Package 'sfaR'

sfaR: A package for estimating stochastic frontier models

Description

The sfaR package provides a set of tools (maximum likelihood - ML and maximum simulated likelihood - MSL) for various specifications of stochastic frontier analysis (SFA).

Details

Three categories of functions are available: sfacross, sfalcmcross, sfaselectioncross, which estimate different types of frontiers and offer eleven alternative optimization algorithms (i.e., "bfgs", "bhhh", "nr", "nm", "cg", "sann", "ucminf", "mla", "sr1", "sparse", "nlminb").

sfacross

sfacross estimates the basic stochastic frontier analysis (SFA) for cross-sectional or pooled data and allows for ten different distributions for the one-sided error term. These distributions include the exponential, the gamma, the generalized exponential, the half normal, the lognormal, the truncated normal, the truncated skewed Laplace, the Rayleigh, the uniform, and the Weibull distributions. In the case of the gamma, lognormal, and Weibull distributions, maximum simulated likelihood (MSL) is used with the possibility of four specific distributions to construct the draws: halton, generalized halton, sobol and uniform. Heteroscedasticity in both error terms can be implemented, in addition to heterogeneity in the truncated mean parameter in the case of the truncated normal and lognormal distributions. In addition, in the case of the truncated normal distribution, the scaling property can be estimated.

sfalcmcross

sfalcmcross estimates latent class stochastic frontier models (LCM) for cross-sectional or pooled data. It accounts for technological heterogeneity by splitting the observations into a maximum number of five classes. The classification operates based on a logit functional form that can be specified using some covariates (namely, the separating variables allowing the separation of observations in several classes). Only the half normal distribution is available for the one-sided error term. Heteroscedasticity in both error terms is possible. The choice of the number of classes can be guided by several information criteria (i.e., AIC, BIC, or HQIC).

sfaselectioncross

sfaselectioncross estimates the frontier for cross-sectional or pooled data in the presence of sample selection. The model solves the selection bias due to the correlation between the two-sided error terms in both the selection and the frontier equations. The likelihood can be estimated using five different possibilities: gauss-kronrod quadrature, adaptive integration over hypercubes (hcubature and pcubature), gauss-hermite quadrature, and maximum simulated likelihood. Only the half normal distribution is available for the one-sided error term. Heteroscedasticity in both error terms is possible.

Bugreport

Any bug or suggestion can be reported using the sfaR tracker facilities at: https://github.com/hdakpo/sfaR/issues

Author(s)

K Hervé Dakpo, Yann Desjeux, Arne Henningsen and Laure Latruffe

---

Extract coefficients of stochastic frontier models

Description

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

From on object of class 'sfacross', 'sfalcmcross', or 'sfaselectioncross', it extracts only the estimated coefficients.

Usage

## S3 method for class 'sfacross'
coef(object, extraPar = FALSE, ...)

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

## S3 method for class 'sfalcmcross'
coef(object, extraPar = FALSE, ...)

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

## S3 method for class 'sfaselectioncross'
coef(object, extraPar = FALSE, ...)

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

Arguments

object	A stochastic frontier model returned by sfacross, sfalcmcross, or sfaselectioncross, or an object of class 'summary.sfacross', 'summary.sfalcmcross', or 'summary.sfaselectioncross'.
extraPar	Logical (default = FALSE). If TRUE, additional parameters are returned: sigmaSq = sigmauSq + sigmavSq lambdaSq = sigmauSq/sigmavSq sigmauSq = $\exp{(Wu)}$ = $\exp{(\bm{\delta}' \mathbf{Z}_u)}$ sigmavSq = $\exp{(Wv)}$ = $\exp{(\bm{\phi}' \mathbf{Z}_v)}$ sigma = sigmaSq^0.5 lambda = lambdaSq^0.5 sigmau = sigmauSq^0.5 sigmav = sigmavSq^0.5 gamma = sigmauSq/(sigmauSq + sigmavSq)
...	Currently ignored.

Value

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

For objects of class 'sfacross', 'sfalcmcross', or 'sfaselectioncross', coef returns a numeric vector of the estimated coefficients. If extraPar = TRUE, additional parameters, detailed in the section ‘Arguments’, are also returned. In the case of object of class 'sfalcmcross', each additional parameter ends with '#' that represents the class number.

See Also

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

sfalcmcross, for the latent class 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')
coef(tl_u_ts, extraPar = TRUE)
coef(summary(tl_u_ts))

## End(Not run)

---

Data on Norwegian dairy farms

Description

This dataset contains nine years (1998-2006) of information on Norwegian dairy farms.

Format

A data frame with 2,727 observations on the following 23 variables.

farmid

Farm identification.

year

Year identification.

y1

Milk sold (1000 liters).

y2

Meat (1000 NOK).

y3

Support payments (1000 NOK).

y4

Other outputs (1000 NOK).

p1

Milk price (NOK/liter).

p2

Meat price (cattle index).

p3

Support payments price (CP index).

p4

Other outputs price index.

x1

Land (decare (daa) = 0.1 ha).

x2

Labour (1000 hours).

x3

Purchase feed (1000 NOK).

x4

Other variable costs (1000 NOK).

x5

Cattle capital (1000 NOK).

x6

Other capital (1000 NOK).

w1

Land price (NOK/daa).

w2

Labour price (NOK/hour).

w3

Feed price index.

w4

Other variable cost index.

w5

Cattle capital rent.

w6

Other capital rent and depreciation.

tc

Total cost.

Source

https://sites.google.com/view/sfbook-stata/home

References

Kumbhakar, S.C., H.J. Wang, and A. Horncastle. 2014. A Practitioner's Guide to Stochastic Frontier Analysis Using Stata. Cambridge University Press.

Examples

str(dairynorway)
summary(dairynorway)

---

Data on Spanish dairy farms

Description

This dataset contains six years of observations on 247 dairy farms in northern Spain, drawn from 1993-1998. The original data consist in the farm and year identifications, plus measurements on one output (i.e. milk), and four inputs (i.e. cows, land, labor and feed).

Format

A data frame with 1,482 observations on the following 29 variables.

FARM

Farm identification.

AGEF

Age of the farmer.

YEAR

Year identification.

COWS

Number of milking cows.

LAND

Agricultural area.

MILK

Milk production.

LABOR

Labor.

FEED

Feed.

YIT

Log of MILK.

X1

Log of COWS.

X2

Log of LAND.

X3

Log of LABOR.

X4

Log of FEED.

X11

1/2 * X1^2.

X22

1/2 * X2^2.

X33

1/2 * X3^2.

X44

1/2 * X4^2.

X12

X1 * X2.

X13

X1 * X3.

X14

X1 * X4.

X23

X2 * X3.

X24

X2 * X4.

X34

X3 * X4.

YEAR93

Dummy for YEAR = 1993.

YEAR94

Dummy for YEAR = 1994.

YEAR95

Dummy for YEAR = 1995.

YEAR96

Dummy for YEAR = 1996.

YEAR97

Dummy for YEAR = 1997.

YEAR98

Dummy for YEAR = 1998.

Details

This dataset has been used in Alvarez et al. (2004). The data have been normalized so that the logs of the inputs sum to zero over the 1,482 observations.

Source

https://pages.stern.nyu.edu/~wgreene/Text/Edition7/tablelist8new.htm

References

Alvarez, A., C. Arias, and W. Greene. 2004. Accounting for unobservables in production models: management and inefficiency. Econometric Society, 341:1–20.

Examples

str(dairyspain)
summary(dairyspain)

---

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):

• 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 $\Phi_2^* \left(\mathbf{s}; \tilde{\bm{\kappa}}, \ddot{\bm{\Delta}}\right)= \frac{\partial \Phi_2\left(\mathbf{s}; \tilde{\bm{\kappa}}, \ddot{\bm{\Delta}} \right)}{\partial \mathbf{s}}$

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)

---

Data on U.S. electric power generation

Description

This dataset is on electric power generation in the United States.

Format

A data frame with 123 observations on the following 9 variables.

firm

Firm identification.

cost

Total cost in 1970, MM USD.

output

Output in million KwH.

lprice

Labor price.

lshare

Labor's cost share.

cprice

Capital price.

cshare

Capital's cost share.

fprice

Fuel price.

fshare

Fuel's cost share.

Details

The dataset is from Christensen and Greene (1976) and has also been used in Greene (1990).

Source

https://pages.stern.nyu.edu/~wgreene/Text/Edition7/tablelist8new.htm

References

Christensen, L.R., and W.H. Greene. 1976. Economies of scale in US electric power generation. The Journal of Political Economy, 84:655–676.

Greene, W.H. 1990. A Gamma-distributed stochastic frontier model. Journal of Econometrics, 46:141–163.

Examples

str(electricity)
summary(electricity)

---

Extract frontier information to be used with texreg package

Description

Extract coefficients and additional information for stochastic frontier models returned by sfacross, sfalcmcross, or sfaselectioncross.

Usage

extract.sfacross(model, ...)

extract.sfalcmcross(model, ...)

extract.sfaselectioncross(model, ...)

Arguments

model	objects of class 'sfacross', 'sfalcmcross', or 'sfaselectioncross'
...	Currently ignored

Value

A texreg object representing the statistical model.

See Also

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

sfalcmcross, for the latent class 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 data.

Examples

hlf <- 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 = 'hnormal', uhet = ~ regu, data = utility, S = -1, method = 'bfgs')
trnorm <- 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, data = utility, S = -1, method = 'bfgs')

tscal <- 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, method = 'bfgs', scaling = TRUE)

expo <- 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 = 'exponential', uhet = ~ regu, data = utility, S = -1, method = 'bfgs')

texreg::screenreg(list(hlf, trnorm, tscal, expo))

---

Extract fitted values of stochastic frontier models

Description

fitted returns the fitted frontier values from stochastic frontier models estimated with sfacross, sfalcmcross, or sfaselectioncross.

Usage

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

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

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

Arguments

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

Value

In the case of an object of class 'sfacross', or 'sfaselectioncross', a vector of fitted values is returned.

In the case of an object of class 'sfalcmcross', a data frame containing the fitted values for each class is returned where each variable ends with '_c#', '#' being the class number.

Note

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

See Also

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

sfalcmcross, for the latent class 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 eighty-two countries production (GDP)
# LCM Cobb Douglas (production function) half normal distribution
cb_2c_h <- sfalcmcross(formula = ly ~ lk + ll + yr, udist = 'hnormal', 
data = worldprod)
fit.cb_2c_h <- fitted(cb_2c_h)
head(fit.cb_2c_h)

## End(Not run)

---

Extract information criteria of stochastic frontier models

Description

ic returns information criterion from stochastic frontier models estimated with sfacross, sfalcmcross, or sfaselectioncross.

Usage

## S3 method for class 'sfacross'
ic(object, IC = "AIC", ...)

## S3 method for class 'sfalcmcross'
ic(object, IC = "AIC", ...)

## S3 method for class 'sfaselectioncross'
ic(object, IC = "AIC", ...)

Arguments

object	A stochastic frontier model returned by sfacross, sfalcmcross, or sfaselectioncross.
IC	Character string. Information criterion measure. Three criteria are available: 'AIC' for Akaike information criterion (default) 'BIC' for Bayesian information criterion 'HQIC' for Hannan-Quinn information criterion .
...	Currently ignored.

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 the value of the information criterion (AIC, BIC or HQIC) of the maximum likelihood coefficients.

See Also

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

sfalcmcross, for the latent class 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 Swiss railway
# LCM (cost function) half normal distribution
cb_2c_u <- sfalcmcross(formula = LNCT ~ LNQ2 + LNQ3 + LNNET + LNPK + LNPL,
udist = 'hnormal', uhet = ~ 1, data = swissrailways, S = -1, method='ucminf')
ic(cb_2c_u)
ic(cb_2c_u, IC = 'BIC')
ic(cb_2c_u, IC = 'HQIC')

## End(Not run)

---

Extract log-likelihood value of stochastic frontier models

Description

logLik extracts the log-likelihood value(s) from stochastic frontier models estimated with sfacross, sfalcmcross, or sfaselectioncross.

Usage

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

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

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

Arguments

object	A stochastic frontier model returned by sfacross, sfalcmcross, or sfaselectioncross.
individual	Logical. If FALSE (default), the sum of all observations' log-likelihood values is returned. If TRUE, a vector of each observation's log-likelihood value is returned.
...	Currently ignored.

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

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

sfalcmcross, for the latent class 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')
logLik(tl_u_ts)

## Using data on eighty-two countries production (GDP)
# LCM Cobb Douglas (production function) half normal distribution
cb_2c_h <- sfalcmcross(formula = ly ~ lk + ll + yr, udist = 'hnormal', 
data = worldprod, S = 1)
logLik(cb_2c_h, individual = TRUE)

## End(Not run)

---

Marginal effects of the inefficiency drivers in stochastic frontier models

Description

This function returns marginal effects of the inefficiency drivers from stochastic frontier models estimated with sfacross, sfalcmcross, or sfaselectioncross.

Usage

## S3 method for class 'sfacross'
marginal(object, newData = NULL, ...)

## S3 method for class 'sfalcmcross'
marginal(object, newData = NULL, ...)

## S3 method for class 'sfaselectioncross'
marginal(object, newData = NULL, ...)

Arguments

object	A stochastic frontier model returned by sfacross, sfalcmcross, or sfaselectioncross.
newData	Optional data frame that is used to calculate the marginal effect of $Z$ variables on inefficiency. If NULL (the default), the marginal estimates are calculated for the observations that were used in the estimation.
...	Currently ignored.

Details

marginal operates in the presence of exogenous variables that explain inefficiency, namely the inefficiency drivers ($uhet = ~ Z_u$ or $muhet = ~ Z_{mu}$).

Two components are computed for each variable: the marginal effects on the expected inefficiency ($\frac{\partial E[u]}{\partial Z_{mu}}$) and the marginal effects on the variance of inefficiency ($\frac{\partial V[u]}{\partial Z_{mu}}$).

The model also allows the Wang (2002) parametrization of $\mu$ and $\sigma_u^2$ by the same vector of exogenous variables. This double parameterization accounts for non-monotonic relationships between the inefficiency and its drivers.

Value

marginal returns a data frame containing the marginal effects of the $Z_u$ variables on the expected inefficiency (each variable has the prefix 'Eu_') and on the variance of the inefficiency (each variable has the prefix 'Vu_').

In the case of the latent class stochastic frontier (LCM), each variable ends with '_c#' where '#' is the class number.

References

Wang, H.J. 2002. Heteroscedasticity and non-monotonic efficiency effects of a stochastic frontier model. Journal of Productivity Analysis, 18:241–253.

See Also

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

sfalcmcross, for the latent class 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 + wl, uhet = ~ regu + wl, data = utility, 
S = -1, scaling = TRUE, method = 'mla')
marg.tl_u_ts <- marginal(tl_u_ts)
summary(marg.tl_u_ts)

## Using data on eighty-two countries production (GDP)
# LCM Cobb Douglas (production function) half normal distribution
cb_2c_h <- sfalcmcross(formula = ly ~ lk + ll + yr, udist = 'hnormal',
    data = worldprod, uhet = ~ initStat + h, S = 1, method = 'mla')
  marg.cb_2c_h <- marginal(cb_2c_h)
  summary(marg.cb_2c_h)
  
## End(Not run)

---

Extract total number of observations used in frontier models

Description

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

Usage

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

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

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

Arguments

object	a sfacross, sfalcmcross, or sfaselectioncross object for which the number of total observations is to be extracted.
...	Currently ignored.

Details

nobs gives the number of observations actually used by the estimation procedure. It is not necessarily the number of observations of the model frame (number of rows in the model frame), because sometimes the model frame is further reduced by the estimation procedure especially in the presence of NA. In the case of sfaselectioncross, nobs returns the number of observations used in the frontier equation.

Value

A single number, normally an integer.

See Also

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

sfalcmcross, for the latent class 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 (cost function) half normal with heteroscedasticity
tl_u_h <- 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 = 'hnormal', uhet = ~ regu, data = utility, S = -1, method = 'bfgs')
nobs(tl_u_h)

## End(Not run)

---

Extract residuals of stochastic frontier models

Description

This function returns the residuals' values from stochastic frontier models estimated with sfacross, sfalcmcross, or sfaselectioncross.

Usage

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

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

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

Arguments

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

Value

When the object is of class 'sfacross', or 'sfaselectioncross', residuals returns a vector of residuals values.

When the object is of 'sfalcmcross', residuals returns a data frame containing the residuals values for each latent class, where each variable ends with '_c#', '#' being the class number.

Note

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

See Also

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

sfalcmcross, for the latent class 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')
resid.tl_u_ts <- residuals(tl_u_ts)
head(resid.tl_u_ts)

## Using data on eighty-two countries production (GDP)
# LCM Cobb Douglas (production function) half normal distribution
cb_2c_h <- sfalcmcross(formula = ly ~ lk + ll + yr, udist = 'hnormal', 
data = worldprod, S = 1)
resid.cb_2c_h <- residuals(cb_2c_h)
head(resid.cb_2c_h)

## End(Not run)

---

Data on rice production in the Philippines

Description

This dataset contains annual data collected from 43 smallholder rice producers in the Tarlac region of the Philippines between 1990 and 1997.

Format

A data frame with 344 observations on the following 17 variables.

YEARDUM

Time period (1= 1990, ..., 8 = 1997).

FARMERCODE

Farmer code (1, ..., 43).

PROD

Output (tonnes of freshly threshed rice).

AREA

Area planted (hectares).

LABOR

Labor used (man-days of family and hired labor).

NPK

Fertiliser used (kg of active ingredients).

OTHER

Other inputs used (Laspeyres index = 100 for Farm 17 in 1991).

PRICE

Output price (pesos per kg).

AREAP

Rental price of land (pesos per hectare).

LABORP

Labor price (pesos per hired man-day).

NPKP

Fertiliser price (pesos per kg of active ingredient).

OTHERP

Price of other inputs (implicit price index).

AGE

Age of the household head (years).

EDYRS

Education of the household head (years).

HHSIZE

Household size.

NADULT

Number of adults in the household.

BANRAT

Percentage of area classified as bantog (upland) fields.

Details

This dataset is published as supplement to Coelli et al. (2005). While most variables of this dataset were supplied by the International Rice Research Institute (IRRI), some were calculated by Coelli et al. (2005, see p. 325–326). The survey is described in Pandey et al. (1999).

References

Coelli, T. J., Rao, D. S. P., O'Donnell, C. J., and Battese, G. E. 2005. An Introduction to Efficiency and Productivity Analysis, Springer, New York.

Pandey, S., Masciat, P., Velasco, L, and Villano, R. 1999. Risk analysis of a rainfed rice production system system in Tarlac, Central Luzon, Philippines. Experimental Agriculture, 35:225–237.

Examples

str(ricephil)
summary(ricephil)

---

Stochastic frontier estimation using cross-sectional data

Description

sfacross is a symbolic formula-based function for the estimation of stochastic frontier models in the case of cross-sectional or pooled cross-sectional data, using maximum (simulated) likelihood - M(S)L.

The function accounts for heteroscedasticity in both one-sided and two-sided error terms as in Reifschneider and Stevenson (1991), Caudill and Ford (1993), Caudill et al. (1995) and Hadri (1999), but also heterogeneity in the mean of the pre-truncated distribution as in Kumbhakar et al. (1991), Huang and Liu (1994) and Battese and Coelli (1995).

Ten distributions are possible for the one-sided error term and eleven optimization algorithms are available.

The truncated normal - normal distribution with scaling property as in Wang and Schmidt (2002) is also implemented.

Usage

sfacross(
  formula,
  muhet,
  uhet,
  vhet,
  logDepVar = TRUE,
  data,
  subset,
  weights,
  wscale = TRUE,
  S = 1L,
  udist = "hnormal",
  scaling = FALSE,
  start = NULL,
  method = "bfgs",
  hessianType = 1L,
  simType = "halton",
  Nsim = 100,
  prime = 2L,
  burn = 10,
  antithetics = FALSE,
  seed = 12345,
  itermax = 2000,
  printInfo = FALSE,
  tol = 1e-12,
  gradtol = 1e-06,
  stepmax = 0.1,
  qac = "marquardt"
)

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

## S3 method for class 'sfacross'
bread(x, ...)

## S3 method for class 'sfacross'
estfun(x, ...)

Arguments

formula	A symbolic description of the model to be estimated based on the generic function formula (see section ‘Details’).
muhet	A one-part formula to consider heterogeneity in the mean of the pre-truncated distribution (see section ‘Details’).
uhet	A one-part formula to consider heteroscedasticity in the one-sided error variance (see section ‘Details’).
vhet	A one-part formula to consider heteroscedasticity in the two-sided error variance (see section ‘Details’).
logDepVar	Logical. Informs whether the dependent variable is logged (TRUE) or not (FALSE). Default = TRUE.
data	The data frame containing the data.
subset	An optional vector specifying a subset of observations to be used in the optimization process.
weights	An optional vector of weights to be used for weighted log-likelihood. Should be NULL or numeric vector with positive values. When NULL, a numeric vector of 1 is used.
wscale	Logical. When weights is not NULL, a scaling transformation is used such that the weights sum to the sample size. Default TRUE. When FALSE no scaling is used.
S	If S = 1 (default), a production (profit) frontier is estimated: $\epsilon_i = v_i-u_i$. If S = -1, a cost frontier is estimated: $\epsilon_i = v_i+u_i$.
udist	Character string. Default = 'hnormal'. Distribution specification for the one-sided error term. 10 different distributions are available: 'hnormal', for the half normal distribution (Aigner et al. 1977, Meeusen and Vandenbroeck 1977) 'exponential', for the exponential distribution 'tnormal' for the truncated normal distribution (Stevenson 1980) 'rayleigh', for the Rayleigh distribution (Hajargasht 2015) 'uniform', for the uniform distribution (Li 1996, Nguyen 2010) 'gamma', for the Gamma distribution (Greene 2003) 'lognormal', for the log normal distribution (Migon and Medici 2001, Wang and Ye 2020) 'weibull', for the Weibull distribution (Tsionas 2007) 'genexponential', for the generalized exponential distribution (Papadopoulos 2020) 'tslaplace', for the truncated skewed Laplace distribution (Wang 2012).
scaling	Logical. Only when udist = 'tnormal' and scaling = TRUE, the scaling property model (Wang and Schmidt 2002) is estimated. Default = FALSE. (see section ‘Details’).
start	Numeric vector. Optional starting values for the maximum likelihood (ML) estimation.
method	Optimization algorithm used for the estimation. Default = 'bfgs'. 11 algorithms are available: 'bfgs', for Broyden-Fletcher-Goldfarb-Shanno (see maxBFGS) 'bhhh', for Berndt-Hall-Hall-Hausman (see maxBHHH) 'nr', for Newton-Raphson (see maxNR) 'nm', for Nelder-Mead (see maxNM) 'cg', for Conjugate Gradient (see maxCG) 'sann', for Simulated Annealing (see maxSANN) 'ucminf', for a quasi-Newton type optimisation with BFGS updating of the inverse Hessian and soft line search with a trust region type monitoring of the input to the line search algorithm (see ucminf) 'mla', for general-purpose optimization based on Marquardt-Levenberg algorithm (see mla) 'sr1', for Symmetric Rank 1 (see trust.optim) 'sparse', for trust regions and sparse Hessian (see trust.optim) 'nlminb', for optimization using PORT routines (see nlminb)
hessianType	Integer. If 1 (Default), analytic Hessian is returned for all the distributions. If 2, bhhh Hessian is estimated ($g'g$).
simType	Character string. If simType = 'halton' (Default), Halton draws are used for maximum simulated likelihood (MSL). If simType = 'ghalton', Generalized-Halton draws are used for MSL. If simType = 'sobol', Sobol draws are used for MSL. If simType = 'uniform', uniform draws are used for MSL. (see section ‘Details’).
Nsim	Number of draws for MSL. Default 100.
prime	Prime number considered for Halton and Generalized-Halton draws. Default = 2.
burn	Number of the first observations discarded in the case of Halton draws. Default = 10.
antithetics	Logical. Default = FALSE. If TRUE, antithetics counterpart of the uniform draws is computed. (see section ‘Details’).
seed	Numeric. Seed for the random draws.
itermax	Maximum number of iterations allowed for optimization. Default = 2000.
printInfo	Logical. Print information during optimization. Default = FALSE.
tol	Numeric. Convergence tolerance. Default = 1e-12.
gradtol	Numeric. Convergence tolerance for gradient. Default = 1e-06.
stepmax	Numeric. Step max for ucminf algorithm. Default = 0.1.
qac	Character. Quadratic Approximation Correction for 'bhhh' and 'nr' algorithms. If 'stephalving', the step length is decreased but the direction is kept. If 'marquardt' (default), the step length is decreased while also moving closer to the pure gradient direction. See maxBHHH and maxNR.
x	an object of class sfacross (returned by the function sfacross).
...	additional arguments of frontier are passed to sfacross; additional arguments of the print, bread, estfun, nobs methods are currently ignored.

Details

The stochastic frontier model for the cross-sectional data is defined as:

$y_i = \alpha + \mathbf{x_i^{\prime}}\bm{\beta} + v_i - Su_i$

with

$\epsilon_i = v_i -Su_i$

where $i$ is the observation, $y$ is the output (cost, revenue, profit), $\mathbf{x}$ is the vector of main explanatory variables (inputs and other control variables), $u$ is the one-sided error term with variance $\sigma_{u}^2$, and $v$ is the two-sided error term with variance $\sigma_{v}^2$.

S = 1 in the case of production (profit) frontier function and S = -1 in the case of cost frontier function.

The model is estimated using maximum likelihood (ML) for most distributions except the Gamma, Weibull and log-normal distributions for which maximum simulated likelihood (MSL) is used. For this latter, several draws can be implemented namely Halton, Generalized Halton, Sobol and uniform. In the case of uniform draws, antithetics can also be computed: first Nsim/2 draws are obtained, then the Nsim/2 other draws are obtained as counterpart of one (1-draw).

To account for heteroscedasticity in the variance parameters of the error terms, a single part (right) formula can also be specified. To impose the positivity to these parameters, the variances are modelled as: $\sigma^2_u = \exp{(\bm{\delta}'\mathbf{Z}_u)}$ or $\sigma^2_v = \exp{(\bm{\phi}'\mathbf{Z}_v)}$, where $\mathbf{Z}_u$ and $\mathbf{Z}_v$ are the heteroscedasticity variables (inefficiency drivers in the case of $\mathbf{Z}_u$) and $\bm{\delta}$ and $\bm{\phi}$ the coefficients. In the case of heterogeneity in the truncated mean $\mu$, it is modelled as $\mu=\bm{\omega}'\mathbf{Z}_{\mu}$. The scaling property can be applied for the truncated normal distribution: $u \sim h(\mathbf{Z}_u, \delta)u$ where $u$ follows a truncated normal distribution $N^+(\tau, \exp{(cu)})$.

In the case of the truncated normal distribution, the convolution of $u_i$ and $v_i$ is:

$f(\epsilon_i)=\frac{1}{\sqrt{\sigma_u^2 + \sigma_v^2}}\phi\left(\frac{S\epsilon_i + \mu}{\sqrt{ \sigma_u^2 + \sigma_v^2}}\right)\Phi\left(\frac{ \mu_{i*}}{\sigma_*}\right)\Big/\Phi\left(\frac{ \mu}{\sigma_u}\right)$

where

$\mu_{i*}=\frac{\mu\\\sigma_v^2 - S\epsilon_i\sigma_u^2}{\sigma_u^2 + \sigma_v^2}$

and

$\sigma_*^2 = \frac{\sigma_u^2 \sigma_v^2}{\sigma_u^2 + \sigma_v^2}$

In the case of the half normal distribution the convolution is obtained by setting $\mu=0$.

sfacross allows for the maximization of weighted log-likelihood. When option weights is specified and wscale = TRUE, the weights are scaled as:

$new_{weights} = sample_{size} \times \frac{old_{weights}}{\sum(old_{weights})}$

For complex problems, non-gradient methods (e.g. nm or sann) can be used to warm start the optimization and zoom in the neighborhood of the solution. Then a gradient-based methods is recommended in the second step. In the case of sann, we recommend to significantly increase the iteration limit (e.g. itermax = 20000). The Conjugate Gradient (cg) can also be used in the first stage.

A set of extractor functions for fitted model objects is available for objects of class 'sfacross' including methods to the generic functions print, summary, coef, fitted, logLik, residuals, vcov, efficiencies, ic, marginal, skewnessTest, estfun and bread (from the sandwich package), lmtest::coeftest() (from the lmtest package).

Value

sfacross returns a list of class 'sfacross' containing the following elements:

call	The matched call.
formula	The estimated model.
S	The argument 'S'. See the section ‘Arguments’.
typeSfa	Character string. 'Stochastic Production/Profit Frontier, e = v - u' when S = 1 and 'Stochastic Cost Frontier, e = v + u' when S = -1.
Nobs	Number of observations used for optimization.
nXvar	Number of explanatory variables in the production or cost frontier.
nmuZUvar	Number of variables explaining heterogeneity in the truncated mean, only if udist = 'tnormal' or 'lognormal'.
scaling	The argument 'scaling'. See the section ‘Arguments’.
logDepVar	The argument 'logDepVar'. See the section ‘Arguments’.
nuZUvar	Number of variables explaining heteroscedasticity in the one-sided error term.
nvZVvar	Number of variables explaining heteroscedasticity in the two-sided error term.
nParm	Total number of parameters estimated.
udist	The argument 'udist'. See the section ‘Arguments’.
startVal	Numeric vector. Starting value for M(S)L estimation.
dataTable	A data frame (tibble format) containing information on data used for optimization along with residuals and fitted values of the OLS and M(S)L estimations, and the individual observation log-likelihood. When weights is specified an additional variable is also provided in dataTable.
olsParam	Numeric vector. OLS estimates.
olsStder	Numeric vector. Standard errors of OLS estimates.
olsSigmasq	Numeric. Estimated variance of OLS random error.
olsLoglik	Numeric. Log-likelihood value of OLS estimation.
olsSkew	Numeric. Skewness of the residuals of the OLS estimation.
olsM3Okay	Logical. Indicating whether the residuals of the OLS estimation have the expected skewness.
CoelliM3Test	Coelli's test for OLS residuals skewness. (See Coelli, 1995).
AgostinoTest	D'Agostino's test for OLS residuals skewness. (See D'Agostino and Pearson, 1973).
isWeights	Logical. If TRUE weighted log-likelihood is maximized.
optType	Optimization algorithm used.
nIter	Number of iterations of the ML estimation.
optStatus	Optimization algorithm termination message.
startLoglik	Log-likelihood at the starting values.
mlLoglik	Log-likelihood value of the M(S)L estimation.
mlParam	Parameters obtained from M(S)L estimation.
gradient	Each variable gradient of the M(S)L estimation.
gradL_OBS	Matrix. Each variable individual observation gradient of the M(S)L estimation.
gradientNorm	Gradient norm of the M(S)L estimation.
invHessian	Covariance matrix of the parameters obtained from the M(S)L estimation.
hessianType	The argument 'hessianType'. See the section ‘Arguments’.
mlDate	Date and time of the estimated model.
simDist	The argument 'simDist', only if udist = 'gamma', 'lognormal' or , 'weibull'. See the section ‘Arguments’.
Nsim	The argument 'Nsim', only if udist = 'gamma', 'lognormal' or , 'weibull'. See the section ‘Arguments’.
FiMat	Matrix of random draws used for MSL, only if udist = 'gamma', 'lognormal' or , 'weibull'.

Note

For the Halton draws, the code is adapted from the mlogit package.

References

Aigner, D., Lovell, C. A. K., and Schmidt, P. 1977. Formulation and estimation of stochastic frontier production function models. Journal of Econometrics, 6(1), 21–37.

Battese, G. E., and Coelli, T. J. 1995. A model for technical inefficiency effects in a stochastic frontier production function for panel data. Empirical Economics, 20(2), 325–332.

Caudill, S. B., and Ford, J. M. 1993. Biases in frontier estimation due to heteroscedasticity. Economics Letters, 41(1), 17–20.

Caudill, S. B., Ford, J. M., and Gropper, D. M. 1995. Frontier estimation and firm-specific inefficiency measures in the presence of heteroscedasticity. Journal of Business & Economic Statistics, 13(1), 105–111.

Coelli, T. 1995. Estimators and hypothesis tests for a stochastic frontier function - a Monte-Carlo analysis. Journal of Productivity Analysis, 6:247–268.

D'Agostino, R., and E.S. Pearson. 1973. Tests for departure from normality. Empirical results for the distributions of $b_2$ and $\sqrt{b_1}$. Biometrika, 60:613–622.

Greene, W. H. 2003. Simulated likelihood estimation of the normal-Gamma stochastic frontier function. Journal of Productivity Analysis, 19(2-3), 179–190.

Hadri, K. 1999. Estimation of a doubly heteroscedastic stochastic frontier cost function. Journal of Business & Economic Statistics, 17(3), 359–363.

Hajargasht, G. 2015. Stochastic frontiers with a Rayleigh distribution. Journal of Productivity Analysis, 44(2), 199–208.

Huang, C. J., and Liu, J.-T. 1994. Estimation of a non-neutral stochastic frontier production function. Journal of Productivity Analysis, 5(2), 171–180.

Kumbhakar, S. C., Ghosh, S., and McGuckin, J. T. 1991) A generalized production frontier approach for estimating determinants of inefficiency in U.S. dairy farms. Journal of Business & Economic Statistics, 9(3), 279–286.

Li, Q. 1996. Estimating a stochastic production frontier when the adjusted error is symmetric. Economics Letters, 52(3), 221–228.

Meeusen, W., and Vandenbroeck, J. 1977. Efficiency estimation from Cobb-Douglas production functions with composed error. International Economic Review, 18(2), 435–445.

Migon, H. S., and Medici, E. V. 2001. Bayesian hierarchical models for stochastic production frontier. Lacea, Montevideo, Uruguay.

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

Papadopoulos, A. 2021. Stochastic frontier models using the generalized exponential distribution. Journal of Productivity Analysis, 55:15–29.

Reifschneider, D., and Stevenson, R. 1991. Systematic departures from the frontier: A framework for the analysis of firm inefficiency. International Economic Review, 32(3), 715–723.

Stevenson, R. E. 1980. Likelihood Functions for Generalized Stochastic Frontier Estimation. Journal of Econometrics, 13(1), 57–66.

Tsionas, E. G. 2007. Efficiency measurement with the Weibull stochastic frontier. Oxford Bulletin of Economics and Statistics, 69(5), 693–706.

Wang, K., and Ye, X. 2020. Development of alternative stochastic frontier models for estimating time-space prism vertices. Transportation.

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, 18:129–144.

Wang, J. 2012. A normal truncated skewed-Laplace model in stochastic frontier analysis. Master thesis, Western Kentucky University, May.

See Also

print for printing sfacross object.

summary for creating and printing summary results.

coef for extracting coefficients of the estimation.

efficiencies for computing (in-)efficiency estimates.

fitted for extracting the fitted frontier values.

ic for extracting information criteria.

logLik for extracting log-likelihood value(s) of the estimation.

marginal for computing marginal effects of inefficiency drivers.

residuals for extracting residuals of the estimation.

skewnessTest for conducting residuals skewness test.

vcov for computing the variance-covariance matrix of the coefficients.

bread for bread for sandwich estimator.

estfun for gradient extraction for each observation.

skewnessTest for implementing skewness test.

Examples

## Using data on fossil fuel fired steam electric power generation plants in the U.S.
# Translog (cost function) half normal with heteroscedasticity
tl_u_h <- 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 = 'hnormal', uhet = ~ regu, data = utility, S = -1, method = 'bfgs')
summary(tl_u_h)

# Translog (cost function) truncated normal with heteroscedasticity
tl_u_t <- 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, data = utility, S = -1, method = 'bhhh')
summary(tl_u_t)

# Translog (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')
summary(tl_u_ts)

## Using data on Philippine rice producers
# Cobb Douglas (production function) generalized exponential, and Weibull 
# distributions

cb_p_ge <- sfacross(formula = log(PROD) ~ log(AREA) + log(LABOR) + log(NPK) +
log(OTHER), udist = 'genexponential', data = ricephil, S = 1, method = 'bfgs')
summary(cb_p_ge)

## Using data on U.S. electric utility industry
# Cost frontier Gamma distribution
tl_u_g <- sfacross(formula = log(cost/fprice) ~ log(output) + I(log(output)^2) +
I(log(lprice/fprice)) + I(log(cprice/fprice)), udist = 'gamma', uhet = ~ 1,
data = electricity, S = -1, method = 'bfgs', simType = 'halton', Nsim = 200,
hessianType = 2)
summary(tl_u_g)

---

Latent class stochastic frontier using cross-sectional data

Description

sfalcmcross is a symbolic formula based function for the estimation of the latent class stochastic frontier model (LCM) in the case of cross-sectional or pooled cross-sectional data. The model is estimated using maximum likelihood (ML). See Orea and Kumbhakar (2004), Parmeter and Kumbhakar (2014, p282).

Only the half-normal distribution is possible for the one-sided error term. Eleven optimization algorithms are available.

The function also accounts for heteroscedasticity in both one-sided and two-sided error terms, as in Reifschneider and Stevenson (1991), Caudill and Ford (1993), Caudill et al. (1995) and Hadri (1999).

The model can estimate up to five classes.

Usage

sfalcmcross(
  formula,
  uhet,
  vhet,
  thet,
  logDepVar = TRUE,
  data,
  subset,
  weights,
  wscale = TRUE,
  S = 1L,
  udist = "hnormal",
  start = NULL,
  whichStart = 2L,
  initAlg = "nm",
  initIter = 100,
  lcmClasses = 2,
  method = "bfgs",
  hessianType = 1,
  itermax = 2000L,
  printInfo = FALSE,
  tol = 1e-12,
  gradtol = 1e-06,
  stepmax = 0.1,
  qac = "marquardt"
)

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

## S3 method for class 'sfalcmcross'
bread(x, ...)

## S3 method for class 'sfalcmcross'
estfun(x, ...)

Arguments

formula	A symbolic description of the model to be estimated based on the generic function formula (see section ‘Details’).
uhet	A one-part formula to account for heteroscedasticity in the one-sided error variance (see section ‘Details’).
vhet	A one-part formula to account for heteroscedasticity in the two-sided error variance (see section ‘Details’).
thet	A one-part formula to account for technological heterogeneity in the construction of the classes.
logDepVar	Logical. Informs whether the dependent variable is logged (TRUE) or not (FALSE). Default = TRUE.
data	The data frame containing the data.
subset	An optional vector specifying a subset of observations to be used in the optimization process.
weights	An optional vector of weights to be used for weighted log-likelihood. Should be NULL or numeric vector with positive values. When NULL, a numeric vector of 1 is used.
wscale	Logical. When weights is not NULL, a scaling transformation is used such that the weights sums to the sample size. Default TRUE. When FALSE no scaling is used.
S	If S = 1 (default), a production (profit) frontier is estimated: $\epsilon_i = v_i-u_i$. If S = -1, a cost frontier is estimated: $\epsilon_i = v_i+u_i$.
udist	Character string. Distribution specification for the one-sided error term. Only the half normal distribution 'hnormal' (Aigner et al., 1977, Meeusen and Vandenbroeck, 1977) is currently implemented.
start	Numeric vector. Optional starting values for the maximum likelihood (ML) estimation.
whichStart	Integer. If 'whichStart = 1', the starting values are obtained from the method of moments. When 'whichStart = 2' (Default), the model is initialized by solving the homoscedastic pooled cross section SFA model.
initAlg	Character string specifying the algorithm used for initialization and obtain the starting values (when 'whichStart = 2'). Only maxLik package algorithms are available: 'bfgs', for Broyden-Fletcher-Goldfarb-Shanno (see maxBFGS) 'bhhh', for Berndt-Hall-Hall-Hausman (see maxBHHH) 'nr', for Newton-Raphson (see maxNR) 'nm', for Nelder-Mead - Default - (see maxNM) 'cg', for Conjugate Gradient (see maxCG) 'sann', for Simulated Annealing (see maxSANN)
initIter	Maximum number of iterations for initialization algorithm. Default 100.
lcmClasses	Number of classes to be estimated (default = 2). A maximum of five classes can be estimated.
method	Optimization algorithm used for the estimation. Default = 'bfgs'. 11 algorithms are available: 'bfgs', for Broyden-Fletcher-Goldfarb-Shanno (see maxBFGS) 'bhhh', for Berndt-Hall-Hall-Hausman (see maxBHHH) 'nr', for Newton-Raphson (see maxNR) 'nm', for Nelder-Mead (see maxNM) 'cg', for Conjugate Gradient (see maxCG) 'sann', for Simulated Annealing (see maxSANN) 'ucminf', for a quasi-Newton type optimization with BFGS updating of the inverse Hessian and soft line search with a trust region type monitoring of the input to the line search algorithm (see ucminf) 'mla', for general-purpose optimization based on Marquardt-Levenberg algorithm (see mla) 'sr1', for Symmetric Rank 1 (see trust.optim) 'sparse', for trust regions and sparse Hessian (see trust.optim) 'nlminb', for optimization using PORT routines (see nlminb)
hessianType	Integer. If 1 (default), analytic Hessian is returned. If 2, bhhh Hessian is estimated ($g'g$).
itermax	Maximum number of iterations allowed for optimization. Default = 2000.
printInfo	Logical. Print information during optimization. Default = FALSE.
tol	Numeric. Convergence tolerance. Default = 1e-12.
gradtol	Numeric. Convergence tolerance for gradient. Default = 1e-06.
stepmax	Numeric. Step max for ucminf algorithm. Default = 0.1.
qac	Character. Quadratic Approximation Correction for 'bhhh' and 'nr' algorithms. If 'qac = stephalving', the step length is decreased but the direction is kept. If 'qac = marquardt' (default), the step length is decreased while also moving closer to the pure gradient direction. See maxBHHH and maxNR.
x	an object of class sfalcmcross (returned by the function sfalcmcross).
...	additional arguments of frontier are passed to sfalcmcross; additional arguments of the print, bread, estfun, nobs methods are currently ignored.

Details

LCM is an estimation of a finite mixture of production functions:

$y_i = \alpha_j + \mathbf{x_i^{\prime}} \bm{\beta_j} + v_{i|j} - Su_{i|j}$

$\epsilon_{i|j} = v_{i|j} - Su_{i|j}$

where $i$ is the observation, $j$ is the class, $y$ is the output (cost, revenue, profit), $x$ is the vector of main explanatory variables (inputs and other control variables), $u$ is the one-sided error term with variance $\sigma_{u}^2$, and $v$ is the two-sided error term with variance $\sigma_{v}^2$.

S = 1 in the case of production (profit) frontier function and S = -1 in the case of cost frontier function.

The contribution of observation $i$ to the likelihood conditional on class $j$ is defined as:

$P(i|j) = \frac{2}{\sqrt{\sigma_{u|j}^2 + \sigma_{v|j}^2}}\phi\left(\frac{S\epsilon_{i|j}}{\sqrt{ \sigma_{u|j}^2 +\sigma_{v|j}^2}}\right)\Phi\left(\frac{ \mu_{i*|j}}{\sigma_{*|j}}\right)$

where

$\mu_{i*|j}=\frac{- S\epsilon_{i|j} \sigma_{u|j}^2}{\sigma_{u|j}^2 + \sigma_{v|j}^2}$

and

$\sigma_*^2 = \frac{\sigma_{u|j}^2 \sigma_{v|j}^2}{\sigma_{u|j}^2 + \sigma_{v|j}^2}$

The prior probability of using a particular technology can depend on some covariates (namely the variables separating the observations into classes) using a logit specification:

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

with $\mathbf{Z}_h$ the covariates, $\bm{\theta}$ the coefficients estimated for the covariates, and $\exp(\bm{\theta}_J'\mathbf{Z}_h)=1$.

The unconditional likelihood of observation $i$ is simply the average over the $J$ classes:

$P(i) = \sum_{m=1}^{J}\pi(i,m)P(i|m)$

The number of classes to retain can be based on information criterion (see for instance ic).

Class assignment is based on the largest posterior probability. This probability is obtained using Bayes' rule, as follows for class $j$:

$w\left(j|i\right)=\frac{P\left(i|j\right) \pi\left(i,j\right)}{\sum_{m=1}^JP\left(i|m\right) \pi\left(i, m\right)}$

To accommodate heteroscedasticity in the variance parameters of the error terms, a single part (right) formula can also be specified. To impose the positivity on these parameters, the variances are modelled respectively as: $\sigma^2_{u|j} = \exp{(\bm{\delta}_j'\mathbf{Z}_u)}$ and $\sigma^2_{v|j} = \exp{(\bm{\phi}_j'\mathbf{Z}_v)}$, where $Z_u$ and $Z_v$ are the heteroscedasticity variables (inefficiency drivers in the case of $\mathbf{Z}_u$) and $\bm{\delta}$ and $\bm{\phi}$ the coefficients. 'sfalcmcross' only supports the half-normal distribution for the one-sided error term.

sfalcmcross allows for the maximization of weighted log-likelihood. When option weights is specified and wscale = TRUE, the weights are scaled as:

$new_{weights} = sample_{size} \times \frac{old_{weights}}{\sum(old_{weights})}$

For complex problems, non-gradient methods (e.g. nm or sann) can be used to warm start the optimization and zoom in the neighborhood of the solution. Then a gradient-based methods is recommended in the second step. In the case of sann, we recommend to significantly increase the iteration limit (e.g. itermax = 20000). The Conjugate Gradient (cg) can also be used in the first stage.

A set of extractor functions for fitted model objects is available for objects of class 'sfalcmcross' including methods to the generic functions print, summary, coef, fitted, logLik, residuals, vcov, efficiencies, ic, marginal, estfun and bread (from the sandwich package), lmtest::coeftest() (from the lmtest package).

Value

sfalcmcross returns a list of class 'sfalcmcross' containing the following elements:

call	The matched call.
formula	Multi parts formula describing the estimated model.
S	The argument 'S'. See the section ‘Arguments’.
typeSfa	Character string. 'Latent Class Production/Profit Frontier, e = v - u' when S = 1 and 'Latent Class Cost Frontier, e = v + u' when S = -1.
Nobs	Number of observations used for optimization.
nXvar	Number of main explanatory variables.
nZHvar	Number of variables in the logit specification of the finite mixture model (i.e. number of covariates).
logDepVar	The argument 'logDepVar'. See the section ‘Arguments’.
nuZUvar	Number of variables explaining heteroscedasticity in the one-sided error term.
nvZVvar	Number of variables explaining heteroscedasticity in the two-sided error term.
nParm	Total number of parameters estimated.
udist	The argument 'udist'. See the section ‘Arguments’.
startVal	Numeric vector. Starting value for ML estimation.
dataTable	A data frame (tibble format) containing information on data used for optimization along with residuals and fitted values of the OLS and ML estimations, and the individual observation log-likelihood. When weights is specified an additional variable is also provided in dataTable.
initHalf	When start = NULL and whichStart == 2L. Initial ML estimation with half normal distribution for the one-sided error term. Model to construct the starting values for the latent class estimation. Object of class 'maxLik' and 'maxim' returned.
isWeights	Logical. If TRUE weighted log-likelihood is maximized.
optType	The optimization algorithm used.
nIter	Number of iterations of the ML estimation.
optStatus	An optimization algorithm termination message.
startLoglik	Log-likelihood at the starting values.
nClasses	The number of classes estimated.
mlLoglik	Log-likelihood value of the ML estimation.
mlParam	Numeric vector. Parameters obtained from ML estimation.
mlParamMatrix	Double. Matrix of ML parameters by class.
gradient	Numeric vector. Each variable gradient of the ML estimation.
gradL_OBS	Matrix. Each variable individual observation gradient of the ML estimation.
gradientNorm	Numeric. Gradient norm of the ML estimation.
invHessian	The covariance matrix of the parameters obtained from the ML estimation.
hessianType	The argument 'hessianType'. See the section ‘Arguments’.
mlDate	Date and time of the estimated model.

Note

In the case of panel data, sfalcmcross estimates a pooled cross-section where the probability of belonging to a class a priori is not permanent (not fixed over time).

References

Aigner, D., Lovell, C. A. K., and P. Schmidt. 1977. Formulation and estimation of stochastic frontier production function models. Journal of Econometrics, 6(1), 21–37.

Caudill, S. B., and J. M. Ford. 1993. Biases in frontier estimation due to heteroscedasticity. Economics Letters, 41(1), 17–20.

Caudill, S. B., Ford, J. M., and D. M. Gropper. 1995. Frontier estimation and firm-specific inefficiency measures in the presence of heteroscedasticity. Journal of Business & Economic Statistics, 13(1), 105–111.

Hadri, K. 1999. Estimation of a doubly heteroscedastic stochastic frontier cost function. Journal of Business & Economic Statistics, 17(3), 359–363.

Meeusen, W., and J. Vandenbroeck. 1977. Efficiency estimation from Cobb-Douglas production functions with composed error. International Economic Review, 18(2), 435–445.

Orea, L., and S.C. Kumbhakar. 2004. Efficiency measurement using a latent class stochastic frontier model. Empirical Economics, 29, 169–183.

Parmeter, C.F., and S.C. Kumbhakar. 2014. Efficiency analysis: A primer on recent advances. Foundations and Trends in Econometrics, 7, 191–385.

Reifschneider, D., and R. Stevenson. 1991. Systematic departures from the frontier: A framework for the analysis of firm inefficiency. International Economic Review, 32(3), 715–723.

See Also

print for printing sfalcmcross object.

summary for creating and printing summary results.

coef for extracting coefficients of the estimation.

efficiencies for computing (in-)efficiency estimates.

fitted for extracting the fitted frontier values.

ic for extracting information criteria.

logLik for extracting log-likelihood value(s) of the estimation.

marginal for computing marginal effects of inefficiency drivers.

residuals for extracting residuals of the estimation.

vcov for computing the variance-covariance matrix of the coefficients.

bread for bread for sandwich estimator.

estfun for gradient extraction for each observation.

Examples

## Using data on eighty-two countries production (GDP)
# LCM Cobb Douglas (production function) half normal distribution
# Intercept and initStat used as separating variables
cb_2c_h1 <- sfalcmcross(formula = ly ~ lk + ll + yr, thet = ~initStat, 
data = worldprod)
summary(cb_2c_h1)

# summary of the initial ML model
summary(cb_2c_h1$InitHalf)

# Only the intercept is used as the separating variable
# and only variable initStat is used as inefficiency driver
cb_2c_h3 <- sfalcmcross(formula = ly ~ lk + ll + yr, uhet = ~initStat, 
data = worldprod)
summary(cb_2c_h3)

---

Deprecated functions of sfaR

Description

These functions are provided for compatibility with older versions of ‘sfaR’ only, and could be defunct at a future release.

Usage

lcmcross(
  formula,
  uhet,
  vhet,
  thet,
  logDepVar = TRUE,
  data,
  subset,
  weights,
  wscale = TRUE,
  S = 1L,
  udist = "hnormal",
  start = NULL,
  whichStart = 2L,
  initAlg = "nm",
  initIter = 100,
  lcmClasses = 2,
  method = "bfgs",
  hessianType = 1,
  itermax = 2000L,
  printInfo = FALSE,
  tol = 1e-12,
  gradtol = 1e-06,
  stepmax = 0.1,
  qac = "marquardt"
)

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

## S3 method for class 'lcmcross'
bread(x, ...)

## S3 method for class 'lcmcross'
estfun(x, ...)

## S3 method for class 'lcmcross'
coef(object, extraPar = FALSE, ...)

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

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

## S3 method for class 'lcmcross'
ic(object, IC = "AIC", ...)

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

## S3 method for class 'lcmcross'
marginal(object, newData = NULL, ...)

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

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

## S3 method for class 'lcmcross'
summary(object, grad = FALSE, ci = FALSE, ...)

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

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

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

Arguments

formula	A symbolic description of the model to be estimated based on the generic function formula (see section ‘Details’).
uhet	A one-part formula to account for heteroscedasticity in the one-sided error variance (see section ‘Details’).
vhet	A one-part formula to account for heteroscedasticity in the two-sided error variance (see section ‘Details’).
thet	A one-part formula to account for technological heterogeneity in the construction of the classes.
logDepVar	Logical. Informs whether the dependent variable is logged (TRUE) or not (FALSE). Default = TRUE.
data	The data frame containing the data.
subset	An optional vector specifying a subset of observations to be used in the optimization process.
weights	An optional vector of weights to be used for weighted log-likelihood. Should be NULL or numeric vector with positive values. When NULL, a numeric vector of 1 is used.
wscale	Logical. When weights is not NULL, a scaling transformation is used such that the weights sums to the sample size. Default TRUE. When FALSE no scaling is used.
S	If S = 1 (default), a production (profit) frontier is estimated: $\epsilon_i = v_i-u_i$. If S = -1, a cost frontier is estimated: $\epsilon_i = v_i+u_i$.
udist	Character string. Distribution specification for the one-sided error term. Only the half normal distribution 'hnormal' (Aigner et al., 1977, Meeusen and Vandenbroeck, 1977) is currently implemented.
start	Numeric vector. Optional starting values for the maximum likelihood (ML) estimation.
whichStart	Integer. If 'whichStart = 1', the starting values are obtained from the method of moments. When 'whichStart = 2' (Default), the model is initialized by solving the homoscedastic pooled cross section SFA model. 'whichStart = 1' can be fast.
initAlg	Character string specifying the algorithm used for initialization and obtain the starting values (when 'whichStart = 2'). Only maxLik package algorithms are available: 'bfgs', for Broyden-Fletcher-Goldfarb-Shanno (see maxBFGS) 'bhhh', for Berndt-Hall-Hall-Hausman (see maxBHHH) 'nr', for Newton-Raphson (see maxNR) 'nm', for Nelder-Mead - Default - (see maxNM) 'cg', for Conjugate Gradient (see maxCG) 'sann', for Simulated Annealing (see maxSANN)
initIter	Maximum number of iterations for initialization algorithm. Default 100.
lcmClasses	Number of classes to be estimated (default = 2). A maximum of five classes can be estimated.
method	Optimization algorithm used for the estimation. Default = 'bfgs'. 11 algorithms are available: 'bfgs', for Broyden-Fletcher-Goldfarb-Shanno (see maxBFGS) 'bhhh', for Berndt-Hall-Hall-Hausman (see maxBHHH) 'nr', for Newton-Raphson (see maxNR) 'nm', for Nelder-Mead (see maxNM) 'cg', for Conjugate Gradient (see maxCG) 'sann', for Simulated Annealing (see maxSANN) 'ucminf', for a quasi-Newton type optimization with BFGS updating of the inverse Hessian and soft line search with a trust region type monitoring of the input to the line search algorithm (see ucminf) 'mla', for general-purpose optimization based on Marquardt-Levenberg algorithm (see mla) 'sr1', for Symmetric Rank 1 (see trust.optim) 'sparse', for trust regions and sparse Hessian (see trust.optim) 'nlminb', for optimization using PORT routines (see nlminb)
hessianType	Integer. If 1 (default), analytic Hessian is returned. If 2, bhhh Hessian is estimated ($g'g$).
itermax	Maximum number of iterations allowed for optimization. Default = 2000.
printInfo	Logical. Print information during optimization. Default = FALSE.
tol	Numeric. Convergence tolerance. Default = 1e-12.
gradtol	Numeric. Convergence tolerance for gradient. Default = 1e-06.
stepmax	Numeric. Step max for ucminf algorithm. Default = 0.1.
qac	Character. Quadratic Approximation Correction for 'bhhh' and 'nr' algorithms. If 'qac = stephalving', the step length is decreased but the direction is kept. If 'qac = marquardt' (default), the step length is decreased while also moving closer to the pure gradient direction. See maxBHHH and maxNR.
x	an object of class lcmcross (returned by the function lcmcross).
...	additional arguments of frontier are passed to lcmcross; additional arguments of the print, bread, estfun, nobs methods are currently ignored.
object	an object of class lcmcross (returned by the function lcmcross).
extraPar	Logical (default = FALSE). If TRUE, additional parameters are returned (see coef or vcov).
IC	Character string. Information criterion measure. Three criteria are available: 'AIC' for Akaike information criterion (default) 'BIC' for Bayesian information criterion 'HQIC' for Hannan-Quinn information criterion .
individual	Logical. If FALSE (default), the sum of all observations' log-likelihood values is returned. If TRUE, a vector of each observation's log-likelihood value is returned.
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.
grad	Logical. Default = FALSE. If TRUE, the gradient for the maximum likelihood (ML) estimates of the different parameters is returned.
ci	Logical. Default = FALSE. If TRUE, the 95% confidence interval for the different parameters (OLS or/and ML estimates) is returned.
digits	Numeric. Number of digits displayed in values.
level	A number between between 0 and 0.9999 used for the computation of (in-)efficiency confidence intervals (defaut = 0.95). Not used in the case of lcmcross.

Details

The following functions are deprecated and could be removed from sfaR in a near future. Use the replacement indicated below:

• lcmcross: sfalcmcross

• bread.lcmcross: bread.sfalcmcross

• coef.lcmcross: coef.sfalcmcross

• coef.summary.lcmcross: coef.summary.sfalcmcross

• efficiencies.lcmcross: efficiencies.sfalcmcross

• estfun.lcmcross: estfun.sfalcmcross

• fitted.lcmcross: fitted.sfalcmcross

• ic.lcmcross: ic.sfalcmcross

• logLik.lcmcross: logLik.sfalcmcross

• marginal.lcmcross: marginal.sfalcmcross

• nobs.lcmcross: nobs.sfalcmcross

• print.lcmcross: print.sfalcmcross

• print.summary.lcmcross: print.summary.sfalcmcross

• residuals.lcmcross: residuals.sfalcmcross

• summary.lcmcross: summary.sfalcmcross

• vcov.lcmcross: vcov.sfalcmcross

---