Standard SFA Metafrontier (groupType = "sfacross") • metafrontieR

Overview

When observed group labels are available (e.g., farm size, region, ownership type), groupType = "sfacross" fits a separate cross-sectional stochastic frontier for each group using sfaR::sfacross(). The group-specific results are then used to estimate the common metafrontier using any of four methods.

Data Preparation

We use the ricephil dataset from sfaR, which contains 344 Filipino rice farms. We create three technology groups based on farm area terciles.

library(metafrontieR)
data("ricephil", package = "sfaR")

ricephil$group <- cut(
  ricephil$AREA,
  breaks        = quantile(ricephil$AREA, probs = c(0, 1/3, 2/3, 1), na.rm = TRUE),
  labels        = c("small", "medium", "large"),
  include.lowest = TRUE
)

table(ricephil$group)
#>  small medium  large
#>    125    104    115

Method 1: LP Metafrontier

The linear programming (LP) envelope minimises the sum of absolute deviations from group frontier predictions while satisfying a convexity constraint. No stochastic parameters are estimated for the metafrontier itself.

meta_lp <- sfametafrontier(
  formula    = log(PROD) ~ log(AREA) + log(LABOR) + log(NPK),
  data       = ricephil,
  group      = "group",
  S          = 1,
  udist      = "hnormal",
  groupType  = "sfacross",
  metaMethod = "lp"
)
summary(meta_lp)

Note: Since the LP metafrontier is estimated via linear programming, no estimated parameters are returned for the metafrontier level. The LP envelope is fully determined by the group frontier predictions.

Method 2: QP Metafrontier

The quadratic programming (QP) envelope minimises the sum of squared deviations from group frontier predictions. Unlike LP, QP produces a smooth envelope that is differentiable everywhere, and it returns estimated coefficients with standard errors.

meta_qp <- sfametafrontier(
  formula    = log(PROD) ~ log(AREA) + log(LABOR) + log(NPK),
  data       = ricephil,
  group      = "group",
  S          = 1,
  udist      = "hnormal",
  groupType  = "sfacross",
  metaMethod = "qp"
)
summary(meta_qp)

Method 3: Stochastic Metafrontier — Huang et al. (2014)

The two-stage stochastic metafrontier of Huang, Huang & Liu (2014) uses the group-specific fitted frontier values as the dependent variable in a second-stage pooled SFA. The technology gap U and noise V are estimated stochastically, which naturally bounds the metatechnology ratio MTR ∈ (0, 1].

meta_huang <- sfametafrontier(
  formula     = log(PROD) ~ log(AREA) + log(LABOR) + log(NPK),
  data        = ricephil,
  group       = "group",
  S           = 1,
  udist       = "hnormal",
  groupType   = "sfacross",
  metaMethod  = "sfa",
  sfaApproach = "huang"
)
summary(meta_huang)

Method 4: Stochastic Envelope — O’Donnell et al. (2008)

The O’Donnell et al. (2008) approach uses the LP deterministic envelope as the dependent variable in the second-stage SFA, rather than the group-specific fitted values. This mixed deterministic–stochastic approach embeds the envelope within an SFA framework.

Warning: MTR values > 1 can arise with this method when the second-stage SFA estimates near-zero inefficiency. If this occurs, consider using metaMethod = "lp" or sfaApproach = "huang" instead.

meta_ordonnell <- sfametafrontier(
  formula     = log(PROD) ~ log(AREA) + log(LABOR) + log(NPK),
  data        = ricephil,
  group       = "group",
  S           = 1,
  udist       = "hnormal",
  groupType   = "sfacross",
  metaMethod  = "sfa",
  sfaApproach = "ordonnell"
)
summary(meta_ordonnell)

Comparing Methods

All four methods use identical group-level estimates. The differences arise only in how the metafrontier is computed:

Method	Metafrontier Coefficients Returned	MTR Bounded ≤ 1?
LP	No (envelope rule)	Yes
QP	Yes (with SE)	Yes
SFA (huang)	Yes (with SE)	Yes
SFA (ordonnell)	Yes (with SE)	Not guaranteed

Extracting Efficiencies

All models return firm-level efficiency estimates via efficiencies():

eff <- efficiencies(meta_lp)
head(eff)

# Subset for a specific group
eff_small <- eff[eff$group == "small", ]
summary(eff_small[, c("TE_group_BC", "TE_meta_BC", "MTR_BC")])

Other Extractors

coef(meta_qp)          # metafrontier coefficients
vcov(meta_qp)          # variance-covariance matrix
logLik(meta_lp)        # log-likelihood
ic(meta_lp)            # AIC, BIC, HQIC
nobs(meta_lp)          # number of observations
fitted(meta_lp)        # fitted values
residuals(meta_lp)     # residuals