Type:	Package
Title:	Analysis of Metafrontier Models for Efficiency and Productivity
Version:	0.2.2
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.
License:	GPL (≥ 3)
Encoding:	UTF-8
RoxygenNote:	7.3.3
Depends:	R (≥ 4.0.0)
Imports:	stats, graphics, grDevices, Formula, numDeriv, lpSolveAPI, methods
Suggests:	testthat (≥ 3.0.0), knitr, rmarkdown, sfaR, frontier, Benchmarking, plm, ggplot2, parallel
VignetteBuilder:	knitr
URL:	https://github.com/iik1/metafrontier
BugReports:	https://github.com/iik1/metafrontier/issues
Config/testthat/edition:	3
NeedsCompilation:	no
Packaged:	2026-04-14 07:18:02 UTC; s14454
Author:	Erik Enstad [aut, cre]
Maintainer:	Erik Enstad <erik.enstad@nhh.no>
Repository:	CRAN
Date/Publication:	2026-04-21 08:30:24 UTC

metafrontier: 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.

Author(s)

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

See Also

Useful links:

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

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

Convert a Fitted Frontier Model to Metafrontier Format

Description

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

Usage

as_metafrontier_model(x, ...)

Arguments

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)

Autoplot Method for Bootstrap TGR Objects

Description

Autoplot Method for Bootstrap TGR Objects

Usage

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

Arguments

Value

A ggplot object.

Examples

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

Autoplot Method for Malmquist Meta Objects

Description

Autoplot Method for Malmquist Meta Objects

Usage

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

Arguments

Value

A ggplot object.

Examples

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

Autoplot Method for Metafrontier Objects

Description

Creates diagnostic and summary plots for metafrontier objects using ggplot2.

Usage

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

Arguments

Value

A ggplot object.

Examples

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

Bootstrap Confidence Intervals for the Technology Gap Ratio

Description

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

Usage

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

Arguments

Value

An object of class "boot_tgr" containing:

tgr_boot

R x n matrix of bootstrapped TGR values

tgr_original

original TGR estimates

ci

n x 2 matrix of observation-level confidence intervals

ci_group

data frame of group-level mean TGR intervals

R_effective

number of successful replications

R

requested number of replications

type

bootstrap type used

ci_type

CI type used

level

confidence level

Examples

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

Confidence Intervals for Metafrontier Coefficients

Description

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

Usage

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

Arguments

Value

A matrix with columns for the lower and upper bounds.

Examples

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

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

Extract Efficiency Scores from a Metafrontier Model

Description

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

Usage

efficiencies(object, ...)

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

Arguments

Details

The fundamental metafrontier decomposition is:

TE^*_i = TE_i \times TGR_i

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

Value

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

See Also

technology_gap_ratio, metafrontier

Examples

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

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

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

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

Latent Class Metafrontier

Description

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

Usage

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

Arguments

Details

The EM algorithm iterates between:

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

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

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

Value

An object of class "lc_metafrontier" containing:

class_assignment

MAP class assignment per observation

posterior

n x C matrix of posterior probabilities

pi

class mixing proportions

class_params

list of class-specific parameter vectors

class_models

list of class-specific model summaries

metafrontier

the fitted metafrontier object on MAP classes

marginal_ll

marginal log-likelihood at convergence

BIC

Bayesian Information Criterion

n_classes

number of classes

n_iter

number of EM iterations used

Examples

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

Metafrontier Malmquist Productivity Index

Description

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

Usage

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

Arguments

Details

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

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

where:

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

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

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

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)

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(),
  ...
)

Arguments

Details

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

TE^*_i = TE_i \times TGR_i

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

The deterministic metafrontier (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

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

Examples

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

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

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

Plot a Metafrontier Object

Description

Produces diagnostic and summary plots for a fitted metafrontier model.

Usage

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

Arguments

Value

Invisibly returns x.

Examples

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

Test Poolability of Group Frontiers

Description

Tests the null hypothesis that all groups share a common frontier (i.e., the metafrontier coincides with all group frontiers) against the alternative that group-specific frontiers differ. Uses a likelihood ratio test.

Usage

poolability_test(object, ...)

Arguments

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)

Predict Frontier Values from a Metafrontier Model

Description

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

Usage

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

Arguments

Value

A numeric vector of predicted frontier values.

Examples

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

Print a Metafrontier Object

Description

Print a Metafrontier Object

Usage

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

Arguments

Value

Invisibly returns x.

Examples

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

Select Number of Latent Classes via BIC

Description

Select Number of Latent Classes via BIC

Usage

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

Arguments

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)

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
)

Arguments

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

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
)

Arguments

Value

A list with components:

data

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

params

list of true parameter values used in generation

Examples

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, ...)

Arguments

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)

Extract Technology Gap Ratios

Description

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

Usage

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

Arguments

Details

The technology gap ratio is defined as:

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

for SFA-based metafrontiers, and

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

for DEA-based metafrontiers.

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

A TGR of 1 means the group frontier coincides with the metafrontier at that input mix. Values less than 1 indicate a technology gap.

Value

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

See Also

metafrontier, efficiencies.metafrontier

Examples

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

Summary of Technology Gap Ratios

Description

Prints a summary table of TGR statistics by group.

Usage

tgr_summary(object, ...)

Arguments

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"), ...)

Arguments

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")