Stochastic frontier models with dependent errors based on normal and exponential margins

REVISTA DE M´ ETODOS CUANTITATIVOS PARA LA ECONOM´ IA Y LA EMPRESA (23). P´aginas 3–23. Junio de 2017. ISSN: 1886-516X. D.L: SE-2927-06. www.upo.es/revistas/index.php/RevMetCuant/article/view/2684 Stochastic Frontier Models with Dependent Errors based on Normal and Exponential Margins G´ omez–D´ eniz, Emilio Department of Quantitative Methods in Economics and TiDES Institute University of Las Palmas de Gran Canaria (Spain) E-mail: [email protected] P´ erez–Rodr´ ıguez, Jorge V. Department of Quantitative Methods in Economics University of Las Palmas de Gran Canaria (Spain) E-mail: [email protected] ABSTRACT Following the recent work of G´omez–D´eniz and P´erez–Rodr´ıguez (2014), this paper extends the results obtained there to the normal–exponential distribution with dependence. Accordingly, the main aim of the present paper is to enhance stochastic production frontier and stochastic cost frontier modelling by proposing a bivariate distribution for dependent errors which allows us to nest the classical models. Closed–form expressions for the error term and technical efficiency are provided. An illustration using real data from the econometric literature is provided to show the applicability of the model proposed. Keywords: Technical and cost efficiencies; stochastic frontier; marginal distribution; dependence; Sarmanov model. JEL classification: C01; C13; C21; C51. MSC2010: 91B70; 62P20; 91G70. Art´ıculo recibido el 25 de octubre de 2016 y aceptado el 19 de mayo de 2017. 3 Modelos de frontera estoc´astica con errores dependientes basados en m´argenes normal y exponencial RESUMEN Continuando el reciente trabajo de G´omez–D´eniz y P´erez–Rodr´ıguez (2014), el presente art´ıculo extiende los resultados obtenidos a la distribuci´on normal– exponencial con dependencia. En consecuencia, el principal prop´osito de este art´ıculo es mejorar el modelado de la frontera estoc´astica tanto de producci´on como de coste proponiendo para ello una distribuci´on bivariante para errores dependientes que nos permitan encajar los modelos cl´asicos. Se obtienen las expresiones en forma cerrada para el t´ermino de error y la eficiencia t´ecnica. Se ilustra la aplicabilidad del modelo propouesto usando datos reales existentes en la literatura econom´etrica. Palabras claves: eficiencias t´ecnica y de coste; frontera estoc´astica; distribuci´on marginal; dependencia; modelo de Sarmanov. Clasificaci´on JEL: C01; C13; C21; C51. MSC2010: 91B70; 62P20; 91G70. 4 1 Introduction In general, the methods used for estimating technical and cost efficiency can be considered either parametric or non–parametric. The former involves the estimation of a stochastic production frontier (SPF) or a stochastic cost frontier (SCF) by imposing an explicit functional form and distribution assumption on the data (Aigner et al., 1977; Meeusen and van den Broeck, 1977; Battese and Corra, 1977; Stevenson, 1980; Greene, 1980a, 1980b; Jondron et al., 1982; Lee, 1983; Greene, 1990, 2003; Smith, 2008), where the output of a firm is a function of a set of inputs, plus inefficiency and random error. The second approach is the linear programming technique of data envelopment analysis (DEA), a non–parametric approach which does not impose any assumptions regarding functional form and which does not take into account random error (see Lovell and Schmidt, 1988, for an early survey). Both techniques have advantages and disadvantages; for example, SPF and SCF require the analyst to assume an underlying distribution about the error term, and independence between the inefficiency term and random error. On the other hand, DEA cannot take into account such statistical noise, and efficiency estimates may be biased if the production process is largely characterised by stochastic elements. Between these two alternatives of modelling, our main interest is based on the the stochastic frontier model in a cross–section framework. The model in this scenario can be written as yi=f(xi;β) + νi±ui, i = 1,2, . . . , n,ui≥0, where the sign of the last term depends on whether the frontier describes costs (positive) or production (negative). For example, if we assume that f(xi;β) takes the log–linear Cobb–Douglas form, then the stochastic production frontier (SPF) model can be written as: log yi=β0+Pk j=1 βjlog xij +νi−ui, i = 1,2, . . . , n, where log yiis the natural logarithm of the production of the i-th firm; log xiis a k×1 vector of (natural log transformations of the) input quantities of the i-th firm; βis a vector of unknown parameters, and the disturbance term εi=νi±ui(which is asymmetric) is assumed to have two components: one with a strictly non–negative distribution, ui(which is a non–negative component often referred to as the inefficiency term), and another with a symmetric distribution, νi(which is termed the idiosyncratic error). Although it is not an assumption of the model, independence of νand umakes it easy to obtain the density of ε. The density of εis then used to conduct maximum likelihood estimation of the model parameters. In addition, it is possible to obtain the conditional density of u|εand E(u|ε). These serve as a basis to obtain estimates 5 for firm–specific inefficiency. The maximum likelihood method can be used to estimate βand ui, the variances of the errors and the technical efficiency of each firm. Therefore, distributional assumptions are required for νiand ui. In terms of vi, and in general, these random variables are assumed to be independently and identically distributed (iid) N0, σ2 ν. On the other hand, in terms of ui, various assumptions may be made; for example, Meeusen and van den Broeck (1977) assigned the exponential distribution to ui, Battese and Corra (1977) assumed a half–normal distribution, while Aigner et al. (1977) considered both distributions. However, since the half–normal and exponential distributions are both single–parameter specifications with modes at zero, some scepticism has been expressed regarding their generality. Thus, Stevenson (1980) suggested the truncated normal and gamma distribution for ui. Greene (1980a, 1980b) proposed the gamma distribution, Lee (1983) proposed a four–parameter Pearson family of distributions and Greene (1990, 2003) proposed the two–parameter gamma density as a more general alternative. More recently, another way to model SPF and SCF are based on dependence of error terms such as Smith (2008) and Wiboonpongse et al. (2015) with copulas and El Mehdia and Hafner (2014) and G´omez–D´eniz and P´erez–Rodr´ıguez (2014) with closed–form solutions by using bivariate distributions. On the other hand, Tran and Tsionas (2015) and Amsler et al. (2016) study the correlation between the inputs and statistical noise or inefficiency. The former one proposes an approach which is based on copula function to directly model the correlation between the endogenous regressors and the composed errors assumed to be independent and identically distributed. Accordingly, the main aim of the present paper is to enhance SPF and SCF modelling by proposing a closed form of a bivariate distribution for dependent errors which allows us to nest the classical models. In particular, we follow G´omez–D´eniz and P´erez–Rodr´ıguez (2014) and extend their results by using Sarmanov’s family of distributions (Sarmanov, 1966; Lee, 1996; G´omez–D´eniz and P´erez–Rodr´ıguez, 2014; among others) to obtain closed–form expressions for the error term and technical efficiency. More specifically, we built a bivariate dependent SPF and SCF models by using normal and exponential distributions (NE), and thus we construct a general extension of the classical stochastic frontier model with these distributions. The remainder of this paper is structured as follows. Section 2 introduces a brief note on the Sarmanov family of distributions which is used to estimate the technical (cost) efficiency in a 6 cross–section framework. We analysed one parametric form, deriving in closed–form expression the log likelihood functions and technical (cost) efficiencies, based on the classical pdf distributions, by including the dependence structure. An application of the new model is discussed in Section 3. Finally, the main conclusions drawn are presented in Section 4. 2 Modelling the dependence In addition to the distributional assumptions on the error terms, νiand ui, in stochastic parametric frontier models, another important characteristic of the above cited models is the independence between them to construct the density and marginal distributions. The classical stochastic frontier model with normal and exponential assumptions is described by the following stochastic representation: (i)vi∼iid N(0, σ2 ν); (ii)ui∼iid exponential with parameter σu>0; and (iii)uiand viare distributed independently of each other and of the regressors. The probability density functions of viand uiare as follows fσν(ν) = 1 σν√2πe−ν2 2σ2 ν, fσu(u) = 1 σu e−u σu, where −∞ < ν < ∞,σν>0, u > 0 and σu>0. In this case, we have fσu,σν(ε) = 1 σu Φ−ε σν−σν σuexp (ε σu +σ2 ν 2σ2 u),(1) f(u|ε) = 1 √2π σνΦ(e µ/σν)exp −1 2σ2 ν (u−e µ)2,(2) where e µ=−ε−σ2 ν/σu. The marginal f(ε) is asymmetrically distributed with given by E(ε) = −σuand the variance by var(ε) = σ2 u+σ2 ν. On the other hand, u|εfollows a half–normal distribution, N+(e µ, σ2 ν), with mean E(u|ε) = e µ+σν φ(−e µ/σν) Φ(−e µ/σν)=σνφ(A) Φ(−A)−A, and where A=−e µ/σν. Following G´omez–D´eniz and P´erez–Rodr´ıguez (2014), we obtain closed–form expression for the likelihood function and technical efficiency for SPF and SCF likelihoods based on the classical 7 mixture of normal and exponential distributions. We propose a broader, more general and flexible range of dependence, which is also easy to handle, for testing the independence between the inefficiency term and random error (the idiosyncratic component). This family of distributions is implemented by assuming that f1(x1) and f2(x2) are univariate probability density functions, with supports defined on A1⊆IR and A2⊆IR, respectively. Let ϕs(t), s= 1,2, be bounded nonconstant functions (the mixing functions) such that Z∞ −∞ ϕs(t)fs(t)dt = 0, then the function defined by f(x1, x2) = f1(x1)f2(x2) [1 + ω ϕ1(x1)ϕ2(x2)] (3) is a bivariate joint density with margins f1(x1) and f2(x2), provided ωis a real number which satisfies the condition 1 + ω ϕ1(x1)ϕ2(x2)≥0, for all x1and x2. Some methods to obtain the mixing function ϕwhen fs(xs), s = 1,2, are members of the natural exponential family of distributions are described in Lee (1996). The Farlie–Gumbel–Morgernstern (FGM) family of copulas can be viewed as a special case of the above–mentioned construction, by setting ϕ(xs) = 1 −2F(xs), s = 1,2 and therefore one of the models proposed in Smith (2008). Here F(·) represents the cumulative distribution function of the random variables with pdf f(·). As we will see this family provide analytical expressions for the marginal pdf of the random variable ε, the conditional pdf of u|ε,cov(u, ν) and technical efficiency are provided in the SPF model, for the NE model. The classical independent models are derived as a particular case when ω= 0 while ω6= 0 measures the dependence structure. As in the classical SPF model, let ν=u+ε. Using (3) we get fσu,σν,ω(u, ε) = fσu(u)fσν(u+ε) [1 + ω ϕσu(u)ϕσν(u+ε)] ,(4) by taking ϕσu(u) = e−u−δu(σu), ϕσν(ν) = e−ν2−2ν−δν(σν), with δu(σu) = 1 1 + σu ,(5) δν(σν) = 1 p1+2σ2 ν exp (2σ2 ν 1+2σ2 ν),(6) defines a bivariate distribution of (u, ν) with marginal distributions fσν(ν) and fσu(u) as in the classical model and where ω1≤ω≤ω2, being 8 ω1= max −1 δu(σu)δν(σν),−1 (1 −δu(σu))(e−δν(σν)),(7) ω2= min 1 δu(σu)(e−δν(σν)),1 (1 −δu(σu))δν(σν).(8) To see this, observe that because d du ϕσu(u)<0 we have that ϕσu(u) is a decreasing function on uand the range of variation of ϕσu(u) is (−δu(σu),1−δu(σu)). In the same way, it is simple to see that ϕσν(ν) has a maximum in ν=−1 and therefore the range of variation of ϕσν(ν) results (−δν(σν), e −δν(σν)). Now, we have that fσu,σν,ω(u, ν) represents a probability density function if 1 + ω ϕσu(u)ϕσν(ν)≥0, and this occurs if ω≥−1 ϕσu(u)ϕσν(ν),for ϕσu(u)ϕσν(ν)>0, ω≤−1 ϕσu(u)ϕσν(ν),for ϕσu(u)ϕσν(ν)<0, from which it is a simple exercise to see that range of ωis given by (ω1, ω2). Although any other mixing functions satisfying that R∞ 0fσu(u)ϕσu(u)du = 0 and that R∞ −∞ fσν(ν)ϕσν(ν)dν = 0 can be considered we have chosen the mixing functions above since: (i) The presence of the exponential term which is also present in the probability density function of the normal and exponential distributions facilitates the computations in order to obtain closed–form expressions for the marginal of εas we will see in the next section; (ii) The square term in the exponential part of ϕσν(ν) is important to ensure appropriate bounds for the ωparameter. Dependence assumption is now depending on ωand the Sarmanov family with normal and half normal marginals studied here can also be considered as an extension of the classical Sarmanov family of distributions dealt in Lee (1996) for the normal case. Some algebra provides the correlation coefficient, which is given by ρ=2ω σuσν (1 + σu)2(1 + 2σ2 ν)3/2exp (2σ2 ν 1+2σ2 ν), This correlation coefficient is bounded by (see Lee, 1996) |ρ| ≤ |ω|hEϕ2 σu(u)Eϕ2 σν(ν)i1/2. Now, we have the following result which can be used to build the likelihood function for SPF (see Appendix 4 for the SCF model). 9 Theorem 1 In the SPF model for NE distributions and assuming dependence, the marginal pdf of εis given by fσu,σν,ω(ε) = Υ1 σu,σν(ε) σup1+2σ2 ν        Φe µ σν   q1+2σ2 ν+ω e 2σ2 ν 1+2σ2 ν 1 + σu    −ωΥ2 σu,σν(ε) Φ e µ σν−σν −ω 1 + σu Υ3 σu,σνΦ −σν σu 1+2σu p1+2σ2 ν−ε σνq1+2σ2 ν! +ωΥ4 σu,σν(ε) Φ −σν σu 1+3σu p1+2σ2 ν−ε σνq1+2σ2 ν!),(9) where Υ1 σu,σν(ε) = exp (ε σu +σ2 ν 2σ2 u), Υ2 σu,σν(ε) = exp (ε+σ2 ν σu +σ2 ν 2+2σ2 ν 1+2σ2 ν), Υ3 σu,σν= exp (σ2 ν 2σ2 u 4σu−2σ2 ν 1+2σ2 ν +2σ2 ν 1+2σ2 ν), Υ4 σu,σν(ε) = exp (ε+σ2 ν 2σ2 u 5σ2 u+ 6σu−2σ2 ν 1+2σ2 ν +2σ2 ν 1+2σ2 ν). Proof: See Appendix 1. Observe that when ω= 0, i.e. the independence case, pdf (9) reduces to (1). Simple computations provide that the mean for the marginal pdf given in (9) is equal to E(ε) = −σu while the variance results var(ε) = σ2 u+σ2 ν"1−4ω σ2 u (1 + σ2 u)(1 + 2σ2 ν)3/2e 2σ2 ν 1+2σ2 ν#. Figure 1 shows examples of the marginal pdf (9) for different values of the model parameters. Having obtained the main result of the likelihood function, we now show the conditional distribution of ugiven ε. 10 Figure 1: Marginal distribution in the SPF–NE model for selected parameter values. Proposition 1 In the SPF model under NE distributions, assuming dependence, the conditional pdf of ugiven εis given by fσu,σν,ω(u|ε) = fσu,σν(u|ε) + ωΨ0 σu,σν(u|ε) 1 + ωΨ1 σu,σν(ε),(10) for u > 0, where fσu,σν(u|ε)is given in (2) and Ψ0 σu,σν(u|ε) = ϕσu(u)ϕσν(u+ε)fσu,σν(u|ε),(11) Ψ1 σu,σν(ε) = 1 p1+2σ2 ν 1 Φeµ σν   Φe µ σνe 2σ2 ν 1+2σ2 ν 1 + σu−Υ2 σu,σν(ε) Φ e µ σν−σν −1 1 + σu Υ3 σu,σνΦ −σν σu 1+2σu p1+2σ2 ν−ε σνq1+2σ2 ν! + Υ4 σu,σν(ε) Φ −σν σu 1+3σu p1+2σ2 ν−ε σνq1+2σ2 ν!#.(12) Proof: See Appendix 2. 11 Battese, G. E. and T. J. Coelli. (1988). Prediction of firm–level technical efficiencies with a generalized frontier production function and panel data. Journal of Econometrics, 38, 387–99. Bonett, D.G. (2000). Sample size requirements for estimating Pearson, Kendall and Spearman correlations. Psychometrika, 65:1, 23–28. Dingman, H.F. and Perry, N.C. (1956). 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Journal of Econometrics, 13, 57–66. Tran, K. and Tsionas, M. (2015). Endogeneity in stochastic frontier models: Copulas approach without external instruments. Economics Letters, 133, 85–88. Wiboonpongse, A.; Liu, J.; Sriboonchitta, S. and Denoeux, T. (2015). Modeling dependence between error components of the stochastic frontier model using copula: Application to intercrop coffee production in Northern Thailand. International Journal of Approximate Reasoning, 65, 34–44. 19 Appendix 1. Proof of Theorem 1 From (4) we have f(ε) = Z∞ 0 fσu,σν,ω(u, ε)du =Z∞ 0 fσu(u)fσν(u+ε)du +ωZ∞ 0 fσu(u)fσν(u+ε)ϕσu(u)ϕσν(u+ε)du. (14) The first integral in (14) coincides with (1). Then, we have f(ε) = [1 + ω δu(σu)δν(σν)] 1 σu Φ−ε σν−σν σuexp (ε σu +σ2 ν 2σ2 u) −ω(J1+J2+J3), where δu(σu), δν(σν) are given in (5), (6), respectively, while J1=Z∞ 0 e−u−(u+ε)2−2(u+ε)fσu(u)fσν(u+ε)du, J2=δν(σν)Z∞ 0 e−ufσu(u)fσν(u+ε)du, J3=δu(σu)Z∞ 0 e−(u+ε)2−2(u+ε)fσu(u)fσν(u+ε)du. Again, simple but tedious computations lead to the following: J1=1 σup1+2σ2 ν Υ1 σu,σν(ε) Υ4 σu,σν(ε) Φ −σν σu 1+3σu p1+2σ2 ν−ε σνq1+2σ2 ν!, J2=1 σup1+2σ2 ν Υ1 σu,σν(ε) Υ2 σu,σν(ε) Φ −ε σν−σν σu−σν, J3=1 σup1+2σ2 ν Υ1 σu,σν(ε) Υ3p σu,σνΦ −σν σu 1+2σu p1+2σ2 ν−ε σνq1+2σ2 ν!, from which (9) is obtained. Appendix 2. Proof of Proposition 1 We start with the wellknown relation fσu,σν,ω(u|ε) = fσu,σν,ω(u, ε) fσu,σν,ω(ε) and replace in this expression the numerator and the denominator by (4) and (9), respectively. Now by taking the common factor for ωin both, numerator and denominator, we get the result after some computations. 20 Appendix 3. Proof of Proposition 2 From (9) it is easy to see Ee−u|ε∝Z∞ 0 e−ufσu,σν(u|ε)du +ωZ∞ 0 e−uΨ0 σu,σν(u|ε)du, (15) where the proportionality factor is given by h1 + ωΨ1 σu,σν(ε)i−1. The first integral in (15) is simple to solve by using the pdf of the half.normal provided in (2). For the second integral we use the expression given in (11) and with simple but tedious computations we get the result. Appendix 4. The stochastic cost frontier model The corresponding expressions for the stochastic cost frontier (SCF) are derived easily from the fact that now we assume v=−u+ε. In this case and under the classical model we have that, fσu,σν(ε) = 1 σu Φε σν−σν σuexp (−ε σu +σ2 ν 2σ2 u), fσu,σν(u|ε) = 1 √2π σνΦ(ee µ/σν)exp −1 2σ2 ν (u−ee µ)2,(16) where ee µ=ε−σ2 ν/σu. Again, the marginal f(ε) is asymmetrically distributed with mean E(ε) = σuand variance var(ε) = σ2 u+σ2 ν. The corresponding expressions under the dependence assumption are given bellow. The marginal pdf of εis given by fσu,σν,ω(ε) = Ξ1 σu,σν(ε) σup1+2σ2 ν(Φ ee µ σν!"q1+2σ2 ν+ω 1 + σu exp 2σ2 ν 1+2σ2 ν!# −ω"q1+2σ2 νΞ2 σu,σν(ε)δν(σν) Φ ee µ σν−σν! + Ξ3 σu,σν(ε)δu(σu) Φ σν(2σu−1) σup1+2σ2 ν +εp1+2σ2 ν σν! −Ξ4 σu,σν(ε) Φ σν(σu−1) σup1+2σ2 ν +εp1+2σ2 ν σν!#),(17) where Ξ1 σu,σν(ε) = exp (−ε σu +σ2 ν 2σ2 u),(18) 21 Ξ2 σu,σν(ε) = exp (−ε+σ2 ν σu +σ2 ν 2),(19) Ξ3 σu,σν= exp (σ2 ν 2σ2 u 4σu(σu−1) −2σ2 ν 1+2σ2 ν),(20) Ξ4 σu,σν(ε) = exp (−ε+σ2 ν 2σ2 u σu(σu−2) −2σ2 ν 1+2σ2 ν).(21) The conditional pdf of ugiven εis given by fσu,σν,ω(u|ε) = fσu,σν(u|ε) + ω=0 σu,σν(u|ε) 1 + ω=1c σu,σν(ε),(22) for u > 0, where fσu,σν(u|ε) is the pdf given in (16) and =0 σu,σν(u|ε) = ϕσu(u)ϕσν(u+ε)fc σu,σν(u|ε), =1 σu,σν(ε) = δu(σu)δν(σν)−1 p1+2σ2 νΦeeµ σν×"q1+2σ2 νΞ2 σu,σν(ε)δν(σν) Φ ee µ σν−σν! + Ξ3 σu,σν(ε)δu(σu) Φ σν(2σu−1) σup1+2σ2 ν +εp1+2σ2 ν σν! −Ξ4 σu,σν(ε) Φ σν(σu−1) σup1+2σ2 ν +εp1+2σ2 ν σν!#. The point estimation for the efficiency in the SCF model under NE distributions, assuming dependence, is given by CEi==01 σu,σν,ω(εi) + ω=02 σu,σν(εi) 1 + ω=1 σu,σν(εi), i = 1,2, . . . , n, where =01 σu,σν,ω(εi) = [1 + ω δu(σu)δν(σν)] Φeeµ σν−σν Φeeµ σνe−εi+σ2 ν/σu+σ2 ν/2 and =02 σu,σν(εi) = 1 =eeµ σν"1 p1+2σ2 ν Ξ11 σu,σν(εi) Φ εi σνq1+2σ2 ν−σν σup1+2σ2 ν! −δν(σν) Ξ12 σu,σν(εi) Φ ee µ σν−2σν! −δu(σu) p1+2σ2 ν Ξ13 σu,σν(εi) Φ εi σνq1+2σ2 ν+σu−1 σu σν p1+2σ2 ν!#, 22 with Ξ11 σu,σν(εi) = exp (−2εi−σ4 ν (1 + 2σ2 ν)σ2 u), Ξ12 σu,σν(εi) = exp (−2εi+2σ2 ν σu (1 + σu)), Ξ13 σu,σν(εi) = exp (−εi+σ2 ν(σu(σu−2) −2σ2 ν) 2σ2 u(1 + 2σ2 ν)), while δu(σu) and δν(σν) are given in (5), (6), respectively. 23