efficiencies returns (in-)efficiency estimates of models
estimated with sfacross, sfalcmcross, or
sfaselectioncross.
# S3 method for sfacross
efficiencies(object, level = 0.95, newData = NULL, ...)# S3 method for sfalcmcross
efficiencies(object, level = 0.95, newData = NULL, ...)
# S3 method for sfaselectioncross
efficiencies(object, level = 0.95, newData = NULL, ...)
A data frame that contains individual (in-)efficiency estimates. These are ordered in the same way as the corresponding observations in the dataset used for the estimation.
- For object of class 'sfacross' the following elements are returned:
Conditional inefficiency. In the case argument udist of
sfacross is set to 'uniform', two conditional inefficiency
estimates are returned: u1 for the classic conditional inefficiency
following Jondrow et al. (1982), and u2 which is obtained when
\(\theta/\sigma_v \longrightarrow \infty\) (see Nguyen, 2010).
Lower bound for conditional inefficiency. Only when the argument
udist of sfacross is set to 'hnormal',
'exponential', 'tnormal' or 'uniform'.
Upper bound for conditional inefficiency. Only when the argument
udist of sfacross is set to 'hnormal',
'exponential', 'tnormal' or 'uniform'.
\(\exp{(-E[u|\epsilon])}\). When the argument udist of
sfacross is set to 'uniform', teJLMS1 =
\(\exp{(-E[u_1|\epsilon])}\) and teJLMS2 =
\(\exp{(-E[u_2|\epsilon])}\). Only when logDepVar = TRUE.
Conditional model. Only when the argument udist of
sfacross is set to 'hnormal', 'exponential',
'tnormal', or 'rayleigh'.
\(\exp{(-m)}\). Only when, in the function sfacross,
logDepVar = TRUE and udist = 'hnormal', 'exponential',
'tnormal', 'uniform', or 'rayleigh'.
Battese and Coelli (1988) conditional efficiency. Only when, in
the function sfacross, logDepVar = TRUE.
In the case udist = 'uniform', two
conditional efficiency estimates are returned:
teBC1 which is the classic conditional efficiency following
Battese and Coelli (1988) and teBC2 when
\(\theta/\sigma_v \longrightarrow \infty\) (see Nguyen, 2010).
Reciprocal of Battese and Coelli (1988) conditional
efficiency. Similar to teBC except that it is computed as
\(E\left[\exp{(u)}|\epsilon\right]\).
Lower bound for Battese and Coelli (1988) conditional
efficiency. Only when, in the function sfacross, logDepVar = TRUE and
udist = 'hnormal', 'exponential', 'tnormal',
or 'uniform'.
Upper bound for Battese and Coelli (1988) conditional
efficiency. Only when, in the function sfacross, logDepVar = TRUE and
udist = 'hnormal', 'exponential', 'tnormal',
or 'uniform'.
In the case udist = 'uniform'. \(u \in [0, \theta]\).
- For object of class 'sfalcmcross' the following elements are returned:
Most probable class for each observation.
Highest posterior probability.
Conditional inefficiency of the most probable class given the posterior probability.
\(\exp{(-E[u_c|\epsilon_c])}\). Only when, in the function
sfalcmcross logDepVar = TRUE.
\(E\left[\exp{(-u_c)}|\epsilon_c\right]\). Only when, in the
function sfalcmcross logDepVar = TRUE.
\(E\left[\exp{(u_c)}|\epsilon_c\right]\). Only
when, in the function sfalcmcross logDepVar = TRUE.
Posterior probability of class #.
Prior probability of class #.
Conditional inefficiency associated to class #, regardless of
Group_c.
Conditional efficiency
(\(E\left[\exp{(-u_c)}|\epsilon_c\right]\)) associated to class #,
regardless of Group_c. Only when, in the function
sfalcmcross logDepVar = TRUE.
Reciprocal conditional efficiency
(\(E\left[\exp{(u_c)}|\epsilon_c\right]\)) associated to class #,
regardless of Group_c. Only when, in the function
sfalcmcross logDepVar = TRUE.
Conditional inefficiency (u_c) for observations in
class # only.
Conditional efficiency (teBC_c) for observations in
class # only.
Reciprocal conditional efficiency (teBC_reciprocal_c)
for observations in class # only.
In the case udist = 'uniform'. \(u \in [0, \theta_{c\#}]\).
- For object of class 'sfaselectioncross' the following elements are returned:
Conditional inefficiency.
\(\exp{(-E[u|\epsilon])}\). Only when logDepVar = TRUE.
Battese and Coelli (1988) conditional efficiency. Only when, in
the function sfaselectioncross,
logDepVar = TRUE.
Reciprocal of Battese and Coelli (1988) conditional
efficiency. Similar to teBC except that it is computed as
\(E\left[\exp{(u)}|\epsilon\right]\).
A stochastic frontier model returned
by sfacross, sfalcmcross, or
sfaselectioncross.
A number between between 0 and 0.9999 used for the computation
of (in-)efficiency confidence intervals (defaut = 0.95). Only used
when udist = 'hnormal', 'exponential', 'tnormal'
or 'uniform' in sfacross.
Optional data frame that is used to calculate the efficiency
estimates. If NULL (the default), the efficiency estimates are calculated
for the observations that were used in the estimation. In the case of object of
class sfaselectioncross
Currently ignored.
In general, the conditional inefficiency is obtained following Jondrow et al. (1982) and the conditional efficiency is computed following Battese and Coelli (1988). In some cases the conditional mode is also returned (Jondrow et al. 1982). The confidence interval is computed following Horrace and Schmidt (1996), Hjalmarsson et al. (1996), or Berra and Sharma (1999) (see ‘Value’ section).
In the case of the half normal distribution for the one-sided error term,
the formulae are as follows (for notations, see the ‘Details’ section
of sfacross or sfalcmcross):
The conditional inefficiency is:
$$E\left\lbrack u_i|\epsilon_i\right \rbrack=\mu_{i\ast} + \sigma_\ast\frac{\phi \left(\frac{\mu_{i\ast}}{\sigma_\ast}\right)}{ \Phi\left(\frac{\mu_{i\ast}}{\sigma_\ast}\right)}$$
where
$$\mu_{i\ast}=\frac{-S\epsilon_i\sigma_u^2}{ \sigma_u^2 + \sigma_v^2}$$
and
$$\sigma_\ast^2 = \frac{\sigma_u^2 \sigma_v^2}{\sigma_u^2 + \sigma_v^2}$$
The Battese and Coelli (1988) conditional efficiency is obtained with:
$$E\left\lbrack\exp{\left(-u_i\right)} |\epsilon_i\right\rbrack = \exp{\left(-\mu_{i\ast}+ \frac{1}{2}\sigma_\ast^2\right)}\frac{\Phi\left( \frac{\mu_{i\ast}}{\sigma_\ast}-\sigma_\ast\right)}{ \Phi\left(\frac{\mu_{i\ast}}{\sigma_\ast}\right)}$$
The reciprocal of the Battese and Coelli (1988) conditional efficiency is obtained with:
$$E\left\lbrack\exp{\left(u_i\right)} |\epsilon_i\right\rbrack = \exp{\left(\mu_{i\ast}+ \frac{1}{2}\sigma_\ast^2\right)} \frac{\Phi\left( \frac{\mu_{i\ast}}{\sigma_\ast}+\sigma_\ast\right)}{ \Phi\left(\frac{\mu_{i\ast}}{\sigma_\ast}\right)}$$
The conditional mode is computed using:
$$M\left\lbrack u_i|\epsilon_i\right \rbrack= \mu_{i\ast} \quad \hbox{For} \quad \mu_{i\ast} > 0$$
and
$$M\left\lbrack u_i|\epsilon_i\right \rbrack= 0 \quad \hbox{For} \quad \mu_{i\ast} \leq 0$$
The confidence intervals are obtained with:
$$\mu_{i\ast} + I_L\sigma_\ast \leq E\left\lbrack u_i|\epsilon_i\right\rbrack \leq \mu_{i\ast} + I_U\sigma_\ast $$
with \(LB_i = \mu_{i*} + I_L\sigma_*\) and \(UB_i = \mu_{i*} + I_U\sigma_*\)
and
$$I_L = \Phi^{-1}\left\lbrace 1 - \left(1-\frac{\alpha}{2}\right)\left\lbrack 1- \Phi\left(-\frac{\mu_{i\ast}}{\sigma_\ast}\right) \right\rbrack\right\rbrace $$
and
$$I_U = \Phi^{-1}\left\lbrace 1- \frac{\alpha}{2}\left\lbrack 1-\Phi \left(-\frac{\mu_{i\ast}}{\sigma_\ast}\right) \right\rbrack\right\rbrace$$
Thus
$$\exp{\left(-UB_i\right)} \leq E\left \lbrack\exp{\left(-u_i\right)}|\epsilon_i\right\rbrack \leq\exp{\left(-LB_i\right)}$$
In the case of the sample selection, as underlined in Greene (2010), the conditional inefficiency could be computed using Jondrow et al. (1982). However, here the conditionanl (in)efficiency is obtained using the properties of the closed skew-normal (CSN) distribution (Lai, 2015). The conditional efficiency can be obtained using the moment generating functions of a CSN distribution (see Gonzalez-Farias et al. (2004)). We have:
$$E\left\lbrack\exp{\left(tu_i\right)} |\epsilon_i\right\rbrack = M_{u|\epsilon}(t)=\frac{\Phi_2\left(\tilde{\mathbf{D}} \tilde{\bm{\Sigma}}t; \tilde{\bm{\kappa}}, \tilde{\bm{\Delta}} + \tilde{\mathbf{D}}\tilde{\bm{\Sigma}}\tilde{\mathbf{D}}' \right)}{ \Phi_2\left(\mathbf{0}; \tilde{\bm{\kappa}}, \tilde{\bm{\Delta}} + \tilde{\mathbf{D}}\tilde{\bm{\Sigma}}\tilde{\mathbf{D}}'\right)}\exp{ \left(t\tilde{\bm{\pi}} + \frac{1}{2}t^2\tilde{\bm{\Sigma}}\right)}$$
where \(\tilde{\bm{\pi}} = \frac{-S\epsilon_i\sigma_u^2}{\sigma_v^2 + \sigma_u^2}\), \(\tilde{\bm{\Sigma}} = \frac{\sigma_v^2\sigma_u^2}{\sigma_v^2 + \sigma_u^2}\), \(\tilde{\mathbf{D}} = \begin{pmatrix} \frac{S\rho}{\sigma_v} \\ 1 \end{pmatrix}\), \(\tilde{\bm{\kappa}} = \begin{pmatrix} - \mathbf{Z}'_{si}\bm{\gamma} - \frac{\rho\sigma_v\epsilon_i}{\sigma_v^2 + \sigma_u^2}\\ \frac{S\sigma_u^2\epsilon_i}{\sigma_v^2 + \sigma_u^2} \end{pmatrix}\), \(\tilde{\bm{\Delta}} = \begin{pmatrix}1-\rho^2 & 0 \\ 0 & 0\end{pmatrix}\).
The derivation of the efficiency and the reciprocal efficiency is obtained by replacing \(t = -1\) and \(t =1\), respectively. To obtain the inefficiency as \(E\left[u_i|\epsilon_i\right]\) is more complicated as it requires the derivation of a multivariate normal cdf. We have:
$$E\left[u_i|\epsilon_i\right] = \left. \frac{\partial M_{u|\epsilon}(t)}{\partial t}\right\rvert_{t = 0}$$
Then
$$E\left[u_i|\epsilon_i\right] = \tilde{\bm{\pi}} + \left(\tilde{\mathbf{D}}\tilde{\bm{\Sigma}}\right)'\frac{\Phi_2^* \left(\mathbf{0}; \tilde{\bm{\kappa}}, \ddot{\bm{\Delta}}\right)}{ \Phi_2\left(\mathbf{0}; \tilde{\bm{\kappa}}, \ddot{\bm{\Delta}}\right)}$$
where \(\Phi_2^* \left(\mathbf{s}; \tilde{\bm{\kappa}}, \ddot{\bm{\Delta}}\right)= \frac{\partial \Phi_2\left(\mathbf{s}; \tilde{\bm{\kappa}}, \ddot{\bm{\Delta}} \right)}{\partial \mathbf{s}}\)
Battese, G.E., and T.J. Coelli. 1988. Prediction of firm-level technical efficiencies with a generalized frontier production function and panel data. Journal of Econometrics, 38:387--399.
Bera, A.K., and S.C. Sharma. 1999. Estimating production uncertainty in stochastic frontier production function models. Journal of Productivity Analysis, 12:187-210.
Gonzalez-Farias, G., Dominguez-Molina, A., Gupta, A. K., 2004. Additive properties of skew normal random vectors. Journal of Statistical Planning and Inference. 126: 521-534.
Greene, W., 2010. A stochastic frontier model with correction for sample selection. Journal of Productivity Analysis. 34, 15--24.
Hjalmarsson, L., S.C. Kumbhakar, and A. Heshmati. 1996. DEA, DFA and SFA: A comparison. Journal of Productivity Analysis, 7:303-327.
Horrace, W.C., and P. Schmidt. 1996. Confidence statements for efficiency estimates from stochastic frontier models. Journal of Productivity Analysis, 7:257-282.
Jondrow, J., C.A.K. Lovell, I.S. Materov, and P. Schmidt. 1982. On the estimation of technical inefficiency in the stochastic frontier production function model. Journal of Econometrics, 19:233--238.
Lai, H. P., 2015. Maximum likelihood estimation of the stochastic frontier model with endogenous switching or sample selection. Journal of Productivity Analysis, 43: 105-117.
Nguyen, N.B. 2010. Estimation of technical efficiency in stochastic frontier analysis. PhD Dissertation, Bowling Green State University, August.
sfalcmcross, for the latent class stochastic frontier analysis
model fitting function using cross-sectional or pooled data.
sfacross, for the stochastic frontier analysis model
fitting function using cross-sectional or pooled data.
sfaselectioncross for sample selection in stochastic frontier
model fitting function using cross-sectional or pooled data.
if (FALSE) {
## Using data on fossil fuel fired steam electric power generation plants in the U.S.
# Translog SFA (cost function) truncated normal with scaling property
tl_u_ts <- sfacross(formula = log(tc/wf) ~ log(y) + I(1/2 * (log(y))^2) + log(wl/wf) +
log(wk/wf) + I(1/2 * (log(wl/wf))^2) + I(1/2 * (log(wk/wf))^2) + I(log(wl/wf) *
log(wk/wf)) + I(log(y) * log(wl/wf)) + I(log(y) * log(wk/wf)), udist = 'tnormal',
muhet = ~ regu, uhet = ~ regu, data = utility, S = -1, scaling = TRUE, method = 'mla')
eff.tl_u_ts <- efficiencies(tl_u_ts)
head(eff.tl_u_ts)
summary(eff.tl_u_ts)
}
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