Efficiency Measurement in Norwegian Electricity Distribution: A Generalized Four-Way-Error-Component Stochastic Frontier Model.

• Author: Tsionas, Mike G.

1. INTRODUCTION AND THE LITERATURE

   Efficiency plays an important role in any production process because (in)efficiency increases costs and therefore reduces profits. Efficiency is also intimately related to management. However, to make the production process more efficient, a production manager needs to know the degree of inefficiency, its sources and perhaps the factors explaining it. Furthermore, inefficiency in a longitudinal (panel) model can have a time-invariant persistent (structural) component, which cannot be changed in the short-run. Another source of inefficiency is transient, which can be remedied in the short-run if one knows the factors causing it. Thus, from a production management point of view, it is important to know the overall efficiency of the production process so that policy changes can be made to improve it. This is especially true for electricity distribution companies, which are regulated. Separation of transient and persistent inefficiency is useful from the regulator's point of view because they want to set a benchmark against which performance of all of the companies can be measured. In defining the benchmark, the regulator should know not only the overall inefficiency but also the part that is persistent and cannot be altered in the short-run. This can give incentive to eliminate transient inefficiency, while setting a target. If the persistent inefficiency is non-negligible, then the regulator can think about making structural changes to reduce it. Thus, we feel that a decomposition of overall inefficiency is important. It is equally important to examine its dynamic nature, as well as the factors that can explain the different components of inefficiency.

   If a production unit (firm) is observed for several time periods (i.e., the data is panel), then one can use models that are much richer than those used in cross-sectional situations (where each firm is observed only once). Panel data can give us a reliable measure of efficiency because it can control for unobserved factors (known as heterogeneity) that are time-invariant but which vary across firms. Similarly, inefficiency can have persistent (time-invariant) and transient (time-variant) components. The most recent models that are used for this purpose are Colombi et al. (2014), Tsionas and Kumbhakar (2014), Kumbhakar et al. (2014), Badunenko and Kumbhakar (2017), Lai and Kumbhakar (2018, 2020), Kumbhakar and Lien (2019), Filippini and Greene (2016), Filippini et al. (2018) and Musau et al. (2021). These models are popularly known as four-component stochastic frontier models because the error term in the econometric model has four-components, which can be separated when panel data are available.

   In this paper, we start with a four-components panel stochastic frontier model (SFM). These four-components are: i) the statistical noise term, which can take both positive and negative values; ii) a transient inefficiency component, which is non-negative and is allowed to vary over time and firms; iii) a non-negative structural or persistent inefficiency, which is time-invariant but firm-specific; and iv) a random firm heterogeneity. Inefficiency components are non-negative (onesided) by definition in a production context because, given everything else, inefficiency reduces output. Alternatively, the presence of inefficiency means that the production process is not fully efficient and there is scope for improvement, which is what a production manager wants to know.

   To generalize the state-of-the-art SFM, we allow: (i) the transient inefficiency to vary over time and firms by assuming that follows an autoregressive process to capture "persistence" in the short-term inefficiency; (ii) the structural or persistent inefficiency term that is time-invariant but firm-specific and depends on exogenous/endogenous covariates; and (iii) fixed or stochastic firm heterogeneity. Why are these generalizations important to a production manager? In the existing models, transient inefficiency is either assumed to be independently and identically distributed (Colombi et al. 2014; Tsionas and Kumbhakar, 2014; Kumbhakar et al. 2014; Filippini and Greene 2016) or its variance is allowed to depend on some exogenous variables (viewed as determinants of transient inefficiency), such as in Badunenko and Kumbhakar (2017), Lai and Kumbhakar (2018, 2020), Musau et al. (2020), and Kumbhakar et al. (2020). However, an inefficient producer is likely to adjust their production plan to make them more efficient over time. Thus, it might be more realistic to allow transient inefficiency to be related to its past values (dynamic). It might also depend on exogenous variables. Ignoring such factors can lead to biased estimates of production parameters, as well as inefficiency. Note that these generalizations are testable and it is not mandatory to use all of the components.

   In many situations (mostly for firms under regulations), one component of inefficiency is likely to be persistent (time-invariant). This might be associated with spending resources to deal with the regulators during a price review (to give an example). This is common in industries such as electricity supply, water and gas distribution, transportation, post offices, and so on. Thus, because of the regulated nature of an industry, all of the firms under the same regulator might have what we call structural inefficiency, which can only be changed if there is a change in regulation regime. Consequently, it is more realistic to allow for a component of inefficiency that is time-invariant (at least in a short panel where time-length is not too long). If there are variables that can explain this structural inefficiency, then we can use those variables and test their presence. This is why we consider (ii). One problem in modeling persistent inefficiency is that the panel models also allow time-invariant firm heterogeneity, and the issue is how to separate two time-invariant components. Consequently, the earlier panel models either ignored firm heterogeneity and/or persistent inefficiency. Ignoring one or the other is likely to affect estimates of inefficiency. In our extension (iii), we separate firm heterogeneity from (ii) by allowing a formulation of firm heterogeneity suggested by Mundlak (1961). In this formulation, firm heterogeneity (also called firm-effects) is specified as a parametric function of some exogenous variables (which can be, for example, the sample means of the input variables) along with a random term. In econometric terminology, this is correlated random effects. This formulation helps us to reduce the number of the fixed effects that are treated as parameters. It also allows correlation of the firm effects with the input variables in the production function. This allows us to avoid the inconsistency problem arising from the correlation between firm effects and the inputs in a production function.

   To summarize, the original four-component (or four-way-error-component) panel SFM (Colombi et al. 2014; Tsionas and Kumbhakar 2014; Kumbhakar et al. 2014) has been generalized previously by Badunenko and Kumbhakar (2017), Lai and Kumbhakar (2018, 2019), Musau et al. (2020) and Kumbhakar et al. (2019). These generalizations include determinants of both persistent and transient inefficiency (Badunenko and Kumbhakar 2017; Kumbhakar et al. 2019; Balezentis and Sun 2020), as well as determinants and endogeneity of inefficiency (Lai and Kumbhakar et al. 2018, 2020; and Lien et al. 2018). However, these generalizations do not consider dynamic adjustment of transient inefficiency, which is addressed in the present paper. (1) We also introduce determinants of both persistent and transient inefficiency. Both the mean and variances of transient and persistent inefficiency are functions of determinants in our specifications, whereas the previous formulations allowed heteroskedastic variances. Furthermore, some of the determinants in our model can be endogenous. Instead of assuming firm effects to be either fixed or random, our specification makes it random but includes some covariates to allow for possible correlation between firm effects and some or all of the covariates in estimating (production/cost/distance) function. Thus, our formulation is much more general than any of the existing four-component SFMs. Given that the transient inefficiency is stochastic and autoregressive, the likelihood function is not available in a closed form because it involves multivariate integrals. However, the Maximum Simulated Likelihood (MSL) method can be used to estimate the parameters of the model and several other quantities of interest, including transient and persistent inefficiency. In addition, for robustness purposes, we use the (Simulated) Bayes Generalized Method of Moments (SGMM) approach.

2. EFFICIENCY MODEL

   We start with a stochastic production frontier function model with the four-components:

   [y.sub.it] = [x'.sub.it][beta] + [v.sub.it] + [[alpha].sub.i] - [u.sup.o.sub.i] - [u.sub.it],i = 1,...,n,t = 1,...,T, (1)

   where [y.sub.it] is output in natural logarithm, [x.sub.it] [member of] [R.sup.k] is a set of explanatory variables (some or all of which can be endogenous) in natural logarithms, [[alpha].sub.i] is the firm effect, [u.sup.o.sub.i] [greater than or equal to] 0 denotes persistent or structural technical inefficiency, [v.sub.it] is a two-sided error term and [u.sub.it] [greater than or equal to] 0 denotes transient technical inefficiency. Transient efficiency is defined as [Please download the PDF to view the mathematical expression]. Similarly, persistent efficiency is [Please download the PDF to view the mathematical expression]. Because for small values of [u.sub.it], [Please download the PDF to view the mathematical expression], transient efficiency is 1 minus transient inefficiency. Similarly, for small values of [u.sup.o.sub.i], [Please download the PDF to view the mathematical...