In this section, the simulation results are presented. First, the estimated mean values of sigma v and sigma u for various models are summarised in Appendix 3A. The standard deviations and squared mean errors are used to assess the performance of different SFA models. The results are presented in Tables 3.1 to 3.3. The density curve of the estimated efficiency term, u, is illustrated figure 3.2 for Normal- Burr x and Normal-Rayleigh.
Table 3.1: Standard Deviation of 𝝈𝒖 for various SFA Models
SD of u for Normal-Half Normal SFA
n/scale 0.2 0.3 0.4 0.5 0.6 20 0.41 0.4 0.38 0.33 0.26 50 0.36 0.34 0.32 0.29 0.17 100 0.3 0.29 0.27 0.24 0.1 250 0.25 0.25 0.23 0.18 0.06 500 0.22 0.22 0.20 0.11 0.04
SD of u for Normal-Exponential SFA
n/scale 0.2 0.3 0.4 0.5 0.6 20 0.23 0.23 0.22 0.22 0.21 50 0.17 0.16 0.16 0.15 0.13 100 0.13 0.14 0.13 0.12 0.09 250 0.11 0.11 0.11 0.09 0.05 500 0.10 0.10 0.09 0.06 0.03
SD of u for Normal-Rayleigh SFA
n/scale 0.2 0.3 0.4 0.5 0.6 20 0.42 0.39 0.36 0.33 0.21 50 0.37 0.36 0.34 0.29 0.14 100 0.32 0.32 0.31 0.26 0.08 250 0.28 0.27 0.25 0.18 0.04 500 0.24 0.23 0.22 0.12 0.03
SD of u for Normal-Burr x SFA
n/scale 0.2 0.3 0.4 0.5 0.6 20 0.21 0.49 0.45 0.31 0.28 50 0.28 0.25 0.39 0.36 0.5 100 0.37 0.42 0.52 0.46 0.29 250 0.29 0.36 0.15 0.44 0.28 500 0.31 0.28 0.39 0.54 0.31
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Table 3.2: Standard Deviation of 𝝈𝒗 for various SFA Models
SD of v for Normal-Half Normal SFA
n/scale 0.2 0.3 0.4 0.5 0.6 20 0.22 0.21 0.2 0.18 0.14 50 0.14 0.13 0.13 0.12 0.1 100 0.09 0.08 0.09 0.08 0.06 250 0.06 0.06 0.06 0.06 0.03 500 0.05 0.05 0.05 0.04 0.02
SD of v for Normal-Exponential SFA
n/scale 0.2 0.3 0.4 0.5 0.6 20 0.17 0.17 0.16 0.16 0.15 50 0.09 0.09 0.09 0.08 0.09 100 0.06 0.06 0.06 0.06 0.06 250 0.04 0.04 0.04 0.04 0.03 500 0.03 0.03 0.03 0.03 0.02
SD of v for Normal-Rayleigh SFA
n/scale 0.2 0.3 0.4 0.5 0.6 20 0.27 0.26 0.24 0.22 0.13 50 0.21 0.21 0.19 0.18 0.11 100 0.14 0.14 0.14 0.14 0.09 250 0.08 0.08 0.08 0.08 0.06 500 0.06 0.06 0.06 0.06 0.04
SD of v for Normal-Burr x SFA
n/scale 0.2 0.3 0.4 0.5 0.6 20 0.16 0.23 0.12 0.17 0.16 50 0.14 0.15 0.13 0.07 0.07 100 0.16 0.14 0.13 0.16 0.15 250 0.12 0.09 0.07 0.11 0.11 500 0.11 0.13 0.12 0.1 0.12
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Table 3.3: Mean Squared Error of Efficiency Estimates for various SFA Models
n/scale MSE of efficiency estimates of Normal-Half Normal SFA
0.2 0.3 0.4 0.5 0.6 20 0.07 0.07 0.07 0.07 0.04 50 0.06 0.06 0.06 0.06 0.02 100 0.04 0.05 0.05 0.05 0.01 250 0.03 0.04 0.05 0.03 0.01 500 0.03 0.04 0.04 0.03 0.01
n/scale MSE of efficiency estimates of Normal-Exponential SFA
0.2 0.3 0.4 0.5 0.6 20 0.04 0.05 0.07 0.08 0.05 50 0.03 0.04 0.06 0.06 0.04 100 0.02 0.04 0.05 0.05 0.03 250 0.02 0.03 0.05 0.04 0.02 500 0.02 0.03 0.03 0.04 0.02
MSE of efficiency estimates of Normal-Rayleigh SFA
n/scale 0.2 0.3 0.4 0.5 0.6 20 0.12 0.10 0.10 0.09 0.05 50 0.11 0.10 0.10 0.08 0.04 100 0.09 0.09 0.09 0.07 0.03 250 0.07 0.07 0.07 0.06 0.03 500 0.06 0.06 0.06 0.05 0.03
n/scale MSE of efficiency estimates of Normal-Burr SFA
0.2 0.3 0.4 0.5 0.6 20 0.04 0.07 0.04 0.07 0.04 50 0.06 0.06 0.07 0.03 0.04 100 0.05 0.06 0.07 0.08 0.05 250 0.05 0.04 0.04 0.06 0.04 500 0.04 0.06 0.08 0.05 0.05
*scale=sigmau, sigmav=sqrt (0.4-sigmau^2)
Tables 3.1-3.3 are some of the outputs of the simulation exercise. In appendices 3A, 3B and 3C, the mean estimates of sigma u, sigma v, BO and B1 are presented. For BO and B1, it is observed that all the models approximate 1 which was fixed in the simulation process. The Rayleigh model has relatively higher estimates of these parameters while the Burr x model has relatively lower values close to that of Normal-Half Normal.
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Using the tables above, the SD and MSE are analysed in order to arrive at the performance of the different SFA models. Since the inefficiency term was simulated using half-normal distribution, it is not surprising the normal-half normal SFA model has the minimum SD and MSE for all cases considered as shown in Tables 3.1 to 3.3. Therefore the estimates of the normal-half normal SFA model are considered benchmark estimates while the other three are the ones subjected to the test.
According to Table 3.3, it is obvious to see that the Rayleigh model’s performance is out of place when compared with the rest of the models. Both Burr X and Exponential models are quite competitive on the MSE test and have very low MSEs that nearly approximates that of the Normal-Half Normal model. This means they perform nearly the same in terms of accuracy of the efficiency estimation. In terms of increasing sample sizes, there is no clear trend and so it cannot be concluded that the models perform better or worse on MSE values when samples sizes decrease or increase. However, with the Rayleigh model, the MSEs of the efficiency estimates seems to improve with increasing sample size and may even get better as samples sizes exceed 500. It is important to note that Burr X SFA’s performance on the standard deviation criterion is generally not the best especially, for sigma u, the performance of Burr X on the standard deviation is probably the worst.
Tables 3.1 and 3.2 show the standard deviations of sigma v and sigma u. For the SDs of sigma v, the exponential, half-normal and Rayleigh models generally have lower values. That of Burr are generally relatively higher. Using Half-normal as a benchmark, it can be concluded that both exponential and Rayleigh seems to fit the simulated data better than the Burr model. In terms of the SDs of sigma u, the trend is quite different as it is rather Burr X and the Rayleigh models that fit the values of the Half-normal model better than that of the exponential model which are generally low.
However, for the means square error measure, the Burr X seems to perform better in both lower samples and larger samples. The Rayleigh, exponential and Burr x are compared in terms of their densities of the inefficiency term, u, as seen below:
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Figure 3.2: Density Plot for Normal-Half Normal Efficiency Estimates (Median Efficiency of 0.9)
Figure 3.3: Density Plot for Normal-Burr X Efficiency Estimates (Median Efficiency of 0.87)
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Figure 3.5: Kernel Density Plot of the Efficiency Estimates for Normal- Half Normal and Normal - Burr SFA Model*1
The Rayleigh seems to relatively overestimate the efficiency term more than that of the Burr X. In the plots above, there is evidence of skewness which is very important in the SFA efficiency technique.
Figure 3.6: Correlation between Normal- Half Normal Residuals and the True Residual (v-u)
1Mean efficiency for Normal-Burr X =0.7809778, Normal-Half Normal =0.8733968, Normal-
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Figure 3.7: Correlation between Normal-Rayleigh Residual and the True Residual (v-u)
Figure 3.8: Correlation between Normal- Burr X Residuals and the True Residual (v-u)
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Table 3.4 Spearman Rank Correlations between Estimated Efficiencies of Various Models (σu=0.2, σv=0.6, n=100)
Half-Normal Exponential Rayleigh Burr –type x
Half-Normal 1
Exponential 0.9992559 1
Rayleigh 1 0.9992559 1
Burr –type x 0.9962196 0.9940594 0.9962196 1
Table 3.4 shows the Spearman rank correlations between estimated efficiencies of various models (𝜎𝑢=0.2, 𝜎𝑣=0.6, n=100). It is obvious to see that the rank correlation for the efficiency measurements is really high in all the distribution types. It does suggest that applying different methods to the same data does not affect efficiency rankings. This observation is consistent with most of the literature aimed at comparing the different SFA distributions, the ranking of the firms rarely change and this study confirms it too.
In summary, the Burr SFA is a competitive candidate for measuring technical efficiency. The experiments conducted used cross-sectional data and there is a possibility the behaviour of the model in question may even perform better in panel models. This could be an area of further research.
3.10 Summary
The objective of the chapter was to derive an SFA model based on Burr X distribution and to test it against other existing SFA models with different distributions. First, the Maximum likelihood approach was used to determine the parameters of the new SFA-Burr X model. Simulation experiments were carried out to assess the performance of the proposed Burr x in terms of mean values, SDs and MSEs and compared with that for half-normal, exponential and Rayleigh. The exponential and half-normal distributions have a mode at zero, suggesting that a high percentage of the firms under study are perfectly efficient. The Burr x like truncated normal and gamma distribution capture wide range of shapes and also can have non-zero mode.
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A comparison was done and a conclusion was arrived at regarding the competiveness of the SFA –Burr X model. There are opportunities to use other ways of testing for the robustness of the Burr X SFA model. A neutral distribution like a Laplace could have been used to also assess the performance of the half-normal SFA model which was not possible in this study as the data generated was biased towards the latter model. Another testing methodology would be to test the different models against their distribution types in order to judge which model performs well in almost all the distribution types. Also, the performance of Burr X SFA could be enhanced in the panel form and therefore another study could explore the construction and testing of the panel forms of the Burr X SFA against existing panel models.
In the following chapter, there is a practical application of the SFA panel models in testing the efficiency of the electricity distribution sector of West Africa. It would have been apt to try the newly constructed Burr X model in the applications to West Africa EDCs but unfortunately its panel counterpart has as yet to be developed.
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Chapter 4
Stochastic Cost Frontier Analysis: Efficiency of West African Electricity
Distribution Companies
4.1 Introduction
Power sector reforms in Africa have been on-going for the last two decades, although at various levels of implementation across the continent. Undeniably, challenges facing the African power sector and other sectors vary across the continent but with significant differences between sub-Saharan Africa and North Africa primarily due to disproportions in the levels of power sector development. The various sub-sectors of the electricity industry have evolved through the many reforms that have been deployed in the past. In spite of these reforms, the power sector of Sub-Saharan Africa is saddled with inadequate and unreliable power supply, frequent power outages and high technical and commercial losses. On the contrary, according to AfDB (2013) electricity prices in Africa are relatively high with an average of 0.17 USD/KWh compared to the East Asian and South Asian regions with an average of 0.07 USD/KWh and 0.04 USD/KWh respectively. The above presupposes that consumers are made to procure poor services at relatively higher prices while suppliers in the continue to post high technical and commercial losses midst of limited electricity supply Given the above and other issues discussed in detail in chapter two, it was therefore not out of place to see the involvement of the West African regional community in attempting to address this regional problem. Other multilateral funding agencies and donor agencies equally have been involved in assisting some West African countries to come out of this predicament. The Economic Community of West African States (ECOWAS) in addressing the energy issues of the Region established two regulatory institutions namely the West African Power Pool (WAPP) and ECOWAS Region Regulatory Authority (ERERA) in 2006 and 2008 respectively by the decision of heads of state and governments (ECOWAS, 2008) in order to address energy security issues. These institutions have been charged to establish a regulated regional market as well as increasing inter-country connections and power generation. Regional markets require regulation and there are various types of this task depending on the objective intended to achieve. Nonetheless, agreeing with Farsi and Filippini (2008) and Jamasb and Pollitt (2001), benchmarking in incentive regulation is becoming increasingly accepted as a core component of all regulation types (rate of return or price cap) and it is
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important in price setting or price control. In particular, efficiency targets are set for utility companies with the objective that their operations would be improved upon to reduce inefficiencies and make electricity more affordable. The motive is that they will perform to achieve such targets to justify price increases during the tariff review period. It incorporates the reward and punish scheme that encourages efficient operations in utilities.
Relative efficiency measurement is a principal component of most price control models in the electricity market and adequately measuring it provides accuracy in setting targets to regulate prices for the purposes of preventing utility companies from passing on unjustified cost to consumers. It follows therefore that in order to have an effective regulatory regime nationally or regionally, it is prudent for regulators to have an initial efficiency measuring exercise conducted on the existing utility companies in new markets with the motive that a benchmark of performance can be established.
In the same vein and considering the establishment of a regional market in West Africa, where most EDCs are perceived inefficient, conducting an empirical study with a technique that holistically measures efficiency would be apt for two reasons. First, it provides an opportunity to estimate the baseline relative efficiencies of the participating EDCs. Secondly, it could confirm or otherwise the many narratives including that of Eberhard and Shkaratan (2012) and Kumi (2017)) that regard most EDCs in West Africa as inefficient. The level of (in)efficiency of EDCs in the region is normally “estimated” using uni-directional indicators involving simple ratios such as labour productivity, customer density, output per staff, technical and non- technical losses, commercial losses and others. These indicators though relevant, do not measure the totality of technical efficiency, allocative efficiency or the combination of both referred to as economic efficiency. Therefore, the voluminous literature reviewed in chapter two offers an array of techniques that could be used to measure relative efficiency in a much holistic way.
The main aim of this work therefore is to model the relative efficiencies of the EDCs of West Africa using an approach that captures both technical and allocative efficiency. The stochastic frontier approach is largely used as one of the techniques for analysing efficiency is employed. In this work, the stochastic cost frontier model is applied to estimate the level of inefficiency or otherwise in a sample of 14 West African EDCs and 1 East African EDC (Comparator firm).
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By this technique, a stochastic cost frontier is randomly constructed for each firm with which their observed costs can be compared.
The results of this research could be very important to the regional regulator (ERERA) as well as regulators of the individual host countries of the EDCs included in this research. Specifically, the estimated frontier cost model could be used to regulate prices for distribution networks that would access the Grid of the Power Pool. Jamasb and Pollitt (2001) indicate that the endogenous measurement of efficiency is much appreciated in markets that are very young and there is need to set up initial performance benchmarks. This suggestion could take a central-stage given that the regionalisation of the West African Electricity Market is young and thus requires an initial efficiency estimates.
The chapter is organised as follows: an overview of the stochastic cost frontier model is given followed by its econometric estimations and analyses of the various panel models. It is followed with an estimation and analysis of relative efficiency scores and subsequently concluding and offering some policy recommendations. An explanation or exposition to the stochastic cost frontier model and other sub-models is made available in the next section.