efficiencies {sfaR}R Documentation

Compute conditional (in-)efficiency estimates of stochastic frontier models

Description

efficiencies returns (in-)efficiency estimates of models estimated with sfacross, sfalcmcross, or sfaselectioncross.

Usage

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

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

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

Arguments

object

A stochastic frontier model returned by sfacross, sfalcmcross, or sfaselectioncross.

level

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.

newData

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.

Details

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):

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}

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)}

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)}

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

\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}}

Value

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:

u

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).

uLB

Lower bound for conditional inefficiency. Only when the argument udist of sfacross is set to 'hnormal', 'exponential', 'tnormal' or 'uniform'.

uUB

Upper bound for conditional inefficiency. Only when the argument udist of sfacross is set to 'hnormal', 'exponential', 'tnormal' or 'uniform'.

teJLMS

\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.

m

Conditional model. Only when the argument udist of sfacross is set to 'hnormal', 'exponential', 'tnormal', or 'rayleigh'.

teMO

\exp{(-m)}. Only when, in the function sfacross, logDepVar = TRUE and udist = 'hnormal', 'exponential', 'tnormal', 'uniform', or 'rayleigh'.

teBC

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).

teBC_reciprocal

Reciprocal of Battese and Coelli (1988) conditional efficiency. Similar to teBC except that it is computed as E\left[\exp{(u)}|\epsilon\right].

teBCLB

Lower bound for Battese and Coelli (1988) conditional efficiency. Only when, in the function sfacross, logDepVar = TRUE and udist = 'hnormal', 'exponential', 'tnormal', or 'uniform'.

teBCUB

Upper bound for Battese and Coelli (1988) conditional efficiency. Only when, in the function sfacross, logDepVar = TRUE and udist = 'hnormal', 'exponential', 'tnormal', or 'uniform'.

theta

In the case udist = 'uniform'. u \in [0, \theta].

- For object of class 'sfalcmcross' the following elements are returned:

Group_c

Most probable class for each observation.

PosteriorProb_c

Highest posterior probability.

u_c

Conditional inefficiency of the most probable class given the posterior probability.

teJLMS_c

\exp{(-E[u_c|\epsilon_c])}. Only when, in the function sfalcmcross logDepVar = TRUE.

teBC_c

E\left[\exp{(-u_c)}|\epsilon_c\right]. Only when, in the function sfalcmcross logDepVar = TRUE.

teBC_reciprocal_c

E\left[\exp{(u_c)}|\epsilon_c\right]. Only when, in the function sfalcmcross logDepVar = TRUE.

PosteriorProb_c#

Posterior probability of class #.

PriorProb_c#

Prior probability of class #.

u_c#

Conditional inefficiency associated to class #, regardless of Group_c.

teBC_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.

teBC_reciprocal_c#

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.

ineff_c#

Conditional inefficiency (u_c) for observations in class # only.

effBC_c#

Conditional efficiency (teBC_c) for observations in class # only.

ReffBC_c#

Reciprocal conditional efficiency (teBC_reciprocal_c) for observations in class # only.

theta_c#

In the case udist = 'uniform'. u \in [0, \theta_{c\#}].

- For object of class 'sfaselectioncross' the following elements are returned:

u

Conditional inefficiency.

teJLMS

\exp{(-E[u|\epsilon])}. Only when logDepVar = TRUE.

teBC

Battese and Coelli (1988) conditional efficiency. Only when, in the function sfaselectioncross, logDepVar = TRUE.

teBC_reciprocal

Reciprocal of Battese and Coelli (1988) conditional efficiency. Similar to teBC except that it is computed as E\left[\exp{(u)}|\epsilon\right].

References

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.

See Also

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.

Examples


## Not run: 
## 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)

## End(Not run)


[Package sfaR version 1.0.0 Index]