CHAPTER 6: EMPIRICAL RESULTS OF TECHNICAL AND SCALE EFFICIENCY
6.2 Choice of functional form and distributional assumption
In an effort to minimize potential bias resulting from the imposition of an unsuitable functional form, five common functional forms were modelled, four of which represent second-order Taylor series expansions, commonly referred to as flexible functional forms. Furthermore, since the choice of distributional assumption is another requirement of stochastic frontier analysis in which researchers do not seem to invest much time or effort (Mbaga et al., 2003), the suitability of two distributional assumptions was formally assessed for each of the five functional forms. In addition, the assumption of either time variant or time invariant efficiencies was modelled for each of the above-mentioned models, resulting in a total of 20 possible milk production functions. Table 6.1
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presents a summary of the resulting 20 model specifications to be subjected to formal model testing.
Each of these 20 models was specified and estimated within the stochastic frontier analysis software package frontier (Coelli & Henningsen, 2013), using the R statistical package (R Core Team, 2015). A series of likelihood ratio tests were then conducted to determine the most suitable model for the data. These likelihood ratio tests can be classified into two categories. The first may be classified as within-model likelihood ratio tests, conducted to determine the most suitable model within each functional form. The second, summarized in Table 6.2 may be classified as between- model likelihood ratio tests, used to compare the most suitable models from each functional form and ultimately determine the most appropriate functional form for the given data set.
Table 6.1: Summary of modelled milk production technologies.
Model Functional form One-sided distribution Nature of efficiency
1
Cobb-Douglass (CD)
HN Time invariant
2 TN Time invariant
3 HN Time variant
4 TN Time variant
5
Simplified Translog (STL)
HN Time invariant
6 TN Time invariant
7 HN Time variant
8 TN Time variant
9
Translog (TL)
HN Time invariant
10 TN Time invariant
11 HN Time variant
12 TN Time variant
13
Generalized Leontief (GL)
HN Time invariant
14 TN Time invariant
15 HN Time variant
16 TN Time variant
17
Normalized Quadratic (NQ)
HN Time invariant
18 TN Time invariant
19 HN Time variant
20 TN Time variant
*HN = half normal, TN = truncated normal
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The within-model likelihood ratio tests, presented in Appendix 3, involve comparisons of each model to the ordinary least squares (OLS) equivalent. This is done to confirm if any significant production inefficiencies are present. The results, shown in Appendix 3, indicate that all 20 models are significantly different from their OLS equivalents. This implies that most of the sampled milk producers operated below the efficient production frontier. Therefore, the average production function, with no inefficiency, is considered to be an inadequate representation of the milk production technology (Karagiannis & Sarris, 2005). It is worth noting that the TN distribution appears to be significantly better than the HN distribution for each and every functional form. TL and GL functional forms provided some interesting results in that time variant and time invariant models were not significantly different. It was expected that models able to account for variation of efficiencies over time would be more appropriate than the more rudimentary time invariant models.
The results presented in Table 6.2 indicate that the simplified translog (STL), Normalized Quadratic (NQ) and the Generalized Leontief (GL) were not significantly different from the Cobb- Douglas (CD) functional form. The translog (TL) models with TN distributions were, however, significantly different from the CD models with TN distributions and are, therefore, considered an improvement. Furthermore, results indicate that the TL models with truncated normal distribution were significantly better than CD, STL, NQ and GL models with truncated normal distribution.
This is true for both time variant and time invariant efficiency models.
In addition to subjecting each of the above-mentioned models to likelihood ratio tests, each of the five functional forms were modelled in R using linear equation methods to determine the relative fit of the various functional forms to the data. In an attempt to gauge the suitability of each functional form, several residual plots (histograms) were generated to provide a graphical representation of goodness of fit (see Appendix 4). In total, ten separate plots were generated, with two for each functional form. For each plot, the Y-axis represents function specific (fitted) output and the X-axis represents actual output. Plots therefore represent the ability of a particular functional form (production technology) to model actual dairy output. Referring to Appendix 4, plots depicted on the right-hand side of the page are simply logarithmic representations of those on the left. This comparison of “fitted” output to actual output provides insight into the goodness of fit of each possible functional form.
87 Table 6.2: Between-model likelihood ratio tests
Likelihood Ratio Tests
Model # DF LogLik Df Chisq Pr>(Chisq) Decision
CD VS STL
CDtnVAR 12 205,1
STLtn 17 209,13 5 8,0631 0,1528 NSD
CDtnVAR 12 205,1
STLtnVAR 18 210,67 6 11,13 0,08443 . NSD
CD VS TL
CDtnVAR 12 205,1
TLtnVAR 33 228,13 21 46,056 0,001256 ** TLtnVAR
CDtnVAR 12 205,1
TLtn 32 226,78 20 43,357 0,001833 ** TLtn
CD VS GL
CDtnVAR 12 205,1
GLtn 32 199,63 20 10,932 0,948 NSD
CDtnVAR 12 205,1
GLtnVAR 33 200,57 21 9,0577 0,9888 NSD
CD VS NQ
CDtnVAR 11 201,02
NQtn 32 204,99 21 7,9447 0,9954 NSD
CDtnVAR 12 205,1
NQtnVAR 33 206,41 21 2,6211 1 NSD
STL VS TL
STLtn 17 209,13
TLtn 32 226,78 15 35,294 0,002233 ** TLtn
STLtn 17 209,13
TLtnVAR 33 228,13 16 37,993 0,001517 ** TLtnVAR
STLtnVAR 18 210,67
TLtn 32 226,78 14 32,227 0,003719 ** TLtn
STLtnVAR 18 210,67
TLtnVAR 33 228,13 15 34,926 0,002519 ** TLtnVAR
NQ VS TL
NQtn 32 204,99
TLtn 32 226,78 0 43,572 2,20E-16 *** TLtn
NQtn 32 204,99
TLtnVAR 33 228,13 1 46,272 1,03E-11 *** TLtnVAR
NQtnVAR 33 206,41
TLtn 32 226,78 -1 40,736 1,74E-10 *** TLtn
NQtnVAR 33 206,41
TLtnVAR 33 228,13 0 43,435 2,20E-16 *** TLtnVAR
GL VS TL
GLtn 32 199,63
TLtn 32 226,78 0 54,288 2,20E-16 *** TLtn
GLtn 32 199,63
TLtnVAR 33 228,13 1 56,988 4,39E-14 *** TLtnVAR
GLtnVAR 33 200,57
TLtn 32 226,78 -1 52,415 4,49E-13 *** TLtn
GLtnVAR 33 200,57
TLtnVAR 33 228,13 0 55,114 2,20E-16 *** TLtnVAR
Significance codes: ***=<0.0001, **=0.001, *=0.05, "."=0.1 NSD = No significant difference, VAR = time variant inefficiencies
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Visual inspection of the plots reveals that CD, STL, and TL specifications appear to display acceptable levels of fit, with all three production technologies representing the actual data reasonably well. GL and NQ specifications displayed very poor levels of fit. Both production technologies failed to model the actual data with any reasonable degree of accuracy.