efficiencies: Compute conditional (in-)efficiency estimates of classic or...

In sfaR: Stochastic Frontier Analysis using R

Compute conditional (in-)efficiency estimates of classic or latent class stochastic models

Description

efficiencies returns (in-)efficiency estimates from classic or latent class stochastic frontier models estimated with sfacross or lcmcross.

Usage

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

## S3 method for class 'lcmcross'
efficiencies(object, level = 0.95, ...)

Arguments

object	A classic or latent class stochastic frontier model returned by sfacross or lcmcross.
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 or lcmcross.
...	Currently ignored.

Details

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 lcmcross):

• The conditional inefficiency is

E≤ft[u_i|ε_i\right]=μ_{i*} + σ_*\frac{φ≤ft(\frac{μ_{i*}}{σ_*}\right)}{Φ≤ft(\frac{μ_{i*}}{σ_*}\right)}

where

μ_{i*}=\frac{-Sε_iσ_u^2}{σ_u^2 + σ_v^2}

and

σ_*^2 = \frac{σ_u^2 σ_v^2}{σ_u^2 + σ_v^2}

• The Battese and Coelli (1988) conditional efficiency is obtained by:

E≤ft[\exp{≤ft(-u_i\right)}|ε_i\right] = \exp{≤ft(-μ_{i*}+\frac{1}{2}σ_*^2\right)} \frac{Φ≤ft(\frac{μ_{i*}}{σ_*}-σ_*\right)}{Φ≤ft(\frac{μ_{i*}}{σ_*}\right)}

• The conditional mode is computed using:

M≤ft[u_i|ε_i\right]= μ_{i*} \quad For \quad μ_{i*} > 0

and

M≤ft[u_i|ε_i\right]= 0 \quad For \quad μ_{i*} ≤q 0

• The confidence intervals are obtained with:

μ_{i*} + I_Lσ_* ≤q E≤ft[u_i|ε_i\right] ≤q μ_{i*} + I_Uσ_*

with LB_i = μ_{i*} + I_Lσ_* and UB_i = μ_{i*} + I_Uσ_*

and

I_L = Φ^{-1}≤ft\{1 - ≤ft(1-\frac{α}{2}\right)≤ft[1-Φ≤ft(-\frac{μ_{i*}}{σ_*}\right)\right]\right\}

and

I_U = Φ^{-1}≤ft\{1-\frac{α}{2}≤ft[1-Φ≤ft(-\frac{μ_{i*}}{σ_*}\right)\right]\right\}

Thus

\exp{≤ft(-UB_i\right)} ≤q E≤ft[\exp{≤ft(-u_i\right)}|ε_i\right] ≤q \exp{≤ft(-LB_i\right)}

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 θ/σ_v \longrightarrow ∞ (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{(-u)}. When the argument udist of sfacross is set to "uniform", teJLMS1 = \exp{(-u1)} and teJLMS2 = \exp{(-u2)}. 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 and udist = "hnormal", "exponential", "tnormal", "genexponential", "rayleigh", or "tslaplace". 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 θ/σ_v \longrightarrow ∞ (see Nguyen, 2010).
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".

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

Group_c	Most probable class of each observation.
PosteriorProb_c	Highest posterior probability.
PosteriorProb_c#	Posterior probability associated to class #, regardless of Group_c.
PriorProb_c#	Prior probability associated to class #, regardless of Group_c.
u_c	Conditional inefficiency of the most probable class given the posterior probability.
teJLMS_c	\exp{(-u_c)}. Only when, in the function lcmcross, logDepVar = TRUE.
u_c#	Conditional inefficiency associated to class #, regardless of Group_c.
ineff_c#	Conditional inefficiency (u_c) for observations in class # only.

Author(s)

K Hervé Dakpo, Yann Desjeux and Laure Latruffe

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.

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.

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

See Also

sfacross, for the stochastic frontier analysis model fitting function.

lcmcross, for the latent class stochastic frontier analysis model fitting function.

Examples

## 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)
  
cb_2c_h1 <- lcmcross(formula = ly ~ lk + ll + yr, thet = ~initStat, data = worldprod)
  eff.ccb_2c_h1 <- efficiencies(cb_2c_h1)
  table(eff.ccb_2c_h1$Group_c)
  summary(eff.ccb_2c_h1[,c("ineff_c1", "ineff_c2")])
  summary(eff.ccb_2c_h1[eff.ccb_2c_h1$Group_c == 1, c("PosteriorProb_c1", "PriorProb_c1")])
  summary(eff.ccb_2c_h1[eff.ccb_2c_h1$Group_c == 2, c("PosteriorProb_c2", "PriorProb_c2")])

sfaR documentation built on May 3, 2022, 3 p.m.