Compute efficiency estimates and metatechnology ratios from stochastic metafrontier models

efficiencies returns all efficiency estimates and metatechnology ratio (MTR) measures for objects of class "sfametafrontier" returned by sfametafrontier. 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 'sfametafrontier'
efficiencies(object, level = 0.95, newData = NULL, ...)

Arguments

object: An object of class "sfametafrontier" returned by sfametafrontier.
level: A number strictly between 0 and 0.9999 specifying the nominal coverage for (in-)efficiency confidence intervals. Default 0.95. This argument is passed to the underlying efficiencies method of the group-level model (class "sfacross", "sfalcmcross", or "sfaselectioncross").
newData: Optional data frame for out-of-sample prediction of efficiency estimates. When NULL (default), efficiencies are computed for the observations used in the estimation.
...: Further arguments (currently ignored).

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 sfametafrontier. 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.

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 sfametafrontier. 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.

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. https://doi.org/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. https://doi.org/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. https://doi.org/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. https://doi.org/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. https://doi.org/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. https://doi.org/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