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In metafrontier: Analysis of Metafrontier Models for Efficiency and Productivity

metafrontier 0.3.1

Bug fixes

• BC88/BC92 efficiencies are now evaluated on the log scale (pnorm(log.p = TRUE)), with an asymptotic closed form in the far left tail of the conditional distribution. The previous natural-scale ratio Phi(mu*/sigma* - sigma*) / Phi(mu*/sigma*) could return 0, Inf, or values above one on platforms without extended-precision long doubles (the CRAN noLD additional check) when both Phi terms were subnormal, which occurred in degenerate fits with sigma_u near zero. On standard platforms results are unchanged to near machine precision; in the affected tail region the new evaluation is also more accurate than the previous fallback to the JLMS value.

metafrontier 0.3.0

Breaking changes

• The default technical efficiency estimator for SFA models is now the Battese-Coelli (1988) conditional expectation E[exp(-u)|eps] (estimator = "bc88"), which provides consistent efficiency estimates. The previous JLMS estimator remains available via estimator = "jlms", and both are stored on every fit; efficiencies(fit, estimator = ) switches without refitting. Deterministic and stochastic TGRs are unaffected (they do not depend on the estimator), but TE and TE* values change slightly relative to 0.2.x.
• malmquist_meta() output id column now carries the user-supplied firm identifier (previously a within-group loop index).
• The as_metafrontier_model() method for frontier::sfa() fits is now registered for the correct class "frontier"; the previous registration ("sfa") never dispatched.

Bug fixes

• Panel SFA (BC92/BC95): fixed a row-alignment bug where observation-level efficiencies were returned in internally sorted (id, time) order but assigned back in input-row order, scrambling te_group/te_meta whenever the input rows were not already in string-sorted order. Coefficients, TGRs, and group means were unaffected. Results are now row-order invariant (regression-tested).
• BC92 decay is now anchored at the panel-wide final period T (previously the firm-specific last period), matching Battese and Coelli's (1992) unbalanced formulation and the package's own simulator. Balanced panels are numerically unchanged.
• Panel SFA now aligns the firm/time index with rows dropped by na.action, and uses the same BFGS-to-Nelder-Mead fallback cascade as the cross-sectional path.
• DEA: removed an erroneous lower bound (phi >= 1) in the output-oriented LPs that made cross-period evaluation of super-efficient DMUs infeasible even under CRS, silently biasing Malmquist TC and MPI means. Cross-period scores with phi < 1 now solve correctly.
• as_metafrontier_model() is now idempotent (converting an already converted object is a no-op), so the previously documented pre-conversion workflow works.
• poolability_test() now derives data.name from the passed expression instead of deparsing the stored call.
• autoplot(boot) (and the base plot() method) now actually draw the dashed CI bound lines promised by the documentation.
• autoplot(malm, which = "mpi_trend") now plots each transition at its end period (axis 2..T), consistent with the caption convention "change relative to the previous period".
• Removed a redundant marginal likelihood computation in the latent class EM loop.

New features

• estimator = c("bc88", "jlms") on metafrontier() and malmquist_meta() (see Breaking changes).
• objective = c("lp", "qp") on metafrontier(): both identification criteria of Battese, Rao and O'Donnell (2004) for the deterministic metafrontier. The default LP minimises the sum of absolute deviations (O'Donnell, Rao and Battese, 2008, Eqs. 23-25); the QP minimises the sum of squared deviations, solved exactly via quadprog (new in Suggests) with a constrOptim() barrier fallback. The bootstrap respects the choice.
• engine = c("internal", "sfaR", "frontier", "Benchmarking") on metafrontier(): delegate group-frontier estimation to external packages; engine = "Benchmarking" also delegates the pooled DEA metafrontier via XREF/YREF.
• check_convergence(): new exported diagnostic reporting one row per estimation stage (group frontiers and metafrontier). print() and summary() methods now include convergence status; every stage warns on non-zero optimiser codes.
• malmquist_meta(id = ): explicit firm matching across periods, with errors on duplicated (id, period) pairs, warnings counting dropped observations on unbalanced panels, and a message when falling back to positional matching. Cross-period infeasible programmes (possible under vrs/drs/irs/fdh) are now counted and reported in a consolidated warning, stored as n_infeasible, and shown by print()/summary(); the SFA path announces its pointwise-maximum approximation in a message and in the documentation.
• Unbalanced panels are fully supported in BC92 (firm-specific period sets), and simulate_panel_metafrontier() gains an attrition argument for generating unbalanced test panels.
• DEA: rts = "fdh" (free disposable hull; exact enumeration for radial measures, binary MIP for DDF), type = "hyperbolic" (graph efficiency; closed form under CRS, bisection otherwise; always feasible cross-period), user-supplied direction vectors or firm-specific direction matrices for DDF (reported via the additive ddf_gap), and slack = TRUE two-stage slack maximisation.
• poolability_test() now dispatches a permutation test for DEA fits (group labels exchangeable under the pooled-technology null), with B and seed arguments.
• coef(), vcov(), and summary() expose all estimated parameters: extraPar = TRUE returns variance parameters (and eta for BC92) with back-transformed values; vcov(which = "group") returns full per-group covariance matrices; group summary tables now include the variance parameters and eta with standard errors.
• simulate_metafrontier() gains beta_groups (group-specific slope coefficients; the true TGR is then computed against the pointwise maximum over group frontiers and varies within groups), input_means (group-specific input distributions), and input_corr (correlated log inputs).
• Latent class: the EM now records whether the best start met the convergence tolerance (em_converged) and warns when it did not.

metafrontier 0.2.2

Bug fixes

• boot_tgr(): Fixed orientation/rts not propagating to bootstrap replicates (always defaulted to output/CRS).
• boot_tgr(): Fixed hardcoded group column name; now respects the user's original group variable.
• Latent class .loglik_to_u_hat() now respects the dist argument with correct JLMS formulas for half-normal, truncated-normal, and exponential distributions.
• Latent class parameter count now correct for truncated-normal (k+3 instead of k+2).
• Fixed two incorrect DOIs: Huang et al. (2014) and O'Donnell et al. (2008) now link to the correct papers.

Improvements

• autoplot methods now use proper conditional S3 registration (@exportS3Method ggplot2::autoplot) instead of direct export().
• .extract_benchmarking() now attempts to retrieve XREF/YREF from Farrell objects and .estimate_from_models() gives a clear error when DEA models lack the required X/y/beta.
• technology_gap_ratio() documentation now correctly notes that TGR can exceed 1 under the stochastic metafrontier.
• Pre-built vignettes included in inst/doc for reliable R CMD check.

metafrontier 0.2.1

New features

Panel SFA and time-varying inefficiency

• simulate_panel_metafrontier(): Simulate balanced panel data with time-varying inefficiency and technical change for Monte Carlo studies.
• Panel SFA estimation with BC92 (Battese & Coelli, 1992) and BC95 (Battese & Coelli, 1995) specifications, supporting time-varying inefficiency via the eta parameter.

Bootstrap inference

• boot_tgr(): Bootstrap confidence intervals for technology gap ratios. Supports parametric (residual resampling) and nonparametric (case resampling) approaches, with percentile and BCa interval types.
• Parallel bootstrap via ncores argument using the parallel package.
• S3 methods: print, confint, and plot for boot_tgr objects.

Latent class metafrontier

• latent_class_metafrontier(): EM algorithm for estimating latent class stochastic frontier models within the metafrontier framework.
• select_n_classes(): Automatic selection of the optimal number of latent classes via BIC.
• S3 methods: print, summary, coef, and efficiencies for lc_metafrontier objects.

ggplot2 visualisation

• autoplot.metafrontier(): Visualise TGR distributions and efficiency decompositions using ggplot2.
• autoplot.malmquist_meta(): Plot Malmquist index components over time.
• autoplot.boot_tgr(): Visualise bootstrap distributions and confidence intervals for TGR.

Model interoperability

• as_metafrontier_model(): Convert pre-fitted model objects from sfaR (sfacross), frontier (sfa), and Benchmarking (Farrell) packages into metafrontier-compatible format.

Directional distance functions

• DEA-based metafrontier now supports directional distance functions alongside radial efficiency measurement.

Improvements

• Murphy-Topel variance correction: Stochastic metafrontier standard errors now account for the generated-regressor problem using the Murphy & Topel (1985) correction with PSD enforcement.
• Improved numerical stability: Safe Mills ratio computation (.safe_mills()) prevents NaN in extreme tail regions for both panel SFA and latent class models.
• Memory-efficient latent class: EM algorithm uses lm.wfit() for weighted regression instead of explicit diagonal weight matrices.
• DEA batch solver: LP object reuse in dea_batch_fast() for improved performance on large datasets.
• Input validation: boot_tgr() now validates R >= 1 with an informative error message.
• All summary() methods now return invisible S3 objects for programmatic access, with print() methods for display.
• Expanded @examples on all exported S3 methods and autoplot functions.
• Three vignettes added: introduction, Malmquist, and methods.

Documentation

• Comprehensive vignette: "Introduction to metafrontier" covering estimation, inference, bootstrap CIs, panel SFA, latent class, and directional distance functions.
• Vignette: "Metafrontier Malmquist Productivity Index" with worked panel data examples and identity verification.
• Vignette: "Metafrontier Methods: Theory and Computation" with mathematical details and Monte Carlo comparisons.
• Expanded @param documentation for control argument in metafrontier() with optim options.

metafrontier 0.1.0

Initial CRAN release.

Estimation

• metafrontier(): Main estimation function supporting SFA- and DEA-based metafrontiers.
• Deterministic metafrontier (Battese, Rao, and O'Donnell, 2004) via constrained optimisation.
• Stochastic metafrontier (Huang, Huang, and Liu, 2014) via second-stage SFA.
• DEA-based metafrontier with CRS, VRS, DRS, and IRS technology assumptions.
• Half-normal, truncated-normal, and exponential inefficiency distributions.
• Heteroscedastic SFA via formula interface (y ~ x1 + x2 | z1 + z2).
• Pre-fitted model support via models= argument (sfaR, frontier).

Productivity

• malmquist_meta(): Metafrontier Malmquist TFP index with three-way decomposition (TEC x TGC x TC*) following O'Donnell, Rao, and Battese (2008).

Analysis tools

• technology_gap_ratio() / tgr_summary(): Extract and summarise TGR by group.
• efficiencies(): Extract group, metafrontier, and TGR efficiency scores.
• poolability_test(): Likelihood ratio test for technology heterogeneity.
• simulate_metafrontier(): Data-generating process for Monte Carlo studies.

Methods

• Full S3 method suite: print, summary, coef, vcov, logLik, fitted, residuals, nobs, plot, confint, predict.
• Four built-in plot types: TGR distributions, efficiency scatter, decomposition, and frontier comparison.

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