Package {metafrontier}

Type:	Package
Title:	Analysis of Metafrontier Models for Efficiency and Productivity
Version:	0.3.0
Description:	Implements metafrontier production function models for estimating technical efficiencies and technology gaps for groups of firms that face different restrictions of a common underlying metatechnology (group-specific technologies in the sense of Battese, Rao, and O'Donnell, 2004). Supports both stochastic frontier analysis (SFA) and data envelopment analysis (DEA) based metafrontiers. Includes the deterministic metafrontier of Battese, Rao, and O'Donnell (2004) <doi:10.1023/B:PROD.0000012454.06094.29>, the stochastic metafrontier of Huang, Huang, and Liu (2014) <doi:10.1007/s11123-014-0402-2>, and the metafrontier Malmquist productivity index of O'Donnell, Rao, and Battese (2008) <doi:10.1007/s00181-007-0119-4>. The deterministic metafrontier can be identified by either the minimum sum of absolute deviations (LP) or the minimum sum of squared deviations (QP) criterion. Additional features include panel SFA with time-varying inefficiency, bootstrap confidence intervals for technology gap ratios, a DEA poolability permutation test, latent class metafrontier estimation via the EM algorithm, Murphy-Topel corrected standard errors, convergence diagnostics, import of pre-fitted models from external estimation engines ('sfaR', 'frontier', 'Benchmarking'), and 'ggplot2' visualisation methods.
License:	GPL (≥ 3)
Encoding:	UTF-8
RoxygenNote:	7.3.3
Depends:	R (≥ 4.0.0)
Imports:	stats, graphics, grDevices, Formula, numDeriv, lpSolveAPI, methods
Suggests:	testthat (≥ 3.0.0), knitr, rmarkdown, sfaR, frontier, Benchmarking, quadprog, plm, ggplot2, parallel
VignetteBuilder:	knitr
URL:	https://github.com/iik1/metafrontier
BugReports:	https://github.com/iik1/metafrontier/issues
Config/testthat/edition:	3
NeedsCompilation:	no
Packaged:	2026-07-15 10:39:15 UTC; enstad
Author:	Erik Enstad [aut, cre]
Maintainer:	Erik Enstad <erik.enstad@nhh.no>
Repository:	CRAN
Date/Publication:	2026-07-15 12:00:02 UTC

metafrontier: Analysis of Metafrontier Models for Efficiency and Productivity

Description

Implements metafrontier production function models for estimating technical efficiencies and technology gaps for groups of firms that face different restrictions of a common underlying metatechnology (group-specific technologies in the sense of Battese, Rao, and O'Donnell, 2004). Supports both stochastic frontier analysis (SFA) and data envelopment analysis (DEA) based metafrontiers. Includes the deterministic metafrontier of Battese, Rao, and O'Donnell (2004) doi:10.1023/B:PROD.0000012454.06094.29, the stochastic metafrontier of Huang, Huang, and Liu (2014) doi:10.1007/s11123-014-0402-2, and the metafrontier Malmquist productivity index of O'Donnell, Rao, and Battese (2008) doi:10.1007/s00181-007-0119-4. The deterministic metafrontier can be identified by either the minimum sum of absolute deviations (LP) or the minimum sum of squared deviations (QP) criterion. Additional features include panel SFA with time-varying inefficiency, bootstrap confidence intervals for technology gap ratios, a DEA poolability permutation test, latent class metafrontier estimation via the EM algorithm, Murphy-Topel corrected standard errors, convergence diagnostics, import of pre-fitted models from external estimation engines ('sfaR', 'frontier', 'Benchmarking'), and 'ggplot2' visualisation methods.

Author(s)

Maintainer: Erik Enstad erik.enstad@nhh.no (ORCID)

See Also

Useful links:

• https://github.com/iik1/metafrontier

• Report bugs at https://github.com/iik1/metafrontier/issues

Convert a Fitted Frontier Model to Metafrontier Format

Description

Generic function that extracts the components needed by metafrontier from a pre-fitted frontier model. Methods are provided for sfaR ("sfacross"), frontier ("frontier"), and Benchmarking ("Farrell") objects, as well as plain lists with the required fields.

Usage

as_metafrontier_model(x, ...)

Arguments

Details

metafrontier(models = ...) calls this function internally on each supplied model, so fitted sfaR or frontier objects can be passed to metafrontier() directly. Manual conversion is only needed for hand-built list models. Converting an object that has already been converted is a no-op, so it is safe to pass converted objects to metafrontier() as well.

Note that Benchmarking::dea() ("Farrell") objects do not store the inputs, outputs, or frontier coefficients, so the converted model carries only efficiency scores and cannot be used with metafrontier(models = ...); use the formula interface with method = "dea" instead. Converting a Farrell object therefore raises a warning.

Value

A list of class "metafrontier_model" with components: beta, te, X, y, sigma_v, sigma_u, logLik, hessian, n, dist.

Examples

# Using a named list:
mod <- as_metafrontier_model(list(
  coefficients = c("(Intercept)" = 2, log_x1 = 0.5, log_x2 = 0.3),
  efficiency = runif(50, 0.7, 1),
  X = matrix(rnorm(150), 50, 3),
  y = rnorm(50, 5)
))
str(mod)

Autoplot Method for Bootstrap TGR Objects

Description

Autoplot Method for Bootstrap TGR Objects

Usage

## S3 method for class 'boot_tgr'
autoplot(object, which = c("distribution", "ci"), ...)

Arguments

Value

A ggplot object.

Examples

if (requireNamespace("ggplot2", quietly = TRUE)) {
  sim <- simulate_metafrontier(n_groups = 2, n_per_group = 50, seed = 42)
  fit <- metafrontier(log_y ~ log_x1 + log_x2, data = sim$data,
                      group = "group", meta_type = "stochastic")
  boot <- boot_tgr(fit, R = 50, seed = 1, progress = FALSE)
  ggplot2::autoplot(boot)
}

Autoplot Method for Malmquist Meta Objects

Description

Autoplot Method for Malmquist Meta Objects

Usage

## S3 method for class 'malmquist_meta'
autoplot(object, which = c("decomposition", "tgr_evolution", "mpi_trend"), ...)

Arguments

Value

A ggplot object.

Examples

if (requireNamespace("ggplot2", quietly = TRUE)) {
  panels <- lapply(1:3, function(t) {
    sim <- simulate_metafrontier(n_groups = 2, n_per_group = 30,
                                 seed = 42 + t)
    sim$data$time <- t
    sim$data$id <- seq_len(nrow(sim$data))
    sim$data
  })
  pdata <- do.call(rbind, panels)
  malm <- malmquist_meta(log_y ~ log_x1 + log_x2, data = pdata,
                         group = "group", time = "time")
  ggplot2::autoplot(malm, which = "decomposition")
}

Autoplot Method for Metafrontier Objects

Description

Creates diagnostic and summary plots for metafrontier objects using ggplot2.

Usage

## S3 method for class 'metafrontier'
autoplot(
  object,
  which = c("tgr", "efficiency", "decomposition", "frontier"),
  ...
)

Arguments

Value

A ggplot object.

Examples

if (requireNamespace("ggplot2", quietly = TRUE)) {
  sim <- simulate_metafrontier(n_groups = 2, n_per_group = 50, seed = 42)
  fit <- metafrontier(log_y ~ log_x1 + log_x2, data = sim$data,
                      group = "group")
  ggplot2::autoplot(fit, which = "tgr")
  ggplot2::autoplot(fit, which = "decomposition")
}

Bootstrap Confidence Intervals for the Technology Gap Ratio

Description

Computes bootstrap confidence intervals for TGR estimates from a fitted metafrontier model. Supports both parametric (residual resampling) and nonparametric (case resampling) bootstraps.

Usage

boot_tgr(
  object,
  R = 999,
  type = c("parametric", "nonparametric"),
  level = 0.95,
  ci_type = c("percentile", "bca"),
  seed = NULL,
  progress = TRUE,
  ncores = 1L,
  ...
)

Arguments

Value

An object of class "boot_tgr" containing:

tgr_boot

R x n matrix of bootstrapped TGR values

tgr_original

original TGR estimates

ci

n x 2 matrix of observation-level confidence intervals

ci_group

data frame of group-level mean TGR intervals

R_effective

number of successful replications

R

requested number of replications

type

bootstrap type used

ci_type

CI type used

level

confidence level

Examples

sim <- simulate_metafrontier(n_groups = 2, n_per_group = 100,
                             seed = 42)
fit <- metafrontier(log_y ~ log_x1 + log_x2,
                    data = sim$data, group = "group",
                    meta_type = "stochastic")
boot <- boot_tgr(fit, R = 50, seed = 1)
print(boot)
confint(boot)

Check Convergence of All Estimation Stages

Description

Reports the convergence status of every estimation stage of a fitted metafrontier model: each group-level frontier and the metafrontier itself. This makes it easy to verify that all optimisers (MLE) and mathematical programmes (LP/QP) finished successfully before interpreting technology gap ratios, confidence intervals, or efficiency decompositions.

Usage

check_convergence(object, ...)

## S3 method for class 'metafrontier'
check_convergence(object, ...)

## S3 method for class 'lc_metafrontier'
check_convergence(object, ...)

## S3 method for class 'malmquist_meta'
check_convergence(object, ...)

Arguments

Details

For groups supplied via the models argument of metafrontier the convergence status of the external fitter is not tracked, and the corresponding rows carry NA with an explanatory note.

Value

A data frame of class "metafrontier_convergence" with one row per estimation stage and columns:

stage

stage label, e.g. "group: G1" or "metafrontier"

method

how the stage was estimated: "MLE", "LP", "QP", "QP (barrier)", "DEA", or "external"

code

the integer convergence code returned by the optimiser (0 indicates success); NA for DEA stages and externally fitted groups

converged

logical convergence indicator; for DEA stages TRUE unless any efficiency score is NA (infeasible programme); NA for externally fitted groups

note

additional detail, e.g. the number of infeasible DEA programmes

Examples

sim <- simulate_metafrontier(n_groups = 2, n_per_group = 50, seed = 42)
fit <- metafrontier(log_y ~ log_x1 + log_x2, data = sim$data,
                    group = "group")
check_convergence(fit)

Extract Coefficients from a Metafrontier Model

Description

Extract Coefficients from a Metafrontier Model

Usage

## S3 method for class 'metafrontier'
coef(object, which = c("meta", "group"), extraPar = FALSE, ...)

Arguments

Value

A named numeric vector (which = "meta") or a named list of numeric vectors (which = "group").

Examples

sim <- simulate_metafrontier(n_groups = 2, n_per_group = 50, seed = 42)
fit <- metafrontier(log_y ~ log_x1 + log_x2, data = sim$data,
                    group = "group", meta_type = "stochastic")
coef(fit)
coef(fit, extraPar = TRUE)
coef(fit, which = "group", extraPar = TRUE)

Confidence Intervals for Metafrontier Coefficients

Description

Computes Wald-type confidence intervals for the metafrontier coefficients using the variance-covariance matrix from the stochastic metafrontier.

Usage

## S3 method for class 'metafrontier'
confint(
  object,
  parm,
  level = 0.95,
  correction = c("none", "murphy-topel"),
  ...
)

Arguments

Value

A matrix with columns for the lower and upper bounds.

Examples

sim <- simulate_metafrontier(n_groups = 2, n_per_group = 50, seed = 42)
fit <- metafrontier(log_y ~ log_x1 + log_x2, data = sim$data,
                    group = "group", meta_type = "stochastic")
confint(fit)
# With Murphy-Topel correction (requires numDeriv):

confint(fit, correction = "murphy-topel")

Extract Efficiency Scores from a Metafrontier Model

Description

Extracts technical efficiency scores from a fitted metafrontier model. Returns either group-specific efficiency (TE), metafrontier efficiency (TE^* = TE \times TGR), or the technology gap ratio (TGR).

Usage

efficiencies(object, ...)

## S3 method for class 'metafrontier'
efficiencies(object, type = c("meta", "group", "tgr"), estimator = NULL, ...)

Arguments

Details

The fundamental metafrontier decomposition is:

TE^*_i = TE_i \times TGR_i

where TE_i is efficiency relative to the group frontier (returned by type = "group") and TGR_i is the technology gap ratio (returned by type = "tgr").

Value

A numeric vector of efficiency scores of length nobs(object).

See Also

technology_gap_ratio, metafrontier

Examples

set.seed(42)
sim <- simulate_metafrontier(n_groups = 2, n_per_group = 100)
fit <- metafrontier(log_y ~ log_x1 + log_x2,
                    data = sim$data, group = "group")

# Group-level efficiency
te <- efficiencies(fit, type = "group")

# Metafrontier efficiency
te_star <- efficiencies(fit, type = "meta")

# Verify decomposition: TE* = TE x TGR
tgr <- efficiencies(fit, type = "tgr")
all.equal(te_star, te * tgr)

Latent Class Metafrontier

Description

Estimates a metafrontier model where group membership is unobserved, using an EM algorithm to jointly estimate class membership probabilities, class-specific frontier parameters, and the metafrontier.

Usage

latent_class_metafrontier(
  formula,
  data,
  n_classes = 2,
  dist = c("hnormal", "tnormal", "exponential"),
  meta_type = c("deterministic", "stochastic"),
  n_starts = 10,
  max_iter = 200,
  tol = 1e-06,
  seed = NULL,
  control = list(),
  ...
)

Arguments

Details

Latent class estimation is available for SFA-based metafrontiers only: the EM posterior class probabilities require a parametric observation-level likelihood, which DEA does not provide. For DEA fits with observed groups, see poolability_test.

The EM algorithm iterates between:

• E-step: compute posterior class membership probabilities for each observation using Bayes' rule

• M-step: update class-specific frontier parameters via weighted MLE, and update class proportions

Multiple random starts (n_starts) are used to avoid local optima. The run with the highest marginal log-likelihood is selected. After convergence, observations are assigned to classes via MAP (maximum a posteriori), and a standard metafrontier is fitted on the MAP classes.

Value

An object of class "lc_metafrontier" containing:

class_assignment

MAP class assignment per observation

posterior

n x C matrix of posterior probabilities

pi

class mixing proportions

class_params

list of class-specific parameter vectors

class_models

list of class-specific model summaries

metafrontier

the fitted metafrontier object on MAP classes

marginal_ll

marginal log-likelihood at convergence

BIC

Bayesian Information Criterion

n_classes

number of classes

n_iter

number of EM iterations used

Examples

sim <- simulate_metafrontier(n_groups = 2, n_per_group = 80, seed = 42)
lc <- latent_class_metafrontier(
  log_y ~ log_x1 + log_x2,
  data = sim$data, n_classes = 2, n_starts = 3, seed = 123
)
print(lc)
summary(lc)
coef(lc, which = "meta")
efficiencies(lc, type = "tgr")

Metafrontier Malmquist Productivity Index

Description

Computes the metafrontier Malmquist total factor productivity (TFP) index and its three-way decomposition into technical efficiency change (TEC), technology gap change (TGC), and metafrontier technical change (TC*) for panel data, following O'Donnell, Rao, and Battese (2008).

Usage

malmquist_meta(
  formula = NULL,
  data = NULL,
  group = NULL,
  time = NULL,
  id = NULL,
  method = c("dea", "sfa"),
  dist = c("hnormal", "tnormal", "exponential"),
  estimator = c("bc88", "jlms"),
  orientation = c("output", "input"),
  rts = c("crs", "vrs", "drs", "irs", "fdh"),
  control = list(),
  ...
)

Arguments

Details

The metafrontier Malmquist TFP index decomposes productivity change into three components:

M^* = TEC \times TGC \times TC^*

where:

• TEC = TE^{group}_{t+1} / TE^{group}_t: technical efficiency change relative to the group frontier

• TGC = TGR_{t+1} / TGR_t: technology gap change, capturing whether a group's frontier is catching up to or falling behind the metafrontier

• TC^*: metafrontier technical change, measuring the shift of the global production possibility frontier

Firm matching: when id is supplied, firms are matched across consecutive periods by identifier within each technology group. Duplicated (id, period) combinations within a group are an error. Observations without a within-group match in the adjacent period, either because the panel is unbalanced or because a firm switches group between periods, are dropped, and a single consolidated warning reports the number dropped per period pair. When id is NULL, firms are matched by row position within each group; this is valid only for balanced panels sorted identically in every period, so a message is emitted as a reminder, and a warning is issued when group sizes differ across a period pair (the unmatched observations are dropped). Supplying id is recommended.

DEA-based computation (method = "dea"): for each consecutive pair of periods (s, t), eight sets of LP problems are solved: within-group and pooled efficiencies at each period, plus cross-period evaluations for the geometric mean formulation of technical change. Distances to the metafrontier are exact distances to the pooled-data frontier, as in O'Donnell, Rao and Battese (2008).

SFA-based computation is an approximation (method = "sfa"): period-specific group SFA frontiers are estimated, and each observation's metafrontier distance is approximated by the pointwise maximum of the estimated group frontier functions evaluated at its inputs; no enveloping metafrontier is re-estimated. This coincides with the O'Donnell et al. (2008) metafrontier wherever a single group frontier dominates, but can understate the metafrontier where group frontiers cross, which affects TGC and TC*. Prefer method = "dea" when an exact decomposition is required.

Infeasible cross-period programs: under rts = "vrs", "drs", "irs", or "fdh", cross-period LPs can be genuinely infeasible because the reference technology cannot reach the evaluated observation. Such cases yield NA (never Inf), are excluded from the reported means, and are counted in a single consolidated warning; the counts are stored in the n_infeasible and infeasible_by_period components. rts = "crs" avoids the issue, as does the hyperbolic orientation available in metafrontier.

Note that the standard Malmquist index is not a ‘proper’ (multiplicatively complete and transitive) TFP index in the sense of O'Donnell (2012), so chained comparisons of index levels across more than two periods should be avoided.

Value

An object of class "malmquist_meta", a list with components:

malmquist

data frame with columns: id, group, period_from, period_to, MPI (metafrontier Malmquist TFP index), TEC (technical efficiency change), TGC (technology gap change), TC (metafrontier technical change). The id column holds the supplied firm identifiers when id is given, and the within-group match position otherwise.

group_malmquist

data frame with the within-group Malmquist index decomposition: MPI_group, EC_group, TC_group

meta_malmquist

data frame with the metafrontier Malmquist index: MPI_meta, EC_meta, TC_meta

tgr

data frame with technology gap ratios at each period endpoint: id, group, period_from, period_to, TGR_from (TGR at the start period), TGR_to (TGR at the end period), and TGC (technology gap change, TGR_to / TGR_from)

call

the matched function call

method

the estimation method used ("dea" or "sfa")

orientation

the orientation used

rts

the returns to scale assumption

groups

group labels

periods

time periods

n_infeasible

total number of infeasible cross-period DEA programs (always 0 for method = "sfa")

infeasible_by_period

data frame with the number of infeasible cross-period DEA programs per period pair (method = "dea" only)

References

O'Donnell, C.J., Rao, D.S.P. and Battese, G.E. (2008). Metafrontier frameworks for the study of firm-level efficiencies and technology ratios. Empirical Economics, 34(2), 231–255. doi:10.1007/s00181-007-0119-4

O'Donnell, C.J. (2012). An aggregate quantity framework for measuring and decomposing productivity change. Journal of Productivity Analysis, 38(3), 255–272. doi:10.1007/s11123-012-0275-1

Examples

# Simulate panel data for 2 groups, 3 time periods
set.seed(42)
panels <- lapply(1:3, function(t) {
  sim <- simulate_metafrontier(
    n_groups = 2, n_per_group = 30,
    tech_gap = c(0, 0.3 + 0.05 * t),
    sigma_u = c(0.2, 0.3),
    seed = 42 + t
  )
  sim$data$time <- t
  sim$data$id <- seq_len(nrow(sim$data))
  sim$data
})
panel_data <- do.call(rbind, panels)

# Compute metafrontier Malmquist index, matching firms by id
malm <- malmquist_meta(
  log_y ~ log_x1 + log_x2,
  data = panel_data,
  group = "group",
  time = "time",
  id = "id"
)
summary(malm)

Estimate a Metafrontier Production Function

Description

Estimates group-specific frontiers and a metafrontier that envelops all group technologies. Supports both SFA-based (parametric) and DEA-based (nonparametric) approaches, with deterministic (Battese, Rao, and O'Donnell, 2004) or stochastic (Huang, Huang, and Liu, 2014) metafrontier estimation.

Usage

metafrontier(
  formula = NULL,
  data = NULL,
  group = NULL,
  method = c("sfa", "dea"),
  meta_type = c("deterministic", "stochastic"),
  dist = c("hnormal", "tnormal", "exponential"),
  orientation = c("output", "input"),
  rts = c("crs", "vrs", "drs", "irs", "fdh"),
  models = NULL,
  panel = NULL,
  panel_dist = c("bc92", "bc95"),
  type = c("radial", "directional", "hyperbolic"),
  direction = c("proportional", "output", "input"),
  control = list(),
  estimator = c("bc88", "jlms"),
  objective = c("lp", "qp"),
  engine = c("internal", "sfaR", "frontier", "Benchmarking"),
  slack = FALSE,
  ...
)

Arguments

Details

The metafrontier framework decomposes efficiency relative to a global technology into two components:

TE^*_i = TE_i \times TGR_i

where TE_i is efficiency relative to the group frontier and TGR_i is the technology gap ratio measuring how close the group frontier is to the metafrontier.

The deterministic metafrontier is identified by one of the two criteria proposed by Battese, Rao, and O'Donnell (2004), subject in both cases to the constraint that the metafrontier envelops all group frontiers: minimising the sum of absolute deviations, which reduces to a linear programme because the envelope constraints force every deviation to be non-negative (O'Donnell, Rao, and Battese, 2008, Eqs. 23-25), or minimising the sum of squared deviations, a convex quadratic programme. The LP (objective = "lp", the default) is solved via lpSolveAPI; the QP (objective = "qp") is solved exactly via quadprog when available, with an adaptive-barrier fallback via constrOptim(). The stochastic metafrontier (Huang, Huang, and Liu, 2014) replaces this with a second-stage SFA, providing a distributional framework for inference on the TGR.

Convergence and failure handling: estimation stops with an error only when no usable estimate exists (for example, when both the BFGS and Nelder-Mead optimisers fail for a group frontier). When an optimiser stops at a non-zero convergence code, the fitted object is returned with a warning and the code is recorded; use check_convergence or summary() to verify all estimation stages before interpreting technology gap ratios, confidence intervals, or productivity decompositions. Infeasible DEA programmes yield NA efficiency scores, accompanied by a warning and counted by check_convergence.

Note on standard errors (stochastic metafrontier): The stochastic metafrontier is a two-stage estimator. Stage 2 treats the fitted group frontier values as data, so the reported standard errors, confidence intervals, and variance-covariance matrix do not account for estimation uncertainty from Stage 1 (the generated-regressor problem; see Murphy and Topel, 1985). Use vcov(fit, correction = "murphy-topel") or bootstrap-based confidence intervals via boot_tgr for corrected inference.

Note on frontier orientation (SFA path): The SFA estimation path assumes a production frontier (\varepsilon = v - u). Cost frontiers (\varepsilon = v + u) are not currently supported via the SFA path. The DEA path supports both orientation = "output" and orientation = "input".

Value

An object of class "metafrontier" (with subclass "metafrontier_sfa" or "metafrontier_dea"), containing:

call

the matched function call

group_models

list of fitted group-specific models

meta_coef

estimated metafrontier parameters

group_coef

list of group-specific coefficient vectors

tgr

technology gap ratios for each observation

te_group

group-specific technical efficiency

te_meta

metafrontier technical efficiency (TE* = TE x TGR)

logLik_groups

log-likelihoods of group models

nobs

number of observations per group and total

groups

group labels

method

estimation method used

meta_type

metafrontier type used

meta_convergence

integer convergence code for the metafrontier stage (0 = success; optim codes for the stochastic metafrontier and the QP barrier fallback; 0 for a successful LP or DEA solution). Each SFA group model in group_models additionally carries its own convergence code. Use check_convergence to inspect all stages.

estimator, objective, engine, meta_solver

the estimation choices used for the fit

References

Battese, G.E., Rao, D.S.P. and O'Donnell, C.J. (2004). A metafrontier production function for estimation of technical efficiencies and technology gaps for firms operating under different technologies. Journal of Productivity Analysis, 21(1), 91–103. doi:10.1023/B:PROD.0000012454.06094.29

Huang, C.J., Huang, T.-H. and Liu, N.-H. (2014). A new approach to estimating the metafrontier production function based on a stochastic frontier framework. Journal of Productivity Analysis, 42(3), 241–254. doi:10.1007/s11123-014-0402-2

O'Donnell, C.J., Rao, D.S.P. and Battese, G.E. (2008). Metafrontier frameworks for the study of firm-level efficiencies and technology ratios. Empirical Economics, 34(2), 231–255. doi:10.1007/s00181-007-0119-4

Examples

# Simulate metafrontier data
set.seed(42)
sim <- simulate_metafrontier(n_groups = 2, n_per_group = 100)

# Estimate deterministic SFA metafrontier (BRO 2004)
fit <- metafrontier(log_y ~ log_x1 + log_x2,
                    data = sim$data,
                    group = "group",
                    method = "sfa",
                    meta_type = "deterministic")
summary(fit)

# Technology gap ratios
tgr <- technology_gap_ratio(fit)
summary(tgr)

Plot a Metafrontier Object

Description

Produces diagnostic and summary plots for a fitted metafrontier model.

Usage

## S3 method for class 'metafrontier'
plot(x, which = c("tgr", "efficiency", "decomposition", "frontier"), ...)

Arguments

Value

Invisibly returns x.

Examples

set.seed(42)
sim <- simulate_metafrontier(n_groups = 2, n_per_group = 100)
fit <- metafrontier(log_y ~ log_x1 + log_x2,
                    data = sim$data, group = "group")
plot(fit, which = "tgr")
plot(fit, which = "decomposition")

Test Poolability of Group Frontiers

Description

Tests the null hypothesis that all groups share a common frontier (i.e., the metafrontier coincides with all group frontiers) against the alternative that group-specific frontiers differ. For SFA-based metafrontiers a likelihood ratio test is used; for DEA-based metafrontiers a permutation test is used.

Usage

poolability_test(object, B = 199, seed = NULL, ...)

Arguments

Details

Likelihood ratio test (SFA). The LR statistic is:

LR = -2 [LL_{pooled} - \sum_j LL_j]

where LL_{pooled} is the log-likelihood of the pooled (single frontier) model and LL_j are the group-specific log-likelihoods. Under H0, the statistic follows a chi-squared distribution with degrees of freedom equal to df = k_{groups} - k_{pooled}, where k_{groups} is the total number of parameters across all group-specific models and k_{pooled} is the number of parameters in the pooled model. For J groups each with p frontier parameters plus distributional parameters, this equals (J - 1) \times p_{total} where p_{total} includes frontier coefficients, \sigma_v, and \sigma_u (and \mu for truncated-normal). This test requires a likelihood and is therefore only available for SFA-based metafrontiers.

Permutation test (DEA). DEA has no likelihood, so the poolability hypothesis is assessed by a permutation test. Under the null of a single pooled technology, group labels are exchangeable: reassigning observations to groups at random should not systematically change the distance between the group frontiers and the metafrontier. The observed statistic is the mean technology gap, S_{obs} = \mathrm{mean}(1 - TGR_i), and its null distribution is approximated by refitting the metafrontier on B random permutations of the group labels. The p-value is (1 + \#\{S_b \ge S_{obs}\}) / (B + 1), following the aggregate-efficiency inference logic of Simar and Zelenyuk (2007). The smoothed subsampling approach of Kneip, Simar, and Wilson (2016) is the asymptotically rigorous alternative for testing hypotheses in nonparametric production models; the permutation test offered here is a computationally simple approximation. The default B = 199 is a pragmatic choice; p-values have resolution 1/(B + 1), so increase B for finer resolution.

Value

A list of class "htest" with components:

statistic

the test statistic (LR statistic for SFA; the mean technology gap, \bar{S} = \mathrm{mean}(1 - TGR), for DEA)

parameter

degrees of freedom (SFA) or the effective number of permutation replicates (DEA)

p.value

p-value of the test

method

description of the test

References

Simar, L. and Zelenyuk, V. (2007). Statistical inference for aggregates of Farrell-type efficiencies. Journal of Applied Econometrics, 22(7), 1367–1394. doi:10.1002/jae.991

Kneip, A., Simar, L. and Wilson, P.W. (2016). Testing hypotheses in nonparametric models of production. Journal of Business & Economic Statistics, 34(3), 435–447. doi:10.1080/07350015.2015.1049747

Examples

set.seed(42)
sim <- simulate_metafrontier(n_groups = 2, n_per_group = 200,
                             tech_gap = c(0, 0.5))
fit <- metafrontier(log_y ~ log_x1 + log_x2,
                    data = sim$data, group = "group")
poolability_test(fit)


# DEA permutation test
fit_dea <- metafrontier(log_y ~ log_x1 + log_x2,
                        data = sim$data, group = "group",
                        method = "dea")
poolability_test(fit_dea, B = 99, seed = 1)

Predict Frontier Values from a Metafrontier Model

Description

Computes predicted frontier values at given input levels using either the metafrontier or a group-specific frontier.

Usage

## S3 method for class 'metafrontier'
predict(object, newdata = NULL, type = c("meta", "group"), ...)

Arguments

Value

A numeric vector of predicted frontier values.

Examples

sim <- simulate_metafrontier(n_groups = 2, n_per_group = 50, seed = 42)
fit <- metafrontier(log_y ~ log_x1 + log_x2, data = sim$data,
                    group = "group", meta_type = "stochastic")
pred <- predict(fit)
# Out-of-sample prediction:
newdata <- data.frame(log_x1 = c(1, 2), log_x2 = c(1.5, 2.5))
predict(fit, newdata = newdata)

Print a Metafrontier Object

Description

Prints a compact overview of a fitted metafrontier model: the estimation method and metafrontier type, the efficiency estimator and identification objective (where applicable), the groups and their sample sizes, group log-likelihoods, mean technology gap ratio by group, and a one-line convergence status.

Usage

## S3 method for class 'metafrontier'
print(x, ...)

Arguments

Value

Invisibly returns x.

Examples

sim <- simulate_metafrontier(n_groups = 2, n_per_group = 50, seed = 42)
fit <- metafrontier(log_y ~ log_x1 + log_x2, data = sim$data, group = "group")
print(fit)

Select Number of Latent Classes via BIC

Description

Fits a latent class metafrontier model for each value in n_classes_range and tabulates the Bayesian information criterion (BIC) and marginal log-likelihood of each fit. The optimal number of classes minimises BIC. Fits that fail are silently dropped from the table. Because the EM algorithm can converge to local optima, the ranking is sensitive to the number of random starts: pass n_starts (forwarded to latent_class_metafrontier) and increase it for a more reliable comparison across class counts.

Usage

select_n_classes(formula, data, n_classes_range = 2:5, ...)

Arguments

Value

A data frame with columns n_classes, BIC, and marginal_ll.

Examples

sim <- simulate_metafrontier(n_groups = 2, n_per_group = 80, seed = 42)
bic_table <- select_n_classes(
  log_y ~ log_x1 + log_x2,
  data = sim$data, n_classes_range = 2:3,
  n_starts = 3, seed = 42
)
print(bic_table)

Simulate Metafrontier Data

Description

Generates synthetic data from a known metafrontier data-generating process. Each group frontier lies weakly below the metafrontier, consistent with groups facing different restrictions of a common metatechnology (Battese, Rao and O'Donnell, 2004). Useful for Monte Carlo simulations, package testing, and teaching.

Usage

simulate_metafrontier(
  n_groups = 2L,
  n_per_group = 100L,
  n_inputs = 2L,
  beta_meta = NULL,
  tech_gap = NULL,
  sigma_u = NULL,
  sigma_v = 0.2,
  seed = NULL,
  beta_groups = NULL,
  input_means = NULL,
  input_corr = NULL
)

Arguments

Details

By default the group frontiers share the metafrontier slopes and differ only in their intercepts, so the true technology gap ratio (TGR) is constant within each group and equals exp(-tech_gap[g]). When beta_groups is supplied the group frontiers may differ in their slopes, in which case no single log-linear metafrontier envelops all groups: the tightest log-linear envelope is then a pseudo-true quantity. The returned true_tgr is instead computed observation by observation against the pointwise maximum over the group frontiers, TGR_i = \exp(x_i^\top \beta_g - \max_j x_i^\top \beta_j), which is guaranteed to lie in (0, 1]. The true group frontier for each firm is x_i^\top \beta_g, true_te is generated exactly as in the default design, and true_te_star = true_te * true_tgr. In this case params$beta_meta is NULL and params$beta_groups holds the supplied coefficients.

By default the log-inputs are drawn i.i.d. from a uniform distribution on [0, 5], identically across groups. Supplying input_means and/or input_corr switches to normal log-inputs with standard deviation 5 / sqrt(12) (matching the spread of the uniform draws), centred at input_means[g, ] (2.5 for every group and input when input_means is NULL). When input_corr is supplied the draws are multivariate normal with that correlation matrix; when it is NULL but input_means is given, the inputs are drawn independently.

Value

A list with components:

data

a data frame with columns log_y, log_x1, log_x2, ..., group, and the true underlying values

params

a list of the true parameters used for generation

Examples

sim <- simulate_metafrontier(n_groups = 3, n_per_group = 200,
                             sigma_u = c(0.2, 0.4, 0.3))
str(sim$data)
table(sim$data$group)

# The true metafrontier coefficients
sim$params$beta_meta

# Group-specific slopes: per-observation true TGR
sim2 <- simulate_metafrontier(
  beta_groups = rbind(c(1.0, 0.5, 0.2), c(0.9, 0.6, 0.1))
)
range(sim2$data$true_tgr)

Simulate Panel Metafrontier Data

Description

Generates a simulated panel dataset with known metafrontier parameters for Monte Carlo studies and testing. Implements the Battese-Coelli 1992 DGP with time-varying inefficiency.

Usage

simulate_panel_metafrontier(
  n_groups = 2,
  n_firms_per_group = 30,
  n_periods = 5,
  beta_meta = c(1, 0.5, 0.3),
  tech_gap = NULL,
  sigma_u = 0.3,
  sigma_v = 0.2,
  eta = 0.05,
  seed = NULL,
  attrition = 0
)

Arguments

Value

A list with components:

data

data frame with columns: firm, year, group, log_y, log_x1, log_x2, true_te, true_u, true_v

params

list of true parameter values used in generation, including attrition and the realised attrition_share

Examples

sim <- simulate_panel_metafrontier(
  n_groups = 2, n_firms_per_group = 20,
  n_periods = 5, seed = 42
)
head(sim$data)
str(sim$params)

# An unbalanced panel with roughly 20% attrition
sim_unbal <- simulate_panel_metafrontier(seed = 42, attrition = 0.2)
table(table(sim_unbal$data$firm))

Summary of a Metafrontier Model

Description

Computes group-level summaries of technical efficiency (TE), technology gap ratio (TGR), and metafrontier efficiency (TE*), full coefficient tables for each group frontier (including variance parameters and, for BC92 panels, eta, all with standard errors where a Hessian is available), the metafrontier coefficient table (with Murphy-Topel corrected standard errors where applicable), and a per-stage convergence table.

Usage

## S3 method for class 'metafrontier'
summary(object, ...)

Arguments

Value

An object of class "summary.metafrontier": a list with components

call

the matched call of the original fit

method

estimation method ("sfa" or "dea")

meta_type

metafrontier type ("deterministic" or "stochastic")

groups

character vector of group labels

nobs

named vector of observation counts (total and per group)

group_tables

named list of coefficient matrices, one per group, with columns Estimate, Std. Error, z value, and Pr(>|z|) where standard errors are available (empty list for DEA fits)

meta_table

metafrontier coefficient matrix in the same format, or NULL for DEA fits

tgr_summary

data frame of TGR statistics by group, as returned by tgr_summary

efficiency_summary

data frame with mean TE, mean TGR, and mean TE* by group

logLik_groups

named vector of group log-likelihoods, or NULL

meta_logLik

Stage 2 log-likelihood of the stochastic metafrontier, or NULL

convergence

data frame with columns stage, code, and converged recording the optimiser status of each estimation stage, or NULL if unavailable; see check_convergence

Examples

sim <- simulate_metafrontier(n_groups = 2, n_per_group = 50, seed = 42)
fit <- metafrontier(log_y ~ log_x1 + log_x2, data = sim$data,
                    group = "group", meta_type = "stochastic")
s <- summary(fit)
print(s)

Extract Technology Gap Ratios

Description

Extracts the technology gap ratios (TGR) from a fitted metafrontier model. The TGR measures how close a group's production frontier is to the global metafrontier at each observation's input mix.

Usage

technology_gap_ratio(object, by_group = TRUE, ...)

Arguments

Details

The technology gap ratio is defined as:

TGR_i = \frac{f(x_i; \hat\beta_j)}{f(x_i; \hat\beta^*)}

for SFA-based metafrontiers, and

TGR_i = \frac{TE^*_i}{TE^{group}_i}

for DEA-based metafrontiers.

Under the deterministic metafrontier and DEA, TGR lies in (0, 1] by construction. Under the stochastic metafrontier of Huang, Huang, and Liu (2014), TGR can exceed 1 for some observations because the metafrontier need not envelop the group frontiers at every point.

A TGR of 1 means the group frontier coincides with the metafrontier at that input mix. Values less than 1 indicate a technology gap. Since each group technology is a restricted subset of the common metatechnology (Battese, Rao and O'Donnell, 2004), the gap reflects the restrictions a group faces (regulation, environment, endowments) rather than a fundamentally different technology.

Value

If by_group = TRUE, a named list of numeric vectors. If by_group = FALSE, a numeric vector of length nobs(object).

See Also

metafrontier, efficiencies.metafrontier

Examples

set.seed(42)
sim <- simulate_metafrontier(n_groups = 2, n_per_group = 100)
fit <- metafrontier(log_y ~ log_x1 + log_x2,
                    data = sim$data, group = "group")
tgr <- technology_gap_ratio(fit)
lapply(tgr, summary)

Summary of Technology Gap Ratios

Description

Prints a summary table of TGR statistics by group. The underlying observation-level TGR values are the same as those returned by efficiencies(object, type = "tgr").

Usage

tgr_summary(object, ...)

Arguments

Value

A data frame with columns: Group, N, Mean, SD, Min, Q1, Median, Q3, Max.

See Also

efficiencies.metafrontier, technology_gap_ratio, boot_tgr

Examples

sim <- simulate_metafrontier(n_groups = 2, n_per_group = 50,
                             seed = 42)
fit <- metafrontier(log_y ~ log_x1 + log_x2, data = sim$data,
                    group = "group", method = "sfa",
                    meta_type = "deterministic")
tgr_summary(fit)

Variance-Covariance Matrix for Metafrontier Coefficients

Description

Returns the variance-covariance matrix of the Stage 2 (metafrontier) coefficients, or, with which = "group", the per-group matrices from the Stage 1 maximum likelihood fits. NULL is returned when no Stage 2 Hessian exists: the deterministic metafrontier is fitted by LP/QP optimisation and has no sampling variance in this framework, so which = "meta" returns NULL with a warning; with which = "group", list entries are NULL for groups without a stored Hessian (e.g. externally fitted models). DEA-based metafrontiers are nonparametric and vcov() signals an error; use boot_tgr for inference instead.

Usage

## S3 method for class 'metafrontier'
vcov(
  object,
  correction = c("none", "murphy-topel"),
  which = c("meta", "group"),
  extraPar = FALSE,
  ...
)

Arguments

Value

A variance-covariance matrix (which = "meta"), a named list of matrices (which = "group"), or NULL if unavailable.

References

Murphy, K.M. and Topel, R.H. (1985). Estimation and inference in two-step econometric models. Journal of Business & Economic Statistics, 3(4), 370–379.

Examples

sim <- simulate_metafrontier(n_groups = 2, n_per_group = 50, seed = 42)
fit <- metafrontier(log_y ~ log_x1 + log_x2, data = sim$data,
                    group = "group", meta_type = "stochastic")
vcov(fit)

vcov(fit, correction = "murphy-topel")