4.4 Method
4.4.2 Data analysis
to these observations is evaluated in section (4.5.3) (Upton & Cook, 1996, p.56).
Figure 4.1: General framework - Input-output combinations of a vessel
4.4.2.2 Sector-specific frontiers
Consider observations on k= 1, ..., N transportation units, with each unit k comprising a vector of inputs xk = (x1k, ..., xpk)∈Rp+ and a vector of outputsyk = (y1k, ..., yqk)∈Rq+. The sample is composed of observations fromg= 1, ..., G distinct sectors (groups) of sizeNg and PG
g=1Ng =N has to hold. Further, the combination of inputs and outputs of unitkis denoted as (xk, yk). The underlying technology setT(g) for each sectorg is defined by,
T(g) ={(x, y)∈Rp+×Rq+|x can producey in sector g}. (4.3) Note that the true technology sets are unknown; thus, the nonparametric data envelope estimator proposed by Charnes et al. (1978) is utilized to derive an estimate ˆT(g) of T(g) for each sector.
The approach is based on the minimum extrapolation principle of the observed input-output com-binations, satisfying free disposability and convexity as technological assumptions. Further, no rescaling is assumed, i.e., varying returns to scale, resulting in a flexible inner approximation of the true technology set based on a minimum set of assumptions. In short, ˆT(g) is a piece-wise linear, non-decreasing, and concave estimate ofT(g) with ˆT(g) ⊆T(g). The estimate for ˆT(g) is then given by,
Tˆ(g) ={(x, y)∈Rp+×Rq+| ∃λ∈RN+g :x≥ X
k∈Ng
λkxk, y≤ X
k∈Ng
λkyk, X
k∈Ng
λk = 1}. (4.4)
Intuitively, equation (4.4) describes the observed best practices for transportation units in sectorg and will be referred to as the estimated efficient sector-specific frontier. To quantify the technical efficiency of the individual transportation units relative to the efficient frontiers, a radial output-based efficiency measure is utilized. More formally, the output-output-based efficiency Fkg of firm kwith (xk, yk) relative to technology set T(g) is defined as,
Dˆgk=max{F ∈R+|(xk, yk/F)∈Tˆ(g)}. (4.5)
This function indicates the maximum radial expansion of the output vector yk given the inputs xk for unitk. Note that inserting equation (4.4) into equation (4.5) yields for the DEA approach a typical linear programming problem, which is thoroughly outlined in Charnes et al. (1978) and Bogetoft (2013). Further, a unit can be considered efficient with respect to the sector-specific frontier if and only if ˆDgk= 1 (O’Donnell et al., 2008).
4.4.2.3 Industry metafrontier
Under the stated formulation, each sector is associated with a different technology set. This is due to the fact that the transportation units (i.e., vessels) are designed for different purposes like, e.g., the kind of cargo they are transporting or the routes on which they are deployed. This restricts the technology choice at the design stage, and the boundaries of the restricted technology setsT(g) de-termine the sector-specific frontiers3. Thus, the observed transportation units form distinct sector groups, each of which has their respective technological conditions. Relying on a traditional DEA approach considering only one common technology set does not reflect the different technological conditions across sectors and might yield unrealistic estimates of efficiency scores. The metafron-tier approach provides an appropriate methodology to compare the efficiency of transportation units belonging to different sectors (Battese et al., 2004; O’Donnell et al., 2008). From the re-stricted sector-specific technology sets, it is possible to formulate an industry metatechnology set containing all feasible input-ouput combinations defined by,
T ={(x, y)∈Rp+×Rq+|x can producey}. (4.6)
3Note that the constraints are not limited to technical constraints but can also be due to any other characteristic of the physical, social, and economic environment influencing ship designs (O’Donnell et al., 2008).
Based on the same set of assumptions as for ˆT(g), it is then possible to derive an estimate ˆT, which I will refer to as the metatechnology frontier. Similarly, the output efficiency of observationkwith respect to the metafrontier can be measured by,
Dˆk=max{F ∈R+|(xk, yk/F)∈Tˆ}. (4.7)
Intuitively, the metafrontier measures the efficiency of a transportation unit given that there are no restrictions in the technology choice at the design stage and under the assumption that all units have access to the same technologies. This allows the formulation of a measure of how close the sector-specific frontiers are to the industry metafrontier. This measure is referred to as the metatechnology ratio, and for units belonging to sectorg with input-output combinations (x, y) is defined by,
M T R(x, y) = D(x, y)
Dg(x, y). (4.8)
The ratio indicates how limiting the technological conditions in sector g are when compared to the unrestricted metatechnology containing all observed ship design technologies in the sample.
To illustrate, a ratio of 0.8 means that, given the input vector, the maximum output that can be achieved by a transportation unit in sectorgis 80% of the feasible output using the metatechnology.
A lower metatechnology ratio indicates that the technology set in sectorg is more restricted and, thus, the technological conditions are more limiting.
4.4.2.4 Sample bias
I remark that while a unit with an estimated efficiency score ˆDkg >1 is objectively inefficient, such a conclusion cannot be made for the efficient units with ˆDkg = 1 spanning the estimated efficient frontiers. As highlighted, the true technology setsT(g) and, thus, the true efficiency scoresDkg are unknown and are estimated by the DEA estimator. However, because the estimated technology set ˆT(g) is a subset of T(g), the estimator is biased (but consistent) and yields upward-biased efficiency score estimates ˆDgk. The same argument applies to the efficiency scores with respect to the metatechnologyT. This is especially problematic for the comparison of different sectors, each having their individual technology set estimate (Zhang & Bartels, 1998). To illustrate, Simar and Wilson (2000) remark that the bias decreases in sample size and density around a frontier point
and increases in the curvature of the frontier. Intuitively, one can expect the bias to be relatively small in a situation where a large sample with homogeneous units faces a frontier with a mild curvature.
Note that the sample size varies between the five sectors, and this is likely also the case for the density and frontier curvature. Thus, the bias cannot be expected to be uniform across sectors, and a comparison of efficiency scores derived from the idiosyncratic sector models would be highly questionable in this application. Results from the asymptotic theory about the sampling distribu-tion can only be applied to derive a bias-corrected DEA estimator for a simple single input and output model. Therefore, the study resorts to bootstrapping techniques to derive bias-corrected efficiency measures and to construct confidence intervals for the (true) efficiency scores Dkg. In particular, the study follows the algorithm for bootstrapping in nonparametric frontier models proposed by Simar and Wilson (1998) (please refer to Daraio and Simar (2007) on page 61–62 for a detailed description of the procedure). The required bandwidth choice for the kernel density estimator has been determined according to the robust normal reference rule (Silverman, 1986).
Lastly, this procedure should also mitigate the effects of the reduced sample size due to focusing on the EU-MRV subsample, and it yields 95% confidence intervals for the unobservable true efficiency scores.
4.4.2.5 Empirical models
Based on the general framework presented in section (4.4.2.1), various multiple inputs-outputs models can be formulated. Table (4.4) summarizes the main candidate models with the respective input and output sets. From a conceptual viewpoint, the study hypothesizes that ”Model 3” is most appropriate in this application for the following reasons. First, in the data collection process, several assumptions had to be made for determinants of the auxiliary engines. Using the aggregate measure for the auxiliary engines mitigates the potential influence of idiosyncratic errors in the decomposed determinants. Second, the main engine is responsible for around 80% of a vessel’s carbon emissions and is of key importance for a vessel’s energy efficiency. Hence, a granular repre-sentation of the main engine’s characteristics in the benchmarking model is conceptually feasible.
Table 4.4: Empirical models
Model 1 Model 2 Model 3 Model 4
Outputs
Capacity Capacity Capacity Capacity
Speed Speed Speed Speed
Inputs
ME Carbon Emissions ME Power ME Power ME Power
AE Carbon Emissions ME SFOC ME SFOC ME SFOC
ME Carbon Factor ME Carbon Factor ME Carbon Factor AE Carbon Emissions AE Power
AE SFOC
AE Carbon Factor
To ensure the obtained results are not driven by the model formulation and are robust to alterna-tive specifications, a sensitivity analysis is conducted in section (4.5.3).