“Combining aggregate production function and technical efficiency: Indonesian case study”

Besides frontiers and Data Enveloped Analysis approaches, a combination between the aggregate production function and technical efficiency can be performed using the Corrected Ordinary Least Square approach. Unfortunately, there are no studies in Indonesia that use this approach. This paper mainly studies how the Corrected Ordinary Least Square approach combines aggregate production function and technical efficiency. The methods are aggregate production function modeling, aggregate production function correcting, and technical efficiency measuring. The data are Gross Regional Domestic Product at a constant price, the number of workers, investment expenditure, education, and health data of Indonesian provinces from 2015 to 2020. There are three results. First, the Indonesian fixed effect panel data aggregate production function is the best model. In this model, Gross Regional Domestic Product at a constant price is influenced by the number of workers, investment expenditure, and human capital. Human capital consists of education and health level. Second, the deterministic frontier aggregate production function shifts the best-fixed effect model so that the constant becomes –15.36. Third, the Indonesian technical efficiency when no factors influence inefficiency is on average 0.9936. All the results indicate that human capital, aggregate production and technical efficiency combination, and the Corrected Ordinary Least Square approach are practical values.


INTRODUCTION
Many studies discussed the aggregate production function and technical efficiency.They learned aggregate production function development, aggregate technical efficiency calculation, convergency, and cluster to promote economic growth and efficiency.They also learned to reduce poverty, economic disparity, environmental damage, and crime.
However, studies about Indonesian aggregate production function and technical efficiency simultaneously are limited.Among them are Muchdie (2016) All of the studies raise questions about the role of the Corrected Ordinary Least Square (COLS) approach to combine aggregate production function and technical efficiency and bridging between the parametric and nonparametric approaches.In that perspective, it is essential to study Indonesian aggregate production function and technical efficiency simultaneously based on the Corrected Ordinary Least Square (COLS) approach.

LITERATURE REVIEW
Various literatures discussed the aggregate production function, methods of combining the aggregate production function and technical efficiency, and techniques for measuring technical efficiency.An organized review is essential in order to obtain the proper framework in combining aggregate production function and technical efficiency.The beginning of this section discusses previous studies on the aggregate production function in order to establish the aggregate production function model.After that, this section discusses previous studies on an appropriate approach to combine aggregate production functions and technical efficiency.Finally, this section reviews studies on measuring technical efficiency at the aggregate level to ensure the appropriateness of the COLS approach to measure aggregate technical efficiency.There are also differences in the chosen variables.All of them operate the number of workers.However, they are different in their aggregate output, investment, and human capital variables.Some studies prefer to approach aggregate output by Gross Regional Domestic Product (GRDP) at a constant price (Mendez, 2020 (Anwar, 2017(Anwar, , 2018;;Kustanto, 2020;Wibowo, 2019).
All of the Indonesian aggregate production function studies above open an opportunity to configure the Indonesian aggregate production function based on the panel data regression equation.In addition, they open an opportunity to specify and signify that GRDP at a constant price is positively influenced by the number of workers, investment expenditure, and human capital.The configuration, specification, and significance are useful to establish the frontier aggregate production function, which is important to run the COLS approach.
Corrected Ordinary Least Square (COLS) is an approach to measure technical efficiency.It has three steps.The first is specifying the initial production function equation.The specification applies the ordinary least square (OLS) approach that all inputs affect significantly output.The second is forming the frontier production function equation by adding the constant of the initial production function equation with its maximum error term.The third is measuring the technical efficiency by dividing the initial production function with the frontier equations (Alarenan et al., 2019;Julien et al., 2020;Vasanthi et al., 2017).
The COLS approach differs from the Stochastic Frontier Analysis (SFA) and Data Envelopment Analysis (DEA) approaches.All of them are approaches to measuring technical efficiency.However, the COLS approach applies a least square estimation, which is a parametric approach, while the SFA is a maximum likelihood estimation, which is a parametric-stochastic approach, and the DEA is a linear programming estimate, which is a non-parametric and non-stochastic approach (Julien et al., 2020;Vasanthi et al., 2017).
The COLS approach can also calculate cost efficiency.The calculation steps are similar to the technical efficiency calculation.The initial step is specifying the cost regression equation.The second is establishing the cost frontier equation by subtracting the constant of the initial cost equation with its highest error term.The last is estimating the cost efficiency by dividing the cost regression equation by its frontier cost equation (Alarenan et al., 2019).
All of the studies above point out that COLS can be an appropriate approach to measure technical efficiency at an aggregate level.The reason is the aggregate level applies the aggregate production function, which has similar equations and estimations to the technical efficiency measurement.The aggregate production function and the technical efficiency measurement are input-output equations.Moreover, they apply the OLS estimation.In other words, the COLS can combine aggregate production function and technical efficiency.
Technical efficiency is production efficiency where all inputs produce optimal output.If a production causes all inputs to produce optimal output, then the production is efficient.The problem is when the production leaves residue and is inefficient.Technical efficiency measurement provides a solution to reduce residue and inefficiency so that the institutional capacity strengthens ( Previous studies on the aggregate production function, COLS approach, and technical efficiency measurement show that the Indonesian aggregate production function and technical efficiency combination on the COLS approach is a specific topic.It starts from the Indonesian aggregate production function model and tests the hypothesis that GRDP at a constant price is influenced positively by the number of workers, investment expendi-ture, and human capital.After that, it applies the COLS approach by correcting to the frontier the Indonesian aggregate production function model and measuring the Indonesian technical efficiency by comparison between the tested Indonesian aggregate production and frontier.
Therefore, this paper aims to model the Indonesian aggregate production function, correct and shift the aggregate production function to its frontier, and estimate Indonesia's level of technical efficiency.

METHODS
This study applied three methods.The first is the Indonesian aggregate production modeling.The second is the COLS approach.The third is the technical efficiency measurement.
The Indonesian aggregate production modeling is panel data modeling.Data in this study is the annual data of 34 provinces in Indonesia from 2015 to 2020.The data consists of provincial GRDP at a constant price, the number of over 15 years old labor force, investment expenditure, higher education gross participation, and life expectancy rates data.GRDP at a constant price is an approach to aggregate output, the number of over 15 years old labor is an approach to the number of workers, investment expenditure is an approach to the amount of investment, the higher education gross participation is an approach to education human capital, and life expectancy is an approach to health human capital.
Modeling consists of model selection and significance testing.Selection is between common, fixed, and random effect models based on Lagrange Multiplier, Chow, and Hausman tests.Significance testing examines totally and partially the relationship between the Indonesian provincial GRDP at a constant price to the number of workers, investment expenditure, education, and health.
The panel data regression equation general form and its specification are: , , , , where GRDP is the gross regional domestic product at a constant price, L is the number of workers, K is investment expenditure, E is education, H is health, i is cross-section data, t is time-series data, β 0 is an intercept, 1-4 is the coefficient of X 1-4 , Ln is the natural logarithm, ε is the regression equation residual.
The COLS approach shifts the aggregate production function to its frontier.There are three steps.First, calculating the maximum error term of the Indonesian aggregate production function as in Equation 2. Second, adding the constant of Equation 2, β 0I,t , by the maximum error term so that the Indonesian aggregate production becomes the frontier Indonesian aggregate production function.Equation 3shows the frontier aggregate production function.The f symbol in Equation 3 means that the equation is the frontier equation.Third, calculating the GRDP at a constant price based on the initial and frontier aggregate production function.
The technical efficiency measurement is by the division between the GRDP at a constant price obtained from the initial aggregate production function to the GRDP at a constant price obtained from the frontier equation.The long formula for calculating technical efficiency (ET) is: where the upper equation is the initial aggregate production function, whilst the lower is the frontier aggregate production function.
Technical efficiency ranges from 0 to 1. Zero means extremely inefficient.Various factors cause inefficient production.One means very efficient where there are no factors that cause inefficient conditions.

RESULTS
This section presents all of the results.The Indonesian fixed effect panel data aggregate production function has three important features.
The first is all inputs influence significantly the aggregate output.The second is the significance of the constant term.The third is human capital consists of education and health variables.
Equation 5 shows the frontier Indonesian aggregate production function.The origin of this equation is the Indonesian fixed effect aggregate production function model.Equation 5has three characteristics.The first is this equation is a frontier equation.This is shown by the symbol on the upper left side of Equation 5.The second is the constant coefficient.It is a sum of the initial aggregate production function constant and its maximum error.It is -15.36=-15.48 + 0.12.The third is the equation is without an error term.It means that the equation is non-stochastic or deterministic. , The Indonesian technical efficiency consists of provincial efficiency.The average value is 0.9936.The lowest technical efficiency is Bengkulu province technical efficiency, 0.9928, and the highest technical efficiency is DKI Jakarta, 0.9943.All provincial efficiencies are close to one and show that the Indonesian economy is technically efficient (see Table 2).

CONCLUSION
The central concept of this paper is combining aggregate production function and technical efficiency based on the COLS approach.Indonesian case shows that setting the aggregate production function model and shifting the model to its frontier deterministically leads to the highest level of technical efficiency.It means that the COLS approach works properly to integrate the aggregate production function and technical efficiency.
There are three novelties from this paper.The first is the role of human capital, especially the health factor to form the Indonesian aggregate production function.The second is the combination of the aggregate production function and the technical efficiency.The third is the role of the COLS approach.
The results and novelties provide two implications.The first is the practical use implication.The economy should pay attention to the aggregate production function and technical efficiency.This attention includes attention to health level and negative externalities that affect technical efficiency such as divergency, poverty, disparity, environmental damage, and crime.The second is the implication of future research.Future studies have a big opportunity to apply the COLS approach to test human capital that affects aggregate output and negative externalities that decrease aggregate technical efficiency.These studies will provide a valuable academic base for economic policy.

Table 1 .
Indonesian aggregate production function model el (REM) is more appropriate than the common effect model (CEM), the Chow test decides that FEM is more appropriate than CEM, and the Hausman test decides that FEM is more appropriate than REM.Therefore, the best model is the fixed effect aggregate production function (Note: The process and results of the model selection are not shown in this paper).Source: Statistical estimation.Note: Dependent variable: LNGRDP, * = Significant at α = 5%.

Table 2 .
Average value of Indonesian technical efficiency from 2015 to 2020 Source: Statistical estimation.

Table 2 (
cont.).Average value of Indonesian technical efficiency from 2015 to 2020Problems and Perspectives in Management, Volume 20, Issue 3, 2022 the same.As a result, the efficiency level becomes very close to one and indicates that the aggregate production runs efficiently.It means that future studies have an opportunity to develop aggregate production function and technical efficiency combination based on the COLS approach by incorporating factors such as cluster and convergency.Another prospect is to examine negative externality factors such as poverty, disparity, environmental damage, and crime.All of them enrich the COLS development and are vital for macroeconomic decision making.