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Latent Class SFA Metafrontier (groupType = "sfalcmcross")

Source: vignettes/sfalcmcross-metafrontier.Rmd
sfalcmcross-metafrontier.Rmd

Overview

In many applications, the technology groups that firms belong to are unobserved — we cannot directly observe which firms operate under which technology type. The latent class model (LCM) addresses this by:

  1. Fitting a pooled latent class SFA on the entire dataset, simultaneously estimating class-specific frontier parameters and class membership probabilities for each firm.
  2. Assigning each firm to a class based on its highest posterior class probability.
  3. Using these assignments as the technology groups for the metafrontier.

This approach is appropriate when: - No observed group variable is available. - Technology heterogeneity is suspected but not directly measurable. - A priori group boundaries are unclear or arbitrary.

Data Preparation

We use the utility dataset from sfaR, which contains 791 observations from US electricity utilities. We estimate a cost frontier with no explicit group variable.

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

Method 1: LCM + LP Metafrontier

Fit a 2-class latent class pooled SFA, then estimate the LP deterministic envelope over the inferred class frontiers.

meta_lcm_lp <- sfametafrontier(
  formula    = log(tc/wf) ~ log(y) + log(wl/wf) + log(wk/wf),
  data       = utility,
  S          = -1,          # cost frontier (S = -1)
  groupType  = "sfalcmcross",
  lcmClasses = 2,           # number of latent classes
  metaMethod = "lp"
)
summary(meta_lcm_lp)

Note: The group argument is not needed when groupType = "sfalcmcross" — the latent classes are identified automatically by the LCM. The lcmClasses argument controls the number of classes.

Method 2: LCM + QP Metafrontier 

meta_lcm_qp <- sfametafrontier(
  formula    = log(tc/wf) ~ log(y) + log(wl/wf) + log(wk/wf),
  data       = utility,
  S          = -1,
  groupType  = "sfalcmcross",
  lcmClasses = 2,
  metaMethod = "qp"
)
summary(meta_lcm_qp)

Method 3: LCM + SFA (Huang) 

meta_lcm_huang <- sfametafrontier(
  formula     = log(tc/wf) ~ log(y) + log(wl/wf) + log(wk/wf),
  data        = utility,
  S           = -1,
  groupType   = "sfalcmcross",
  lcmClasses  = 2,
  metaMethod  = "sfa",
  sfaApproach = "huang"
)
summary(meta_lcm_huang)

Method 4: LCM + SFA (O’Donnell) 

meta_lcm_odonnell <- sfametafrontier(
  formula     = log(tc/wf) ~ log(y) + log(wl/wf) + log(wk/wf),
  data        = utility,
  S           = -1,
  groupType   = "sfalcmcross",
  lcmClasses  = 2,
  metaMethod  = "sfa",
  sfaApproach = "ordonnell"
)
summary(meta_lcm_odonnell)

Choosing the Number of Classes

The number of latent classes (lcmClasses) should be guided by economic theory and information criteria. You can compare models with different numbers of classes:

meta_lcm_2 <- sfametafrontier(
  formula    = log(tc/wf) ~ log(y) + log(wl/wf) + log(wk/wf),
  data       = utility, S = -1,
  groupType  = "sfalcmcross", lcmClasses = 2, metaMethod = "lp"
)
meta_lcm_3 <- sfametafrontier(
  formula    = log(tc/wf) ~ log(y) + log(wl/wf) + log(wk/wf),
  data       = utility, S = -1,
  groupType  = "sfalcmcross", lcmClasses = 3, metaMethod = "lp"
)
# Compare information criteria
ic(meta_lcm_2)
ic(meta_lcm_3)

Prefer the model with the lower AIC/BIC.

Extracting Efficiencies and Posterior Probabilities

For LCM models, efficiencies() returns extra columns for posterior class membership probabilities, which can be used for robustness checks or classification:

eff_lcm <- efficiencies(meta_lcm_lp)
head(eff_lcm)

# Key LCM-specific columns:
# Group_c          — most likely class assignment
# PosteriorProb_c  — posterior probability of assigned class
# PosteriorProb_c1 — posterior probability of Class 1
# PosteriorProb_c2 — posterior probability of Class 2

Class membership summary 

# Proportion assigned to each class and mean posterior probability
with(eff_lcm, table(Group_c)) / nrow(eff_lcm) * 100   # % in each class

Key Difference from sfacross

Feature sfacross sfalcmcross
Group variable Required Not required
Group estimation Separate SFA per group Pooled LCM simultaneously
Output Group-level SFA summaries Pooled LCM summary with class-specific parameters
Extra efficiency columns Confidence bounds Posterior class probabilities