Empirical model

volume of a rectangular block defined by these three variables. A higher block coefficient leads to a higher hull resistance, and in turn, more power is required for propulsion and lower fuel ef-ficiency. Therefore, controlling for these ship design features in the empirical analysis is imperative.

The cargo-carrying capacity of a ship is reflected by the variable Capacity, which is measured in thousand deadweight tonnes (DWT). DWT states the vessel’s maximum weight-carrying capacity, excluding its own weight, and is in practice indicated by the certified load line marking amidships.

Capacity is an important part of the EEDI formula, as it reflects, together with a ship’s speed, the benefit part of the equation. Further, we control for the type of vessel due to the structural differences in ship design. The ship type is reflected by a categorical variable divided into seven categories: chemical tanker, container ship, gas carrier, general cargo ship, oil tanker, and other ship type, and the omitted categorical level is bulk carrier. We have chosen to follow the classifi-cation in the EU-MRV data set to determine the ship type.

Bassett (1978). For a quantileτ ∈(0,1), the conditional quantile function can be formulated as,

q_y|x(τ) :=xβ_τ. (2.3)

Similar to OLS, the conditional quantile function is assumed linear in the parameters β_τ. An important distinction of OLS is that the marginal effects depicted by βτ are allowed to vary in τ. In fact, only in the case of y = xβ_τ +u_τ, with homoscedastic error term u_τ, will the slopes of parameters βτ be the same across τ. This also leads to the fact that effects can change signs across quantiles, which might suggest no effect of a covariate if investigated with the conditional mean model. For example, under these circumstances, one might conclude that a covariate has no effect on the dependent variable, which might not be true (Fitzenberger & Wilke, 2015). The empirically observed quantileτ of a random variable’s ydistributionqy(τ) can be expressed as the solution to the following minimization problem,

q_y(τ) := arg min

q

h

τ X

i:yi>q

|y_i−q|+ (1−τ) X

i:yi<q

|y_i−q|i

. (2.4)

This expression is the so-called check function and can be interpreted as the (asymmetrically) weighted absolute difference from the location parameterq. In a linear QR, the location parameter is stated as a function of the covariates x_i, and the conditional quantile of the response variable q_y|xi(τ) is a linear function ofxiβτ. Thus, the check function in Equation (2.4) can also be stated as,

βτ := arg min

β

h

τ X

i:yi>xiβ

|y_i−xiβ|+ (1−τ) X

i:yi<xiβ

|y_i−xiβ|i

. (2.5)

Note that due to the piece-wise linear form of Equation (2.5), special techniques to provide asymp-totic refinements of the standard errors are required (Horowitz, 2001). Two feasible features of the QR estimator, when compared to the OLS estimator, are that it is more robust to outliers in the dependent variable due to its focus on the median and that it makes no assumption regarding the distribution of the error term (Koenker, 2005). The interpretation of the linear QR parametersβ_τ is analogous to the OLS case: they describe the change in the conditional quantile of the dependent variable when the respective covariate changes by one unit.

The basic model formulation for testing H1(a), H2(a), H2(c), and H3(a) can be written as:

log(EEDI_i) =β₀+β₁Age_i+β₂Speed_i+β₃VLS fuel_i+β₄Alternative fuel_i +β₅Main efficiency_i+β₆Main power_i+β_controlsZ_i+ϵ_i Similarly, our basic empirical model for testing H1(b), H2(b), H2(c), and H3(b) is:

log(EERS_i) =β₀+β₁Age_i+β₂Speed_i+β₃VLS fuel_i+β₄Alternative fuel_i +β₅Main efficiency_i+β₆Main power_i+β_controlsZ_i+ϵ_i

Before turning to the empirical results, we address two important points. First, we formally test our initial presumption that the relationship between technology and operational levers and environmental performance is not adequately captured by a single point-wise estimate. Second, we briefly outline the strategy for a heteroscedasticity-robust inference in the empirical models.

2.5.2.1 Test for location shift hypothesis

We test our presumption that the effects of technology and operational levers on environmental performance are heterogeneous across the range of environmental performance. In formal terms, we test the so-called location shift hypothesis, stating that the estimated slopes are equal across a specified set of quantilesτ ∈ {0.1,0.25,0.5,0.75,0.9}of the conditional distribution. If the hypoth-esis is rejected, it is an indication of the presence of heteroscedasticity in the data. To test this hypothesis, a variant of the Wald-test proposed in Koenker and Bassett (1982) is conducted with the energy efficiency measure (i.e., EEDI rating) as the dependent variable. To obtain a robust inference, the kernel method for the sandwich estimator proposed by Powell (1991) is performed with the kernel function stated in Powell (1991) and the bandwidth described in Koenker (2005) as parameter choices. The following tests are conducted.

Joint test of equality of slopes for all slope parameters across quantiles:

This first test checks for any statistically significant differences in the estimated coefficients βτ

across two of the specified quantile levels. For this purpose, the joint test is performed for each pairwise combination of the five quantiles of interest and for the full set of quantiles. The results are reported in the appendix in Table (2.5). In all pairwise tests, the null hypothesis of the joint

equality of slopes is rejected at the 1% significance level, indicating that the slopes of the condi-tional quantile functions differ significantly across quantiles.

Test of equality of slopes for individual slope parameters across quantiles:

This test examines whether, given a certain covariate, the estimated coefficient significantly changes across the specified set of quantiles. For example, the null hypothesis for the variable Age is β_τ=0.1^AGE =β_τ=0.25^AGE =β^AGE_τ=0.50=β_τ=0.75^AGE =β_τ=0.90^AGE . The results of the test are depicted in Table (2.2).

Table 2.2: Test of equality of distinct slopes

Covariate F-statistic p-value

Age 16.999 .000

Speed 3.072 .015

VLS fuel 3.07 .016

Alternative fuel 3.78 .004

Main efficiency 0.87 .481

Main power 7.389 .000

Capacity 2.873 .022

Draught 6.703 .000

LOA 2.623 .033

Beam 0.878 .476

Note: Test of equality of slopes for individual slope param-eters across quantiles according to the Wald-test with omit-ted ship type controls. Heteroscedasticity-robust standard errors are derived using the kernel method of the sandwich estimator. The null hypothesis in the Wald-test is the equal-ity of distinct slopes across quantiles.

Except for the main engine’s fuel efficiency and beam variable, the effects of the individual covari-ates change significantly along the conditional energy efficiency distribution. To conclude, both the joint and distinct Wald test overall reject the location shift hypothesis. This is an important finding, as it has two implications for our further analysis. First, there is an indication of the

presence of heteroscedasticity, as the conditional variation in energy efficiency changes with the covariates. This must be addressed in the empirical analysis to ensure a heteroscedasticity-robust inference. Second, the presence of changing effects confirms our initial presumption that technology and operational levers have heterogeneous impacts across the range of environmental performance.

Therefore, we find evidence that explaining variations in performance across different levels of performance might yield important additional insights into our empirical context.

2.5.2.2 Estimation of asymptotic standard errors

To address the presence of heteroscedasticity in our data, we consider resampling techniques, such as the bootstrap, to approximate the asymptotic distributions of the estimators without having to make assumptions about them. The standard approach for QR is to use a full-sample (x, y)-pair bootstrap that is replicatedBtimes to obtain a heteroscedasticy-robust inference (Koenker, 2005).

In this approach, samples of the (xi, yi) pairs are drawn from the empirically observed joint dis-tribution of the sample (with replacement), and then a QR is estimated at the specified different quantiles, yielding estimates of the covariance matrix. It is worth noting that the accuracy of the approximation depends on the choice of B; hence, a sufficiently high B must be chosen. There is no clear guidance in the literature about the precise value of B, and recommendations range from 200 to 1,000 replications depending on the application (see Cameron & Trivedi, 2005, p.361, for a more thorough discussion of the literature on this topic). In line with this literature, we consider for each quantile estimator 700 bootstrap replications in our analysis, and the obtained standard errors are reported in Table (2.3)⁴. Note that we did not consider approaches relying on asymptotic theory, due to their reliance on assumptions to estimate the (unknown) density of the error term.

4Note that as a robustness check, we have conducted the alternative QR gradient condition bootstrap method proposed by Parzen et al. (1994). The differences in estimated standard errors are marginal.