Stochastic Energy Performance Evaluation Using a Bayesian Approach

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Introduction
Te theoretical defnition of the production function describing the maximum amount of output that can be obtained from a given amount of input has been accepted for years.Te estimation of frontier production functions and the results of theoretical and empirical studies converge.Stochastic frontier analysis (SFA), which was frst introduced by Aigner et al. [1] and Meeusen and van den Broeck [2], has been used extensively in the determination of energy efciency over the last two decades (see [3][4][5][6][7][8][9]).Some signifcant shortcomings of previous studies in this feld can be overcome by the fact that the error term in estimating the frontier production function consists of two components.SFA is frequently used as a parametric method to estimate the boundary functions and measure production efciency.It establishes a functional relationship between output variables such as cost, proft, and production and input variables such as explosive and environmental factors.
In recent years, there has been a remarkable increase in the use of SFA in the energy sector.Some studies in the literature regarding this situation are discussed.First, Huntington [10] described the relationship between energy efciency and productivity using SFA.In addition to the standard randomly distributed error term, the econometric approach specifes a second error term with a skewed distribution to allow decision-making units to be above the limit rather than below it.Estimated productivity varies according to the assumed essential production function's form and researchers' deterministic or stochastic approach.
Buck and Young [11] discussed a parametric approach to estimate a stochastic boundary function for energy use in Canadian commercial plants.Te stochastic frontier approach explicitly acknowledges that not all decision-making units will use energy efectively, given the building activities and the level of technology.Boyd [12] used a similar methodology in moist maize milling plants, while Filippini and Hunt [3] used panel stochastic boundary analysis to calculate the energy efciency of 29 countries.When panel data are used with the stochastic frontier model, the estimated measurement error productivity is considered, and a more reliable measure of productivity is calculated.Te maximum likelihood method, one of the most popular estimation methods, is frequently used in deriving parameter estimates of SFA.In recent years, there has been an increase in the use of the Bayesian approach for SFA parameter estimation.Te Bayesian approach expresses the results in terms of probability density functions and provides a direct probability interpretation of unknown parameters.Te signifcant advantage of using a Bayesian approach is the preliminary distribution, which allows one to express uncertainty about unknown parameters before considering some evidence.
Because of its many advantages, several studies on stochastic frontier analysis with a Bayesian approach exist.Te Bayesian approach to SFA was frst introduced by Van den Broeck et al. [13].Tey reconsidered the error model on productivity with diferent sampling distributions.According to them, given a particular model, all efciencies are assumed to be derived from the same distribution, but which one is unknown.Mixing diferent distributions for each efciency in the sampling model is generally avoided, making analysis easier.Instead, simple models are mixed at the fnal stage.If we want to choose a particular distribution to calculate efciency, it may be an excellent alternative to use Bayes factors as a criterion for model selection.Koop et al. [14] described the Markov chain Monte Carlo (MCMC) as a numeric integration method in a stochastic frontier frame.Te advantage of Bayesian methods is that they provide precise fnite sample results for any feature of interest and that parameter uncertainty is fully considered.Te posterior features were evaluated using Monte Carlo integration.Tey explained how Gibbs sampling methods used the Bayesian approach to signifcantly reduce the computational burdens of stochastic boundary models.In addition, current developments in Bayesian stochastic frontier analysis (see [15][16][17][18][19]) can also be viewed.More recently, Grifn and Steel [20] described MCMC methods for Bayesian analysis of stochastic frontier models using the WinBUGS package software.Tsionas and Papadakis [21] provided a Bayesian approach to the problem organized around simulation techniques.Tabak and Tacles [22] used a Bayesian stochastic frontier for the cost and proft efciency of the Indian banking industry.Tonini [23] estimated the total factor productivity growth in agriculture for the European Union and candidate countries using SFA with a Bayesian approach.Feng and Zhang [24] compared the effciency of large and community banks in the United States from 1997 to 2006 using a Bayesian approach.Assaf and Josiassen [25] estimated the efciency of healthcare food service operations with Bayesian SFA.Assaf et al. [26] analyzed the efciency of Turkish banks from 2002 to 2010 using a Bayesian stochastic frontier approach.Barros [27] studied airport efciency in Mozambique, estimating a cost function with random and fxed-efects stochastic frontier models with a Bayesian stochastic frontier model.
Despite its rapid growth across several disciplines, the use of the Bayesian approach to measuring energy performance has yet to gain strong attention in energy research.Te authors in [1,2] used exponential and half-normal distributions, respectively.Gamma distributions were used in [28], and log-normal distributions were studied by Migon and Medrano [29].Grifn and Steel [20] described a semiparametric modeling technique to estimate the inefciency distribution.Alghalith [30] described an alternative method for specifying the distribution of the inefciency term.Each of these inefciency terms can cause diferent behaviors in the distribution of technical efciencies [31].Tere are no apparent reasons for selecting one distributional form over the other; each has its pros and cons [32].
Here, we propose diferent models with diferent inefciency components as exponential, half normal, truncated normal, and gamma in a formal Bayesian framework.Bayesian methods appear suitable for stochastic frontier models because they provide precise small-sample results (inference of efciencies), allow prior knowledge and regularities conditions to be incorporated during the estimation, and more accurately represent parameter uncertainties through kernel densities.Stochastic frontier models require numeric integration methods because they are so complex; the most appropriate method is MCMC.Efciency measurement with stochastic frontier models is troublesome in many situations because decomposing the overall error term into a two-sided and a one-sided disturbance term may be problematic.Te reason is that when the noise-to-signal ratio is relatively high, the overall error term would appear to be approximately symmetric, in which case identifcation of the efciency component would be problematic [19].Te main aim is to compare these models for diferent distributions of the inefciency term using the MCMC method and to rank countries according to their technical efciency with the best model.A comparison of models with diferent distributional assumptions was performed using the DIC.
While there are numerous advantages to employing the Bayesian approach, its application in the context of energy efciency has been limited.Our research seeks to fll this gap by conducting a comprehensive analysis of 29 OECD countries' energy-based development performance using SFA with a Bayesian approach.One specifc aspect that has received insufcient attention in the literature is the selection of the distribution for the inefciency term.Te inefciency term represents the unexplained deviation from the production frontier and plays a crucial role in accurately measuring energy efciency.Despite its importance, no apparent method for selecting the distribution of the inefciency term has been established in the literature.To address this gap, we propose and compare diferent models with various inefciency components, including the half normal, truncated normal, exponential distribution, and gamma distribution.By considering these alternative models, we aim to explore the impact of diferent distributional assumptions on the measurement of energy efciency.To conduct our analysis, we utilized a panel dataset spanning from 2004 to 2010.Te Bayesian implementation of the proposed models is performed using the WinBUGS package, employing the Markov chain Monte Carlo (MCMC) method.
Te primary objective of our study is to compare the performance of these models, each assuming a diferent distribution for the inefciency term, using the deviance information criterion (DIC).Te DIC provides a robust basis for model comparison, enabling us to identify the model that best fts the data and captures the true energy efciency scores of the countries.By addressing this research gap and employing a Bayesian SFA approach, we contribute to the existing body of knowledge on energy efciency analysis.Our fndings will not only shed light on the most appropriate model for measuring energy efciency but also allow us to rank countries based on their technical efciency using the identifed best model.
Te paper proceeds as follows: Section 2 describes the Bayesian stochastic frontier model.Section 3 shows diferent models with diferent inefciency components, such as the half normal, truncated normal, exponential distribution, and gamma distribution.Section 4 presents the result from the model estimation.Section 5 provides further discussion of the results, and the conclusion summarises the results and provides directions for future research.

Bayesian Stochastic Frontier Model
Te SFA approach [1] can be illustrated as the following equation: where y i is the log of output for DMU i (i = 1, 2, . .., N), X i is a vector of input variables, β is the vector of coefcients, v i is a symmetric disturbance capturing measurement error in the stochastic frontier, the error term is independent and identically distributed (IID), and u i is a nonnegative disturbance capturing the level of DMU inefciency (u i ≥ 0).Te error term ε i � v i − u i has a symmetric distribution.Tere needs to be more clarity about the inefciency of term distribution.A particular distributional assumption on u is needed.In the literature on efciency estimation, four distributional assumptions have been proposed, namely, an exponential distribution [2], a half-normal distribution [1], a halftruncated normal distribution [24], and a gamma distribution [19].Te posterior distribution is shown in the following equation: where β is the set of coefcients in the production function, θ is the set of parameters in the prior distribution, and X is the matrix with logarithms of the input variables.Te complete conditional distributions are given by (3)

Models
In this study, we adopt a preferred aggregate energy demand model for a panel of OECD countries, which was previously utilized in [21].Te model, as depicted in equation ( 5), relies on an unbalanced dataset encompassing a sample of 29 OECD countries from 2004 to 2010.Te dataset is sourced from the International Energy Agency (IEA) database and the OECD database.We use energy consumption (EC) as the dependent variable and gross domestic product (GDP), the accurate price of energy for households and industry (RPE), the area size of a country measured in squared km (ASC), the share of value added of the industrial sector (SVAIS), and the share of value added for the service sector (SVASS) as independent variables.We focus on comparing these production functions and diferent distributions of the efciency term.We adopt a Bayesian approach and use Markov chain Monte Carlo (MCMC) simulation to estimate parameters and compare models.Terefore, we propose diferent models with diferent inefciency components as the half normal, truncated normal, exponential distribution, and gamma distribution.Te proposed model is specifed as the following equations: where for ith country in tth year, Y it is the logarithm of energy consumption (EC), X1 it is the logarithm of GDP, X2 it is the logarithm of the accurate price of energy for households and industry (RPE) (2005 � 100), X3 it is the logarithm of the area size of a country measured in squared km (ASC), X4 it is the share of value added of the industrial sector (SVAIS), X5 it is the share of value added for the service sector (SVASS), t is a time trend, v it is a symmetric disturbance representing the efect of noise, and u it is a term for inefcient energy use.Descriptive statistics for the variables used in the model are presented in Table 1.

Journal of Mathematics
Appropriate prior specifcations for the parameters need to be included.Various suggestions for prior choices have been made in the literature (e.g.[33,34]).We used the same priors proposed by Grifn and Steel [20].We defne the prior distribution for the parameters in θ that all parameters are independent and β j ∼N(0, σ 2 β ).We assigned prior λ∼Exp (−log r * ), (r * ∈ (0, 1)) for a half-normal distribution supposing that u i ∼ Exp(λ).We also used a Gamma prior for the precision; that is, λ − 1 ∼Ga(c, d), a truncated normal distribution is assumed.We enclosed a normal prior for the location, that is, ε ′ ∼N(0, ε ′ 2 ), and prefer the same prior for λ as in λ − 1 ∼Ga(c, d).Our model used the WinBUGS package program to implement the Bayesian implementation.For all our applications, the MCMC algorithm involved 32,001 MCMC iterations where the frst 15,000 were discarded in a burn-in phase.We used the deviance information criterion (DIC), which was introduced in [35] and commonly used in Bayesian analysis, to evaluate the models defning the deviance of a model with parameters θ as follows: then the DIC is where D is the expected deviance and pD is a complex term such that where θ is the mean of the posterior parameter distribution.Te DIC can be evaluated automatically within the WinBUGS setup, and a good description of its use in stochastic frontier models can be seen in the study by Grifn and Steel [34].Before using the Bayesian approach results, it is necessary to check the convergence assessment which involves checking whether the chain is converged.Tis study considers several statistical diagnostic tests for Markov chain convergence, such as Gelman, Rubin, Geweke, and Raftery-Lewis.Te diagnostic statistics indicate that the Markov chain has reached convergence for each parameter for all models using diferent convergence methods such as Geweke, Gelman, Rubin, and Raftery-Lewis diagnostics.

Results and Discussion
Posterior summaries and densities for the frontier model in equation (4a), after running the MCMC algorithm for 47,000 iterations and discarding the initials 15,000, are shown in Table 1.It presents the posterior mean, standard deviation (SD), and the 95% prediction intervals of the parameters β's in model 1.We also obtain the MC (Monte Carlo) error to see if the convergence is satisfed and simulate that the MC error for each parameter is less than 5% of the sample SD.One way to assess the accuracy of the posterior estimates is by calculating the MC error for each parameter.Tis is an estimate of the diference between the mean of the sampled values and the true posterior mean.As seen in Table 2, the MC error for each parameter is less than 5% of the sample SD.
If the prediction interval passes through zero, one can conclude that the parameter is not signifcant.
From the table, it is clear that the only four signifcant coefcients are the ones associated with the gross domestic product (β 1 ), the accurate price of energy for households and industry (β 2 ), the area size of a country measured in squared km (β 3 ), and time trend (β 6 ), confrming the previous results by Table 3 which compares the posterior results from all models.
Te average annual efciency of countries in terms of distributions with the TIME variant is presented in Table 4.It shows a summary of the posterior distribution for the countries in the sample (high rank corresponds to high efciencies).Te posterior distribution clearly demonstrates a large spread of the rankings.From these, it is observed that the mean efciency values are in the range of 0.482-0.921for the exponential distribution, 0.489-0.868for the gamma distribution, 0.493-0.914for half-normal distribution, and 0.372-0.901for the truncated normal distribution.Te average technical efciency scores imply that on average, the countries were producing about 49.6%, 72.0%, 74.5%, and 66.2% of the outputs that could be produced using the observed input quantities using exponential, gamma, half-normal, and truncated normal distributions, respectively.Te half-normal distribution gave higher technical efciency estimates than the other distributions.
In this study, 29 OECD countries' energy-based development performances were measured using stochastic frontier analysis with a Bayesian approach.To do this, four diferent models have been considered.According to these models, the energy efciency scores of the countries have been estimated, and according to these scores, rankings of efciency have been done.In Table 4, " * " indicates countries with diferent average energy efciency rankings for different distributions.Table 5 examines the correlation among efciency rankings for all the models using Spearman's rank correlation coefcient.

Journal of Mathematics
Correlation coefcients between the efciency rankings for all models are high and positive (minimum 0.995).In this case, there is a strong positive relationship between the orders of all models, and this relationship is statistically signifcant at even a 1% signifcance level.
Table 6 compares the DIC scores for the error and the diferent inefciency distributions.Te DIC is an attractive alternative to the Bayes factor; it is highly reliable and can handle complicated models (see [35,36]).A lower value of DIC indicates a better ftting model.Overall, the results favour the half-normal distribution.Terefore, we use this model for the fnal decision to rank countries' energy efciency scores for the whole period.
All models generally gave similar efciency scores and orders of efciency except for some minor changes, such as Canada and Greece shared values with similar events.Based on technical efciency, the most infuential country was Finland and the lowest efective was Mexico.

Conclusion
In our study, we aimed to address a notable gap in the existing literature on energy efciency analysis.Over the past two decades, stochastic frontier analysis (SFA) has been widely used to assess energy efciency.While there are numerous advantages to employing the Bayesian approach, its application in the context of energy efciency has been limited.Our research seeks to fll this gap by conducting a comprehensive analysis of 29 OECD countries' energy-based development performance using SFA with a Bayesian approach.One specifc aspect that has received insufcient attention in the literature is the selection of the distribution for the inefciency term.Te inefciency term represents the unexplained deviation from the production frontier and plays a crucial role in accurately measuring energy efciency.Despite its importance, no apparent method for selecting the distribution of the inefciency term has been established in the literature.To address this gap, we proposed and compared diferent models with various inefciency components, including the half normal, truncated normal, exponential distribution, and gamma distribution.By considering these alternative models, we explored the impact of diferent distributional assumptions on the measurement of energy efciency.We utilized a panel dataset spanning from 2004 to 2010.Te Bayesian implementation of the proposed models is performed using the WinBUGS package, employing the Markov chain Monte Carlo (MCMC) method.
We compared the performance of these models, each assuming a diferent distribution for the inefciency term, using the deviance information criterion (DIC).Te DIC provides a robust basis for model comparison, enabling us to identify the model that best fts the data and captures the true energy efciency scores of the countries.By addressing this research gap and employing a Bayesian SFA approach, we contributed to the existing body of knowledge on energy efciency analysis.Our fndings not only shed light on the most appropriate model for measuring energy efciency but also allow us to rank countries based on their technical efciency using the identifed best model.
In recent literature, several attempts have been made to overcome the main weaknesses of the feld by evolving more specifcation and estimation procedures.Te use of Bayesian techniques endows the researcher with the tools to use more fexible models, and it is not needed to impose a priori distributional assumptions on the efciency term in the framework of stochastic frontier approaches.
In summary, our study extended the application of Bayesian SFA in energy efciency analysis, addressed the gap in the selection of the inefciency term distribution, and provided valuable insights into the relative energy efciency performance of 29 OECD countries.
According to the convergence criteria such as Gelman, Rubin, Geweke, and Raftery-Lewis, the convergence of all parameters of each model was granted.For all models, GDP, RPE, ASC, and TIME were found as statistically signifcant parameters, while the others were found to be insignifcant.Te model with half-normal inefciency distribution gave the most minimum DIC score.Terefore, we use this model to rank countries' energy efciency scores.According to the model with half-normal inefciency, based on technical efciency, the most infuential country was Finland and the lowest efective was Mexico.Since Mexico and Turkey are the lowest efcient countries, they should reconsider their energy policy and take precautions to improve energy effciency.For instance, energy intensity and losses in the industry should be reduced.Energy should be used efectively and efciently in the public sector.Future studies may consider assessing the productivity of more countries over a broader period and identify future strategies for improvement.Tis study covers only OECD countries.A more comprehensive study can be conducted using a larger dataset, or more detailed research can be conducted by considering the leading countries in the feld of energy.In summary, the results of this study provide a reference for managers and policymakers in the energy-based development performance and show the way forward for future strategies and investments.

Table 1 :
Descriptive statistics for the variables.

Table 3 :
Summary of posterior results from all models.

Table 4 :
Comparison of the average energy efciency score and rankings for all models of the whole period.

Table 2 :
Bayesian estimated parameters of the stochastic production frontier.