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1、Electronic copy available at: http:/ Observed and unobserved heterogeneity in stochastic frontier models: an application to the electricity distribution industry Maria Kopsakangas-Savolainen Assistant Professor and Rauli Svento Professor, Vice President University of Oulu University of Oulu Faculty
2、of Economics and Faculty of Economics and Business Administration Business Administration Department of Economics Department of Economics P.O.Box 4600 P.O.Box 4600 90014 University of Oulu 90014 University of Oulu Tel: +358 8 5532910 Tel: +358 8 5532912 Email: maria.kopsakangasoulu.fi Email: rauli.s
3、ventooulu.fi Electronic copy available at: http:/ Observed and unobserved heterogeneity in stochastic frontier models: an application to the electricity distribution industry Abstract In this study we combine different possibilities to model firm level heterogeneity in Stochastic Frontier Analysis.
4、We show that both observed and unobserved heterogeneity cause serious biases in inefficiency results if left unmodelled. Modelling observed and unobserved heterogeneity treats individual firms in different ways and even though the mean inefficiency scores in both cases diminish the firm level effici
5、ency rank orders turn out to be very different. The best fit with the data is obtained by modelling unobserved heterogeneity through randomising frontier parameters and at the same time explicitly modelling the observed heterogeneity into the inefficiency distribution. These results are obtained by
6、using data of Finnish electricity distribution utilities and the results are relevant in relation to electricity distribution pricing and regulation. Keywords: Cost efficiency, heterogeneity, electricity distribution, benchmarking, random parameter JEL Classifications: C13, C23, D24, L51, L94 2 3 1.
7、 Introduction The literature concerning different frontier based efficiency (productivity or cost) measuring methods have increased rapidly during the last decade. There are two main ways to model frontier efficiency, namely the data envelopment analysis (DEA) and stochastic frontier analysis (SFA).
8、1 In this paper we concentrate on modeling firm specific heterogeneity by using the stochastic frontier analysis. The traditional models of stochastic frontiers have been extended such that firm specific heterogeneity can be better taken into account. When the heterogeneity accounting literature sta
9、rted to develop it was first assumed that in the models time invariant parts are representing inefficiency whereas time variant parts can be seen as firm or unit specific heterogeneity. However recently (see e.g. Greene (2004), (2005a) and (2005b) this interpretation has radically changed. In recent
10、 papers it has been assumed that such parts of firm specific effects which are not changing in time are mainly due to firm specific heterogeneity while the time variant part should be seen as inefficiency. Which one of these views is rights is not an easy question. It is understandable that there ar
11、e firm specific heterogeneity factors which do not change in time and which are beyond the managerial effort. These should of course be interpreted as time invariant heterogeneity. However, it is also possible that only part of the inefficiency is time variant. This is more likely to be the case if
12、the industry under consideration is (local) regulated monopoly and hence there may not exists full incentives to minimize costs. If firm specific heterogeneity is not accounted for it can create considerable bias in the inefficiency estimates. There have been, however, rapid developments in various
13、forms of econometric methods during the past two decades which can, especially if we have panel data, identify the unobserved heterogeneity. The literature of panel data models in stochastic frontier analysis starts form Pitt and Lee (1981) and is followed by Schmidt and Sickles (1984) among others.
14、 During the last few years many authors (see e.g. Jha and Singh (2001), and Dalen and Gomez-Lobo (2003) have included also exogenous variables in the model to explain better the inefficiency component in the model.2 1 For comparisons of DEA and SFA methods see e.g. Hjalmarsson et al. (1996), Lee (20
15、05) and Odeck (2007). 2 See Kumbhakar and Lovell (2000) for an extensive survey of stochastic frontier models. 4 In these conventional fixed and random effect panel data models firm-specific heterogeneity can be taken into account but it is still more or less considered as inefficiency. Farsi and Fi
16、lippini (2004) studied cost-efficiency with panel data models in the Swiss electricity distribution utilities. In that paper, they utilised original random effects and fixed effects models and reported that different model specifications could lead to different individual efficiency estimates. Kopsa
17、kangas-Savolainen and Svento (2008) utilised the variations of conventional random effects models in measuring cost-effectiveness of Finnish electricity distribution utilities. According to their results it seems that even though part or the heterogeneity can be explained by network characteristic v
18、ariables the unobserved heterogeneity still appears as inefficiency in the conventional random effects models. Greene (2005a) proposed an approach that integrates an additional stochastic term in both fixed and random effects models in order to distinguish unobserved heterogeneities from cost ineffi
19、ciencies. Farsi, Filippini and Greene (2006) applied stochastic frontier models in cost efficiency measuring to the electricity distribution sector. In that paper they focus on three panel data models: GLS model (Schmidt and Sickles), MLE model (Pitt and Lee (1981) , and the true random effects (TRE
20、) model (Greene (2005a). According to their results it is very important to model heterogeneity and inefficiency separately. In their paper (2005) Farsi, Filippini and Greene studied network industries and compared different stochastic frontier models in very comprehensive and detailed manner. It se
21、ems that the true random effects model gives significantly lower inefficiency values than the other models they utilised. However, they point out a shortcoming of that model, namely that the firm specific heterogeneity terms are assumed to be uncorrelated with the explanatory variables. Farsi, Filip
22、pini and Kuenzle (2006) have found similar results connected to different model specifications in measuring regional bus companies cost efficiencies. According to their results the true random effects model seems to give much more plausible results than the other model specifications. In their paper
23、 concerning the efficiency of Swiss gas distribution sector Farsi, Filippini and Keunzle (2007) pointed out the importance of taking into account the output characteristics (such as customer density and network size) in the cost efficiency measuring process. To overcome the well known problems relat
24、ed to the basic fixed effect (FE) model (Schmidt and Sickles 1984), especially the fact that in the FE model any time invariant unobserved heterogeneity appears in the inefficiency component, Greene (2005a) proposes an extended 5 model which he called the “true fixed effects model” (TFE) to underlin
25、e the difference with the FE framework commonly used. In the TFE model, fixed effects represent the unobserved heterogeneity, not the inefficiency as in the original FE model. The basic difference between the true fixed effects model and true random effects model is that in TFE model any correlation
26、 among the effects and explanatory variables are allowed. The purpose of this paper is to study the different ways how the firm specific heterogeneity can be taken into account in the stochastic frontier models framework. Observed heterogeneity can be taken into account by incorporating firm specifi
27、c heterogeneity either in the estimated distribution of inefficiency or in the cost function itself. It is important to include observed firm specific effects to the model because otherwise e.g. Hausman test can reject the model because of the presence of such heterogeneity which is correlated with
28、the regressors but not necessarily related to inefficiency in the model as such. Unobserved heterogeneity can be taken into account by randomizing some of the parameters of the model in which case it is assumed that this randomization captures all time invariant unobserved heterogeneity. We are espe
29、cially interested in which kind of differences in inefficiency scores and firm rankings occur if we compare models which take into account only the observed heterogeneity to those which take into account also the unobserved heterogeneity. We take heterogeneity into account both through the inclusion
30、 of those effects in the cost function and in the mean of the distribution of inefficiency (observed heterogeneity) and by randomizing some parameters of the stochastic frontier model (unobserved heterogeneity). We also estimate a combined model where we have randomized the frontier constant term an
31、d at the same time explained the mean of the inefficiency distribution by a covariate. We also estimate the “true fixed effects model” (see Greene 2005a), where the unobserved heterogeneity is represented by the individual fixed effects. The models are estimated for 76 Finnish electricity distributi
32、on utilities. Our results indicate that in all heterogeneity accounting models mean inefficiency decreases significantly compared to the basic random effects model. According to our results randomizing some of the parameters seems to help to capture the unobserved heterogeneity and hence this kind o
33、f firm specific heterogeneity does not appear as inefficiency in our estimation results. The model which accounts observed heterogeneity and the models which account unobserved heterogeneity produce clearly different rank orders. It is, however, notable that (especially) the true fixed effects model
34、 may be 6 v u i overspecified so that if there exists persistent inefficiency it is appearing completely in the firm specific constant term of this model. The rest of the paper is organized as follow. Section 2 gives an introduction of the heterogeneity in stochastic frontier models and in section 3
35、 the data is presented. The estimated model specifications and the estimation technique is described in section 4. Section 5 gives the estimation results and provides brief discussion on their implications. Section 6 summarizes the findings. 2. Heterogeneity in stochastic frontier models 2.1. Observ
36、ed heterogeneity The first model that we use is the basic random effects (RE) specification proposed by Pitt and Lee (1981). In this model it is assumed that the firm specific inefficiency (in proportional terms) is the same every year cit xit vit ui , vit N (0,2 ), u N 0,2 , (1) where cit are the c
37、osts to be explained, xit are the explaining variables, and are the parameters to be estimated, ui is the inefficiency term and vit is the error term capturing the effect of noise. It is assumed that ui and vit are independent and, moreover, ui is independent of xit. Equation (1) can be estimated by
38、 maximum likelihood. There are some recognized problems connected to this model. One of them is that this model not only absorbs all unmeasured heterogeneity in ui, but it also assumes that inefficiency is uncorrelated with included variables (Greene 2005a). This problem can be reduced through the i
39、nclusion of those effects in the mean and/or variance of the distribution of ui or to the variance of the distribution of vit. Another problem connected to the basic RE model is that the inefficiency term is time invariant. Consequently it is possible that the time variant error term (vit) of the fr
40、ontier can in fact capture major part of inefficiencies whereas the time invariant term (ui) in fact captures time invariant firm specific heterogeneity. 7 v u In the following the observed heterogeneity3 is resided to the mean of the inefficiency distribution. This formulation reduces the implicit
41、assumption that the effects are not correlated with the included variables in the basic RE model. This model specification is called REH in the following. It can be written as c x v u v N (0,2 ), u N ,2 , it it it i, it v i i u (2) i 0 1hi , where hi is heterogeneity summarising covariate explaining
42、 the mean of the inefficiency distribution and 0 and 1 are new parameters to be estimated. One problem connected to this specification is that even though the observed heterogeneity is now modelled out from the inefficiency distribution it does not recognise the unobserved heterogeneity which still
43、remains in ui. However, the second problem of the basic RE model is now reduced by allowing correlation between inefficiency explaining variables and frontier explaining variables through inclusion of the heterogeneity characterising covariate. Another positive feature related to this model is that
44、it enables a more precise estimation of the frontier. It is important to note that the REH model accounts only for the observed heterogeneity and it is difficult to evaluate beforehand any kind of superiority of these models in inefficiency measurements. One must also be careful in making interpreta
45、tions, as the unobserved heterogeneity still remains in the inefficiency distributions. 2.2. Unobserved heterogeneity Fixed effects model To overcome the well known problems related to the basic fixed effect (FE) model (Schmidt and Sickles 1984), especially the fact that in the FE model any time inv
46、ariant unobserved heterogeneity appears in the inefficiency component, Greene (2005a) proposes the following model where firm specific constant terms are placed in the stochastic frontier cit i xit vit uit vit N (0,2 ), uit N 0,2 . (3) 3 See Greene (2004) for incorporating measured heterogeneity in
47、the production function. 8 v u Greene refers to this extended model as “true fixed effects model” (TFE) to underline the difference with the FE framework commonly used. In the TFE model, fixed effects represent the unobserved cross firm heterogeneity, not the inefficiency as in the original FE model
48、. In other words it places unmeasured heterogeneity in the cost function and hence if the model is log linear it produces a neutral shift of the function specific to each firm. This approach will become impractical, however, as the number of firms in the sample, and the number of parameters in the m
49、odel, becomes large (see Greene 2005a). Greene shows, by using simulated samples, that although the fixed effects may be largely biased, as far as the structural parameters and inefficiency estimates are concerned, the model performs reasonably well. The model can be fit by maximum likelihood. Random effects modelling Greene (2005a) proposes an extension also to the random effects
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