Stochastic Cost Frontier Estimation Results

Before proceeding to present and discuss the estimation results, it is prudent to show the descriptive statistics of the variables used in the mode. Table 4.2 therefore indicates the logged variables that were used to model efficiencies of the firms under review.

Table 4.2: Descriptive Statistics of Variables included in the Stochastic Cost Frontier Model

Variable (ln) Description Standard Deviation

Mean Minimum Maximum

TC/PC Total Cost/Capital Price (PPP$) 0.6920 4.8450 3.0853 6.4583 Y Distributed Electricity(GWh) 0.7982 7.9930 6.0753 9.0870 PL/PC Labour Price/Capital Price(PPP$) 0.4661 4.6127 3.0777 5.6445 LF Load factor 0.3346 3.8696 3.2222 4.2777 CD Customer density 1.0505 3.3780 0.2999 4.6968 lnTC PC= ln𝐓𝐂 and ln PL PC= ln 𝐏𝐋

In the above table, it can be seen that the variables employed show some level of diversity. The customer density (CD) variable shows significant diversity with a mean of 3.3780, a maximum 4.6968 and a minimum as low as 0.2999. The suspicion of heterogeneity in between the firms for which the variable was included is becoming stronger and confirmed by a high standard deviation.

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Table 4.3: SFA Cost Models Independent

Variables (ln)

Dependent Variable (ln𝐓𝐂)

Coefficients of Normal -Half Normal SFA Panel Models Pooled Model Pitt & Lee Model

Constant 0.67212 -3.73207** Y 0.34302*** 0.36084 PL 0.50578*** 0.51326*** LF -0.40447** 0.72599 CD -0.02061 -0.09722 Lambda 3.03739*** 2.75095 sigma 0.96158*** 1.12986 Independent Variables (ln) Dependent Variable (ln𝐓𝐂)

Coefficients of SFA Normal-Exponential Panel Models

Pooled Model P&L Model

Constant 1.19069 -3.16358** Y 0.29917*** 0.35938 PL 0.53605*** 0.53090*** LF -0.38129* 0.57666 CD -0.05260* -0.08235 Lambda 2.25446* 1.17994 Sigma 0.45526*** 0.41981*** Independent Variables (ln) Dependent Variable (ln𝐓𝐂)

Coefficients of SFA Normal-Truncated Normal Panel Models

Pooled Model P&L Model

Constant 0.55896 -5.52837 Y 0.34065*** 0.56347* PL 0.51095*** 0.48323*** LF -0.39004 0.75130 CD -0.02240 -0.16951 Lambda 2.79863 1.80428 sigma 0.85208** 0.70905

Note: ***, ** and * denote 1%, 5% and 10% significant levels respectively.

Also, as discussed in Jondrow et.al. (1982), lambda for cost frontier, λ=_𝑢

𝑣 and sigma for cost frontier, =√𝑣2+𝑢2

The results of the estimation are set out in Table 4.3 above. The four stochastic cost frontier panel models discussed in the methodology were estimated using a half normal, exponential

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and truncated normal distributions. Gamma was one of the chosen distribution but failed to yield meaningful results and as suggested earlier, stochastic frontier models based on gamma distribution are known to be problematic in its application in various data sets.

Also, out of the four panel models used, results produced by TFEM and TREM was inconsistent and not robust with insignificant Log likelihood ratios. Generally they overshot the efficiency levels in excess of 80% and sometimes 99% in the case of the TREM. That apart, the estimations of TFEM did not produce estimates that conform to the apriori signs (from theory and application) expected for most of the variables. Due to these shortcomings of the TFEM and TREM models, they have been eliminated and will not be included. The strict assumptions of these models may be violated as well as the panel problem of small size and could be a cause for the behaviour of the TFEM and TREM.

The Pitt and Lee model is essentially a random effects model constructed for the stochastic frontier technique with a time-invariant inefficiency term. Similar to a standard random effect, it is assumed that there is no correlation between the error term and the independent variables. The Pitt and Lee model produced consistent estimates but not robust enough compared to the pooled model.

The pooled model is a basic panel data model which does not make any assumptions of effects but constructed to accommodate the stochastic frontier technique. In terms of the estimation results, the pooled model fits the data better and shows more robust and significant results with relatively higher efficiency levels than that of the Pitt and Lee model. It appears that certain unobserved variables in the model possess some random effect which creates the high inefficiency estimates captured by the Pitt and Lee model.

In terms of the performance of the two chosen models on the assumed distribution types, the half-normal and the exponential seem to outperform the truncated-normal in terms of robustness and significant statistical estimates.

The sub-models (with the various assumed distributions) of the two preferred models (pooled and Pitt & Lee) generally produced estimates with their expected signs. However, not all of the sub-models yielded significant variables. The Normal-Half Normal Pooled model and the Normal –Exponential model were selected to be the most competitive sub models with the

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former being more robust and a better fit. Using the log-likelihood ratio test3, AIC4 (Akaike information criterion AIC) goodness of fit test and not ignoring discussion above, the Normal- Half Normal Pooled model exhibits a better fit and it is the preferred model for analysis. Appendices 4A1 and 4A2 show the regression and test outputs of the two models compared.

𝑻𝑪 = 0.67212 + 0.34302𝒀 + 0.50578 𝑷_𝒍 − 0.02061𝐶𝐷 − 0.40447𝐿𝐹 (4.14) Where 𝑻𝑪 =𝑇𝐶_𝑃 𝑐 and 𝑷𝒍= 𝑃𝑙 𝑃𝑐

Inference from the most preferred model in equation 4.10 can now be made to explain how the variables drive the cost of electricity distribution.

As evidenced in Filippini, Hrovatin and Zoric (2004) the positive coefficient of the labour and capital cost shares indicate that the cost function is monotonically increasing in input prices. The coefficient of output is also positive which suggest that total cost (not unit cost) increases with increasing output and this is expected regardless of the modification of the model in the linearization task as discussed earlier during the model specification. For the two factors controlling for heterogeneity (CD and LF), their negative coefficient suggest that as these variables increase, the total cost of electricity distribution decreases.

The model was estimated using logged dependent and independent variables and so the estimated coefficients can be interpreted as cost elasticities. Therefore, for the the pooled half-normal model, a 1% increase in output could increase total distribution costs by approximately 0.30% while a 1% increase in labour cost could drive a 0.51% increase in the total distribution cost. For the factors accounting for heterogeneity, a 1% increase customer density could reduce the total distribution cost by approximately 0.1%.while a 1% increase in the load factor could effect a 0.38% reduction in total distribution cost. The load factor has a

3 _{The null that OLS (u=o) as a preferred model is rejected given that the λ< C (Kodde-Palm critical static). The} pooled normal-half normal and the pooled normal-exponential models both were better models than the OLS alternatives.

4_{Between the two models the pooled normal-half normal model had a lower AIC and hence could be}

suggested to fit the data better. It is therefore chosen as the preferred model from which further analysis can be executed

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relationship with the capacity of plants and the peak demand of a distribution system and for a capital intensive sector it is not unusual to see the high impact it has on the total cost of electricity distribution.