Getting Started with smfa

What is smfa?

smfa is an R package for implementing metafrontier analysis — a framework for productivity and performance benchmarking of firms (or farms, utilities, hospitals, etc.) that operate under different technologies. This is also referred to as “heterogeneous technology” analysis.

The classic stochastic frontier analysis (SFA) estimates a single frontier for all firms. Metafrontier analysis extends this by:

Estimating group-specific frontiers for each technology sub-group.
Estimating a common metafrontier that envelops all group frontiers.
Computing metatechnology ratios (MTR), which quantify how far each group’s frontier lies below the best-practice metafrontier.

This allows researchers to disentangle:

Technical efficiency relative to the group’s own frontier (TE_group).
Metafrontier efficiency relative to the common best-practice frontier (TE_meta).
Metatechnology ratio (MTR = TE_meta / TE_group), which captures how technologically advanced a group’s production possibilities are.

Conceptual Framework

\[\text{MTR}_i = \frac{\text{TE\_meta}_i}{\text{TE\_group}_i}\]

A MTR close to 1 means the group’s technology is near the metafrontier (advanced technology). A MTR far below 1 means the group operates under a less advanced technology.

Methods Supported

smfa supports four metafrontier estimation methods (metaMethod):

Method	Description
`"lp"`	Linear programming deterministic envelope — Battese, Rao & O’Donnell (2004)
`"qp"`	Quadratic programming deterministic envelope
`"sfa"` (`sfaApproach = "huang"`)	Two-stage stochastic metafrontier — Huang, Huang & Liu (2014)
`"sfa"` (`sfaApproach = "ordonnell"`)	Two-stage SFA on LP envelope — O’Donnell, Rao & Battese (2008)

And three group frontier types (groupType):

Group Type	When to Use
`"sfacross"`	Technology groups are observed (e.g., a group variable exists)
`"sfalcmcross"`	Technology groups are unobserved — latent class model identifies them (Dakpo et al. 2021)
`"sfaselectioncross"`	Sample selection bias is present (Dakpo et al. 2022)

Installation

# Install smfa from CRAN
install.packages("smfa")

# Or install the development version from GitHub
# if (!require("devtools")) install.packages("devtools")
# devtools::install_github("SulmanOlieko/smfa")

Note: sfaR is automatically installed as a dependency — you do not need to install it separately.

Quick-Start Example

This minimal example demonstrates the core workflow using the ricephil dataset from the sfaR package, with three Filipino rice farm-size groups (small, medium, large).

Step 1: Load data and create groups

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

# Create technology groups based on farm area terciles
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

Step 2: Fit the metafrontier model

meta_lp <- smfa(
  formula    = log(PROD) ~ log(AREA) + log(LABOR) + log(NPK),
  data       = ricephil,
  group      = "group",
  S          = 1, # production frontier (S=1) or cost frontier (S=-1)
  udist      = "hnormal",
  groupType  = "sfacross",
  metaMethod = "lp"
)

Step 3: Summarise results

summary(meta_lp)
#> ============================================================ 
#> Stochastic Metafrontier Analysis
#> Metafrontier method: Linear Programming (LP) Metafrontier 
#> Stochastic Production/Profit Frontier, e = v - u 
#> Group approach     : Stochastic Frontier Analysis 
#> Group estimator    : sfacross 
#> Group optim solver : BFGS maximization 
#> Groups ( 3 ): small, medium, large 
#> Total observations : 344 
#> Distribution       : hnormal 
#> ============================================================ 
#> 
#> ------------------------------------------------------------ 
#> Group: small (N = 125)  Log-likelihood: -50.98578
#> ------------------------------------------------------------ 
#> -------------------------------------------------------------------------------- 
#> Normal-Half Normal SF Model 
#> Dependent Variable:                                                    log(PROD) 
#> Log likelihood solver:                                         BFGS maximization 
#> Log likelihood iter:                                                          42 
#> Log likelihood value:                                                  -50.98578 
#> Log likelihood gradient norm:                                        9.40653e-06 
#> Estimation based on:                                         N =  125 and K =  6 
#> Inf. Cr:                                           AIC  =  114.0 AIC/N  =  0.912 
#>                                                    BIC  =  130.9 BIC/N  =  1.048 
#>                                                    HQIC =  120.9 HQIC/N =  0.967 
#> -------------------------------------------------------------------------------- 
#> Variances: Sigma-squared(v)   =                                          0.05318 
#>            Sigma(v)           =                                          0.05318 
#>            Sigma-squared(u)   =                                          0.23435 
#>            Sigma(u)           =                                          0.23435 
#> Sigma = Sqrt[(s^2(u)+s^2(v))] =                                          0.53622 
#> Gamma = sigma(u)^2/sigma^2    =                                          0.81504 
#> Lambda = sigma(u)/sigma(v)    =                                          2.09921 
#> Var[u]/{Var[u]+Var[v]}        =                                          0.61558 
#> -------------------------------------------------------------------------------- 
#> Average inefficiency E[ui]     =                                         0.38626 
#> Average efficiency E[exp(-ui)] =                                         0.70643 
#> -------------------------------------------------------------------------------- 
#> Stochastic Production/Profit Frontier, e = v - u 
#> -----[ Tests vs. No Inefficiency ]-----
#> Likelihood Ratio Test of Inefficiency
#> Deg. freedom for inefficiency model                                            1 
#> Log Likelihood for OLS Log(H0) =                                       -54.80277 
#> LR statistic:  
#> Chisq = 2*[LogL(H0)-LogL(H1)]  =                                         7.63398 
#> Kodde-Palm C*:       95%: 2.70554                                   99%: 5.41189 
#> Coelli (1995) skewness test on OLS residuals
#> M3T: z                         =                                        -3.57676 
#> M3T: p.value                   =                                         0.00035 
#> Final maximum likelihood estimates 
#> -------------------------------------------------------------------------------- 
#>                          Deterministic Component of SFA 
#> -------------------------------------------------------------------------------- 
#>                Coefficient Std. Error z value  Pr(>|z|)    
#> (Intercept)       -1.58745    0.51274 -3.0960  0.001962 ** 
#> log(AREA)          0.24014    0.11834  2.0292  0.042440 *  
#> log(LABOR)         0.43464    0.12292  3.5361  0.000406 ***
#> log(NPK)           0.30516    0.05701  5.3523 8.682e-08 ***
#> -------------------------------------------------------------------------------- 
#>                   Parameter in variance of u (one-sided error) 
#> -------------------------------------------------------------------------------- 
#>                Coefficient Std. Error z value  Pr(>|z|)    
#> Zu_(Intercept)    -1.45093    0.29867  -4.858 1.186e-06 ***
#> -------------------------------------------------------------------------------- 
#>                  Parameters in variance of v (two-sided error) 
#> -------------------------------------------------------------------------------- 
#>                Coefficient Std. Error z value  Pr(>|z|)    
#> Zv_(Intercept)    -2.93406    0.35401  -8.288 < 2.2e-16 ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> -------------------------------------------------------------------------------- 
#> Model was estimated on : Apr Fri 24, 2026 at 17:25 
#> Log likelihood status: successful convergence  
#> --------------------------------------------------------------------------------  
#> 
#> ------------------------------------------------------------ 
#> Group: medium (N = 104)  Log-likelihood: -15.28164
#> ------------------------------------------------------------ 
#> -------------------------------------------------------------------------------- 
#> Normal-Half Normal SF Model 
#> Dependent Variable:                                                    log(PROD) 
#> Log likelihood solver:                                         BFGS maximization 
#> Log likelihood iter:                                                          41 
#> Log likelihood value:                                                  -15.28164 
#> Log likelihood gradient norm:                                        3.83566e-05 
#> Estimation based on:                                         N =  104 and K =  6 
#> Inf. Cr:                                            AIC  =  42.6 AIC/N  =  0.409 
#>                                                     BIC  =  58.4 BIC/N  =  0.562 
#>                                                     HQIC =  49.0 HQIC/N =  0.471 
#> -------------------------------------------------------------------------------- 
#> Variances: Sigma-squared(v)   =                                          0.01058 
#>            Sigma(v)           =                                          0.01058 
#>            Sigma-squared(u)   =                                          0.22010 
#>            Sigma(u)           =                                          0.22010 
#> Sigma = Sqrt[(s^2(u)+s^2(v))] =                                          0.48030 
#> Gamma = sigma(u)^2/sigma^2    =                                          0.95412 
#> Lambda = sigma(u)/sigma(v)    =                                          4.56034 
#> Var[u]/{Var[u]+Var[v]}        =                                          0.88314 
#> -------------------------------------------------------------------------------- 
#> Average inefficiency E[ui]     =                                         0.37433 
#> Average efficiency E[exp(-ui)] =                                         0.71330 
#> -------------------------------------------------------------------------------- 
#> Stochastic Production/Profit Frontier, e = v - u 
#> -----[ Tests vs. No Inefficiency ]-----
#> Likelihood Ratio Test of Inefficiency
#> Deg. freedom for inefficiency model                                            1 
#> Log Likelihood for OLS Log(H0) =                                       -21.11323 
#> LR statistic:  
#> Chisq = 2*[LogL(H0)-LogL(H1)]  =                                        11.66318 
#> Kodde-Palm C*:       95%: 2.70554                                   99%: 5.41189 
#> Coelli (1995) skewness test on OLS residuals
#> M3T: z                         =                                        -2.91021 
#> M3T: p.value                   =                                         0.00361 
#> Final maximum likelihood estimates 
#> -------------------------------------------------------------------------------- 
#>                          Deterministic Component of SFA 
#> -------------------------------------------------------------------------------- 
#>                Coefficient Std. Error z value  Pr(>|z|)    
#> (Intercept)       -0.08182    0.50668 -0.1615 0.8717190    
#> log(AREA)          0.47410    0.13984  3.3903 0.0006981 ***
#> log(LABOR)         0.17935    0.10201  1.7581 0.0787310 .  
#> log(NPK)           0.20255    0.08130  2.4913 0.0127289 *  
#> -------------------------------------------------------------------------------- 
#>                   Parameter in variance of u (one-sided error) 
#> -------------------------------------------------------------------------------- 
#>                Coefficient Std. Error z value  Pr(>|z|)    
#> Zu_(Intercept)    -1.51367    0.23549 -6.4276 1.296e-10 ***
#> -------------------------------------------------------------------------------- 
#>                  Parameters in variance of v (two-sided error) 
#> -------------------------------------------------------------------------------- 
#>                Coefficient Std. Error z value  Pr(>|z|)    
#> Zv_(Intercept)    -4.54846    0.76429 -5.9512 2.661e-09 ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> -------------------------------------------------------------------------------- 
#> Model was estimated on : Apr Fri 24, 2026 at 17:25 
#> Log likelihood status: successful convergence  
#> --------------------------------------------------------------------------------  
#> 
#> ------------------------------------------------------------ 
#> Group: large (N = 115)  Log-likelihood: -8.02197
#> ------------------------------------------------------------ 
#> -------------------------------------------------------------------------------- 
#> Normal-Half Normal SF Model 
#> Dependent Variable:                                                    log(PROD) 
#> Log likelihood solver:                                         BFGS maximization 
#> Log likelihood iter:                                                          68 
#> Log likelihood value:                                                   -8.02197 
#> Log likelihood gradient norm:                                        4.01301e-05 
#> Estimation based on:                                         N =  115 and K =  6 
#> Inf. Cr:                                            AIC  =  28.0 AIC/N  =  0.244 
#>                                                     BIC  =  44.5 BIC/N  =  0.387 
#>                                                     HQIC =  34.7 HQIC/N =  0.302 
#> -------------------------------------------------------------------------------- 
#> Variances: Sigma-squared(v)   =                                          0.01399 
#>            Sigma(v)           =                                          0.01399 
#>            Sigma-squared(u)   =                                          0.16751 
#>            Sigma(u)           =                                          0.16751 
#> Sigma = Sqrt[(s^2(u)+s^2(v))] =                                          0.42602 
#> Gamma = sigma(u)^2/sigma^2    =                                          0.92293 
#> Lambda = sigma(u)/sigma(v)    =                                          3.46063 
#> Var[u]/{Var[u]+Var[v]}        =                                          0.81315 
#> -------------------------------------------------------------------------------- 
#> Average inefficiency E[ui]     =                                         0.32656 
#> Average efficiency E[exp(-ui)] =                                         0.74195 
#> -------------------------------------------------------------------------------- 
#> Stochastic Production/Profit Frontier, e = v - u 
#> -----[ Tests vs. No Inefficiency ]-----
#> Likelihood Ratio Test of Inefficiency
#> Deg. freedom for inefficiency model                                            1 
#> Log Likelihood for OLS Log(H0) =                                       -16.96836 
#> LR statistic:  
#> Chisq = 2*[LogL(H0)-LogL(H1)]  =                                        17.89279 
#> Kodde-Palm C*:       95%: 2.70554                                   99%: 5.41189 
#> Coelli (1995) skewness test on OLS residuals
#> M3T: z                         =                                        -4.12175 
#> M3T: p.value                   =                                         0.00004 
#> Final maximum likelihood estimates 
#> -------------------------------------------------------------------------------- 
#>                          Deterministic Component of SFA 
#> -------------------------------------------------------------------------------- 
#>                Coefficient Std. Error z value  Pr(>|z|)    
#> (Intercept)       -1.31194    0.41859 -3.1342 0.0017234 ** 
#> log(AREA)          0.38278    0.14297  2.6772 0.0074236 ** 
#> log(LABOR)         0.42105    0.10992  3.8303 0.0001280 ***
#> log(NPK)           0.23143    0.06065  3.8160 0.0001356 ***
#> -------------------------------------------------------------------------------- 
#>                   Parameter in variance of u (one-sided error) 
#> -------------------------------------------------------------------------------- 
#>                Coefficient Std. Error z value  Pr(>|z|)    
#> Zu_(Intercept)    -1.78673    0.20176 -8.8555 < 2.2e-16 ***
#> -------------------------------------------------------------------------------- 
#>                  Parameters in variance of v (two-sided error) 
#> -------------------------------------------------------------------------------- 
#>                Coefficient Std. Error z value  Pr(>|z|)    
#> Zv_(Intercept)    -4.26963    0.40584 -10.521 < 2.2e-16 ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> -------------------------------------------------------------------------------- 
#> Model was estimated on : Apr Fri 24, 2026 at 17:25 
#> Log likelihood status: successful convergence  
#> --------------------------------------------------------------------------------  
#> 
#> ------------------------------------------------------------ 
#> Metafrontier Coefficients (lp):
#>   (LP: deterministic envelope - no estimated parameters)
#> 
#> ------------------------------------------------------------ 
#> Efficiency Statistics (group means):
#> ------------------------------------------------------------ 
#>        N_obs N_valid TE_group_BC TE_group_JLMS TE_meta_BC TE_meta_JLMS  MTR_BC
#> small    125     125     0.71065       0.70090    0.64126      0.63244 0.89981
#> medium   104     104     0.71253       0.70965    0.68204      0.67929 0.95597
#> large    115     115     0.74772       0.74406    0.72186      0.71834 0.96521
#>        MTR_JLMS
#> small   0.89981
#> medium  0.95597
#> large   0.96521
#> 
#> Overall:
#> TE_group_BC=0.7236  TE_group_JLMS=0.7182
#> TE_meta_BC=0.6817   TE_meta_JLMS=0.6767
#> MTR_BC=0.9403     MTR_JLMS=0.9403
#> ------------------------------------------------------------ 
#> Total Log-likelihood: -74.28939 
#> AIC: 184.5788   BIC: 253.7103   HQIC: 212.113 
#> ------------------------------------------------------------ 
#> Model was estimated on : Apr Fri 24, 2026 at 17:25

Step 4: Extract firm-level efficiencies

eff <- efficiencies(meta_lp)
head(eff)
#>   id  group       u_g TE_group_JLMS TE_group_BC TE_group_BC_reciprocal
#> 1  1 medium 0.2697165     0.7635959   0.7673345               1.316036
#> 2  2  large 0.3515642     0.7035867   0.7080897               1.430406
#> 3  3  large 0.2774565     0.7577085   0.7623358               1.327899
#> 4  4 medium 0.1710417     0.8427864   0.8461331               1.191355
#> 5  5  large 0.2119629     0.8089947   0.8133556               1.242901
#> 6  6  small 0.1987499     0.8197549   0.8275685               1.232467
#>         uLB_g     uUB_g        m_g TE_group_mode  teBCLB_g  teBCUB_g    u_meta
#> 1 0.077581942 0.4657010 0.26858570     0.7644599 0.6276949 0.9253512 0.3944439
#> 2 0.130356248 0.5739174 0.35118207     0.7038556 0.5633144 0.8777827 0.3779836
#> 3 0.065447909 0.4980807 0.27501606     0.7595599 0.6076959 0.9366478 0.3049531
#> 4 0.018022507 0.3583190 0.15885675     0.8531186 0.6988501 0.9821389 0.1710417
#> 5 0.027125654 0.4268531 0.20231520     0.8168374 0.6525594 0.9732389 0.2379271
#> 6 0.009050601 0.5251973 0.07998025     0.9231346 0.5914386 0.9909902 0.3295263
#>   TE_meta_JLMS TE_meta_BC  MTR_JLMS    MTR_BC
#> 1    0.6740548  0.6773549 0.8827375 0.8827375
#> 2    0.6852418  0.6896274 0.9739266 0.9739266
#> 3    0.7371580  0.7416598 0.9728780 0.9728780
#> 4    0.8427864  0.8461331 1.0000000 1.0000000
#> 5    0.7882601  0.7925093 0.9743700 0.9743700
#> 6    0.7192644  0.7261201 0.8774139 0.8774139

Key output columns:

Column	Description
`TE_group_BC`	Group TE (Battese & Coelli 1988)
`TE_meta_BC`	Metafrontier TE
`MTR_BC`	Metatechnology ratio

What Next?

Standard SFA groups → see vignette("sfacross-metafrontier")
Unobserved/latent groups → see vignette("sfalcmcross-metafrontier")
Sample selection → see vignette("sfaselectioncross-metafrontier")
Extracting all outputs → see vignette("efficiency-extraction")