Title:	Stochastic Metafrontier Analysis
Version:	1.0.0
Description:	Implements stochastic metafrontier analysis for productivity and performance benchmarking across firms operating under different technologies. Contains routines for the deterministic metafrontier envelope of O'Donnell et al. (2008) <doi:10.1007/s00181-007-0119-4> via linear and quadratic programming, and the stochastic metafrontier of Huang et al. (2014) <doi:10.1007/s11123-014-0402-2>. Also supports latent class stochastic metafrontier analysis and sample selection correction stochastic metafrontier models. Depends on the 'sfaR' package by Dakpo et al. (2023) https://CRAN.R-project.org/package=sfaR.
License:	GPL (≥ 3)
URL:	https://github.com/SulmanOlieko/smfa, https://SulmanOlieko.github.io/smfa/
BugReports:	https://github.com/SulmanOlieko/smfa/issues
Depends:	R (≥ 3.5.0), sfaR
Imports:	stats
Suggests:	knitr, rmarkdown, pkgdown, lmtest
VignetteBuilder:	knitr
Config/Needs/website:	pkgdown
Encoding:	UTF-8
Language:	en-US
RoxygenNote:	7.3.3
NeedsCompilation:	no
Packaged:	2026-04-24 14:26:20 UTC; macbookpro
Author:	Sulman Olieko Owili [aut, cre]
Maintainer:	Sulman Olieko Owili <oliekosulman@gmail.com>
Repository:	CRAN
Date/Publication:	2026-04-28 19:00:02 UTC

Extract coefficients of stochastic metafrontier models

Description

From an object of class 'summary.smfa', coef extracts the coefficients, their standard errors, z-values, and (asymptotic) P-values.

From on object of class 'smfa', it extracts only the estimated coefficients.

Usage

## S3 method for class 'smfa'
coef(object, ...)

## S3 method for class 'summary.smfa'
coef(object, ...)

Arguments

Value

For objects of class 'summary.smfa', coef returns a matrix with four columns. Namely, the estimated coefficients, their standard errors, z-values, and (asymptotic) P-values.

For objects of class 'smfa', coef returns a numeric vector of the estimated coefficients.

See Also

smfa, for the stochastic metafrontier analysis model fitting function using cross-sectional or pooled data.

Compute efficiency estimates and metatechnology ratios from stochastic metafrontier models

Description

efficiencies returns all efficiency estimates and metatechnology ratio (MTR) measures for objects of class "smfa" returned by smfa. The function supports models estimated via linear programming (LP), quadratic programming (QP), and stochastic second-stage SFA ("sfa"), and for each observation it computes the group-specific technical efficiency, the metafrontier technical efficiency, and the metatechnology ratio (MTR), using both the Jondrow, Lovell, Materov, and Schmidt (1982) (JLMS) and the Battese and Coelli (1988) (BC) estimators. Additional model-specific columns are returned depending on groupType.

Usage

## S3 method for class 'smfa'
efficiencies(object, level = 0.95, newData = NULL, ...)

Arguments

Details

Group-specific efficiency estimates

For each group, the group-level frontier model is estimated by maximising the log-likelihood using the distribution specified by udist in smfa. Given the estimated composite error \varepsilon_i = v_i - Su_i, the conditional distribution of u_i \mid \varepsilon_i is used to compute:

• the JLMS estimator (Jondrow, Lovell, Materov, and Schmidt, 1982): \hat{u}_i = E[u_i \mid \varepsilon_i], and TE^g_{JLMS,i} = \exp(-\hat{u}_i);

• the BC estimator (Battese and Coelli, 1988): TE^g_{BC,i} = E[\exp(-u_i) \mid \varepsilon_i];

• the mode estimator: m_i = \mathrm{mode}[u_i \mid \varepsilon_i], and TE^g_{\mathrm{mode},i} = \exp(-m_i);

• confidence bounds on u_i and TE^g_{BC,i} at the nominal level level.

For groupType = "sfaselectioncross", all estimates are NA for observations not selected into the sample (binary selection indicator equal to 0). For groupType = "sfalcmcross", the composite efficiencies u_g, TE_group_JLMS, and TE_group_BC are computed using the posterior-probability-weighted class assignments.

Metatechnology ratio and metafrontier efficiency

The MTR measures how far the group frontier lies below the metafrontier for each observation. Let \ln \hat{y}^g_i be the group-specific fitted frontier value and \ln \hat{y}^*_i the metafrontier fitted value.

• For metaMethod = "lp" or "qp" (Battese, Rao, and O'Donnell, 2004):

  MTR_i = \exp\!\bigl( -\max\!\bigl\{S \cdot (\ln \hat{y}^*_i - \ln \hat{y}^g_i),\, 0\bigr\} \bigr)

  where S = 1 for production/profit frontiers and S = -1 for cost frontiers. The technology gap U_i = \max\{S \cdot (\ln \hat{y}^*_i - \ln \hat{y}^g_i), 0\} is stored in u_meta.

• For metaMethod = "sfa" with sfaApproach = "huang" (Huang, Huang, and Liu, 2014):

  MTR_i = TE^*_i = \exp(-U_i)

  where U_i is the one-sided error term from the second-stage SFA.

• For metaMethod = "sfa" with sfaApproach = "ordonnell" (O'Donnell, Rao, and Battese, 2008): MTR_i = TE^{*,\mathrm{sfa}}_i / TE^g_i, where TE^{*,\mathrm{sfa}}_i is the technical efficiency from the second-stage SFA fitted on the LP envelope values.

The metafrontier technical efficiency is then:

TE^*_i = TE^g_i \times MTR_i

computed separately for the JLMS and BC group efficiency estimators. Both MTR_JLMS and MTR_BC are reported, distinguishing which group-level efficiency estimator was used as the basis.

Value

A data frame with one row per observation (in the original row order), containing the following columns. The exact set of columns depends on groupType:

Columns present for all model types:

id

Observation identifier. Contains the row name of each observation as it appeared in the data supplied to smfa. When the data frame has no explicit row names, sequential integers ("1", "2", ...) are used. This column is always the first column of the returned data frame.

<group> or Group_c

The technology group identifier for each observation. For groupType = "sfacross" and "sfaselectioncross", this column takes the name of the user-supplied group variable and contains the group label to which each observation belongs. For groupType = "sfalcmcross", it is named Group_c and contains the integer index of the latent class assigned by the maximum posterior probability criterion.

u_g

Group-specific conditional mean of the inefficiency term, computed as E[u_i \mid \varepsilon_i]. This is the JLMS (Jondrow, Lovell, Materov, and Schmidt, 1982) point estimate of the inefficiency at the group-frontier level. For groupType = "sfaselectioncross", u_g is NA for observations not selected into the sample (selection indicator = 0).

TE_group_JLMS

Group-specific technical efficiency using the Jondrow, Lovell, Materov, and Schmidt (1982) estimator: TE^g_i = \exp(-E[u_i \mid \varepsilon_i]). For groupType = "sfaselectioncross", NA for non-selected observations.

TE_group_BC

Group-specific technical efficiency using the Battese and Coelli (1988) estimator: TE^g_i = E[\exp(-u_i) \mid \varepsilon_i]. For groupType = "sfaselectioncross", NA for non-selected observations.

TE_group_BC_reciprocal

Reciprocal of the Battese and Coelli (1988) group technical efficiency: 1 / TE^{g,BC}_i. For production frontiers this equals the cost-efficiency index implied by the BC estimator. Present for all three model types. For groupType = "sfaselectioncross", NA for non-selected observations.

u_meta

Metafrontier inefficiency, measuring the technology-gap component U_i \ge 0 that separates the group frontier from the global metafrontier. Computed from the second-stage SFA when metaMethod = "sfa", or derived from the LP/QP gap as U_i = \max\{S \cdot (\ln \hat{y}^*_i - \ln \hat{y}^g_i), 0\} when metaMethod = "lp" or "qp".

TE_meta_JLMS

Metafrontier technical efficiency based on the JLMS group efficiency: TE^*_{JLMS,i} = TE^g_{JLMS,i} \times MTR_{JLMS,i}.

TE_meta_BC

Metafrontier technical efficiency based on the Battese and Coelli (1988) group efficiency: TE^*_{BC,i} = TE^g_{BC,i} \times MTR_{BC,i}.

MTR_JLMS

Metatechnology ratio computed using the JLMS group efficiency: MTR_{JLMS,i} = TE^*_{JLMS,i} / TE^g_{JLMS,i} = \exp(-U_i). Values range from 0 to 1. A value of 1 indicates that the group frontier for this observation coincides with the metafrontier.

MTR_BC

Metatechnology ratio computed using the Battese and Coelli (1988) group efficiency: MTR_{BC,i} = TE^*_{BC,i} / TE^g_{BC,i} = \exp(-U_i).

Additional columns for groupType = "sfacross" only:

uLB_g, uUB_g

Lower and upper bounds of the level confidence interval for the conditional mean inefficiency u_g, constructed using the asymptotic distribution of the conditional estimator. Available for distributions with closed-form expressions for the confidence bounds, such as udist = "hnormal" and udist = "tnormal".

m_g

Mode of the conditional distribution of the one-sided error term u_i \mid \varepsilon_i. This is an alternative point estimate of inefficiency. Available for distributions for which the conditional mode has a closed-form expression.

TE_group_mode

Group-specific technical efficiency evaluated at the conditional mode: TE^g_{\mathrm{mode},i} = \exp(-m_i).

teBCLB_g, teBCUB_g

Lower and upper bounds of the level confidence interval for the Battese and Coelli (1988) group technical efficiency TE_group_BC. Constructed from the corresponding bounds on the conditional distribution of \exp(-u_i \mid \varepsilon_i).

Additional columns for groupType = "sfalcmcross" only:

PosteriorProb_c

Posterior probability that observation i belongs to its assigned class (the one with the highest posterior probability). Computed via Bayes' rule as P(j \mid y_i, x_i) \propto \pi(i,j) \, P(i \mid j), where \pi(i,j) is the prior class probability and P(i \mid j) is the class-conditional likelihood.

PosteriorProb_cJ (per class, J = 1, 2, \ldots)

Posterior probability of belonging to latent class J, computed via Bayes' rule for each class separately. One column is produced for each estimated class.

PriorProb_cJ (per class, J = 1, 2, \ldots)

Prior (unconditional) probability of belonging to latent class J, given by the logistic specification \pi(i,J) = \exp(\bm{\theta}_J'\mathbf{Z}_{hi}) / \sum_m \exp(\bm{\theta}_m'\mathbf{Z}_{hi}).

u_cJ (per class, J = 1, 2, \ldots)

Conditional mean of the inefficiency term for class J: E[u_{i \mid J} \mid \varepsilon_{i \mid J}].

teBC_cJ (per class, J = 1, 2, \ldots)

Battese and Coelli (1988) technical efficiency for class J: E[\exp(-u_{i \mid J}) \mid \varepsilon_{i \mid J}].

teBC_reciprocal_cJ (per class, J = 1, 2, \ldots)

Reciprocal of the class-J Battese and Coelli (1988) efficiency: 1/TE^{BC}_{i \mid J}.

ineff_cJ (per class, J = 1, 2, \ldots)

Inefficiency estimate for the observation restricted to class J (i.e. the value assigned to the class to which the observation does belong; NA for other classes).

effBC_cJ (per class, J = 1, 2, \ldots)

Battese and Coelli (1988) efficiency for the observation's assigned class; NA for non-assigned classes.

ReffBC_cJ (per class, J = 1, 2, \ldots)

Reciprocal Battese and Coelli (1988) efficiency for the observation's assigned class; NA for non-assigned classes.

References

Battese, G. E., and Coelli, T. J. 1988. Prediction of firm-level technical efficiencies with a generalized frontier production function and panel data. Journal of Econometrics, 38(3), 387–399. doi:10.1016/0304-4076(88)90053-X

Battese, G. E., Rao, D. S. P., and O'Donnell, C. J. 2004. A metafrontier production function for estimation of technical efficiencies and technology gaps for firms operating under different technologies. Journal of Productivity Analysis, 21(1), 91–103. doi:10.1023/B:PROD.0000012454.06094.29

Huang, C. J., Huang, T.-H., and Liu, N.-H. 2014. A new approach to estimating the metafrontier production function based on a stochastic frontier framework. Journal of Productivity Analysis, 42(3), 241–254. doi:10.1007/s11123-014-0402-2

Jondrow, J., Lovell, C. A. K., Materov, I. S., and Schmidt, P. 1982. On the estimation of technical inefficiency in the stochastic frontier production function model. Journal of Econometrics, 19(2-3), 233–238. doi:10.1016/0304-4076(82)90004-5

O'Donnell, C. J., Rao, D. S. P., and Battese, G. E. 2008. Metafrontier frameworks for the study of firm-level efficiencies and technology ratios. Empirical Economics, 34(2), 231–255. doi:10.1007/s00181-007-0119-4

Orea, L., and Kumbhakar, S. C. 2004. Efficiency measurement using a latent class stochastic frontier model. Empirical Economics, 29(1), 169–183. doi:10.1007/s00181-003-0184-2

Dakpo, K. H., Desjeux, Y., and Latruffe, L. 2023. sfaR: Stochastic Frontier Analysis using R. R package version 1.0.1. https://CRAN.R-project.org/package=sfaR

See Also

smfa, for the stochastic metafrontier analysis model fitting function using cross-sectional or pooled data; efficiencies, for the underlying group-level efficiency extractor.

Extract fitted values of stochastic metafrontier models

Description

fitted returns the fitted frontier values from stochastic metafrontier models estimated with smfa.

Usage

## S3 method for class 'smfa'
fitted(object, ...)

Arguments

Value

A vector of fitted values is returned.

Note

The fitted values are ordered in the same way as the corresponding observations in the dataset used for the estimation.

See Also

smfa, for the stochastic metafrontier analysis model fitting function using cross-sectional or pooled data.

Extract information criteria of stochastic metafrontier models

Description

ic returns information criterion from stochastic metafrontier models estimated with smfa.

Usage

## S3 method for class 'smfa'
ic(object, IC = NULL, ...)

Arguments

Details

The different information criteria are computed as follows:

• AIC: -2 \log{LL} + 2 * K

• BIC: -2 \log{LL} + \log{N} * K

• HQIC: -2 \log{LL} + 2 \log{\left[\log{N}\right]} * K

where LL is the maximum likelihood value, K the number of parameters estimated and N the number of observations.

Value

ic returns a data frame with the values of the information criteria (AIC, BIC and HQIC) of the maximum likelihood coefficients. If the IC argument is provided, it returns only the requested criterion as a numeric value.

See Also

smfa, for the stochastic metafrontier analysis model fitting function using cross-sectional or pooled data.

Extract log-likelihood value of stochastic metafrontier models

Description

logLik extracts the log-likelihood value(s) from stochastic metafrontier models estimated with smfa.

Usage

## S3 method for class 'smfa'
logLik(object, individual = FALSE, ...)

Arguments

Value

logLik returns either an object of class 'logLik', which is the log-likelihood value with the total number of observations (nobs) and the number of free parameters (df) as attributes, when individual = FALSE, or a list of elements, containing the log-likelihood of each observation (logLik), the total number of observations (Nobs) and the number of free parameters (df), when individual = TRUE.

See Also

smfa, for the stochastic metafrontier analysis model fitting function using cross-sectional or pooled data.

Extract total number of observations used in frontier models

Description

This function extracts the total number of 'observations' from a fitted point frontier model.

Usage

## S3 method for class 'smfa'
nobs(object, ...)

Arguments

Details

nobs gives the number of observations actually used by the estimation procedure.

Value

A single number, normally an integer.

See Also

smfa, for the stochastic metafrontier analysis model fitting function using cross-sectional or pooled data

Extract residuals of stochastic metafrontier models

Description

This function returns the residuals' values from stochastic metafrontier models estimated with smfa.

Usage

## S3 method for class 'smfa'
residuals(object, ...)

Arguments

Value

residuals returns a vector of residuals values.

Note

The residuals values are ordered in the same way as the corresponding observations in the dataset used for the estimation.

See Also

smfa, for the stochastic metafrontier analysis model fitting function using cross-sectional or pooled data.

Stochastic metafrontier estimation

Description

smfa estimates a stochastic metafrontier model for cross-sectional or pooled data. The function follows the theoretical frameworks of Battese, Rao, and O'Donnell (2004) and O'Donnell, Rao, and Battese (2008), and additionally implements the two-stage stochastic approach of Huang, Huang, and Liu (2014). Three types of group-level frontier models are supported: standard stochastic frontier analysis (sfacross), sample selection stochastic frontier analysis (sfaselectioncross), and latent class stochastic frontier analysis (sfalcmcross).

Usage

smfa(
  formula,
  muhet,
  uhet,
  vhet,
  thet,
  logDepVar = TRUE,
  data,
  subset,
  weights,
  wscale = TRUE,
  group = NULL,
  S = 1L,
  udist = "hnormal",
  start = NULL,
  scaling = FALSE,
  modelType = "greene10",
  groupType = "sfacross",
  metaMethod = "lp",
  sfaApproach = "ordonnell",
  selectionF = NULL,
  lcmClasses = 2L,
  whichStart = 2L,
  initAlg = "nm",
  initIter = 100L,
  lType = "ghermite",
  Nsub = 100L,
  uBound = Inf,
  intol = 1e-06,
  method = "bfgs",
  hessianType = NULL,
  simType = "halton",
  Nsim = 100L,
  prime = 2L,
  burn = 10L,
  antithetics = FALSE,
  seed = 12345L,
  itermax = 2000L,
  printInfo = FALSE,
  tol = 1e-12,
  gradtol = 1e-06,
  stepmax = 0.1,
  qac = "marquardt",
  ...
)

## S3 method for class 'smfa'
print(x, ...)

Arguments

Details

Standard stochastic frontier (groupType = "sfacross")

The stochastic frontier model is defined as:

y_i = \alpha + \mathbf{x}_i'\bm{\beta} + v_i - Su_i

where y is the output (cost, revenue, or profit), \mathbf{x} is the vector of frontier regressors, u_i \ge 0 is the one-sided inefficiency term with variance \sigma^2_u, and v_i is the symmetric noise term with variance \sigma^2_v.

Estimation is by ML for all distributions except "gamma", "lognormal", and "weibull", for which MSL is used with Halton, Generalised-Halton, Sobol, or uniform draws. Antithetic draws are available for the uniform case.

To account for heteroscedasticity, the variances are modelled as \sigma^2_u = \exp(\bm{\delta}'\mathbf{Z}_u) and \sigma^2_v = \exp(\bm{\phi}'\mathbf{Z}_v). For the truncated normal distribution, heterogeneity in the pre-truncation mean is modelled as \mu_i = \bm{\omega}'\mathbf{Z}_{\mu}. The scaling property (Wang and Schmidt, 2002) can also be imposed for the truncated normal.

Sample selection frontier (groupType = "sfaselectioncross")

This model extends the Heckman (1979) selection framework to the stochastic frontier setting (Greene, 2010; Dakpo et al., 2021). The selection and frontier equations are:

y_{1i}^* = \mathbf{Z}_{si}'\bm{\gamma} + w_i, \quad w_i \sim \mathcal{N}(0,1)

y_{2i}^* = \mathbf{x}_i'\bm{\beta} + v_i - Su_i

where y_{1i} = \mathbf{1}(y_{1i}^* > 0) is the binary selection indicator and y_{2i} = y_{2i}^* is observed only when y_{1i} = 1. Selection bias arises from \rho = \mathrm{Corr}(w_i, v_i) \ne 0. Only selected observations enter the frontier and metafrontier estimation; efficiency estimates for non-selected observations are NA.

Latent class frontier (groupType = "sfalcmcross")

The latent class model (Orea and Kumbhakar, 2004) estimates a finite mixture of J frontier models:

y_i = \alpha_j + \mathbf{x}_i'\bm{\beta}_j + v_{i|j} - Su_{i|j}

The prior class probability follows a logit specification:

\pi(i,j) = \frac{\exp(\bm{\theta}_j'\mathbf{Z}_{hi})} {\sum_{m=1}^{J}\exp(\bm{\theta}_m'\mathbf{Z}_{hi})}

Class assignment is based on the maximum posterior probability computed via Bayes' rule. When group is omitted, a single pooled model is estimated and class assignments serve as technology groups.

Metafrontier estimation

The global metafrontier f(x_i, \bm{\beta}^*) envelopes all group frontiers. With LP (Battese et al., 2004), \bm{\beta}^* minimises \sum_i |\ln \hat{f}(x_i, \bm{\beta}^*) - \ln \hat{f}(x_i, \hat{\bm{\beta}}_{(g)})| subject to \ln \hat{f}(x_i, \bm{\beta}^*) \ge \ln \hat{f}(x_i, \hat{\bm{\beta}}_{(g)}). QP minimises the squared analogue. The stochastic approaches (Huang et al., 2014; O'Donnell et al., 2008) treat the technology gap U_i as a one-sided error in a second-stage SFA. Group and metafrontier efficiencies are:

TE_i^g = \exp(-u_i^g), \quad MTR_i = \exp(-U_i), \quad TE_i^* = TE_i^g \times MTR_i

Both Jondrow et al. (1982) and Battese and Coelli (1988) estimators are provided for each measure. See efficiencies for details.

Value

smfa returns an object of class "smfa", which is a list containing:

References

Aigner, D. J., Lovell, C. A. K., and Schmidt, P. 1977. Formulation and estimation of stochastic frontier production function models. Journal of Econometrics, 6(1), 21–37. doi:10.1016/0304-4076(77)90052-5

Battese, G. E., and Coelli, T. J. 1988. Prediction of firm-level technical efficiencies with a generalized frontier production function and panel data. Journal of Econometrics, 38(3), 387–399. doi:10.1016/0304-4076(88)90053-X

Battese, G. E., Rao, D. S. P., and O'Donnell, C. J. 2004. A metafrontier production function for estimation of technical efficiencies and technology gaps for firms operating under different technologies. Journal of Productivity Analysis, 21(1), 91–103. doi:10.1023/B:PROD.0000012454.06094.29

Greene, W. 2003. Simulated likelihood estimation of the normal-gamma stochastic frontier function. Journal of Productivity Analysis, 19(2-3), 179–190. doi:10.1023/A:1022853416499

Greene, W. 2010. A stochastic frontier model with correction for sample selection. Journal of Productivity Analysis, 34(1), 15–24. doi:10.1007/s11123-009-0159-1

Hajargasht, G. 2015. Stochastic frontiers with a Rayleigh distribution. Journal of Productivity Analysis, 44(2), 199–208. doi:10.1007/s11123-014-0417-8

Heckman, J. J. 1979. Sample selection bias as a specification error. Econometrica, 47(1), 153–161. doi:10.2307/1912352

Huang, C. J., Huang, T.-H., and Liu, N.-H. 2014. A new approach to estimating the metafrontier production function based on a stochastic frontier framework. Journal of Productivity Analysis, 42(3), 241–254. doi:10.1007/s11123-014-0402-2

Jondrow, J., Lovell, C. A. K., Materov, I. S., and Schmidt, P. 1982. On the estimation of technical inefficiency in the stochastic frontier production function model. Journal of Econometrics, 19(2-3), 233–238. doi:10.1016/0304-4076(82)90004-5

Li, Q. 1996. Estimating a stochastic production frontier when the adjusted error is symmetric. Economics Letters, 52(3), 221–228. doi:10.1016/S0165-1765(96)00857-9

Meeusen, W., and van den Broeck, J. 1977. Efficiency estimation from Cobb-Douglas production functions with composed error. International Economic Review, 18(2), 435–444. doi:10.2307/2525757

Migon, H. S., and Medici, E. 2001. Bayesian inference for generalised exponential models. Working paper, Universidade Federal do Rio de Janeiro.

Nguyen, N. B. 2010. Estimation of technical efficiency in stochastic frontier analysis. PhD thesis, Bowling Green State University.

O'Donnell, C. J., Rao, D. S. P., and Battese, G. E. 2008. Metafrontier frameworks for the study of firm-level efficiencies and technology ratios. Empirical Economics, 34(2), 231–255. doi:10.1007/s00181-007-0119-4

Orea, L., and Kumbhakar, S. C. 2004. Efficiency measurement using a latent class stochastic frontier model. Empirical Economics, 29(1), 169–183. doi:10.1007/s00181-003-0184-2

Dakpo, K. H., Jeanneaux, P., and Latruffe, L. 2016. Modelling pollution-generating technologies in performance benchmarking: Recent developments, limits and future prospects in the nonparametric framework. European Journal of Operational Research, 250(2), 347–359. doi:10.1016/j.ejor.2015.07.024

Papadopoulos, A. 2015. The half-normal specification for the two-tier stochastic frontier model. Journal of Productivity Analysis, 43(2), 225–230. doi:10.1007/s11123-014-0389-8

Stevenson, R. E. 1980. Likelihood functions for generalised stochastic frontier estimation. Journal of Econometrics, 13(1), 57–66. doi:10.1016/0304-4076(80)90042-1

Dakpo, K. H., Latruffe, L., Desjeux, Y., and Jeanneaux, P. 2021. Latent class modelling for a robust assessment of productivity: Application to French grazing livestock farms. Journal of Agricultural Economics, 72(3), 760–781. doi:10.1111/1477-9552.12422

Dakpo, K. H., Latruffe, L., Desjeux, Y., and Jeanneaux, P. 2022. Modeling heterogeneous technologies in the presence of sample selection: The case of dairy farms and the adoption of agri-environmental schemes in France. Agricultural Economics, 53(3), 422–438. doi:10.1111/agec.12683

Tsionas, E. G. 2007. Efficiency measurement with the Weibull stochastic frontier. Oxford Bulletin of Economics and Statistics, 69(5), 693–706. doi:10.1111/j.1468-0084.2007.00475.x

Wang, H.-J. 2012. Stochastic frontier models. In A Companion to Theoretical Econometrics, ed. B. H. Baltagi, Blackwell, Oxford.

Wang, H.-J., and Schmidt, P. 2002. One-step and two-step estimation of the effects of exogenous variables on technical efficiency levels. Journal of Productivity Analysis, 18(2), 129–144. doi:10.1023/A:1016565719882

Dakpo, K. H., Desjeux, Y., and Latruffe, L. 2023. sfaR: Stochastic Frontier Analysis using R. R package version 1.0.1. https://CRAN.R-project.org/package=sfaR

See Also

sfacross, sfaselectioncross, sfalcmcross, efficiencies, summary.smfa, ic

Examples

###########################################################################
## -------- SECTION 1: Standard SFA Group Frontier ----------------------##
## Using the rice production dataset (ricephil) from Battese et al.      ##
## Groups are formed based on farm area terciles (small/medium/large).   ##
###########################################################################

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
)

## 1a. sfacross groups + LP metafrontier
##     Deterministic envelope via linear programming (Battese et al., 2004).
meta_sfacross_lp <- smfa(
  formula    = log(PROD) ~ log(AREA) + log(LABOR) + log(NPK),
  data       = ricephil,
  group      = "group",
  S          = 1,
  udist      = "hnormal",
  groupType  = "sfacross",
  metaMethod = "lp"
)
summary(meta_sfacross_lp)
# Retrieve individual efficiency and metatechnology ratio estimates:
ef_lp <- efficiencies(meta_sfacross_lp)
head(ef_lp)

## 1b. sfacross groups + QP metafrontier
##     Deterministic envelope via quadratic programming.
meta_sfacross_qp <- smfa(
  formula    = log(PROD) ~ log(AREA) + log(LABOR) + log(NPK),
  data       = ricephil,
  group      = "group",
  S          = 1,
  udist      = "hnormal",
  groupType  = "sfacross",
  metaMethod = "qp"
)
summary(meta_sfacross_qp)


## 1c. sfacross groups + Two-stage SFA metafrontier (Huang et al., 2014)
##     The group-specific fitted frontier values serve as the dependent
##     variable in the second-stage SFA, yielding a stochastic technology gap.
meta_sfacross_huang <- smfa(
  formula     = log(PROD) ~ log(AREA) + log(LABOR) + log(NPK),
  data        = ricephil,
  group       = "group",
  S           = 1,
  udist       = "hnormal",
  groupType   = "sfacross",
  metaMethod  = "sfa",
  sfaApproach = "huang"
)
summary(meta_sfacross_huang)
ef_huang <- efficiencies(meta_sfacross_huang)
head(ef_huang)

## 1d. sfacross groups + O'Donnell et al. (2008) stochastic metafrontier
##     The LP deterministic envelope is used as the second-stage dependent
##     variable: the metafrontier is estimated stochastically around the
##     envelope.
meta_sfacross_odonnell <- smfa(
  formula     = log(PROD) ~ log(AREA) + log(LABOR) + log(NPK),
  data        = ricephil,
  group       = "group",
  S           = 1,
  udist       = "hnormal",
  groupType   = "sfacross",
  metaMethod  = "sfa",
  sfaApproach = "ordonnell"
)
summary(meta_sfacross_odonnell)


###########################################################################
## -------- SECTION 2: Latent Class (LCM) Group Frontier ---------------##
## No observed group variable: a pooled sfalcmcross model assigns       ##
## observations to 2 latent technology classes; these classes become the ##
## technology groups for the metafrontier.                               ##
###########################################################################

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

## 2a. sfalcmcross (pooled, 2 classes) + LP metafrontier
meta_lcm_lp <- smfa(
  formula    = log(tc / wf) ~ log(y) + log(wl / wf) + log(wk / wf),
  data       = utility,
  S          = -1,
  groupType  = "sfalcmcross",
  lcmClasses = 2,
  metaMethod = "lp"
)
summary(meta_lcm_lp)
ef_lcm_lp <- efficiencies(meta_lcm_lp)
head(ef_lcm_lp)


## 2b. sfalcmcross (pooled, 2 classes) + QP metafrontier
meta_lcm_qp <- smfa(
  formula    = log(tc / wf) ~ log(y) + log(wl / wf) + log(wk / wf),
  data       = utility,
  S          = -1,
  groupType  = "sfalcmcross",
  lcmClasses = 2,
  metaMethod = "qp"
)
summary(meta_lcm_qp)

## 2c. sfalcmcross (pooled, 2 classes) + Two-stage SFA metafrontier
##     (Huang et al., 2014)
meta_lcm_huang <- smfa(
  formula     = log(tc / wf) ~ log(y) + log(wl / wf) + log(wk / wf),
  data        = utility,
  S           = -1,
  groupType   = "sfalcmcross",
  lcmClasses  = 2,
  metaMethod  = "sfa",
  sfaApproach = "huang"
)
summary(meta_lcm_huang)
ef_lcm_huang <- efficiencies(meta_lcm_huang)
head(ef_lcm_huang)

## 2d. sfalcmcross (pooled, 2 classes) + O'Donnell et al. (2008)
meta_lcm_odonnell <- smfa(
  formula     = log(tc / wf) ~ log(y) + log(wl / wf) + log(wk / wf),
  data        = utility,
  S           = -1,
  groupType   = "sfalcmcross",
  lcmClasses  = 2,
  metaMethod  = "sfa",
  sfaApproach = "ordonnell"
)
summary(meta_lcm_odonnell)


###########################################################################
## -------- SECTION 3: Sample Selection SFA Group Frontier -------------##
###########################################################################

## 3a. Small toy example for automatic testing (< 5s)
N <- 100
set.seed(12345)
z1 <- rnorm(N); v1 <- rnorm(N); g <- rnorm(N)
ds <- z1 + v1; d <- ifelse(ds > 0, 1, 0)
group <- ifelse(g > 0, 1, 0)
x1 <- rnorm(N); y <- x1 + rnorm(N) - abs(rnorm(N))
dat <- data.frame(y = y, x1 = x1, z1 = z1, d = d, group = group)

meta_toy <- smfa(
  formula    = y ~ x1,
  selectionF = d ~ z1,
  data       = dat,
  group      = "group",
  groupType  = "sfaselectioncross",
  lType      = "ghermite",
  Nsub       = 10,
  itermax    = 100,
  metaMethod = "lp"
)
summary(meta_toy)


## 3b. More complex selection models
## Simulated dataset with a Heckman selection mechanism.

N <- 2000
set.seed(12345)
z1 <- rnorm(N); z2 <- rnorm(N); v1 <- rnorm(N); v2 <- rnorm(N); g <- rnorm(N)
e1 <- v1; e2 <- 0.7071 * (v1 + v2)
ds <- z1 + z2 + e1; d <- ifelse(ds > 0, 1, 0)
group <- ifelse(g > 0, 1, 0)
u <- abs(rnorm(N)); x1 <- rnorm(N); x2 <- rnorm(N)
y <- x1 + x2 + e2 - u
dat <- data.frame(y = y, x1 = x1, x2 = x2, z1 = z1, z2 = z2, d = d, group = group)

meta_sel_lp <- smfa(
  formula    = y ~ x1 + x2,
  selectionF = d ~ z1 + z2,
  data       = dat,
  group      = "group",
  S          = 1L,
  udist      = "hnormal",
  groupType  = "sfaselectioncross",
  modelType  = "greene10",
  lType      = "kronrod",
  Nsub       = 100,
  metaMethod = "lp"
)
summary(meta_sel_lp)

Summary of results for stochastic metafrontier models

Description

Create and print summary results for stochastic metafrontier models returned by smfa.

Usage

## S3 method for class 'smfa'
summary(object, ...)

## S3 method for class 'summary.smfa'
print(x, digits = max(3, getOption("digits") - 2), ...)

Arguments

Value

The summary method returns a list of class 'summary.smfa' that contains the same elements as an object returned by smfa with the following additional elements:

See Also

smfa, for the stochastic metafrontier analysis model fitting function for cross-sectional or pooled data.

Compute variance-covariance matrix of stochastic metafrontier models

Description

vcov computes the variance-covariance matrix of the maximum likelihood (ML) coefficients from stochastic metafrontier models estimated with smfa.

Usage

## S3 method for class 'smfa'
vcov(object, ...)

Arguments

Details

The variance-covariance matrix is obtained by the inversion of the negative Hessian matrix. Depending on the distribution and the 'hessianType' option, the analytical/numeric Hessian or the bhhh Hessian is evaluated.

Value

The variance-covariance matrix of the maximum likelihood coefficients is returned.

See Also

smfa, for the stochastic metafrontier analysis model fitting function using cross-sectional or pooled data.