Sample Selection SFA Metafrontier (groupType = "sfaselectioncross")
Source:vignettes/sfaselectioncross-metafrontier.Rmd
sfaselectioncross-metafrontier.RmdOverview
Sample selection bias arises when the observed sample is not a random draw from the population. For example:
- Only firms above a revenue threshold are surveyed.
- Only farms that adopted a technology are observed using it.
- Participation in a programme is voluntary, so only volunteers are observed.
If selection into the sample is correlated with firm efficiency,
ignoring this leads to biased frontier estimates.
sfaselectioncross implements the two-step ML
estimator of Greene (2010), leveraging the sample selection
correction provided via sfaR (Dakpo et al. 2022), which
corrects for this bias using a probit selection equation (Heckman 1979
correction).
The selection model requires: - A binary selection
indicator d (1 = selected/observed, 0 = not
selected). - A selection equation formula
selectionF specifying which variables drive selection. At
least one variable must appear in selectionF but not in the
main frontier formula. - Only selected observations
(d == 1) participate in the frontier and receive efficiency
estimates. Efficiency for non-selected observations is
NA.
Data Preparation (Simulated Example)
We simulate data following the approach in the sfaR
documentation:
library(smfa)
#> Loading required package: sfaR
#> **** *******
#> /**/ /**////**
#> ****** ****** ****** /** /**
#> **//// ///**/ //////** /*******
#> //***** /** ******* /**///**
#> /////** /** **////** /** //**
#> ****** /** //********/** //**
#> ////// // //////// // // version 1.0.1
#>
#> * Please cite the 'sfaR' package as:
#> Dakpo KH., Desjeux Y., Henningsen A., and Latruffe L. (2024). sfaR: Stochastic Frontier Analysis Using R. R package version 1.0.1.
#>
#> See also: citation("sfaR")
#>
#> * For any questions, suggestions, or comments on the 'sfaR' package, you can contact directly the authors or visit: https://github.com/hdakpo/sfaR/issues
#> .d888
#> d88P"
#> 888
#> .d8888b 88888b.d88b. 888888 8888b.
#> 88K 888 "888 "88b 888 "88b
#> Y8888b. 888 888 888 888 .d888888
#> X88 888 888 888 888 888 888
#> 88888P' 888 888 888 888 "Y888888
#> version 1.0.0
#>
#> * Please cite the 'smfa' package as:
#> Owili, S. O. (2026). smfa: Stochastic Metafrontier Analysis. R package version 1.0.0.
#>
#> See also: citation("smfa")
#>
#> * For any questions, suggestions, or comments on the 'smfa' package, you can contact the authors directly or visit:
#> https://github.com/SulmanOlieko/smfa/issues
N <- 500; 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) # 1 = selected into the sample
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
#> 237 263
#> 0 1
#> 1013 987Method 1: sfaselectioncross + LP Metafrontier
meta_sel_lp <- smfa(
formula = log(y) ~ log(x1) + log(x2),
selectionF = d ~ z1 + z2, # selection equation: d is the binary indicator
data = dat,
group = "group",
S = 1L,
udist = "hnormal",
groupType = "sfaselectioncross",
modelType = "greene10", # Greene (2010) two-step ML correction
lType = "kronrod", # integration method for the selection likelihood
Nsub = 20, # number of sub-intervals for numerical integration
uBound = Inf,
method = "bfgs",
itermax = 2000,
metaMethod = "lp"
)
#> First step probit model...
#> Second step Frontier model...
#> First step probit model...
#> Second step Frontier model...
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 : 500
#> Distribution : hnormal
#> ============================================================
#>
#> ------------------------------------------------------------
#> Group: 0 (N = 252) Log-likelihood: -225.33119
#> ------------------------------------------------------------
#> --------------------------------------------------------------------------------
#> Sample Selection Correction Stochastic Frontier Model
#> Dependent Variable: log(y)
#> Log likelihood solver: BFGS maximization
#> Log likelihood iter: 67
#> Log likelihood value: -225.33119
#> Log likelihood gradient norm: 5.77769e-07
#> Estimation based on: N = 131 of 252 obs. and K = 6
#> Inf. Cr: AIC = 462.7 AIC/N = 3.532
#> BIC = 479.9 BIC/N = 3.663
#> HQIC = 469.7 HQIC/N = 3.585
#> --------------------------------------------------------------------------------
#> Variances: Sigma-squared(v) = 0.01981
#> Sigma(v) = 0.01981
#> Sigma-squared(u) = 2.94034
#> Sigma(u) = 2.94034
#> Sigma = Sqrt[(s^2(u)+s^2(v))] = 1.72051
#> Gamma = sigma(u)^2/sigma^2 = 0.99331
#> Lambda = sigma(u)/sigma(v) = 12.18326
#> Var[u]/{Var[u]+Var[v]} = 0.98180
#> --------------------------------------------------------------------------------
#> Average inefficiency E[ui] = 1.36817
#> Average efficiency E[exp(-ui)] = 0.37581
#> --------------------------------------------------------------------------------
#> 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) 1.36885 0.11843 11.5584 < 2.2e-16 ***
#> log(x1) 0.17123 0.06350 2.6963 0.007011 **
#> log(x2) 0.07768 0.05238 1.4829 0.138102
#> --------------------------------------------------------------------------------
#> Parameter in variance of u (one-sided error)
#> --------------------------------------------------------------------------------
#> Coefficient Std. Error z value Pr(>|z|)
#> Zu_(Intercept) 1.07853 0.10204 10.569 < 2.2e-16 ***
#> --------------------------------------------------------------------------------
#> Parameters in variance of v (two-sided error)
#> --------------------------------------------------------------------------------
#> Coefficient Std. Error z value Pr(>|z|)
#> Zv_(Intercept) -3.9216 1.1265 -3.4813 0.000499 ***
#> --------------------------------------------------------------------------------
#> Selection bias parameter
#> --------------------------------------------------------------------------------
#> Coefficient Std. Error z value Pr(>|z|)
#> rho 0.27413 1.03629 0.2645 0.7914
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> --------------------------------------------------------------------------------
#> Model was estimated on : Apr Fri 24, 2026 at 15:14
#> Log likelihood status: successful convergence
#> --------------------------------------------------------------------------------
#>
#> ------------------------------------------------------------
#> Group: 1 (N = 248) Log-likelihood: -197.67108
#> ------------------------------------------------------------
#> --------------------------------------------------------------------------------
#> Sample Selection Correction Stochastic Frontier Model
#> Dependent Variable: log(y)
#> Log likelihood solver: BFGS maximization
#> Log likelihood iter: 68
#> Log likelihood value: -197.67108
#> Log likelihood gradient norm: 2.70510e-06
#> Estimation based on: N = 132 of 248 obs. and K = 6
#> Inf. Cr: AIC = 407.3 AIC/N = 3.086
#> BIC = 424.6 BIC/N = 3.217
#> HQIC = 414.4 HQIC/N = 3.139
#> --------------------------------------------------------------------------------
#> Variances: Sigma-squared(v) = 0.03365
#> Sigma(v) = 0.03365
#> Sigma-squared(u) = 1.84490
#> Sigma(u) = 1.84490
#> Sigma = Sqrt[(s^2(u)+s^2(v))] = 1.37060
#> Gamma = sigma(u)^2/sigma^2 = 0.98209
#> Lambda = sigma(u)/sigma(v) = 7.40483
#> Var[u]/{Var[u]+Var[v]} = 0.95221
#> --------------------------------------------------------------------------------
#> Average inefficiency E[ui] = 1.08374
#> Average efficiency E[exp(-ui)] = 0.43864
#> --------------------------------------------------------------------------------
#> 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) 1.22609 0.09780 12.5368 < 2.2e-16 ***
#> log(x1) 0.14445 0.03802 3.7994 0.000145 ***
#> log(x2) 0.11056 0.03775 2.9290 0.003401 **
#> --------------------------------------------------------------------------------
#> Parameter in variance of u (one-sided error)
#> --------------------------------------------------------------------------------
#> Coefficient Std. Error z value Pr(>|z|)
#> Zu_(Intercept) 0.61243 0.12841 4.7694 1.847e-06 ***
#> --------------------------------------------------------------------------------
#> Parameters in variance of v (two-sided error)
#> --------------------------------------------------------------------------------
#> Coefficient Std. Error z value Pr(>|z|)
#> Zv_(Intercept) -3.39184 0.69675 -4.8681 1.127e-06 ***
#> --------------------------------------------------------------------------------
#> Selection bias parameter
#> --------------------------------------------------------------------------------
#> Coefficient Std. Error z value Pr(>|z|)
#> rho 0.78787 0.60643 1.2992 0.1939
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> --------------------------------------------------------------------------------
#> Model was estimated on : Apr Fri 24, 2026 at 15:14
#> 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
#> 0 252 131 0.39953 0.39621 0.39953 0.39621 1.00000
#> 1 248 132 0.43254 0.42840 0.37567 0.37207 0.86741
#> MTR_JLMS
#> 0 1.00000
#> 1 0.86741
#>
#> Overall:
#> TE_group_BC=0.4160 TE_group_JLMS=0.4123
#> TE_meta_BC=0.3876 TE_meta_JLMS=0.3841
#> MTR_BC=0.9337 MTR_JLMS=0.9337
#> ------------------------------------------------------------
#> Total Log-likelihood: -423.0023
#> AIC: 870.0045 BIC: 920.5798 HQIC: 889.8502
#> ------------------------------------------------------------
#> Model was estimated on : Apr Fri 24, 2026 at 15:14Note: The
selectionFargument is compulsory forgroupType = "sfaselectioncross". The left-hand side must be the binary selection variable (d). At least one regressor in the selection equation should not appear in the main frontier formula (exclusion restriction for identification).
Method 2: sfaselectioncross + QP Metafrontier
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 = 20,
uBound = Inf,
method = "bfgs",
itermax = 2000,
metaMethod = "qp"
)
#> First step probit model...
#> Second step Frontier model...
#> First step probit model...
#> Second step Frontier model...
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 : 500
#> Distribution : hnormal
#> ============================================================
#>
#> ------------------------------------------------------------
#> Group: 0 (N = 252) Log-likelihood: -225.33119
#> ------------------------------------------------------------
#> --------------------------------------------------------------------------------
#> Sample Selection Correction Stochastic Frontier Model
#> Dependent Variable: log(y)
#> Log likelihood solver: BFGS maximization
#> Log likelihood iter: 67
#> Log likelihood value: -225.33119
#> Log likelihood gradient norm: 5.77769e-07
#> Estimation based on: N = 131 of 252 obs. and K = 6
#> Inf. Cr: AIC = 462.7 AIC/N = 3.532
#> BIC = 479.9 BIC/N = 3.663
#> HQIC = 469.7 HQIC/N = 3.585
#> --------------------------------------------------------------------------------
#> Variances: Sigma-squared(v) = 0.01981
#> Sigma(v) = 0.01981
#> Sigma-squared(u) = 2.94034
#> Sigma(u) = 2.94034
#> Sigma = Sqrt[(s^2(u)+s^2(v))] = 1.72051
#> Gamma = sigma(u)^2/sigma^2 = 0.99331
#> Lambda = sigma(u)/sigma(v) = 12.18326
#> Var[u]/{Var[u]+Var[v]} = 0.98180
#> --------------------------------------------------------------------------------
#> Average inefficiency E[ui] = 1.36817
#> Average efficiency E[exp(-ui)] = 0.37581
#> --------------------------------------------------------------------------------
#> 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) 1.36885 0.11843 11.5584 < 2.2e-16 ***
#> log(x1) 0.17123 0.06350 2.6963 0.007011 **
#> log(x2) 0.07768 0.05238 1.4829 0.138102
#> --------------------------------------------------------------------------------
#> Parameter in variance of u (one-sided error)
#> --------------------------------------------------------------------------------
#> Coefficient Std. Error z value Pr(>|z|)
#> Zu_(Intercept) 1.07853 0.10204 10.569 < 2.2e-16 ***
#> --------------------------------------------------------------------------------
#> Parameters in variance of v (two-sided error)
#> --------------------------------------------------------------------------------
#> Coefficient Std. Error z value Pr(>|z|)
#> Zv_(Intercept) -3.9216 1.1265 -3.4813 0.000499 ***
#> --------------------------------------------------------------------------------
#> Selection bias parameter
#> --------------------------------------------------------------------------------
#> Coefficient Std. Error z value Pr(>|z|)
#> rho 0.27413 1.03629 0.2645 0.7914
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> --------------------------------------------------------------------------------
#> Model was estimated on : Apr Fri 24, 2026 at 15:14
#> Log likelihood status: successful convergence
#> --------------------------------------------------------------------------------
#>
#> ------------------------------------------------------------
#> Group: 1 (N = 248) Log-likelihood: -197.67108
#> ------------------------------------------------------------
#> --------------------------------------------------------------------------------
#> Sample Selection Correction Stochastic Frontier Model
#> Dependent Variable: log(y)
#> Log likelihood solver: BFGS maximization
#> Log likelihood iter: 68
#> Log likelihood value: -197.67108
#> Log likelihood gradient norm: 2.70510e-06
#> Estimation based on: N = 132 of 248 obs. and K = 6
#> Inf. Cr: AIC = 407.3 AIC/N = 3.086
#> BIC = 424.6 BIC/N = 3.217
#> HQIC = 414.4 HQIC/N = 3.139
#> --------------------------------------------------------------------------------
#> Variances: Sigma-squared(v) = 0.03365
#> Sigma(v) = 0.03365
#> Sigma-squared(u) = 1.84490
#> Sigma(u) = 1.84490
#> Sigma = Sqrt[(s^2(u)+s^2(v))] = 1.37060
#> Gamma = sigma(u)^2/sigma^2 = 0.98209
#> Lambda = sigma(u)/sigma(v) = 7.40483
#> Var[u]/{Var[u]+Var[v]} = 0.95221
#> --------------------------------------------------------------------------------
#> Average inefficiency E[ui] = 1.08374
#> Average efficiency E[exp(-ui)] = 0.43864
#> --------------------------------------------------------------------------------
#> 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) 1.22609 0.09780 12.5368 < 2.2e-16 ***
#> log(x1) 0.14445 0.03802 3.7994 0.000145 ***
#> log(x2) 0.11056 0.03775 2.9290 0.003401 **
#> --------------------------------------------------------------------------------
#> Parameter in variance of u (one-sided error)
#> --------------------------------------------------------------------------------
#> Coefficient Std. Error z value Pr(>|z|)
#> Zu_(Intercept) 0.61243 0.12841 4.7694 1.847e-06 ***
#> --------------------------------------------------------------------------------
#> Parameters in variance of v (two-sided error)
#> --------------------------------------------------------------------------------
#> Coefficient Std. Error z value Pr(>|z|)
#> Zv_(Intercept) -3.39184 0.69675 -4.8681 1.127e-06 ***
#> --------------------------------------------------------------------------------
#> Selection bias parameter
#> --------------------------------------------------------------------------------
#> Coefficient Std. Error z value Pr(>|z|)
#> rho 0.78787 0.60643 1.2992 0.1939
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> --------------------------------------------------------------------------------
#> Model was estimated on : Apr Fri 24, 2026 at 15:14
#> Log likelihood status: successful convergence
#> --------------------------------------------------------------------------------
#>
#> ------------------------------------------------------------
#> Metafrontier Coefficients (qp):
#> Estimate Std. Error z value Pr(>|z|)
#> (Intercept) 1.36736720 0.00042531 3215.00 < 2.2e-16 ***
#> log(x1) 0.16870720 0.00027176 620.79 < 2.2e-16 ***
#> log(x2) 0.07759335 0.00030299 256.09 < 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
#> 0 252 131 0.39953 0.39621 0.39914 0.39582 0.99892
#> 1 248 132 0.43254 0.42840 0.37556 0.37196 0.86709
#> MTR_JLMS
#> 0 0.99892
#> 1 0.86709
#>
#> Overall:
#> TE_group_BC=0.4160 TE_group_JLMS=0.4123
#> TE_meta_BC=0.3874 TE_meta_JLMS=0.3839
#> MTR_BC=0.9330 MTR_JLMS=0.9330
#> ------------------------------------------------------------
#> Total Log-likelihood: -423.0023
#> AIC: 876.0045 BIC: 939.2237 HQIC: 900.8116
#> ------------------------------------------------------------
#> Model was estimated on : Apr Fri 24, 2026 at 15:14Method 3: sfaselectioncross + SFA (Huang)
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,
method = "bfgs",
itermax = 2000,
metaMethod = "sfa",
sfaApproach = "huang"
)
#> First step probit model...
#> Second step Frontier model...
#> First step probit model...
#> Second step Frontier model...
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 : 500
#> Distribution : hnormal
#> ============================================================
#>
#> ------------------------------------------------------------
#> Group: 0 (N = 252) Log-likelihood: -225.33119
#> ------------------------------------------------------------
#> --------------------------------------------------------------------------------
#> Sample Selection Correction Stochastic Frontier Model
#> Dependent Variable: log(y)
#> Log likelihood solver: BFGS maximization
#> Log likelihood iter: 67
#> Log likelihood value: -225.33119
#> Log likelihood gradient norm: 5.77769e-07
#> Estimation based on: N = 131 of 252 obs. and K = 6
#> Inf. Cr: AIC = 462.7 AIC/N = 3.532
#> BIC = 479.9 BIC/N = 3.663
#> HQIC = 469.7 HQIC/N = 3.585
#> --------------------------------------------------------------------------------
#> Variances: Sigma-squared(v) = 0.01981
#> Sigma(v) = 0.01981
#> Sigma-squared(u) = 2.94034
#> Sigma(u) = 2.94034
#> Sigma = Sqrt[(s^2(u)+s^2(v))] = 1.72051
#> Gamma = sigma(u)^2/sigma^2 = 0.99331
#> Lambda = sigma(u)/sigma(v) = 12.18326
#> Var[u]/{Var[u]+Var[v]} = 0.98180
#> --------------------------------------------------------------------------------
#> Average inefficiency E[ui] = 1.36817
#> Average efficiency E[exp(-ui)] = 0.37581
#> --------------------------------------------------------------------------------
#> 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) 1.36885 0.11843 11.5584 < 2.2e-16 ***
#> log(x1) 0.17123 0.06350 2.6963 0.007011 **
#> log(x2) 0.07768 0.05238 1.4829 0.138102
#> --------------------------------------------------------------------------------
#> Parameter in variance of u (one-sided error)
#> --------------------------------------------------------------------------------
#> Coefficient Std. Error z value Pr(>|z|)
#> Zu_(Intercept) 1.07853 0.10204 10.569 < 2.2e-16 ***
#> --------------------------------------------------------------------------------
#> Parameters in variance of v (two-sided error)
#> --------------------------------------------------------------------------------
#> Coefficient Std. Error z value Pr(>|z|)
#> Zv_(Intercept) -3.9216 1.1265 -3.4813 0.000499 ***
#> --------------------------------------------------------------------------------
#> Selection bias parameter
#> --------------------------------------------------------------------------------
#> Coefficient Std. Error z value Pr(>|z|)
#> rho 0.27413 1.03629 0.2645 0.7914
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> --------------------------------------------------------------------------------
#> Model was estimated on : Apr Fri 24, 2026 at 15:14
#> Log likelihood status: successful convergence
#> --------------------------------------------------------------------------------
#>
#> ------------------------------------------------------------
#> Group: 1 (N = 248) Log-likelihood: -197.67108
#> ------------------------------------------------------------
#> --------------------------------------------------------------------------------
#> Sample Selection Correction Stochastic Frontier Model
#> Dependent Variable: log(y)
#> Log likelihood solver: BFGS maximization
#> Log likelihood iter: 68
#> Log likelihood value: -197.67108
#> Log likelihood gradient norm: 2.70510e-06
#> Estimation based on: N = 132 of 248 obs. and K = 6
#> Inf. Cr: AIC = 407.3 AIC/N = 3.086
#> BIC = 424.6 BIC/N = 3.217
#> HQIC = 414.4 HQIC/N = 3.139
#> --------------------------------------------------------------------------------
#> Variances: Sigma-squared(v) = 0.03365
#> Sigma(v) = 0.03365
#> Sigma-squared(u) = 1.84490
#> Sigma(u) = 1.84490
#> Sigma = Sqrt[(s^2(u)+s^2(v))] = 1.37060
#> Gamma = sigma(u)^2/sigma^2 = 0.98209
#> Lambda = sigma(u)/sigma(v) = 7.40483
#> Var[u]/{Var[u]+Var[v]} = 0.95221
#> --------------------------------------------------------------------------------
#> Average inefficiency E[ui] = 1.08374
#> Average efficiency E[exp(-ui)] = 0.43864
#> --------------------------------------------------------------------------------
#> 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) 1.22609 0.09780 12.5368 < 2.2e-16 ***
#> log(x1) 0.14445 0.03802 3.7994 0.000145 ***
#> log(x2) 0.11056 0.03775 2.9290 0.003401 **
#> --------------------------------------------------------------------------------
#> Parameter in variance of u (one-sided error)
#> --------------------------------------------------------------------------------
#> Coefficient Std. Error z value Pr(>|z|)
#> Zu_(Intercept) 0.61243 0.12841 4.7694 1.847e-06 ***
#> --------------------------------------------------------------------------------
#> Parameters in variance of v (two-sided error)
#> --------------------------------------------------------------------------------
#> Coefficient Std. Error z value Pr(>|z|)
#> Zv_(Intercept) -3.39184 0.69675 -4.8681 1.127e-06 ***
#> --------------------------------------------------------------------------------
#> Selection bias parameter
#> --------------------------------------------------------------------------------
#> Coefficient Std. Error z value Pr(>|z|)
#> rho 0.78787 0.60643 1.2992 0.1939
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> --------------------------------------------------------------------------------
#> Model was estimated on : Apr Fri 24, 2026 at 15:14
#> Log likelihood status: successful convergence
#> --------------------------------------------------------------------------------
#>
#> ------------------------------------------------------------
#> Metafrontier Coefficients (sfa):
#> Meta-optim solver : BFGS maximization
#> Estimate Std. Error z value Pr(>|z|)
#> (Intercept) 1.2984590 0.2752514 4.7174 2.389e-06 ***
#> log(x1) 0.1557503 0.0038319 40.6458 < 2.2e-16 ***
#> log(x2) 0.0921453 0.0042741 21.5589 < 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: 608
#> Log likelihood value: 305.40441
#> Log likelihood gradient norm: 2.74614e-05
#> Estimation based on: N = 263 and K = 5
#> Inf. Cr: AIC = -600.8 AIC/N = -2.284
#> BIC = -582.9 BIC/N = -2.217
#> HQIC = -593.6 HQIC/N = -2.257
#> --------------------------------------------------------------------------------
#> Variances: Sigma-squared(v) = 0.00573
#> Sigma(v) = 0.00573
#> Sigma-squared(u) = 0.00002
#> Sigma(u) = 0.00002
#> Sigma = Sqrt[(s^2(u)+s^2(v))] = 0.07583
#> Gamma = sigma(u)^2/sigma^2 = 0.00295
#> Lambda = sigma(u)/sigma(v) = 0.05438
#> Var[u]/{Var[u]+Var[v]} = 0.00107
#> --------------------------------------------------------------------------------
#> Average inefficiency E[ui] = 0.00329
#> Average efficiency E[exp(-ui)] = 0.99672
#> --------------------------------------------------------------------------------
#> 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) = 305.40442
#> LR statistic:
#> Chisq = 2*[LogL(H0)-LogL(H1)] = -0.00000
#> Kodde-Palm C*: 95%: 2.70554 99%: 5.41189
#> Coelli (1995) skewness test on OLS residuals
#> M3T: z = -0.06529
#> M3T: p.value = 0.94794
#> Final maximum likelihood estimates
#> --------------------------------------------------------------------------------
#> Deterministic Component of SFA
#> --------------------------------------------------------------------------------
#> Coefficient Std. Error z value Pr(>|z|)
#> (Intercept) 1.29846 0.27525 4.7174 2.389e-06 ***
#> .X2 0.15575 0.00383 40.6458 < 2.2e-16 ***
#> .X3 0.09215 0.00427 21.5589 < 2.2e-16 ***
#> --------------------------------------------------------------------------------
#> Parameter in variance of u (one-sided error)
#> --------------------------------------------------------------------------------
#> Coefficient Std. Error z value Pr(>|z|)
#> Zu_(Intercept) -10.985 167.527 -0.0656 0.9477
#> --------------------------------------------------------------------------------
#> Parameters in variance of v (two-sided error)
#> --------------------------------------------------------------------------------
#> Coefficient Std. Error z value Pr(>|z|)
#> Zv_(Intercept) -5.16142 0.20002 -25.805 < 2.2e-16 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> --------------------------------------------------------------------------------
#> Model was estimated on : Apr Fri 24, 2026 at 15:14
#> 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
#> 0 252 131 0.39953 0.39621 0.39825 0.39494 0.99680
#> 1 248 132 0.43254 0.42840 0.43109 0.42696 0.99665
#> MTR_JLMS
#> 0 0.99680
#> 1 0.99664
#>
#> Overall:
#> TE_group_BC=0.4160 TE_group_JLMS=0.4123
#> TE_meta_BC=0.4147 TE_meta_JLMS=0.4109
#> MTR_BC=0.9967 MTR_JLMS=0.9967
#> ------------------------------------------------------------
#> Total Log-likelihood: -117.5979
#> AIC: 269.1957 BIC: 340.844 HQIC: 297.3104
#> ------------------------------------------------------------
#> Model was estimated on : Apr Fri 24, 2026 at 15:14Method 4: sfaselectioncross + SFA (O’Donnell)
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"
)
#> First step probit model...
#> Second step Frontier model...
#> First step probit model...
#> Second step Frontier model...
#> Warning: 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)
#> Warning: 263 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.
#> ============================================================
#> 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 : 500
#> Distribution : hnormal
#> ============================================================
#>
#> ------------------------------------------------------------
#> Group: 0 (N = 252) Log-likelihood: -225.33119
#> ------------------------------------------------------------
#> --------------------------------------------------------------------------------
#> Sample Selection Correction Stochastic Frontier Model
#> Dependent Variable: log(y)
#> Log likelihood solver: BFGS maximization
#> Log likelihood iter: 67
#> Log likelihood value: -225.33119
#> Log likelihood gradient norm: 5.77769e-07
#> Estimation based on: N = 131 of 252 obs. and K = 6
#> Inf. Cr: AIC = 462.7 AIC/N = 3.532
#> BIC = 479.9 BIC/N = 3.663
#> HQIC = 469.7 HQIC/N = 3.585
#> --------------------------------------------------------------------------------
#> Variances: Sigma-squared(v) = 0.01981
#> Sigma(v) = 0.01981
#> Sigma-squared(u) = 2.94034
#> Sigma(u) = 2.94034
#> Sigma = Sqrt[(s^2(u)+s^2(v))] = 1.72051
#> Gamma = sigma(u)^2/sigma^2 = 0.99331
#> Lambda = sigma(u)/sigma(v) = 12.18326
#> Var[u]/{Var[u]+Var[v]} = 0.98180
#> --------------------------------------------------------------------------------
#> Average inefficiency E[ui] = 1.36817
#> Average efficiency E[exp(-ui)] = 0.37581
#> --------------------------------------------------------------------------------
#> 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) 1.36885 0.11843 11.5584 < 2.2e-16 ***
#> log(x1) 0.17123 0.06350 2.6963 0.007011 **
#> log(x2) 0.07768 0.05238 1.4829 0.138102
#> --------------------------------------------------------------------------------
#> Parameter in variance of u (one-sided error)
#> --------------------------------------------------------------------------------
#> Coefficient Std. Error z value Pr(>|z|)
#> Zu_(Intercept) 1.07853 0.10204 10.569 < 2.2e-16 ***
#> --------------------------------------------------------------------------------
#> Parameters in variance of v (two-sided error)
#> --------------------------------------------------------------------------------
#> Coefficient Std. Error z value Pr(>|z|)
#> Zv_(Intercept) -3.9216 1.1265 -3.4813 0.000499 ***
#> --------------------------------------------------------------------------------
#> Selection bias parameter
#> --------------------------------------------------------------------------------
#> Coefficient Std. Error z value Pr(>|z|)
#> rho 0.27413 1.03629 0.2645 0.7914
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> --------------------------------------------------------------------------------
#> Model was estimated on : Apr Fri 24, 2026 at 15:14
#> Log likelihood status: successful convergence
#> --------------------------------------------------------------------------------
#>
#> ------------------------------------------------------------
#> Group: 1 (N = 248) Log-likelihood: -197.67108
#> ------------------------------------------------------------
#> --------------------------------------------------------------------------------
#> Sample Selection Correction Stochastic Frontier Model
#> Dependent Variable: log(y)
#> Log likelihood solver: BFGS maximization
#> Log likelihood iter: 68
#> Log likelihood value: -197.67108
#> Log likelihood gradient norm: 2.70510e-06
#> Estimation based on: N = 132 of 248 obs. and K = 6
#> Inf. Cr: AIC = 407.3 AIC/N = 3.086
#> BIC = 424.6 BIC/N = 3.217
#> HQIC = 414.4 HQIC/N = 3.139
#> --------------------------------------------------------------------------------
#> Variances: Sigma-squared(v) = 0.03365
#> Sigma(v) = 0.03365
#> Sigma-squared(u) = 1.84490
#> Sigma(u) = 1.84490
#> Sigma = Sqrt[(s^2(u)+s^2(v))] = 1.37060
#> Gamma = sigma(u)^2/sigma^2 = 0.98209
#> Lambda = sigma(u)/sigma(v) = 7.40483
#> Var[u]/{Var[u]+Var[v]} = 0.95221
#> --------------------------------------------------------------------------------
#> Average inefficiency E[ui] = 1.08374
#> Average efficiency E[exp(-ui)] = 0.43864
#> --------------------------------------------------------------------------------
#> 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) 1.22609 0.09780 12.5368 < 2.2e-16 ***
#> log(x1) 0.14445 0.03802 3.7994 0.000145 ***
#> log(x2) 0.11056 0.03775 2.9290 0.003401 **
#> --------------------------------------------------------------------------------
#> Parameter in variance of u (one-sided error)
#> --------------------------------------------------------------------------------
#> Coefficient Std. Error z value Pr(>|z|)
#> Zu_(Intercept) 0.61243 0.12841 4.7694 1.847e-06 ***
#> --------------------------------------------------------------------------------
#> Parameters in variance of v (two-sided error)
#> --------------------------------------------------------------------------------
#> Coefficient Std. Error z value Pr(>|z|)
#> Zv_(Intercept) -3.39184 0.69675 -4.8681 1.127e-06 ***
#> --------------------------------------------------------------------------------
#> Selection bias parameter
#> --------------------------------------------------------------------------------
#> Coefficient Std. Error z value Pr(>|z|)
#> rho 0.78787 0.60643 1.2992 0.1939
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> --------------------------------------------------------------------------------
#> Model was estimated on : Apr Fri 24, 2026 at 15:14
#> Log likelihood status: successful convergence
#> --------------------------------------------------------------------------------
#>
#> ------------------------------------------------------------
#> Metafrontier Coefficients (sfa):
#> Meta-optim solver : BFGS maximization
#> Estimate Std. Error z value Pr(>|z|)
#> (Intercept) 1.36739085 0.00198573 688.61 < 2.2e-16 ***
#> log(x1) 0.16870720 0.00027021 624.36 < 2.2e-16 ***
#> log(x2) 0.07759335 0.00030126 257.57 < 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: 333
#> Log likelihood value: 1002.79269
#> Log likelihood gradient norm: 3.27261e-03
#> Estimation based on: N = 263 and K = 5
#> Inf. Cr: AIC = -1995.6 AIC/N = -7.588
#> BIC = -1977.7 BIC/N = -7.520
#> HQIC = -1988.4 HQIC/N = -7.560
#> --------------------------------------------------------------------------------
#> Variances: Sigma-squared(v) = 0.00003
#> Sigma(v) = 0.00003
#> Sigma-squared(u) = 0.00000
#> Sigma(u) = 0.00000
#> Sigma = Sqrt[(s^2(u)+s^2(v))] = 0.00534
#> Gamma = sigma(u)^2/sigma^2 = 0.00003
#> Lambda = sigma(u)/sigma(v) = 0.00555
#> Var[u]/{Var[u]+Var[v]} = 0.00001
#> --------------------------------------------------------------------------------
#> Average inefficiency E[ui] = 0.00002
#> Average efficiency E[exp(-ui)] = 0.99998
#> --------------------------------------------------------------------------------
#> 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) = 1002.79271
#> 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 = 68.20600
#> M3T: p.value = 0.00000
#> Final maximum likelihood estimates
#> --------------------------------------------------------------------------------
#> Deterministic Component of SFA
#> --------------------------------------------------------------------------------
#> Coefficient Std. Error z value Pr(>|z|)
#> (Intercept) 1.36739 0.00199 688.61 < 2.2e-16 ***
#> .X2 0.16871 0.00027 624.36 < 2.2e-16 ***
#> .X3 0.07759 0.00030 257.57 < 2.2e-16 ***
#> --------------------------------------------------------------------------------
#> Parameter in variance of u (one-sided error)
#> --------------------------------------------------------------------------------
#> Coefficient Std. Error z value Pr(>|z|)
#> Zu_(Intercept) -20.852 164.024 -0.1271 0.8988
#> --------------------------------------------------------------------------------
#> Parameters in variance of v (two-sided error)
#> --------------------------------------------------------------------------------
#> Coefficient Std. Error z value Pr(>|z|)
#> Zv_(Intercept) -10.46369 0.08722 -119.97 < 2.2e-16 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> --------------------------------------------------------------------------------
#> Model was estimated on : Apr Fri 24, 2026 at 15:14
#> 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
#> 0 252 131 0.39953 0.39621 0.99998 0.99998 16.55549
#> 1 248 132 0.43254 0.42840 0.99998 0.99998 5.29814
#> MTR_JLMS
#> 0 16.71157
#> 1 5.35380
#>
#> Overall:
#> TE_group_BC=0.4160 TE_group_JLMS=0.4123
#> TE_meta_BC=1.0000 TE_meta_JLMS=1.0000
#> MTR_BC=10.9268 MTR_JLMS=11.0327
#> ------------------------------------------------------------
#> Total Log-likelihood: 579.7904
#> AIC: -1125.581 BIC: -1053.933 HQIC: -1097.466
#> ------------------------------------------------------------
#> Model was estimated on : Apr Fri 24, 2026 at 15:14Interpreting the Selection Correction
The first-stage probit model estimates the selection probability. The
key additional parameter in the frontier model is rho — the
correlation between the selection equation error and the frontier
equation noise.
# The rho parameter appears in the summary output:
# ----------------------------------------------------------------
# Selection bias parameter
# ----------------------------------------------------------------
# Coefficient Std. Error z value Pr(>|z|)
# rho 0.89550 0.28696 3.1207 0.001804 **
# A significant rho indicates selection bias IS present and the
# correction is important.
rho value |
Interpretation |
|---|---|
| ≈ 0, p > 0.05 | No significant selection bias; standard SFA may be sufficient |
| > 0, p < 0.05 | Positive selection — efficient firms are more likely selected |
| < 0, p < 0.05 | Negative selection — inefficient firms are more likely selected |
Extracting Efficiencies
Only selected observations (those with d == 1) receive
efficiency estimates:
eff_sel <- efficiencies(meta_sel_lp)
# Non-selected observations have NA efficiencies
sum(is.na(eff_sel$TE_group_BC)) # count of non-selected obs
#> [1] 237
# Subset for selected observations in group 1
sel_grp1 <- eff_sel[eff_sel$group == 1 & !is.na(eff_sel$TE_group_BC), ]
summary(sel_grp1[, c("TE_group_BC", "TE_meta_BC", "MTR_BC")])
#> TE_group_BC TE_meta_BC MTR_BC
#> Min. :0.01365 Min. :0.01189 Min. :0.7502
#> 1st Qu.:0.21013 1st Qu.:0.18598 1st Qu.:0.8450
#> Median :0.40784 Median :0.34783 Median :0.8686
#> Mean :0.43254 Mean :0.37567 Mean :0.8674
#> 3rd Qu.:0.64468 3rd Qu.:0.57565 3rd Qu.:0.8873
#> Max. :0.93622 Max. :0.86071 Max. :1.0000