k (Fy (y), Fk (k)) · cy,l (Fy (y), Fl (l)) ·

(E3 )

ck,l|y (Fk|y (k|y), Fl|y (l|y)) ck,l (Fk (k), Fl (l)

With this, the conditional expectation in (9) can be computing using Gaussian Quadrature. 6 The

log-likelihood function for V2 and V3 are quite similar.

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4.5. Stochastic vine frontier and technical efficiency estimation

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With the development of the frontier method, the regression technique has been widely used to analyze firm efficiency. The current practice is based on the assumption of normality and concentrated in the center of the distribution. However, we are more interested in modeling the data distributions and analyzing the non-linear dependence and the tails of the distribution. The data show asymmetry and heavy tails, which make the existing methods inappropriate for measuring effectiveness. To solve this problem, we propose a model based on vine copulas when companies are timeinvariant. The objective is to present a stochastic frontier model that accommodates the nonlinear analysis and a firm’s efficiency tails. yi = α + f (xi ) + (vi − ui ), u  0

(12)

yi = E[yi |ki , li ] + (vi − ui ), u  0

(13)

Here, yi is the total product observed for the company i, and f (xi ) denotes the frontier vine production function built with copula specifications. The random error of the stochastic frontier model consists of two parts: vi represents the statistical noise and ui is a non-negative random variable that represents the inefficiency of the ith firm. The estimation for the stochastic vine frontier function is represented as well

R

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where α + f (xi ) is replaced by E[yi |ki , li ] = ℜ+ yi f (yi |ki , li )dyi , which represents the theoretical and empirical contribution of this work. In order to calculate the technical efficiency in (3), additional assumption are required, i.e., distributional assumptions on vi and ui have to be made. Since the main goal of this paper is model a stochastic frontier implied by a vine copula based production function, will be used a Normal-Exponential model: vi ∼ N(0.σ2v ), ui ∼ Exponential(λ) and ui and vi are independent of each other and of the variables of the in the production function. This specification is chosen due to its simplicity and be widely used in the literature alike7 . To estimate all the distributional parameters a MLE is performed. 5. Empirical analysis

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The empirical analysis shows the description of the data and the results of the performed calculations and estimations. Tables 1 and 2 show the results of technical efficiency scores and their standard deviations. Table 3 presents the copulas parameters, with statistically significant results. Table 4 presents the results of dependencies for each pair-copula. Tables 5, 6 and 7 show the descriptive statistics of the data, and Tables 8, 9 and 10 present the estimated parameters for the marginal distributions. 5.1. Data description and vine production results

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The data were obtained from the financial terminal Thomson Reuters Eikon and are annual data of financial statements for listed companies in the United States, Germany and England for the years between 2005 and 2012. This is the biggest time span where the data for each firm in 7 See

for instance Aigner et al. [19] and Greene [55].

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the sample is consistent and complete. It is important to notice that the estimation is carried out year by year. This is need because there are different firms for each year and a in a cross-sectional estimation allow one to observe how the production function parameters evolve over time. The variables used for analysis were from the production function, product (y), capital (k) and labor (l), which are, respectively, y = total revenue, k = fixed assets and l = labor costs. These variables reach consensus in the economic literature that uses applications with the production function. The first result from the vine copula specification procedure, described in section 4.3, the selected vine structure was V3 . Which implies the following three pair-copula to be analyzed: (y − k), (y − l) and (k − l|y). For the US, companies were selected from the S&P500 (Standard & Poor’s 500). Companies from Germany were selected from the Frankfurt Stock Exchange (Xetra system), for England, companies were selected from the London Stock Exchange. The total number of observation along 8 years used in the analysis was 13.857 for the three countries, with 2.617 for the S&P500, 4.155 for Germany and 7.085 for England. Firms that presented data with negative values were excluded. This action is justified, first, because the data are log linearized for the construction of marginal distributions, and second, because negative production, capital or labor may represent an inactive state or acquisition by another company. The same procedure is used in works with copulas; see Shi & Zhang [14] and Iyetomi et al.[30]. Another inference in the database was not using companies in the banking and insurance sectors because the calculations of economic indicators of product (y), capital (k) and labor (l) differ for these firms. Additionally, in the banking sector, other variables are relevant such as ownership, as discussed in [56]. To calculate the marginal distributions the Normal distribution function is used8 . The results are very similar to the parameters and the standard deviation for the three countries in all years. The Kolmogorov-Smirnov (KS) and Anderson-Darling (AD) tests are performed in order to test the transformed marginal (Fi (i), i = y, k, l), and there is no evidence that they are not uniform (0,1). The results for the marginal are shown in Tables 8, 9 and 10. Once the marginal distributions and the vine structure are obtained, one can proceed to the pair-copula family specification and copula parameters estimation. At this point, it is worth to that the copula family is chosen among six different copula functions: Gumbel (asymmetric to the right - right tailed), Clayton (asymmetric to the left - left tailed), Normal (symmetric with no tails), Student-t (symmetric with equal tails), BB1 and BB7 (asymmetric with possibly different right and left tails - right and left tailed). Table 3 shows the estimated parameters for each selected copula. It can be seen that for the pair y, l the selected copula is Gumbel. This result implies that the relationship between output and labor is asymmetric to the left, suggesting that for bigger firms this relationship is stronger. This is true for all countries in all years. For German and England, the Gumbel copula is selected for the pair y, k suggesting that for bigger firms the output-capital relationship is stronger. For US, the Student-t copula is selected for y, k, indicating in this case a symmetric with tails relationship between output and capital. For the conditional pair k, l|y the selected copula is Student-t for German and England and Normal9 for US. It is worth to notice that this relation is negative, 8 Gamma distribution was used too. However, the results were not different from the Normal distribution and it will not be reported here 9 Except for 2005 and 2007 that the selected copula is Student-t.

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which implies that conditional on some output level the capital-labor relationship is negative. The dependence coefficients can be found in Table 4. All these results are in line with the production function definition, however, there are evidences of asymmetries in that relation. This suggest that using the vine copula approach can add to the production function analysis. With the estimated joint distribution V3 , the conditional distribution f (y|k, l) can be obtained and thus, the conditional expectation in (9) can be computed. This conditional expectation is in fact the vine production function the will be used in the stochastic frontier estimation and technical efficiency measure. 5.2. Stochastic vine frontier results

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The results of the technical efficiency indexes measured by the frontier vine method are shown in Table 1 and shown by the average efficiency of the companies in each year and for each country. For the traditional stochastic frontier method, the results are shown in Table 2. To complement the efficiency analysis, we calculated the variances of errors σ2u and σ2v for the new vine production and traditional models.

Average σ2u σ2v Average σ2u σ2v Average σ2u σ2v

Table 1: Average Scores of Technical Efficiency Stochastic Frontier Vine S&P500 2005 2006 2007 2008 2009 2010 2011 0.56647 0.51702 0.56542 0.53898 0.57273 0.59915 0.61647 0.71937 1.10920 0.75646 0.99877 0.70030 0.58992 0.55294 0.68923 0.69267 0.78552 0.60682 0.71017 0.69393 0.70922 Germany 0.24933 0.25679 0.27051 0.25042 0.26401 0.25509 0.24096 2.82278 2.66148 2.40732 2.36945 2.39728 2.46319 2.63712 1.67917 1.64489 1.57562 1.53919 1.65550 1.73752 1.88413 England 0.21299 0.22692 0.24079 0.25042 0.25063 0.24485 0.25288 2.99210 2.79723 2.40426 2.30903 2.32615 2.31358 2.19966 2.17253 2.13259 2.15475 2.06186 2.02898 2.12531 2.06366

11

2012 0.55311 0.79552 1.14846 0.20306 3.08736 2.55243 0.25374 2.15179 2.08111

320

325

330

The companies listed on the S&P500 between the years 2005 and 2012 showed relative efficiency ratios with an average of 0.56, as shown in Table 1. The companies in general are very close to this average, which may indicate relative efficiency stability among firms. In Germany, the listed companies had an average efficiency coefficient around 0.25, a low number, showing that, despite being very close to the average over time, German companies have a high degree of inefficiency, as measured by the frontier vine. Following Germany’s inefficiency behavior, companies in England had average efficiency scores around 0.24, also well below expectations for developed economies and companies listed on the stock exchange. Among the companies of the three countries, the US showed higher relative efficiency compared to Germany and England between the years 2005 and 2012. It is noticeable that the variance of errors σ2u and σ2v for the S&P500 is much smaller than the variance for Germany and England, showing firms’ heterogeneity and volatile behavior over time. To complement this study, data were estimated using the traditional stochastic frontier method, and the ui error term was calculated with the exponential distribution. The results are shown in Table 2 and included divergent results from the frontier vine method.

Average σ2u σ2v Average σ2u σ2v Average σ2u σ2v 335

340

345

350

Table 2: Average Scores of Technical Efficiency - Traditional Stochastic Frontier SP500 2005 2006 2007 2008 2009 2010 2011 0.98162 0.98362 0.98555 0.98555 0.98328 0.98353 0.98344 0.16975 0.13845 0.10859 0.10859 0.14559 0.14253 0.14414 0.01018 0.01213 0.10686 0.10686 0.02304 0.01622 0.01549 Germany 0.97333 0.97375 0.97304 0.97375 0.97304 0.97390 0.97501 0.22254 0.21931 0.23213 0.21931 0.23213 0.21532 0.19856 0.08165 0.17295 0.11551 0.17295 0.11551 0.08626 0.07556 England 0.98215 0.98163 0.97531 0.97136 0.96854 0.97383 0.97748 0.09255 0.09922 0.17786 0.24698 0.29046 0.20402 0.15253 0.45285 0.37108 0.25007 0.18946 0.18228 0.27271 0.47045

2012 0.98314 0.15111 0.01536 0.97792 0.15872 0.18302 0.97487 0.19145 0.32947

The efficiency ratios were high, resulting in an average efficiency around 0.98 for all countries. This implies that there is a change in efficiency analysis when linearity of the data is not assumed. This difference between the two approaches shows evidence of the deepening of traditional methods and the new vine production. The traditional frontier model showed that technical efficiency indices for S&P500, German and English companies were high and not very volatile over time. The small inefficiency presented and the low variance of the errors in the traditional stochastic frontier differ from the frontier vine results. The difference in technical efficiency is estimated by the traditional and frontier vine mode with the ui error term using the exponential distribution. The series estimated by traditional OLS are estimates in log and in level, which have a common tendency and may lead to spurious results with very little inefficiency. Because the vine-copula method used is invariant to any monotonic transformation, the results show more inefficiency than the traditional results. Several families of copulas were found, including Gumbel, t-student, BB7, Gaussian (Normal) and Clayton that characterized the changes in distributions and dependence of each pair-copula, showed that firms are heterogeneous and have different dependence structures in each country and each year. 12

6. Conclusions

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The proposal was to model and analyze the Farrel efficiency of firms listed through the vine copula methodology. The results showed that firms are heterogeneous and have different dependence structures in each country and each year. Several families of copulas were found, including Gumbel, tstudent, BB7, Gaussian (Normal) and Clayton that characterized the changes in distributions and dependence of each pair-copula, confirming the inappropriate use of the multivariate normality hypothesis and the symmetry of conventional works. It should also be noted that with the new approach, the variance of errors σ2u and σ2v for the S&P500 is much smaller than the variance for Germany and England, showing firms’ heterogeneity and volatile behavior over time. The new stochastic efficiency frontier proved different from traditional Kumbhakar model, concluding that copulas can be an interesting instrument, complementary and useful for applications in stochastic frontiers. The divergence between traditional and vine efficiency scores opens new possibilities for research and discussion of measurement methods for efficiency frontiers. This study expects new comparisons and tests to be applied from the presented method, adding new perspectives and new conclusions for empirical research. Acknowledgements FUNDECT - Research Foundation of the State of Mato Grosso do Sul. Benjamin M. Tabak and Osvaldo Candido gratefully acknowledges financial support from the CNPq Foundation. References

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7. Annex

Table 3: Estimated copula parameters - Normal marginal distributions Year

Pair-copula y, k - Student-t

2005

y, l Gumbel k, l|y Student-t y, k Student-t

2006

y, l Gumbel k, l|y Normal y, k Student-t

2007

y, l Gumbel k, l|y Student-t y, k Student-t

2008

y, l Gumbel k, l|y Normal y, k Student-t

2009

y, l Gumbel k, l|y Normal y, k Student-t

2010 -

y, l Gumbel k, l|y Normal y, k Student-t

2011

y, l Gumbel k, l|y Normal y, k Student-t

2012

y, l Gumbel k, l|y Normal

SP500 pair1 0.75885 [0.02679] 4.82445 [0.22563] -0.1387726 [0.06166] 0.73686 [0.22424] 4.83597 [0.03487] -0.21344 [0.03963] 0.73036 [0.21843] 4.69493 [0.03385] -0.18523 [0.04075] 0.73651 [0.20871] 4.48614 [0.03450] -0.22113 [0.03996] 0.70160 [0.20012] 4.30559 [0.03698] -0.16593 [0.04160] 0.71094 [0.20105] 4.32585 [0.03610] -0.16455 [0.04169] 0.71448 [0.20144] 4.32774 [0.03598] -0.13988 [0.04321] 0.99064 [0.19779] 4.24236 [0.03590] -0.12177 [0.04442]

pair2 2.67848 [0.64802] 9.41692 [4.02678] 2.38817 [0.60375] 3.01688 [1.01366] 13.14097 [0.00071] 3.48470 [1.45753] 3.20776 [1.23538] 3.30895 [1.26958] 3.20937 [1.12115] 1.37071 [0.99921] -

Pair-copula y, k Gumbel y, l Gumbel k, l|y Student t y, k Gumbel y, l Gumbel k, l|y Student-t y, k Gumbel y, l Gumbel k, l|y Student-t y, k Gumbel y, l Gumbel k, l|y Student-t y, k Gumbel y, l Gumbel k, l|y Student-t y, k Gumbel y, l Gumbel k, l|y Student-t y, k Gumbel y, l Gumbel k, l|y Student-t y, k Gumbel y, l Gumbel k, l|y Student-t

16

German pair1 3.21644 [0.12149] 7.24322 [0.27372] -0.1035665 [0.04872] 3.19760 [0.11803] 7.58242 [0.28123] -0.08786 [0.04817] 3.15823 [0.11505] 7.57408 [0.27901] -0.04616965 [0.04686] 3.15330 [0.11622] 7.23800 [0.26908] -0.04330 [0.04764] 3.18619 [0.11666] 7.23421 [0.27001] -0.00371 [0.04702] 3.11713 [0.11339] 7.54522 [0.27666] -0.00976 [0.04661] 3.20401 [0.11671] 8.05028 [0.29535] -0.03755 [0.04738] 3.10747 [0.11311] 15.78964 [9.58409] -0.03990 [0.04590]

pair2 10.93458 [4.17504] 10.2208 [4.11625] 12.64496 [5.74913] 10.74928 [3.52165] 16.06468 [7.91225] 12.15357 [5.32640] 11.09154 [4.77258] 15.78964 [9.58409]

Pair-copula y, k Gumbel y, l Gumbel k, l|y Student-t y, k Gumbel y, l Gumbel k, l|y Student-t y, k Gumbel y, l Gumbel k, l|y Student-t y, k Gumbel y, l Gumbel k, l|y Student-t y, k Gumbel y, l Gumbel k, l|y Student-t y, k Gumbel y, l Gumbel k, l|y Student-t y, k Gumbel y, l Gumbel k, l|y Student-t y, k Gumbel y, l Gumbel k, l|y Student-t

England pair1 2.95175 [0.09083] 2.95175 [0.09083] -0.05236 [0.03840] 2.83462 [0.08426] 8.04421 [0.24182] -0.07616 [0.03766] 2.72301 [0.07877] 8.08286 [0.23919] -0.05267 [0.03789] 2.63314 [0.03493] 7.44744 [0.21568] -0.03292 [0.03493] 2.58660 [0.03454] 7.3737 [0.20872] 0.00416 [0.03454] 2.61396 [0.07001] 7.51985 [0.20597] -0.03324 [0.03346] 2.54521 [0.06744] 7.06924 [0.19217] -0.02316 [0.03328] 2.53003 [0.06607] 7.01502 [0.18977] -0.01517 [0.03249]

pair2 11.5973 [4.22109] 10.3629 [3.2476] 8.80610 [2.64369] 18.46495 [10.27706] 17.26667 [8.70078] 15.40423 [6.93198] 14.47233 [5.84454] 22.63839 [14.38905]

17

2012

2011

2010

2009

2008

2007

2006

2005

Year

Pair-copula y, k - Student-t y, l - Gumbel k, l|y - Student-t y, k - Student-t y, l - Gumbel k, l|y - Normal y, k - Student-t y, l - Gumbel k, l|y - Normal y, k - Student-t y, l - Gumbel k, l|y - Normal y, k - Student-t y, l - Gumbel k, l|y - Normal y, k - Student-t y, l - Gumbel k, l|y - Normal y, k - Student-t y, l - Gumbel k, l|y - Normal y, k - Student-t y, l - Gumbel k, l|y - Normal

SP500 τ 0.54848 0.79272 -0.08863 0.52739 0.79321 -0.14325 0.52129 0.78700 -0.11860 0.52706 0.77709 -0.14195 0.49591 0.76774 -0.10612 0.50346 0.76883 -0.10524 0.50668 0.76508 -0.08934 0.51211 0.76428 -0.07771 λU 0.52001 0.72298 0.00376 0.51995 0.84588 0.47294 0.84090 0.45105 0.83291 0.43652 0.82532 0.43837 0.82621 0.44736 0.82629 0.44164 0.82250 -

Table 4: Dependence measures for the estimated pair-copulas German λL Pair-copula τ λU λL 0.52001 y, k - Gumbel 0.68909 0.75951 y, l - Gumbel 0.86194 0.89957 0.00376 k, l|y - Student-t -0.06605 0.00240 0.00240 0.51995 y, k - Gumbel 0.68726 0.75794 y, l - Gumbel 0.86811 0.90427 k, l|y - Student t -0.05601 0.00364 0.00364 0.47294 y, k - Gumbel 0.68336 0.75457 y, l - Gumbel 0.86797 0.90416 k, l|y - Student-t -0.02940 0.00178 0.00178 0.45105 y, k - Gumbel 0.68287 0.75415 y, l - Gumbel 0.86184 0.89949 k, l|y - Student-t -0.02757 0.00390 0.00390 0.43652 y, k - Gumbel 0.68614 0.75697 y, l - Gumbel 0.86176 0.89944 k, l|y - Student-t -0.00236 0.00293 0.00293 0.43837 y, k - Gumbel 0.67919 0.75097 y, l - Gumbel 0.86746 0.90378 k, l|y - Student-t -0.00621 0.00281 0.00281 0.44736 y, k - Gumbel 0.68789 0.75847 y, l - Gumbel 0.87578 0.91008 k, l|y - Student-t -0.02391 0.00353 0.00353 0.44164 y, k - Gumbel 0.67819 0.75010 y, l - Gumbel 0.87415 0.95512 k, l|y - Student-t -0.02541 0.00053 0.00053 Pair-copula y, k - Gumbel y, l - Gumbel k, l|y - Student-t y, k - Gumbel y, l - Gumbel k, l|y - Student-t y, k - Gumbel y, l - Gumbel k, l|y - Student-t y, k - Gumbel y, l - Gumbel k, l|y - Student-t y, k - Gumbel y,l - Gumbel k, l|y - Student-t y, k - Gumbel y, l - Gumbel k, l|y - Student-t y, k - Gumbel y, l - Gumbel k, l|y - Student-t y, k - Gumbel y, l - Gumbel k, l|y - Student-t

England τ 0.66121 0.87214 -0.03335 0.64721 0.87568 -0.04853 0.63276 0.87628 -0.03354 0.62022 0.86572 -0.02096 0.61339 0.86438 0.00265 0.61743 0.86701 -0.02117 0.60710 0.85854 -0.01474 0.60474 0.85744 -0.00965 λU 0.72298 0.90733 0.00260 0.72298 0.91001 0.00370 0.71011 0.91046 0.00821 0.69886 0.90245 0.00020 0.69268 0.90143 0.00046 0.69634 0.90344 0.00066 0.68697 0.89698 0.00103 0.68482 0.89614 0.00005

λL 0.00260 0.00370 0.00821 0.00020 0.00046 0.00066 0.00103 0.00005

Year 2005 2006 2007 2008 2009 2010 2011 2012

Year 2005 2006 2007 2008 2009 2010 2011 2012

variable y k l y k l y k l y k l y k l y k l y k l y k l

variable y k l y k l y k l y k l y k l y k l y k l y k l

mean 345000 185000 280000 357000 198000 293000 357000 198000 293000 371000 208000 300000 366000 366000 313000 390000 207000 328000 389000 208000 345000 403000 215000 333000

Table 5: Descriptive statistics: SP500 std deviation variance asymmetry 248000 6.15E+28 0.85268 224000 5.04E+28 1.46030 248000 6.17E+28 0.86627 245000 6.01E+28 0.82381 233000 5.44E+28 1.35315 251000 6.29E+28 0.83714 245000 6.01E+28 0.82381 233000 5.44E+28 1.35315 251000 6.29E+28 0.83714 250000 6.26E+28 0.83051 237000 5.63E+28 1.33133 247000 6.15E+28 0.85268 240000 5.76E+28 0.73981 240000 6.02E+28 1.35669 256000 6.58E+28 0.86728 248000 6.15E+28 0.67422 233000 5.44E+28 1.38328 245000 6.01E+28 0.85364 242000 5.84E+28 0.64138 234000 5.49E+28 1.41306 245000 5.99E+28 0.79900 250000 6.27E+28 0.65181 229000 5.23E+28 1.39822 250000 6.25E+28 0.78499

Table 6: Descriptive statistics: Germany mean std desviation variance asymmetry 549000 155000 2.39E+28 3.42918 246000 109000 1.19E+28 5.72412 396000 124000 1.53E+28 3.93263 571000 158000 2.51E+28 3.44769 254000 111000 1.23E+28 5.61691 450000 137000 1.88E+28 3.66022 506000 139000 1.93E+28 3.35634 231000 104000 1.07E+28 5.87795 460000 141000 1.99E+28 3.70807 585000 157000 2.47E+28 3.44135 243000 104000 1.09E+28 5.71843 475000 147000 2.16E+28 3.83170 543000 146000 2.13E+28 3.29375 238000 983000 9.66E+27 5.29172 464000 145000 2.09E+28 3.80671 616000 163000 2.65E+28 3.39284 241000 100000 1.01E+28 5.46435 475000 142000 2.02E+28 3.75795 616000 163000 2.65E+28 3.39284 241000 100000 1.01E+28 5.46435 475000 142000 2.02E+28 3.75795 585000 146000 2.14E+28 3.26449 276000 108000 1.16E+28 5.00897 493000 146000 2.14E+28 3.79432

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kurtosis 2.84953 4.73914 2.96216 2.74709 4.20395 2.91861 2.74709 4.20395 2.91861 2.74671 4.09818 2.84953 2.58549 4.15728 2.94366 2.44902 4.43521 3.05400 2.30117 2.84953 2.87871 2.29861 4.47787 2.84953

kurtosis 15.0294 40.0367 20.0792 15.2078 38.0237 16.7874 14.4563 41.8605 17.2563 15.5255 40.9009 18.4321 14.0729 35.1110 18.2498 15.2233 38.1402 17.9829 15.2233 38.1402 17.9829 14.7730 31.4195 18.38780

obs 324 324 324 324 324 324 324 324 324 328 328 328 329 329 329 328 328 328 328 328 328 328 328 328

obs 494 494 494 512 512 512 523 523 523 524 524 524 519 519 519 529 529 529 529 529 529 527 527 527

Year 2005 2006 2007 2008 2009 2010 2011 2012

Year 2005 2006 2007 2008 2009 2010 2011 2012

variable y k l y k l y k l y k l y k l y k l y k l y k l

variable y k l y k l y k l y k l y k l y k l y k l y k l

µ 22.604 21.127 21.938 22.730 21.246 22.071 22.820 21.346 22.161 22.923 21.415 22.259 22.823 21.440 22.145 22.919 21.466 22.225 23.039 21.527 22.351 23.092 21.611 22.409

Table 7: Descriptive statistics: England mean std deviation variance asymmetry 481000 140000 1.96E+28 3.73756 254000 103000 1.05E+28 5.72412 432000 140000 1.96E+28 4.27189 509000 145000 2.1E+28 3.68043 246000 980000 9.59E+27 5.11413 378000 120000 1.45E+28 4.11891 506000 139000 1.93E+28 3.35634 231000 104000 1.07E+28 5.87795 460000 141000 1.99E+28 3.70807 522000 142000 2.01E+28 3.44956 289000 117000 1.36E+28 5.13201 360000 109000 1.2E+28 3.84909 531000 148000 2.19E+28 3.60111 284000 115000 1.32E+28 5.13957 389000 125000 1.56E+28 4.03078 552000 153000 2.34E+28 3.56884 275000 108000 1.17E+28 5.24241 398000 129000 1.66E+28 4.15186 552000 153000 2.34E+28 3.56884 275000 108000 1.17E+28 5.24241 398000 129000 1.66E+28 4.15186 544000 145000 2.1E+28 3.55592 333000 124000 1.55E+28 5.04287 407000 129000 1.68E+28 4.17996

Table 8: Parameters of Margins: SP500 std σ std ` KS 0.072 1.304 0.970 -545.237 0.027 0.086 1.558 0.993 -603.051 0.023 0.086 1.562 0.976 -603.794 0.026 0.069 1.242 0.937 -529.558 0.029 0.085 1.532 0.986 -597.532 0.025 0.081 1.473 0.988 -591.601 0.024 0.067 1.223 0.847 -531.010 0.033 0.084 1.533 0.977 -605.156 0.026 0.081 1.471 0.999 -591.601 0.020 0.066 1.195 0.339 -521.980 0.052 0.083 1.510 0.945 -598.269 0.029 0.080 1.458 0.849 -586.857 0.033 0.064 1.178 0.334 -520.255 0.052 0.082 1.494 0.887 -598.627 0.032 0.079 1.450 0.798 -588.582 0.035 0.063 1.149 0.332 -510.675 0.052 0.082 1.502 0.851 -598.499 0.033 0.078 1.427 0.830 -581.676 0.034 0.062 1.136 0.234 -507.008 0.057 0.082 1.503 0.467 -598.574 0.046 0.078 1.4178 0.767 -579.459 0.036 0.061 1.117 0.249 -501.339 0.056 0.081 1.483 0.512 -594.174 0.045 0.077 1.402 0.730 -575.876 0.038

19

kurtosis 18.4180 35.6261 23.3154 17.9550 32.4695 22.4553 14.4563 41.8605 17.2563 15.9813 32.0506 19.3470 17.2506 32.0237 20.5295 16.7063 33.7373 21.9419 16.7063 33.7373 21.9419 17.1425 31.5437 22.8057

p-value 0.051 0.061 0.061 0.048 0.060 0.058 0.047 0.060 0.057 0.046 0.059 0.057 0.046 0.058 0.056 0.044 0.058 0.055 0.044 0.058 0.055 0.043 0.058 0.054

obs 744 744 744 788 788 788 820 820 820 869 869 869 907 907 907 963 963 963 963 963 963 1004 1004 1004

AD 0.329 0.288 0.317 0.439 0.272 0.177 0.596 0.287 0.219 0.892 0.333 0.284 0.299 0.403 0.411 1.438 0.514 0.367 1.754 0.897 0.358 1.993 0.981 0.383

p-value 0.914 0.946 0.924 0.808 0.957 0.995 0.651 0.947 0.984 0.418 0.910 0.949 1.120 0.844 0.837 0.191 0.732 0.880 0.125 0.897 0.888 0.092 0.367 0.864

Year 2005 2006 2007 2008 2009 2010 2011 2012

Year 2005 2006 2007 2008 2009 2010 2011 2012

variable y k l y k l y k l y k l y k l y k l y k l y k l

µ 18.431 16.319 17.687 18.546 16.415 17.829 18.607 16.463 17.891 18.716 16.614 18.012 18.621 16.613 17.935 18.700 16.607 17.986 18.786 16.704 18.121 18.798 16.701 18.157

Table 9: Parameters of Margins: Germany std σ std ` KS 0.108 2.410 0.076 -1135.06 0.047 0.137 3.065 0.097 -1253.85 0.029 0.123 2.736 0.087 -1197.80 0.036 0.106 2.404 0.075 -1175.19 0.050 0.132 3.065 0.097 -1287.51 0.036 0.119 2.705 0.084 -1235.62 0.041 0.106 2.428 0.075 -1205.72 0.044 0.130 2.983 0.092 -1313.24 0.030 0.118 2.718 0.084 -1264.56 0.032 0.105 2.426 0.075 -1207.49 0.043 0.128 2.949 0.091 -1309.75 0.034 0.118 2.718 0.084 -1267.14 0.034 0.106 2.432 0.075 -1197.37 0.047 0.133 3.047 0.094 -1314.26 0.034 0.117 2.682 0.083 -1247.99 0.030 0.106 2.439 0.075 -1221.98 0.040 0.132 3.046 0.093 -1339.42 0.030 0.119 2.743 0.084 -1283.98 0.028 0.106 2.449 0.075 -1219.30 0.043 0.132 3.036 0.093 -1332.68 0.032 0.118 2.722 0.083 -1275.12 0.032 0.106 2.455 0.075 -1220.76 0.056 0.133 3.073 0.094 -1338.98 0.027 0.117 2.699 0.083 -1270.65 0.033

p-value 0.206 0.797 0.512 0.144 0.516 0.347 0.248 0.720 0.631 0.267 0.562 0.579 0.193 0.570 0.703 0.341 0.710 0.793 0.268 0.619 0.647 0.071 0.834 0.589

AD 1.747 0.376 0.827 1.654 0.637 0.804 1.835 0.457 0.616 1.876 0.468 0.683 1.614 0.416 0.652 1.289 0.266 0.519 1.243 0.317 0.727 1.868 0.253 0.814

p-value 0.127 0.871 0.461 0.143 0.613 0.478 0.113 0.790 0.632 0.107 0.779 0.572 0.151 0.831 0.600 0.235 0.961 0.727 0.251 0.924 0.536 0.108 0.968 0.470

variable y k l y k l y k l y k l y k l y k l y k l y k l

µ 17.808 16.012 17.186 17.928 16.125 17.283 17.971 16.236 17.325 18.119 16.419 17.457 18.010 16.309 17.338 18.029 16.275 17.345 18.112 16.355 17.449 18.170 16.357 17.518

Table 10: Parameters of Margins: England std σ std ` KS 0.108 2.946 0.076 -1859.10 0.027 0.120 3.288 0.085 -1940.90 0.023 0.110 3.014 0.078 -1876.13 0.026 0.103 2.900 0.073 -1956.68 0.029 0.116 3.269 0.082 -2051.19 0.025 0.106 2.982 0.075 -1978.84 0.024 0.102 2.921 0.072 -2042.12 0.033 0.114 3.193 0.076 -2133.82 0.026 0.105 3.026 0.074 -2071.23 0.020 0.095 2.807 0.067 -2129.7 0.052 0.108 3.193 0.076 -2241.63 0.029 0.099 2.933 0.070 -2167.67 0.033 0.095 2.864 0.067 -2241.1 0.052 0.109 3.285 0.077 -2365.43 0.032 0.100 3.014 0.070 -2287.43 0.035 2.986 2.872 0.065 -2382.12 0.052 0.106 3.311 0.075 -2518.94 0.033 0.096 2.986 0.068 -2419.66 0.034 0.091 2.867 0.064 -2447.05 0.057 0.105 3.311 0.074 -2589.77 0.046 0.092 2.924 0.065 -2466.56 0.036 0.087 2.782 0.062 -2451.69 0.056 0.106 3.360 0.075 -2640.98 0.045 0.090 2.864 0.063 -2480.82 0.038

p-value 0.970 0.993 0.976 0.937 0.988 0.988 0.847 0.977 0.999 0.339 0.945 0.849 0.334 0.887 0.798 0.332 0.851 0.830 0.234 0.467 0.767 0.249 0.512 0.730

AD 0.329 0.288 0.317 0.439 0.272 0.177 0.596 0.287 0.219 0.892 0.468 0.284 1.120 0.403 0.411 1.438 0.514 0.367 1.754 0.897 0.358 1.993 0.981 0.383

p-value 0.914 0.946 0.924 0.808 0.957 0.995 0.651 0.947 0.984 0.418 0.779 0.949 0.299 0.844 0.837 0.191 0.732 0.880 0.125 0.415 0.888 0.092 0.367 0.864

20

Highlights

This paper proposes and new approach to model efficiency. We use Vine Copulas and show an empirical application. Our model can be used to datasets that has asymmetry and the usual normality hypothesis does not hold. We show that on average S&P companies are more efficient than the ones listed in England and Germany. We compare the results of our model with traditional stochastic frontier.