README.md

In metafrontier: Analysis of Metafrontier Models for Efficiency and Productivity

metafrontier

Analysis of Metafrontier Models for Efficiency and Productivity

Overview

metafrontier provides a unified R implementation of metafrontier production function models for estimating technical efficiencies and technology gaps across groups of firms that face different restrictions of a common underlying metatechnology (group-specific technologies in the sense of Battese, Rao & O'Donnell, 2004).

Estimation methods

• Deterministic metafrontier (Battese, Rao & O'Donnell, 2004) identified by minimum sum of absolute deviations (LP, default) or minimum sum of squared deviations (QP)
• Stochastic metafrontier (Huang, Huang & Liu, 2014) via second-stage SFA with Murphy-Topel corrected standard errors
• DEA-based metafrontier with CRS, VRS, DRS, IRS, and FDH technology assumptions
• Latent class metafrontier via EM algorithm with BIC-based class selection
• Efficiency estimators: BC88 (default) and JLMS, both stored so efficiencies(fit, estimator = ) switches without refitting

Productivity analysis

• Metafrontier Malmquist TFP index (O'Donnell, Rao & Battese, 2008) with three-way decomposition (TEC x TGC x TC*), firm matching via id =, and explicit accounting of cross-period infeasibilities
• Panel SFA with time-varying inefficiency (BC92 and BC95 specifications), including unbalanced panels
• Directional distance functions for DEA-based efficiency measurement, including hyperbolic orientation, custom numeric direction vectors, and two-stage slack computation

Inference and diagnostics

• Bootstrap confidence intervals for TGR (parametric and nonparametric; percentile and BCa)
• Murphy-Topel variance correction for stochastic metafrontier standard errors
• Poolability tests for common vs group-specific frontiers: likelihood ratio (SFA) and a permutation test (DEA)
• Convergence diagnostics via check_convergence(), with convergence status reported in print() and summary()
• Half-normal, exponential, and truncated-normal inefficiency distributions

Visualisation

• Base R plot() with four plot types: TGR distributions, efficiency scatter, frontier decomposition, and frontier comparison
• ggplot2 integration via autoplot() methods for metafrontier, Malmquist, and bootstrap results

Interoperability

• Import pre-fitted models from sfaR, frontier, and Benchmarking via as_metafrontier_model(), or delegate group estimation directly with engine = c("internal", "sfaR", "frontier", "Benchmarking")
• Formula interface with heteroscedastic SFA support (y ~ x1 + x2 | z1 + z2)
• Full S3 method suite: print, summary, coef, vcov, logLik, fitted, residuals, nobs, confint, predict, plot; coef() and vcov() expose auxiliary parameters (e.g. eta, variance parameters) via extraPar = TRUE

Installation

Install the development version from GitHub:

# install.packages("devtools")
devtools::install_github("iik1/metafrontier")

Quick start

library(metafrontier)

# Simulate metafrontier data with two technology groups
sim <- simulate_metafrontier(n_groups = 2, n_per_group = 200, seed = 42)

# Estimate a deterministic SFA-based metafrontier
fit_det <- metafrontier(log_y ~ log_x1 + log_x2,
                        data = sim$data, group = "group")

# Estimate a stochastic metafrontier (with Murphy-Topel SEs)
fit_sto <- metafrontier(log_y ~ log_x1 + log_x2,
                        data = sim$data, group = "group",
                        meta_type = "stochastic")

# DEA-based metafrontier (requires level-scale inputs/outputs)
dat_lev <- within(sim$data, { y <- exp(log_y); x1 <- exp(log_x1); x2 <- exp(log_x2) })
fit_dea <- metafrontier(y ~ x1 + x2,
                        data = dat_lev, group = "group",
                        method = "dea", rts = "vrs")

# Inspect results
summary(fit_det)
tgr_summary(fit_det)
confint(fit_det)

Bootstrap confidence intervals

boot <- boot_tgr(fit_det, R = 999, seed = 1)
confint(boot)

# Parallel bootstrap
boot_par <- boot_tgr(fit_det, R = 999, ncores = 4, seed = 1)

Malmquist productivity index

# Simulate panel data
panel <- simulate_panel_metafrontier(n_groups = 3, n_firms_per_group = 50,
                                     n_periods = 5, seed = 42)

# Three-way Malmquist decomposition
malm <- malmquist_meta(log_y ~ log_x1 + log_x2,
                       data = panel$data, group = "group",
                       time = "year")
summary(malm)

Latent class metafrontier

# Automatic class selection via BIC (discovers groups endogenously)
lc <- latent_class_metafrontier(log_y ~ log_x1 + log_x2,
                                data = sim$data,
                                n_classes = 3)
summary(lc)

Visualisation

# Base R
plot(fit_det, which = "tgr")
plot(fit_det, which = "decomposition")

# ggplot2
library(ggplot2)
autoplot(fit_det)
autoplot(boot)
autoplot(malm)

Using pre-fitted models

library(sfaR)

# Fit group-specific SFA models externally
sfa_g1 <- sfacross(log_y ~ log_x1 + log_x2,
                   data = subset(sim$data, group == "G1"))
sfa_g2 <- sfacross(log_y ~ log_x1 + log_x2,
                   data = subset(sim$data, group == "G2"))

# Pass to metafrontier
fit <- metafrontier(models = list(G1 = sfa_g1, G2 = sfa_g2))

Vignettes

The package includes three vignettes:

• Introduction to metafrontier -- end-to-end walkthrough covering estimation, inference, bootstrap CIs, panel SFA, latent class, and directional distance functions
• Metafrontier Malmquist Productivity Index -- panel data productivity decomposition with worked examples
• Metafrontier Methods: Theory and Computation -- mathematical details, comparison of deterministic/stochastic/DEA approaches, and Monte Carlo evidence

browseVignettes("metafrontier")

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

License

GPL (>= 3)

metafrontier documentation built on Aug. 19, 2026, 5:08 p.m.