Every estimator in pgt solves one linear programme per
evaluated unit \(i\) against a
reference (peer) set. This vignette states each programme as solved, so
the mapping from the cited papers to the code can be checked without
reading the source. Throughout, \(\lambda \ge
0\) are intensity weights over the peers, \(x\) the \(N\) material inputs, \(y\) the \(M\) good outputs (written as a single
output where the original model is scalar; the output rows repeat per
column of \(y\)), \(b\) the bad output of the selected
pollutant, and the variable-returns row \(\sum_l \lambda_l = 1\) is dropped under
constant returns. The materials-balance cap of unit \(i\) for pollutant \(p\) is \(c_{pi} =
u_{pi}'x_i - v_{pi}'y_i\), with DMU-specific coefficients
\(u_{pi}\) (a row of the \(L \times N \times P\) array) and \(v_{pi}\) (a row of the \(L \times M \times P\) array).
model = "wgd")Equation 6 of Rødseth (2025) minimises emissions over peer mixes that
hold only the intended outputs at the evaluated unit’s levels: the
inputs are decision variables, and the weak-G summing-up condition \(u_i'\varepsilon_x + v_i'\varepsilon_y =
\varepsilon_b\) ties the disposal slacks of inputs, outputs and
the bad output together. Substituting the condition and setting the
input slacks to zero collapses the programme to the reduced form solved
by the package, \[
b^*_i = \min_{\lambda}\ \textstyle\sum_l \lambda_l
\bigl(b_l + v_i'y_l\bigr) - v_i'y_i
\quad\text{s.t.}\quad
\textstyle\sum_l \lambda_l y_{ml} \ge y_{mi},\qquad
\textstyle\sum_l \lambda_l = 1,\qquad \lambda \ge 0:
\] the peer emission envelope plus the retained content of good
output produced beyond the unit’s own levels, valued at the unit’s own
coefficients \(v_i\). Self-reference is
always feasible, so every score \(b^*_i/b_i\) lies in \((0, 1]\), and a materials-balance violation
in the data does not affect the programme (the audit in
mb_check() remains a data-quality check). When the
abatement output is observed the optimal peer mix also delivers the
projection’s implied uncontrolled emission \(z^* = \sum_l \lambda_l z_l\), with \(z_l = u_l'x_l - v_l'y_l\) the
peer’s materials-balance cap (equal to \(b_l +
a_l\) when the peer’s account closes), and abatement \(a^* = \sum_l \lambda_l a_l\). Faithfulness
to the paper is exercised numerically: the programme reproduces the
Table 2 minima of Rødseth (2025) and agrees with an independent
reference implementation in the test suite. The reported output duals
are the total derivatives \(\partial
b^*_i/\partial y_{mi} = \mu_m - v_{mi}\), which can be negative
when the retained content is large: producing more output binds more
pollutant into the product.
model = "wgd_input_fixed")The model estimated as "wgd" before version 0.6.0 holds
the evaluated unit’s inputs fixed and adds the unit’s own
materials-balance cap: \[
\begin{aligned}
\min_{\lambda,\, b_q}\quad & b_q \\
\text{s.t.}\quad
& \textstyle\sum_l \lambda_l y_{ml} \ge y_{mi}, \qquad m = 1, \dots,
M,\\
& \textstyle\sum_l \lambda_l x_{nl} \le x_{ni}, \qquad n = 1, \dots,
N,\\
& \textstyle\sum_l \lambda_l b_l \le b_q, \\
& b_q \le c_{pi}, \\
& \textstyle\sum_l \lambda_l b_{ql} \le c_{qi}, \qquad q \ne p,\\
& \textstyle\sum_l \lambda_l = 1, \qquad \lambda \ge 0,\ b_q \ge 0.
\end{aligned}
\] This is not Eq. 6, whose inputs are free; it is the package’s
own current-input companion: the emission reduction attainable at the
unit’s observed input endowment, with the cap row keeping the projection
consistent with the unit’s own pollutant potential.
With several pollutants, each remaining pollutant \(q \ne p\) contributes the single row shown above. This collapse is exact: pollutant \(q\) carries no objective term, so introducing its own emission variable \(b_q'\) with \(\sum_l \lambda_l b_{ql} \le b_q' \le c_{qi}\) is feasible exactly when the combined row \(\sum_l \lambda_l b_{ql} \le c_{qi}\) holds (take \(b_q' = \sum_l \lambda_l b_{ql}\)).
A unit whose own account violates a cap (\(b_{qi} > c_{qi}\) for some \(q\)) loses its self-reference: its
programme is feasible only if some peer mix satisfies every row, it may
be infeasible (an NA score), and its projection can emit
more of pollutant \(p\) than observed,
so its score can exceed 1. Units satisfying every account always solve
with scores in \((0, 1]\).
model = "envelope")Setting \(v = 0\) in the
"wgd" reduced form drops the retained-content term and
leaves the convex lower envelope of the \((y,
b)\) scatter: \[
\min_{\lambda}\ \textstyle\sum_l \lambda_l b_l
\quad\text{s.t.}\quad
\textstyle\sum_l \lambda_l y_{ml} \ge y_{mi},\qquad
\textstyle\sum_l \lambda_l = 1,\qquad \lambda \ge 0.
\] The two programmes coincide on data whose products retain no
pollutant. Self-reference is always feasible, so every score lies in
\((0, 1]\).
model = "fdmo")The factorially determined multi-output form of Rødseth (2025, Eq.
13), with the abatement output \(a\)
held at the evaluated unit’s own level: \[
\begin{aligned}
\max_{\lambda,\, \theta_y,\, \theta_b}\quad & \theta_y + \theta_b \\
\text{s.t.}\quad
& \textstyle\sum_l \lambda_l y_l - \theta_y \ge y_i, \\
& \textstyle\sum_l \lambda_l a_l \ge a_i, \\
& \textstyle\sum_l \lambda_l x_{nl} \le x_{ni}, \qquad n = 1, \dots,
N,\\
& v_i\,\theta_y - \theta_b = c_{pi} - b_i - a_i, \\
& \textstyle\sum_l \lambda_l = 1, \qquad
\lambda \ge 0,\ \theta_y \ge 0,\ \theta_b \ge 0.
\end{aligned}
\] The materials-balance row is an exact equality, so the model
requires accounts that close exactly (\(c_{pi}
= b_i + a_i\)); open accounts either make the programme
infeasible or shift the closure gap into the scores, and
pgt() warns when it detects them. When the account closes,
the row forces \(\theta_b =
v_i\,\theta_y\). The gross inefficiency \(\theta_y + \theta_b\) adds good-output
units to bad-output units (direction \((1,
1)\)), so it is unit-dependent. The model is defined for a single
intended output.
pgt_decompose(type = "rodseth"))The five components of Rødseth (2025, Eq. 11) are ratios of stage
minima of the extended representation (Eq. 9). Write \(w_l = u_i'x_l - v_i'y_l\) for peer
\(l\)’s account valued at the evaluated
unit’s coefficients, \(z_l\) for its
uncontrolled emission, \(q_l = z_l -
w_l\) for its quality gap, and \(a_l\) for its abatement. Each stage solves
\[
\begin{aligned}
\min\quad & z - a \\
\text{s.t.}\quad
& \textstyle\sum_l \lambda_l y_{ml} - \varepsilon_{y,m} = y_{mi},
\qquad m = 1, \dots, M,\\
& \textstyle\sum_l \lambda_l x_{nl} + \varepsilon_{x,n} = x_{ni}
\qquad \text{for each held input } n,\\
& \rho = \textstyle\sum_l \lambda_l q_l, \qquad
z = \textstyle\sum_l \lambda_l w_l + \rho + \varepsilon_z, \qquad
a = \textstyle\sum_l \lambda_l a_l - \varepsilon_a, \\
& \varepsilon_z + \varepsilon_a = u_i'\varepsilon_x +
v_i'\varepsilon_y, \\
& \textstyle\sum_l \lambda_l = 1, \qquad \lambda \ge 0,\
\varepsilon \ge 0,
\end{aligned}
\] where the summing-up row relaxes to \(\ge\) when an input with positive material
content is freed (a freed input’s disposal slack can absorb any amount).
Holding the quality variable fixes \(\rho =
0\); holding the abatement output fixes \(a = a_i\); holding an input keeps its row,
and freeing it deletes the row. The stages hold everything first and
then free, in order, the quality variable, the production inputs, the
abatement output and the pollution-control inputs; the ratios of
successive minima are the five components reported by
pgt_decompose(), and the fully freed stage reproduces the
reduced form of Eq. 6, so the components multiply to the
wgd efficiency exactly.
When the abatement output is unobserved, the implied abatement \(a_l = z_l - b_l\) closes each account by construction and abatement is left endogenous in every stage, which collapses the two abatement components to 1. When abatement is observed but an account does not close exactly, the stages use the measurement-consistent uncontrolled emission \(z_l = b_l + a_l\), so that peers’ controlled emissions equal their measured \(b_l\); the two constructions coincide when every account closes, the case of Rødseth (2025).
model = "mb_cost")Coelli, Lauwers and Van Huylenbroeck (2007), with the material flow
coefficients in the role of prices. Two programmes: radial technical
efficiency, \[
TE = \min\{\theta :\ \textstyle\sum_l \lambda_l x_{nl} \le \theta
x_{ni}\ \forall n,\ \sum_l \lambda_l y_l \ge y_i,\ \sum_l \lambda_l =
1\},
\] and the material-inflow minimisation over a free input point
\(\chi\), \[
EE = \frac{\min\{u_{pi}'\chi :\ \sum_l \lambda_l x_{nl} \le \chi_n\
\forall n,\ \sum_l \lambda_l y_l \ge y_i,\ \sum_l \lambda_l = 1\}}
{u_{pi}'x_i},
\] with environmental-allocative efficiency \(EAE = EE / TE\). The principal score is
\(EE\), a material-inflow ratio rather
than an emission ratio; the implied minimal controlled emission \(u_{pi}'\chi^* - v_{pi}'y_i\) is
reported as b_star and can be negative under DMU-specific
coefficients, in which case pgt() warns.
model = "byprod")Murty, Russell and Levkoff (2012): the technology is the intersection
of an intended-production sub-technology and a residual-generation
sub-technology over the emission-causing inputs \(\mathcal{P}\). Output efficiency inverts
the maximal radial expansion of \(y\),
\[
E_1 = 1 \big/ \max\{\phi :\ \textstyle\sum_l \lambda_l y_l \ge \phi
y_i,\ \sum_l \lambda_l x_{nl} \le x_{ni}\ \forall n,\ \sum_l \lambda_l
= 1\},
\] and emission efficiency is the maximal radial contraction of
\(b\) with the polluting inputs bounded
below, \[
E_2 = \min\{\psi :\ \textstyle\sum_l \mu_l b_l \le \psi b_i,\ \sum_l
\mu_l x_{nl} \ge x_{ni}\ \forall n \in \mathcal{P},\ \sum_l \mu_l =
1\}.
\] \(E_2\) is the principal
environmental efficiency; mean_eff is the arithmetic mean
\((E_1 + E_2)/2\), in the spirit of the
Färe-Grosskopf-Lovell graph measure. If either sub-programme fails,
every score of that unit is NA.
model = "wd")The Kuosmanen (2005) variable-returns form, a reference system with no materials balance. The intensity weights split into an active part \(z\) (scaling good and bad outputs together) and an abatement part \(w\) (scaling inputs only): \[ \begin{aligned} \min_{z,\, w,\, \phi}\quad & \phi \\ \text{s.t.}\quad & \textstyle\sum_l z_l y_l \ge y_i, \qquad \textstyle\sum_l z_l b_l = \phi\, b_i, \\ & \textstyle\sum_l (z_l + w_l)\, x_{nl} \le x_{ni}, \qquad n = 1, \dots, N,\\ & \textstyle\sum_l (z_l + w_l) = 1, \qquad z, w \ge 0. \end{aligned} \] The equality on the bad output is weak disposability: emissions shrink only with a proportional scaling of the active production plan.
pgt_ml())The global Malmquist-Luenberger index scores every observation \(k\) on a directional distance with
direction \(g = (y_k, b_k)\) against a
reference set \(R\) (a period or the
pooled panel); the index is defined for a single good output. Under
technology = "wd" the distance uses the Kuosmanen
weak-disposability technology above with \(\sum_{l \in R} z_l b_l = (1 - \beta) b_k\)
and \(\sum_{l \in R} z_l y_l \ge (1 + \beta)
y_k\); this is the technology under which Chung, Färe and
Grosskopf (1997) and Oh (2010) define the index. Under
technology = "input_envelope" and "envelope"
the bad-output row is the free-disposal envelope \(\sum_{l \in R} \lambda_l b_l \le (1 - \beta)
b_k\), with and without the input rows; these exploratory
variants bound emissions by the peer envelope rather than imposing weak
disposability.