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 firms operating under different technologies. 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>. Additional features include panel SFA with time-varying inefficiency, bootstrap confidence intervals for technology gap ratios, latent class metafrontier estimation via the EM algorithm, Murphy-Topel corrected standard errors, 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:	0.2.2
Built:	2026-05-23 09:04:37 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, frontier, and Benchmarking 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.

Value

A list with components: coefficients, efficiency, 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.

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).

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.

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)


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)

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

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

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.

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

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

method

character. "dea" (default) for DEA-based distance functions or "sfa" for SFA-based parametric distance functions.

dist

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

orientation

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

rts

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

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

Computation uses DEA-based distance functions. 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.

Balanced panel assumption: Firms are matched across periods by position within each group. The data should contain a balanced panel (the same firms observed in every period) with consistent ordering. If group sizes differ across periods, only the first min(n_s, n_t) firms per group are paired and unmatched observations are silently dropped.

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)
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
orientation: the orientation used
rts: the returns to scale assumption
groups: group labels
periods: time periods

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

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
malm <- malmquist_meta(
  log_y ~ log_x1 + log_x2,
  data = panel_data,
  group = "group",
  time = "time"
)
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
malm <- malmquist_meta(
  log_y ~ log_x1 + log_x2,
  data = panel_data,
  group = "group",
  time = "time"
)
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"),
  models = NULL,
  panel = NULL,
  panel_dist = c("bc92", "bc95"),
  type = c("radial", "directional"),
  direction = c("proportional", "output", "input"),
  control = list(),
  ...
)
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"),
  models = NULL,
  panel = NULL,
  panel_dist = c("bc92", "bc95"),
  type = c("radial", "directional"),
  direction = c("proportional", "output", "input"),
  control = list(),
  ...
)

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), or "irs" (increasing). Ignored when method = "sfa".

models

an optional named list of pre-fitted group-specific frontier models (objects from sfaR, frontier, or Benchmarking). If 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 or "directional" for directional distance functions.

direction

character. Direction vector for DDF: "proportional" (default), "output", or "input". 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).

...

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 (Battese, Rao, and O'Donnell, 2004) is estimated by solving a linear program that minimises the total envelope overshoot subject to the constraint that the metafrontier envelops all group frontiers. BRO (2004) originally proposed a constrained least-squares (QP) formulation; the LP yields the tightest envelope and is solved via lpSolveAPI, with a QP fallback via constrOptim() when the LP is infeasible. The stochastic metafrontier (Huang, Huang, and Liu, 2014) replaces this with a second-stage SFA, providing a distributional framework for inference on the TGR.

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
convergence: convergence status

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. Uses a likelihood ratio test.

Usage

poolability_test(object, ...)
poolability_test(object, ...)

Arguments

object

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

...

additional arguments (currently unused).

Details

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).

Value

A list of class "htest" with components:

statistic: the LR test statistic
parameter: degrees of freedom
p.value: p-value of the test
method: description of the test

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)

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)

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

Print a Metafrontier Object

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

Select Number of Latent Classes via BIC

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.

Details

Fits latent class metafrontier models for each value in n_classes_range and returns BIC values. The optimal number of classes minimises BIC.

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. 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
)
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
)

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, 0.5, 0.3).

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.

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.

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

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

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
)
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
)

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.

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

Examples

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

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

Summary of a Metafrontier Model

Description

Summary of a Metafrontier Model

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".

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.

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.

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.

Variance-Covariance Matrix for Metafrontier Coefficients

Description

Variance-Covariance Matrix for Metafrontier Coefficients

Usage

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

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.

...

additional arguments (currently unused).

Value

A variance-covariance matrix, 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

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

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

Arguments

Details

Value

References

Examples

Plot a Metafrontier Object

Description

Usage

Arguments

Value

Examples

Test Poolability of Group Frontiers

Description

Usage

Arguments

Details