Package 'metafrontier' reference manual

Title:	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.
Authors:	Erik Enstad [aut, cre] (ORCID: <https://orcid.org/0009-0006-1053-3849>)
Maintainer:	Erik Enstad <[email protected]>
License:	GPL (>= 3)
Version:	0.3.1.9000
Built:	2026-09-24 10:21:23 UTC
Source:	https://github.com/iik1/metafrontier

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, ...)
as_metafrontier_model(x, ...)

Arguments

x

a fitted frontier model object.

...

additional arguments passed to methods.

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)

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

Arguments

object

a "boot_tgr" object.

which

character. "distribution" (default) or "ci".

...

additional 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)
}


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

Arguments

object

a "malmquist_meta" object.

which

character. Which plot: "decomposition" (default), "tgr_evolution", or "mpi_trend".

...

additional 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")
}


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

Arguments

object

a "metafrontier" object.

which

character. Which plot to produce: "tgr" (default) for TGR density distributions by group (degenerate groups with zero-variance TGR are annotated and excluded from the density), "efficiency" for the TE/TGR/TE* decomposition (boxplots), "decomposition" for grouped bars of mean TE, TGR, and TE*, "frontier" for metafrontier vs group frontiers scatter plot.

...

additional arguments (currently unused).

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")
}


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

Arguments

object

a "metafrontier" object fitted from formula, data and group. Metafrontiers built from pre-fitted models are not supported, because the group models cannot be re-estimated on bootstrap samples.

R

integer. Number of bootstrap replications (default 999).

type

character. "parametric" resamples from estimated error distributions; "nonparametric" resamples rows within groups with replacement.

level

numeric. Confidence level (default 0.95).

ci_type

character. "percentile" (default) or "bca" (bias-corrected and accelerated). Applies to the observation-level intervals of the parametric bootstrap; the group-level intervals are always percentile intervals.

seed

optional integer seed for reproducibility.

progress

logical. Show progress bar (default TRUE).

ncores

integer. Number of CPU cores for parallel bootstrap (default 1, sequential). Requires the parallel package.

...

additional arguments passed to metafrontier.

Details

The parametric bootstrap keeps the design fixed and redraws the response, so column i of tgr_boot is the TGR of DMU i in every replicate. It returns observation-level intervals (ci) as well as group-level intervals.

The nonparametric bootstrap resamples rows with replacement within each group, stacks the resampled rows group by group in the order of groups, and refits the model. Column j of tgr_boot is then the j-th resampled unit, a random draw from group boot_group[j], not DMU j. The group-level intervals are computed over these position blocks and do not depend on the row order of the data. No observation-level intervals are returned (ci is NULL): a column is not tied to any DMU, and evaluating the original DMUs against each bootstrap frontier instead is, for DEA, the naive bootstrap, which is inconsistent (Kneip, Simar and Wilson, 2008). For SFA fits, observation-level intervals are available from the parametric bootstrap.

Group-level intervals are percentile intervals of the within-group mean (ci_group) and median (ci_group_median) of the TGR across replicates. Rows dropped from the fit because of missing values are left out of both bootstraps.

Value

An object of class "boot_tgr" containing:

tgr_boot: R x n matrix of bootstrapped TGR values. For the parametric bootstrap, column i is DMU i; for the nonparametric bootstrap, column j is the j-th resampled unit (see Details).
tgr_original: original TGR estimates
ci: n x 2 matrix of observation-level confidence intervals (parametric bootstrap), or NULL (nonparametric bootstrap)
ci_group: data frame of group-level mean TGR intervals
ci_group_median: data frame of group-level median 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
group_vec: group of each element of tgr_original
boot_group: group of each column of tgr_boot
groups: group labels

References

Kneip, A., Simar, L. and Wilson, P.W. (2008). Asymptotics and consistent bootstraps for DEA estimators in nonparametric frontier models. Econometric Theory, 24(6), 1663–1697. doi:10.1017/S0266466608080651

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)

# Nonparametric bootstrap: group-level intervals only
boot_np <- boot_tgr(fit, R = 50, type = "nonparametric", seed = 1)
boot_np$ci_group
boot_np$ci_group_median


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)

# Nonparametric bootstrap: group-level intervals only
boot_np <- boot_tgr(fit, R = 50, type = "nonparametric", seed = 1)
boot_np$ci_group
boot_np$ci_group_median

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, ...)
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

object

a fitted model object.

...

additional arguments passed to methods.

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)

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

Arguments

object

a "metafrontier" object.

which

character. "meta" (default) returns the metafrontier coefficients; "group" returns a named list of group-specific coefficient vectors.

extraPar

logical. If TRUE, auxiliary parameters are included alongside the frontier coefficients. For which = "group" the variance parameters are back-transformed to their natural scale (sigmaV, sigmaU), mu and eta are kept as estimated, and heteroscedastic Z coefficients are labelled with their column names. For which = "meta" the Stage 2 variance parameters are appended for stochastic metafrontiers.

...

additional arguments (currently unused).

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)

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

Arguments

object

a "metafrontier" object.

parm

a character or integer vector of parameter names or indices. If missing, all frontier coefficients are used.

level

the confidence level (default 0.95).

correction

character. Passed to vcov.metafrontier. "none" (default) uses the Stage 2 variance-covariance matrix; "murphy-topel" applies the Murphy and Topel (1985) correction for first-stage estimation uncertainty.

...

additional arguments (currently unused).

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")


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, ...)
efficiencies(object, ...)

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

Arguments

object

a fitted "metafrontier" object.

...

additional arguments (currently unused).

type

character. The type of efficiency to return: "group" for efficiency relative to the group frontier, "meta" (default) for efficiency relative to the metafrontier, or "tgr" for the technology gap ratio.

estimator

optional character. Override the efficiency estimator used at fit time: "bc88" for the Battese-Coelli (1988) conditional expectation $E[\exp(-u)|\varepsilon]$ or "jlms" for $\exp(-E[u|\varepsilon])$ (Jondrow et al., 1982). Both are stored on SFA fits, so no refitting is needed; type = "meta" is recomputed as $TE \times TGR$ . The TGR itself does not depend on the estimator. Ignored (with a warning) for DEA fits and externally fitted group models that do not carry both estimators. Default NULL returns the scores selected at fit time.

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).

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)

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(),
  ...
)
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

formula

a Formula object (y ~ x1 + x2).

data

a data frame.

n_classes

integer. Number of latent classes (default 2).

dist

distribution of the inefficiency term.

meta_type

metafrontier type for Stage 2.

n_starts

integer. Number of random initializations (default 10). The best is selected by log-likelihood.

max_iter

integer. Maximum EM iterations (default 200).

tol

numeric. Convergence tolerance on marginal LL (default 1e-6).

seed

optional integer seed.

control

list of control parameters for the optimiser.

...

additional 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")


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(),
  ...
)
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

formula

an object of class Formula. Left-hand side specifies the output(s); right-hand side specifies the inputs. Example: y ~ x1 + x2.

data

a data frame containing all variables, plus the grouping and time variables.

group

a character string naming the column in data that identifies technology groups, or a vector of group indicators.

time

a character string naming the column in data that identifies time periods, or a vector of time indicators. Periods must be consecutive integers or sortable.

id

optional. A character string naming the column in data that identifies firms across periods, or a vector of firm identifiers. When supplied, firms are matched across consecutive periods by identifier within each group. When NULL (default), firms are matched by row position within each group, which is valid only for balanced panels sorted identically in every period (see Details).

method

character. "dea" (default) for DEA-based distance functions or "sfa" for SFA-based parametric distance functions (an approximation; see Details).

dist

character. Distribution of the inefficiency term when method = "sfa": "hnormal" (default), "tnormal", or "exponential".

estimator

character. Technical efficiency estimator used when method = "sfa": "bc88" (default) for the Battese and Coelli (1988) estimator $E[\exp(-u)|\varepsilon]$ , or "jlms" for the Jondrow et al. (1982) estimator $\exp(-E[u|\varepsilon])$ . Passed to the group SFA fitter.

orientation

character. "output" (default) or "input".

rts

character. Returns to scale assumption: "crs" (default), "vrs", "drs", "irs", or "fdh".

control

a list of control parameters for the SFA optimiser.

...

additional arguments (currently unused).

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)

# 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,
  ...
)
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

formula

an object of class Formula. The left-hand side specifies the (log) output variable. The first right-hand side part specifies inputs for the frontier. An optional second part (separated by |) specifies inefficiency determinants. Example: log_y ~ log_x1 + log_x2 | z1 + z2. Ignored if models is provided.

data

a data frame containing all variables in the formula and the grouping variable. Ignored if models is provided.

group

a character string naming the column in data that identifies technology groups, or a vector of group indicators of length nrow(data). Ignored if models is provided.

method

character. The frontier estimation method for group-specific models: "sfa" (default) for stochastic frontier analysis or "dea" for data envelopment analysis.

meta_type

character. The method for estimating the metafrontier: "deterministic" (default) uses the linear programming approach of Battese, Rao, and O'Donnell (2004); "stochastic" uses the second-stage SFA approach of Huang, Huang, and Liu (2014).

dist

character. Distribution of the one-sided inefficiency term in SFA models. One of "hnormal" (half-normal, default), "tnormal" (truncated normal), or "exponential". Ignored when method = "dea".

orientation

character. For DEA: "output" (default) or "input" orientation. Ignored when method = "sfa".

rts

character. Returns to scale for DEA: "crs" (constant, default), "vrs" (variable), "drs" (decreasing), "irs" (increasing), or "fdh" (free disposable hull, i.e. no convexity). Ignored when method = "sfa".

models

an optional named list of pre-fitted group-specific frontier models (objects from sfaR or frontier, or hand-built lists). Fitted model objects are converted automatically via as_metafrontier_model, so no manual conversion is required (pre-converting is harmless, the conversion is idempotent). Farrell objects from Benchmarking store neither coefficients nor data and cannot be used here; use the formula interface with method = "dea" instead. If models is provided, formula, data, and group are ignored.

panel

an optional list with components id and time naming the panel identifier and time columns in data. When non-NULL, panel SFA models (BC92/BC95) are used at the group level.

panel_dist

character. Panel SFA model: "bc92" (Battese and Coelli 1992, time-varying inefficiency, default) or "bc95" (Battese and Coelli 1995, observation-specific mean). Only used when panel is non-NULL.

type

character. For DEA: "radial" (default) for standard radial DEA, "directional" for directional distance functions, or "hyperbolic" for hyperbolic (graph) efficiency, which contracts inputs and expands outputs simultaneously.

direction

direction vector for DDF. Either a character preset ("proportional" (default), "output", or "input"), a numeric vector of length m + s giving a common direction (first m elements for inputs, last s for outputs), or a numeric n x (m + s) matrix of firm-specific directions. With numeric directions the ratio-based TGR is not defined; the additive gap (ddf_gap) is reported instead. Only used when type = "directional".

control

a named list of control parameters passed to optim. Common options include maxit (maximum iterations, default 5000), reltol (relative convergence tolerance, default 1e-10), and fnscale (set to -1 internally for maximisation).

estimator

character. Technical efficiency estimator for SFA models: "bc88" (default) for the conditional expectation $E[\exp(-u)|\varepsilon]$ of Battese and Coelli (1988), which is the consistent estimator of technical efficiency, or "jlms" for $\exp(-E[u|\varepsilon])$ following Jondrow et al. (1982). Both are stored on the fitted object; see efficiencies. Ignored when method = "dea".

objective

character. Identification criterion for the deterministic metafrontier: "lp" (default) minimises the sum of absolute deviations (a linear programme), "qp" minimises the sum of squared deviations (a quadratic programme, solved exactly via quadprog when available). Both criteria are proposed in Battese, Rao, and O'Donnell (2004). Only used when method = "sfa" and meta_type = "deterministic".

engine

character. Estimation backend for the group frontiers: "internal" (default) uses the package's own estimators; "sfaR" or "frontier" delegate the SFA group frontiers to sfacross or sfa (cross-sectional, single-part formulas only); "Benchmarking" delegates the DEA group frontiers and the pooled metafrontier to dea (radial only), using its XREF/YREF external-reference facility for the metafrontier stage. The metafrontier stage for SFA methods is always estimated internally (the Murphy-Topel correction requires the internal likelihood).

slack

logical. For radial DEA, compute second-stage input and output slacks (with the radial score held fixed) against both the group and the pooled reference sets. Default FALSE.

...

additional arguments passed to the group-level estimation functions.

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

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)

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

Arguments

x

a "metafrontier" object.

which

character. Type of plot to produce: "tgr" (default) for TGR distributions by group, "efficiency" for TE* vs TE scatter coloured by group, "decomposition" for side-by-side boxplots of TE, TGR, and TE* by group, or "frontier" for group frontier vs metafrontier (only for single-input SFA models).

...

additional graphical parameters passed to base plotting functions.

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")

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

Arguments

object

a fitted "metafrontier" object with method = "sfa" or method = "dea".

B

integer. Number of permutation replicates for the DEA permutation test (default 199). Ignored for SFA objects.

seed

integer or NULL. Random seed for the DEA permutation test, for reproducibility. Ignored for SFA objects.

...

additional arguments (currently unused).

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), 435–456. 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)


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

Arguments

object

a "metafrontier" object.

newdata

an optional data frame of new inputs. If omitted, the training data predictions are returned.

type

character. "meta" (default) for metafrontier predictions, or "group" for group-specific frontier predictions (requires a group column in newdata).

...

additional arguments (currently unused).

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)

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

Arguments

x

a "metafrontier" object.

...

additional arguments (currently unused).

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$data, group = "group")
print(fit)
sim <- simulate_metafrontier(n_groups = 2, n_per_group = 50, seed = 42)
fit <- metafrontier(log_y ~ log_x1 + log_x2, data = sim$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, ...)
select_n_classes(formula, data, n_classes_range = 2:5, ...)

Arguments

formula

formula.

data

data frame.

n_classes_range

integer vector of class counts to try.

...

additional arguments passed to latent_class_metafrontier, notably n_starts.

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)


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
)
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

n_groups

integer. Number of technology groups (default 2).

n_per_group

integer or integer vector. Number of observations per group. If a single value, the same number is used for all groups. If a vector, must be of length n_groups.

n_inputs

integer. Number of input variables (default 2).

beta_meta

numeric vector. Metafrontier coefficients (including intercept). Length must be n_inputs + 1. Default: c(1.0, seq(0.5, 0.2, length.out = n_inputs)), i.e. c(1.0, 0.5, 0.2) for the default two inputs. Ignored when beta_groups is supplied.

tech_gap

numeric vector of length n_groups. The technology gap for each group, defined as the reduction in the intercept relative to the metafrontier. Default: evenly spaced from 0 to 0.5. Ignored (with a warning) when beta_groups is supplied.

sigma_u

numeric vector of length n_groups. Standard deviation of the half-normal inefficiency term for each group. Default: rep(0.3, n_groups).

sigma_v

numeric. Standard deviation of the symmetric noise term. Default: 0.2.

seed

integer or NULL. Random seed for reproducibility.

beta_groups

optional group-specific frontier coefficients, including slopes: either an n_groups x (n_inputs + 1) numeric matrix (one row per group) or a list of n_groups numeric vectors of length n_inputs + 1. When supplied, it replaces the intercept-shift construction based on tech_gap; see Details. Default NULL (intercept-shift design).

input_means

optional n_groups x n_inputs numeric matrix of per-group mean log-input levels. When supplied, the log-inputs for group g are drawn from a normal distribution centred at input_means[g, ]; see Details. Default NULL (identical uniform inputs across groups).

input_corr

optional n_inputs x n_inputs correlation matrix for the log-inputs. When supplied, the log-inputs are drawn from a multivariate normal distribution with this correlation structure; see Details. Default NULL (independent inputs).

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)

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
)
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

n_groups

integer. Number of technology groups.

n_firms_per_group

integer. Number of firms per group.

n_periods

integer. Number of time periods.

beta_meta

numeric vector. Metafrontier coefficients.

tech_gap

numeric vector of group technology gaps.

sigma_u

numeric. Standard deviation of the firm effect.

sigma_v

numeric. Standard deviation of noise.

eta

numeric. Time-decay parameter for BC92.

seed

integer or NULL. Random seed.

attrition

numeric in [0, 0.5]. Probability that each firm-period observation after a firm's first period is dropped independently, producing an unbalanced panel. Every firm's first period is always kept, so all firms remain in the data. The attrition draws are made after all other random numbers, so attrition = 0 (the default) reproduces legacy balanced datasets exactly for the same seed. The realised share of at-risk observations dropped is stored in params$attrition_share.

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))

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

Arguments

object

a "metafrontier" object.

...

additional arguments (currently unused).

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

Arguments

object

a fitted "metafrontier" object.

by_group

logical. If TRUE (default), returns a named list of TGR vectors, one per group. If FALSE, returns a single numeric vector.

...

additional arguments (currently unused).

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).

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)

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, ...)
tgr_summary(object, ...)

Arguments

object

a fitted "metafrontier" object.

...

additional arguments (currently unused).

Value

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

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)

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

Arguments

object

a "metafrontier" object.

correction

character. "none" (default) returns the Stage 2 variance-covariance matrix. "murphy-topel" applies the Murphy and Topel (1985) correction for first-stage estimation uncertainty (the generated-regressor problem). Only available for stochastic metafrontiers.

which

character. "meta" (default) returns the metafrontier (Stage 2) variance-covariance matrix; "group" returns a named list with one full variance-covariance matrix per group (from the inverse negative Hessian of the group MLE), with NULL entries for groups without a stored Hessian.

extraPar

logical. If TRUE and which = "meta", the full Stage 2 matrix is returned, including the rows and columns for the auxiliary parameters (raw MLE parameterisation, e.g. log_sigma_v); the default returns only the block for the frontier coefficients.

...

additional arguments (currently unused).

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")


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")

Package 'metafrontier'

Help Index

Convert a Fitted Frontier Model to Metafrontier Format

Description

Usage

Arguments

Details

Value

Examples

Autoplot Method for Bootstrap TGR Objects

Description

Usage

Arguments

Value

Examples

Autoplot Method for Malmquist Meta Objects

Description

Usage

Arguments

Value

Examples

Autoplot Method for Metafrontier Objects

Description

Usage

Arguments

Value

Examples

Bootstrap Confidence Intervals for the Technology Gap Ratio

Description

Usage

Arguments

Details

Value

References

Examples

Check Convergence of All Estimation Stages

Description

Usage

Arguments

Details

Value

Examples

Extract Coefficients from a Metafrontier Model

Description

Usage

Arguments

Value

Examples

Confidence Intervals for Metafrontier Coefficients

Description

Usage

Arguments

Value

Examples

Extract Efficiency Scores from a Metafrontier Model

Description

Usage

Arguments

Details

Value

See Also

Examples

Latent Class Metafrontier

Description

Usage

Arguments

Details

Value

Examples

Metafrontier Malmquist Productivity Index

Description

Usage

Arguments

Details

Value

References

Examples

Estimate a Metafrontier Production Function

Description

Usage