Standard SFA Metafrontier (groupType = "sfacross")
Source:vignettes/sfacross-metafrontier.Rmd
sfacross-metafrontier.RmdOverview
When observed group labels are available (e.g., farm size, region,
ownership type), groupType = "sfacross" fits a separate
cross-sectional stochastic frontier for each group using
sfaR::sfacross(). The group-specific results are then used
to estimate the common metafrontier using any of four methods.
Data Preparation
We use the ricephil dataset from sfaR,
which contains 344 Filipino rice farms. We create three technology
groups based on farm area terciles.
library(smfa)
#> Loading required package: sfaR
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#> /////** /** **////** /** //**
#> ****** /** //********/** //**
#> ////// // //////// // // 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.
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#> 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
data("ricephil", package = "sfaR")
ricephil$group <- cut(
ricephil$AREA,
breaks = quantile(ricephil$AREA, probs = c(0, 1/3, 2/3, 1), na.rm = TRUE),
labels = c("small", "medium", "large"),
include.lowest = TRUE
)
table(ricephil$group)
#>
#> small medium large
#> 125 104 115
#> small medium large
#> 125 104 115Method 1: LP Metafrontier
The linear programming (LP) envelope minimises the sum of absolute deviations from group frontier predictions while satisfying a convexity constraint. No stochastic parameters are estimated for the metafrontier itself.
meta_lp <- smfa(
formula = log(PROD) ~ log(AREA) + log(LABOR) + log(NPK),
data = ricephil,
group = "group",
S = 1,
udist = "hnormal",
groupType = "sfacross",
metaMethod = "lp"
)
summary(meta_lp)
#> ============================================================
#> Stochastic Metafrontier Analysis
#> Metafrontier method: Linear Programming (LP) Metafrontier
#> Stochastic Production/Profit Frontier, e = v - u
#> Group approach : Stochastic Frontier Analysis
#> Group estimator : sfacross
#> Group optim solver : BFGS maximization
#> Groups ( 3 ): small, medium, large
#> Total observations : 344
#> Distribution : hnormal
#> ============================================================
#>
#> ------------------------------------------------------------
#> Group: small (N = 125) Log-likelihood: -50.98578
#> ------------------------------------------------------------
#> --------------------------------------------------------------------------------
#> Normal-Half Normal SF Model
#> Dependent Variable: log(PROD)
#> Log likelihood solver: BFGS maximization
#> Log likelihood iter: 42
#> Log likelihood value: -50.98578
#> Log likelihood gradient norm: 9.40653e-06
#> Estimation based on: N = 125 and K = 6
#> Inf. Cr: AIC = 114.0 AIC/N = 0.912
#> BIC = 130.9 BIC/N = 1.048
#> HQIC = 120.9 HQIC/N = 0.967
#> --------------------------------------------------------------------------------
#> Variances: Sigma-squared(v) = 0.05318
#> Sigma(v) = 0.05318
#> Sigma-squared(u) = 0.23435
#> Sigma(u) = 0.23435
#> Sigma = Sqrt[(s^2(u)+s^2(v))] = 0.53622
#> Gamma = sigma(u)^2/sigma^2 = 0.81504
#> Lambda = sigma(u)/sigma(v) = 2.09921
#> Var[u]/{Var[u]+Var[v]} = 0.61558
#> --------------------------------------------------------------------------------
#> Average inefficiency E[ui] = 0.38626
#> Average efficiency E[exp(-ui)] = 0.70643
#> --------------------------------------------------------------------------------
#> Stochastic Production/Profit Frontier, e = v - u
#> -----[ Tests vs. No Inefficiency ]-----
#> Likelihood Ratio Test of Inefficiency
#> Deg. freedom for inefficiency model 1
#> Log Likelihood for OLS Log(H0) = -54.80277
#> LR statistic:
#> Chisq = 2*[LogL(H0)-LogL(H1)] = 7.63398
#> Kodde-Palm C*: 95%: 2.70554 99%: 5.41189
#> Coelli (1995) skewness test on OLS residuals
#> M3T: z = -3.57676
#> M3T: p.value = 0.00035
#> Final maximum likelihood estimates
#> --------------------------------------------------------------------------------
#> Deterministic Component of SFA
#> --------------------------------------------------------------------------------
#> Coefficient Std. Error z value Pr(>|z|)
#> (Intercept) -1.58745 0.51274 -3.0960 0.001962 **
#> log(AREA) 0.24014 0.11834 2.0292 0.042440 *
#> log(LABOR) 0.43464 0.12292 3.5361 0.000406 ***
#> log(NPK) 0.30516 0.05701 5.3523 8.682e-08 ***
#> --------------------------------------------------------------------------------
#> Parameter in variance of u (one-sided error)
#> --------------------------------------------------------------------------------
#> Coefficient Std. Error z value Pr(>|z|)
#> Zu_(Intercept) -1.45093 0.29867 -4.858 1.186e-06 ***
#> --------------------------------------------------------------------------------
#> Parameters in variance of v (two-sided error)
#> --------------------------------------------------------------------------------
#> Coefficient Std. Error z value Pr(>|z|)
#> Zv_(Intercept) -2.93406 0.35401 -8.288 < 2.2e-16 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> --------------------------------------------------------------------------------
#> Model was estimated on : Apr Fri 24, 2026 at 15:13
#> Log likelihood status: successful convergence
#> --------------------------------------------------------------------------------
#>
#> ------------------------------------------------------------
#> Group: medium (N = 104) Log-likelihood: -15.28164
#> ------------------------------------------------------------
#> --------------------------------------------------------------------------------
#> Normal-Half Normal SF Model
#> Dependent Variable: log(PROD)
#> Log likelihood solver: BFGS maximization
#> Log likelihood iter: 41
#> Log likelihood value: -15.28164
#> Log likelihood gradient norm: 3.83566e-05
#> Estimation based on: N = 104 and K = 6
#> Inf. Cr: AIC = 42.6 AIC/N = 0.409
#> BIC = 58.4 BIC/N = 0.562
#> HQIC = 49.0 HQIC/N = 0.471
#> --------------------------------------------------------------------------------
#> Variances: Sigma-squared(v) = 0.01058
#> Sigma(v) = 0.01058
#> Sigma-squared(u) = 0.22010
#> Sigma(u) = 0.22010
#> Sigma = Sqrt[(s^2(u)+s^2(v))] = 0.48030
#> Gamma = sigma(u)^2/sigma^2 = 0.95412
#> Lambda = sigma(u)/sigma(v) = 4.56034
#> Var[u]/{Var[u]+Var[v]} = 0.88314
#> --------------------------------------------------------------------------------
#> Average inefficiency E[ui] = 0.37433
#> Average efficiency E[exp(-ui)] = 0.71330
#> --------------------------------------------------------------------------------
#> Stochastic Production/Profit Frontier, e = v - u
#> -----[ Tests vs. No Inefficiency ]-----
#> Likelihood Ratio Test of Inefficiency
#> Deg. freedom for inefficiency model 1
#> Log Likelihood for OLS Log(H0) = -21.11323
#> LR statistic:
#> Chisq = 2*[LogL(H0)-LogL(H1)] = 11.66318
#> Kodde-Palm C*: 95%: 2.70554 99%: 5.41189
#> Coelli (1995) skewness test on OLS residuals
#> M3T: z = -2.91021
#> M3T: p.value = 0.00361
#> Final maximum likelihood estimates
#> --------------------------------------------------------------------------------
#> Deterministic Component of SFA
#> --------------------------------------------------------------------------------
#> Coefficient Std. Error z value Pr(>|z|)
#> (Intercept) -0.08182 0.50668 -0.1615 0.8717190
#> log(AREA) 0.47410 0.13984 3.3903 0.0006981 ***
#> log(LABOR) 0.17935 0.10201 1.7581 0.0787310 .
#> log(NPK) 0.20255 0.08130 2.4913 0.0127289 *
#> --------------------------------------------------------------------------------
#> Parameter in variance of u (one-sided error)
#> --------------------------------------------------------------------------------
#> Coefficient Std. Error z value Pr(>|z|)
#> Zu_(Intercept) -1.51367 0.23549 -6.4276 1.296e-10 ***
#> --------------------------------------------------------------------------------
#> Parameters in variance of v (two-sided error)
#> --------------------------------------------------------------------------------
#> Coefficient Std. Error z value Pr(>|z|)
#> Zv_(Intercept) -4.54846 0.76429 -5.9512 2.661e-09 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> --------------------------------------------------------------------------------
#> Model was estimated on : Apr Fri 24, 2026 at 15:13
#> Log likelihood status: successful convergence
#> --------------------------------------------------------------------------------
#>
#> ------------------------------------------------------------
#> Group: large (N = 115) Log-likelihood: -8.02197
#> ------------------------------------------------------------
#> --------------------------------------------------------------------------------
#> Normal-Half Normal SF Model
#> Dependent Variable: log(PROD)
#> Log likelihood solver: BFGS maximization
#> Log likelihood iter: 68
#> Log likelihood value: -8.02197
#> Log likelihood gradient norm: 4.01301e-05
#> Estimation based on: N = 115 and K = 6
#> Inf. Cr: AIC = 28.0 AIC/N = 0.244
#> BIC = 44.5 BIC/N = 0.387
#> HQIC = 34.7 HQIC/N = 0.302
#> --------------------------------------------------------------------------------
#> Variances: Sigma-squared(v) = 0.01399
#> Sigma(v) = 0.01399
#> Sigma-squared(u) = 0.16751
#> Sigma(u) = 0.16751
#> Sigma = Sqrt[(s^2(u)+s^2(v))] = 0.42602
#> Gamma = sigma(u)^2/sigma^2 = 0.92293
#> Lambda = sigma(u)/sigma(v) = 3.46063
#> Var[u]/{Var[u]+Var[v]} = 0.81315
#> --------------------------------------------------------------------------------
#> Average inefficiency E[ui] = 0.32656
#> Average efficiency E[exp(-ui)] = 0.74195
#> --------------------------------------------------------------------------------
#> Stochastic Production/Profit Frontier, e = v - u
#> -----[ Tests vs. No Inefficiency ]-----
#> Likelihood Ratio Test of Inefficiency
#> Deg. freedom for inefficiency model 1
#> Log Likelihood for OLS Log(H0) = -16.96836
#> LR statistic:
#> Chisq = 2*[LogL(H0)-LogL(H1)] = 17.89279
#> Kodde-Palm C*: 95%: 2.70554 99%: 5.41189
#> Coelli (1995) skewness test on OLS residuals
#> M3T: z = -4.12175
#> M3T: p.value = 0.00004
#> Final maximum likelihood estimates
#> --------------------------------------------------------------------------------
#> Deterministic Component of SFA
#> --------------------------------------------------------------------------------
#> Coefficient Std. Error z value Pr(>|z|)
#> (Intercept) -1.31194 0.41859 -3.1342 0.0017234 **
#> log(AREA) 0.38278 0.14297 2.6772 0.0074236 **
#> log(LABOR) 0.42105 0.10992 3.8303 0.0001280 ***
#> log(NPK) 0.23143 0.06065 3.8160 0.0001356 ***
#> --------------------------------------------------------------------------------
#> Parameter in variance of u (one-sided error)
#> --------------------------------------------------------------------------------
#> Coefficient Std. Error z value Pr(>|z|)
#> Zu_(Intercept) -1.78673 0.20176 -8.8555 < 2.2e-16 ***
#> --------------------------------------------------------------------------------
#> Parameters in variance of v (two-sided error)
#> --------------------------------------------------------------------------------
#> Coefficient Std. Error z value Pr(>|z|)
#> Zv_(Intercept) -4.26963 0.40584 -10.521 < 2.2e-16 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> --------------------------------------------------------------------------------
#> Model was estimated on : Apr Fri 24, 2026 at 15:13
#> Log likelihood status: successful convergence
#> --------------------------------------------------------------------------------
#>
#> ------------------------------------------------------------
#> Metafrontier Coefficients (lp):
#> (LP: deterministic envelope - no estimated parameters)
#>
#> ------------------------------------------------------------
#> Efficiency Statistics (group means):
#> ------------------------------------------------------------
#> N_obs N_valid TE_group_BC TE_group_JLMS TE_meta_BC TE_meta_JLMS MTR_BC
#> small 125 125 0.71065 0.70090 0.64126 0.63244 0.89981
#> medium 104 104 0.71253 0.70965 0.68204 0.67929 0.95597
#> large 115 115 0.74772 0.74406 0.72186 0.71834 0.96521
#> MTR_JLMS
#> small 0.89981
#> medium 0.95597
#> large 0.96521
#>
#> Overall:
#> TE_group_BC=0.7236 TE_group_JLMS=0.7182
#> TE_meta_BC=0.6817 TE_meta_JLMS=0.6767
#> MTR_BC=0.9403 MTR_JLMS=0.9403
#> ------------------------------------------------------------
#> Total Log-likelihood: -74.28939
#> AIC: 184.5788 BIC: 253.7103 HQIC: 212.113
#> ------------------------------------------------------------
#> Model was estimated on : Apr Fri 24, 2026 at 15:13Note: Since the LP metafrontier is estimated via linear programming, no estimated parameters are returned for the metafrontier level. The LP envelope is fully determined by the group frontier predictions.
Method 2: QP Metafrontier
The quadratic programming (QP) envelope minimises the sum of squared deviations from group frontier predictions. Unlike LP, QP produces a smooth envelope that is differentiable everywhere, and it returns estimated coefficients with standard errors.
meta_qp <- smfa(
formula = log(PROD) ~ log(AREA) + log(LABOR) + log(NPK),
data = ricephil,
group = "group",
S = 1,
udist = "hnormal",
groupType = "sfacross",
metaMethod = "qp"
)
summary(meta_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 : Apr Fri 24, 2026 at 15:13
#> Log likelihood status: successful convergence
#> --------------------------------------------------------------------------------
#>
#> ------------------------------------------------------------
#> Group: medium (N = 104) Log-likelihood: -15.28164
#> ------------------------------------------------------------
#> --------------------------------------------------------------------------------
#> Normal-Half Normal SF Model
#> Dependent Variable: log(PROD)
#> Log likelihood solver: BFGS maximization
#> Log likelihood iter: 41
#> Log likelihood value: -15.28164
#> Log likelihood gradient norm: 3.83566e-05
#> Estimation based on: N = 104 and K = 6
#> Inf. Cr: AIC = 42.6 AIC/N = 0.409
#> BIC = 58.4 BIC/N = 0.562
#> HQIC = 49.0 HQIC/N = 0.471
#> --------------------------------------------------------------------------------
#> Variances: Sigma-squared(v) = 0.01058
#> Sigma(v) = 0.01058
#> Sigma-squared(u) = 0.22010
#> Sigma(u) = 0.22010
#> Sigma = Sqrt[(s^2(u)+s^2(v))] = 0.48030
#> Gamma = sigma(u)^2/sigma^2 = 0.95412
#> Lambda = sigma(u)/sigma(v) = 4.56034
#> Var[u]/{Var[u]+Var[v]} = 0.88314
#> --------------------------------------------------------------------------------
#> Average inefficiency E[ui] = 0.37433
#> Average efficiency E[exp(-ui)] = 0.71330
#> --------------------------------------------------------------------------------
#> Stochastic Production/Profit Frontier, e = v - u
#> -----[ Tests vs. No Inefficiency ]-----
#> Likelihood Ratio Test of Inefficiency
#> Deg. freedom for inefficiency model 1
#> Log Likelihood for OLS Log(H0) = -21.11323
#> LR statistic:
#> Chisq = 2*[LogL(H0)-LogL(H1)] = 11.66318
#> Kodde-Palm C*: 95%: 2.70554 99%: 5.41189
#> Coelli (1995) skewness test on OLS residuals
#> M3T: z = -2.91021
#> M3T: p.value = 0.00361
#> Final maximum likelihood estimates
#> --------------------------------------------------------------------------------
#> Deterministic Component of SFA
#> --------------------------------------------------------------------------------
#> Coefficient Std. Error z value Pr(>|z|)
#> (Intercept) -0.08182 0.50668 -0.1615 0.8717190
#> log(AREA) 0.47410 0.13984 3.3903 0.0006981 ***
#> log(LABOR) 0.17935 0.10201 1.7581 0.0787310 .
#> log(NPK) 0.20255 0.08130 2.4913 0.0127289 *
#> --------------------------------------------------------------------------------
#> Parameter in variance of u (one-sided error)
#> --------------------------------------------------------------------------------
#> Coefficient Std. Error z value Pr(>|z|)
#> Zu_(Intercept) -1.51367 0.23549 -6.4276 1.296e-10 ***
#> --------------------------------------------------------------------------------
#> Parameters in variance of v (two-sided error)
#> --------------------------------------------------------------------------------
#> Coefficient Std. Error z value Pr(>|z|)
#> Zv_(Intercept) -4.54846 0.76429 -5.9512 2.661e-09 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> --------------------------------------------------------------------------------
#> Model was estimated on : Apr Fri 24, 2026 at 15:13
#> Log likelihood status: successful convergence
#> --------------------------------------------------------------------------------
#>
#> ------------------------------------------------------------
#> Group: large (N = 115) Log-likelihood: -8.02197
#> ------------------------------------------------------------
#> --------------------------------------------------------------------------------
#> Normal-Half Normal SF Model
#> Dependent Variable: log(PROD)
#> Log likelihood solver: BFGS maximization
#> Log likelihood iter: 68
#> Log likelihood value: -8.02197
#> Log likelihood gradient norm: 4.01301e-05
#> Estimation based on: N = 115 and K = 6
#> Inf. Cr: AIC = 28.0 AIC/N = 0.244
#> BIC = 44.5 BIC/N = 0.387
#> HQIC = 34.7 HQIC/N = 0.302
#> --------------------------------------------------------------------------------
#> Variances: Sigma-squared(v) = 0.01399
#> Sigma(v) = 0.01399
#> Sigma-squared(u) = 0.16751
#> Sigma(u) = 0.16751
#> Sigma = Sqrt[(s^2(u)+s^2(v))] = 0.42602
#> Gamma = sigma(u)^2/sigma^2 = 0.92293
#> Lambda = sigma(u)/sigma(v) = 3.46063
#> Var[u]/{Var[u]+Var[v]} = 0.81315
#> --------------------------------------------------------------------------------
#> Average inefficiency E[ui] = 0.32656
#> Average efficiency E[exp(-ui)] = 0.74195
#> --------------------------------------------------------------------------------
#> Stochastic Production/Profit Frontier, e = v - u
#> -----[ Tests vs. No Inefficiency ]-----
#> Likelihood Ratio Test of Inefficiency
#> Deg. freedom for inefficiency model 1
#> Log Likelihood for OLS Log(H0) = -16.96836
#> LR statistic:
#> Chisq = 2*[LogL(H0)-LogL(H1)] = 17.89279
#> Kodde-Palm C*: 95%: 2.70554 99%: 5.41189
#> Coelli (1995) skewness test on OLS residuals
#> M3T: z = -4.12175
#> M3T: p.value = 0.00004
#> Final maximum likelihood estimates
#> --------------------------------------------------------------------------------
#> Deterministic Component of SFA
#> --------------------------------------------------------------------------------
#> Coefficient Std. Error z value Pr(>|z|)
#> (Intercept) -1.31194 0.41859 -3.1342 0.0017234 **
#> log(AREA) 0.38278 0.14297 2.6772 0.0074236 **
#> log(LABOR) 0.42105 0.10992 3.8303 0.0001280 ***
#> log(NPK) 0.23143 0.06065 3.8160 0.0001356 ***
#> --------------------------------------------------------------------------------
#> Parameter in variance of u (one-sided error)
#> --------------------------------------------------------------------------------
#> Coefficient Std. Error z value Pr(>|z|)
#> Zu_(Intercept) -1.78673 0.20176 -8.8555 < 2.2e-16 ***
#> --------------------------------------------------------------------------------
#> Parameters in variance of v (two-sided error)
#> --------------------------------------------------------------------------------
#> Coefficient Std. Error z value Pr(>|z|)
#> Zv_(Intercept) -4.26963 0.40584 -10.521 < 2.2e-16 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> --------------------------------------------------------------------------------
#> Model was estimated on : Apr Fri 24, 2026 at 15:13
#> 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
#> small 125 125 0.71065 0.70090 0.64037 0.63156 0.89972
#> medium 104 104 0.71253 0.70965 0.66998 0.66727 0.94053
#> large 115 115 0.74772 0.74406 0.72290 0.71937 0.96676
#> MTR_JLMS
#> small 0.89972
#> medium 0.94053
#> large 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 : Apr Fri 24, 2026 at 15:13Method 3: Stochastic Metafrontier — Huang et al. (2014)
The two-stage stochastic metafrontier of Huang, Huang & Liu (2014) uses the group-specific fitted frontier values as the dependent variable in a second-stage pooled SFA. The technology gap U and noise V are estimated stochastically, which naturally bounds the metatechnology ratio MTR ∈ (0, 1].
meta_huang <- smfa(
formula = log(PROD) ~ log(AREA) + log(LABOR) + log(NPK),
data = ricephil,
group = "group",
S = 1,
udist = "hnormal",
groupType = "sfacross",
metaMethod = "sfa",
sfaApproach = "huang"
)
#> 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_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 : Apr Fri 24, 2026 at 15:13
#> Log likelihood status: successful convergence
#> --------------------------------------------------------------------------------
#>
#> ------------------------------------------------------------
#> Group: medium (N = 104) Log-likelihood: -15.28164
#> ------------------------------------------------------------
#> --------------------------------------------------------------------------------
#> Normal-Half Normal SF Model
#> Dependent Variable: log(PROD)
#> Log likelihood solver: BFGS maximization
#> Log likelihood iter: 41
#> Log likelihood value: -15.28164
#> Log likelihood gradient norm: 3.83566e-05
#> Estimation based on: N = 104 and K = 6
#> Inf. Cr: AIC = 42.6 AIC/N = 0.409
#> BIC = 58.4 BIC/N = 0.562
#> HQIC = 49.0 HQIC/N = 0.471
#> --------------------------------------------------------------------------------
#> Variances: Sigma-squared(v) = 0.01058
#> Sigma(v) = 0.01058
#> Sigma-squared(u) = 0.22010
#> Sigma(u) = 0.22010
#> Sigma = Sqrt[(s^2(u)+s^2(v))] = 0.48030
#> Gamma = sigma(u)^2/sigma^2 = 0.95412
#> Lambda = sigma(u)/sigma(v) = 4.56034
#> Var[u]/{Var[u]+Var[v]} = 0.88314
#> --------------------------------------------------------------------------------
#> Average inefficiency E[ui] = 0.37433
#> Average efficiency E[exp(-ui)] = 0.71330
#> --------------------------------------------------------------------------------
#> Stochastic Production/Profit Frontier, e = v - u
#> -----[ Tests vs. No Inefficiency ]-----
#> Likelihood Ratio Test of Inefficiency
#> Deg. freedom for inefficiency model 1
#> Log Likelihood for OLS Log(H0) = -21.11323
#> LR statistic:
#> Chisq = 2*[LogL(H0)-LogL(H1)] = 11.66318
#> Kodde-Palm C*: 95%: 2.70554 99%: 5.41189
#> Coelli (1995) skewness test on OLS residuals
#> M3T: z = -2.91021
#> M3T: p.value = 0.00361
#> Final maximum likelihood estimates
#> --------------------------------------------------------------------------------
#> Deterministic Component of SFA
#> --------------------------------------------------------------------------------
#> Coefficient Std. Error z value Pr(>|z|)
#> (Intercept) -0.08182 0.50668 -0.1615 0.8717190
#> log(AREA) 0.47410 0.13984 3.3903 0.0006981 ***
#> log(LABOR) 0.17935 0.10201 1.7581 0.0787310 .
#> log(NPK) 0.20255 0.08130 2.4913 0.0127289 *
#> --------------------------------------------------------------------------------
#> Parameter in variance of u (one-sided error)
#> --------------------------------------------------------------------------------
#> Coefficient Std. Error z value Pr(>|z|)
#> Zu_(Intercept) -1.51367 0.23549 -6.4276 1.296e-10 ***
#> --------------------------------------------------------------------------------
#> Parameters in variance of v (two-sided error)
#> --------------------------------------------------------------------------------
#> Coefficient Std. Error z value Pr(>|z|)
#> Zv_(Intercept) -4.54846 0.76429 -5.9512 2.661e-09 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> --------------------------------------------------------------------------------
#> Model was estimated on : Apr Fri 24, 2026 at 15:13
#> Log likelihood status: successful convergence
#> --------------------------------------------------------------------------------
#>
#> ------------------------------------------------------------
#> Group: large (N = 115) Log-likelihood: -8.02197
#> ------------------------------------------------------------
#> --------------------------------------------------------------------------------
#> Normal-Half Normal SF Model
#> Dependent Variable: log(PROD)
#> Log likelihood solver: BFGS maximization
#> Log likelihood iter: 68
#> Log likelihood value: -8.02197
#> Log likelihood gradient norm: 4.01301e-05
#> Estimation based on: N = 115 and K = 6
#> Inf. Cr: AIC = 28.0 AIC/N = 0.244
#> BIC = 44.5 BIC/N = 0.387
#> HQIC = 34.7 HQIC/N = 0.302
#> --------------------------------------------------------------------------------
#> Variances: Sigma-squared(v) = 0.01399
#> Sigma(v) = 0.01399
#> Sigma-squared(u) = 0.16751
#> Sigma(u) = 0.16751
#> Sigma = Sqrt[(s^2(u)+s^2(v))] = 0.42602
#> Gamma = sigma(u)^2/sigma^2 = 0.92293
#> Lambda = sigma(u)/sigma(v) = 3.46063
#> Var[u]/{Var[u]+Var[v]} = 0.81315
#> --------------------------------------------------------------------------------
#> Average inefficiency E[ui] = 0.32656
#> Average efficiency E[exp(-ui)] = 0.74195
#> --------------------------------------------------------------------------------
#> Stochastic Production/Profit Frontier, e = v - u
#> -----[ Tests vs. No Inefficiency ]-----
#> Likelihood Ratio Test of Inefficiency
#> Deg. freedom for inefficiency model 1
#> Log Likelihood for OLS Log(H0) = -16.96836
#> LR statistic:
#> Chisq = 2*[LogL(H0)-LogL(H1)] = 17.89279
#> Kodde-Palm C*: 95%: 2.70554 99%: 5.41189
#> Coelli (1995) skewness test on OLS residuals
#> M3T: z = -4.12175
#> M3T: p.value = 0.00004
#> Final maximum likelihood estimates
#> --------------------------------------------------------------------------------
#> Deterministic Component of SFA
#> --------------------------------------------------------------------------------
#> Coefficient Std. Error z value Pr(>|z|)
#> (Intercept) -1.31194 0.41859 -3.1342 0.0017234 **
#> log(AREA) 0.38278 0.14297 2.6772 0.0074236 **
#> log(LABOR) 0.42105 0.10992 3.8303 0.0001280 ***
#> log(NPK) 0.23143 0.06065 3.8160 0.0001356 ***
#> --------------------------------------------------------------------------------
#> Parameter in variance of u (one-sided error)
#> --------------------------------------------------------------------------------
#> Coefficient Std. Error z value Pr(>|z|)
#> Zu_(Intercept) -1.78673 0.20176 -8.8555 < 2.2e-16 ***
#> --------------------------------------------------------------------------------
#> Parameters in variance of v (two-sided error)
#> --------------------------------------------------------------------------------
#> Coefficient Std. Error z value Pr(>|z|)
#> Zv_(Intercept) -4.26963 0.40584 -10.521 < 2.2e-16 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> --------------------------------------------------------------------------------
#> Model was estimated on : Apr Fri 24, 2026 at 15:13
#> Log likelihood status: successful convergence
#> --------------------------------------------------------------------------------
#>
#> ------------------------------------------------------------
#> Metafrontier Coefficients (sfa):
#> Meta-optim solver : BFGS maximization
#> Estimate Std. Error z value Pr(>|z|)
#> (Intercept) -1.0031443 0.0568887 -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: 579
#> Log likelihood value: 553.35240
#> Log likelihood gradient norm: 5.37373e-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 -17.633 < 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) -14.75 174.37 -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 : Apr Fri 24, 2026 at 15:13
#> 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
#> small 125 125 0.71065 0.70090 0.71030 0.70055 0.99950
#> medium 104 104 0.71253 0.70965 0.71217 0.70930 0.99950
#> large 115 115 0.74772 0.74406 0.74734 0.74369 0.99950
#> MTR_JLMS
#> small 0.99950
#> medium 0.99950
#> large 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 : Apr Fri 24, 2026 at 15:13Method 4: Stochastic Envelope — O’Donnell et al. (2008)
The O’Donnell et al. (2008) approach uses the LP deterministic envelope as the dependent variable in the second-stage SFA, rather than the group-specific fitted values. This mixed deterministic–stochastic approach embeds the envelope within an SFA framework.
Warning: MTR values > 1 can arise with this method when the second-stage SFA estimates near-zero inefficiency. If this occurs, consider using
metaMethod = "lp"orsfaApproach = "huang"instead.
meta_ordonnell <- smfa(
formula = log(PROD) ~ log(AREA) + log(LABOR) + log(NPK),
data = ricephil,
group = "group",
S = 1,
udist = "hnormal",
groupType = "sfacross",
metaMethod = "sfa",
sfaApproach = "ordonnell"
)
#> 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_ordonnell)
#> Warning: 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.
#> ============================================================
#> 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 : Apr Fri 24, 2026 at 15:13
#> Log likelihood status: successful convergence
#> --------------------------------------------------------------------------------
#>
#> ------------------------------------------------------------
#> Group: medium (N = 104) Log-likelihood: -15.28164
#> ------------------------------------------------------------
#> --------------------------------------------------------------------------------
#> Normal-Half Normal SF Model
#> Dependent Variable: log(PROD)
#> Log likelihood solver: BFGS maximization
#> Log likelihood iter: 41
#> Log likelihood value: -15.28164
#> Log likelihood gradient norm: 3.83566e-05
#> Estimation based on: N = 104 and K = 6
#> Inf. Cr: AIC = 42.6 AIC/N = 0.409
#> BIC = 58.4 BIC/N = 0.562
#> HQIC = 49.0 HQIC/N = 0.471
#> --------------------------------------------------------------------------------
#> Variances: Sigma-squared(v) = 0.01058
#> Sigma(v) = 0.01058
#> Sigma-squared(u) = 0.22010
#> Sigma(u) = 0.22010
#> Sigma = Sqrt[(s^2(u)+s^2(v))] = 0.48030
#> Gamma = sigma(u)^2/sigma^2 = 0.95412
#> Lambda = sigma(u)/sigma(v) = 4.56034
#> Var[u]/{Var[u]+Var[v]} = 0.88314
#> --------------------------------------------------------------------------------
#> Average inefficiency E[ui] = 0.37433
#> Average efficiency E[exp(-ui)] = 0.71330
#> --------------------------------------------------------------------------------
#> Stochastic Production/Profit Frontier, e = v - u
#> -----[ Tests vs. No Inefficiency ]-----
#> Likelihood Ratio Test of Inefficiency
#> Deg. freedom for inefficiency model 1
#> Log Likelihood for OLS Log(H0) = -21.11323
#> LR statistic:
#> Chisq = 2*[LogL(H0)-LogL(H1)] = 11.66318
#> Kodde-Palm C*: 95%: 2.70554 99%: 5.41189
#> Coelli (1995) skewness test on OLS residuals
#> M3T: z = -2.91021
#> M3T: p.value = 0.00361
#> Final maximum likelihood estimates
#> --------------------------------------------------------------------------------
#> Deterministic Component of SFA
#> --------------------------------------------------------------------------------
#> Coefficient Std. Error z value Pr(>|z|)
#> (Intercept) -0.08182 0.50668 -0.1615 0.8717190
#> log(AREA) 0.47410 0.13984 3.3903 0.0006981 ***
#> log(LABOR) 0.17935 0.10201 1.7581 0.0787310 .
#> log(NPK) 0.20255 0.08130 2.4913 0.0127289 *
#> --------------------------------------------------------------------------------
#> Parameter in variance of u (one-sided error)
#> --------------------------------------------------------------------------------
#> Coefficient Std. Error z value Pr(>|z|)
#> Zu_(Intercept) -1.51367 0.23549 -6.4276 1.296e-10 ***
#> --------------------------------------------------------------------------------
#> Parameters in variance of v (two-sided error)
#> --------------------------------------------------------------------------------
#> Coefficient Std. Error z value Pr(>|z|)
#> Zv_(Intercept) -4.54846 0.76429 -5.9512 2.661e-09 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> --------------------------------------------------------------------------------
#> Model was estimated on : Apr Fri 24, 2026 at 15:13
#> Log likelihood status: successful convergence
#> --------------------------------------------------------------------------------
#>
#> ------------------------------------------------------------
#> Group: large (N = 115) Log-likelihood: -8.02197
#> ------------------------------------------------------------
#> --------------------------------------------------------------------------------
#> Normal-Half Normal SF Model
#> Dependent Variable: log(PROD)
#> Log likelihood solver: BFGS maximization
#> Log likelihood iter: 68
#> Log likelihood value: -8.02197
#> Log likelihood gradient norm: 4.01301e-05
#> Estimation based on: N = 115 and K = 6
#> Inf. Cr: AIC = 28.0 AIC/N = 0.244
#> BIC = 44.5 BIC/N = 0.387
#> HQIC = 34.7 HQIC/N = 0.302
#> --------------------------------------------------------------------------------
#> Variances: Sigma-squared(v) = 0.01399
#> Sigma(v) = 0.01399
#> Sigma-squared(u) = 0.16751
#> Sigma(u) = 0.16751
#> Sigma = Sqrt[(s^2(u)+s^2(v))] = 0.42602
#> Gamma = sigma(u)^2/sigma^2 = 0.92293
#> Lambda = sigma(u)/sigma(v) = 3.46063
#> Var[u]/{Var[u]+Var[v]} = 0.81315
#> --------------------------------------------------------------------------------
#> Average inefficiency E[ui] = 0.32656
#> Average efficiency E[exp(-ui)] = 0.74195
#> --------------------------------------------------------------------------------
#> Stochastic Production/Profit Frontier, e = v - u
#> -----[ Tests vs. No Inefficiency ]-----
#> Likelihood Ratio Test of Inefficiency
#> Deg. freedom for inefficiency model 1
#> Log Likelihood for OLS Log(H0) = -16.96836
#> LR statistic:
#> Chisq = 2*[LogL(H0)-LogL(H1)] = 17.89279
#> Kodde-Palm C*: 95%: 2.70554 99%: 5.41189
#> Coelli (1995) skewness test on OLS residuals
#> M3T: z = -4.12175
#> M3T: p.value = 0.00004
#> Final maximum likelihood estimates
#> --------------------------------------------------------------------------------
#> Deterministic Component of SFA
#> --------------------------------------------------------------------------------
#> Coefficient Std. Error z value Pr(>|z|)
#> (Intercept) -1.31194 0.41859 -3.1342 0.0017234 **
#> log(AREA) 0.38278 0.14297 2.6772 0.0074236 **
#> log(LABOR) 0.42105 0.10992 3.8303 0.0001280 ***
#> log(NPK) 0.23143 0.06065 3.8160 0.0001356 ***
#> --------------------------------------------------------------------------------
#> Parameter in variance of u (one-sided error)
#> --------------------------------------------------------------------------------
#> Coefficient Std. Error z value Pr(>|z|)
#> Zu_(Intercept) -1.78673 0.20176 -8.8555 < 2.2e-16 ***
#> --------------------------------------------------------------------------------
#> Parameters in variance of v (two-sided error)
#> --------------------------------------------------------------------------------
#> Coefficient Std. Error z value Pr(>|z|)
#> Zv_(Intercept) -4.26963 0.40584 -10.521 < 2.2e-16 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> --------------------------------------------------------------------------------
#> Model was estimated on : Apr Fri 24, 2026 at 15:13
#> 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: 5.34744e-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 : Apr Fri 24, 2026 at 15:13
#> 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
#> small 125 125 0.71065 0.70090 0.99966 0.99966 1.49276
#> medium 104 104 0.71253 0.70965 0.99966 0.99966 1.50575
#> large 115 115 0.74772 0.74406 0.99966 0.99966 1.41180
#> MTR_JLMS
#> small 1.51673
#> medium 1.51248
#> large 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 : Apr Fri 24, 2026 at 15:13Comparing Methods
All four methods use identical group-level estimates. The differences arise only in how the metafrontier is computed:
| Method | Metafrontier Coefficients Returned | MTR Bounded ≤ 1? |
|---|---|---|
| LP | No (envelope rule) | Yes |
| QP | Yes (with SE) | Yes |
| SFA (huang) | Yes (with SE) | Yes |
| SFA (ordonnell) | Yes (with SE) | Not guaranteed |
Extracting Efficiencies
All models return firm-level efficiency estimates via
efficiencies():
eff <- efficiencies(meta_lp)
head(eff)
#> id group u_g TE_group_JLMS TE_group_BC TE_group_BC_reciprocal
#> 1 1 medium 0.2697165 0.7635959 0.7673345 1.316036
#> 2 2 large 0.3515642 0.7035867 0.7080897 1.430406
#> 3 3 large 0.2774565 0.7577085 0.7623358 1.327899
#> 4 4 medium 0.1710417 0.8427864 0.8461331 1.191355
#> 5 5 large 0.2119629 0.8089947 0.8133556 1.242901
#> 6 6 small 0.1987499 0.8197549 0.8275685 1.232467
#> uLB_g uUB_g m_g TE_group_mode teBCLB_g teBCUB_g u_meta
#> 1 0.077581942 0.4657010 0.26858570 0.7644599 0.6276949 0.9253512 0.3944439
#> 2 0.130356248 0.5739174 0.35118207 0.7038556 0.5633144 0.8777827 0.3779836
#> 3 0.065447909 0.4980807 0.27501606 0.7595599 0.6076959 0.9366478 0.3049531
#> 4 0.018022507 0.3583190 0.15885675 0.8531186 0.6988501 0.9821389 0.1710417
#> 5 0.027125654 0.4268531 0.20231520 0.8168374 0.6525594 0.9732389 0.2379271
#> 6 0.009050601 0.5251973 0.07998025 0.9231346 0.5914386 0.9909902 0.3295263
#> TE_meta_JLMS TE_meta_BC MTR_JLMS MTR_BC
#> 1 0.6740548 0.6773549 0.8827375 0.8827375
#> 2 0.6852418 0.6896274 0.9739266 0.9739266
#> 3 0.7371580 0.7416598 0.9728780 0.9728780
#> 4 0.8427864 0.8461331 1.0000000 1.0000000
#> 5 0.7882601 0.7925093 0.9743700 0.9743700
#> 6 0.7192644 0.7261201 0.8774139 0.8774139
# Subset for a specific group
eff_small <- eff[eff$group == "small", ]
summary(eff_small[, c("TE_group_BC", "TE_meta_BC", "MTR_BC")])
#> TE_group_BC TE_meta_BC MTR_BC
#> Min. :0.1737 Min. :0.1179 Min. :0.5907
#> 1st Qu.:0.6217 1st Qu.:0.5678 1st Qu.:0.8313
#> Median :0.7427 Median :0.6683 Median :0.9204
#> Mean :0.7107 Mean :0.6413 Mean :0.8998
#> 3rd Qu.:0.8165 3rd Qu.:0.7509 3rd Qu.:0.9908
#> Max. :0.9275 Max. :0.8774 Max. :1.0000Other Extractors
coef(meta_qp) # metafrontier coefficients
#> (Intercept) log(AREA) log(LABOR) log(NPK)
#> -0.6117795 0.3937843 0.2791273 0.2409454
vcov(meta_qp) # variance-covariance matrix
#> (Intercept) `log(AREA)` `log(LABOR)` `log(NPK)`
#> (Intercept) 8.514304e-04 1.954064e-04 -1.729963e-04 -3.730091e-05
#> `log(AREA)` 1.954064e-04 5.359537e-05 -3.976453e-05 -9.599635e-06
#> `log(LABOR)` -1.729963e-04 -3.976453e-05 5.962116e-05 -1.454127e-05
#> `log(NPK)` -3.730091e-05 -9.599635e-06 -1.454127e-05 2.194543e-05
logLik(meta_lp) # log-likelihood
#> 'log Lik.' -74.28939 (df=18)
ic(meta_lp) # AIC, BIC, HQIC
#> AIC BIC HQIC
#> 1 184.5788 253.7103 212.113
nobs(meta_lp) # number of observations
#> [1] 344
fitted(meta_lp) # fitted values
#> [1] 2.4692860 2.7439118 2.6260597 1.7413418 2.4130830 0.8386720 2.1838447
#> [8] 2.1368249 2.5632590 2.6811266 1.0256316 0.1866309 1.6584919 2.3673143
#> [15] 0.5318491 1.0749268 2.9226606 3.1820088 2.7526696 2.7875600 2.2775942
#> [22] 1.8328112 2.9712349 2.3411190 2.9540079 1.5904844 2.4159250 1.8695405
#> [29] 1.8806492 1.8081081 1.2053265 1.2472296 1.9714412 1.1387651 2.7063760
#> [36] 1.9160249 1.6961237 2.8670880 1.2103088 2.1644129 1.7155060 2.0399982
#> [43] 1.8390740 2.5101720 2.5445910 2.7921145 1.6829644 2.5266572 0.7143551
#> [50] 2.1361437 2.0008358 2.4644202 2.6488824 1.2647850 0.3109477 1.6718514
#> [57] 2.3258381 0.5434242 1.0306822 2.9736508 3.1803910 2.7121716 2.7667310
#> [64] 2.5782398 1.8760712 3.0852801 2.2654800 2.8031443 1.5769829 2.1311538
#> [71] 1.9129112 1.8138741 1.7296852 1.2309918 1.2858102 1.9978788 1.0892187
#> [78] 2.0766095 1.7471685 1.8009587 2.8613923 1.1336461 2.1069233 1.6561833
#> [85] 2.0988490 1.9150465 2.5787879 2.8104161 2.7846665 1.5953777 2.5187777
#> [92] 0.9488430 2.2119053 2.1542782 2.5159164 2.6740634 1.1166023 0.3722135
#> [99] 1.6928734 2.2298570 0.5843291 1.2561007 3.0662891 3.2740149 3.0130060
#> [106] 2.8966776 2.3924492 1.9195720 3.1038424 2.4698633 2.8910529 1.7264018
#> [113] 2.4273869 1.9378162 1.6742073 0.5639574 1.1764958 1.2496804 1.8637001
#> [120] 1.1691521 2.0872688 1.5744022 1.7298469 2.9304411 1.0201563 1.9811044
#> [127] 1.6681942 2.1348045 1.5313581 2.5150745 2.9973843 2.7681912 1.7154370
#> [134] 2.5772810 0.8951080 2.0540639 2.2076712 2.5922775 2.5457456 1.6160185
#> [141] 0.2355537 1.8541395 2.0494468 0.4914381 0.9626502 2.8953205 3.2430506
#> [148] 2.8505586 2.8138232 2.3029447 1.9745591 2.9895696 2.1532577 2.8004837
#> [155] 1.7566981 1.7815008 2.0389834 2.0122922 0.6748837 1.4306932 1.5142029
#> [162] 1.1333442 1.1276370 2.1006493 2.1675322 1.8864125 2.5926241 1.0516188
#> [169] 2.0762856 1.4716730 2.0082786 1.8555625 2.5722335 2.9254124 2.8315667
#> [176] 1.7095770 2.5310052 0.9156910 2.1895030 2.1133067 2.4433770 2.5284215
#> [183] 1.5917408 0.3365432 1.9200143 1.9170048 0.5100709 1.0241596 2.8920102
#> [190] 3.3345192 2.7927954 2.8126605 2.4130604 1.9143897 3.0940329 2.4190965
#> [197] 3.0585010 1.6222408 1.6625476 1.8857861 1.9242456 0.6377106 1.1021552
#> [204] 1.3132326 1.7753533 0.9128345 2.4091480 2.1650832 1.4916809 2.8787436
#> [211] 0.6548046 2.1711145 1.4429747 1.8281572 1.9867232 2.3518294 2.8197699
#> [218] 2.5347746 1.8711220 2.4972608 0.8320827 2.9844363 2.2310199 2.5788511
#> [225] 2.5037044 1.7868914 0.2655194 1.7010836 1.9617781 0.4734649 0.9704104
#> [232] 3.0409983 3.5790913 2.8954003 2.9146311 2.6674014 2.0420367 3.2798123
#> [239] 2.6144439 3.1474022 1.6457397 1.9044427 2.0077175 2.3988297 0.6098552
#> [246] 1.2576153 1.2932868 1.1942484 1.4741438 2.4985791 2.0472037 1.6638331
#> [253] 1.1832652 0.7223914 2.2743936 1.4444894 2.3832867 2.0192976 2.4856384
#> [260] 2.8492927 2.6693680 1.8159416 2.5786298 0.8569476 2.7205239 2.3290874
#> [267] 2.3625518 2.6763391 1.7500781 0.4880103 1.7336934 1.7571496 0.4816027
#> [274] 0.8310833 3.0787333 3.4816827 3.0615238 2.8771885 2.5014493 2.0924022
#> [281] 3.4050940 2.3637466 2.7953274 1.6079663 1.9153483 1.1776235 1.8736594
#> [288] 0.7990109 1.1506511 1.2063827 1.0593765 1.5096871 2.4415489 2.0734100
#> [295] 1.3592284 1.1583390 0.8507941 1.9451625 1.4728146 2.1092525 1.7678016
#> [302] 2.1872556 2.4621500 2.5665104 1.7351918 2.5951799 1.0868779 2.2386175
#> [309] 2.1667252 2.2906645 2.7338493 1.7816010 0.4911352 1.8329253 1.8120439
#> [316] 0.4770658 0.9030706 2.7378122 3.4560616 2.9363898 2.8635826 2.5768058
#> [323] 1.9027268 3.3379239 2.6178754 2.7099038 1.6341968 2.3195787 1.1575454
#> [330] 2.0290368 0.1534083 1.1206897 1.3624220 1.1015321 1.5564064 2.3988119
#> [337] 2.1502764 1.4867585 1.1304626 1.0437816 1.9765226 1.5564463 2.1815665
#> [344] 2.0019192
residuals(meta_lp) # residuals
#> [1] 5.40071399 7.60608821 7.35394035 3.08865822 6.32691704 1.00132804
#> [7] 5.17615529 4.53317508 6.63674105 6.60887340 -0.19563164 0.73336915
#> [13] 2.25150806 5.08268567 0.38815087 0.07507323 7.88733941 17.88799120
#> [19] 10.81733041 9.45244004 2.83240578 3.59718878 6.68876505 4.87888104
#> [25] 9.51599215 1.57951565 2.64407501 3.65045952 2.25935083 0.88189188
#> [31] 2.61467351 -0.09722964 3.08855880 -0.44876507 9.94362401 0.15397506
#> [37] 3.36387628 14.15291201 0.16969124 1.79558706 2.42449404 6.56000176
#> [43] -0.04907398 4.66982802 7.66540903 7.64788553 1.44703560 6.66334280
#> [49] 0.25564491 5.68385626 3.83916415 7.42557979 5.91111760 -0.16478498
#> [55] 0.60905228 1.17814859 3.97416190 0.37657580 1.08931777 3.00634923
#> [61] 14.93960900 12.60782835 10.34326896 3.26176019 2.40392875 9.79471993
#> [67] 3.66451996 10.94685569 0.95301713 3.84884622 4.47708879 0.99612587
#> [73] 1.44031477 0.55900818 1.01418985 3.06212118 -0.44921865 5.46339053
#> [79] 1.33283151 3.48904129 10.84860767 -0.21364610 1.25307665 3.40381667
#> [85] 5.54115104 -0.16504651 6.34121212 10.47958390 7.61533351 4.24462232
#> [91] 7.32122234 0.75115697 5.37809466 2.39572179 7.37408362 9.51593665
#> [97] 0.26339774 0.54778649 2.26712661 4.16014297 0.24567094 1.09389932
#> [103] 13.40371088 18.57598513 12.85699397 11.73332239 7.95755082 4.01042799
#> [109] 13.45615763 7.60013666 12.00894706 2.59359823 6.49261310 4.59218376
#> [115] 1.36579271 0.30604264 1.63350416 1.56031962 2.68629994 -0.01915215
#> [121] 4.86273120 1.36559777 3.79015309 13.30955889 0.35984371 2.01889564
#> [127] 3.39180576 5.40519554 1.17864190 4.79492550 15.58261574 6.15180881
#> [133] 3.16456303 8.37271904 0.34489200 5.58593613 4.14232878 6.37772255
#> [139] 8.67425440 1.42398147 1.05444631 1.59586055 3.93055318 0.51856192
#> [145] 0.87734977 15.09467954 16.99694940 3.35944136 9.60617679 2.15705528
#> [151] 4.46544092 5.29043041 4.39674232 12.05951625 1.41330189 1.94849916
#> [157] 3.62101659 4.88770778 0.52511627 1.09930676 2.01579715 1.16665577
#> [163] 0.53236303 4.38935065 2.66246782 3.86358755 9.57737588 0.78838120
#> [169] 2.38371439 2.20832700 3.10172137 1.45443752 4.96776654 14.14458758
#> [175] 4.52843330 2.75042302 5.46899481 0.64430897 4.39049702 3.72669331
#> [181] 5.97662295 3.17157851 0.38825921 0.44345679 1.11998566 3.37299521
#> [187] 0.49992913 1.18584036 17.02798983 17.73548082 10.13720459 10.11733955
#> [193] 5.45693963 1.21561029 12.08596709 1.49090349 5.54149905 2.46775920
#> [199] 1.09745241 3.40421394 1.24575439 0.42228944 1.24784484 1.06676742
#> [205] 3.37464671 -0.31283447 3.20085200 2.66491685 1.67831914 7.01125643
#> [211] 0.95519543 2.65888548 0.85702530 1.76184276 3.62327679 2.15817061
#> [217] 13.79023007 7.63522540 2.49887800 6.42273922 0.63791727 10.07556365
#> [223] 2.67898013 5.24114893 8.07629561 2.21310856 0.90448056 1.79891642
#> [229] 3.23822187 0.49653508 1.32958960 16.68900167 22.96090874 13.10459974
#> [235] 9.45536891 11.49259857 1.63796331 14.72018770 5.98555608 8.90259777
#> [241] 2.49426034 3.20555730 3.37228245 0.26117034 0.34014481 1.87238465
#> [247] 0.21671324 1.34575157 0.52585618 5.27142095 2.78279626 2.24616690
#> [253] 1.71673480 0.80760862 2.83560641 3.15551060 4.83671332 5.76070235
#> [259] 1.88436164 9.43070726 2.76063198 2.32405839 6.62137016 0.15305242
#> [265] 7.21947612 1.95091263 5.45744816 5.92366094 0.17992192 0.94198971
#> [271] 1.91630659 3.30285039 -0.11160275 0.95891668 7.04126671 19.23831733
#> [277] 3.37847620 8.34281148 8.76855073 1.81759779 7.17490605 2.51625341
#> [283] 7.64467261 1.61203371 1.39465165 1.35237647 1.75634061 0.26098908
#> [289] -0.09065105 0.63361732 0.27062353 0.38031288 6.46845106 1.74659000
#> [295] 0.85077164 1.19166097 0.24920592 0.81483750 1.28718539 2.86074746
#> [301] 1.93219844 5.08274437 11.73785003 8.38348964 2.17480822 8.44482011
#> [307] 0.75312209 6.04138253 4.45327478 6.35933553 9.82615068 2.04839897
#> [313] 1.20886477 2.90707466 3.06795612 0.21293418 1.07692936 11.29218781
#> [319] 27.64393839 18.76361023 11.94641737 6.85319417 2.69727323 15.75207609
#> [325] 2.53212457 13.34009620 2.64580325 7.02042126 1.60245457 4.02096320
#> [331] -0.06340833 2.49931032 4.79757796 1.69846793 0.37359360 8.32118806
#> [337] 5.48972358 1.73324152 2.54953741 0.47621842 3.52347738 3.73355374
#> [343] 5.49843347 5.72808077