Stochastic Metafrontier Analysis
An R package for implementing various deterministic and stochastic metafrontier analyses for efficiency and performance benchmarking, assessing technical efficiencies (TE), metafrontier technical efficiencies (MTE), and computing metatechnology ratios (MTRs) for firms operating under different technologies.
smfa provides routines for:
- Deterministic envelope metafrontier via linear programming (LP) and quadratic programming (QP), following Battese, Rao & O'Donnell (2004) and O'Donnell, Rao & Battese (2008).
- Stochastic two-stage metafrontier following Huang, Huang & Liu (2014).
In addition, the package implements:
- Latent class stochastic metafrontier analysis — when technology groups are unobserved, the latent class model (LCM) robustly identifies classes and routes them to the metafrontier for benchmarking following Greene and Hensher (2003), Orea and Kumbhakar (2004), Greene (2005), Parmeter and Kumbhakar (2014).
- Sample selection correction metafrontier models — corrects for sample selection bias following Heckman (1979), Greene (2010), Greene (2003).
Dependency:
smfadepends on thesfaRpackage by Dakpo, Desjeux & Latruffe (2023), which provides the underlying stochastic frontier estimation routines for all group-level models.
# Install devtools if not already installed
if (!require("devtools")) install.packages("devtools")
# Install smfa from GitHub
devtools::install_github("SulmanOlieko/smfa")Note You do not need to install
sfaRmanually,smfatakes care of that automatically.
The following sections provide comprehensive examples covering all three group-level model types and all four metafrontier methods. Each section demonstrates the full workflow: data preparation → group frontier estimation → metafrontier → efficiency and MTR extraction.
Let's use the ricephil data from sfaR. In this data, group boundaries are observed (a farm-size variable). If we assume that the production technology varies by farm size, we can try to estimate three frontiers that correspond to three types of farm sizes, namely small, medium and large. We can create a group variable group that captures these groups. We can then estimate each group's frontier separately using sfacross from the sfaR package. So, we will specify the option groupType = "sfacross" in the smfa().
library(smfa)
data("ricephil", package = "sfaR")
# Create three 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)This is the distrubition of the various farm types:
small medium large
125 104 115
We can begin by estimating a deterministic linear programming envelope (Battese, Rao & O'Donnell, 2004) over the three group frontiers. The metafrontier parameter vector minimises the sum of absolute deviations while staying at or above all group frontier predictions. We will be using a Cobb-Douglas functional form with rice production PROD as the response variable, and AREA, LABOR and NPK as the inputs.
meta_sfacross_lp <- smfa(
formula = log(PROD) ~ log(AREA) + log(LABOR) + log(NPK),
data = ricephil,
group = "group",
S = 1,
udist = "hnormal",
groupType = "sfacross",
metaMethod = "lp"
)
summary(meta_sfacross_lp)Toggle to see the output
> meta_sfacross_lp <- smfa(
+ formula = log(PROD) ~ log(AREA) + log(LABOR) + log(NPK),
+ data = ricephil,
+ group = "group",
+ S = 1,
+ udist = "hnormal",
+ groupType = "sfacross",
+ metaMethod = "lp"
+ )
Estimating group-specific stochastic frontiers (sfacross) ...
Group: small
Group: medium
Group: large
Group frontiers estimated.
Estimating metafrontier using method: Linear Programming (LP) Metafrontier
> summary(meta_sfacross_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 : Mar Tue 03, 2026 at 00:27
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 : Mar Tue 03, 2026 at 00:27
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 : Mar Tue 03, 2026 at 00:27
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 MTR_JLMS
small 125 125 0.71065 0.70090 0.64126 0.63244 0.89981 0.89981
medium 104 104 0.71253 0.70965 0.68204 0.67929 0.95597 0.95597
large 115 115 0.74772 0.74406 0.72186 0.71834 0.96521 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 : Mar Tue 03, 2026 at 00:27
Note: Since the metafrontier is estimated via linear programming, no estimated parameters are returned.
To harvest individual efficiency, metafrontier efficiency and MTR estimates:
efficiencies(meta_sfacross_lp)
head(efficiencies(meta_sfacross_lp))
#To subset only for small farms
#head(efficiencies(meta_sfacross_lp)[efficiencies(meta_sfacross_lp)$group == "small", ])Toggle to see the output
> head(efficiencies(meta_sfacross_lp))
id group u_g TE_group_JLMS TE_group_BC TE_group_BC_reciprocal uLB_g uUB_g m_g TE_group_mode teBCLB_g teBCUB_g u_meta TE_meta_JLMS TE_meta_BC MTR_JLMS MTR_BC
1 1 medium 0.2697165 0.7635959 0.7673345 1.316036 0.077581942 0.4657010 0.26858570 0.7644599 0.6276949 0.9253512 0.3944439 0.6740548 0.6773549 0.8827375 0.8827375
2 2 large 0.3515642 0.7035867 0.7080897 1.430406 0.130356248 0.5739174 0.35118207 0.7038556 0.5633144 0.8777827 0.3779836 0.6852418 0.6896274 0.9739266 0.9739266
3 3 large 0.2774565 0.7577085 0.7623358 1.327899 0.065447909 0.4980807 0.27501606 0.7595599 0.6076959 0.9366478 0.3049531 0.7371580 0.7416598 0.9728780 0.9728780
4 4 medium 0.1710417 0.8427864 0.8461331 1.191355 0.018022507 0.3583190 0.15885675 0.8531186 0.6988501 0.9821389 0.1710417 0.8427864 0.8461331 1.0000000 1.0000000
5 5 large 0.2119629 0.8089947 0.8133556 1.242901 0.027125654 0.4268531 0.20231520 0.8168374 0.6525594 0.9732389 0.2379271 0.7882601 0.7925093 0.9743700 0.9743700
6 6 small 0.1987499 0.8197549 0.8275685 1.232467 0.009050601 0.5251973 0.07998025 0.9231346 0.5914386 0.9909902 0.3295263 0.7192644 0.7261201 0.8774139 0.8774139
We can also estimate a quadratic programming envelope that minimises the sum of squared deviations from group frontier predictions subject to the envelope constraint. We now switch to metaMethod = "qp".
meta_sfacross_qp <- smfa(
formula = log(PROD) ~ log(AREA) + log(LABOR) + log(NPK),
data = ricephil,
group = "group",
S = 1,
udist = "hnormal",
groupType = "sfacross",
metaMethod = "qp"
)
summary(meta_sfacross_qp)Toggle to see the output
> meta_sfacross_qp <- smfa(
+ formula = log(PROD) ~ log(AREA) + log(LABOR) + log(NPK),
+ data = ricephil,
+ group = "group",
+ S = 1,
+ udist = "hnormal",
+ groupType = "sfacross",
+ metaMethod = "qp"
+ )
Estimating group-specific stochastic frontiers (sfacross) ...
Group: small
Group: medium
Group: large
Group frontiers estimated.
Estimating metafrontier using method: Quadratic Programming (QP) Metafrontier
> summary(meta_sfacross_qp)
============================================================
Stochastic Metafrontier Analysis
Metafrontier method: Quadratic Programming (QP) 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 : Mar Tue 03, 2026 at 00:34
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 : Mar Tue 03, 2026 at 00:34
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 : Mar Tue 03, 2026 at 00:34
Log likelihood status: successful convergence
--------------------------------------------------------------------------------
------------------------------------------------------------
Metafrontier Coefficients (qp):
Estimate Std. Error z value Pr(>|z|)
(Intercept) -0.6117795 0.0291793 -20.966 < 2.2e-16 ***
log(AREA) 0.3937843 0.0073209 53.789 < 2.2e-16 ***
log(LABOR) 0.2791273 0.0077215 36.150 < 2.2e-16 ***
log(NPK) 0.2409454 0.0046846 51.434 < 2.2e-16 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
------------------------------------------------------------
Efficiency Statistics (group means):
------------------------------------------------------------
N_obs N_valid TE_group_BC TE_group_JLMS TE_meta_BC TE_meta_JLMS MTR_BC MTR_JLMS
small 125 125 0.71065 0.70090 0.64037 0.63156 0.89972 0.89972
medium 104 104 0.71253 0.70965 0.66998 0.66727 0.94053 0.94053
large 115 115 0.74772 0.74406 0.72290 0.71937 0.96676 0.96676
Overall:
TE_group_BC=0.7236 TE_group_JLMS=0.7182
TE_meta_BC=0.6777 TE_meta_JLMS=0.6727
MTR_BC=0.9357 MTR_JLMS=0.9357
------------------------------------------------------------
Total Log-likelihood: -74.28939
AIC: 192.5788 BIC: 277.0729 HQIC: 226.2318
------------------------------------------------------------
Model was estimated on : Mar Tue 03, 2026 at 00:34
As expected, the two approaches produce almost identical outputs.
Note: The estimation of the group frontiers use the same method,
sfacross, and the only difference is in how we compute the metafrontier. QP estimates a deterministic frontier, no stochastic variance parameters are returned.
In this approach, the group-specific fitted frontier values are pooled together and serve as the dependent variable in a second-stage pooled SFA. The technology gap U and noise V are estimated stochastically.
meta_sfacross_huang <- smfa(
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_sfacross_huang)Toggle to see the output
> meta_sfacross_huang <- smfa(
+ formula = log(PROD) ~ log(AREA) + log(LABOR) + log(NPK),
+ data = ricephil,
+ group = "group",
+ S = 1,
+ udist = "hnormal",
+ groupType = "sfacross",
+ metaMethod = "sfa",
+ sfaApproach = "huang"
+ )
Estimating group-specific stochastic frontiers (sfacross) ...
Group: small
Group: medium
Group: large
Group frontiers estimated.
Estimating metafrontier using method: SFA Metafrontier [Huang et al. (2014), two-stage]
Warning message:
The residuals of the OLS are right-skewed. This may indicate the absence of inefficiency or
model misspecification or sample 'bad luck'
> summary(meta_sfacross_huang)
============================================================
Stochastic Metafrontier Analysis
Metafrontier method: SFA Metafrontier [Huang et al. (2014), two-stage]
Stochastic Production/Profit Frontier, e = v - u
SFA approach : huang
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 : Mar Tue 03, 2026 at 00:36
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 : Mar Tue 03, 2026 at 00:36
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 : Mar Tue 03, 2026 at 00:36
Log likelihood status: successful convergence
--------------------------------------------------------------------------------
------------------------------------------------------------
Metafrontier Coefficients (sfa):
Meta-optim solver : BFGS maximization
Estimate Std. Error z value Pr(>|z|)
(Intercept) -1.0031443 0.0568874 -17.634 < 2.2e-16 ***
log(AREA) 0.3670206 0.0091533 40.097 < 2.2e-16 ***
log(LABOR) 0.3297853 0.0096542 34.160 < 2.2e-16 ***
log(NPK) 0.2648079 0.0058572 45.211 < 2.2e-16 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Meta-frontier model details:
--------------------------------------------------------------------------------
Normal-Half Normal SF Model
Dependent Variable: group_fitted_values
Log likelihood solver: BFGS maximization
Log likelihood iter: 582
Log likelihood value: 553.35240
Log likelihood gradient norm: 5.34979e-04
Estimation based on: N = 344 and K = 6
Inf. Cr: AIC = -1094.7 AIC/N = -3.182
BIC = -1071.7 BIC/N = -3.115
HQIC = -1085.5 HQIC/N = -3.156
--------------------------------------------------------------------------------
Variances: Sigma-squared(v) = 0.00235
Sigma(v) = 0.00235
Sigma-squared(u) = 0.00000
Sigma(u) = 0.00000
Sigma = Sqrt[(s^2(u)+s^2(v))] = 0.04844
Gamma = sigma(u)^2/sigma^2 = 0.00017
Lambda = sigma(u)/sigma(v) = 0.01294
Var[u]/{Var[u]+Var[v]} = 0.00006
--------------------------------------------------------------------------------
Average inefficiency E[ui] = 0.00050
Average efficiency E[exp(-ui)] = 0.99950
--------------------------------------------------------------------------------
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) = 553.35242
LR statistic:
Chisq = 2*[LogL(H0)-LogL(H1)] = -0.00003
Kodde-Palm C*: 95%: 2.70554 99%: 5.41189
Coelli (1995) skewness test on OLS residuals
M3T: z = 4.16139
M3T: p.value = 0.00003
Final maximum likelihood estimates
--------------------------------------------------------------------------------
Deterministic Component of SFA
--------------------------------------------------------------------------------
Coefficient Std. Error z value Pr(>|z|)
(Intercept) -1.00314 0.05689 -17.634 < 2.2e-16 ***
.X2 0.36702 0.00915 40.097 < 2.2e-16 ***
.X3 0.32979 0.00965 34.160 < 2.2e-16 ***
.X4 0.26481 0.00586 45.211 < 2.2e-16 ***
--------------------------------------------------------------------------------
Parameter in variance of u (one-sided error)
--------------------------------------------------------------------------------
Coefficient Std. Error z value Pr(>|z|)
Zu_(Intercept) -14.749 174.352 -0.0846 0.9326
--------------------------------------------------------------------------------
Parameters in variance of v (two-sided error)
--------------------------------------------------------------------------------
Coefficient Std. Error z value Pr(>|z|)
Zv_(Intercept) -6.05510 0.07698 -78.658 < 2.2e-16 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
--------------------------------------------------------------------------------
Model was estimated on : Mar Tue 03, 2026 at 00:36
Log likelihood status: successful convergence
--------------------------------------------------------------------------------
Log likelihood status: successful convergence
------------------------------------------------------------
Efficiency Statistics (group means):
------------------------------------------------------------
N_obs N_valid TE_group_BC TE_group_JLMS TE_meta_BC TE_meta_JLMS MTR_BC MTR_JLMS
small 125 125 0.71065 0.70090 0.71030 0.70055 0.99950 0.99950
medium 104 104 0.71253 0.70965 0.71217 0.70930 0.99950 0.99950
large 115 115 0.74772 0.74406 0.74734 0.74369 0.99950 0.99950
Overall:
TE_group_BC=0.7236 TE_group_JLMS=0.7182
TE_meta_BC=0.7233 TE_meta_JLMS=0.7178
MTR_BC=0.9995 MTR_JLMS=0.9995
------------------------------------------------------------
Total Log-likelihood: 479.063
AIC: -910.126 BIC: -817.9506 HQIC: -873.4137
------------------------------------------------------------
Model was estimated on : Mar Tue 03, 2026 at 00:36
In this approach, the LP deterministic envelope is used as the response variable in the second stage and the SFA quantifies the stochastic variation around this envelope.
meta_sfacross_odonnell <- smfa(
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_sfacross_odonnell)Toggle to see the output
> meta_sfacross_odonnell <- smfa(
+ formula = log(PROD) ~ log(AREA) + log(LABOR) + log(NPK),
+ data = ricephil,
+ group = "group",
+ S = 1,
+ udist = "hnormal",
+ groupType = "sfacross",
+ metaMethod = "sfa",
+ sfaApproach = "ordonnell"
+ )
Estimating group-specific stochastic frontiers (sfacross) ...
Group: small
Group: medium
Group: large
Group frontiers estimated.
Estimating metafrontier using method: SFA Metafrontier [O'Donnell et al. (2008), envelope]
Warning message:
The residuals of the OLS are right-skewed. This may indicate the absence of inefficiency or
model misspecification or sample 'bad luck'
> summary(meta_sfacross_odonnell)
============================================================
Stochastic Metafrontier Analysis
Metafrontier method: SFA Metafrontier [O'Donnell et al. (2008), envelope]
Stochastic Production/Profit Frontier, e = v - u
SFA approach : ordonnell
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 : Mar Tue 03, 2026 at 00:37
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 : Mar Tue 03, 2026 at 00:37
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 : Mar Tue 03, 2026 at 00:37
Log likelihood status: successful convergence
--------------------------------------------------------------------------------
------------------------------------------------------------
Metafrontier Coefficients (sfa):
Meta-optim solver : BFGS maximization
Estimate Std. Error z value Pr(>|z|)
(Intercept) -0.6114342 0.0414990 -14.734 < 2.2e-16 ***
log(AREA) 0.3937848 0.0072782 54.105 < 2.2e-16 ***
log(LABOR) 0.2791270 0.0076764 36.361 < 2.2e-16 ***
log(NPK) 0.2409454 0.0046573 51.735 < 2.2e-16 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Meta-frontier model details:
--------------------------------------------------------------------------------
Normal-Half Normal SF Model
Dependent Variable: lp_envelope
Log likelihood solver: BFGS maximization
Log likelihood iter: 436
Log likelihood value: 632.20951
Log likelihood gradient norm: 5.34711e-02
Estimation based on: N = 344 and K = 6
Inf. Cr: AIC = -1252.4 AIC/N = -3.641
BIC = -1229.4 BIC/N = -3.574
HQIC = -1243.2 HQIC/N = -3.614
--------------------------------------------------------------------------------
Variances: Sigma-squared(v) = 0.00148
Sigma(v) = 0.00148
Sigma-squared(u) = 0.00000
Sigma(u) = 0.00000
Sigma = Sqrt[(s^2(u)+s^2(v))] = 0.03851
Gamma = sigma(u)^2/sigma^2 = 0.00013
Lambda = sigma(u)/sigma(v) = 0.01121
Var[u]/{Var[u]+Var[v]} = 0.00005
--------------------------------------------------------------------------------
Average inefficiency E[ui] = 0.00034
Average efficiency E[exp(-ui)] = 0.99966
--------------------------------------------------------------------------------
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) = 632.20952
LR statistic:
Chisq = 2*[LogL(H0)-LogL(H1)] = -0.00003
Kodde-Palm C*: 95%: 2.70554 99%: 5.41189
Coelli (1995) skewness test on OLS residuals
M3T: z = 6.24028
M3T: p.value = 0.00000
Final maximum likelihood estimates
--------------------------------------------------------------------------------
Deterministic Component of SFA
--------------------------------------------------------------------------------
Coefficient Std. Error z value Pr(>|z|)
(Intercept) -0.61143 0.04150 -14.734 < 2.2e-16 ***
.X2 0.39378 0.00728 54.105 < 2.2e-16 ***
.X3 0.27913 0.00768 36.361 < 2.2e-16 ***
.X4 0.24095 0.00466 51.735 < 2.2e-16 ***
--------------------------------------------------------------------------------
Parameter in variance of u (one-sided error)
--------------------------------------------------------------------------------
Coefficient Std. Error z value Pr(>|z|)
Zu_(Intercept) -15.496 172.306 -0.0899 0.9283
--------------------------------------------------------------------------------
Parameters in variance of v (two-sided error)
--------------------------------------------------------------------------------
Coefficient Std. Error z value Pr(>|z|)
Zv_(Intercept) -6.51356 0.07665 -84.978 < 2.2e-16 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
--------------------------------------------------------------------------------
Model was estimated on : Mar Tue 03, 2026 at 00:37
Log likelihood status: successful convergence
--------------------------------------------------------------------------------
Log likelihood status: successful convergence
------------------------------------------------------------
Efficiency Statistics (group means):
------------------------------------------------------------
N_obs N_valid TE_group_BC TE_group_JLMS TE_meta_BC TE_meta_JLMS MTR_BC MTR_JLMS
small 125 125 0.71065 0.70090 0.99966 0.99966 1.49276 1.51673
medium 104 104 0.71253 0.70965 0.99966 0.99966 1.50575 1.51248
large 115 115 0.74772 0.74406 0.99966 0.99966 1.41180 1.41943
Overall:
TE_group_BC=0.7236 TE_group_JLMS=0.7182
TE_meta_BC=0.9997 TE_meta_JLMS=0.9997
MTR_BC=1.4701 MTR_JLMS=1.4829
------------------------------------------------------------
Total Log-likelihood: 557.9201
AIC: -1067.84 BIC: -975.6648 HQIC: -1031.128
------------------------------------------------------------
Model was estimated on : Mar Tue 03, 2026 at 00:37
Warning message:
344 MTR value(s) > 1 detected in O'Donnell SFA approach. This typically occurs when the second-stage SFA estimates near-zero inefficiency (sigma_u -> 0), causing TE_meta ~= 1 and MTR = TE_meta/TE_group > 1. Consider using metaMethod='lp' or sfaApproach='huang' instead.
O'Donnell et al. (2008) approach involves taking the deterministic envelope (the maximum) of all group frontier values at each data point (yMeta = apply(groupFrontierMat, 1, max)). Because this new yMeta surface is the maximum of several hyperplanes, its shape is purely convex and completely lacks statistical noise. When the second-stage sfacross tries to fit a single straight line (hyperplane) through this convex envelope, the residuals predominantly curve upward away from the line. In production frontiers, this results in right-skewed OLS residuals. SFA models naturally interpret right-skewed residuals as having near-zero inefficiency (sigma_u -> 0). Because the meta-inefficiency is estimated as near zero (u_meta -> 0), the meta-efficiency approaches 1 (TE_meta ~= 1). Because TE_meta = TE_group * MTR, we get MTR = TE_meta / TE_group. Since TE_meta is ~1 and TE_group < 1, this mathematically forces MTR > 1. Why does this happen? Because fitting a stochastic frontier via Maximum Likelihood onto a purely mathematical LP envelope does not strictly enforce the bounding constraint MTR <= 1 at every individual data point. The SFA line will inevitably cut through the LP envelope rather than sitting strictly above it for all points. This is a well-known theoretical and computational limitation of the producing MTR value(s) > 1, which is partly why Huang et al. (2014) proposed their alternative method.
How to solve it? This is exactly why the warning message was added to the code. If you require MTR <= 1 bounds: (1) Use metaMethod = "lp" or "qp" (which explicitly enforce the mathematical envelope constraints). (2) Use sfaApproach = "huang", which avoids this "wrong skewness" problem by using the actual observations' own-group fitted values (yhat_group) as the SFA dependent variable, and directly estimating the technology gap U_i, inherently bounding MTR = exp(-U_i) <= 1.
When technology groups are unobserved, a pooled latent class model (sfalcmcross) is fitted on all data. The resulting class assignments (by maximum posterior probability) serve as the technology groups for the metafrontier.
data("utility", package = "sfaR")
# No group variable needed for pooled LCM (groupType = "sfalcmcross" with no group argument)meta_lcm_lp <- smfa(
formula = log(tc/wf) ~ log(y) + log(wl/wf) + log(wk/wf),
data = utility,
S = -1,
groupType = "sfalcmcross",
lcmClasses = 2,
metaMethod = "lp"
)
summary(meta_lcm_lp)Toggle to see the output
> meta_lcm_lp <- smfa(
+ formula = log(tc/wf) ~ log(y) + log(wl/wf) + log(wk/wf),
+ data = utility,
+ S = -1,
+ groupType = "sfalcmcross",
+ lcmClasses = 2,
+ metaMethod = "lp"
+ )
Fitting pooled sfalcmcross (2 classes) on all data ...
Initialization: SFA + halfnormal - normal distributions...
LCM 2 Classes Estimation...
Warning: hessian is singular for 'qr.solve' switching to 'ginv'
Pooled LCM estimated.
Estimating metafrontier using method: Linear Programming (LP) Metafrontier
> summary(meta_lcm_lp)
============================================================
Stochastic Metafrontier Analysis
Metafrontier method: Linear Programming (LP) Metafrontier
Stochastic Cost Frontier, e = v + u
Group approach : Latent Class Stochastic Frontier Analysis
Group estimator : sfalcmcross
Group optim solver : BFGS maximization
(Pooled LCM - latent classes used as groups)
Groups ( 2 ): Class_1, Class_2
Total observations : 791
Distribution : hnormal
============================================================
------------------------------------------------------------
Pooled LCM (2 classes) on all data (N = 791) Log-likelihood: 61.35325
------------------------------------------------------------
-- Latent Class 1 --
Frontier:
Coefficient Std. Error z value Pr(>|z|)
(Intercept) -1.4472e+00 3.9123e-05 -36992 < 2.2e-16 ***
log(y) 8.4541e-01 2.3364e-06 361846 < 2.2e-16 ***
log(wl/wf) 3.5408e-01 4.4754e-06 79118 < 2.2e-16 ***
log(wk/wf) 4.2883e-01 1.3682e-05 31343 < 2.2e-16 ***
Var(u):
Coefficient Std. Error z value Pr(>|z|)
Zu_(Intercept) -1.7658e+00 5.8803e-08 -30029863 < 2.2e-16 ***
Var(v):
Coefficient Std. Error z value Pr(>|z|)
Zv_(Intercept) -3.8759e+01 3.2680e-13 -1.186e+14 < 2.2e-16 ***
Sigma_u=0.4136 Sigma_v=0.0000 Sigma=0.4136 Gamma=1.0000 Lambda=107911523.6712
-- Latent Class 2 --
Frontier:
Coefficient Std. Error z value Pr(>|z|)
(Intercept) -2.0490e+00 2.5608e-05 -80011.07 < 2.2e-16 ***
log(y) 1.0079e+00 4.1082e-04 2453.37 < 2.2e-16 ***
log(wl/wf) -2.5916e-02 6.3375e-05 -408.93 < 2.2e-16 ***
log(wk/wf) 8.8450e-01 6.9315e-05 12760.74 < 2.2e-16 ***
Var(u):
Coefficient Std. Error z value Pr(>|z|)
Zu_(Intercept) -3.0117e+00 1.1348e-06 -2653956 < 2.2e-16 ***
Var(v):
Coefficient Std. Error z value Pr(>|z|)
Zv_(Intercept) -4.9663e+00 5.3865e-07 -9219998 < 2.2e-16 ***
Sigma_u=0.2218 Sigma_v=0.0835 Sigma=0.2370 Gamma=0.8759 Lambda=2.6573
-- Class Membership (logit) --
Coefficient Std. Error z value Pr(>|z|)
Cl1_(Intercept) -6.0163e-01 5.0453e-07 -1192458 < 2.2e-16 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
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 MTR_JLMS
Class_1 202 202 0.68291 0.68291 0.68291 0.68291 1.00000 1.00000
Class_2 589 589 0.85142 0.84951 0.85142 0.84951 1.00000 1.00000
Overall:
TE_group_BC=0.7672 TE_group_JLMS=0.7662
TE_meta_BC=0.7672 TE_meta_JLMS=0.7662
MTR_BC=1.0000 MTR_JLMS=1.0000
------------------------------------------------------------
Posterior Class Membership (pooled LCM):
------------------------------------------------------------
% assigned Mean post. prob.
Class 1 25.5 0.354
Class 2 74.5 0.646
------------------------------------------------------------
Total Log-likelihood: 61.35325
AIC: -96.70649 BIC: -35.95362 HQIC: -73.35552
------------------------------------------------------------
Model was estimated on : Mar Tue 03, 2026 at 00:55
Retrieve efficiencies including per-class posterior probabilities
# Retrieve efficiencies including per-class posterior probabilities
head(efficiencies(meta_lcm_lp))
# Columns include: Group_c, u_g, TE_group_JLMS, TE_group_BC, TE_group_BC_reciprocal,
# PosteriorProb_c, PosteriorProb_c1, PosteriorProb_c2,
# PriorProb_c1, PriorProb_c2, u_c1, u_c2,
# teBC_c1, teBC_c2, u_meta, TE_meta_JLMS, TE_meta_BC, MTR_JLMS, MTR_BCToggle to see the output
> head(efficiencies(meta_lcm_lp))
id Group_c u_g TE_group_JLMS TE_group_BC TE_group_BC_reciprocal PosteriorProb_c PosteriorProb_c1 PriorProb_c1 u_c1 teBC_c1 teBC_reciprocal_c1 PosteriorProb_c2
1 1 2 0.15842759 0.8534848 0.8557686 1.174838 0.7249992 0.2750008 0.3539715 0.19428008 0.8234272 1.214436 0.7249992
2 2 2 0.11418774 0.8920905 0.8939942 1.123406 0.7334016 0.2665984 0.3539715 0.16086081 0.8514106 1.174521 0.7334016
3 3 2 0.08540291 0.9181423 0.9196136 1.090950 0.7039780 0.2960220 0.3539715 0.10301513 0.9021133 1.108508 0.7039780
4 4 2 0.08020641 0.9229258 0.9243036 1.085175 0.6909119 0.3090881 0.3539715 0.07745685 0.9254670 1.080536 0.6909119
5 5 2 0.05774132 0.9438941 0.9448245 1.060519 0.5808752 0.4191248 0.3539715 0.05410185 0.9473356 1.055592 0.5808752
6 6 2 0.08181796 0.9214397 0.9228469 1.086964 0.6927818 0.3072182 0.3539715 0.05781993 0.9438199 1.059524 0.6927818
PriorProb_c2 u_c2 teBC_c2 teBC_reciprocal_c2 ineff_c1 ineff_c2 effBC_c1 effBC_c2 ReffBC_c1 ReffBC_c2 u_meta TE_meta_JLMS TE_meta_BC MTR_JLMS MTR_BC
1 0.6460285 0.15842759 0.8557686 1.174838 NA 0.15842759 NA 0.8557686 NA 1.174838 0.15842759 0.8534848 0.8557686 1 1
2 0.6460285 0.11418774 0.8939942 1.123406 NA 0.11418774 NA 0.8939942 NA 1.123406 0.11418774 0.8920905 0.8939942 1 1
3 0.6460285 0.08540291 0.9196136 1.090950 NA 0.08540291 NA 0.9196136 NA 1.090950 0.08540291 0.9181423 0.9196136 1 1
4 0.6460285 0.08020641 0.9243036 1.085175 NA 0.08020641 NA 0.9243036 NA 1.085175 0.08020641 0.9229258 0.9243036 1 1
5 0.6460285 0.05774132 0.9448245 1.060519 NA 0.05774132 NA 0.9448245 NA 1.060519 0.05774132 0.9438941 0.9448245 1 1
6 0.6460285 0.08181796 0.9228469 1.086964 NA 0.08181796 NA 0.9228469 NA 1.086964 0.08181796 0.9214397 0.9228469 1 1
meta_lcm_qp <- smfa(
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)Toggle to see the output
> meta_lcm_qp <- smfa(
+ formula = log(tc/wf) ~ log(y) + log(wl/wf) + log(wk/wf),
+ data = utility,
+ S = -1,
+ groupType = "sfalcmcross",
+ lcmClasses = 2,
+ metaMethod = "qp"
+ )
Fitting pooled sfalcmcross (2 classes) on all data ...
Initialization: SFA + halfnormal - normal distributions...
LCM 2 Classes Estimation...
Warning: hessian is singular for 'qr.solve' switching to 'ginv'
Pooled LCM estimated.
Estimating metafrontier using method: Quadratic Programming (QP) Metafrontier
> summary(meta_lcm_qp)
============================================================
Stochastic Metafrontier Analysis
Metafrontier method: Quadratic Programming (QP) Metafrontier
Stochastic Cost Frontier, e = v + u
Group approach : Latent Class Stochastic Frontier Analysis
Group estimator : sfalcmcross
Group optim solver : BFGS maximization
(Pooled LCM - latent classes used as groups)
Groups ( 2 ): Class_1, Class_2
Total observations : 791
Distribution : hnormal
============================================================
------------------------------------------------------------
Pooled LCM (2 classes) on all data (N = 791) Log-likelihood: 61.35325
------------------------------------------------------------
-- Latent Class 1 --
Frontier:
Coefficient Std. Error z value Pr(>|z|)
(Intercept) -1.4472e+00 3.9123e-05 -36992 < 2.2e-16 ***
log(y) 8.4541e-01 2.3364e-06 361846 < 2.2e-16 ***
log(wl/wf) 3.5408e-01 4.4754e-06 79118 < 2.2e-16 ***
log(wk/wf) 4.2883e-01 1.3682e-05 31343 < 2.2e-16 ***
Var(u):
Coefficient Std. Error z value Pr(>|z|)
Zu_(Intercept) -1.7658e+00 5.8803e-08 -30029863 < 2.2e-16 ***
Var(v):
Coefficient Std. Error z value Pr(>|z|)
Zv_(Intercept) -3.8759e+01 3.2680e-13 -1.186e+14 < 2.2e-16 ***
Sigma_u=0.4136 Sigma_v=0.0000 Sigma=0.4136 Gamma=1.0000 Lambda=107911523.6712
-- Latent Class 2 --
Frontier:
Coefficient Std. Error z value Pr(>|z|)
(Intercept) -2.0490e+00 2.5608e-05 -80011.07 < 2.2e-16 ***
log(y) 1.0079e+00 4.1082e-04 2453.37 < 2.2e-16 ***
log(wl/wf) -2.5916e-02 6.3375e-05 -408.93 < 2.2e-16 ***
log(wk/wf) 8.8450e-01 6.9315e-05 12760.74 < 2.2e-16 ***
Var(u):
Coefficient Std. Error z value Pr(>|z|)
Zu_(Intercept) -3.0117e+00 1.1348e-06 -2653956 < 2.2e-16 ***
Var(v):
Coefficient Std. Error z value Pr(>|z|)
Zv_(Intercept) -4.9663e+00 5.3865e-07 -9219998 < 2.2e-16 ***
Sigma_u=0.2218 Sigma_v=0.0835 Sigma=0.2370 Gamma=0.8759 Lambda=2.6573
-- Class Membership (logit) --
Coefficient Std. Error z value Pr(>|z|)
Cl1_(Intercept) -6.0163e-01 5.0453e-07 -1192458 < 2.2e-16 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Log likelihood status: successful convergence
------------------------------------------------------------
Metafrontier Coefficients (qp):
Estimate Std. Error z value Pr(>|z|)
(Intercept) -1.3923607 0.0310122 -44.897 < 2.2e-16 ***
log(y) 0.8649986 0.0012419 696.519 < 2.2e-16 ***
log(wl/wf) 0.2909728 0.0047965 60.664 < 2.2e-16 ***
log(wk/wf) 0.5028978 0.0069500 72.359 < 2.2e-16 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
------------------------------------------------------------
Efficiency Statistics (group means):
------------------------------------------------------------
N_obs N_valid TE_group_BC TE_group_JLMS TE_meta_BC TE_meta_JLMS MTR_BC MTR_JLMS
Class_1 202 202 0.68291 0.68291 0.67786 0.67786 0.99326 0.99326
Class_2 589 589 0.85142 0.84951 0.84548 0.84359 0.99285 0.99285
Overall:
TE_group_BC=0.7672 TE_group_JLMS=0.7662
TE_meta_BC=0.7617 TE_meta_JLMS=0.7607
MTR_BC=0.9931 MTR_JLMS=0.9931
------------------------------------------------------------
Posterior Class Membership (pooled LCM):
------------------------------------------------------------
% assigned Mean post. prob.
Class 1 25.5 0.354
Class 2 74.5 0.646
------------------------------------------------------------
Total Log-likelihood: 61.35325
AIC: -88.70649 BIC: -9.26042 HQIC: -58.17061
------------------------------------------------------------
Model was estimated on : Mar Tue 03, 2026 at 01:01
meta_lcm_huang <- smfa(
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)Toggle to see the output
> meta_lcm_huang <- smfa(
+ formula = log(tc/wf) ~ log(y) + log(wl/wf) + log(wk/wf),
+ data = utility,
+ S = -1,
+ groupType = "sfalcmcross",
+ lcmClasses = 2,
+ metaMethod = "sfa",
+ sfaApproach = "huang"
+ )
Fitting pooled sfalcmcross (2 classes) on all data ...
Initialization: SFA + halfnormal - normal distributions...
LCM 2 Classes Estimation...
Warning: hessian is singular for 'qr.solve' switching to 'ginv'
Pooled LCM estimated.
Estimating metafrontier using method: SFA Metafrontier [Huang et al. (2014), two-stage]
> summary(meta_lcm_huang)
============================================================
Stochastic Metafrontier Analysis
Metafrontier method: SFA Metafrontier [Huang et al. (2014), two-stage]
Stochastic Cost Frontier, e = v + u
SFA approach : huang
Group approach : Latent Class Stochastic Frontier Analysis
Group estimator : sfalcmcross
Group optim solver : BFGS maximization
(Pooled LCM - latent classes used as groups)
Groups ( 2 ): Class_1, Class_2
Total observations : 791
Distribution : hnormal
============================================================
------------------------------------------------------------
Pooled LCM (2 classes) on all data (N = 791) Log-likelihood: 61.35325
------------------------------------------------------------
-- Latent Class 1 --
Frontier:
Coefficient Std. Error z value Pr(>|z|)
(Intercept) -1.4472e+00 3.9123e-05 -36992 < 2.2e-16 ***
log(y) 8.4541e-01 2.3364e-06 361846 < 2.2e-16 ***
log(wl/wf) 3.5408e-01 4.4754e-06 79118 < 2.2e-16 ***
log(wk/wf) 4.2883e-01 1.3682e-05 31343 < 2.2e-16 ***
Var(u):
Coefficient Std. Error z value Pr(>|z|)
Zu_(Intercept) -1.7658e+00 5.8803e-08 -30029863 < 2.2e-16 ***
Var(v):
Coefficient Std. Error z value Pr(>|z|)
Zv_(Intercept) -3.8759e+01 3.2680e-13 -1.186e+14 < 2.2e-16 ***
Sigma_u=0.4136 Sigma_v=0.0000 Sigma=0.4136 Gamma=1.0000 Lambda=107911523.6712
-- Latent Class 2 --
Frontier:
Coefficient Std. Error z value Pr(>|z|)
(Intercept) -2.0490e+00 2.5608e-05 -80011.07 < 2.2e-16 ***
log(y) 1.0079e+00 4.1082e-04 2453.37 < 2.2e-16 ***
log(wl/wf) -2.5916e-02 6.3375e-05 -408.93 < 2.2e-16 ***
log(wk/wf) 8.8450e-01 6.9315e-05 12760.74 < 2.2e-16 ***
Var(u):
Coefficient Std. Error z value Pr(>|z|)
Zu_(Intercept) -3.0117e+00 1.1348e-06 -2653956 < 2.2e-16 ***
Var(v):
Coefficient Std. Error z value Pr(>|z|)
Zv_(Intercept) -4.9663e+00 5.3865e-07 -9219998 < 2.2e-16 ***
Sigma_u=0.2218 Sigma_v=0.0835 Sigma=0.2370 Gamma=0.8759 Lambda=2.6573
-- Class Membership (logit) --
Coefficient Std. Error z value Pr(>|z|)
Cl1_(Intercept) -6.0163e-01 5.0453e-07 -1192458 < 2.2e-16 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Log likelihood status: successful convergence
------------------------------------------------------------
Metafrontier Coefficients (sfa):
Meta-optim solver : BFGS maximization
Estimate Std. Error z value Pr(>|z|)
(Intercept) -2.2495079 0.0686629 -32.7616 < 2.2e-16 ***
log(y) 0.9909918 0.0024687 401.4291 < 2.2e-16 ***
log(wl/wf) 0.0399586 0.0106166 3.7638 0.0001674 ***
log(wk/wf) 0.7890986 0.0143801 54.8745 < 2.2e-16 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Meta-frontier model details:
--------------------------------------------------------------------------------
Normal-Half Normal SF Model
Dependent Variable: group_fitted_values
Log likelihood solver: BFGS maximization
Log likelihood iter: 58
Log likelihood value: 759.62892
Log likelihood gradient norm: 2.06059e-03
Estimation based on: N = 791 and K = 6
Inf. Cr: AIC = -1507.3 AIC/N = -1.906
BIC = -1479.2 BIC/N = -1.870
HQIC = -1496.5 HQIC/N = -1.892
--------------------------------------------------------------------------------
Variances: Sigma-squared(v) = 0.00068
Sigma(v) = 0.00068
Sigma-squared(u) = 0.02713
Sigma(u) = 0.02713
Sigma = Sqrt[(s^2(u)+s^2(v))] = 0.16676
Gamma = sigma(u)^2/sigma^2 = 0.97554
Lambda = sigma(u)/sigma(v) = 6.31586
Var[u]/{Var[u]+Var[v]} = 0.93546
--------------------------------------------------------------------------------
Average inefficiency E[ui] = 0.13141
Average efficiency E[exp(-ui)] = 0.88105
--------------------------------------------------------------------------------
Stochastic Cost 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) = 588.58962
LR statistic:
Chisq = 2*[LogL(H0)-LogL(H1)] = 342.07861
Kodde-Palm C*: 95%: 2.70554 99%: 5.41189
Coelli (1995) skewness test on OLS residuals
M3T: z = 10.23625
M3T: p.value = 0.00000
Final maximum likelihood estimates
--------------------------------------------------------------------------------
Deterministic Component of SFA
--------------------------------------------------------------------------------
Coefficient Std. Error z value Pr(>|z|)
(Intercept) -2.24951 0.06866 -32.7616 < 2.2e-16 ***
.X2 0.99099 0.00247 401.4291 < 2.2e-16 ***
.X3 0.03996 0.01062 3.7638 0.0001674 ***
.X4 0.78910 0.01438 54.8745 < 2.2e-16 ***
--------------------------------------------------------------------------------
Parameter in variance of u (one-sided error)
--------------------------------------------------------------------------------
Coefficient Std. Error z value Pr(>|z|)
Zu_(Intercept) -3.60721 0.05642 -63.932 < 2.2e-16 ***
--------------------------------------------------------------------------------
Parameters in variance of v (two-sided error)
--------------------------------------------------------------------------------
Coefficient Std. Error z value Pr(>|z|)
Zv_(Intercept) -7.29334 0.12966 -56.25 < 2.2e-16 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
--------------------------------------------------------------------------------
Model was estimated on : Mar Tue 03, 2026 at 01:03
Log likelihood status: successful convergence
--------------------------------------------------------------------------------
Log likelihood status: successful convergence
------------------------------------------------------------
Efficiency Statistics (group means):
------------------------------------------------------------
N_obs N_valid TE_group_BC TE_group_JLMS TE_meta_BC TE_meta_JLMS MTR_BC MTR_JLMS
Class_1 202 202 0.68291 0.68291 0.51861 0.51845 0.76693 0.76669
Class_2 589 589 0.85142 0.84951 0.80511 0.80308 0.94559 0.94532
Overall:
TE_group_BC=0.7672 TE_group_JLMS=0.7662
TE_meta_BC=0.6619 TE_meta_JLMS=0.6608
MTR_BC=0.8563 MTR_JLMS=0.8560
------------------------------------------------------------
Posterior Class Membership (pooled LCM):
------------------------------------------------------------
% assigned Mean post. prob.
Class 1 25.5 0.354
Class 2 74.5 0.646
------------------------------------------------------------
Total Log-likelihood: 820.9822
AIC: -1603.964 BIC: -1515.172 HQIC: -1569.836
------------------------------------------------------------
Model was estimated on : Mar Tue 03, 2026 at 01:03
meta_lcm_odonnell <- smfa(
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)Toggle to see the output
> meta_lcm_odonnell <- smfa(
+ formula = log(tc/wf) ~ log(y) + log(wl/wf) + log(wk/wf),
+ data = utility,
+ S = -1,
+ groupType = "sfalcmcross",
+ lcmClasses = 2,
+ metaMethod = "sfa",
+ sfaApproach = "ordonnell"
+ )
Fitting pooled sfalcmcross (2 classes) on all data ...
Initialization: SFA + halfnormal - normal distributions...
LCM 2 Classes Estimation...
Warning: hessian is singular for 'qr.solve' switching to 'ginv'
Pooled LCM estimated.
Estimating metafrontier using method: SFA Metafrontier [O'Donnell et al. (2008), envelope]
Warning: hessian is singular for 'qr.solve' switching to 'ginv'
> summary(meta_lcm_odonnell)
============================================================
Stochastic Metafrontier Analysis
Metafrontier method: SFA Metafrontier [O'Donnell et al. (2008), envelope]
Stochastic Cost Frontier, e = v + u
SFA approach : ordonnell
Group approach : Latent Class Stochastic Frontier Analysis
Group estimator : sfalcmcross
Group optim solver : BFGS maximization
(Pooled LCM - latent classes used as groups)
Groups ( 2 ): Class_1, Class_2
Total observations : 791
Distribution : hnormal
============================================================
------------------------------------------------------------
Pooled LCM (2 classes) on all data (N = 791) Log-likelihood: 61.35325
------------------------------------------------------------
-- Latent Class 1 --
Frontier:
Coefficient Std. Error z value Pr(>|z|)
(Intercept) -1.4472e+00 3.9123e-05 -36992 < 2.2e-16 ***
log(y) 8.4541e-01 2.3364e-06 361846 < 2.2e-16 ***
log(wl/wf) 3.5408e-01 4.4754e-06 79118 < 2.2e-16 ***
log(wk/wf) 4.2883e-01 1.3682e-05 31343 < 2.2e-16 ***
Var(u):
Coefficient Std. Error z value Pr(>|z|)
Zu_(Intercept) -1.7658e+00 5.8803e-08 -30029863 < 2.2e-16 ***
Var(v):
Coefficient Std. Error z value Pr(>|z|)
Zv_(Intercept) -3.8759e+01 3.2680e-13 -1.186e+14 < 2.2e-16 ***
Sigma_u=0.4136 Sigma_v=0.0000 Sigma=0.4136 Gamma=1.0000 Lambda=107911523.6712
-- Latent Class 2 --
Frontier:
Coefficient Std. Error z value Pr(>|z|)
(Intercept) -2.0490e+00 2.5608e-05 -80011.07 < 2.2e-16 ***
log(y) 1.0079e+00 4.1082e-04 2453.37 < 2.2e-16 ***
log(wl/wf) -2.5916e-02 6.3375e-05 -408.93 < 2.2e-16 ***
log(wk/wf) 8.8450e-01 6.9315e-05 12760.74 < 2.2e-16 ***
Var(u):
Coefficient Std. Error z value Pr(>|z|)
Zu_(Intercept) -3.0117e+00 1.1348e-06 -2653956 < 2.2e-16 ***
Var(v):
Coefficient Std. Error z value Pr(>|z|)
Zv_(Intercept) -4.9663e+00 5.3865e-07 -9219998 < 2.2e-16 ***
Sigma_u=0.2218 Sigma_v=0.0835 Sigma=0.2370 Gamma=0.8759 Lambda=2.6573
-- Class Membership (logit) --
Coefficient Std. Error z value Pr(>|z|)
Cl1_(Intercept) -6.0163e-01 5.0453e-07 -1192458 < 2.2e-16 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Log likelihood status: successful convergence
------------------------------------------------------------
Metafrontier Coefficients (sfa):
Meta-optim solver : BFGS maximization
Estimate Std. Error z value Pr(>|z|)
(Intercept) -1.4655e+00 1.4682e-05 -99822 < 2.2e-16 ***
log(y) 8.5052e-01 2.0413e-06 416655 < 2.2e-16 ***
log(wl/wf) 3.4196e-01 8.2433e-06 41483 < 2.2e-16 ***
log(wk/wf) 4.4325e-01 1.1455e-05 38693 < 2.2e-16 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Meta-frontier model details:
--------------------------------------------------------------------------------
Normal-Half Normal SF Model
Dependent Variable: lp_envelope
Log likelihood solver: BFGS maximization
Log likelihood iter: 1139
Log likelihood value: 1949.16740
Log likelihood gradient norm: 3.76958e+02
Estimation based on: N = 791 and K = 6
Inf. Cr: AIC = -3886.3 AIC/N = -4.913
BIC = -3858.3 BIC/N = -4.878
HQIC = -3875.6 HQIC/N = -4.900
--------------------------------------------------------------------------------
Variances: Sigma-squared(v) = 0.00000
Sigma(v) = 0.00000
Sigma-squared(u) = 0.00170
Sigma(u) = 0.00170
Sigma = Sqrt[(s^2(u)+s^2(v))] = 0.04117
Gamma = sigma(u)^2/sigma^2 = 1.00000
Lambda = sigma(u)/sigma(v) = 5005571.02335
Var[u]/{Var[u]+Var[v]} = 1.00000
--------------------------------------------------------------------------------
Average inefficiency E[ui] = 0.03285
Average efficiency E[exp(-ui)] = 0.96798
--------------------------------------------------------------------------------
Stochastic Cost 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) = 1595.35974
LR statistic:
Chisq = 2*[LogL(H0)-LogL(H1)] = 707.61532
Kodde-Palm C*: 95%: 2.70554 99%: 5.41189
Coelli (1995) skewness test on OLS residuals
M3T: z = 30.29500
M3T: p.value = 0.00000
Final maximum likelihood estimates
--------------------------------------------------------------------------------
Deterministic Component of SFA
--------------------------------------------------------------------------------
Coefficient Std. Error z value Pr(>|z|)
(Intercept) -1.46554 0.00001 -99822 < 2.2e-16 ***
.X2 0.85052 0.00000 416655 < 2.2e-16 ***
.X3 0.34196 0.00001 41483 < 2.2e-16 ***
.X4 0.44325 0.00001 38693 < 2.2e-16 ***
--------------------------------------------------------------------------------
Parameter in variance of u (one-sided error)
--------------------------------------------------------------------------------
Coefficient Std. Error z value Pr(>|z|)
Zu_(Intercept) -6.3799 0.0000 -1.8575e+13 < 2.2e-16 ***
--------------------------------------------------------------------------------
Parameters in variance of v (two-sided error)
--------------------------------------------------------------------------------
Coefficient Std. Error z value Pr(>|z|)
Zv_(Intercept) -37.232 0.000 -4.2838e+13 < 2.2e-16 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
--------------------------------------------------------------------------------
Model was estimated on : Mar Tue 03, 2026 at 01:04
Log likelihood status: successful convergence
--------------------------------------------------------------------------------
Log likelihood status: successful convergence
------------------------------------------------------------
Efficiency Statistics (group means):
------------------------------------------------------------
N_obs N_valid TE_group_BC TE_group_JLMS TE_meta_BC TE_meta_JLMS MTR_BC MTR_JLMS
Class_1 202 202 0.68291 0.68291 0.98665 0.98665 1.53705 1.53705
Class_2 589 589 0.85142 0.84951 0.98091 0.98091 1.16150 1.16424
Overall:
TE_group_BC=0.7672 TE_group_JLMS=0.7662
TE_meta_BC=0.9838 TE_meta_JLMS=0.9838
MTR_BC=1.3493 MTR_JLMS=1.3506
------------------------------------------------------------
Posterior Class Membership (pooled LCM):
------------------------------------------------------------
% assigned Mean post. prob.
Class 1 25.5 0.354
Class 2 74.5 0.646
------------------------------------------------------------
Total Log-likelihood: 2010.521
AIC: -3983.041 BIC: -3894.249 HQIC: -3948.913
------------------------------------------------------------
Model was estimated on : Mar Tue 03, 2026 at 01:04
Warning message:
761 MTR value(s) > 1 detected in O'Donnell SFA approach. This typically occurs when the second-stage SFA estimates near-zero inefficiency (sigma_u -> 0), causing TE_meta ~= 1 and MTR = TE_meta/TE_group > 1. Consider using metaMethod='lp' or sfaApproach='huang' instead.
When the observed sample is not random (e.g., only firms above a revenue threshold are surveyed), sample selection bias can distort frontier estimates. sfaselectioncross corrects for this using the two-step approach of Greene (2010). Only selected observations (d == 1) participate in the frontier and metafrontier; efficiency estimates for non-selected observations are NA. Here is a simulated example (adapted from sfaR):
N <- 2000; set.seed(12345)
z1 <- rnorm(N); z2 <- rnorm(N)
v1 <- rnorm(N); v2 <- rnorm(N)
g <- rnorm(N)
e1 <- v1
e2 <- 0.7071 * (v1 + v2)
ds <- z1 + z2 + e1
d <- ifelse(ds > 0, 1, 0) # binary selection indicator: 1 = selected
group <- ifelse(g > 0, 1, 0) # two technology groups
u <- abs(rnorm(N))
x1 <- abs(rnorm(N)); x2 <- abs(rnorm(N))
y <- abs(x1 + x2 + e2 - u)
dat <- as.data.frame(cbind(y=y, x1=x1, x2=x2, z1=z1, z2=z2, d=d, group=group))
# About 50% of observations are selected:
table(dat$d) 0 1
1013 987
meta_sel_lp <- smfa(
formula = log(y) ~ log(x1) + log(x2),
selectionF = d ~ z1 + z2,
data = dat,
group = "group",
S = 1L,
udist = "hnormal",
groupType = "sfaselectioncross",
modelType = "greene10",
lType = "kronrod",
Nsub = 100,
uBound = Inf,
method = "bfgs",
itermax = 2000,
metaMethod = "lp"
)
summary(meta_sel_lp)Toggle to see the output
> meta_sel_lp <- smfa(
+ formula = log(y) ~ log(x1) + log(x2),
+ selectionF = d ~ z1 + z2,
+ data = dat,
+ group = "group",
+ S = 1L,
+ udist = "hnormal",
+ groupType = "sfaselectioncross",
+ modelType = "greene10",
+ lType = "kronrod",
+ Nsub = 100,
+ uBound = Inf,
+ method = "bfgs",
+ itermax = 2000,
+ metaMethod = "lp"
+ )
Estimating group-specific stochastic frontiers (sfaselectioncross) ...
Group: 0
First step probit model...
Second step Frontier model...
Group: 1
First step probit model...
Second step Frontier model...
Group frontiers estimated.
Estimating metafrontier using method: Linear Programming (LP) Metafrontier
> summary(meta_sel_lp)
============================================================
Stochastic Metafrontier Analysis
Metafrontier method: Linear Programming (LP) Metafrontier
Stochastic Production/Profit Frontier, e = v - u
Group approach : Sample Selection Stochastic Frontier Analysis
Group estimator : sfaselectioncross
Group optim solver : BFGS maximization
Groups ( 2 ): 0, 1
Total observations : 2000
Distribution : hnormal
============================================================
------------------------------------------------------------
Group: 0 (N = 994) Log-likelihood: -799.94522
------------------------------------------------------------
--------------------------------------------------------------------------------
Sample Selection Correction Stochastic Frontier Model
Dependent Variable: log(y)
Log likelihood solver: BFGS maximization
Log likelihood iter: 349
Log likelihood value: -799.94522
Log likelihood gradient norm: 3.41571e+01
Estimation based on: N = 489 of 994 obs. and K = 6
Inf. Cr: AIC = 1611.9 AIC/N = 3.296
BIC = 1637.0 BIC/N = 3.348
HQIC = 1621.8 HQIC/N = 3.317
--------------------------------------------------------------------------------
Variances: Sigma-squared(v) = 0.05814
Sigma(v) = 0.05814
Sigma-squared(u) = 2.29587
Sigma(u) = 2.29587
Sigma = Sqrt[(s^2(u)+s^2(v))] = 1.53428
Gamma = sigma(u)^2/sigma^2 = 0.97530
Lambda = sigma(u)/sigma(v) = 6.28402
Var[u]/{Var[u]+Var[v]} = 0.93485
--------------------------------------------------------------------------------
Average inefficiency E[ui] = 1.20897
Average efficiency E[exp(-ui)] = 0.40883
--------------------------------------------------------------------------------
Stochastic Production/Profit Frontier, e = v - u
Estimator is 2 step Maximum Likelihood
Final maximum likelihood estimates
--------------------------------------------------------------------------------
Deterministic Component of SFA
--------------------------------------------------------------------------------
Coefficient Std. Error z value Pr(>|z|)
(Intercept) 1.27249 0.05191 24.5133 < 2.2e-16 ***
log(x1) 0.13811 0.02289 6.0339 1.601e-09 ***
log(x2) 0.18668 0.02252 8.2892 < 2.2e-16 ***
--------------------------------------------------------------------------------
Parameter in variance of u (one-sided error)
--------------------------------------------------------------------------------
Coefficient Std. Error z value Pr(>|z|)
Zu_(Intercept) 0.83111 0.06438 12.91 < 2.2e-16 ***
--------------------------------------------------------------------------------
Parameters in variance of v (two-sided error)
--------------------------------------------------------------------------------
Coefficient Std. Error z value Pr(>|z|)
Zv_(Intercept) -2.84491 0.33126 -8.5882 < 2.2e-16 ***
--------------------------------------------------------------------------------
Selection bias parameter
--------------------------------------------------------------------------------
Coefficient Std. Error z value Pr(>|z|)
rho 0.89550 0.28696 3.1207 0.001804 **
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
--------------------------------------------------------------------------------
Model was estimated on : Mar Tue 03, 2026 at 01:08
Log likelihood status: successful convergence
--------------------------------------------------------------------------------
------------------------------------------------------------
Group: 1 (N = 1006) Log-likelihood: -851.19119
------------------------------------------------------------
--------------------------------------------------------------------------------
Sample Selection Correction Stochastic Frontier Model
Dependent Variable: log(y)
Log likelihood solver: BFGS maximization
Log likelihood iter: 75
Log likelihood value: -851.19119
Log likelihood gradient norm: 7.36354e-06
Estimation based on: N = 498 of 1006 obs. and K = 6
Inf. Cr: AIC = 1714.4 AIC/N = 3.443
BIC = 1739.6 BIC/N = 3.493
HQIC = 1724.3 HQIC/N = 3.462
--------------------------------------------------------------------------------
Variances: Sigma-squared(v) = 0.06106
Sigma(v) = 0.06106
Sigma-squared(u) = 2.36720
Sigma(u) = 2.36720
Sigma = Sqrt[(s^2(u)+s^2(v))] = 1.55829
Gamma = sigma(u)^2/sigma^2 = 0.97485
Lambda = sigma(u)/sigma(v) = 6.22643
Var[u]/{Var[u]+Var[v]} = 0.93372
--------------------------------------------------------------------------------
Average inefficiency E[ui] = 1.22760
Average efficiency E[exp(-ui)] = 0.40470
--------------------------------------------------------------------------------
Stochastic Production/Profit Frontier, e = v - u
Estimator is 2 step Maximum Likelihood
Final maximum likelihood estimates
--------------------------------------------------------------------------------
Deterministic Component of SFA
--------------------------------------------------------------------------------
Coefficient Std. Error z value Pr(>|z|)
(Intercept) 1.33000 0.06335 20.9937 < 2.2e-16 ***
log(x1) 0.18342 0.02349 7.8098 5.727e-15 ***
log(x2) 0.09987 0.01918 5.2061 1.928e-07 ***
--------------------------------------------------------------------------------
Parameter in variance of u (one-sided error)
--------------------------------------------------------------------------------
Coefficient Std. Error z value Pr(>|z|)
Zu_(Intercept) 0.86171 0.06328 13.618 < 2.2e-16 ***
--------------------------------------------------------------------------------
Parameters in variance of v (two-sided error)
--------------------------------------------------------------------------------
Coefficient Std. Error z value Pr(>|z|)
Zv_(Intercept) -2.79590 0.33567 -8.3292 < 2.2e-16 ***
--------------------------------------------------------------------------------
Selection bias parameter
--------------------------------------------------------------------------------
Coefficient Std. Error z value Pr(>|z|)
rho 0.40516 0.35322 1.147 0.2514
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
--------------------------------------------------------------------------------
Model was estimated on : Mar Tue 03, 2026 at 01:08
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 MTR_JLMS
0 994 489 0.38742 0.38227 0.35455 0.34983 0.91286 0.91286
1 1006 498 0.41484 0.40585 0.41112 0.40220 0.99226 0.99226
Overall:
TE_group_BC=0.4011 TE_group_JLMS=0.3941
TE_meta_BC=0.3828 TE_meta_JLMS=0.3760
MTR_BC=0.9526 MTR_JLMS=0.9526
------------------------------------------------------------
Total Log-likelihood: -1651.136
AIC: 3326.273 BIC: 3393.484 HQIC: 3350.951
------------------------------------------------------------
Model was estimated on : Mar Tue 03, 2026 at 01:08
# Efficiencies: non-selected observations have NA
ef_sel_lp <- efficiencies(meta_sel_lp)
head(ef_sel_lp)
# Selected observations in group 0:
#head(ef_sel_lp[ef_sel_lp$group == 0 & !is.na(ef_sel_lp$TE_group_BC), ])Toggle to see the output
> ef_sel_lp <- efficiencies(meta_sel_lp)
> head(ef_sel_lp)
id group u_g TE_group_JLMS TE_group_BC TE_group_BC_reciprocal u_meta TE_meta_JLMS TE_meta_BC MTR_JLMS MTR_BC
1 1 0 NA NA NA NA NA NA NA NA NA
2 2 0 3.0260520 0.04850676 0.04908457 20.872707 3.2369372 0.03928403 0.03975198 0.8098671 0.8098671
3 3 1 NA NA NA NA NA NA NA NA NA
4 4 0 NA NA NA NA NA NA NA NA NA
5 5 1 0.9405135 0.39042730 0.40073806 2.629065 0.9405135 0.39042730 0.40073806 1.0000000 1.0000000
6 6 0 NA NA NA NA NA NA NA NA NA
meta_sel_qp <- smfa(
formula = log(y) ~ log(x1) + log(x2),
selectionF = d ~ z1 + z2,
data = dat,
group = "group",
S = 1L,
udist = "hnormal",
groupType = "sfaselectioncross",
modelType = "greene10",
lType = "kronrod",
Nsub = 100,
uBound = Inf,
method = "bfgs",
itermax = 2000,
metaMethod = "qp"
)
summary(meta_sel_qp)Toggle to see the output
> meta_sel_qp <- smfa(
+ formula = log(y) ~ log(x1) + log(x2),
+ selectionF = d ~ z1 + z2,
+ data = dat,
+ group = "group",
+ S = 1L,
+ udist = "hnormal",
+ groupType = "sfaselectioncross",
+ modelType = "greene10",
+ lType = "kronrod",
+ Nsub = 100,
+ uBound = Inf,
+ method = "bfgs",
+ itermax = 2000,
+ metaMethod = "qp"
+ )
Estimating group-specific stochastic frontiers (sfaselectioncross) ...
Group: 0
First step probit model...
Second step Frontier model...
Group: 1
First step probit model...
Second step Frontier model...
Group frontiers estimated.
Estimating metafrontier using method: Quadratic Programming (QP) Metafrontier
> summary(meta_sel_qp)
============================================================
Stochastic Metafrontier Analysis
Metafrontier method: Quadratic Programming (QP) Metafrontier
Stochastic Production/Profit Frontier, e = v - u
Group approach : Sample Selection Stochastic Frontier Analysis
Group estimator : sfaselectioncross
Group optim solver : BFGS maximization
Groups ( 2 ): 0, 1
Total observations : 2000
Distribution : hnormal
============================================================
------------------------------------------------------------
Group: 0 (N = 994) Log-likelihood: -799.94522
------------------------------------------------------------
--------------------------------------------------------------------------------
Sample Selection Correction Stochastic Frontier Model
Dependent Variable: log(y)
Log likelihood solver: BFGS maximization
Log likelihood iter: 349
Log likelihood value: -799.94522
Log likelihood gradient norm: 3.41571e+01
Estimation based on: N = 489 of 994 obs. and K = 6
Inf. Cr: AIC = 1611.9 AIC/N = 3.296
BIC = 1637.0 BIC/N = 3.348
HQIC = 1621.8 HQIC/N = 3.317
--------------------------------------------------------------------------------
Variances: Sigma-squared(v) = 0.05814
Sigma(v) = 0.05814
Sigma-squared(u) = 2.29587
Sigma(u) = 2.29587
Sigma = Sqrt[(s^2(u)+s^2(v))] = 1.53428
Gamma = sigma(u)^2/sigma^2 = 0.97530
Lambda = sigma(u)/sigma(v) = 6.28402
Var[u]/{Var[u]+Var[v]} = 0.93485
--------------------------------------------------------------------------------
Average inefficiency E[ui] = 1.20897
Average efficiency E[exp(-ui)] = 0.40883
--------------------------------------------------------------------------------
Stochastic Production/Profit Frontier, e = v - u
Estimator is 2 step Maximum Likelihood
Final maximum likelihood estimates
--------------------------------------------------------------------------------
Deterministic Component of SFA
--------------------------------------------------------------------------------
Coefficient Std. Error z value Pr(>|z|)
(Intercept) 1.27249 0.05191 24.5133 < 2.2e-16 ***
log(x1) 0.13811 0.02289 6.0339 1.601e-09 ***
log(x2) 0.18668 0.02252 8.2892 < 2.2e-16 ***
--------------------------------------------------------------------------------
Parameter in variance of u (one-sided error)
--------------------------------------------------------------------------------
Coefficient Std. Error z value Pr(>|z|)
Zu_(Intercept) 0.83111 0.06438 12.91 < 2.2e-16 ***
--------------------------------------------------------------------------------
Parameters in variance of v (two-sided error)
--------------------------------------------------------------------------------
Coefficient Std. Error z value Pr(>|z|)
Zv_(Intercept) -2.84491 0.33126 -8.5882 < 2.2e-16 ***
--------------------------------------------------------------------------------
Selection bias parameter
--------------------------------------------------------------------------------
Coefficient Std. Error z value Pr(>|z|)
rho 0.89550 0.28696 3.1207 0.001804 **
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
--------------------------------------------------------------------------------
Model was estimated on : Mar Tue 03, 2026 at 01:14
Log likelihood status: successful convergence
--------------------------------------------------------------------------------
------------------------------------------------------------
Group: 1 (N = 1006) Log-likelihood: -851.19119
------------------------------------------------------------
--------------------------------------------------------------------------------
Sample Selection Correction Stochastic Frontier Model
Dependent Variable: log(y)
Log likelihood solver: BFGS maximization
Log likelihood iter: 75
Log likelihood value: -851.19119
Log likelihood gradient norm: 7.36354e-06
Estimation based on: N = 498 of 1006 obs. and K = 6
Inf. Cr: AIC = 1714.4 AIC/N = 3.443
BIC = 1739.6 BIC/N = 3.493
HQIC = 1724.3 HQIC/N = 3.462
--------------------------------------------------------------------------------
Variances: Sigma-squared(v) = 0.06106
Sigma(v) = 0.06106
Sigma-squared(u) = 2.36720
Sigma(u) = 2.36720
Sigma = Sqrt[(s^2(u)+s^2(v))] = 1.55829
Gamma = sigma(u)^2/sigma^2 = 0.97485
Lambda = sigma(u)/sigma(v) = 6.22643
Var[u]/{Var[u]+Var[v]} = 0.93372
--------------------------------------------------------------------------------
Average inefficiency E[ui] = 1.22760
Average efficiency E[exp(-ui)] = 0.40470
--------------------------------------------------------------------------------
Stochastic Production/Profit Frontier, e = v - u
Estimator is 2 step Maximum Likelihood
Final maximum likelihood estimates
--------------------------------------------------------------------------------
Deterministic Component of SFA
--------------------------------------------------------------------------------
Coefficient Std. Error z value Pr(>|z|)
(Intercept) 1.33000 0.06335 20.9937 < 2.2e-16 ***
log(x1) 0.18342 0.02349 7.8098 5.727e-15 ***
log(x2) 0.09987 0.01918 5.2061 1.928e-07 ***
--------------------------------------------------------------------------------
Parameter in variance of u (one-sided error)
--------------------------------------------------------------------------------
Coefficient Std. Error z value Pr(>|z|)
Zu_(Intercept) 0.86171 0.06328 13.618 < 2.2e-16 ***
--------------------------------------------------------------------------------
Parameters in variance of v (two-sided error)
--------------------------------------------------------------------------------
Coefficient Std. Error z value Pr(>|z|)
Zv_(Intercept) -2.79590 0.33567 -8.3292 < 2.2e-16 ***
--------------------------------------------------------------------------------
Selection bias parameter
--------------------------------------------------------------------------------
Coefficient Std. Error z value Pr(>|z|)
rho 0.40516 0.35322 1.147 0.2514
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
--------------------------------------------------------------------------------
Model was estimated on : Mar Tue 03, 2026 at 01:15
Log likelihood status: successful convergence
--------------------------------------------------------------------------------
------------------------------------------------------------
Metafrontier Coefficients (qp):
Estimate Std. Error z value Pr(>|z|)
(Intercept) 1.33523296 0.00074356 1795.74 < 2.2e-16 ***
log(x1) 0.16970553 0.00054744 310.00 < 2.2e-16 ***
log(x2) 0.10714467 0.00050675 211.44 < 2.2e-16 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
------------------------------------------------------------
Efficiency Statistics (group means):
------------------------------------------------------------
N_obs N_valid TE_group_BC TE_group_JLMS TE_meta_BC TE_meta_JLMS MTR_BC MTR_JLMS
0 994 489 0.38742 0.38227 0.35331 0.34860 0.90889 0.90889
1 1006 498 0.41484 0.40585 0.41052 0.40162 0.98932 0.98932
Overall:
TE_group_BC=0.4011 TE_group_JLMS=0.3941
TE_meta_BC=0.3819 TE_meta_JLMS=0.3751
MTR_BC=0.9491 MTR_JLMS=0.9491
------------------------------------------------------------
Total Log-likelihood: -1651.136
AIC: 3332.273 BIC: 3416.286 HQIC: 3363.121
------------------------------------------------------------
Model was estimated on : Mar Tue 03, 2026 at 01:15
meta_sel_huang <- smfa(
formula = log(y) ~ log(x1) + log(x2),
selectionF = d ~ z1 + z2,
data = dat,
group = "group",
S = 1L,
udist = "hnormal",
groupType = "sfaselectioncross",
modelType = "greene10",
lType = "kronrod",
Nsub = 100,
uBound = Inf,
simType = "halton",
Nsim = 300,
prime = 2L,
burn = 10,
antithetics = FALSE,
seed = 12345,
method = "bfgs",
itermax = 2000,
metaMethod = "sfa",
sfaApproach = "huang"
)
summary(meta_sel_huang)Toggle to see the output
> meta_sel_huang <- smfa(
+ formula = log(y) ~ log(x1) + log(x2),
+ selectionF = d ~ z1 + z2,
+ data = dat,
+ group = "group",
+ S = 1L,
+ udist = "hnormal",
+ groupType = "sfaselectioncross",
+ modelType = "greene10",
+ lType = "kronrod",
+ Nsub = 100,
+ uBound = Inf,
+ simType = "halton",
+ Nsim = 300,
+ prime = 2L,
+ burn = 10,
+ antithetics = FALSE,
+ seed = 12345,
+ method = "bfgs",
+ itermax = 2000,
+ metaMethod = "sfa",
+ sfaApproach = "huang"
+ )
Estimating group-specific stochastic frontiers (sfaselectioncross) ...
Group: 0
First step probit model...
Second step Frontier model...
Group: 1
First step probit model...
Second step Frontier model...
Group frontiers estimated.
Estimating metafrontier using method: SFA Metafrontier [Huang et al. (2014), two-stage]
> summary(meta_sel_huang)
============================================================
Stochastic Metafrontier Analysis
Metafrontier method: SFA Metafrontier [Huang et al. (2014), two-stage]
Stochastic Production/Profit Frontier, e = v - u
SFA approach : huang
Group approach : Sample Selection Stochastic Frontier Analysis
Group estimator : sfaselectioncross
Group optim solver : BFGS maximization
Groups ( 2 ): 0, 1
Total observations : 2000
Distribution : hnormal
============================================================
------------------------------------------------------------
Group: 0 (N = 994) Log-likelihood: -799.94522
------------------------------------------------------------
--------------------------------------------------------------------------------
Sample Selection Correction Stochastic Frontier Model
Dependent Variable: log(y)
Log likelihood solver: BFGS maximization
Log likelihood iter: 349
Log likelihood value: -799.94522
Log likelihood gradient norm: 3.41571e+01
Estimation based on: N = 489 of 994 obs. and K = 6
Inf. Cr: AIC = 1611.9 AIC/N = 3.296
BIC = 1637.0 BIC/N = 3.348
HQIC = 1621.8 HQIC/N = 3.317
--------------------------------------------------------------------------------
Variances: Sigma-squared(v) = 0.05814
Sigma(v) = 0.05814
Sigma-squared(u) = 2.29587
Sigma(u) = 2.29587
Sigma = Sqrt[(s^2(u)+s^2(v))] = 1.53428
Gamma = sigma(u)^2/sigma^2 = 0.97530
Lambda = sigma(u)/sigma(v) = 6.28402
Var[u]/{Var[u]+Var[v]} = 0.93485
--------------------------------------------------------------------------------
Average inefficiency E[ui] = 1.20897
Average efficiency E[exp(-ui)] = 0.40883
--------------------------------------------------------------------------------
Stochastic Production/Profit Frontier, e = v - u
Estimator is 2 step Maximum Likelihood
Final maximum likelihood estimates
--------------------------------------------------------------------------------
Deterministic Component of SFA
--------------------------------------------------------------------------------
Coefficient Std. Error z value Pr(>|z|)
(Intercept) 1.27249 0.05191 24.5133 < 2.2e-16 ***
log(x1) 0.13811 0.02289 6.0339 1.601e-09 ***
log(x2) 0.18668 0.02252 8.2892 < 2.2e-16 ***
--------------------------------------------------------------------------------
Parameter in variance of u (one-sided error)
--------------------------------------------------------------------------------
Coefficient Std. Error z value Pr(>|z|)
Zu_(Intercept) 0.83111 0.06438 12.91 < 2.2e-16 ***
--------------------------------------------------------------------------------
Parameters in variance of v (two-sided error)
--------------------------------------------------------------------------------
Coefficient Std. Error z value Pr(>|z|)
Zv_(Intercept) -2.84491 0.33126 -8.5882 < 2.2e-16 ***
--------------------------------------------------------------------------------
Selection bias parameter
--------------------------------------------------------------------------------
Coefficient Std. Error z value Pr(>|z|)
rho 0.89550 0.28696 3.1207 0.001804 **
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
--------------------------------------------------------------------------------
Model was estimated on : Mar Tue 03, 2026 at 01:17
Log likelihood status: successful convergence
--------------------------------------------------------------------------------
------------------------------------------------------------
Group: 1 (N = 1006) Log-likelihood: -851.19119
------------------------------------------------------------
--------------------------------------------------------------------------------
Sample Selection Correction Stochastic Frontier Model
Dependent Variable: log(y)
Log likelihood solver: BFGS maximization
Log likelihood iter: 75
Log likelihood value: -851.19119
Log likelihood gradient norm: 7.36354e-06
Estimation based on: N = 498 of 1006 obs. and K = 6
Inf. Cr: AIC = 1714.4 AIC/N = 3.443
BIC = 1739.6 BIC/N = 3.493
HQIC = 1724.3 HQIC/N = 3.462
--------------------------------------------------------------------------------
Variances: Sigma-squared(v) = 0.06106
Sigma(v) = 0.06106
Sigma-squared(u) = 2.36720
Sigma(u) = 2.36720
Sigma = Sqrt[(s^2(u)+s^2(v))] = 1.55829
Gamma = sigma(u)^2/sigma^2 = 0.97485
Lambda = sigma(u)/sigma(v) = 6.22643
Var[u]/{Var[u]+Var[v]} = 0.93372
--------------------------------------------------------------------------------
Average inefficiency E[ui] = 1.22760
Average efficiency E[exp(-ui)] = 0.40470
--------------------------------------------------------------------------------
Stochastic Production/Profit Frontier, e = v - u
Estimator is 2 step Maximum Likelihood
Final maximum likelihood estimates
--------------------------------------------------------------------------------
Deterministic Component of SFA
--------------------------------------------------------------------------------
Coefficient Std. Error z value Pr(>|z|)
(Intercept) 1.33000 0.06335 20.9937 < 2.2e-16 ***
log(x1) 0.18342 0.02349 7.8098 5.727e-15 ***
log(x2) 0.09987 0.01918 5.2061 1.928e-07 ***
--------------------------------------------------------------------------------
Parameter in variance of u (one-sided error)
--------------------------------------------------------------------------------
Coefficient Std. Error z value Pr(>|z|)
Zu_(Intercept) 0.86171 0.06328 13.618 < 2.2e-16 ***
--------------------------------------------------------------------------------
Parameters in variance of v (two-sided error)
--------------------------------------------------------------------------------
Coefficient Std. Error z value Pr(>|z|)
Zv_(Intercept) -2.79590 0.33567 -8.3292 < 2.2e-16 ***
--------------------------------------------------------------------------------
Selection bias parameter
--------------------------------------------------------------------------------
Coefficient Std. Error z value Pr(>|z|)
rho 0.40516 0.35322 1.147 0.2514
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
--------------------------------------------------------------------------------
Model was estimated on : Mar Tue 03, 2026 at 01:17
Log likelihood status: successful convergence
--------------------------------------------------------------------------------
------------------------------------------------------------
Metafrontier Coefficients (sfa):
Meta-optim solver : BFGS maximization
Estimate Std. Error z value Pr(>|z|)
(Intercept) 1.3636175 0.0018936 720.12 < 2.2e-16 ***
log(x1) 0.1610981 0.0014891 108.19 < 2.2e-16 ***
log(x2) 0.1094264 0.0010363 105.59 < 2.2e-16 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Meta-frontier model details:
--------------------------------------------------------------------------------
Normal-Half Normal SF Model
Dependent Variable: group_fitted_values
Log likelihood solver: BFGS maximization
Log likelihood iter: 130
Log likelihood value: 1332.99378
Log likelihood gradient norm: 1.60687e-04
Estimation based on: N = 987 and K = 5
Inf. Cr: AIC = -2656.0 AIC/N = -2.691
BIC = -2631.5 BIC/N = -2.666
HQIC = -2646.7 HQIC/N = -2.682
--------------------------------------------------------------------------------
Variances: Sigma-squared(v) = 0.00025
Sigma(v) = 0.00025
Sigma-squared(u) = 0.01270
Sigma(u) = 0.01270
Sigma = Sqrt[(s^2(u)+s^2(v))] = 0.11381
Gamma = sigma(u)^2/sigma^2 = 0.98050
Lambda = sigma(u)/sigma(v) = 7.09083
Var[u]/{Var[u]+Var[v]} = 0.94811
--------------------------------------------------------------------------------
Average inefficiency E[ui] = 0.08992
Average efficiency E[exp(-ui)] = 0.91607
--------------------------------------------------------------------------------
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) = 1227.26017
LR statistic:
Chisq = 2*[LogL(H0)-LogL(H1)] = 211.46722
Kodde-Palm C*: 95%: 2.70554 99%: 5.41189
Coelli (1995) skewness test on OLS residuals
M3T: z = -1.05865
M3T: p.value = 0.28976
Final maximum likelihood estimates
--------------------------------------------------------------------------------
Deterministic Component of SFA
--------------------------------------------------------------------------------
Coefficient Std. Error z value Pr(>|z|)
(Intercept) 1.36362 0.00189 720.12 < 2.2e-16 ***
.X2 0.16110 0.00149 108.19 < 2.2e-16 ***
.X3 0.10943 0.00104 105.59 < 2.2e-16 ***
--------------------------------------------------------------------------------
Parameter in variance of u (one-sided error)
--------------------------------------------------------------------------------
Coefficient Std. Error z value Pr(>|z|)
Zu_(Intercept) -4.36616 0.05277 -82.743 < 2.2e-16 ***
--------------------------------------------------------------------------------
Parameters in variance of v (two-sided error)
--------------------------------------------------------------------------------
Coefficient Std. Error z value Pr(>|z|)
Zv_(Intercept) -8.28377 0.17628 -46.992 < 2.2e-16 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
--------------------------------------------------------------------------------
Model was estimated on : Mar Tue 03, 2026 at 01:17
Log likelihood status: successful convergence
--------------------------------------------------------------------------------
Log likelihood status: successful convergence
------------------------------------------------------------
Efficiency Statistics (group means):
------------------------------------------------------------
N_obs N_valid TE_group_BC TE_group_JLMS TE_meta_BC TE_meta_JLMS MTR_BC MTR_JLMS
0 994 489 0.38742 0.38227 0.34425 0.33963 0.88461 0.88451
1 1006 498 0.41484 0.40585 0.39861 0.38993 0.95990 0.95980
Overall:
TE_group_BC=0.4011 TE_group_JLMS=0.3941
TE_meta_BC=0.3714 TE_meta_JLMS=0.3648
MTR_BC=0.9223 MTR_JLMS=0.9222
------------------------------------------------------------
Total Log-likelihood: -318.1426
AIC: 670.2853 BIC: 765.5006 HQIC: 705.2464
------------------------------------------------------------
Model was estimated on : Mar Tue 03, 2026 at 01:17
meta_sel_odonnell <- smfa(
formula = log(y) ~ log(x1) + log(x2),
selectionF = d ~ z1 + z2,
data = dat,
group = "group",
S = 1L,
udist = "hnormal",
groupType = "sfaselectioncross",
modelType = "greene10",
lType = "kronrod",
Nsub = 100,
uBound = Inf,
method = "bfgs",
itermax = 2000,
metaMethod = "sfa",
sfaApproach = "ordonnell"
)
summary(meta_sel_odonnell)Toggle to see the output
> meta_sel_odonnell <- smfa(
+ formula = log(y) ~ log(x1) + log(x2),
+ selectionF = d ~ z1 + z2,
+ data = dat,
+ group = "group",
+ S = 1L,
+ udist = "hnormal",
+ groupType = "sfaselectioncross",
+ modelType = "greene10",
+ lType = "kronrod",
+ Nsub = 100,
+ uBound = Inf,
+ method = "bfgs",
+ itermax = 2000,
+ metaMethod = "sfa",
+ sfaApproach = "ordonnell"
+ )
Estimating group-specific stochastic frontiers (sfaselectioncross) ...
Group: 0
First step probit model...
Second step Frontier model...
Group: 1
First step probit model...
Second step Frontier model...
Group frontiers estimated.
Estimating metafrontier using method: SFA Metafrontier [O'Donnell et al. (2008), envelope]
Warning message:
The residuals of the OLS are right-skewed. This may indicate the absence of inefficiency or
model misspecification or sample 'bad luck'
> summary(meta_sel_odonnell)
============================================================
Stochastic Metafrontier Analysis
Metafrontier method: SFA Metafrontier [O'Donnell et al. (2008), envelope]
Stochastic Production/Profit Frontier, e = v - u
SFA approach : ordonnell
Group approach : Sample Selection Stochastic Frontier Analysis
Group estimator : sfaselectioncross
Group optim solver : BFGS maximization
Groups ( 2 ): 0, 1
Total observations : 2000
Distribution : hnormal
============================================================
------------------------------------------------------------
Group: 0 (N = 994) Log-likelihood: -799.94522
------------------------------------------------------------
--------------------------------------------------------------------------------
Sample Selection Correction Stochastic Frontier Model
Dependent Variable: log(y)
Log likelihood solver: BFGS maximization
Log likelihood iter: 349
Log likelihood value: -799.94522
Log likelihood gradient norm: 3.41571e+01
Estimation based on: N = 489 of 994 obs. and K = 6
Inf. Cr: AIC = 1611.9 AIC/N = 3.296
BIC = 1637.0 BIC/N = 3.348
HQIC = 1621.8 HQIC/N = 3.317
--------------------------------------------------------------------------------
Variances: Sigma-squared(v) = 0.05814
Sigma(v) = 0.05814
Sigma-squared(u) = 2.29587
Sigma(u) = 2.29587
Sigma = Sqrt[(s^2(u)+s^2(v))] = 1.53428
Gamma = sigma(u)^2/sigma^2 = 0.97530
Lambda = sigma(u)/sigma(v) = 6.28402
Var[u]/{Var[u]+Var[v]} = 0.93485
--------------------------------------------------------------------------------
Average inefficiency E[ui] = 1.20897
Average efficiency E[exp(-ui)] = 0.40883
--------------------------------------------------------------------------------
Stochastic Production/Profit Frontier, e = v - u
Estimator is 2 step Maximum Likelihood
Final maximum likelihood estimates
--------------------------------------------------------------------------------
Deterministic Component of SFA
--------------------------------------------------------------------------------
Coefficient Std. Error z value Pr(>|z|)
(Intercept) 1.27249 0.05191 24.5133 < 2.2e-16 ***
log(x1) 0.13811 0.02289 6.0339 1.601e-09 ***
log(x2) 0.18668 0.02252 8.2892 < 2.2e-16 ***
--------------------------------------------------------------------------------
Parameter in variance of u (one-sided error)
--------------------------------------------------------------------------------
Coefficient Std. Error z value Pr(>|z|)
Zu_(Intercept) 0.83111 0.06438 12.91 < 2.2e-16 ***
--------------------------------------------------------------------------------
Parameters in variance of v (two-sided error)
--------------------------------------------------------------------------------
Coefficient Std. Error z value Pr(>|z|)
Zv_(Intercept) -2.84491 0.33126 -8.5882 < 2.2e-16 ***
--------------------------------------------------------------------------------
Selection bias parameter
--------------------------------------------------------------------------------
Coefficient Std. Error z value Pr(>|z|)
rho 0.89550 0.28696 3.1207 0.001804 **
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
--------------------------------------------------------------------------------
Model was estimated on : Mar Tue 03, 2026 at 01:19
Log likelihood status: successful convergence
--------------------------------------------------------------------------------
------------------------------------------------------------
Group: 1 (N = 1006) Log-likelihood: -851.19119
------------------------------------------------------------
--------------------------------------------------------------------------------
Sample Selection Correction Stochastic Frontier Model
Dependent Variable: log(y)
Log likelihood solver: BFGS maximization
Log likelihood iter: 75
Log likelihood value: -851.19119
Log likelihood gradient norm: 7.36354e-06
Estimation based on: N = 498 of 1006 obs. and K = 6
Inf. Cr: AIC = 1714.4 AIC/N = 3.443
BIC = 1739.6 BIC/N = 3.493
HQIC = 1724.3 HQIC/N = 3.462
--------------------------------------------------------------------------------
Variances: Sigma-squared(v) = 0.06106
Sigma(v) = 0.06106
Sigma-squared(u) = 2.36720
Sigma(u) = 2.36720
Sigma = Sqrt[(s^2(u)+s^2(v))] = 1.55829
Gamma = sigma(u)^2/sigma^2 = 0.97485
Lambda = sigma(u)/sigma(v) = 6.22643
Var[u]/{Var[u]+Var[v]} = 0.93372
--------------------------------------------------------------------------------
Average inefficiency E[ui] = 1.22760
Average efficiency E[exp(-ui)] = 0.40470
--------------------------------------------------------------------------------
Stochastic Production/Profit Frontier, e = v - u
Estimator is 2 step Maximum Likelihood
Final maximum likelihood estimates
--------------------------------------------------------------------------------
Deterministic Component of SFA
--------------------------------------------------------------------------------
Coefficient Std. Error z value Pr(>|z|)
(Intercept) 1.33000 0.06335 20.9937 < 2.2e-16 ***
log(x1) 0.18342 0.02349 7.8098 5.727e-15 ***
log(x2) 0.09987 0.01918 5.2061 1.928e-07 ***
--------------------------------------------------------------------------------
Parameter in variance of u (one-sided error)
--------------------------------------------------------------------------------
Coefficient Std. Error z value Pr(>|z|)
Zu_(Intercept) 0.86171 0.06328 13.618 < 2.2e-16 ***
--------------------------------------------------------------------------------
Parameters in variance of v (two-sided error)
--------------------------------------------------------------------------------
Coefficient Std. Error z value Pr(>|z|)
Zv_(Intercept) -2.79590 0.33567 -8.3292 < 2.2e-16 ***
--------------------------------------------------------------------------------
Selection bias parameter
--------------------------------------------------------------------------------
Coefficient Std. Error z value Pr(>|z|)
rho 0.40516 0.35322 1.147 0.2514
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
--------------------------------------------------------------------------------
Model was estimated on : Mar Tue 03, 2026 at 01:19
Log likelihood status: successful convergence
--------------------------------------------------------------------------------
------------------------------------------------------------
Metafrontier Coefficients (sfa):
Meta-optim solver : BFGS maximization
Estimate Std. Error z value Pr(>|z|)
(Intercept) 1.33534799 0.00637899 209.34 < 2.2e-16 ***
log(x1) 0.16970553 0.00054661 310.47 < 2.2e-16 ***
log(x2) 0.10714469 0.00050598 211.76 < 2.2e-16 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Meta-frontier model details:
--------------------------------------------------------------------------------
Normal-Half Normal SF Model
Dependent Variable: lp_envelope
Log likelihood solver: BFGS maximization
Log likelihood iter: 336
Log likelihood value: 2562.82127
Log likelihood gradient norm: 7.45486e-02
Estimation based on: N = 987 and K = 5
Inf. Cr: AIC = -5115.6 AIC/N = -5.183
BIC = -5091.2 BIC/N = -5.158
HQIC = -5106.3 HQIC/N = -5.174
--------------------------------------------------------------------------------
Variances: Sigma-squared(v) = 0.00033
Sigma(v) = 0.00033
Sigma-squared(u) = 0.00000
Sigma(u) = 0.00000
Sigma = Sqrt[(s^2(u)+s^2(v))] = 0.01803
Gamma = sigma(u)^2/sigma^2 = 0.00006
Lambda = sigma(u)/sigma(v) = 0.00799
Var[u]/{Var[u]+Var[v]} = 0.00002
--------------------------------------------------------------------------------
Average inefficiency E[ui] = 0.00012
Average efficiency E[exp(-ui)] = 0.99988
--------------------------------------------------------------------------------
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) = 2562.82131
LR statistic:
Chisq = 2*[LogL(H0)-LogL(H1)] = -0.00007
Kodde-Palm C*: 95%: 2.70554 99%: 5.41189
Coelli (1995) skewness test on OLS residuals
M3T: z = 25.86056
M3T: p.value = 0.00000
Final maximum likelihood estimates
--------------------------------------------------------------------------------
Deterministic Component of SFA
--------------------------------------------------------------------------------
Coefficient Std. Error z value Pr(>|z|)
(Intercept) 1.33535 0.00638 209.34 < 2.2e-16 ***
.X2 0.16971 0.00055 310.47 < 2.2e-16 ***
.X3 0.10714 0.00051 211.76 < 2.2e-16 ***
--------------------------------------------------------------------------------
Parameter in variance of u (one-sided error)
--------------------------------------------------------------------------------
Coefficient Std. Error z value Pr(>|z|)
Zu_(Intercept) -17.689 110.175 -0.1606 0.8724
--------------------------------------------------------------------------------
Parameters in variance of v (two-sided error)
--------------------------------------------------------------------------------
Coefficient Std. Error z value Pr(>|z|)
Zv_(Intercept) -8.03105 0.04509 -178.12 < 2.2e-16 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
--------------------------------------------------------------------------------
Model was estimated on : Mar Tue 03, 2026 at 01:19
Log likelihood status: successful convergence
--------------------------------------------------------------------------------
Log likelihood status: successful convergence
------------------------------------------------------------
Efficiency Statistics (group means):
------------------------------------------------------------
N_obs N_valid TE_group_BC TE_group_JLMS TE_meta_BC TE_meta_JLMS MTR_BC MTR_JLMS
0 994 489 0.38742 0.38227 0.99989 0.99988 8.54748 8.65155
1 1006 498 0.41484 0.40585 0.99988 0.99988 8.59832 8.82540
Overall:
TE_group_BC=0.4011 TE_group_JLMS=0.3941
TE_meta_BC=0.9999 TE_meta_JLMS=0.9999
MTR_BC=8.5729 MTR_JLMS=8.7385
------------------------------------------------------------
Total Log-likelihood: 911.6848
AIC: -1789.37 BIC: -1694.154 HQIC: -1754.409
------------------------------------------------------------
Model was estimated on : Mar Tue 03, 2026 at 01:19
Warning message:
987 MTR value(s) > 1 detected in O'Donnell SFA approach. This typically occurs when the second-stage SFA estimates near-zero inefficiency (sigma_u -> 0), causing TE_meta ~= 1 and MTR = TE_meta/TE_group > 1. Consider using metaMethod='lp' or sfaApproach='huang' instead.
The efficiencies() function returns a data frame with one row per observation containing group-specific and metafrontier efficiency estimates and MTRs. The columns present depend on groupType:
| Column | Description |
|---|---|
id |
Observation identifier |
group / Group_c |
Technology group identifier |
u_g |
Group-specific inefficiency — Jondrow et al. (1982) |
TE_group_JLMS |
Group TE — Jondrow et al. (1982): exp(−u) |
TE_group_BC |
Group TE — Battese & Coelli (1988): E[exp(−u)|ε] |
TE_group_BC_reciprocal |
Reciprocal of Battese & Coelli (1988) group TE |
uLB_g, uUB_g |
Confidence bounds for u (sfacross only) |
m_g, TE_group_mode |
Mode-based inefficiency and TE (sfacross only) |
PosteriorProb_c, PosteriorProb_c1 … |
Posterior class probabilities (sfalcmcross only) |
u_meta |
Metafrontier technology gap U |
TE_meta_JLMS |
Metafrontier TE (JLMS basis): TE_group_JLMS × MTR |
TE_meta_BC |
Metafrontier TE (BC basis): TE_group_BC × MTR |
MTR_JLMS |
Metatechnology ratio (JLMS basis) |
MTR_BC |
Metatechnology ratio (BC basis) |
# Example: extract and print for group 1 selected farms only
ef_sel_lp <- efficiencies(meta_sel_lp)
sel_grp1 <- ef_sel_lp[ef_sel_lp$group == 1 & !is.na(ef_sel_lp$TE_group_BC), ]
summary(sel_grp1[, c("TE_group_BC", "TE_meta_BC", "MTR_BC")])- Battese, G. E., & Coelli, T. J. (1988). Prediction of firm-level technical efficiencies with a generalized frontier production function and panel data. Journal of Econometrics, 38(3), 387–399. https://doi.org/10.1016/0304-4076(88)90053-X
- Battese, G. E., Rao, D. S. P., & 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. https://doi.org/10.1023/B:PROD.0000012454.06094.29
- Dakpo, K. H., Desjeux, Y., & Latruffe, L. (2023). sfaR: Stochastic Frontier Analysis using R. R package version 1.0.1. https://CRAN.R-project.org/package=sfaR
- Greene, W. (2010). A stochastic frontier model with correction for sample selection. Journal of Productivity Analysis, 34(1), 15–24. https://doi.org/10.1007/s11123-009-0159-1
- Huang, C. J., Huang, T.-H., & 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. https://doi.org/10.1007/s11123-014-0402-2
- Jondrow, J., Lovell, C. A. K., Materov, I. S., & Schmidt, P. (1982). On the estimation of technical inefficiency in the stochastic frontier production function model. Journal of Econometrics, 19(2–3), 233–238. https://doi.org/10.1016/0304-4076(82)90004-5
- O'Donnell, C. J., Rao, D. S. P., & Battese, G. E. (2008). Metafrontier frameworks for the study of firm-level efficiencies and technology ratios. Empirical Economics, 34(2), 231–255. https://doi.org/10.1007/s00181-007-0119-4
- Orea, L., & Kumbhakar, S. C. (2004). Efficiency measurement using a latent class stochastic frontier model. Empirical Economics, 29(1), 169–183. https://doi.org/10.1007/s00181-003-0184-2